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## Material Information- Title:
- A dynamic model of input demand for agriculture in the Southeastern United States
- Creator:
- Monson, Michael James, 1956-
- Publication Date:
- 1986
- Language:
- English
- Physical Description:
- vii, 148 leaves : ill. ; 28 cm.
## Subjects- Subjects / Keywords:
- Agricultural land ( jstor )
Agriculture ( jstor ) Capital shortages ( jstor ) Capital stocks ( jstor ) Elasticity of demand ( jstor ) Financial investments ( jstor ) Labor demand ( jstor ) Mathematical variables ( jstor ) Net investment ( jstor ) Price elasticity ( jstor ) Agriculture -- Economic aspects -- Southern States ( lcsh ) Dissertations, Academic -- Food and Resource Economics -- UF Duality theory (Mathematics) ( lcsh ) Food and Resource Economics thesis Ph. D - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis (Ph. D.)--University of Florida, 1986.
- Bibliography:
- Bibliography: leaves 141-147.
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Michael James Monson.
## Record Information- Source Institution:
- University of Florida
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- University of Florida
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- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 029576254 ( ALEPH )
15464499 ( OCLC ) AEH5769 ( NOTIS ) AA00004868_00001 ( sobekcm )
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A DYNAMIC MODEL OF INPUT DEMAND FOR AGRICULTURE IN THE SOUTHEASTERN UNITED STATES By MICHAEL JAMES MONSON 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 1986 Copyright 1986 by Michael James Monson ACKNOWLEDGEMENTS Custom alone does not dictate the need for me to express my gratitude to the members of my committee, as each member has provided valuable assistance in the completion of this dissertation. I thank Dr. Boggess for enabling me to pursue a variety of topics and enhancing the breadth of my graduate research experience, as well as keeping a down-to-earth perspective. Dr. Taylor has served me well as motivator and mentor of my dissertational research. I also express my appreciation to Dr. Majthay for the classroom instruction in optimal control theory and tolerance of an agricultural economist's employment of the theory, Dr. Langham for some timely advice, and Dr. Emerson for restoring my faith in nonlinear models at a crucial moment. Additionally, thanks to the staff of the FARM lab, particularly Rom Alderman, for technical support in preparing the manuscript, and to Pat Smart for doing her best to see that I turned the required forms in on time. Finally, I thank my wife, Sandra, son, Jeffery, and my parents. Sorry it took so long. in TABLE OF CONTENTS PAGE ACKNOWLEDGEMENTS iii ABSTRACT V CHAPTER I INTRODUCTION 1 Background 3 Objectives 15 II THEORETICAL MODEL 18 Dynamic Models Using Static Optimization 20 Dynamic Optimization 27 Theoretical Model 35 The Flexible Accelerator 42 III EMPIRICAL MODEL AND DATA 46 Empirical Model 46 Data Construction 60 IV RESULTS 75 Theoretical Consistency 75 Quasi-fixed Input Adjustment 99 V SUMMARY AND CONCLUSIONS 110 APPENDIX A INPUT DEMAND EQUATIONS 119 B REGIONAL EXPENDITURE, PRICE, AND INPUT DATA 12 3 C EVALUATION OF CONVEXITY OF THE VALUE FUNCTION 130 D ANNUAL SHORT- AND LONG-RUN PRICE ELASTICITY ESTIMATES 132 REFERENCES 141 BIOGRAPHICAL SKETCH 148 iv 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 A DYNAMIC MODEL OF INPUT DEMAND FOR AGRICULTURE IN THE SOUTHEASTERN UNITED STATES By Michael James Monson August 1986 Chairman: William G. Boggess Cochairman: Timothy G. Taylor Major Department: Food and Resource Economics The current crisis in U.S. agriculture has focused attention on the need to adjust to lower output prices as a result of a variety of factors. The ability of agriculture to adjust is linked to the adjustment of inputs used in production. Static models of input demand ignore dynamic processes of adjustment. This analysis utilizes a model based on dynamic optimization to specify and estimate a system of input demands for Southeastern U.S. agriculture. Dynamic duality theory is used to derive a system of variable input demand and net investment equations. v The duality between a value function representing the maximized present value of future profits of the firm and the technology are reflected in a set of regularity conditions appropriate to the value function. Labor and materials are taken to be variable inputs, while land and capital are treated as potentially quasi-fixed. The equations for optimal net investment yield short- and long-run demand equations for capital and land. Thus, estimates of input demand price elasticities for both the short run and long run are readily obtained. The estimated model indicates that the data for the Southeast support the assumption of dynamic optimizing behavior as the regularity conditions of the value function are generally satisfied. The model indicates that labor and capital, and land and capital are short-run substitutes. Land and labor, and materials and capital are short-run complements. Similar relationships are obtained in the long run as well. Estimated adjustment rates of capital and land indicate that both are slow to adjust to changes in equilibrium levels in response to relative price changes. The estimated own rate of adjustment in the difference between actual and equilibrium levels of vi capital within one time period is 54 percent. The corresponding adjustment rate for land is 16 percent. However, the rate of adjustment for capital is dependent on the difference in actual and equilibrium levels of land. vii CHAPTER I INTRODUCTION If one were to characterize the current situation in U.S. agriculture, one might say that agriculture faces a period of adjustment. Attention has focused on the need to adjust to lower output prices as a result of a variety of factors including increasing foreign competition, trade regulations, a declining share of the world market, strength of the U.S. dollar, changes in consumer demand, high real rates of interest and governmental policies. The adjustments facing agriculture as a result of low farm incomes have become an emotional and political issue as farmers face bankruptcy or foreclosure. The ability of the agricultural sector to respond to these changing economic conditions is ultimately linked to the sector's ability to adjust the inputs used in production. Traditionally the analysis of input use in agriculture has been through the derivation of input demands based on static or single period optimizing behavior. While static models maintain some inputs as fixed in the short-run yet variable in the long-run, there is nothing in the theory of static optimizing behavior to explain a 1 2 divergence in the short- and long-run levels of such inputs. Previous models based on static optimization recognize that some inputs are slow to adjust, but they lack a theoretical foundation for less-than- instantaneous input adjustment. A theoretical model of dynamic optimization has been recently proposed by Epstein (1981). Epstein has established a full dynamic duality between a dual function representing the present value of the firm and the firm's technology. This value function can be used in conjunction with a generalized version of Hotelling's Lemma to obtain expressions for variable input demands and optimal net investment in less-than- variable inputs consistent with dynamic optimizing behavior. While the theoretical implications of dynamic duality are substantial, there are a limited number of empirical applications upon which an evaluation of the methodology can be based. An empirical application to agriculture presenting a complete system of factor demands is absent in the literature. Chambers and Vasavada (1983b) present only net investment equations for the factors of land and capital in U.S. agriculture, focusing on the rates of adjustment. This same approach is used by Karp, Fawson, and Shumway (1984). The rates of adjustment in less-than- 3 variable inputs are important, but the methodology is richer than current literature would indicate. The purpose of this study is to utilize dynamic duality to specify and estimate a dynamic model of aggregate Southeastern U.S. agriculture. The model enables a clear distinction between the short- and long-run. Thus, input demands and elasticities for the short- and long-run may be determined. A key part of the methodology based on the explicit assumption of dynamic optimizing behavior is whether the data support such an assumption. This analysis intends to explore the regularity conditions necessary for a duality between the dynamic behavioral objective function and the underlying technology. Background The term "period of adjustment" is an appropriate recognition of the fact that agriculture does not always adjust instantaneously to changes in its environment. Agriculture has faced and endured periods of adjustment in the past. The Depression and Dust Bowl eras and world wars produced dramatic changes in the agricultural sector (Cochrane, 1979). In more recent history, agricultural economists have noted other periods of adjustment in response to less spectacular stimuli. Examples include the exodus of labor from the agricultural sector in the 1950s that 4 stemmed from the substitution of other inputs for labor (Tweeten, 1969), and the introduction of hybrid seed (Griliches, 1957) and low-cost fertilizers (Huffman, 1974). Two points are crucial. First, the period of adjustment in each case is less than instantaneous if not protracted, and second, the adjustments are ultimately reflected in the demand for inputs underlying production. Quasi-fixed Inputs The assumption of short-run input fixity is questionable. In a dynamic or long-run setting, input fixity is, of course, inconsistent with the definition of the long run as the time period in which all inputs are variable. The recognition that some inputs are neither completely variable or fixed dictates using the term "quasi-fixed" to describe the pattern of change in the levels of such inputs. Given a change in relative prices, net investment propels the level of the input towards the long-run optimum. The input does not remain fixed in the short-run, nor is there an immediate adjustment to the long-run optimum. Alternatively, the cost of changing the level of an input does not necessarily preclude a change as is the case for fixed inputs, nor is the cost solely the marginal cost of additional units in the variable input case. No inputs are absolutely fixed, but 5 rather fixed at a cost per unit time. The quasi fixity of an input is limited by the cost of adjusting that input. Short-run Input Fixity In light of the notion of quasi-fixity, input fixity in short-run static models is based on the assumption that the cost of adjusting a fixed input exceeds the returns in the current period. In the static model, the choice is either no adjustment in the short-run initial level or a complete adjustment to a long-run equilibrium position. The static model is unable to evaluate this adjustment as a gradual or partial transition from one equilibrium state to another. Given such a restricted horizon, it is understandable that static models of the short run often treat some inputs as fixed. The assumption of input fixity in the short run casts agriculture in the framework of a putty-clay technology, where firms have complete freedom to choose input combinations ex ante but once a choice is made, the technology becomes one of fixed coefficients (Bischoff, 1971). Chambers and Vasavda (1983a) tested the assumption of the putty-clay hypothesis for 6 aggregate U.S. agriculture with respect to capital, labor, and an intermediate materials input. Land was maintained as fixed and not tested. The methodology, developed by Fuss (1978), presumes that uncertainty with respect to relative prices determines the fixed behavior of some inputs. The underlying foundation for Chambers and Vasavada's test of asset fixity is a trade-off between flexibility of input combinations in response to relative input price changes and short-run efficiency with respect to output. Essentially, the farmer is faced with choosing input combinations in anticipation of uncertain future prices. If the prices in the future differ from expectations and the input ratios do not adjust, the technology is putty-clay. If the input ratios can be adjusted such that static economic conditions for allocative efficiency are fulfilled (marginal rate of technical substitution equal to the factor price ratio), the technology is putty-putty. Finally, if the input ratios do adjust, but not to the point of maximum efficient output, the technology is said to be putty-tin. Chambers and Vasavda concluded that the data did not support the assumption of short-run fixity for capital, labor, or materials. However, the effects of the maintained assumptions on the conclusions is 7 unclear. The assumption of short-run fixed stocks of land makes it difficult to determine if the degree of fixity exhibited by the other inputs is inherent to each input or a manifestation of the fixity of land. In addition to the assumption that land is fixed, the test performed by Chambers and Vasavada also relied on the assumption of constant returns to scale, an assumption used in previous studies at the aggregate level (Binswanger, 1974; Brown, 1978). The measure of efficiency, output, is dependent on the constant returns to scale assumption. If there were in fact increasing returns to scale presumably inefficient input combinations may actually be efficient. The Cost of Adjustment Hypothesis The cost of adjustment hypothesis put forth by Penrose (1959) provides an intuitive understanding of why some inputs are quasi-fixed. Simply stated, the firm must incur a cost in order to change or adjust the level of some inputs. The assumption of short-run fixity is predicated on the a priori notion that these costs preclude a change. The cost of search model by Stigler (1961) and the transactions cost models of Barro (1969) can be interpreted as specialized cases of adjustment costs characterized by a "bang-bang" investment policy. Such models lead to discrete jumps rather than a 8 continous or gradual pattern of investment. For a model based on static optimization, this appears to be the only adjustment mechanism available. Inputs remain at the short-run fixed level until the returns to investment justify a complete jump to the long-run equilibrium. Land is most often considered fixed in the short- run (Hathaway, 1963; Tweeten, 1969; Brown, 1978; Chambers and Vasavada, 1983a). Transaction costs appear to be the basis for short-run fixity of land and capital in static models. Galbraith and Black (1938) reasoned that high fixed costs prohibit substitution or investment in the short run. G. L. Johnson (1956) and Edwards (1959) hypothesized that a divergence in acquisition costs and salvage value could effectively limit the movement of capital inputs in agriculture. While labor and materials are usually considered as variable inputs in the short run, a search or information cost approach may be used to rationalize fixity in the short run. Tweeten (1969) has suggested that labor may be trapped in agriculture as farmers and farm laborers are prohibited from leaving by the costs of relocation, retraining, or simply finding an alternative job. However, the treatment of 9 agriculture as a residual employer of unskilled labor seems inappropriate given the technical knowledge and skills required by modern practices. These same technical requirements can be extended to material inputs such as fertilizers, chemicals, and feedstuffs, in order to justify short-run fixity. The cost of obtaining information on new material inputs could exceed the benefits in the short run. Alternatively, some material inputs may be employed by force of habit, such that the demand for these inputs is analagous to the habit persistance models of consumer demand (Pope, Green, and Eales, 1980). Recognizing the problems of such "bang-bang" investment policies, the adjustment cost hypothesis has been extended to incorporate a wider variety of potential costs. The adjustment costs in the first dynamic models may be considered as external (Eisner and Strotz, 1963; Lucas, 1967; Gould, 1968). External costs of adjustment are based on rising supply prices of some inputs to the individual firm and are inconsistent with the notion of competitive markets. Imperfect credit markets and wealth constraints may also be classified as external costs of adjustment. Internal costs of adjustment (Treadway 1969, 1974) reflect some foregone output by the firm in the present in order to invest in or acquire additional 10 units of a factor for future production. The assumption of increasing adjustment costs, where the marginal increment of output foregone increases for an incremental increase in a quasi-fixed factor, leads to a continous or smooth form of investment behavior. Quasi-fixed inputs adjust to the point that the present value of future changes in output are equal to the present value of acquisition and foregone output. The adjustment cost hypothesis, particularly internal adjustment costs, is important in a model of dynamic optimizing behavior. The costs of adjustment can be reflected by including net investment as an argument of the underlying production function. The constraint of input fixity in a short-run static model is relaxed to permit at least a partial adjustment of input levels in the current period. Assuming that these adjustment costs are increasing and convex in the level of net investment in the current period precludes the instantaneous adjustment of inputs in the long-run static models. Therefore, a theoretical foundation for quasi-fixed inputs can be established. Models of Quasi-fixed Input Demand Econometric models which allow quasi-fixity of inputs provide a compromise between maintaining some inputs as completely fixed or freely variable in the short-run. The empirical attraction of such models is 11 evident in the significant body of research in factor demand analysis consistent with quasi-fixity surveyed in the following chapter. Unfortunately, a theoretical foundation based on dynamic optimizing behavior is generally absent in these models. By far the most common means of incorporating dynamic elements in the analysis of input demand has been through the use of the partial adjustment model (Nerlove, 1956) or other distributed lag specifications. Such models typically focus on a single input. The coefficient of adjustment then is a statistical estimate of the change in the actual level of the input as a proportion of the complete adjustment that would be expected if the input was freely variable. The principle shortcoming of the single-equation partial adjustment model is that such a specification ignores the effect of, and potential for, quasi-fixity in the demands for the remaining inputs. The recent payment-in-kind (PIK) program is an excellent example of the significance of the interrelationships among factors of production. The program attempted to reduce the amount of land in crops as a means of reducing commodity surpluses. The effect of the reduction in one input, land, reduced the demand for other inputs, such as machinery, fertilizer and 12 chemicals, and the labor for operation of equipment and application of materials. The consequences of a reduction in this single input extended beyond the farm gate into the industries supporting agriculture as well. The partial adjustment model has been the foundation for many so-called dynamic optimization models cast in the framework of a series of static problems with the imposition of an adjustment coefficient as the linkage between the individual production periods (e.g., Day, 1962; Langham, 1968; Zinser, Miranowski, Shortle, and Monson, 1985) This effectively ignores the potential for instantaneous adjustment. Adjustments and relationships among inputs are determined arbitrarily. Others maintain potentially quasi-fixed inputs as variable (e.g., McConnell, 1983). The actual rate of adjustment in the system may be slower than the model would indicate as the adjustment of the maintained quasi-fixed inputs depends on the adjustments in supposedly variable inputs. The multivariate flexible accelerator (Eisner and Strotz, 1963) is an extension of the partial adjustment model to a system of input demands rather than a single equation. Lucas (1967), Treadway (1969, 1974), and Mortensen (1973) have demonstrated that 13 under certain restrictive assumptions concerning the production technology and adjustment cost structure, a flexible accelerator mechanism of input adjustment can be derived from the solution of a dynamic optimization problem. The empirical usefulness of this approach is limited, however, as the underlying input demand equations are expressed as derivatives of the production function. Thus, any restrictions inherent in the production function employed in the specification of the dynamic objective function are manifest in the demand equations. Berndt, Fuss, and Waverman (1979), and Denny, Fuss, and Waverman (1979) derived systems of input demand equations consistent with the flexible accelerator by introducing static duality concepts into the dynamic problem. The derivation of input demands through the use of a static dual function reduces the restrictions imposed by the primal approach utilizing a production function. Given the assumption of quadratic costs of adjustment as an approximation of the true underlying cost structure, these analyses obtained systems of variable input demand functions and net investment equations by solving the Euler equation corresponding to a dynamic objective function. This methodology, however, is 14 generally tractable for only one quasi-fixed input and critically relies on the quadratic adjustment cost structure. Duality is a convenience in models of static optimizing behavior. However, the application of static duality concepts to a dynamic objective function is somewhat limited. McClaren and Cooper (1980) first explored a dynamic duality between the firm's technology and a value function representing the maximum value of the integral of discounted future profits. Epstein (1981) established a full characterization of this dynamic duality using the Bellman equation corresponding to the dynamic problem. The optimal control theory underlying the solution to a dynamic optimization problem is consistent with quasi-fixity and the cost of adjustment hypothesis. The initial state is characterized by those inputs assumed fixed in the short run. Net investment in these inputs serves as the control (optimal in that the marginal benefit equals the cost of investment) that adjusts these input levels towards a desired or optimal state. This optimal state corresponds to the optimal levels of a long-run static optimization in which all inputs are variable. 15 Objectives The objective of this analysis is to utilize dynamic duality to specify and estimate a system of variable input demands and net investment equations for aggregate southeastern U.S. agriculture. Inherent to this effort is a recognition of the empirical applicability of dynamic duality theory to a small portion of U.S. agriculture. While the study has no pretense of determining the acceptance or rejection of the methodology for aggregate economic analysis, a presentation of the methods employed and difficulties encountered may provide some basis for further research. In addition to obtaining estimates of the optimal rates of net investment in land and capital, appropriate regularity conditions are evaluated. The clear distinction between the short- and long-run permits derivation of short- and long-run price elasticities for all inputs. Furthermore, the specificiation used for the value function permits the testing of hypotheses concerning the degree of fixity of land and capital and the degree of interdependence in the rates of net investment in these inputs. The potential significance of this interdependence with respect to policy is briefly explored. 16 Scope The dynamic objective function is expressed in terms of quasi-fixed factor stocks, net investment, the discount rate, and relative input prices. Endogenizing the factors conjectured as responsible for lower output prices in the introduction is no less difficult in a dynamic setting than in a static model. This study is content to explore the effects of relative price changes on factor demands and adjustment. Labor and materials are taken to be variable inputs while land and capital are considered as potentially quasi-fixed. This treatment is dictated by the available data at the regional level consistent with measurement of variable inputs rather than factor stocks. The development of appropriate stock measures for labor should involve a measure of human capital (Ball, 1985). In fact, the incorporation of human capital in a model of dynamic factor demands is a logical extension of the methodology as a means of not only determining but explaining estimated rates of adjustment. However, such a model exceeds the scope of this analysis. Additionally, a method of incorporating policy measures in the theory remains for the future. Noting this limitation, the Southeast is perhaps best suited 17 for an initial exploration of the methodology. The diversity of product mix in the components of total output in the Southeast reduces the influence of governmental policies directed at specific commodities or commodity groups. In 1980, the revenue share of cash receipts for the commodity groups typically subject to governmental price support in the U.S., namely dairy, feed grains, food grains, cotton, tobacco, and peanuts, was nearly 36 percent of total cash receipts, while the share of those commodities in the Southeast was 18 percent. Overview A review of previous models incorporating dynamic elements in the analysis of factor demands leads naturally to the theoretical model developed in Chapter II. An empirical model potentially consistent with dynamic duality theory and construction of the data measures follows in Chapter III. The estimation results and their consistency with the regularity conditions, the measures of short- and long-run factor demands and price elasticities, and hypotheses tests are presented in Chapter IV. The final chapter discusses the implications of the results and the methodology. CHAPTER II THEORETICAL DEVELOPMENT The primary objective of this study is to specify and estimate a system of dynamic input demands for southeastern U.S. agriculture. In order to explore the adjustment process of agricultural input use, the model should be consistent with dynamic optimizing behavior, quasi-fixity, and the adjustment cost hypothesis. This entails an exploration of the empirical applicability of a theory of dynamic optimization capable of yielding such a system. Yet models of input adjustment, hence dynamic input demands, based on static optimization generally lack a theoretical foundation. Treadway summarizes the incorporation of theory in these models. A footnote is often included on the adjustment-cost literature as if that literature had fully rationalized the econometric specification. And other adjustment mechanisms continue to appear with no discernible anxiety about optimality exhibited by their users. Furthermore, it is still common for economists to publish studies of production functions separately from studies of dynamic factor demand without so much as mentioning that the two are theoretically linked. (Treadway, 1974, p. 18) In retrospect, the search for a theoretical foundation for empirical models rationalized on the notions of quasi-fixity and adjustment costs culminate in the very 18 19 theory to be empirically explored. While these prior empirical models of input demand and investment are not necessarily consistent with the theoretical model finally developed, they are important elements in its history. This chapter includes a review of dynamic input demand models, dynamic in the sense that changes in input levels are characterized by an adjustment process of some form. The alternative models are evaluated with respect to empirical tractability and adherence to economic theory. In the first section, an adjustment mechanism is imposed on an input demand derived from static optimization. These models, whether the adjustment mechanism is a single coefficient or a matrix of coefficients, are empirically attractive but lack a firm theoretical foundation. The models in the second group are based on dynamic optimization. These models are theoretically consistent, yet limited by the form of the underlying production function or the number of inputs which may be quasi-fixed, even though static duality concepts are incorporated. The advantages presented by static duality lead to the development of dynamic duality theory. The application of dynamic duality theory to a problem of dynamic optimizing behavior permits the derivation of a 20 system of dynamic input demands explicitly related to the underlying production technology and offers a means of empirically analyzing the adjustment process in the demand for agricultural inputs in a manner consistent with optimizing behavior. Dynamic Models Using Static Optimization The input demand equations derived from a static objective function with at least one input held constant provide limited information relative to input adjustment. The derived demands are conditional on the level of the fixed input(s). The strict fixity of some inputs makes such models inappropriate for a dynamic analysis. The demands derived from the static approach without constraints on factor levels characterize equilibrium or optimal demands if the factors are in fact freely variable. However, a full adjustment to a new equilibrium given a change in prices is inconsistent with the observed demand for some inputs. Input demand equations derived from a static optimization problem, whether cost minimization or profit maximization, characterize input demand for a single period and permit either no adjustment in some inputs or instantaneous adjustment of all inputs. There is nothing in static theory to reflect adjustment in input 21 demands over time. Models of dynamic input demand based on static optimization attempt to mimic rather than explain this adjustment process. Distributed Lags and Investment The investigation of capital investment through distributed lag models seems to represent a much greater contribution to econometric modeling and estimation techniques than to a dynamic theory of the demand for inputs. However, models of capital investment characterize an early empirical approach to quasi-fixity of inputs, recognizing that net investment in capital is actually an adjustment in the dynamic demand for capital as a factor of production. Additionally, these models developed an implicit rental price or user cost of a quasi-fixed input as a function of depreciation, the discount rate, and tax rate as the appropriate measure of the quasi-fixed input price (e.g., Hall and Jorgenson, 1967). Particular lag structures identified and employed in the analysis of capital investment include the geometric lag (Koyck, 1954), inverted V lag (DeLeeuw, 1962), polynomial lag (Almon, 1965), and rational lag (Jorgenson, 1966). The statistical methods and problems 22 of estimating these lag forms has been addressed extensively1. Lag models inherently recognize the dynamic process, as current levels of capital are assumed to be related to previous stocks. A problem with these models is that lag structures are arbitrarily imposed rather than derived on the basis of some theory. Griliches (1967, p. 42) deems such methods "theoretical ad-hockery." The basic approach in distributed lag models is to derive input demand equations from static optimization with all inputs freely variable. The demand equations obtained characterize the long-run equilibrium position of the firm. If an input is quasi-fixed, it will be slow to adjust to a new equilibrium position. The amount of adjustment, net investment, depends on the difference in the equilibrium demand level and the current level of the input. The underlying rationale of a distributed lag is that the current level of a quasi-fixed input is a result of the partial adjustments to previous equilibrium positions. The various forms of the distributed-lag determine how important these past adjustments are in determining the current response. The finite lag distributions proposed by Fisher (1937), 1 Maddala (1977, pg. 355-76) presents econometric estimation methods and problems. 