UFDC Home  Search all Groups  UF Institutional Repository  UF Institutional Repository  UF Theses & Dissertations  Vendor Digitized Files   Help 
Material Information
Subjects
Notes
Record Information

Full Text 
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 downtoearth 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. iii TABLE OF CONTENTS ACKNOWLEDGEMENTS...................... ABSTRACT................................. CHAPTER I II III IV V APPENDIX A B C D INTRODUCTION.......................... Background.. ........................ Objectives. ........................... THEORETICAL MODEL.................... Dynamic Models Using Static Optimization....................... Dynamic Optimization................. Theoretical Model..................... The Flexible Accelerator.............. EMPIRICAL MODEL AND DATA............... Empirical Model....................... Data Construction.................. RESULTS............................. . Theoretical Consistency............... Quasifixed Input Adjustment........... SUMMARY AND CONCLUSIONS............... INPUT DEMAND EQUATIONS................ REGIONAL EXPENDITURE, PRICE, AND INPUT DATA......................... EVALUATION OF CONVEXITY OF THE VALUE FUNCTION..................... ANNUAL SHORT AND LONGRUN PRICE ELASTICITY ESTIMATES............ REFERENCES...................................... BIOGRAPHICAL SKETCH............................... iv PAGE iii v 1 3 15 18 20 27 35 42 46 46 60 75 75 99 110 119 123 130 132 141 148 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. 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 quasifixed. The equations for optimal net investment yield short and longrun 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 shortrun substitutes. Land and labor, and materials and capital are shortrun 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 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 shortrun yet variable in the longrun, there is nothing in the theory of static optimizing behavior to explain a 1 divergence in the short and longrun 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 lessthan 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 lessthan 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 lessthan 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 longrun. Thus, input demands and elasticities for the short and longrun 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 lowcost 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. Quasifixed Inputs The assumption of shortrun input fixity is questionable. In a dynamic or longrun 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 "quasifixed" 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 longrun optimum. The input does not remain fixed in the shortrun, nor is there an immediate adjustment to the longrun 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 rather fixed at a cost per unit time. The quasi fixity of an input is limited by the cost of adjusting that input. Shortrun Input Fixity In light of the notion of quasifixity, input fixity in shortrun 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 shortrun initial level or a complete adjustment to a longrun 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 puttyclay 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 puttyclay 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 tradeoff between flexibility of input combinations in response to relative input price changes and shortrun 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 puttyclay. 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 puttyputty. Finally, if the input ratios do adjust, but not to the point of maximum efficient output, the technology is said to be puttytin. Chambers and Vasavda concluded that the data did not support the assumption of shortrun fixity for capital, labor, or materials. However, the effects of the maintained assumptions on the conclusions is 7 unclear. The assumption of shortrun 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 quasifixed. Simply stated, the firm must incur a cost in order to change or adjust the level of some inputs. The assumption of shortrun 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 "bangbang" investment policy. Such models lead to discrete jumps rather than a 8 continuous 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 shortrun fixed level until the returns to investment justify a complete jump to the longrun 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 shortrun 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 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 shortrun 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 persistence models of consumer demand (Pope, Green, and Eales, 1980). Recognizing the problems of such "bangbang" 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 quasifixed factor, leads to a continuous or smooth form of investment behavior. Quasifixed 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 shortrun 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 longrun static models. Therefore, a theoretical foundation for quasifixed inputs can be established. Models of Quasifixed Input Demand Econometric models which allow quasifixity of inputs provide a compromise between maintaining some inputs as completely fixed or freely variable in the shortrun. The empirical attraction