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A‘ III, uh)?- II:I I I J ,J'JJJ' J 4.3., 5, , I .r- . 2.31wé5 “.5""\VFIOIAI ‘ufllv 1 0' _ A. v Q ..~ '7 TH'ESYE lIHIIIIIINIIHIQIIIIINIINiall/III!!!)HIIIHIIIII 1mm L ..... a 312 31 04 2604 --.—-1~-‘ «an... - This is to certify that the dissertation entitled A TRANSLOG COST FUNCTION STUDY OF PURCHASED INPUTS USED IN U.S. FARM PRODUCTION presented by HS IN-HUI HSU has been accepted towards fulfillment of the requirements for PH.D. degreem AGRICULTURAL ECONOMICS Robert Gustafson Major professor Date /- 30/ MS U i: an Affirmative Action/Equal Opportunity Institution 0- 1 2771 33:9 '9 Smog if, MSU RETURNING MATERIAL§: Place in book drop to LJBRARJES remove this checkout from —_ your record. FINES will be charged if book is returned after the date stamped below. I ‘ r ‘ I a. , ’ ‘- AFié'fggfggs é—ggsy “”3 3 3 20023 A TRANSLOG COST FUNCTION STUDY OF PURCHASED INPUTS USED IN U.S. FARM PRODUCTION BY Hsin-Hui Hsu A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Economics 198h ii Copyright by Hsin-hui Hsu l98b ABSTRACT A TRANSLOG COST FUNCTION STUDY OF PURCHASED INPUTS USED IN U.S. FARM PRODUCTION By Hsin-Hui Hsu Agricultural economists encounter several recurring difficulties when they study the demand structure for purchased farm inputs. A partial list of potential difficulties includes interdependent relationships between inputs, the linear homogeneity condition assumed in Euler's Theorem, the limitations of existing functional forms, and the impracticability of obtaining insight into the input demand structure when a production function is unknown. These probl-ms indicate that an improved analytical framework could lead to resolutions which would enable the public and the private sectors to make better decisions. This paper presents an empirical demand estimation system by using duality theory and a transcendental logarithmic cost function to measure the interrelationships between U.S. farm inputs. By using time-series data (l9lO-l98l) on five input subgroups and applying Zellner's seemingly unrelated regression technique, a complete set of demand equations can be estimated in quantity dependent form. A comparison of own price demand elasticities and pairwise elasticities of substitution was made between these and other research results. Hsin-hui Hsu Capital was found to be a substitute for all other inputs. Labor is a substitute for capital, feed, seed. livestock purchased. and miscellaneous inputs, but there is a weak complemetary relationship between labor and fertilizer. The specified translog cost function has passed the tests of linear homogenity and monotonicity regularity conditions implied by the duality theory. However, it fails the test of the symmetry condition and the concavity condition is indefinite. Another important conclusion is that factor-augmenting technological change in U.S. agriculture has been mainly labor-saving and capital-using. This confirms previous empirical studies. A 2.92 annual growth rate of agricultural productivity sustained over the last seven decades is quite impressive. Finally, an attempt was made to investigate the question of economies of scale, but the results were inconclusive. ACKNOWLEDGMENTS I wish to express my sincere gratitude to Professor Robert Gustafson, my dissertation and guidance committee chairman, for his invaluable advice, encouragement, and guidance, kindness, and understanding during the difficult time when this study was being undertaken. Special appreciation is extended to Professors John Ferris, Lester Manderscheid, Anthony Koo, and Peter Schmidt, for their ideas, suggestions, and criticisms in the development and completion of this dissertation. I wish to thank the Departments of Agricultural Economics and Mathematics at Michigan State University for continued financial support. I would like to acknowledge to Drs. Dale Heien (University of California - Davis), James Chalfant (University of California - Berkeley), and Douglas Young (Washington State University) for their valuable encouragements and suggestions. I must extend my appreciation to my fellow graduate students, William Rockwell, Douglas Krieger, John Staatz, Larry Lev, and David Trechter for their patience and diligence in editorial assistances as well as comments. Also my great editor, Ms. Tammy Jantz. My great debt is owed to my wife, Jing-Ling, who sacrificed much and encouraged me at every step. I also would like to thank the members of my family for their moral support through the years. 111 Table of Contents Chapter I. Introduction . . . . l.l Problem Statement . . . . . . . . . . I. Research Objectives and Procedures . . lu3 Summary and Dissertation Organization . . . Input Demand in a Theoretical Setting 2.l Duality Theory in Application . 2.l.l What is Duality Theory . 2.1.2 To Dual or Not to Dual . . . 2.l.3 Cost Function vs Profit Function . 2.2 'Flexible Functional Forms . 2.2.l 2.2.2 The Translog Cost Function . Derivation of Elasticity Measures . Technological L‘ange . . . . . Economies of Scale Summary . . . . . . NNNN 0‘er Empirical Input Demand System Estimation . 3.l Input Definitions and Data Sources. . 3.l.l Inputs 3.l.2 Output . . . 3.l.3 Input Prices . WW I 0 WM Testing Hypotheses . . . . . . . . . . . . 3 3.l Symmetry . . . . . . . . . . . . . 3.3.2 Homogeneity . . . . . . . . . . 3 3.3 Technological Change . . . . . . . . . 3.h Elasticities of Demand and Substitution . 3.h.l Own-Price Deamnd Elasticities . . . 3.h.2 Elasticities of Substitution 3.5 Economies of Scale . . . . . . . . . . 3.6 Summary . . . . . . . . . . iv Choice among Flexible Functional Forms . Statistical Model Specification and Procedure . page Il l2 IA lh . I5 . 22 . 2h . 25 . 29 . 3I - 35 . Al . Ah . A6 . A7 . 5O . 5i . 5h . 63 . 76 . 80 . 9] A. Summary and Conclusion . . . . . . . 97 h.l Summary and Implication of Results . . 97 h.2 Future Research Needs . . . . . . . . 98 Appendices A Translog Function as a Second-Order Approximation of Any Function . . . . . . . . . . . . . . . . . . .lOl B The Second Order Partial Derivatives of the Translog Cost Function . . . . . . . . . . . . . . . . . . . .lO3 C Divisia Index . . . . . . . . . . . . . . . . . . . .l06 D A Computer Program for Calculating the Eigenvalues. .llO Bibliography . . . . . .llh Table l.l 1.2 3.l 3.2 3.3 3.1. 