23 DeLeeuw (1962), and Almon (1965) limit the number of prior adjustments that determine the current response. The infinite lag distributions (Koyck, 1954; Jorgenson, 1966) are consistent with the notion that the current adjustment depends on all prior adjustments. While such models may characterize the adjustment process of a single factor, there is little economic information to be gained. There is no underlying foundation for a less-than-complete adjustment or existence of a divergence of the observed and optimal level of the factor. The individual factor demand is a component of a system of demands derived from static optimization, yet its relationship to this system is often ignored. The Partial-Adjustment Model The partial adjustment model put forth by Nerlove (1956) provides an empirical recognition of input demand consistent with quasi-fixity. The partial adjustment model as a dynamic model has been widely employed (Askari and Cummings (1977) cite over 600 studies), and continues to be applied in agricultural input demand analyses (e.g., Kolajo and Adrian, 1984). The partial adjustment model recognizes that some inputs are neither fixed nor variable, but rather quasi- 24 fixed in that they are slow to adjust to equilibrium levels. In its simplest form, the partial adjustment model may be represented by (2.1) xt xt_x = a ( x*t xt_x), where x^ is the observed level of some input x in period t, and x*t is the equilibrium input level in period t defined as a function of exogenous factors. The observed change in the input level represented by x^ - x-(-_i in (2.1) is consistent with a model of net investment demand for the input. The observed change in the input is proportional to the difference in the actual and equilibrium input levels. Assuming the firm seeks to maximize profit, x*-(- becomes a function of input prices and output price such that (2.2) x t f (Py, w^ , w^j) . where output price is denoted by Py, and the w, i = 1 to n, are the input prices. Equation (2.2) allows the unobserved variable, x*^, in (2.1) to be expressed as observations in the current and the prior period. The parameter a in (2.1) represents the coefficient of adjustment of observed input demand to the equilibrium level. The model is a departure from static optimization theory in that x*^ is no longer derived from the first- 25 order conditions of an optimal solution for a static objective function. While static theory does not directly determine the form of the adjustment process, there is an explicit recognition that certain inputs are slow to adjust to long-run equilibrium levels. Interrelated Factor Demands Coen and Hickman (1970) extended the approach of distributed-lag models to a system of demand equations for each input of the production function employed in the static optimization. Essentially, the input demand equations derived from the production function under static maximization conditions are taken as a system of long-run or equilibrium demand equations. A geometric lag is arbitrarily imposed on the differences in actual and equilibrium input levels. The shared parameters from the underlying production function are restricted to be identical across equations. Coen and Hickman apply this model to labor and capital demands derived from a Cobb-Douglas production function. However, this method becomes untractable when applied to a more complex functional form or a much greater number of inputs. Additionally, the adjustment rate or lag structure for each input is not only arbitrary but remains independent of the disequilibrium in the other factors. 26 Nadiri and Rosen (1969) formulate an alternative approach to a system of interrelated factor demands where the adjustment in one factor depends explicitly on the degree of disequilibrium in other factors. The model is a generalization of the partial adjustment model to n inputs such that (2.6) x.(- = B [x t - This specification is similar to that in (2.1), except that x-t and x*t are n x 1 vectors of actual and equilibrium levels of inputs, and the adjustment coefficent becomes an n x n matrix, B. Individual input demand equations in (2.6) are of the form n (2.7) xi,t = ^ bij*(x j,t xj,t-l) + bfi* (x*j^t x-^t-l) + xi t-1* This representation permits disequilibrium in one input to affect demand for another input. This interdependence allows inputs to "overshoot" equilibrium levels in the short run. For example, assume an input is initially below its equilibrium level. Depending on the sign and magnitude of the coefficients in (2.7), the adjustment produced by disequilibria in other inputs may drive the observed level of the input beyond the long- run level before falling back to the optimal level. 27 Nadiri and Rosen (1969) derived expressions for equilibrium factor levels using a Cobb-Douglas production function in a manner similiar to Coen and Hickman (1968). However, they failed to consider the implied cross-equation restrictions on the parameters implied by the production function. Additionally, stability of such a system requires that the characteristic roots of B should be within the unit circle, yet appropriately restricting each bj is impossible. Neither Coen and Hickman, or Nadiri and Rosen provide a distinction between variable and quasi-fixed inputs. All inputs are treated as quasi-fixed and the adjustment mechanism is extended to all inputs in the system. They do provide key elements to a model of dynamic factor demands in that Coen and Hickman recognize the relationship between the underlying technology and the derived demands and Nadiri and Rosen incorporate an interdependence of input adjustment. Dynamic Optimization Recently, there has been a renewed interest in optimal control theory and its application to dynamic economic behavior. As Dorfman (1969, pg. 817) notes, although economists in the past have employed the calculus of variations in studies of investment (Hotelling, 1938; Ramsey, 1942), the modern version of 28 the calculus of variations, optimal control theory, has been able to address numerous practical and theoretical issues that previously could not even be formulated in static theory. Primal Approach Eisner and Strotz (1963) developed a theoretical model of input demand consistent with dynamic optimizing behavior and a single quasi-fixed input. Lucas (1967) and Gould (1968) extended this model to an arbitrary number of quasi-fixed inputs. However, these extensions are limited by the nature of adjustment costs external to the firm. Thus, the potential interdependence of adjustment among quasi-fixed inputs is ignored. Treadway (1969) introduced interdependence of quasi-fixed inputs by internalizing adjustment costs in the production function of a representative firm. The firm foregoes output in order to invest in or adjust quasi-fixed inputs. Assuming all inputs are quasi- fixed, the underlying structure of this model is shown by CO (2.8) V = max / e-rt(f(x,x) p'x) dt, X 0 where V represents the present value of current and future profits, x is a vector of quasi-fixed inputs, and x denotes net investment in these inputs. The vector p represents the user costs or implicit rental prices of 29 the quasi-fixed inputs normalized by output price. The current levels of the quasi-fixed inputs serve as the intitial conditions for the dynamic problem. Assuming a constant real rate of discount, r, and static price expectations, the Euler equations corresponding to an optimal adjustment path2 for the quasi-fixed inputs are given by (2.9) [fxx]^ + [fxx]* = + P- Treadway assumed the existence of an equilibrium solution to (2.9), where x = x = 0, in order to derive a system of long-run or equilibrium demand equations for the quasi-fixed inputs. However, derivation of a demand equation for net investment is more complicated, in that an explicit solution for x exists for only restricted forms of the production function (e.g., Treadway, 1974). Yet, net investment demand is the key to characterizing the dynamic adjustment of quasi-fixed inputs. The difficulty of deriving input demand equations from a primal dynamic optimization problem are apparent. While input demand equations derived from a primal static optimization involve first-order derivatives of 2 Additionally, the optimality of x depends on a system of transversality conditions where v*+* , lim e [f.] = 0, and a Legendre condition that t-> OO [f..] negative semi-definite (Treadway, 1971 p. 847). 30 the production function, derivation of input demands from (2.9) involve second-order derivatives as well. Treadway (1974) shows that the introduction of variable inputs further increases the difficulty in deriving input demands from a primal dynamic optimization. This derivation is necessarily in terms of a general production function, owing to the primal specification of the objective function. In terms of empirical interest, estimation of such a system is nonexistent. However, the establishment of necessary conditions for an optimal solution to the dynamic problem in terms of the technology provides the foundation for the use of duality that follows. Application of Static Duality Concepts The primary difficulty in estimating a system of dynamic factor demands from the direct or primal approach is that the characteristic equations underlying the dynamic optimization problem in (2.8) are necessarily expressed in terms of first- and second- order derivatives of the dynamic production function. Thus, unless a truly flexible functional form of the production function (e.g., Christensen, Jorgenson, and Lau, 1973) is employed, restrictions on the underlying technology are imposed a priori. In a static model, a behavioral function such as the profit or cost function with well-defined properties 31 can serve as a dual representation of the underlying technology (Fuss and McFadden, 1978). A system of factor demands is readily derived from the profit function by Hotelling's Lemma (Sir (p,Py)/3Pj = -x*j(p,Py)) or from the cost function by Shephard's Lemma (9C(p,y)/9pj = Xj(p,y)). However, all inputs are necessarily variable. These static models had been extended to the restricted variable profit and cost functions that hold some factors as fixed (e.g., Lau, 1976). Berndt, Fuss, and Waverman (1979) incorporated a restricted variable profit function into the primal dynamic problem in order to simplify the dynamic objective function. Berndt, Morrison-White, and Watkins (1979) derive an alternative method employing the restricted variable cost function as a component of the dynamic problem of minimizing the present value of current and future costs. The advantages of static duality reduce the explicit dependence on the form of the production function and facilitate the incorporation of variable inputs in the dynamic problem. Berndt, Fuss, and Waverman specified a normalized restricted variable profit function presumably dual to 32 the technology in (2.8) based on the conditions for such a static duality as presented by Lau (1976). This function may be written as (2.10)it(w,x,x) = max f(L,x,x) w'L L > 0 Assuming that the level of net investment is optimal for the problem in (2.8), the remaining short-run problem as reflected in (2.10) is to determine the optimal level of the variable factor L dependent on its price, w, and on the quasi-fixed factor stock and net investment. Current revenues are tt(w,x,x) p'x, which can be substituted directly into (2.8). The use of static duality in the dynamic problem allows the production function to be replaced by the restricted variable profit function. A general functional form for it, quadratic in (x,x), can be hypothesized as (2.11) tl(w,x,x)= aQ + a'x + b'x + l/2[x' x'] A C X C B J LXJ where ag, the vectors a, b, and matrices A, B, and C will be dependent on w in a manner determined by an exact specification of tt(w,x,x). The Euler equation for the dynamic problem in (2.8) after substitution of (2.11) is (2.12) Bx + (C- C rB) x (A + rC') x = rb p + a . Note that this solution is now expressed in terms of the 33 parameters of the restricted variable profit function instead of the production function. A steady state or equilibrium for the quasi-fixed factors denoted as x(p,w,r) can be computed from (2.12) evaluated at x = x = 0, such that (2.13) x (p,w,r) = -[A + rC']_1(rb p + a). Applying Hotelling's Lemma to the profit function yields a system of optimal variable input demands, L*, conditional on the quasi-fixed factors and net investment of the form (2.14) L*(x,p,w,r) = -jrw(w,x,x) . The system of optimal net investment equations can expressed as (2.15) x(x,p,w,r) = M(w,r)[x x(p,w,r)]. The exact form of the matrix M is uniquely determined by the specification of the profit function in the solution of (2.10). Only in the case of one quasi-fixed factor has this matrix been expressed explicitly in terms of the parameters of the profit function, where M = r/2 - (r2/4 + (A + rCJ/B)1/2, where A, B, and C are scalars. In order to generalize this methodology to more than one quasi-fixed factor, LeBlanc and Hrubovcak (1984) specified a quadratic form such that the optimal 34 levels of variable inputs depend only on factor stocks and are independent of investment. Therefore, they rely on external adjustment costs reflected by rising supply prices of the factors. In addition, the adjustment mechanism for each input is assumed independent of the degree of disequilibrium in other quasi-fixed factors such that the implied adjustment matrix is diagonal. This facilitates expression of net investment demand equations for more than one quasi-fixed input in terms of parameters of the profit function, but at considerable expense to the generality of their approach. Dynamic Duality The use of static duality in these models of dynamic factor demands leads naturally to the development of a dual relationship of dynamic optimizing behavior and an underlying technology. Such a general dynamic duality was conjectured by McLaren and Cooper (1980). Epstein (1981) establishes the duality of a technology and a behavioral function consistent with maximizing the present value of an infinite stream of future profits termed the value function. 35 Theoretical Model The firm's problem of maximizing the present value of current and future profits3 may be written as OO (2.16) J0(K,p,w) = max./ ert [F (L, K, K) w'L p'K] dt L > 0, K < 0 subject to K(0) = K0 > 0. The production function F(L,K,K) yields the maximum amount of output that can be produced from the vectors of variable inputs, L, and quasi-fixed inputs, K, given that net investment K is taking place. The vectors w and p are the rental prices or user costs corresponding to L and K respectively, normalized with respect to output price. Additionally, r > 0 is the constant real rate of discount, and K0 is the initial quasi-fixed input stock. J(K0,p,w,r) then characterizes a value function reflecting current and discounted future profits of the firm. The following regularity conditions are imposed on the technology represented by F(L,K,K) in (2.16): T.l. F maps variable and quasi-fixed inputs and net investment in the positive orthant; F, Fl, and Fj< are continuously differentiable. T.2. Fl, Fk > 0, F < 0 as K > 0. T.3. F is strongly concave in (L, K) . 3 The exposition of dynamic duality draws heavily on the theory developed by Epstein (1981). 36 T.4. For each combination of K, p, and w in the domain of J, a unique solution for (2.16) exists. The functions of optimal net investment, K*(K,p,w), variable input demand, L*(K,p,w), and supply, y*(K,p,w) are continuously differentiable in prices, the shadow price function for the quasi-fixed inputs, A*(K,p,w), is twice continuously differentiable. T.5. ^*p(K,p,w) is nonsingular for each combination of quasi-fixed inputs and input prices. T.6. For each combination of inputs and net investment, there exists a corresponding set of input prices such that the levels of the inputs and net investment are optimal. T.7. The problem in (2.16) has a unique steady state solution for the quasi-fixed inputs that is globally stable. Condition T.l requires that output be positive for positive levels of inputs. Declining marginal products of the inputs characterize the first requirements of T.2. Internal adjustment costs are reflected in the requirements of FÂ£# The extension allowing for positive and negative levels of investment requires that the adjustment process be symmetric in the sense that when net investment is positive some current output is foregone but when investment is negative current output is augmented. Consistent with assumed optimizing behavior, points that violate T.6 would never be observed. 37 Assuming price expectations are static, inputs adjust to "fixed" rather than "moving" targets of long- run or equilibrium values. However, prices are not treated as fixed. In each subsequent period a new set of prices is observed which redefine the equilibrium. As the decision period changes, expectations are altered and previous decisions are no longer optimal. Only that part of the decision optimal under the initial price expectations is actually implemented. Given the assumption of static price expectations and a constant real discount rate, the value function in (2.16) can be viewed as resulting from the static optimization of a dynamic objective funtion. Under these assumptions and the regularity conditions imposed on F(L,K,K), the value function J(K,p,w) is at a maximum in any period t if it satisfies the Bellman (Hamilton- Jacobi) equation for an optimal control (e.g., Intriligator, 1971, p. 329) problem such that (2.17) rJ*(K,p,w) = max {F(L,K,K) w'L p'K + JK(k/P'w) K*}, where JK(K,p,w) denotes the vector of shadow values corresponding to the quasi-fixed inputs, and K* represents the optimal rate of net investment. 38 Through the Bellman equation in (2.17), the dynamic optimization problem in (2.16) may be transformed into a static optimization problem. In particular, (2.17) implies that the value function may be defined as the maximized value of current profit plus the discounted present value of the marginal benefit stream of an optimal adjustment in net investment. Thus, through the Maximum Principle (e.g., Intriligator, 1971, p. 344) the maximizing values of L and K in (2.17) when K = K0 are precisely the optimal values of L and K in (2.16) at t = 0. Utilizing (2.17), Epstein (1981) has demonstrated that the value function is dual to F(L,K,K) in the dynamic optimization problem of (2.16) in that, conditional on the hypothesized optimizing behavior, properties of F(L,K,K) are manifest in the properties of J(K,p,w). Conversely, specific properties of J(K,p,w) may be related to properties on F(L,K,K). Thus, a full dynamic duality can be shown to exist between J(K,p,w) and F(L,K,K) in the sense that each function is theoretically obtainable from the other by solving the appropriate static optimization problem as expressed in (2.17). The dual problem can be represented by (2.18) F*(L,K,K) = min (rJ(K,p,w) + w'L + p'K - p,w JK*K}. 39 The static representation of the value function in (2.17) also permits derivation of demand functions for variable inputs and net investment in quasi-fixed factors. Application of the envelope theorem by differentiating (2.17) with respect to w yields the system of variable factor demand equations (2.19) L*(K,p,w) = -rJw' + Jwk'K*, and differentiation with respect to p yields a system of optimal net investment equations for the quasi-fixed inputs, (2.20) K*(K,p,w) = JpK-1'(rJp'+ K) . This generalized version of Hotelling's Lemma permits the direct derivation of a complete system of input demand equations theoretically consistent with dynamic optimizing behavior. The ability to derive an equation for net investment is crucial to understanding the short-run changes or adjustments in the demand for quasi-fixed inputs. The system is simultaneous in that 40 the optimal variable input demands depend on the optimal levels of net investment, K*. In the short-run, when K* f 0, the demand for variable inputs is conditional on net investment and the stock of quasi-fixed factors. In addition, a supply function for output is endogenous to the system. The optimal supply equation derived by solving (2.17) for F(L,K,K) where (L,K)=(L*,K*) may be expressed as (2.21) y*(K,p,w) = rJ + w'L* + p'K JKK*. As for the variable input equation, optimal supply depends on the optimal level of net investment. This is consistent with internal adjustment costs as the cost of adjusting quasi-fixed factors through net investment is reflected in foregone output. The regularity conditions implied by the properties of the production function are manifested in (2.19)- (2.21) and provide an empirically verifiable set of conditions on which to evaluate the theoretical consistency of the model. Consistency with the notion of duality dictates that the previously noted properties of the technology be reflected in the value function. The properties (V) manifest in J from the technology are V.1. J is a real-valued, bounded-from-below function defined in prices and quasi-fixed inputs. J and JK are twice-continously differentiable. V. 2 . V. 3 . 41 V. 4 . V. 5. V. 6. V. 7 . rJK + P JKK(K*), Jk > JK< as K*<0. For each element in the domain of J, y*>0; for such K in the domain of J, (L*, K, K*) maps the domain of $ onto the domain of F. The dynamic system K*, K(0) = K0, in the domain of J defines a profile K(t) such that (K(t),p,w) is in the domain of J for all t and K(t) approaches K(p,w), a globally steady state also in the domain of J. JpK is nonsingular. For the element (K,p,w) in the domain of J, a minimum in (2.18) is attained at (p,w) if (K,L) = (K*,L*) . The matrix Lp is nonsingular for Kw KÂ£ each element, (K,p,w), in the domain of J. These regularity conditions are essential in establishing the dynamic duality between the technology and the value function. In fact, the properties of J are a reflection of the properties of F. The definition of the domain of F implies V.l. Condition V.2 reflects in (p,w) the restrictions imposed on the marginal products of the inputs, FL and F, and net investment, Fk, in T.2. The conditions in V.3 with respect to an optimal solution in price space, (p,w), are dual to the conditions for an optimal solution in input space, (L,K), maintained in T.6. V.4 is the assumption of the global steady state solution as in T.7. Given JK =A * noted earlier, V.5 is the dual of T.5. V.7 is a reflection of the concavity requirement of T.3. 42 Condition V.6 may be interpreted as a curvature restriction requiring that first-order conditions are sufficient for a global minimum in (2.18). Epstein (1981) has demonstrated that if Jj< is linear in (p,w) , V.6 is equivalent to the convexity of J in (p,w). An advantage of dynamic duality is that these conditions can be readily evaluated using the parameters of the empirically specified value function. The specification of a functional form for J must be potentially consistent with these properties. The Flexible Accelerator Dynamic duality in conjunction with the value function permits the theoretical derivation of input demand systems consistent with dynamic optimizing behavior. Such a theoretical foundation establishes the relationship between quasi-fixed and variable input demand and an adjustment process in the levels of quasi- fixed inputs as a consequence of the underlying production technology. One may note that the net investment demand equation for a single quasi-fixed input derived from the incorporation of the restricted variable profit function in the primal dynamic problem yields a coefficient of adjustment as a function of the discount rate and the parameters of the profit function similar to the constant adjustment coefficient employed in the partial 43 adjustment model. However, an explicit solution of the system of net investment equations with two or more quasi-fixed inputs in terms of an adjustment matrix is difficult. Nadiri and Rosen (1969) considered their model as an approximate representation of an adjustment matrix derived from dynamic optimization. Dynamic duality provides a theoretical means of deriving a wide variety of adjustment mechanisms. The difficulties in relating a specific functional form of the production function to the adjustment mechanism in the direct or primal approach and the limited applicability of the adjustment mechanism derived from incorporating the restricted variable profit function in the dynamic objective function are alleviated considerably. However, the functional form of the value function is critical in determining the adjustment mechanism. The adjustment mechanism of interest in this analysis is the multivariate flexible accelerator. Although the theoretical model relies on a constant real discount rate, it is not unreasonable to hypothesize that this constant rate of discount is partially responsible for the rates of adjustment in quasi-fixed inputs to their equilibrium levels. Noting the potential number of parameters and non-linearities in the demand equations, an adjustment matrix of 44 coefficents as a function of the discount rate and the parameters of the value function may be the desired form of the adjustment process for empirical purposes. Epstein (1981) establishes a general form of the value function from which a number of globally optimal adjustment mechanisms may be derived. The adjustment mechanism of constant coefficents is a special case.4 The flexible accelerator [r + G] is globally optimal if the value function takes the general form (2.22) J(K,p,w) = g(K,w) + h(p,w) + p'G-1K. This form yields JpK = G_1 and Jp = hp(p,w) + G-1K. Substituting in (2.20) yields the optimal net investment equations of the form (2.23) K*(K,p,w) = G[rhp(p,w)] + [r + G]K. Solving (2.23) for K(p,w) at K*=0, (2.24) K(p,w) = -[r + Grl-Gtrhp^w) ] . Multiplying (2.24) by [r + G] and substituting directly in (2.23) yields (2.25) K*(K,p,w) = -[r + G]K(p,w) + [r + G]K = [r + G] [K K(P,W)]. 4 The derivation and proof of global optimality of a general flexible accelerator is provided by Epstein, 1981, p. 92. 45 Thus, the flexible accelerator derived in (2.25) is globally optimal given a value function of the form specified in (2.25). While the accelerator is dependent on the real discount rate, the assumption that this rate is constant implies a flexible accelerator of constants. The linearity of Jj^ in (p,w) which implies the convexity of J in (p,w), is crucial in the derivation of a globally optimal flexible accelerator of fixed coefficients. CHAPTER III EMPIRICAL MODEL AND DATA Empirical Model The specification of the value function J is taken to be log-quadratic in normalized prices and quadratic in the quasi-fixed inputs. The specific form of the value function J(K,p,w) is thus given by (3.1) J(K,p,w) = a0 + a'K + b'log p + c'log w + 1/2(K1AK + log p'B log p + log w'C log w) + + log p'D log w + p' G-1K + w'NK + p'G'^-VkT - w'VlT where K = [K, A], a vector of the quasi-fixed inputs, capital and land, p = [pk, pa] denotes the vector of normalized (with respect to output price) prices for the quasi-fixed inputs, and w = [pw, pm], the vector of normalized variable input prices for labor and materials respectively. Thus, log p = [log pk, log pa] and log w = [log pw, log pm]. T denotes a time trend variable. Parameter vectors are defined by a = [aK, aA], b = [bk, ba], c = [cw, cm], VK = [vK, vA], and VL = [vL, vM]. The vectors VK and VL are technical change parameters for the quasi-fixed inputs and variable 46 47 inputs. The variable input vector is defined by L = [L, M], where L denotes labor and M denotes materials. Parameter matrices are defined as: A = aKK aKA aAK &AA_ ... bkk bka~ bak baa cww cwm cmw cmm ! D = dkw dkm daw dam ... nKw nKm nAw nAm , and G -1 = [gKK gKA " gAK gAA Let G = gKK ^kaT The matrices A, B, and C are _9AK 9aaJ symmetric. The incorporation of some measure of technical change is perhaps as much a theoretical as empirical issue. The assumption of static expectations applies not only to relative prices but the technology as well. The literature contains two approaches to the problem of technical change in dynamic analysis: detrending the data (Epstein and Denny, 1983) or incorporating an unrestricted time trend (Chambers and Vasavada, 1983b; Karp, Fawson, and Shumway, 1984). An argument for the former (Sargent, 1978, p. 1027) is that the dynamic model should explain the indeterminate component of the data seriesthat which is not simply explained by the passage of time. However, as Karp, Fawson, and Shumway (1984, p. 3) note, the restrictions of dynamic model reflected in the investment equations involve real rather than detrended economic variables so the restrictions may not be appropriate for detrended values. 48 The latter approach is adopted in the above specification of the value function in (3.1). Thus, investment and variable input demand equations derived from the value function in (3.1) include an unrestricited time trend. This form allows the technical change parameters to measure in part the relative effect of technical change with respect to factor use or savings over time. Note that the presence of G-1 in the interaction of p, VK, and T in the interaction of p, VK, and T in (3.1) ensures that the technical change parameters enter the investment demand equations without restriction. The incorporation of technical change in the value function serves as an illustration of the difficulty in incorporating policy, human capital, and other variables besides prices into the value function. In static optimization, the input demand equations are determined by first-order derivatives of the objective function. Therefore, the interpretation of parameters in terms of their effects on the objective function is straightforward. The demand equations derived from dynamic optimization contain first- and second-order derivatives of the value function. The value function can be specified to permit a direct interpretation of the parameters in terms of the underlying demand equations. However, 49 relating these parameters to the dynamic objective function becomes difficult. Without estimating the value function directly, one must rely on the regularity conditions implied by dynamic duality to ensure consistency of the empirical specification and underlying theory. Input Demand Equations Utilizing the generalized version of Hotelling's Lemma in (2.20), the demand equations for optimal net investment in the quasi-fixed inputs are given by A (3.2) K*(K,p,w) = G[rp-1(b + B log p + D log w) + rVKT] + [r + G]K, where K*(K,p,w) = [K*(K,p,w), *(K,p,w)] signifies that optimal net investment in capital and land, is a function of factor stocks and input prices. r is a diagonal matrix of the discount rate, and p is a diagonal matrix of the quasi-fixed factor prices. The specification of G-1 in (3.1) permits direct estimation of the parameters of G in the net investment equations. Net investment, or the rate of change in the quasi-fixed factors, is determined by the relative input prices and the initial levels of the quasi-fixed factors, as evidenced by the presence of K in (3.2). The premultiplication by G (G = JpK-1 from (2.20)) 50 yields a system of net investment demand equations that are nonlinear in parameters. The technical change component for the quasi-fixed inputs in the value function (3.1) enters the net investment demand equations in a manner consistent with the assumption of disembodied technical change. The optimal short-run demand equations for the variable inputs are derived using (2.19), and are given by (3.3) L*(K,p,w) = -rw-1(c + D log p + C log w) + rVwT - rNK + NK*(K,p,w), where L*(K,p,w) = [L*(K,p,w), M*(K,p,w)], the optimal short-run input demands for the variable inputs, labor and materials, r is again a diagonal matrix of the discount rate, and w is a diagonal matrix of the variable input prices. The short-run variable input demand equations depend not only on the initial quasi-fixed input stocks but the optimal rate of net investment in these inputs as well. While variable inputs adjust instantaneously, the adjustments are conditioned by both K and K*. The presence of K*(K,p,w) in the variable input demand equations dictates that net investment and variable input demands are determined jointly, requiring a simultaneous equations approach. 