of such models is 11 evident in the significant body of research in factor demand analysis consistent with quasifixity 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 singleequation partial adjustment model is that such a specification ignores the effect of, and potential for, quasifixity in the demands for the remaining inputs. The recent paymentinkind (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 socalled 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 quasifixed 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 quasifixed 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 quasifixed 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 quasifixity 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 longrun 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 longrun permits derivation of short and longrun 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. Scope The dynamic objective function is expressed in terms of quasifixed 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 quasifixed. 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 longrun 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, quasifixity, 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 adjustmentcost 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 quasifixity and adjustment costs culminate in the very 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 quasifixed, 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 quasifixity 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 quasifixed input as a function of depreciation, the discount rate, and tax rate as the appropriate measure of the quasifixed 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 extensively. 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 adhockery." 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 longrun equilibrium position of the firm. If an input is quasifixed, 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 quasifixed input is a result of the partial adjustments to previous equilibrium positions. The various forms of the distributedlag determine how important these past adjustments are in determining the current response. The finite lag distributions proposed by Fisher (1937), 1 Maddala (1977, pg. 35576) 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 lessthancomplete 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 PartialAdjustment Model The partial adjustment model put forth by Nerlove (1956) provides an empirical recognition of input demand consistent with quasifixity. 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 Xtl = a ( x*t xt1) where xt 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 xt  xtl 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 t becomes a function of input prices and output price such that (2.2) x*t = f(Py, w1 *.... wn). where output price is denoted by py, and the wi, i = 1 to n, are the input prices. Equation (2.2) allows the unobserved variable, x*t, 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*t 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 longrun equilibrium levels. Interrelated Factor Demands Coen and Hickman (1970) extended the approach of distributedlag 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 longrun 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 CobbDouglas 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) xt xtl = B [x*t Xt1]. This specification is similar to that in (2.1), except that xt 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) xit = z bij*(x*j,t xj,t1) + jfi bi'*(x*i,t xi'tl) + xit1* 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. Nadiri and Rosen (1969) derived expressions for equilibrium factor levels using a CobbDouglas production function in a manner similar to Coen and Hickman (1968). However, they failed to consider the implied crossequation 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 bij is impossible. Neither Coen and Hickman, or Nadiri and Rosen provide a distinction between variable and quasifixed inputs. All inputs are treated as quasifixed 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 quasifixed input. Lucas (1967) and Gould (1968) extended this model to an arbitrary number of quasifixed inputs. However, these extensions are limited by the nature of adjustment costs external to the firm. Thus, the potential interdependence of adjustment among quasifixed inputs is ignored. Treadway (1969) introduced interdependence of quasifixed inputs by internalizing adjustment costs in the production function of a representative firm. The firm foregoes output in order to invest in or adjust quasifixed inputs. Assuming all inputs are quasi fixed, the underlying structure of this model is shown by (2.8) V = max f ert(f(x,x) p'x) dt, where V represents the present value of current and future profits, x is a vector of quasifixed inputs, and x denotes net investment in these inputs. The vector p represents the user costs or implicit rental prices of 29 the quasifixed inputs normalized by output price. The current levels of the quasifixed inputs serve as the initial 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 quasifixed inputs are given by (2.9) [fkk]i + [f.xJk = [f (x,*)] + r[f.