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F20101202_AABDHR salois_m_Page_144thm.jpg 1dd72f366eadc58c29b14deb8e770d87 bd7547b7073d3977a74ee21268286ca93186dbf5 83493 F20101202_AABBYV salois_m_Page_032.jpg e78d658b713d2c1748e67bd103b863b4 566a537813cde9923ce0b4df95d8f3fdc143fa07 92299 F20101202_AABCFE salois_m_Page_013.jpg f5dfefe903ded37f165b64bfe571e962 d1d50802452912a3ef28669185c201de30131e8f 89542 F20101202_AABCEP salois_m_Page_161.jpg 4e9480560989e284de82fb93798a8fc8 84b99df8c544069b149a22e737c9c024a0082947 F20101202_AABBZJ salois_m_Page_085.tif f61baa1fb3ed4512d86fc60067f936a2 0d807e425dac4e604422dba17c25441b40010db9 INTERTEMPORAL PREFERENCES AND TIMEINCONSISTENCY: THE CASE OF FARMLAND VALUES AND RURALURBAN LAND CONVERSION By MATTHEW JUDE SALOIS A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2008 2008 Matthew Jude Salois To my close friends, my dear family, and my lovely fiancee. ACKNOWLEDGMENTS The work presented here could not have been accomplished without the creative talent and the sincere dedication of my dissertation committee. I have Professor Andrew Schmitz to thank for his brilliant mind and his worldly view of economics. My conversations with Dr. Schmitz were always fun, perceptive, and memorable. Professor Jonathan Hamilton provided much guidance both inside and outside the classroom. Meetings with Dr. Hamilton always imparted me with the kind of feedback that left me in awe of his ability. I owe a great debt to Professor Timothy Taylor for his gracious mentorship and his honest view of economics. Never have I seen a more gifted teacher than Dr. Taylor, whose singular wit kept me motivated. Finally, I express my most sincere gratitude to my committee chair and mentor, Professor Charles Moss. Working with Dr. Moss has been an inspiring, edifying, and immensely rewarding experience. Dr. Moss is a true renascence man of economics and to whom I owe the sum of my graduate career. I would also like to thank my parents, Michael and Carol, whose unwavering support of my past and present academic accomplishments have made my future a bright one. My sisters, Aimee and Carolyn, and brother, Jeremiah, provided a ready supply of needed distraction and back to reality moments. Finally, and most importantly, I struggle to convey the true pillar that my fiancee, Alisa Beth, has been for me to lean on. Thank you for providing the will and the way towards the completion of this academic endeavor. May we prosper together in what matters most in this world, our lives together. Above all else I would like to thank the One who has made every past, present, and future endeavor possible. Any success is owed to Him and I am humbled by the chance He has given to me in this pursuit of knowledge. TABLE OF CONTENTS page A C K N O W L E D G M E N T S ..............................................................................................................4 L IS T O F T A B L E S ........................................................................................................................... 7 LIST OF FIGURES .................................. .. ..... ..... ................. .8 L IST O F A B B R E V IA TIO N S ....................................................................... ... ....................... 9 A B S T R A C T ............ ................... ............................................................ 10 CHAPTER 1 IN TR OD U CTION .......................................................................... .. ... .... 12 Overview and Purpose ...................................................... .......... ..... ......... 12 Tim e Preferences and Econom ics................................................. .............................. 15 D discourse on D discounting ..................................... ................ ....... .......... .... 18 Application to RuralUrban D evelopm ent......................................... ......................... 22 Motivation: The Importance of Discounting ...............................................................23 Relevance: The TimeInconsistent Landowner....................... ...................... 25 Theoretical Heuristic: The Development Decision.......................................................27 Study O bj ectives ...................................... .................................................... 3 1 2 L ITE R A TU R E R E V IE W ........................................................................ .. .......................36 In tro d u ctio n ................... ...................3...................6.......... T h e o ry ................................ ............................................................................................... 3 8 Land V values and U se ........................................................ ........ .. ............ 39 A Sim ple Static M odel .......................... .............. ................. .... ....... 43 A Sim ple D ynam ic M odel .............................................................................. ........ 45 Modeling the Land Development Decision................................................................. 48 The C apitalization A pproach................................................. .............................. 50 T he discount rate .......................................................................53 Land rents and conversion costs......................................... .......................... 55 L a n d tax e s ............... .......... .......................................................5 8 Market and information imperfections ............... .............................................65 U rb an g row th ................................................... ................ 6 8 U n c e rta in ty ...............................................................................................................7 0 The R eal O options A approach .................................................. .............................. 72 Introduction to real options theory ........................................ ........ ............... 72 A brief account of real options.................................... ...................... ..................74 Application of real options to land development ...................................................76 The Transactions Cost Approach ............................................................................. 84 Land develop ent and institutions ........................................ ....... ............... 84 M odels of land values with transaction costs.........................................................88 E m pirical M odels of L and C hange........................................................................... ......91 C apitalization Em pirical M ethods.............................................................. ... ............ 91 O ption V alue E m pirical M ethods...................................................................... ...... 100 Chapter Summary .............................. ...................... ............ 105 3 TH E O R Y A N D EM PIR IC S ................................................................ ........................112 Theoretical Fram ew ork .................. ............................. ............ .. ........ .... 112 Economy etric Procedure ................................................... ............ ........... 117 Instrum ent Selection and Identification........................................................... ............... 121 Chapter Summary .............................. ...................... ............ 122 4 D A T A A N D R E SU L T S ............................................................................ .....................123 D ata and V variable D escription............... ................................................... ............... 123 Estimation Results .............. ..... ................ ......... ..... 125 A ppalachian States ............................ ........................ .... .... ......... .. .... .. 126 Corn B elt States ................... .... .......... ...... ....... .......... ............. 128 D elta States ......... ........ .........................................130 G re at P lain S tate s .................................................................................................... 13 1 L ake States................................................... 13 1 Mountain States ............... ................... ........ .........................132 Northeast States ............... ................... ........ .........................133 P pacific S states ...............................................................13 4 Southeast States ................... ..... ..... ... ....... .............. ............ 134 C chapter Sum m ary ...............................................................135 5 CONCLUSION AND FUTURE WORK ... ........................... .......... 147 Comparisons and Limitations ............... ....... ................... 147 Im portance and Im plications .......................................................................... ...............159 F future W ork ......................................................165 Sum m ary .............................. ............ ...... ................. ............. ............. 166 APPENDIX PROCEDURE FOR RCODE ............................. ............. ........ ...............169 L IST O F R E FE R E N C E S ......... ............... ................. ............................ ..............................206 B IO G R A PH IC A L SK E T C H ......... ................. ............................................ ...........................2 18 6 LIST OF TABLES Table page 21 Selected comparative static results from capitalization papers.................... ......... 111 41 A ppalachian states result ........................................................... .. ............... 143 42 C orn B elt states result ......... ........................................................ .... ....... 143 43 Delta states result ............ .... .............. .................... .. ........ .. 143 44 G great Plain states results .............. .................................................................... .... ...... 144 45 Lake states results ................................. .. .. ... ... .. ....... ......... .144 46 Mountain states result .............................. .... ..................... ........145 47 N northeast states result .............. ....................................................................... ........... 145 48 Pacific states result............... .. ................ ................ .. 146 49 Southeast states result .............................................................. .................. 146 410 D discount rates by region ......................................................................... ................... 146 LIST OF FIGURES Figure page 11 E xponential discounting ........................................................................... ....................34 12 H yperbolic discounting .............................................................................. ............... 34 13 Quasihyperbolic discounting .......................................................................... 35 14 C om prison of discount factors .............................................................. .....................35 21 Land allocation and bidrent model ...........................................................................108 22 O ptim al land allocation .......................................................................... ....................108 23 Optim al conversion tim e ........................................................... .. ............... 109 24 Change in value of agricultural land awaiting conversion......................................... 109 25 T im ing of conversion decision ........................................................................... .... 110 26 Call option payoff ..................................................... ............ ............... 110 41 Change in farmland values for the Appalachian states...................................................138 42 Change in farmland values for the Corn Belt states ................................ ...............138 43 Change in farmland values for the Delta states .................................... ............... 139 44 Change in farmland values for the Great Plain states................... ...................................139 45 Change in farmland values for the Lake states ......................................... ...........140 46 Change in farmland values for the Mountain states ............. ......................................140 48 Change in farmland values for the Pacific states ......................................................141 49 Change in farmland values for the Southeast states ............. ......................................142 8 LIST OF ABBREVIATIONS BP BreuschPagan BL BoxLjung CBD Central business district CES Constant elasticity of substitution CPI Consumer price index CRP Conservation Reserve Program CRRA Coefficient of relative risk aversion DVT Development value tax ERS Economic Research Service GIS Graphical information system GMM Generalized method of moments IRR Internal rate of return LVT Land value tax NLSY National Longitudinal Survey of Youth NPV Net present value NRI National Resources Inventory PCE Personal consumption expenditure PPI Producer price index PV Present value PDV Present discounted value USDA United States Department of Agriculture Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy INTERTEMPORAL PREFERENCES AND TIMEINCONSISTENCY: THE CASE OF FARMLAND VALUES AND RURALURBAN LAND CONVERSION By Matthew Jude Salois August 2008 Chair: Charles B. Moss Major: Food and Resource Economics Many studies have rejected the standard present value model under rational expectations as a viable model for explaining farmland values in both domestic and international data. Current models of farmland values inadequately explain why land prices rise and fall faster than land rents, particularly in the shortrun. Previous inquiries into the nature of farmland values assume a timeconsistent discount factor and do not seriously investigate the role of intertemporal preferences. This dissertation introduces timeinconsistent preferences into a model of farmland values by including a quasihyperbolic discount parameter in the asset equation of nine agricultural regions in the United States. Strong evidence is found in favor of quasihyperbolic discounting as a more appropriate way of describing the discount structure in models of farmland values. By introducing quasi hyperbolic discounting to the presentvalue model of farmland values, the discount factor can be broken down into two timespecific rates, the shortrun discount rate and the longrun discount rate. Using a timeinconsistent discount factor, like a quasihyperbolic one, allows time preferences in the shortrun to be different than in the longrun Thus, the model offers an explanation as to why shortrun and longrun land values do not follow the same path. The theoretical formulation in this paper generalizes time preferences in the asset equation and allows values of the exponential and quasihyperbolic discount parameters to be obtained. A hypothesis test is constructed, permitting for a direct test on the discount parameters. Using the linear panel Generalized Method of Moments estimator, problems of heteroskedasticity and serial correlation are reduced. The results of the hypothesis tests imply a formal rejection that shortrun discount rates are equal to longrun discount rates, and that the shortrun discount rates are substantially larger than the longrun discount rates, a new result in the literature on farmland values. The results presented in this dissertation are not only important because of their unique insights into intertemporal preferences, but also because of the implications generated by the results. First, the results imply that landuse decisions may be made with a greater interest in shortrun gains than longrun returns. Second, the results offer an appealing explanation as to why land values and rents do not follow the same path in the shortrun. Finally, the results also suggest that farmland serves as a golden egg to farmers, landowners, or developers who may demand commitment devices to constrain themselves from hasty land investment decisions. The results have importance policy and extension relevance. Knowing that landuse decisions may be dominated by shortrun thinking, extension efforts should address this tendency to insure that future consequences of present choices are fully considered. The results also lend support to policy instruments that act as commitment devices to constrain landowners to their present choices. Future work should focus on the potential consequences of time inconsistency in other common agricultural economics models such as ruralurban land conversion and food demand, as well as extending the results in this dissertation to account for risk, inflation, and adaptive expectations. CHAPTER 1 INTRODUCTION Overview and Purpose The capitalization approach or net present value model has dominated the literature on farmland values and the general literature on asset pricing as a whole. The method generally states that the value of farmland is determined by the discounted expected future return to farmland (Melichar 1979; Alston 1986; Burt 1986; Featherstone and Baker, 1987). However, a number of empirical issues have arisen from the use of the present value technique, particularly in the arena of farmland pricing. A burgeoning literature has appeared criticizing the capitalization technique for oversimplifying the market fundamental process, leading to empirical rejection of the present value model for farmland (Falk 1991; Clark, Fulton, and Scott 1993; Lloyd 1994). One serious flaw in the present value model is the inability to explain why land prices rise and fall faster than land rents, particularly in the shortrun (Schmitz 1995; Falk and Lee 1998). These socalled boom/bust cycles represent the tendency of markets to overvalue land in periods of prosperity while undervaluing land in periods of relative decline (Schmitz and Moss 1996). As pointed out by Featherstone and Moss (2003), any sustained time period of overvaluation or undervaluation is inconsistent with a rational farmland market. Not only is the presence of boom/bust cycles well documented within the land market, but the causes for these cycles has been the topic of considerable debate in the literature (Lavin and Zorn 2001). Recent studies on models of land values have attributed the failure of the standard present value method to a variety of causes including the presence of transaction costs (Just and Miranowski 1993; Chavas and Thomas 1999; de Fontnouvelle and Lence,2002), a timevarying risk premium (Hanson and Myers 1995), fads and overreaction (Falk and Lee 1998), or inadequate econometric methods (Gutierrez, Westerlund, and Erickson 2007). However, few studies examine the role of time preferences within the context of agricultural land values or have seriously considered the shape and form of the discount factor in particular. The literature to date has overwhelmingly relied on the apriori assumption that individuals, such as farmers and landowners, are time consistent and are described by standard exponential discounting, which implies a constant rate of discount. While such an assumption invokes intertemporal consistency in preferences, this rationale should be called into question. Even Paul Samuelson (1937, p. 156), in his seminal work on the discounted utility model, was keen to state that the assumption of constant discounting was purely arbitrary and "is in the nature of an h)',,pthei', subject to refutation by observable facts." Substantial empirical evidence in experimental, behavioral, psychological, and financial economics has shown that individuals tend to be time inconsistent (Thaler 1981; Benzion, Rapoport, and Yagil 1989; Benzion, Shachmurove, and Yagil 2004) and that time preferences are better modeled by quasihyperbolic discounting (Loewenstein and Prelec1992; Eisenhauer and Ventura 2006). This dissertation introduces intertemporal inconsistency into a model of land values and tests for the presence of quasihyperbolic discounting in the farmland asset equation. Unlike exponential discounting, which implies that individuals apply the same constant rate of discount each year, quasihyperbolic discounting implies a nonconstant rate of discount that declines over time. Hence, discounting is heavier in earlier time periods with the discount rate falling across the time horizon. This implies that individuals are more impatient when they make short run decisions than when they make longrun decisions. The introduction of quasihyperbolic discounting allows the discount factor to be decomposed into shortrun and longrun discount rates. Thus the model offers an explanation into the apparent disconnect between shortrun and longrun land values and explains why significant shortrun deviations from the discounted formulation may occur. This dissertation makes three important contributions. The first contribution is theoretical, employing a net present value model that generalizes intertemporal preferences to allow for quasihyperbolic discounting. The literature often uses intricate and rigorous game theoretic approaches to demonstrate the effect of timeinconsistent preferences on consumption and savings decisions. The approach in this dissertation focuses on a simplified modification of the asset investment equation to account for quasihyperbolic discounting. This method of measuring the economic impact of time inconsistent preferences by obtaining the quasi hyperbolic discount parameter in a reducedform model is unique in the literature. The second contribution is empirical. Most studies that estimate the quasihyperbolic discount parameter use experimental data. Furthermore, many studies rely on calibration methods rather than estimation methods to obtain the discount parameter (Laibson 1997, 1998; Angeletos et al. 2001). This dissertation uses aggregate field level data and empirically estimates the discount parameter, joining the small but growing number of studies that use field data to structurally estimate discount factors (Paserman 2004; Ahumada and Garegnani 2007; Fang and Silverman 2007; Laibson, Repetto, and Tobacman 2007). Finally, within the context of land values, and farmland values in particular, the dissertation examines the importance of timeinconsistent preferences in an important area of economics. The dissertation offers an explanation for the observed inconsistency in farmland markets between shortrun and longrun farmland values through presence of quasihyperbolic discounting. The results in the dissertation also yields insights into landuse decisions and the tendency of landowners to make present decisions for instant gains at the sacrifice of future returns. The shortrun discount rates obtained also provide quantitative support for the Golden Eggs hypothesis posited by Laibson (1997) and suggest that landowners may desire commitment devices to help constrain their future selves. To the best knowledge available, this dissertation contributes to the literature on land values with the first theoretical and empirical study of the relationship between timeinconsistent preferences and land values. The remainder of this chapter is organized as follows. The second section provides a cursory review of the literature on intertemporal preferences and time inconsistency. The third section discusses the nature of hyperbolic discounting in more detail and provides a useful example. The fourth section introduces an important application of intertemporal preferences in the land values literature, the economic decision to develop rural land to urban use. The fifth and last section of this chapter reiterates the study objectives and provides a summary. Time Preferences and Economics A growing body of literature on the economics of intertemporal decisionmaking and time preferences has led to a significant body of research on time inconsistent preferences. For excellent reviews of the literature see Loewenstein and Prelec (1992), Laibson (1997), and Frederick, Loewenstein, and O'Donoghue (2002). There are different ways to model time inconsistent preferences with a nonconstant discount rate, but the most common method is through the use of a hyperbolic discount factor. Hyperbolic discounting originally developed in the psychological literature and was used to model intertemporal discounting in experiments on pigeon behavior in Chung and Herrnstein (1961) and later applied to discounting by people in Ainslie (1975). The financial, behavioral, and experimental economics literature has taken recent note of hyperbolic discounting, being applied to models of savings, investment, economic growth, and addiction. Unlike time consistent preferences, time inconsistent preferences are characterized by two phenomena. The first are preference reversals and the second are intrapersonal games involving a tussle between desires to act patiently against desires for instantaneous gratification (Laibson 1997). A preference reversal occurs when an individual's present self makes a decision and then the future self makes a different decision. A common example is when someone sets their alarm clock before going to bed only to hit the snooze button once, or several times more, when they awake the next morning. The internal tussle comes into play when an individual is torn between the longrun desire to act patiently and the shortrun desire to be impatient. An example here would be someone who wants to have a better looking physique next year but still indulges in fastfood that evening for dinner. Exponential discounting, representing timeconsistent preferences, discounts at a constant rate across each time period in the horizon and so cannot account for either preference reversals or internal tussles. Hyperbolic discounting, and the discrete form case of quasihyperbolic discounting, accounts for the timeinconsistencies discussed above by imparting a nonconstant discount rate, one that declines over time. Robert Strotz (1956) was the first to suggest that individuals may exhibit time inconsistency through an "intertemporal tussle" where the future self may have different preferences from the present self. Strotz (1956) proposes a discount function based on the time distance of a future date from the present moment rather than just the future date as more descriptive of individual behavior. Over the years, several elegant but general models of timeinconsistent preferences have been proposed by Pollak (1968), Peleg and Yaari (1973), and Goldman (1979, 1980). The simple time inconsistency introduced in the previous studies is based upon an individual valuing wellbeing or utility more at the present time than at some future time, but values future wellbeing or utility at the same rate. The application of time inconsistent preferences to important economic models reveals noteworthy conclusions. Barro (1999) adapts the neoclassical growth model of Ramsey to account for a variable rate of time preference. While the basic properties of the neoclassical growth model are invariant under timeinconsistent preferences, the timevarying model yields important welfare implications depending on the ability and level of commitment from households to their future choices of consumption. Gruber and Koszegi (2001) apply time inconsistent preferences to the rational addiction model of Becker and Murphy. While the prediction of their model was equivalent to the BeckerMurphy model, that current consumption of an addictive substance is sensitive to future price expectations, the time inconsistent model reveals substantially different optimal level of government taxation. Laibson (1996, 1997), in his pioneering work on applying hyperbolic discounting to economic models of savings and investment, finds convincing evidence that rates of time preference are not constant. Ahumada and Garegnani (2007) recently found evidence of hyperbolic discounting in consumptionsavings decisions using aggregate consumer expenditure data in Argentina. Laibson, Repetto, and Tobacman (2007) use individual level data on credit card borrowing, consumption, income, and retirement savings and strongly reject the constant discount rate model in several specifications. Intertemporal preferences have been largely ignored in the agricultural economics literature, with few exceptions, ignoring the role time preferences play in farmer or landowner decisions. Flora (1966) represents one of the only attempts to determine how landowners' time preferences affect the discount factor. He finds investment decisions in forest lands can be affected by time preferences. Analysis of survey data finds that some individuals place a higher time priority than the prevailing interest rate (Flora 1966). Barry, Lindon, and Nartea (1996) make a valuable contribution by establishing time attitude measures analogous to the Arrow Pratt measures of risk attitudes of increasing, decreasing, and constant absolute time aversion. They point out that farmer time attitudes may change over time, directly affecting choices involving consumption, savings, and investment. Lence (2000) is one of the only studies known that estimates the farmer's rate of time preference based on consumption data using Euler equations. He points out that the literature has largely ignored farmer's intertemporal preferences despite the tremendous benefit such knowledge would endow towards a greater understanding of how agricultural policy can optimally allocate resources across time. To gain a better understanding of how constant and timeconsistent discounting differs from nonconstant timeinconsistent discount, the following section offers a primer on discount factors. Discourse on Discounting Time inconsistency is generated in the hyperbolic discount factor by a rate of discount that falls as the discounted event is moved further away in time. Events in the near future are discounted at a higher implicit discount rate than events in the distant future. The generalized hyperbolic discount factor developed in Loewenstein and Prelec (1992) is 3(t)= (1 + at) a, where a, y > 0. The parameter a determines how much the function departs from constant discounting. As explained by Luttmer and Mariotti (2003), a is the parameter in the discount factor that controls how fast the rate of time preference changes between short run and long run values. The limiting case, as a goes to zero, is the exponential discount factor 3(t) = '. The y parameter represents a first period immediacy effect, discounting the initial period more heavily. As noted in Weitzman (2001), the hyperbolic discount function generalizes to the wellknown gamma distribution, where the exponential function is simply a more special case. The instantaneous discount rate at time t is for the hyperbolic discount factor is given by (t)/3(t) = y/(1 + at). The noteworthy property of hyperbolic discounting is that as t increases, the instantaneous rate of discount decreases, meaning the hyperbolic discount rate is not constant, as in exponential discounting, but rather is a function of time, declining over the time interval. The percent change in the hyperbolic discount factor depends on the time horizon, being steeper for the near future and flatter for the distant future, implying a discount factor that declines at a faster rate in the short run than in the long run. An alternative type of discounting is quasihyperbolic discounting, originally proposed by Phelps and Pollak (1968) and developed further by Laibson (1997, 1998, 2007). The quasi hyperbolic discount factor is a discretevalue time function and maintains the declining property of generalized hyperbolic discounting. At the same time, the discrete quasihyperbolic formulation keeps the analytical simplicity of the timeconsistent model by still incorporating certain qualitative aspects of exponential discounting. The actual values of the discount function under a discrete setup are with discount values {1, 2 3, 3 52,..., /3. 3' }. To obtain a better understanding of how the form of the discount factor can affect economic analysis, and in particular the economic model of land development, consider a comparison of the following discount factors: Exponential discount factor t (11) Hyperbolic discount factor (I + at) /" (12) Quasi hyperbolic discount factor = ', 2. 3, Y. 32,..., 3'T } (13) The exponential discount factor is graphed in Figure 11 for two values of 3, 0.959 for exponential factor 1 and 0.951 for exponential factor 2. As can be discerned from graphical comparison, exponential discount factor 2, with a value of 0.951, discounts more heavily than exponential factor 1, with a value of 0.959. In discrete time, the exponential discount factor is described by the functional form (t) = ( + r) ', where r is the rate of time preference, often taken to be the prevailing interest rate. Intuitively, exponential factor 2, having a lower value than exponential factor 1, must have a higher rate of time preference. A higher rate of time preference means a greater preference for consumption today than tomorrow. Hence, we can refer to 3 as descriptor of individual impatience. The greater the value of 3, the more patient the individual. The generalized hyperbolic discount factor is graphed in Figure 12 for different values of a and y. Hyperbolic factor 1 has a = 500,000 and y = 10900 while hyperbolic factor 2 has a = 250,000 and y = 10000. Upon comparison of the two hyperbolic factors, it can be seen that hyperbolic factor 2 discounts the future more than hyperbolic factor 1. The initial jump in the value of hyperbolic factor 2 between the first and second time periods is notably greater than the jump in hyperbolic factor 1. The parameters a and y have counterbalancing effects on the value of the hyperbolic discount factor. Smaller values of a result in a smaller jump between the initial periods while smaller values of y result in a bigger jump. The parameter y behaves much like the rate of time preference in the exponential discount factor, while a determines how much the hyperbolic factor departs from constant discounting. Finally, the quasihyperbolic discount factor, graphed in Figure 13 is presented for two different value of / and 3. Quasihyperbolic factor 1 has / = 0.91 and 3 = 0.971, while quasihyperbolic factor 2 has / = 0.85 and 3 = 0.964. Similar to the cases above, quasi hyperbolic factor 2 discounts more heavily than quasihyperbolic factor 1, which has higher values for both / and 3. The / parameter captures the essence of hyperbolic discounting and contains a first period immediacy effect in the individual's time preference. Changes in / determine how much the quasihyperbolic factor will deviate form exponential discounting. Higher values of / will result in a larger jump between the first two time periods. This jump in the value of the discount factor is what creates dynamic time inconsistent preferences. The 3 parameter behaves similar to the exponential discount factor. When /7 > 3, the quasihyperbolic factor becomes highly convex, discounting the near term much greater than more distant time period. Smaller values of 3 will result in a more bowedshaped discount factor implying a greater preference for immediate consumption. The quasihyperbolic discount function marries the qualitative properties of the exponential and generalized hyperbolic discount functions. Comparison of the three discount factors is depicted in Figure 14. Each of the discount functions have been calibrated so they approximately cross at t = 8. As can be seen from the figure, the generalized hyperbolic discount factor is the most convex of the three, with the exponential as the least convex up until time period 8. The greater convexity or more bowed shape of the hyperbolic factor implies that the hyperbolic factor discounts more heavily than either the exponential or the quasihyperbolic up until time period 8. After that point, the exponential discount factor is the most bowed of three implying that the exponential factor discounts more heavily than either the hyperbolic or quasihyperbolic factors. With the given calibration of parameters, dynamic timeinconsistent preferences are modeled with the near term discounted more heavily than the more distant term. Here, the near term is time periods up to period 8, with the distant term being time periods after 8. The figures illustrate the sensitivity of discounting to the selection of parameter values with the choice of near term and distant term being subjective in regards to the economic situation being examined. One can easily imagine how the discount factor can affect a wide range of economic decisions, particularly those that are intertemporal in nature. A very important and relevant economic decision is the choice to convert rural land to an urban use. This is the topic of the next section. Application to RuralUrban Development This section discusses the role of time preferences and in particular the impact of time inconsistent preferences as modeled by hyperbolic discounting in a model of land conversion. The act of ruralurban land conversion remains an important consideration since models of land development and the actual development decision are derivative of models of land values. From a logical standpoint, if one does not have an understanding of how intertemporal preferences affect land values, then one cannot begin to understand how time preferences affect the land development decision. This section attempts to support and promote such an understanding. Loss of agricultural land to developed uses has been a public policy issue for decades. For many years, economists have analyzed the structure of agricultural land prices and the timing of development in an effort to understand alternative uses to agriculture posed by land development. A specific aim of such research is to identify policies to prevent or discourage what may be considered suboptimal landuse changes. Numerous studies have examined the hedonic characteristics of the land itself as factors in land conversion (Taylor and Brester 2005). However, many land development policies are directed at the developer or landowner, including tax structures aimed at either accelerating or decelerating the rate of land conversion. Despite policies directed at the individual, little account is taken of the individual traits of the decisionmaker, particularly excluding any motivational or behavioral forces of the landowner in the decision to convert land from rural to urban use. One behavioral aspect of the landowner is the time preference involved in the intertemporal decision to develop land. Of special interest is relaxing the assumption of time consistent preferences in intertemporal decisionmaking to allow for dynamic inconsistency using nonconstant discounting. Motivation: The Importance of Discounting The choice of the discount rate used in the model is a key variable in the determination of land values and development times. The rate of time preference, given by the discount rate, is one central component of intertemporal choice, and is an aspect overlooked in the land economics literature. Typically, the landowner is assumed to be time consistent with a constant discount rate formulated in an exponential discount factor. In theoretical models of optimal development times, the discount rate has been generally found to have negative effects on land values, for clear reasons, and tends to accelerate the development process (Ellson and Roberts 1983; Capozza and Helsley 1989). The rather "conclusive" effects of the discount rate in the land values literature may however stem from the arbitrary nature in the choice of the actual discount factor used. As far is known, all capitalization approaches to modeling land values and the optimal development time have used a single, constant discount rate in their discounted cash flow analyses, implying that farmers and landowners have time consistent preferences (Rose 1973; Markusen and Scheffman 1978, Capozza and Helsley 1989; Arnott 2005). As noted earlier however, recent evidence suggests that individuals are time inconsistent. The random and potentially capricious nature of a constant discount rate in models of land development was recognized early by Shoup (1970). The potential error of a constant discount rate is compounded when uncertainty is brought into the analysis. The more distant the expected development time, the more uncertain landowners, developers, or investors are regarding the value of land. If uncertainty is the case, then the discount rate used in the present value formulation of undeveloped lands may be higher in time periods closer to the present (Shoup 1970). The actual discount rate would then fall as the conversion time approaches. A declining discount rate through time would imply that the value of land appreciates faster in time periods before development. This reasoning seems highly probable, given the uncertain nature of the land market and makes a particular case against a constant discount rate in favor of one that is a declining function of time. The internal tussle described by Strotz (1956) was recognized by Mills (1981). Owners of undeveloped land "exercise restraint in foregoing development opportunities n i/h high immediate returns in favor offuture options that are, in the final analysis, more remunerative," (Mills 1981, p.246). The use of a quasihyperbolic discount factor accounts for the propensity of landowners to have both a short run preference for instantaneous gratification and a long run preference to act patiently. The restraint alluded to by Mills (1981) could involve the use of commitment devices by farmers or landowners to prevent future selves from reversing a decision by the present self to not convert land in future time periods. Examples might involve the farmer enrolling in a cooperative agreement or a resource conservation contract, such as the Conservation Reserve Program, which requires landowners to commit their land to some rural use for a contractual period of time (Albaek and Schultz 1998; Gulati and Vercammen 2006). Modeling the traditional development model with timeinconsistent preferences has important policy implications. Consider the fact that expected net present discounted values for long term projects are infamously hypersensitive to the discount rate being used in the evaluation. Projects involving land development are hence acutely susceptible to this hypersensitivity. Not only does this affect land use policy, but the effects are important to policy makers who wish to maintain lands in either a developed or undeveloped capacity. Numerous studies have argued that the only effective deterrent to farmland conversion may be a policy of compensation to landowners for foregone development rent (Lopez, Adelaja, and Andrews 1988; Plantinga, Lubowski, and Stavins 2002). These rents and the compensation required could be greatly misrepresented under an exponential discount factor if preferences are timeinconsistent. Relevance: The TimeInconsistent Landowner There are many reasons why a landowner might be characterized by time inconsistent preferences. First, landowners like any other consumer or investor might exhibit preference reversals. Consider the following example: a landowner may prefer to contract his land to a developer for $1.01 million in 21 years, rather than for $1 million in 20 years. But when the contract is brought forward in time, preferences exhibit a reversal, reflecting impatience. The same landowner prefers to contract his land to a developer at $1 million today rather than sell for $1.01 million next year. The primary assumption driving the reversal is the discount factor for a fixed time interval decreasing as the interval becomes more remote. A nonconstant or decreasing rate means the discount rate in the short run is much higher than discount rates in the long run. Time inconsistent preferences imply the percent change in the discount factor depends on the time horizon, being steeper for the near future and flatter for the distant future. Second, a variety of institutional and government policies, such as growth management policies, may create a degree of impatience upon the landowner. A number of empirical studies have examined the effects of various development pressures on the timing of agricultural land transition and also on land values (Bell and Irwin 2002; Irwin and Bockstael 2002; Carrion Flores and Irwin 2004; Cho and Newman 2005; Livanis et al. 2006). The establishing of priority funding areas, for example, is seen to effect development times. Priority funding areas are growth areas that are designated by the county and receive financial support from the state for infrastructure development. The presence of growth areas could introduce a time inconsistency into a landowner's preference for conversion by creating a sense of impatience. Expectations of land reform may also affect the time consistency of preferences, which may include zoning, taxation, development rights, and clear definitions on the boundaries of urban growth. Third, landowners' characteristics have been found to influence their land use decisions. Barnard and Butcher (1989) hypothesize that landowner age, education, years of land ownership, income net of taxes, and expected increases in value and development time will have an affect on time preferences, thus influencing the landowners perceived net present value of land and their decision to sell. The authors conclude that not only are landowner characteristics significant but that they are more explanatory than the characteristics of the land itself (i.e., parcel size, distance from CBD, soil quality, etc.) in determining parcel level land sales at the urban fringe. Factor analysis indicates that the expected time until development is the single most important factor for distinguishing between landowners selling versus holding the land with those expecting a shorter wait being more likely to sell. While this result is not evidence of inconsistent preferences, it does suggest that the psychology of the landowner is a critical characteristic in the timing of land development as an intertemporal decision. Finally, since the land development decision is intertemporal in nature, knowledge regarding future returns to land in competing uses, as well as conversion costs, may be imperfect and uncertain. As time unfolds, landowners may rethink and revise the development decision. Acknowledging future revisions to the landowner's current decision could imply one of two things. First, the current decision might permit flexibility in the plan so that future revisions can be made. For example, the landowner may decide to sell rural land parcels only if an escape clause is written in the contract that allows the landowner to opt out with limited or no financial penalty. Second, the landowner may make the current decision to maintain land in rural use under a commitment device to avoid tempting offers that may present themselves at a future date, thus playing a strategic intertemporal game with himself. For example, the landowner might enroll in a conservation reserve program for a specified number of years, disallowing the conversion of land for a specified time period in exchange for some pecuniary payment. Since the landowner can continually update and revise the optimal plan, time inconsistent preferences offer an attractive method towards analyzing landowner behavior and the optimal development strategy across time. Theoretical Heuristic: The Development Decision This subsection aims to first provide a cursory perspective on how the use of a hyperbolic discount rate can affect the traditional model of land development. A simple net present value model is compared between the time consistent case and the time inconsistent case using the model outlined in Irwin and Bockstael (2002). This approach affords an intuitive understanding of the land development model and the effect of discounting regimes on conversion times. The time discounted path of the conversion value of land is the central question the model addresses. Suppose the landowner is in an infiniteperiod decision model and owns a quantity of land, 1, in some rural use at time t. The landowner receives a rent on rural land, R(1, t), from a use such as agriculture, forestry, or open space. For simplicity, the time path of rural returns is assumed to be constant over time, rather than increasing. The decision facing the landowner is to either keep land in rural use or to sell to a developer for conversion to an urban use for a one time return, V(l,T), at time T, equal to the sales price. The time path of the gross development return, V(l, T), is assumed to be rising over time due to, for example, rising population and increasing income per capital concurrent with a diminishing supply of unconverted land (Irwin and Bockstael 2002). The landowner may decide to keep land in a rural use for development in future periods, when the future urban return might be higher than the current urban return. There is a cost to the landowner of converting land at time T, C(I, T), which could include administrative fees, permit expenses, institutional costs, or necessary infrastructure expenditures (Irwin and Bockstael, 2002). The discount factor, 3, is taken to be equivalent to 1/(1 + r) for the time consistent case. The actual discount rate or rate of time preference is r, usually assumed to be the prevailing interest rate in the real estate market. Proceeding to solve for the optimal development time, the net returns to the landowner from converting land in time period T is given by: V(l,T) 8 R (,T +i)C (l,T) (14) 1=0 Equation (14) subtracts the costs of converting land, C(1, T), and the present value of forgone returns from rural land use, R(I, t), from the one time return to development, V(l, T) at time T. On the other hand, if land is kept in rural use in period T and conversion is postponed to T +1, the net returns from delaying conversion discounted to time period T are given by: R (1, T)+ V (l, T + 1) R (/l, T + i) C (l, T + ) (15) The first term in Equation (15) is the return from rural land in period T. The second term is the value of returns from conversion in period T +1 discounted to period T. The last term is the costs of converting rural land in period T +1 discounted to period T. In words, Equation (15) represents the expected dividends from rural use in time T plus the discounted net capital gains from conversion less foregone rural returns and conversion costs. Two conditions are required in order for the optimal conversion time to occur in period T. First, Equation (14) must be strictly greater than zero, meaning net returns from conversion in period T are positive. Second, Equation (14) must be weakly greater than Equation (15), meaning returns from converting rural land are greater in period T than in period T +1. In mathematical notation, these two conditions can be written as: V(l,T) 3{,' R(T+i) C(l,T)>0 (16) i=0 V(1, T) 2R(, T) c(1, T) 6V (l, T + i) c(, T + 1)} (17) The discount factor 3 can be substituted with 1/(1 + r) in Equation (17), where r is initially assumed to be the interest rate. The standard discounting case yields the result that the landowner will only convert rural land in period T when: V(1, T + 1) C(, T + 1) V(1, T) 2R(, T) C(I, T)} Equation (18) has the simple interpretation that conversion will occur when the rate of time preference, given by the interest rate, is greater than the percent change of development returns between the two periods. The discount rate, r, is assumed to be constant across time, implying the discount factor declines with the length of the time horizon involved, approaching zero asymptotically. If the assumption of constant discounting holds, the individual is said to exhibit a timeconsistent discounting pattern. More generally, however, r must not necessarily remain constant across periods and, in fact, there is no well founded reason to assume so other than for computational ease. Time inconsistency can be introduced into the model by with quasihyperbolic intertemporal preferences, implying a nonconstant rate of discount given by the discrete discount function {, / ., 8/. 2,..., /" j By substituting quasihyperbolic preferences into Equation (17) the inequality becomes: V (, T) 2R(1, T) C (,) >(/.).{V(l,T + 1) C (, + 1)} (19) Simplifying gives: p.{r(/,r +) c(,r+)} {(/, T)2R(1, T)C(,T)) r (1 While there is a similarity between the timeconsistent optimal conversion rule in Equation (18) and the timeinconsistent optimal conversion rule in Equation (110), note that the net returns received in period T +1 is discounted by the quasihyperbolic parameter f8, reflecting an immediate impatience effect on the landowner. The standard discount factor, 5, and the implied rate of constant rate of discount, r, represent consistent time preferences. However, the presence of / gives a measure of how intense immediate rewards associated with conversion are valued to more distant rewards. Clearly, the optimal time to conversion will depend on not only the values of the parameters in the discount factor, but also on the time horizon. The form of Equation (110) expresses the sensitivity a landowner may have to time delay, an effect not expressed in the standard exponential discounting case. The formulation also expresses quite simply how the landowner would be engaged in an intertemporal tussle as to the development decision. This tussle can be seen by the fact that the parameters / and r appear on different sides of the inequality, in effect revealing the counterbalancing force each parameter has on the discount formulation. The opposing individual tradeoffs between present and future conversion are represented by the different values the discount parameters may have. While the simple theoretical heuristic presented above is intuitive, more intricate theoretical methods are needed to solve the land development model with hyperbolic discounting in order to circumvent the time inconsistency problem, which violates traditional neoclassical assumptions and optimization theory. The mainstream approach to solving economic problems involving time inconsistent preferences uses dynamic game theory to find sets of subgame perfect equilibrium. The decisionmaker is viewed as a collection of subindividuals, having different rates of time preference at distinct points in time, and so plays a series of multiplayer intrapersonal games. The game theoretic approach has been prevalent in the economic literature being used by Peleg and Yaari (1973), Laibson (1996, 1997), Barro (1999), Gruber and Koszegi (2001), and Harris and Laibson (2001, 2004). More recently, Caplin and Leahy (2006) have proposed a recursive dynamic programming approach to obtain optimal strategies under time inconsistency in finite horizon problems. There approach could be generalized for the infinite horizon case to apply to the farmland values problem. Study Objectives With the insights, evidence, and questions discussed above, it is important to know whether the traditional model of land values and derivative models of ruralurban development can continue to be depended upon by economists. Despite the large body of research examining the net present value model of farmland values, few investigate the role of time preferences and none consider the presence of timeinconsistent preferences. The fundamental objective of the current study is to partially fill this void. This dissertation has three central components. First, the notion of timeinconsistency and hyperbolic discounting is related to a critical issue in the agricultural economics literature, rural to urban land conversion and development. Models of land conversion that describe the timing and intensity of development are derivative of models of land values. Therefore, an understanding of the role that time preferences play in a model of land values has a direct impact on the understanding of land development models. The current chapter presented the intuition of why one might suspect the farmer or landowner to be characterized by quasihyperbolic discounting and discussed the relevance and motivation in the context of land conversion. Second, the paper discusses and reviews the vast literature pertaining to the determination of land values and the development decision from an economic perspective. Chapter II presents an examination of the literature on land values and ruralurban land conversion to foster a greater understanding of traditional models of land values and development before timeinconsistencies are introduced. Such a review of the current state of knowledge is lacking in the literature. The theoretical models are described with an emphasis on the three most prevalent approaches to modeling the development decision: capitalization, option value, and transaction cost methodologies. The most common econometric techniques used are discussed followed by an assessment of the potential future research. Third, the paper enters the ensuing debate regarding the reliability of net present models to adequately explain farmland values, which is the focus of the remaining chapters. Chapter III presents the fundamental theoretical model of this paper and develops a structural model of land values with quasihyperbolic discounting and describes the econometric procedure used in the analysis. Many studies have rejected the standard present value model under rational expectations as a viable model for explaining farmland values using both domestic and international data. Previous inquiries into the nature of land values assume a timeconsistent discount factor. This dissertation introduces intertemporal inconsistency into a model of land values by including a quasihyperbolic discount parameter in the asset equation. Significant evidence is found in favor of quasihyperbolic discounting in U.S. agriculture. The Generalized Method of Moments estimator is used to obtain estimates of the parameters, overcoming problems of heteroskedasticity and serial correlation, and provides consistent and robust estimates. The data used in the analysis and the estimation results are discussed in Chapter IV. Aggregate panel data are used for agricultural asset values from 19602002 for nine agricultural regions of the United States including the Appalachian states, Corn Belt states, Delta states, Great Plain states, Lake states, Mountain states, Northeast states, Pacific states, and the Southeast states. The results for each region are discussed distinctly and then comparisons are offered with other studies that have estimated structural models under quasihyperbolic discounting. Finally, Chapter V will conclude the main content and key findings. Intuition on why an agricultural asset, such as land, may be characterized by timeinconsistency in general and quasi hyperbolic discounting is discussed. Ideas for potential future research will be offered. 1 : 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0  Time (years) Figure 11. Exponential discounting *Hyperbolic 1  Hyperbolic 2 Time (years) Figure 12. Hyperbolic discounting X  Exponential 1 SExponential 2 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 z L ~ 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 O Figure 13. 1 0.9 =0.8 0 .7 0.6 0.5 e0.4 S0.3 >0.2 0.1 Time (years) Quasihyperbolic discounting  Exponential 1  Hyperbolic 1 QuasiHyperbolic 0 Time (Years) Figure 14. Comparison of discount factors  QuasiHyperbolic 1  QuasiHyperbolic 2 CHAPTER 2 LITERATURE REVIEW Introduction The economic question on the rate and timing of land conversion has been the subject of much analysis over the past forty years. Development of rural lands to urban uses, such as converting farmland to residential housing, has been a prominent issue in the economics literature. Once soaring and now falling residential land prices, rapidly growing urban areas, and the sprawling and discontinuous nature of land development around the urban fringe have all served to stimulate interest in the subject by academic researchers from various fields such as urban and regional economics, rural sociology, and urban planning. The timing of land development is also of key interest to many beyond the research arena such as local chambers of commerce, planning authorities, developers, and landownersall of whom are stakeholders in any future project to convert land. The critical question the decisionmaker faces is: what is the optimal time to develop land? Not only does the decisionmaker have a choice as to the form of land use, such as agricultural farming or residential housing, but also a choice as to when the land should be put to such use. Economic doctrine says that this decision is made so that the use of land and timing of conversion maximize the net wealth of the landowner. Since land use and land values are intertwined in the development decision, any model of land development must examine how land values and use affect the development decision. The approach in this chapter emphasizes the role of value and use in land development models. The purposes of investigating land development models and obtaining optimal conversion rules are many. First, they provide a better understanding of land values in terms of prices, rents, and also capital appreciation (Clarke and Reed 1988). This is important since, unlike many other investments, land development tends to be irreversible. Furthermore, unlike many other resources and commodities, there is only a finite availability of land. Any decision to develop land, residential housing for example, depletes a fixed supply of land available for an alternative use. Second, governments often wish to create policies to encourage or discourage the conversion and development of land. Government policies take the form of property taxes, easement requirements, and conservation initiatives, just to name a few. Any government policy can have unintended consequences. For example, Titman (1985) shows that a policy restricting building height may actually result in a greater number of buildings constructed. Accurate and efficient policy instruments cannot be designed without an understanding of land development models and a thoughtful consideration of the timing of development decisions. Third, a better knowledge of the conversion process facilitates an improved understanding of seemingly random outcomes of conversion, such as discontinuous urban development patterns (also known as urban sprawl), heterogeneous spatial patterns in land use, and the creative destruction of capital structures. Still in need of investigation is the appropriate timing of land conversion and the optimal intensity of development. When is it optimal to convert land from a lower intensity rural use to a higher intensity urban use? What is the optimal intensity of capital that should be applied to the land development project? How do government policies, such as property taxes and growth controls affect the optimal timing and intensity? What effects do changing discount rates and expected returns imply for land development? How does an uncertain world alter matters? Many factors affect the optimal timing rule, such as the addition of a property tax or interim rent received from temporary use of the land. Further, both uncertainty and imperfections in the land market will affect the intertemporal conversion decision. For example, landowners tend to make decisions under imperfect information, especially when it comes to forecasting prices or interim rental rates. Many studies have also relaxed the assumption of certainty (Ellson and Roberts 1983; Clarke and Reed 1988; Capozza and Li 2002; Schatzski 2003) allowing land rents, conversion costs, or discount rates to be stochastic. Effects on rents, costs, discounting and other complications on the conversion decision are examined in more detail later. Temporal variation in decisions regarding conversion by owners of neighboring land is of particular concern. When returns from development appear to be high, some landowners defer development while other owners in similar situations decide to develop their land. This variation creates scattered and discontinuous development patterns, especially along the ruralurban fringe, often referred to as urban sprawl. These questions are addressed in this review, presenting a survey of models and methods in the land conversion literature spanning the past 40 years. The next section discusses the theory in land development models, addressing economics of land values and land use. The static and dynamic theory of land values and development are explained with an emphasis on how competing uses of land affect how and when land is developed. In the third section, the most pervasive approaches to modeling land values and development are discussed including the capitalization approach, the option values approach, and the transactions cost approach. The fourth section reviews the empirical approaches and methodologies to modeling the land development process. The final section concludes. Theory You do not have to be a farmer, or even an astute economist, to know that land is a productive input. Not only is land a factor of production for agriculture, but also for openspace and urban development. Landowner wealth has shifted over time as land prices, and in particular farmland prices, have fluctuated. As Schmitz (1995) notes, farmland values have been subject to 38 boom/bust cycles, just as the stock market has, creating wide variations in the wealth of land owners. Variations in wealth are also caused by competing uses for land, particularly from agricultural and urban uses. The rapid rate of urban growth has led to soaring rates of land values and landowner wealth causing land initially used for farming, openspace, or other rural uses to be bid out of such use and converted to more capital intensive urban uses (Moss and Schmitz, 2003). Agriculture has been fading away from the landscape because returns from farming are unable to compensate for skyrocketing land prices. This has been the case for land located along the ruralurban fringe, which is especially sensitive to increased urbanization. This section presents a broad perspective on the economics of land values with a particular emphasis on land located along the ruralurban fringe. Land is discussed in terms of utilization, with rural use referring to agricultural, forestry, or open space use, while urban use referring to residential, industrial, or commercial use. An excellent and extensive examination can be found in Moss and Schmitz (2003) in regards to farmland. Indeed, there is no dearth of studies on land markets and the factors that affect land values. Rather than provide an interminable account on the plethora of economic models and empirical methods, a succinct discussion on the most relevant and informative issues will be presented. A basic theoretical treatment will be given first, followed by a comparative discussion of the most contemporary used models in the literature. Land Values and Use The key to understanding how land is valued means a proper understanding of the land market, and in particular the market for rural land. The following traits best characterize the rural land market: heterogeneity, localization, segmentation, high transaction costs, and imperfect information. First, land is not homogeneous; parcels of land differ in size, geographic location, landscape, quality, and other physical characteristics. In this sense, a parcel of land is 39 very unique and cannot serve as a perfect substitute for another. Second, the land market is highly localized and permanent since the location of land itself is fixedit cannot be moved or transported. In other words, land is a permanent asset. Further, the buyers and sellers of land tend to be limited to a geographic region, though distance buying of land has become more common. Also, land is durable and in some contexts can be considered indestructible. Once land has been allocated to a particular use, such as housing, it tends to remain in that use. As such, land development is often assumed to be irreversible. Third, land has a finite and fixed supply since new land cannot be produced. In an extreme longrun view, total land supply is perfectly inelastic and thus is characterized by a vertical supply curve. However, particular uses of land may be described by a downward sloping supply curve. Also, countries, states, cities, and other geographic bodies can increase the supply of land by extending their borders. However, this means that some other geographic body must reduce their supply of land by an equivalent amount. Further, land can be used in varying levels of intensity thereby increasing the effective supply. Fourth, land use tends to be mutually exclusive. For example, the allocation of land to agriculture precludes land from being used in residential housing or industry. While there are some exceptions to this (e.g. high rise buildings combing commercial and residential space), generally a unit of land cannot be allocated to multiple uses. Fifth, the land market is segmented since it is divided into many submarkets or market segments depending upon geography, land use, ownership, and property rights. Within each market segment, a parcel of land may have a different price. On the demand side, users of land require very specific land types and locations. For example, a farmer cannot use a parcel of land downtown or a land zoned for commercial use anymore than a retail store has use for land hundreds of miles from a residential area. Sixth, the land market is also characterized by high transaction costs. A buyer and developer of land may face zoning restrictions, titling costs, and survey fees before development can be begin. Seventh, land is both a consumption good and an investment good (Kivell, 1993). An individual may purchase land solely for the utility of owning, possibly for open space or recreational purposes. Likewise, an individual may purchase land for the intention of earning a return from uses such as farming or forestry, or may hold onto the land for later development with the expectation of a higher rate of return. Eighth, uncertainty and less than perfect information are inherent in land transactions. Future rents and returns from land use and development are not known with certainty and both future demand and supply must be forecasted. Risk perceptions and preferences vary widely between landowners. Last, land development tends to be characterized by long delays. For example, the time between the purchase of vacant land and completion of high intensity urban construction can extend years. How land should be used, or the land allocation problem, has been the subject of much research. Land is used is postulated to be based on its value or the rent it accrues to the owner. David Ricardo first suggested that land would be allocated according to the soil quality. The more productive land would be allocated to farming and agriculture, while less quality land would be allocated to more urban uses. Differences in fertility or productivity determine how much rent the land generates. The most productive lands are used first with increases in demand forcing less fertile lands into use. This places an advantage on owners of more fertile land over owners of less fertile land, equivalent to the value of the difference in the productivity of land. Location theories of land rent were developed by Johann von Thunen who believed that the distance of land from the core of a city, or the central business district (CBD), would be the primary incentive behind land allocation. The closer a unit of land is to the CBD the more valuable it becomes. The idea is best described as a bidrent model, which assumes different uses of land have unique bidrent curves. Each bidrent curve is unique since the slope depends on each parcel's location from the center of the city. Proximal land units have lower transportation costs, resulting in a savings over more distant land units, resulting in a bid for locations. Both Ricardo and von Thunen were speaking in terms of agricultural uses for land. However, notable contributions by Alonso (1964) and Muth (1969) have extended these notions to account for urban uses. The bidrent theory of land uses and values can be described graphically in Figure 21. In a free market, the land between the city center and location A will go to urban uses and more distant land will go to rural uses. Thus, point A represents the socially optimal allocation between the two uses. The bidrent model emphasizes the tradeoff between land uses based on the high rents of land in the central region and the costs of transportation incurred by more distant locations. The equilibrium described in Figure 21 is the result of demand and supply forces in the land market at work. In a perfectly competitive market, the supply of land is given by the quantity of prospective individual parcels that compete against one another for potential users or tenants. On the demand side, potential users compete against each other for use of each parcel. In equilibrium, the optimal allocation maximizes the total value of all land with each parcel being used at its highest and best use. In other words, each parcel is used in a manner that is most productive for that given geographic location. However, the value of land is based not only on returns from the land itself, but also on products, such as crops or urban dwellings, produced using land as an input. For example, higher prices for residential housing or agricultural crops will drive up the price for land. Since land offers both production and consumption of goods (e.g., forestry products) and services (e.g. recreation) the demand for land is a derived demand.1 A common theoretical approach to modeling land valuation is based on the residual value of land, defining the market value by the net land residual income or rents. The difference between the value of what is produced on the land and the production costs yields the net land value, if the development of a hypothetical project is the highest and best use for a given parcel. A Simple Static Model The nature of the development decision can be illustrated using a simple static maximization problem where a landowner must decide between two alternative uses of land. Suppose the landowner has a fixed amount of land, L, and two competing uses. Further suppose that land is currently in agricultural use yielding a return of a per unit of land and using l land units. Alternatively, the landowner can convert his land to residential housing, yielding a per unit return of h, which uses 1 units of land. Similar in spirit to the analysis in Bell, Boyle, and Rubin (2006), linear returns are assumed implying returns to a particular land use do not depend on the amount of land being used. The assumption of linearity allows net returns to agriculture to be defined as A = a* l and net returns to residential housing to be defined as H = h 1 The maximization problem is: Max, ,1 V= A+H (21) subject to (a + = L) & (0 The solution to the maximization problem given by Equation (21) is a corner solution, implying land should be wholly dedicated to the use yielding the highest net return. If agriculture is more 1 Derived demand is when one good or service occurs as a result of demand for another. For example, the demand for housing leads to derived demand for land since land must be developed for housing to be consumed. profitable than housing, meaning a > h, then all land should remain in agriculture, la = L, and none in housing, = 0. Likewise, if farming yields a lower return, then the land should be converted into residential housing. Diminishing returns, rather than linear returns to land use, may be a more realistic assumption, especially in farming where the returns per a unit of land depend on how many parcels currently in use. One can also easily see diminishing marginal returns to many urban projects. The classical example, given by Alfred Marshall and reiterated by Shoup (1970), involves the construction of a skyscraper. As the height of the building increases, certain production and structural elements of the building become necessary, such as additional equipment like specialized cranes and elevators. Additional production and structural items cause the price per square foot of rental space to increase relative to future returns. Assuming declining marginal returns implies declining net returns with the number of units in use. Mathematically, diminishing returns to agriculture is given by / > 0, /2 <0, and diminishing returns to housing is given by H/ > 0, O2H/ <0 The conversion decision now becomes one of allocating the land between the two uses until the marginal returns are equal, or when: 0A =QH (22) Graphically, the solution represented by Equation (22) is depicted in Figure 22 and emphasizes the point of land being shifted to the use providing the highest marginal value product. The static models of both linear and diminishing returns are overly simplistic. The conversion decision in reality involves a number of factors, discussed in more detail in the next section. One factor affecting conversion is the intertemporal nature of the development decision. While the static model describes how will be land be developed, the model does not address when land will be developed. Other factors complicate the decision, for example property taxes can be introduced into the problem affecting the timing of the conversion decision. The issue of land taxation and land conversion has been addressed by many (Skouras 1978; Bentick 1979; Arnott and Lewis 1979; Anderson 1986). Costs also affect the conversion decision since conversion requires an investment in capital development. Most importantly, the environment surrounding the conversion decision is in reality an uncertain environment. The returns from land are seldom known with certainty, which can have a large impact on development decisions. The issue of uncertainty has been at the very crux of recent research (Capozza and Helsley 1990; Capozza and Li 1994). Models of land conversion under uncertainty inevitably draw upon the investment under uncertainty literature and, in particular, option pricing theory. Models of land conversion can only be examined in a serious way in an intertemporal dynamic context. A Simple Dynamic Model Dynamic theories of land development acknowledge the effect future economic expectations have on current land values and how land is allocated for certain uses. This inevitably results in a more focused discussion on the timing of land use change. In a dynamic model, the value of land for a particular use is represented by the discounted present value of expected returns from that use. The simple static model can be extended to account for the intertemporal nature of land allocation and conversion. Let A(t) be the net return from a given unit of land in period t for land in preconverted or agricultural use. The term A(t) can equal zero if the land is vacant. The net return per unit of land in period t for land in postconversion or residential housing use is H(t, T), which depends on the time of conversion given by T. Conversion time is chosen to maximize the value of land, given by V(T): T Max V(T)= JA(t)e dt + fH(t, T)e dt (23) 0 T The landowner's discount rate is given by i, often taken to be the prevailing interest rate in the real estate market. The first term on the right hand side of Equation (23) is the present discounted value (PDV) of agricultural returns from the start time to the conversion time. The second term on the right hand side of Equation (23) is the PDV of housing returns from the time of conversion onward. The level of postconversion returns depends on both the time when housing returns occur, but also on when the conversion itself occurs. The first order condition with respect to the conversion time is given by: A(t) H(t, T) + 8 H(t, T) (T)dt = 0 (24) aT The third term on the left hand side of Equation (24) is equivalent to the PDV of expected future changes in housing net returns. If the time of conversion is irrelevant to the returns from housing development, then the third term on the left hand side vanishes and the conversion rule is equivalent to the static case. In other words, the optimal conversion time T* occurs when the net return to agricultural use is equal to the net return from housing use, mathematically represented by A(t) = H(t, T*). There are reasons why the third term on the left hand side of Equation (24) is nonzero, as explained by Bell, Boyle, and Rubin (2006). For example, costs may increase over time, delaying conversion until a later time, given by T**. The optimal rule with conversion costs is to convert when A(t) = H(t, T**) i H(T**). The accounting for cost results in a downward shift of the housing net return function, illustrated in Figure 23. The case where an increase in housing returns, perhaps due to population growth, results in an upward shift, hastening conversion. The flow of returns can also be viewed in conjunction with the capitalized value of land in competing uses. In a dynamic model, the value of rural or unconverted land is determined by the expectation of future returns. Figure 24 describes Equation (24), representing the maximum value of V(T) which occurs when RA equals RH. The line RA represents the returns from land in agricultural use assumed to be constant.2 The return from housing or postconversion use is given by RH. Assuming that conversion takes place at time T, VA represents the capital value of land in current use (agriculture) and VH represents postconversion use (housing) value. The curve V, is the sum of values before and expected values after conversion. Although the development value of housing is greater than the value of agriculture after time t, conversion occurs until the income returns from housing exceed that of farming at time T. Conversion occurs before time T because RH is expected to exceed RA prior to time T, implying that VH exceeds V The expectation of returns explains why rural lands along the urban fringe have a greater potential value to urban uses over farmland uses (Neutze 1987). At time zero, the present value of returns from agriculture is given by the vertical distance OVA in Figure 24. This value declines over time as the optimal date of conversion approaches. As Goodchild and Munton (1985) and Neutze (1987) explain, conversion can be delayed but at an opportunity cost of the forfeited postconversion housing rent, RH. 2 This implies that agricultural products are sold in large markets and is both a reasonable and simplifying assumption. Similar graphical analysis can be found in Goodchild and Munton (1985) and Neutze (1987). 