51 The derivation of optimal net investment and variable input demands in (3.2) and (3.3) are presented as systems in matrix notation. The precise forms of the individual net investment and variable input demands used in estimation are presented in Appendix Table A-l. Long-run Demand Equations In the dynamic model, the quasi-fixed inputs gradually adjust toward an equilibrium or steady state. The long-run level of demand for an input is defined by this steady state, such that there are no more adjustments in the input level. In other words, net investment is zero. The long-run or steady state demands for the quasi-fixed inputs are derived by solving (3.2) for K when K*(K,p,w) = 0. The long-run demand equation for the quasi-fixed factors is thus given by A (3.4) K(p,w) = [I + rG-1]-1[rp-1(b + B log p + D log w) + rVj^T] , where K(p,w) =[K(p,w), A(p,w)]. Note that these long- run demand equations are functions of input prices alone. Noting that the short-run demand equations for the variable inputs in (3.3) are conditional on K and 52 K*, substitution of K(p,w) for K and K*(K,p,w) = 0 in the short-run equations yields (3.5) L(K,p,w) = -rw-1(c + D log p + C log w) + rVLT - rNK(p,w), where L(K,p,w)=[L(K,p,w), M(K,p,w)]. The long-run variable input demands are no longer conditional on net investment, but are determined by the long-run levels of the quasi-fixed inputs. The individual long-run demand equations for all inputs are presented in Appendix Table A-2. Short-run Demands The short-run variable input demands were presented in (3.3). The variable input demands are conditional on the initial levels of the quasi-fixed inputs and optimal net investments. The short-run demand for the quasi-fixed inputs requires the explicit introduction of time subscripts in order to define optimal net investment in discrete form as (3.6) Kt*(K,p,w) = K*t Kt_1# where K*-(- = [K*^, A*tl / the vector of quasi-fixed inputs at the end of period t. Therefore, the short- run demand for capital at the beginning of period t is (3.7) K*t(K,p,w) = Kt_x + K*t(K,p,w), 53 where K*t(K,p,w) is the optimal demand for the quasi- fixed inputs in period t, K^-i is the initial stock at the beginning of the period, and K*-j- is net investment during the previous period. The short-run demand equations for the quasi-fixed inputs are optimal in the sense that the level of the quasi-fixed input, K*t, is the sum of the previous quasi-fixed input level and optimal net investment during the prior period. Returning to (3.2), the short-run demand equation for the quasi-fixed input vector can be written (3.8) K*t(K,p,w) = G[rp-1(b + B log p + D log w)] + rVKT + [I + r + G]Kt_!, where the time subscripts are added to clarify the distinction between short-run demand and initial stocks of the quasi-fixed inputs. The individual short-run demand equations for the quasi-fixed inputs are presented in Appendix Table A-3. The Flexible Accelerator The flexible accelerator matrix M = [r + G] was shown to be consistent with the general form of the value function in (3.1) in the previous chapter. 54 Rewriting (3.2) and multiplying both sides of the equation by G-1 yields (3.9) G-1K*(K,p,w) = rp-1(b + B log p + D log w) + rVKT + [I + rG-1]K. Multiplying both sides by [I + rG-1]-1 and noting that [I + rG-1]-1 = [r + G]-1G, then (3.9) can be written as (3.10) [r + G]-1K*(K,p,w) = [I + rG-1]-1[rp-1(b + B log p + D log w) + rVKT] + K. The first term on the right hand side of (3.10) is identical to the negative of the long-run quasi-fixed input demand equation in (3.4). Substituting -K(p,w) in (3.10) and solving for K*(K,p,w) yields (3.11) K*(K,p,w) = [r + G] [K K(p,w]. As may be noted, this is precisely the form of the multivariate flexible accelerator. Solving (3.11) for the individual equations, the optimal net investment in capital is (3.12) K* = (r + gKK)(K K) + gKA(A A), and optimal net investment in land may be written (3.13) * = gAK(K K) + (r + g^) (A A) . 55 Thus, gj^ and the parameters associated with land in the value function appear in the net investment equation for capital. Likewise, gAK and the parameters associated with capital in the value function appear in the net investment equation for land. Hypotheses Tests The form of the flexible accelerator in (3.11) permits direct testing of hypotheses on the adjustment matrix in terms of nested parameter restrictions. The appropriateness of these tests are based on Chambers and Vasavada (1983b). Of particular interest is the hypothesis of independent rates of adjustment for capital and land which can be tested via the restrictions gj^ = gAK = o. Independent rates of adjustment indicate that the rate of adjustment to long-run equilibrium for one quasi-fixed factor is independent of the level of the other quasi-fixed factors. The hypothesis of an instantaneous rate of adjustment for the quasi-fixed inputs relys on independent rates of adjustment. Thus, a sequential testing procedure is dictated. Given that the hypothesis of independent rates of adjustment is not rejected, instantaneous adjustment for land and capital requires r + gK^ = r + g^ = -1, in addition 56 to 9KA = 9ak = 0. If both inputs adjust instantaneously, the adjustment matrix takes the form of a negative identity matrix. Capital and land would adjust immediately to long-run equilibrium levels in each time period. Regularity Conditions An attractive feature of the theoretical model is the regularity conditions that establish the duality of the value function and technology. Even so, little focus has been given to these conditions in previous empirical studies beyond the recognition of the existence of steady states for the quasi-fixed factors and a stable adjustment matrix required by condition V. 4. Without estimating the supply function or value function directly it is impossible to verify the regularity conditions stated in V.I., V.2 and Y*>0, the first part of condition V.3. One can note with slight satisfaction, however, that these conditions are likely to be satisfied if a0, aK, and aA are sufficiently large positive (Epstein, 1980, pg 88). The differentiability of J and JK are, of course, implicitly maintained in the choice of the value function. The conditions in V.4 are readily verified by determining if the long-run or equilibrium factor demands at each data point are positive to ensure the 57 existence and uniqueness of the steady states. Furthermore, the stability of these long-run demands is ensured if the implied adjustment matrix is nonsingular and negative definite. The nonsingularity of the adjustment matrix is related to condition V.5, the nonsingularity of JpK, as JpK = [M r]-1 demonstrated in the previous chapter. Regularity condition V.7 is easily verified by the calculation of demand price elasticities for the inputs. Condition V.6 may be viewed as a curvature restriction ensuring a global minimum to the dual problem. Since JK is linear in prices, this condition is equivalent to the convexity of the value function J in input prices. The appropriate Hessian of second- order derivatives is required to be positive definite. Elasticities One particularly attractive aspect of dynamic optimization is the clear distinction between the short run, where quasi-fixed inputs only partially adjust to relative price changes along the optimal investment paths, and the long run, where quasi-fixed inputs fully adjust to their equilibrium levels. However, expectations with respect to the signs of price elasticities based on static theory are not necessarily valid in a dynamic framework. 58 Treadway (1970) and Mortensen (1973) have shown that positive own-price elasticities are an indication that some inputs contribute not only to production but to the adjustment activities of the firm. Thus in the short-run, the firm may employ more of the input in response to a relative price increase in order to facilitate adjustment towards a long-run equilibrium. However, this does not justify a positive own-price elasticity in the long-run. This same contribution to the adjustment process may also indicate short-run effects which exhibit greater elasticity than the long run. The firm may utilize more of an input in the short-run in order to enhance adjustment than in the long-run in response to a given price change. Short-run variable input demand elasticities may be calculated from (3.3). For example, the elasticity of labor demand with respect to the price of the jth input, e L,pj' is . * 9 L 3 L * * * 3K + 3 L * * 3 A pj. * 3 Pj 3 K 3Pj 3 A 3Pj L The short-run elasticity of demand for a variable input depends not only on the direct effect of a price change, but the also on the indirect effects of a price change on the short-run demand for the quasi- fixed inputs. 59 The short-run price elasticity for a quasi-fixed input is obtained from (3.7). The short-run demand elasticity for capital with respect to a change in the price of the jth input, Â£K,pj i-s * <3-16> e5,Pj 9K Pj * 9Pj K The short-run elasticity of demand for a quasi-fixed input depends only on the direct price effect in the short-run demand equation. The long-run elasticity of demand for a variable input can be obtained from (3.5). In the long run, all inputs are at equilibrium levels. Thus, the long- run elasticity of demand for labor with respect to the price of input j is (3.17) 9 L 3 L 3p_. 3 K 3 K 3 L 3 Pj 3 A 3_^_ _Â£j 9 Pj L The long-run elasticity of a variable input is conditional on the effect of a price change in the equilibrium levels of the quasi-fixed inputs. The long-run demand elasticity for a quasi-fixed input is determined from (3.4). The long-run 60 elasticity of demand for capital with respect to the jth input price is (3.18) 8 K 3-pT In contrast to the short-run demand for a quasi-fixed input, where the short-run demand for one quasi-fixed input is determined in part by the level of the other quasi-fixed input, the long-run demand for a quasi- fixed input is solely an argument of prices. Data Construction The data requirements for the model consist of stock levels and net investment in land and capital, quantities of the variable inputs, labor and materials, as well as normalized (with respect to output price) rental prices for the inputs for the Southeast region. This region corresponds to the states of Alabama, Florida, Georgia, and South Carolina. The appropriate data are constructed for the period from 1949 through 1981. Data Sources Indices of output and input categories for the the Southeast are provided in Production and Efficiency Statistics (USDA, 1982). The inputs consist of farm power and machinery, farm labor, feed, seed, and livestock purchases, agricultural chemicals, 61 and a miscellaneous category. These indices provide a comprehensive coverage of output and input items used in agriculture for the respective categories. Annual expenditures for livestock, seed, feed, fertilizer, hired labor, depreciation, repairs and operations, and miscellaneous inputs for each state were obtained from the State Income and Balance Sheet Statistics (USDA) series. The expenditures for each of the Southeastern states are summed to form regional expenditures corresponding to the appropriate regional input indices cited above. This same series also contains revenue data for each state in the categories of cash receipts from farm marketing, value of home consumption, government payments, and net change in farm inventories. These data are aggregated across states to form a regional measure of total receipts. These sources provide the data for the construction of capital, materials, and labor quantity indices and capital and materials price indices. A GNP deflator is used to convert all expenditures and receipts to 1977 dollars. Additional data is drawn from Farm Labor (USDA) in order to construct a labor price index. Farm Real Estate Market Developments (USDA) provides quantity and price data for land. The undeflated regional expenditure and input data are provided in Appendix B. 62 Capital Capital equipment stocks and investment data are not available below the national level prior to 1970. Therefore, the mechanical power and machinery index was taken as a measure of capital stocks. As Ball (1985) points out, this index is intended to measure the service flow derived from capital rather than the actual capital stock. The validity of the mechanical power and machinery index as a measure of capital stock rests on the assumption that the service flow is proportional to the underlying capital stock. It is possible that the service flow from capital could increase temporarily without an increase in the capital stock if farmers used existing machinery more intensely without replacing worn-out equipment. Eventually, worn-out capital would have to replaced. Ball relies on a similar assumption of proportionality in employing the perpetual inventory method (Jorgenson, 1974) in deriving capital stocks. This method relies on the assumption of a constant rate of replacement in using gross investment to determine capital stocks such that (3.19) Ait = Iit + (1 6i)Ai/t-l, where Ait is capital stock i in period t, lit is gross investment, and 6i is the rate of replacement. Even 63 the regional level, the perpetual inventory approach appears to share the potential weakness of the mechanical power and machinery index. Determining the appropriate price of capital presents additional difficulty. Hall and Jorgenson (1967) and Jorgenson (1967) define the user cost or implicit rental price of unit of capital as the cost of the capital service internally supplied by the firm. This actual cost is complicated by the discount rate, service life of the asset, marginal tax rate, allowable depreciation, interest deductions, and degree of equity financing. An alternate measure of user cost is provided by expenditure data representing actual depreciation or consumption of capital in terms of replacement cost and repairs and operation of capital items (Appendix Table B-l). By combining these expenditure categories in each time period to represent the user cost of the capital stock in place during the period, these expenditures and the machinery index can be used to construct an implicit price index for the region. 64 An implicit price index for capital is constructed using Fisher's weak factor reversal test (Diewert, 1976). The implicit price index may be calculated by (3.20) Pit = (Eit/Eib) / it' where and Pit denote the quantity and price indices corresponding to the ith input in period t, and expenditures on the ith input in the same time period are denoted by E-^, and b denotes the index base period. Fisher's weak factor reversal test for price and quantity indices is satisfied if the ratio of expenditures in the current time period to the base is equal to the product of the price and quantity indices in the current time period. Since the machinery index and expenditure data are based in 1977, the resulting implicit price index for capital is also based in 1977. Land The land index represents the total acres in farms in the Southeast. The regional total is the sum of the total in each state. Hence, farmland is assumed homogeneous in quality within each state. An adjustment in these totals is necessary for the years 65 after 1975 as the USDA definition of a farm changed.5 Observations after 1975 are adjusted by the ratio of total acres under the old definition to total acres using the new definition. A regional land price index is constructed by weighting the deflated index of the average per acre value of farmland in each state by that state's share of total acres in the region. Unlike most price indices, the published index of farmland prices is not expressed in constant dollars. As rental prices are not available for the region, the use of an index of price per acre implicitly assumes that the rental rate is proportional to this price. The regional acreage total, quantity index, and price index may be found in Appendix Table B-2. Labor Beyond the additional parameters needed in the empirical model to treat labor as quasi-fixed, the farm labor index reflects the quantity employed, not necessarily the stock or quality of labor available. Hence, the regional labor index by definition represents a variable input. The USDA index of labor weights all hours equally, regardless of the human 5 Prior to 1975, a farm was defined as any unit with annual sales of at least $250 of agricultural products or at least 10 acres with annual sales of at least $50. After 1975, a farm is defined as any unit with annual sales of at least $1000. 66 capital characteristics of the workers. Additionally, this quantity index is not determined by a survey of hours worked but calculated based on estimated quantities required for various production activities. This presents some difficulties. The USDA farm labor quantity index includes owner-operator and unpaid family labor as well as hired labor, while the corresponding expenditures include wages and perquisites paid to hired labor, and social security taxes for hired labor and the owner- operator. Derivation of a price index as in (3.20) using these quantity and expenditure data treats owner-operator and family labor as if they were free. Instead, the USDA expenditures on hired labor and a regional quantity index of hired labor for the region calculated from Farm Labor (USDA) are used to calculate a labor price index. This assumes that owner-operators value their own time as they would hired labor. While this may seem inappropriate, the relative magnitude of hired labor to owner-operator labor in the Southeast reduces the impact of such an assumption. The regional total for expenditures on hired labor, the hired labor quantity index, and labor price index are presented in Appendix B, Table B-3. 67 Materials Expenditure data on feed, livestock, seed, fertilizer, and miscellaneous inputs are used to construct budget shares that provide the appropriate weights for each input in constructing an aggregate index. The indices represent quantities used rather than stocks, so the materials index characterizes a variable input. Some part of the livestock expenditure goes toward breeding stock, which is potentially quasi-fixed. The impact of investment in breeding stock is minimal, as the relative share of expenditures on livestock in the region is quite small. Again, Fisher's weak factor reversal test as shown in (3.20) can be readily applied to derive an implicit price index for materials. The expenditures on each of the inputs are aggregated and deflated. The ratio of aggregate materials expenditures in each time period to expenditures in 1977 is divided by the corresponding ratio of the aggregate materials input index. The regional expenditures for material inputs, aggregate materials index, and materials price index are presented in Appendix Table B-4. Output Price Equation (3.20) can also be used to construct an implicit output price index for the Southeast region 68 in order to normalize input prices. By combining the value of cash receipts, government payments, net inventory change, and the value of home consumption as a measure of output value for each region, this value and the aggregate output quantity index can be used to derive an implicit output price index. The output price of the prior year is used to normalize input prices to reflect that current price is not generally observed by producers when production and investment decisions are made. Regional total receipts, output quantity index, and output price index are found in Appendix Table B-5. Net Investment The observations on the USDA input indices correspond to quantities used during the production period. This is satisfactory for the variable inputs, labor and materials. However, the mechanical power and machinery index in effect reflects stock in place at the end of the production period. Therefore, this index is lagged one time period to reflect an initial level of available capital stock. The same procedure applies to the index of total acres in farms for the Southeast, as total acres are measured at year's end. As noted earlier, it is not possible to obtain estimates of gross investment in capital for the Southeast region over the entire data period. A 69 measure of net investment in capital and land for each time period can be defined for each of the inputs by (3.21) Ki/t = Kift Ki t.lf where is net investment in the quasi-fixed input i during period t, is the level of the input stock in place at the end of period t, and is the level of input stock in place at the beginning of period t. By developing the model in terms of net investment, the need for gross investment and depreciation rate data in the determination of quasi- fixed factor stocks via (3.19) is elimimated. Since the estimated variable is actual net investment, it has been common practice in previous studies (Chambers and Vasavada, 1983b; Karp, Fawson, and Shumway, 1984) to assume constant rates of actual depreciation in order to calculate net investment from gross investment data. However, it is possible that the rate of depreciation could vary over observations. By using the difference of a quasi-fixed input index between two time periods as a measure of net investment, this problem can be at least partially avoided. 70 Data Summary Before proceeding to the estimation results of the empirical model, a brief examination of input use in the Southeast is in order. The quantity indices for capital, land, labor, and materials inputs used in the Southeast region for the years 1949 through 1981 are presented in Appendix Table B-6. The base year for the quantity and price indices is 1977. Figure 3-1 depicts the quantity indices for the 1949-1981 period. During the early years of the data period, agricultural production in the Southeast was characterized by a substantial reliance on labor and land relative to materials and capital. The quantity index of labor in 1949 was over three-and-one-half times the quantity index in 1981. Except for a short period of increase from 1949 to 1952, the quantity of land in farms has gradually declined from a high of 774 million acres in 1952 to 517 million in 1981, a decrease of nearly 35 percent. On the other hand, capital stocks nearly doubled, from 52 to 105, and the use of aggregate materials rose 250 percent, 44 to 112, from 1949 to 1981. 71 Quantity Capital + Land ^ Labor A Materials Figure 3-1. Observed Input Demand for Southeastern Agriculture, 1949-1981. 72 Turning to the normalized input prices, Figure 3-2 charts these input prices over the period of analysis. Not suprisingly, the same inputs whose quantities have dropped the most, labor and land, correspond to the inputs whose normalized prices have increased dramatically, labor increasing seventeen fold, from 0.10 to 1.71, and land six-fold, 0.21 to 1.35, over the data period. The most dramatic increase in the labor price index begins in 1968, such that nearly eighty percent of the increase in the labor price index occurs from 1967 to 1981, jumping from 0.39 to 1.71. The increase in the normalized land price index is more gradual, such that 50 percent of the increase occurs prior, 0.21 to 0.66, and 50 percent, 0.66 to 1.35, after 1966, the midpoint of the data period. The normalized price of capital doubled between 1949 and 1981, from 0.68 to 1.21, while the materials price increased only 10 percent, from 0.913 to 1.04. Interpretation of these changes in the normalized price indices should be tempered by recognizing that the indices are normalized with respect to output price. A drop in the output price would produce an increase in the normalized input price, everything else constant. However, examination of the actual output price index in Appendix Table B-5 shows only a 73 Capital, Land Index Year n Capital Land Labor Materials U Price T Price v Price a Price Figure 3-2. Normalized Input Prices for the Southeast, 1949-1981. 74 12 percent change in the output price index from endpoint, 1.08 in 1949, to endpoint, 0.94 in 1981. The rapid increase in output price of nearly 25 percent from 1972 to 1973, 0.89 to 1.112, produced a substantial drop in the normalized price indices for capital, land, and labor. The materials price index, however, rose even faster than the output price index, so the normalized price of materials increased. These data indicate that the Southeast has undergone some substantial changes from 1949 to 1981.6 The normalized price of labor has risen as dramatically as the quantity index has fallen. The Southeast has come to rely substantially more on materials and capital than in the past. The quantity of land in farms has gradually declined. It remains for the next chapter to see what light a dynamic model of factor demands can shed on these changes. 6. McPherson and Langham (1983) provide a historical perspective of southern agriculture. CHAPTER IV RESULTS Theoretical Consistency This chapter presents the results of estimating net investment demand equations for capital and land and variable input demand equations for labor and materials. The consistency of the data with the assumption of dynamic optimizing behavior is considered by evaluating the regularity conditions of the value function. Estimated short- and long-run levels of demand are obtained from the parameters of the estimated equations and compared to observed input demand. Estimates of short- and long-run price elasticities are computed in order to identify gross substitute/complement relationships among the inputs. Method of Estimation The system of equations presented in the previous chapter were estimated using iterated nonlinear three- stage least squares.7 For purposes of estimation, a disturbance term was appended to the net investment and variable input demand equations to reflect errors in optimizing behavior. This convention is consistent 7. The model was estimated using the LSQ option of the Time Series Processor (TSP) Version 4.0 as coded by Hall and Hall, 1983. 75 76 with other empirical applications (Chambers and Vasavada, 1983b; Karp, Fawson, and Shumway, 1984), although Epstein and Denny assume a first order autoregressive process in the error term for the quasi-fixed input demand equations.8 The iterated nonlinear three stage least squares estimation technique is a minimum distance estimator with the distance function D expressed as (4.1) D = f(y,b)'[S-1S H (H'H)1H'] f(y,b) where f(y,b) is the stacked vector of residuals from the nonlinear system, S is the residual covariance matrix, and H is the Kronecker product of an identity matrix dimensioned by the number of equations and a matrix of instrumental variables. For this system, the instruments consist of the normalized prices and their logarithms, quasi-fixed factor levels, and the time trend. Although the system is nonlinear in parameters, it is linear in variables. Hence, the minimum distance estimator is asympotically equivalent to full information maximum likelihood (Hausman, 1975) and provides consistent and asymptotically efficient parameter estimates. 8. Such an assumption necessitates estimation of a matrix of autocorrelation parameters. For two quasi-fixed inputs, this would require estimation of four additional parameters. 77 A constant real discount rate of five percent was employed in the estimation. This rate is consistent with the estimates derived by Hoffman and Gustafson (1983) of 4.4 percent reflecting the average twenty year current return to farm assets, 4.3 percent obtained by Tweeten (1981), and 4.25 percent by Melichar (1979).9 The parameter estimates and associated standard errors are presented in Table 4-1. Thirteen of the twenty-six parameters are at least twice their asymptotic standard errors. Given the nonlinear and simultaneous nature of the system, it is difficult to evaluate the theoretical and economic consistency of the model solely on the structural parameters. Thus, one must consider the underlying regularity conditions and the consistency of the derived input demand equations with observed behavior in order to assess the empirical model. Regularity Conditions An important feature of the dual approach, whether applied to static or dynamic optimization, is that the relevant conditions (V in Chapter II) are easy to check. Lau (1976) notes the difficulty of statistically testing the conditions for a static 9. The parameter estimates are fairly insensitive but not invariant to the choice of discount rates. 78 Table 4-1. Parameter Estimates Treating Materials and Labor as Variable Inputs, Capital and Land as Quasi-Fixed. Parameter Standard Parameter Estimate Errora bK 1913.160 496.327* bA -423.746 229.533 CW -677.077 212.572* cM -343.822 658.473 bkk 2472.608 478.233* bka -243.214 153.757 baa -123.888 98.104 cww -105.878 123.786 cwm -233.275 72.084* cmm 406.767 636.244 dwk -49.625 110.041 dwa 98.263 102.696 dmk 152.451 79.123 dma -111.878 74.397 nwK 1.900 0.731* nwA -1.879 2.283 nmK 0.826 0.309* nmA -0.159 0.530 VK 18.639 5.808* VA 4.155 0.707* VW 17.267 1.507* VM 23.045 8.312* -0.588 0.160* 9KA 0.490 0.242* -0.023 0.015 9aa -0.213 0.056* a indicates parameter estimate two times its standard error. 79 duality, concluding that such tests are limited to dual functions linear in parameters. Statistical testing of the regularity conditions underlying dynamic duality is even more difficult. However, these conditions can be numerically evaluated. Since one of the objectives of this study is to obtain estimates of the adjustment rates of the quasi- fixed inputs and since the elements of the adjustment matrix M=[r+G] can be determined readily from the parameter estimates, the regularity conditions of nonsingularity of JpK-1 and stability of M are examined first. The nonsingularity of JpK-1 is determined from the estimates of the elements of G, as JpK-1 = G. The determinant of G is -0.334, thus satisfying the nonsingularity of JpK-1* The stability of the adjustment matrix requires that the eigenvalues of M have negative real parts and lie within the unit circle. The eigenvalues of G are -0.196 and -0.505, which satisfy the necessary stability criteria. The equilibrium demand levels for capital and land are positive at all data points. The existence and uniqueness of equilibrium or steady state levels of capital, K(p,w), and land, A(p,w)> as a theoretical requirement are also established. 80 It was shown in Chapter II that convexity of the value function in normalized input prices is sufficient to verify the necessary curvature properties of the underlying technology when JK is linear in prices, as is the empirical specification used to derive the current estimates. In fact, the linearity of JK in prices is necessary to generate an accelerator matrix consistent with net investment equations of the form K* = M (K-K). The elements of the matrix of second-order derivatives of the value function with respect to prices in this model are dependent upon the exogenous variables (prices) in the system. Thus, the Hessian must be evaluated for positive definiteness at each data point. This regularity condition was satisfied at thirty-one of the thirty-three observations (See Appendix C-l for numerical results). The only exceptions were the years 1949 and 1950. Given that these observations immediately follow the removal of World War II agricultural policies, the return of a large number of the potential agricultural work force, and rapidly changing production practices incorporating newly available materials, it is perhaps not surprising that the data are inconsistent with dynamic optimizing behavior at these points. 81 Technical Change The parameters representing technical change in the system of equations indicate that technical change has stimulated the demand for all inputs in the Southeast. Incorporation of these parameters as a linear function of time implicitly assumes technical change is disembodied. The relative magnitutude of these estimates indicates that technical change has been material-using relative to labor, and capital using relative to land. While some studies of U.S. agriculture have found technical change to be labor- and land-saving (Chambers and Vasavada, 1983b), the estimated positive values for these inputs is not surprising given the rebirth of agriculture in the Southeast over the past quarter century. At least some portion of technical change has aided in maintaining the demand for labor in the face of rising labor prices by increasing productivity for many crops in the Southeast that rely on hand-harvesting, such as fresh vegetables and citrus. Consistency with Observed Behavior Evaluation of the empirical model relies on more than the theoretical consistency of the parameter estimates with respect to the regularity conditions. In addition, the economic consistency of the model is determined by the correspondence of observed net 82 investment and input use with the estimates or predicted values obtained from the derived demand equations. Satisfaction of the regularity conditions alone is not verification that dynamic optimizing behavior is an appropriate assumption. The observed and estimated values of K* in Table 4-2 show that the Southeast has been characterized by a steady increase in net capital investment, with only a few periods of net disinvestment. The estimated JL , values of K correspond fairly closely to observed net investment. Observed capital stocks and the estimated short-run demand for the stock of capital correspond closely with never more than a two percent difference. However, there is a notable divergence of observed and equilibrium capital stock demand from 1949 to 1973. Contrary to the concerns of overcapitalization today, the Southeast only initially exhibited an excess of capital. However, the equilibrium level of capital rises in response to changing relative prices such that by 1974 observed and equilibrium levels are in close correspondence. The observed capital use and short- and long-run demand levels for capital presented in Figure 4-1 further illustrate this convergence. However, one should note the adjustment 83 Table 4-2. Comparison of Observed and Estimated Levels of Net Investment and Demand for Capital. Net Investment + K K observed optimal Year Capital Demand K K*(K,p, w) observed short- run K(p,w) long- run 49 8.00 7.13 52.00 53.81 16. 58 50 5.00 5.22 60.00 60.34 17. 33 51 6.00 5.69 65.00 65.60 22. 46 52 5.00 5.53 71.00 71.56 29. 16 53 2.00 4.65 76.00 75.98 32. 54 54 1.00 0.29 78.00 75.13 26. 29 55 2.00 0.94 79.00 76.56 28 . 64 56 2.00 0.68 81.00 78.59 33. 48 57 -1.00 -1.67 83.00 79.24 34. 22 58 0.00 -0.42 82.00 79.32 40. 18 59 2.00 1.27 82.00 80.69 47. 80 60 -2.00 -0.92 84.00 81.48 49. 57 61 -1.00 0.33 82.00 80.43 52 . 08 62 -1.00 0.44 81.00 79.68 54 . 