(x,k)] p. Treadway assumed the existence of an equilibrium solution to (2.9), where x = x = 0, in order to derive a system of longrun or equilibrium demand equations for the quasifixed 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 quasifixed 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 firstorder derivatives of 2 Additionally, the optimality of x depends on a system of transversality conditions where rt lim e [f] = 0, and a Legendre condition that [f..] negative semidefinite (Treadway, 1971 p. 847). xxfK 30 the production function, derivation of input demands from (2.9) involve secondorder 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 welldefined 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 (7r (p,py)/apj = x* (p,py)) or from the cost function by Shephard's Lemma (aC(p,y)/apj = 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, MorrisonWhite, 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) 7r(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 shortrun 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 quasifixed factor stock and net investment. Current revenues are 7(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 7, quadratic in (x,x), can be hypothesized as (2.11) IL(w,x,k)= a0 + a'x + b'x + l/2[x' x'] CA x where ao, the vectors a, b, and matrices A, B, and C will be dependent on w in a manner determined by an exact specification of 7(w,x,k). 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 quasifixed 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 quasifixed factors and net investment of the form (2.14) L*(x,p,w,r) = !w(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 quasifixed factor has this matrix been expressed explicitly in terms of the parameters of the profit function, where M = r/2  (r2/4 + (A + rC)/B)1/2, where A, B, and C are scalars. In order to generalize this methodology to more than one quasifixed 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 quasifixed factors such that the implied adjustment matrix is diagonal. This facilitates expression of net investment demand equations for more than one quasifixed 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 (2.16) JO(K,p,w) = max.f e [F(L,K,K) w'L p'K] dt 0 L > 0, K 0 subject to K(O) = KO > 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 quasifixed 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 KO is the initial quasifixed input stock. J(Ko,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 quasifixed inputs and net investment in the positive orthant; F, FL, and FK are continuously differentiable. T.2. FL, FK > 0, Fk > 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 quasifixed inputs, (K,p,w), is twice continuously differentiable. T.5. *p(K,p,w) is nonsingular for each combination of quasifixed 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 quasifixed inputs that is globally stable. Condition T.1 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 Fk. 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 function. 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 quasifixed 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 = KO 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 JKK). 39 The static representation of the value function in (2.17) also permits derivation of demand functions for variable inputs and net investment in quasifixed 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' + JwKK'*, and differentiation with respect to p yields a system of optimal net investment equations for the quasifixed inputs, (2.20) K*(K,p,w) = JpK1.(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 shortrun changes or adjustments in the demand for quasifixed inputs. The system is simultaneous in that 40 the optimal variable input demands depend on the optimal levels of net investment, K*. In the shortrun, when K* f 0, the demand for variable inputs is conditional on net investment and the stock of quasifixed 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 quasifixed 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 realvalued, boundedfrombelow function defined in prices and quasifixed inputs. J and JK are twicecontinously differentiable. 41 V.2. rJK + P JKK(k*) JK > o, J0O as k*>0. V.3. 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 q onto the domain of F. V.4. The dynamic system K*, K(0) = KO, 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 R(p,w), a globally steady state also in the domain of J. V.5. JpK is nonsingular. V.6. For the element (K,p,w) in the domain of J, a minimum in (2.18) is attained at ( if (K,L)=(K*,L*). V.7. The matrix Kw Lp is nonsingular for 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.1. Condition V.2 reflects in (p,w) the restrictions imposed on the marginal products of the inputs, FL and Fk, 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 =X " 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 firstorder conditions are sufficient for a global minimum in (2.18). Epstein (1981) has demonstrated that if JK 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 quasifixed 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 quasifixed 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 quasifixed 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 quasifixed inputs to their equilibrium levels. Noting the potential number of parameters and nonlinearities 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'GlK. This form yields JpK = G1 and Jp = hp(p,w) + GlK. 