47 The dynamic analysis underscores two key points. First, the optimal time to convert land is based on the premise that a decision to convert now is balanced against a possible decision to convert in the future at a higher return. Second, allocations of land to particular uses and land values depend on both the present economic conditions as well as future expectations. Dynamic models of the land development decision are the mainstream method to explain the forces that compel a landowner to convert land. The dynamic model so far assumes a simple setting without any taxes, land use laws, or growth controls. Further, the model does not explain the capital intensity of development nor does the model address conversion costs. Perhaps most paramount to the development decision is the discount rate and the presence of uncertainty, since both have profound affects on the timing of conversion and intensity of development. More complicated dynamic models are examined later as well as the primary models of the development decision in the current literature: the capitalization approach, the option value approach, and the transaction cost approach. Modeling the Land Development Decision The theoretical foundation for the land development and conversion literature did not receive rigorous economic analysis until the late 1970s and early 1980s. Most currently used models, largely based on the seminal papers by Shoup (1970) and Arnott and Lewis (1979), describe the optimal timing problem in a Wicksellian framework on the optimal timing of wine maturation.3 Wicksellianbased partial equilibrium models utilize discounted cash flows to describe land values and the timing of conversion and focus primarily the effects land taxes have on land values and on the development process.4 The development decision is also influenced 3 Wicksell, K. (1934). Lectures on Political Economy, Volume I: General Theory (translated by E. Chassen). New York, NY: The MacMillen Company, pages 178183. 4 The partial equilibrium models also include Rose (1973), Skouras (1978), Douglas (1980), Mills (1981b), Anderson (1986), Bentick and Pogue (1988), Anderson (1993), and Arnott (2005). 48 by conversion costs, the discount or interest rate, land rents from rural and urban uses, and assessed property value taxes and capital gains taxes. More recent theoretical models focus on a general equilibrium approach (Markusen and Scheffman 1978; Ellson and Roberts 1983; Capozza and Helsley 1989; Kanemoto 1985; McMillen 1990), questioning the conclusions drawn from a ceterisparibus partial equilibrium framework. For instance, the simplifying assumption of a Marshallian land demand curve in Shoup (1970) implies monotonically growing demand over time. Therefore, the land value function V(T) is strictly concave by assumption, presupposing results not guaranteed in a general equilibrium model. For example, the presence of a property tax will reduce the length of time an individual will hold land, thereby hastening land conversion (Markusen and Scheffman 1978). A general equilibrium model has additional advantages over a partial equilibrium model, allowing the effects of uncertainty to enter into the land development decision. Ellson and Roberts (1983) introduce uncertainty in the context of a government's land zoning decision and find slower rates of conversion compared to the certainty case. As a disadvantage, the added complexity of a general equilibrium model leaves many comparative static results untenable, or in the very least ambiguous, reducing the model's predictive power. A brief summary of the main theoretical results from both partial and general equilibrium capitalization models is provided in Table 21. While some results appear heterogeneous, disagreement on the effects of particular exogenous variables, such as taxation, is noticeable. Inevitably, introducing uncertainty to the land development process involves a foray into the investment literature. Once uncertainty is accounted for in the land conversion process, the value of land and the decision to develop becomes an option value. The option value approach has gained recent attention in the literature and shows the most promise for modeling the land development process. Transactions costs can have effects on the development decision and land values as well. Such approaches account for the pecuniary and nonpecuniary costs associated with conversion. Nonpecuniary costs include legal fees for petitioning public officials to rezone the land, promises to dedicate portions of developable land for public use, openspace, or conservation programs, or costs associated with lobbying efforts and campaign contributions. The Capitalization Approach The capitalization approach forms the bulk of the literature on the land development decision and assumes land values are solely determined on the basis of expected future income. The simple static and dynamic land allocation theory outlined in the previous section does not make a clear distinction between the price and rent of the land. Evans (1983) distinguishes the price of land and the rent received by land. Land price is what is paid for the ownership of land, while land rent is paid by the tenant or occupier for the prevailing usage of the land (Evans 1983). The relationship between the two concepts of price and rent is fundamentally based on individual expectations. The price paid for land will be a function of the discounted rents expected to be received in the future between land uses. Put simply, the value of land is a capitalized value of future rents. The capitalization model is equivalent to the dynamic model discussed earlier where current and expected future returns are discounted according to some discount factor, usually the interest rate. The analysis is made simpler by the use of summation signs instead of integral signs. The most general model of the capitalization approach is given by: V=, {E{[]E J} (25) 1=0 The value or price of land at time tis V, the discount factor is 8', and E, [R], ] is the expectation at time t for returns (or rents) from land in rural use at time t + i. To obtain the optimal conversion decision, the model is extended to account for a competing use, such as recreation or urban development. Assuming conversion is irreversible and extending Equation (25) to account for competing using, the model becomes: c =Max (3 E[RE, ])+ ( (3' E, [U, ) (26) S=0 j=T+1 J The value of land considering conversion potential is VJc, T is the time land is converted to urban use, and E, [U, j is the expectation at t for the rental rate from converted urban land at t + j. By comparing Equation (25) to Equation (26), the value of land with conversion potential Vy, exceeds the value dedicated to rural use Vy. The key assumptions implicit in Equation (26) include costless conversion, a constant discount rate, risk neutrality, perfect information, and no uncertainty. The formulation for conversion potential given by Vy, in Equation (26) is an oversimplifying case that defines a point of reference for studying further models with varying levels complexity. The time path of land conversion for land on the urban fringe is often described graphically as an Sshaped curve, depicted by curve VV in Figure 25. The rate of increase in the value of land, or its conversion value, is declining, as depicted in Figure 25. A declining rate of increase in the conversion value implies the base value of land is always greater than the additional value that accrues to the land through time, a necessary condition for conversion to occur. Otherwise, the landowner or decisionmaker would delay conversion indefinitely. The slope of the PP curve in Figure 25 is the discount rate. The PP curve slope can also be modified to account for property taxes, interim rents, and other effects. At the optimal time to develop, the VV curve is tangent to the PP curve. Before the optimal time is reached, there exists an incentive to hold land and delay conversion, possibly for a more capitalintensive and profitable development or due to profits from interim rents (Fredland 1975). Delineating from simplifying assumptions implies, however, that not all land may be converted at the optimal time. The presence of imperfect markets or knowledge, transactions costs, uncertainty, barriers to entry, and varying rates of time preference described in the discount rate will cause a degree of heterogeneity and dispersion in the actual conversion times. Further, the lumpy nature of the VV curve, represented by various rapid and slow increases, reflects a variety of factors affecting the demand for land and, more specifically, for land development (Shoup 1970). For example, the rezoning of land or the construction of nearby highway systems or utility services may suddenly make the land more palatable for conversion. More private amenities, such as local shops or golf ranges, could have the same effect. Conversely, the conversion value may rise more slowly or even decline due to negative developments such as prisons, reclamation facilities, or lowincome housing. The key insight is that the probability of conversion increases with rises in land value, population, and income. Dynamic models of land value serve as the basis for most of the theoretical and empirical work on land conversion and development, and vary in complexity. Shoup (1970) explains that growing numbers of inhabitants and rising incomes will escalate the demand for vacant land in a way that increasingly more intensive use of land will be needed. For example, population and wealth may increase requiring a capitalintensive development project, such as highrise construction, to satisfy future demands. Shoup (1970) also explains the effect of irreversibility on most land developments. Converting land from lower uses, like agriculture or open space, to higher uses, like residential housing, requires lasting changes. For example, zoning regulation, land subdivision, building construction, and changes in the geographic landscape can usually not be undone. The irreversibility of land development makes converting back to a lower use costly, if not impossible. Once conversion has occurred, land and capital are committed to the higher use (Shoup 1970). These facets of land demand and conversion imply that the form of land development, especially in terms of density and intensity, are intertemporal in nature. As time progresses, and the decisionmaker waits to convert land to some later time, an intertemporal tradeoff occurs between low capitalintensive conversion and low land payments versus high capitalintensive conversion and high land payments (Shoup 1970). In other words, the decisionmaker will be in an internal tussle to either delay or hasten conversion. The observation of a tradeoff makes keenly apparent the importance of certain behavioral aspects of the decisionmaker. In the context of land development, the decisionmaker can take on a number of roles: the developer, the investor, or the landowner. The intention of the decisionmaker is to conserve the land, holding onto it to ensure it is not converted too hastily to some lower use when a future date may call for more capital intensive development due to demand increases (Shoup 1970). Thus, expectations regarding future demand for more intensive uses of the land will play an important part in determining the optimal time of development. The discount rate Perhaps no other facet of the land development decision than the choice of the discount rate used in the model is so vital to the determination of accurate land values and the optimal conversion time (Gunterman 1994). In Shoup (1970), the problem of the decisionmaker is to maximize the present discounted value of land. The simplest analysis begins by assuming no conversion costs and that land is vacant. The vacancy assumption implies that the land is not 53 being used for any temporary revenue generating activity before conversion. The present value of land can be described as: P(t, T)= V(T)e r(Tt) (27) The present value of land at time t for a future conversion time Tis P(t, T) The vacant land's conversion value at a future date T is V(T), if the land is converted to its optimal use at date T. The discount rate is r, which assumes the role of opportunity costs to the owner of vacant land. Although not explicitly defined in the analysis by Shoup (1970), the discount rate is often taken to be equal to the interest rate in the real estate market. The decisionmaker then wishes to maximize the land's present value in regards to the conversion time: P(t,T) [V(T)e )]= 0 (28) OT OT Solving Equation (28) yields the first order condition that the optimal conversion time occurs when the following is satisfied: V(T) = r (29) V(T) The left hand side of Equation (29) is the rate of change in the conversion value. The solution states land is converted when the rate of increase in the conversion value of land equals the interest forgone on other possible investments, given by the discount rate. The discount rate has been generally found to have negative effects on land values (for clear reasons) and tends to accelerate the land conversion process (Ellson and Roberts 1983; Capozza and Helsley 1989). Arnott and Lewis (1979) found an indeterminate effect of the discount rate, assumed to be the interest rate. The ambiguous nature of the discount rate in the Arnott and Lewis (1979) analysis stems from the simultaneous relationship between the output elasticity of capital, the interest rate, and the expected rate of growth of rental rates. Still, critical questions regarding the impact of the discount rate have not been seriously examined. Will the optimal timing of conversion be different under a nonconstant discount rate? What are the property tax and other welfare implications? These remain important questions yet to be addressed by the literature. Land rents and conversion costs Often times land development along the ruralurban fringe occurs not on vacant land, but on land that is currently engaged in some interim use, such as from agriculture or forestry. This case is more realistic than the case of vacant land since undeveloped land is often held by farmers or investors who temporarily use the land until the optimal conversion time. Now the value of the land is increased by rents received from the temporary improvement. In terms of the model described by Shoup (1970), the optimal time for conversion with interim rent is given by: T F(T, T) +ert' e F(i, T)/dt di V(T) =r (210) V(T) V(T) The second term on the right hand side of Equation (210) is the rent received from the interim use, F (i, T), over the whole time period. The effect of a positive interim use is to delay the conversion time since the capitalized present value of the land is higher when current rents are received. This model can be extended to include the future rents from an urban development at time T. Interim rental rate are received until the terminal time. Due to the irreversible nature of most development projects, conversion is assumed to be permanent so that at the conversion time no more rent can be generated from the rural use. Rents received either before or after conversion will increase land values. Generally, increases in the preconversion rent will slow the development process, while higher postconversion rents will speed up the conversion time. There are some notable exceptions, for example in Markusen and Scheffman (1978) post conversion rents do not affect the development decision since their model only has twoperiods. In Ellson and Roberts (1983), Capozza and Helsley (1989), and Anderson (1986, 1993), increases in the urban use rental rate have an ambiguous effect on the conversion time. In Anderson (1986, 1993), the indeterminacy arises from the model accounting for both pre conversion and postconversion tax rates. The tax rate parameters enter into the discount rate, requiring the magnitudes of the parameters be known in order to sign the postconversion rental rate effect. Additionally, Anderson (1986, 1993) assumes that the partial derivative of the urban rental function with respect to the conversion is zero, implying that income received from a developed property is independent of the time of development. The assumption of independence of developed rental income from development time is not realistic. Both Ellson and Roberts (1983), and Capozza and Helsley (1989) utilize a general equilibrium approach to the land development decision, which requires simulation in order to determine the direction of partial derivatives. The effects of conversion costs are also quite similar across studies. With higher costs of capital, the price of land will increase to offset the higher conversion expenditure. However, many studies do not explicitly account for conversion costs, usually assuming that capital expenditures are implicitly accounted for in the rental function for urban developments (Shoup 1970; Anderson 1986). Ellson and Roberts (1983) find that conversion costs in the presence of uncertainty will increase the time land is rezoned for development and an urban infrastructure is put place, but reduces the duration between rezoning time and the point developed land is actually consumed or converted. Another exception is Arnott and Lewis (1979) who take the vacant land conversion model from Shoup (1970) and develop it more rigorously by adding construction costs and residential density (i.e., capital intensity) to the problem. The key assumptions the authors make are: zero interim rents from temporary land use, zero fluctuations in capital costs and rental rates, zero property taxes, and zero building depreciation costs. Like Shoup (1970), Arnott and Lewis (1979) assume once conversion occurs the land remains in the developed use permanently and that perfect foresight exits. Unlike Shoup (1970), however, Arnott and Lewis (1979) treat rents and conversion costs in terms of capital, distinctly. The conversion that Arnott and Lewis (1979) consider is a more specific one of vacant land being converted to residential housing. The objective of the landowner in their model is to maximize the difference between the present value of housing rents and the present value of conversion expenditures with respect to the time of conversion and the amount of capital needed: max L(T,K) = r(t)Q(K)e dt pKe (211) T,K T The present value of each land unit developed at time T with stock of capital K is given by L(T, K). The output of residential housing on each land unit with capital K is given by Q(K). The rental rate on a housing unit at time t is given by r(t). The discount rate, equivalent to the interest rate, is given by i, and the price per a capital unit is given by p .5 Optimal conversion time is delayed when the capital price p increases but is hastened when the current housing 5 Arnott and Lewis (1979) assume rental rates on housing units do not fluctuate, but rather are constant through time, and define 7 as the housing rental rates expected rate of change. Solving the above partial equilibrium model, the authors conclude that land is developed optimally when: V(T) _i P(T) i That is, land is converted from a lower use to a higher use when the ratio of the value of land at time T given by V(T) if land is optimally developed, to the value of the property (including the land and buildings) equals the ratio of the growth rate of housing rental rates to the discount rate. rental rate r(t) at time t = 0 increases (Arnott and Lewis 1979). The density of housing construction is not affected by higher capital outlays, as developers delay construction to compensate for the extra costs. The expected growth rate on the housing rental rate has an ambiguous effect on timing, but increases the density of development. Many of these results change when uncertainty is brought into the analysis. Clarke and Reed (1988) investigate the development decision under uncertainty and find that conversion is hastened when construction costs are expected to increase, however the conversion time is indeterminate when either rental rates or the discount rate is expected to increase. When increased rental growth is expected or a decrease in growth of costs or discounting is expected land values, housing density, and the ratio of the value of undeveloped land to the value of developed land all increase (Clarke and Reed 1988). Simulations based on parameter estimates from Arnott and Lewis (1979) reveal that for the most part their model offers a reasonable description of the land development process. However, both simulations on structural density and the discount rate based on results from Arnott and Lewis (1979) reveal that a much higher discount rate (i.e., 4.1%) would be needed to replicate results under higher levels of uncertainty. Models that account for uncertainty are not adequately examined in a capitalization context and require more sophisticated analysis, however. More complicated models involve the use of real options theory and the investment under uncertainty literaturethese topics will be addressed later. Land taxes One of the most investigated and controversial issues have been on the effects of a property tax and a capital gains tax. The property tax is often referred to as an ad valorem or land value tax (LVT) and is assessed based on the value of the property site, ignoring any improvements to land such as buildings and personal property.6 The capital gains tax, or development value tax (DVT) is levied based upon the actual developments and improvements made to the land. A significant debate ensued in the literature regarding whether or not a tax on land is neutral, meaning a land tax does not alter the allocation of land to different uses. This is the view originally taken by David Ricardo. However, Henry George purported that a land tax would remove land from speculators, transitioning land from future uses to current uses, hastening the conversion process (Bentick and Pogue 1988). This section reviews the tax effects in the theoretical land development decision models and how the effect is sensitive to the specific nature of each model. Shoup's (1970) analysis indicates that value of land is lower in the presence of a property tax but that the rate of increase in the conversion value is higher. The effect of an ad valorem property tax would simply imply a conversion rule of the form: V(T)= + r (212) V(T) The left hand side of Equation (212) is the rate of change in the conversion value. On the right hand side, the tax is represented by r, and the discount rate is given by 3. According to Equation (212), land is developed when the rate of increase in the conversion value of land equals the interest forgone on other possible investments given by the discount rate.7 In this simple case, an increase in either the discount rate or tax rate would hasten conversion, while earning a positive rate of return from an interim use would delay conversion. 6 The nineteenth century economist Henry George was amongst an influential proponent for this type land tax. 7 This assumes costless conversion and no interim rents received from the land in the Shoup model. If land is not vacant, as assumed above, but rather receives interim rent from some revenue generating activity, like farming or forestry, then the rate of return earned in the interim is subtracted from the rate of increase in the conversion value. Hence, an interim use of the land will delay the optimal date of conversion. 59 One of the earliest examinations of a DVT is attributed to Rose (1973), who uses a partial equilibrium Wickselliantype model to distinguish between two types of capital gains taxes. The first DVT is levied at the time the land is rezoned from rural to urban use. The second DVT is levied when the land is actually converted. Rose (1973) finds that a levy at the development time yields an indeterminate derivative of development time with respect to the tax, depending upon the magnitude of the price and tax parameters and the functional form of the rural and urban rent functions. A DVT imposed at the rezoning time is found not to have any effect on the conversion decision, thus supporting Ricardo's view of a land tax as being neutral. Countering the result in Rose (1973), Skouras (1978) models the development time as an implicit function of the capital gains tax and finds the imposition of a DVT to accelerate the conversion time, owing to the fact that the present value of land falls as a result of the tax. Bentick (1979) supports the result in Skouras (1978) and demonstrates how a tax on the capitalized value of land from urban conversion can change the preferences of the landowner in a manner that may alter decisions to convert land. Hence, a tax on land values (i.e., a property tax) is nonneutral in the sense that the tax distorts the allocation of land as pointed out by Mills (1981b) and more recently by Arnott (2005), supporting George's view of a land tax being non neutral. Markusen and Scheffman (1978) and Arnott and Lewis (1979) find contrasting results from the authors above. Using a twoperiod general equilibrium model of land conversion, Markusen and Sheffman (1978) find that an ad valorem tax, or LVT, increases land demand in the first period and decreases it in the final period. However, the effect on the actual timing of land conversion is ambiguous and depends on the magnitude of price changes in the land market. In examining the imposition of a DVT, Markusen and Sheffman (1978) state that such a tax will clearly cause land price appreciation and therefore result in a higher rate of land conversion. However, this outcome is sensitive to the fact that the DVT is anticipated ahead of time.8 Arnott and Lewis (1979) model land values and the development decision with land value taxes assessed both before and after conversion. Their preconversion tax, equivalent to a LVT, has the same effect as Shoup (1970) and Bentick (1979). However, their postconversion tax, similar to a DVT, results in a slower rate of conversion. The contrasting results between the pre conversion and postconversion tax in Arnott and Lewis (1979) stems from their analysis accounting for conversion costs and the density of land development. The preconversion tax reduced density while the postconversion tax does not affect development density. Anderson (1986) extends the analysis by Bentick (1979) and Mills (1981b) and generalizes the comparative static results of the property tax in Arnott and Lewis (1979). He finds the post conversion tax has an ambiguous result on the conversion decision, with the direction of the effect depending on the magnitudes of the pre and postconversion tax rates which are influenced by prevailing market conditions. However, Anderson (1986) assumes that the rent from developed property is independent of the conversion time, a restrictive and unrealistic assumption which affects the comparative static results. None of the studies above distinguish between anticipated and unanticipated taxation and all analyze tax effects in a partial equilibrium framework, with the exception of Markusen and Sheffman (1978)9. Kanemoto (1985) models the anticipatory effects of a capital gains tax in a general equilibrium framework. If unanticipated, a DVT will result in a higher price of land and a lower allocation of land to developed uses and hence a slower conversion process. The effect 8 This special case is pointed out by Kanemoto (1985) who examines the anticipatory effects of a land tax more carefully. 9 The theoretical models of Shoup (1970) and Arnott and Lewis (1979) appear to examine land taxation from an unanticipated approach. on capital intensity is ambiguous, depending upon the magnitude of the elasticity of substitution between land and capital. The analysis of anticipated capital gains taxes is similar to the analysis of Markusen and Sheffman (1978), with such a tax resulting in a hastening of land development. Kanemoto (1985) determines that a LVT has an ambiguous effect on the amount of land allocated to developed uses depending again on the elasticity of substitution. If the capitalland substitution elasticity is greater than the rent elasticity of demand for developed uses (i.e., residential housing), then an anticipated capital gains tax will result in a speedier conversion process (Kanemoto, 1985).10 McMillen (1990) extends the Kanemoto (1990) model to account for the duration as well as uncertainty in regards to the timing of a capital gains tax increase. Under conditions of certainty, the effect of an unanticipated and permanent tax increase is the same as in Kanemoto (1985): a slower rate of conversion and smaller steady state allocation of land to urban uses. If the unanticipated tax is viewed as being temporary with certainty, the rate of conversion also slows, but by more than a permanent tax increase of the same size. However, once the tax rate returns to its prior level, a period of more rapid development ensuesthe shorter the interim tax period, the more the conversion rate slows. The end result in the steady state, the total amount of land allocated to urban uses, is the same between an unanticipated permanent tax and a temporary one. However, the time paths of development are much different. As noted by McMillen (1990), a temporary capital gains tax may be useful from a policy making standpoint if the intended goal is a reduction of rural land conversion in the shortrun. If a permanent tax increase is anticipated, the end result may be completely opposite of what the policymaking authority intended. In this case, the amount of developed land in the steady state 10 The effect on the price of land from an anticipated DVT could not be determined from the model due to analytical intractability, however, in the steady state land prices will be lower as a result of such a tax. is smaller, just as in the unanticipated cases, however, the rate of conversion once the tax is anticipated (usually before the actual levy of the tax), results in a faster rate of conversion due to landowners trying to avoid the tax. The earlier the tax rate is anticipated, the faster rate of conversion will last even longer. If the anticipated tax is viewed as being temporary with certainty, the conversion rate also increases, but at a slower pace than the anticipated permanent tax casethe shorter the duration of the tax, the smaller the rate of increase in land conversion. These results underscore the importance of understanding the land development decision, as intended goals of government policies may cause the very outcome they are attempting to prevent, usually a loss of rural lands. Uncertainty over the timing and duration of the above tax systems does not alter the effects described above, but does magnify them (McMillen 1990). The steady state is the same regardless of whether the world is certain or not, however, the land development path of convergence will be different. Much of the prevailing disagreement on the effects of a land tax is attributable to the imprecise definition between a land's market value and development value, as explained by Douglas (1980). The market value depends on whether the land development decision is immediate or delayed for a higher net present value whereas the development value is independent of this effect. This distinction has a profound implication on how land is valued and on the development decision, especially when real options valuation is used. The use of real options theory allows for the value in delay to be explicitly accounted. Also, mixed predictions are largely attributed to the individual intricacies of the theoretical model. As explained by Bentick and Pogue (1988), there are three types of partial equilibrium models prevalent in the literatureone assumes urban rents are constant, the second does not account for redevelopment options, and the third accounts for both of these. Each type of partial equilibrium model is uniquely sensitive to how the tax enters the formulation and so each differs in their implications. For example, when urban rents do not grow, a capital gains tax changes the land values in rural and urban uses proportionately, so in this regard the tax is neutral. However, when redevelopment is considered the model imparts an option value in the development decision, possibly resulting in delay due to the value of waiting. Another reason for the heterogeneity of tax results is due to the fact that partial equilibrium models are unable to carefully account for two distinctive features of taxation: the expectation of its arrival and the duration of its levy. The results from general equilibrium approaches suggest that expectations regarding the timing of a tax will affect whether land conversion increases or decreases, depending on whether the tax is anticipated or not. Expectations regarding the duration of a tax will affect the magnitude of the change, with rates of conversion being slower or faster depending on the permanent or temporary nature of the tax. Despite the abundance of theoretical land development models that examine the effects of taxation, only one study is known that empirically tests theoretical prediction. Zax and Skidmore (1994) use data from Douglas Country, Colorado from 1986 to 1991 and examine how changes in a property tax affect the duration of time a parcel remains undeveloped. At the initial time period, parcels of land described in the data were undeveloped, with subsequent tax changes and conversions recorded until 1991. Since the dependent variable is the length of time until a parcel is development, the authors use a duration or hazard function to determine the effect of tax on the conversion time. The general form is given by: S= P(D,, JD, = 0,j > 0, X,) (213) The D, term in Equation (213) is a dummy variable representing whether or not parcel i has been development in year t, and a vector of other conversion factors for parcel i in year t is given by XA The main results suggest that a relative modest property tax increase that is anticipated will not only increase the probability of development, but substantially increase the number of developed parcels. The results are generally consistent with the theory outlined by Kanemoto (1985) and McMillen (1990) and underscore the potentially significant impacts even a modest property tax may have. Market and information imperfections Aside from the use of a general equilibrium model, the Markusen and Scheffman (1978) study is also of interest since it is one of the first to examine the effects of a monopoly market on land development. Interestingly, they show that contrary to the standard view, a monopoly developer may not result in slower conversion rates and higher price appreciation rates. Rather the effect on development and prices will largely depend on the elasticity of demand (Markusen and Scheffman 1978). Further, the property tax and capital gains tax effects outlined above for the competitive case are indeterminate for the monopoly case. In the competitive case, a landowner converts when the price of land equals the forgone opportunity cost. A monopolist, however, is willing to convert land up to the point where marginal revenue net of capital outlays is equal to the opportunity cost of the rural use (Markusen and Scheffman 1978). Generally, the monopoly price will exceed the competitive price, but this may not necessarily lead to changes in the rate of conversion since under a monopoly market, rate increases in land supply and land price go in opposite directions. The impact of a capital gains tax in a monopoly market will largely depend on demand and supply responses and, in particular, on the magnitude of the elasticity of demand. One already noted limitations of the Markusen and Scheffman (1978) study involves the use of only two time periods when, in reality, the land 11 Zax and Skidmore (1994) also examine the effects of an unanticipated tax increase, but find inconsistent significance across multiple equations and thus could not make an explicit conclusion on their effect. development decision involves a greater number of periods in continuous time. Further, the model unrealistically assumes perfect information. The paper by Mills (1981a) represents an effort to relax the perfect information assumption and examines three types of informational settings of the decisionmaker: perfect foresight, zero foresight or myopia, and imperfect foresight. Generally, economic outcomes under cases of uncertainty tend be inefficient, as is the case with land development. The Mills (1981a) model is of the general equilibrium and perfect competition type, and envisions three types of decision makers. The key decision makers are the landowners who earn rent, R], on unconverted land and decide when to develop the parcel for housing. Landowners also decide the housing type, of which n types are assumed. Each type of housing has its own requirement for parcel size given by the vector a = (a,,..., a,). The second types of decisionmakers are the housing construction firms who incur costs to building each type of housing, given by the vector c = (c,,...cj). The firms supply the housing units at time t according to the vector x(t) = (x (t),..., x(t)). The residents, or socalled tenants, compose the final type and demand housing according to the inverse demand function, f (x( tt), i = 1,...n, and pay rent R(t) = (R(t),...,R (t)) . Under the assumption of perfect foresight, the landowner is able to forecast exactly both the supply of housing, x(t), and the demand for housing, f (x(t)). Given a discount rate defined by r, the landowner will not develop a unit of land unless it generates a profit in excess of the present discounted value of unimproved land, given by R /r Therefore, the landowner's maximization problem is: maxjy = e t (x(r),r)aR]dre (214) 't la I t As noted by Mills (1981a), the landowner has a set of infinite options given by the type of housing development and the timing of housing development: (i, t). An option by the landowner will not be exercised if a more profitable one exists, implying not only that all options are equally profitable, but that the solution is a competitive equilibrium and is efficient (Mills, 1981a). Under perfect foresight, in either the land or housing markets, prices represent profit expectations. These expectations are shared equally among all types of landowners meaning they each have the same base of information. Since all landowners base profit expectations on the same set of information, no individual landowner is better off than another in terms of informational advantages. The conversion rule resulting from the Mills (1981a) analysis states the rent received from building a type of housing will be maintained over any time horizon until the rental rate equals the opportunity cost. The opportunity cost is defined as the point where the revenues received from the housing market equal the combined costs of construction and opportunity landcost (Mills, 1981a). In the case where landowners are myopic, decisions are made using only the current stock of information and do not consider future expectations. As a result, the myopic landowner will not heed the opportunity cost level and will continue converting land until the supply is exhausted. In this case, not all options are equally profitable, and so not only will some landowners gain more than others, but the equilibrium is not competitive and thus inefficient. The case of imperfect information lies between these two extremes. Landowners do not have perfect foresight regarding prices and rents, but neither do they have complete disregard for future outcomes. In this case, landowners are speculative investors, with some having better forecasts than others. Options are still not all equally profitable, like in the perfect information case, but the land will not be converted until the point that supply is exhausted, like in the myopic case. Rather, landowners behave much like speculative investors, and the conversion rates are moderated between the two extreme cases of perfect information and myopia. One of the key insights about the Mills (1981 a) paper is the fact that operating under a circumstance of uncertainty and of less than perfect information will reduce economic efficiency. Urban growth Urban growth models are based on dynamic constructs of spatial structure accounting for population increases and higher demands for urban developments. One of the earliest contributions is Anas (1978) who shows under circumstances of population growth that as the commuting distance to employment centers increase, the density of housing developments will also increase. Other studies in this spirit include Arnott (1980), Brueckner (1980), and Wheaton (1982). Capozza and Helsley (1989) model the land conversion process under conditions of a growing urban area. Like Arnott and Lewis (1979), the authors assume perfect foresight and irreversibility. Next, the value of land to owners is defined as four components: rent received from undeveloped agricultural land, expected rent from developed urban land, the development cost, and the value associated with how accessible the land is viewed. This final assumption is largely based on the von Thunen concept of proximity in determining land values. The closer a unit of land is to the central business district (CBD), the more valuable the land unit is due to greater transportation and commuting costs for more distant locations. Models of this type are often described as a monocentric urban area. In regards to the timing of conversion, the landowner chooses the optimal time, t*, to maximize the present value of undeveloped agricultural land: P (t, z)= fR(r, z)e r("dr+(AY) Ie 1  eCe *(215) (* Rent received from urban land uses is given by R(r, z), where z is the boundary of the urban area which indexes the distance of a unit of land from the CBD (Capozza and Helsley 1989). Agricultural land rents are given by A is, C is the conversion cost for a unit of land, r is the discount rate, and t* is the time of conversion. The firstorder condition solved using Leibnitz' rule is of the following familiar form: R(t*,z) = A + rC (216) Conversion is optimal at the time period when the sums of the opportunity costs from capitalization and from agriculture equal the urban use rental rate. A similar expression is obtained for the average price of developed land. Comparative static results indicate that the average price of urban land rises with higher agricultural land rents, conversion costs, commuting costs, the size of the city, and population growth. The average price of urban land falls with the discount rate. The Capozza and Helsley (1989) model explains the substantial gap between the value of agricultural rent and the price of land at the urban fringe, which may be explained by land rental growth expectations in the future. High growth rates of land rentals are due in part to the large degree of speculation by land developers on undeveloped land. In the context of uncertainty, discussed in more detail below, Capozza and Li (1994) find generally higher growth rates raise the hurdle or reservation rent leading to delayed conversion times and a larger, denser, urban area. However, density levels are also dependent upon the degree of uncertainty and the elasticity of capital. Higher growth rates can have a negative effect on rents under conditions of high uncertainty and low capital elasticity, leading to a less dense urban area. Therefore, the effect of growth rates on conversion timing is ambiguous overall. Uncertainty Ellson and Roberts (1983) also investigate the effects of uncertainty on the timing of land rezoning and infrastructure development. Using a dynamic model, the key decisionmakers analyzed are governments and planning agencies that make the conversion decision rezone parcels for urban uses. An aspect not considered in earlier studies, Ellson and Roberts (1983) model the planner's problem as one of consumer surplus maximization as a way of finding the socially optimal rezoning time. Using a simulation of a translog utility function, they conclude uncertainty tends to slow the rate of conversion, which is sensitive to the discount rate and the elasticity of demand. Increases in the discount rate, assumed to be the interest rate, tend to speed up the rate of conversion or rezoning (Ellson and Roberts 1983). As the time to development advances further into the future, the uncertainty on the conversion value of land becomes greater. This may imply the presence of a declining discount rate in the present value problem, with discount rates being greater in periods before development, declining as the conversion time approaches, as noted in the discussion on the discount rate (Shoup 1970). One creative approach to modeling uncertainty has been through the use of stochastic processes. When returns are uncertain, they exhibit a type of random walk. Clarke and Reed (1988) describe the evolution of capital prices and rental rates on housing units as stochastic differential equations following a geometric Brownian motion with a Weiner process drift. 12 Like some earlier studies, their analysis does not allow for any interim uses of the land, but does maintain the assumption of irreversibility of development. The optimal conversion rule obtained is familiar: "develop land when the ratio of unit rentals to unit construction costs exceeds a critical barrier, otherwise do not," (Clarke and Reed 1988). This type of conversion rule would 12 The authors examine the implication of defining the stochastic equations based on an Ito versus Stratonovich solution but, for the purposes of this review, this difference is not critical. lend itself to econometric estimation under a hurdle or hazard function (Capozza and Li 2001; Irwin and Bockstael 2002). Despite the use of complicated stochastic differential calculus, their development rule is quite similar to others, and more specifically to the one obtained by Arnott and Lewis (1979). Simply put, conversion takes place when: expected land value 1+ r (217) sum of conversion costs r The value r in Equation (217) is defined as a stochastic parameter in the equation for the market value of land (Clarke and Reed 1988). The effects obtained through comparative statics are revealing. Greater uncertainty in either construction costs or housing rentals results in a greater land value and an increase in density of construction. However, while increasing uncertainty in construction costs raises the socalled "critical barrier," and hence, delays conversion, the effect of uncertainty on rental rates is indeterminate (Clarke and Reed 1988). Similar to Mills (1981a), the Clarke and Reed (1988) describe the decision to develop as an option value. This makes sense, especially since the nature of conversion is assumed to be irreversible in the land development literature. With assumed irreversibility, the conversion problem is related to the work in financial economics on irreversible investment projects under uncertainty. The development decision is predominately modeled in a stochastic uncertain framework in the literature (Capozza and Helsley 1990; Capozza and Li 1994, 2001, 2002; Majd and Pindyck 1987; Titman 1985). However, the capitalization approach is not able to accurately model land values and conversion timing in an option value context. Primarily, the failure of standard capitalization approaches is due to the incapability of the standard net present value calculations to explicitly account for the value in waiting to develop. This socalled option value is absent in the standard models of Shoup (1970), Arnott and Lewis (1979) and Capozza and Helsley (1989). The Real Options Approach The introduction of uncertainty changes the analysis of the development process considerably. The study of investment under uncertainty has spawned a whole body of literature culminating in the influential work of Dixit and Pindyck (1994). 13 The traditional investment rule in finance is to undertake a project when its net present value is positive. Likewise, if two projects are being considered and are mutually exclusive in the sense that both cannot be undertaken at the same time, then the project with the higher NPV is the optimal choice. Such basic rules form the foundation of the analysis in the seminal papers by Shoup (1970) and Arnott and Lewis (1979). However, all investment projects, especially those involving land conversion and development, come with an ability to delay or postpone investment until some later time. This means that the project competes with itself through time, imparting a value in the option to postpone investment. The idea of an investment opportunity, and land conversion in particular, as an option has acquired recent interest. The groundbreaking work of Dixit and Pindyck (1994) in investment under uncertainty underscores the value of waiting to invest. In reality, the land development decision, or any investment decision for that matter, is rarely a now or never decision since the individual can exercise the option to delay development or investment. Introduction to real options theory Although options theory has a long history, the first rigorous theoretical treatment can be found in the works of Black and Scholes (1973) and in Merton (1973). In finance, options can be categorized as either a call or a put option. A call option is the opportunity to buy or sell a 13 For an excellent review of the investment under uncertainty literature, see Pindyck (1991). commodity or unit of stock at some future time at a specified price. Another type of option is a put option where the holder has the right to sell a stock or unit of commodity at a stated price. In this sense, options are contingent assets since they only have a value contingent on certain outcomes in the economy. The exercise price of an option is usually referred to as the striking price, while the current value or quoted price of the stock or commodity is referred to as the spot price. The date at which an option is exercised is also referred to as the terminal time, or the date of expiration or maturity. A call option has a positive value when the spot price exceeds the strike price, or is "in the money." Put options are "in the money" when the strike price exceeds the spot price. Options are only exercised in this range. An important characteristic of an option is that it will always have a positive or zero value, never negative. This is because the point at which the option is worthless the holder of the option will simply disregard it, or "abandon the option." When an option has zero value it is referred to as "out of the money." The payoff function from a call option can be represented graphically in Figure 26. Two types of options in the finance literature are European options, which can only be exercised on the final date of expiration, and American options, which can be exercised at any point in time until the final data of expiration. Land development and conversion decisions are best represented by an American call option since the option to convert can be exercised at any time until the terminal date. The value of an option is made up of two basic parts: the intrinsic value and the time value. The intrinsic value of an option is the difference between the strike price option and the spot price. The time value represents the possibility that the option may increase in value over time due to volatility in the stock or commodity price. At the terminal or expiration time of the option, the time value is zero, declining over time, at which point the option is equal to its intrinsic value. The value of an option can be obtained by different methods. An often used numerical procedure for options pricing is presented in Cox, Ross, and Rubinstein (1979) and is based on a discrete binomial process. An alternative method is the wellknown BlackScholes method presented in Black and Scholes (1973), which utilizes stochastic drifts to model the option price. Various extensions of the current methodologies are present in the literature (Alvarez and Koskela 2006; Rodrigo and Mamon 2006). An excellent discussion of option pricing theory and its applications can be found in Merton (1998). A brief account of real options Abel (1983) is one of the earlier studies on optimal investment under uncertainty. Additional studies have examined the irreversibility of some investments and have shown this to impart an option value in the investment decision (Bernanke 1983). McDonald and Siegel (1986) rigorously derive the value in the option to delay investment. Assuming investors are risk averse and hold a diverse portfolio, the authors examine the value in waiting to invest and develop rules on the optimal timing of investment for an irreversible project. Using the firm as the decisionmaking unit, the key component of the model is the choice between two mutually exclusive projects with only one being able to be undertaken at any given time. This mutual exclusiveness, concurrent with assumptions of uncertainty in project payoffs and investment costs, irreversibility of the investment, and risk averseness of the firm, impart a value in the firm's choice to delay investment. This option value is higher under conditions of greater uncertainty due to the increased variance in possible values of project returns. The standard assumption when uncertainty is considered in investment returns is to model the payments from an investment according to a standard Brownian motion process. McDonald and Siegel (1986) also consider a form of the Poisson process which allows for the present value of future returns to take a discrete jump to a zero value. Ingersoll and Ross (1992) generalize McDonald and Siegel (1986) and show that option values exist in nearly all investment projects regardless of whether or not there is uncertainty in the expected payoffs or cash flows as long as there is some uncertainty in the interest rate. In Ingersoll and Ross (1992), there is an optimal acceptance rate of interest at which point the project is undertaken. As the real interest rate moves away from the optimal acceptance rate, greater postponement of investment occurs, causing the project value to decline (Ingersoll and Ross 1992). Depending on the length of delay and the average rate of discount, the cost of waiting is the forfeited present value of the project. Their results underscore the flaw in using the traditional NPV approach to decide on investment projects which dictates investment in all projects with positive net present values. Uncertainty effects on land conversion decisions are even more important due to the long duration of most land development projects. Projects which tend to have a longer timespan have more volatile present values which make the investment option more valuable. The focus in the literature has often been on the effects of uncertainty on the timing of investment, however the effect on intensity of investment is investigated by BarIlan and Strange (1999). In particular, they look at how price uncertainty affects the intensity and timing of investment when capital investment can be either lumpy or incremental. When investment is lumpy, the level of capital required for a project is decided at the moment the investment occurs. However, incremental investment involves units of capital that may be added over time. Some examples might include a rental car agency updating its fleet of vehicles or the addition of books in a library. Lumpy investment, like the construction of a building or a roadway, best describes the type of investment in land conversion. Under lumpy investment decisions when both timing and intensity of investment are considered, uncertainty tends to increase the trigger price. A higher trigger price indicates the value to delay is higher, postponing investment. However, once investment does occur it tends to occur with higher capital intensity when uncertainty is present. This stands in contrast to the incremental case when only intensity is a factor in the decision where greater uncertainty tends reduce the intensity of capital. Application of real options to land development One of the first studies to formally model land prices and development using an option value approach to investment decisions is Titman (1985). Under conditions of uncertain future real estate prices, the option to delay construction on vacant land becomes valuable since future development may be more profitable than current development given current prices. When the landowner is assumed to be risk neutral, the option value in the conversion decision will increase under conditions of greater uncertainty since the expected value of vacant land increases, resulting in current vacant land values to increase under uncertainty (Titman 1985). Using a BlackScholes model of options values, several interesting comparative static results are obtained from the Titman (1985) model. For example, if the interest rate rises, the value of vacant land will increase, resulting in a greater incentive to hold onto the vacant land for future construction. An increase in rental rates will have the opposite effect, decreasing the value of vacant land resulting in a greater attractiveness to initiate building construction. The results in Titman (1985) are interesting when compared to the studies of land conversion when certainty is assumed. For example, in Arnott and Lewis (1979), an increase in the interest rate has an indeterminate effect on timing. The impact of uncertainty on land development underscores the importance of a more robust understanding of land development models, particularly when government policies are issued with the objective of shaping landowner behavior and land development. For example, Titman (1985) shows that under a building regulation stipulating the maximum allowable height 76 of a new building, the actual number of new buildings could actually increase as a result. No doubt part of this effect is due to the fact that more buildings are needed since each new building must be necessarily smaller. However, this effect is also due in part to the reduction of uncertainty in the optimal building size. A height restriction mitigates the uncertainty of future prices from the decision to build now or later, and since lower uncertainty means a lower option value from delaying construction, vacant land could be developed sooner than it would have if there was no height restriction (Titman 1985). Extending Capozza and Helsley (1989) to account for uncertainty, Capozza and Helsley (1990) models household income, land rents, and prices as stochastic processes. The introduction of uncertainty is shown to impart an option value to the price of agricultural land. This has the effect of delaying the time to conversion, with the option value falling as the urban size grows and the distance from the boundary of the fringe region increases. Clarke and Reed (1988) also find that uncertainty adds an option value to the conversion decision, but do not examine the effects on urban growth and city size. The introduction of uncertainty in Capozza and Helsley (1990) required a reformulation of the standard problem of the landowner maximizing the value of land. Thus, Capozza and Helsley (1990) form the landowners' problem as a hitting time problem. A specific type of the stochastic optimal stopping time problem, a hitting time refers to the point, or first hit time, where an outcome is optimal. In this problem, the first hit time is defined as a reservation or hurdle rent level. Once land rents reach the reservation rent level conversion is optimal. This hurdle rent is given by: R* =A+rC+r ag (218) ar Agricultural returns in Equation (218) are given by A, conversion costs are given by C, and the discount rate is r, which have the same interpretation as in Capozza and Helsley (1989). Additional parameters include, a and g, which is the drift parameter in the Brownian motion process for household income. When compared to the reservation rent in the certain case given by Capozza and Helsley (1989), the uncertain case has a higher trigger level (Capozza and Helsley 1990). The last term in Equation (218) is defined as the irreversibility premium. Despite risk neutrality of landowners, the presence of uncertainty affects equilibrium land rents and prices due to the permanence of land conversion. The authors' equations on expected prices of agricultural and urban lands show that uncertainty increases land price, but only if city size is exogenous. If the size of the city is endogenously determined, then the effect of uncertainty on the price of agricultural land is ambiguous and largely depends on the degree of uncertainty (Capozza and Helsley 1990). In recent years, the land development problem has been described using concepts from the financial economics literature, namely investment theory under uncertainty, and in particular, options theory. Capozza and Li (1994) represent one of the more rigorous attempts to model the timing and intensity of land conversion as an investment decision. The timing of urban residential development is framed as an option in the model developed by Mills (1981a), but focuses on the effects of conversion decisions under conditions of myopia and perfect foresight. Clarke and Reed (1988) more carefully examine the conversion decision as a perpetual option framework using stochastic calculus, however they ignore the interaction that capital intensity has on the timing decision and also on rents and property taxes. Capozza and Li (1994) fill in these gaps by describing how urban areas are affected by capital intensity, particularly in the land's spatial patterns, as well as obtaining effects on the discount rate, conversion costs, rental rates, capital elasticity, and expected growth rates. Using the theory of optimalstopping, Capozza and Li (1994) frame the land conversion decision as an American option value with varying levels of intensity. In a general sense, the time required to take some specific action is described by the theory of optimal stopping based on a series of random variables which are randomly observed. Often times this is done for the purpose of maximizing an expected reward or minimizing an expected cost (Kamien and Schwartz 1991). The presence of the option is based on the decision to invest between two different activities. Suppose the land has two revenue generating activities per unit or parcel, R, and R2, with output per parcel, q (k,) and q, (k,), where K = (k,, k,) is the capitalland ratio (i.e., capital intensity). Further, suppose the initial case is activity one with revenue R1, then the landowner has the option at any time, t, to convert to activity two by replacing the current capital intensity, k,, with intensity k,. Thus, not only does the landowner choose the time t, but also the optimal capitalland ratio (Capozza and Li 1994). In the case of vacant land in the initial period, here period one, it is assumed that there are no rents and that no capital is applied to the land. The decision becomes one of choosing a benchmark or hurdle rent, given by R*, and corresponding capital intensity, k*, at some time t. Capozza and Li (1994) make the following assumption: positive variable conversion costs, c; the discount rate r is taken to be the interest rate; and the net rental rate follows a normal diffusion process with a constant drift g and standard deviation a, which are both constants. The equation of motion for rents is given by: dR = gdt + rdB (219) The price per parcel of land is given by the present value of future cash flows: P = R +gr (220) The value of the conversion option is: P(t) = W(R) = max E, {V (R(t))e r( t (221) The stopping time is T, W(R) is the value of a perpetual warrant moving from no capital to capital intensity k*, and the intrinsic value of the warrant, given by V(R), is the value of the warrant if exercised at time t. The intrinsic value of conversion is defined as: V(R*) = q(k*)p* ck* (222) Using the fundamental differential equation of optimal stopping time, the model is solved to obtain the following conversion rule: q(k*)R* =rck* + q(k)(Y (223) In Equation (223), a is a parameter. Thus, the option to convert will take place when the sum of the cost of capital for conversion and a risk premium from uncertainty equals the rent given up by delaying conversion (Capozza and Li 1994). By assuming a CobbDouglas functional form of the production function, the authors obtain equations for the optimal hurdle rent and capital intensity. Like in other models, greater levels of uncertainty increase the conversion or option value. As a result of uncertainty the hurdle rent rises, delaying conversion, increasing the necessary capitalland ratio, and reducing the structural density of the urban area (Capozza and Li 1994). The costs of conversion have a negative effect on the intensity of capital which in turn will lead to lower density and lower land values. Increasing costs, however, have no effect on either the hurdle rent or the conversion time. The elasticity of capital, on the other hand, has a positive effect on the both the reservation rent and the conversion time, but has an ambiguous effect on 80 the capital intensity. The discount rate has an overall negative effect on the endogenous variables. Reservation rents are lower, and so conversion times tend to be delayed. Land values and capital intensity are also lower under higher discount rates. If a property tax is set on post development land, then the model predicts decreasing capital intensity and land values, but a higher hurdle rent, implying delayed conversion. A predevelopment property tax has the opposite effect on hurdle rents, implying more hasty conversion. Capozza and Li (2002) simplify their earlier analysis by assuming only one possible investment project. Assuming net rents from the project grow exponentially and that capital intensity is variable, the authors obtain rules for the optimal timing of land conversion under both certain and uncertain growth rates using internal rate of return (IRR) and net present value (NPV) principles. Under certainty, the conventional rule for undertaking a project arising from the real options approach in investment theory is to delay investment until the current yield, or IRR, equals the cost of capital (Capozza and Li 2002). This level of yield is often referred to as the reservation or hurdle IRR since investment only occurs until this level is reached. In the case of land development, not only is conversion irreversible, or assumed to be so, but the added assumption of increasing rents over time further delays the optimal time to invest. Under conditions of perfect foresight, the hurdle rate is given by: IRR* = r +g (224) The discount rate is r, taken the prevailing interest rate in the real estate market, and g is the growth rate of rents. In this case, the internal rate of return must not only exceed the interest rate but also the rate of increasing cash flows. Assuming a constant elasticity of substitution production function, explicit solutions for the optimal hurdle level of rents and capital intensity are obtained: X =r [ a (225) and, ) 1Y K*= (226) The asset's coefficient of distribution is given by a, and p is the elasticity of substitution. The growth rate has a positive effect on optimal capital intensity while intensity is decreasing in the interest rate. Unlike in Capozza and Li (1994), the effect of growth rates on conversion time is clear. A higher rate of increase in cash flows implies a higher hurdle rent and hence greater delay in conversion. Exponentially increasing rents tend to have an effect on future option values rather than on current options values (Capozza and Li 2002). Further, the interest rate in Capozza and Li (1994) has an unambiguous affect on timing whereas in the present case the effect of interest rates is unclear. Note however, that in Capozza and Li (1994) a CobbDouglas form of the production function is assumed. In the stochastic case, a continuous time version of the Capital Asset Pricing Model is utilized to describe the option value to invest. The rental rate is assumed to grow with uncertainty according to a Brownian motion with a standard Weiner process. Under uncertainty the IRR hurdle level for the timing of investment in land development is made when: 2 IRR* =r+g +a (227) 2 The variance of the growth rate of cash flows is given by 2, and a is a parameter that approaches one as a approaches co, and approaches / as a approaches zero. This is equivalent to the certainty IRR hurdle except for the addition of a term accounting for the value 82 of waiting under conditions of uncertainty (Capozza and Li 2002). Using again the CES production function, explicit solutions for the optimal hurdle level of rents and capital intensity are obtained: X a r1aa (228) 2 )lX K* = (229) (a1) The effects of the growth rate are the same as in the certain case. A higher rate of growth of cash flows will delay conversion and increase the level of capital. The uncertain option value increases the variance of rental flows, increasing the time horizon of the investment and the capital intensity required when the decision to invest is made (Capozza and Li 2002). The effect of the interest rate is a bit more complicated due to the limiting nature of the a and a parameters. There are two offsetting effects to increased interest rates in an option value under uncertainty with positive growth rates. Increases in capital costs from higher interest rates serve to delay investment, while lower waiting option values serve to hasten investment. Generally, however, for any level of growth, if the uncertainty is high enough so that a is large, then an increase in the interest rate will have a declining affect on the optimal hurdle rent, hastening conversion. As noted by Capozza and Li (2001), when the world is uncertain, irreversibility impacts the optimal timing of investment in a real options framework. The IRR has a higher hurdle level when investment is irreversible resulting in increased delay of a project. When capital intensity is variable and the economy is growing, delay also occurs. When a landowner commits to a certain lower level of capital in the current time period, some amount of revenue is forfeited in a future time period from an optimal project that requires higher capital than if the landowner had waited (Capozza and Li 2001). The Transactions Cost Approach The capitalization approach or present value method has dominated the literature on land values and the general literature on asset pricing as a whole. However, a number of empirical issues have arisen from the use of the present value techniques, particularly in the arena of land pricing. Indeed, a burgeoning literature has appeared criticizing the capitalization technique for oversimplifying the valuation of land and leading to empirical rejection of present value methods, particularly for farmland (Falk 1991; Clark, Fulton, and Scott 1993; Lloyd 1994). One flaw in present value techniques is the inability to explain why land prices rise and fall faster than land rents, especially during boombust cycles (Schmitz 1995). While the option value approach discussed above is one alternative to the failure of capitalization methods, another approach has called for the incorporation of transaction costs in the model. As noted, the land market is particular prone to costly transactions, which have been estimated by some authors. Wunderlich (1989) estimates transaction costs from the transfer of land between buyer and seller around 3 percent of the total land value net of brokerage fees. This estimate is in line with the 2.5 percent estimated transaction costs given by Moyer and Daugherty (1982). However, neither of these estimates includes the cost of brokerage firms, which most land transactions occur through. Once accounted for, Wunderlich (1989) estimates the transaction cost to be as high as 15 percent. Clearly, such costs are not trivial and the need for more recent estimates is also clearly warranted. Land development and institutions The notion of transaction costs and institutions is not new, beginning with the work of Ronald Coase in 1937 and expounded on by Williamson (1985) and North (1990). Transaction 84 costs can take on many forms in the land development process. While the mainstream literature has not yet fully developed a framework for the land development process under market frictions, some enlightening initial investigations are available (Healey 1991; Alexander 1992; Lai 1994; Benjamin and Phimister 1997; Buitelaar 2004). For example, zoning restrictions, titling cost, survey fees, and brokerage fees are all embedded in the transfer of land from rural to urban use. Any cost not accounting for in the physical production of urban land can be considered a transaction cost. As stated by Coase, transaction costs arise from two key failures in neoclassical economics: perfect information and rationality. Models of the land development decision with imperfect information have already been described. The landowner or developer may attempt to close the information gap by trying to acquire new information. For example, the conversion to residential property may involve research into housing preferences (Buitelaar 2004). The presence of uncertainty also introduces transaction costs in the land market as landowners and developers will attempt to gain information to reduce uncertainty. For example, a landowner might be in possession of a substantial size of unzoned property. The zoning of her land into residential or commercial use has a profound implication on its potential value and the decision to convert. In an attempt to eliminate such uncertainty, she may lobby the local municipality to zone the property to its valuemaximizing use. One purpose for the creation of institutions is to reduce uncertainty. There is also a degree of institutional cost in the land market and in the development process. Largely based on the work of North (1990), institutions can be described as the rules of the game in a society. Institutions define the constraints conceived by people and shape human interaction. By providing rules or constraints, institutions provide a structure to human interaction and reduce uncertainty in everyday lives. Local zoning restrictions are an example of institutional constraints in the land development process. As noted by North (1990), the creation of institutions involves a transaction cost itself, referred to as institutional cost. For example, the planning agency is the organization which places an institutional cost, zoning, on the land development process. Transactions tend to be costly due to the fact that information itself is costly. For example, the costs of measuring the attributes of value to the individual of what is being exchanged and also the costs of protecting the rights of the individual as well as the costs of enforcing agreements. These costs constitute the source of social, political, and economic institutions. Indeed, stark implications for the land conversion decision are implied by the very notion of an exchange process with transaction costs. Since these costs are embedded in the costs of production, North believes that an entire restatement of the production relationship is necessary. This restatement must recognize that the costs of production are "the sum of transformation and transactions costs." Before one can understand however the implications on a theory of institutions, one must understand why transacting can be costly. Consider the following statement by North: We get utility from the diverse attributes of a good or service, or in the case of the performance of an agent, from the multitude of separate activities that constitute performance... The value of an exchange to the parties, then, is the value of the different attributes lumped into the good or service. It takes resources to measure these attributes and additional resources to define and to measure rights that are transferred. The underlying aspect of transactions costs stem from both parties involved in the exchange trying to ascertain the value of the individual attributes of the unit being exchanged. The seller or owner of a tract of rural land would likely have full information on the quality of land and its suitability for urban development, whereas the potential buyer or developer would have to approximate that information. Enforcement is another factor that adds to the costs of transacting. As mentioned earlier, land transactions are infrequent and occur over long time periods. Land 86 transactions are often specified according to some written contract as to development dates, limitations to urban uses, etc. The developer might later find out that the land is not well suited for a commercial property, despite having already contracted to purchase the land and therefore would want to opt out of the contract. The enforcement of such a contract involves a cost. This additional cost would not present an issue if it is in the best interests of either party to concede to the original agreement. However, as North points out (as did Adam Smith 250 years ago) individuals are very much self interested which invokes feelings of uncertainty in either party that the other will not renege on the agreement. Uncertainty about possible reneging produces a premium on the risk that the other party will in fact renege, presenting a cost to the losing party. Information costs and uncertainty, conjoined with the behavior of the individual, presents challenges both to traditional economic theory and institutional theory. North (1990) obtains a better understanding of how individual behavior and society's institutional structure are related. Property rights are the rights individuals appropriate over the labor, goods and services they own. This "appropriation" is a function of the institutional framework, such as legal rules, organizational forms, enforcement and norms of behavior. North describes that because of the presence of transaction costs and illdefined property rights, certain attributes valued by the individual, "remain in the public domain." Individuals gain then by devoting resources to try to obtain such attributes. How this plays out is a function of the institutional structure, which facilitates exchange and determines the cost of transacting. Now how well this "game is played," North describes, depends on the extent that the rules institutionscan solve the problems of coordination and production. The outcome of the game is determined by the motivation of the players, the complexity of the environment, and the ability of players to decipher and order the environment. Despite the need for an institutional model in the land development literature, to date none have been formally described. However, a number of studies have both theoretically and empirically described the impact of quantifiable transaction costs on land values. While notions of transaction costs in these papers are limited to exchange costs (i.e., brokerage fees) and do not examine institutional costs, they remain a revealing and potentially promising alternative to the capitalization approach to modeling land values and the land conversion decision. Models of land values with transaction costs Borrowing from the notation in Lence and Miller (1999), the capitalization model given by Equation (25) can be extended to account for transactions costs. Let Tp and Ts denote the transaction costs on the purchase and sale of land, respectively, defined in terms of a percentage of the total price of land. Then define the purchasing and selling conditions of land as: (1i+T,) <1 E, R 15 3 t (230) ,I n=0 (1 T) > E, Rt (231) =1 n=0 As in Lence and Miller (1999), the condition being tested is: Ts The term g, is a stochastic variable that denotes the excess return yielded by holding the land indefinitely beginning at time t discounted at the rate 3. The excess return is defined as: g, C~, n 1 (233) 9 t =1l n=O The formulation given by Equations (330) through (333) imply the transfer of land will only occur if the transactions costs do not exceed the expected excess returns of the land (Lence and Miller 1999). The conversion rule in Equation (230) and Equation (231) collapse to the familiar capitalization formula given by Equation (25) if Ts = Tp = 0, which is equivalent to assuming a perfectly frictionless transfer of land. Therefore, one way of explaining the gap between land prices and land rentals is through the presence of transactions costs. The sum of the land's expected discounted rent may differ from the current value or price, but not by an amount greater than the transaction costs associated with the transfer from seller to buyer. The capitalization model of the land development decision does not account for these transaction costs and assumes that the land market is frictionless. Just and Miranowski (1993) is one of the first known studies to include such frictions in a structural model of land prices. Their analysis empirically demonstrates that real land values do not closely follow land rentals. The theoretical model accounts for transaction costs by including parameters on the sales commissions incurred in selling land. The