34 63 1.00 0.49 80.00 78.90 56. 71 64 0.00 0.44 81.00 80.03 60. 29 65 1.00 0.85 81.00 80.36 62. 11 66 1.00 0.92 82.00 81.44 63 . 89 67 3.00 1.22 83.00 82.83 68 . 77 68 1.00 0.22 86.00 85.19 70. 21 69 0.00 -0.23 87.00 86.05 72 . 96 70 -1.00 0.58 87.00 86.70 76. 50 71 4.00 4.80 86.00 88.55 84. 92 72 -2.00 2.52 90.00 91.06 84. 73 73 3.00 3.37 88.00 89.82 87. 85 74 4.00 3.75 91.00 93.28 95. 12 75 2.00 0.38 95.00 95.09 93. 03 76 2.00 1.67 97.00 97.86 96. 18 77 1.00 0.96 99.00 99.44 97. 72 78 6.00 2.07 100.00 101.22 101. 59 79 6.00 -0.12 106.00 105.88 105. 04 80 -5.00 -2.54 112.00 110.29 106. 43 81 -2.00 2.58 107.00 108.57 109. 84 84 of an excess capital stock to equilibrium levels is not achieved by a disinvestment in capital, but by an increase in the equilibrium level of capital demand. An examination of net investment and demand levels for land in Table 4-3 reveals a situation completely opposite from that of capital. Apart from a short period initially, the Southeast has exhibited a gradual reduction in the stock of land in farms. While estimated long-run demand exceeds observed and short-run demand for land stocks until 1961, the equilibrium level of demand falls at a faster rate than the short-run and observed levels. After 1961, the Southeast was marked by a slight degree of overinvestment in land stocks, owing primarily to an increase in the relative price of land. Observed stocks of land and estimates of short- run demand correspond closely over the data range. While the equilibrium level of capital increased in response to the increasing relative price of labor, the long-run demand for land has declined in response to an increase in the relative price of land as well as increase in the relative price of labor. Observed and estimated short- and long-run demand levels for land are presented graphically in Figure 4-2. 85 Capital Index Observed Estimated Estimated c Demand ^ Short-run ^ Long-run Demand Demand Figure 4-1. Comparison of Observed and Estimated Demand for Capital in Southeastern Agriculture, 1949-1981. 86 Table 4-3. Comparison of Observed and Estimated Levels of Net Investment and Demand for Land. Net Investment Demand for Land Year A observed A* optimal A observed A*(K,p,w) short- run A(p,w) long- run 49 3.45 2.64 142.32 146.78 193.99 50 1.14 2.11 145.77 150.01 198.58 51 0.76 1.01 146.91 149.98 189.99 52 0.19 -0.16 147.68 149.35 177.78 53 -0.95 -0.92 147.87 148.79 172.26 54 -0.57 -1.85 146.91 147.27 170.28 55 -2.86 -2.68 146.34 145.87 163.14 56 -3.05 -3.18 143.48 142.29 152.68 57 -3.63 -3.52 140.43 138.86 146.27 58 -3.82 -3.02 136.80 135.44 141.64 59 -3.44 -2.70 132.99 131.66 135.40 60 -2.48 -2.83 129.55 128.00 129.24 61 -2.86 -2.77 127.07 125.47 125.47 62 -2.86 -2.65 124.21 122.61 121.53 63 -3.63 -2.24 121.35 120.04 119.86 64 -1.72 -2.21 117.72 116.35 114.83 65 -1.53 -2.21 116.00 114.60 112.55 66 -1.53 -2.34 114.48 112.94 109.86 67 -1.91 -1.48 112.95 112.14 112.78 68 -2.29 -2.19 111.04 109.49 104.95 69 -1.91 -1.74 108.75 107.59 105.01 70 -1.34 -1.45 106.85 105.87 103.60 71 -0.95 -1.63 105.51 104.11 96.77 72 -0.76 -1.56 104.56 103.31 97.74 73 -1.15 -0.87 103.79 103.09 99.82 74 -0.57 -0.38 102.65 102.27 99.84 75 0.19 -0.28 102.08 101.90 101.64 76 -1.23 -0.61 102.27 101.76 99.29 77 -1.04 -0.58 101.04 100.54 97.83 78 -1.23 -0.48 100.00 99.52 96.33 79 0.87 -0.38 98.77 98.37 95.52 80 0.22 -0.48 99.63 99.20 96.70 81 -1.62 -0.48 99.85 99.32 95.36 87 Land Index Observed Estimated Estimated ^ Demand + Short-run *' Long-run Demand Demand Figure 4-2. Comparison of Observed and Estimated Demand for Land in Southeastern Agriculture, 1949-1981. 88 The observed use of labor and estimated short- and long-run demands for labor as presented in Table 4-4 indicate almost complete adjustment of observed labor demand to the estimated equilibrium within one time period. This is consistent with the assumption that labor is a variable input. Returning to the parameter estimates in Table 4-1 and the short-run demand equation for labor in Table A-l, capital stocks slightly reduce the short-run demand for labor. Capital investment increases the short-run demand for labor. This indicates that labor facilitates adjustment in capital. The effect of land stocks on the short-demand for labor indicates an increase in land increases the short-run demand for labor. The effect of net investment in land decreases the short- run demand for labor. Labor appears to have a negative effect on the adjustment of land. The long-run demand for labor depends on the equilibrium levels of capital and land to the same degree that short-run labor demand depends on capital and land stocks. An increase in the equilibrium level of capital decreases the long-run demand for labor. Conversely, an increase in the equilibrium level of land increases the long-run demand for labor. Disequilibrium in the quasi-fixed inputs could potentially cause a divergence in the short- and long- 89 Table 4-4. Comparison of Observed and Estimated Short- and Long-Run Demands for Labor. Year Labor observed Labor Demand L*(K,p,w) short-run L (K, p, w) long-run 49 351.195 351.812 351.592 50 320.518 336.175 339.369 51 336.653 320.176 319.446 52 311.155 306.751 302.778 53 295.817 294.246 290.103 54 267.331 270.240 273.312 55 265.139 253.921 253.412 56 239.841 232.346 230.377 57 205.578 217.464 219.113 58 192.231 211.425 210.896 59 196.813 204.972 200.875 60 189.442 189.117 188.722 61 181.873 181.008 177.777 62 179.681 173.914 170.290 63 175.299 169.490 166.351 64 161.355 155.314 151.941 65 146.813 148.204 143.835 66 136.454 138.309 133.380 67 138.048 143.208 139.392 68 128.685 126.861 123.177 69 129.283 124.008 122.116 70 122.908 121.029 117.851 71 120.319 115.411 102.439 72 114.542 108.599 100.668 73 113.147 113.642 105.218 74 109.761 116.091 107.571 75 106.375 107.527 106.405 76 103.785 105.591 101.054 77 100.000 101.159 98.056 78 96.016 99.784 94.424 79 92.430 92.872 92.140 80 95.817 89.070 93.235 81 91.434 95.881 89.346 90 run demand for labor. However, the magnitudes of the parameter estimates associated with the dependence of labor demand on the quasi-fixed factors are small. Thus, the short-and long-run demands for labor are similiar. This is also true for the materials input as shown in Table 4-5. The degree of correspondence of observed and short- and long-run demands for materials is even greater than for labor. The short- and long- run demands for materials depend on quasi-fixed input stocks and equilibrium levels only slightly. Materials appear to facilitate adjustment in capital and slow adjustment in land. The substantial disequilibrium in the Southeast with respect to capital and land during the first part of the sample period may be interpreted from at least two viewpoints, one empirical and one intuitive. Empirically, the specification of the adjustment mechanism in the model is only indirectly dependent on factor prices through the determination of equilibrium levels of the quasi-fixed inputs. The accelerator itself is a matrix of constants. Yet the degree of adjustment in each factor level depends on the disequilibrium between actual and equilibrium input levels, which in turn are a function of the input prices. Changes in relative prices of the inputs, 91 Table 4-5. Comparison of Observed and Estimated Short- and Long-Run Demands for Materials. Materials Demand Materials M*(K,p,w) M(K,p,w) Year observed short-run long-run 49 44.237 44.877 41.464 50 42.623 44.431 43.123 51 49.666 47.821 45.440 52 50.194 50.111 47.166 53 51.917 52.350 49.974 54 51.345 49.255 51.308 55 54.943 53.015 53.908 56 55.006 55.332 56.009 57 54.630 54.572 57.477 58 57.657 58.220 59.693 59 63.332 64.191 63.653 60 63.839 63.245 64.828 61 66.081 67.067 67.159 62 68.409 69.178 69.021 63 69.281 70.373 70.176 64 76.910 75.303 75.007 65 78.817 78.569 77.764 66 79.390 79.980 78.993 67 81.210 83.445 82.368 68 82.814 85.197 84.826 69 87.717 88.000 88.212 70 94.124 93.462 92.798 71 98.432 97.382 91.829 72 101.150 98.821 95.828 73 101.801 94.462 90.681 74 97.131 96.437 92.264 75 86.274 92.328 91.928 76 96.985 100.289 98.369 77 100.000 101.008 99.833 78 105.649 107.951 105.538 79 114.375 109.014 108.972 80 105.367 103.573 106.169 81 112.323 114.331 111.323 92 especially labor, have caused the equilibrium level of capital to rise more rapidly than observed or short run capital demand. A complementary relationship between land and labor and substitute relation between capital and labor contribute to a high demand for land in the long-run and a low equilibrium for capital given the low initial prices of land and labor. Intuitively, these low prices may be attributed to a share-cropper economy, itself a vestige of the old plantations. While the relative prices of labor and land in 1949 reflect this notion, the observed levels of land and capital do not. It thus appears very plausible that during the initial postwar period, agriculture in the Southeast anticipated a change in this system and had already begun investing in capital and reducing land stocks. Elasticity Measures Given the inability to estimate the supply equation, only Marshallian (uncompensated) input demand elasticities were estimated. The explicit recognition of dynamic optimization provides a clear distinction between the short run, where quasi-fixed inputs adjust only partially to relative price changes along an optimal investment path, and the long run, where quasi-fixed input stocks are fully adjusted to equilibrium levels. |

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A DYNAMIC MODEL OF INPUT DEMAND FOR AGRICULTURE IN THE SOUTHEASTERN UNITED STATES By MICHAEL JAMES MONSON 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 1986 Copyright 1986 by Michael James Monson ACKNOWLEDGEMENTS Custom alone does not dictate the need for me to express my gratitude to the members of my committee, as each member has provided valuable assistance in the completion of this dissertation. I thank Dr. Boggess for enabling me to pursue a variety of topics and enhancing the breadth of my graduate research experience, as well as keeping a down-to-earth perspective. Dr. Taylor has served me well as motivator and mentor of my dissertational research. I also express my appreciation to Dr. Majthay for the classroom instruction in optimal control theory and tolerance of an agricultural economist's employment of the theory, Dr. Langham for some timely advice, and Dr. Emerson for restoring my faith in nonlinear models at a crucial moment. Additionally, thanks to the staff of the FARM lab, particularly Rom Alderman, for technical support in preparing the manuscript, and to Pat Smart for doing her best to see that I turned the required forms in on time. Finally, I thank my wife, Sandra, son, Jeffery, and my parents. Sorry it took so long. in TABLE OF CONTENTS PAGE ACKNOWLEDGEMENTS iii ABSTRACT V CHAPTER I INTRODUCTION 1 Background 3 Objectives 15 II THEORETICAL MODEL 18 Dynamic Models Using Static Optimization 20 Dynamic Optimization 27 Theoretical Model 35 The Flexible Accelerator 42 III EMPIRICAL MODEL AND DATA 46 Empirical Model 46 Data Construction 60 IV RESULTS 75 Theoretical Consistency 75 Quasi-fixed Input Adjustment 99 V SUMMARY AND CONCLUSIONS 110 APPENDIX A INPUT DEMAND EQUATIONS 119 B REGIONAL EXPENDITURE, PRICE, AND INPUT DATA 12 3 C EVALUATION OF CONVEXITY OF THE VALUE FUNCTION 130 D ANNUAL SHORT- AND LONG-RUN PRICE ELASTICITY ESTIMATES 132 REFERENCES 141 BIOGRAPHICAL SKETCH 148 iv 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 A DYNAMIC MODEL OF INPUT DEMAND FOR AGRICULTURE IN THE SOUTHEASTERN UNITED STATES By Michael James Monson August 1986 Chairman: William G. Boggess Cochairman: Timothy G. Taylor Major Department: Food and Resource Economics The current crisis in U.S. agriculture has focused attention on the need to adjust to lower output prices as a result of a variety of factors. The ability of agriculture to adjust is linked to the adjustment of inputs used in production. Static models of input demand ignore dynamic processes of adjustment. This analysis utilizes a model based on dynamic optimization to specify and estimate a system of input demands for Southeastern U.S. agriculture. Dynamic duality theory is used to derive a system of variable input demand and net investment equations. v The duality between a value function representing the maximized present value of future profits of the firm and the technology are reflected in a set of regularity conditions appropriate to the value function. Labor and materials are taken to be variable inputs, while land and capital are treated as potentially quasi-fixed. The equations for optimal net investment yield short- and long-run demand equations for capital and land. Thus, estimates of input demand price elasticities for both the short run and long run are readily obtained. The estimated model indicates that the data for the Southeast support the assumption of dynamic optimizing behavior as the regularity conditions of the value function are generally satisfied. The model indicates that labor and capital, and land and capital are short-run substitutes. Land and labor, and materials and capital are short-run complements. Similar relationships are obtained in the long run as well. Estimated adjustment rates of capital and land indicate that both are slow to adjust to changes in equilibrium levels in response to relative price changes. The estimated own rate of adjustment in the difference between actual and equilibrium levels of vi capital within one time period is 54 percent. The corresponding adjustment rate for land is 16 percent. However, the rate of adjustment for capital is dependent on the difference in actual and equilibrium levels of land. vii CHAPTER I INTRODUCTION If one were to characterize the current situation in U.S. agriculture, one might say that agriculture faces a period of adjustment. Attention has focused on the need to adjust to lower output prices as a result of a variety of factors including increasing foreign competition, trade regulations, a declining share of the world market, strength of the U.S. dollar, changes in consumer demand, high real rates of interest and governmental policies. The adjustments facing agriculture as a result of low farm incomes have become an emotional and political issue as farmers face bankruptcy or foreclosure. The ability of the agricultural sector to respond to these changing economic conditions is ultimately linked to the sector's ability to adjust the inputs used in production. Traditionally the analysis of input use in agriculture has been through the derivation of input demands based on static or singleÂ¬ period optimizing behavior. While static models maintain some inputs as fixed in the short-run yet variable in the long-run, there is nothing in the theory of static optimizing behavior to explain a 1 2 divergence in the short- and long-run levels of such inputs. Previous models based on static optimization recognize that some inputs are slow to adjust, but they lack a theoretical foundation for less-than- instantaneous input adjustment. A theoretical model of dynamic optimization has been recently proposed by Epstein (1981). Epstein has established a full dynamic duality between a dual function representing the present value of the firm and the firm's technology. This value function can be used in conjunction with a generalized version of Hotelling's Lemma to obtain expressions for variable input demands and optimal net investment in less-than- variable inputs consistent with dynamic optimizing behavior. While the theoretical implications of dynamic duality are substantial, there are a limited number of empirical applications upon which an evaluation of the methodology can be based. An empirical application to agriculture presenting a complete system of factor demands is absent in the literature. Chambers and Vasavada (1983b) present only net investment equations for the factors of land and capital in U.S. agriculture, focusing on the rates of adjustment. This same approach is used by Karp, Fawson, and Shumway (1984). The rates of adjustment in less-than- 3 variable inputs are important, but the methodology is richer than current literature would indicate. The purpose of this study is to utilize dynamic duality to specify and estimate a dynamic model of aggregate Southeastern U.S. agriculture. The model enables a clear distinction between the short- and long-run. Thus, input demands and elasticities for the short- and long-run may be determined. A key part of the methodology based on the explicit assumption of dynamic optimizing behavior is whether the data support such an assumption. This analysis intends to explore the regularity conditions necessary for a duality between the dynamic behavioral objective function and the underlying technology. Background The term "period of adjustment" is an appropriate recognition of the fact that agriculture does not always adjust instantaneously to changes in its environment. Agriculture has faced and endured periods of adjustment in the past. The Depression and Dust Bowl eras and world wars produced dramatic changes in the agricultural sector (Cochrane, 1979). In more recent history, agricultural economists have noted other periods of adjustment in response to less spectacular stimuli. Examples include the exodus of labor from the agricultural sector in the 1950s that 4 stemmed from the substitution of other inputs for labor (Tweeten, 1969), and the introduction of hybrid seed (Griliches, 1957) and low-cost fertilizers (Huffman, 1974). Two points are crucial. First, the period of adjustment in each case is less than instantaneous if not protracted, and second, the adjustments are ultimately reflected in the demand for inputs underlying production. Quasi-fixed Inputs The assumption of short-run input fixity is questionable. In a dynamic or long-run setting, input fixity is, of course, inconsistent with the definition of the long run as the time period in which all inputs are variable. The recognition that some inputs are neither completely variable or fixed dictates using the term "quasi-fixed" to describe the pattern of change in the levels of such inputs. Given a change in relative prices, net investment propels the level of the input towards the long-run optimum. The input does not remain fixed in the short-run, nor is there an immediate adjustment to the long-run optimum. Alternatively, the cost of changing the level of an input does not necessarily preclude a change as is the case for fixed inputs, nor is the cost solely the marginal cost of additional units in the variable input case. No inputs are absolutely fixed, but 5 rather fixed at a cost per unit time. The quasiÂ¬ fixity of an input is limited by the cost of adjusting that input. Short-run Input Fixity In light of the notion of quasi-fixity, input fixity in short-run static models is based on the assumption that the cost of adjusting a fixed input exceeds the returns in the current period. In the static model, the choice is either no adjustment in the short-run initial level or a complete adjustment to a long-run equilibrium position. The static model is unable to evaluate this adjustment as a gradual or partial transition from one equilibrium state to another. Given such a restricted horizon, it is understandable that static models of the short run often treat some inputs as fixed. The assumption of input fixity in the short run casts agriculture in the framework of a putty-clay technology, where firms have complete freedom to choose input combinations ex ante but once a choice is made, the technology becomes one of fixed coefficients (Bischoff, 1971). Chambers and Vasavda (1983a) tested the assumption of the putty-clay hypothesis for 6 aggregate U.S. agriculture with respect to capital, labor, and an intermediate materials input. Land was maintained as fixed and not tested. The methodology, developed by Fuss (1978), presumes that uncertainty with respect to relative prices determines the fixed behavior of some inputs. The underlying foundation for Chambers and Vasavada's test of asset fixity is a trade-off between flexibility of input combinations in response to relative input price changes and short-run efficiency with respect to output. Essentially, the farmer is faced with choosing input combinations in anticipation of uncertain future prices. If the prices in the future differ from expectations and the input ratios do not adjust, the technology is putty-clay. If the input ratios can be adjusted such that static economic conditions for allocative efficiency are fulfilled (marginal rate of technical substitution equal to the factor price ratio), the technology is putty-putty. Finally, if the input ratios do adjust, but not to the point of maximum efficient output, the technology is said to be putty-tin. Chambers and Vasavda concluded that the data did not support the assumption of short-run fixity for capital, labor, or materials. However, the effects of the maintained assumptions on the conclusions is 7 unclear. The assumption of short-run fixed stocks of land makes it difficult to determine if the degree of fixity exhibited by the other inputs is inherent to each input or a manifestation of the fixity of land. In addition to the assumption that land is fixed, the test performed by Chambers and Vasavada also relied on the assumption of constant returns to scale, an assumption used in previous studies at the aggregate level (Binswanger, 1974; Brown, 1978). The measure of efficiency, output, is dependent on the constant returns to scale assumption. If there were in fact increasing returns to scale presumably inefficient input combinations may actually be efficient. The Cost of Adjustment Hypothesis The cost of adjustment hypothesis put forth by Penrose (1959) provides an intuitive understanding of why some inputs are quasi-fixed. Simply stated, the firm must incur a cost in order to change or adjust the level of some inputs. The assumption of short-run fixity is predicated on the a priori notion that these costs preclude a change. The cost of search model by Stigler (1961) and the transactions cost models of Barro (1969) can be interpreted as specialized cases of adjustment costs characterized by a "bang-bang" investment policy. Such models lead to discrete jumps rather than a 8 continous or gradual pattern of investment. For a model based on static optimization, this appears to be the only adjustment mechanism available. Inputs remain at the short-run fixed level until the returns to investment justify a complete jump to the long-run equilibrium. Land is most often considered fixed in the short- run (Hathaway, 1963; Tweeten, 1969; Brown, 1978; Chambers and Vasavada, 1983a). Transaction costs appear to be the basis for short-run fixity of land and capital in static models. Galbraith and Black (1938) reasoned that high fixed costs prohibit substitution or investment in the short run. G. L. Johnson (1956) and Edwards (1959) hypothesized that a divergence in acquisition costs and salvage value could effectively limit the movement of capital inputs in agriculture. While labor and materials are usually considered as variable inputs in the short run, a search or information cost approach may be used to rationalize fixity in the short run. Tweeten (1969) has suggested that labor may be trapped in agriculture as farmers and farm laborers are prohibited from leaving by the costs of relocation, retraining, or simply finding an alternative job. However, the treatment of 9 agriculture as a residual employer of unskilled labor seems inappropriate given the technical knowledge and skills required by modern practices. These same technical requirements can be extended to material inputs such as fertilizers, chemicals, and feedstuffs, in order to justify short-run fixity. The cost of obtaining information on new material inputs could exceed the benefits in the short run. Alternatively, some material inputs may be employed by force of habit, such that the demand for these inputs is analagous to the habit persistance models of consumer demand (Pope, Green, and Eales, 1980). Recognizing the problems of such "bang-bang" investment policies, the adjustment cost hypothesis has been extended to incorporate a wider variety of potential costs. The adjustment costs in the first dynamic models may be considered as external (Eisner and Strotz, 1963; Lucas, 1967; Gould, 1968). External costs of adjustment are based on rising supply prices of some inputs to the individual firm and are inconsistent with the notion of competitive markets. Imperfect credit markets and wealth constraints may also be classified as external costs of adjustment. Internal costs of adjustment (Treadway 1969, 1974) reflect some foregone output by the firm in the present in order to invest in or acquire additional 10 units of a factor for future production. The assumption of increasing adjustment costs, where the marginal increment of output foregone increases for an incremental increase in a quasi-fixed factor, leads to a continous or smooth form of investment behavior. Quasi-fixed inputs adjust to the point that the present value of future changes in output are equal to the present value of acquisition and foregone output. The adjustment cost hypothesis, particularly internal adjustment costs, is important in a model of dynamic optimizing behavior. The costs of adjustment can be reflected by including net investment as an argument of the underlying production function. The constraint of input fixity in a short-run static model is relaxed to permit at least a partial adjustment of input levels in the current period. Assuming that these adjustment costs are increasing and convex in the level of net investment in the current period precludes the instantaneous adjustment of inputs in the long-run static models. Therefore, a theoretical foundation for quasi-fixed inputs can be established. Models of Quasi-fixed Input Demand Econometric models which allow quasi-fixity of inputs provide a compromise between maintaining some inputs as completely fixed or freely variable in the short-run. The empirical attraction of such models is 11 evident in the significant body of research in factor demand analysis consistent with quasi-fixity surveyed in the following chapter. Unfortunately, a theoretical foundation based on dynamic optimizing behavior is generally absent in these models. By far the most common means of incorporating dynamic elements in the analysis of input demand has been through the use of the partial adjustment model (Nerlove, 1956) or other distributed lag specifications. Such models typically focus on a single input. The coefficient of adjustment then is a statistical estimate of the change in the actual level of the input as a proportion of the complete adjustment that would be expected if the input was freely variable. The principle shortcoming of the single-equation partial adjustment model is that such a specification ignores the effect of, and potential for, quasi-fixity in the demands for the remaining inputs. The recent payment-in-kind (PIK) program is an excellent example of the significance of the interrelationships among factors of production. The program attempted to reduce the amount of land in crops as a means of reducing commodity surpluses. The effect of the reduction in one input, land, reduced the demand for other inputs, such as machinery, fertilizer and 12 chemicals, and the labor for operation of equipment and application of materials. The consequences of a reduction in this single input extended beyond the farm gate into the industries supporting agriculture as well. The partial adjustment model has been the foundation for many so-called dynamic optimization models cast in the framework of a series of static problems with the imposition of an adjustment coefficient as the linkage between the individual production periods (e.g., Day, 1962; Langham, 1968; Zinser, Miranowski, Shortle, and Monson, 1985). This effectively ignores the potential for instantaneous adjustment. Adjustments and relationships among inputs are determined arbitrarily. Others maintain potentially quasi-fixed inputs as variable (e.g., McConnell, 1983). The actual rate of adjustment in the system may be slower than the model would indicate as the adjustment of the maintained quasi-fixed inputs depends on the adjustments in supposedly variable inputs. The multivariate flexible accelerator (Eisner and Strotz, 1963) is an extension of the partial adjustment model to a system of input demands rather than a single equation. Lucas (1967), Treadway (1969, 1974), and Mortensen (1973) have demonstrated that 13 under certain restrictive assumptions concerning the production technology and adjustment cost structure, a flexible accelerator mechanism of input adjustment can be derived from the solution of a dynamic optimization problem. The empirical usefulness of this approach is limited, however, as the underlying input demand equations are expressed as derivatives of the production function. Thus, any restrictions inherent in the production function employed in the specification of the dynamic objective function are manifest in the demand equations. Berndt, Fuss, and Waverman (1979), and Denny, Fuss, and Waverman (1979) derived systems of input demand equations consistent with the flexible accelerator by introducing static duality concepts into the dynamic problem. The derivation of input demands through the use of a static dual function reduces the restrictions imposed by the primal approach utilizing a production function. Given the assumption of quadratic costs of adjustment as an approximation of the true underlying cost structure, these analyses obtained systems of variable input demand functions and net investment equations by solving the Euler equation corresponding to a dynamic objective function. This methodology, however, is 14 generally tractable for only one quasi-fixed input and critically relies on the quadratic adjustment cost structure. Duality is a convenience in models of static optimizing behavior. However, the application of static duality concepts to a dynamic objective function is somewhat limited. McClaren and Cooper (1980) first explored a dynamic duality between the firm's technology and a value function representing the maximum value of the integral of discounted future profits. Epstein (1981) established a full characterization of this dynamic duality using the Bellman equation corresponding to the dynamic problem. The optimal control theory underlying the solution to a dynamic optimization problem is consistent with quasi-fixity and the cost of adjustment hypothesis. The initial state is characterized by those inputs assumed fixed in the short run. Net investment in these inputs serves as the control (optimal in that the marginal benefit equals the cost of investment) that adjusts these input levels towards a desired or optimal state. This optimal state corresponds to the optimal levels of a long-run static optimization in which all inputs are variable. 15 Objectives The objective of this analysis is to utilize dynamic duality to specify and estimate a system of variable input demands and net investment equations for aggregate southeastern U.S. agriculture. Inherent to this effort is a recognition of the empirical applicability of dynamic duality theory to a small portion of U.S. agriculture. While the study has no pretense of determining the acceptance or rejection of the methodology for aggregate economic analysis, a presentation of the methods employed and difficulties encountered may provide some basis for further research. In addition to obtaining estimates of the optimal rates of net investment in land and capital, appropriate regularity conditions are evaluated. The clear distinction between the short- and long-run permits derivation of short- and long-run price elasticities for all inputs. Furthermore, the specificiation used for the value function permits the testing of hypotheses concerning the degree of fixity of land and capital and the degree of interdependence in the rates of net investment in these inputs. The potential significance of this interdependence with respect to policy is briefly explored. 