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 X(p,w) at K*=0, (2.24) K(p,w) = [r + G]1G[rhp(p,w)]. Multiplying (2.24) by [r + G] and substituting directly in (2.23) yields (2.25) K*(K,p,w) = [r + G]R(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 JK 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 logquadratic in normalized prices and quadratic in the quasifixed inputs. The specific form of the value function J(K,p,w) is thus given by (3.1) J(K,p,w) = ag + a'K + b'log p + c'log w + 1/2(K'AK + log p'B log p + log w'C log w) + + log p'D log w + p'G'lK + w'NK + p'G'VKT  W'VLT where K = [K, A], a vector of the quasifixed inputs, capital and land, p = [Pk, Pa] denotes the vector of normalized (with respect to output price) prices for the quasifixed 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 py, log pm]. T denotes a time trend variable. Parameter vectors are defined by a = [aK, aA], b = [bk, ba, c = [cw cm), VK = IVK, VA], and VL = [vL, vM]. The vectors VK and VL are technical change parameters for the quasifixed 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 B = [bkk bka C = c wm , aAK aAA bak baa cmw CmJ D = dkw dkm N = nKw nKm and G1= gKK KA. daw dam nAw nAmj AK AA Let G = [gKK gKA The matrices A, B, and C are 19AK 9AA 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 unrestricted 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 G1 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 firstorder 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 secondorder 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 Eauations Utilizing the generalized version of Hotelling's Lemma in (2.20), the demand equations for optimal net investment in the quasifixed 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), A*(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 quasifixed factor prices. The specification of G1 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 quasifixed factors, is determined by the relative input prices and the initial levels of the quasifixed factors, as evidenced by the presence of K in (3.2). The premultiplication by G (G = JpK1 from (2.20)) 50 yields a system of net investment demand equations that are nonlinear in parameters. The technical change component for the quasifixed 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 shortrun 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 shortrun 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 shortrun variable input demand equations depend not only on the initial quasifixed 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 A1. Longrun Demand Equations In the dynamic model, the quasifixed inputs gradually adjust toward an equilibrium or steady state. The longrun 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 longrun or steady state demands for the quasifixed inputs are derived by solving (3.2) for K when K*(K,p,w) = 0. The longrun demand equation for the quasifixed factors is thus given by A (3.4) K(p,w) = [I + rG1]1[rp1(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 shortrun demand equations for the variable inputs in (3.3) are conditional on K and K*, substitution of K(p,w) for K and K*(K,p,w) = 0 in the shortrun 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 longrun variable input demands are no longer conditional on net investment, but are determined by the longrun levels of the quasifixed inputs. The individual longrun demand equations for all inputs are presented in Appendix Table A2. Shortrun Demands The shortrun variable input demands were presented in (3.3). The variable input demands are conditional on the initial levels of the quasifixed inputs and optimal net investments. The shortrun demand for the quasifixed 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 Kt11 where K*t = [K*t, A*t], the vector of quasifixed 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) = Kt1 + K*t(K,p,w), 53 where K*t(K,p,w) is the optimal demand for the quasi fixed inputs in period t, Kt1 is the initial stock at the beginning of the period, and K*t is net investment during the previous period. The shortrun demand equations for the quasifixed inputs are optimal in the sense that the level of the quasifixed input, K *t, is the sum of the previous quasifixed input level and optimal net investment during the prior period. Returning to (3.2), the shortrun demand equation for the quasifixed input vector can be written (3.8) K*t(K,p,w) = G[rpl(b + B log p + D log w)] + rVKT + [I + r + G]Ktl, where the time subscripts are added to clarify the distinction between shortrun demand and initial stocks of the quasifixed inputs. The individual shortrun demand equations for the quasifixed inputs are presented in Appendix Table A3. 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. Rewriting (3.2) and multiplying both sides of the equation by G1 yields (3.9) Glk*(K,p,w) = rp"1(b + B log p + D log w) + rVKT + [I + rG'1]K. Multiplying both sides by [I + rG1]1 and noting that [I + rG1]1 = [r + G]1G, then (3.9) can be written as (3.10) [r + G]1k*(K,p,w) = [I + rG1]lr1^(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 longrun quasifixed 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) A* = gAK(K X) + (r + gAA)(A A). 