values of these parameters are simply imposed in the econometric model as those given by Wunderlich (1989). The results in Just and Miranowski (1993) imply a superior fit over the capitalization method. Taking a different approach, Chavas and Thomas (1999) use a dynamic model of land prices, relaxing the assumption of timeadditive dynamic preferences, risk neutrality, and zero transaction costs. The model is an extension of Epstein and Zin (1991) but allows for frictions in the transfer of land. Rather than impute the parameters describing transaction costs, like Just and Miranowski (1993), Chavas and Thomas (1999) estimate the marginal transaction costs as a proportion of changes in land quantity as a result of buying and selling land using the generalized method of moments. Not only do the results in Chavas and Thomas (1999) also support a strong rejection of the capitalization approach, but also provide substantial statistical evidence that transaction costs have a significant effect on land prices. However, the result from both studies should be taken with caution as Lence (2001) notes serious flaws with both Just and Miranowski (1993) and Chavas and Thomas (1999). According to Lence (2001), incorrect first order conditions on the expression describing land values leads to imprecise theoretical predictions and possibly inconsistent econometric results. Lence (2001) also notes the inherent flaw in assuming a representative agent when using aggregated data, as in Chavas and Thomas (1999), since such an assumption is invalid when transaction costs are present. Further, the complicated structures inherent in the Just and Miranowski (1993) and Chavas and Thomas (1999) models lend themselves to difficult estimation procedures and sometime vague intuition whereas Lence and Miller (1999) and de Fontnouvelle and Lence (2002) model land values with frictions in a very analytically tractable way.14 The method of Shiha and Chavas (1995) modifies the traditional capitalization model in a manner similar to Lence and Miller (1999) and de Fontnouvelle and Lence (2002) by modeling market frictions as barriers to investment in agriculture. Such barriers are postulated to result in market segmentation and include legal fees, information and search costs, and represent the transaction costs in the model. Their model imposes these barriers on nonfarm investors attempting to hold equity in the farmland market. Similar to Lence and Miller (1999) the transactions costs are assumed to be proportional to the value of land holdings. To estimate the transaction cost parameter in their model, the authors use iterative nonlinear seemingly unrelated regressions. The Shiha and Chavas (1995) result suggest that not only is the farmland real estate market segmented, but that this segmentation is the result of transaction costs. In fact, the 14 The theoretical model of de Fontnouvelle and Lence (2002) is the same as Lence and Miller (1999), however the former uses a kernel estimation method and an expanded data set while the latter relies ordinary least squares. estimates are somewhat larger than those reported by Moyer and Daugherty (1982) and Wunderlich (1989) reaching a peak of 6.18 percent per year during the 19491983 period (Shiha and Chavas 1995). Empirical Models of Land Change There is voluminous empirical work that attempts to estimate land values. An historical and comprehensive look at the literature, with particular attention to farmland values, can be found in Moss and Schmitz (2007). The particular attention in this section is on how the land valuation approaches described affect the empirics behind the land development decision. Capitalization Empirical Methods In one of the first and simplest empirical tests of the conversion model, Arnott and Lewis (1979) examine the real estate data for 21 metropolitan areas in Canada from 19611975 from the Central Mortgage and Housing Corporation. Actual ratios of land values to property values for the periods 19611971 and 19721975 are compared to those predicted by the model. Results indicate that the model explains 60 percent of the variation in the land value to property value ratio for the given areas. An elasticity of substitution between land and capital is also estimated. Using a CES form of the housing production function Q(K), the authors perform a regression on the following equation V(T) 7 In T)= In pInK (234) p(T)K \1Y) The elasticity of substitution between land and capital in Equation (234) is given by p, y is a coefficient describing the distribution of land, p(T) is the price of a unit of capital at time T, and K is assumed be equal to the average area of floor space to the average size of the residential lot (Arnott and Lewis 1979). Using data for 23 Canadian metropolitan areas from 19751976 on new single family homes, results imply an elasticity of substitution of 0.372 for 1975 and 0.342 91 for 1976. The authors take the low values as evidence that the model does in fact produce an optimum. However, a weakness in the Arnott and Lewis (1979) model involves the assumption that on the urban periphery, the supply of land is perfectly elastic and developable. This may not be realistic given the heterogeneity of spatial land characteristics and the location of some parcels to more amenities. More rigorous approaches to modeling the conversion decision can be found in the literature and generally fall into two categories: Probit models and duration models. The general intuition behind the Probit specification is an attempt to capture the effects of variables that increase or decrease the probability of a landowner to convert land from rural to urban uses. Many of these variables will be measurable and relate to certain characteristics of the land, such as land rentals, current land use, proximity to other rural or urban areas, estimated nearby land prices, and spatial characteristics. Other measurable variables may be unique to the landowner such as age, gender, income, and occupation. However, many characteristics of both the land and the landowner are not directly measurable or observable and so a stochastic framework is necessary. Suppose we have a landuse decision rule derived from the capitalization approach given by Capozza and Helsley (1989) of the form: R(w,,T) = A(x,,T)+rC(z,,T) (235) The functions R(*), A(*), and C(*) in Equation (235) represent the returns from conversion, rural land rent, and conversion costs, respectively. The discount rate is given by r and the optimal time of conversion is T. Vectors of observable characteristics describing conversion returns, rural rents, and development costs for parcel i are given by w,, x, and z, respectively. If we define a vector of unobservable characteristics for parcel i as y,, the probabilistic model of the development decision can be formulated as in CarrionFlores and Irwin (2004): Pr(D(i,T) = R(w,,T) A(x,,T) rC (z,, T) + E (,,T)) > 0 (236) The decision to develop rural parcel i at the optimal time T is given by D (i, T). The error term E (\,, T) associated with the development decision is assumed to follow a normal distribution. The parameterization of the model can be made explicit: Pr (D(i, T) = 9(i, T)' + (,, T)) > 0 (237) The vector p(i, T) denotes the vector of observable characteristics (w, x, z,) and the vector of parameters to be estimated is given by p The equation becomes more specific depending on the choice of variables to include. Further, if spatial characteristics are included, the model may require correction for spatial error autocorrelation in the error term E (i,, T). This specification was used by CarrionFlores and Irwin (2004) to determine the factors associated with rural land conversion to residential uses in Medina County, Ohio. Using parcel level data from a Geographic Information System (GIS), the authors use the model in Equation (237) to explain the conversion pattern between 1991 and 1996. Among the variables included in the Probit function are: distance to Cleveland, distance from nearest town, population, neighboring residential, agricultural, commercial, and other areas, population density, size of parcel, and soil quality. Results show the probability of conversion decreases with distance from Cleveland, population density, size of parcel in acres, and if the parcel is considered large. Conversely, conversion is more probable with greater distances to the nearest town, better soil quality, and greater neighboring residential and commercial areas. Of interest is the variable on distance to Cleveland, a highly urbanized location. For parcel located within 14 miles of the 93 Cleveland fringe, the probability of development decreases at a decreasing rate, but outside this 14 mile boundary the probability of conversion increases. Of contrary expectation is the result that larger parcels are less likely to be developed. The authors state, however, this may be a result of the limited data available since the model is unable to distinguish between undeveloped land and land that is undeveloped but zoned for development (CarrionFlores and Irwin 2004). Ding (2001) also utilizes GIS data for Washington County, Oregon and estimates the probability that a parcel of vacant land is developed into an urban use between 1990 and 1994. His data set is comprised of nearly 14,000 identified vacant parcels with almost 5,000 converted into an urban use during the studied time period. Variables in the Probit analysis include access time to the central business district for four urban areas (Beaverton, Forest Grove, Hillsboro, and Portland), dummy variables indicating what urban area the parcel is located in, dummy variables indicating adjacency to major roads, adjacency to existing urban land, and adjacency to parcels also being converted. Dummy variables are also included to indicate if the parcel is located in a flood plain, a growth boundary, and a one mile zone of light rail. Two continuous variables are included for distance to the urban growth boundary and a tax rate. Ding (2001) also finds that the likelihood of conversion decreases with parcel size like in CarrionFlores and Irwin (2004). Similarly, he finds that land is more likely to be converted if it is closer to Portland but further away than the less urban areas of Beaverton, Forest Grove, and Hillsboro. Interestingly, Ding (2001) finds the probability of conversion greater in areas with a higher tax rate. While he explains this as a possibility of higher tax regions producing better amenities such as schools, he does not investigate this further. Another likely possibility, as is the fact that higher tax rates, when they are anticipated, tend to speed up the conversion process as developers try to avoid the higher tax penalties. A lagged and lead tax variable ought to capture this effect, but Ding (2001) does not include these in his Probit model. Claassen and Tegene (1999) take a slightly different approach and use a Probit equation to model the conversion of pastureland to cropland, rather than to urban uses. Although the end development is different, the concept is the same: conversion from a lower to higher use. Another alternative use is given by the Conservation Reserve Program (CRP), established by the 1985 farm bill. The CRP pays landowners to keep their land out of any productive uses, such as forestry or agriculture, in order to preserve the land. One of the variables in Claassen and Tegene (1999) is the difference in rental rates between cropland and pastureland. The estimate on the rental rate difference is positive and significant suggesting that when cropland rents are large relative to pastureland rents, then the probability of conversion to cropland is greater. Estimates on the first and second lag of the rental rate difference are positive and negative, respectively, but not significant. The estimated interest rate effect suggests that the probability of conversion from pastureland to cropland conversion is smaller with higher interest rates. The estimated coefficient on the rent received if land is enrolled in the CRP program is negative, suggests that as the rent received under CRP increases the likelihood of land remaining in pastureland decreases. Cho and Newman (2005) provide an innovative threestage analysis of the development process. In the first stage they estimate a hedonic regression of land values. The results of this equation are used in the second stage in which a Probit equation is used to estimate the probability of a parcel of land being converted to a developed use. Finally, a thirdstage Probit equation is used to estimate the density of development. Their data on vacant land parcels for Macon County, North Carolina was obtained from the land records division of the tax administration department for over 40,000 parcels, of which nearly 16,000 were converted between 1967 and 2003.15 Cho and Newman (2005) find that an undeveloped parcel is more likely to be developed if it is located near a parcel that is already developed. The predicated value of land, quantity of roadways in the area, and degree of flatness of the land all are estimated to increase the probability of conversion. The Cho and Newman (2005) results are intuitive and follow from the theoretical analysis. For example, as a parcel of land increases in value it becomes too costly for the landowner to keep the land in vacant or agricultural use, as the rentals received from agriculture are not enough to compensate for the opportunity cost of conversion. Further, the flatter a parcel of land is, the more amenable it is for residential housing and commercial projects. The probability of conversion and the density of development declines with the size of the parcel, distances to roadways, and the median elevation on the land. Large lots are less likely to be developed since residential developments occur after rezoning, which will break a larger lot size into many smaller lots. The presence of a large lot, say 10 or more acres, indicates the possibility that the land may not have been zoned for development yet. In fact, Cho and Newman (2005) find that parcel sizes greater than 10 acres have nearly a zero chance for high density development. Another common econometric method of modeling the conversion process is through the use of duration or survival models. Duration models are often used to answer questions regarding the duration of unemployment spells, or time intervals between human conceptions. In general, duration models are concerned with how a variable changes through time, from one state to another and are particularly well suited to the land conversion decision. Two key questions are addressed by estimating the conversion process through a duration model. First, what is the 15 According to the authors, the data set was updated every 4 years. 96 length of time a parcel of land will remain undeveloped? Second, what is the likelihood that it will be developed in the next time period? Typically the duration of time a parcel spends in an undeveloped state, the key variable of interest, is described by a hazard function. Other variables may change during the conversion duration such as population growth rates, conversion of nearby parcels, capital costs, interest rates, and even the landowner's discount rate. 16 Such time varying covariates represent additional complications and can also be included in the hazard function. The hazard function is the probability density of the duration of being undeveloped and is a function of time. To provide intuition for this method, modify Equation (236) so that we have: Pr (sy,, T) < R(w, T) A(x,T) rC(z, T)) (238) Landowner's who currently engage in farming receive rent A(x,, T) and are characterized by a vector of unobservables given by e (yi, T). Individuals who are better farmers or who place higher value on land if it is in farming use will have a later conversion time than individuals who are not as able farmers or who place less value on the land in farming use. Therefore, the probability that a parcel will be converted at time T can be defined as the hazard rate for that time period for a given set of characteristics (Irwin and Bockstael 2002). This hazard rate is given by: ) E*(T+1) G(T)] (239) h(T}= (239) 1 G l E (T)] 16 A burgeoning literature has spawned beginning with Strotz (1956) on how individual time preferences may change through time possibly exhibiting declining preferences or even preference reversals. This may suggest the need to model the conversion process with a nonconstant discount rate, such as the hyperbolic discount rate. For more discussion on this literature see Frederick, Loewenstein, and O'Donoghue (2002). The cumulative distribution function for the unobservables is given by G [*] and e* is the unobservable value that makes Equation (239) hold with equality. As noted by Irwin and Bockstael (2002), this is the value that makes the landowner indifferent between keeping the land in an undeveloped use or convert to a developed use. There are multiple methods of calculating the hazard function. One involves parametric estimation which requires an assumption on the distribution of Equation (239). Common distributions include the exponential, Weibull, and variants of the logarithmic and normal distributions such as the lognormal and loglogistic (Bell and Irwin 2002). However, as noted by Greene (2003), the choice of distribution has profound implications on the answers to the questions regarding the timing of conversion. For example, the hazard function can slope either upwards or downwards, depending on whether the duration length increases or decreases the likelihood of the parcel not being converted in the next time period. One way of avoiding this problem is to estimate the hazard function through semi parametric methods.17 This is often referred to as a proportional hazard model or a Cox regression model. Two distinct parts comprise the hazard function in a Cox model: the baseline hazard function and the explanatory function. The general form of the proportional hazard function is: A(t) = A (t) ex (240) The baseline hazard in Equation (240) is A0 (t) and represents the heterogeneity among individual observations, the explanatory component is given by ex'p. The vector of exogenous variables is defined as x' and the vector of estimated parameters is /7. The key feature of the 1 Another approach would be through a fully nonparametric estimation method. However, this method introduces its own complexities and thus the Cox method serves as a nice median between parametric and nonparametric estimation methods. formulation above is that time is distinct from the vector of explanatory variables. An implication of this distinction of time is that the hazard function for each individual observation is a proportion of the baseline hazard. Being set up as a proportion implies that as values in the explanatory variables change, the function A, (t) shifts so that a value of the hazard A (t) is attained (Greene 2003). Cox (1972) defines a partial likelihood function which is maximized to obtain estimates of the parameters without having to estimate the baseline hazard. The proportional hazard approach is used in Irwin and Bockstael (2002) to obtain undeveloped parcel conversion rates. In particular, the authors focus on the effects of neighboring parcel conversions by measuring surrounding developments as a way of identifying potential spillover effects. They use an intricate data set on a seven county region of Maryland including Washington, D.C. and Baltimore as major urban areas. The data set was obtained from the state's panning office. Variables in the analysis include an index of zoning potential, a variable measuring the maximum permitted development density, distance measures to Washington, D.C. and Baltimore, an indicator for parcels that are relatively more costly to develop due to steep slopes or poor drainage, and an indicator for prime agricultural land. Estimates from Irwin and Bockstael (2002) fully specified model imply a reduced hazard of development with greater commutes to Washington, D.C., steeper and more poorly drained soils, being prime agricultural land, and greater allowable density. If greater density is allowed and development returns are increasing over time with a concave production function, then a landowner will find it optimal to delay conversion until a later time. Of particular interest is the result of negative spillover effects, implying a negative interaction between undeveloped and developed parcels in the decision to convert land (Irwin and Bockstael 2002). While this negative interaction might seem conflicting with the theory, the authors note that the analysis is of land along the ruralurban fringe where amenities such as open space are more highly regarded. Further, the magnitude of the negative interaction is based on commuting distances between Baltimore and D.C. being held constant. Option Value Empirical Methods While options value theory is a relatively recent development in the financial economics literature, empirical testing of real options models is even more recent. The empirical models of the capitalization approach have a firm ground in econometric theory. Optionbased econometric approaches are not afforded the same luxury. However, in recent years, a number of empirical studies have emerged, particularly in the real estate economics literature, due in part to the attractive nature of the theoretical aspects of the option value model. A review of some of the most noteworthy studies is discussed in this section. One of the first such tests is Shilling et. al (1990) who use a simple ttest to determine if a time premium is present in a real option model of land development. The authors estimate both the current market value of developable land, V, given by the appraisal value and the discounted exercise price Xe The discount rate is given by r, the development time is T and the exercise price is X. The authors then calculate the mean difference between the option premium and the intrinsic value. A ttest on the null hypothesis of the presence of a time premium is given by: t 0 (241) [(s s)/Nl5 Shilling et. al (1990) define c as the mean option price, and Q as the mean intrinsic value given by the mean difference of V Xe ". The respective sample variances are s2 and s and the number of observations is N. The null hypothesis could not be rejected at the 90 percent confidence level, indicating a zero mean difference between the option premium and the intrinsic value (Shilling et. al 1990). However, this test is far from rigorous and does not explicitly examine how an empirical model of option pricing affects the conversion decision. The first rigorous empirical study is Quigg (1993) and examines the optionbased value of undeveloped land by directly incorporating the value of waiting to invest in land development into a simultaneous equations model. Relaying on the hedonic methods of Rosen (1974), the author specifies a hedonic price function, p(Z), on how market prices of land characteristics, Z, affect the price of land. The estimates from the hedonic price function are used to estimate another equation describing the potential value of construction on an undeveloped parcel. The data comes from the Real Estate Monitor Corporation and consists of a substantial number of land transactions in Seattle, Washington from most of 1976 through 1979 and includes 2,700 transactions of undeveloped land parcels (Quigg 1993). The central conclusion is that the development option represents a premium in the market price for undeveloped land at an average of 6 percent of the land's value. Further, the model does well at predicting transaction prices unlike the net present value methods discussed earlier. Capozza and Li (2001) investigate the effects of positive interest rate changes in an uncertain real option with variable capital intensity and a project that is irreversible. The general view from real options model of investment is that increases in the interest rate tend to increase investment (Ingersoll and Ross 1992; Capozza and Li 1994; Capozza and Li 2002). In the context of real estate development, Capozza and Li (2001) test this response using panel data on residential building permits, to empirically test the presence of a positive relationship between the interest rate with the land investment decision. The response on the hurdle level of net rents to interest rate changes varies with the level of growth and uncertainty. When either growth rates or uncertainty, measured by the variance or volatility of growth, are high, positive responses to development or conversion from interest rate changes tend to occur. Data from the U.S. Department of Commerce on building permits and population growth is obtained for 56 metropolitan areas from 1980 to 1989. Population growth rates are used as a proxy variable for net rental growth rates. The primary home mortgage rate is used as the nominal interest rate variable. The authors estimate a regression of the form: GHPCp AGPOP  =a +P + pAGPOP + P3AGPOl2 + 4,SGPOP +F (year)+ ,, (242) GRMA RM The term GHPC is the annual growth rate of building permits per capital in percent for single family homes in area i for year t. GRM is the annual percentage change in the real mortgage rate for year t. RM is the annualized real mortgage rate in percent terms defined as the beginning of the year yield on the primary conventional mortgage minus the current CPI inflation rate. AGPOP is the average annual growth rate of population in area i. SGPOP is the standard deviation or volatility of population in area i. The ratio of GHPC to GRM measures the elasticity of residential investment in terms of building permits (Capozza and Li 2001). The authors also include variables for government regulations, tax rates, and monetary policies. Several forms of the regression above are estimated. In particular, deterministic and stochastic versions are obtained to estimate effects in a certain and uncertain world given by variable growth rates and volatility. In the certain case, the ratio AGPOP/RM, serving as a proxy for the ratio of growth rates to the interest rate, is positive and significant, supporting the Capozza and Li (1994, 2001, 2002) model. In the uncertain case, AGPOP is used as the explanatory variable and is also positive and significant suggesting again that growth rates respond positively to interest rate increases. Regressions including the variable SGPOP for population growth volatility reveal positive and significant 102 estimates, also indicating positive responses to interest rate changes under greater uncertain conditions. The Capozza and Li (2001) regression results point to the importance of accounting for growth rates in any policy with a purpose of affecting investment rates. Since metropolitan areas are quite heterogeneous in terms of the population growth, interest rate policies can have very different effects between localities. Further, extreme care is warranted in any changes to the interest rate, since investment can be hastened beyond which the monetary authority intended depending on the volatility of growth rates. Further empirical evidence on the importance of option values on the decision to convert land under uncertainty is presented in Schatzki (2003). The standard expected net present value model is compared to a real option investment model with uncertain returns. Under expected NPV models, the decision to convert land is made by the landowner once the discounted stream of returns from the converted use exceed that of the unconverted use after accounting for conversion costs. However, conversion decisions under a real options framework with uncertainty tend to have higher necessary returns to induce conversion than under the expected NPV method. The higher returns needed are due to the option to delay and sunk costs which cannot be recovered after conversion (Schatzki 2003). Since landowners have an incentive to delay conversion from additional information about future returns, there is value in the option to convert land. Schatzki (2003) conducts empirical tests on the effect of uncertainty in the decision to convert land from agriculture to forested using panel data from the National Resources Inventory (NRI) from 19821992. The NRI is a statistical survey of land use and landuse changes on parcellevel nonfederal lands with a particular focus on the conversion of croplands to forests. In the Schatzki (2003) model, the problem of the agricultural landowner is to choose the maximum of two alternative uses. The first is the sum of the expected returns from cropland in the first period with the expected value of land in agriculture while the second is the expected value of bare forest minus conversion costs (Schatzki 2003): maxV =max {E[Ra] +e rtE[V],e rtE[V ] Cj (243) The expected value operator is given by E [], the annual agricultural returns are R,, the value of land in agriculture is V,', the value of land in forests is VT the cost of conversion is C" and the discount rate is given by r. If returns are uncertain and assumed to follow a Brownian motion with drift, the conversion rule can be written as: = RF (jat ,oftatft ,tCa, g,(t)) (244) Ra The annual return to forests is R The variances of the motion process are given by oa, or, and the Brownian motion drift parameters are p/a,/ p The correlation between agricultural and forest returns is p,; and the growth rate of forests is g (t). Thus, the landowner will convert land from agriculture to forests when the relative return of forest to agriculture is greater than a threshold based on the set of variables. Due to the uncertain nature of returns, the threshold necessary to induce investment in the conversion decisions is higher than in the standard expected NPV model without an option value. This is because the option to delay conversion has value. To determine the probability of a parcel being converted from cropland to forests, the author estimates a limited dependent variable regression model of the form: Pr(conversion) = Pr (ln Rf, In R, > In R ) (245) Parcellevel returns to forests and agriculture are given by In Rf, and In R,, respectively, and the conversion threshold is defined by R Relevant explanatory variables in the NRI data set that are in the regression include indicators of land quality, conservation practices, population density, irrigation, uncertainty, agricultural revenue and forest return trends, current revenues and returns, and the correlation between agricultural revenues and forest returns. Schatzki (2003) finds the probability of conversion falls with increased uncertainty of returns in agricultural revenues and of forest returns, suggesting that option values affect landowner decisions. Thus, as either agricultural revenues or forest returns increase, the conversion threshold also increases. Interestingly, the coefficient estimate on the correlation of agricultural revenues and forest returns is positive and significant, indicating that the likelihood of conversion is greater with a higher correlation. Finally, conversion is more likely when agricultural returns are low and forest returns are high, the land is not irrigated, population density is low, and no conservation practices are used (Schatzki 2003). Chapter Summary The literature review contained in this chapter has examined both theoretical and econometric models of land use and land development. The land development decision in regards to conversion use and timing can be modeled by three approaches: capitalization, option value pricing, and transaction costs. Within the capitalization framework, the standard net present value of land is the primary method of obtaining land values and the decision time to develop land. Capitalization approaches use both partial and general equilibrium models of the development process. However, capitalization methods have failed to accurately predict land values (Falk 1991) and do not explain the gap between land rents and prices (Capozza and Helsley 1989). Two alternative approaches were discussed. The option value approach accounts for the inherent value of waiting to invest in land development arising from both uncertainty and irreversibility. While option value models have the advantage of being more realistic, they are conceptually more difficult to employ. The transaction costs approach accounts for the fact that land transactions are long, infrequent, and costly. Typical costs such as informational and institutional costs, not captured by either the capitalization or option value approaches, are expressed in a transaction costs framework. While this method holds particular merit, the lack of a sophisticated conceptual model prevents it from mainstream utilization. Recall the questions posited at the beginning of this discourse. When is it optimal to convert land from a lower rural use to a higher urban use? What is the optimal intensity of capital that should be applied to the land development project? How do government policies, such as property taxes and growth controls affect the optimal timing and intensity? What affects to changing discount rates and expected returns imply for land development? How does an uncertain world alter matters? This review has attempted to compare and contrast how answers to these questions differ depending on the choice of theoretical and empirical model. Further, even within the same approach, disagreement occurs between authors as to the correct answer due to the unique nuances within a particular framework. This disagreement not only remains problematic for academic researchers, but also especially for policy makers who require a keen understanding of the development process in order to formulate policies intended for rural conservation or urban growth. Accurate and efficient policy instruments cannot be designed without an understanding of land development models and a knowledge regarding timing decisions. A better knowledge of the conversion process will facilitate an improved understanding of numerous outcomes of development that seem random such as discontinuous urban development, heterogeneous spatial patterns in land use, and the creative destruction of capital structures. This review contributes to the understanding of the development process through a comprehensive critique of the currently used conceptual and empirical models, something which is lacking in the literature. Recommendations for future work include continued advancement of real options and transaction costs models, as these remain the most encouraging in terms of accurately describing the development process. Further, since the development process is an intertemporal decision, it is recommended that models from the economics of time literature be applied. Methods proved fruitful in other fields, such as dynamic programming, have yet to be applied to the land conversion problem. Since models of land development are derivative of models of land values, a complete understanding of the development decision must come from a thorough knowledge on the nature of land values. This paper serves to enhance the body of knowledge by introducing time inconsistent preferences to a model of land values. The next chapter will present the theoretical model. Land value Rural bidrent unction Urban bid rent function City center A Figure 21. Land allocation and bidrent model Net rent I Distance from center [(lh) Figure 22. Optimal land allocation  Land (L) H(t,T*) A(t) ** Figure 23. Optimal conversion time Return VH I, pI RH RA 0 t T Time Figure 24. Change in value of agricultural land awaiting conversion Time T Time Figure 25. Timing of conversion decision Payoff Exercise price .....450 ........ 5 Stock or commodity price Figure 26. Call option payoff Table 21. Selected comparative static results from capitalization papers Exogenous variables Capital / Pre Post Pre Post Endogenous conversion Discount development development development development Study variables costs rate rental rate rental rate property tax property tax Land value / N/A + N/A N/A Shoup pnce (1970) Conversion N/A + N/A N/A time Land value / Landvalue N/A + +  Skouras price (1978) Conversion N/A +  time Land value / Markusen price + 0 & Scheff man (1978) Conversion + + 0 ? + time Land value / N/A +  Arnott & price Conversion Lewis Conversion + ? N/A + (1979) time (1979) CapitalN/A 0 tn / 0 N/A 0 0 intensity Land value / Ellson & price+ + N/A N/A Roberts (1983) Conversion + + ? N/A N/A time Land value / A N/A + +  Anderson price (1986) Conversion N/A + ?  time Capozza & Land value/ + + + N/A N/A Helsley pnce (1989) Conversion + + ? N/A N/A time CHAPTER 3 THEORY AND EMPIRICS This chapter presents a presentvalue framework for land price determination using rational expectations in the context of farmland values. The theoretical model is formulated, generalizing time preferences to permit for quasihyperbolic discounting. The empirical model is developed from the theory presented and allows the exponential and quasihyperbolic discount parameters to be obtained. A hypothesis test is constructed, permitting for a direct test on the discount parameters. Finally, the econometric procedure is presented. The parameters are estimated using the linear panel Generalized Method of Moments estimator. The discussion pays particular attention to the selection of instruments. Theoretical Framework The main assumption underlying presentvalue models of farmland values is that the expected rents received from land exclusively determine the value of land, holding other factors fixed, such as taxes. While this is a strong assumption, present value models of asset prices under rational expectations rely on this principle of rental determination. The other implied assumptions are riskneutral landowners and a timeconsistent discount factor with a constant rate of discount. The timeconsistent assumption will be relaxed, permitting for quasihyperbolic discounting in the asset value equation. Specifically, a model for values based on changes in asset valuation over time will be derived and will nest both the standard case of exponential discounting and the case of quasihyperbolic discounting. Financial theory typically stipulates that a firm or individual decisionmaker should adopt a project, such as converting or developing land, if the net present value of the project is positive. Under simplifying assumptions of risk neutrality and timeconsistency, the value of land today can be written as the discounted stream of expected future rents: V =Z K,Et[R (31) s=0 The price or value of land at time t in Equation (31) is V,. The expected rental rate or nominal cash flow from land use at time t + s is Et [R ] and is based on the information available to the landowner in period t. The standard exponential discount factor is defined as: t+ = i (1+ pt+,). (32) 1=1 The discount factor is taken to be 3 e [0,1]. The rate of time preference in period t is given by the constant rate of discount, p, ,, which is often assumed to be the nominal or real interest rate. The formulation given in Equation (31) and Equation (32) is the basic expression for land values examined by many authors to test how well the PV model can explain movements in farmland values when a rational expectations framework is assumed (Falk 1991; Clark, Fulton, and Scott 1993; Tegene and Kuchler 1993; Schmitz 1995; Schmitz and Moss 1996). Based on this formulation, the expectations in the land market that both buyers and sellers have on discounted future returns play a central role in determining the value of land. Some studies have rejected the present value model as a rejection of rational expectations (Lloyd, Rayner, and Orme 1991; Tegene and Kuchler 1991, 1993; Engsted 1998). However, this rejection may be due in part to a potentially anomalous assumption of exponential discounting. An alternative formulation takes note of potential time inconsistency and assumes a quasi hyperbolic discount factor implying a nonconstant rate of discount. Quasihyperbolic discounting, developed by Laibson (1997), is a discretevalue time function and maintains the declining property of generalized hyperbolic discounting. At the same time, the discrete quasi hyperbolic formulation keeps the analytical simplicity of the timeconsistent model by still incorporating certain qualitative aspects of exponential discounting. The actual values of the discount factor under a discrete setup are 1, /8. 8. 2,.. .,/. 8o" with the time periods defined as t = 0,1, 2,..., o When t = 0, the discrete discount function is normalized to one. The 6 parameter behaves similar to the exponential discount factor. The /3 parameter captures the essence of hyperbolic discounting and contains a first period immediacy effect in the individual's time preference. Discount rates under a quasihyperbolic discount function clearly decline over time as the shortrun discount rate, given by In (/ ) is greater than the long runs discount rate, given by In(3) as computed in Laibson (2007). Setting up the problem under quasihyperbolic intertemporal preferences implies that Equation (31) takes the form: V, = 8. E ,]. (33) s=l Changes in the / parameter in Equation (33) determine how much the discount factor will deviate from exponential discounting. If discounting is timeconsistent then / = 1, and is time inconsistent if p e (0,1) Following the literature, /8.8 represents the discount factor between the current time period and the next time period, while 3 represents the discount factor between any two future time periods (DellaVigna and Paserman 2005). The expression in Equation (33) can be modified to derive a model for asset values based on changes in the asset valuation over time. Specifically, taking the time difference of Equation (33) yields: AVT = V, V, =i Eq u.St E, [(3,4 ] c b .e si_,mE [' a ,r]. (34) s=1 s=0 The expression in Equation (34) can be simplified by aggregating over like exponents to: AV, = 6 E, t] +P. 6 1..{ E[/.1 E E E ]}. (35) s=1 Under rational expectations there is a forecast error in the second term on the right hand side of Equation (35) between the two expectation operators, meaning E, [R~ ] = E,1 [R ] + e, where e, is an uncorrelated residual term. Equation (35) can be written as: A = 3. E,[,] +] 3. (, 1'1)Et[R,,J+es (36) s=1 The residual e, represents a "white noise" forecast error if expectations are in fact rational, implying that no information is present in the error term. Simplifying gives: AV = 36 E [R, +] +(1 ). t+sE,t [R,+e,. (37) s=1 The observed cash flow in the next period is assumed to proxy the expected return in the next period, that is, E, [R, ] R,. Equation (33) can also be substituted in for the second term on the right hand side of Equation (37). Making these changes yields the structural model: AV, = p R, + (1 8)V +e,. (38) To test for hyperbolic discounting, the structural equation given by Equation (38) can be parameterized into an identified reducedform linear panel regression. Doing so gives: AV, = a, +a,R +a2t e,. (39) The reduced form model given by Equation (39) includes a constant term, ao, which is included in most regressions on land values. The reduced form specification in Equation (39) has also been extended to account for observations over time and space. Observations come from a sample of i geographic regions over a period of t years. The nice feature of Equation (39) is that the formulation nests both the exponential and hyperbolic discount factors. The discount parameters in the structural specification in Equation (38) can be obtained once the reducedform coefficients in Equation (39) have been estimated. Since a, = ,8 and a2 =(1 ), then: = (12), (310) = a (311) 1 2 If the assumption of exponential discounting is true, then one would expect a, to be close to 1 and a, to be close to zero. The constant term, a,, should not be statistically different from zero under rational expectations regardless of the shape of the discount factor. Based on the reducedform parameter estimates, two types of hypothesis tests can be devised to test for the presence of hyperbolic discounting. The assumption of / = 1 is equivalent to assuming exponential discounting, which implies a2 a = 1 using Equation (310) and Equation (311). Hence, an appropriate test for exponential discounting would test the null hypothesis of Ho : a2 a = 1 against the alternative hypothesis of Ha : a2 a1 # 1. This test will be referred to as the implicit test of hyperbolic discounting, since the value of / is assumed and since the test basically amounts to a test of exponential discounting. A second test is a direct nonlinear hypothesis test on Equation (311) with the null hypothesis of Ho : a1/(1 a,) = 1 against the alternative of H : al/(1 a2) < 1. This test is referred at as the explicit test of hyperbolic discounting since the standard error on the hyperbolic parameter is computed. While both tests are equivalent in theory, the implicit test is more efficient while the explicit test introduces some noise due to the Taylor series approximation. Overall, the explicit test is better empirically since it provides an exact test on hyperbolic discounting. Since / is a nonlinear function of the parameters, a linear Taylor series approximation is used to obtain the standard error: h(&) = h(a) + ah a). (312) The hyperbolic parameter is defined as a vector function of the estimated coefficients, h(d), in Equation (39) to obtain the linear Taylor series approximation. The formulation in Equation (3 12) appeals to the Central Limit Theorem and asymptotic theory for consistency. If the Law of Large Numbers holds and if the data are independently and identically distributed then obtaining the standard errors through a Taylor series approximation is appropriate. If these assumptions hold, then the standard error of h(d) is given by: S.E. [h(&)] {= (VAR []d (313) The partial derivatives in Equation (313), while functions of the unknown parameters, can be computed using the sample estimates. Econometric Procedure Given the nature of the data on land values, using a simple estimator such as least squares for estimating the panel regressions in Equation (39) would not be appropriate for a number of reasons. First, while the error term, e,, is assumed to be independently and identically distributed with zero mean, heteroskedasticity across years and farms remains a possibility. Second, while the error term is not correlated with the dependent variable AV,, the first difference of farmland values, et is serially correlated across time. For these reasons, estimation by the Generalized Method of Moments (GMM) is preferred. Originally proposed by Hansen (1982) and Hansen and Singleton (1982) for estimating consumptionbased asset pricing models, GMM provides consistent estimates of the parameters. The consistency of the estimates however is largely determined by the selection of instruments, which often remains a difficult task. Despite the benefits of GMM, however, surprisingly little work has been done applying GMM to the econometric problems inherit in land values data (Chavas and Thomas 1999; Lence and Mishra 2003). The regressors in Equation (39) have both timevarying and timeinvariant components, with observations assumed to be independent over i. The time period T is fixed, and the variables can be stacked for the ith region over all T. Rewriting Equation (39) in more general terms yields: y,= Xa+ (314) The dependent variable and the error term in Equation (314) have been reduced to Tx 1 vectors, while the matrix of independent variables, XA, is Tx K, with K denoting the number of regressors: y, ; = : ; = (315) LAVJT VT LeT Panel GMM estimation of Equation (39) follows from the sample moment condition: E[Z, (a)]= 0, i= ,...,N. (316) The matrix of instruments, Z, = [z' z' ] has dimension Tx r, with r denoting the number of instruments and T denoting the number of time periods. In the residual vector, E, (a), a is a K x 1 vector of K parameters to be estimated. The foundation of GMM estimation is the specification of the orthogonality or moment condition in Equation (316). The goal of the GMM estimator is to find a vector of parameter estimates so that the residuals are orthogonal to the set of instruments. Given this background, if there are more instruments than parameters, that is if r > q, the form of the minimand for the linear panel GMM model, QN (a), can be expressed as (Cameron and Trivedi, 2005): QN(a)= [Z ,(a) WN, JZ;,(a) (317) The matrix WN is a r x r positive semidefinite weighting matrix, akin to a variance matrix, which depends on the data. The weighting or distance matrix, WN, converges in probability to a nonstochastic positive definite matrix of constants, W. The orthogonality conditions in Equation (316) are in a way being emulated by minimizing the function QN (a), which is a quadratic form of the sample means across both time and space. The panel GMM estimator aGmc can be defined as: aGmu = argmin,.QN (a) (318) Equation (318) states that GMM estimation selects the value of the vector a from a subset of the parameter space 0D, which is itself a subset of the qdimensional Euclidean space 9V, so that the value of the function QN (a) is minimized (Hall 2005). The GMM estimator is asymptotically normal with variance matrix consistently estimated by: (aGm) =N[Xi= ZWNZ'] [X ZWN (i)wZ' X[ZWZ'X] (319) where S = N Z 1Z, Z Z. A Whiterobust estimate of S is obtained by assuming independence over i in the residual vector e, where = y, Xa The standard errors obtained from Equation (319) are robust to heteroskedasticity and serial correlation over time (Cameron and Trivedi 2005). The choice of the weighting matrix remains an integral component of the estimation procedure. The matrix WN determines how the information in the instruments based upon the moment condition is weighted in the estimation of the parameters. There is little guidance on how to select WN, and so estimation of Equation (317) proceeds in two stages. First, the weighting matrix is computed as W = (ZZ,) in the initial stage of the GMM estimator. Then V GM ) is estimated next based on the first stage estimation of aGM Thus, in the second stage, the first stage estimate of V (aGMv) is used in the second stage GMM estimator based on the weighting matrix now computed as WN = S1. A numerical optimization routine is typically necessary to estimate the panel GMM estimator. While the NewtonRaphson method is the usual choice, as noted by Hall (2005), this routine and others does not guarantee a global optimum has been achieved, even after a large number of iterations. The numerical procedure used here is that of Nelder and Mead, which performs well for nonlinear and nondifferentiable functions. Since this method uses only function values, computing a large number of iterations can be slow, but is also robust. The Nelder and Mead simplex algorithm is also preferred because of the degree of nonconvexities in quasihyperbolic functions, as noted by Laibson, Repetto, and Tobacman (2007). All parameter estimates were obtained using R version 2.6.0 software. The procedural code for each panel region is available in the Appendix. Instrument Selection and Identification The choice of instruments remains a crucial, but problematic, aspect of GMM. Nearly any variable that is correlated with a regressor but independent of the error term can be selected as an instrument, thus ensuring the overidentifying restrictions introduced by Hansen (1982) are in fact met. A benefit of the two step procedure of panel GMM estimation outlined above is that the estimator allows for the instrumental variable matrix Z to include regressors in the X matrix as well, thus aiding identification. Additionally, a benefit of panel data over cross sectional data is the availability of a weak exogeneity assumption: E [zselt ] = 0, s < t, and t = 1,..., T. As noted by Cameron and Trivedi (2005), this condition arises in models involving rational expectations and in models of intertemporal choice, like in the present paper. The condition given by E [zse] = 0 allows for lagged or lead values of regressors as instruments. The number of overidentifying restrictions is given by r K. A test of the over identifying restrictions is given by: JN = (YN) 'G )(QN) ( G YN ) (a G)'M Z, SZ's, (aGM )(YN) (320) The JN statistic converges to a 2 distribution with (r K) degrees of freedom. The null hypothesis is that the moment condition in Equation (316) is true, that is, Ho : E[Z;, (a)] = 0. If the value of the test statistic results in rejection of the hypothesis, then model misspecification may be the case. If the J, statistic is large, then another possibility could be the presence of endogenous instruments, that is, instruments that are correlated with the error term. Problems of weak identification occur when the moment condition in Equation (316) is nonzero but very small. Under such a situation the moment condition provides very little information about the parameter vector, calling into question the reliability of the estimates. However, one cannot take an insignificant statistic to imply that weak identification is not a problem. Although recent work has examined possible procedures for identifying and handling weak instruments, such as Stock, Wright, and Yogo (2002), there is no firm method to correct for weakly identified instruments and so the J, test remains the best available tool for examining model specification. Clearly, caution must be exercised regardless of the test results. Chapter Summary This chapter laid the theoretical foundation for empirical analysis. A net present value model for farmland is developed. The structural equation for the annual change in farmland values is obtained. The reduced form parameters for the quasihyperbolic and exponential discount factors, / and 3, respectively, are solved in an identified framework. The empirical strategy is to first estimate the reducedform in Equation (39). Second, the structural parameters identifying the exponential and quasihyperbolic discount factors are obtained through Equation (310) and Equation (311), and then appropriate hypothesis tests are conducted. Hence, testing for hyperbolic discounting is a twostep process. This chapter also worked out the econometric method used in the analysis, describing the linear panel GMM. Particular attention was paid to the choice of the weighting matrix and the numerical procedure used to obtain the estimated. The choice of instruments was described as well as the overidentification test explained. In the next chapter, the data are described and the primary estimation results are presented for the major U.S. agricultural regions. CHAPTER 4 DATA AND RESULTS To examine the possibility of hyperbolic discounting in U.S. farmland, annual observations on asset values and returns to agricultural assets are used. The data represent a 43 year panel from 1960 to 2002 of the nine major agricultural regions of the United States. The data come from the U.S. Department of Agriculture's (USDA) Economic Research Service (ERS). The primary source of data is the USDA's state level farm balance sheet and income statement. A detailed description of the agricultural regions and the variables are discussed first. Then the estimation results including estimates of the discount parameters are presented. Data and Variable Description The nine regions investigated include the Appalachian states (Kentucky, North Carolina, Tennessee, Virginia, and West Virginia), Corn Belt states (Illinois, Indiana, Iowa, Missouri, and Ohio), the Delta states (Arkansas, Louisiana, and Missouri), the Great Plain states (Kansas, Nebraska, North Dakota, Oklahoma, South Dakota, and Texas), the Lake states (Michigan, Minnesota, and Wisconsin), the Mountain states (Arizona, Colorado, Idaho, Montana, Nevada, New Mexico, Utah, and Wyoming), the Northeast states (Connecticut, Delaware, Maine, Maryland, Massachusetts, New Hampshire, New Jersey, New York, Pennsylvania, Rhode Island, and Vermont), the Pacific states (California, Oregon, and Washington), and the Southeast states (Alabama, Florida, Georgia, and South Carolina). Other regional definitions exist, such as those defined by Theil and Moss (2000). However, the USDA/ERS defined regions are used since they remain the most commonly used. The dependent variable is the first difference of farmland values, which represents the annual change in the value of farmland for each state. The independent variables are annual farmland values and annual farmland returns. Farmland values are defined as the value of farmland per acre. The definition of farmland returns is gross revenues per acre less the expenditure on variable inputs and follows Mishra, Moss and Erickson (2007). While many studies opt for a more complete specification of returns, such as the aggregate series of imputed returns as defined in Melichar (1979), the definition used in this dissertation is preferred. As revealed in Mishra, Moss, and Erickson (2004), measurement error problems are possible in more complete formulations of imputed returns if quasifixed assets are present. The data used in the analysis are the nominal data. The use of nominal data is preferred over the real data for two main reasons. First, the deflated real data has a higher variance, indicating more radical changes in both farmland values and returns between years. Secondly, and most importantly, the results are sensitive to the choice of deflator used. The estimation results were obtained under three different deflators: the consumer price index (CPI), the producer price index (PPI), and personal consumption expenditure index (PCE). The values of the discount parameters vary depending upon the choice of deflator used. In order to focus on the rate of time preference and avoid a debate on the best choice of deflator, the nominal data is used. Figure 41 through Figure 49 show the historical trend in the annual change of farmland values from the 1960 to 2002 time period for nine farm production regions (Appalachian, Corn Belt, Delta, Lake, Great Plains, Mountain, Pacific, Northeast, and Southeast). Within each region, states tend to follow the same pattern in the change of land values over time, that is, the year to year fluctuations by state within each region tend to be closely followed. However, notable exceptions can be discerned. Virginia departs substantially from the other Appalachian states between 1988 and 1991. Both California in Figure 44 and Florida in Figure 45, deviate from the other states in their panel region over the time horizon. Florida and California are quite distinct in the agricultural products each state produces, as well as the urban pressures generated from within each state compared to others the Southeast and Pacific regions. The overall pattern between agricultural regions is similar, namely the clear representation of the boom/bust cycle, which occurred from around 1970 to 1985. However, the magnitude of the fluctuations for each region are notably different, hitting the Corn Belt and Delta states the hardest followed by the Lake and Pacific states next. Land values for the Appalachian, Great Plain, and Southeast states were comparatively the least hurt during the boom/bust cycle. The Northeast states have the widest variation in land values over the 43 year time period. This is attributed to several reasons, but primarily because urban pressures are greatest for this region. Further, farmland and agriculture are not the highest value activity for land in the Northeast states. Also, the agricultural industries within the Northeast vary widely. As mentioned earlier, the selection of instruments can be a challenging task. Instruments ideally satisfy the orthogonality condition in Equation (316). Good instruments represent the nature of how land is valued and the expectations of the economic agent. For these reasons, lagged values of farmland values and return on assets are preferred instruments. Not only do they reflect the expectations of the agent, but they are known to both the econometrician and the farmland agent at the current time, making for a good choice of instrument (Chavas and Thomas, 1999). In addition to the regressors, the instrument set includes squared terms and lagged terms of land returns and land values, for a total of four overidentifying restrictions. Estimation Results Equation (39) is estimated using annual aggregate panel data for nine agricultural regions of the United States (Appalachian, Corn Belt, Delta, Great Plains, Lake, Mountain, Pacific, Northeast, and Southeast) over a 43 year time period from 1960 to 2002. Since the dependent variable is a first difference, an observation is lost for each state per a panel reducing the sample size by the number of states in the panel. A fixed effects dummy variable approach is used in the panel GMM. While this approach does have the unfortunate effect of losing degrees of freedom depending upon the number of states in each panel, the inclusion of dummy variables for controlling state fixed effects remains a simple way of acquiring estimates of the parameters. Tests for heteroskedasticity and serial correlation are conducted. The BreuschPagan test for heteroskedasticity is used in both the farmland values and returns series with the null hypothesis of no heteroskedasticity, and is distributed Chisquare with two degrees of freedom providing a critical value of 5.991 for each panel. The BoxLjung test for serial correlation is used on the change in farmland values series with the null hypothesis of no autocorrelation, and is distributed Chisquare with one degree of freedom providing a critical value of 3.841. Finally, Hansen's overidentifying restrictions Jtest is used to test whether the model is correctly specified with the null hypothesis that the overidentifying restrictions hold. Since there are four overidentifying restrictions and the test statistic has a Chisquare distribution, the critical value is 9.488. The critical values of all three test statistics are based on a 5% level of significance. The results for each regional panel are discussed first, followed by a general discussion of the results overall. Appalachian States The Appalachian states consist of Kentucky, North Carolina, Tennessee, Virginia, and West Virginia. The total sample used in the analysis is 210, with the omitted dummy variable being Virginia. The BreuschPagan (BP) test resulted in a test statistic of 16.405, implying rejection of the null hypothesis and indicates heteroskedasticity in the farmland values and returns series. The BoxLjung (BL) test is next conducted and results in a test statistic of 42.290, which also implies rejection of the null hypothesis, indicating the presence of serial correlation in the change in farmland values series. Finally the test statistic for the overidentifying restrictions test is 0.069, hence the overidentifying restrictions cannot be rejected. Since heteroskedasticity and serial correlation are present in the data, and since the overidentifying restrictions cannot be rejected, the discussion will focus on the GMM results. The parameter estimates are summarized in Table 41. The constant term is not significantly different from zero, as expected for a net present value model under rational expectations. The estimated coefficients on farmland returns (a,) and farmland values (a,) are of the expected sign with a, being statistically significant while a, is not. None of the estimated coefficients on the state dummy variables are significant. Based on the parameter estimates in Table 41, values of the exponential and hyperbolic discount parameters can be obtained through Equation (310) and Equation (311). The exponential discount factor is S= 0.939, and significantly different from zero. The hyperbolic discount parameter is S= 0.060, and is also significantly different from zero. However, if hyperbolic discounting is in fact present, / will be significantly different from one. To test whether the estimates represent exponential or hyperbolic discounting, the implicit test of Ho : a a = 1 is first conducted. The calculated Fstatistic is 119.963, which implies the null hypothesis of exponential discounting can be rejected at any conventional level of significance. The explicit test of hyperbolic discounting, H : a /(1 a2 ) = 1, is next conducted resulting an Fstatistic of 102.152, implying the null hypothesis of f = 1 can also be rejected. Again, the null hypothesis can be rejected at any significance level. The results lead to the conclusion that discounting is not exponential but hyperbolic in the Appalachian panel. Based on the estimates of / and 3, shortrun and longrun discount rates can be computed. Given the values of the discount parameters, the longrun rate of discount is ln(0.939) = 6.3% and the shortrun discount rate is ln(0.060 0.939) = 287.6% An interpretation of the estimated discount rates can be offered. The longrun discount rate of 6.3% says that the value of a dollar is worth 6.3% less in the long run than the present time or that a dollar in the long run is worth .063 cents less to you now. In other words, you value a dollar in the long run at about .94 cents right now. The shortrun discount rate can be interpreted similarly, though discount rates in excess of 100% are difficult to interpret. Suppose the short run discount rate was 75%. In this case you view a dollar in the shortrun 75% less than you do right now. In other words, if you discount rate is 75%, then you value a dollar in the shortrun as 0.25 cents to you right now. With a shortrun discount rate of 287.6%, you view a dollar in the short run at 287.6% less than you do right now. Such a high discount rate attaches a negative value to a dollar in the present. The key result, however, is that not only are the shortrun and longrun discount rates significantly different from each other, but the magnitudes starkly contrast one another and suggest that the initial time periods in the farmland values market are critical. Corn Belt States The Corn Belt states consist of Illinois, Iowa, Missouri, and Ohio. The total sample size used in the analysis is 210, with the omitted dummy variable being Missouri. The BP test resulted in a test statistic of 10.418, implying rejection of the null hypothesis and indicating the presence of heteroskedasticity. The BL test is next conducted, resulting in a test statistic of 86.902, which also implies rejection of the null hypothesis and indicates the presence of serial correlation. Finally, the test statistic for the overidentifying restrictions test is 0.069, implying that the moment conditions cannot be rejected. The parameter estimates are summarized in Table 42. The constant term is not significantly different from zero, as expected for a net present value model under rational expectations. The estimated coefficients on farmland returns (a ) and farmland values (a,) are of the expected sign and both are statistically significant from zero. None of the state dummies are statistically significant, though all are negative in sign. Based on the parameter estimates in Table 41, values of the exponential and hyperbolic discount parameters can be obtained through Equation (310) and Equation (311). The exponential discount factor is 3 = 0.929, and significantly different from zero. The hyperbolic discount parameter is / = 0.584, and is also significantly different from zero. To test whether the estimates represent exponential or hyperbolic discounting, the implicit test of Ho : a, a = 1 is first conducted. The calculated Fstatistic is 1.637, which does not exceed the critical value at (1, oc) degrees of freedom of 3.840 at 0.05 level of significance. Hence, the null hypothesis of exponential discounting cannot be rejected. The explicit test of hyperbolic discounting, H0 : al/(i a2) = 1, is next conducted resulting an Fstatistic of 1.417, implying the null hypothesis of / = 1 cannot be rejected. The results lead to the conclusion that discounting is not hyperbolic but exponential in the Corn Belt panel. However, longrun and shortrun discount rates can still be computed based on the parameter estimates. Given the values of the discount parameters, the longrun rate of discount is ln(0.929) = 7.4% and the shortrun discount rate is ln(0.584 0.929) = 61.2% However, these discount rates are not significantly different from each other since / = 1 could not be rejected. Delta States The Delta states consist of Arkansas, Louisiana, and Missouri. Land returns and land values are characterized by both heteroskedasticity since the BP test statistic is valued at 18.210. The change in land values variable is serial correlated as evidenced by the LB test statistic value of 46.401. Table 43 lists the parameter estimates. The total sample size is 126 and Louisiana is the omitted dummy variable in the analysis. The overidentifying restrictions cannot be rejected in the GMM regression since the value of the Jtest statistic is 0.089, which is less than the critical value of 9.488 for a Chisquare distributed test statistic with 4 degrees of freedom. In regards to the parameter estimates, the constant term is positive, but not significant. The coefficient estimate on farmland returns (a, ) is negative, as anticipated by the theoretical model, but also is not significantly different from zero. The coefficient estimate on farmland values (a2) is positive and significant. Neither of the coefficient estimates for the two state dummy variables are significant, though both are positive and similar in magnitude. Based on the estimates of a1 and a2, the quasihyperbolic discount factor is 8 = 0.155, and is significantly different from one based on the explicit test of hyperbolic discounting since the value of the Fstatistic is 11.199. The exponential discount factor is 3 = 0.963, with the null hypothesis of no hyperbolic discounting for the implicit test being rejected since the value of the Fstatistic is 13.869. Thus, in addition to the Appalachian states, evidence of quasihyperbolic discounting is found in the Delta states. The results imply that the longrun rate of discount is given by ln(0.963) = 3.8% and the shortrun rate of discount is ln(0.155 0.963) = 190.2% . Clearly, the shape of discounting is different for the Delta states between the longrun and short run, with this difference being statistically significant. Great Plain States The Great Plain states consist of Kansas, Nebraska, North Dakota, Oklahoma, South Dakota, and Texas. Based on the value of the BP test statistic of 15.104, and on the value of the LB test statistic of 89.157, heteroskedasticity and serial correlation are in fact present in this panel series. The parameter estimates are summarized in Table 44, noting that the total sample size is 252 and that South Dakota is the omitted dummy variable. The critical value of the Hansen Jtest is 0.063, which indicates that the moment conditions are not incorrectly specified. Results are similar to the Corn Belt states: the constant term is not significant and none of the dummy variables are significant. Also, the parameter estimates on farmland returns and farmland values are of the same sign and similar in magnitude. Based on the parameter estimates, the value of the exponential discount factor is also 3 = 0.929 and the value of the hyperbolic discount parameter is / = 0.508, however the standard errors are notably smaller in magnitude than the Corn Belt results. Conducting the implicit test of hyperbolic discounting yields an Fstatistic of 6.389 while the explicit test yields an Fstatistic of 5.064, both reject exponential discounting at the 5% level for (1, oc) degrees of freedom and imply that / is significantly different from one. Therefore, discounting is better represented by a hyperbolic discount factor for the Great Plain states. More importantly, the results imply a significant difference between the longrun and shortrun rates of discount. The longrun discount rate is ln(0.929) = 7.4% and the shortrun discount rate is ln(0.508 0.929) = 75.1% . Lake States The Lake states consist of Michigan, Minnesota, and Wisconsin. Heteroskedasticity and autocorrelation are indicated based the BP test statistic of 14.463 for the values and returns series and the LB test statistic is 64.008 for the change in land values series. The parameter estimates are summarized in Table 45. Total sample size for the Lake states is 126, with the omitted dummy variable being Michigan. Based on the choice of instrument, the Jtest statistic for this panel is 0.089, again suggesting that the overidentifying restrictions cannot be rejected. The calculated value of the exponential discount factor is 3 = 0.890. The hyperbolic discount parameter is / = 0.546, with a critical value of the Fstatistic of 3.920 based on (1, 121) degrees of freedom and a 5% level of significance. The implicit Ftest statistic is 5.090 and the explicit Ftest statistic is 4.187, both suggesting a rejection of exponential discounting and implying that again / is significantly different from one. Thus the Lake states panel is better described by quasihyperbolic discounting. The longrun rate of discount is ln(0.890) = 11.6% while the shortrun rate of discount is given by ln(0.546 0.890) = 72.2% . Mountain States The Mountain states consist of Arizona, Colorado, Idaho, Montana, Nevada, New Mexico, Utah, and Wyoming. The omitted dummy variable in the analysis is Arizona and the total sample size is 336 observations. The value of the BP test statistic, 13.434, implies that the land returns and values variables are heteroskedastic. The BL statistic, with a value of 42.712, suggests that the change in land values series is serially correlated. The GMM regression estimates, presented in Table 46, yield a Jtest statistic of 0.055, and so overidentification is met. The constant term is negative and insignificant. The parameter estimate of farmland returns is negative and significant, while the estimate of farmland values is positive and significant. None of the state dummy estimates are significant, though they differ in sign and magnitude. The value of the exponential discount factor is = 0.897 and the value of the quasi hyperbolic discount factor is / = 0.637. The implicit test, which has a null hypothesis of exponential discounting, has an Fstatistic of 3.949, and so the null hypothesis is rejected at the 5% level of significance. The explicit test, which has no hyperbolic discounting as the null hypothesis, has an Fstatistic of 3.305, implying that the null hypothesis cannot be rejected at the 5% level, but can be rejected at the 10% level. Based on the values of the discount parameters, the longrun rate of discount is ln(0.897) = 10.9% and the shortrun discount rate is given by ln(0.637* 0.890) = 56.8%. Again, evidence of quasihyperbolic discounting over exponential discounting is found. Northeast States The Northeast states consist of Connecticut, Delaware, Maine, Maryland, Massachusetts, New Hampshire, New Jersey, New York, Pennsylvania, Rhode Island, and Vermont. Rhode Island is the omitted dummy variable in the analysis, with 462 total observations in the sample. The BP test statistic of 82.065 indicates heteroskedasticity in the independent variables. The dependent variable is serially correlated, as evidence by the BL test with a statistic of 106.067. The overidentifying restrictions cannot be rejected, based on a Jtest statistic of 0.047. The parameter estimates are summarized in table 4.7. The constant term is positive and insignificant. The estimate on farmland values is positive and significant, which is contrary to the theoretical model prediction of a negative coefficient. The coefficient estimate is just barely significant at the 5% level, though significant nonetheless. The estimate on farmland returns is positive and significant. None of the state dummy variables are statistically significant. Based on the estimates of a, and a the exponential discount parameter is 3 = 0.966 and the quasi hyperbolic discount parameter is /7 = .107. The negative number on the quasihyperbolic parameter is a strange result, and is a statistically significant one. The implicit test yields a F statistic of 315.572 and the explicit test yields a Fstatistic of 278.590, which rejects exponential discounting at any level of significance. However, based on the negative value of the quasi hyperbolic parameter, the implication of the hypothesis tests is questionable. The results imply a negative shortrun discount rate of 227.07% and a long run discount rate of 3.5%. Most studies of farmland values ignore the Northeast region completely, owing to the diverse nature of the land market and the weak agricultural sector in these states. Pacific States The Pacific states consist of California, Oregon, and Washington. The BP test statistic of 22.462 implies heteroskedasticity in the independent variables and the BL test statistic of 61.317 implies autocorrelation in the dependent variable. The parameter estimates are summarized in Table 48, noting that the omitted dummy variable is California and that the total sample size is 126 observations. The exponential discount parameter is 3 = 0.933 while the hyperbolic discount parameter is / = 0.361. The implicit and explicit tests of hyperbolic discounting yield Ftest statistics of 8.881 and 6.563, respectively. Hence, at 5% level of significance and (1, 126) degrees of freedom, the null hypothesis of exponential discounting, that is the null hypothesis of / = 1, is strongly rejected. Again, evidence of hyperbolic discounting is found over exponential discounting. The implied longrun and shortrun discount rates are ln(0.933) = 6.9% and ln(0.361*0.933)= 108.8%, respectively. Southeast States The Southeast states consist of Alabama, Florida, and Georgia. The total sample size used in the analysis is 168, with the omitted dummy variable being Florida. The BreuschPagan test for heteroskedasticity resulted in a test statistic of 33.245, indicating the presence of heteroskedasticity. To test for autocorrelation, a LjungBox test is conducted, resulting in a test statistic of 56.392, which indicates the presence of serial correlation. Hansen's test of over identifying restrictions resulted in a test statistic of 0.077. Since the critical value is 9.488, the overidentifying restrictions cannot be rejected. The parameter estimates are summarized in Table 49. The constant term is not significantly different from zero, as expected. The estimated coefficients on farmland returns (a, ) and farmland values (a, ) are of the expected sign and both are statistically significant from zero. None of the state dummies are statistically significant. The exponential discount factor is S= 0.910. To test whether these calculations represent exponential discounting, the implicit test of Ho : a, a = 1 is conducted. The calculated Fstatistic is 9.900, which exceeds the critical value at (1, 162) degrees of freedom of 3.840 at 0.05 level of significance. Hence, the null hypothesis of exponential discounting is rejected. The hyperbolic discount factor is / = 0.313 with a computed standard error of 0.259. The explicit test of hyperbolic discounting, H0 : a'/(1 a) = 1, is next conducted resulting an Fstatistic of 7.043, hence rejecting the null hypothesis of no hyperbolic discounting at any conventional level of significance. Further, the values of / and 3 imply a shortrun discount rate of ln(0.313 0.920)= 124% and a longrun discount rate of ln(0.920) = 8.4% . Chapter Summary The data set used in the analysis was described and the exact definition of the variables utilized was also provided. This chapter also presented the estimation results from the quasi hyperbolic farmland value asset equation. Three key remarks regarding the nature of the results are warranted. First, a clear case has been made not only against the standard exponential discounting model, but for the presence of hyperbolic discounting as a viable alternative. Second, an important consideration is that the discount parameters were obtained without constraining 3 e [0,1] or 1 e [0,1], a fact which should lend additional credibility to the estimates. Based on the values of the exponential and quasihyperbolic discount parameters, strong evidence has been found that the shortrun discount rate is both different and substantially larger than the longrun discount rate. Lastly, estimates of the discount parameters suggest that P < 8 is true for farmland values. This result is concordant with the existing body of studies that estimate or calibrate a quasi hyperbolic discount parameter (Laibson et al. 1998; Angeletos et al. 2001; Eisenhauer and Venture 2006; Ahumada and Garegnani 2007). Smaller values of both / and 3 imply a greater tendency of the farmerlandowner to consume now rather than in future periods and therefore tend to save and invest less. Lower values of / will result in a larger jump between the first two time periods. This jump in the value of the discount factor is what creates dynamic time inconsistent preferences. As P > 1, the discount factor converges to the standard exponential case. Smaller values of 3 will result in a more bowedshaped discount factor implying a greater preference for immediate consumption. As 3 gets closer to zero, meaning the rate of time preference p increases, the shape of the discount factor becomes more convex to the origin. The values of the shortrun and longrun discount rates for each region are summarized in Table 410, along with the value of the Fstatistic for statistical difference from the explicit test of hyperbolic discounting. Most of the long run discount rates are below 10%, which is reasonable when compared to prevailing interest rates in the land market, although on the higher end. Data obtained from the USDA/ERS on the average nominal interest rate in the U.S. on farm business debt yields a national average of about 3.92% for 19602002 time period. As reflected in Table 410, the shortrun discount rates are extremely high and cover a wide range in values from 57% for the Mountain states to 288% for the Appalachian states. Four regions have short run discount rates under 100%, which include the Corn Belt, Great Plain, Lake, and Mountain states. Three regions have shortrun discount rates above 100%, which include the Delta, Pacific, and Southeast states. The Appalachian region has a shortrun discount rate of over 200% and the Northeast region has a negative discount rate below 200%. Note that the difference in the shortrun and longrun discount rate is not statistically significant for the Corn Belt region and is only statistically significant at the 10% level for the Mountain region. In the next and final chapter, a more indepth commentary on the meaning of the results is presented along with concluding statements. 