16 Scope The dynamic objective function is expressed in terms of quasi-fixed factor stocks, net investment, the discount rate, and relative input prices. Endogenizing the factors conjectured as responsible for lower output prices in the introduction is no less difficult in a dynamic setting than in a static model. This study is content to explore the effects of relative price changes on factor demands and adjustment. Labor and materials are taken to be variable inputs while land and capital are considered as potentially quasi-fixed. This treatment is dictated by the available data at the regional level consistent with measurement of variable inputs rather than factor stocks. The development of appropriate stock measures for labor should involve a measure of human capital (Ball, 1985). In fact, the incorporation of human capital in a model of dynamic factor demands is a logical extension of the methodology as a means of not only determining but explaining estimated rates of adjustment. However, such a model exceeds the scope of this analysis. Additionally, a method of incorporating policy measures in the theory remains for the future. Noting this limitation, the Southeast is perhaps best suited 17 for an initial exploration of the methodology. The diversity of product mix in the components of total output in the Southeast reduces the influence of governmental policies directed at specific commodities or commodity groups. In 1980, the revenue share of cash receipts for the commodity groups typically subject to governmental price support in the U.S., namely dairy, feed grains, food grains, cotton, tobacco, and peanuts, was nearly 36 percent of total cash receipts, while the share of those commodities in the Southeast was 18 percent. Overview A review of previous models incorporating dynamic elements in the analysis of factor demands leads naturally to the theoretical model developed in Chapter II. An empirical model potentially consistent with dynamic duality theory and construction of the data measures follows in Chapter III. The estimation results and their consistency with the regularity conditions, the measures of short- and long-run factor demands and price elasticities, and hypotheses tests are presented in Chapter IV. The final chapter discusses the implications of the results and the methodology. CHAPTER II THEORETICAL DEVELOPMENT The primary objective of this study is to specify and estimate a system of dynamic input demands for southeastern U.S. agriculture. In order to explore the adjustment process of agricultural input use, the model should be consistent with dynamic optimizing behavior, quasi-fixity, and the adjustment cost hypothesis. This entails an exploration of the empirical applicability of a theory of dynamic optimization capable of yielding such a system. Yet models of input adjustment, hence dynamic input demands, based on static optimization generally lack a theoretical foundation. Treadway summarizes the incorporation of theory in these models. A footnote is often included on the adjustment-cost literature as if that literature had fully rationalized the econometric specification. And other adjustment mechanisms continue to appear with no discernible anxiety about optimality exhibited by their users. Furthermore, it is still common for economists to publish studies of production functions separately from studies of dynamic factor demand without so much as mentioning that the two are theoretically linked. (Treadway, 1974, p. 18) In retrospect, the search for a theoretical foundation for empirical models rationalized on the notions of quasi-fixity and adjustment costs culminate in the very 18 19 theory to be empirically explored. While these prior empirical models of input demand and investment are not necessarily consistent with the theoretical model finally developed, they are important elements in its history. This chapter includes a review of dynamic input demand models, dynamic in the sense that changes in input levels are characterized by an adjustment process of some form. The alternative models are evaluated with respect to empirical tractability and adherence to economic theory. In the first section, an adjustment mechanism is imposed on an input demand derived from static optimization. These models, whether the adjustment mechanism is a single coefficient or a matrix of coefficients, are empirically attractive but lack a firm theoretical foundation. The models in the second group are based on dynamic optimization. These models are theoretically consistent, yet limited by the form of the underlying production function or the number of inputs which may be quasi-fixed, even though static duality concepts are incorporated. The advantages presented by static duality lead to the development of dynamic duality theory. The application of dynamic duality theory to a problem of dynamic optimizing behavior permits the derivation of a 20 system of dynamic input demands explicitly related to the underlying production technology and offers a means of empirically analyzing the adjustment process in the demand for agricultural inputs in a manner consistent with optimizing behavior. Dynamic Models Using Static Optimization The input demand equations derived from a static objective function with at least one input held constant provide limited information relative to input adjustment. The derived demands are conditional on the level of the fixed input(s). The strict fixity of some inputs makes such models inappropriate for a dynamic analysis. The demands derived from the static approach without constraints on factor levels characterize equilibrium or optimal demands if the factors are in fact freely variable. However, a full adjustment to a new equilibrium given a change in prices is inconsistent with the observed demand for some inputs. Input demand equations derived from a static optimization problem, whether cost minimization or profit maximization, characterize input demand for a single period and permit either no adjustment in some inputs or instantaneous adjustment of all inputs. There is nothing in static theory to reflect adjustment in input 21 demands over time. Models of dynamic input demand based on static optimization attempt to mimic rather than explain this adjustment process. Distributed Lags and Investment The investigation of capital investment through distributed lag models seems to represent a much greater contribution to econometric modeling and estimation techniques than to a dynamic theory of the demand for inputs. However, models of capital investment characterize an early empirical approach to quasi-fixity of inputs, recognizing that net investment in capital is actually an adjustment in the dynamic demand for capital as a factor of production. Additionally, these models developed an implicit rental price or user cost of a quasi-fixed input as a function of depreciation, the discount rate, and tax rate as the appropriate measure of the quasi-fixed input price (e.g., Hall and Jorgenson, 1967). Particular lag structures identified and employed in the analysis of capital investment include the geometric lag (Koyck, 1954), inverted V lag (DeLeeuw, 1962), polynomial lag (Almon, 1965), and rational lag (Jorgenson, 1966). The statistical methods and problems 22 of estimating these lag forms has been addressed extensively1. Lag models inherently recognize the dynamic process, as current levels of capital are assumed to be related to previous stocks. A problem with these models is that lag structures are arbitrarily imposed rather than derived on the basis of some theory. Griliches (1967, p. 42) deems such methods "theoretical ad-hockery." The basic approach in distributed lag models is to derive input demand equations from static optimization with all inputs freely variable. The demand equations obtained characterize the long-run equilibrium position of the firm. If an input is quasi-fixed, it will be slow to adjust to a new equilibrium position. The amount of adjustment, net investment, depends on the difference in the equilibrium demand level and the current level of the input. The underlying rationale of a distributed lag is that the current level of a quasi-fixed input is a result of the partial adjustments to previous equilibrium positions. The various forms of the distributed-lag determine how important these past adjustments are in determining the current response. The finite lag distributions proposed by Fisher (1937), 1 Maddala (1977, pg. 355-76) presents econometric estimation methods and problems. 23 DeLeeuw (1962), and Almon (1965) limit the number of prior adjustments that determine the current response. The infinite lag distributions (Koyck, 1954; Jorgenson, 1966) are consistent with the notion that the current adjustment depends on all prior adjustments. While such models may characterize the adjustment process of a single factor, there is little economic information to be gained. There is no underlying foundation for a less-than-complete adjustment or existence of a divergence of the observed and optimal level of the factor. The individual factor demand is a component of a system of demands derived from static optimization, yet its relationship to this system is often ignored. The Partial-Adjustment Model The partial adjustment model put forth by Nerlove (1956) provides an empirical recognition of input demand consistent with quasi-fixity. The partial adjustment model as a dynamic model has been widely employed (Askari and Cummings (1977) cite over 600 studies), and continues to be applied in agricultural input demand analyses (e.g., Kolajo and Adrian, 1984). The partial adjustment model recognizes that some inputs are neither fixed nor variable, but rather quasi- 24 fixed in that they are slow to adjust to equilibrium levels. In its simplest form, the partial adjustment model may be represented by (2.1) xt - xt_! = a( x*t - xt_x), where x-Â¡- is the observed level of some input x in period t, and x*t is the equilibrium input level in period t defined as a function of exogenous factors. The observed change in the input level represented by x^ - x-)-_i in (2.1) is consistent with a model of net investment demand for the input. The observed change in the input is proportional to the difference in the actual and equilibrium input levels. Assuming the firm seeks to maximize profit, x*^ becomes a function of input prices and output price such that (2.2) x t â€” f (Py, w^ , , w^j) . where output price is denoted by Py, and the wÂ¿, i = 1 to n, are the input prices. Equation (2.2) allows the unobserved variable, x*^, in (2.1) to be expressed as observations in the current and the prior period. The parameter a in (2.1) represents the coefficient of adjustment of observed input demand to the equilibrium level. The model is a departure from static optimization theory in that x*^ is no longer derived from the first- 25 order conditions of an optimal solution for a static objective function. While static theory does not directly determine the form of the adjustment process, there is an explicit recognition that certain inputs are slow to adjust to long-run equilibrium levels. Interrelated Factor Demands Coen and Hickman (1970) extended the approach of distributed-lag models to a system of demand equations for each input of the production function employed in the static optimization. Essentially, the input demand equations derived from the production function under static maximization conditions are taken as a system of long-run or equilibrium demand equations. A geometric lag is arbitrarily imposed on the differences in actual and equilibrium input levels. The shared parameters from the underlying production function are restricted to be identical across equations. Coen and Hickman apply this model to labor and capital demands derived from a Cobb-Douglas production function. However, this method becomes untractable when applied to a more complex functional form or a much greater number of inputs. Additionally, the adjustment rate or lag structure for each input is not only arbitrary but remains independent of the disequilibrium in the other factors. 26 Nadiri and Rosen (1969) formulate an alternative approach to a system of interrelated factor demands where the adjustment in one factor depends explicitly on the degree of disequilibrium in other factors. The model is a generalization of the partial adjustment model to n inputs such that (2.6) x.(- â€” = B [x t - This specification is similar to that in (2.1), except that x-t and x*t are n x 1 vectors of actual and equilibrium levels of inputs, and the adjustment coefficent becomes an n x n matrix, B. Individual input demand equations in (2.6) are of the form n (2.7) xi,t = ^ bij-(x j,t â€œ xj,t-l) + bfi* (x*j^t - x-^t-l) + xi,t-l* This representation permits disequilibrium in one input to affect demand for another input. This interdependence allows inputs to "overshoot" equilibrium levels in the short run. For example, assume an input is initially below its equilibrium level. Depending on the sign and magnitude of the coefficients in (2.7), the adjustment produced by disequilibria in other inputs may drive the observed level of the input beyond the long- run level before falling back to the optimal level. 27 Nadiri and Rosen (1969) derived expressions for equilibrium factor levels using a Cobb-Douglas production function in a manner similiar to Coen and Hickman (1968). However, they failed to consider the implied cross-equation restrictions on the parameters implied by the production function. Additionally, stability of such a system requires that the characteristic roots of B should be within the unit circle, yet appropriately restricting each bÂ¿j is impossible. Neither Coen and Hickman, or Nadiri and Rosen provide a distinction between variable and quasi-fixed inputs. All inputs are treated as quasi-fixed and the adjustment mechanism is extended to all inputs in the system. They do provide key elements to a model of dynamic factor demands in that Coen and Hickman recognize the relationship between the underlying technology and the derived demands and Nadiri and Rosen incorporate an interdependence of input adjustment. Dynamic Optimization Recently, there has been a renewed interest in optimal control theory and its application to dynamic economic behavior. As Dorfman (1969, pg. 817) notes, although economists in the past have employed the calculus of variations in studies of investment (Hotelling, 1938? Ramsey, 1942), the modern version of 28 the calculus of variations, optimal control theory, has been able to address numerous practical and theoretical issues that previously could not even be formulated in static theory. Primal Approach Eisner and Strotz (1963) developed a theoretical model of input demand consistent with dynamic optimizing behavior and a single quasi-fixed input. Lucas (1967) and Gould (1968) extended this model to an arbitrary number of quasi-fixed inputs. However, these extensions are limited by the nature of adjustment costs external to the firm. Thus, the potential interdependence of adjustment among quasi-fixed inputs is ignored. Treadway (1969) introduced interdependence of quasi-fixed inputs by internalizing adjustment costs in the production function of a representative firm. The firm foregoes output in order to invest in or adjust quasi-fixed inputs. Assuming all inputs are quasi- fixed, the underlying structure of this model is shown by 00 (2.8) V = max / e-rt(f(x,x) - p'x) dt, X 0 where V represents the present value of current and future profits, x is a vector of quasi-fixed inputs, and x denotes net investment in these inputs. The vector p represents the user costs or implicit rental prices of 29 the quasi-fixed inputs normalized by output price. The current levels of the quasi-fixed inputs serve as the intitial conditions for the dynamic problem. Assuming a constant real rate of discount, r, and static price expectations, the Euler equations corresponding to an optimal adjustment path2 for the quasi-fixed inputs are given by (2.9) [fxx]^ + [fxx]* = + _ P- Treadway assumed the existence of an equilibrium solution to (2.9), where x = x = 0, in order to derive a system of long-run or equilibrium demand equations for the quasi-fixed inputs. However, derivation of a demand equation for net investment is more complicated, in that an explicit solution for x exists for only restricted forms of the production function (e.g., Treadway, 1974). Yet, net investment demand is the key to characterizing the dynamic adjustment of quasi-fixed inputs. The difficulty of deriving input demand equations from a primal dynamic optimization problem are apparent. While input demand equations derived from a primal static optimization involve first-order derivatives of 2 Additionally, the optimality of x depends on a system of transversality conditions where â€” Vâ€œt* , lim e [f.] = 0, and a Legendre condition that t-*- oo [f..] negative semi-definite (Treadway, 1971 p. 847). 30 the production function, derivation of input demands from (2.9) involve second-order derivatives as well. Treadway (1974) shows that the introduction of variable inputs further increases the difficulty in deriving input demands from a primal dynamic optimization. This derivation is necessarily in terms of a general production function, owing to the primal specification of the objective function. In terms of empirical interest, estimation of such a system is nonexistent. However, the establishment of necessary conditions for an optimal solution to the dynamic problem in terms of the technology provides the foundation for the use of duality that follows. Application of Static Duality Concepts The primary difficulty in estimating a system of dynamic factor demands from the direct or primal approach is that the characteristic equations underlying the dynamic optimization problem in (2.8) are necessarily expressed in terms of first- and second- order derivatives of the dynamic production function. Thus, unless a truly flexible functional form of the production function (e.g., Christensen, Jorgenson, and Lau, 1973) is employed, restrictions on the underlying technology are imposed a priori. In a static model, a behavioral function such as the profit or cost function with well-defined properties 31 can serve as a dual representation of the underlying technology (Fuss and McFadden, 1978). A system of factor demands is readily derived from the profit function by Hotelling's Lemma (Sir (p,Py)/3Pj = -x*j(p,Py)) or from the cost function by Shephard's Lemma (9C(p,y)/9pj = Xj(p,y)). However, all inputs are necessarily variable. These static models had been extended to the restricted variable profit and cost functions that hold some factors as fixed (e.g., Lau, 1976). Berndt, Fuss, and Waverman (1979) incorporated a restricted variable profit function into the primal dynamic problem in order to simplify the dynamic objective function. Berndt, Morrison-White, and Watkins (1979) derive an alternative method employing the restricted variable cost function as a component of the dynamic problem of minimizing the present value of current and future costs. The advantages of static duality reduce the explicit dependence on the form of the production function and facilitate the incorporation of variable inputs in the dynamic problem. Berndt, Fuss, and Waverman specified a normalized restricted variable profit function presumably dual to 32 the technology in (2.8) based on the conditions for such a static duality as presented by Lau (1976). This function may be written as (2.10)it(w,x,x) = max f(L,x,x) - w'L L > 0 Assuming that the level of net investment is optimal for the problem in (2.8), the remaining short-run problem as reflected in (2.10) is to determine the optimal level of the variable factor L dependent on its price, w, and on the quasi-fixed factor stock and net investment. Current revenues are tt(w,x,x) - p'x, which can be substituted directly into (2.8). The use of static duality in the dynamic problem allows the production function to be replaced by the restricted variable profit function. A general functional form for it, quadratic in (x,x), can be hypothesized as (2.11) tl(w,x,x)= aQ + a'x + b'x + l/2[x' x'] A C X C B J â€¢ LXJ where ag, the vectors a, b, and matrices A, B, and C will be dependent on w in a manner determined by an exact specification of tt(w,x,x). The Euler equation for the dynamic problem in (2.8) after substitution of (2.11) is (2.12) Bx + (C- C - rB) x - (A + rC') x = rb - p + a . Note that this solution is now expressed in terms of the 33 parameters of the restricted variable profit function instead of the production function. A steady state or equilibrium for the quasi-fixed factors denoted as x(p,w,r) can be computed from (2.12) evaluated at x = x = 0, such that (2.13) x (p,w,r) = -[A + rC']_1(rb - p + a). Applying Hotelling's Lemma to the profit function yields a system of optimal variable input demands, L*, conditional on the quasi-fixed factors and net investment of the form (2.14) L* (x, p, w, r) = -ttw(w,x,x) . The system of optimal net investment equations can expressed as (2.15) x(x,p,w,r) = M(w,r)[x - x(p,w,r)]. The exact form of the matrix M is uniquely determined by the specification of the profit function in the solution of (2.10). Only in the case of one quasi-fixed factor has this matrix been expressed explicitly in terms of the parameters of the profit function, where M = r/2 - (r2/4 + (A + rCJ/B)1/2, where A, B, and C are scalars. In order to generalize this methodology to more than one quasi-fixed factor, LeBlanc and Hrubovcak (1984) specified a quadratic form such that the optimal 34 levels of variable inputs depend only on factor stocks and are independent of investment. Therefore, they rely on external adjustment costs reflected by rising supply prices of the factors. In addition, the adjustment mechanism for each input is assumed independent of the degree of disequilibrium in other quasi-fixed factors such that the implied adjustment matrix is diagonal. This facilitates expression of net investment demand equations for more than one quasi-fixed input in terms of parameters of the profit function, but at considerable expense to the generality of their approach. Dynamic Duality The use of static duality in these models of dynamic factor demands leads naturally to the development of a dual relationship of dynamic optimizing behavior and an underlying technology. Such a general dynamic duality was conjectured by McLaren and Cooper (1980). Epstein (1981) establishes the duality of a technology and a behavioral function consistent with maximizing the present value of an infinite stream of future profits termed the value function. 35 Theoretical Model The firm's problem of maximizing the present value of current and future profits3 may be written as OO (2.16) J0(K,p,w) = max./ eâ€œrt [F (L, K, K) - w'L - p'K] dt L > 0, K < 0 subject to K(0) = K0 > 0. The production function F(L,K,K) yields the maximum amount of output that can be produced from the vectors of variable inputs, L, and quasi-fixed inputs, K, given that net investment K is taking place. The vectors w and p are the rental prices or user costs corresponding to L and K respectively, normalized with respect to output price. Additionally, r > 0 is the constant real rate of discount, and K0 is the initial quasi-fixed input stock. J(K0,p,w,r) then characterizes a value function reflecting current and discounted future profits of the firm. The following regularity conditions are imposed on the technology represented by F(L,K,K) in (2.16): T.l. F maps variable and quasi-fixed inputs and net investment in the positive orthant; F, Fl, and Fj< are continuously differentiable. T.2. Fl, Fk > 0, FÂ¿ < 0 as K > 0. T.3. F is strongly concave in (L, K) . 3 The exposition of dynamic duality draws heavily on the theory developed by Epstein (1981). 36 T.4. For each combination of K, p, and w in the domain of J, a unique solution for (2.16) exists. The functions of optimal net investment, K*(K,p,w), variable input demand, L*(K,p,w), and supply, y*(K,p,w) are continuously differentiable in prices, the shadow price function for the quasi-fixed inputs, A*(K,p,w), is twice continuously differentiable. T.5. ^*p(K,p,w) is nonsingular for each combination of quasi-fixed inputs and input prices. T.6. For each combination of inputs and net investment, there exists a corresponding set of input prices such that the levels of the inputs and net investment are optimal. T.7. The problem in (2.16) has a unique steady state solution for the quasi-fixed inputs that is globally stable. Condition T.l requires that output be positive for positive levels of inputs. Declining marginal products of the inputs characterize the first requirements of T.2. Internal adjustment costs are reflected in the requirements of FÂ£# The extension allowing for positive and negative levels of investment requires that the adjustment process be symmetric in the sense that when net investment is positive some current output is foregone but when investment is negative current output is augmented. Consistent with assumed optimizing behavior, points that violate T.6 would never be observed. 37 Assuming price expectations are static, inputs adjust to "fixed" rather than "moving" targets of long- run or equilibrium values. However, prices are not treated as fixed. In each subsequent period a new set of prices is observed which redefine the equilibrium. As the decision period changes, expectations are altered and previous decisions are no longer optimal. Only that part of the decision optimal under the initial price expectations is actually implemented. Given the assumption of static price expectations and a constant real discount rate, the value function in (2.16) can be viewed as resulting from the static optimization of a dynamic objective funtion. Under these assumptions and the regularity conditions imposed on F(L,K,K), the value function J(K,p,w) is at a maximum in any period t if it satisfies the Bellman (Hamilton- Jacobi) equation for an optimal control (e.g., Intriligator, 1971, p. 329) problem such that (2.17) rJ*(K,p,w) = max {F(L,K,K) - w'L - p'K + JK(K,P,u) K*}, where JK(K,p,w) denotes the vector of shadow values corresponding to the quasi-fixed inputs, and K* represents the optimal rate of net investment. 38 Through the Bellman equation in (2.17), the dynamic optimization problem in (2.16) may be transformed into a static optimization problem. In particular, (2.17) implies that the value function may be defined as the maximized value of current profit plus the discounted present value of the marginal benefit stream of an optimal adjustment in net investment. Thus, through the Maximum Principle (e.g., Intriligator, 1971, p. 344) the â€¢ maximizing values of L and K in (2.17) when K = K0 are Â» precisely the optimal values of L and K in (2.16) at t = 0. Utilizing (2.17), Epstein (1981) has demonstrated that the value function is dual to F(L,K,K) in the dynamic optimization problem of (2.16) in that, conditional on the hypothesized optimizing behavior, properties of F(L,K,K) are manifest in the properties of J(K,p,w). Conversely, specific properties of J(K,p,w) may be related to properties on F(L,K,K). Thus, a full dynamic duality can be shown to exist between J(K,p,w) and F(L,K,K) in the sense that each function is theoretically obtainable from the other by solving the appropriate static optimization problem as expressed in (2.17). The dual problem can be represented by (2.18) F*(L,K,K) = min (rJ(K,p,w) + w'L + p'K - p,w JK*K}. 39 The static representation of the value function in (2.17) also permits derivation of demand functions for variable inputs and net investment in quasi-fixed factors. Application of the envelope theorem by differentiating (2.17) with respect to w yields the system of variable factor demand equations (2.19) L*(K,p,w) = -rJw' + JwK*K*, and differentiation with respect to p yields a system of optimal net investment equations for the quasi-fixed inputs, (2.20) K*(K,p,w) = JpK_1-(rJp'+ K) . This generalized version of Hotelling's Lemma permits the direct derivation of a complete system of input demand equations theoretically consistent with dynamic optimizing behavior. The ability to derive an equation for net investment is crucial to understanding the short-run changes or adjustments in the demand for quasi-fixed inputs. The system is simultaneous in that 40 the optimal variable input demands depend on the optimal levels of net investment, K*. In the short-run, when K* f 0, the demand for variable inputs is conditional on net investment and the stock of quasi-fixed factors. In addition, a supply function for output is endogenous to the system. The optimal supply equation derived by solving (2.17) for F(L,K,K) where (L,K)=(L*,K*) may be expressed as (2.21) y*(K,p,w) = rJ + w'L* + p'K - JKK*. As for the variable input equation, optimal supply depends on the optimal level of net investment. This is consistent with internal adjustment costs as the cost of adjusting quasi-fixed factors through net investment is reflected in foregone output. The regularity conditions implied by the properties of the production function are manifested in (2.19)- (2.21) and provide an empirically verifiable set of conditions on which to evaluate the theoretical consistency of the model. Consistency with the notion of duality dictates that the previously noted properties of the technology be reflected in the value function. The properties (V) manifest in J from the technology are V.l. J is a real-valued, bounded-from-below function defined in prices and quasi-fixed inputs. J and JK are twice-continously differentiable. V. 2 . V. 3 . 41 V. 4 . V. 5. V. 6. V. 7 . rJK + P â€œ JKk(K*), Jk > JK<Â° as K*<0. For each element in the domain of J, y*>0; for such K in the domain of J, (L*, K, K*) maps the domain of $ onto the domain of F. The dynamic system K*, K(0) = K0, in the domain of J defines a profile K(t) such that (K(t),p,w) is in the domain of J for all t and K(t) approaches K(p,w), a globally steady state also in the domain of J. JpK is nonsingular. For the element (K,p,w) in the domain of J, a minimum in (2.18) is attained at (p,w) if (K,L) = (K*,L*) . The matrix Lp is nonsingular for Kw KÂ£ each element, (K,p,w), in the domain of J. These regularity conditions are essential in establishing the dynamic duality between the technology and the value function. In fact, the properties of J are a reflection of the properties of F. The definition of the domain of F implies V.l. Condition V.2 reflects in (p,w) the restrictions imposed on the marginal products of the inputs, FL and FÂ¿, and net investment, Fk, in T.2. The conditions in V.3 with respect to an optimal solution in price space, (p,w), are dual to the conditions for an optimal solution in input space, (L,K), maintained in T.6. V.4 is the assumption of the global steady state solution as in T.7. Given JK =A * noted earlier, V.5 is the dual of T.5. V.7 is a reflection of the concavity requirement of T.3. 42 Condition V.6 may be interpreted as a curvature restriction requiring that first-order conditions are sufficient for a global minimum in (2.18). Epstein (1981) has demonstrated that if Jj< is linear in (p,w) , V.6 is equivalent to the convexity of J in (p,w). An advantage of dynamic duality is that these conditions can be readily evaluated using the parameters of the empirically specified value function. The specification of a functional form for J must be potentially consistent with these properties. The Flexible Accelerator Dynamic duality in conjunction with the value function permits the theoretical derivation of input demand systems consistent with dynamic optimizing behavior. Such a theoretical foundation establishes the relationship between quasi-fixed and variable input demand and an adjustment process in the levels of quasi- fixed inputs as a consequence of the underlying production technology. One may note that the net investment demand equation for a single quasi-fixed input derived from the incorporation of the restricted variable profit function in the primal dynamic problem yields a coefficient of adjustment as a function of the discount rate and the parameters of the profit function similar to the constant adjustment coefficient employed in the partial 43 adjustment model. However, an explicit solution of the system of net investment equations with two or more quasi-fixed inputs in terms of an adjustment matrix is difficult. Nadiri and Rosen (1969) considered their model as an approximate representation of an adjustment matrix derived from dynamic optimization. Dynamic duality provides a theoretical means of deriving a wide variety of adjustment mechanisms. The difficulties in relating a specific functional form of the production function to the adjustment mechanism in the direct or primal approach and the limited applicability of the adjustment mechanism derived from incorporating the restricted variable profit function in the dynamic objective function are alleviated considerably. However, the functional form of the value function is critical in determining the adjustment mechanism. The adjustment mechanism of interest in this analysis is the multivariate flexible accelerator. Although the theoretical model relies on a constant real discount rate, it is not unreasonable to hypothesize that this constant rate of discount is partially responsible for the rates of adjustment in quasi-fixed inputs to their equilibrium levels. Noting the potential number of parameters and non-linearities in the demand equations, an adjustment matrix of 44 coefficents as a function of the discount rate and the parameters of the value function may be the desired form of the adjustment process for empirical purposes. Epstein (1981) establishes a general form of the value function from which a number of globally optimal adjustment mechanisms may be derived. The adjustment mechanism of constant coefficents is a special case.4 The flexible accelerator [r + G] is globally optimal if the value function takes the general form (2.22) J(K,p,w) = g(K,w) + h(p,w) + p'G-1K. This form yields JpK = G_1 and Jp = hp(p,w) + G-1K. Substituting in (2.20) yields the optimal net investment equations of the form (2.23) K*(K,p,w) = G[rhp(p,w)] + [r + G]K. Solving (2.23) for K(p,w) at K*=0, (2.24) K(p,w) = -[r + Gr^-Gtrhp^w) ] . Multiplying (2.24) by [r + G] and substituting directly in (2.23) yields (2.25) K*(K,p,w) = -[r + G]K(p,w) + [r + G]K = [r + G] [K - K(p,w)]. 