3-5 3.6 3-7 3.8 3-9 C.l 0.] LIST OF TABLES page Index Number of Total Farm Inputs and Inputs in Major Subgrbups, U.S., Selected Year, 19lO-8O (l967-IOO). . . . . . 3 Index Number of Farm Output, Input, and Productivity, Selected Years, l9lO-80 (l967=lOO). . . . . . . . . . . . . 5 Common Linear-in-Parameters Flexible Functional Forms . . . . 28 Correlation Coefficients of Interest Rates (l965-8l). . . . . 53 Estimated Coefficients of Translog Function, l9lO-8I, with Homogeneity and Symmetry Restrictions. . . . . . . . . . 59 Results of Alternative Model Specification. . . . . . . . . . 65 Statistical Test Results of Symmetry by Previous Researchers on Flexible Functional Forms. . . . . . . . . . . 68 Estimated Measures of Annual Technological Advancement . . . 75 The Own-Price Input Demand Elasticity and Allen's Partial Elasticity of Substitution under Various Technology Specifications. . . . . . . . . . . . . . . . . . . . . . . . 78 Own-Price Demand Elasticity of Demand for Inputs, l9lO-8l. . 8l Allen's partial Elasticity of Substitution between Inputs, l9lO-8l. . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Alternative Estimates of Scale Economies, with and without Scale Trend. . . . . . . . . . . . . . . . . . . ... . . . . 93 Constructed Divisia Price Index for Feed, Seed, and Livestock Purchased, l9lO-8l (l977-l.0). . . . . . . . . . .l08 The Computed Characteristic Roots of the Hessian Matrix Based on Model 9. . . . . . . . . . . . . . . . . . . . . .ll2 vi LIST OF FIGURES page 2.l The Cost Function and the ”Passive” Cost Function. . . . . . 2l vii CHAPTER I Introduction A farm produces output from various combinations of inputs. Inputs are materials and factor services fed in at one end of the production process and used in the process of production. Farmers use land, labor, capital equipment, and other inputs to grow crops and to raise livestock. In economics, it is difficult to describe the characteristics of farm production and it is also not easy to measure farm input demand when interaction between inputs exists, especially if we want to measure the aggregate input demand or describe the nature of production at the industry level. The agricultural industry is a system consisting of suppliers of farm inputs, farmers, and various businesses which are engaged in buying, processing and distributing farm products. The farm input demand system is part of the agricultural sector and is interlinked with, and affected by, industrial developments in the rest of the economy. The U.S. farming system is also of central importance in an increasingly interdependent world agricultural economy. The combination of scientific industrialization, increasing interdependence and a great deal of production uncertainty creats the need for an improved analytical system, that will help farmers, businessmen and public officials make better judgements on production planning and policy decisions. This paper broadly investigates the demand for inpUts and the research results are potentially valuable for various decision makers. Estimation of the elasticities of substitution between inputs and the own-price elasticities of input demand were of primary interest in this study. The subject-matter of elasticity measurements has many policy implications, such as the following: Is labor a substitute for all other inputs? and to what degree? Is the substitutability between farm labor and machinery declining? As the results emerge from the empirical study, the structure of U.S. farm production can be analyzed. For instance, a slowing down of the rate of mechanization and an intensifying use of agricultural chemicals may change farmers' long-term investment strategies. I am not suggesting that this paper is the only study which deals with the purchased inputs used in U.S. farm production. Agricultural economists have long recognized the importance of farm input markets. Although there are thousands of economic studies of inputs in U.S. farm production (see references in Dahl and Spinks, l981), one rarely finds a study which adequately handles the relationships among different groups of inputs and/or within a specific group of inputs. Empirical studies have derived numerous elasticity measures for single farm inputs. However, to the extent that changes in the quantities of inputs occur simultaneously, the estimates of elasticity measures obtained directly from single-equation models are likely to be biased. Furthermore, the basic relationships of demand-supply-price structure, in either an optimization or a behavioral context, need quantitative reestimation as factor and product markets, technology, and institutions change. This is one of the reasons that a new empirical study of the farm input demand system is important. Let us briefly review U.S. farm input utilization and its economic implications. A major change in the structure of U.S. farm inputs has been the shift from inputs of farm origin (e.g., number of acres and labor), to purchased inputs of nonfarm origin [Table l.l]. Table l.l Index Numbers of Total Farm Inputs and Inputs in Major Subgroups, U.S., Selected Years, l9lO-8I (l967-IOO). Total Inputs Non- Farm Real Farm Feed Year All purchased Purchased Labor Machinery estate chemicals seed I9I0 86 l58 38 32I 20 98 5 l9 I920 98 I80 A3 3AI 3I I02 7 25 I930 IOI I76 50 326 39 IOI I0 30 I9AO I00 I59 58 293 A2 I03 l3 A2 I950 IOA ISO 70 2I7 8A I05 29 63 I960 IOI II9 86 IA5 97 I00 A9 8A I970 I00 97 I02 89 I00 IOI II5 IOA I980 I06 85 I27 65 I28 96 I7A II9 I98I I05 89 I2A 63 I2I 98 l83 II3 Source: USDA, Economic Indicators of the Farm Sector: Production and EfficiencyiStatistics, I980. Pp. 6A-65, and I982, p. 59. This change is quite significant and continuing. Since the volume of farm real estate has remained stable, the major adjustment has been substitution of agricultural chemicals (e.g., fertilizer), machinery, and other capital inputs purchased from the nonfarm sector. Farm machinery, feed, and fertilizer were the most important in value, accounting for more than 60% of total farm inputs (Dahl and Hammond, I977). The volume of nonpurchased inputs declined by nearly one-half from l9lO to I981. Meanwhile, the volume of