250 200 150 Co100 50  50 50 100 150  Kentucky  North Carolina A Tennessee Virginia  West Virginia Figure 41. Change in farmland values for the Appalachian states 400 300 200 S100 o 0 ~100 0 200 300 400 500  Illinois  Indiana A Iowa Missouri  Ohio Figure 42. Change in farmland values for the Corn Belt states 500 400 300 200 100 S100 0 200 300 400 500 4c (02l By C 8 9~ b ` Arkansas Louisiana Figure 43. Change in farmland values for the Delta states 150 100 2 50 50 1 50 50 0 100 150 200 Mississippi t\` r\%~e~lj rtBC ow"`,u q ~ \cK SKansas South Dakota  Nebraska  Oklahoma North Dakota  Texas Figure 44. Change in farmland values for the Great Plain states 139 250 200  150  100 50 S50 S100 150 200 250  Michigan *Minnesota Wisconsin Figure 45. Change in farmland values for the Lake states 500  400  S300  200  10 100 100  200 F Arizona v Colorado Idaho Montana x Nevada *New Mexico  Utah Wyoming Figure 46. Change in farmland values for the Mountain states 140 1200 1000 800 600 400 200 200 400 1K f?f ____________________ ^ f^^^ s^^^  Connecticut  Massachusetts Pennslyvania  Delaware Maine  New Hampshire New Jersey  Rhode Island Vermont Maryland New York Figure 47. Change in farmland values for the Northeast states 300 250 200 150 100 50 50 100 150 200 V VV * California  Oregon Washington Figure 48. Change in farmland values for the Pacific states S~ 250 200 S150  S100  50  0 50  100 Figlabamal Florida Georgia South Carolina Figure 49. Change in farmland values for the Southeast states Table 41. Appalachian states result Variable Parameter estimate Standard error ao 0.168 10.807 a, 0.056 0.080 a2 0.061 0.011 a3 (KY) 0.508 10.791 a4 (TN) 1.923 12.310 a, (VA) 0.140 12.259 a6 (WV) 0.086 12.798 3 0.939 0.011 P 0.060 0.086 N=210 Corn Belt states result Parameter estimate 19.499 0.543 0.071 13.529 2.722 1.576 3.057 0.929 0.584 Standard error 17.525 0.316 0.020 23.407 23.020 24.489 23.066 0.020 0.349 Table 43. Delta states result Variable Parameter estimate Standard error tstatistic pvalue ao 3.306 15.068 0.219 0.413 a, 0.150 0.240 0.624 0.266 a2 0.037 0.028 1.333 0.091 a3 (AR) 10.922 14.785 0.739 0.230 a4 (MS) 10.467 13.815 0.758 0.224 3 0.963 0.028 P 0.155 0.252 N= 126 tstatistic 0.016 0.701 5.771 0.047 0.156 0.011 0.007 pvalue 0.494 0.242 0.000 0.481 0.438 0.495 0.497 Table 42. Variable ao a, a2 a3 (IL) a4 (IN) a, (IA) a6 (OH) N=210 N= 210 tstatistic 1.113 1.717 3.615 0.578 0.118 0.064 0.133 pvalue 0.133 0.043 0.000 0.282 0.453 0.474 0.447 Table 44. Variable ao a, a2 a3 (KS) a4 (NE) a, (ND) a6 (OK) a7 (TX) P N= 252 Great Plain states results Parameter estimate Standard error 0.242 5.472 0.472 0.195 0.071 0.019 4.162 6.662 0.572 6.798 0.073 6.452 0.354 7.000 0.460 7.177 0.929 0.019 0.508 0.218 tstatistic 0.044 2.419 3.714 0.625 0.084 0.011 0.051 0.064 pvalue 0.482 0.008 0.000 0.266 0.466 0.495 0.480 0.474 Table 45. Lake states results Variable Parameter estimate Standard error tstatistic pvalue ao 0.163 14.588 0.011 0.496 a, 0.486 0.191 2.551 0.005 a2 0.110 0.019 5.940 0.000 a3 (MN) 11.146 12.858 0.867 0.193 a4 (WI) 10.203 14.100 0.723 0.235 3 0.890 0.019 P 0.546 0.222 N= 126 Table 46. Mountain states result Variable Parameter estimate a, 1.865 a, 0.571 a2 0.103 a3 (CO) 2.711 a4 (ID) 4.871 a5 (MT) 2.140 a6 (NV) 0.154 a7 (NM) 0.987 a8 (UT) 0.825 a9 (WY) 1.280 3 0.897 P 0.637 N=336 Table 47. Variable a0 a, a2 a3 (CT) a4 (DE) a, (ME) a6 (MD) a7 (MA) a, (NH) a9 (NJ) a10 (NY) a11 (PA) a12 (VT) S462 N = 462 Northeast states result Parameter estimate 8.676 0.103 0.034 3.469 5.109 2.288 2.850 2.379 0.355 2.204 0.689 6.380 9.405 0.966 0.107 Standard error 5.707 0.173 0.013 7.035 8.854 7.032 7.031 7.128 6.820 7.124 0.013 0.200 Standard error 20.627 0.065 0.006 23.041 23.685 24.870 24.014 23.496 24.993 23.982 24.768 24.306 24.744 0.006 0.066 tstatistic 0.327 3.296 8.209 0.385 0.550 0.304 0.022 0.138 0.121 0.180 tstatistic 0.421 1.604 5.605 0.151 0.216 0.092 0.119 0.101 0.014 0.092 0.028 0.263 0.380 pvalue 0.372 0.000 0.000 0.350 0.291 0.380 0.491 0.445 0.452 0.429 pvalue 0.337 0.054 0.000 0.440 0.414 0.463 0.453 0.460 0.494 0.463 0.489 0.396 0.352 Table 48. Pacific states result Variable Parameter estimate Standard error tstatistic pvalue ao 18.803 16.802 1.119 0.132 a, 0.337 0.224 1.507 0.066 a2 0.067 0.027 2.517 0.006 a3 (OR) 19.901 16.794 1.185 0.118 a4 (WA) 11.209 15.045 0.745 0.228 3 0.933 0.027 / 0.361 0.249 N= 126 Table 49. Southeast states result Variable Parameter estimate Standard error tstatistic pvalue a, 10.841 11.657 0.930 0.176 a, 0.280 0.142 1.969 0.024 a2 0.078 0.021 3.652 0.000 a3 (AL) 3.429 11.492 0.298 0.383 a4 (GA) 0.356 11.436 0.031 0.488 a, (SC) 5.252 11.394 0.461 0.322 3 0.922 0.021 / 0.304 0.161 N= 168 Table 410. Discount rates by region Region Shortrun discount rate Longrun discount rate Fstatistic Appalachian 287.6% 6.3% 102.152 Corn Belt2 61.2% 7.4% 1.417 Delta 190.2% 3.8% 11.199 Great Plain 75.1% 7.4% 5.064 Lake 72.2% 11.6% 4.187 Mountain3 56.8% 10.9% 3.305 Northeast 227.0% 3.5% 278.590 Pacific 108.8% 6.9% 6.563 Southeast 124.0% 8.4% 9.900 1 The critical value of the Fstatistic with (1, o0 ) degrees of freedom and 5% level of significance is 3.840. 2 Discount rates are not statistically different from one another at conventional levels 3 Discount rates are statistically different from one another at the 10% level of significance. CHAPTER 5 CONCLUSION AND FUTURE WORK This chapter presents a more thorough interpretation of the empirical results with three main goals. The first goal of this chapter is to foster a greater understanding of the results and offers a comparison to other relevant studies on time discounting while also focusing on the major weaknesses and limitations of the study. Second, a discussion on the importance and intuition of the results is also presented with an emphasis on the practical implications of the results and the major relevance to policy and extension. Finally, suggestions for future research are provided. The dissertation concludes with a summary. Comparisons and Limitations The empirical results prompt three key questions regarding the estimates of the shortrun and longrun discount rates presented in Table 410. First, why are there differences in discount rates within regions? Second, why are there differences in discount rates across regions? Lastly, and most importantly, why are the shortrun discount rates so high compared to the longrun discount rates? The first two questions can be addressed rather succinctly while the last one requires more thoughtful consideration. The first question regarding differences in shortrun and longrun discount rates within a specific region relates to the crux of the dissertation regarding timeinconsistency and non constant discounting. Evidence in numerous fields of economics has found that individuals are not timeconsistent in their intertemporal preferences but rather timeinconsistent. This time inconsistency is the result of two related but distinct phenomena, preference reversals and intra personal games, as explained in Chapter 1. Time inconsistency implies a nonconstant and declining discount rate through time, meaning that discount rates in the shortrun will be larger than discount rates in the longrun. Quasihyperbolic discounting is a declining discount function and represents one way of modeling timeinconsistent preferences. The question regarding why differences in the shortrun and longrun discount rates exist is answered by the presence of timeinconsistent preferences. Chapter 1 presented the reasoning and intuition as to why a landowner might be characterized by timeinconsistency and why quasihyperbolic discounting might be present in the land market with a particular emphasis on land conversion models. The second question pertaining to why the shortrun and longrun discount rates differ between regions is best explained by the heterogeneity of the land market, the landowner, and the regional economy. Variables affecting the value of land not accounted for in this analysis will have a direct effect on the discount rate whether such variables are landspecific, landowner specific, or economyspecific. Hedonic characteristics of the land, such as parcel size, location, and soil quality differ greatly both within and between major agricultural regions, and have been found in a number of studies to directly affect farmland values (CarrionFlores and Irwin 2004; Taylor and Brester 2005; Livanis et al. 2006). For example, Livanis et al. (2006) estimate a positive affect of median singlefamily house value on U.S. farmland values. Landownerspecific characteristics also vary substantially between regions. Barnard and Butcher (1989) show how the demographic characteristics of the landowner, such as age, education, income, and other traits affect the perceived present value of undeveloped land. Since demographic variables affect the perceived present value, the discount rate is also going to differ not only between individuals, but between regions as well. Eisenhauer and Ventura (2006, p.1229), using international survey data, find that hyperbolic discounters tend to be "younger, poorer, less educated, bluecollar, unemployed, individuals from larger cities, and those in southern regions working in agriculture." In a study using survey data from a developing country, Robberstad and Cairns (2007) find that individuals who obtain most of their income from farming will have higher discount rates than individuals whose main income is nonfarm related. To the extent these demographic traits exist and differ between regions, differences in the discount rate, particularly the shortrun discount rate, are to be expected. Lastly, the state of the economy will affect intertemporal preferences and the implied discount rate. In an influential study, Lawrance (1991) uses data from the Panel Study of Income Dynamics and finds evidence that time preferences are positively related to household income. In particular, the author concludes that poor households tend to have relatively higher discount rates. In a metaanalytic review of time discounting studies, Percoco and Nijkamp (2007) find that the discount rate is negatively related to the level of per capital GDP implying that as GDP rises on a per capital basis, the discount rate falls. Since both household income and state level GDP differ between the agricultural regions studied in this dissertation, and even between states in each region for that matter, differences between both the shortrun and longrun discount rate are to be expected. This brings us to the third and final question regarding the magnitudes of the shortrun discount rates and why they are so stark compared to the longrun discount rates. The results of hypothesis tests imply a formal rejection that the shortrun discount rate is equal to the longrun discount rate, a new result in the literature on land values. The results in this dissertation are unique and there are no baseline estimates for which to make exact comparisons. Although one would be hardpressed to defend the estimated shortrun discount rates given in Table 410 in actual financial analysis, the presence of large discount rates is not without precedent in the empirical literature on both timeconsistent discounting and timeinconsistent discounting. Important studies on constant discounting in energy consumption and household durables have revealed large exponential discount rates. Hausman (1979), using data on air conditioner purchases, finds personal discount rates ranging from 5% to 89% depending on household income, with higher income households having lower discount rates. Houston (1983), using a survey method on energy appliance demand, finds discount rates in the range of 20% to 25% also depending on income. Gately (1980), using data on refrigerator purchases, finds much higher discount rates ranging from 45% to 300% with most of the discount rates exceeding 100%, depending upon the energy efficiency and brand name of the refrigerator. Using data on military personnel and retirement decisions, Warner and Pleeter (2000) estimate personal discount rates in the range of 0% to 59%, again depending on the individual characteristics. One of the only known studies to estimate the rate of time preference in an agricultural context is Lence (2000). Using consumption and asset return data for U.S. farmers, he estimates the standard discount factor to be 3 = 0.962, implying a longrun discount rate of 3.92%. However, Lence (2000) assumes timeconsistency and only examines time preference with standard exponential discounting. While there are no existing studies that examine land values in this context for which to compare estimates, the values of the quasihyperbolic discount parameter, /, obtained here in the range of 0.06 to 0.64 (including even one negative value for the Northeast) is lower than many estimates presented in the experimental economics literature, typically calculated between 0.80 and 0.90 (Benzion, Rapoport, and Yagil 1989; Coller and Williams 1999; Eisenhauer and Ventura 2006). The findings in this dissertation are similar with the few studies that have estimated quasi hyperbolic time preferences from field data using a structural model. Laibson, Repetto, and Tobacman (2007) use individual level data on credit card borrowing, consumption, income, and retirement savings to estimate the discount factors finding that 3 = 0.958 and / = 0.703 both statistically significant. Their results imply a shortrun discount rate of 39.53% and a longrun discount rate of 4.29%. Fang and Silverman (2004) use panel data from the National Longitudinal Survey of Youth (NLSY) on welfare participation for single mothers and estimate how timeinconsistency affects the decision to takeup welfare. The authors estimate the discount factors to be 3 = 0.875 and / = 0.338, implying a shortrun discount rate of 121.82% and a longrun discount rate of 12.78%. Paserman (2004) uses data on unemployment spells and job search duration from the NLSY to estimate time preferences. In a sample of lowwage workers his estimates of the discount parameters, 3 = 0.996 and / = 0.402, imply shortrun and longrun discount rates of 91.53% and 0.40%, respectively. His findings show the rate of time preference increases as the wage level increases. The highwage sample in Paserman (2004) yields discount parameter estimates of 3 = 0.999 and / = 0.894, implying a shortrun discount rate of 11.31% and a longrun discount rate of 0.10%. Clearly, the literature offers anecdotal explanations as to why the shortrun discount rates stand in such sharp contrast to the longrun discount rates. Given that the primary data set in this dissertation is agricultural land values, the evidence from both Ventura (2006) and Robberstad and Cairns (2007) regarding higher discount rates for agriculture lends additional explanation to the high shortrun discount rates that were obtained. Further, to the extent that landowners and farmers have lower than average incomes, higher shortrun discount rates may be expected, as described in Paserman (2004). The durable goods and energy demand literature also gave precedent for high discount rates. Since land can be considered a durable good itself, if not an infinitely lived asset, then high discount rates also make sense. Despite the evidence of high shortrun discount rates in the literature, the question remains as to why such high shortrun discount rates have been found in this dissertation on farmland values. A simple mathematical explanation is offered first. The rent generated from land will affect the discount factor, and the regions differ greatly in terms of the income generated from agricultural use. Careful reexamination of the parameter estimates in Table 41 through Table 4 9 with the estimated discount rates in Table 410 reveals that as the estimated coefficient on farmland returns increases in magnitude the value of the shortrun discount rate decreases in size. This of course is not coincidence since the relationship is an artifact of the theoretical model. In Equation (311), the quasihyperbolic discount parameter increases as the estimated coefficient on farmland returns increases. A higher quasihyperbolic discount parameter reflects greater individual patience, implying a lower shortrun discount rate. When land returns are high, the shortrun discount rate is lower, implying a greater level of shortrun patience. When land returns are low, the shortrun discount rate is high, implying a lower level of shortrun patience, or a stronger desire for instantaneous gratification. Generally, land returns have been on an upward trend since 1960, however for many of the regions farmland returns have decreased substantially over the past 10 to 20 years, depending on the region. To the extent returns have fluctuated and fallen, the value of the quasihyperbolic parameter and the implied shortrun interest rate will be biased upward. Future work should examine more closely the relationship between expected returns from land and the rate of timepreference. However, more rigorous explanations for the relatively high shortrun discount rates stem from the formulation of the net present value model in Chapter 3. The simplistic assumptions made in the theoretical and empirical model regarding risk neutrality, rational expectations, and no inflation are possible explanations for the high shortrun discount rates reported in Table 4 10. First, the presence of inflation will result in an upward bias in the estimated discount rate. Although inflation is not accounted for in the empirical analysis, there is reason to suspect that inflation and expectations of inflation will affect not only the present value of land, but both the shortrun and longrun discount rates. Consider the argument outlined in Frederick, Loewenstein, and O'Donoghue (2002) stating that the presence of inflation creates an upward bias in discount rate estimates. The authors reason, quite simply, that a sum of money today is not worth the same sum if inflation is expected, in fact that sum of money will be worth less. In other words, should you expect inflation to occur in the next five years, spending $500 today will generate more utility from consumption than spending $500 in the future. Since inflation gives an incentive to value future rewards less than present rewards, the discount rate will be higher. As Frederick, Loewenstein, and O'Donoghue (2002) correctly state, the magnitude of this upward bias in the discount rate will depend on the individual expectations regarding the extent of inflation. In the farmland markets, inflation has been nonconstant across the time horizon and as high as 13% in some cases (Moss 2003). Lloyd (1994) argues that land is purchased as a hedge against inflation based on investor perceptions of land as a resilient asset, capable of holding value in real terms amidst inflationary periods. In fact, there is evidence to suggest that inflation may be the most important factor influencing farmland values. Moss (1997) examines the sensitivity of farmland values to changes in inflation, asset returns, and the cost of capital by employing Theil's (1987) statistical formulation of information. The basic premise of Moss (1997) states that changes in an independent variable that account for more volatility in farmland prices will imply larger fluctuations in farmland prices than an independent variable that accounts for less volatility, where the total amount of information is described in bits. According to the empirical results in Moss (1997), inflation contributes the most to the explanation of changes in U.S. farmland values. Even more revealing is how inflation contributes to the explanation of farmland prices by region. The Appalachian region had inflation as the largest contribution of information to farmland values while the Northeast region had the lowest contribution of information to farmland values (Moss 1997). The results in Table 410 would seem to underscore the importance of inflation in discount rate estimates. The Appalachian region had the highest short run discount rate, while the Northeast region had a negative discount rate, providing at least anecdotal evidence that inflation has resulted in an upward bias in the estimated discount rates in this dissertation. Further, the information contribution of inflation is far more uniform in the five states that compose the Appalachian region than the eleven states that compose the Northeast region. As mentioned earlier, the data used in the analysis to obtain the discount rates in Table 4 10 are based on nominal rather than real data and so inflationary affects are present in the data. In fact, this represents an advantage to using nominal data over real data since real data loses the information present in nominal data from inflation. Further, the use of real data involves a debate regarding which deflator to use, the PPI, the CPI, or the PCE index. Unreported results reflected a sensitivity of the discount rates to the choice of deflator used and so nominal data were preferred to detract from a debate on the best method of deflating the data. At the present time, there is no well defined land price deflator and none of the current price deflators seem appropriate for the land market. Regardless, future work should attempt to extract the effects of inflation from the farmland values data and determine the impact on shortrun and longrun discount rates. Second, the presence of risk and uncertainty will also result in an upward bias in the estimated discount rate. Recall that the empirical results obtained in this dissertation and the discount rates presented in Table 410 are generated under the assumption of risk neutrality. Frederick, Loewenstein, and O'Donoghue (2002) point out that since a future reward received with some degree of delay is often associated with at least some degree of uncertainty, the exact effect of the rate of time preference on the size of the discount rate is a complicated relationship that is difficult to measure. However, there is evidence in both experimental data and field data to suggest that accounting for risk aversion reduces the size of estimated shortrun and longrun discount rates. Using experimental data from the 2000 Bank of Italy Survey of Household Income and Wealth, Eisenhauer and Ventura (2006) find that controlling for risk aversion reduces estimates of the hyperbolic discount rate by several orders of magnitude. Anderhub et al. (2001) utilize an experiment on college undergraduates to elicit discount rates and examine how risk and time preferences are interrelated. The authors find clear evidence that higher degrees of risk aversion are associated with lower values of the discount factor, which implies a higher discount rate. Hence, the authors find a positive relationship between the value of the risk aversion coefficient and the rate of time preference. Hence, higher risk aversion means heavier discounting, meaning risk averse individuals tend to be more impatient. Anderhub et al. (2001) argue that the heavier discounting seen in more risk averse individuals is the result of the uncertainty that surrounds future payoffs as opposed to immediate payoffs. Andersen et al. (forthcoming), using experimental data from Denmark, jointly estimate the coefficient of relative risk aversion (CRRA) and the discount rates assuming both a time consistent and timeinconsistent discount structure. In both cases, joint estimation of risk and time preference results in significantly lower discount rates congruent with market interest rates. Assuming risk neutrality and timeconsistency, the authors obtain a discount rate of 25.2%, but when risk aversion is accounted for, the estimated discount rate falls to 10.1%. When time inconsistent preferences are modeled using a hyperbolic discount factor evidence of declining discount rates are still obtained, but the magnitudes between the shortrun and longrun discount rates is substantially softened. Thus, in contrast to Anderhub et al. (2001), who find a positive relationship, Andersen et al. (forthcoming) find a negative relationship between the degree of risk aversion and the rate of time preference. The authors attribute this finding to the fact that Anderhub et al. (2001) uses data on risk attitudes to impute the discount rate over money rather than over utility as in Andersen et al. (forthcoming). Estimated discount rates will be lower when defined over utility than when defined over money (Andersen et al. forthcoming). The high shortrun discount rates in this dissertation may be a result of the data, which are based on dollar values of farmland rather than utility. Using field data, Laibson, Repetto, and Tobacman (2007) estimate several versions of the lifecycle consumption model in which the CRRA is either jointly estimated or imposed on the model. The estimates of / and 3 vary substantially in regards to both magnitude and statistical significance depending upon the risk assumption imposed. When the CRRA is jointly estimated the shortrun discount rate falls from 39.53% to 14.63% and the longrun discount rate falls from 4.29% to 3.87%, with the estimated CRRA being about 0.22, which is quite low. The authors also impose values of the relative risk aversion coefficient on the estimates, ranging from a CRRA of three to a CRRA of one. As the imposed value of the CRRA falls, the estimated value of both P and 3 rise, which implies both lower shortrun and longrun discount rates. The results in Laibson, Repetto, and Tobacman (2007) would seem to imply that the greater the level of risk aversion, the lower the rate of time preference, meaning that risk averse individuals are more impatient. This would seem to make sense if future rewards are received with a greater amount of uncertainty than immediate rewards and is in accord with the results in Anderhub et al. (2001). Farmland markets and farmland values are especially sensitive to risk, and farmland is in general considered a risky investment (Hanson 1995). As explained in Moss, Shonkwiler, and Schmitz (2003), changes in risk affect the value of farmland over time, as evidenced by estimates of the certaintyequivalence parameter in a data set on U.S. farmland returns, interest rates, and values. Both Lence (2000) and Chavas and Thomas (1999) reject the hypothesis of risk neutrality and find evidence of risk aversion in the U.S. farmland market. Antle (1987) finds evidence to suggest that agricultural producers are risk averse with a risk premium as high as 25% of expected returns. The large shortrun discount rates presented in Table 410 could in fact be the result of not accounting for risk in the analysis. Future work should attempt to incorporate risk in models of farmland values and determine the extent that shortrun and longrun discount rates are affected. Third, and finally, the high shortrun discount rates in Table 410 may be in part due to the possibly unrealistic assumption of rational expectations. Under rational expectations, landowners are assumed to correctly forecast, on average, future land rents using all relevant economic information without any systematic bias. However, some authors have attributed the shortrun failure of NPV models of farmland to be a violation of rational expectations (Lloyd, Rayner, and Orme 1991; Tegene and Kuchler 1991; Engsted 1998). Furthermore, the shortrun failure of the NPV model has also been noted in the literature on housing prices with strong rejection of the rational expectations hypothesis (Meese and Wallace 1994; Clayton 1996). A possible alternative to rational expectations is an adaptive expectations formulation which assumes that landowners would base future expectations on land rents based on past land rents with the possibility of systematic bias due to stochastic economic shocks. Chow (1989) compared both rational and adaptive expectations in a NPV model of stock prices and dividends. Based on the parameters estimates, the NPV under rational expectations are not consistent with the theory. Obtained values of the longrun discount rates, for example, are above 100% when rational expectations is assumed. The inconsistent discount rates persist under rational expectations even when homoskedasticity is corrected for and a timevarying discount factor included in the model. When adaptive expectations is assumed, the obtained longrun discount rates become much more reasonable and congruent with market interest rates. Chow (1989) concludes that when a correct model incorrectly assumes rational expectations, then unreasonable parameter estimates are often the case There is empirical evidence of atypical longrun discount rates under assumptions of rational expectations in the farmland values literature. Lloyd, Rayner, and Orme (1991) use a structural model similar to the one in this dissertation to obtain values of the longrun discount rate from a reducedform parameterization of the structural model. The authors examine the NPV model of real farmland values in England and Wales using both rational and adaptive expectations. The rational expectations model uncovers a real longrun discount rate of 37.74% with a rejection of the null hypothesis of a positive real rate of discount. In the authors' adaptive expectations model, the real long run rate of discount is 2.38% and is much more in line with market interest rates. Tegene and Kuchler (1991) compare the NPV model of U.S. farmland values under both rational and adaptive expectations for the Corn Belt, Lake States, and Northern Plain regions. The estimated coefficients of land prices and rents in the rational expectations model imply discount rates above 100% and are inconsistent with the rational expectation hypothesis. In the authors' adaptive expectation model, longrun discount rates are found to be less than 7% and estimated parameters are consistent with the adaptive expectations hypothesis. The authors conclude with a strong rejection of rational expectations in favor of adaptive expectations. Promising future work remains in uncovering the relationship between expectation formulations and time preferences. Importance and Implications Although the magnitudes of the shortrun discount rates presented in Table 410 are surprising, the results are not being recommended for use in financial analysis or in forecasting farmland values. As discussed above, the incorporation of inflation, risk, and adaptive expectations is suggested before such a recommendation is even contemplated. The results then are not so much important for the specific magnitudes of the shortrun and longrun discount rates presented, but are important for what the differences in shortrun and longrun discount rates imply for the land market. In this section, added intuition regarding the nature and importance of the results is offered. The value and practical use of the results are based on what they imply for landuse decisionmaking, what they offer for explaining the methodological failure of the NPV of farmland, and what they suggest in regards to landowner behavior. The results also have important policy and extension implications. First, the results are important for their implication and description of landuse decisions and investments. The fundamental interpretation of the large shortrun discount rates presented in Table 410 reflect the fact that landowners have a tendency to be shortrun in their thinking. Such a nearterm or shortrun dominate way of thinking would tend to result in decisions that sacrifice the future for the sake of the present. For example, suppose a landowner is facing a decision of converting farmland to a more capital intensive use such as residential housing or an ethanol refinery plant with a large onetime monetary gain. The shortrun and longrun discount rates obtained would imply that the landowner may be inclined to hastily accept such an offer without full consideration of more remunerative future returns from farmland. In essence, a landowner with a higher shortrun discount rate may prefer to sell his land now to an investor for a substantial instant return, rather than be forwardthinking in his behavior and wait 20 years for a more remunerative return. In a sense, the large shortrun discount rates suggest a sort of shortrun bandwagon that landowners have a tendency to ride. Recent investments in residential real estate and ethanol refineries have either failed to generate substantial returns or have resulted in financial losses. In some regard, a high shortrun discount rate would provide an explanation into the occasional tendency of landowners to make shortsighted decisions regarding land use. For example, the longrun future rewards of grainbased ethanol production is in a precarious state, however, the shortrun desire for instant returns may drive the decision of landowners to either plant more corn or to construct an ethanol refinery on their land. The notion of quasihyperbolic discounting would seem quite appropriate given the frenzy on biofuels and the ensuing land rush that has accompanied the phenomenon. A 2006 editorial in Nature Biotechnology described the surge of investment in biofuels, which figures in the billions of dollars in the United States, as "irrational exuberance." However, as of yet, no company has produced and sold ethanol in U.S. on any sort of mass level, raising doubts regarding the success of the current business model (Waltz 2008). Furthermore, the importance of landuse decisions in the bioenergy industry is critical, underscoring the importance of the results in this dissertation. Land is the largest and most valued productive input in ethanol production, and so time preferences regarding the value of land remain an important area of investigation. Given that land used in production of energy cannot be used in the production of food or other uses, the decisions made for every parcel over time are of great importance not only to economists, but to landowners, policy makers, and consumers. Hence, one of the largest contributions of the results in this paper is a numerical description of the impulsiveness that may characterize many landuse decisions. Second, the results provide an explanation for the significant shortrun deviations that occur in the discounted value formulation of farmland values (Falk and Lee 1998; Featherstone and Moss 2003). One explanation for the apparent disconnect between shortrun and longrun values is the presence of boom/bust cycles in which farmland values change more dramatically than would be expected in response to an increase or decrease in returns (Schmitz 1995). Of course, the prevailing question remains: what causes boom/bust cycles in farmland values? In addition to transaction costs and timevarying risk premia, alternative explanations include the presence of quasirationality or rational bubbles (Featherstone and Baker 1987), market overreaction (Burt 1986), fads (Falk and Lee 1998), and option values and hysteresis (Titman 1985; Turvey 2002). A possible explanation offered here is the presence of quasihyperbolic discounting. The mix of longrun rationality with shortrun irrationality makes quasihyperbolic discounting an appealing explanation. The quasihyperbolic discount factor explicitly models disconnects between the shortrun and longrun. Unlike other farm assets, farmland is unique in that approximately 80% of the asset portfolio in U.S. agriculture is accounted for by farmland values (Moss and Schmitz 2003). Unlike assets that have a definite lifespan, such as capital equipment that depreciate substantially like tractors, farmland is typically characterized by an infinite horizon. Since the life span of land as an asset is infinite, the choice of the discount factor plays a premier role, and may be subject to the shortrun jumps described by quasihyperbolic discounting. With quasihyperbolic discounting, the shortrun bubble nature of farmland values can persist, but equilibrium in the long run can be achieved. Thus, during the life of the asset, especially during boom/bust cycles, the discount rate for farmland may change over time. During periods of economic boom, the rate of time preference decreases as available capital increases, implying more patience. During periods of economic bust, the rate of time preference decreases as borrowing increases to cover financial losses, implying a greater level of impatience. The results are also important for their description of landowner behavior and, in particular, what they imply for farmland investment and savings. The results also further the Golden Eggs investment hypothesis presented by Laibson (1997). In an agricultural economics framework, the story of the goose that laid the golden egg can be thought of as a farmer who owns a region of land. The land represents the farmer's socalled goose, whose lifespan depends on whether he continues to keep the land in agriculture or convert the land to an urban use. The farmer has two options: one, he can sell the goose to a developer for some immediate sum and thus increase his stock of liquid assets; or two, he can keep the goose in the hopes of a more remunerative return and keep the illiquid asset. The timeinconsistent model presented in Laibson (1997) would suggest that hyperbolic discounters sell the goose and increase the stock of liquid assets to fund immediate consumption in the shortrun. This is where the notion of commitment devices come into play. The farmer could alternatively decide to keep the goose, his farmland, in the current period and constrain his future self from developing the land in the next period. This may lend some support to the notion that the Conservation Reserve Program (CRP) may serve as a commitment device to a farmer or landowner who potentially discounts hyperbolically. The CRP pays billions of dollars annually to landowners and farmers to keep land out of cropuse under 10 to 15 year contracts in order to sustain the land for future use. Farmland may even serve as a commitment device itself to some landowners. Hyperbolic discounters tend to hold relatively little wealth in liquid assets and hold much more of there wealth portfolio in illiquid assets (Laibson 1997). According to the USDA Farm Balance Sheet published by the Economic Research Service, over $3 trillion of the portfolio (about 80% of the total) is attributed to land. Farmland also represents one of the most illiquid of assets in the farm investment portfolio. This may imply land is a golden egg to the farmerlandowner. In the literature on timeinconsistent preferences and hyperbolic discounting, illiquid assets serve two fundamental roles. The first role is as a commitment device, preventing one's future self from capriciously spending his wealth. The second role is as a golden egg, generating substantial benefits but only after a long period of time has elapsed. Although a less transparent instrument for commitment than other devices, land as an illiquid asset that generates a stream of income can serve as a mechanism to constrain future choices. Further, with an asset such as land, whose sale typically must be initiated a period or more before the actual revenues from the sale are obtained, farmland promises substantial benefits in the longrun but immediate benefits are hard to realize in the shortrun. The difficult to sell nature of farmland can be attributed to a variety of factors including transaction costs, information asymmetries, and incomplete markets. Finally, the results have important extension and policy relevance. The results underline the shortrun nature of thinking of many farmers, growers, ranchers, and developers. The high calculated shortrun discount rates compared to the low longrun discount rates imply the relative importance of the two time periods to a decisionmaker, with the priority being the shortrun. If the shortrun discount rates suggest a tendency of individuals to sacrifice future returns for immediate gains, then extension efforts should attempt to curve or address this behavior. In essence, extension work must recognize that there is a shortrun bandwagon that individuals have a tendency to focus on. Many agricultural investments that offer high shortrun returns such as biofuel programs, commodity speculation, and urbanization offer dubious longrun security. Extension efforts should address this behavioral tendency and emphasize the longrun opportunities in postponing or delaying land development. The greatest potential for policy may come from the design of precommitment devices for landowners. Christmas clubs were once a popular way of constraining individuals from spending money so that a stock of liquid cash would be available for the purchase of holiday gifts. Landowners may desire a similar style of commitment device so that their future self is constrained from making decisions against their present self, such as hasty land conversion or commodity speculation. As mentioned, the Conservation Reserve Program may serve as a commitment device for some landowners. However, the efficiency of the CRP is in doubt if landowners are timeinconsistent, resulting in a future decision to drop out of the program for higher monetary gain than the CRP offers. Gulati and Vercammen (2006) model resource conservation contracts under time inconsistent preferences. The authors note that commitment and timeinconsistency issues are important in the implementation of the CRP and other similar contracts. Since the CRP pre determines payment schedules over the length of the contract according to a discounted present value model, problems of landowners dropping out of the contract are common. To address this problem, the authors suggest instituting penalties for dropping out of the CRP or for increasing the payment schedule over time in order to combat the hyperbolic nature of landowner time preferences. Since the results in this dissertation imply that landowners are described by quasi hyperbolic time preferences, the results in this dissertation would seem to confirm the suggestions made by Gulati and Vercammen (2006). Future Work Given the evidence presented in this dissertation, critical attention should be devoted to the investigation of intertemporal preference in the land values literature specifically, and in the agricultural economics literature more generally. In regards to this dissertation, several potential areas for future research are offered. First, NPV models of land values under adaptive expectations should be explored and test for the presence of hyperbolic discounting. Another interesting extension would involve the incorporation of risk into the NPV model and simultaneously estimate both risk and time preference parameters. Such an extension may involve other models than the NPV model used here, such as Euler equations for investment and savings decisions. Also, including a better control of inflation in the NPV model offers an interesting and important area of exploration. In regards to the general land literature, investigation on how the shape of the discount factor affects the comovement of land price and land rents, as well as the affect on development and rural land conversion times and intensity are important and interesting questions. How does the timing of rural land conversion differ under hyperbolic time preferences? What does time inconsistency imply for land taxes, conversion costs, and capital intensity of development efforts? Also, survey methods directed at landowners offer an attractive method of obtaining discount rates and testing for hyperbolic discounting. There is also a range of interesting questions yet to be examined in the context of potential timeinconsistency and hyperbolic discounting in other research areas as well. Food demand, for example, offers a good opportunity for testing hyperbolic preferences and could also lend itself to experimental methods, which have become recently popular. Policy research on time inconsistency and resource conservation contracts has just begun. Data on CRP participation offers a viable avenue of potential research. The model and the results in this dissertation could be generalized into a model of land conversion and seek implications on development timing under timeinconsistent preferences. Indeed, there is no shortage of important and engaging questions to examine under the umbrella of intertemporal preferences. The investigation of nonconstant discounting and time inconsistency is waiting to be uncovered in the agricultural economics profession. It is hoped that this dissertation serves to fuel an enthusiastic interest in the topic. Summary This dissertation estimates the parameters of time preference in the land values asset equation, generalizing the standard discount factor to include quasihyperbolic discounting. While the presentvalue model has been used almost exclusively to estimate land values, empirical inquiry has revealed two serious flaws. First, presentvalue models are not able to explain the presence of rational bubbles and the apparent boom/bust nature of land prices. Second, there are apparent disconnects in the movement between land rents and land prices in the shortrun versus the longrun. The poor performance of the PV model to estimate land values in other papers may of course be due to a variety of other potential issues, hyperbolic discounting being just one possibility. Failure of the standard discounting model may be attributed to transaction costs (Chavas and Tomas 1999) or timevarying risk premia in farmland returns (Hanson and Myers 1995). Another possible explanation for the poor performance of the present value model and its failure to stand up to empirical scrutiny may involves a priori assumptions on the shape of the discount factor. In particular, this dissertation modifies the standard model to incorporate quasihyperbolic discounting; a generalized form of exponential discounting that has received considerable but recent attention in the financial and experimental economics literature. Statistically significant evidence of quasihyperbolic discounting over standard exponential discounting is uncovered in a dataset of U.S. farmland values and returns for the major agricultural regions of the United States. The results not only suggest a significant quasihyperbolic parameter, but significantly different shortrun and longrun discount rates in eight out of the nine panel sets examined. The results show that individuals discount land far more heavily in the shortrun than the longrun, with obtained shortrun discount rates several orders of magnitudes larger than the obtained longrun discount rates. The shortrun discount rates obtained are not important for use in actual financial analysis, but for what they imply about landuse decisions, theoretical models of land values, and for landowner behavior. First, the high shortrun discount rates suggest that landowner decisions tend to be dominated by shortrun thinking. This may result in possible land investment projects that offer a high instant return, but offer little longrun gain. The shortrun discount rates imply land decisions that tend to sacrifice the future for the sake of the present. Second, the presence of quasihyperbolic discounting offers an explanation into the apparent shortrun and longrun disconnects in NPV models of farmland. Farmland values and returns tend to be well co integrated in the longrun but not in the shortrun. If individuals have a higher shortrun discount rate than the longrun, then this might explain why the shortrun dynamic relationship tends to break down. Third, the results offer evidence that farmland may acts as the socalled goose that laid the golden egg, with landowners caught in an internal tussle as whether to keep land in farming or to sell a developer for a more capital intensive use. In this case, commitment devices may be a helpful policy tool for the timeinconsistent landowner. Some limitations to the results apply since the high shortrun discount rates may be affected by the simplistic assumptions maintained in the theoretical model. First, individuals were assumed to be riskneutral. Evidence in the literature suggests that not accounting for risk aversion results in shortrun discount rates that are biased upward. Second, rational expectations were assumed, which has been shown to result in unreasonable discount parameter estimates in models of stock prices and farmland values. Third, inflation was not accounted for in this analysis which may also result in an upward bias in both the shortrun and longrun discount rates. Although the assumptions made in this dissertation are limiting, they also offer a starting point for the analysis which is congruent with the assumptions made in the farmland values literature. While more complicated assumptions may result in lower discount rates, it is believed that significant differences between the shortrun and longrun discount rate would still be uncovered. Future work should attempt to obtain the time preference parameters under more relaxed assumptions of risk, expectations, and inflation. There is also tremendous potential for future work into time preferences and time inconsistency and other economic models such as land conversion and food demand. It is hoped that this dissertation has not only served to bridge the gap between the literature on intertemporal preferences and agricultural economics, but that it has sparked an interest into further investigation on time inconsistency in the profession. APPENDIX PROCEDURE FOR RCODE # TITLE: Appalachian State Panel OLS & Linear GMM # RESET ALL WORK rm(list = ls)) # Call needed libraries library(lmtest); library(stats); library(tseries) library(car); library(sandwich); library(systemfit) # Download CPI Deflator dta0 < read.table("Deflators.dta") cpi < as.numeric(dta0[2:44,1]) lagcpi < as.numeric(dta0[3:44,1]) # Download data sets # Column Order: YKYNCTNVAWVVR~IDV dta < read.table("AppalachiaNom.dta") # Create lagged variables obs < nrow(dta) lagVky < as.matrix(dta[l:(431),7]) lagRky < as.matrix(dta[ :(431),8]) lagVnc < as.matrix(dta[44:(861),7]) lagRnc < as.matrix(dta[44:(861),8]) lagVtn < as.matrix(dta[87:(1291),7]) lagRtn < as.matrix(dta[87:(1291),8]) lagVva < as.matrix(dta[130:(1721),7]) lagRva < as.matrix(dta[130:(1721),8]) lagVwv < as.matrix(dta[173:(2151),7]) lagRwv < as.matrix(dta[173:(2151),8]) lagV < rbind(lagVky,lagVnc,lagVtn,lagVva,lagVwv) lagR < rbind(lagRky,lagRnc,lagRtn,lagRva,lagRwv) # Define dep var, indep var, & scale dvky < as.matrix(dta[2:43,10]) dvnc < as.matrix(dta[45:86,10]) dvtn < as.matrix(dta[88:129,10]) dvva < as.matrix(dta[131:172,10]) dvwv < as.matrix(dta[174:215,10]) dv < rbind(dvky,dvnc,dvtn,dvva,dvwv) n < nrow(dv) constant < as.matrix(seq(length=n,from=l,by=0)) vky < as.matrix(dta[2:43,7]) vnc < as.matrix(dta[45:86,7]) vtn < as.matrix(dta[88:129,7]) vva < as.matrix(dta[131:172,7]) vwv < as.matrix(dta[174:215,7]) v < rbind(vky,vnc,vtn,vva,vwv) rky < as.matrix(dta[2:43,8]) rnc < as.matrix(dta[45:86,8]) rtn < as.matrix(dta[88:129,8]) rva < as.matrix(dta[131:172,8]) rwv < as.matrix(dta[174:215,8]) r < rbind(rky,rc,rtn,rva,rwv) one < as.matrix(seq(length=42,from= ,by=0)) zero < as.matrix(seq(length=42,from=0,by=0)) ky < rbind(one,zero,zero,zero,zero) nc < rbind(zero,one,zero,zero,zero) tn < rbind(zero,zero,one,zero,zero) va < rbind(zero,zero,zero,one,zero) wv < rbind(zero,zero,zero,zero,one) x < cbind(constant,r,v,ky,tn,va,wv) y < dv # OLS estimation and test results ols < lm(dv~r+v+ky+tn+va+wv) output < summary(ols); output NWvcov < as.matrix(NeweyWest(ols)) bptest(ols) # Null Hypothesis of BP test is homoscedasticity # Critical value is Chisquare with alp=.05, dof=2: 5.991 durbin.watson(ols,max.lag=2) # Null hypothsis is no autocorrelation (rho=0) # If d
# Critcal value is DW with k=2, n=41: dl=1.449, du=1.549 Box.test(dv, type = c("LjungBox")) # Null hypothesis is no autocorrelation (rho=0) # Critical value is Chisquare with alp=.05, dof=l: 3.841 adf.test(dv) # Null hypothesis is nonstationary (unit root) # Calculate Discount Parameters coeff < as.matrix(ols$coefficients) al < coeff[2,1]; a2 < coeff[3,1] alse < sqrt(NWvcov[2,2]); a2se < sqrt(NWvcov[3,3]); ala2cov < NWvcov[2,3] delta < 1a2; beta < al/(1a2) # Conduct linear hypothsis test on exponential discounting q < a2al1 qse < sqrt(alseA2+a2seA2+2*ala2cov) exptest < (q/qse)A2 deltase < a2se print("Value of Exponential Factor"); print(cbind(delta,deltase)) print("Exponential Discounting Ftest"); print(exptest) # This is a twosided Ftest with dof=1,203 and alpha=0.05 # Null hypothesis is exponential discounting # Critical value is 3.84 # Conduct nonlinear hypothesis test on hyperbolic discounting gal < l/(la2); ga2 < al/((la2)A2) betase < sqrt((galA2)*(alseA2)+(ga2A2)*(a2seA2)+2*(gal)*(ga2)*(ala2cov)) hyptest < ((beta1)/betase)A2 print("Value of QuasiHyperbolic Factor"); print(cbind(beta,betase)) print(" QuasiHyperbolic Discounting Ftest"); print(hyptest) # This is onesided Ftest with dof=1,203 and alpha=0.05 # Null hypothesis is no hyperbolic discounting # Critical value is 3.84 # Define IV matrix and scale z < as.matrix(cbind(constant,v,r,ky,tn,va,wv,rA2,1agV,lagR)) k < ncol(z) # Define initial weighting matrix w < nrow(z)*solve(t(z)%*%z) # Define GMM function fr < function(b){ (l/n)*t(yx%*%b)%*%z%*%w%*%((l/n)*t(z)%*%(yx%*%b))} # Define gradient gfr < function(b){ 2*t(yx%*%b)%*%z%*%w%*%t(z)%*%x } # Conduct first step GMM gmm < optim(c(0,0,0,0,0,0,0),fr,gfr) sse < as.numeric(t(yx%*%gmm$par)%*%(yx%*%gmm$par)/nrow(y)) shat < (sse/nrow(z))*(t(z)%*%z) mhat < nrow(z)*solve((t(x)%*%z)%*%w%*%(t(z)%*%x))%*%(t(x)%*%z)%*%w vhat < mhat%*%shat%*%t(mhat) print(cbind(gmm$par,sqrt(diag(vhat)/nrow(x)),gmm$par/sqrt(diag(vhat)/nrow(x)), 1pnorm(abs(gmm$par/sqrt(diag(vhat)/nrow(x)))))) # Conduct second step GMM w < solve(shat) gmm < optim(c(0,0,0,0,0,0,0),fr,gfr) sse < as.numeric(t(yx%*%gmm$par)%*%(yx%*%gmm$par)/nrow(y)) shat < (sse/nrow(z))*(t(z)%*%z) mhat < nrow(z)*solve((t(x)%*%z)%*%w%*%(t(z)%*%x))%*%(t(x)%*%z)%*%w vhat < mhat%*%shat%*%t(mhat) print(cbind(gmm$par, sqrt(diag(vhat)/nrow(x)),gmm$par/sqrt(diag(vhat)/nrow(x)), 1 pnorm(abs(gmm$par/sqrt(diag(vhat)/nrow(x)))))) # Overidentifying restrictions / Specification test statistic (JTest) Jdof < ncol(z)ncol(x) sse < as.numeric(t(yx%*%gmm$par)%*%(yx%*%gmm$par)) shat < (sse)*(t(z)%*%z) print("Jtest Degrees of Freedom"); Jdof jt < (nA0.5)*t(yx%*%gmm$par)%*%z%*%solve(shat)%*%t(z)%*%(yx%*%gmm$par) print("Jt Test Statistic"); jt # Jt converges to ChiSquare with qp degrees of freedom # The degrees of freedom equal the number of overidentifying restrictions # The critical value with alpha=0.05 and qp=4 is 9.488 # Rejection indicates problems with the GMM estimator # Calculate Discount Parameters coeff < as.matrix(gmm$par) al < coeff[2,1]; a2 < coeff[3,1] alse < sqrt(vhat[2,2]/nrow(x)); a2se < sqrt(vhat[3,3]/nrow(x)); ala2cov < vhat[2,3]/nrow(x) delta < 1a2; beta < al/(1a2) # Conduct linear hypothsis test on exponential discounting q < a2al1 qse < sqrt(alseA2+a2seA2+2*ala2cov) exptest < (q/qse)A2 deltase < a2se print("Value of Exponential Factor"); print(cbind(delta,deltase)) print("Exponential Discounting Ftest"); print(exptest) # This is a twosided Ftest with dof=1,203 and alp=0.05 # Null hypothesis is exponential discounting # Critical value is 3.84 # Conduct nonlinear hypothesis test on hyperbolic discounting gal < 1/(1a2); ga2 < al/((1a2)A2) betase < sqrt((galA2)*(alseA2)+(ga2A2)*(a2seA2)+2*(gal)*(ga2)*(ala2cov)) hyptest < ((beta1)/betase)A2 print("Value of QuasiHyperbolic Factor"); print(cbind(beta,betase)) print(" QuasiHyperbolic Discounting Ftest"); print(hyptest) # This is onesided Ftest with dof=1,203 and alp=0.05 # Null hypothesis is no hyperbolic discounting # Critical value is 3.84 # END PROGRAM # TITLE: Cornbelt States Panel OLS & Linear GMM # RESET ALL WORK rm(list = ls)) # Call needed libraries library(lmtest); library(stats); library(tseries) library(car); library(sandwich); library(systemfit) # Download CPI Deflator dta0 < read.table("Deflators.dta") cpi < as.numeric(dta0[2:44,1]) lagcpi < as.numeric(dta0[3:44,1]) # Download data sets # Column Order: YILIN~IAMOOHVR~IDV dta < read.table("CornbeltNom.dta") # Create lagged variables obs < nrow(dta) lagVil < as.matrix(dta[l:(431),7]) lagRil < as.matrix(dta[1:(431),8]) lagVin < as.matrix(dta[44:(861),7]) lagRin < as.matrix(dta[44:(861),8]) lagVia < as.matrix(dta[87:(1291),7]) lagRia < as.matrix(dta[87:(1291),8]) lagVmo < as.matrix(dta[130:(1721),7]) lagRmo < as.matrix(dta[130:(1721),8]) lagVoh < as.matrix(dta[173:(2151),7]) lagRoh < as.matrix(dta[173:(2151),8]) lagV < rbind(lagVil,lagVin,lagVia,lagVmo,lagVoh) lagR < rbind(lagRil,lagRin,lagRia,lagRmo,lagRoh) # Define dep var, indep var, & scale dvil < as.matrix(dta[2:43,10]) dvin < as.matrix(dta[45:86,10]) dvia < as.matrix(dta[88:129,10]) dvmo < as.matrix(dta[131:172,10]) dvoh < as.matrix(dta[174:215,10]) dv < rbind(dvil,dvin,dvia, dvmo,dvoh) n < nrow(dv) constant < as.matrix(seq(length=n,from=l,by=0)) vil < as.matrix(dta[2:43,7]) vin < as.matrix(dta[45:86,7]) via < as.matrix(dta[88:129,7]) vmo < as.matrix(dta[131:172,7]) voh < as.matrix(dta[174:215,7]) v < rbind(vil,vin,via,vmo,voh) ril < as.matrix(dta[2:43,8]) rin < as.matrix(dta[45:86,8]) ria < as.matrix(dta[88:129,8]) rmo < as.matrix(dta[131:172,8]) roh < as.matrix(dta[174:215,8]) r < rbind(ril,rin,ria,rmo,roh) one < as.matrix(seq(length=42,from= ,by=0)) zero < as.matrix(seq(length=42,from=0,by=0)) il < rbind(one,zero,zero,zero,zero) inn < rbind(zero,one,zero,zero,zero) ia < rbind(zero,zero,one,zero,zero) mo < rbind(zero,zero,zero,one,zero) oh < rbind(zero,zero,zero,zero,one) x < cbind(constant,r,v,il,inn,ia,oh) y < dv # OLS estimation and test results ols < lm(dv~r+v+il+inn+ia+oh) output < summary(ols); output NWvcov < as.matrix(NeweyWest(ols)) bptest(ols) # Null Hypothesis of BP test is homoscedasticity # Critical value is Chisquare with alp=.05, dof=2: 5.991 durbin.watson(ols,max.lag=2) # Null hypothsis is no autocorrelation (rho=0) # If d
# Critcal value is DW with k=2, n=41: dl=1.449, du=1.549 Box.test(dv, type = c("LjungBox")) # Null hypothesis is no autocorrelation (rho=0) # Critical value is Chisquare with alp=.05, dof=l: 3.841 adf.test(dv) # Null hypothesis is nonstationary (unit root) # Calculate Discount Parameters coeff < as.matrix(ols$coefficients) al < coeff[2,1]; a2 < coeff[3,1] alse < sqrt(NWvcov[2,2]); a2se < sqrt(NWvcov[3,3]); ala2cov < NWvcov[2,3] delta < 1a2; beta < al/(1a2) # Conduct linear hypothsis test on exponential discounting q < a2al1 qse < sqrt(alseA2+a2seA2+2*ala2cov) exptest < (q/qse)A2 deltase < a2se print("Value of Exponential Factor"); print(cbind(delta,deltase)) print("Exponential Discounting Ftest"); print(exptest) # This is a twosided Ftest with dof=1,203 and alpha=0.05 # Null hypothesis is exponential discounting # Critical value is 3.84 # Conduct nonlinear hypothesis test on hyperbolic discounting gal < l/(la2); ga2 < al/((1a2)A2) betase < sqrt((galA2)*(alseA2)+(ga2A2)*(a2seA2)+2*(gal)*(ga2)*(ala2cov)) hyptest < ((beta1)/betase)A2 print("Value of QuasiHyperbolic Factor"); print(cbind(beta,betase)) print(" QuasiHyperbolic Discounting Ftest"); print(hyptest) # This is onesided Ftest with dof=1,203 and alpha=0.05 # Null hypothesis is no hyperbolic discounting # Critical value is 3.84 # Define IV matrix and scale z < as.matrix(cbind(constant,v,r,il,inn,ia,oh,rA2,vA2,lagV,lagR)) k < ncol(z) # Define initial weighting matrix w < nrow(z)*solve(t(z)%*%z) # Define GMM function fr < function(b){ (l/n)*t(yx%*%b)%*%z%*%w%*%((l/n)*t(z)%*%(yx%*%b)) } # Define gradient gfr < function(b){ 2*t(yx%*%b)%*%z%*%w%*%t(z)%*%x } # Conduct first step GMM gmm < optim(c(0,0,0,0,0,0,0),fr,gfr) sse < as.numeric(t(yx%*%gmm$par)%*%(yx%*%gmm$par)/nrow(y)) shat < (sse/nrow(z))*(t(z)%*%z) mhat < nrow(z)*solve((t(x)%*%z)%*%w%*%(t(z)%*%x))%*%(t(x)%*%z)%*%w vhat < mhat%*%shat%*%t(mhat) print(cbind(gmm$par,sqrt(diag(vhat)/nrow(x)),gmm$par/sqrt(diag(vhat)/nrow(x)), 1pnorm(abs(gmm$par/sqrt(diag(vhat)/nrow(x)))))) # Conduct second step GMM w < solve(shat) gmm < optim(c(0,0,0,0,0,0,0),fr,gfr) sse < as.numeric(t(yx%*%gmm$par)%*%(yx%*%gmm$par)/nrow(y)) shat < (sse/nrow(z))*(t(z)%*%z) mhat < nrow(z)*solve((t(x)%*%z)%*%w%*%(t(z)%*%x))%*%(t(x)%*%z)%*%w vhat < mhat%*%shat%*%t(mhat) print(cbind(gmm$par, sqrt(diag(vhat)/nrow(x)),gmm$par/sqrt(diag(vhat)/nrow(x)), 1 pnorm(abs(gmm$par/sqrt(diag(vhat)/nrow(x)))))) # Overidentifying restrictions / Specification test statistic (JTest) Jdof < ncol(z)ncol(x) sse < as.numeric(t(yx%*%gmm$par)%*%(yx%*%gmm$par)) shat < (sse)*(t(z)%*%z) print("Jtest Degrees of Freedom"); Jdof jt < (nA0.5)*t(yx%*%gmm$par)%*%z%*%solve(shat)%*%t(z)%*%(yx%*%gmm$par) print("Jt Test Statistic"); jt # Jt converges to ChiSquare with qp degrees of freedom # The degrees of freedom equal the number of overidentifying restrictions # The critical value with alpha=0.05 and qp=4 is 9.488 # Rejection indicates problems with the GMM estimator # Calculate Discount Parameters coeff < as.matrix(gmm$par) al < coeff[2,1]; a2 < coeff[3,1] alse < sqrt(vhat[2,2]/nrow(x)); a2se < sqrt(vhat[3,3]/nrow(x)); ala2cov < vhat[2,3]/nrow(x) delta < 1a2; beta < al/(1a2) # Conduct linear hypothsis test on exponential discounting q < a2al1 qse < sqrt(alseA2+a2seA2+2*ala2cov) exptest < (q/qse)A2 deltase < a2se print("Value of Exponential Factor"); print(cbind(delta,deltase)) print("Exponential Discounting Ftest"); print(exptest) # This is a twosided Ftest with dof=1,203 and alp=0.05 # Null hypothesis is exponential discounting # Critical value is 3.84 # Conduct nonlinear hypothesis test on hyperbolic discounting gal < 1/(1a2); ga2 < al/((1a2)A2) betase < sqrt((galA2)*(alseA2)+(ga2A2)*(a2seA2)+2*(gal)*(ga2)*(ala2cov)) hyptest < ((beta1)/betase)A2 print("Value of QuasiHyperbolic Factor"); print(cbind(beta,betase)) print(" QuasiHyperbolic Discounting Ftest"); print(hyptest) # This is onesided Ftest with dof=1,203 and alp=0.05 # Null hypothesis is no hyperbolic discounting # Critical value is 3.84 # END PROGRAM # TITLE: Delta States Panel OLS & Linear GMM # Last Modified: 02/22/2008 # RESET ALL WORK rm(list = ls)) # Call needed libraries library(lmtest); library(stats); library(tseries) library(car); library(sandwich); library(systemfit) # Download data sets # Column Order: YARLAMSVR~IDV dta < read.table("DeltaNom.dta") # Create lagged variables obs < nrow(dta) lagVar < as.matrix(dta[ :(431),5]) lagRar < as.matrix(dta[l:(431),6]) lagVla < as.matrix(dta[44:(861),5]) lagRla < as.matrix(dta[44:(861),6]) lagVms < as.matrix(dta[87:(1291),5]) lagRms < as.matrix(dta[87:(1291),6]) lagV < rbind(lagVar,lagVla,lagVms) lagR < rbind(lagVar,lagVla,lagVms) # Define dep var, indep var, & scale dvar < as.matrix(dta[2:43,8]) dvla < as.matrix(dta[45:86,8]) dvms < as.matrix(dta[88:129,8]) dv < rbind(dvar,dvla,dvms) n < nrow(dv) constant < as.matrix(seq(length=n,from=l,by=0)) var < as.matrix(dta[2:43,5]) vla < as.matrix(dta[45:86,5]) vms < as.matrix(dta[88:129,5]) v < rbind(var,vla,vms) rar < as.matrix(dta[2:43,6]) rla < as.matrix(dta[45:86,6]) rms < as.matrix(dta[88:129,6]) r < rbind(rar,rla,rms) one < as.matrix(seq(length=42,from= ,by=0)) zero < as.matrix(seq(length=42,from=0,by=0)) ar < rbind(one,zero,zero) la < rbind(zero,one,zero) ms < rbind(zero,zero,one) x < cbind(constant,r,v,ar,ms) y < dv # OLS estimation and test results ols < lm(dv~r+v+ar+ms) output < summary(ols); output NWvcov < as.matrix(NeweyWest(ols)) bptest(ols) # Null Hypothesis of BP test is homoscedasticity # Critical value is Chisquare with alp=.05, dof=2: 5.991 durbin.watson(ols,max.lag=2) # Null hypothsis is no autocorrelation (rho=0) # If d
# Critcal value is DW with k=2, n=41: dl=1.449, du=1.549 Box.test(dv, type = c("LjungBox")) # Null hypothesis is no autocorrelation (rho=0) # Critical value is Chisquare with alp=.05, dof=l: 3.841 adf.test(dv) # Null hypothesis is nonstationary (unit root) # Calculate Discount Parameters coeff < as.matrix(ols$coefficients) al < coeff[2,1]; a2 < coeff[3,1] alse < sqrt(NWvcov[2,2]); a2se < sqrt(NWvcov[3,3]); ala2cov < NWvcov[2,3] delta < 1a2; beta < al/(1a2) # Conduct linear hypothsis test on exponential discounting q < a2al1 qse < sqrt(alseA2+a2seA2+2*ala2cov) exptest < (q/qse)A2 deltase < a2se print("Value of Exponential Factor"); print(cbind(delta,deltase)) print("Exponential Discounting Ftest"); print(exptest) # This is a twosided Ftest with dof=l,121 and alpha=0.05 # Null hypothesis is exponential discounting # Critical value is 3.92 # Conduct nonlinear hypothesis test on hyperbolic discounting gal < 1/(1a2); ga2 < al/((1a2)A2) betase < sqrt((galA2)*(alseA2)+(ga2A2)*(a2seA2)+2*(gal)*(ga2)*(ala2cov)) hyptest < ((beta1)/betase)A2 print("Value of QuasiHyperbolic Factor"); print(cbind(beta,betase)) print(" QuasiHyperbolic Discounting Ftest"); print(hyptest) # This is onesided Ftest with dof=l,121 and alpha=0.05 # Null hypothesis is no hyperbolic discounting # Critical value is 3.92 # Define IV matrix and scale z < as.matrix(cbind(constant,v,r,ar,ms,vA2,rA2,lagV)) k < ncol(z) # Define initial weighting matrix w < nrow(z)*solve(t(z)%*%z) # Define GMM function fr < function(b){ (1/n)*t(yx%*%b)%*%z%*%w%*%((1/n)*t(z)%*%(yx%*%b)) } # Define gradient gfr < function(b){ 2*t(yx%*%b)%*%z%*%w%*%t(z)%*%x } # Conduct first step GMM gmm < optim(c(0,0,0,0,0),fr,gfr) sse < as.numeric(t(yx%*%gmm$par)%*%(yx%*%gmm$par)/nrow(y)) shat < (sse/nrow(z))*(t(z)%*%z) mhat < nrow(z)*solve((t(x)%*%z)%*%w%*%(t(z)%*%x))%*%(t(x)%*%z)%*%w vhat < mhat%*%shat%*%t(mhat) print(cbind(gmm$par,sqrt(diag(vhat)/nrow(x)),gmm$par/sqrt(diag(vhat)/nrow(x)), 1pnorm(abs(gmm$par/sqrt(diag(vhat)/nrow(x)))))) # Conduct second step GMM w < solve(shat) gmm < optim(c(0,0,0,0,0),fr,gfr) sse < as.numeric(t(yx%*%gmm$par)%*%(yx%*%gmm$par)/nrow(y)) shat < (sse/nrow(z))*(t(z)%*%z) mhat < nrow(z)*solve((t(x)%*%z)%*%w%*%(t(z)%*%x))%*%(t(x)%*%z)%*%w vhat < mhat%*%shat%*%t(mhat) print(cbind(gmm$par,sqrt(diag(vhat)/nrow(x)),gmm$par/sqrt(diag(vhat)/nrow(x)), 1pnorm(abs(gmm$par/sqrt(diag(vhat)/nrow(x)))))) # Overidentifying restrictions / Specification test statistic (JTest) Jdof < ncol(z)ncol(x) sse < as.numeric(t(yx%*%gmm$par)%*%(yx%*%gmm$par)) shat < (sse)*(t(z)%*%z) print("Jtest Degrees of Freedom"); Jdof jt < (nA0.5)*t(yx%*%gmm$par)%*%z%*%solve(shat)%*%t(z)%*%(yx%*%gmm$par) print("Jt Test Statistic"); jt # Jt converges to ChiSquare with qp degrees of freedom # The degrees of freedom equal the number of overidentifying restrictions # The critical value with alpha=0.05 and qp=4 is 9.488 # Rejection indicates problems with the GMM estimator # Calculate Discount Parameters coeff < as.matrix(gmm$par) al < coeff[2,1]; a2 < coeff[3,1] alse < sqrt(vhat[2,2]/nrow(x)); a2se < sqrt(vhat[3,3]/nrow(x)); ala2cov < vhat[2,3]/nrow(x) delta < 1a2; beta < al/(1a2) # Conduct linear hypothsis test on exponential discounting q < a2al1 qse < sqrt(alseA2+a2seA2+2*ala2cov) exptest < (q/qse)A2 deltase < a2se print("Value of Exponential Factor"); print(cbind(delta,deltase)) print("Exponential Discounting Ftest"); print(exptest) # This is a twosided Ftest with dof=1,121 and alp=0.05 # Null hypothesis is exponential discounting # Critical value is 3.92 # Conduct nonlinear hypothesis test on hyperbolic discounting gal < 1/(1a2); ga2 < al/((1a2)A2) betase < sqrt((galA2)*(alseA2)+(ga2A2)*(a2seA2)+2*(gal)*(ga2)*(ala2cov)) hyptest < ((beta1)/betase)A2 print("Value of QuasiHyperbolic Factor"); print(cbind(beta,betase)) print(" QuasiHyperbolic Discounting Ftest"); print(hyptest) # This is onesided Ftest with dof=l,121 and alp=0.05 # Null hypothesis is no hyperbolic discounting # Critical value is 3.92 # END PROGRAM # TITLE: Plain States Panel OLS & Linear GMM # Last Modified: 02/26/2008 # RESET ALL WORK rm(list = ls)) # Call needed libraries library(lmtest); library(stats); library(tseries) library(car); library(sandwich); library(systemfit) # Download CPI Deflator dta0 < read.table("Deflators.dta") cpi < as.numeric(dta0[2:44,1]) lagcpi < as.numeric(dta0[3:44,1]) # Download data sets # Column Order: YKSNENDOKSDTXVRIDV dta < read.table("PlainsNom.dta") # Create lagged variables obs < nrow(dta) lagVks < as.matrix(dta[1:(431),8]) lagRks < as.matrix(dta[1:(431),9]) lagVne < as.matrix(dta[44:(861),8]) lagRne < as.matrix(dta[44:(861),9]) lagVnd < as.matrix(dta[87:(1291),8]) lagRnd < as.matrix(dta[87:(1291),9]) lagVok < as.matrix(dta[130:(1721),8]) lagRok < as.matrix(dta[130:(1721),9]) lagVsd < as.matrix(dta[173:(2151),8]) lagRsd < as.matrix(dta[173:(2151),9]) lagVtx < as.matrix(dta[216:(2581),8]) lagRtx < as.matrix(dta[216:(2581),9]) lagV < rbind(lagVks,lagVne,lagVnd,lagVok,lagVsd,lagVtx) lagR < rbind(lagRks,lagRne,lagRnd,lagRok,lagRsd,lagRtx) # Define dep var, indep var, & scale dvks < as.matrix(dta[2:43,11]) dvne < as.matrix(dta[45:86,11]) dvnd < as.matrix(dta[88:129,11]) dvok < as.matrix(dta[131:172,11]) dvsd < as.matrix(dta[174:215,11]) dvtx < as.matrix(dta[217:258,11]) dv < rbind(dvks,dvne,dvnd,dvok,dvsd,dvtx) n < nrow(dv) constant < as.matrix(seq(length=n,from=l,by=0)) vks < as.matrix(dta[2:43,8]) vne < as.matrix(dta[45:86,8]) vnd < as.matrix(dta[88:129,8]) vok < as.matrix(dta[131:172,8]) vsd < as.matrix(dta[174:215,8]) vtx < as.matrix(dta[217:258,8]) v < rbind(vks,vne,vnd,vok,vsd,vtx) rks < as.matrix(dta[2:43,9]) rne < as.matrix(dta[45:86,9]) rnd < as.matrix(dta[88:129,9]) rok < as.matrix(dta[131:172,9]) rsd < as.matrix(dta[174:215,9]) rtx < as.matrix(dta[217:258,9]) r < rbind(rks,rne,rnd,rok,rsd,rtx) one < as.matrix(seq(length=42,from= ,by=0)) zero < as.matrix(seq(length=42,from=0,by=0)) ks < rbind(one,zero,zero,zero,zero,zero) ne < rbind(zero,one,zero,zero,zero,zero) nd < rbind(zero,zero,one,zero,zero,zero) ok < rbind(zero,zero,zero,one,zero,zero) sd < rbind(zero,zero,zero,zero,one,zero) tx < rbind(zero,zero,zero,zero,zero,one) x < cbind(constant,r,v,ks,ne,nd,ok,tx) y < dv # OLS estimation and test results ols < lm(dv~r+v+ks+ne+nd+ok+tx) output < summary(ols); output NWvcov < as.matrix(NeweyWest(ols)) bptest(ols) # Null Hypothesis of BP test is homoscedasticity # Critical value is Chisquare with alp=.05, dof=2: 5.991 durbin.watson(ols,max.lag=2) # Null hypothsis is no autocorrelation (rho=0) # If d
# Critcal value is DW with k=2, n=41: dl=1.449, du=1.549 Box.test(dv, type = c("LjungBox")) # Null hypothesis is no autocorrelation (rho=0) # Critical value is Chisquare with alp=.05, dof=l: 3.841 adf.test(dv) # Null hypothesis is nonstationary (unit root) # Calculate Discount Parameters coeff < as.matrix(ols$coefficients) al < coeff[2,1]; a2 < coeff[3,1] alse < sqrt(NWvcov[2,2]); a2se < sqrt(NWvcov[3,3]); ala2cov < NWvcov[2,3] delta < 1a2; beta < al/(1a2) # Conduct linear hypothsis test on exponential discounting q < a2al1 qse < sqrt(alseA2+a2seA2+2*ala2cov) exptest < (q/qse)A2 print("Value of Exponential Factor"); print(delta) print("Exponential Discounting Ftest"); print(exptest) # This is a twosided Ftest with dof=1,244 # Null hypothesis is exponential discounting # Critical value is 3.84 (alp=0.05) & 2.70 (alp=0.10) # Conduct nonlinear hypothesis test on hyperbolic discounting gal < 1/(1a2); ga2 < al/((1a2)A2) betase < sqrt((galA2)*(alseA2)+(ga2A2)*(a2seA2)+2*(gal)*(ga2)*(ala2cov)) hyptest < ((beta1)/betase)A2 print("Value of QuasiHyperbolic Factor"); print(cbind(beta,betase)) print(" QuasiHyperbolic Discounting Ftest"); print(hyptest) # This is onesided Ftest with dof=1,244 # Null hypothesis is no hyperbolic discounting # Critical value is 3.84 (alp=0.05) & 2.70 (alp=0.10) # Define IV matrix and scale z < as.matrix(cbind(constant,v,r,ks,ne,nd,ok,tx,v^2,r^2,lagV,lagR)) k < ncol(z) # Define initial weighting matrix w < nrow(z)*solve(t(z)%*%z) # Define GMM function fr < function(b){ (1/n)*t(yx%*%b)%*%z%*%w%*%((1/n)*t(z)%*%(yx%*%b)) } # Define gradient gfr < function(b){ 2*t(yx%*%b)%*%z%*%w%*%t(z)%*%x } # Conduct first step GMM gmm < optim(c(0,0,0,0,0,0,0,0),fr,gfr) sse < as.numeric(t(yx%*%gmm$par)%*%(yx%*%gmm$par)/nrow(y)) shat < (sse/nrow(z))*(t(z)%*%z) mhat < nrow(z)*solve((t(x)%*%z)%*%w%*%(t(z)%*%x))%*%(t(x)%*%z)%*%w vhat < mhat%*%shat%*%t(mhat) print(cbind(gmm$par,sqrt(diag(vhat)/nrow(x)),gmm$par/sqrt(diag(vhat)/nrow(x)), 1pnorm(abs(gmm$par/sqrt(diag(vhat)/nrow(x)))))) # Conduct second step GMM w < solve(shat) gmm < optim(c(0,0,0,0,0,0,0,0),fr,gfr) sse < as.numeric(t(yx%*%gmm$par)%*%(yx%*%gmm$par)/nrow(y)) shat < (sse/nrow(z))*(t(z)%*%z) mhat < nrow(z)*solve((t(x)%*%z)%*%w%*%(t(z)%*%x))%*%(t(x)%*%z)%*%w vhat < mhat%*%shat%*%t(mhat) print(cbind(gmm$par,sqrt(diag(vhat)/nrow(x)),gmm$par/sqrt(diag(vhat)/nrow(x)), 1pnorm(abs(gmm$par/sqrt(diag(vhat)/nrow(x)))))) # Overidentifying restrictions / Specification test statistic (JTest) Jdof < ncol(z)ncol(x) sse < as.numeric(t(yx%*%gmm$par)%*%(yx%*%gmm$par)) shat < (sse)*(t(z)%*%z) print("Jtest Degrees of Freedom"); Jdof jt < (nA0.5)*t(yx%*%gmm$par)%*%z%*%solve(shat)%*%t(z)%*%(yx%*%gmm$par) print("Jt Test Statistic"); jt # Jt converges to ChiSquare with qp degrees of freedom # The degrees of freedom equal the number of overidentifying restrictions # The critical value with alpha=0.05 and qp=4 is 9.488 # Rejection indicates problems with the GMM estimator # Calculate Discount Parameters coeff < as.matrix(gmm$par) al < coeff[2,l]; a2 < coeff[3,1] alse < sqrt(vhat[2,2]/nrow(x)); a2se < sqrt(vhat[3,3]/nrow(x)); ala2cov < vhat[2,3]/nrow(x) delta < la2; beta < al/(la2) # Conduct linear hypothsis test on exponential discounting q < a2al1 qse < sqrt(alseA2+a2seA2+2*ala2cov) exptest < (q/qse)A2 deltase < a2se print("Value of Exponential Factor"); print(cbind(delta,deltase)) print("Exponential Discounting Ftest"); print(exptest) # This is a twosided Ftest with dof=1,244 # Null hypothesis is exponential discounting # Critical value is 3.84 (alp=0.05) & 2.70 (alp=0.10) # Conduct nonlinear hypothesis test on hyperbolic discounting gal < l/(la2); ga2 < al/((1a2)A2) betase < sqrt((galA2)*(alseA2)+(ga2A2)*(a2seA2)+2*(gal)*(ga2)*(ala2cov)) hyptest < ((beta1)/betase)A2 print("Value of QuasiHyperbolic Factor"); print(cbind(beta,betase)) print(" QuasiHyperbolic Discounting Ftest"); print(hyptest) # This is onesided Ftest with dof=1,244 # Null hypothesis is no hyperbolic discounting # Critical value is 3.84 (alp=0.05) & 2.70 (alp=0.10) # END PROGRAM # TITLE: Lake States Panel OLS & Linear GMM # RESET ALL WORK rm(list = ls)) # Call needed libraries library(lmtest); library(stats); library(tseries) library(car); library(sandwich); library(systemfit) # Download CPI Deflator dta0 < read.table("Deflators.dta") cpi < as.numeric(dta0[2:44,1]) lagcpi < as.numeric(dta0[3:44,1]) # Download data sets # Column Order: YMIMN~WIVR~IDV dta < read.table("LakeNom.dta") # Create lagged variables obs < nrow(dta) lagVmi < as.matrix(dta[l:(431),5]) lagRmi < as.matrix(dta[1:(431),6]) lagVmn < as.matrix(dta[44:(861),5]) lagRmn < as.matrix(dta[44:(861),6]) lagVwi < as.matrix(dta[87:(1291),5]) lagRwi < as.matrix(dta[87:(1291),6]) lagV < rbind(lagVmi,lagVmn,lagVwi) lagR < rbind(lagVmi,lagVmn,lagVwi) # Define dep var, indep var, & scale dvmi < as.matrix(dta[2:43,8]) dvmn < as.matrix(dta[45:86,8]) dvwi < as.matrix(dta[88:129,8]) dv < rbind(dvmi,dvmn,dvwi) n < nrow(dv) constant < as.matrix(seq(length=n,from=l,by=0)) vmi < as.matrix(dta[2:43,5]) vmn < as.matrix(dta[45:86,5]) vwi < as.matrix(dta[88:129,5]) v < rbind(vmi,vmn,vwi) rmi < as.matrix(dta[2:43,6]) rmn < as.matrix(dta[45:86,6]) rwi < as.matrix(dta[88:129,6]) r < rbind(rmi,rmn,rwi) one < as.matrix(seq(length=42,from= ,by=0)) zero < as.matrix(seq(length=42,from=0,by=0)) mi < rbind(one,zero,zero) mn < rbind(zero,one,zero) wi < rbind(zero,zero,one) x < cbind(constant,r,v,mn,wi) y < dv # OLS estimation and test results ols < lm(dv~r+v+mn+wi) output < summary(ols); output NWvcov < as.matrix(NeweyWest(ols)) bptest(ols) # Null Hypothesis of BP test is homoscedasticity # Critical value is Chisquare with alp=.05, dof=2: 5.991 durbin.watson(ols,max.lag=2) # Null hypothsis is no autocorrelation (rho=0) # If d
# Critcal value is DW with k=2, n=41: dl=1.449, du=1.549 Box.test(dv, type = c("LjungBox")) # Null hypothesis is no autocorrelation (rho=0) # Critical value is Chisquare with alp=.05, dof=l: 3.841 adf.test(dv) # Null hypothesis is nonstationary (unit root) # Calculate Discount Parameters coeff < as.matrix(ols$coefficients) al < coeff[2,1]; a2 < coeff[3,1] alse < sqrt(NWvcov[2,2]); a2se < sqrt(NWvcov[3,3]); ala2cov < NWvcov[2,3] delta < 1a2; beta < al/(1a2) # Conduct linear hypothsis test on exponential discounting q < a2al1 qse < sqrt(alseA2+a2seA2+2*ala2cov) exptest < (q/qse)A2 deltase < a2se print("Value of Exponential Factor"); print(cbind(delta,deltase)) print("Exponential Discounting Ftest"); print(exptest) # This is a twosided Ftest with dof=l,121 and alpha=0.05 # Null hypothesis is exponential discounting # Critical value is 3.92 # Conduct nonlinear hypothesis test on hyperbolic discounting gal < 1/(1a2); ga2 < al/((1a2)^2) betase < sqrt((gal 2)*(alseA2)+(ga2A2)*(a2seA2)+2*(gal)*(ga2)*(ala2cov)) hyptest < ((beta1)/betase)^2 print("Value of QuasiHyperbolic Factor"); print(cbind(beta,betase)) print(" QuasiHyperbolic Discounting Ftest"); print(hyptest) # This is onesided Ftest with dof=l,121 and alpha=0.05 # Null hypothesis is no hyperbolic discounting # Critical value is 3.92 # Define IV matrix and scale z < as.matrix(cbind(constant,v,r,mn,wi,v^2,r2,agV)) k < ncol(z) # Define initial weighting matrix w < nrow(z)*solve(t(z)%*%z) # Define GMM function fr < function(b){ (1/n)*t(yx%*%b)%*%z%*%w%*%((1/n)*t(z)%*%(yx%*%b)) } # Define gradient gfr < function(b){ 2*t(yx%*%b)%*%z%*%w%*%t(z)%*%x } # Conduct first step GMM gmm < optim(c(0,0,0,0,0),fr,gfr) sse < as.numeric(t(yx%*%gmm$par)%*%(yx%*%gmm$par)/nrow(y)) shat < (sse/nrow(z))*(t(z)%*%z) mhat < nrow(z)*solve((t(x)%*%z)%*%w%*%(t(z)%*%x))%*%(t(x)%*%z)%*%w vhat < mhat%*%shat%*%t(mhat) print(cbind(gmm$par,sqrt(diag(vhat)/nrow(x)),gmm$par/sqrt(diag(vhat)/nrow(x)), 1pnorm(abs(gmm$par/sqrt(diag(vhat)/nrow(x)))))) # Conduct second step GMM w < solve(shat) gmm < optim(c(0,0,0,0,0),fr,gfr) sse < as.numeric(t(yx%*%gmm$par)%*%(yx%*%gmm$par)/nrow(y)) shat < (sse/nrow(z))*(t(z)%*%z) mhat < nrow(z)*solve((t(x)%*%z)%*%w%*%(t(z)%*%x))%*%(t(x)%*%z)%*%w vhat < mhat%*%shat%*%t(mhat) print(cbind(gmm$par,sqrt(diag(vhat)/nrow(x)),gmm$par/sqrt(diag(vhat)/nrow(x)), 1pnorm(abs(gmm$par/sqrt(diag(vhat)/nrow(x)))))) # Overidentifying restrictions / Specification test statistic (JTest) Jdof < ncol(z)ncol(x) sse < as.numeric(t(yx%*%gmm$par)%*%(yx%*%gmm$par)) shat < (sse)*(t(z)%*%z) print("Jtest Degrees of Freedom"); Jdof jt < (nA0.5)*t(yx%*%gmm$par)%*%z%*%solve(shat)%*%t(z)%*%(yx%*%gmm$par) print("Jt Test Statistic"); jt # Jt converges to ChiSquare with qp degrees of freedom # The degrees of freedom equal the number of overidentifying restrictions # The critical value with alpha=0.05 and qp=4 is 9.488 # Rejection indicates problems with the GMM estimator # Calculate Discount Parameters coeff < as.matrix(gmm$par) al < coeff[2,1]; a2 < coeff[3,1] alse < sqrt(vhat[2,2]/nrow(x)); a2se < sqrt(vhat[3,3]/nrow(x)); ala2cov < vhat[2,3]/nrow(x) delta < 1a2; beta < al/(1a2) # Conduct linear hypothsis test on exponential discounting q < a2al1 qse < sqrt(alseA2+a2seA2+2*ala2cov) exptest < (q/qse)A2 deltase < a2se print("Value of Exponential Factor"); print(cbind(delta,deltase)) print("Exponential Discounting Ftest"); print(exptest) # This is a twosided Ftest with dof=1,121 and alp=0.05 # Null hypothesis is exponential discounting # Critical value is 3.92 # Conduct nonlinear hypothesis test on hyperbolic discounting gal < l/(la2); ga2 < al/((la2)A2) betase < sqrt((galA2)*(alseA2)+(ga2A2)*(a2seA2)+2*(gal)*(ga2)*(ala2cov)) hyptest < ((beta1)/betase)A2 print("Value of QuasiHyperbolic Factor"); print(cbind(beta,betase)) print(" QuasiHyperbolic Discounting Ftest"); print(hyptest) # This is onesided Ftest with dof=l,121 and alp=0.05 # Null hypothesis is no hyperbolic discounting # Critical value is 3.92 # END PROGRAM # TITLE: Mountain States Panel OLS & Linear GMM # RESET ALL WORK rm(list = ls)) # Call needed libraries library(lmtest); library(stats); library(tseries) library(car); library(sandwich); library(systemfit) # Download CPI Deflator dta0 < read.table("Deflators.dta") cpi < as.numeric(dta0[2:44,1]) lagcpi < as.numeric(dta0[3:44,1]) # Download data sets # Column Order: YAZCO~ID~MTNVNMUTWYVR~IDV dta < read.table("MountainNom.dta") # Create lagged variables obs < nrow(dta) lagVaz < as.matrix(dta[l:(431),10]) lagRaz < as.matrix(dta[1:(431),11]) lagVco < as.matrix(dta[44:(861),10]) lagRco < as.matrix(dta[44:(861), 11]) lagVid < as.matrix(dta[87:(1291),10]) lagRid < as.matrix(dta[87:(1291), 11]) lagVmt < as.matrix(dta[130:(1721),10]) lagRmt < as.matrix(dta[130:(1721),11]) lagVnv < as.matrix(dta[173:(2151),10]) lagRnv < as.matrix(dta[173:(2151),11]) lagVnm < as.matrix(dta[216:(2581),10]) lagRnm < as.matrix(dta[216:(2581),11]) lagVut < as.matrix(dta[259:(3011),10]) lagRut < as.matrix(dta[259:(3011),11]) lagVwy < as.matrix(dta[302:(3441),10]) lagRwy < as.matrix(dta[302:(3441), 11]) lagV < rbind(lagVaz,lagVco,lagVid,lagVmt,lagVnv,lagVnm,lagVut,lagVwy) lagR < rbind(lagRaz,lagRco,lagRid,lagRmt,lagRnv,lagRnm,lagRut,lagRwy) # Define dep var, indep var, & scale dvaz < as.matrix(dta[2:43,13]) dvco < as.matrix(dta[45:86,13]) dvid < as.matrix(dta[88:129,13]) dvmt < as.matrix(dta[131:172,13]) dvnv < as.matrix(dta[174:215,13]) dvnm < as.matrix(dta[217:258,13]) dvut < as.matrix(dta[260:301,13]) dvwy < as.matrix(dta[303:344,13]) dv < rbind(dvaz,dvco,dvid, dvmt, dvnv,dvnm,dvut, dvwy) n < nrow(dv) constant < as.matrix(seq(length=n,from=l,by=0)) vaz < as.matrix(dta[2:43,10]) vco < as.matrix(dta[45:86,10]) vid < as.matrix(dta[88:129,10]) vmt < as.matrix(dta[131:172,10]) vnv < as.matrix(dta[174:215,10]) vnm < as.matrix(dta[217:258,10]) vut < as.matrix(dta[260:301,10]) vwy < as.matrix(dta[303:344,10]) v < rbind(vaz,vco,vid,vmt,vnv,vnm,vut,vwy) raz < as.matrix(dta[2:43,11]) rco < as.matrix(dta[45:86,11]) rid < as.matrix(dta[88:129,11]) rmt < as.matrix(dta[131:172,11]) rnv < as.matrix(dta[174:215,11]) rnm < as.matrix(dta[217:258,11]) rut < as.matrix(dta[260:301,11]) rwy < as.matrix(dta[303:344,11]) r < rbind(raz,rco,rid,rmt,mrv,rnm,rut,rwy) one < as.matrix(seq(length=42,from= ,by=0)) zero < as.matrix(seq(length=42,from=0,by=0)) az < rbind(one,zero,zero,zero,zero,zero,zero,zero) co < rbind(zero,one,zero,zero,zero,zero,zero,zero) id < rbind(zero,zero,one,zero,zero,zero,zero,zero) mt < rbind(zero,zero,zero,one,zero,zero,zero,zero) nv < rbind(zero,zero,zero,zero,one,zero,zero,zero) nm < rbind(zero,zero,zero,zero,zero, one,zero,zero) ut < rbind(zero,zero,zero,zero,zero,zero,one,zero) wy < rbind(zero,zero,zero,zero,zeroero,zero,zero, one) x < cbind(constant,r,v,co,id,mt,nv,nm,ut,wy) y < dv # OLS estimation and test results ols < lm(dv~r+v+co+id+mt+nv+nm+ut+wy) output < summary(ols); output NWvcov < as.matrix(NeweyWest(ols)) bptest(ols) # Null Hypothesis of BP test is homoscedasticity # Critical value is Chisquare with alp=.05, dof=2: 5.991 durbin.watson(ols,max.lag=2) # Null hypothsis is no autocorrelation (rho=0) # If d
# Critcal value is DW with k=2, n=41: dl=1.449, du=1.549 Box.test(dv, type = c("LjungBox")) # Null hypothesis is no autocorrelation (rho=0) # Critical value is Chisquare with alp=.05, dof=l: 3.841 adf.test(dv) # Null hypothesis is nonstationary (unit root) # Calculate Discount Parameters coeff < as.matrix(ols$coefficients) al < coeff[2,1]; a2 < coeff[3,1] alse < sqrt(NWvcov[2,2]); a2se < sqrt(NWvcov[3,3]); ala2cov < NWvcov[2,3] delta < 1a2; beta < al/(1a2) # Conduct linear hypothsis test on exponential discounting q < a2al1 qse < sqrt(alseA2+a2seA2+2*ala2cov) exptest < (q/qse)A2 deltase < a2se print("Value of Exponential Factor"); print(cbind(delta,deltase)) print("Exponential Discounting Ftest"); print(exptest) # This is a twosided Ftest with dof=1,203 and alpha=0.05 # Null hypothesis is exponential discounting # Critical value is 3.84 # Conduct nonlinear hypothesis test on hyperbolic discounting gal < l/(la2); ga2 < al/((1a2)A2) betase < sqrt((galA2)*(alseA2)+(ga2A2)*(a2seA2)+2*(gal)*(ga2)*(ala2cov)) hyptest < ((beta1)/betase)A2 print("Value of QuasiHyperbolic Factor"); print(cbind(beta,betase)) print(" QuasiHyperbolic Discounting Ftest"); print(hyptest) # This is onesided Ftest with dof=1,203 and alpha=0.05 # Null hypothesis is no hyperbolic discounting # Critical value is 3.84 # Define IV matrix and scale z < as.matrix(cbind(constant,v,r,co,id,mt,nv,nm,ut,wy,rA2,vA2,lagV,1agR)) k < ncol(z) # Define initial weighting matrix w < nrow(z)*solve(t(z)%*%z) # Define GMM function fr < function(b){ (l/n)*t(yx%*%b)%*%z%*%w%*%((l/n)*t(z)%*%(yx%*%b)) } # Define gradient gfr < function(b){ 2*t(yx%*%b)%*%z%*%w%*%t(z)%*%x } # Conduct first step GMM gmm < optim(c(00,0,0,0,0,00,0,0,0),fr,gfr) sse < as.numeric(t(yx%*%gmm$par)%*%(yx%*%gmm$par)/nrow(y)) shat < (sse/nrow(z))*(t(z)%*%z) mhat < nrow(z)*solve((t(x)%*%z)%*%w%*%(t(z)%*%x))%*%(t(x)%*%z)%*%w vhat < mhat%*%shat%*%t(mhat) print(cbind(gmm$par,sqrt(diag(vhat)/nrow(x)),gmm$par/sqrt(diag(vhat)/nrow(x)), 1pnorm(abs(gmm$par/sqrt(diag(vhat)/nrow(x)))))) # Conduct second step GMM w < solve(shat) gmm < optim(c(00,0,0,0,0,00,0,0,0),fr,gfr) sse < as.numeric(t(yx%*%gmm$par)%*%(yx%*%gmm$par)/nrow(y)) shat < (sse/nrow(z))*(t(z)%*%z) mhat < nrow(z)*solve((t(x)%*%z)%*%w%*%(t(z)%*%x))%*%(t(x)%*%z)%*%w vhat < mhat%*%shat%*%t(mhat) print(cbind(gmm$par,sqrt(diag(vhat)/nrow(x)),gmm$par/sqrt(diag(vhat)/nrow(x)), 1pnorm(abs(gmm$par/sqrt(diag(vhat)/nrow(x)))))) # Overidentifying restrictions / Specification test statistic (JTest) Jdof < ncol(z)ncol(x) sse < as.numeric(t(yx%*%gmm$par)%*%(yx%*%gmm$par)) shat < (sse)*(t(z)%*%z) print("Jtest Degrees of Freedom"); Jdof jt < (nA0.5)*t(yx%*%gmm$par)%*%z%*%solve(shat)%*%t(z)%*%(yx%*%gmm$par) print("Jt Test Statistic"); jt # Jt converges to ChiSquare with qp degrees of freedom # The degrees of freedom equal the number of overidentifying restrictions # The critical value with alpha=0.05 and qp=4 is 9.488 # Rejection indicates problems with the GMM estimator # Calculate Discount Parameters coeff < as.matrix(gmm$par) al < coeff[2,1]; a2 < coeff[3,1] alse < sqrt(vhat[2,2]/nrow(x)); a2se < sqrt(vhat[3,3]/nrow(x)); ala2cov < vhat[2,3]/nrow(x) delta < 1a2; beta < al/(1a2) # Conduct linear hypothsis test on exponential discounting q < a2al1 qse < sqrt(alseA2+a2seA2+2*ala2cov) exptest < (q/qse)A2 deltase < a2se print("Value of Exponential Factor"); print(cbind(delta,deltase)) print("Exponential Discounting Ftest"); print(exptest) # This is a twosided Ftest with dof=1,203 and alp=0.05 # Null hypothesis is exponential discounting # Critical value is 3.84 # Conduct nonlinear hypothesis test on hyperbolic discounting gal < l/(la2); ga2 < al/((1a2)A2) betase < sqrt((galA2)*(alseA2)+(ga2A2)*(a2seA2)+2*(gal)*(ga2)*(ala2cov)) hyptest < ((beta1)/betase)A2 print("Value of QuasiHyperbolic Factor"); print(cbind(beta,betase)) print(" QuasiHyperbolic Discounting Ftest"); print(hyptest) # This is onesided Ftest with dof=1,203 and alp=0.05 # Null hypothesis is no hyperbolic discounting # Critical value is 3.84 # END PROGRAM # TITLE: Northeast States Panel OLS & Linear GMM # RESET ALL WORK rm(list = ls)) # Call needed libraries library(lmtest); library(stats); library(tseries) library(car); library(sandwich); library(systemfit) # Download CPI Deflator dta0 < read.table("Deflators.dta") cpi < as.numeric(dta0[2:44,1]) lagcpi < as.numeric(dta0[3:44,1]) # Download data sets # Column Order: YCTDEMEMDMANHNJNYPARIVTVR~IDV dta < read.table("NortheastNom.dta") # Create lagged variables obs < nrow(dta) lagVct < as.matrix(dta[l:(431),13]) lagRct < as.matrix(dta[ :(431),14]) lagVde < as.matrix(dta[44:(861),13]) lagRde < as.matrix(dta[44:(861),14]) lagVme < as.matrix(dta[87:(1291),13]) lagRme < as.matrix(dta[87:(1291),14]) lagVmd < as.matrix(dta[130:(1721),13]) lagRmd < as.matrix(dta[130:(1721),14]) lagVma < as.matrix(dta[173:(2151),13]) lagRma < as.matrix(dta[173:(2151),14]) lagVnh < as.matrix(dta[216:(2581),13]) lagRnh < as.matrix(dta[216:(2581),14]) lagVnj < as.matrix(dta[259:(3011),13]) lagRnj < as.matrix(dta[259:(3011),14]) lagVny < as.matrix(dta[302:(3441),13]) lagRny < as.matrix(dta[302:(3441),14]) lagVpa < as.matrix(dta[345:(3871),13]) lagRpa < as.matrix(dta[345:(3871),14]) lagVri < as.matrix(dta[388:(4301),13]) lagRri < as.matrix(dta[388:(4301),14]) lagVvt < as.matrix(dta[431:(4731),13]) lagRvt < as.matrix(dta[431:(4731),14]) lagV < rbind(lagVct,lagVde,lagVme,agVmd,agVma,lagVnh,lagVnj,lagVny,lagVpa,lagVri,lagVvt) lagR < rbind(lagRct,lagRde,lagRme,lagRmd,lagRma,lagRnh,lagRnj,lagRny,lagRpa,lagRri,lagRvt) # Define dep var, indep var, & scale dvct < as.matrix(dta[2:43,16]) dvde < as.matrix(dta[45:86,16]) dvme < as.matrix(dta[88:129,16]) dvmd < as.matrix(dta[131:172,16]) dvma < as.matrix(dta[174:215,16]) dvnh < as.matrix(dta[217:258,16]) dvnj < as.matrix(dta[260:301,16]) dvny < as.matrix(dta[303:344,16]) dvpa < as.matrix(dta[346:387,16]) dvri < as.matrix(dta[389:430,16]) dvvt < as.matrix(dta[432:473,16]) dv < rbind(dvct,dvde,dvme,dvmd,dvma,dvnh,dvnj,dvny,dvpa,dvri,dvvt) n < nrow(dv) constant < as.matrix(seq(length=n,from=l,by=O)) vet < as.matrix(dta[2:43,13]) vde < as.matrix(dta[45:86,13]) vme < as.matrix(dta[88:129,13]) vmd < as.matrix(dta[131:172,13]) vma < as.matrix(dta[174:215,13]) vnh < as.matrix(dta[217:258,13]) vnj < as.matrix(dta[260:301,13]) vny < as.matrix(dta[303:344,13]) vpa < as.matrix(dta[346:387,13]) vri < as.matrix(dta[389:430,13]) vvt < as.matrix(dta[432:473,13]) v < rbind(vct,vde,vme,vmd,vma,vnh,vnj,vny,vpa,vri,vvt) rct < as.matrix(dta[2:43,14]) rde < as.matrix(dta[45:86,14]) rme < as.matrix(dta[88:129,14]) rmd < as.matrix(dta[131:172,14]) rma < as.matrix(dta[174:215,14]) rnh < as.matrix(dta[217:258,14]) rnj < as.matrix(dta[260:301,14]) rny < as.matrix(dta[303:344,14]) rpa < as.matrix(dta[346:387,14]) rri < as.matrix(dta[389:430,14]) rvt < as.matrix(dta[432:473,14]) r < rbind(rct,rde,rme,rmd,rma,rnh,mj,rny,rpa,rri,rvt) one < as.matrix(seq(length=42,from= ,by=O)) zero < as.matrix(seq(length=42,from=0,by=0)) ct < rbind(one,zero,zero,zero,zero,zero,zero,zero,zero,zero,zero) de < rbind(zero, one,zero,zero,zero,zero,zero,zero,zero,zero,zero) me < rbind(zero,zero,one,zero,zero,zero,zero,zero,zero,zero,zero) md < rbind(zero,zero,zero, one,zero,zero,zero,zero,zero,zero,zero) ma < rbind(zero,zero,zero,zero, one,zero,zero,zero,zero,zero,zero) nh < rbind(zero,zero,zero,zero,zero, one,zero,zero,zero,zero,zero) nj < rbind(zero,zero,zero,zero,zero,zero, one,zero,zero,zero,zero) ny < rbind(zero,zero,zero,zero,zero,zero,zero, one,zero,zero,zero) pa < rbind(zero,zero,zero,zero,zero,zero,zero,zero, one,zero,zero) ri < rbind(zero,zero,zero,zero,ro,zero,zo,zero,zero,ero, one,zero) vt < rbind(zero,zero,zero,zero,zero,zero,zero,zero,zero,zero,one) x < cbind(constant,r,v,ct,de,me,md,ma,nh,nj,ny,pa,vt) y < dv # OLS estimation and test results ols < lm(dv~r+v+ct+de+me+md+ma+nh+nj+ny+pa+vt) output < summary(ols); output NWvcov < as.matrix(NeweyWest(ols)) bptest(ols) # Null Hypothesis of BP test is homoscedasticity # Critical value is Chisquare with alp=.05, dof=2: 5.991 durbin.watson(ols,max.lag=2) # Null hypothsis is no autocorrelation (rho=0) # If d
# Critcal value is DW with k=2, n=41: dl=1.449, du=1.549 Box.test(dv, type = c("LjungBox")) # Null hypothesis is no autocorrelation (rho=0) # Critical value is Chisquare with alp=.05, dof=l: 3.841 adf.test(dv) # Null hypothesis is nonstationary (unit root) # Calculate Discount Parameters coeff < as.matrix(ols$coefficients) al < coeff[2,1]; a2 < coeff[3,1] alse < sqrt(NWvcov[2,2]); a2se < sqrt(NWvcov[3,3]); ala2cov < NWvcov[2,3] delta < 1a2; beta < al/(1a2) # Conduct linear hypothsis test on exponential discounting q < a2al1 qse < sqrt(alseA2+a2seA2+2*ala2cov) exptest < (q/qse)A2 deltase < a2se print("Value of Exponential Factor"); print(cbind(delta,deltase)) print("Exponential Discounting Ftest"); print(exptest) # This is a twosided Ftest with dof=1,203 and alpha=0.05 # Null hypothesis is exponential discounting # Critical value is 3.84 # Conduct nonlinear hypothesis test on hyperbolic discounting gal < 1/(1a2); ga2 < al/((1a2)A2) betase < sqrt((galA2)*(alseA2)+(ga2A2)*(a2seA2)+2*(gal)*(ga2)*(ala2cov)) hyptest < ((beta1)/betase)A2 print("Value of QuasiHyperbolic Factor"); print(cbind(beta,betase)) print(" QuasiHyperbolic Discounting Ftest"); print(hyptest) # This is onesided Ftest with dof=1,203 and alpha=0.05 # Null hypothesis is no hyperbolic discounting # Critical value is 3.84 # Define IV matrix and scale z < as.matrix(cbind(constant,r,v,ct,de,me,md,ma,nh,nj,ny,pa,vt,rA2,lagR,lagV)) k < ncol(z) # Define initial weighting matrix w < nrow(z)*solve(t(z)%*%z) # Define GMM function fr < function(b){ (1/n)*t(yx%*%b)%*%z%*%w%*%((1/n)*t(z)%*%(yx%*%b)) } # Define gradient gfr < function(b){ 2*t(yx%*%b)%*%z%*%w%*%t(z)%*%x } # Conduct first step GMM gmm < optim(c(0,0,0,0,0,0,0,0,0,0,0,0,0),fr,gfr) sse < as.numeric(t(yx%*%gmm$par)%*%(yx%*%gmm$par)/nrow(y)) shat < (sse/nrow(z))*(t(z)%*%z) mhat < nrow(z)*solve((t(x)%*%z)%*%w%*%(t(z)%*%x))%*%(t(x)%*%z)%*%w vhat < mhat%*%shat%*%t(mhat) print(cbind(gmm$par,sqrt(diag(vhat)/nrow(x)),gmm$par/sqrt(diag(vhat)/nrow(x)), 1pnorm(abs(gmm$par/sqrt(diag(vhat)/nrow(x)))))) # Conduct second step GMM w < solve(shat) gmm < optim(c(0,0,0,0,0,0,0,0,0,0,0,0,0),fr,gfr) sse < as.numeric(t(yx%*%gmm$par)%*%(yx%*%gmm$par)/nrow(y)) shat < (sse/nrow(z))*(t(z)%*%z) mhat < nrow(z)*solve((t(x)%*%z)%*%w%*%(t(z)%*%x))%*%(t(x)%*%z)%*%w vhat < mhat%*%shat%*%t(mhat) print(cbind(gmm$par,sqrt(diag(vhat)/nrow(x)),gmm$par/sqrt(diag(vhat)/nrow(x)), 1pnorm(abs(gmm$par/sqrt(diag(vhat)/nrow(x)))))) # Overidentifying restrictions / Specification test statistic (JTest) Jdof < ncol(z)ncol(x) sse < as.numeric(t(yx%*%gmm$par)%*%(yx%*%gmm$par)) shat < (sse)*(t(z)%*%z) print("Jtest Degrees of Freedom"); Jdof jt < (nA0.5)*t(yx%*%gmm$par)%*%z%*%solve(shat)%*%t(z)%*%(yx%*%gmm$par) print("Jt Test Statistic"); jt # Jt converges to ChiSquare with qp degrees of freedom # The degrees of freedom equal the number of overidentifying restrictions # The critical value with alpha=0.05 and qp=4 is 9.488 # Rejection indicates problems with the GMM estimator # Calculate Discount Parameters coeff < as.matrix(gmm$par) al < coeff[2,1]; a2 < coeff[3,1] alse < sqrt(vhat[2,2]/nrow(x)); a2se < sqrt(vhat[3,3]/nrow(x)); ala2cov < vhat[2,3]/nrow(x) delta < 1a2; beta < al/(1a2) # Conduct linear hypothsis test on exponential discounting q < a2al1 qse < sqrt(alseA2+a2seA2+2*ala2cov) exptest < (q/qse)A2 deltase < a2se print("Value of Exponential Factor"); print(cbind(delta,deltase)) print("Exponential Discounting Ftest"); print(exptest) # This is a twosided Ftest with dof=1,203 and alp=0.05 # Null hypothesis is exponential discounting # Critical value is 3.84 # Conduct nonlinear hypothesis test on hyperbolic discounting gal < 1/(1a2); ga2 < al/((1a2)A2) betase < sqrt((galA2)*(alseA2)+(ga2A2)*(a2seA2)+2*(gal)*(ga2)*(ala2cov)) hyptest < ((beta1)/betase)A2 print("Value of QuasiHyperbolic Factor"); print(cbind(beta,betase)) print(" QuasiHyperbolic Discounting Ftest"); print(hyptest) # This is onesided Ftest with dof=1,203 and alp=0.05 # Null hypothesis is no hyperbolic discounting # Critical value is 3.84 # END PROGRAM # TITLE: Pacific States Panel OLS & Linear GMM # RESET ALL WORK rm(list = ls)) # Call needed libraries library(lmtest); library(stats); library(tseries) library(car); library(sandwich); library(systemfit) # Download CPI Deflator dta0 < read.table("Deflators.dta") cpi < as.numeric(dta0[2:44,1]) lagcpi < as.numeric(dta0[3:44,1]) # Download data sets # Column Order: YCAORWAVR~IDV dta < read.table("PacificNom.dta") # Create lagged variables obs < nrow(dta) lagVca < as.matrix(dta[ :(431),5]) lagRca < as.matrix(dta[1:(431),6]) lagVor < as.matrix(dta[44:(861),5]) lagRor < as.matrix(dta[44:(861),6]) lagVwa < as.matrix(dta[87:(1291),5]) lagRwa < as.matrix(dta[87:(1291),6]) lagV < rbind(lagVca,lagVor,lagVwa) lagR < rbind(lagVca,lagVor,lagVwa) # Define dep var, indep var, & scale dvca < as.matrix(dta[2:43,8]) dvor < as.matrix(dta[45:86,8]) dvwa < as.matrix(dta[88:129,8]) dv < rbind(dvca,dvor,dvwa) n < nrow(dv) constant < as.matrix(seq(length=n,from=l,by=0)) vca < as.matrix(dta[2:43,5]) vor < as.matrix(dta[45:86,5]) vwa < as.matrix(dta[88:129,5]) v < rbind(vca,vor,vwa) rca < as.matrix(dta[2:43,6]) ror < as.matrix(dta[45:86,6]) rwa < as.matrix(dta[88:129,6]) r < rbind(rca,ror,rwa) one < as.matrix(seq(length=42,from= ,by=0)) zero < as.matrix(seq(length=42,from=0,by=0)) ca < rbind(one,zero,zero) or < rbind(zero,one,zero) wa < rbind(zero,zero,one) x < cbind(constant,r,v,or,wa) y < dv # OLS estimation and test results ols < lm(dv~r+v+or+wa) output < summary(ols); output NWvcov < as.matrix(NeweyWest(ols)) bptest(ols) # Null Hypothesis of BP test is homoscedasticity # Critical value is Chisquare with alp=.05, dof=2: 5.991 durbin.watson(ols,max.lag=2) # Null hypothsis is no autocorrelation (rho=0) # If d
# Critcal value is DW with k=2, n=41: dl=1.449, du=1.549 Box.test(dv, type = c("LjungBox")) # Null hypothesis is no autocorrelation (rho=0) # Critical value is Chisquare with alp=.05, dof=l: 3.841 adf.test(dv) # Null hypothesis is nonstationary (unit root) # Calculate Discount Parameters coeff < as.matrix(ols$coefficients) al < coeff[2,1]; a2 < coeff[3,1] alse < sqrt(NWvcov[2,2]); a2se < sqrt(NWvcov[3,3]); ala2cov < NWvcov[2,3] delta < 1a2; beta < al/(1a2) # Conduct linear hypothsis test on exponential discounting q < a2al1 qse < sqrt(alseA2+a2seA2+2*ala2cov) exptest < (q/qse)A2 deltase < a2se print("Value of Exponential Factor"); print(cbind(delta,deltase)) print("Exponential Discounting Ftest"); print(exptest) # This is a twosided Ftest with dof=l,121 and alpha=0.05 # Null hypothesis is exponential discounting # Critical value is 3.92 # Conduct nonlinear hypothesis test on hyperbolic discounting gal < 1/(1a2); ga2 < al/((1a2)A2) betase < sqrt((galA2)*(alseA2)+(ga2A2)*(a2seA2)+2*(gal)*(ga2)*(ala2cov)) hyptest < ((beta1)/betase)A2 print("Value of QuasiHyperbolic Factor"); print(cbind(beta,betase)) print(" QuasiHyperbolic Discounting Ftest"); print(hyptest) # This is onesided Ftest with dof=l,121 and alpha=0.05 # Null hypothesis is no hyperbolic discounting # Critical value is 3.92 # Define IV matrix and scale z < as.matrix(cbind(constant,v,r,or,wa,rA2,lagR)) k < ncol(z) # Define initial weighting matrix w < nrow(z)*solve(t(z)%*%z) # Define GMM function fr < function(b){ (1/n)*t(yx%*%b)%*%z%*%w%*%((1/n)*t(z)%*%(yx%*%b))} # Define gradient gfr < function(b){ 2*t(yx%*%b)%*%z%*%w%*%t(z)%*%x } # Conduct first step GMM gmm < optim(c(0,0,0,0,0),fr,gfr) sse < as.numeric(t(yx%*%gmm$par)%*%(yx%*%gmm$par)/nrow(y)) shat < (sse/nrow(z))*(t(z)%*%z) mhat < nrow(z)*solve((t(x)%*%z)%*%w%*%(t(z)%*%x))%*%(t(x)%*%z)%*%w vhat < mhat%*%shat%*%t(mhat) print(cbind(gmm$par,sqrt(diag(vhat)/nrow(x)),gmm$par/sqrt(diag(vhat)/nrow(x)), 1pnorm(abs(gmm$par/sqrt(diag(vhat)/nrow(x)))))) # Conduct second step GMM w < solve(shat) gmm < optim(c(0,0,0,0,0),fr,gfr) sse < as.numeric(t(yx%*%gmm$par)%*%(yx%*%gmm$par)/nrow(y)) shat < (sse/nrow(z))*(t(z)%*%z) mhat < nrow(z)*solve((t(x)%*%z)%*%w%*%(t(z)%*%x))%*%(t(x)%*%z)%*%w vhat < mhat%*%shat%*%t(mhat) print(cbind(gmm$par,sqrt(diag(vhat)/nrow(x)),gmm$par/sqrt(diag(vhat)/nrow(x)), 1pnorm(abs(gmm$par/sqrt(diag(vhat)/nrow(x)))))) # Overidentifying restrictions / Specification test statistic (JTest) Jdof < ncol(z)ncol(x) sse < as.numeric(t(yx%*%gmm$par)%*%(yx%*%gmm$par)) shat < (sse)*(t(z)%*%z) print("Jtest Degrees of Freedom"); Jdof jt < (nA0.5)*t(yx%*%gmm$par)%*%z%*%solve(shat)%*%t(z)%*%(yx%*%gmm$par) print("Jt Test Statistic"); jt # Jt converges to ChiSquare with qp degrees of freedom # The degrees of freedom equal the number of overidentifying restrictions # The critical value with alpha=0.05 and qp=4 is 9.488 # Rejection indicates problems with the GMM estimator # Calculate Discount Parameters coeff < as.matrix(gmm$par) al < coeff[2,1]; a2 < coeff[3,1] alse < sqrt(vhat[2,2]/nrow(x)); a2se < sqrt(vhat[3,3]/nrow(x)); ala2cov < vhat[2,3]/nrow(x) delta < 1a2; beta < al/(1a2) # Conduct linear hypothsis test on exponential discounting q < a2al1 qse < sqrt(alseA2+a2seA2+2*ala2cov) exptest < (q/qse)A2 deltase < a2se print("Value of Exponential Factor"); print(cbind(delta,deltase)) print("Exponential Discounting Ftest"); print(exptest) 200 # This is a twosided Ftest with dof=1,121 and alp=0.05 # Null hypothesis is exponential discounting # Critical value is 3.92 # Conduct nonlinear hypothesis test on hyperbolic discounting gal < 1/(la2); ga2 < al/((la2)A2) betase < sqrt((galA2)*(alseA2)+(ga2A2)*(a2seA2)+2*(gal)*(ga2)*(ala2cov)) hyptest < ((beta1)/betase)A2 print("Value of QuasiHyperbolic Factor"); print(cbind(beta,betase)) print(" QuasiHyperbolic Discounting Ftest"); print(hyptest) # This is onesided Ftest with dof=l,121 and alp=0.05 # Null hypothesis is no hyperbolic discounting # Critical value is 3.92 # END PROGRAM # TITLE: Southeast Panel OLS & Linear GMM # RESET ALL WORK rm(list = ls)) # Call needed libraries library(lmtest); library(stats); library(tseries) library(car); library(sandwich); library(systemfit) # Download CPI Deflator dta0 < read.table("Deflators.dta") cpi < as.numeric(dta0[2:44,1]) lagcpi < as.numeric(dta0[3:44,1]) # Download data sets # Column Order: YALFLGASCVRIDV dta < read.table("SoutheastNom.dta") # Create lagged variables obs < nrow(dta) lagVal < as.matrix(dta[l:(431),6]) lagRal < as.matrix(dta[ :(431),7]) lagVfl < as.matrix(dta[44:(861),6]) lagRfl < as.matrix(dta[44:(861),7]) lagVga < as.matrix(dta[87:(1291),6]) lagRga < as.matrix(dta[87:(1291),7]) lagVsc < as.matrix(dta[130:(1721),6]) lagRsc < as.matrix(dta[130:(1721),7]) lagV < rbind(lagVal,lagVfl,lagVga,lagVsc) lagR < rbind(lagRal,lagRfl,lagRga,lagRsc) # Define dep var, indep var, & scale dval < as.matrix(dta[2:43,9]) dvfl < as.matrix(dta[45:86,9]) dvga < as.matrix(dta[88:129,9]) dvsc < as.matrix(dta[131:172,9]) dv < rbind(dval,dvfl,dvga,dvsc) n < nrow(dv) constant < as.matrix(seq(length=n,from=l,by=0)) val < as.matrix(dta[2:43,6]) vfl < as.matrix(dta[45:86,6]) vga < as.matrix(dta[88:129,6]) vsc < as.matrix(dta[131:172,6]) v < rbind(val,vfl,vga,vsc) ral < as.matrix(dta[2:43,7]) rfl < as.matrix(dta[45:86,7]) rga < as.matrix(dta[88:129,7]) rsc < as.matrix(dta[131:172,7]) r < rbind(ral,rfl,rga,rsc) one < as.matrix(seq(length=42,from= ,by=0)) zero < as.matrix(seq(length=42,from=0,by=0)) al < rbind(one,zero,zero,zero) fl < rbind(zero,one,zero,zero) ga < rbind(zero,zero,one,zero) sc < rbind(zero,zero,zero,one) x < cbind(constant,r,v,al,ga,sc) y < dv # OLS estimation and test results ols < lm(dv~r+v+al+ga+sc) output < summary(ols); output NWvcov < as.matrix(NeweyWest(ols)) bptest(ols) # Null Hypothesis of BP test is homoscedasticity # Critical value is Chisquare with alp=.05, dof=2: 5.991 durbin.watson(ols,max.lag=2) # Null hypothsis is no autocorrelation (rho=0) # If d
# Critcal value is DW with k=2, n=41: dl=1.449, du=1.549 Box.test(dv, type = c("LjungBox")) # Null hypothesis is no autocorrelation (rho=0) # Critical value is Chisquare with alp=.05, dof=l: 3.841 adf.test(dv) # Null hypothesis is nonstationary (unit root) # Calculate Discount Parameters 202 coeff < as.matrix(ols$coefficients) al < coeff[2,l]; a2 < coeff[3,1] alse < sqrt(NWvcov[2,2]); a2se < sqrt(NWvcov[3,3]); ala2cov < NWvcov[2,3] delta < 1a2; beta < al/(1a2) # Conduct linear hypothsis test on exponential discounting q < a2al1 qse < sqrt(alseA2+a2seA2+2*ala2cov) exptest < (q/qse)A2 print("Value of Exponential Factor"); print(delta) print("Exponential Discounting Ftest"); print(exptest) # This is a twosided Ftest with dof=l,162 and alpha=0.05 # Null hypothesis is exponential discounting # Critical value is 3.84 # Conduct nonlinear hypothesis test on hyperbolic discounting gal < l/(la2); ga2 < al/((1a2)A2) betase < sqrt((galA2)*(alseA2)+(ga2A2)*(a2seA2)+2*(gal)*(ga2)*(ala2cov)) hyptest < ((beta1)/betase)A2 print("Value of QuasiHyperbolic Factor"); print(cbind(beta,betase)) print(" QuasiHyperbolic Discounting Ftest"); print(hyptest) # This is onesided Ftest with dof=l,162 and alpha=0.05 # Null hypothesis is no hyperbolic discounting # Critical value is 3.84 # Define IV matrix and scale z < as.matrix(cbind(constant,v,r,ga,al,sc,vA2,rA2,lagV,lagR)) k < ncol(z) # Define initial weighting matrix w < nrow(z)*solve(t(z)%*%z) # Define GMM function fr < function(b){ (1/n)*t(yx%*%b)%*%z%*%w%*%((1/n)*t(z)%*%(yx%*%b))} # Define gradient gfr < function(b){ 2*t(yx%*%b)%*%z%*%w%*%t(z)%*%x } # Conduct first step GMM gmm < optim(c(0,0,0,0,0,0),fr,gfr) sse < as.numeric(t(yx%*%gmm$par)%*%(yx%*%gmm$par)/nrow(y)) shat < (sse/nrow(z))*(t(z)%*%z) mhat < nrow(z)*solve((t(x)%*%z)%*%w%*%(t(z)%*%x))%*%(t(x)%*%z)%*%w vhat < mhat%*%shat%*%t(mhat) 203 print(cbind(gmm$par, sqrt(diag(vhat)/nrow(x)),gmm$par/sqrt(diag(vhat)/nrow(x)), 1 pnorm(abs(gmm$par/sqrt(diag(vhat)/nrow(x)))))) # Conduct second step GMM w < solve(shat) gmm < optim(c(0,0,0,0,0,0),fr,gfr) sse < as.numeric(t(yx%*%gmm$par)%*%(yx%*%gmm$par)/nrow(y)) shat < (sse/nrow(z))*(t(z)%*%z) mhat < nrow(z)*solve((t(x)%*%z)%*%w%*%(t(z)%*%x))%*%(t(x)%*%z)%*%w vhat < mhat%*%shat%*%t(mhat) print(cbind(gmm$par,sqrt(diag(vhat)/nrow(x)),gmm$par/sqrt(diag(vhat)/nrow(x)), 1pnorm(abs(gmm$par/sqrt(diag(vhat)/nrow(x)))))) # Overidentifying restrictions / Specification test statistic (JTest) Jdof < ncol(z)ncol(x) sse < as.numeric(t(yx%*%gmm$par)%*%(yx%*%gmm$par)) shat < (sse)*(t(z)%*%z) print("Jtest Degrees of Freedom"); Jdof jt < (nA0.5)*t(yx%*%gmm$par)%*%z%*%solve(shat)%*%t(z)%*%(yx%*%gmm$par) print("Jt Test Statistic"); jt # Jt converges to ChiSquare with qp degrees of freedom # The degrees of freedom equal the number of overidentifying restrictions # The critical value with alpha=0.05 and qp=4 is 9.488 # Rejection indicates problems with the GMM estimator # Calculate Discount Parameters coeff < as.matrix(gmm$par) al < coeff[2,1]; a2 < coeff[3,1] alse < sqrt(vhat[2,2]/nrow(x)); a2se < sqrt(vhat[3,3]/nrow(x)); ala2cov < vhat[2,3]/nrow(x) delta < 1a2; beta < al/(1a2) # Conduct linear hypothsis test on exponential discounting q < a2al1 qse < sqrt(alseA2+a2seA2+2*ala2cov) exptest < (q/qse)A2 deltase < a2se print("Value of Exponential Factor"); print(cbind(delta,deltase)) print("Exponential Discounting Ftest"); print(exptest) # This is a twosided Ftest with dof=1,162 and alp=0.05 # Null hypothesis is exponential discounting # Critical value is 3.84 # Conduct nonlinear hypothesis test on hyperbolic discounting gal < 1/(1a2); ga2 < al/((1a2)A2) betase < sqrt((galA2)*(alseA2)+(ga2A2)*(a2seA2)+2*(gal)*(ga2)*(ala2cov)) hyptest < ((beta1)/betase)A2 204 print("Value of QuasiHyperbolic Factor"); print(cbind(beta,betase)) print(" QuasiHyperbolic Discounting Ftest"); print(hyptest) # This is onesided Ftest with dof=l,162 and alp=0.05 # Null hypothesis is no hyperbolic discounting # Critical value is 3.84 # END PROGRAM 205 LIST OF REFERENCES Abel, A.B. 1983. 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Zax, J., and M. Skidmore. 1994. "Property Tax Rate Changes and Rates of Development." Journal of Urban Economics 36: 314332. 217 BIOGRAPHICAL SKETCH Matthew J. Salois finished his Master of Arts in applied economics at the University of Central Florida in December 2003. He then joined the Food and Resource Economics Department as a UF Presidential predoctoral fellow. He has also worked with faculty in the Department of Epidemiology and Health Policy Research. His current research topics include time preferences and land values, optimal timing of land conversion from rural to urban use, tax effects on alcohol consumption, household willingness to pay for child health, and the application of nonparametric methods to econometric problems. Matthew was born in Providence, Rhode Island and later moved to Florida where he began his college career. After graduating from with his Bachelor of Science degree in health services administration, he was admitted to graduate studies in the applied economics program, where he also minored in statistical computing. In fall 2004, he came to the University of Florida and began his doctoral degree in food and resource economics. His major fields of interest are applied microeconomic theory and applied econometrics with a research emphasis in production theory, environmental economics, urban and regional economics, risk and uncertainty, and health economics. PAGE 1 1 INTERTEMPORAL PREFERENCES AND TIMEINCONSISTENCY: THE CASE OF FARMLAND VALUES AND RURALURBAN LAND CONVERSION By MATTHEW JUDE SALOIS A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2008 PAGE 2 2 2008 Matthew Jude Salois PAGE 3 3 To my close friends, my dear family, and my lovely fiance. PAGE 4 4 ACKNOWLEDGMENTS The work presented here could not have been accom plished without th e creative talent and the sincere dedication of my dissertation committee. I have Professor Andrew Schmitz to thank for his brilliant mind and his worldly view of economics. My conversations with Dr. Schmitz were always fun, perceptive, and memorable. Professor Jonathan Hamilton provided much guidance both inside and outside the classroom. Meetings wi th Dr. Hamilton always imparted me with the kind of feedback that left me in aw e of his ability. I owe a great debt to Professor Timothy Taylor for his gracious mentorship and his honest view of economics. Never have I seen a more gifted teacher than Dr. Taylor, w hose singular wit kept me motivated. Finally, I express my most sincere gratit ude to my committee chair and me ntor, Professor Charles Moss. Working with Dr. Moss has been an inspiri ng, edifying, and immensely rewarding experience. Dr. Moss is a true renascence man of economics and to whom I owe the sum of my graduate career. I would also like to thank my parents, Michael and Carol, whose unwavering support of my past and present academic accomplishments have made my future a bright one. My sisters, Aimee and Carolyn, and brother, Jeremiah, provi ded a ready supply of needed distraction and back to reality moments. Finall y, and most importantly, I struggle to convey th e true pillar that my fiance, Alisa Beth, has been for me to lean on. Thank you for providing the will and the way towards the completion of this academic e ndeavor. May we prosper together in what matters most in this world, our lives together. Above all else I would like to thank the One w ho has made every past, present, and future endeavor possible. Any success is owed to Hi m and I am humbled by the chance He has given to me in this pursuit of knowledge. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ............................................................................................................... 4 LIST OF TABLES ...........................................................................................................................7 LIST OF FIGURES .........................................................................................................................8 LIST OF ABBREVIATIONS ......................................................................................................... .9 ABSTRACT ...................................................................................................................... .............10 CHAP TER 1 INTRODUCTION .................................................................................................................. 12 Overview and Purpose .......................................................................................................... ..12 Time Preferences and Economics ........................................................................................... 15 Discourse on Discounting .......................................................................................................18 Application to RuralUrban Development .............................................................................. 22 Motivation: The Importance of Discounting ................................................................... 23 Relevance: The TimeInconsistent Landowner ............................................................... 25 Theoretical Heuristic: Th e Developm ent Decision ......................................................... 27 Study Objectives .....................................................................................................................31 2 LITERATURE REVIEW .......................................................................................................36 Introduction .................................................................................................................. ...........36 Theory ........................................................................................................................ .............38 Land Values and Use .......................................................................................................39 A Simple Static Model .................................................................................................... 43 A Simple Dynamic Model ...............................................................................................45 Modeling the Land Development Decision ............................................................................ 48 The Capitalization Approach ...........................................................................................50 The discount rate ......................................................................................................53 Land rents and conversion costs ............................................................................... 55 Land taxes ................................................................................................................58 Market and information imperfections ..................................................................... 65 Urban growth ............................................................................................................68 Uncertainty ............................................................................................................... 70 The Real Options Approach ............................................................................................72 Introduction to real options theory ...........................................................................72 A brief account of real options ................................................................................. 74 Application of real optio ns to land developm ent ..................................................... 76 The Transactions Cost Approach .................................................................................... 84 PAGE 6 6 Land development a nd institu tions .......................................................................... 84 Models of land values wi th transaction costs ........................................................... 88 Empirical Models of Land Change ......................................................................................... 91 Capitalization Empirical Methods ................................................................................... 91 Option Value Empirical Methods .................................................................................. 100 Chapter Summary .................................................................................................................105 3 THEORY AND EMPIRICS ................................................................................................. 112 Theoretical Framework .........................................................................................................112 Econometric Procedure ......................................................................................................... 117 Instrument Selection and Identification ................................................................................ 121 Chapter Summary .................................................................................................................122 4 DATA AND RESULTS ....................................................................................................... 123 Data and Variable Description ..............................................................................................123 Estimation Results ............................................................................................................ ....125 Appalachian States ........................................................................................................126 Corn Belt States .............................................................................................................128 Delta States ....................................................................................................................130 Great Plain States .......................................................................................................... 131 Lake States .....................................................................................................................131 Mountain States ............................................................................................................. 132 Northeast States ............................................................................................................. 133 Pacific States .................................................................................................................134 Southeast States ............................................................................................................. 134 Chapter Summary .................................................................................................................135 5 CONCLUSION AND FUTURE WORK ............................................................................. 147 Comparisons and Limitations ............................................................................................... 147 Importance and Implications ................................................................................................ 159 Future Work ..........................................................................................................................165 Summary ....................................................................................................................... ........166 APPENDIX PROCEDURE FOR RCODE ................................................................................ 169 LIST OF REFERENCES .............................................................................................................206 BIOGRAPHICAL SKETCH .......................................................................................................218 PAGE 7 7 LIST OF TABLES Table page 21 Selected comparative static re sults f rom capitalization papers ........................................ 111 41 Appalachian states result..................................................................................................143 42 Corn Belt states result ......................................................................................................143 43 Delta states result ....................................................................................................... ......143 44 Great Plain states results ................................................................................................ ..144 45 Lake states results ............................................................................................................144 46 Mountain states result ......................................................................................................145 47 Northeast states result ................................................................................................... ...145 48 Pacific states result ...........................................................................................................146 49 Southeast states result ......................................................................................................146 410 Discount rates by region ..................................................................................................146 PAGE 8 8 LIST OF FIGURES Figure page 11 Exponential discounting ................................................................................................... ..34 12 Hyperbolic discounting .................................................................................................... ..34 13 Quasihyperbolic discounting ............................................................................................ 35 14 Comparison of discount factors ......................................................................................... 35 21 Land allocation and bidrent model ................................................................................. 108 22 Optimal land allocation ................................................................................................... .108 23 Optimal conversion time .................................................................................................. 109 24 Change in value of agricu ltural land awaiting conversion ............................................... 109 25 Timing of conversion decision .........................................................................................110 26 Call option payoff ........................................................................................................ ....110 41 Change in farmland values for the Appalachian states .................................................... 138 42 Change in farmland values for the Corn Belt states ........................................................ 138 43 Change in farmland values for the Delta states ............................................................... 139 44 Change in farmland values for the Great Plain states ...................................................... 139 45 Change in farmland values for the Lake states ................................................................ 140 46 Change in farmland values for the Mountain states ......................................................... 140 48 Change in farmland values for the Pacific states ............................................................. 141 49 Change in farmland values for the Southeast states ......................................................... 142 PAGE 9 9 LIST OF ABBREVIATIONS BP BreuschPagan BL BoxLjung CBD Central business district CES Constant elasticity of substitution CPI Consumer price index CRP Conservation Reserve Program CRRA Coefficient of relative risk aversion DVT Development value tax ERS Economic Research Service GIS Graphical information system GMM Generalized method of moments IRR Internal rate of return LVT Land value tax NLSY National Longitudinal Survey of Youth NPV Net present value NRI National Resources Inventory PCE Personal consumption expenditure PPI Producer price index PV Present value PDV Present discounted value USDA United States Department of Agriculture PAGE 10 10 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy INTERTEMPORAL PREFERENCES AND TIMEINCONSISTENCY: THE CASE OF FARMLAND VALUES AND RURALURBAN LAND CONVERSION By Matthew Jude Salois August 2008 Chair: Charles B. Moss Major: Food and Resource Economics Many studies have rejected the standard pres ent value model under rational expectations as a viable model for explaining farmland values in both domestic and international data. Current models of farmland values inadequately explain why land prices rise a nd fall faster than land rents, particularly in th e shortrun. Previous inquiries into the nature of farmland values assume a timeconsistent discount factor and do not se riously investigate the role of intertemporal preferences. This dissertation introduces timeinconsistent pref erences into a model of farmland values by including a quasihyper bolic discount parameter in the asset equation of nine agricultural regions in the United States. Strong evidence is found in favor of quasihype rbolic discounting as a more appropriate way of describing the discount structure in models of farmland values. By introducing quasihyperbolic discounting to the presentvalue model of farmland values, the discount factor can be broken down into two timespecific rates, the shortrun discount rate and the longrun discount rate. Using a timeinconsistent discount fact or, like a quasihyperbol ic one, allows time preferences in the shortrun to be different than in the longrun Thus, the model offers an explanation as to why shortrun and longrun land values do not follow the same path. PAGE 11 11 The theoretical formulation in this paper gene ralizes time preferences in the asset equation and allows values of the exponential and quasihype rbolic discount parameters to be obtained. A hypothesis test is constructed, permitting for a dir ect test on the discount parameters. Using the linear panel Generalized Method of Moments estimator, problems of heteroskedasticity and serial correlation are reduced. The results of th e hypothesis tests imply a formal rejection that shortrun discount rates are equal to longrun disc ount rates, and that the shortrun discount rates are substantially larger than the longrun discount rates, a new re sult in the literature on farmland values. The results presented in this dissertation ar e not only important because of their unique insights into intertemporal preferences, but also because of the impli cations generated by the results. First, the results imply that landuse de cisions may be made with a greater interest in shortrun gains than longrun retu rns. Second, the results offer an appealing explanation as to why land values and rents do not follow the same pa th in the shortrun. Fi nally, the results also suggest that farmland serves as a golden egg to farmers, landowners, or developers who may demand commitment devices to constrain themselv es from hasty land investment decisions. The results have importance policy and extens ion relevance. Knowing that landuse decisions may be dominated by shortrun thin king, extension effort s should address this tendency to insure that future c onsequences of present choices ar e fully considered. The results also lend support to policy instru ments that act as commitment devices to constrain landowners to their present choices. Future work should focus on the potential consequences of timeinconsistency in other common agricultural economics models such as ruralurban land conversion and food demand, as well as extending th e results in this dissertation to account for risk, inflation, and adaptive expectations. PAGE 12 12 CHAPTER 1 INTRODUCTION Overview and Purpose The capitalization appro ach or net present va lue model has dominated the literature on farmland values and the general literature on as set pricing as a whole. The method generally states that the value of farmland is determined by the discounted expected future return to farmland (Melichar 1979; Alston 198 6; Burt 1986; Featherstone and Baker, 1987). However, a number of empirical issues have arisen from the use of the present value technique, particularly in the arena of farmland pricing. A burgeoni ng literature has app eared criticizing the capitalization technique for ove rsimplifying the market fundamental process, leading to empirical rejection of the present value mode l for farmland (Falk 1991; Clark, Fulton, and Scott 1993; Lloyd 1994). One serious flaw in the present value model is the inability to explain why land prices rise and fall faster than land rents, particularly in the shortrun (Schmitz 1995; Falk and Lee 1998). These socalled boom/bust cycles represent the te ndency of markets to overvalue land in periods of prosperity while undervaluing land in period s of relative decline (Schmitz and Moss 1996). As pointed out by Featherstone and Moss (2003), any sustained time period of overvaluation or undervaluation is inconsistent with a rational farmland market. Not only is the presence of boom/bust cycles well documented within the land market, but the causes for these cycles has been the topic of considerable debate in the literature (Lavin and Zorn 2001). Recent studies on models of land values have a ttributed the failure of the standard present value method to a variety of causes including th e presence of transaction costs (Just and Miranowski 1993; Chavas and Thomas 1999; de Fontnouvelle and Lence,2002), a timevarying risk premium (Hanson and Myers 1995), fa ds and overreaction (Falk and Lee 1998), or PAGE 13 13 inadequate econometric methods (Gutierrez, Westerlund, and Erickson 2007). However, few studies examine the role of time preferences within the context of agricultural land values or have seriously considered the shape and form of the discount factor in pa rticular. The literature to date has overwhelmingly relied on the a priori assumption that individuals, such as farmers and landowners, are time consistent and are described by standard exponential discounting, which implies a constant rate of discount. While such an assumption invokes intertemporal consistency in preferences, this rationale should be called into question. Even Paul Samuelson (1937, p. 156), in his seminal work on the discounted utility model, was keen to state that the assumption of constant discount ing was purely arbitrary and is in the nature of an hypothesis, subject to refutation by observable facts Substantial empirical evidence in experimental, behavioral, psychological, and fi nancial economics has shown that individuals tend to be timeinconsistent (Thaler 1981; Be nzion, Rapoport, and Yagil 1989; Benzion, Shachmurove, and Yagil 2004) and that time preferences are be tter modeled by quasihype rbolic discounting (Loewenstein and Prelec1992; Eisenhauer and Ventura 2006). This dissertation introduces intertemporal inc onsistency into a model of land values and tests for the presence of quasihyperbolic disc ounting in the farmland asset equation. Unlike exponential discounting, which implie s that individuals apply the sa me constant rate of discount each year, quasihyperbolic discounting implies a nonconstant rate of discount that declines over time. Hence, discounting is heavier in earlier time periods with the discount rate falling across the time horizon. This implies that individua ls are more impatient when they make short run decisions than when they make longrun decisions. The introducti on of quasihyperbolic discounting allows the discount factor to be de composed into shortrun and longrun discount rates. Thus the model offers an explanation into the apparent disconnect between shortrun and PAGE 14 14 longrun land values and explai ns why significant shortrun devi ations from the discounted formulation may occur. This dissertation makes three important contributions. The first contribution is theoretical, employing a net present value model that genera lizes intertemporal preferences to allow for quasihyperbolic discounting. The literature often uses intricate and rigor ous game theoretic approaches to demonstrate the effect of tim einconsistent preferences on consumption and savings decisions. The approach in this disser tation focuses on a simplified modification of the asset investment equation to account for qua sihyperbolic discounting. This method of measuring the economic impact of time inc onsistent preferences by obtaining the quasihyperbolic discount parameter in a reducedfor m model is unique in the literature. The second contribution is empirical. Most studies that estimate the quasihyperbolic discount parameter use experimental data. Fu rthermore, many studies rely on calibration methods rather than estimation methods to obt ain the discount parameter (Laibson 1997, 1998; Angeletos et al. 2001). This disse rtation uses aggregate field leve l data and empirically estimates the discount parameter, joining the small but grow ing number of studies th at use field data to structurally estimate discount factors (Paserman 2004; Ahumada and Garegnani 2007; Fang and Silverman 2007; Laibson, Repetto, and Tobacman 2007). Finally, within the context of land values and farmland values in particular, the dissertation examines the importance of timeincons istent preferences in an important area of economics. The dissertation offers an explana tion for the observed inc onsistency in farmland markets between shortrun and longrun farmla nd values through presen ce of quasihyperbolic discounting. The results in the dissertation also yields insights into la nduse decisions and the tendency of landowners to make pr esent decisions for instant gain s at the sacrifice of future PAGE 15 15 returns. The shortrun discount rates obtained also provide quan titative support for the Golden Eggs hypothesis posited by Laibson (1997) and sugge st that landowners may desire commitment devices to help constrain their future selves. To the best knowledge available, this dissertation contributes to the literature on land values with the first theoretical and empirical study of the relationship between timeinconsiste nt preferences and land values. The remainder of this chapter is organized as follows. The second section provides a cursory review of the literatur e on intertemporal preferences and time inconsistency. The third section discusses the nature of hyperbolic disc ounting in more detail and provides a useful example. The fourth section introduces an important application of intertemporal preferences in the land values literature, the economic decision to develop rural land to ur ban use. The fifth and last section of this chapter reiterates the study object ives and provides a summary. Time Preferences and Economics A growing body of literature on the econom ics of intertemporal decisionmaking and time preferences has led to a signifi cant body of research on time in consistent preferences. For excellent reviews of the literature see Loewenstein and Prelec (1992), Laibson (1997), and Frederick, Loewenstein, and ODonoghue (2002). There are different ways to model timeinconsistent preferences with a nonconstant discount rate, but the most common method is through the use of a hyperbolic di scount factor. Hyperbolic disc ounting originally developed in the psychological litera ture and was used to model intertemporal discounting in experiments on pigeon behavior in Chung and Herrnstein (1961) and later app lied to discounting by people in Ainslie (1975). The financial, behavioral, and experimental economics literature has taken recent note of hyperbolic discounting, being applie d to models of savings, investment, economic growth, and addiction. PAGE 16 16 Unlike time consistent preferences, time inc onsistent preferences ar e characterized by two phenomena. The first are preference reversals and the second are intrapersonal games involving a tussle between desires to act patiently against desires for instantaneou s gratification (Laibson 1997). A preference reversal occurs when an indi viduals present self makes a decision and then the future self makes a different decision. A co mmon example is when someone sets their alarm clock before going to bed only to hit the snooze button once, or several times more, when they awake the next morning. The internal tussle come s into play when an individual is torn between the longrun desire to act patiently and the shortrun desire to be impatient. An example here would be someone who wants to have a better looking physique ne xt year but still indulges in fastfood that evening for dinner. Exponential discounting, re presenting timeconsistent prefer ences, discounts at a constant rate across each time period in th e horizon and so cannot account for either pref erence reversals or internal tussles. Hyperbolic discounting, an d the discrete form case of quasihyperbolic discounting, accounts for the timeinconsistencies discussed above by imparting a nonconstant discount rate, one that declines over time. Robert Strotz (1956) was the first to suggest that individuals may exhibit time inconsistency throu gh an intertemporal tussle where the future self may have different preferences from the present self. Strotz (1956) proposes a discount function based on the time distance of a future date from the present moment rather than just the future date as more descriptive of individual behavior. Over the years, several elegant but general models of timeinconsistent preference s have been proposed by Pollak (1968), Peleg and Yaari (1973), and Goldman (1979, 1980). The si mple time inconsistenc y introduced in the previous studies is based upon an individual valu ing wellbeing or utilit y more at the present time than at some future time, but values fu ture wellbeing or utility at the same rate. PAGE 17 17 The application of time inconsistent prefer ences to important ec onomic models reveals noteworthy conclusions. Barro (1999) adapts the neoclassical growth model of Ramsey to account for a variable rate of time preference. While the basi c properties of the neoclassical growth model are invariant under timeinconsistent preferences, the timevarying model yields important welfare implications depending on the ability and level of commitment from households to their future choices of cons umption. Gruber and Koszegi (2001) apply time inconsistent preferences to the rational addiction model of Becker and Murphy. While the prediction of their model was equivalent to the BeckerMurphy model, th at current consumption of an addictive substance is sensitive to future price expectations, the time inconsistent model reveals substantially different optimal level of government taxation. Laibson (1996, 1997), in his pioneering work on applying hyperbolic discoun ting to economic models of savings and investment, finds convincing eviden ce that rates of time preference are not constant. Ahumada and Garegnani (2007) recently f ound evidence of hyperbolic disc ounting in consumptionsavings decisions using aggregate consumer expenditure data in Argentina. Laibson, Repetto, and Tobacman (2007) use individual level data on credit card borrowing, consumption, income, and retirement savings and strongly reje ct the constant discount rate mode l in several specifications. Intertemporal preferences have been larg ely ignored in the ag ricultural economics literature, with few exceptions, ignoring the role time preferences play in farmer or landowner decisions. Flora (1966) represents one of the only a ttempts to determine how landowners time preferences affect the discount f actor. He finds investment deci sions in forest lands can be affected by time preferences. Analysis of survey data finds that some individuals place a higher time priority than the prevaili ng interest rate (Flora 1966). Barry, Lindon, and Nartea (1996) make a valuable contribution by establishing tim e attitude measures analogous to the Arrow PAGE 18 18 Pratt measures of risk attitudes of increasing, decreasing, and constant absolute time aversion. They point out that farmer time attitudes may change over time, directly affecting choices involving consumption, savings, a nd investment. Lence (2000) is one of the only studies known that estimates the farmers rate of time pr eference based on consumption data using Euler equations. He points out that the literature has largely ignored farmers intertemporal preferences despite the treme ndous benefit such knowledge w ould endow towards a greater understanding of how agricultural po licy can optimally allocate resources across time. To gain a better understanding of how constant and timecons istent discounting differs from nonconstant timeinconsistent discount, the following sect ion offers a primer on discount factors. Discourse on Discounting Tim e inconsistency is generated in the hyperbol ic discount factor by a rate of discount that falls as the discounted event is moved further aw ay in time. Events in the near future are discounted at a higher implicit disc ount rate than events in the distant future. The generalized hyperbolic discount factor developed in Loewenstein and Prelec (1992) is tt 1, where 0, The parameter determines how much the f unction departs from constant discounting. As explained by Luttmer and Mariotti (2003), is the parameter in the discount factor that controls how fast the rate of time preference changes between short run and long run values. The limiting case, as goes to zero, is the e xponential discount factor tt. The parameter represents a first peri od immediacy effect, discounting the initial period more heavily. As noted in Weitzman (2001), the hyperbolic di scount function generalizes to the wellknown gamma distribution, where the exponential func tion is simply a more special case. The instantaneous disc ount rate at time t is for the hyperbolic di scount factor is given by t tt 1'. The noteworthy property of hype rbolic discounting is that as t PAGE 19 19 increases, the instantaneous rate of discount d ecreases, meaning the hyperbolic discount rate is not constant, as in exponential di scounting, but rather is a func tion of time, declining over the time interval. The percent change in the hyperb olic discount factor depends on the time horizon, being steeper for the near future and flatter for the distant future, implyi ng a discount factor that declines at a faster rate in the short run than in the long run. An alternative type of discounting is quasihyperbolic discounting, originally proposed by Phelps and Pollak (1968) and developed fu rther by Laibson (1997, 1998, 2007). The quasihyperbolic discount factor is a discretevalue time function and maintains the declining property of generalized hyperbolic discounting. At th e same time, the discrete quasihyperbolic formulation keeps the analytical simplicity of the timeconsistent model by still incorporating certain qualitative aspects of e xponential discounting. The actual values of the discount function under a discrete setup are with discount values T,,,,12. To obtain a better understanding of how the form of the discount factor can aff ect economic analysis, and in particular the economic model of land development, consider a comparison of the following discount factors: tfactor discountlExponentia (11) t factor discount Hyperbolic1 (12) Tfactor discount hyperbolic Quasi ,,,,12 (13) The exponential discount f actor is graphed in Figur e 11 for two values of 0.959 for exponential factor 1 and 0.951 for exponential factor 2. As can be discerned from graphical comparison, exponential discount factor 2, with a value of 0.951, discounts more heavily than exponential factor 1, with a value of 0.959. In discrete time, the exponen tial discount factor is described by the functional form trt 1, where r is the rate of tim e preference, often PAGE 20 20 taken to be the prevailing interest rate. Intuitively, exponential factor 2, having a lower value than exponential factor 1, must have a higher rate of time pref erence. A higher rate of time preference means a greater preference for consum ption today than tomorrow. Hence, we can refer to as descriptor of individual impa tience. The greater the value of the more patient the individual. The generalized hyperbolic discount factor is graphed in Figure 12 for different values of and Hyperbolic factor 1 has 500,000 and 10900 while hyperbolic factor 2 has 000,250 and 10000 Upon comparison of the two hyperbol ic factors, it can be seen that hyperbolic factor 2 discounts the future more than hyperbolic factor 1. The initial jump in the value of hyperbolic factor 2 betw een the first and second time peri ods is notably greater than the jump in hyperbolic factor 1. The parameters and have counterbalancing effects on the value of the hyperbolic discount factor. Smaller values of result in a smaller jump between the initial periods while smaller values of result in a bigger jump. The parameter behaves much like the rate of time preference in the exponential discount factor, while determines how much the hyperbolic factor depart s from constant discounting. Finally, the quasihyperbolic disc ount factor, graphed in Figure 13 is presented for two different value of and Quasihyperbolic factor 1 has 91 .0 and 971.0 while quasihyperbolic factor 2 has 85 .0 and 964.0 Similar to the cases above, quasihyperbolic factor 2 discounts more heavily than quasihyperbolic factor 1, which has higher values for both and The parameter captures the essenc e of hyperbolic discounting and contains a first period immediacy effect in the individuals ti me preference. Changes in determine how much the quasihyperbolic factor will deviate form exponential discounting. Higher values of will result in a larger jump between the first two time periods. This jump in PAGE 21 21 the value of the discount factor is what creates dynamic time in consistent preferences. The parameter behaves similar to the ex ponential discount factor. When the quasihyperbolic factor becomes highly convex, disc ounting the near term much greater than more distant time period. Smaller values of will result in a more bowedsha ped discount factor implying a greater preference for immediate consumption. The quasihyperbolic discount function marries the qualitative propertie s of the exponential and generalized hyperbolic discount functions. Comparison of the three discount factors is depicted in Figure 14. Each of the discount functions have been calibrated so they approximately cross at 8 t. As can be seen from the figure, the generalized hyperbolic discount factor is the most convex of the three, with the exponential as the least convex up until time period 8. The greater convexity or more bowed shape of the hyperbolic factor im plies that the hyperbolic factor discounts more heavily than either the exponential or the quasihyperbolic up until time period 8. After that point, the exponential discount factor is th e most bowed of three implyi ng that the exponential factor discounts more heavily than either the hyperbolic or qua sihyperbolic factors. With the given calibration of parameters, dynamic timeinconsistent preferences are modeled with the near term discounted more heavily than the more distant term. Here, the near term is time periods up to period 8, with the distant term being time periods after 8. The figures illustrate the sensitivity of discounting to the se lection of parameter values with the choice of near term and distant term bei ng subjective in regards to the ec onomic situation being examined. One can easily imagine how the discount factor can affect a wide range of economic decisions, particularly those that are intertemporal in na ture. A very importan t and relevant economic decision is the choice to convert rura l land to an urban use. This is the topic of the next section. PAGE 22 22 Application to RuralUrban Development This section discusses the role of time preferences and in particular the impact of timeinconsistent preferences as m odeled by hyperbolic discounting in a model of land conversion. The act of ruralurban land conversion remains an important consideration since models of land development and the actual development decision ar e derivative of models of land values. From a logical standpoint, if one does not have an understanding of how intertemporal preferences affect land values, then one cannot begin to understand how time prefer ences affect the land development decision. This section attempts to support and promote such an understanding. Loss of agricultural land to developed uses has been a public policy issue for decades. For many years, economists have analyzed the structure of agricultural land pri ces and the timing of development in an effort to understand alte rnative uses to agriculture posed by land development. A specific aim of such research is to identify policies to prevent or discourage what may be considered suboptimal landuse changes. Numerous studies have examined the hedonic characteristics of the land itself as factors in land conversion (Taylor and Br ester 2005). However, many land development policies are directed at the developer or la ndowner, including tax structures aimed at either accelerating or decelerating the rate of land conversion. Desp ite policies directed at the individual, little account is taken of the individua l traits of the decisionmake r, particularly excluding any motivational or behavioral forces of the landowne r in the decision to conv ert land from rural to urban use. One behavioral as pect of the landowner is the time preference involved in the intertemporal decision to develop land. Of speci al interest is relaxing the assumption of time consistent preferences in intertemporal deci sionmaking to allow for dynamic inconsistency using nonconstant discounting. PAGE 23 23 Motivation: The Importance of Discounting The choice of the discount rate used in the model is a key variable in the determination of land values and development times. The rate of time preference, given by the discount rate, is one central component of intertemporal choi ce, and is an aspect overlooked in the land economics literature. Typically, th e landowner is assumed to be tim e consistent with a constant discount rate formulated in an exponential discount f actor. In theoretical models of optimal development times, the discount rate has been generally found to have negative effects on land values, for clear reasons, and tends to acceler ate the development process (Ellson and Roberts 1983; Capozza and Helsley 1989). The rather conclusive effects of the discount rate in the land values literature may however stem from th e arbitrary nature in the choice of the actual discount factor used. As far is known, all capita lization approaches to modeling land values and the optimal development time have used a single, constant discount rate in their discounted cash flow analyses, implying that farmers and landowne rs have time consistent preferences (Rose 1973; Markusen and Scheffman 1978, Capozza and Helsley 1989; Arnot t 2005). As noted earlier however, recent evidence suggests that individuals are time inconsistent. The random and potentially caprici ous nature of a constant disc ount rate in models of land development was recognized early by Shoup (1970). The potential error of a constant discount rate is compounded when uncertainty is brought into the analysis. The more distant the expected development time, the more uncertain landowners developers, or invest ors are regarding the value of land. If uncertainty is the case, th en the discount rate used in the present value formulation of undeveloped lands may be higher in time periods closer to the present (Shoup 1970). The actual discount rate would then fall as the conversion time approaches. A declining discount rate through time would im ply that the value of land apprec iates faster in time periods before development. This reasoning seems highl y probable, given the uncertain nature of the PAGE 24 24 land market and makes a particular case against a c onstant discount rate in favor of one that is a declining function of time. The internal tussle described by Strotz (1956) was recognized by Mill s (1981). Owners of undeveloped land exercise restraint in foregoing development opportunities with high immediate returns in favor of future options that ar e, in the final analysis, more remunerative, (Mills 1981, p.246). The use of a quasihyperbolic discount factor accounts for the propensity of landowners to have both a short run preference for instantaneous gratification and a long run preference to act patiently. Th e restraint alluded to by Mills (1981) could involve the use of commitment devices by farmers or landowners to pr event future selves fr om reversing a decision by the present self to not convert land in future time periods. Ex amples might involve the farmer enrolling in a cooperative agr eement or a resource conservation contract, such as the Conservation Reserve Program, which requires la ndowners to commit their land to some rural use for a contractual period of time (Albaek a nd Schultz 1998; Gulati and Vercammen 2006). Modeling the traditional development model with timeinconsistent preferences has important policy implications. Consider the fact that expected net present discounted values for long term projects are infamously hypersensitive to the discount rate being used in the evaluation. Projects involv ing land development are hence acutely susceptible to this hypersensitivity. Not only does this affect land us e policy, but the effects are important to policy makers who wish to maintain lands in either a developed or undevel oped capacity. Numerous studies have argued that the onl y effective deterrent to farmla nd conversion may be a policy of compensation to landowners for foregone developm ent rent (Lopez, Adelaja, and Andrews 1988; Plantinga, Lubowski, and Stavins 2002). These rents and the compensa tion required could be greatly misrepresented under an exponential discount f actor if preferences are timeinconsistent. PAGE 25 25 Relevance: The TimeInconsistent Landowner There are many reasons why a landowner might be characterized by time inconsistent preferences. First, landowner s like any other consumer or i nvestor might exhibit preference reversals. Consider the following example: a landowner may prefer to contract his land to a developer for $1.01 million in 21 years, rather than for $1 million in 20 years. But when the contract is brought forward in tim e, preferences exhibit a reversal, reflecting impatience. The same landowner prefers to contract his land to a developer at $1 million today rather than sell for $1.01 million next year. The primary assumption driv ing the reversal is the discount factor for a fixed time interval decreasing as the interval becomes more remote. A nonconstant or decreasing rate means the discount rate in the short run is much higher than discount rates in the long run. Time inconsistent preferences imply the per cent change in the discount factor depends on the time horizon, being steeper for the near future and flatter for the distant future. Second, a variety of institutional and governme nt policies, such as growth management policies, may create a degree of impa tience upon the landowner. A number of empirical studies have examined the effects of various developmen t pressures on the timing of agricultural land transition and also on land values (Bell a nd Irwin 2002; Irwin and Bockstael 2002; CarrionFlores and Irwin 2004; Cho and Ne wman 2005; Livanis et al. 2006). The establishing of priority funding areas, for example, is seen to effect development tim es. Priority funding areas are growth areas that are designate d by the county and receive financ ial support from the state for infrastructure development. The presence of growth areas could intr oduce a time inconsistency into a landowners preference for conversion by crea ting a sense of impatience. Expectations of land reform may also affect the time consistenc y of preferences, which may include zoning, taxation, development rights, and clear defini tions on the boundaries of urban growth. PAGE 26 26 Third, landowners characteristics have been found to influence their land use decisions. Barnard and Butcher (1989) hypothesi ze that landowner age, education, years of land ownership, income net of taxes, and expected increases in value and development time will have an affect on time preferences, thus influenci ng the landowners perceived net pr esent value of land and their decision to sell. The authors conc lude that not only are landowner characteristics significant but that they are more explanatory th an the characteristics of the land itself (i.e., parcel size, distance from CBD, soil quality, etc.) in determining parcel level land sales at the urban fringe. Factor analysis indicates that the expect ed time until development is the single most important factor for distinguishing between landowners selling versus holding the land w ith those expec ting a shorter wait being more likely to sell. While this result is not evidence of inconsistent preferences, it does suggest that the psychology of the landowner is a critical characteristic in the timing of land development as an intertemporal decision. Finally, since the land development decision is intertemporal in nature, knowledge regarding future returns to land in competing uses as well as conversion costs, may be imperfect and uncertain. As time unfolds, landowners may rethink and revi se the development decision. Acknowledging future revisions to the landowners current deci sion could imply one of two things. First, the current decision might permit flex ibility in the plan so th at future revisions can be made. For example, the landowner may decide to sell rural land parcels only if an escape clause is written in the contract that allows the landow ner to opt out with limited or no financial penalty. Second, the landowner may make the cu rrent decision to mainta in land in rural use under a commitment device to avoid tempting offers that may present themselves at a future date, thus playing a strategic intertemporal ga me with himself. For example, the landowner might enroll in a conservation reserve program fo r a specified number of years, disallowing the PAGE 27 27 conversion of land for a specified time period in exchange for some pecuniary payment. Since the landowner can continually update and revise th e optimal plan, time inconsistent preferences offer an attractive method towards analyzing la ndowner behavior and th e optimal development strategy across time. Theoretical Heuristic: The Development Decision This subsection aims to first provide a curs ory perspective on how th e use of a hyperbolic discount rate can affect the trad itional model of land development. A simple net present value model is compared between the time consistent case and the time inconsistent case using the model outlined in Irwin and Bockstael (2002). Th is approach affords an intuitive understanding of the land development model and the effect of discounting regimes on conversion times. The time discounted path of the conversion value of land is the central que stion the model addresses. Suppose the landowner is in an infiniteperiod decision model and owns a quantity of land, l, in some rural use at time t The landowner receives a rent on rural land, tlR ,, from a use such as agriculture, forestry, or open space. For simplicity, the time path of rural returns is assumed to be constant over time, rather than in creasing. The decision f acing the landowner is to either keep land in rural use or to sell to a de veloper for conversion to an urban use for a onetime return, ),( TlV, at time T equal to the sales price. The time path of the gross development return, ),(TlV, is assumed to be rising over time due to, for example, rising population and increasing income per capita concurrent with a diminishing supply of unconverted land (Irwin and Bockstael 2002). The landowner may decide to keep land in a rural use for development in future periods, when the future urban return might be higher than the curr ent urban return. There is a cost to the landowner of converting land at time T TlC ,, which could include administrative fees, permit expenses, institutional costs, or necessary in frastructure expenditures PAGE 28 28 (Irwin and Bockstael, 2002). The discount factor, is taken to be equivalent to )1(1 r for the time consistent case. The actual disc ount rate or rate of time preference is r usually assumed to be the prevailing interest rate in the real estate market. Proceeding to solve for the optimal developm ent time, the net returns to the landowner from converting land in time period T is given by: 0,,,i iVlTRlTiClT (14) Equation (14) subtracts the costs of converting land, TlC ,, and the present value of forgone returns from rural land use, tlR ,, from the one time return to development, ) ,(TlV at time T On the other hand, if land is kept in rural use in period T and conversion is postponed to 1 T the net returns from delaying c onversion discounted to time period T are given by: 1,,1,,1i iRlTVlT RlTiClT (15) The first term in Equation (15) is the return from rural land in period T The second term is the value of returns from conversion in period 1 T discounted to period T The last term is the costs of converting rural land in period 1 T discounted to period T In words, Equation (15) represents the expected dividends from rural use in time T plus the discounted net capital gains from conversion less foregone rura l returns and conversion costs. Two conditions are required in order for the optimal conversion time to occur in period T First, Equation (14) must be strictly greater than zero, meaning net retu rns from conversion in period T are positive. Second, Equation (14) must be weakly greater than Equation (15), meaning returns from converting rural land are greater in period T than in period 1 T In mathematical notation, these two conditions can be written as: PAGE 29 29 0,, 0i iVlTRTiClT (16) 1,1, ,,2, TlCTlVTlCTlRTlV (17) The discount factor can be substituted with )1(1 r in Equation (17), where r is initially assumed to be the interest rate. The standard discounting case yields the result that the landowner will only convert rural land in period T when: r TlCTlRTlV TlCTlRTlVTlCTlV ,,2, ,,2,1,1, (18) Equation (18) has the simple interpretation that conversion will occur when the rate of time preference, given by the interest rate, is greater than the percent change of development returns between the two periods. The discount rate, r is assumed to be constant across time, implying the discount factor declines w ith the length of the time horiz on involved, approaching zero asymptotically. If the assumption of constant di scounting holds, the individual is said to exhibit a timeconsistent discounting patte rn. More generally, however, r must not necessarily remain constant across periods and, in fact, there is no well founded reason to assume so other than for computational ease. Time inconsistency can be introduced into the model by with quasihyperbolic intertemporal preferences, implying a nonconsta nt rate of discount given by the discrete discount function 21,,,, By substituting quasihyperbolic preferences into Equation (17) the inequality becomes: ,2,, ,1,1 VlTRlTClTVlTClT (19) Simplifying gives: PAGE 30 30 ,1,1,2,, ,2,, VlTClTVlTRlTClT r VlTRlTClT (110) While there is a similarity between the timecons istent optimal conversion rule in Equation (18) and the timeinconsistent optimal conversion rule in Equation (110), note that the net returns received in period 1 T is discounted by the quasihyperbolic parameter reflecting an immediate impatience effect on the landow ner. The standard discount factor, and the implied rate of constant rate of discount, r, represent consistent time preferences. However, the presence of gives a measure of how intense immediat e rewards associated with conversion are valued to more distant rewards. Clearly, the optimal time to conversion will depend on not only the values of the parameters in the discount factor, but also on the time horizon. The form of Equation (110) expresses the sensitivity a landowner may have to time delay, an effect not expressed in the standard exponential discounting case. The fo rmulation also expresses quite simply how the landowner would be engaged in an intertemporal tussle as to the development decision. This tussle can be seen by the f act that the parameters and r appear on different sides of the inequality, in effect revealin g the counterbalancing force eac h parameter has on the discount formulation. The opposing individual tradeoffs between present and future conversion are represented by the different values the discount parameters may have. While the simple theoretical heuristic pres ented above is intuitive, more intricate theoretical methods are needed to solve the land development model with hyperbolic discounting in order to circumvent the time inconsistency problem, which violates traditional neoclassical assumptions and optimization theory. The main stream approach to solving economic problems involving time inconsistent preferences uses dy namic game theory to find sets of subgame PAGE 31 31 perfect equilibrium. The decisionmaker is view ed as a collection of su bindividuals, having different rates of time preference at distinct points in time, and so plays a series of multiplayer