4 The derivation and proof of global optimality of a general flexible accelerator is provided by Epstein, 1981, p. 92. 45 Thus, the flexible accelerator derived in (2.25) is globally optimal given a value function of the form specified in (2.25). While the accelerator is dependent on the real discount rate, the assumption that this rate is constant implies a flexible accelerator of constants. The linearity of Jj^ in (p,w) , which implies the convexity of J in (p,w), is crucial in the derivation of a globally optimal flexible accelerator of fixed coefficients. CHAPTER III EMPIRICAL MODEL AND DATA Empirical Model The specification of the value function J is taken to be log-quadratic in normalized prices and quadratic in the quasi-fixed inputs. The specific form of the value function J(K,p,w) is thus given by (3.1) J(K,p,w) = a0 + a'K + b'log p + c'log w + 1/2(K1AK + log p'B log p + log w'C log w) + + log p'D log w + p' G-1K + w'NK + p'G'^-VkT - w'VlT where K = [K, A], a vector of the quasi-fixed inputs, capital and land, p = [pk, pa] denotes the vector of normalized (with respect to output price) prices for the quasi-fixed inputs, and w = [pw, pm], the vector of normalized variable input prices for labor and materials respectively. Thus, log p = [log pk, log pa] and log w = [log pw, log pm]. T denotes a time trend variable. Parameter vectors are defined by a = [aK, aA], b = [bk, ba], c = [cw, cm], VK = [vK, vA], and VL = [vL, vM]. The vectors VK and VL are technical change parameters for the quasi-fixed inputs and variable 46 47 inputs. The variable input vector is defined by L = [L, M], where L denotes labor and M denotes materials. Parameter matrices are defined as: A = aKK aKA aAK &AA_ ... â€™bkk bka~ bak baa cww cwm cmw cmm r D = dkw dkm daw dam ... nKw nKm nAw nAm , and G -1 = [gKK gKA " gAK gAA Let G = gKK 9kaT The matrices A, B, and C are _9AK 9aaJ symmetric. The incorporation of some measure of technical change is perhaps as much a theoretical as empirical issue. The assumption of static expectations applies not only to relative prices but the technology as well. The literature contains two approaches to the problem of technical change in dynamic analysis: detrending the data (Epstein and Denny, 1983) or incorporating an unrestricted time trend (Chambers and Vasavada, 1983b; Karp, Fawson, and Shumway, 1984). An argument for the former (Sargent, 1978, p. 1027) is that the dynamic model should explain the indeterminate component of the data seriesâ€”that which is not simply explained by the passage of time. However, as Karp, Fawson, and Shumway (1984, p. 3) note, the restrictions of dynamic model reflected in the investment equations involve real rather than detrended economic variables so the restrictions may not be appropriate for detrended values. 48 The latter approach is adopted in the above specification of the value function in (3.1). Thus, investment and variable input demand equations derived from the value function in (3.1) include an unrestricited time trend. This form allows the technical change parameters to measure in part the relative effect of technical change with respect to factor use or savings over time. Note that the presence of G-1 in the interaction of p, VK, and T in the interaction of p, VK, and T in (3.1) ensures that the technical change parameters enter the investment demand equations without restriction. The incorporation of technical change in the value function serves as an illustration of the difficulty in incorporating policy, human capital, and other variables besides prices into the value function. In static optimization, the input demand equations are determined by first-order derivatives of the objective function. Therefore, the interpretation of parameters in terms of their effects on the objective function is straightforward. The demand equations derived from dynamic optimization contain first- and second-order derivatives of the value function. The value function can be specified to permit a direct interpretation of the parameters in terms of the underlying demand equations. However, 49 relating these parameters to the dynamic objective function becomes difficult. Without estimating the value function directly, one must rely on the regularity conditions implied by dynamic duality to ensure consistency of the empirical specification and underlying theory. Input Demand Equations Utilizing the generalized version of Hotelling's Lemma in (2.20), the demand equations for optimal net investment in the quasi-fixed inputs are given by A (3.2) K*(K,p,w) = G[rp-1(b + B log p + D log w) + rVKT] + [r + G]K, where K*(K,p,w) = [K*(K,p,w), Ã*(K,p,w)] signifies that optimal net investment in capital and land, is a function of factor stocks and input prices. r is a diagonal matrix of the discount rate, and p is a diagonal matrix of the quasi-fixed factor prices. The specification of G-1 in (3.1) permits direct estimation of the parameters of G in the net investment equations. Net investment, or the rate of change in the quasi-fixed factors, is determined by the relative input prices and the initial levels of the quasi-fixed factors, as evidenced by the presence of K in (3.2). The premultiplication by G (G = JpK-1 from (2.20)) 50 yields a system of net investment demand equations that are nonlinear in parameters. The technical change component for the quasi-fixed inputs in the value function (3.1) enters the net investment demand equations in a manner consistent with the assumption of disembodied technical change. The optimal short-run demand equations for the variable inputs are derived using (2.19), and are given by (3.3) L*(K,p,w) = -rw-1(c + D log p + C log w) + rVwT - rNK + NK*(K,p,w), where L*(K,p,w) = [L*(K,p,w), M*(K,p,w)], the optimal short-run input demands for the variable inputs, labor and materials, r is again a diagonal matrix of the discount rate, and w is a diagonal matrix of the variable input prices. The short-run variable input demand equations depend not only on the initial quasi-fixed input stocks but the optimal rate of net investment in these inputs as well. While variable inputs adjust instantaneously, the adjustments are conditioned by both K and K*. The presence of K*(K,p,w) in the variable input demand equations dictates that net investment and variable input demands are determined jointly, requiring a simultaneous equations approach. 51 The derivation of optimal net investment and variable input demands in (3.2) and (3.3) are presented as systems in matrix notation. The precise forms of the individual net investment and variable input demands used in estimation are presented in Appendix Table A-l. Long-run Demand Equations In the dynamic model, the quasi-fixed inputs gradually adjust toward an equilibrium or steady state. The long-run level of demand for an input is defined by this steady state, such that there are no more adjustments in the input level. In other words, net investment is zero. The long-run or steady state demands for the quasi-fixed inputs are derived by solving (3.2) for K when K*(K,p,w) = 0. The long-run demand equation for the quasi-fixed factors is thus given by A (3.4) K(p,w) = - [I + rG-1]-1[rp-1(b + B log p + D log w) + rVKT], where K(p,w) =[K(p,w), A(p,w)]. Note that these long- run demand equations are functions of input prices alone. Noting that the short-run demand equations for the variable inputs in (3.3) are conditional on K and 52 K*, substitution of K(p,w) for K and K*(K,p,w) = 0 in the short-run equations yields (3.5) L(K,p,w) = -rw-1(c + D log p + C log w) + rVLT - rNK(p,w), where L(K,p,w)=[L(K,p,w), M(K,p,w)]. The long-run variable input demands are no longer conditional on net investment, but are determined by the long-run levels of the quasi-fixed inputs. The individual long-run demand equations for all inputs are presented in Appendix Table A-2. Short-run Demands The short-run variable input demands were presented in (3.3). The variable input demands are conditional on the initial levels of the quasi-fixed inputs and optimal net investments. The short-run demand for the quasi-fixed inputs requires the explicit introduction of time subscripts in order to define optimal net investment in discrete form as (3.6) Kt*(K,p,w) = K*t - Kt_1# where K*-(- = [K*^, A*tl < the vector of quasi-fixed inputs at the end of period t. Therefore, the short- run demand for capital at the beginning of period t is (3.7) K*t(K,p,w) = Kt_! + K*t(K,p,w), 53 where K*t(K,p,w) is the optimal demand for the quasi- fixed inputs in period t, K^-i is the initial stock at the beginning of the period, and K*-j- is net investment during the previous period. The short-run demand equations for the quasi-fixed inputs are optimal in the sense that the level of the quasi-fixed input, K*t, is the sum of the previous quasi-fixed input level and optimal net investment during the prior period. Returning to (3.2), the short-run demand equation for the quasi-fixed input vector can be written (3.8) K*t(K,p,w) = G[rp-1(b + B log p + D log w)] + rVKT + [I + r + G]Kt_!, where the time subscripts are added to clarify the distinction between short-run demand and initial stocks of the quasi-fixed inputs. The individual short-run demand equations for the quasi-fixed inputs are presented in Appendix Table A-3. The Flexible Accelerator The flexible accelerator matrix M = [r + G] was shown to be consistent with the general form of the value function in (3.1) in the previous chapter. 54 Rewriting (3.2) and multiplying both sides of the equation by G-1 yields (3.9) G-1K*(K,p,w) = rp-1(b + B log p + D log w) + rVKT + [I + rG-1]K. Multiplying both sides by [I + rG-1]-1 and noting that [I + rG-1]-1 = [r + G]-1G, then (3.9) can be written as (3.10) [r + G]-1K*(K,p,w) = [I + rG-1]-1[rp-1(b + B log p + D log w) + rVKT] + K. The first term on the right hand side of (3.10) is identical to the negative of the long-run quasi-fixed input demand equation in (3.4). Substituting -K(p,w) in (3.10) and solving for K*(K,p,w) yields (3.11) K*(K,p,w) = [r + G] [K - K(p,w]. As may be noted, this is precisely the form of the multivariate flexible accelerator. Solving (3.11) for the individual equations, the optimal net investment in capital is (3.12) K* = (r + gKK)(K - K) + gKA(A - A), and optimal net investment in land may be written (3.13) Ã* = gAK(K - K) + (r + g^) (A - A) . 55 Thus, gj^ and the parameters associated with land in the value function appear in the net investment equation for capital. Likewise, gAK and the parameters associated with capital in the value function appear in the net investment equation for land. Hypotheses Tests The form of the flexible accelerator in (3.11) permits direct testing of hypotheses on the adjustment matrix in terms of nested parameter restrictions. The appropriateness of these tests are based on Chambers and Vasavada (1983b). Of particular interest is the hypothesis of independent rates of adjustment for capital and land which can be tested via the restrictions gj^ = gAK = o. Independent rates of adjustment indicate that the rate of adjustment to long-run equilibrium for one quasi-fixed factor is independent of the level of the other quasi-fixed factors. The hypothesis of an instantaneous rate of adjustment for the quasi-fixed inputs relys on independent rates of adjustment. Thus, a sequential testing procedure is dictated. Given that the hypothesis of independent rates of adjustment is not rejected, instantaneous adjustment for land and capital requires r + gK^ = r + g^ = -1, in addition 56 to 9KA = 9ak = 0* If both inputs adjust instantaneously, the adjustment matrix takes the form of a negative identity matrix. Capital and land would adjust immediately to long-run equilibrium levels in each time period. Regularity Conditions An attractive feature of the theoretical model is the regularity conditions that establish the duality of the value function and technology. Even so, little focus has been given to these conditions in previous empirical studies beyond the recognition of the existence of steady states for the quasi-fixed factors and a stable adjustment matrix required by condition V. 4. Without estimating the supply function or value function directly it is impossible to verify the regularity conditions stated in V.I., V.2 and Y*>0, the first part of condition V.3. One can note with slight satisfaction, however, that these conditions are likely to be satisfied if a0, aK, and aA are sufficiently large positive (Epstein, 1980, pg 88). The differentiability of J and JK are, of course, implicitly maintained in the choice of the value function. The conditions in V.4 are readily verified by determining if the long-run or equilibrium factor demands at each data point are positive to ensure the 57 existence and uniqueness of the steady states. Furthermore, the stability of these long-run demands is ensured if the implied adjustment matrix is nonsingular and negative definite. The nonsingularity of the adjustment matrix is related to condition V.5, the nonsingularity of JpK, as JpK = [M - r]-1 demonstrated in the previous chapter. Regularity condition V.7 is easily verified by the calculation of demand price elasticities for the inputs. Condition V.6 may be viewed as a curvature restriction ensuring a global minimum to the dual problem. Since JK is linear in prices, this condition is equivalent to the convexity of the value function J in input prices. The appropriate Hessian of second- order derivatives is required to be positive definite. Elasticities One particularly attractive aspect of dynamic optimization is the clear distinction between the short run, where quasi-fixed inputs only partially adjust to relative price changes along the optimal investment paths, and the long run, where quasi-fixed inputs fully adjust to their equilibrium levels. However, expectations with respect to the signs of price elasticities based on static theory are not necessarily valid in a dynamic framework. 58 Treadway (1970) and Mortensen (1973) have shown that positive own-price elasticities are an indication that some inputs contribute not only to production but to the adjustment activities of the firm. Thus in the short-run, the firm may employ more of the input in response to a relative price increase in order to facilitate adjustment towards a long-run equilibrium. However, this does not justify a positive own-price elasticity in the long-run. This same contribution to the adjustment process may also indicate short-run effects which exhibit greater elasticity than the long run. The firm may utilize more of an input in the short-run in order to enhance adjustment than in the long-run in response to a given price change. Short-run variable input demand elasticities may be calculated from (3.3). For example, the elasticity of labor demand with respect to the price of the jth input, z l,pj' is . * * 3 L 3 L â€¢ * * * 9K + 9 L * * 9 A pj. * 9 Pj 9 K 9Pj 9 A 3Pj L The short-run elasticity of demand for a variable input depends not only on the direct effect of a price change, but the also on the indirect effects of a price change on the short-run demand for the quasi- fixed inputs. 59 The short-run price elasticity for a quasi-fixed input is obtained from (3.7). The short-run demand elasticity for capital with respect to a change in the price of the jth input, Â£K,pj i-s * <3-16> e5,Pj 9K _ Pj * 9Pj K The short-run elasticity of demand for a quasi-fixed input depends only on the direct price effect in the short-run demand equation. The long-run elasticity of demand for a variable input can be obtained from (3.5). In the long run, all inputs are at equilibrium levels. Thus, the long- run elasticity of demand for labor with respect to the price of input j is (3.17) 9 L 9 L 3p_. 3 K 3 K 3 L 3 Pj 3 A 3_^_ . _Â£j 9 Pj L The long-run elasticity of a variable input is conditional on the effect of a price change in the equilibrium levels of the quasi-fixed inputs. The long-run demand elasticity for a quasi-fixed input is determined from (3.4). The long-run 60 elasticity of demand for capital with respect to the jth input price is (3.18) 8 K 3-pT In contrast to the short-run demand for a quasi-fixed input, where the short-run demand for one quasi-fixed input is determined in part by the level of the other quasi-fixed input, the long-run demand for a quasi- fixed input is solely an argument of prices. Data Construction The data requirements for the model consist of stock levels and net investment in land and capital, quantities of the variable inputs, labor and materials, as well as normalized (with respect to output price) rental prices for the inputs for the Southeast region. This region corresponds to the states of Alabama, Florida, Georgia, and South Carolina. The appropriate data are constructed for the period from 1949 through 1981. Data Sources Indices of output and input categories for the the Southeast are provided in Production and Efficiency Statistics (USDA, 1982). The inputs consist of farm power and machinery, farm labor, feed, seed, and livestock purchases, agricultural chemicals, 61 and a miscellaneous category. These indices provide a comprehensive coverage of output and input items used in agriculture for the respective categories. Annual expenditures for livestock, seed, feed, fertilizer, hired labor, depreciation, repairs and operations, and miscellaneous inputs for each state were obtained from the State Income and Balance Sheet Statistics (USDA) series. The expenditures for each of the Southeastern states are summed to form regional expenditures corresponding to the appropriate regional input indices cited above. This same series also contains revenue data for each state in the categories of cash receipts from farm marketing, value of home consumption, government payments, and net change in farm inventories. These data are aggregated across states to form a regional measure of total receipts. These sources provide the data for the construction of capital, materials, and labor quantity indices and capital and materials price indices. A GNP deflator is used to convert all expenditures and receipts to 1977 dollars. Additional data is drawn from Farm Labor (USDA) in order to construct a labor price index. Farm Real Estate Market Developments (USDA) provides quantity and price data for land. The undeflated regional expenditure and input data are provided in Appendix B. 62 Capital Capital equipment stocks and investment data are not available below the national level prior to 1970. Therefore, the mechanical power and machinery index was taken as a measure of capital stocks. As Ball (1985) points out, this index is intended to measure the service flow derived from capital rather than the actual capital stock. The validity of the mechanical power and machinery index as a measure of capital stock rests on the assumption that the service flow is proportional to the underlying capital stock. It is possible that the service flow from capital could increase temporarily without an increase in the capital stock if farmers used existing machinery more intensely without replacing worn-out equipment. Eventually, worn-out capital would have to replaced. Ball relies on a similar assumption of proportionality in employing the perpetual inventory method (Jorgenson, 1974) in deriving capital stocks. This method relies on the assumption of a constant rate of replacement in using gross investment to determine capital stocks such that (3.19) Ait = Iit + (1 - 6i)Ai/t_1, where A-j^ is capital stock i in period t, Ij^ is gross investment, and 6j_ is the rate of replacement. Even 63 the regional level, the perpetual inventory approach appears to share the potential weakness of the mechanical power and machinery index. Determining the appropriate price of capital presents additional difficulty. Hall and Jorgenson (1967) and Jorgenson (1967) define the user cost or implicit rental price of unit of capital as the cost of the capital service internally supplied by the firm. This actual cost is complicated by the discount rate, service life of the asset, marginal tax rate, allowable depreciation, interest deductions, and degree of equity financing. An alternate measure of user cost is provided by expenditure data representing actual depreciation or consumption of capital in terms of replacement cost and repairs and operation of capital items (Appendix Table B-l). By combining these expenditure categories in each time period to represent the user cost of the capital stock in place during the period, these expenditures and the machinery index can be used to construct an implicit price index for the region. 64 An implicit price index for capital is constructed using Fisher's weak factor reversal test (Diewert, 1976). The implicit price index may be calculated by (3.20) Pit = (Eit/Eib) / Â®it' where and Pit denote the quantity and price indices corresponding to the ith input in period t, and expenditures on the ith input in the same time period are denoted by E-^, and b denotes the index base period. Fisher's weak factor reversal test for price and quantity indices is satisfied if the ratio of expenditures in the current time period to the base is equal to the product of the price and quantity indices in the current time period. Since the machinery index and expenditure data are based in 1977, the resulting implicit price index for capital is also based in 1977. Land The land index represents the total acres in farms in the Southeast. The regional total is the sum of the total in each state. Hence, farmland is assumed homogeneous in quality within each state. An adjustment in these totals is necessary for the years 65 after 1975 as the USDA definition of a farm changed.5 Observations after 1975 are adjusted by the ratio of total acres under the old definition to total acres using the new definition. A regional land price index is constructed by weighting the deflated index of the average per acre value of farmland in each state by that state's share of total acres in the region. Unlike most price indices, the published index of farmland prices is not expressed in constant dollars. As rental prices are not available for the region, the use of an index of price per acre implicitly assumes that the rental rate is proportional to this price. The regional acreage total, quantity index, and price index may be found in Appendix Table B-2. Labor Beyond the additional parameters needed in the empirical model to treat labor as quasi-fixed, the farm labor index reflects the quantity employed, not necessarily the stock or quality of labor available. Hence, the regional labor index by definition represents a variable input. The USDA index of labor weights all hours equally, regardless of the human 5 Prior to 1975, a farm was defined as any unit with annual sales of at least $250 of agricultural products or at least 10 acres with annual sales of at least $50. After 1975, a farm is defined as any unit with annual sales of at least $1000. 66 capital characteristics of the workers. Additionally, this quantity index is not determined by a survey of hours worked but calculated based on estimated quantities required for various production activities. This presents some difficulties. The USDA farm labor quantity index includes owner-operator and unpaid family labor as well as hired labor, while the corresponding expenditures include wages and perquisites paid to hired labor, and social security taxes for hired labor and the owner- operator. Derivation of a price index as in (3.20) using these quantity and expenditure data treats owner-operator and family labor as if they were free. Instead, the USDA expenditures on hired labor and a regional quantity index of hired labor for the region calculated from Farm Labor (USDA) are used to calculate a labor price index. This assumes that owner-operators value their own time as they would hired labor. While this may seem inappropriate, the relative magnitude of hired labor to owner-operator labor in the Southeast reduces the impact of such an assumption. The regional total for expenditures on hired labor, the hired labor quantity index, and labor price index are presented in Appendix B, Table B-3. 67 Materials Expenditure data on feed, livestock, seed, fertilizer, and miscellaneous inputs are used to construct budget shares that provide the appropriate weights for each input in constructing an aggregate index. The indices represent quantities used rather than stocks, so the materials index characterizes a variable input. Some part of the livestock expenditure goes toward breeding stock, which is potentially quasi-fixed. The impact of investment in breeding stock is minimal, as the relative share of expenditures on livestock in the region is quite small. Again, Fisher's weak factor reversal test as shown in (3.20) can be readily applied to derive an implicit price index for materials. The expenditures on each of the inputs are aggregated and deflated. The ratio of aggregate materials expenditures in each time period to expenditures in 1977 is divided by the corresponding ratio of the aggregate materials input index. The regional expenditures for material inputs, aggregate materials index, and materials price index are presented in Appendix Table B-4. Output Price Equation (3.20) can also be used to construct an implicit output price index for the Southeast region 68 in order to normalize input prices. By combining the value of cash receipts, government payments, net inventory change, and the value of home consumption as a measure of output value for each region, this value and the aggregate output quantity index can be used to derive an implicit output price index. The output price of the prior year is used to normalize input prices to reflect that current price is not generally observed by producers when production and investment decisions are made. Regional total receipts, output quantity index, and output price index are found in Appendix Table B-5. Net Investment The observations on the USDA input indices correspond to quantities used during the production period. This is satisfactory for the variable inputs, labor and materials. However, the mechanical power and machinery index in effect reflects stock in place at the end of the production period. Therefore, this index is lagged one time period to reflect an initial level of available capital stock. The same procedure applies to the index of total acres in farms for the Southeast, as total acres are measured at year's end. As noted earlier, it is not possible to obtain estimates of gross investment in capital for the Southeast region over the entire data period. A 69 measure of net investment in capital and land for each time period can be defined for each of the inputs by (3.21) KÂ¿/t = Kift - where is net investment in the quasi-fixed input i during period t, is the level of the input stock in place at the end of period t, and is the level of input stock in place at the beginning of period t. By developing the model in terms of net investment, the need for gross investment and depreciation rate data in the determination of quasi- fixed factor stocks via (3.19) is elimimated. Since the estimated variable is actual net investment, it has been common practice in previous studies (Chambers and Vasavada, 1983b; Karp, Fawson, and Shumway, 1984) to assume constant rates of actual depreciation in order to calculate net investment from gross investment data. However, it is possible that the rate of depreciation could vary over observations. By using the difference of a quasi-fixed input index between two time periods as a measure of net investment, this problem can be at least partially avoided. 70 Data Summary Before proceeding to the estimation results of the empirical model, a brief examination of input use in the Southeast is in order. The quantity indices for capital, land, labor, and materials inputs used in the Southeast region for the years 1949 through 1981 are presented in Appendix Table B-6. The base year for the quantity and price indices is 1977. Figure 3-1 depicts the quantity indices for the 1949-1981 period. During the early years of the data period, agricultural production in the Southeast was characterized by a substantial reliance on labor and land relative to materials and capital. The quantity index of labor in 1949 was over three-and-one-half times the quantity index in 1981. Except for a short period of increase from 1949 to 1952, the quantity of land in farms has gradually declined from a high of 774 million acres in 1952 to 517 million in 1981, a decrease of nearly 35 percent. On the other hand, capital stocks nearly doubled, from 52 to 105, and the use of aggregate materials rose 250 percent, 44 to 112, from 1949 to 1981. 71 Quantity Capital + Land ^ Labor A Materials Figure 3-1. Observed Input Demand for Southeastern Agriculture, 1949-1981. 72 Turning to the normalized input prices, Figure 3-2 charts these input prices over the period of analysis. Not suprisingly, the same inputs whose quantities have dropped the most, labor and land, correspond to the inputs whose normalized prices have increased dramatically, labor increasing seventeenÂ¬ fold, from 0.10 to 1.71, and land six-fold, 0.21 to 1.35, over the data period. The most dramatic increase in the labor price index begins in 1968, such that nearly eighty percent of the increase in the labor price index occurs from 1967 to 1981, jumping from 0.39 to 1.71. The increase in the normalized land price index is more gradual, such that 50 percent of the increase occurs prior, 0.21 to 0.66, and 50 percent, 0.66 to 1.35, after 1966, the midpoint of the data period. The normalized price of capital doubled between 1949 and 1981, from 0.68 to 1.21, while the materials price increased only 10 percent, from 0.913 to 1.04. Interpretation of these changes in the normalized price indices should be tempered by recognizing that the indices are normalized with respect to output price. A drop in the output price would produce an increase in the normalized input price, everything else constant. However, examination of the actual output price index in Appendix Table B-5 shows only a 73 Capital, Land Index Year n Capital , Land , Labor Materials U Price T Price v Price a Price Figure 3-2. Normalized Input Prices for the Southeast, 1949-1981. 74 12 percent change in the output price index from endpoint, 1.08 in 1949, to endpoint, 0.94 in 1981. The rapid increase in output price of nearly 25 percent from 1972 to 1973, 0.89 to 1.112, produced a substantial drop in the normalized price indices for capital, land, and labor. The materials price index, however, rose even faster than the output price index, so the normalized price of materials increased. These data indicate that the Southeast has undergone some substantial changes from 1949 to 1981.6 The normalized price of labor has risen as dramatically as the quantity index has fallen. The Southeast has come to rely substantially more on materials and capital than in the past. The quantity of land in farms has gradually declined. It remains for the next chapter to see what light a dynamic model of factor demands can shed on these changes. 6. McPherson and Langham (1983) provide a historical perspective of southern agriculture. CHAPTER IV RESULTS Theoretical Consistency This chapter presents the results of estimating net investment demand equations for capital and land and variable input demand equations for labor and materials. The consistency of the data with the assumption of dynamic optimizing behavior is considered by evaluating the regularity conditions of the value function. Estimated short- and long-run levels of demand are obtained from the parameters of the estimated equations and compared to observed input demand. Estimates of short- and long-run price elasticities are computed in order to identify gross substitute/complement relationships among the inputs. Method of Estimation The system of equations presented in the previous chapter were estimated using iterated nonlinear three- stage least squares.7 For purposes of estimation, a disturbance term was appended to the net investment and variable input demand equations to reflect errors in optimizing behavior. This convention is consistent 7. The model was estimated using the LSQ option of the Time Series Processor (TSP) Version 4.0 as coded by Hall and Hall, 1983. 75 76 with other empirical applications (Chambers and Vasavada, 1983b; Karp, Fawson, and Shumway, 1984), although Epstein and Denny assume a first order autoregressive process in the error term for the quasi-fixed input demand equations.8 The iterated nonlinear three stage least squares estimation technique is a minimum distance estimator with the distance function D expressed as (4.1) D = f(y,b)'[S-1S H (H'H)â€1H'] f(y,b) where f(y,b) is the stacked vector of residuals from the nonlinear system, S is the residual covariance matrix, and H is the Kronecker product of an identity matrix dimensioned by the number of equations and a matrix of instrumental variables. For this system, the instruments consist of the normalized prices and their logarithms, quasi-fixed factor levels, and the time trend. Although the system is nonlinear in parameters, it is linear in variables. Hence, the minimum distance estimator is asympotically equivalent to full information maximum likelihood (Hausman, 1975) and provides consistent and asymptotically efficient parameter estimates. 