purchased inputs more than tripled. Labor declined drastically. Part of the decline in labor use is attributable to shorter working hours on the farm, but most is due to outmigration. Farm numbers have declined continuously since l9lO, and still appear to be declining. All other input categories in Table l.l increased except real estate. The land resource base is essentially limited, although expansion can be achieved by reclamation of wetlands or irrigation of irrigable land. Both produce only minor changes in cultivatable land. In spite of the possibilities for increasing cropland, the U.S. has actually reduced cropland use between l93O and the early l970's. The total acreage.of cropland decreased from 382 million in I930 to 332 million in I970. (Clearly the number of "real estate" used by the USDA in Table l.l is not equivalent to either total cultivatable cropland or to cropland used, but the three are related.) However, the composite measure of all inputs, calculated by the USDA, indicates that the index of total inputs has remained remarkably stable since I930. Out-migration from farms and reduced cropland use in agriculture have been offset by substitution of other inputs. Mechanical power has substituted for manual labor and animal power. Fertilizer and lime have substituted for land inputs. But because a given dollar value of purchased inputs was more productive than a given dollar value of farm originated inputs (e.g., labor) which was replaced, output more than tripled from l9lO to l98l [Table l.2]. Although the long-term growth of output has been strongly upward, it is somewhat erratic. Output was essentially stable between I920 and I930. The largest decennial gains have been made since l9AO. Table l.2 Index Number of Farm Output, Input, and Productivity, Selected Years, l9lO-l98l (l967-IOO). Farm Output Production Inputs Productivity (O/I) I9I0 A3 86 50 I920 SI 98 52 I930 52 IOI 5I I9AO 60 I00 60 I950 7A IOA 7I I960 9I IOI 90 I970 IOI I00 I02 I980 I22 I06 II5 I98I IAZ I05 I3A Source: USDA, Economic Indicators of the Farm Sector: Production and Efficiency Statistics, I980, p. 77. and I982, p. 7l. In the above review of the U.S. farm input structure, we can not avoid the important issue of technological change. Many agricultural production economists speak of efficiency and gains in efficiency. Commonly used measures of farm productivity are output per acre, output per worker-hour, and output per farmer. These measures are somewhat misleading if they ignore the contribution from the nonfarm sector. The total farm productivity index, which is presented in the last column of Table l.2, probably shows the increase in farm productivity more accurately. The productivity index is defined as the ratio of the total output index to the total input index expressed as a percent. These figures tell us that farm productivity has more than doubled since l9lO. However, there is no consensus on the meaning of this farm productivity index. Some argue that the index is basically correct, and some argue that the index underestimates the overall gain of productivity. One of the important issues is how depreciation is calculated. Whereas the farm productivity index can serve as an indicator of technological change, there are four basic ingredients in the analysis of technological advancement (Yotopoulos and Nugent, I976): (l) the technical efficiency of production, (2) the scale of operation of production, (3) the bias of technological change, and (A) the elasticity of substitution. An increase in the technical efficiency of production refers to a reduction in the quantity of all factors used in producing the same unit of output, or equivalently, an increase of the quantity of output with inputs held constant. Increasing, decreasing, or constant returns to scale depend on whether total output increases more, less, or equally in proportion to the increase in all inputs. Bias in technological change may be thought of a change in the ratios of marginal products of factor (at given factor levels). The fourth characteristic of a technology -- the elasticity of substitution -- is the ease with which one input can be substituted for another. The combination of these four elements gives us a composite picture of the changes in technology which are reflected in the actual production process. The increase in farm production can be roughly decomposed into the results of ”scale effect”, "technological change", and substitution due to exogenous price or input supply effects. Technological change incorporates technical efficiency and bias of technological change, and effects of the third category are determined by the elasticity of substitution. For example, from I970 to I980, total input use in agriculture increased approximately 6%. Total output increased by 21%. Consequently, if we assume constant returns to scale, approximately 30% (6/2l) of the increase in output was the result of increased input and 702 was attributable to technological change or substitution effects. Agricultural economists are not always in agreement with this explanation. The difference in opinion may stem from disagreement about the mix of input substitutes, scale effect, embodied and disembodied technological change, and the input costs considered. Time~series econometric models can be useful in predicting real world responses of farmers to input prices and whether, and how, these may be changing with the structure and growing commercialization of agriculture. This research aims at providing an empirical analysis of the structure of demand for U.S. purchased farm inputs.. Embodied technological changes and economies of scale are part of the analysis due to the close relationship between input demand and farm production. In production economic theory, input demand is a derived demand. Usually we consider the marginal revenue product schedule for an input, Xi, to be the firm's demand for Xi, if Xi is the only variable resource employed. There are many ways to derive an input demand function from a given technology and a given endowment of fixed factors of production in the neoclassical framework. The most popular approach is the constrained optimization approach. When a study employs the constrained maximization or