intrapersonal games. The game theoretic approach has been prevalent in the economic literature being used by Peleg and Yaari (1973), Laibson (1996, 1997), Barro (1999), Gruber and Koszegi (2001), and Harris and Laibson (2001, 2004). Mo re recently, Caplin and Leahy (2006) have proposed a recursive dynamic programming appro ach to obtain optimal strategies under time inconsistency in finite horizon problems. There approach could be generalized for the infinite horizon case to apply to the farmland values problem. Study Objectives With the insights, evidence, and questions discussed above, it is important to know whether the traditional model of land values and derivative models of ruralurban development can continue to be depended upon by economists. Despite the large body of research examining the net present value model of farmland values, few investigate the role of time preferences and none consider the presence of timeinconsistent preferences. The fundamental objective of the current study is to partially fill this void. This dissertation has three central components. First, the notion of timeinconsistency and hyperbolic discounting is related to a critical issue in the agricu ltural economics literature, rural to urban land conversion and development. M odels of land conversion th at describe the timing and intensity of development are derivative of models of land values. Therefore, an understanding of the role that time preferences play in a model of la nd values has a direct impact on the understanding of land development models. The current chapter pres ented the intuition of why one might suspect the farmer or landowner to be characterized by quasihyperbolic discounting and discussed the relevance and motivation in th e context of land conversion. PAGE 32 32 Second, the paper discusses and reviews the va st literature pertaining to the determination of land values and the development decision from an economic perspective. Chapter II presents an examination of the literature on land values and ruralurban land conversion to foster a greater understanding of traditional models of land valu es and development before timeinconsistencies are introduced. Such a review of the current stat e of knowledge is lacking in the literature. The theoretical models are described with an emphas is on the three most pr evalent approaches to modeling the development decision: capitalization, option value, and transaction cost methodologies. The most common econometric techniques used are discussed followed by an assessment of the potential future research. Third, the paper enters the ensuing debate rega rding the reliability of net present models to adequately explain farmland values, which is the focus of the remaining chapters. Chapter III presents the fundamental theoretical model of this paper and develops a structural model of land values with quasihyperbolic discounting and describes the econometric procedure used in the analysis. Many studies have rejected the standard present value model under rational expectations as a viable model for explai ning farmland values us ing both domestic and international data. Previous inquiries into the nature of land values assume a timeconsistent discount factor. This dissertation introduces intertemporal inconsistency into a model of land values by including a quasihype rbolic discount parameter in the asset equation. Significant evidence is found in favor of quasihyperbolic disc ounting in U.S. agriculture. The Generalized Method of Moments estimator is used to obtain estimates of the parameters, overcoming problems of heteroskedasticity and serial correl ation, and provides consistent and robust estimates. PAGE 33 33 The data used in the analysis and the estim ation results are discussed in Chapter IV. Aggregate panel data are used for agricultural asset values from 19602002 for nine agricultural regions of the United States including the Appalachian states, Corn Belt states, Delta states, Great Plain states, Lake states, Mo untain states, Northeast states, Pacific states, and the Southeast states. The results for each region are discussed distinctly and then comparisons are offered with other studies that have estimated structural models under quasihyperbolic discounting. Finally, Chapter V will conclude the main content and key findings. Intuition on why an agricultural asset, such as land, may be characte rized by timeinconsistency in general and quasihyperbolic discounting is discussed. Ideas for potential future research will be offered. PAGE 34 34 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10 4 8 1 2 1 6 2 0 2 4 2 8 3 2 3 6 4 0 4 4 4 8Time (years)Discount Value Exponential 1 Exponential 2 Figure 11. Exponential discounting 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10 4 8 12 1 6 20 2 4 2 8 3 2 3 6 40 4 4 48Time (years)Discount Value Hyperbolic 1 Hyperbolic 2 Figure 12. Hyperbolic discounting PAGE 35 35 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10 4 8 12 1 6 20 2 4 28 3 2 36 4 0 44 4 8Time (years)Discount Value QuasiHyperbolic 1 QuasiHyperbolic 2 Figure 13. Quasihyperbolic discounting 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10 4 8 1 2 16 2 0 24 2 8 32 3 6 40 4 4 48Time (Years)Value of Discount Function Exponential 1 Hyperbolic 1 QuasiHyperbolic 1 Figure 14. Comparison of discount factors PAGE 36 36 CHAPTER 2 LITERATURE REVIEW Introduction The economic question on the rate and timing of land conversion has been the subject of much analysis over the past forty years. Development of rural lands to urban uses, such as converting farmland to residential housing, has been a prominent issue in the economics literature. Once soaring and now falling residentia l land prices, rapidly gr owing urban areas, and the sprawling and discontinuous nature of land development around the urban fringe have all served to stimulate interest in the subject by academic researchers from various fields such as urban and regional economics, rural sociology, and urban pla nning. The timing of land development is also of key interest to many beyond the research arena such as local chambers of commerce, planning authorities, developers, an d landownersall of whom are stakeholders in any future project to convert land. The critical question the decisionmaker faces is : what is the optimal time to develop land? Not only does the decisionmaker have a choice as to the form of land use, such as agricultural farming or residential housing, but also a choice as to when the land should be put to such use. Economic doctrine says that this decision is made so that the use of land and timing of conversion maximize the net wealth of the landow ner. Since land use and land values are intertwined in the development decision, any mo del of land development must examine how land values and use affect the development decision. The approach in this chapter emphasizes the role of value and use in land development models. The purposes of investigating land developm ent models and obtaining optimal conversion rules are many. First, they provide a better underst anding of land values in terms of prices, rents, and also capital appreciation (Clarke and Reed 1988 ). This is important since, unlike many other PAGE 37 37 investments, land development tends to be irreversible. Furthermore, unlike many other resources and commodities, there is only a finite availability of land. Any decision to develop land, residential housing for example, depletes a fixed supply of land available for an alternative use. Second, governments often wish to create policies to encourage or discourage the conversion and development of land. Government policies take the form of property taxes, easement requirements, and conservation initiatives, just to name a few. Any government policy can have unintended consequences. For example, Titman (1985) shows that a policy restricting building height may actually result in a greater nu mber of buildings constr ucted. Accurate and efficient policy instruments cannot be designed without an understandin g of land development models and a thoughtful consideration of the timin g of development decisions. Third, a better knowledge of the conversion pro cess facilitates an improved und erstanding of seemingly random outcomes of conversion, such as discontinuous urban development patterns (also known as urban sprawl), heterogeneous spatial patterns in land use, and the creative de struction of capital structures. Still in need of investigation is the approp riate timing of land conversion and the optimal intensity of development. When is it optimal to convert land from a lower intensity rural use to a higher intensity urban use? What is the optimal in tensity of capital that should be applied to the land development project? How do government policies, such as property taxes and growth controls affect the optimal timi ng and intensity? What effects do changing discount rates and expected returns imply for land development? How does an uncertain world alter matters? Many factors affect the optimal timing rule, su ch as the addition of a property tax or interim rent received from temporary use of the land. Further, both uncertainty and imperfections in the land market will affect the intertemporal conversion decision. For example, PAGE 38 38 landowners tend to make decisions under imperfect information, especially when it comes to forecasting prices or interim rental rates. Ma ny studies have also relaxed the assumption of certainty (Ellson and Roberts 1983; Clarke and Reed 1988; Capozza and Li 2002; Schatzski 2003) allowing land rents, conversion costs, or di scount rates to be stochastic. Effects on rents, costs, discounting and other complications on the conversion decision are examined in more detail later. Temporal variation in decisions regarding conversion by owners of neighboring land is of particular concern. When returns from development appear to be high, some landowners defer development while other owners in similar situations decide to develop their land. This variation creates scattered and discontinuous development patt erns, especially along the ruralurban fringe, often referred to as urban sprawl. These questi ons are addressed in this review, presenting a survey of models and methods in the land conversi on literature spanning the past 40 years. The next section discusses the theory in land development models, addressing economics of land values and land use. The static and dynamic theory of land values and development are explained with an emphasis on how competing uses of land affect how and when land is developed. In the third section, the most pe rvasive approaches to modeling land values and development are discussed including the capitaliza tion approach, the option values approach, and the transactions cost approach. The fourth section reviews the empirical approaches and methodologies to modeling the land developmen t process. The final section concludes. Theory You do not have to be a farmer, or even an astute economist, to know that land is a productive input. Not only is land a factor of production for agriculture, but also for openspace and urban development. Landowner wealth has sh ifted over time as land prices, and in particular farmland prices, have fluctuated. As Schmitz (199 5) notes, farmland values have been subject to PAGE 39 39 boom/bust cycles, just as the stock market has, creating wide variations in the wealth of landowners. Variations in wealth are also caused by competing uses for land, particularly from agricultural and urban uses. The rapid rate of urban growth has led to soaring rates of land values and landowner wealth causi ng land initially used for farm ing, openspace, or other rural uses to be bid out of such use and converted to more capital intensive urban uses (Moss and Schmitz, 2003). Agriculture has been fading away from the landscape because returns from farming are unable to compensate for skyrocketi ng land prices. This has been the case for land located along the ruralurban fringe, which is es pecially sensitive to increased urbanization. This section presents a broad pe rspective on the econ omics of land values with a particular emphasis on land located along the ruralurban fri nge. Land is discussed in terms of utilization, with rural use referring to agricultural, forestry or open space use, while urban use referring to residential, industrial, or commercial use. An excellent a nd extensive examination can be found in Moss and Schmitz (2003) in regards to farmland. Indeed, there is no dearth of studies on land markets and the factors that affect land values. Rather than provide an interminable account on the plethora of economic models and empirical methods, a succinct di scussion on the most relevant and informative issues will be presente d. A basic theoretical treatment will be given first, followed by a comparative discussion of the most contemporary used models in the literature. Land Values and Use The key to understanding how land is valued means a proper understanding of the land market, and in particular the market for rural land. The following traits best characterize the rural land market: heterogeneity, localization, segmentation, high transaction costs, and imperfect information. First, land is not homo geneous; parcels of land differ in size, geographic location, landscape, quality, and other physical charac teristics. In this sense, a parcel of land is PAGE 40 40 very unique and cannot serve as a perfect substitu te for another. Second, the land market is highly localized and permanent since the location of land itself is fixedit cannot be moved or transported. In other words, land is a permanent asset. Further, the buyers and sellers of land tend to be limited to a geographic region, th ough distance buying of land has become more common. Also, land is durable and in some contex ts can be considered indestructible. Once land has been allocated to a particular use, such as ho using, it tends to remain in that use. As such, land development is often assumed to be irreversible. Third, land has a finite and fixed supply sinc e new land cannot be produced. In an extreme longrun view, total land supply is perfectly inelastic and thus is characterized by a vertical supply curve. However, particular uses of la nd may be described by a downward sloping supply curve. Also, countries, states, cities, and other geographic bodies can increase the supply of land by extending their borders. However, this m eans that some other geographic body must reduce their supply of land by an equivalent amount. Further, land can be used in varying levels of intensity thereby increasing the effective supply Fourth, land use tends to be mutually exclusive. For example, the allocation of land to agriculture precludes land from being used in residential housing or industry. While there are some exceptions to this (e.g. high rise buildings combing commercial and residential space), gene rally a unit of land cannot be allocated to multiple uses. Fifth, the land market is segmented since it is divided into many submarkets or market segments depending upon geography, land use, ownership, and property rights. Within each market segment, a parcel of land may have a different price. On the demand side, users of land require very specific land types a nd locations. For example, a farmer cannot use a parcel of land downtown or a land zoned for commercial use anymore than a retail store has use for land PAGE 41 41 hundreds of miles from a residential area. Sixth, the land market is also characterized by high transaction costs. A buyer and developer of land may face zoning restrictions, titling costs, and survey fees before development can be begin. Seventh, land is both a consumption good and an investment good (Kivell, 1993). An individual may purchase land solely for the utility of owning, po ssibly for open space or recreational purposes. Likewise an individual may purchase land for the intention of earning a return from uses such as farming or forestry or may hold onto the land for later development with the expectation of a higher rate of return Eighth, uncertainty and less than perfect information are inherent in land transactions. Future rents and returns from land use and development are not known with certainty an d both future demand and supply must be forecasted. Risk perceptions and preferences vary widely between landowners. Last, land development tends to be characterized by long delays. For example, the time between the purchase of vacant land and comple tion of high intensity urban construction can extend years. How land should be used, or the land allo cation problem, has been the subject of much research. Land is used is postulat ed to be based on its value or th e rent it accrues to the owner. David Ricardo first suggested that land would be allocated according to the soil quality. The more productive land would be allocated to farming and agri culture, while less quality land would be allocated to more urba n uses. Differences in fertility or productivity determine how much rent the land generates. The most productive lands are used first with increases in demand forcing less fertile lands into use. This places an advantage on owners of more fertile land over owners of less fertile land, equivalent to the valu e of the difference in the productivity of land. Location theories of land rent were develo ped by Johann von Thunen who believed that the distance of land from the core of a city, or th e central business district (CBD), would be the PAGE 42 42 primary incentive behind land allocation. The closer a unit of land is to the CBD the more valuable it becomes. The idea is best describe d as a bidrent model, which assumes different uses of land have unique bidrent curves. Each bidrent curve is unique since the slope depends on each parcels location from the center of the city. Proximal land units have lower transportation costs, resulting in a savings over more distant land units, resulting in a bid for locations. Both Ricardo and von Thunen were speaking in terms of agricultural uses for land. However, notable contributions by Alonso (1964) and Muth (1969) have extended these notions to account for urban uses. The bidrent theory of land uses and values can be described graphically in Figure 21. In a free market the land between the city center and location A will go to urban uses and more distant land will go to rural uses. Thus, point A represents the socially optimal allocation between the two uses. The bidrent model emphasizes the tradeoff between land uses based on the high rents of land in the central region and the costs of transportation incurred by more distant locations. The equilibrium described in Figure 21 is th e result of demand and supply forces in the land market at work. In a perfectly competitiv e market, the supply of land is given by the quantity of prospective individual parcels that compete against one another for potential users or tenants. On the demand side, potential users co mpete against each other for use of each parcel. In equilibrium, the optimal allo cation maximizes the total value of all land with each parcel being used at its highest and best use. In othe r words, each parcel is used in a manner that is most productive for that given ge ographic location. However, the value of land is based not only on returns from the land itself, but also on prod ucts, such as crops or urban dwellings, produced using land as an input. For example, higher pr ices for residential housin g or agricultural crops PAGE 43 43 will drive up the price for land. Since land offers both production and consumption of goods (e.g., forestry products) and services (e.g. recr eation) the demand for land is a derived demand.1 A common theoretical approach to modeling land valuation is based on the residual value of land, defining the market value by the net land resi dual income or rents. The difference between the value of what is produced on the land and the production costs yields the net land value, if the development of a hypothetical project is the highest and best use for a given parcel. A Simple Static Model The nature of the development decision can be illustrated using a simple static maximization problem where a landowner must deci de between two alternative uses of land. Suppose the landowner has a fixed amount of land, L and two competing uses. Further suppose that land is currently in agricu ltural use yielding a return of a per unit of land and using al land units. Alternatively, the landowner can convert hi s land to residential housing, yielding a per unit return of h, which uses hl units of land. Similar in spirit to the analysis in Bell, Boyle, and Rubin (2006), linear returns are assumed implying returns to a particular land use do not depend on the amount of land being used. The assumption of linearity allows net returns to agriculture to be defined as aAal and net returns to residential housing to be defined as hHhl The maximization problem is: a,Max ()&(0,)hll ahahVAH subjecttollLllL (21) The solution to the maximization problem given by Equation (21) is a co rner solution, implying land should be wholly dedicated to the use yielding the highest net return. If agriculture is more 1 Derived demand is when one good or service occurs as a result of demand for another. For example, the demand for housing leads to derived demand for land since land must be developed for housing to be consumed. PAGE 44 44 profitable than housing, meaning ah then all land should remain in agriculture, alL and none in housing, 0hl Likewise, if farming yields a lower return, then the land should be converted into residential housing. Diminishing returns, rather than linear retu rns to land use, may be a more realistic assumption, especially in farming where the returns per a unit of land depend on how many parcels currently in use. One can also easily see diminishing marginal returns to many urban projects. The classical example, given by Alfred Marshall and re iterated by Shoup (1970), involves the construction of a skyscraper. As the height of the building increases, certain production and structural elements of the bu ilding become necessary, such as additional equipment like specialized cranes and elevator s. Additional production and structural items cause the price per square foot of rental space to increase relative to future returns. Assuming declining marginal returns implies declining net returns with the number of units in use. Mathematically, diminishing returns to agriculture is given by 2 20,0a aAA l l and diminishing returns to housing is given by 2 20,0h hHH l l The conversion decision now becomes one of allocating the la nd between the two uses until the marginal returns are equal, or when: ahAH ll (22) Graphically, the solution represented by Equation (2 2) is depicted in Figure 22 and emphasizes the point of land being shifte d to the use providin g the highest margin al value product. The static models of both linear and diminishing returns are overly simplistic. The conversion decision in reality involves a number of factors, discussed in more detail in the next section. One factor affecting c onversion is the intertemporal nature of the development decision. PAGE 45 45 While the static model describes how will be land be developed, the model does not address when land will be developed. Other factors co mplicate the decision, for example property taxes can be introduced into the problem affecting the timing of the conversion decision. The issue of land taxation and land conversion has been ad dressed by many (Skouras 1978; Bentick 1979; Arnott and Lewis 1979; Anderson 1986). Costs al so affect the conversion decision since conversion requires an investment in capital de velopment. Most impor tantly, the environment surrounding the conversion decision is in realit y an uncertain environment. The returns from land are seldom known with certainty, which can have a large impact on development decisions. The issue of uncertainty has been at the very crux of recent research (Capozza and Helsley 1990; Capozza and Li 1994). Models of land conv ersion under uncertainty inevitably draw upon the investment under uncertainty literature and, in pa rticular, option pricing theory. Models of land conversion can only be examined in a seriou s way in an intertemporal dynamic context. A Simple Dynamic Model Dynamic theories of land development acknowledge the effect future economic expectations have on current land values and how land is allocated for certain uses. This inevitably results in a more focused discussion on the timing of land use change. In a dynamic model, the value of land for a particular use is represented by the discounted present value of expected returns from that use. The simple static model can be extended to account for the intertemporal nature of land allocation and conversion. Let () A t be the net return from a given unit of land in period t for land in preconverted or agricultural use. The term () A t can equal zero if the land is vaca nt. The net return per unit of land in period t for land in postconversion or residential housing use is (,) HtT, which depends PAGE 46 46 on the time of conversion given by T Conversion time is chosen to maximize the value of land, given by () VT : 0Max()()(,)T itit TVTAtedtHtTedt (23) The landowners discount rate is given by i, often taken to be the prevailing interest rate in the real estate market. The first term on the right hand side of Equati on (23) is the present discounted value (PDV) of agricultural returns from the start time to th e conversion time. The second term on the right hand side of Equation (2 3) is the PDV of housing returns from the time of conversion onward. The level of postconve rsion returns depends on both the time when housing returns occur, but also on when the conv ersion itself occurs. The first order condition with respect to the conversion time is given by: 0(,) ()(,)0itTHtT AtHtTedt T (24) The third term on the left hand side of Equation (24) is equivalent to the PDV of expected future changes in housing net returns. If the time of conversion is irreleva nt to the returns from housing development, then the third term on the le ft hand side vanishes and the conversion rule is equivalent to the static case. In other words, the optimal conversion time Toccurs when the net return to agricultural use is equal to the net return from housing use, mathematically represented by ()(,) AtHtT There are reasons why the third term on the left hand side of Equation (24) is nonzero, as explained by Bell, Boyle, and Rubin (2006). For example, costs may increase over time, delaying conversion until a later time, given by T The optimal rule with conversion costs is to convert when ()(,)() AtHtTiHT The accounting for cost re sults in a downward shift PAGE 47 47 of the housing net return function, illustrated in Figure 23. The case where an increase in housing returns, perhaps due to population growth, results in an upward shift, hastening conversion. The flow of returns can also be viewed in conjunction with the capitalized value of land in competing uses. In a dynamic model, the value of rural or unconverted land is determined by the expectation of future returns. Figure 24 de scribes Equation (24), representing the maximum value of () VT which occurs when A R equals H R The line A R represents the returns from land in agricultural use assumed to be constant.2 The return from housing or postconversion use is given by H R Assuming that conversion takes place at time T AV represents the capital value of land in current use (agriculture) and H V represents postconversion use (housing) value. The curve tV is the sum of values before and expected values after conve rsion. Although the development value of housing is greater than the value of agriculture after time t conversion occurs until the income returns from hous ing exceed that of farming at time T. Conversion occurs before time T because H R is expected to exceed A R prior to time T, implying that H V exceeds AV The expectation of returns explains w hy rural lands along the urban fringe have a greater potential value to urban uses over farmland uses (Neutze 1987). At time zero, the present value of returns from agriculture is given by the vertical distance 0AV in Figure 24. This value declines over time as the optimal date of c onversion approaches. As Goodchild and Munton (1985) and Neutze (1987) e xplain, conversion can be delayed but at an opportunity cost of the forfeited postconversion housing rent, H R 2 This implies that agricultural products are sold in large markets and is both a reasonable and simplifying assumption. Similar graphical analysis can be found in Goodchild and Munton (1985) and Neutze (1987). PAGE 48 48 The dynamic analysis underscores two key points. First, the optimal time to convert land is based on the premise that a decision to convert now is balanced agains t a possible decision to convert in the future at a higher return. Second, allocations of land to particular uses and land values depend on both the present economic conditions as well as future expectations. Dynamic models of the land development decision are the mainstream met hod to explain the forces that compel a landowner to convert land. The dyna mic model so far assumes a simple setting without any taxes, land use laws, or growth controls. Further, the model does not explain the capital intensity of development nor does the m odel address conversion costs. Perhaps most paramount to the development decision is the disc ount rate and the presence of uncertainty, since both have profound affects on the timing of conve rsion and intensity of development. More complicated dynamic models are examined later as well as the primary models of the development decision in the current literature : the capitalization appr oach, the option value approach, and the transaction cost approach. Modeling the Land Development Decision The theoretical foundation for the land develo pment and conversion literature did not receive rigorous economic analysis until the la te 1970s and early 1980s. Most currently used models, largely based on the seminal pape rs by Shoup (1970) and Arnott and Lewis (1979), describe the optimal timing problem in a Wickse llian framework on the op timal timing of wine maturation.3 Wicksellianbased partial equilibrium models utilize discounted cash flows to describe land values and the timing of conversion and focus primarily the effects land taxes have on land values and on the development process.4 The development decision is also influenced 3 Wicksell, K. (1934). Lectures on Political Economy, Volume I: General Theory (translated by E. Chassen). New York, NY: The MacMillen Company, pages 178183. 4 The partial equilibrium models also include Rose (1 973), Skouras (1978), Douglas (1980), Mills (1981b), Anderson (1986), Bentick and Pogue (1988), Anderson (1993), and Arnott (2005). PAGE 49 49 by conversion costs, the discount or interest rate, land rents from rural and urban uses, and assessed property value taxes and capital gains taxes. More recent theoretical models focus on a general equilib rium approach (Markusen and Scheffman 1978; Ellson and Roberts 1983; Capozza and Hels ley 1989; Kanemoto 1985; McMillen 1990), questioning th e conclusions drawn from a ceteris paribus partial equilibrium framework. For instance, the simplifying assu mption of a Marshallian land demand curve in Shoup (1970) implies monotonically growing demand over time. Therefore, the land value function () VT is strictly concave by assumption, pr esupposing results not guaranteed in a general equilibrium model. For example, the pr esence of a property tax will reduce the length of time an individual will hold land, thereby hast ening land conversion (Markusen and Scheffman 1978). A general equilibrium model has additional a dvantages over a partial equilibrium model, allowing the effects of uncertainty to enter in to the land development decision. Ellson and Roberts (1983) introduce uncertain ty in the context of a governments land zoning decision and find slower rates of conversion compared to the certainty case. As a disadvantage, the added complexity of a general equilibrium model leaves many comparative static results untenable, or in the very least ambiguous, reducing the model s predictive power. A brief summary of the main theoretical results from both partial a nd general equilibrium capitalization models is provided in Table 21. While so me results appear heterogeneous, disagreement on the effects of particular exogenous variables, such as taxa tion, is noticeable. Inevitably, introducing uncertainty to the land development process involv es a foray into the investment literature. Once uncertainty is accounted for in the land conversion process, the value of land and the decision to develop becomes an option value. The option valu e approach has gained recent attention in the PAGE 50 50 literature and shows the most promise for modeli ng the land development process. Transactions costs can have effects on the development decision and land values as well. Such approaches account for the pecuniary and nonpecuniary costs associated with convers ion. Nonpecuniary costs include legal fees for petitioning public officials to rezone the land, promises to dedicate portions of developable land for public use, ope nspace, or conservation programs, or costs associated with lobbying effort s and campaign contributions. The Capitalization Approach The capitalization approach forms the bulk of the literature on the land development decision and assumes land values are solely determ ined on the basis of expected future income. The simple static and dynamic land allocation theo ry outlined in the previous section does not make a clear distinction between the price and rent of the land. Evans (1983) distinguishes the price of land and the rent received by land. Land price is what is paid for the ownership of land, while land rent is paid by the tenant or occ upier for the prevailing usage of the land (Evans 1983). The relationship between th e two concepts of price and re nt is fundamentally based on individual expectations. The price paid for land will be a function of the discounted rents expected to be received in the future between land uses. Put simply, the value of land is a capitalized value of future rents. The capitalization model is equivalent to the dynamic model disc ussed earlier where current and expected future returns are discounte d according to some discount factor, usually the interest rate. The analysis is made simpler by the use of summation signs instead of integral signs. The most general model of the capitalization approach is given by: 0 i ttti iVER (25) PAGE 51 51 The value or price of land at time t is tV the discount factor is i and ttiER is the expectation at time t for returns (or rents) from land in rural use at time ti To obtain the optimal conversion decision, the model is extend ed to account for a competing use, such as recreation or urban development. Assuming co nversion is irreversible and extending Equation (25) to account for competing using, the model becomes: 01 T Cii tttittj n ij TVMaxER EU (26) The value of land consideri ng conversion potential is C tV T is the time land is converted to urban use, and ttjEU is the expectation at t for the rental rate from converted urban land at tj By comparing Equation (25) to Equati on (26), the value of land with conversion potential C tV exceeds the value dedicated to rural use tV The key assumptions implicit in Equation (26) include costless c onversion, a constant discount rate, risk neutrality, perfect information, and no uncertainty. The form ulation for conversi on potential given by C tV in Equation (26) is an oversimplif ying case that defines a point of reference for studying further models with varying levels complexity. The time path of land conversion for land on the urban fringe is often described graphically as an Sshaped curve, depicted by curve VV in Fi gure 25. The rate of increase in the value of land, or its conversion value, is d eclining, as depicted in Figure 25. A declining rate of increase in the conversion value implies the base value of land is always greater than the additional value that accrues to the land through tim e, a necessary condition for conversion to occur. Otherwise, the landowner or decisionmaker would delay conversion indefinite ly. The slope of the PP curve in Figure 25 is the discount rate. The PP curv e slope can also be modified to account for PAGE 52 52 property taxes, interim rents, and other effects. At the optimal time to develop, the VV curve is tangent to the PP curve. Before the optimal time is reached, there exists an incentive to hold land and delay conversion, possibly for a more capitalintensive and pr ofitable development or due to profits from interim rents (Fredland 1975). Delineating from simplifying assumptions imp lies, however, that not all land may be converted at the optimal time. The presence of imperfect markets or knowledge, transactions costs, uncertainty, barriers to entry, and varying rates of time preference described in the discount rate will cause a degree of heterogeneity and dispersion in the actual conversion times. Further, the lumpy nature of the VV curve, re presented by various rapid and slow increases, reflects a variety of factors affecting the de mand for land and, more specifically, for land development (Shoup 1970). For example, the re zoning of land or the construction of nearby highway systems or utility services may suddenly make the land more palatable for conversion. More private amenities, such as local shops or golf ranges, could have the same effect. Conversely, the conversion value may rise more slowly or even decline due to negative developments such as prisons, reclamation facili ties, or lowincome housing. The key insight is that the probability of conversion increases with rises in land value, population, and income. Dynamic models of land value serve as the basis for most of the theoretical and empirical work on land conversion and development, and va ry in complexity. Shoup (1970) explains that growing numbers of inhabitants and rising income s will escalate the demand for vacant land in a way that increasingly more intensive use of land will be needed. For example, population and wealth may increase requiring a capitalinten sive development project, such as highrise construction, to satisfy future demands. Shoup (1970) also explains the eff ect of irreversibility on most land developments. Converting land from lower uses, like agriculture or open space, to PAGE 53 53 higher uses, like residential housi ng, requires lasting changes. For example, zoning regulation, land subdivision, building construction, and change s in the geographic landscape can usually not be undone. The irreversibility of land developmen t makes converting back to a lower use costly, if not impossible. Once convers ion has occurred, land and capital are committed to the higher use (Shoup 1970). These facets of land demand and conversion im ply that the form of land development, especially in terms of density and intensity, are intertemporal in nature. As time progresses, and the decisionmaker waits to convert land to some later time, an intertemporal tradeoff occurs between low capitalintensive conversion and low land payments versus high capitalintensive conversion and high land payments (Shoup 1970). In other words, the decisionmaker will be in an internal tussle to either de lay or hasten conversion. The observation of a tradeoff makes keenly apparent the importance of certain behavioral aspects of the decisionmaker. In the context of land development, the decisionmaker can take on a number of roles: the developer, the investor, or the landowner. The intention of the decisionmaker is to conserve the land, holding onto it to ensure it is not converted too hastily to some lo wer use when a future date may call for more capital intensiv e development due to demand increases (Shoup 1970). Thus, expectations regarding future demand for more intensive uses of the land will play an important part in determining the op timal time of development. The discount rate Perhaps no other facet of the land development decision than the choice of the discount rate used in the model is so vital to the dete rmination of accurate land values and the optimal conversion time (Gunterman 1994). In Shoup (1970) the problem of the decisionmaker is to maximize the present discounted value of land. The simplest analysis begins by assuming no conversion costs and that land is vacant. The vacancy assumption implies that the land is not PAGE 54 54 being used for any temporary revenue generating activity before conversion. The present value of land can be described as: ()(,)()rTtPtTVTe (27) The present value of land at time t for a future conversion time T is (,) PtT The vacant lands conversion value at a future date T is () VT if the land is converted to its optimal use at date T The discount rate is r which assumes the role of opportunity costs to the owner of vacant land. Although not explicitly defined in the analysis by Shoup (1970), the discount rate is often taken to be equal to the interest rate in the real estate market. The decisionmaker then wishes to maximize the lands present value in regards to the conversion time: ()(,) [()]0rTtPtT VTe TT (28) Solving Equation (28) yields the first order condition that the optimal c onversion time occurs when the following is satisfied: () () VT r VT (29) The left hand side of Equation (29) is the rate of change in the convers ion value. The solution states land is converted when th e rate of increase in the conv ersion value of land equals the interest forgone on other possible investments, gi ven by the discount rate. The discount rate has been generally found to have negative effects on land values (for clear reasons) and tends to accelerate the land conversion process (Ellson an d Roberts 1983; Capozza and Helsley 1989). Arnott and Lewis (1979) found an indeterminate eff ect of the discount rate, assumed to be the interest rate. The ambiguous natu re of the discount ra te in the Arnott and Lewis (1979) analysis PAGE 55 55 stems from the simultaneous relationship between th e output elasticity of capital, the interest rate, and the expected rate of growth of rental rates. Still, critical questions regarding the impact of the disc ount rate have not been seriously examined. Will the optimal timing of conversion be different under a nonconstant discount rate? What are the property tax and other we lfare implications? These remain important questions yet to be addr essed by the literature. Land rents and conversion costs Often times land development along the ruralur ban fringe occurs not on vacant land, but on land that is currently engaged in some interim use, such as from agriculture or forestry. This case is more realistic than the case of vacan t land since undeveloped land is often held by farmers or investors who temporarily use the la nd until the optimal conversion time. Now the value of the land is increased by rents received from the temporary improvement. In terms of the model described by Shoup (1970), the optimal time fo r conversion with interim rent is given by: 0(,)(,) () ()()T rtrtFTTeeFiTtdi VT r VTVT (210) The second term on the right hand side of Equa tion (210) is the rent received from the interim use, F iT, over the whole time period. The effect of a positive interim use is to delay the conversion time since the capita lized present value of the land is higher when current rents are received. This model can be extended to include the future rents from an urban development at time T. Interim rental rate are recei ved until the terminal time. Due to the irreversible nature of most development projects, conversion is assu med to be permanent so that at the conversion time no more rent can be generated from the rural use. Rents received either before or after PAGE 56 56 conversion will increase land values. Generally, increases in the preconversion rent will slow the development process, while higher postconv ersion rents will speed up the conversion time. There are some notable exceptions, for exampl e in Markusen and Scheffman (1978) postconversion rents do not affect the development decision since their model only has twoperiods. In Ellson and Roberts (1983), Capozza and Helsley (1989), and Anderson (1986, 1993), increases in the urban use rent al rate have an ambiguous effect on the conversion time. In Anderson (1986, 1993), the indeterminacy arises from the model accounting for both preconversion and postconversion tax ra tes. The tax rate parameters enter into the discount rate, requiring the magnitudes of the parameters be know n in order to sign the postconversion rental rate effect. Additiona lly, Anderson (1986, 1993) assumes that the partial derivative of the urban rental function with respect to the conversion is zero, implyi ng that income received from a developed property is independent of the time of development. The assumption of independence of developed rental income from development ti me is not realistic. Both Ellson and Roberts (1983), and Capozza and Helsley (1989) utilize a general equilibrium approach to the land development decision, which requires simulation in order to determine the direction of partial derivatives. The effects of conversion costs are also quite similar across studies. With higher costs of capital, the price of land will increase to offs et the higher conversion expenditure. However, many studies do not explicitly account for conve rsion costs, usually assuming that capital expenditures are implicitly accounted for in the rental function for urban developments (Shoup 1970; Anderson 1986). Ellson and Robe rts (1983) find that conversion costs in the presence of uncertainty will increase the time land is rezoned for development and an urban infrastructure is put place, but reduces the duration between re zoning time and the point developed land is PAGE 57 57 actually consumed or converted. Another excep tion is Arnott and Lewis (1979) who take the vacant land conversion model from Shoup (1970) and develop it more rigorously by adding construction costs and residential density (i.e., capital intensity) to the problem. The key assumptions the authors make are: zero interim rents from temporary land use, zero fluctuations in capital costs and rental rates, zero property taxes, and zero bu ilding depreciation costs. Like Shoup (1970), Arnott and Lewis (1979) assume once conversion occurs the land remains in the developed use permanently and that perfect foresight exits. Unlike Shoup (1970), however, Arnott and Lewis (1979) treat rent s and conversion costs in term s of capital, distinctly. The conversion that Arnott and Lewis (1979) co nsider is a more specific one of vacant land being converted to residentia l housing. The objective of the landowner in their model is to maximize the difference between the present valu e of housing rents and the present value of conversion expenditures with resp ect to the time of conversion a nd the amount of capital needed: ,max(,)()()it it TK TLTKrtQKedtpKe (211) The present value of each la nd unit developed at time T with stock of capital K is given by (,) LTK. The output of residential housi ng on each land unit with capital K is given by () QK. The rental rate on a housing unit at time t is given by () rt The discount rate, equivalent to the interest rate, is given by i and the price per a capital unit is given by p .5 Optimal conversion time is delayed when the capital price p increases but is hastened when the current housing 5 Arnott and Lewis (1979) assume rental rates on housing units do not fluctuate, but rather are constant through time, and define as the housing rental rates expected rate of change. Solving the above partial equilibrium model, the authors conclude that land is developed optimally when: () () VT PTi That is, land is converted from a lower use to a higher use when the ratio of the value of land at time T, given by () VT if land is optimally developed, to the value of the property (including the land and buildings) equals the ratio of the growth rate of housing rental rates to the discount rate. PAGE 58 58 rental rate () rt at time 0 t increases (Arnott and Lewis 1979). The density of housing construction is not affected by higher capital outlays, as developers delay construction to compensate for the extra costs. The expected growth rate on the housing rental rate has an ambiguous effect on timing, but increases the density of development. Many of these results change when uncertainty is brought into the analysis. Clarke and Reed (1988) investigate the development decision under uncertainty and find that conversion is hastened when construction costs are expected to increase, however the conversion time is indeterminate when either rental rates or the di scount rate is expected to increase. When increased rental growth is expected or a decreas e in growth of costs or discounting is expected land values, housing density, and the ratio of th e value of undeveloped la nd to the value of developed land all increase (Clarke and Reed 1988). Simulations based on parameter estimates from Arnott and Lewis (1979) reveal that for the most part their model offers a reasonable descri ption of the land development process. However, both simulations on structural density and the discount rate based on results from Arnott and Lewis (1979) reveal that a much higher discount rate (i.e., 4.1%) would be needed to replicate results under higher levels of uncer tainty. Models that account for uncertainty are not adequately examined in a capitalization cont ext and require more sophisticat ed analysis, however. More complicated models involve the use of real options theory and the investment under uncertainty literaturethese topics will be addressed later. Land taxes One of the most investigated and controversial issues have been on the effects of a property tax and a capital gains tax. The property tax is often referred to as an ad valorem or land value tax (LVT) and is a ssessed based on the value of th e property site, ignoring any PAGE 59 59 improvements to land such as buildings and personal property.6 The capital gains tax, or development value tax (DVT) is levied based upon the actual developments and improvements made to the land. A significant debate ensued in the literature regardi ng whether or not a tax on land is neutral, meaning a land tax does not alter the allocation of land to di fferent uses. This is the view originally taken by David Ricardo. Ho wever, Henry George pu rported that a land tax would remove land from speculators, transitioni ng land from future uses to current uses, hastening the conversion process (Bentick and Pogue 1988). This se ction reviews the tax effects in the theoretical land development decision mode ls and how the effect is sensitive to the specific nature of each model. Shoups (1970) analysis indicate s that value of land is lower in the presence of a property tax but that the rate of increase in the conve rsion value is higher. The effect of an ad valorem property tax would simply imply a conversion rule of the form: () () VT VT (212) The left hand side of Equation (212) is the rate of change in the conversion value. On the right hand side, the tax is represented by and the discount rate is given by According to Equation (212), land is developed when the rate of increase in the c onversion value of land equals the interest forgone on other possib le investments given by the discount rate.7 In this simple case, an increase in eith er the discount rate or tax rate would hasten conversion, while earning a positive rate of return from an interim use would delay conversion. 6 The nineteenth century economist Henry George was amongst an influential proponent for this type land tax. 7 This assumes costless conversion and no interim rents received from the land in the Shoup model. If land is not vacant, as assumed above, but rather receives interim rent from some reve nue generating activity, like farming or forestry, then the rate of return earned in the interim is subt racted from the rate of increa se in the conversion value. Hence, an interim use of the land will delay the optimal date of conversion. PAGE 60 60 One of the earliest examinations of a DVT is attributed to Rose (1973), who uses a partial equilibrium Wickselliantype model to distinguish between two types of capital gains taxes. The first DVT is levied at the time the land is rezo ned from rural to urban use. The second DVT is levied when the land is actually converted. Rose (1973) finds that a levy at the development time yields an indeterminate deri vative of development time with respect to the tax, depending upon the magnitude of the price and tax paramete rs and the functional form of the rural and urban rent functions. A DVT imposed at the rez oning time is found not to have any effect on the conversion decision, thus suppor ting Ricardos view of a land tax as being neutral. Countering the result in Rose (1973), Skouras (1978) models the development time as an implicit function of the capital gains tax and finds the imposition of a DVT to accelerate the conversion time, owing to the fact that the presen t value of land falls as a result of the tax. Bentick (1979) supports the re sult in Skouras (19 78) and demonstrates how a tax on the capitalized value of land from urban conversion ca n change the preferences of the landowner in a manner that may alter decisions to convert land. Hence, a tax on land values (i.e., a property tax) is nonneutral in the sense that the tax distorts the allocation of land as pointed out by Mills (1981b) and more recently by Arnott (2005), suppor ting Georges view of a land tax being nonneutral. Markusen and Scheffman (1978) and Arnott and Lewis (1979) find contrasting results from the authors above. Using a twoperiod ge neral equilibrium mode l of land conversion, Markusen and Sheffman (1978) find that an ad valorem tax, or LVT, increases land demand in the first period and decreases it in the final period. However, the effect on the actual timing of land conversion is ambiguous and depends on the magnitude of price changes in the land market. In examining the imposition of a DVT, Markusen a nd Sheffman (1978) state that such a tax will PAGE 61 61 clearly cause land price appreciati on and therefore result in a higher rate of land conversion. However, this outcome is sensitive to the fact that the DVT is anticipated ahead of time.8 Arnott and Lewis (1979) model land values and the development decision with land value taxes assessed both before and af ter conversion. Their preconvers ion tax, equivalent to a LVT, has the same effect as Shoup (1970) and Bentic k (1979). However, their postconversion tax, similar to a DVT, results in a slower rate of c onversion. The contrasti ng results between the preconversion and postconversion tax in Arnott an d Lewis (1979) stems from their analysis accounting for conversion costs and the density of land development. The preconversion tax reduced density while the postconversion tax does not affect development density. Anderson (1986) extends the analysis by Bentic k (1979) and Mills (1981b) and generalizes the comparative static results of the property ta x in Arnott and Lewis (1979). He finds the postconversion tax has an ambiguous result on the conversion decision, with the dir ection of the effect depending on the magnitudes of the pr eand postconversion tax rates which are influenced by prevailing market conditions. Ho wever, Anderson (1986) assumes that the rent from developed property is independent of th e conversion time, a restrictive and unrealistic assumption which affects the comparative static results. None of the studies above di stinguish between anticipated and unanticipated taxation and all analyze tax effects in a partial equilibrium framework, with the exception of Markusen and Sheffman (1978)9. Kanemoto (1985) models the anticipator y effects of a capital gains tax in a general equilibrium framework. If unanticipated, a DVT will result in a higher price of land and a lower allocation of land to developed uses and hence a slower conversion process. The effect 8 This special case is pointed out by Kanemoto (1985) wh o examines the anticipatory ef fects of a land tax more carefully. 9 The theoretical models of Shoup (1970) and Arnott and Lewis (1979) appear to examine land taxation from an unanticipated approach. PAGE 62 62 on capital intensity is ambiguous, depending upon th e magnitude of the elasticity of substitution between land and capital. The analysis of anticipated capital gains taxes is similar to the analysis of Markusen and Sheffman (1978), w ith such a tax resulting in a hastening of land development. Kanemoto (1985) determines that a LVT has an ambiguous effect on the amount of land allocated to developed uses depending again on the elasticity of substitution. If the capitalland substitution elasticity is greater than the rent elasticity of demand for developed uses (i.e., residential housing), then an anticipated capital gains tax will result in a speedier conversion process (Kanemoto, 1985).10 McMillen (1990) extends the Kanemoto (1990) model to account for the duration as well as uncertainty in regards to the timing of a capital gains tax increase. Under conditions of certainty, the effect of an unanticipated and perm anent tax increase is the same as in Kanemoto (1985): a slower rate of conversion and smaller st eady state allocation of land to urban uses. If the unanticipated tax is viewed as being temporar y with certainty, the ra te of conversion also slows, but by more than a permanent tax increase of the same size. However, once the tax rate returns to its prior level, a period of more rapi d development ensuesthe shorter the interim tax period, the more the conversion rate slows. The end result in the steady state, the total amount of land allocated to urban uses, is the same be tween an unanticipated permanent tax and a temporary one. However, the time paths of development are much different. As noted by McMillen (1990), a temporary capi tal gains tax may be useful from a policymaking standpoint if the intended go al is a reduction of rural land conversion in the shortrun. If a permanent tax increase is anticipated, the e nd result may be completely opposite of what the policymaking authority intended. In this case, the amount of developed land in the steady state 10 The effect on the price of land from an anticipated DVT could not be determ ined from the model due to analytical intractability, however, in the steady state land prices will be lower as a result of such a tax. PAGE 63 63 is smaller, just as in the unanticipated cases however, the rate of c onversion once the tax is anticipated (usually before the actual levy of the ta x), results in a faster ra te of conversion due to landowners trying to avoid the tax. The earlier the tax rate is anticipated, the faster rate of conversion will last even longer. If the antici pated tax is viewed as being temporary with certainty, the conversion rate also increases, but at a slower pace than the anticipated permanent tax casethe shorter the dur ation of the tax, the smaller the rate of increase in land conversion. These results underscore the im portance of understanding the la nd development decision, as intended goals of government policies may cause the very outcome they are attempting to prevent, usually a loss of rural lands. Uncertain ty over the timing and duration of the above tax systems does not alter the effects described a bove, but does magnify them (McMillen 1990). The steady state is the same regardless of whether th e world is certain or not, however, the land development path of convergence will be different. Much of the prevailing disagreement on the e ffects of a land tax is attributable to the imprecise definition between a lands market va lue and development value, as explained by Douglas (1980). The market value depends on whether the land development decision is immediate or delayed for a higher net presen t value whereas the development value is independent of this effect. This distinction has a profound implication on how land is valued and on the development decision, especially when real options valuation is used. The use of real options theory allows for the valu e in delay to be explicitly ac counted. Also, mixed predictions are largely attributed to the individual intricaci es of the theoretical model. As explained by Bentick and Pogue (1988), there are three types of partial equilibrium mo dels prevalent in the literatureone assumes urban rents are constant the second does not account for redevelopment options, and the third accounts for both of these. Each type of partia l equilibrium model is PAGE 64 64 uniquely sensitive to how the tax enters the formulat ion and so each differs in their implications. For example, when urban rents do not grow, a cap ital gains tax changes the land values in rural and urban uses proportionately, so in this re gard the tax is neutral. However, when redevelopment is considered th e model imparts an option value in the development decision, possibly resulting in delay due to the value of waiting. Another reason for the heterogeneity of tax results is due to the fact th at partial equilibrium models ar e unable to carefully account for two distinctive features of taxation: the expectation of its arrival and the duration of its levy. The results from general equilibrium approaches sugg est that expectations regarding the timing of a tax will affect whether land conversion increase s or decreases, depending on whether the tax is anticipated or not. Expectations regarding the du ration of a tax will affect the magnitude of the change, with rates of conversion being slower or faster depending on the permanent or temporary nature of the tax. Despite the abundance of theoretical land develo pment models that examine the effects of taxation, only one study is known that empirically te sts theoretical prediction. Zax and Skidmore (1994) use data from Douglas Country, Colorado from 1986 to 1991 and examine how changes in a property tax affect the duration of time a pa rcel remains undeveloped. At the initial time period, parcels of land described in the data we re undeveloped, with subsequent tax changes and conversions recorded until 1991. Since the dependen t variable is the length of time until a parcel is development, the authors use a duration or hazard function to de termine the effect of tax on the conversion time. The general form is given by: 1,0,0,it ititj itPPDDjX (213) The itD term in Equation (213) is a dummy vari able representing whether or not parcel i has been development in year t and a vector of other c onversion factors for parcel i in year t is PAGE 65 65 given by it X The main results suggest that a rela tive modest property tax increase that is anticipated will not only increase the probability of development, but substantially increase the number of developed parcels.11 The results are generally consistent with the theory outlined by Kanemoto (1985) and McMillen (1 990) and underscore the potential ly significant impacts even a modest property tax may have. Market and information imperfections Aside from the use of a general equilibrium model, the Markusen and Scheffman (1978) study is also of interest since it is one of the first to examine the effects of a monopoly market on land development. Interestingly, they show that contrary to the standard view, a monopoly developer may not result in slow er conversion rates and higher pr ice appreciation rates. Rather the effect on development and prices will larg ely depend on the elasticity of demand (Markusen and Scheffman 1978). Further, th e property tax and capital gains tax effects outlined above for the competitive case are indete rminate for the monopoly case. In the competitive case, a landowner converts when the price of land equals the forgone opportunity cost. A monopolist, however, is willing to convert land up to the poin t where marginal revenue net of capital outlays is equal to the opportunity cost of the rural use (Markusen and Scheffman 1978). Generally, the monopoly price will exceed the competitive price, but this may not necessarily lead to changes in the rate of conversion since under a monopoly market, rate increases in land supply and land price go in opposite directions. The impact of a capital gains tax in a monopoly market will largely depend on de mand and supply responses and, in particular, on the magnitude of the elasticity of demand. One already noted limitations of the Markusen and Scheffman (1978) study involves th e use of only two time periods when, in reality, the land 11 Zax and Skidmore (1994) also exam ine the effects of an un anticipated tax increase, but find inconsistent significance across multiple equations and thus could not make an explicit conclusion on their effect. PAGE 66 66 development decision involves a greater number of periods in continuous time. Further, the model unrealistically assumes perfect information. The paper by Mills (1981a) represents an effort to relax the perfect information assumption and examines three types of informational settings of the decisionmaker: perfect foresight, zero foresight or myopia, and imperfect foresight. Generally, economic outcomes under cases of uncertainty tend be inefficient, as is the case with land development. The M ills (1981a) model is of the general equilibrium and perfect competition type, and envisions three types of decisionmakers. The key decision makers are the landowners who earn rent, u R on unconverted land and decide when to develop the parcel for housi ng. Landowners also decide the housing type, of which n types are assumed. Each type of housing has its own requirement for parcel size given by the vector 1(,...,)n The second types of decisionma kers are the housing construction firms who incur costs to building each type of housing, given by the vector 1(,...)nccc The firms supply the housing units at time t according to the vector 1()(),...,()n x txtxt. The residents, or socalled tenants, compose th e final type and demand housing according to the inverse demand function, (),,1,...i f xttin, and pay rent 1()(),...,()n R tRtRt. Under the assumption of perfect foresight, the landowner is able to forecast exactly both the supply of housing, () x t and the demand for housing, ()i f xt. Given a discount rate defined by r, the landowner will not develop a unit of land unless it generates a profit in excess of the present discounted value of unimproved land, given by u R r Therefore, the landowners maximization problem is: ,1 () max(),rt rt i ii ui it i i tVt efxRdec (214) PAGE 67 67 As noted by Mills (1981a), the landowner has a se t of infinite options given by the type of housing development and the timing of housing development: (,) it An option by the landowner will not be exercised if a more profitable one exists, implying not only that all options are equally profitable, but that the solution is a competitive equilibrium and is efficient (Mills, 1981a). Under perfect foresight, in either the land or housing ma rkets, prices represent profit expectations. These expectations are shared equally among all types of landowners meaning they each have the same base of information. Since all landowners base profit expectations on the same set of information, no individual landow ner is better off than another in terms of informational advantages. The conversion rule re sulting from the Mills (1981a) analysis states the rent received from building a type of hous ing will be maintained over any time horizon until the rental rate equals the opportuni ty cost. The opportunity cost is defined as the point where the revenues received from the housing market e qual the combined costs of construction and opportunity landcost (Mills, 1981a). In the case where landowners are myopic, deci sions are made using only the current stock of information and do not consider future expect ations. As a result, the myopic landowner will not heed the opportunity cost level and will continue converting land until the supply is exhausted. In this case, not all options are equally profitable, and so not only will some landowners gain more than others, but the equilib rium is not competitive and thus inefficient. The case of imperfect information lies between these two extremes. Landowners do not have perfect foresight regarding prices and rents, but neither do they have complete disregard for future outcomes. In this case, landowners ar e speculative investors, with some having better forecasts than others. Options are still not all equally profitable, like in the perfect information case, but the land will not be converted until the point that supply is exhausted, like in the PAGE 68 68 myopic case. Rather, landowners behave much like speculative investors, and the conversion rates are moderated between the two extreme cases of perfect information and myopia. One of the key insights about the Mills (1981a) paper is the fact that operating under a circumstance of uncertainty and of less than perfect in formation will reduce economic efficiency. Urban growth Urban growth models are based on dynamic c onstructs of spatial structure accounting for population increases and higher demands for ur ban developments. One of the earliest contributions is Anas (1978) who shows under circumstances of population growth that as the commuting distance to employment centers increase, the density of housing developments will also increase. Other studies in this spirit include Arnott ( 1980), Brueckner (1980), and Wheaton (1982). Capozza and Helsley (1989) model the la nd conversion process under conditions of a growing urban area. Like Arnott and Lewis (197 9), the authors assume perfect foresight and irreversibility. Next, the value of land to owne rs is defined as four components: rent received from undeveloped agricultural land, expected rent from developed urban land, the development cost, and the value associated with how accessible the land is viewed. This final assumption is largely based on the von Thunen concept of proxim ity in determining land values. The closer a unit of land is to the central business district ( CBD), the more valuable the land unit is due to greater transportation and commuting co sts for more distant locations. Models of this type are ofte n described as a monocentric urban area. In regards to the timing of conversion, the landowner chooses the optimal time, t to maximize the present value of undeveloped agricultural land: () ()()(,)(,)()1ar tr t t r t t tA PtzRzedeCe r (215) PAGE 69 69 Rent received from urban land uses is given by (,) R z where z is the boundary of the urban area which indexes the distance of a unit of land from the CBD (Capozza and Helsley 1989). Agricultural land rents are given by A is, C is the conversion cost for a unit of land, r is the discount rate, and t is the time of conversion. The firs torder condition solved using Leibnitz rule is of the following familiar form: (,) R tzArC (216) Conversion is optimal at the time period wh en the sums of the opportunity costs from capitalization and from agriculture equal the urba n use rental rate. A similar expression is obtained for the average price of developed land. Comparative sta tic results indicate that the average price of urban land rises with highe r agricultural land rents, conversion costs, commuting costs, the size of the city, and populat ion growth. The average price of urban land falls with the discount rate. The Capozza and Hels ley (1989) model explains the substantial gap between the value of agricultural rent and the price of land at the urban fringe, which may be explained by land rental growth ex pectations in the future. High growth rates of land rentals are due in part to the large degree of specul ation by land developers on undeveloped land. In the context of uncertainty, discussed in more detail below, Capozza and Li (1994) find generally higher growth ra tes raise the hurdle or reservation rent leading to delayed conversion times and a larger, denser, urban area. Howeve r, density levels are also dependent upon the degree of uncertainty and the elasti city of capital. Higher growth rates can have a negative effect on rents under conditions of high uncertainty and low capital elas ticity, leading to a less dense urban area. Therefore, the effect of growth rates on conversion timing is ambiguous overall. PAGE 70 70 Uncertainty Ellson and Roberts (1983) also investigate the effects of un certainty on the timing of land rezoning and infrastructure development. Us ing a dynamic model, the key decisionmakers analyzed are governments and planning agencies that make the conversion decision rezone parcels for urban uses. An aspect not consider ed in earlier studies, Ellson and Roberts (1983) model the planners problem as one of consumer surplus maximization as a way of finding the socially optimal rezoning time. Using a simulation of a translog utility f unction, they conclude uncertainty tends to slow the rate of conversion, which is sensitive to the discount rate and the elasticity of demand. Increases in the discount rate assumed to be the interest rate, tend to speed up the rate of conversion or rezoning (Ellson a nd Roberts 1983). As the time to development advances further into the future, the uncertainty on the conversion value of land becomes greater. This may imply the presence of a declining discount rate in the present value problem, with discount rates being greater in periods before development, d eclining as the conversion time approaches, as noted in the discus sion on the discount rate (Shoup 1970). One creative approach to modeling uncertainty has been through the use of stochastic processes. When returns are uncertain, they exhibit a type of random walk. Clarke and Reed (1988) describe the evolution of capital prices and rental rate s on housing units as stochastic differential equations following a geometric Brownian motion with a Weiner process drift.12 Like some earlier studies, their analysis does not allow for any interim uses of the land, but does maintain the assumption of irreversibility of de velopment. The optimal conversion rule obtained is familiar: develop land when the ratio of unit rentals to unit construction costs exceeds a critical barrier, otherwise do not, (Clarke and Reed 1988). This type of conversion rule would 12 The authors examine the implication of defining the stochastic equations based on an Ito versus Stratonovich solution but, for the purposes of this review, this difference is not critical. PAGE 71 71 lend itself to econometric estimation under a hur dle or hazard function (Capozza and Li 2001; Irwin and Bockstael 2002). Despite the use of complicated stochast ic differential calculus, their development rule is quite similar to others, a nd more specifically to the one obtained by Arnott and Lewis (1979). Simply put, conversion takes place when: expected land value1 sum of conversion costs (217) The value in Equation (217) is defined as a stochastic parameter in the equation for the market value of land (Clarke and Reed 1988). Th e effects obtained through comparative statics are revealing. Greater uncertainty in either construction costs or housing rentals results in a greater land value and an increase in density of construction. However, while increasing uncertainty in construction costs raises the so called critical barrier and hence, delays conversion, the effect of uncerta inty on rental rates is indete rminate (Clarke and Reed 1988). Similar to Mills (1981a), the Clar ke and Reed (1988) describe th e decision to develop as an option value. This makes sense, especially since the nature of conversion is assumed to be irreversible in the land development literature. With assumed irreversibility, the conversion problem is related to the work in financial economics on irreversible investment projects under uncertainty. The development decision is predominately m odeled in a stochastic uncertain framework in the literature (Capozza and Helsley 1990; Capozza and Li 1994, 2001, 2002; Majd and Pindyck 1987; Titman 1985). However, the capitali zation approach is no t able to accurately model land values and conversion timing in an option value contex t. Primarily, the failure of standard capitalization approaches is due to the incapability of the standard net present value calculations to explicitly account for the value in waiting to develop. This socalled option value PAGE 72 72 is absent in the standard models of S houp (1970), Arnott and Lewis (1979) and Capozza and Helsley (1989). The Real Options Approach The introduction of uncertain ty changes the analysis of the development process considerably. The study of investment under unc ertainty has spawned a whole body of literature culminating in the influential wo rk of Dixit and Pindyck (1994).13 The traditional investment rule in finance is to undertake a project when its net present value is positive. Likewise, if two projects are being considered and are mutually exclusive in the sense that both cannot be undertaken at the same time, then the project wi th the higher NPV is the optimal choice. Such basic rules form the foundation of the analysis in the semina l papers by Shoup (1970) and Arnott and Lewis (1979). However, all investment proj ects, especially those involving land conversion and development, come with an ability to dela y or postpone investment until some later time. This means that the project competes with itsel f through time, imparting a value in the option to postpone investment. The idea of an investment opportunity, and land conversion in particular, as an option has acquired recent interest. The groundbreaking work of Dixit and Pindyck (1994) in investment under uncertainty und erscores the value of waiting to invest. In reality, the land development decision, or any investment decision for that matter, is rarely a now or never decision since the individual can exercise the option to delay development or investment. Introduction to real options theory Although options theory has a long history, the first rigorous theoretical treatment can be found in the works of Black and Scholes (1973) and in Merton (1973) In finance, options can be categorized as either a call or a put option. A call option is the opport unity to buy or sell a 13 For an excellent review of the investment under uncertainty literature, see Pindyck (1991). PAGE 73 73 commodity or unit of stock at some future time at a specified price. Anot her type of option is a put option where the holder has the right to sell a stoc k or unit of commodity at a stated price. In this sense, options are contingent assets sin ce they only have a value contingent on certain outcomes in the economy. The exercise price of an option is usually refe rred to as the striking price, while the current value or quoted price of th e stock or commodity is referred to as the spot price. The date at which an option is exercised is also referred to as the terminal time, or the date of expiration or maturity. A call option has a positive value when the spot price exceeds the strike price, or is in the money. Put options are in the money when the strike price exceeds the spot price. Options are only exercised in this range. An important char acteristic of an option is that it will always have a positive or zero valu e, never negative. This is because the point at which the option is worthless the holder of the option will simply disr egard it, or abandon the option. When an option has zero value it is referred to as out of the money. The payoff function from a call option can be repr esented graphically in Figure 26. Two types of options in the finance literat ure are European options, which can only be exercised on the final date of expiration, and American options, which can be exercised at any point in time until the final data of expiration. Land development and conversion decisions are best represented by an American call option since the op tion to convert can be exercised at any time until the terminal date. The value of an opti on is made up of two basic parts: the intrinsic value and the time value. The intrinsic value of an option is the difference between the strike price option and the spot price. The time value represents the possibility that the option may increase in value over time due to volatility in th e stock or commodity price. At the terminal or expiration time of the option, the time value is zero, declining over time, at which point the option is equal to its intrinsic value. The value of an option can be obtained by different PAGE 74 74 methods. An often used numerical procedure for options pricing is presen ted in Cox, Ross, and Rubinstein (1979) and is based on a discrete binomial process. An alternative method is the wellknown BlackScholes method presented in Black and Scholes (1973), which utilizes stochastic drifts to model the option price. Va rious extensions of the current methodologies are present in the literatu re (Alvarez and Koskela 2006; Rodri go and Mamon 2006). An excellent discussion of option pricing theory and its applications can be found in Merton (1998). A brief account of real options Abel (1983) is one of the earlier studies on optimal i nvestment under uncertainty. Additional studies have examined the irreversibility of some investments and have shown this to impart an option value in the investment de cision (Bernanke 1983). McDonald and Siegel (1986) rigorously derive the value in the option to delay investment. Assuming investors are risk averse and hold a diverse portfolio, the author s examine the value in waiting to invest and develop rules on the optimal timing of investment fo r an irreversible project. Using the firm as the decisionmaking unit, the ke y component of the model is th e choice between two mutually exclusive projects with only one