8. Such an assumption necessitates estimation of a matrix of autocorrelation parameters. For two quasi-fixed inputs, this would require estimation of four additional parameters. 77 A constant real discount rate of five percent was employed in the estimation. This rate is consistent with the estimates derived by Hoffman and Gustafson (1983) of 4.4 percent reflecting the average twenty year current return to farm assets, 4.3 percent obtained by Tweeten (1981), and 4.25 percent by Melichar (1979).9 The parameter estimates and associated standard errors are presented in Table 4-1. Thirteen of the twenty-six parameters are at least twice their asymptotic standard errors. Given the nonlinear and simultaneous nature of the system, it is difficult to evaluate the theoretical and economic consistency of the model solely on the structural parameters. Thus, one must consider the underlying regularity conditions and the consistency of the derived input demand equations with observed behavior in order to assess the empirical model. Regularity Conditions An important feature of the dual approach, whether applied to static or dynamic optimization, is that the relevant conditions (V in Chapter II) are easy to check. Lau (1976) notes the difficulty of statistically testing the conditions for a static 9. The parameter estimates are fairly insensitive but not invariant to the choice of discount rates. 78 Table 4-1. Parameter Estimates Treating Materials and Labor as Variable Inputs, Capital and Land as Quasi-Fixed. Parameter Standard Parameter Estimate Errora bK 1913.160 496.327* bA -423.746 229.533 CW -677.077 212.572* cM -343.822 658.473 bkk 2472.608 478.233* bka -243.214 153.757 baa -123.888 98.104 cww -105.878 123.786 cwm -233.275 72.084* cmm 406.767 636.244 dwk -49.625 110.041 dwa 98.263 102.696 dmk 152.451 79.123 dma -111.878 74.397 nwK 1.900 0.731* nwA -1.879 2.283 nmK 0.826 0.309* nmA -0.159 0.530 VK 18.639 5.808* VA 4.155 0.707* VW 17.267 1.507* VM 23.045 8.312* -0.588 0.160* 9KA 0.490 0.242* -0.023 0.015 9aa -0.213 0.056* a * indicates parameter estimate two times its standard error. 79 duality, concluding that such tests are limited to dual functions linear in parameters. Statistical testing of the regularity conditions underlying dynamic duality is even more difficult. However, these conditions can be numerically evaluated. Since one of the objectives of this study is to obtain estimates of the adjustment rates of the quasi- fixed inputs and since the elements of the adjustment matrix M=[r+G] can be determined readily from the parameter estimates, the regularity conditions of nonsingularity of JpK-1 and stability of M are examined first. The nonsingularity of JpK-1 is determined from the estimates of the elements of G, as JpK-1 = G. The determinant of G is -0.334, thus satisfying the nonsingularity of JpK-1* The stability of the adjustment matrix requires that the eigenvalues of M have negative real parts and lie within the unit circle. The eigenvalues of G are -0.196 and -0.505, which satisfy the necessary stability criteria. The equilibrium demand levels for capital and land are positive at all data points. The existence and uniqueness of equilibrium or steady state levels of capital, K(p,w), and land, A(p,w)> as a theoretical requirement are also established. 80 It was shown in Chapter II that convexity of the value function in normalized input prices is sufficient to verify the necessary curvature properties of the underlying technology when JK is linear in prices, as is the empirical specification used to derive the current estimates. In fact, the linearity of JK in prices is necessary to generate an accelerator matrix consistent with net investment equations of the form K* = M (K-K). The elements of the matrix of second-order derivatives of the value function with respect to prices in this model are dependent upon the exogenous variables (prices) in the system. Thus, the Hessian must be evaluated for positive definiteness at each data point. This regularity condition was satisfied at thirty-one of the thirty-three observations (See Appendix C-l for numerical results). The only exceptions were the years 1949 and 1950. Given that these observations immediately follow the removal of World War II agricultural policies, the return of a large number of the potential agricultural work force, and rapidly changing production practices incorporating newly available materials, it is perhaps not surprising that the data are inconsistent with dynamic optimizing behavior at these points. 81 Technical Change The parameters representing technical change in the system of equations indicate that technical change has stimulated the demand for all inputs in the Southeast. Incorporation of these parameters as a linear function of time implicitly assumes technical change is disembodied. The relative magnitutude of these estimates indicates that technical change has been material-using relative to labor, and capitalÂ¬ using relative to land. While some studies of U.S. agriculture have found technical change to be labor- and land-saving (Chambers and Vasavada, 1983b), the estimated positive values for these inputs is not surprising given the rebirth of agriculture in the Southeast over the past quarter century. At least some portion of technical change has aided in maintaining the demand for labor in the face of rising labor prices by increasing productivity for many crops in the Southeast that rely on hand-harvesting, such as fresh vegetables and citrus. Consistency with Observed Behavior Evaluation of the empirical model relies on more than the theoretical consistency of the parameter estimates with respect to the regularity conditions. In addition, the economic consistency of the model is determined by the correspondence of observed net 82 investment and input use with the estimates or predicted values obtained from the derived demand equations. Satisfaction of the regularity conditions alone is not verification that dynamic optimizing behavior is an appropriate assumption. The observed and estimated values of K* in Table 4-2 show that the Southeast has been characterized by a steady increase in net capital investment, with only a few periods of net disinvestment. The estimated â€¢ JL , values of K correspond fairly closely to observed net investment. Observed capital stocks and the estimated short-run demand for the stock of capital correspond closely with never more than a two percent difference. However, there is a notable divergence of observed and equilibrium capital stock demand from 1949 to 1973. Contrary to the concerns of overcapitalization today, the Southeast only initially exhibited an excess of capital. However, the equilibrium level of capital rises in response to changing relative prices such that by 1974 observed and equilibrium levels are in close correspondence. The observed capital use and short- and long-run demand levels for capital presented in Figure 4-1 further illustrate this convergence. However, one should note the adjustment 83 Table 4-2. Comparison of Observed and Estimated Levels of Net Investment and Demand for Capital. Net Investment â€¢ â€¢ + K K observed optimal Year Capital Demand K K* (K, p, w) observed short- run K(p,w) long- run 49 8.00 7.13 52.00 53.81 16. 58 50 5.00 5.22 60.00 60.34 17. 33 51 6.00 5.69 65.00 65.60 22. 46 52 5.00 5.53 71.00 71.56 29. 16 53 2.00 4.65 76.00 75.98 32. 54 54 1.00 0.29 78.00 75.13 26. 29 55 2.00 0.94 79.00 76.56 28. 64 56 2.00 0.68 81.00 78.59 33. 48 57 -1.00 -1.67 83.00 79.24 34. 22 58 0.00 -0.42 82.00 79.32 40. 18 59 2.00 1.27 82.00 80.69 47. 80 60 -2.00 -0.92 84.00 81.48 49. 57 61 -1.00 0.33 82.00 80.43 52 . 08 62 -1.00 0.44 81.00 79.68 54 . 34 63 1.00 0.49 80.00 78.90 56. 71 64 0.00 0.44 81.00 80.03 60. 29 65 1.00 0.85 81.00 80.36 62. 11 66 1.00 0.92 82.00 81.44 63 . 89 67 3.00 1.22 83.00 82.83 68 . 77 68 1.00 0.22 86.00 85.19 70. 21 69 0.00 -0.23 87.00 86.05 72 . 96 70 -1.00 0.58 87.00 86.70 76. 50 71 4.00 4.80 86.00 88.55 84. 92 72 -2.00 2.52 90.00 91.06 84. 73 73 3.00 3.37 88.00 89.82 87. 85 74 4.00 3.75 91.00 93.28 95. 12 75 2.00 0.38 95.00 95.09 93. 03 76 2.00 1.67 97.00 97.86 96. 18 77 1.00 0.96 99.00 99.44 97. 72 78 6.00 2.07 100.00 101.22 101. 59 79 6.00 -0.12 106.00 105.88 105. 04 80 -5.00 -2.54 112.00 110.29 106. 43 81 -2.00 2.58 107.00 108.57 109. 84 84 of an excess capital stock to equilibrium levels is not achieved by a disinvestment in capital, but by an increase in the equilibrium level of capital demand. An examination of net investment and demand levels for land in Table 4-3 reveals a situation completely opposite from that of capital. Apart from a short period initially, the Southeast has exhibited a gradual reduction in the stock of land in farms. While estimated long-run demand exceeds observed and short-run demand for land stocks until 1961, the equilibrium level of demand falls at a faster rate than the short-run and observed levels. After 1961, the Southeast was marked by a slight degree of overinvestment in land stocks, owing primarily to an increase in the relative price of land. Observed stocks of land and estimates of short- run demand correspond closely over the data range. While the equilibrium level of capital increased in response to the increasing relative price of labor, the long-run demand for land has declined in response to an increase in the relative price of land as well as increase in the relative price of labor. Observed and estimated short- and long-run demand levels for land are presented graphically in Figure 4-2. 85 Capital Index Observed Demand Estimated Short-run Demand Estimated Long-run Demand Figure 4-1. Comparison of Observed and Estimated Demand for Capital in Southeastern Agriculture, 1949-1981. 86 Table 4-3. Comparison of Observed and Estimated Levels of Net Investment and Demand for Land. Net Investment Demand for Land Year â€¢ A observed A* optimal A observed A*(K,p,w) short- run A(p,w) long- run 49 3.45 2.64 142.32 146.78 193.99 50 1.14 2.11 145.77 150.01 198.58 51 0.76 1.01 146.91 149.98 189.99 52 0.19 -0.16 147.68 149.35 177.78 53 -0.95 -0.92 147.87 148.79 172.26 54 -0.57 -1.85 146.91 147.27 170.28 55 -2.86 -2.68 146.34 145.87 163.14 56 -3.05 -3.18 143.48 142.29 152.68 57 -3.63 -3.52 140.43 138.86 146.27 58 -3.82 -3.02 136.80 135.44 141.64 59 -3.44 -2.70 132.99 131.66 135.40 60 -2.48 -2.83 129.55 128.00 129.24 61 -2.86 -2.77 127.07 125.47 125.47 62 -2.86 -2.65 124.21 122.61 121.53 63 -3.63 -2.24 121.35 120.04 119.86 64 -1.72 -2.21 117.72 116.35 114.83 65 -1.53 -2.21 116.00 114.60 112.55 66 -1.53 -2.34 114.48 112.94 109.86 67 -1.91 -1.48 112.95 112.14 112.78 68 -2.29 -2.19 111.04 109.49 104.95 69 -1.91 -1.74 108.75 107.59 105.01 70 -1.34 -1.45 106.85 105.87 103.60 71 -0.95 -1.63 105.51 104.11 96.77 72 -0.76 -1.56 104.56 103.31 97.74 73 -1.15 -0.87 103.79 103.09 99.82 74 -0.57 -0.38 102.65 102.27 99.84 75 0.19 -0.28 102.08 101.90 101.64 76 -1.23 -0.61 102.27 101.76 99.29 77 -1.04 -0.58 101.04 100.54 97.83 78 -1.23 -0.48 100.00 99.52 96.33 79 0.87 -0.38 98.77 98.37 95.52 80 0.22 -0.48 99.63 99.20 96.70 81 -1.62 -0.48 99.85 99.32 95.36 87 Land Index Observed Estimated Estimated ^ Demand + Short-run ^ Long-run Demand Demand Figure 4-2. Comparison of Observed and Estimated Demand for Land in Southeastern Agriculture, 1949-1981. 88 The observed use of labor and estimated short- and long-run demands for labor as presented in Table 4-4 indicate almost complete adjustment of observed labor demand to the estimated equilibrium within one time period. This is consistent with the assumption that labor is a variable input. Returning to the parameter estimates in Table 4-1 and the short-run demand equation for labor in Table A-l, capital stocks slightly reduce the short-run demand for labor. Capital investment increases the short-run demand for labor. This indicates that labor facilitates adjustment in capital. The effect of land stocks on the short-demand for labor indicates an increase in land increases the short-run demand for labor. The effect of net investment in land decreases the short- run demand for labor. Labor appears to have a negative effect on the adjustment of land. The long-run demand for labor depends on the equilibrium levels of capital and land to the same degree that short-run labor demand depends on capital and land stocks. An increase in the equilibrium level of capital decreases the long-run demand for labor. Conversely, an increase in the equilibrium level of land increases the long-run demand for labor. Disequilibrium in the quasi-fixed inputs could potentially cause a divergence in the short- and long- 89 Table 4-4. Comparison of Observed and Estimated Short- and Long-Run Demands for Labor. Year Labor observed Labor Demand L*(K,p,w) short-run L (K, p, w) long-run 49 351.195 351.812 351.592 50 320.518 336.175 339.369 51 336.653 320.176 319.446 52 311.155 306.751 302.778 53 295.817 294.246 290.103 54 267.331 270.240 273.312 55 265.139 253.921 253.412 56 239.841 232.346 230.377 57 205.578 217.464 219.113 58 192.231 211.425 210.896 59 196.813 204.972 200.875 60 189.442 189.117 188.722 61 181.873 181.008 177.777 62 179.681 173.914 170.290 63 175.299 169.490 166.351 64 161.355 155.314 151.941 65 146.813 148.204 143.835 66 136.454 138.309 133.380 67 138.048 143.208 139.392 68 128.685 126.861 123.177 69 129.283 124.008 122.116 70 122.908 121.029 117.851 71 120.319 115.411 102.439 72 114.542 108.599 100.668 73 113.147 113.642 105.218 74 109.761 116.091 107.571 75 106.375 107.527 106.405 76 103.785 105.591 101.054 77 100.000 101.159 98.056 78 96.016 99.784 94.424 79 92.430 92.872 92.140 80 95.817 89.070 93.235 81 91.434 95.881 89.346 90 run demand for labor. However, the magnitudes of the parameter estimates associated with the dependence of labor demand on the quasi-fixed factors are small. Thus, the short-and long-run demands for labor are similiar. This is also true for the materials input as shown in Table 4-5. The degree of correspondence of observed and short- and long-run demands for materials is even greater than for labor. The short- and long- run demands for materials depend on quasi-fixed input stocks and equilibrium levels only slightly. Materials appear to facilitate adjustment in capital and slow adjustment in land. The substantial disequilibrium in the Southeast with respect to capital and land during the first part of the sample period may be interpreted from at least two viewpoints, one empirical and one intuitive. Empirically, the specification of the adjustment mechanism in the model is only indirectly dependent on factor prices through the determination of equilibrium levels of the quasi-fixed inputs. The accelerator itself is a matrix of constants. Yet the degree of adjustment in each factor level depends on the disequilibrium between actual and equilibrium input levels, which in turn are a function of the input prices. Changes in relative prices of the inputs, 91 Table 4-5. Comparison of Observed and Estimated Short- and Long-Run Demands for Materials. Materials Demand Materials M*(K,p,w) M(K,p,w) Year observed short-run long-run 49 44.237 44.877 41.464 50 42.623 44.431 43.123 51 49.666 47.821 45.440 52 50.194 50.111 47.166 53 51.917 52.350 49.974 54 51.345 49.255 51.308 55 54.943 53.015 53.908 56 55.006 55.332 56.009 57 54.630 54.572 57.477 58 57.657 58.220 59.693 59 63.332 64.191 63.653 60 63.839 63.245 64.828 61 66.081 67.067 67.159 62 68.409 69.178 69.021 63 69.281 70.373 70.176 64 76.910 75.303 75.007 65 78.817 78.569 77.764 66 79.390 79.980 78.993 67 81.210 83.445 82.368 68 82.814 85.197 84.826 69 87.717 88.000 88.212 70 94.124 93.462 92.798 71 98.432 97.382 91.829 72 101.150 98.821 95.828 73 101.801 94.462 90.681 74 97.131 96.437 92.264 75 86.274 92.328 91.928 76 96.985 100.289 98.369 77 100.000 101.008 99.833 78 105.649 107.951 105.538 79 114.375 109.014 108.972 80 105.367 103.573 106.169 81 112.323 114.331 111.323 92 especially labor, have caused the equilibrium level of capital to rise more rapidly than observed or short run capital demand. A complementary relationship between land and labor and substitute relation between capital and labor contribute to a high demand for land in the long-run and a low equilibrium for capital given the low initial prices of land and labor. Intuitively, these low prices may be attributed to a share-cropper economy, itself a vestige of the old plantations. While the relative prices of labor and land in 1949 reflect this notion, the observed levels of land and capital do not. It thus appears very plausible that during the initial postwar period, agriculture in the Southeast anticipated a change in this system and had already begun investing in capital and reducing land stocks. Elasticity Measures Given the inability to estimate the supply equation, only Marshallian (uncompensated) input demand elasticities were estimated. The explicit recognition of dynamic optimization provides a clear distinction between the short run, where quasi-fixed inputs adjust only partially to relative price changes along an optimal investment path, and the long run, where quasi-fixed input stocks are fully adjusted to equilibrium levels. 93 Average short-run gross elasticities for capital, land, labor and materials for selected periods are presented in Table 4-6. The short run own-price elasticities were negative at each data point for all inputs. Short-run elasticity estimates for each year are presented in Appendix Tables D-l - D-4. As noted earlier, positive short-run own-price elasticities are not inconsistent with the adjustment cost model. However, such a result did not occur in this analysis. The signs of the elasticity measures indicate that land and labor are short-run complements. This is not surprising given the labor-intensive crops that characterize production in the Southeast. Labor and capital and labor and materials are short-run substitutes. Materials and capital are short-run complements while materials and land are substitutes. Finally, capital and land are short-run substitutes. The short-run own-price elasticities for all four inputs are inelastic. Given capital and land are quasi-fixed inputs and the short-run demands for labor and materials are conditional on these inputs, inelastic short-run demands should be expected. The short-run own price elasticity of land is the most inelastic of the four inputs, ranging from an average of -0.08 in 1949-1955 to -0.02 in 1976-1981. Capital generally has next lowest own-price elasticity. The 94 Table 4-6. Estimated Short-run Average Uncompensated Input Demand Elasticities for Southeastern U.S. Agriculture for Selected Subperiods, 1949-1981. Elasticity with Respect to the Input Period price of: Labor Materials Capital Land Labor 1949-55 1956-60 1961-65 1966-70 1971-75 1976-81 Materials 1949-55 1956-60 1961-65 1966-70 1971-75 1976-81 Capital 1949-55 1956-60 1961-65 1966-70 1971-75 1976-81 Land 1949-55 1956-60 1961-65 1966-70 1971-75 1976-81 -0.703 0.311 -0.629 0.272 -0.542 0.242 -0.436 0.196 -0.335 0.134 -0.248 0.100 0.280 -0.170 0.195 -0.234 0.166 -0.299 0.148 -0.367 0.112 -0.325 0.104 -0.358 0.062 -0.071 0.046 -0.052 0.043 -0.049 0.035 -0.040 0.034 -0.039 0.026 -0.029 -0.041 0.029 -0.028 0.020 -0.022 0.016 -0.018 0.013 -0.015 0.011 -0.012 0.009 0.073 -0.207 0.065 -0.181 0.059 -0.161 0.046 -0.131 0.039 -0.090 0.025 -0.068 -0.098 0.130 -0.073 0.091 -0.064 0.078 -0.060 0.070 -0.043 0.053 -0.043 0.050 -0.344 0.066 -0.191 0.048 -0.154 0.045 -0.074 0.037 -0.126 0.036 -0.041 0.027 0.023 -0.082 0.016 -0.057 0.012 -0.044 0.010 -0.033 0.009 -0.024 0.007 -0.018 95 own-price elasticity of demand for capital declines in absolute value throughout, with the exception of an increase from 1966 to 1970. The demand for labor is more elastic than the demand for materials until the 1976-81 time period, when the own-price elasticity of labor demand is -0.25 and the corresponding elasticity for materials is -0.36. The estimated long-run uncompensated demand elasticities for selected periods are presented in Table 4-7 (See Appendix Tables D-5 through D-8 for annual estimates). All long-run own-price demand elasticities are negative. The substitute/complement relationships of the long run are identical to those of the short run. Labor and capital and land and capital are long-run substitutes. Land and labor are long-run complements, as well as land and materials. Finally, materials and labor are long-run substitutes. With the exception of labor, the own-price elasticities are more elastic in the long run than the short run, consistent with the Le Chatilier principle which states that long-run own price elasticities should be at least a large as the corresponding short- run elasticities. This difference is especially obvious in the short- and long-run elasticities for the quasi-fixed inputs. 96 Table 4-7. Estimated Long-run Average Uncompensated Input Demand Elasticities for Southeastern U.S. Agriculture for Selected Subperiods, 1949-1981. Elasticity with Respect to the Input Period price of: Labor Materials Capital Land Labor Materials Capital Land 1949-55 -0.636 0.325 0.074 -0.166 1956-60 -0.577 0.270 0.066 -0.140 1961-65 -0.505 0.235 0.062 -0.120 1966-70 -0.410 0.184 0.046 -0.096 1971-75 -0.318 0.120 0.049 -0.062 1976-81 -0.239 0.087 0.026 -0.046 1949-55 0.275 -0.196 -0.115 0.129 1956-60 0.195 -0.245 -0.091 0.095 1961-65 0.168 -0.306 -0.084 0.084 1966-70 0.150 -0.369 -0.085 0.075 1971-75 0.115 -0.330 -0.057 0.059 1976-81 0.107 -0.360 -0.062 0.055 1949-55 0.044 -0.215 -3.105 0.804 1956-60 0.024 -0.118 -1.370 0.441 1961-65 0.016 -0.079 -0.844 0.294 1966-70 0.011 -0.056 -0.413 0.209 1971-75 0.009 -0.047 -0.468 0.177 1976-81 0.007 -0.035 -0.197 0.131 1949-55 -0.229 0.169 0.398 -0.555 1956-60 -0.187 0.138 0.326 -0.474 1961-65 -0.154 0.114 0.268 -0.391 1966-70 -0.127 0.094 0.222 -0.316 1971-75 -0.112 0.083 0.195 -0.236 1976-81 -0.087 0.064 0.152 -0.189 97 For the variable inputs, the direct effect of a change in the own price of the input is identical in the short-run and long-run. But as noted earlier, the elasticity of a variable input is also conditional on the levels of the quasi-fixed inputs and net investment in the short-run. The greater short-run elasticity of labor arises as labor enhances the adjustment of capital. This is evident in the elasticity of capital with respect to a change in the price of labor. The estimated short-run cross-price elasticity is more elastic than the long-run. The absolute value of the long-run own-price elasticity of demand for capital is greater than one through 1960. This may be attributed to the relatively small value of equilibrium capital demand during that time period. The other factor demands are inelastic throughout the time span of the data. After 1965 the long-run demand for land is the most own price inelastic, and is consistently less elastic than the long-run demand for capital. The own-price elasticity of materials is the least elastic of the four inputs until 1966. However, by 1981, materials demand is the most elastic. The own- and cross-price elasticities for the short- and long-run trend downward with the exception of the own-price elasticity of materials which becomes 98 more elastic over time. One may also note that the normalized prices of labor, capital, and land trend upward over time which increase the numerators in the various elasticity formulae. However, Epstein and Denny also noted a trending in elasticity measures in their study of U.S. manufacturing using detrended data. While the rise in the normalized price of materials over the data period is not as great as that of the other inputs, the price does increase so that attributing the downward trend in elasticity measures to an upward trend in normalized prices would be somewhat inconsistent with the increasing own- price elasticity of materials demand. The Multivariate Flexible Accelerator The adjustment matrix M may be obtained almost directly from the estimated parameters by adding the discount rate to the diagonal elements of G. The implied rate of adjustment for capital, r + gKK, is â€œ -0.538. This implies that nearly 54 percent of the optimal net investment in capital occurs in the first year following a change in relative prices given an equilibrium level of land. The adjustment rate for land, r + g^A' i-s â€œ0*163. Thus, net investment in land reduces the difference in the equilibrium and actual level of land by 16 percent in the first year in response to a change in relative prices. The rate 99 of adjustment for one quasi-fixed input also depends on the degree of disequilibrium in the other input. Capital is especially dependent on the disequilibrium in land stocks given the estimated magnitude of g^ at 0.49. An overinvestment in land would significantly reduce the net adjustment of an excess capital stock. Net investment in land is much less dependent on an equilibrium in capital stocks. This is reflected by gAK = -0.0213 not significantly different from zero. The negative value of gAK implies that overinvestment in capital stocks speeds the reduction of land stocks given an overinvestment in land. The accelerator mechanism derived from the value function is amenable to a sequential test of independent rates of adjustment and instantaneous rates of adjustment, as both hypotheses are nested within the unrestricted model. The test of independent rates of adjustment is relevant given the number of studies previously cited that assume the rate of adjustment of one quasi-fixed factor is unaffected by the degree of disequilibrium in other quasi-fixed input levels. The hypothesis of instantaneous adjustment for both quasi-fixed inputs is conditional on independent rates of adjustment. Instantaneous rates of adjustment indicate that the quasi-fixed inputs, capital and land, adjust 100 completely to equilibrium levels within one time period. This does not imply that the quasi-fixed factors are in fact variable inputs, as the demands for labor and materials are still conditional on net investment and the levels of capital and land. The results of the sequential hypotheses tests are presented in Table 4-8. The hypothesis of independent rates of adjustment, H1, imposed by restricting the adjustment matrix to be diagonal, is rejected at a significance level of 0.05. Rejection of this hypothesis should terminate the testing procedure at this point, implicitly rejecting the hypothesis of instantaneous rates of adjustment. This explains the large magnitude of the test statistic for H2, where the adjustment matrix is restricted to the negative of an identity matrix. These results support the empirical specification of the multivariate flexible accelerator treating capital and land as quasi-fixed factors interdependent on the degree of disequilibrium in their respective levels. Beyond the result that capital and land do appear to be quasi-fixed, exhibiting a less-than-complete adjustment to equilibrium levels given a change in relative prices, the interdependence of adjustment has some significant implications. Were the adjustments of land and capital to equilibrium levels independent 101 Table 4-8. Sequential Hypothesis Tests of Independent and Instantaneous Rates of Adjustment for Capital and Land. Hypothesis Test Statistic3 Critical Value : Independent rates of adjustment <%A= 9aK â€œ Â°) 9.444 X2,0.025= 7-378 Â«1 = Unrestricted model Ho : Instantaneous adjustment <9KK+ r = Â§AA+ r = 1 / Â¿KA = Â®AK = 0) 405.459 X3,0.025= 9'348 H1 : Independent rates of adjustment a The test statistic utilized is TÂ° = n(SÂ° - S) where SÂ° denotes the minimized distance of the residual vector under the null hypothesis. S represents this same value for the unrestricted model and n is the sample size. Under the null hypothesis, TÂ° - X2 with degrees of freedom equal to the number of independent restrictions (Gallant and Jorgenson, 1979). 102 or if land was at its equilibrium level, capital would achieve 75 percent of the necessary adjustment in two years, while land would require nearly nine years to achieve the same level of adjustment if the level of capital was fully adjusted to its equilibrium level. It may be noted in Figures 4-1 and 4-2 that changes in the equilibrium level of capital produce a much steeper change in the short-run demand for capital than appears in the short-run demand for land with respect to a corresponding change in the equilibrium level of land. In 1978, the observed level of capital was quite close to the equilibrium level. The index of short-run demand for land exceeded equilibrium demand by 3.3 percent. The predicted disinvestment in land was 15 percent of this disequilibrium, quite close to the own rate of adjustment for land. Recognition of the own rate of response in these figures, especially for capital, is clouded by the interdependence of factor adjustments. Disequilibrium in the stock of land also substantially affects the adjustment of capital. While a scenario of fixed relative prices is somewhat unrealistic, it may be used to illustrate the importance of this interdependent relationship and the hazards of single-equation partial adjustment models. Figure 4-3 and 4-4 illustrate the adjustment of land 103 and capital to equilibrium levels reflecting two alternative starting points. Figure 4-3 corresponds to the observed and equilibrium levels of land and capital based on 1980 relative prices. Initially, observed values of land and capital exceed equilibrium levels. While the own rates of adjustment indicate that capital adjusts over three times as quickly as land, the overinvestment in land significantly reduces the net adjustment in capital so that after five years, 72 percent of the adjustment in capital and nearly 70 percent of the adjustment in land has been achieved. Figure 4-4 corresponds to the relative prices in 1981 and respective observed and equilibrium levels of capital and land. In this case, capital is initially below equilibrium and land stocks exceed equilibrium levels. The overinvestment in land speeds net investment in capital to the point that simulated short-run capital "overshoots" the equilibrium level. Stability of the adjustment matrix ensures that capital levels will eventually approach equilibrium. However, the original disequilibrium was one of underinvestment in capital. With the overinvestment in land present in 1981, it would require three years for capital stocks to peak and then begin to return to an equilibrium level. 104 Capital, Land Index n Initial , Equilibrium. Initial Equilibrium Capital Capital Land Ã¼ Land Figure 4-3. Adjustment Paths of Capital and Land Given 1980 Relative Prices and Initial Capital and Land Stocks. 105 Capital, Land Index Year g Initial Capital , Equilibrium Initial Capital Land Equilibrium Land Figure 4-4. Adjustment Paths of Capital and Land Given 1981 Relative Prices and Initial Capital and Land Stocks. 106 An alternative policy to fixing relative prices is one that artificially maintains an input above its equilibrium level. While such a policy is likely to affect relative prices, identifying that effect is difficult. Figures 4-5 and 4-6 represent the same time periods as the previous graphs with identical initial disequilibria and relative prices. However, no adjustment in land stocks is permitted. For the 1980 conditions, capital initially moves toward its equilibrium level but levels out. In effect, a policy maintaining land stocks also maintains capital above its equilibrium level. Similarly, if capital initially is below equilibrium as shown in the 1981 graph, maintaining land stocks not only causes capital to overshoot its equilibrium but to remain at a excess level. Rather than peaking and returning to the long- run level of demand, capital levels continue to climb above the equilibrium level. The difference in simulated short-run and equilibrium levels of capital has a small effect on the adjustment process of land stocks unless the capital disequilibrium is extremely large. The relative independence of the adjustment of land would indicate that policies directed at land stocks appear to work, yet create undesirable effects in the level of other factors. When such a policy is removed, the 107 Capital, Land Index Year Initial Equilibrium Initial Equilibrium U Capital + Capital 0 Land A Land (fixed) Figure 4-5. Adjustment Path of Capital with 1980 Relative Prices and Initial Capital and Land Fixed. 114 113 112 111 110 109 108 107 106 105 104 103 102 101 100 99 98 97 96 95 ] I: c are Land 0â€”9â€”9â€”0 0 0 0 Â» 0 9 9â€”OOOOOOO 0 A- â– ft- A- - A - -A A -A- A--A--Aâ€”A--Aâ€”ft A ft A ft ft -A 85 90 95 Year .tial )ital + Equilibrium^ Capital Initial Land (fixed) . Equilibrium 4 Lind 4-6. Adjustment Path of Capital with 1981 : Prices and Initial Capital and Land Fixed. 