minimization approach, a particular underlying production function is assumed. It is difficult to gain insight into the input demand structure when the production function is unknown. The latest developments in duality theory provide a simpler econometric approach and assure us that it is in fact theoretically sound. The duality theory is important for reasons other than mathematical elegance. One reason for the increasing popularity of the use of duality in applied economic analysis is that it allows flexibility in the specification of input demand equations and permits a close relationship between economic theory and econometric practice. l.I Problem Statements In analyzing demand for farm inputs, researchers have encountered several recurring difficulties. .The first difficulty arises from the desire to measure the interdependence among farm inputs. Simultaneous relationships and interdependence exist for all agricultural inputs. Most of the traditional empirical studies that used production function specifications adopted a form of either the constant elasticity of substitution (CES) or Cobb-Douglas (C-D) variety. This is not to suggest that the traditional functional forms are not appropriate, but only that their use in studying the details of the structure of the input demand system is restrictive. Input demands are difficult to determine when the production technology is complex. For example, the elasticity of substitution between each two factors must be identically one for a C-D production function. Although the CES function accommodates elasticities of substitution different from zero or unity, they remain constant at all levels of inputs. Flexibility of the analytical model is important for measuring the demand interrelationships. An additional advantage of the duality approach is that by basing it on the cost function instead of the production function, we can evade the problem of requiring linear homogeneity in the production function (as Euler's Theorem implies is required to exhaust the production). Indeed, the cost function will be linearly homogeneous with respect to input price, regardless of the nature of the production IO function. A more detailed discussion can be found in Chapter 2. In a large scale modelling work, for instance, the International Institute for Applied System Analysis (IIASA) and the USDA's National Inter-Region Agricultural Projections (NIRAP) model and the MSU Agriculture Model, quantity-dependent input equations are modelled as standard demand equations (Abkin, l98l). In each of these models, there is no attempt to estimate jointly a system of input demand equations. Rather, each demand equation is estimated separately by a single-equation method. For example, in the IIASA-NIRAPZ model, the demand for hired labor is modelled as a function of price received by farmers, the farm wage, the number of farm family workers and the stock value of machinery on farms. Meanwhile, price-dependent input equations also exist in the system, bLt the prices of input items are determined simply as linear functions of the nonagricultural sector's inflation rate. The primary concern of the estimated price-dependent equations is to generate and to differentiate the total expenditures of farm production rather than to explain the demand behavior. For these reasons, serious attention has been given to develop new approaches which avoid the above-mentioned difficulties. The following section contains the explanations of why a concerned researcher wants to carry out this study and how to address the relevant issues. II l.2 Research Objectives and Procedures A model's specification is influenced by the problem definition. Therefore, the intended application of the analysis must be adequately defined. There are many problems that require information on farm inputs for their solution. No model can attempt to produce all of the information required to solve all of these problems. .To avoid an oversized research project, my ambition is limited to a modest one of providing only the information required to address a rather well defined set of problems. In view of the problems and issues presented in the previous section, the objectives of :his study are geared to two primary interests: I) Specify, describe and analyze the demand system for U.S. farm inputs by employing a flexible cost function approach. Emphasis will be placed on the own-price demand elasticities and the pattern of substitution among the inputs. 2) Examine the impact of the technological changes and investigate the economies of scale for the U.S. farm production structure. To accomplish the objectives of this study, the following research procedures are used: I) Review and summarize existing empirical studies of agricultural input demand. Particular attention is focused on studies which have employed microeconomic duality theory and the cost function approach. The so-called flexible functional form which is a closely related topic I2 to the application of the cost function approach will be scrutinized. 2) Identify and define purchased input subgroups, namely hired labor, capital, fertilizer, feed, seed, livestock purchased, and miscellaneous items. 3) Specify the theoretical and statistical models, namely, the farm input demand system derived from a translog cost function. A) Develop, refine and test the validity of the theoretical regularity restrictions of linear homogeneity and symmetry with respect to factor prices, and also the hypotheses of biased or neutral technological change. l.3 Summarxgand dissertation organization The present study was undertaken in an effort to gain a better understanding of the purchased inputs used in U.S. farm production. The study is different from previous studies on U.S. farm input demand or production structure in four ways: (I) The data were collected over a longer time period, l9lO-l98l, including the war years, than was used in other studies. Assuming a correctly specified model, then more observations imply more reliable estimates: on the other hand, it is true that using a longer time period makes it somewhat more likely that the model may not be completely and correctly specified. (2) The empirical study includes testing the validity of the regularity conditions