being able to be undertaken at any give n time. This mutual exclusiveness, concurrent with assumptions of uncertainty in project payoffs and investment costs, irreversibility of the investment, and risk averseness of the firm, impart a value in the firms choice to delay investment. This option value is higher under conditions of greater uncertainty due to the increased variance in possi ble values of project returns. The standard assumption when uncertainty is considered in inve stment returns is to model the payments from an investment according to a standard Brownian motion process. McDonald and Siegel (1986) also consider a form of the Poisson process which allows for the present value of future returns to take a discrete jump to a zero value. PAGE 75 75 Ingersoll and Ross (1992) generalize McDonald and Siegel (1986) and show that option values exist in nearly all investment projects rega rdless of whether or not there is uncertainty in the expected payoffs or cash flows as long as there is some uncertainty in the interest rate. In Ingersoll and Ross (1992), there is an optimal accep tance rate of interest at which point the project is undertaken. As the real interest rate moves away from th e optimal acceptance rate, greater postponement of investment occurs, causing the project value to decline (Ingersoll and Ross 1992). Depending on the length of delay and the average rate of discount, the cost of waiting is the forfeited present va lue of the project. Their resu lts underscore the flaw in using the traditional NPV approach to decide on invest ment projects which dictates investment in all projects with positive net present values. Uncertainty effects on land conversion decisions are even more important due to the long duration of most land devel opment projects. Projects which tend to have a longer timespan have more volat ile present values which make the investment option more valuable. The focus in the literature has often been on the effects of uncertainty on the timing of investment, however the effect on intensity of investment is investigated by BarIlan and Strange (1999). In particular, they look at how price unc ertainty affects the intensity and timing of investment when capital investment can be either lumpy or incremental. When investment is lumpy, the level of capital required for a project is decided at the moment the investment occurs. However, incremental investment involves units of capital that may be added over time. Some examples might include a rental car agency updating its fl eet of vehicles or the addition of books in a library. Lumpy investment like the construction of a building or a road way, best describes the type of investment in land conversion. U nder lumpy investment decisions when both timing and intensity of investment are considered, uncer tainty tends to increase the trigger price. A PAGE 76 76 higher trigger price indicates the value to delay is higher, postponing investment. However, once investment does occur it tends to occur with high er capital intensity when uncertainty is present. This stands in contrast to the incremental case when only intensity is a factor in the decision where greater uncertainty tends re duce the intensity of capital. Application of real options to land development One of the first studies to formally model land prices and development using an option value approach to investment decisions is Titm an (1985). Under conditions of uncertain future real estate prices, the option to delay construction on vacant land becomes valuable since future development may be more profitable than curren t development given current prices. When the landowner is assumed to be risk neutral, the opt ion value in the conversion decision will increase under conditions of greater uncertainty since the expected value of vacant land increases, resulting in current vacant la nd values to increase under uncer tainty (Titman 1985). Using a BlackScholes model of options va lues, several interesting compara tive static results are obtained from the Titman (1985) model. For example, if the interest rate rises, the value of vacant land will increase, resulting in a greater incentive to hold onto the vacant land for future construction. An increase in rental rates will have the oppos ite effect, decreasing the value of vacant land resulting in a greater attractiveness to initiate building construction. The results in Titman (1985) are interesting when compared to the studies of land conversion when certainty is assumed. For example, in Arnott and Lewis (19 79), an increase in the interest rate has an indeterminate effect on timing. The impact of uncertainty on land developm ent underscores the importance of a more robust understanding of land development models, particularly when government policies are issued with the objective of sh aping landowner behavior and land development. For example, Titman (1985) shows that under a building regulat ion stipulating the maximum allowable height PAGE 77 77 of a new building, the actual number of new buildi ngs could actually increase as a result. No doubt part of this effect is due to the fact that more buildings are needed since each new building must be necessarily smaller. However, this e ffect is also due in pa rt to the reduction of uncertainty in the optimal building size. A height restriction mitigates the uncertainty of future prices from the decision to build now or later, and since lower uncertainty means a lower option value from delaying construction, vacant land co uld be developed sooner than it would have if there was no height re striction (Titman 1985). Extending Capozza and Helsley (1989) to ac count for uncertainty, Capozza and Helsley (1990) models household income, land rents, an d prices as stochastic processes. The introduction of uncertainty is shown to impart an option value to the price of agricultural land. This has the effect of delaying th e time to conversion, with the option value falling as the urban size grows and the distance from the boundary of the fringe region increases. Clarke and Reed (1988) also find that uncertainty adds an opti on value to the conversion decision, but do not examine the effects on urban growth and city si ze. The introduction of uncertainty in Capozza and Helsley (1990) required a reformulation of the standard problem of the landowner maximizing the value of land. Thus, Capozza and Helsley (1990) form the landowners problem as a hitting time problem. A specific type of the stochastic optimal stopping time problem, a hitting time refers to the point, or first hit time, where an outcome is optimal. In this problem, the first hit time is defined as a reservation or hurdle rent level. Once land rents reach the reservation rent level conversion is optimal. This hurdl e rent is given by: rg RArC r (218) PAGE 78 78 Agricultural returns in E quation (218) are given by A conversion costs are given by C, and the discount rate is r, which have the same interpretation as in Capozza and Helsley (1989). Additional parameters include, and g which is the drift parameter in the Brownian motion process for household income. When compared to the reservation rent in the certain case given by Capozza and Helsley (1989), the uncertain case has a higher trigger level (Capozza and Helsley 1990). The last term in Equation (218) is defined as the irreversibility premium. Despite risk neutrality of landowners, the presence of uncerta inty affects equilibrium land rents and prices due to the permanence of land conversion. Th e authors equations on expected prices of agricultural and urban lands show that uncertainty increases la nd price, but only if city size is exogenous. If the size of the ci ty is endogenously determined, then the effect of uncertainty on the price of agricultura l land is ambiguous and largely depe nds on the degree of uncertainty (Capozza and Helsley 1990). In recent years, the land development problem has been described using concepts from the financial economics literature, namely investment theory under uncertainty, and in particular, options theory. Capozza and Li ( 1994) represent one of the more rigorous attempts to model the timing and intensity of land conversion as an investment decision. The timing of urban residential development is framed as an opti on in the model developed by Mills (1981a), but focuses on the effects of conversion decisions un der conditions of myopia and perfect foresight. Clarke and Reed (1988) more car efully examine the conversion decision as a perpetual option framework using stochastic calculu s, however they ignore the inte raction that capital intensity has on the timing decision and also on rents and pr operty taxes. Capozza and Li (1994) fill in these gaps by describing how urban areas are affected by capital intensity, particularly in the PAGE 79 79 lands spatial patterns, as well as obtaining effect s on the discount rate, conversion costs, rental rates, capital elasticity, and e xpected growth rates. Using the theory of optimalstopping, Capozza and Li (1994) frame the land conversion decision as an American option value with varying levels of intensity. In a general sense, the time required to take some specific action is described by the theory of optimal stopping ba sed on a series of rando m variables which are randomly observed. Often times this is done fo r the purpose of maximizing an expected reward or minimizing an expected cost (Kamien and Schwartz 1991). The presence of the option is based on the decision to invest be tween two different activities. Suppose the land has two revenue generating activities pe r unit or parcel, 1 R and 2 R with output per parcel, 11() qk and 22() qk where 12(,) Kkk is the capitalland ratio (i.e., capital intensity). Further, suppose the init ial case is activity one with revenue 1 R then the landowner has the option at any time, t to convert to activity two by repl acing the current cap ital intensity, 1k with intensity 2k Thus, not only does the landowner choose the time t but also the optimal capitalland ratio (Capozza and Li 1994). In the case of vacant land in the initial period, here period one, it is assumed that there are no rents and that no capita l is applied to the land. The decision becomes one of choosing a benchmark or hurdle rent, given by R and corresponding capital intensity, k, at some time t Capozza and Li (1994) make the following assumption: positive variable conversion costs, c; the discount rate r is taken to be the interest rate; and the net rental rate follows a nor mal diffusion process with a constant drift g and standard deviation which are both constants. The equa tion of motion for rents is given by: dRgdtdB (219) The price per parcel of la nd is given by the present value of future cash flows: PAGE 80 80 2g R p r r (220) The value of the conversion option is: ()()()max()cr T t t TPtWREVRte (221) The stopping time is T () WR is the value of a perpetual wa rrant moving from no capital to capital intensity k, and the intrinsic value of the warrant, given by () VR, is the value of the warrant if exercised at time t The intrinsic value of conversion is defined as: ()() VRqkpck (222) Using the fundamental differential equation of op timal stopping time, the model is solved to obtain the following conversion rule: *1 ()() g qkRrckqk r (223) In Equation (223), is a parameter. Thus, the option to convert will take place when the sum of the cost of capital for conversion and a risk pr emium from uncertainty e quals the rent given up by delaying conversion (Capozza an d Li 1994). By assuming a CobbDouglas functional form of the production function, the authors obtain equations for the optimal hurdle rent and capital intensity. Like in other models, greater levels of uncerta inty increase the conversion or option value. As a result of uncertainty the hurdle rent rise s, delaying conversion, increasing the necessary capitalland ratio, and reducing the structural density of the urban area (Capozza and Li 1994). The costs of conversion have a negative effect on the intensity of capital which in turn will lead to lower density and lower land values. Increasi ng costs, however, have no effect on either the hurdle rent or the conversion time. The elasticity of capital, on the other hand, has a positive effect on the both the reservation rent and the conversion time, but has an ambiguous effect on PAGE 81 81 the capital intensity. The di scount rate has an overall ne gative effect on the endogenous variables. Reservation rents are lower, and so conversion times tend to be delayed. Land values and capital intensity are also lower under higher discount rates. If a property tax is set on postdevelopment land, then the model predicts d ecreasing capital intensity and land values, but a higher hurdle rent, implying dela yed conversion. A predevelo pment property tax has the opposite effect on hurdle rents, im plying more hasty conversion. Capozza and Li (2002) simplify their earlie r analysis by assuming only one possible investment project. Assuming net rents from the project grow exponentially and that capital intensity is variable, th e authors obtain rules for the optim al timing of land conversion under both certain and uncertain growth rates using internal rate of return (IRR) and net present value (NPV) principles. Under certainty, the conventi onal rule for undertaking a project arising from the real options approach in investment theory is to delay investment until the current yield, or IRR, equals the cost of capital (Capozza and Li 2002). This level of yield is often referred to as the reservation or hurdle IRR since investment only occurs until this level is reached. In the case of land development, not only is conversion irreversible, or assumed to be so, but the added assumption of increas ing rents over time further delays the optimal time to invest. Under conditions of perfect foresi ght, the hurdle rate is given by: I RRrg (224) The discount rate is r, taken the prevailing interest rate in the real estate market, and g is the growth rate of rents. In this ca se, the internal rate of return mu st not only exceed the interest rate but also the rate of increasing cash flows. Assuming a constant elasticity of substitution production function, explicit solutions for the optim al hurdle level of rents and capital intensity are obtained: PAGE 82 82 11 1g ra Xr (225) and, 111 1a r gK (226) The assets coefficient of distribution is given by a, and is the elasticity of substitution. The growth rate has a positive effect on optimal capital intensity while intensity is decreasing in the interest rate. Unlike in Capozza and Li (1994), the effect of gr owth rates on conversion time is clear. A higher rate of increase in cash flows implies a higher hurdle rent and hence greater delay in conversion. Exponentially increasin g rents tend to have an effect on future option values rather than on current options values (Capozza and Li 2002) Further, the intere st rate in Capozza and Li (1994) has an unambiguous affect on timing whereas in the present case the effect of interest rates is unclear. Note however, that in Ca pozza and Li (1994) a CobbDouglas form of the production function is assumed. In the stochastic case, a continuous time ve rsion of the Capital A sset Pricing Model is utilized to describe the option value to invest The rental rate is assumed to grow with uncertainty according to a Brownian motion with a standard Weiner proce ss. Under uncertainty the IRR hurdle level for the timing of invest ment in land development is made when: 22 IRRrg (227) The variance of the growth ra te of cash flows is given by 2 and is a parameter that approaches one as approaches and approaches r g as approaches zero. This is equivalent to the certainty IRR hurdle except fo r the addition of a term accounting for the value PAGE 83 83 of waiting under conditions of uncertainty (Capozza and Li 2002). Using again the CES production function, explicit solutions for the optim al hurdle level of rents and capital intensity are obtained: 12 11 21aa Xr (228) 111 1aK (229) The effects of the growth rate are the same as in the certain case. A higher rate of growth of cash flows will delay conversion and increase the le vel of capital. The uncertain option value increases the variance of rental flows, increasi ng the time horizon of the investment and the capital intensity required when the decision to invest is made (Capozza and Li 2002). The effect of the interest rate is a bit more complicated due to the limiting nature of the and parameters. There are two offsetting effects to increased interest rates in an option value under uncertainty with positive growth rates. In creases in capital costs from higher interest rates serve to delay investment, while lower waiting option values serve to hasten investment. Generally, however, for any level of growth if the uncertainty is high enough so that is large, then an increase in the interest rate will have a declining affect on the optimal hurdle rent, hastening conversion. As noted by Capozza and Li (2001), when the wo rld is uncertain, irreversibility impacts the optimal timing of investment in a real options framework. The IRR has a higher hurdle level when investment is irreversible resulting in in creased delay of a project. When capital intensity is variable and the economy is growing, delay also occurs. When a landowner commits to a certain lower level of ca pital in the current time period, some amount of revenue is forfeited in a PAGE 84 84 future time period from an optimal project that requires higher capital than if the landowner had waited (Capozza and Li 2001). The Transactions Cost Approach The capitalization approach or present valu e method has dominated the literature on land values and the general literature on asset pricing as a whole. Ho wever, a number of empirical issues have arisen from the use of the present va lue techniques, particularly in the arena of land pricing. Indeed, a burgeoning literature has appe ared criticizing the capitalization technique for oversimplifying the valuation of land and leading to empirical rejection of present value methods, particularly for farmland (Falk 1991; Clark, Fulton, and Scott 1993; Lloyd 1994). One flaw in present value techniques is the inability to explain why land prices rise and fall faster than land rents, especially during boombust cycles (Schmitz 1995). While the option value approach discussed above is one alternative to the failure of capitalization methods, another appr oach has called for the incorpor ation of transaction costs in the model. As noted, the land market is particular prone to costly transactions, which have been estimated by some authors. Wunderlich (1989) es timates transaction costs from the transfer of land between buyer and seller aroun d 3 percent of the total land va lue net of brokerage fees. This estimate is in line with the 2.5 percent estimated transaction costs given by Moyer and Daugherty (1982). However, neither of these estimat es includes the cost of brokerage firms, which most land transactions occur through. Once accounted for, Wunderlich (1989) estimates the transaction cost to be as high as 15 percent. Clearly, such costs are not trivial and the need for more recent estimates is also clearly warranted. Land development and institutions The notion of transaction costs and institutions is not new, beginning with the work of Ronald Coase in 1937 and expounded on by William son (1985) and North (1990). Transaction PAGE 85 85 costs can take on many forms in the land developm ent process. While the mainstream literature has not yet fully developed a framework for the land development process under market frictions, some enlightening ini tial investigations are availabl e (Healey 1991; Alexander 1992; Lai 1994; Benjamin and Phimister 1997; Buitelaar 2004). For example, zoning restrictions, titling cost, survey fees, and brokerage fees are all embedded in th e transfer of land from rural to urban use. Any cost not accounting for in th e physical production of urban land can be considered a transaction cost. As stated by Coase, transaction costs arise from two key failures in neoclassical economics: perfect information a nd rationality. Models of the land development decision with imperfect information have alread y been described. Th e landowner or developer may attempt to close the information gap by trying to acquire new information. For example, the conversion to residential property may involve research into h ousing preferences (Buitelaar 2004). The presence of uncertainty also introdu ces transaction costs in the land market as landowners and developers will attempt to gain information to reduce uncertainty. For example, a landowner might be in possession of a substantial size of unzoned property. The zoning of her land into residential or commercial use has a profound implication on its potential value and the decision to convert. In an attempt to elim inate such uncertainty, she may lobby the local municipality to zone th e property to its valuemaximizing use. One purpose for the creation of institutions is to reduce uncertainty. There is also a degree of institutional cost in the land market and in the development process. Largely based on the work of North (1990) institutions can be described as the rules of the game in a society. Institutions define the constraints conceived by people and shape human interaction. By providing rules or constraints, institutions provide a structure to human interaction and reduce uncertainty in everyday lives. Local zoning restrictions are an example of PAGE 86 86 institutional constraints in the land development process. As noted by North (1990), the creation of institutions involves a transactio n cost itself, referred to as institutional cost. For example, the planning agency is the organi zation which places an instituti onal cost, zoning, on the land development process. Transactions tend to be costly due to the fact that information itself is costly. For example, the costs of measuring the attributes of value to the individual of what is being exchanged and also the costs of protecting the rights of the individual as well as the costs of enforcing agreements. These costs constitute the source of social, political, and eco nomic institutions. Indeed, stark implications for the land conversi on decision are implied by the very notion of an exchange process with transaction costs. Since these costs are em bedded in the costs of production, North believes that an entire restatem ent of the production rela tionship is necessary. This restatement must recognize that the costs of production are the su m of transformation and transactions costs. Before one can understand however the implicat ions on a theory of institutions, one must understand why transacting can be costly. C onsider the following statement by North: We get utility from the divers e attributes of a good or service, or in the case of the performance of an agent, from the multitude of separate activities that constitute performanceThe value of an exchange to the parties, then, is the value of the different attributes lumped into the good or service. It takes resources to measure these attributes and additional resources to define and to measure rights that are transferred. The underlying aspect of transactions costs stem from both parties involved in the exchange trying to ascertain the value of th e individual attributes of the un it being exchanged. The seller or owner of a tract of rural land would likely ha ve full information on the quality of land and its suitability for urban development, whereas th e potential buyer or developer would have to approximate that information. Enforcement is anothe r factor that adds to th e costs of transacting. As mentioned earlier, land transactions are in frequent and occur over long time periods. Land PAGE 87 87 transactions are often specified according to some written contract as to development dates, limitations to urban uses, etc. The developer might later find out that the land is not well suited for a commercial property, despite having already contracted to purchase the land and therefore would want to opt out of the contract. The enforcement of such a contract involves a cost. This additional cost would not present an issue if it is in the best interests of either party to concede to the original agreement. However, as North points out (as did Adam Smith 250 years ago) individuals are very much selfinterested which invokes feelings of uncertainty in either party that the other will not renege on the agreement. Uncertainty about possible reneging produces a premium on the risk that the other party will in fact renege, presenting a cost to the losing party. Information costs and uncertainty, conjoined with the behavior of the individual, presents challenges both to traditional economic theory and institutional theory. North (1990) obtains a better understanding of how individual beha vior and societys institutional structure are related. Property ri ghts are the rights indivi duals appropriate over the labor, goods and services they own. This appr opriation is a functi on of the institutional framework, such as legal rules, organizational forms, enforcement and norms of behavior. North describes that because of the presence of transa ction costs and illdefined property rights, certain attributes valued by the individual, remain in the public domai n. Individuals gain then by devoting resources to try to obtai n such attributes. How this plays out is a function of the institutional structure, which faci litates exchange and determines the cost of transacting. Now how well this game is played, North descri bes, depends on the extent that the rules institutionscan solve the prob lems of coordination and production. The outcome of the game PAGE 88 88 is determined by the motivation of the players, th e complexity of the environment, and the ability of players to decipher an d order the environment. Despite the need for an institutional model in the land development liter ature, to date none have been formally described. However, a number of studies have both theoretically and empirically described the impact of quantifiable transaction costs on land values. While notions of transaction costs in these papers are limited to exchange costs (i.e., brokerage fees) and do not examine institutional costs, they remain a revea ling and potentially promising alternative to the capitalization approach to modeling land values and the land conversion decision. Models of land values with transaction costs Borrowing from the notation in Lence and Mi ller (1999), the capitali zation model given by Equation (25) can be extended to acc ount for transactions costs. Let P T and ST denote the transaction costs on the purchase and sale of land, respectively, defined in terms of a percentage of the total price of land. Then define th e purchasing and selling conditions of land as: 1 1 01i tn Pttti i nTVER (230) 1 1 01i tn Sttti i nTVER (231) As in Lence and Miller (1999), the condition being tested is : SttPTEgT (232) The term tg is a stochastic variable that denotes the excess return yiel ded by holding the land indefinitely beginning at time t discounted at the rate The excess return is defined as: 1 1 01 1i tn tti i n tgR V (233) PAGE 89 89 The formulation given by Equations (330) thro ugh (333) imply the transfer of land will only occur if the transactions cost s do not exceed the expected excess returns of the land (Lence and Miller 1999). The conversion ru le in Equation (230) and E quation (231) collapse to the familiar capitalization formula given by Equation (25) if 0SPTT which is equivalent to assuming a perfectly frictionless transfer of land. Therefore, one way of explaining the gap be tween land prices and land rentals is through the presence of transactions costs. The sum of the lands expected discounted rent may differ from the current value or price, but not by an am ount greater than the tran saction costs associated with the transfer from seller to buyer. The capitalization m odel of the land development decision does not account for these transaction costs and as sumes that the land market is frictionless. Just and Miranowski (1993) is one of the first known studies to include such frictions in a structural model of land prices. Their analysis empirically demonstr ates that real land values do not closely follow land rentals. The theoretical model accounts fo r transaction costs by including parameters on the sales commissions incurred in selling land. The values of these parameters are simply imposed in the econometric model as t hose given by Wunderlich ( 1989). The results in Just and Miranowski (1993) imply a superior fit over the capitaliza tion method. Taking a different approach, Chavas and Thomas (1999) use a dynamic model of land prices, relaxing the assumption of timeadditive dynamic preferences, risk neutrality, and zero transaction costs. The model is an extension of Epstein and Zin (1991) but allows for fric tions in the transfer of land. Rather than impute the parameters describing tr ansaction costs, like Just and Miranowski (1993), Chavas and Thomas (1999) estimate the marginal transaction costs as a proportion of changes in land quantity as a result of buying and selling land using the ge neralized method of moments. Not only do the results in Chavas and Thomas (1999) also support a str ong rejection of the PAGE 90 90 capitalization approach, but also provide substan tial statistical evidence that transaction costs have a significant e ffect on land prices. However, the result from both studies should be taken with caution as Lence (2001) notes serious flaws with both Just and Miranowski (1993) and Chavas and Thomas (1999). According to Lence (2001), incorrect first order conditions on the expression describi ng land values leads to imprecise theoretical predictions and possibly in consistent econometric results. Lence (2001) also notes the inherent flaw in assuming a represen tative agent when using aggregated data, as in Chavas and Thomas (1999), since such an assu mption is invalid when transaction costs are present. Further, the complicated structures inherent in the Just and Miranowski (1993) and Chavas and Thomas (1999) models lend themselves to difficult estimation procedures and sometime vague intuition whereas Lence and Miller (1999) and de Fontnouvelle and Lence (2002) model land values with frictions in a very analytic ally tractable way.14 The method of Shiha and Chavas (1995) modifi es the traditional cap italization model in a manner similar to Lence and Miller (1999) and de Fontnouvelle and Lence (2002) by modeling market frictions as barriers to investment in ag riculture. Such barriers ar e postulated to result in market segmentation and include legal fees, info rmation and search costs, and represent the transaction costs in the mode l. Their model imposes thes e barriers on nonfarm investors attempting to hold equity in the farmland market Similar to Lence and Miller (1999) the transactions costs are assumed to be proportional to the value of land holdings. To estimate the transaction cost parameter in their model, the authors use iterative nonlin ear seemingly unrelated regressions. The Shiha and Chavas (1995) result suggest that not onl y is the farmland real estate market segmented, but that this segmentation is the result of transaction costs. In fact, the 14 The theoretical model of de Fontnouvelle and Lence (200 2) is the same as Lence and Miller (1999), however the former uses a kernel estimation method and an expanded data set while the latter relies ordinary least squares. PAGE 91 91 estimates are somewhat larger than thos e reported by Moyer a nd Daugherty (1982) and Wunderlich (1989) reaching a peak of 6.18 percent per year during the 19491983 period (Shiha and Chavas 1995). Empirical Models of Land Change There is voluminous empirical wo rk that attempts to estimate land values. An historical and comprehensive look at the literature, with particular attention to farmland values, can be found in Moss and Schmitz (2007). The particular attention in this sect ion is on how the land valuation approaches described affect the em pirics behind the land development decision. Capitalization Empirical Methods In one of the first and simplest empirical tests of the conversion model, Arnott and Lewis (1979) examine the real estate data for 21 metropolitan areas in Canada from 19611975 from the Central Mortgage and Housing Corporation. Actual ratios of land values to property values for the periods 19611971 and 19721975 are compared to those predicted by the model. Results indicate that the model explains 60 percent of the variation in the land value to property value ratio for the given areas. An elasticity of subs titution between land and capital is also estimated. Using a CES form of the housing production function () QK, the authors perform a regression on the following equation () lnlnln ()1 VT K pTK (234) The elasticity of substitution between land and capital in Equation (234) is given by is a coefficient describing th e distribution of land, () p Tis the price of a unit of capital at time T, and K is assumed be equal to the average area of floor space to the average size of the residential lot (Arnott and Lewis 1979). Using data for 23 Canadian metropolitan areas from 19751976 on new single family homes, results imply an elasticity of substitution of 0.372 for 1975 and 0.342 PAGE 92 92 for 1976. The authors take the low values as evidence that the model does in fact produce an optimum. However, a weakness in the Arnott and Lewis (1979) model involves the assumption that on the urban periphery, the supply of land is perfectly elastic and developable. This may not be realistic given the heterogeneity of spatia l land characteristics and the location of some parcels to more amenities. More rigorous approaches to modeling the conversion decision can be found in the literature and generally fall into two categories: Probit models and duration models. The general intuition behind the Probit specification is an attempt to capture the effects of variables that increase or decrease the probabi lity of a landowner to convert land from rural to urban uses. Many of these variables will be measurable and rela te to certain characteristics of the land, such as land rentals, current land use, proximity to other rural or urban area s, estimated nearby land prices, and spatial characteristics. Other meas urable variables may be unique to the landowner such as age, gender, income, and occupation. However, many characte ristics of both the land and the landowner are not directly measurable or observable and so a stochastic framework is necessary. Suppose we have a landuse decision rule de rived from the capitalization approach given by Capozza and Helsley (1989) of the form: ,,,iii R TATrCT wxz (235) The functions R, A, and C in Equation (235) represent the return s from conversion, rural land rent, and conversion costs, respectively. The discount rate is given by r and the optimal time of conversion is T. Vectors of observable charac teristics describing conversion returns, rural rents, and development costs for parcel i are given by iw, ix, and iz, respectively. PAGE 93 93 If we define a vector of unobser vable characteristics for parcel i as i, the probabilistic model of the development decision can be formulat ed as in CarrionFlores and Irwin (2004): Pr,,,,,0iiiiDiTRTATrCTTwxz (236) The decision to develop rural parcel i at the optimal time T is given by DiT. The error term ,iT associated with the development decision is assumed to follow a normal distribution. The parameterization of the model can be made explicit: 'Pr,,,0iDiTiTT (237) The vector ', iT denotes the vector of observable characteristics iiiwxz and the vector of parameters to be estimated is given by The equation becomes more specific depending on the choice of variables to include. Further, if spatial characteristics are included, the model may require correction for spatial error autocorrelation in the error term ,iT. This specification was used by CarrionFlores and Irwin (2004) to determine the factors associated with rural land conversion to resident ial uses in Medina County, Ohio. Using parcel level data from a Geographic Information System (GIS), the authors use the model in Equation (237) to explain the convers ion pattern between 1991 and 1996. Among the variables included in the Probit function are: distance to Cleveland, distance from nearest town, population, neighboring residential, agricu ltural, commercial, and other areas, population density, size of parcel, and soil quality. Results show the probabili ty of conversion decreases with distance from Cleveland, population density, size of parcel in acres, and if the parcel is considered large. Conversely, conversion is more probable with greater distances to the nearest town, better soil quality, and greater ne ighboring residential and commercial areas Of interest is the variable on distance to Cleveland, a highly urbanized location. For parcel located within 14 miles of the PAGE 94 94 Cleveland fringe, the probability of development d ecreases at a decreasing rate, but outside this 14 mile boundary the probability of conversion increases. Of contra ry expectation is the result that larger parcels are less likely to be developed. The authors state, however, this may be a result of the limited data available since the m odel is unable to disti nguish between undeveloped land and land that is undeveloped but zoned fo r development (CarrionFlores and Irwin 2004). Ding (2001) also utilizes GIS data for Wa shington County, Oregon and estimates the probability that a parcel of v acant land is developed into an urban use between 1990 and 1994. His data set is comprised of nearly 14,000 iden tified vacant parcels with almost 5,000 converted into an urban use during the studied time period. Variables in the Probit analysis include access time to the central business district for four ur ban areas (Beaverton, Forest Grove, Hillsboro, and Portland), dummy variables indicating what urban area the parcel is located in, dummy variables indicating adjacency to major roads, adjacency to existing urban land, and adjacency to parcels also being converted. Dummy variab les are also included to indicate if the parc el is located in a flood plain, a growth boundary, and a one mile zone of light rail. Two continuous variables are included for distance to the urban growth boundary and a tax rate. Ding (2001) also finds that the likelihood of c onversion decreases with parcel size like in CarrionFlores and Irwin (2004). Similarly, he finds that land is more likely to be converted if it is closer to Portland but further away than the less urban areas of Beaver ton, Forest Grove, and Hillsboro. Interestingly, Ding (2001) finds the pr obability of conversion gr eater in areas with a higher tax rate. While he explains this as a possibility of higher ta x regions producing better amenities such as schools, he does not investigate th is further. Another likely possibility, as is the fact that higher tax rates, when they are anticipated, te nd to speed up the conversion process PAGE 95 95 as developers try to avoid the higher tax penalti es. A lagged and lead tax variable ought to capture this effect, but Ding (2001) does not include these in his Probit model. Claassen and Tegene (1999) take a slightly different approach and use a Probit equation to model the conversion of pastureland to cropland, rather than to urban uses. Although the end development is different, the concept is the sa me: conversion from a lower to higher use. Another alternative use is give n by the Conservation Reserve Program (CRP), established by the 1985 farm bill. The CRP pays landowners to keep their land out of any productive uses, such as forestry or agriculture, in order to preserve the land. One of the va riables in Claassen and Tegene (1999) is the difference in rental rates between cropland and pastureland. The estimate on the rental rate difference is positive and significant suggesting that when cropland rents are large relative to pastureland rents, then the probability of conversion to cropland is greater. Estimates on the first and second lag of the rent al rate difference are positive and negative, respectively, but not signif icant. The estimated interest rate effect suggests that the probability of conversion from pastureland to cropland conversi on is smaller with higher interest rates. The estimated coefficient on the rent received if la nd is enrolled in the CRP program is negative, suggests that as the rent rece ived under CRP increases the likelihood of land remaining in pastureland decreases. Cho and Newman (2005) provide an innovative threestage an alysis of the development process. In the first stage they estimate a hedoni c regression of land values. The results of this equation are used in the second stage in whic h a Probit equation is used to estimate the probability of a parcel of land being converted to a developed use. Finally, a thirdstage Probit equation is used to estimate the density of deve lopment. Their data on vacant land parcels for Macon County, North Carolina was obtained from the land records division of the tax PAGE 96 96 administration department for over 40,000 parcel s, of which nearly 16,000 were converted between 1967 and 2003.15 Cho and Newman (2005) find that an undeveloped parcel is more likely to be developed if it is lo cated near a parcel th at is already developed. The predicated value of land, quantity of roadways in the ar ea, and degree of flatness of the land all are estimated to increase the pr obability of conversion. The Cho and Newman (2005) results are intuitive and follow from the theoretical analysis. For example, as a parcel of land increases in value it becomes too costly for the landowner to keep the land in vacant or agricultural use, as the rentals received from agriculture are not enough to compensate for the opportunity cost of conve rsion. Further, the fl atter a parcel of land is, the more amenable it is for residential housin g and commercial projects. The probability of conversion and the density of development declines with the si ze of the parcel, distances to roadways, and the median elevation on the land. La rge lots are less likely to be developed since residential developments occur after rezoning, which will break a larger lot size into many smaller lots. The presence of a large lot, say 10 or more acres, indicates the possibility that the land may not have been zoned for development ye t. In fact, Cho and Newman (2005) find that parcel sizes greater than 10 acres have nearly a zero chance for high density development. Another common econometric met hod of modeling the conversi on process is through the use of duration or survival models. Duration models are often used to answer questions regarding the duration of unemployment spells, or time inte rvals between human conceptions. In general, duration models are concerned wi th how a variable changes th rough time, from one state to another and are particularly well suited to the land conversion d ecision. Two key questions are addressed by estimating the conversion process th rough a duration model. First, what is the 15 According to the authors, the data set was updated every 4 years. PAGE 97 97 length of time a parcel of land will remain undeve loped? Second, what is the likelihood that it will be developed in the next time period? Typi cally the duration of time a parcel spends in an undeveloped state, the key variable of interest, is described by a hazard function. Other variables may change during the conversion duration such as population growth rates, conversion of nearby parcels, capital costs, interest rates, and even th e landowners discount rate.16 Such timevarying covariates represent additional complica tions and can also be included in the hazard function. The hazard function is the probability density of the duration of being undeveloped and is a function of time. To provide intu ition for this method, modify Equa tion (236) so that we have: Pr,,,,iiiiTRTATrCT wxz (238) Landowners who currently engage in farming receive rent ,iAT x and are characterized by a vector of unobservables given by ,iT. Individuals who are better farmers or who place higher value on land if it is in farming use will have a later conversion time than individuals who are not as able farmers or who place less value on the land in farming use. Therefore, the probability that a parcel w ill be converted at time T can be defined as the hazard rate for that time period for a given set of char acteristics (Irwin and Bockstael 2002). This hazard rate is given by: 1 1 GTGT hT GT (239) 16 A burgeoning literature has spawned beginning with Strotz (1956) on how individual time preferences may change through time possibly exhibiting declining preferences or even preference reversals. This may suggest the need to model the conversion process with a nonconstant discount rate, such as the hyperbolic discount rate. For more discussion on this literature see Frederick, Loewenstein, and ODonoghue (2002). PAGE 98 98 The cumulative distribution function for the unobservables is given by G and is the unobservable value that makes Equation (239) ho ld with equality. As noted by Irwin and Bockstael (2002), this is the value that make s the landowner indifferent between keeping the land in an undeveloped use or convert to a developed use. There are multiple methods of calculating the hazard function. One involves parametric estimation which requires an assumption on th e distribution of Equa tion (239). Common distributions include the exponential, Weibull, and variants of the logarithmic and normal distributions such as the lognormal and loglogi stic (Bell and Irwin 2002) However, as noted by Greene (2003), the choice of distribution has profound implications on the answers to the questions regarding the timing of conversion. Fo r example, the hazard function can slope either upwards or downwards, depending on whether th e duration length increases or decreases the likelihood of the parcel not being conve rted in the next time period. One way of avoiding this problem is to estimate the hazard function through semiparametric methods.17 This is often referred to as a proportional hazard model or a Cox regression model. Two distinct parts comprise the hazard function in a Cox model: the baseline hazard function and the explanat ory function. The general fo rm of the proportional hazard function is: 0tte x (240) The baseline hazard in Equation (240) is 0t and represents the heterogeneity among individual observations, the e xplanatory component is given by 'e x. The vector of exogenous variables is defined as 'x and the vector of estimated parameters is The key feature of the 17 Another approach would be through a fully nonparametric estimation method. However, this method introduces its own complexities and thus the Cox method serves as a nice median between parametric and nonparametric estimation methods. PAGE 99 99 formulation above is that time is distinct from the vector of explanatory variables. An implication of this distinction of time is that the hazard function for each individual observation is a proportion of the baseline hazard. Being set up as a proportion implies that as values in the explanatory variables change, the function 0t shifts so that a value of the hazard t is attained (Greene 2003). Cox (1972) defines a par tial likelihood function wh ich is maximized to obtain estimates of the parameters without having to estimate the baseline hazard. The proportional hazard approach is used in Irwin and Bockstael (2002) to obtain undeveloped parcel conversion ra tes. In particular, the au thors focus on the effects of neighboring parcel conversions by measuring su rrounding developments as a way of identifying potential spillover effects. They use an intr icate data set on a seven county region of Maryland including Washington, D.C. and Baltimore as major urban areas. The data set was obtained from the states panning office. Variables in the an alysis include an index of zoning potential, a variable measuring the maximum permitted development density, distance measures to Washington, D.C. and Baltimore, an indicator fo r parcels that are relatively more costly to develop due to steep slopes or poor drainage, and an indicator for prime agricultural land. Estimates from Irwin and Bockstael (2002) fully specified model imply a reduced hazard of development with greater commutes to Wash ington, D.C., steeper and more poorly drained soils, being prime agricultural land, and greater allowable density. If greater density is allowed and development returns are increasing over time with a concave produc tion function, then a landowner will find it optimal to delay conversion until a later time. Of partic ular interest is the result of negative spillover effects, implyi ng a negative interaction between undeveloped and developed parcels in the decisi on to convert land (Irwin and Bockstael 2002). While this negative interaction might seem c onflicting with the theory, the aut hors note that the analysis is PAGE 100 100 of land along the ruralurban fringe where amenities such as open space are more highly regarded. Further, the magnitude of the nega tive interaction is based on commuting distances between Baltimore and D.C. being held constant. Option Value Empirical Methods While options value theory is a relatively recent development in the financial economics literature, empirical testing of real options models is even more recent. The empirical models of the capitalization approach have a firm ground in econometric theory. Optionbased econometric approaches are not afforded the same luxury. However, in recent years, a number of empirical studies have emerged, particularly in the real estate economics literature, due in part to the attractive nature of the theoretica l aspects of the option value model. A review of some of the most noteworthy studies is di scussed in this section. One of the first such tests is Shilling et. al (1990) who use a si mple ttest to determine if a time premium is present in a real option model of land development. The authors estimate both the current market value of developable land, V, given by the appraisal value and the discounted exercise price rT X e. The discount rate is given by r the development time is T and the exercise price is X The authors then calculate the mean difference between the option premium and the intrinsic value. A ttest on the null hypothesis of the presence of a time premium is given by: 0.50 221cQcQ t ssN (241) Shilling et. al (1990) define c as the mean option price, and Q as the mean intrinsic value given by the mean difference of rTVXe. The respective sample variances are 2 cs and 2Qs, and the number of observations is N. The null hypothesis could not be rejected at the 90 percent PAGE 101 101 confidence level, indicating a zero mean differe nce between the option premium and the intrinsic value (Shilling et. al 1990). However, this test is far from rigorous and does not explicitly examine how an empirical model of option pricing affects the c onversion decision. The first rigorous empirical study is Quigg ( 1993) and examines the optionbased value of undeveloped land by directly incorporating the va lue of waiting to invest in land development into a simultaneous equations model. Relaying on the hedonic methods of Rosen (1974), the author specifies a hedonic price function, p Z, on how market prices of land characteristics, Z affect the price of land. The estimates from the hedonic price function are used to estimate another equation describing the po tential value of construction on an undeveloped parcel. The data comes from the Real Estate Monitor Corporation and consists of a substantial number of land transactions in Seattle, Washington from most of 1976 through 1979 and includes 2,700 transactions of undeveloped la nd parcels (Quigg 1993). The centr al conclusion is that the development option represents a premium in the market price for undeveloped land at an average of 6 percent of the lands value. Further, the model does well at predicting transaction prices unlike the net present value me thods discussed earlier. Capozza and Li (2001) investigate the effects of positive interest rate changes in an uncertain real option with variable capital intensity and a project that is irreversible. The general view from real options model of i nvestment is that increases in the interest rate tend to increase investment (Ingersoll and Ross 1992; Capozza and Li 1994; Capozza and Li 2002). In the context of real estate development, Capozza and Li (2001) test this response using panel data on residential building permits, to empirically test the presence of a positive relationship between the interest rate with the land investment decisi on. The response on the hur dle level of net rents to interest rate changes varies with the level of growth and uncertainty. When either growth PAGE 102 102 rates or uncertainty, measured by the variance or volatility of grow th, are high, positive responses to development or convers ion from interest rate changes te nd to occur. Data from the U.S. Department of Commerce on building permits and population growth is obtained for 56 metropolitan areas from 1980 to 1989. Population growth rates are used as a proxy variable for net rental growth rates. The primary home mortgage rate is used as the nominal interest rate variable. The authors estima te a regression of the form: 2 1234()it i iii ii i t tGHPCAGPOP AGPOPAGPOPSGPOPyear GRM RM (242) The term GHPC is the annual growth rate of building permits per capita in percent for single family homes in area i for year t GRM is the annual percentage change in the real mortgage rate for year t RM is the annualized real mortgage rate in percent terms defined as the beginning of the year yield on the primary conventional mortgage minus the current CPI inflation rate. AGPOP is the average a nnual growth rate of population in area i SGPOP is the standard deviation or volatility of population in area i The ratio of GHPC to GRM measures the elasticity of residential investment in terms of building permits (Capozza and Li 2001). The au thors also include variables for government regulations, tax rates, and mone tary policies. Several form s of the regression above are estimated. In particular, deterministic and stochas tic versions are obtained to estimate effects in a certain and uncertain world given by variable gr owth rates and volatility. In the certain case, the ratio AGPOP/RM, serving as a proxy for the ratio of growth rates to the interest rate, is positive and significant, supporting the Capozza and Li (1994, 2001, 2002) model. In the uncertain case, AGPOP is used as the explanatory variable and is also positive and significant suggesting again that growth rate s respond positively to interest rate increases. Regressions including the variable SGPOP for population grow th volatility reveal positive and significant PAGE 103 103 estimates, also indicating positive responses to interest rate changes under greater uncertain conditions. The Capozza and Li (2001) regressi on results point to the importance of accounting for growth rates in any policy with a purpose of affecting investment rates. Since metropolitan areas are quite heterogeneous in terms of the popul ation growth, interest rate policies can have very different effects between lo calities. Further, extreme care is warranted in any changes to the interest rate, since investment can be hastened beyond which the moneta ry authority intended depending on the volatility of growth rates. Further empirical evidence on the importance of option values on the decision to convert land under uncertainty is presented in Schatzki (2 003). The standard expe cted net present value model is compared to a real op tion investment model with uncerta in returns. Under expected NPV models, the decision to convert land is ma de by the landowner once the discounted stream of returns from the converted use exceed that of the unconverted us e after accounting for conversion costs. However, conversion deci sions under a real options framework with uncertainty tend to have higher necessary returns to induce conversion than under the expected NPV method. The higher returns needed are du e to the option to dela y and sunk costs which cannot be recovered after convers ion (Schatzki 2003). Since landowners have an incentive to delay conversion from additional information about fu ture returns, there is value in the option to convert land. Schatzki (2003) conducts empirical tests on the e ffect of uncertainty in the decision to convert land from agriculture to forested using panel data from the National Resources Inventory (NRI) from 19821992. The NRI is a statistical su rvey of land use and landuse changes on parcellevel nonfederal lands with a particul ar focus on the conversion of croplands to forests. PAGE 104 104 In the Schatzki (2003) model, the problem of the agricultural landow ner is to choose the maximum of two alternative uses. The first is th e sum of the expected returns from cropland in the first period with the expected value of land in agriculture while the second is the expected value of bare forest minus conversion costs (S chatzki 2003): 11 1maxmax ,aa r t a r t fa tttt tVEReEVeEVC (243) The expected value operator is given by E the annual agricultural returns are a t R the value of land in agriculture is 1a tV, the value of land in forests is 1,1 f tV the cost of conversion is a tC and the discount rate is given by r. If returns are uncertain and assumed to follow a Brownian motion with drift, the conversion rule can be written as: ,,,,,,,()f Fa t atftatfttf a tR R rCgt R (244) The annual return to forests is f t R The variances of the motion process are given by ,atft and the Brownian motion drift parameters are ,atft The correlation between agricultural and forest returns is t ; and the growth rate of forests is ()fgt Thus, the landowner will convert land from agriculture to forests when the relative return of forest to agriculture is greater than a threshold based on the set of variables. Due to the uncertain nature of returns, the threshold necessary to induce investment in the conversio n decisions is higher than in the standard expected NPV model without an option value. This is because the option to delay conversion has value. To determine the probability of a parcel being converted from cropland to forests, the author estimates a limited dependent va riable regression model of the form: Pr()PrlnlnlnF f itaititconversionRRR (245) PAGE 105 105 Parcellevel returns to forest s and agriculture are given by ln f it R and lnait R respectively, and the conversion threshold is defined by F it R Relevant explanatory vari ables in the NRI data set that are in the regression incl ude indicators of land quality, conservation practices, population density, irrigation, uncertainty, agricultural reve nue and forest return trends, current revenues and returns, and the correlation between agricu ltural revenues and forest returns. Schatzki (2003) finds the probability of conversion falls with increased uncertainty of returns in agricultural revenues and of forest returns, s uggesting that option valu es affect landowner decisions. Thus, as either agricultural revenu es or forest returns increase, the conversion threshold also increases. Interestingly, the coefficient estimate on the correlation of agricultural revenues and forest returns is positive and signif icant, indicating that the likelihood of conversion is greater with a hi gher correlation. Finally, conv ersion is more likely when agricultural returns are low and forest returns are high, the land is not irrigated, population density is low, and no conservation practices are used (Schatzki 2003). Chapter Summary The literature review contained in this chapter has examined both theoretical and econometric models of land use and land develo pment. The land development decision in regards to conversion use and timing can be mo deled by three approaches : capitalization, option value pricing, and transaction costs. Within the capitalization framework, the standard net present value of land is the primary method of ob taining land values and the decision time to develop land. Capitalization approaches use both partial and general equilibrium models of the development process. However, capitalization methods have failed to accurately predict land values (Falk 1991) and do not explain the ga p between land rents a nd prices (Capozza and Helsley 1989). Two alternative approaches were discussed. The option value approach accounts PAGE 106 106 for the inherent value of waiting to invest in land development ar ising from both uncertainty and irreversibility. While option valu e models have the advantage of being more realistic, they are conceptually more difficult to employ. The tran saction costs approach acc ounts for the fact that land transactions are long, infrequent, and cost ly. Typical costs such as informational and institutional costs, not captur ed by either the capitalization or option value approaches, are expressed in a trans action costs framework. Wh ile this method holds particular merit, the lack of a sophisticated conceptual model preven ts it from mainst ream utilization. Recall the questions posited at the beginning of this discourse. When is it optimal to convert land from a lower rural use to a higher urban use? What is the optimal intensity of capital that should be applied to the land development project? How do government policies, such as property taxes and growth controls affect the optimal timing and intensity? What affects to changing discount rates and expected return s imply for land development? How does an uncertain world alter matters? This review has attempted to compare and contrast how answers to these questions differ depending on the choice of theoretical and empirical model. Further, even within the same approach, disagreement occu rs between authors as to the correct answer due to the unique nuances with in a particular framework. This disagreement not only remains problem atic for academic researchers, but also especially for policy makers who require a keen understanding of the development process in order to formulate policies intended for rural conservation or urban gr owth. Accurate and efficient policy instruments cannot be designed without an understandin g of land development models and a knowledge regarding timing deci sions. A better knowledge of the conversion process will facilitate an impr oved understanding of numerous outcomes of development that seem random such as discontinuous urban develo pment, heterogeneous spatial patterns in land PAGE 107 107 use, and the creative de struction of capital structures. This review contributes to the understanding of the development process throu gh a comprehensive critique of the currently used conceptual and empirical models, someth ing which is lacking in the literature. Recommendations for future work include co ntinued advancement of real options and transaction costs models, as these remain the most encouraging in terms of accurately describing the development process. Further, since the deve lopment process is an intertemporal decision, it is recommended that models from the economics of time literature be applied. Methods proved fruitful in other fields, such as dynamic progr amming, have yet to be applied to the land conversion problem. Since models of land development are derivati ve of models of land values, a complete understanding of the development decision must come from a thorough k nowledge on the nature of land values. This paper serves to e nhance the body of knowledge by introducing time inconsistent preferences to a m odel of land values. The next chapter will present the theoretical model. PAGE 108 108 Figure 21. Land allocati on and bidrent model Figure 22. Optimal land allocation Net rent A ( la ) H(lh) la lh Land (L) City center A Distance from center Land value Rural bidrent function Urban bidrent function PAGE 109 109 Figure 23. Optimal conversion time Figure 24. Change in value of agricultural land awaiting conversion Time Retur n t T Vt VHVA R H R A 0 Time Retur n T T ** H(t,T*) H(t,T**)iH(T**) A ( t ) PAGE 110 110 Figure 25. Timing of conversion decision Figure 26. Call option payoff 45oStock or commodity price Payoff Exercise price Time Conversion value V V P P T V(T) PAGE 111 111 Table 21. Selected comparative stat ic results from capitalization papers Exogenous variables Study Endogenous variables Capital / conversion costs Discount rate Predevelopment rental rate Postdevelopment rental rate Predevelopment property tax Postdevelopment property tax Shoup (1970) Land value / price N/A + N/A N/A Conversion time N/A + N/A N/A Skouras (1978) Land value / price N/A + + Conversion time N/A + Markusen & Scheffman (1978) Land value / price + 0 Conversion time + + 0 ? + Arnott & Lewis (1979) Land value / price N/A + Conversion time + ? N/A + Capital intensity 0 N/A 0 0 Ellson & Roberts (1983) Land value / price + + 0 N/A N/A Conversion time + + ? N/A N/A Anderson (1986) Land value / price N/A + + Conversion time N/A + ? ? Capozza & Helsley (1989) Land value / price + + + N/A N/A Conversion time + + ? N/A N/A PAGE 112 112 CHAPTER 3 THEORY AND EMPIRICS This chapter presents a p resentvalue fr amework for land price determination using rational expectations in the cont ext of farmland values. The theoretical model is formulated, generalizing time preferences to permit for quasi hyperbolic discounting. The empirical model is developed from the theory presented and allows the exponential and quasihyperbolic discount parameters to be obtained. A hypothesis test is constructed, permitting fo r a direct test on the discount parameters. Finally, th e econometric procedure is pres ented. The parameters are estimated using the linear panel Generalized Method of Moments es timator. The discussion pays particular attention to the selection of instruments. Theoretical Framework The main assumption underlying presentvalue models of farmland values is that the expected rents received from land exclusively de termine the value of land, holding other factors fixed, such as taxes. While this is a strong assumption, present value models of asset prices under rational expectations rely on this principle of re ntal determination. The other implied assumptions are riskneutral landowners and a ti meconsistent discount f actor with a constant rate of discount. The timeconsistent assumpti on will be relaxed, permitting for quasihyperbolic discounting in the asset value e quation. Specifically, a model for values based on changes in asset valuation over time will be derived and w ill nest both the standa rd case of exponential discounting and the case of quasihyperbolic discounting. Financial theory typically stipulates that a firm or individual d ecisionmaker should adopt a project, such as converting or developing land, if the net present value of the project is positive. Under simplifying assumptions of risk neutrality and timeconsistency, the value of land today can be written as the discounted stre am of expected future rents: PAGE 113 1130ttstts sVER (31) The price or value of land at time t in Equation (31) is tV. The expected rental rate or nominal cash flow from land use at time st is sttRE and is based on the information available to the landowner in period t. The standard exponential di scount factor is defined as: 1 11s tsti i (32) The discount factor is taken to be 0,1. The rate of time preference in period t is given by the constant rate of discount,ti which is often assumed to be the nominal or real interest rate. The formulation given in Equation (31) and Eq uation (32) is the basic expression for land values examined by many author s to test how well the PV model can explain movements in farmland values when a rational expectations framework is assumed (Falk 1991; Clark, Fulton, and Scott 1993; Tegene and Kuchler 1993; Sc hmitz 1995; Schmitz and Moss 1996). Based on this formulation, the expecta tions in the land market that both buyers and sellers have on discounted future returns play a cen tral role in determining the value of land. Some studies have rejected the present value model as a rejection of rational expect ations (Lloyd, Rayner, and Orme 1991; Tegene and Kuchler 1991, 1993; Engsted 1998) However, this rejection may be due in part to a potentially anomalous as sumption of exponential discounting. An alternative formulation takes note of potential time inconsistency and assumes a quasihyperbolic discount factor implying a noncons tant rate of discount. Quasihyperbolic discounting, developed by Laibson (1997), is a di scretevalue time function and maintains the declining property of generalized hyperbolic discounting. At the same time, the discrete quasihyperbolic formulation keeps the analytical si mplicity of the timeconsistent model by still PAGE 114 114 incorporating certain qualitative aspects of expo nential discounting. Th e actual values of the discount factor under a discrete setup are 21,,,, with the time periods defined as 0,1,2,,t. When 0t, the discrete discount functi on is normalized to one. The parameter behaves similar to the exponential discount factor. The parameter captures the essence of hyperbolic discounting and contains a first period immedi acy effect in the individuals time preference. Discount rate s under a quasihyperbolic discount function clearly decline over time as the shortrun discount rate, given by ln is greater than the long runs discount rate, given by ln as computed in Laibson (2007). Setting up the problem under quasi hyperbolic intertemporal preferences implies that Equation (31) takes the form: 1tt s tt s sVER (33) Changes in the parameter in Equation (33) determ ine how much the di scount factor will deviate from exponential discounting. If discounting is timeconsistent then 1 and is timeinconsistent if 0,1. Following the literature, represents the discount factor between the current time period and the next time period, while represents the discount factor between any two future time periods (DellaVigna and Pa serman 2005). The expression in Equation (33) can be modified to derive a model for asset va lues based on changes in the asset valuation over time. Specifically, taking the time differ ence of Equation (33) yields: 11 10ttt tstts tstts ssVVVER ER (34) The expression in Equation (34) can be simplified by aggregating over like exponents to: PAGE 115 115 1 11 1t tts tsttstts sVER ERER (35) Under rational expectations there is a forecast e rror in the second term on the right hand side of Equation (35) between the two expectation operators, meaning 1ttsttstERERe where te is an uncorrelated residual term. Equation (35) can be written as: 1 1 11t tts ts ttst sVER ERe (36) The residual te represents a white noise forecast error if expectations are in fact rational, implying that no information is present in the error term. Simplifying gives: 1 1(1)t tts tsttst sVERERe (37) The observed cash flow in the next period is assu med to proxy the expected return in the next period, that is, tttRRE1. Equation (33) can also be substituted in for the second term on the right hand side of Equation (37). Making these change s yields the structural model: (1)ttttVRVe (38) To test for hyperbolic discounting, the structural equation given by Equation (38) can be parameterized into an identified reducedform linear panel regression. Doing so gives: 012it it ititVRVe (39) The reduced form model given by Equa tion (39) includes a constant term, 0 which is included in most regressions on land values. The reduced form specification in Equation (39) has also been extended to account for observations over time and space. Observations come from a PAGE 116 116 sample of i geographic regions over a period of t years. The nice feature of Equation (39) is that the formulation nests both the exponent ial and hyperbolic di scount factors. The discount parameters in the structural sp ecification in Equation (38) can be obtained once the reducedform coefficients in Equation (39) have been estimated. Since 1 and 21 then: 21 (310) 1 21 (311) If the assumption of exponential discoun ting is true, then one would expect 1 to be close to 1 and 2 to be close to zero. The constant term, 0 should not be statistically different from zero under rational expectations regardless of the shape of the discount factor. Based on the reducedform parameter estimate s, two types of hypothesis tests can be devised to test for the presence of hyper bolic discounting. The assumption of 1 is equivalent to assuming exponential discounting, which implies 211 using Equation (310) and Equation (311). Hence, an appropriate test for exponential discounti ng would test the null hypothesis of 21:1oH against the altern ative hypothesis of 21:1aH This test will be referred to as the implicit test of hyperbolic discounting, since the value of is assumed and since the test basically amounts to a test of exponential discounting. A second test is a direct nonlinear hypothesis test on Equation (311) with the null hypothesis of 12:11oH against the alternative of 12:11aH. This test is referred at as the explicit test of hyperbolic discounting since the standard error on the hyperbolic parameter is computed. While both tests are equivalent in theory, the implicit te st is more efficient while the explicit test PAGE 117 117 introduces some noise due to the Ta ylor series approximation. Overa ll, the explicit test is better empirically since it provides an exac t test on hyperbolic discounting. Since is a nonlinear function of the parameters, a linear Taylor series approximation is used to obtain the standard error: ()() h hh (312) The hyperbolic parameter is defined as a vect or function of the estimated coefficients, () h in Equation (39) to obtain the linear Taylor series approximation. The formulation in Equation (312) appeals to the Central Limit Theorem and asympt otic theory for consistency. If the Law of Large Numbers holds and if the data are independ ently and identically distributed then obtaining the standard errors through a Taylor series appr oximation is appropriate. If these assumptions hold, then the standard error of () h is given by: 1 2 ..