55 Thus, gKA 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 gKA = gAK = 0. Independent rates of adjustment indicate that the rate of adjustment to longrun equilibrium for one quasifixed factor is independent of the level of the other quasifixed factors. The hypothesis of an instantaneous rate of adjustment for the quasifixed 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 + gKK = r + gAA = 1, in addition 56 to gKA = =AK = 0. If both inputs adjust instantaneously, the adjustment matrix takes the form of a negative identity matrix. Capital and land would adjust immediately to longrun 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 quasifixed 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.1., 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 ag, 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 longrun 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 longrun 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]l 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 quasifixed inputs only partially adjust to relative price changes along the optimal investment paths, and the long run, where quasifixed 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 ownprice elasticities are an indication that some inputs contribute not only to production but to the adjustment activities of the firm. Thus in the shortrun, the firm may employ more of the input in response to a relative price increase in order to facilitate adjustment towards a longrun equilibrium. However, this does not justify a positive ownprice elasticity in the longrun. This same contribution to the adjustment process may also indicate shortrun effects which exhibit greater elasticity than the long run. The firm may utilize more of an input in the shortrun in order to enhance adjustment than in the longrun in response to a given price change. Shortrun 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,pji is (3.15) E: + + A. L,p * 3 Pj 9K apj 9A a p. L The shortrun 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 shortrun demand for the quasi fixed inputs. 59 The shortrun price elasticity for a quasifixed input is obtained from (3.7). The shortrun demand elasticity for capital with respect to a change in the price of the jth input, sK,pj is (3.16) E s K pj. pj ap K The shortrun elasticity of demand for a quasifixed input depends only on the direct price effect in the shortrun demand equation. The longrun 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 1 L +DL 3 K + L D A pj (3.17) gLlp = + + . Spj DK a p. 3 A D p. L The longrun elasticity of a variable input is conditional on the effect of a price change in the equilibrium levels of the quasifixed inputs. The longrun demand elasticity for a quasifixed input is determined from (3.4). The longrun 60 elasticity of demand for capital with respect to the jth input price is (3.18) = _ a Pj In contrast to the shortrun demand for a quasifixed input, where the shortrun demand for one quasifixed input is determined in part by the level of the other quasifixed input, the longrun 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. 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 wornout equipment. Eventually, wornout 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 = lit + (1 6i)Ai,t1, 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 Bl). 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) / Qit, where Qit 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 Eit, 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 B2. Labor Beyond the additional parameters needed in the empirical model to treat labor as quasifixed, 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 owneroperator 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 owneroperator 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 owneroperators value their own time as they would hired labor. While this may seem inappropriate, the relative magnitude of hired labor to owneroperator 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 B3. 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 quasifixed. 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 B4. 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 B5. 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 = Ki,t Ki,tl, where kit is net investment in the quasifixed input i during period t, Ki,t is the level of the input stock in place at the end of period t, and Kit1 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 eliminated. 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 quasifixed input index between two time periods as a measure of net investment, this problem can be at least partially avoided. 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 B6. The base year for the quantity and price indices is 1977. Figure 31 depicts the quantity indices for the 19491981 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 threeandonehalf 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. Quantity 360 340 320 300 280 260 240 220 200 180 160 140 120 100 80 60 4 0 1 1 1 1 1 1 1 1 ,1 1 1 1 1 1 1 1 1 1 ,1 1 1 1,I T  50 55 60 65 70 75 80 Year Capital + Land Labor A Materials Figure 31. Observed Input Demand for Southeastern Agriculture, 19491981. 72 Turning to the normalized input prices, Figure 32 charts these input prices over the period of analysis. Not surprisingly, 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 sixfold, 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 B5 shows only a Capital, Land Index 1.81 50 55 60 65 70 75 80 Year D Capital Price Land Price Labor Price Materials A Price Figure 32. Normalized Input Prices for the Southeast, 19491981. 