implied by the theory, a point that was neglected by most researchers. (3) Long term factor-augmenting technological changes and their impacts on the derived elasticity measures are examined. (A) The I3 study also compares different measures of economies of scale obtained from various specifications. The plan of the dissertation is as follows. In chapter 2, the input demand relationships are examined by contrasting a primal approach and a dual formulation. The advantages of employing the dual approach are stressed and the properties of the translog cost function and other flexible functional forms are presented. Measures of elasticities of input demand and substitution, returns to scale, and biases in technological change are investigated, based on the neoclassical model of production decisions and assuming the translog cost function is adequately specified. In chapter 3, an empirical estimation is carried out by using Zellner's seemingly unrelated regression technique. Data sources, input subgroup definition, and hypotheses on the theoretical restrictions are discussed. Special attention is paid to the elasticity of substitution between inputs and the own-price demand elasticity of each input. Also, the results of estimating biased technological changes and economies of scale will be reported. Chapter A summarizes the important conclusions based on the findings of the models and further research needs are suggested. IA CHAPTER 2 Input Demand in A Theoretical Setting Presentation of the theoretical background is limited to relevant studies of demand relationships among U.S. farm input subgroups, particularily those studies which apply the duality approach. Two sets of literature are examined. The first set deals with theoretical deveIOpment and application of the duality approach. The second set deals mainly with the so-called flexible functional forms. 2.l _g§lity theory in application A firm produces output from various combinations of inputs. The "production possibilities set" of a firm is a convenient way to summarize the set of all feasible production plans. Since the production possibilities set describes all feasible patterns of input and output, this set implies a complete description of the technological possibilities facing the firm. The traditional starting IS point of production theory is the set of physical technological posssibilities, often described by a production function. The development of production theory then follows the line of a firm's operation, as the firm seeks to achieve its goals subject to limitations of its technology. The results are constructed input demands and output supplies. These demands and supplies are expressed as functions of economic variables, given the technology, and using constrained maximization or minimization. In practical application, duality theory is different from the earlier form of production theory in two aspects. First, duality theory provides a derived system of input demand equations, consistent with the maximizing or minimizing behavior of a producer, by simply differentiating a function instead of solving for the behavior functions (e.g., by the Lagrange multiplier method) explicitly. Second, duality theory reaches the “comparative static” results '(e.g., elasticity of substitution), originally deduced from maximizing behavior, effortlessly (Diewert, I97Aa, p. lO7). By using duality, the technology implied by an economic model can be tested for compatibility with a priori hypotheses. 2.l.l What is the Duality Theory? To say there is a duality between the cost and production functions means that. there exists an invertible, one-to-one relationship between these two functions. In other words, the mapping that yields the cost function from the production function and the mapping that yields the production function from the cost function are I6 mutual inverses. Diewert (I982, p. 535) describes this relationship as the following: Suppose that a production function F is given and that y-F(X) , where y is the maximum amount of output that can be produced by the technology during a certain period if the vector of input quantities X - (xl,x2,...,xn) is utilized during the period. Thus, the production function F describes the technology of the given firm. On the other hand, the firm's minimum total cost of producing at least the output level y given the input prices W=(wl,w2,...,wn) is defined as C(W,y), and it is obviously a function of W, y and the given production function F. ...Thus, there is a duality between cost and production functions in the sense that either of these functions can describe the technology of the firm equally well in certain circumstances. The production function, y-F(X), referred to as the “primal”, describes global output response to all possible combinations of input quantities. The cost function, C(W,y), the ”dual" of the production function, describes the minimum cost of producing any level of output given a set of input prices and production technology. Therefore, the existence of a duality between cost and production functions allows a researcher to use either function in analysis since the same information can be obtained from either function. Duality theory has its roots in the work of Hotelling (I932), Roy (l9A2), Hicks (I9A6), and Samuelson (l9A7), but it is the pioneering work of Shephard (I953) which treats the subject comprehensively. The theoretical background on how to apply duality to empirical studies is rigorously explained and mathematically proven by Shephard (l953, I970), Diewert (l97Aa, I982), Lau (I976, I978), Fuss and McFadden (I978), Blackorby, Primont, and Russell (I978), and Deaton and Muellbauer .