() hh SEhVAR (313) The partial derivatives in Equation (313), wh ile functions of the unknown parameters, can be computed using the sample estimates. Econometric Procedure Given the nature of the data on land values, us ing a simple estimator such as least squares for estimating the panel regressions in Equation (3 9) would not be approp riate for a number of reasons. First, while the error term, ite, is assumed to be independently and identically distributed with zero mean, heteroskedasticity across years and farms remains a possibility. Second, while the error term is not co rrelated with the dependent variable itV, the first PAGE 118 118 difference of farmland values, ite is serially correlated across time. For these reasons, estimation by the Generalized Method of Mo ments (GMM) is preferred. Originally proposed by Hansen (1982) and Hansen and Singleton (1982) for estimating consumptionbased asset pricing models, GMM provides consistent estimates of the para meters. The consistency of the estimates however is largely determined by the selection of instruments, which often remains a difficult task. Despite the benefits of GMM, however, surprisingly little work has been done applying GMM to the econometric problems inherit in land values data (Chavas and Thomas 1999; Lence and Mishra 2003). The regressors in Equation (39) have both timevarying and timeinvariant components, with observations assumed to be independent over i. The time period T is fixed, and the variables can be stacked for the thi region over all T. Rewriting Equation (39) in more general terms yields: iiiyX (314) The dependent variable and the error term in Equation (314) have been reduced to 1T vectors, while the matrix of independent variables, i X is TK with K denoting the number of regressors: 11 11;;.ii ii iii iT iTiT iTVVRe yX VVRe (315) Panel GMM estimation of Equation (39) follo ws from the sample moment condition: '0,1,,.ii E iN (316) The matrix of instruments,' '' 1iiiT Z zz has dimension Tr with r denoting the number of instruments and T denoting the number of time periods. In the residual vector, PAGE 119 119i is a 1 K vector of K parameters to be estimated. The foundation of GMM estimation is the specification of the orthogonality or moment c ondition in Equation (316). The goal of the GMM estimator is to find a vector of parameter estimates so that the residuals are orthogonal to the set of instruments. Given this background, if there are more in struments than parameters, that is if rq the form of the minimand for th e linear panel GMM model, NQ can be expressed as (Cameron and Trivedi, 2005): '' 11 NN NiiNii iiQZWZ (317) The matrix NW is a rr positive semidefinite weighting matr ix, akin to a variance matrix, which depends on the data. The weighting or distance matrix, NW, converges in probability to a nonstochastic positive definite matrix of constants, W. The orthogonality conditions in Equation (316) are in a way being emulated by minimizing the function NQ which is a quadratic form of the sample means across both time and space. The panel GMM estimator GMM can be defined as: argminGMM NQ (318) Equation (318) states that GMM estima tion selects the value of the vector from a subset of the parameter space which is itself a subset of the qdimensional Euclidean space q, so that the value of the function NQ is minimized (Hall 2005). The GMM estimator is asymptotically normal with variance matrix consistently estimated by: 11 '''''' GMM NNNNVNXZWZXXZWSWZXXZWZX (319) PAGE 120 120 where 1' 1 N i iii iSNZZ A Whiterobust estimate of S is obtained by assuming independence over i in the residual vector i where iii y X The standard errors obtained from Equation (319) are robust to hetero skedasticity and serial correlation over time (Cameron and Trivedi 2005). The choice of the weighting matrix remains an integral component of the estimation procedure. The matrix NW determines how the information in the instruments based upon the moment condition is weighted in the estimation of the parameters. There is little guidance on how to select NW, and so estimation of Equation (317) proceeds in two stages. First, the weighting matrix is computed as 1 NiiWZZ in the initial stage of the GMM estimator. Then GMMV is estimated next based on the first stage estimation of GMM. Thus, in the second stage, the first stage estimate of GMMV is used in the second stage GMM estimator based on the weighting matrix now computed as 1 NWS A numerical optimization routine is typical ly necessary to estimate the panel GMM estimator. While the NewtonRaphson method is th e usual choice, as noted by Hall (2005), this routine and others does not guara ntee a global optimum has been achieved, even after a large number of iterations. The numerical procedure used here is th at of Nelder and Mead, which performs well for nonlinear and nondifferentia ble functions. Since this method uses only function values, computing a large number of itera tions can be slow, but is also robust. The Nelder and Mead simplex algorithm is also preferred because of the degree of nonconvexities in quasihyperbolic functions, as noted by Laibson, Repetto, and Tobacman (2007). All parameter PAGE 121 121 estimates were obtained using R version 2.6.0 so ftware. The procedural code for each panel region is available in the Appendix. Instrument Selection and Identification The choice of instruments remains a crucial, but problematic, aspect of GMM. Nearly any variable that is correlated with a regressor but in dependent of the error term can be selected as an instrument, thus ensuring the overidentifying re strictions introduced by Hansen (1982) are in fact met. A benefit of the two step procedure of panel GMM estimation outlined above is that the estimator allows for the instrumental variable matrix Z to include regressors in the X matrix as well, thus aiding identification. Additionally, a benefit of panel data over crosssectional data is the availability of a weak exogeneity assumption: 0isitEze, st and 1,, tT As noted by Cameron and Trivedi (2005), this condition arises in models involving rational expectations and in mode ls of intertemporal choice, lik e in the present paper. The condition given by 0isitEze allows for lagged or lead values of regressors as instruments. The number of overidentifying restrictions is given by rK A test of the overidentifying restrictions is given by: 1' 1111 GMMGMMGMM NNNN NNiiNiiJQ ZSZ (320) The NJ statistic converges to a 2 distribution with rK degrees of freedom. The null hypothesis is that the moment condition in Equation (316) is true, that is, ':0oiiHE If the value of the test statistic results in rejection of the hypothesis, then model misspecification may be the case. If the NJ statistic is large, then another pos sibility could be the presence of endogenous instruments, that is instruments that are correla ted with the error term. PAGE 122 122 Problems of weak identification occur when the moment condition in Equation (316) is nonzero but very small. Under such a situa tion the moment condition provides very little information about the parameter vector, calling in to question the reliability of the estimates. However, one cannot take an insignificant statisti c to imply that weak identification is not a problem. Although recent work has examined pos sible procedures for identifying and handling weak instruments, such as Stock, Wright, and Yogo (2002), there is no firm method to correct for weakly identified instruments and so the NJ test remains the best available tool for examining model specification. Clearly, caution must be exercised regardless of the test results. Chapter Summary This chapter laid the theoretical foundation for empirical analysis. A net present value model for farmland is developed. The structur al equation for the annual change in farmland values is obtained. The reduced form parame ters for the quasihype rbolic and exponential discount factors, and respectively, are solved in an i ndentified framework. The empirical strategy is to first estimate the reducedform in E quation (39). Second, the structural parameters identifying the exponential and quasihyperbolic discount factors are obt ained through Equation (310) and Equation (311), and th en appropriate hypothesis tests ar e conducted. Hence, testing for hyperbolic discounting is a twostep process. This chapter also worked out the econometri c method used in the analysis, describing the linear panel GMM. Particular at tention was paid to the choice of the weighting matrix and the numerical procedure used to obtain the estimated. The choice of instruments was described as well as the overidentification test explained. In the next chapter, the data are described and the primary estimation results are presented for the major U.S. agricultural regions. PAGE 123 123 CHAPTER 4 DATA AND RESULTS To exam ine the possibility of hyperbolic disc ounting in U.S. farmland, annual observations on asset values and returns to agricultural assets are used. The data represent a 43 year panel from 1960 to 2002 of the nine majo r agricultural regions of the United States. The data come from the U.S. Department of Agricultures (U SDA) Economic Research Service (ERS). The primary source of data is the US DAs state level farm balance sh eet and income statement. A detailed description of the agricultural regions and the variables are discussed first. Then the estimation results including estimates of the discount parameters are presented. Data and Variable Description The nine regions investigated include the Appalachian states (Kentucky, North Carolina, Tennessee, Virginia, and West Virginia), Corn Be lt states (Illinois, Indiana, Iowa, Missouri, and Ohio), the Delta states (Arkansas, Louisiana, and Missouri), the Great Plain states (Kansas, Nebraska, North Dakota, Oklahoma, South Dakota, and Texas), the Lake states (Michigan, Minnesota, and Wisconsin), the Mo untain states (Arizona, Colorado, Idaho, Montana, Nevada, New Mexico, Utah, and Wyoming), the Northe ast states (Connecticut, Delaware, Maine, Maryland, Massachusetts, New Hampshire, New Jersey, New York, Pennsylvania, Rhode Island, and Vermont), the Pacific states (California, Or egon, and Washington), a nd the Southeast states (Alabama, Florida, Georgia, and South Carolina). Other regional definitions exist, such as those defined by Theil and Moss (2000). However, th e USDA/ERS defined regions are used since they remain the most commonly used. The dependent variable is the first differen ce of farmland values, which represents the annual change in the value of farmland for each state. The independent variables are annual farmland values and annual farmland returns. Farmland values are defined as the value of PAGE 124 124 farmland per acre. The definition of farmland returns is gross revenues per acre less the expenditure on variable inputs and follows Mi shra, Moss and Erickson (2007). While many studies opt for a more complete specification of re turns, such as the aggregate series of imputed returns as defined in Melichar ( 1979), the definition used in this dissertation is preferred. As revealed in Mishra, Moss, a nd Erickson (2004), measurement e rror problems are possible in more complete formulations of imputed retu rns if quasifixed assets are present. The data used in the analysis are the nominal data. The use of nominal data is preferred over the real data for two main reasons. First, the deflated real data has a higher variance, indicating more radical changes in both farmland values and returns between years. Secondly, and most importantly, the results are sensitive to the choice of deflator used. The estimation results were obtained under three different defl ators: the consumer price index (CPI), the producer price index (PPI), and personal consumption expenditure index (PCE). The values of the discount parameters vary de pending upon the choice of deflator used. In order to focus on the rate of time preference and avoid a debate on the best choice of deflator, the nominal data is used. Figure 41 through Figure 49 s how the historical trend in the annual change of farmland values from the 1960 to 2002 time period for nine farm production regions (Appalachian, Corn Belt, Delta, Lake, Great Plains, Mountain, Paci fic, Northeast, and Southeast). Within each region, states tend to follow the same pattern in the change of land values over time, that is, the year to year fluctuations by st ate within each region tend to be closely followed. However, notable exceptions can be discer ned. Virginia departs substan tially from the other Appalachian states between 1988 and 1991. Both California in Figure 44 and Florida in Figure 45, deviate from the other states in their panel region over the time horizon. Florida and California are quite PAGE 125 125 distinct in the agricultu ral products each state produces, as well as the urban pressures generated from within each state compared to others the Southeast and Pacific regions. The overall pattern between agricultural regions is similar, namely the clear representation of the boom/bust cycle, which occurred from around 1970 to 1985. However, the magnitude of the fluctuations for each region are notably differe nt, hitting the Corn Belt and Delta states the hardest followed by the Lake and Pacific states next. Land values for the Appalachian, Great Plain, and Southeast states were comparatively the least hurt during the boom/bust cycle. The Northeast states have the widest variation in land valu es over the 43 year time period. This is attributed to several reasons, but primarily because urban pressures are greatest for this region. Further, farmland and agriculture are not the highest value activity for land in the Northeast states. Also, the agricultural industrie s within the Northeast vary widely. As mentioned earlier, the selection of instrume nts can be a challenging task. Instruments ideally satisfy the orthogonality condition in Equation (316). Good instruments represent the nature of how land is valued a nd the expectations of the econo mic agent. For these reasons, lagged values of farmland values and return on assets are preferred in struments. Not only do they reflect the expectations of the agent, but they are known to both th e econometrician and the farmland agent at the current time, making for a good choice of instrument (Chavas and Thomas, 1999). In addition to the regressors, the instrume nt set includes squared terms and lagged terms of land returns and land values, for a tota l of four overidentifying restrictions. Estimation Results Equation (39) is estimated using annual aggreg ate panel data for nine agricultural regions of the United States (Appalachia n, Corn Belt, Delta, Great Plai ns, Lake, Mountain, Pacific, Northeast, and Southeast) over a 43 year time period from 1960 to 2002. Since the dependent variable is a first difference, an observation is lost for each state per a panel reducing the sample PAGE 126 126 size by the number of states in the panel. A fixe d effects dummy variable approach is used in the panel GMM. While this approach does have the unfortunate effect of losing degrees of freedom depending upon the number of states in each pa nel, the inclusion of dummy variables for controlling state fixed effects remains a simple way of acquiring estimates of the parameters. Tests for heteroskedasticity and serial correl ation are conducted. Th e BreuschPagan test for heteroskedasticity is used in both the fa rmland values and returns series with the null hypothesis of no heteroskedasticity, and is distri buted Chisquare with two degrees of freedom providing a critical value of 5. 991 for each panel. The BoxLjung test for serial correlation is used on the change in farmland values series with the null hypothesis of no autocorrelation, and is distributed Chisquare with one degree of freedom providing a critic al value of 3.841. Finally, Hansens overidentifying restrictions Jtest is used to test whether the model is correctly specified with the null hypothesis that the overidentifying restrictions hold. Since there are four overidentifying restrictions and the test statistic has a Chisquare distribution, the critical value is 9.488. The critical values of all three test st atistics are based on a 5% level of significance. The results for each regional panel are discussed first, followed by a general discussion of the results overall. Appalachian States The Appalachian states consist of Kentucky, North Carolina, Tennessee, Virginia, and West Virginia. The total sample used in the analysis is 210, with the omitted dummy variable being Virginia. The BreuschPa gan (BP) test resulted in a te st statistic of 16.405, implying rejection of the null hypothesis and indicates heteroskedasticity in the farmland values and returns series. The BoxLjung (BL) test is next conducted and results in a test statistic of 42.290, which also implies rejection of the null hypothesis, indicating the presence of serial correlation in the change in farmland values series. Finally the test statistic for the overidentifying restrictions PAGE 127 127 test is 0.069, hence the overident ifying restrictions cannot be rejected. Sin ce heteroskedasticity and serial correlation are present in the data, and since the overide ntifying restric tions cannot be rejected, the discussion will focus on the GMM results. The parameter estimates are summarized in Table 41. The constant term is not significantly different from zero, as expected for a net present value model under rational expectations. The estimated co efficients on farmland returns (1 ) and farmland values (2 ) are of the expected sign with 2 being statistically significant while 1 is not. None of the estimated coefficients on the state dummy vari ables are significant. Based on the parameter estimates in Table 41, values of the exponent ial and hyperbolic discount parameters can be obtained through Equation (310) and Equation (3 11). The exponential discount factor is 0.939, and significantly different from zero. The hyperbolic discount parameter is 0.060, and is also significantly diffe rent from zero. However, if hyperbolic discounting is in fact present, will be significantly different from one. To test whether the estimates represent expone ntial or hyperbolic disc ounting, the implicit test of 21:1oH is first conducted. The calculated Fstatistic is 119.963, which implies the null hypothesis of exponential discounting can be rejected at any conventional level of significance. The explicit te st of hyperbolic discounting, 012:11H is next conducted resulting an Fstatistic of 102. 152, implying the null hypothesis of 1 can also be rejected. Again, the null hypothesis can be rejected at a ny significance level. The results lead to the conclusion that discounting is not exponential bu t hyperbolic in the Appalachian panel. Based on the estimates of and shortrun and longrun discount rates can be computed. Given the values of the discount parameters, the longrun rate of discount is PAGE 128 128 ln(0.939)6.3%and the shortrun discount rate is ln(0.0600.939)287.6% An interpretation of the estimated discount rates can be offered. Th e longrun discount rate of 6.3% says that the value of a dollar is worth 6.3% less in the long run than the present time or that a dollar in the long run is worth .063 cents less to you now. In other words, you value a dollar in the long run at about .94 cents right now. The shortrun disc ount rate can be interpreted similarly, though discount rates in excess of 100% are difficult to interpret. Suppose the shortrun discount rate was 75%. In this case you vi ew a dollar in the shor trun 75% less than you do right now. In other words, if you discount rate is 75%, then you value a do llar in the shortrun as 0.25 cents to you right now. With a shortrun discount rate of 287.6%, you view a dollar in the short run at 287.6% less than you do right now. Such a high disc ount rate attaches a negative value to a dollar in the present. The key result, however, is that not on ly are the shortrun and longrun discount rates significan tly different from each other, but the magnitudes starkly contrast one another and suggest that the initial time periods in the farmland values market are critical. Corn Belt States The Corn Belt states consist of Illinois, Iowa, Missouri, and Ohio. The total sample size used in the analysis is 210, with the omitted dummy variable being Missouri. The BP test resulted in a test statistic of 10.418, implying rejection of th e null hypothesis and indicating the presence of heteroskedasticity. The BL test is next conducted, resulting in a test statistic of 86.902, which also implies rejec tion of the null hypothe sis and indicates the presence of serial correlation. Finally, the test statistic for the ove ridentifying restrictions test is 0.069, implying that the moment conditions cannot be rejected. PAGE 129 129 The parameter estimates are summarized in Table 42. The constant term is not significantly different from zero, as expected for a net present value model under rational expectations. The estimated co efficients on farmland returns (1 ) and farmland values (2 ) are of the expected sign and both are statistically significant from zer o. None of the state dummies are statistically significant, though all are negative in sign. Based on the parameter estimates in Table 41, values of the exponential and hyperbo lic discount parameters can be obtained through Equation (310) and Equation (311). The exponential discount factor is 0.929, and significantly different from zero. The hyperbolic discount parameter is 0.584, and is also significantly different from zero. To test whether the estimates represent expone ntial or hyperbolic disc ounting, the implicit test of 21:1oH is first conducted. The calculated Fstatistic is 1.637, which does not exceed the critical value at (1, ) degrees of freedom of 3.840 at 0.05 level of significance. Hence, the null hypothesis of exponential discounti ng cannot be rejected. The explicit test of hyperbolic discounting, 012:11H, is next conducted resulti ng an Fstatistic of 1.417, implying the null hypothesis of 1 cannot be rejected. The results lead to the conclusion that discounting is not hyperbolic but exponential in the Corn Belt panel. However, longrun and shortrun discount rates can sti ll be computed based on the para meter estimates. Given the values of the discount parameters, the longrun rate of discount is ln(0.929)7.4% and the shortrun discount rate is ln(0.5840.929)61.2%. However, these discount rates are not significantly different from each other since 1 could not be rejected. PAGE 130 130Delta States The Delta states consist of Arkansas, Louisiana, and Misso uri. Land returns and land values are characterized by both heteroskedasticity since th e BP test statistic is valued at 18.210. The change in land values variable is serial correlated as evidenced by the LB test statistic value of 46.401. Table 43 lists the parameter estimates. The total sample size is 126 and Louisiana is the omitted dummy variable in the analysis. The overidentifying restrictions cannot be rejected in the GMM regression since the value of the Jte st statistic is 0.089, which is less than the critical value of 9.488 for a Chisquare distributed test statisti c with 4 degrees of freedom. In regards to the parameter estimates, the consta nt term is positive, bu t not significant. The coefficient estimate on farmland returns (1 ) is negative, as anticipated by the theoretical model, but also is not significantly di fferent from zero. The coefficient estimate on farmland values (2 ) is positive and significant. Neither of the coefficient es timates for the two state dummy variables are significant, though both are positive and si milar in magnitude. Based on the estimates of 1 and 2 the quasihyperbolic discount factor is 0.155, and is significantly different from one based on the explicit test of hyper bolic discounting since the value of the Fstatistic is 11.199. The exponential disc ount factor is 0.963, with the null hypothesis of no hyperbolic discounting for the implic it test being rejected since the value of the Fstatistic is 13.869. Thus, in addition to the Appalachian states, evidence of quasihyperbolic discounting is found in the Delta states. The resu lts imply that the longrun rate of discount is given by ln(0.963)3.8% and the shortrun rate of discount is ln(0.1550.963)190.2% Clearly, the shape of discounting is different for the Delta states between the longrun and shortrun, with this difference be ing statistically significant. PAGE 131 131Great Plain States The Great Plain states consist of Kansas Nebraska, North Dakota, Oklahoma, South Dakota, and Texas. Based on the value of the BP test statistic of 15.104, and on the value of the LB test statistic of 89.157, heteroskedasticity an d serial correlation are in fact present in this panel series. The parameter estimates are summari zed in Table 44, noting that the total sample size is 252 and that South Dakota is the omitted dummy variable. The critical value of the Hansen Jtest is 0.063, which indicates that the moment conditions are not incorrectly specified. Results are similar to the Corn Belt states: the constant term is not significant and none of the dummy variables are significant. Also, th e parameter estimates on farmland returns and farmland values are of the same sign and similar in magnitude. Based on the parameter estimates, the value of the expone ntial discount f actor is also 0.929 and the value of the hyperbolic discount parameter is 0.508, however the standard errors are notably smaller in magnitude than the Corn Belt results. Conducti ng the implicit test of hyperbolic discounting yields an Fstatistic of 6.389 wh ile the explicit test yields an Fstatistic of 5.064, both reject exponential discounting at the 5% level for (1, ) degrees of freedom and imply that is significantly different from one. Therefore, discounting is better re presented by a hyperbolic discount factor for the Great Plain states. Mo re importantly, the resu lts imply a significant difference between the longrun and shortrun rates of discount. The l ongrun discount rate is ln(0.929)7.4% and the shortrun discount rate is ln(0.5080.929)75.1% Lake States The Lake states consist of Michigan, Minneso ta, and Wisconsin. Heteroskedasticity and autocorrelation are indicated based the BP test st atistic of 14.463 for the va lues and returns series and the LB test statistic is 64.008 for the change in land values series. The parameter estimates PAGE 132 132 are summarized in Table 45. Total sample si ze for the Lake states is 126, with the omitted dummy variable being Michigan. Based on the choice of instrument, the Jtest statistic for this panel is 0.089, again suggesting that the overidentifying restrictions can not be rejected. The calculated value of the expone ntial discount factor is 0.890. The hyperbolic discount parameter is 0.546, with a critical value of the Fstat istic of 3.920 based on (1, 121) degrees of freedom and a 5% level of significance. The implicit Ftest statistic is 5.090 and the explicit Ftest statistic is 4.187, both s uggesting a rejection of exponentia l discounting and implying that again is significantly different from one. Thus the Lake states panel is better described by quasihyperbolic discounting. The long run rate of discount is ln(0.890)11.6% while the shortrun rate of discount is given by ln(0.5460.890)72.2% Mountain States The Mountain states consist of Arizona, Colo rado, Idaho, Montana, Nevada, New Mexico, Utah, and Wyoming. The omitted dummy variable in the analysis is Arizona and the total sample size is 336 observations. Th e value of the BP test statisti c, 13.434, implies that the land returns and values variables ar e heteroskedastic. The BL stat istic, with a value of 42.712, suggests that the change in la nd values series is serially correlated. The GMM regression estimates, presented in Table 46, yield a Jtest statistic of 0.055, and so overidentification is met. The constant term is negative and insignifi cant. The parameter estimate of farmland returns is negative and significant, while the estimate of farmland values is positive and significant. None of the stat e dummy estimates are significan t, though they differ in sign and magnitude. The value of the exponential discount factor is 0.897 and the value of the quasihyperbolic discount factor is 0.637. The implicit test, whic h has a null hypothesis of PAGE 133 133 exponential discounting, has an Fst atistic of 3.949, and so the null hypothesis is rejected at the 5% level of significance. The explicit test, which has no hyper bolic discounting as the null hypothesis, has an Fstatistic of 3.305, implying th at the null hypothesis cannot be rejected at the 5% level, but can be rejected at the 10% level. Based on the values of the discount parameters, the longrun rate of discount is ln(0.897)10.9% and the shortrun disc ount rate is given by ln(0.6370.890)56.8%. Again, evidence of quasihyper bolic discounting over exponential discounting is found. Northeast States The Northeast states consist of Connecticut Delaware, Maine, Maryland, Massachusetts, New Hampshire, New Jersey, New York, Pennsyl vania, Rhode Island, and Vermont. Rhode Island is the omitted dummy variable in the analys is, with 462 total observations in the sample. The BP test statistic of 82.065 i ndicates heteroskedasti city in the independent variables. The dependent variable is serially correlated, as ev idence by the BL test with a statistic of 106.067. The overidentifying restrictions cannot be rejected, based on a Jtest statistic of 0.047. The parameter estimates are summarized in tabl e 4.7. The constant term is positive and insignificant. The estimate on farmland values is positive and significant, which is contrary to the theoretical model prediction of a negative coeffi cient. The coefficient estimate is just barely significant at the 5% level, though significant none theless. The estimate on farmland returns is positive and significant. None of the state dummy variables are statistically significant. Based on the estimates of 1 and 2 the exponential discount parameter is 0.966 and the quasihyperbolic discount parameter is .107. The negative number on the quasihyperbolic parameter is a strange result, and is a statistica lly significant one. The implicit test yields a Fstatistic of 315.572 and the explicit test yields a Fstatistic of 278.590, which rejects exponential PAGE 134 134 discounting at any level of signi ficance. However, based on the negative value of the quasihyperbolic parameter, the implication of the hypothe sis tests is questionable. The results imply a negative shortrun discount rate of 227.07% and a long run discount rate of 3.5%. Most studies of farmland values ignore the Northeast region co mpletely, owing to the diverse nature of the land market and the weak agricultural sector in these states. Pacific States The Pacific states consist of Ca lifornia, Oregon, and Washington. The BP test statistic of 22.462 implies heteroskedasticity in the independen t variables and the BL test statistic of 61.317 implies autocorrelation in the dependent variable The parameter estimates are summarized in Table 48, noting that the omitted dummy variable is California and that the total sample size is 126 observations. The exponential discount parameter is 0.933 while the hyperbolic discount parameter is 0.361. The implicit and explicit te sts of hyperbolic discounting yield Ftest statistics of 8.881 and 6.563, respectively. Hence, at 5% level of significance and (1, 126) degrees of freedom, the null hypothesis of exponentia l discounting, that is the null hypothesis of 1, is strongly rejected. Agai n, evidence of hyperbolic disc ounting is found over exponential discounting. The implied longrun and s hortrun discount rates are ln(0.933)6.9% and ln(0.3610.933)108.8%, respectively. Southeast States The Southeast states consist of Alabama, Florid a, and Georgia. The total sample size used in the analysis is 168, with the omitted dummy va riable being Florida. The BreuschPagan test for heteroskedasticity resulte d in a test statis tic of 33.245, indica ting the presence of heteroskedasticity. To test for autocorrelation, a LjungBox test is conducted resulting in a test statistic of 56.392, which indicate s the presence of serial correlat ion. Hansens test of over PAGE 135 135 identifying restrictions resulted in a test statistic of 0.077. Sin ce the critical value is 9.488, the overidentifying restrictions cannot be rejected. The parameter estimates are summarized in Table 49. The constant term is not significantly different from zero, as expected. The estimated coefficients on farmland returns (1 ) and farmland values (2 ) are of the expected sign and both are statistically significant from zero. None of the state dummies are statistically significant. Th e exponential discount factor is 0.910. To test whether these calculations repres ent exponential discounting, the implicit test of 21:1oH is conducted. The calculated Fstatis tic is 9.900, which exceeds the critical value at (1, 162) degrees of freedom of 3.840 at 0.05 level of significance. Hence, the null hypothesis of exponential discoun ting is rejected. The hyperb olic discount factor is 0.313 with a computed standard error of 0.259. The explicit test of hyperbolic discounting, 012:11H, is next conducted resulting an Fsta tistic of 7.043, hen ce rejecting the null hypothesis of no hyperbolic discounting at any conve ntional level of significance. Further, the values of and imply a shortrun discount rate of ln(0.3130.920)124% and a longrun discount rate of ln(0.920)8.4%. Chapter Summary The data set used in the analysis was descri bed and the exact definition of the variables utilized was also provided. This chapter also presented the estimation results from the quasihyperbolic farmland value asset equation. Three ke y remarks regarding the nature of the results are warranted. First, a clear case has been made not only against the standard exponential discounting model, but for the presence of hyperbo lic discounting as a vi able alternative. Second, an important considera tion is that the discount parame ters were obtained without PAGE 136 136 constraining 0,1 or 0,1, a fact which should lend a dditional credibility to the estimates. Based on the values of the exponent ial and quasihyperbolic discount parameters, strong evidence has been found that the shortrun discount rate is both different and substantially larger than the long run discount rate. Lastly, estimates of the discount parameters suggest that is true for farmland values. This result is concordant with the existing body of studies that estimate or calibrate a quasihyperbolic discount parameter (Laibson et al. 1998; Angeletos et al 2001; Eisenhauer and Venture 2006; Ahumada and Garegnani 2007). Smaller values of both and imply a greater tendency of the farmerlandowner to consume now rather than in future periods and therefore tend to save and invest less. Lower values of will result in a larger jump between the first two time periods. This jump in th e value of the disco unt factor is what creates dynamic time inconsistent preferences. As 1 the discount factor converges to the standard exponential case. Smaller values of will result in a more bowedshaped discount factor implying a greater preference for immediate consumption. As gets closer to zero, m eaning the rate of time preference increases, the shape of the discount factor becomes more convex to the origin. The values of the shortrun and longrun disc ount rates for each regi on are summarized in Table 410, along with the value of the Fstatistic for statistical difference from the explicit test of hyperbolic discounting. Most of the l ong run discount rates ar e below 10%, which is reasonable when compared to prevailing interest rates in the land mark et, although on the higher end. Data obtained from the USDA/ERS on the av erage nominal interest rate in the U.S. on farm business debt yields a national average of a bout 3.92% for 19602002 time period. As reflected in Table 410, the shortrun disc ount rates are extremely high and cover a wide range in values from 57% for the Mountain states to 288% for th e Appalachian states. F our regions have short PAGE 137 137 run discount rates under 100%, wh ich include the Corn Belt, Gr eat Plain, Lake, and Mountain states. Three regions have shortrun discount rates above 100%, which include the Delta, Pacific, and Southeast states. The Appalachia n region has a shortrun discount rate of over 200% and the Northeast region ha s a negative discount rate be low 200%. Note that the difference in the shortrun and l ongrun discount rate is not statis tically significant for the Corn Belt region and is only statistically significant at the 10% level for the Mountain region. In the next and final chapter, a more indepth commen tary on the meaning of the results is presented along with concluding statements. PAGE 138 138 150 100 50 0 50 100 150 200 25019 61 1 964 196 7 19 70 1 973 197 6 19 79 1 982 198 5 1 988 1991 199 4 1 997 2000Dollars per acre Kentucky North Carolina Tennessee Virginia West Virginia Figure 41. Change in farmland va lues for the Appalachian states 500 400 300 200 100 0 100 200 300 40019 61 1 964 196 7 19 70 1 973 197 6 19 79 1 982 198 5 19 88 1 991 199 4 19 97 2 000Dollars per acre Illinois Indiana Iowa Missouri Ohio Figure 42. Change in farmland values for the Corn Belt states PAGE 139 139 500 400 300 200 100 0 100 200 300 400 50019 61 1 964 196 7 19 70 1 973 197 6 19 79 1 982 198 5 19 88 1 991 199 4 19 97 2 000Dollars per acre Arkansas Louisiana Mississippi Figure 43. Change in farmland values for the Delta states 200 150 100 50 0 50 100 15019 6 1 1 96 4 1967 19 7 0 1 97 3 1976 19 7 9 1 98 2 1985 19 8 8 1 99 1 1994 19 9 7 2 00 0Dollars per acre Kansas Nebraska North Dakota South Dakota Oklahoma Texas Figure 44. Change in farmland va lues for the Great Plain states PAGE 140 140 250 200 150 100 50 0 50 100 150 200 25019 6 1 1 96 4 1967 1 97 0 197 3 19 7 6 1 97 9 1982 19 8 5 198 8 1991 1 99 4 199 7 20 0 0Dollars per acre Michigan Minnesota Wisconsin Figure 45. Change in farmland values for the Lake states 200 100 0 100 200 300 400 5001961 1 9 64 19 6 7 1 9 70 19 73 1976 1 9 79 19 8 2 1985 19 88 19 9 1 1 9 94 19 97 2000Dollars per acre Arizona Colorado Idaho Montana Nevada New Mexico Utah Wyoming Figure 46. Change in farmland values for the Mountain states PAGE 141 141 400 200 0 200 400 600 800 1000 1200196 1 1 96 4 1 96 7 19 7 0 19 7 3 1976 197 9 198 2 1 98 5 1 98 8 199 1 1 99 4 1 99 7 20 0 0Dollars per acr e Connecticut Delaware Maine Maryland Massachusetts New Hampshire New Jersey New York Pennslyvania Rhode Island VermontFigure 47. Change in farmland values for the Northeast states 200 150 100 50 0 50 100 150 200 250 30019 6 1 1 96 4 1967 19 7 0 1 97 3 1976 19 7 9 1 98 2 1985 19 8 8 1 99 1 1994 19 9 7 2 00 0Dollars per acre California Oregon Washington Figure 48. Change in farmland values for the Pacific states PAGE 142 142 100 50 0 50 100 150 200 2501 96 1 1964 19 6 7 1 97 0 1973 19 7 6 197 9 1982 1 98 5 198 8 19 9 1 1 99 4 1997 20 0 0Dollars per acre Alabama Florida Georgia South Carolina Figure 49. Change in farmland values for the Southeast states PAGE 143 143 Table 41. Appalachian states result Variable Parameter estimate Standard error tstatistic pvalue 0 0.168 10.807 0.016 0.494 1 0.056 0.080 0.701 0.242 2 0.061 0.011 5.771 0.000 3 (KY) 0.508 10.791 0.047 0.481 4 (TN) 1.923 12.310 0.156 0.438 5 (VA) 0.140 12.259 0.011 0.495 6 (WV) 0.086 12.798 0.007 0.497 0.939 0.011 0.060 0.086 N = 210 Table 42. Corn Belt states result Variable Parameter estimate Standard error tstatistic pvalue 0 19.499 17.525 1.113 0.133 1 0.543 0.316 1.717 0.043 2 0.071 0.020 3.615 0.000 3 (IL) 13.529 23.407 0.578 0.282 4 (IN) 2.722 23.020 0.118 0.453 5 (IA) 1.576 24.489 0.064 0.474 6 (OH) 3.057 23.066 0.133 0.447 0.929 0.020 0.584 0.349 N = 210 Table 43. Delta states result Variable Parameter estimate Standard error tstatistic pvalue 0 3.306 15.068 0.219 0.413 1 0.150 0.240 0.624 0.266 2 0.037 0.028 1.333 0.091 3 (AR) 10.922 14.785 0.739 0.230 4 (MS) 10.467 13.815 0.758 0.224 0.963 0.028 0.155 0.252 N = 126 PAGE 144 144 Table 44. Great Plain states results Variable Parameter estimate Standard error tstatistic pvalue 0 0.242 5.472 0.044 0.482 1 0.472 0.195 2.419 0.008 2 0.071 0.019 3.714 0.000 3 (KS) 4.162 6.662 0.625 0.266 4 (NE) 0.572 6.798 0.084 0.466 5 (ND) 0.073 6.452 0.011 0.495 6 (OK) 0.354 7.000 0.051 0.480 7 (TX) 0.460 7.177 0.064 0.474 0.929 0.019 0.508 0.218 N = 252 Table 45. Lake states results Variable Parameter estimate Standard error tstatistic pvalue 0 0.163 14.588 0.011 0.496 1 0.486 0.191 2.551 0.005 2 0.110 0.019 5.940 0.000 3 (MN) 11.146 12.858 0.867 0.193 4 (WI) 10.203 14.100 0.723 0.235 0.890 0.019 0.546 0.222 N = 126 PAGE 145 145 Table 46. Mountain states result Variable Parameter estimate Standard error tstatistic pvalue 0 1.865 5.707 0.327 0.372 1 0.571 0.173 3.296 0.000 2 0.103 0.013 8.209 0.000 3 (CO) 2.711 7.035 0.385 0.350 4 (ID) 4.871 8.854 0.550 0.291 5 (MT) 2.140 7.032 0.304 0.380 6 (NV) 0.154 7.031 0.022 0.491 7 (NM) 0.987 7.128 0.138 0.445 8 (UT) 0.825 6.820 0.121 0.452 9 (WY) 1.280 7.124 0.180 0.429 0.897 0.013 0.637 0.200 N = 336 Table 47. Northeast states result Variable Parameter estimate Standard error tstatistic pvalue 0 8.676 20.627 0.421 0.337 1 0.103 0.065 1.604 0.054 2 0.034 0.006 5.605 0.000 3 (CT) 3.469 23.041 0.151 0.440 4 (DE) 5.109 23.685 0.216 0.414 5 (ME) 2.288 24.870 0.092 0.463 6 (MD) 2.850 24.014 0.119 0.453 7 (MA) 2.379 23.496 0.101 0.460 8 (NH) 0.355 24.993 0.014 0.494 9 (NJ) 2.204 23.982 0.092 0.463 10 (NY) 0.689 24.768 0.028 0.489 11 (PA) 6.380 24.306 0.263 0.396 12 (VT) 9.405 24.744 0.380 0.352 0.966 0.006 0.107 0.066 N = 462 PAGE 146 146 Table 48. Pacific states result Variable Parameter estimate Standard error tstatistic pvalue 0 18.803 16.802 1.119 0.132 1 0.337 0.224 1.507 0.066 2 0.067 0.027 2.517 0.006 3 (OR) 19.901 16.794 1.185 0.118 4 (WA) 11.209 15.045 0.745 0.228 0.933 0.027 0.361 0.249 N = 126 Table 49. Southeast states result Variable Parameter estimate Standard error tstatistic pvalue 0 10.841 11.657 0.930 0.176 1 0.280 0.142 1.969 0.024 2 0.078 0.021 3.652 0.000 3 (AL) 3.429 11.492 0.298 0.383 4 (GA) 0.356 11.436 0.031 0.488 5 (SC) 5.252 11.394 0.461 0.322 0.922 0.021 0.304 0.161 N = 168 Table 410. Discount rates by region Region Shortrun discount rate L ongrun discount rate Fstatistic1 Appalachian 287.6% 6.3% 102.152 Corn Belt2 61.2% 7.4% 1.417 Delta 190.2% 3.8% 11.199 Great Plain 75.1% 7.4% 5.064 Lake 72.2% 11.6% 4.187 Mountain3 56.8% 10.9% 3.305 Northeast 227.0% 3.5% 278.590 Pacific 108.8% 6.9% 6.563 Southeast 124.0% 8.4% 9.900 1 The critical value of the Fstatistic with (1, ) degrees of freedom and 5% level of significance is 3.840. 2 Discount rates are not statistically different from one another at conventional levels 3 Discount rates are statistically different from one another at the 10% level of significance. PAGE 147 147 CHAPTER 5 CONCLUSION AND FUTURE WORK This chapter presents a more thorou gh interp retation of the empirical results with three main goals. The first goal of this chapter is to foster a greater understa nding of the results and offers a comparison to other relevant studies on time discounting while also focusing on the major weaknesses and limitations of the study. Second, a discussion on the importance and intuition of the results is also presented with an emphasis on the practical implications of the results and the major relevance to policy and extens ion. Finally, suggestions for future research are provided. The dissertati on concludes with a summary. Comparisons and Limitations The empirical results prompt three key questi ons regarding the estima tes of the shortrun and longrun discount rates presente d in Table 410. First, why ar e there differences in discount rates within regions? Second, why are there differe nces in discount rates across regions? Lastly, and most importantly, why are the shortrun discount rates so high comp ared to the longrun discount rates? The first two questions can be addressed rather succinctly while the last one requires more thoughtful consideration. The first question regarding differences in s hortrun and longrun discount rates within a specific region relates to the crux of the dissertation regard ing timeinconsistency and nonconstant discounting. Evidence in numerous fields of economics has found that individuals are not timeconsistent in their intertemporal preferences but rather timeinconsistent. This timeinconsistency is the result of two related but distinct phenomena, preference reversals and intrapersonal games, as explained in Chapter 1. Time inconsistency implies a nonconstant and declining discount rate through time, meaning that discount rates in the shortrun will be larger than discount rates in the longrun. Quasihyperbolic discounting is a declining discount PAGE 148 148 function and represents one way of modeling tim einconsistent preferences. The question regarding why differences in the shortrun and longrun discount rates ex ist is answered by the presence of timeinconsistent preferences. Chap ter 1 presented the reasoni ng and intuition as to why a landowner might be characterized by timeinconsistency and why quasihyperbolic discounting might be present in the land market with a particular emphasis on land conversion models. The second question pertaining to why the s hortrun and longrun discount rates differ between regions is best explai ned by the heterogeneity of th e land market, the landowner, and the regional economy. Variables affecting the va lue of land not accounted for in this analysis will have a direct effect on the discount rate whether such vari ables are landspecific, landownerspecific, or economyspecifi c. Hedonic characteristic s of the land, such as parcel size, location, and soil quality differ greatly both within and be tween major agricultural regions, and have been found in a number of studies to directly affect farmland values (Carri onFlores and Irwin 2004; Taylor and Brester 2005; Livanis et al. 2006). For example, Livanis et al. (2006) estimate a positive affect of median singlefamily house value on U.S. farmland values. Landownerspecific characteristics also vary substantially betw een regions. Barnard and Butcher (1989) show how the demographic char acteristics of the landowner, such as age, education, income, and other traits affect the perceived present va lue of undeveloped land. Since demographic variables affect the perceived present value, the discount rate is also going to differ not only between individuals, but between regi ons as well. Eisenha uer and Ventura (2006, p.1229), using international survey data, find th at hyperbolic discounter s tend to be younger, poorer, less educated, bluecollar, unemployed, individuals from larger cities, and those in southern regions working in agriculture. In a study using survey data from a developing PAGE 149 149 country, Robberstad and Cairns ( 2007) find that individuals who obtain most of their income from farming will have higher discount rates th an individuals whose main income is nonfarm related. To the extent these demographic trai ts exist and differ between regions, differences in the discount rate, particularly the shortr un discount rate, are to be expected. Lastly, the state of the econom y will affect intertemporal preferences and the implied discount rate. In an influential study, Lawrance (1991) uses data from the Panel Study of Income Dynamics and finds evidence that time preferences are positively related to household income. In particular, the author conc ludes that poor households tend to have relatively higher discount rates. In a metaanalytic review of time di scounting studies, Percoco and Nijkamp (2007) find that the discount rate is negatively related to the level of per capita GDP implying that as GDP rises on a per capita basis, the di scount rate falls. Since both household income and state level GDP differ between the agricultura l regions studied in this dissert ation, and even between states in each region for that matter, differences betw een both the shortrun and longrun discount rate are to be expected. This brings us to the third and final ques tion regarding the magnit udes of the shortrun discount rates and why they are so stark compared to the longrun discount rates. The results of hypothesis tests imply a formal reje ction that the shortrun discount rate is equal to the longrun discount rate, a new result in the literature on land values. The results in this dissertation are unique and there are no baseline estimates for which to make exact comparisons. Although one would be hardpressed to defend the estimated s hortrun discount rates given in Table 410 in actual financial analysis, the pres ence of large discount rates is not without precedent in the empirical literature on both timeconsistent discounting and timeinc onsistent discounting. PAGE 150 150 Important studies on constant discounting in energy consumption and household durables have revealed large exponential discount rates. Hausman (1979), using data on air conditioner purchases, finds personal discount rates ra nging from 5% to 89% depending on household income, with higher income households having lower discount rates. Houston (1983), using a survey method on energy appliance demand, finds discount rates in the range of 20% to 25% also depending on income. Gately (1980), usi ng data on refrigerator purchases, finds much higher discount rates ranging from 45% to 300% with most of the discount rates exceeding 100%, depending upon the energy efficiency and brand name of the refrigerator. Using data on military personnel and retirement decisions, Warn er and Pleeter (2000) estimate personal discount rates in the range of 0% to 59%, agai n depending on the individual characteristics. One of the only known studies to estimate the rate of time preferen ce in an agricultural context is Lence (2000). Using c onsumption and asset return data for U.S. farmers, he estimates the standard discount factor to be 0.962 implying a longrun disc ount rate of 3.92%. However, Lence (2000) assumes timeconsiste ncy and only examines time preference with standard exponential discounting. While there are no existing st udies that examine land values in this context for which to compare estimates, the values of the quasihyperbolic discount parameter, obtained here in the range of 0.06 to 0.64 (including even one negative value for the Northeast) is lower than many estimates presented in th e experimental economics literature, typically calculated between 0.80 and 0.90 (Benzion, Rapoport, and Yagil 1989; Coller and Williams 1999; Eisenhauer and Ventura 2006). The findings in this dissertation are similar with the few studies that have estimated quasihyperbolic time preferences from field data using a structural model. Laibson, Repetto, and Tobacman (2007) use individual level data on cr edit card borrowing, consumption, income, and PAGE 151 151 retirement savings to estimate the discount factors finding that 0.958 and 0.703 both statistically significant. Their results imply a shortrun discount rate of 39.53% and a longrun discount rate of 4.29%. Fang and Silverma n (2004) use panel data from the National Longitudinal Survey of Youth (NLSY) on welfar e participation for single mothers and estimate how timeinconsistency affects the decision to takeup welfare. The authors estimate the discount factors to be 0.875 and 0.338 implying a shortrun discount rate of 121.82% and a longrun discount rate of 12.78%. Paserman (2004) uses data on unemployment spells and job search duration from the NLSY to estimate time preferences. In a sample of lowwage workers his estimates of the discount parameters, 0.996 and 0.402 imply shortrun and longrun discount rates of 91.53% and 0.40%, respec tively. His findings s how the rate of time preference increases as the wage level increases The highwage sample in Paserman (2004) yields discount parameter estimates of 0.999 and 0.894 implying a shortrun discount rate of 11.31% and a longrun discount rate of 0.10%. Clearly, the literature offers an ecdotal explanations as to wh y the shortrun discount rates stand in such sharp contrast to th e longrun discount rates. Given th at the primary data set in this dissertation is agricultural land values, the ev idence from both Ventura (2006) and Robberstad and Cairns (2007) regarding highe r discount rates for agriculture lends additional explanation to the high shortrun discount rates th at were obtained. Further, to the extent that landowners and farmers have lower than average incomes, highe r shortrun discount rate s may be expected, as described in Paserman (2004). The durable goods and energy demand literature also gave precedent for high discount rates. Since land can be considered a durable good itself, if not an infinitely lived asset, then high di scount rates also make sense. PAGE 152 152 Despite the evidence of high shor trun discount rates in the l iterature, the question remains as to why such high shortrun discount rates have been found in this dissertation on farmland values. A simple mathematical explanation is o ffered first. The rent generated from land will affect the discount factor, and th e regions differ greatly in terms of the income generated from agricultural use. Careful reexam ination of the parameter estima tes in Table 41 through Table 49 with the estimated discount ra tes in Table 410 reveals that as the estimated coefficient on farmland returns increases in magnitude the value of the shortrun discount ra te decreases in size. This of course is not coincidence since the relationship is an artifact of the theoretical model. In Equation (311), the quasihyperbo lic discount parameter increases as the estimated coefficient on farmland returns increases. A higher quasih yperbolic discount parame ter reflects greater individual patience, implying a lower shortrun discount rate. When land returns are high, the shortrun discount rate is lower, implying a gr eater level of shortrun patience. When land returns are low, the shortrun disc ount rate is high, implying a lo wer level of shortrun patience, or a stronger desire for instanta neous gratification. Generally, land returns have been on an upward trend since 1960, however for many of th e regions farmland returns have decreased substantially over the past 10 to 20 years, depend ing on the region. To the extent returns have fluctuated and fallen, the value of the quasihyperbolic parame ter and the implied shortrun interest rate will be biased upward. Future work should examine more closely the relationship between expected returns from land and the rate of timepreference. However, more rigorous explanations for the relatively high shortrun discount rates stem from the formulation of the net present value model in Chapter 3. The simplistic assumptions made in the theoretical and empirical model regarding risk neutrality, rational expectations, and no inflation are possible explana tions for the high shortrun di scount rates reported in Table 4 PAGE 153 153 10. First, the presence of inflation will result in an upward bias in the estimated discount rate. Although inflation is not accounted for in the empiri cal analysis, there is reason to suspect that inflation and expectations of inflation will affect not only the present value of land, but both the shortrun and longrun discount rates. Cons ider the argument outlined in Frederick, Loewenstein, and ODonoghue (2002) stating that the presence of inflation creates an upward bias in discount rate estimates. The authors r eason, quite simply, that a sum of money today is not worth the same sum if inflation is expected, in fact that sum of money will be worth less. In other words, should you expect infl ation to occur in the next five years, spending $500 today will generate more utility from consumption than sp ending $500 in the future. Since inflation gives an incentive to value future reward s less than present rewards, th e discount rate will be higher. As Frederick, Loewenstein, and ODonoghue (2002) correctly state, th e magnitude of this upward bias in the discount rate will depend on the individual e xpectations regarding the extent of inflation. In the farmland markets, infla tion has been nonconstant across the time horizon and as high as 13% in some cases (Moss 2003). Lloyd (1994) argues that la nd is purchased as a hedge against inflation based on investor percepti ons of land as a resilient asset, capable of holding value in real terms am idst inflationary periods. In fact, there is evidence to suggest that inflation may be the most important factor influencing farmland values. Moss (1997) exam ines the sensitivity of farmland values to changes in inflation, asset returns, and the cost of capital by employing The ils (1987) statistical formulation of information. The basic premis e of Moss (1997) states that changes in an independent variable that accoun t for more volatility in farmland prices will imply larger fluctuations in farmland prices than an independent variable that accounts for less volatility, where the total amount of informati on is described in bits. Accord ing to the empirical results in PAGE 154 154 Moss (1997), inflation contribute s the most to the explanation of changes in U.S. farmland values. Even more revealing is how inflation co ntributes to the explanation of farmland prices by region. The Appalachian region had inflation as the largest contribution of information to farmland values while the Northeast region had the lowest contribution of information to farmland values (Moss 1997). The results in Table 410 would seem to underscore the importance of inflation in discount rate estimates The Appalachian region had the highest shortrun discount rate, while the Northeast region ha d a negative discount ra te, providing at least anecdotal evidence that inflation ha s resulted in an upward bias in the estimated discount rates in this dissertation. Further, the information contribution of inflation is far more uniform in the five states that compose the Appalachian region than the eleven states that compose the Northeast region. As mentioned earlier, the data used in the an alysis to obtain the di scount rates in Table 410 are based on nominal rather than real data and so inflationary affects are present in the data. In fact, this represents an advant age to using nominal data over real data since real data loses the information present in nominal data from inflatio n. Further, the use of real data involves a debate regarding which deflator to use, the PPI, the CPI, or the PCE index. Unreported results reflected a sensitivity of the discount rates to th e choice of deflator used and so nominal data were preferred to detract from a debate on the be st method of deflating th e data. At the present time, there is no well defined land price deflator and none of the current price deflators seem appropriate for the land market. Regardless, future work should attempt to extract the effects of inflation from the farmland values data and determine the impact on shortrun and longrun discount rates. PAGE 155 155 Second, the presence of risk and uncertainty will also result in an upward bias in the estimated discount rate. Recall th at the empirical results obtaine d in this dissertation and the discount rates presented in Table 410 are gene rated under the assumption of risk neutrality. Frederick, Loewenstein, and O Donoghue (2002) point out that since a future reward received with some degree of delay is often associated wi th at least some degree of uncertainty, the exact effect of the rate of time pref erence on the size of the discount ra te is a complicated relationship that is difficult to measure. However, there is evidence in both experimental data and field data to suggest that accounting for risk aversion reduc es the size of estimated shortrun and longrun discount rates. Using experime ntal data from the 2000 Bank of Italy Survey of Household Income and Wealth, Eisenhauer and Ventura (2 006) find that contro lling for risk aversion reduces estimates of the hyper bolic discount rate by severa l orders of magnitude. Anderhub et al. (2001) utilize an experiment on college undergraduate s to elicit discount rates and examine how risk and time preference s are interrelated. The authors find clear evidence that higher degrees of risk aversion are associated with lower values of the discount factor, which implies a higher discount rate. Hence, the authors find a positive relationship between the value of the risk aversion coefficien t and the rate of time preference. Hence, higher risk aversion means heavier discounting, meani ng risk averse individuals tend to be more impatient. Anderhub et al. (2001) argue that the heavier discounti ng seen in more risk averse individuals is the result of th e uncertainty that surrounds future payoffs as opposed to immediate payoffs. Andersen et al. (forthcoming), using experimental data from Denmark, jointly estimate the coefficient of relative risk aversion (CRRA) and the disc ount rates assuming both a timeconsistent and timeinconsistent discount structure. In both case s, joint estimation of risk and PAGE 156 156 time preference results in significan tly lower discount rates congruent with market interest rates. Assuming risk neutrality and timeconsistency, the authors obtain a discount rate of 25.2%, but when risk aversion is accounted for, the estimated discount rate falls to 10.1%. When timeinconsistent preferences are modeled using a hyp erbolic discount factor evidence of declining discount rates are still obtained, but the magnit udes between the shortru n and longrun discount rates is substantially softened. Thus, in cont rast to Anderhub et al. (2001), who find a positive relationship, Andersen et al. (forthcoming) find a ne gative relationship be tween the degree of risk aversion and the rate of time preference. Th e authors attribute this finding to the fact that Anderhub et al. (2001) uses data on risk attitude s to impute the discount rate over money rather than over utility as in Andersen et al. (forthcoming). Estimated discount rates will be lower when defined over utility than when defined ove r money (Andersen et al. forthcoming). The high shortrun discount rates in this dissertation may be a result of the data, which are based on dollar values of farmland rather than utility. Using field data, Laibson, Repetto, and Tobacm an (2007) estimate several versions of the lifecycle consumption model in which the CRRA is either jointly estimated or imposed on the model. The estimates of and vary substantially in regards to both magnitude and statistical significance depending upon the risk assumption imposed. When the CRRA is jointly estimated the shortrun discount rate falls from 39.53% to 14.63% and the longrun di scount rate falls from 4.29% to 3.87%, with the estimated CRRA being about 0.22, which is quite low. The authors also impose values of the relative risk aversi on coefficient on the estimates, ranging from a CRRA of three to a CRRA of one. As the impose d value of the CRRA falls the estimated value of both and rise, which implies both lower shortrun and longrun discount rates. The results in Laibson, Repetto, and Tobacman (2007) would seem to imply that the greater the level PAGE 157 157 of risk aversion, the lower the rate of time pref erence, meaning that risk averse individuals are more impatient. This would seem to make sense if future rewards are re ceived with a greater amount of uncertainty than immedi ate rewards and is in accord w ith the results in Anderhub et al. (2001). Farmland markets and farmland values are especi ally sensitive to risk, and farmland is in general considered a ri sky investment (Hanson 1995). As explained in Moss, Shonkwiler, and Schmitz (2003), changes in risk affect the value of farmland over time, as evidenced by estimates of the certaintyequivalence parameter in a data set on U.S. farmland returns, interest rates, and values. Both Lence (2000) and Chavas and Thomas (1999) reject the hypothesis of risk neutrality and find evidence of risk aversion in the U.S. farmland market. Antle (1987) finds evidence to suggest that agricultu ral producers are risk averse w ith a risk premium as high as 25% of expected returns. The la rge shortrun discount rates presen ted in Table 410 could in fact be the result of not accounting for risk in the anal ysis. Future work should attempt to incorporate risk in models of farmland values and determin e the extent that short run and longrun discount rates are affected. Third, and finally, the high shortrun discount ra tes in Table 410 may be in part due to the possibly unrealistic assumption of rational expectations. Under rational expectations, landowners are assumed to correctly forecast, on average, future land rents using all relevant economic information without any systematic bias. However, some authors have attributed the shortrun failure of NPV models of farmland to be a violation of rationa l expectations (Lloyd, Rayner, and Orme 1991; Tegene and Kuchler 1991; Engsted 1998). Furthermore, the shortrun failure of the NPV model has also been noted in the literature on hous ing prices with strong rejection of the rational exp ectations hypothesis (Meese and Wa llace 1994; Clayton 1996). PAGE 158 158 A possible alternative to rational expectations is an adapti ve expectations formulation which assumes that landowners would base future expectations on land rents based on past land rents with the possibility of systematic bias due to stocha stic economic shocks. Chow (1989) compared both rational and adaptive expectations in a NPV model of stock prices and dividends. Based on the parameters estimates, the NPV under rational expectations ar e not consistent with the theory. Obtained values of the longrun discount rates, for example, are above 100% when rational expectations is assumed. The inc onsistent discount rates persist under rational expectations even when homoskedasticity is co rrected for and a timevarying discount factor included in the model. When adaptive expectations is assumed, the obtained longrun discount rates become much more reasonable and congruent with market interest rates. Chow (1989) concludes that when a correct model incorre ctly assumes rational expectations, then unreasonable parameter estimates are often the case There is empirical evidence of atypical longrun discount rates under assumptions of rational expectations in the farmland values lit erature. Lloyd, Rayner, and Orme (1991) use a structural model similar to the one in this dissertation to obtain values of the longrun discount rate from a reducedform parameterization of the structural model. The authors examine the NPV model of real farmland values in Engl and and Wales using both rational and adaptive expectations. The rational e xpectations model uncovers a re al longrun discount rate of 37.74% with a rejection of the null hypothesis of a positive re al rate of discount. In the authors adaptive expectations model, the real long run rate of discount is 2.38% and is much more in line with market interest rates. Tegene and Kuchler (1991) compare the NPV model of U.S. farmland values under both rational and adaptive expectations for the Corn Be lt, Lake States, and Northern Plain regions. PAGE 159 159 The estimated coefficients of land prices and rents in the rational expectations model imply discount rates above 100% and are inconsistent with the rational expectation hypothesis. In the authors adaptive expectation model, longrun di scount rates are found to be less than 7% and estimated parameters are consistent with the adaptive expectations hypothesis. The authors conclude with a strong rejection of rational expe ctations in favor of adaptive expectations. Promising future work remains in uncovering th e relationship between expectation formulations and time preferences. Importance and Implications Although the magnitudes of the shortrun disc ount rates presented in Table 410 are surprising, the results are not be ing recommended for use in financial analysis or in forecasting farmland values. As discussed above, the inco rporation of inflati on, risk, and adaptive expectations is suggested before such a recomme ndation is even contemplated. The results then are not so much important for the specific ma gnitudes of the shortrun and longrun discount rates presented, but are important for what th e differences in shortrun and longrun discount rates imply for the land market. In this sec tion, added intuition re garding the nature and importance of the results is offered. The value and practical use of the results are based on what they imply for landuse decisionmaking, what they offer for explaining the methodological failure of the NPV of farmland, and what they suggest in regards to la ndowner behavior. The results also have important policy and extension implications. First, the results are important for their imp lication and description of landuse decisions and investments. The fundamental interpretation of the large shortrun discount rates presented in Table 410 reflect the fact that landowners have a tendency to be shortrun in their thinking. Such a nearterm or shortrun dominate way of thinking would tend to re sult in decisions that sacrifice the future for the sake of the present. For example, suppose a landowner is facing a PAGE 160 160 decision of converting farmland to a more capital intensive use such as residential housing or an ethanol refinery plant with a la rge onetime monetary gain. Th e shortrun and longrun discount rates obtained would imply that the landowner may be inclined to hastily accept such an offer without full consideration of more remunerative future return s from farmland. In essence, a landowner with a higher shortrun di scount rate may prefer to sell hi s land now to an investor for a substantial instant return, rather than be forwardthinking in his behavior and wait 20 years for a more remunerative return. In a sense, the large shortr un discount rates suggest a sort of shortrun bandwagon that landowners have a tendency to ride. Recent inve stments in residential real estate and ethanol refineries have either failed to generate substantial retu rns or have resulted in financial losses. In some regard, a high shortrun discount rate w ould provide an explana tion into the occasional tendency of landowners to make shortsighted de cisions regarding land use. For example, the longrun future rewards of grainbased ethanol production is in a pr ecarious state, however, the shortrun desire for instant retu rns may drive the decision of la ndowners to either plant more corn or to construct an ethanol refinery on thei r land. The notion of quasihyperbolic discounting would seem quite appropriate given the frenzy on biofuels and the ensu ing land rush that has accompanied the phenomenon. A 2006 editorial in Nature Biotechnology described the surge of investment in biofuels, which figures in the billions of dollars in the Un ited States, as irrationa l exuberance. However, as of yet, no company has produced and sold etha nol in U.S. on any sort of mass level, raising doubts regarding the success of the current bu siness model (Waltz 2008). Furthermore, the importance of landuse decisions in the bioenergy industry is critical, underscoring the importance of the results in this dissertation. Land is the largest and most valued productive PAGE 161 161 input in ethanol production, and so time preferences regarding the value of land remain an important area of investigation. Given that land used in production of en ergy cannot be used in the production of food or other uses the decisions made for every parcel over time are of great importance not only to economists, but to landowners, policy makers, and consumers. Hence, one of the largest contributions of the results in this paper is a numeri cal description of the impulsiveness that may characte rize many landuse decisions. Second, the results provide an explanation fo r the significant shortrun deviations that occur in the discounted value formulation of fa rmland values (Falk and Lee 1998; Featherstone and Moss 2003). One explanation for the apparent disconnect between shortrun and longrun values is the presence of boom/ bust cycles in which farmland values change more dramatically than would be expected in response to an incr ease or decrease in return s (Schmitz 1995). Of course, the prevailing question remains: what cau ses boom/bust cycles in farmland values? In addition to transaction costs and timevarying risk premia, alternative explanations include the presence of quasirationality or rational bubbles (Featherstone a nd Baker 1987), market overreaction (Burt 1986), fads (Fal k and Lee 1998), and option va lues and hysteresis (Titman 1985; Turvey 2002). A possible explanation offered here is the presence of quasihyperbol ic discounting. The mix of longrun rationality with shortrun irrationality makes quasihyperbolic discounting an appealing explanation. The quasihyperbolic di scount factor explicitly models disconnects between the shortrun and longrun. Unlike other farm assets farmland is unique in that approximately 80% of the asset portfolio in U.S. agriculture is accounted for by farmland values (Moss and Schmitz 2003). Unlike assets that have a definite lifespan, such as capital equipment that depreciate substantially like tractors, farmland is typically characterized by an infinite PAGE 162 162 horizon. Since the life span of land as an asset is infinite, the choice of the discount factor plays a premier role, and may be subject to the shortrun jumps descri bed by quasihyperbolic discounting. With quasihyperbol ic discounting, the shortrun bubbl e nature of farmland values can persist, but equilibrium in th e long run can be achieved. Thus, during the life of the asset, especially during boom/bust cycles, the discount rate for farmland may change over time. During periods of economic boom, the rate of ti me preference decreases as available capital increases, implying more patience. During periods of economic bus t, the rate of time preference decreases as borrowing increases to cover fi nancial losses, implying a greater level of impatience. The results are also important for their de scription of landowner behavior and, in particular, what they imply for farmland investment and savings. The results also further the Golden Eggs investment hypothesis presented by Laibson (1997). In an agricultural economics framework, the story of the goose that laid the golden egg can be thought of as a farmer who owns a region of land. The land represents th e farmers socalled goose, whose lifespan depends on whether he continues to keep the land in agricultu re or convert the land to an urban use. The farmer has two options: one, he can sell the goos e to a developer for some immediate sum and thus increase his stock of liquid assets; or two, he can keep the goose in the hopes of a more remunerative return and keep th e illiquid asset. The timeinc onsistent model presented in Laibson (1997) would suggest that hyperbolic discounters sell the goose and increase the stock of liquid assets to fund immediate consumption in the shortrun. This is where the notion of commitment devices come into play. The farmer could alternatively decide to keep the goose, his farmland, in the current period and constrain his future self from developing the land in the next period. This may lend some support to the notion that the Cons ervation Reserve Program PAGE 163 163 (CRP) may serve as a commitment device to a farmer or landowner who potentially discounts hyperbolically. The CRP pays billions of dolla rs annually to landowners and farmers to keep land out of cropuse under 10 to 15 year contracts in order to sustain the land for future use. Farmland may even serve as a commitment devi ce itself to some landowners. Hyperbolic discounters tend to hold relatively little wealth in liquid assets and hold much more of there wealth portfolio in illiquid a ssets (Laibson 1997). According to the USDA Farm Balance Sheet published by the Economic Research Service, over $3 trillion of the portf olio (about 80% of the total) is attributed to land. Farmland also represents one of the most illiquid of assets in the farm investment portfolio. This may imply land is a golden egg to the farmerlandowner. In the literature on timeinconsistent preferences and hy perbolic discounting, illiquid assets serve two fundamental roles. The first role is as a comm itment device, preventing ones future self from capriciously spending his wealth. The second role is as a golde n egg, generating substantial benefits but only after a long pe riod of time has elapsed. Alth ough a less transparent instrument for commitment than other devices, land as an illi quid asset that generates a stream of income can serve as a mechanism to constrain future choices. Further, with an asset such as land, whose sale typically must be initiated a period or more before the actu al revenues from the sale are obtained, farmland promises substantial benefits in the longrun but immediate benefits are hard to realize in the shortrun. The difficult to sell na ture of farmland can be attributed to a variety of factors including transaction costs, information asymmetries, and incomplete markets. Finally, the results have important extensi on and policy relevance. The results underline the shortrun nature of thinking of many farmer s, growers, ranchers, and developers. The high calculated shortrun discount rates compared to the low longrun discount rates imply the relative importance of the two time periods to a decisionmake r, with the priority being the shortrun. If PAGE 164 164 the shortrun discount rates sugge st a tendency of individuals to sacrifice futu re returns for immediate gains, then extension efforts should at tempt to curve or address this behavior. In essence, extension work must r ecognize that there is a shortr un bandwagon that individuals have a tendency to focus on. Many agricultural investme nts that offer high shortrun returns such as biofuel programs, commodity speculation, and urbanization offer dubious longrun security. Extension efforts should address this beha vioral tendency and emphasize the longrun opportunities in postponing or delaying land development. The greatest potential for policy may come from the design of precommitment devices for landowners. Christmas clubs were once a popul ar way of constraining individuals from spending money so that a stock of liquid cash w ould be available for the purchase of holiday gifts. Landowners may desire a similar style of commitment device so that their future self is constrained from making decisions against their present self, su ch as hasty land conversion or commodity speculation. As mentioned, the Conservation Reserve Program may serve as a commitment device for some landowners. However, the efficiency of the CRP is in doubt if landowners are timeinconsistent, resulting in a fu ture decision to drop out of the program for higher monetary gain than the CRP offers. Gulati and Vercammen (2006) model resour ce conservation contracts under timeinconsistent preferences. The authors note th at commitment and timeinconsistency issues are important in the implementation of the CRP and other similar contracts. Since the CRP predetermines payment schedules over the length of the contract according to a discounted present value model, problems of landowners dropping out of the contract are common. To address this problem, the authors suggest instituting penaltie s for dropping out of the CRP or for increasing the payment schedule over time in order to combat the hyperbolic na ture of landowner time PAGE 165 165 preferences. Since the results in this dissertation imply that landowners are described by quasihyperbolic time preferences, the results in th is dissertation would seem to confirm the suggestions made by Gulati and Vercammen (2006). Future Work Given the evidence presented in th is dissertation, critical atte ntion should be devoted to the investigation of intertemporal preference in the land values lit erature specifically, and in the agricultural economics litera ture more generally. In regards to this dissertation, several potential areas for future research are offered. Firs t, NPV models of land values under adaptive expectations should be explored and test for the presence of hyperbolic discounting. Another interesting extension would involve the inco rporation of risk into the NPV model and simultaneously estimate both risk and time pref erence parameters. Such an extension may involve other models than the NPV model used he re, such as Euler equations for investment and savings decisions. Also, includ ing a better control of inflation in the NPV model offers an interesting and important ar ea of exploration. In regards to the gene ral land literature, i nvestigation on how the shape of the discount factor affects the comovement of land price and land rents, as we ll as the affect on development and rural land conversion times and intensity are important and interest ing questions. How does the timing of rural land conversion differ under hyperbolic time preferences? What does timeinconsistency imply for land taxes, conversion costs, and capital intensity of development efforts? Also, survey methods directed at landowners offer an attr active method of obtaining discount rates and testing fo r hyperbolic discounting. There is also a range of interesting questions yet to be examined in the context of potential timeinconsistency and hyperbolic discounting in other research ar eas as well. Food demand, for example, offers a good opportunity for testing hy perbolic preferences and could also lend itself PAGE 166 166 to experimental methods, which have become recently popular. Policy research on timeinconsistency and resource conservation contract s has just begun. Data on CRP participation offers a viable avenue of potential research. The model and the results in this dissertation could be generalized into a model of land conversi on and seek implications on development timing under timeinconsistent preferences. Indeed, there is no shortage of important and engaging questions to examine under the umbrella of intertemporal pref erences. The investigation of nonconstant discounting and timeinconsistency is waiting to be uncovered in the agricu ltural economics profession. It is hoped that this dissertation serves to fuel an enthusiastic interest in the topic. Summary This dissertation estimates the parameters of time preference in the land values asset equation, generalizing the standa rd discount factor to include quasihyperbolic discounting. While the presentvalue model has been used almost exclusively to estimate land values, empirical inquiry has revealed tw o serious flaws. First, presen tvalue models are not able to explain the presence of rational bubbles and the apparent boom /bust nature of land prices. Second, there are apparent disc onnects in the movement between land rents and land prices in the shortrun versus the longrun. The poor performance of the PV model to estimate land values in other papers may of course be due to a variety of other potential issues, hyperbolic discounting being just one possibility. Failure of the sta ndard discounting model may be attributed to transaction costs (Chavas and Tomas 1999) or timev arying risk premia in farmland returns (Hanson and Myers 1995). Another possible explanation for the poor performance of the present value model and its failure to stand up to empirical scrutiny may involves a priori assumptions on the shape of the discount factor. PAGE 167 167 In particular, this dissertation modifies the standard model to inco rporate quasihyperbolic discounting; a generalized form of exponential di scounting that has receiv ed considerable but recent attention in the financial and experimental economics literature. Statistically significant evidence of quasihyperbolic disc ounting over standard exponentia l discounting is uncovered in a dataset of U.S. farmland values and returns for the major agricultural regions of the United States. The results not only s uggest a significant quasihyperbolic parameter, but significantly different shortrun and longrun disc ount rates in eight ou t of the nine panel sets examined. The results show that individuals discount land far mo re heavily in the shortrun than the longrun, with obtained shortrun discount rates several orders of magnit udes larger than the obtained longrun discount rates. The shortrun discount rates obta ined are not important for use in actual financial analysis, but for what they imply about landuse decisi ons, theoretical models of land values, and for landowner behavior. First, the high shortrun discount rates suggest that landowner decisions tend to be dominated by shortrun thinking. Th is may result in possible land investment projects that offer a high instant return, but offer little longrun gain. The shortrun discount rates imply land decisions that tend to sacrifi ce the future for the sake of the present. Second, the presence of quasihyperbolic discounting offers an explan ation into the apparent shortrun and longrun disconnects in NPV models of farmland. Farm land values and returns tend to be well cointegrated in the longrun but not in the shortrun. If individual s have a higher shortrun discount rate than the longrun, then th is might explain why the shortrun dynamic relationship tends to break down. Third, the results offer evidence th at farmland may acts as the socalled goose that laid the golden egg, with landowne rs caught in an internal tussle as whether to keep land in PAGE 168 168 farming or to sell a developer for a more capital intensive use. In this case, commitment devices may be a helpful policy tool for the timeinconsistent landowner. Some limitations to the results apply since the high shortrun discount rates may be affected by the simplistic assumptions maintained in the theoretical model. First, individuals were assumed to be riskneutral. Evidence in the literature suggests that not accounting for risk aversion results in shortrun disc ount rates that are biased upward Second, rational expectations were assumed, which has been shown to result in unreasonable discount parameter estimates in models of stock prices and farmland values. Third, inflation was not accounted for in this analysis which may also result in an upward bias in both the shortrun and longrun discount rates. Although the assumptions made in this dissertation are limiting, they also offer a starting point for the analysis which is congruent with the assumptions made in the farmland values literature. While more complicated assumptions ma y result in lower discount rates, it is believed that significant differences be tween the shortrun and longrun discount rate would still be uncovered. Future work should attempt to obtain the time preference parameters under more relaxed assumptions of risk, expectations, and inflation. There is also tremendous potential for future work into time preferences and time inconsis tency and other economic models such as land conversion and food demand. It is hoped that this diss ertation has not only served to bridge the gap between the literature on in tertemporal preferences and agri cultural economics, but that it has sparked an interest into further inves tigation on time inconsistency in the profession. PAGE 169 169 APPENDIX PROCEDURE FOR RCODE # TITLE: Appalachian State Panel OLS & Linear GMM # RESET ALL WORK rm (list = ls()) # Call needed libraries library(lmtest); library(stats); library(tseries) library(car); library(sandwich); library(systemfit) # Download CPI Deflator dta0 PAGE 170 170 vtn PAGE 171 171 exptest <(q/qse)^2 deltase PAGE 172 172 shat <(sse/nrow(z))*(t(z)%*%z) mhat PAGE 173 173 # TITLE: Cornbelt States Panel OLS & Linear GMM # RESET ALL WORK rm(list = ls()) # Call needed libraries library(lmtest); library(stats); library(tseries) library(car); library(sandwich); library(systemfit) # Download CPI Deflator dta0 PAGE 174 174 voh PAGE 175 175 print("Value of Exponential Factor "); print(cbind(delta,deltase)) print("Exponential Discounting Ftest"); print(exptest) # This is a twosided Ftest with dof=1,203 and alpha=0.05 # Null hypothesis is exponential discounting # Critical value is 3.84 # Conduct nonlinear hypothesis te st on hyperbolic discounting ga1 <1/(1a2); ga2 <a1/((1a2)^2) betase PAGE 176 176 vhat PAGE 177 177 # TITLE: Delta States Panel OLS & Linear GMM # Last Modified: 02/22/2008 # RESET ALL WORK rm(list = ls()) # Call needed libraries library(lmtest); library(stats); library(tseries) library(car); library(sandwich); library(systemfit) # Download data sets # Column Order: Y~AR~LA~MS~V~R~I~DV dta PAGE 178 178 y PAGE 179 179 # Define IV matrix and scale z PAGE 180 180 # Calculate Discount Parameters coeff PAGE 181 181 dta PAGE 182 182 ne PAGE 183 183 print("QuasiHyperbolic Discoun ting Ftest"); print(hyptest) # This is onesided Ftest with dof=1,244 # Null hypothesis is no hyperbolic discounting # Critical value is 3.84 (alp=0.05) & 2.70 (alp=0.10) # Define IV matrix and scale z PAGE 184 184 # Jt converges to ChiSquare with qp degrees of freedom # The degrees of freedom equal the nu mber of overidentif ying restrictions # The critical value with alpha=0.05 and qp=4 is 9.488 # Rejection indicates problems with the GMM estimator # Calculate Discount Parameters coeff PAGE 185 185 lagcpi PAGE 186 186 # Critical value is Chisqua re with alp=.05, dof=2: 5.991 durbin.watson(ols,max.lag=2) # Null hypothsis is no au tocorrelation (rho=0) # If d