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 longrun levels of demand are obtained from the parameters of the estimated equations and compared to observed input demand. Estimates of short and longrun 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 quasifixed 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 1 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, quasifixed 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 quasifixed 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 41. Thirteen of the twentysix 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. Table 41. Parameter Estimates Treating Materials and Labor as Variable Inputs, Capital and Land as QuasiFixed. Parameter Standard Parameter Estimate Errora 1913.160 423.746 677.077 343.822 2472.608 243.214 123.888 105.878 233.275 406.767 49.625 98.263 152.451 111.878 1.900 1.879 0.826 0.159 18.639 4.155 17.267 23.045 0.588 0.490 0.023 0.213 bkk bka baa cww cwm Cmm dwk dwa dmk dma nwK nwA nmK nmA VK VA VW VM 9KK gKA gAK gAA 496.327* 229.533 212.572* 658.473 478.233* 153.757 98.104 123.786 72.084* 636.244 110.041 102.696 79.123 74.397 0.731* 2.283 0.309* 0.530 5.808* 0.707* 1.507* 8.312* 0.160* 0.242* 0.015 0.056* times its a indicates parameter estimate two 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 JpK1 and stability of M are examined first. The nonsingularity of JpK1 is determined from the estimates of the elements of G, as JpKl = G. The determinant of G is 0.334, thus satisfying the nonsingularity of Jpl1. 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 (KK). The elements of the matrix of secondorder 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 thirtyone of the thirtythree observations (See Appendix C1 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. 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 materialusing relative to labor, and capital using relative to land. While some studies of U.S. agriculture have found technical change to be labor and landsaving (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 handharvesting, 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 42 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 values of K* correspond fairly closely to observed net investment. Observed capital stocks and the estimated shortrun 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 longrun demand levels for capital presented in Figure 41 further illustrate this convergence. However, one should note the adjustment Table 42. Comparison of Observed and Estimated Levels of Net Investment and Demand for Capital. Net Investment Capital Demand K K* K K*(K,p,w) R(p,w) observed optimal observed short long Year run run 49 8.00 5.00 6.00 5.00 2.00 1.00 2.00 2.00 1.00 0.00 2.00 2.00 1.00 1.00 1.00 0.00 1.00 1.00 3.00 1.00 0.00 1.00 4.00 2.00 3.00 4.00 2.00 2.00 1.00 6.00 6.00 5.00 2.00 7.13 5.22 5.69 5.53 4.65 0.29 0.94 0.68 1.67 0.42 1.27 0.92 0.33 0.44 0.49 0.44 0.85 0.92 1.22 0.22 0.23 0.58 4.80 2.52 3.37 3.75 0.38 1.67 0.96 2.07 0.12 2.54 2.58 52.00 60.00 65.00 71.00 76.00 78.00 79.00 81.00 83.00 82.00 82.00 84.00 82.00 81.00 80.00 81.00 81.00 82.00 83.00 86.00 87.00 87.00 86.00 90.00 88.00 91.00 95.00 97.00 99.00 100.00 106.00 112.00 107.00 53.81 60.34 65.60 71.56 75.98 75.13 76.56 78.59 79.24 79.32 80.69 81.48 80.43 79.68 78.90 80.03 80.36 81.44 82.83 85.19 86.05 86.70 88.55 91.06 89.82 93.28 95.09 97.86 99.44 101.22 105.88 110.29 108.57 16.58 17.33 22.46 29.16 32.54 26.29 28.64 33.48 34.22 40.18 47.80 49.57 52.08 54.34 56.71 60.29 62.11 63.89 68.77 70.21 72.96 76.50 84.92 84.73 87.85 95.12 93.03 96.18 97.72 101.59 105.04 106.43 109.84 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 43 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 longrun demand exceeds observed and shortrun demand for land stocks until 1961, the equilibrium level of demand falls at a faster rate than the shortrun 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 longrun 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 longrun demand levels for land are presented graphically in Figure 42. Capital Index 120 110 100 90 80 70 60 50 40 30 20 10 50 55 60 65 70 75 80 Year Observed u Demand Estimated Estimated + Shortrun 0 Longrun Demand Demand Figure 41. Comparison of Observed and Estimated Demand for Capital in Southeastern Agriculture, 19491981. Table 43. Comparison of Observed and Estimated Levels of Net Investment and Demand for Land. Net Investment Demand for Land A A* A A* (K,p,w) A(p,w) observed optimal observed short long Year run run 3.45 1.14 0.76 0.19 0.95 0.57 2.86 3.05 3.63 3.82 3.44 2.48 2.86 2.86 3.63 1.72 1.53 1.53 1.91 2.29 1.91 1.34 0.95 0.76 1.15 0.57 0.19 1.23 1.04 1.23 0.87 0.22 1.62 2.64 142.32 2.11 1.01 0.16 0.92 1.85 2.68 3.18 3.52 3.02 2.70 2.83 2.77 2.65 2.24 2.21 2.21 2.34 1.48 2.19 1.74 1.45 1.63 1.56 0.87 0.38 0.28 0.61 0.58 0.48 0.38 0.48 0.48 145.77 146.91 147.68 147.87 146.91 146.34 143.48 140.43 136.80 132.99 129.55 127.07 124.21 121.35 117.72 116.00 114.48 112.95 111.04 108.75 106.85 105.51 104.56 103.79 102.65 102.08 102.27 101.04 100.00 98.77 99.63 99.85 146.78 150.01 149.98 149.35 148.79 147.27 145.87 142.29 138.86 135.44 131.66 128.00 125.47 122.61 120.04 116.35 114.60 112.94 112.14 109.49 107.59 105.87 104.11 103.31 103.09 102.27 101.90 101.76 100.54 99.52 98.37 99.20 99.32 193.99 198.58 189.99 177.78 172.26 170.28 163.14 152.68 146.27 141.64 135.40 129.24 125.47 121.53 119.86 114.83 112.55 109.86 112.78 104.95 105.01 103.60 96.77 97.74 99.82 99.84 101.64 99.29 97.83 96.33 95.52 96.70 95.36 Land Index 200 50 55 60 65 70 75 80 Year Observed D Demand Estimated Estimated + Shortrun 0 Longrun Demand Demand Figure 42. Comparison of Observed and Estimated Demand for Land in Southeastern Agriculture, 19491981. 