(l980). These researchers show that the cost function 17 contains all of the information on production technology that is present in the production function. Therefore, one can proceed with the cost function approach without prior regard to a functional form for production technology. In brief, assuming that a firm minimizes costs subject to a production function f: (2.l) y 8 f (xl, x2, ..., xn) It can be shown that the cost function which corresponds to f has the following form: for yzO, w20, (i.e., each component of the vectors y and W is nonnegative) (2.2) C(w,y) - min {W'X: f(x)2y, x20 } x where W'X iszwi.xi, the inner product of the vectors W and X. Equation (2.2) simply says that the producing unit (e.g., a firm) takes' factor prices as given, and attempts to minimize the total cost at a specified level of output. The procedure for deriving input demand functions from constrained minimization of total cost, subject to an output constraint, is commonly known. (2.3) min L-E wixi+/\i [y-f(xl,x2,...,xn)] xi's Solving (2.3) yields the n constant output input demands. * it 3': (LA) Xi (my) - [xl (my). ..., xn (MYIJ l8 The asterisk (*) denotes that the variable is the outcome of an optimization process. In (2.A), Xif is the minimum level of input quantity associated with the exogenous input price wi, and an output- level y. The substitution of (2.A) into: wixi provides an expression for the minimum level of cost in terms of input price and output level, C(W.y)*. However, by applying duality theory, the producer's system of input demand functions (i.e., equation (2.A)) can be obtained simply by differentiating the cost function with respect to input prices (Shephard Lemma). {Footnote I} This conceptual simplicity and the ease of generating the farm production expenditure system are the major advantages of adopting a cost function, rather than a production function, to represent production technology. To represent a rational output constrained minimization of cost given a ”well-behaved" {FoOtnote 2} production technology, a cost function must meet the following regularity conditions: (2.5) (I) Continuity: continuous with respect to input prices. (2) Homogeneity: linearly homogeneous in input prices. (3) Monotonicity: nondecreasing in input prices. (A) Concavity: concave with respect to input prices. {Footnote I} Shephard Lemma: The partial derivative of the cost function with respect to the ith input price yields the constant output demand function for input i. dC(W.Y)/dwi-xi. {Footnote 2} "Well-behaved” means, that a unique minimum to the cost minimization problem exists. The empirical validity of these conditions in the context of the present study will be discussed later. The following remarks are intend to explain their meaning and nature (I) Continuity with respect to factor prices. This condition is possibly true for most factor prices, however, in order to apply the theory, it is assumed true for all of them. (2) Linear homogeneity in factor prices. For a given level of output total cost must increase proportionally with a proportional increase in all factor prices. This is intuitively plausible if it is assumed that total cost is made up only the cost of purchased inputs. If all factor prices double, one would expect the minimum cost of producing a given output level to double. (3) Monotonicity with respect to input prices. The cost function must be a non-decreasing function of each factor price. The derivative of the cost function with respect to a factor price, dC/dWi, is expected to be non-negative. If one or more input prices increase and those inputs are used at positive levels, it is necessary to move to a higher isocost line to secure any specified output. (A) Concavity with respect to factor prices is less intuitively apparent. Mathematically speaking, a cost function C(w,y) is concave if the Hessian matrix {footnote 3} is negative semidefinite within the range of factor prices. The Hessian is negative semidefinite if, and only if, the principal minors obtained from the Hessian alternate in {Footnote 3} The Hessian matrix is the matrix of second-order partial derivatives of a particular function F with respect to its arguments. 20 sign (so that all odd-numbered principal minors are negative and all even-numbered ones are positive). For estimated empirical functions, one could numerically check for concavity by evaluating the characteristic roots of the .Hessian of the cost function at each observation point. The Hessian will be negative semidefinite and the cost function will be concave if, and only if, all characteristic roots are nonpositive (Chiang, l97A, p. 3A5). Hll HIZ...HIj...HIn I I I I I HZI H22 I I ”U I H 8 I . HII : : Hil I I . HIJ : : - : I Hnl ............ Hnn : A graphical presentation may help us to better understand. this concept. Suppose we illustrate cost as a function of the price of a single input with all other prices held constant. If the price of a factor rises, cost will never go down ( monotonicity property), but the cost will go up at a decreasing rate. Why? Because as this particular factor becomes more expensive and other factor prices stay the same, the cost-minimizing firm will gradually replace this costly input with less-expensive inputs. 2I Figure 2.l The Cost Function and the ”Passive” Cost Function Cost n I wl-xl*+ 2 wi*xi* (”passive”) L1 i=2 VCIWJI wI wl* Consider Fifure 2.l (Varian, I978, p. 29), let x* be a cost-minimizing bundle at price w*. If the price of factor I changes from wl* to wl (wl>wl*), and we behave passively and continue to use x*, the cost curve is the linear line, c-(wl)~(xl*)+£ (wi*)(xi*). However, the minimal cost of production C(w,y) must be less than this "passive” cost function since the substitution effect is not restricted. Thus, the graph of C(w,y) must lie below the graph of the passive cost function, with both curves coinciding at wl*. Now it is easier to see why the cost function C(w,y) is concave with respect to factor price wl. For a two dimension coordinate system, the geometrical graph of a concave function always lies below its tangent line. In higher dimensions, we say that the graph of a concave function always lies 22 below its tangent hyperplane. In this cost-minimizing situation, the concavity implies that it is necessary for each isoquant to be strictly convex (to rule out perfect substitutes or perfect complements) for the existence of a unique dual cost function. As discussed later, it is possible to conduct statistical tests to find out if estimated cost equations meet some of these regularity conditions. The duality theory can be interpreted either in the producer or the consumer context. I will use producer terminology for the sake of consistency. 