109 new disequilibrium could potentially be worse than the original. Finally, ignoring the relative price effects of a policy not directed specifically at factor stocks may be worse than a failure to recognize the interdependence of the adjustment process. CHAPTER V SUMMARY AND CONCLUSIONS The application of duality theory in static models of economic analysis is somewhat of a convenience. Although primal formulations may be difficult to solve, analytic solutions can be found for a wide variety of objective functions. Within the realm of dynamic economic analysis, however, the use of duality in order to generate theoretically consistent systems of demand equations is a necessity rather than a convenience in all but the simplest of model specifications. In spite of the importance of dynamic duality in generating systems of demand equations, its use in empirical analysis has been limited. The purpose of this study was to specify and estimate a complete system of dynamic input demand equations for southeastern U.S. agriculture by drawing upon the recent developments in the theory of dynamic duality. The results of this study not only provide information on the dynamic adjustment and short- and long-run interrelationships of inputs, but also provide some further insight and evidence concerning the empirical usefulness of dynamic duality theory. 110 Ill The empirical model treated capital and land as quasi-fixed inputs and labor and materials as variable inputs. The estimated system was generally consistent with the regularity conditions of the theory. Equilibrium levels of capital and land are positive at each data point and the matrix that characterizes adjustment of capital and land toward equilibrium levels is stable. Furthermore, the required convexity of the value function in input prices is satisfied by the data at all but two observations. As noted in the analysis, explicit use of a dynamic objective function permits a clear distinction between the short run and long run. The substitute/complement relationships indicated by the short-run price elasticity estimates indicate capital and land are short-run substitutes, as are capital and labor. Capital and materials are estimated to be complements in the short run, while labor and materials are short-run substitutes. The long-run elasticity estimates indicate the same substitute/complement relationships as in the short run. The demand for land is generally more inelastic than the other input demands in the short and long run. Long-run demands are generally more elastic than short-run. Labor appears to facilitate the adjustment of capital and thus labor demand is more elastic in the short-run 112 With the exception of the own-price elasticity of materials, elasticity estimates in the short and long run become less elastic over time. The ability to estimate both short-run and long-run demands permits estimation of the relative amount of disequilibria in inputs, or just how far the short-run observed levels are from the long-run levels. In the initial years of the data period, the model indicates a substantial degree of disequilibrium in capital and land inputs. However, from 1965 the differences in long-run equilibrium levels and observed demands are much less. The model indicates that there is a slight overinvestment in land in the Southeast from 1960 to 1981. Contrary to the notion of "backward" agricultural production practices in the South, in the sense that the substitution of capital for labor indicates technical progress, the data indicate that the Southeast was overcapitalized from 1949 to 1978. The adjustment of capital to near long-run equilibrium was not acheived through disinvestment in capital stocks, however, but was a consequence of the rising relative price of labor, which increased the long-run demand for capital. The parameter estimates of technical change indicate that technical change has stimulated the demand for all inputs in the Southeast. This is consistent with the general growth of agriculture in the South. 113 The differences in short-run and long-run demand for capital and land from 1949 to 1965 are perhaps a result of the maintained assumption that price expectations are stationary. In retrospect, it is easy to recognize the trend of rising prices for land and labor. As noted earlier, perhaps producers in the Southeast perceived this trend and had already invested in capital as a substitute for labor and land. Alternatively, the relatively large difference in long- and short-run levels of demand for capital and land may be a lingering effect of agricultural policies initiated prior and during World War II which were not removed until 1948. The empirical results indicate that capital and land are slow to adjust to changes in equilibrium levels. Given an equilibrium level of land, the model indicates an adjustment of nearly 54 percent of the difference in actual and long-run level within one time period. Land appears to adjust less rapidly, reducing the difference in actual and equilibrium levels by 15 percent in one time period. However, the rate of adjustment for capital appears to be particularly dependent on the degree of disequilibrium in land stocks. 114 The relatively slow adjustment rate of land provides an indication of why some static optimization models maintained land as fixed in the short run. However, the dependence of adjustment of capital on the land is shown to prevent capital from attaining a long- run equilibrium if a disequilibrium in land was fixed. In fact, whether capital is initially above or below its equilibrium position, maintaining land stocks above equilibrium also drives the capital stock above its equilibrium position. Although it may be unrealistic to assume relative prices remain unchanged for an extended period, the simulated adjustments of land and capital based on relative prices and existing disequilibrium in 1980 and 1981 illustrates the importance of interdependent rates of adjustment. The problems of "overshooting" and fixing the level of one input are heightened if relative price changes increase the degree of disequilibrium. The appropriateness of policy directed at alleviating the disequilibrium of a single input is questionable at best, and may be counterproductive to desired goals. It should be noted that the adjustment rates estimated for the Southeast differ considerable from estimates derived from a similar model of U.S agriculture. Chambers and Vasavada (1983b) and Karp, Fawson, and Shumway (1984), conclude that at the 115 national level land adjusts almost instaneously to changes in its long-run or equilibrium position. It is not surprising that they reach the same conclusion as they appear to be using the same data set. However, whether the difference in this analysis and the prior analyses is a result of definitional differences or a real difference in adjustment in land in the Southeast from the U.S. is unclear. If one considers the significance of perennial crops in the Southeast, such as fruits and tree nuts, and the acreage allotment policies for peanuts and tobacco that serve to keep land in agriculture, perhaps it is not surprising that the adjustment of land in the Southeast is slow. At the present time, the application of dynamic duality in economic theory is relatively unexplored. The opportunities for its application extend far beyond the area of input demands. However, there are several issues pertinent to this analysis that deserve mention. Even though a wide variety of functional forms of the value function potentially consistent with the underlying regularity conditions may be specified, there appears to be a trade-off between simplicity of the specification and satisfaction of the regularity conditions in empirical application. Initial efforts using a strictly quadratic form of the value function for the Southeast, other regions, and the U.S. proved to 116 be unsatisfactory. Limitations on time and money prevented application of the log-quadratic model to other regions for this dissertation. Given the number of parameters and nonlinearities inherent in the estimation equations, the multivariate flexible accelerator as a matrix of constants may be the only practical adjustment mechanism for empirical work. Ideally, a general form of input demand equations, for which the flexible accelerator as a matrix of constants is a special case imposed by restricting parameters, could be tested. Even then, the application is limited by the data in that each additional quasi-fixed input considered expands the parameters of the adjustment mechanism. Finally, the test for instantaneous adjustment of an input depends on independent rates of adjustment. Thus, appropriately testing quasi-fixity for a single input does not appear possible. The notion of rational expectations is often considered in determining the appropriate price for decision-making in static models (see Fisher, 1982). Taylor (1984) noted that the derivation of input demands through dynamic duality depends on the certainty of price expectations and incorporation of risk in the dynamic model implies a different set of conditions for an optimal solution. However, Karp, Fawson, and Shumway (1984) estimate a dynamic model under several price 117 expectation hypotheses and conclude that the adjustment rate is fairly insensitive to the underlying price expectation. Finally, Chambers and Lopez (1984) argue that stationary price expectations may be rational given the costs of information and difficulties in formulating alternative expectations in a dynamic setting. Additionally, the treatment of relative input prices as exogenous variables may be not be appropriate. It would seem that the adjustment rates and prices would be related. If a large degree of overinvestment and a "fast" rate of adjustment were indicated for a quasi- fixed input, then a large quantity of the input is potentially for sale, in turn depressing the price of the input. Likewise, if an input was below its long-run equilbrium, it would seem that the price of the input would be bid up as firms adjust toward equilibrium. The treatment of prices as exogenous also has implications for policy based strictly on the rates of adjustment. Policy directed at physical quantities of inputs, such as set-asides, buy-outs, and land retirment, inevitably affect input price. In turn, these price changes determine a new equilibrium position, and the adjustment process starts again. Recent work on joint determination of technolgoy and price expectations by Epstein and Yatchew (1985) may provide some insight to this problem. 118 Another potential issue is the symmetry of adjustment costs. The empirical model assumes that net disinvestment augments output by the same amount that net investment decreases output. It is entirely possible that adjustment costs are not symmetric, in that it may be more or less costly to invest rather than disinvest in a quasi-fixed input. In its current state, dynamic duality theory has little to say about this issue, but appears to be most likely method for exploration. In closing, it should be noted that the application of optimal control theory in the first manned mission to the moon indicated that the Apollo spacecraft was "off course" 90 percent of the time (Garfield, 1986). Yet the theory was vital to the conceptualization of man actually landing on the moon. Perhaps an analogy can be drawn for the application of dynamic duality theory in economic analyses. For the present, emphasis should not be on how "on course" such analyses are, but recognize instead the ability to conceptualize and address issues that were previously outside the possibilities of economic theory. APPENDIX A INPUT DEMAND EQUATIONS Table A-l. Estimation Equations for Optimal Net Investment in Quasi-fixed Inputs and Short-run Variable Input Demands. Optimal Net Investment in Capital: K*(K,p,w) = (r/pK) â€¢ [ (bk-gKK + b^g^) + (gKK*bkk + gKA*bka)â€˜l0(3 PK + (9KK*bka + gKA*baa)'log Pa + (9KK-dkw + gKA'daw)â€™log PL + (9KK*dkm + ^KAâ€™dam)â€™lo9 Pm] + r'vK*T + (r + gKK)*K + 9ka'a Optimal Net Investment in Land: Ã*(K,p,w) = (r/pA) â€¢ [ (bk*gAK + b^g^) + (gAKâ€˜bkk + gAAâ€˜bka)'log PK + (*?AK"bka + gAA,baa)'lo ( 9AA* dam) *lo Short-run Demand for Labor: L* (K, p, w) = (-r/pL) â€¢ (cw + c^-log pL + c^-log pM + dkw'log PK + dawâ€˜log Pa)~r"(nKw* ^ + nAw'A) + (nKwâ€˜K*(K,p,w) + nAw* Ã*(K,p,w)) + r-vL*T Short-run Demand for Materials: M*(K,p,w) = (-r/pM)-(cm + c^* log pL + cmm*log pM + dkm*log PK + damâ€™log PA)-r,(nKmâ€˜K + nAmâ€˜A) + (nKm'K*(K,p,w) + nAm-A*(K,p,w)) + r*vM*T 120 121 Table A-2. Long-run Input Demand Equations. Long-run Demand for Capital: K(p,w) = (-r/pk) â€¢ (c0 + cx * log pk + c2-log pa + c3â€™lÂ°g Pw + c4â€¢log pm) - râ€¢c5â€¢T Long-run Demand for Land: A(p,w) = (-r/pa) â€¢ (d0 + dx * log pk + d2*log pa + d3 * log pw + d4â€™log pm) - r-d5-T Long-run Demand for Labor: L(K;p,w) = (-r/pw) â€¢ (cw + cw'log pw + c^-log pm + dkwâ€˜lo Long-run Demand for Materials: M(K,p,w) = (-r/pm) ' (cm + c^-log pw + c^-log pm + dkmâ€™log Pk + damâ€˜lo where c0 = mxlâ€¢a0 + m12*a0 d0 = C1 = mllâ€™al + m12â€™al dl = c2 = mn" a2 + m12'a2 d2 = Ci = m-i-i'a-i + mi/ai do = c4 = m11,a4 + itii2' a4 d4 = c5 = in-vK + m12-vA d5 = mll = (r + gj^/Cir + gKK) ' (r + m2 2 = (r + gKK)/[(r + gKK)'(r + â„¢12 = â€œ9ka/ [ (r + 9kk) ' (r + ro21 = -9AK/[(r + 9kk)â€˜(r + ?Aa) and m2i'b0 + m22â€¢b0 mo iâ€¢bn + mooâ€¢bi m21 * b2 + m22'b2 mii-bo + moo *bo m2l'b4 + m22*b4 m21â€™vk + â„¢22*VA , 9aa) â€œ ^KAâ€™gAK) 9aa) " 9KAâ€˜9akJ - 9kaâ€˜9ak) _ 9ka"9ak) # a0 â€œ bk* ^KK + ba,(?KA al = bkkâ€™9KK + bka * 9ka a2 = bkaâ€™9kk + baa * 9ka a3 â€œ dkw**?KK + dawâ€™9KA a4 = dkm* *?KK + damâ€˜ 9kA b0 = bk,(?AK + ba * 9aA bl = bkkâ€™9aK + bka* b2 = bka * 9aK + haa'^AA b3 â€œ dkwâ€˜^AK + dawâ€™9AA b4 = dkm, 122 Table A-3. Short-run Demand Equations for the Quasi- fixed Inputs. Short-run Demand for Capital: K*(K,p,w) = (r/pK) * [ (bk*gKK + ba-g^) + (gKK*bkk + gKA*bka)'log PK + (gKKâ€˜bka + gKA-baa)â€˜log Pa + (9KK*dkw + gKA,daw)â€™log Pl + (9KK*dkm + gKA'dam)*log Pm] + r>vK'T + (1 + r + gKK)-K + Short-run Demand for Land: A*(K,p,w) = (r/pA)â€¢[(bk-gAK + ba-gAA) + (gAKâ€˜bkk + gAA'bka)â€˜log PK + ( (dkw + gAA,daw)â€™log PL + (9AK'dkm + gAA*dam)â€™log Pm] + r'vA*T + gAK*K + (1 + r + gAA)-A APPENDIX B REGIONAL EXPENDITURE, PRICE, AND INPUT DATA Table B-l. Expenditures on Capital Use and Capital Price Index for the Southeast. Repairs and Operations Depreciation and Consumption Capital Price Index Year Million dollars 49 139.2 118.3 0.721 50 148.0 136.6 0.744 51 166.7 162.7 0.775 52 181.8 172.3 0.731 53 190.3 183.6 0.735 54 198.0 201.9 0.764 55 203.8 207.4 0.765 56 214.6 204.2 0.734 57 224.1 214.0 0.760 58 228.2 223.5 0.760 59 237.7 236.6 0.760 60 227.4 229.5 0.738 61 222.0 227.6 0.725 62 227.8 237.2 0.746 63 229.9 249.1 0.748 64 227.2 268.5 0.757 65 247.8 292.9 0.807 66 259.2 311.5 0.826 67 278.5 342.8 0.848 68 292.3 369.4 0.847 69 304.5 408.4 0.887 70 312.1 422.3 0.874 71 298.8 441.9 0.740 72 301.0 476.1 0.819 73 323.5 541.8 0.861 74 411.5 637.5 0.959 75 454.1 744.1 0.978 76 533.6 805.6 0.957 77 600.5 868.7 1.000 78 687.1 1123.5 1.081 79 834.9 1274.0 1.060 80 992.8 1367.4 1.165 81 1091.7 1522.4 1.144 124 125 Table B-2. Acres and Acreage Index of Land in Farms and Value of Farm Real Estate Index for the Southeast. Acres in Acreage Value Farms Index Index Year Million acres 48 749 142.909 - 49 767 146.343 0.226 50 770 146.916 0.220 51 773 147.488 0.241 52 774 147.679 0.256 53 771 147.106 0.266 54 770 146.916 0.268 55 752 143.481 0.278 56 736 140.428 0.296 57 717 136.803 0.323 58 697 132.987 0.347 59 679 129.553 0.379 60 666 127.072 0.399 61 651 124.210 0.408 62 636 121.348 0.445 63 617 117.723 0.462 64 608 116.006 0.493 65 600 114.480 0.535 66 592 112.953 0.558 67 582 111.045 0.570 68 570 108.756 0.632 69 560 106.848 0.665 70 553 105.512 0.681 71 548 104.558 0.705 72 544 103.795 0.730 73 538 102.650 0.795 74 535 102.078 0.960 75 536 102.269 0.911 76 530 101.038 0.924 77 524 100.000 1.000 78 518 98.769 1.138 79 522 99.643 1.163 80 523 99.859 1.249 81 517 98.628 1.272 126 Table B-3. Expenditures on Hired Labor and Quantity and Price Index for the Southeast. Hired Labor Hired Labor Index Labor Price Index Year Million dollars 49 189.9 674.98 0.097 50 190.6 582.38 0.110 51 214.8 549.97 0.123 52 215.4 531.82 0.126 53 215.6 516.43 0.128 54 215.6 481.92 0.135 55 225.6 458.74 0.146 56 235.7 410.47 0.165 57 246.4 397.40 0.172 58 249.4 361.86 0.188 59 261.7 347.13 0.201 60 258.3 322.37 0.210 61 265.8 309.04 0.223 62 268.1 287.96 0.237 63 268.9 274.99 0.246 64 277.5 246.10 0.279 65 278.5 212.91 0.317 66 278.8 179.36 0.370 67 298.5 182.95 0.372 68 318.0 166.68 0.416 69 348.7 165.41 0.437 70 365.6 153.28 0.470 71 453.3 124.43 0.683 72 484.8 116.52 0.749 73 568.8 126.42 0.766 74 632.2 113.11 0.875 75 689.6 110.21 0.896 76 711.7 100.93 0.960 77 777.6 100.00 1.000 78 925.3 88.74 1.249 79 1047.5 84.21 1.371 80 1185.2 87.71 1.362 81 1417.3 80.91 1.614 127 Table B-4. Expenditures on Materials and Materials Price Index for the Southeast. Material: Feed LiveÂ¬ stock Seed FertiÂ¬ lizer MiscellÂ¬ aneous Price Index Year Million dollars 49 123.4 22.5 37.1 187.2 103.1 0.969 50 136.3 26.6 36.5 188.3 107.9 1.053 51 182.6 41.4 41.1 205.9 129.8 1.054 52 208.9 42.3 43.7 214.5 132.3 1.037 53 196.8 40.3 43.9 203.1 129.2 0.943 54 222.1 52.4 39.4 202.0 126.8 0.983 55 225.4 59.8 43.9 191.4 136.6 0.937 56 259.2 66.8 40.7 193.5 144.1 0.963 57 283.4 70.3 40.6 190.9 133.7 0.993 58 335.8 87.9 40.4 187.7 141.7 0.990 59 358.5 74.0 36.5 196.8 159.8 0.905 60 365.8 89.7 37.7 201.6 165.7 0.910 61 382.1 82.3 37.8 206.6 174.4 0.878 62 412.4 89.1 36.8 211.9 187.1 0.883 63 457.5 88.9 38.1 207.7 196.1 0.899 64 464.5 83.9 40.4 219.5 214.3 0.814 65 486.6 97.3 47.3 217.2 230.9 0.820 66 544.0 115.6 45.6 225.2 245.6 0.853 67 568.8 116.0 52.4 234.7 254.0 0.854 68 514.9 116.4 48.6 237.9 279.6 0.784 69 562.4 138.0 52.0 231.0 308.5 0.768 70 629.5 137.4 52.4 227.9 339.2 0.721 71 776.1 127.1 84.7 304.4 533.6 0.831 72 776.5 137.1 87.6 329.0 587.9 0.799 73 1260.1 161.5 110.0 412.2 606.4 1.050 74 1410.5 151.4 132.7 680.1 720.5 1.247 75 1213.0 151.0 154.1 699.1 794.6 1.275 76 1307.6 169.6 178.7 586.9 912.1 1.039 77 1268.3 220.4 215.7 612.6 952.9 1.000 78 1274.7 270.2 192.8 633.3 1118.6 1.007 79 1626.0 310.7 216.3 685.2 1364.3 0.968 80 1659.7 319.0 244.8 839.1 1520.9 1.063 81 1637.5 357.4 289.0 843.4 1900.4 0.982 128 Table B-5. Total Gross Receipts, Output Index and Output Price Index for the Southeast. Total Receipts Output Index Output Price Index Year Million dollars 48 2057.7 62.80 1.062 49 1948.7 58.76 1.085 50 2103.0 58.66 1.149 51 2525.5 68.57 1.108 52 2400.9 64.16 1.109 53 2439.4 71.26 0.999 54 2137.0 63.54 0.970 55 2537.8 76.49 0.936 56 2402.1 73.84 0.890 57 2343.1 65.62 0.944 58 2642.4 68.18 1.008 59 2664.8 72.63 0.932 60 2659.0 74.35 0.894 61 2788.1 78.08 0.884 62 2842.2 79.81 0.866 63 3008.4 83.45 0.864 64 3017.4 81.08 0.878 65 3163.9 86.77 0.842 66 3313.9 78.19 0.961 67 3427.4 91.14 0.817 68 3433.0 83.83 0.852 69 3857.1 89.31 0.854 70 3871.2 89.66 0.811 71 4686.8 101.18 0.828 72 5144.9 98.65 0.895 73 6897.3 100.74 1.112 74 7252.6 105.97 1.021 75 7725.2 108.26 0.974 76 7798.2 112.13 0.902 77 8156.8 100.00 1.000 78 9629.2 111.26 0.988 79 11008.2 119.80 0.965 80 10543.1 107.31 0.944 129 Table B-6. Prices of Inputs Normalized with Respect to Output Price Used in Estimations. Normalized Land Price of: Capital Labor Materials Year 49 0.213 0.679 0.091 0.913 50 0.203 0.686 0.101 0.971 51 0.210 0.674 0.107 0.917 52 0.231 0.660 0.114 0.936 53 0.240 0.663 0.115 0.850 54 0.268 0.764 0.135 0.984 55 0.287 0.789 0.150 0.966 56 0.316 0.784 0.176 1.028 57 0.363 0.854 0.193 1.116 58 0.368 0.805 0.199 1.049 59 0.376 0.754 0.199 0.898 60 0.428 0.792 0.225 0.977 61 0.457 0.811 0.250 0.982 62 0.503 0.844 0.269 0.999 63 0.533 0.864 0.284 1.038 64 0.571 0.876 0.323 0.942 65 0.609 0.919 0.361 0.934 66 0.663 0.981 0.439 1.013 67 0.593 0.883 0.387 0.889 68 0.774 1.037 0.510 0.960 69 0.781 1.041 0.514 0.902 70 0.797 1.023 0.550 0.844 71 0.870 0.913 0.843 1.025 72 0.881 0.988 0.905 0.964 73 0.888 0.961 0.856 1.173 74 0.864 0.863 0.787 1.122 75 0.892 0.958 0.877 1.249 76 0.949 0.983 0.985 1.067 77 1.108 1.108 1.108 1.108 78 1.138 1.081 1.249 1.007 79 1.177 1.073 0.388 0.980 80 1.294 1.207 1.411 1.101 81 1.347 1.212 1.709 1.040 APPENDIX C EVALUATION OF CONVEXITY OF THE VALUE FUNCTION Table C-l. Results for Convexity Test of the Value Function with Respect to Prices. Jkk Jkk Jka Jak Jaa Jkk Jka Jkw Jak Jaa Jaw Jwk Jwa Jww Jkk Jka Jkw Jkm Jak Jaa Jaw Jam Jwk Jwa Jww Jwm Jmk Jma Jmw Jmm Year 49 2.246E+03 1.376E+08 7.041E+12 -1.490E+13 * 50 2.113E+03 1.420E+08 6.325E+12 -1.060E+13* 51 2.328E+03 1.458E+08 5.641E+12 1.340E+13 52 2.601E+03 1.349E+08 4.611E+12 1.990E+13 53 2.607E+03 1.250E+08 3.963E+12 3.810E+13 54 1.384E+03 5.375E+07 1.374E+12 8.140E+12 55 1.210E+03 4.094E+07 8.506E+11 7.980E+12 56 1.286E+03 3.598E+07 5.683E+11 6.060E+12 57 8.293E+02 1.7 68E+07 2.411E+11 2.190E+12 58 1.181E+03 2.447E+07 3.048E+11 4.090E+12 59 1.682E+03 3.320E+07 3.790E+11 9.020E+12 60 1.370E+03 2.101E+07 1.968E+11 4.310E+12 61 1.248E+03 1.680E+07 1.293E+11 3.100E+12 62 1.048E+03 1.168E+07 7.83 6E+10 1.980E+12 63 9.373E+02 9.325E+06 5.727E+10 1.370E+12 64 9.168E+02 7.916E+06 3.644E+10 1.23 OE+12 65 7.194E+02 5.449E+06 2.031E+10 7.260E+11 66 4.823E+02 3.067E+06 8.265E+09 2.430E+11 67 9.117E+02 7.255E+06 2.338E+10 9.440E+11 68 3.535E+02 1.640E+06 3.211E+09 1.080E+11 69 3.531E+02 1.606E+06 3.009E+09 1.160E+11 70 4.245E+02 1.858E+06 2.975E+09 1.410E+11 71 8.858E+02 3.273E+06 2.590E+09 9.23 0E+10 72 5.728E+02 2.050E+06 1.395E+09 5.220E+10 73 6.4 63E+02 2.300E+06 1.8 69E+09 5.040E+10 74 1.153E+03 4.340E+06 4.025E+09 1.22 0E+11 75 6.509E+02 2.299E+06 1.829E+09 4.330E+10 76 5.987E+02 1.856E+06 1.115E+09 3.630E+10 77 2.609E+02 5.845E+05 2.818E+08 6.770E+09 78 3.494E+02 7.461E+05 2.773E+08 9.040E+09 79 3.309E+02 6.853E+05 2.029E+09 8.020E+09 80 1.094E+02 1.685E+05 5.066E+07 2.433E+10 81 1.206E+02 1.726E+05 3.564E+07 2.201E+10 * indicates determinant inconsistent with convexity. 131 APPENDIX D ANNUAL SHORT- AND LONG-RUN PRICE ELASTICITY ESTIMATES Table D-l. Short-run Uncompensated Price Elasticities for Capital. Short-run Elasticity of Capital Demand with respect to the price of: Labor Materials Capital Land Year 49 0.080 -0.091 -0.453 0.086 50 0.070 -0.080 -0.388 0.077 51 0.066 -0.075 -0.393 0.070 52 0.062 -0.071 -0.399 0.065 53 0.058 -0.066 -0.379 0.060 54 0.051 -0.058 -0.213 0.055 55 0.048 -0.055 -0.186 0.050 56 0.047 -0.054 -0.196 0.049 57 0.043 -0.049 -0.121 0.046 58 0.046 -0.052 -0.180 0.048 59 0.048 -0.055 -0.253 0.050 60 0.045 -0.052 -0.207 0.048 61 0.045 -0.051 -0.194 0.047 62 0.044 -0.050 -0.166 0.046 63 0.043 -0.049 -0.149 0.045 64 0.042 -0.048 -0.147 0.045 65 0.040 -0.045 -0.115 0.041 66 0.037 -0.042 -0.071 0.038 67 0.040 -0.046 -0.147 0.042 68 0.033 -0.038 -0.045 0.035 69 0.033 -0.037 -0.045 0.035 70 0.033 -0.038 -0.061 0.035 71 0.036 -0.041 -0.149 0.038 72 0.033 -0.037 -0.094 0.034 73 0.034 -0.039 -0.106 0.036 74 0.036 -0.042 -0.179 0.039 75 0.032 -0.037 -0.101 0.034 76 0.030 -0.035 -0.093 0.033 77 0.027 -0.030 -0.032 0.028 78 0.027 -0.031 -0.051 0.028 79 0.026 -0.029 -0.056 0.027 80 0.022 -0.025 -0.003 0.023 81 0.022 -0.025 -0.008 0.024 133 134 Table D-2. Short-run Uncompensated Price Elasticities for Land. Short-run Elasticity of Land Demand with respect to the price of: Labor Materials Capital Land Year 49 -0.045 0.032 0.025 -0.095 50 -0.046 0.033 0.026 -0.093 51 -0.045 0.032 0.025 -0.087 52 -0.041 0.029 0.023 -0.080 53 -0.039 0.028 0.022 -0.076 54 -0.036 0.025 0.020 -0.073 55 -0.033 0.024 0.019 -0.068 56 -0.031 0.022 0.017 -0.062 57 -0.028 0.020 0.016 -0.060 58 -0.028 0.020 0.016 -0.058 59 -0.028 0.020 0.016 -0.054 60 -0.026 0.018 0.014 -0.051 61 -0.024 0.017 0.014 -0.048 62 -0.023 0.016 0.013 -0.046 63 -0.022 0.016 0.012 -0.045 64 -0.021 0.015 0.012 -0.041 65 -0.020 0.014 0.011 -0.038 66 -0.019 0.013 0.010 -0.035 67 -0.021 0.015 0.012 -0.037 68 -0.017 0.012 0.009 -0.031 69 -0.017 0.012 0.009 -0.031 70 -0.017 0.012 0.009 -0.029 71 -0.015 0.011 0.009 -0.023 72 -0.015 0.011 0.009 -0.022 73 -0.015 0.011 0.009 -0.024 74 -0.016 0.011 0.009 -0.025 75 -0.015 0.011 0.009 -0.025 76 -0.015 0.010 0.008 -0.021 77 -0.013 0.009 0.007 -0.020 78 -0.012 0.009 0.007 -0.017 79 -0.012 0.009 0.007 -0.016 80 -0.011 0.008 0.006 -0.017 81 -0.010 0.007 0.006 -0.014 135 Table D-3. Short-run Uncompensated Price Elasticities for Labor. Short-run Elasticity of Labor with respect to the price of: Labor Materials Capital Land Year 49 -0.724 0.330 0.077 -0.219 50 -0.726 0.311 0.073 -0.207 51 -0.714 0.314 0.074 -0.208 52 -0.705 0.312 0.075 -0.207 53 -0.692 0.323 0.078 -0.214 54 -0.688 0.297 0.069 -0.197 55 -0.672 0.291 0.067 -0.194 56 -0.656 0.276 0.065 -0.184 57 -0.646 0.266 0.061 -0.177 58 -0.633 0.268 0.064 -0.179 59 -0.613 0.282 0.069 -0.187 60 -0.598 0.267 0.065 -0.178 61 -0.579 0.257 0.063 -0.171 62 -0.563 0.250 0.061 -0.167 63 -0.554 0.243 0.058 -0.162 64 -0.519 0.236 0.058 -0.157 65 -0.496 0.225 0.054 -0.150 66 -0.468 0.201 0.047 -0.134 67 -0.477 0.217 0.055 -0.145 68 -0.426 0.189 0.043 -0.126 69 -0.415 0.189 0.043 -0.126 70 -0.393 0.184 0.043 -0.123 71 -0.334 0.138 0.041 -0.093 72 -0.317 0.132 0.036 -0.089 73 -0.339 0.132 0.037 -0.089 74 -0.347 0.140 0.044 -0.094 75 -0.337 0.128 0.036 -0.086 76 -0.300 0.121 0.034 -0.081 77 -0.277 0.111 0.027 -0.075 78 -0.249 0.103 0.027 -0.069 79 -0.231 0.093 0.026 -0.064 80 -0.233 0.093 0.020 -0.063 81 -0.199 0.080 0.018 -0.054 136 Table D-4 . Short-run Uncompensated Price Elasticities for Materials. Short-run Elasticity of Materials Demand with respect to the price of: Labor Materials Capital Land Year 49 0.327 -0.128 -0.114 0.152 50 0.296 -0.106 -0.102 0.137 51 0.297 -0.161 -0.102 0.138 52 0.276 -0.173 -0.091 0.128 53 0.290 -0.239 -0.098 0.135 54 0.242 -0.175 -0.090 0.113 55 0.233 -0.206 -0.088 0.109 56 0.208 -0.204 -0.077 0.097 57 0.186 -0.179 -0.073 0.087 58 0.189 -0.215 -0.071 0.089 59 0.207 -0.300 -0.075 0.097 60 0.186 -0.272 -0.068 0.087 61 0.177 -0.281 -0.066 0.083 62 0.169 -0.281 -0.064 0.079 63 0.160 -0.266 -0.061 0.075 64 0.165 -0.328 -0.064 0.078 65 0.160 -0.341 -0.063 0.075 66 0.145 -0.318 -0.059 0.068 67 0.159 -0.364 -0.062 0.075 68 0.142 -0.354 -0.059 0.067 69 0.146 -0.381 -0.060 0.069 70 0.148 -0.416 -0.061 0.070 71 0.121 -0.362 -0.046 0.057 72 0.124 -0.389 -0.049 0.058 73 0.107 -0.297 -0.042 0.051 74 0.110 -0.308 -0.040 0.052 75 0.100 -0.269 -0.039 0.047 76 0.109 -0.338 -0.043 0.052 77 0.104 -0.329 -0.043 0.049 78 0.108 -0.375 -0.044 0.051 79 0.107 -0.392 -0.043 0.051 80 0.099 -0.342 -0.042 0.047 81 0.099 -0.374 -0.042 0.047 137 Table D-5. Long-run Uncompensated Price Elasticities for Capital. Long-run Elasticity of Capital Demand with respect to the price of: Labor Materials Capital Land Year 49 0.060 -0.294 -4.407 1.101 50 0.058 -0.285 -4.107 1.065 51 0.047 -0.230 -3.499 0.861 52 0.038 -0.187 -3.035 0.699 53 0.034 -0.167 -2.734 0.625 54 0.036 -0.180 -2.145 0.674 55 0.033 -0.162 -1.805 0.605 56 0.029 -0.142 -1.658 0.531 57 0.026 -0.129 -1.196 0.481 58 0.024 -0.117 -1.329 0.437 59 0.021 -0.105 -1.446 0.392 60 0.020 -0.097 -1.221 0.363 61 0.018 -0.091 -1.090 0.339 62 0.017 -0.084 -0.930 0.313 63 0.016 -0.079 -0.826 0.294 64 0.015 -0.073 -0.760 0.273 65 0.014 -0.068 -0.616 0.253 66 0.013 -0.062 -0.445 0.232 67 0.013 -0.064 -0.662 0.238 68 0.011 -0.053 -0.321 0.200 69 0.010 -0.051 -0.306 0.191 70 0.010 -0.050 -0.333 0.186 71 0.010 -0.051 -0.543 0.190 72 0.009 -0.047 -0.384 0.175 73 0.009 -0.047 -0.418 0.174 74 0.010 -0.048 -0.598 0.179 75 0.009 -0.044 -0.399 0.165 76 0.008 -0.042 -0.357 0.156 77 0.007 -0.036 -0.187 0.136 78 0.007 -0.036 -0.222 0.134 79 0.007 -0.035 -0.231 0.131 80 0.006 -0.031 -0.091 0.115 81 0.006 -0.030 -0.091 0.111 138 Table D-6. Long-run Uncompensated Price Elasticities for Land. Long-run Elasticity of Land Demand with respect to the price of: Labor Materials Capital Land Year 49 -0.233 0.172 0.405 -0.586 50 -0.239 0.177 0.416 -0.582 51 -0.242 0.178 0.421 -0.564 52 -0.235 0.174 0.409 -0.547 53 -0.234 0.173 0.407 -0.534 54 -0.212 0.156 0.368 -0.543 55 -0.207 0.152 0.359 -0.528 56 -0.201 0.149 0.350 -0.505 57 -0.183 0.135 0.318 -0.499 58 -0.186 0.137 0.324 -0.478 59 -0.191 0.141 0.332 -0.450 60 -0.176 0.130 0.305 -0.436 61 -0.170 0.125 0.295 -0.420 62 -0.159 0.117 0.276 -0.407 63 -0.152 0.112 0.265 -0.399 64 -0.149 0.110 0.258 -0.372 65 -0.142 0.105 0.247 -0.358 66 -0.134 0.099 0.233 -0.343 67 -0.146 0.108 0.254 -0.339 68 -0.120 0.088 0.209 -0.311 69 -0.119 0.088 0.207 -0.303 70 -0.118 0.087 0.205 -0.286 71 -0.116 0.086 0.202 -0.234 72 -0.114 0.084 0.198 -0.232 73 -0.110 0.081 0.191 -0.242 74 -0.113 0.083 0.196 -0.231 75 -0.107 0.079 0.187 -0.239 76 -0.104 0.076 0.180 -0.216 77 -0.090 0.066 0.156 -0.209 78 -0.089 0.066 0.155 -0.188 79 -0.087 0.064 0.151 -0.173 80 -0.078 0.057 0.135 -0.182 81 -0.076 0.056 0.132 -0.163 139 Table D-7. Long-run Uncompensated Price Elasticities for Labor. Long-run Elasticity of Labor Demand with respect to the price of: Labor Materials Capital Land Year 49 -0.656 0.353 0.079 -0.178 50 -0.655 0.328 0.072 -0.169 51 -0.643 0.330 0.075 -0.169 52 -0.639 0.327 0.080 -0.166 53 -0.624 0.339 0.083 -0.170 54 -0.625 0.304 0.067 -0.158 55 -0.610 0.296 0.064 -0.155 56 -0.598 0.277 0.064 -0.145 57 -0.593 0.264 0.057 -0.139 58 -0.581 0.266 0.063 -0.139 59 -0.560 0.281 0.075 -0.142 60 -0.551 0.263 0.070 -0.134 61 -0.535 0.251 0.067 -0.129 62 -0.524 0.244 0.064 -0.125 63 -0.518 0.236 0.062 -0.121 64 -0.485 0.227 0.061 -0.116 65 -0.464 0.215 0.056 -0.111 66 -0.439 0.189 0.045 -0.100 67 -0.444 0.206 0.059 -0.106 68 -0.403 0.176 0.041 -0.092 69 -0.392 0.177 0.041 -0.092 70 -0.371 0.171 0.042 -0.088 71 -0.318 0.124 0.054 -0.063 72 -0.300 0.117 0.040 -0.061 73 -0.323 0.118 0.044 -0.062 74 -0.331 0.126 0.062 -0.063 75 -0.320 0.114 0.043 -0.060 76 -0.286 0.106 0.041 -0.055 77 -0.267 0.097 0.026 -0.052 78 -0.240 0.089 0.029 -0.046 79 -0.221 0.081 0.030 -0.041 80 -0.224 0.079 0.015 -0.043 81 -0.193 0.067 0.014 -0.036 140 Table D-8. Long-run Uncompensated Price Elasticities for Materials. Long-run Elasticity of Materials Demand with respect to the price of: Labor Materials Capital Land Year 49 0.320 -0.164 -0.131 0.148 50 0.289 -0.141 -0.119 0.133 51 0.290 -0.193 -0.117 0.136 52 0.271 -0.202 -0.100 0.129 53 0.283 -0.264 -0.111 0.136 54 0.239 -0.192 -0.112 0.112 55 0.231 -0.219 -0.113 0.109 56 0.207 -0.216 -0.096 0.100 57 0.186 -0.188 -0.094 0.089 58 0.190 -0.225 -0.089 0.093 59 0.207 -0.312 -0.092 0.102 60 0.186 -0.282 -0.085 0.093 61 0.179 -0.290 -0.084 0.089 62 0.171 -0.288 -0.083 0.085 63 0.162 -0.273 -0.080 0.081 64 0.166 -0.334 -0.085 0.083 65 0.162 -0.345 -0.087 0.081 66 0.147 -0.321 -0.082 0.074 67 0.161 -0.370 -0.083 0.081 68 0.144 -0.355 -0.084 0.072 69 0.148 -0.382 -0.087 0.074 70 0.150 -0.418 -0.088 0.075 71 0.123 -0.368 -0.061 0.064 72 0.126 -0.393 -0.069 0.064 73 0.110 -0.302 -0.056 0.057 74 0.113 -0.315 -0.049 0.059 75 0.103 -0.274 -0.051 0.053 76 0.112 -0.342 -0.059 0.058 77 0.106 -0.331 -0.062 0.054 78 0.110 -0.377 -0.064 0.056 79 0.109 -0.394 -0.063 0.056 80 0.101 -0.341 -0.063 0.051 81 0.101 -0.374 -0.063 0.052 REFERENCES Almon, Shirley. 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" American Journal of Agricultural Economics 67(1985):558-562. BIOGRAPHICAL SKETCH Michael James Monson was born August 15, 1956, the first of James and Glenda Monson*s three children. He grew up on a farm near Stratford, Iowa. After graduating from Stratford Community High School in 1974, he entered Iowa State University. In 1978, he received a Bachelor of Science degree in agricultural business, graduating with distinction. He continued his education at Iowa State as a research assistant in agricultural economics from 1978 to 1979, research associate in 1980 and 1981, and taught undergraduate Agricultural Policy in 1981 and 1982. Michael earned a Master of Science degree from the Department of Agricultural Economics specializing in natural resource economics in May, 1982. In August of 1982 he entered the graduate program of the Department of Food and Resource Economics as a research assistant in order to pursue a doctoral degree. On July 2, 1983, he married Sandra Lynn Johnson. On August 16, 1984, Mike and Sandra were blessed with their first child, Jeffery James. Michael James Monson is a member of the Phi Kappa Phi and Phi Eta Sigma honorary societies and the American Agricultural Economics Association. 148 I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. JQ- William G. Boggese, Chairman Associate Professor of Food and Resource Economics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Timothy Taylor, Cdchairman Assistant Professor of Food and Resource Economics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the detjree^>f Doc tot: of Philosophy. ' ' ' * ' â€¢ V Â¿bbert D. Emerson Associate Professor of Food and Resource Economics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Max R. Langham Professor of Food Economics nd Resource I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Antal Majthay Associate Professor of Management This dissertation was submitted to the Graduate Faculty of the College of Agriculture and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. 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