88 The observed use of labor and estimated short and longrun demands for labor as presented in Table 44 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 41 and the shortrun demand equation for labor in Table A1, capital stocks slightly reduce the shortrun demand for labor. Capital investment increases the shortrun demand for labor. This indicates that labor facilitates adjustment in capital. The effect of land stocks on the shortdemand for labor indicates an increase in land increases the shortrun 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 longrun demand for labor depends on the equilibrium levels of capital and land to the same degree that shortrun labor demand depends on capital and land stocks. An increase in the equilibrium level of capital decreases the longrun demand for labor. Conversely, an increase in the equilibrium level of land increases the longrun demand for labor. Disequilibrium in the quasifixed inputs could potentially cause a divergence in the short and long Table 44. Comparison of Observed and Estimated Short and LongRun Demands for Labor. Labor Demand Labor L* (K,p,w) L(K,p,w) Year observed shortrun longrun 49 351.195 320.518 336.653 311.155 295.817 267.331 265.139 239.841 205.578 192.231 196.813 189.442 181.873 179.681 175.299 161.355 146.813 136.454 138.048 128.685 129.283 122.908 120.319 114.542 113.147 109.761 106.375 103.785 100.000 96.016 92.430 95.817 91.434 351.812 336.175 320.176 306.751 294.246 270.240 253.921 232.346 217.464 211.425 204.972 189.117 181.008 173.914 169.490 155.314 148.204 138.309 143.208 126.861 124.008 121.029 115.411 108.599 113.642 116.091 107.527 105.591 101.159 99.784 92.872 89.070 95.881 351.592 339.369 319.446 302.778 290.103 273.312 253.412 230.377 219.113 210.896 200.875 188.722 177.777 170.290 166.351 151.941 143.835 133.380 139.392 123.177 122.116 117.851 102.439 100.668 105.218 107.571 106.405 101.054 98.056 94.424 92.140 93.235 89.346 90 run demand for labor. However, the magnitudes of the parameter estimates associated with the dependence of labor demand on the quasifixed factors are small. Thus, the shortand longrun demands for labor are similar. This is also true for the materials input as shown in Table 45. The degree of correspondence of observed and short and longrun demands for materials is even greater than for labor. The short and long run demands for materials depend on quasifixed 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 quasifixed 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, Table 45. Comparison of Observed and Estimated Short and LongRun Demands for Materials. Materials Demand Materials M*(K,p,w) M(K,p,w) Year observed shortrun longrun 44.237 42.623 49.666 50.194 51.917 51.345 54.943 55.006 54.630 57.657 63.332 63.839 66.081 68.409 69.281 76.910 78.817 79.390 81.210 82.814 87.717 94.124 98.432 101.150 101.801 97.131 86.274 96.985 100.000 105.649 114.375 105.367 112.323 44.877 44.431 47.821 50.111 52.350 49.255 53.015 55.332 54.572 58.220 64.191 63.245 67.067 69.178 70.373 75.303 78.569 79.980 83.445 85.197 88.000 93.462 97.382 98.821 94.462 96.437 92.328 100.289 101.008 107.951 109.014 103.573 114.331 41.464 43.123 45.440 47.166 49.974 51.308 53.908 56.009 57.477 59.693 63.653 64.828 67.159 69.021 70.176 75.007 77.764 78.993 82.368 84.826 88.212 92.798 91.829 95.828 90.681 92.264 91.928 98.369 99.833 105.538 108.972 106.169 111.323 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 longrun and a low equilibrium for capital given the low initial prices of land and labor. Intuitively, these low prices may be attributed to a sharecropper 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 quasifixed inputs adjust only partially to relative price changes along an optimal investment path, and the long run, where quasifixed input stocks are fully adjusted to equilibrium levels. 93 Average shortrun gross elasticities for capital, land, labor and materials for selected periods are presented in Table 46. The short run ownprice elasticities were negative at each data point for all inputs. Shortrun elasticity estimates for each year are presented in Appendix Tables D1 D4. As noted earlier, positive shortrun ownprice 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 shortrun complements. This is not surprising given the laborintensive crops that characterize production in the Southeast. Labor and capital and labor and materials are shortrun substitutes. Materials and capital are shortrun complements while materials and land are substitutes. Finally, capital and land are shortrun substitutes. The shortrun ownprice elasticities for all four inputs are inelastic. Given capital and land are quasifixed inputs and the shortrun demands for labor and materials are conditional on these inputs, inelastic shortrun demands should be expected. The shortrun own price elasticity of land is the most inelastic of the four inputs, ranging from an average of 0.08 in 19491955 to 0.02 in 19761981. Capital generally has next lowest ownprice elasticity. The 