2.l.2 To Dual or not to Dual? One may find neither an absolutely positive nor negative answer to this question. The decision on whether to use the dual (i.e., cost function) or the primal (i.e., production function) is largely a matter of statistical convenience and analytical purpose. The dual approaches allow estimation of the same information of practical value to policy makers (e.g., elasticity of substitution) that applied economists have supplied by traditional primal approaches for years. In some cases, primal approaches may be superior to dual approaches. However, data availability and the convenience of econometric estimation considerations will allow less difficult and costly analysis of many problems with dual approaches. The cost function C(W,y), is expressed in terms of factor prices and the level of output while the production function, y-f(X), is expressed in terms of input quantities. Silberberg (I978) argues that production 23 functions are largely unobservable. The data points of the production surface represent a sampling of input and output levels that have taken place at different times or places, as factor or output prices changed. An instantaneous adjustment process is implicitly assumed in many production function studies. The main statistical issue is whether it is safer to treat factor prices and level of output, or the use of inputs, as exogenous to the firm. Direct estimation of the production function is attractive when the level of output is endogenous and the quantity of factor inputs are exogenous. Estimation of the cost function is more attractive, however, if the level of output and factor prices are exogenous. Neither of these two approaches is completely satisfactory, but I have chosen the latter. In the U.S., output is at least partly exogenous to the farm sector, being determined to some degree by government policy, It may be argue that producers act more like cost minimizers than profit maximizers if they choose to participate in a farm program. Mundlak and Hoch (I965) show that if the firm is a cost minimizer, input choice is necessarily endogenous and direct estimation of the production function will yield inconsistent results. Furthermore, according to Woodland (I975) and Lopez (I980), farm input prices are determined in the nonagricultural sector. Berndt and Wood (I975, p. 26l) also say that "at the level of an individual firm it may be reasonable to assume that the supply of inputs is perfectly elastic and, therefore, the input prices are fixed." Exogeneity of input prices is a convenient assumption, since it permits fairly simple estimation of the cost function and quantity-dependent input demand 2A functions. An alternative would be to assume the farm sector is faced with rising supply curves for its inputs, in practice, this is not usually done. In conclusion, the dual approach has the advantages of theoretical soundness and statistical convenience, therefore it is the approach I have chosen, rather than direct estimation of the production function, to study the farm input demand. 2.l.3 Cost function vs. profit function The duality theory is not only applicable to the cost function approach, it is also applicable to the profit function. {Footnote A} Lopez (I982) makes a distinction between these two approaches. The cost function is used to estimate Hicksian (compensated) input demand while the profit function approach allows researchers to estimate Marshallian (ordinary) input demand. A firm's variable profit function I](p,w) can be simply defined as the maximum revenue minus variable input expenditure. I](p.WI - max {p.y - W'X. F(x.y)20 I X,Y where W'X is the inner product of factor prices and qunatities. {Footnote A} The third method, indirect production function approach, y-f(W,c), can be used to assess own- and cross-price elasticities of input demand for production with constant expenditure (i.e., total budget). We will not discuss this case since it has not been much used. A common feature of a cost function approach is the assumption that output levels are not affected by factor price changes. Therefore, the indirect effects of factor price changes (via output levels) on factor demands are ignored. On the other hand, using the profit function and Hotelling's lemma, the input demand and output supply equations can be deriVed by simple differentiation with respect to input price and output price, respectively. However, a profit function requires a stronger behavioral assumption. The profit maximization assumption may be more difficult to support in agriculture than simple cost minimization. This is caused by the risk-related (instability of output and product price rather than the costs of production. The major differences between the two approaches may be briefly summarized as follows: .(I) the cost function approach assumes cost minimization, and that output quantities are exogenous: (2) the profit function approach assumes profit maximization, and that output prices are exogenous. (Both assume that input prices are exogenous.) As between these two, I (and most other investigators) have chosen the cost function approach. 2.2 Flexible fpnctional forms The econometric applications of the new production theory based on the duality relationship between production and variable cost functions are major steps towards generating appropriate empirical estimates for 26 input demand functions. However, to determine the elasticity of demand for fertilizer, elasticity of substitution between labor and capital, and economies of scale (these are examples of some important instruments for addressing policy issues), it is necessary to estimate the parameters of a production funttion or a cost function. The development of flexible functional forms permits application of the duality theory to a less restrictive analysis of the nature of production than has been previously possible.. A specified flexible cost function suggests a set of derived input demand equations, as indicated by the theory. Flexible functional forms have been developed with two attractive features, namely, they imply derived demand equations which are linear in the parameters, and at the same time, they may represent a very general picture of the production structune even though they are not derived from explicit production functions. We say f is a "flexible functional form" if it' can provide a second-order (differential) approximation to an arbitrary twice continuously differentiable function f* at x* (I982, p. 57A). The term "differential approximation" is defined by Lau (l97A, p. l83): According to Diewert's definition, a function G(y) is a second order approximation to a function F(y) at yO if the first and second order derivatives of the two functions are equal at yO, that is, Glyol-FIYOI. l I . IY'YO . dyi dyi Iy=Y° dyi dyj IY'YO - [for all i and j, and both G and F are assumed to be twice differentiable.] 27 A flexible functional form should be capable of representing a wide range of technology and be tractable with respect to the ease of computation, estimation, and interpretation. The set of flexible functional forms which are suitable candidates for investigating input demand functions has grown rapidly in the past decade. A partial list of these forms includes the transcendental logarithmic function (translog), the generalized Leontief (GL), the generalized Cobb-Douglas (GCD), the generalized square root quadratic (GSRQ), and the generalized Box-Cox (BBC). The GL, GCD and GSRQ forms were introduced by Diewert (l97l, l973, l97Ab). The translog was developed by Christensen, Jorgenson, and Lau (I97I, I975). Berndt and Khaled proposed the CBC form (I979). The advantage of the CBC form is that restrictions on the Box-Cox parameters produce the other flexible functional forms. 28 (Table 2.l) Common Linear-in-Parameters Flexible Functional Forms Functional Forms Formula Translog In C 8 a0 + 2 ai In wi + E 2 aij (In wi)(ln wj) l/2 I/2 Generalized Leontief C = y . [ 2 2 aij (wi) (wj) 3 Generalized C-D In C = a0 + 2 2 aij [ln(wi+wj)/2] Quadratic C = a0 + E ai wi + 2 2 aij (wi)(wj) 2 2 I/2 GSRQ c =2 2 aij [ l/2(wi) + i/2(w.i)] .y_ p/2 p/Z I/p B(y,w) Generalized Box-Cox C = [(2/p)2 2 bij . wi . wj )] . y where B(y,w) = b + r/2 . In y + Z c In wi, and (p/2) wi 8 (Ni -l)/(p/2). Selection of the flexible functional form best suited for use in empirical estimation is of primary concern. Mathematically speaking, the generalization of these flexible forms can become a never-ending pursuit. One may construct a variety of flexible functional forms containing the GBC as a limiting case, test for the restrictions of the GBC form, and so forth. The number of flexible functional forms which could be applied to duality theory is very large. 29 2.2.l Choice among flexible functional forms Since, by definition, all previously mentioned flexible functional forms have the same, or similar, attractive properties, it is unclear how the practitioner should choose among them. The choice of specific functional forms for estimating and deriving input demand equations involves compromise. One form cannot serve all analytical purposes. Lau (l97A, p. I86) provides two principles for choosing the functional form. The first principle is that the functional form must be capable of approximating an arbitrary function to a desired order of accuracy (flexibility). The second principle is that the functional form must result in estimating forms that are linear in parameters (workability of econometric application). Fuss, McFadden, and Mundlak (I978 p. 22A) suggest five more criteria to help distinquish between forms: (I) parsimony in parameters, (2) ease of interpretation, (3) computational ease, (A) interpolative robustness within the sample (e.g., concavity, monotonicity), and (5) extrapolative robustness outside the sample (i.e., forecasting). Four recent papers discuss the issue of choosing among flexible functional forms. Wales (I977) performed a ‘Monte Carlo study to investigate the ability of the GL and translog forms to represent two-product homothetic preference and exhibit constant elasticity of substitution. He found that in some cases the CL performed better, while in other cases the translog performed better. Wales found the performance of the translog form to deteriorate as the true elasticity of substitution departs from unity in either direction, and found the 30 performance of the GL form to deteriorate as 'the true elasticity of substitution increases away from zero. Berndt, Darrough, and Diewert (I977) used postwar Canadian expenditure data to estimate three-product nonhomothetic GL, translog, and GCD forms. On the basis of better fit and conformity to neoclassical restrictions, they concluded that the translog form is the preferred form on Bayesian grounds 3 posteriori. Applebaum (I979) and Berndt and Khaled (I979) showed that the GBC form contains the GL, GSRQ, and translog forms as special or limiting cases. Using I929-7l U.S. manufacturing data, Applebaum found that the GL and GSRQ forms are the best representations for the primal and dual specifications of technology. Using l9A7-7l U.S. manufacturing data, Berndt and Khaled were able to reject the GSRQ model. However, the CL was not rejected and result. regarding the translog were inconclusive. Guilkey, Lovell and Sickles (I983) continue and broaden the line of attack initiated by Wales. The Monte Carlo experiments include the translog, GL, and GCD forms. They conclude that the translog form provides a dependable approximation to reality and outperforms all other flexible functional forms. They also conducted limited experiments on the GBC form and the results were not fruitful. The translog and the GL are the most popular forms in previous applications. However, the translog is superior to the CL in analyzing technological change. (The translog can incorperate both neutral and factor-augmenting technological changes while the GL can only assume factor-augmenting technological change). Thus, I decided to employ the translog functional form because it is flexible enough to fit my research interests. 3I 2.2.2 The Translog Cost Function The cost function specified is an adaptation of Christenson, Jorgenson and Lau's (l97l,l973) transcendental logarithmic cost function. Expressed as a second-order polynomial in logarithms of input prices and output, it is a generalization of the Cobb-Douglas (which is linear in logarithms) functional form. Note that the translog function places no 3 priori restrictions upon homotheticity, returns to scale, or the elasticity of substitution between pairs of inputs. (2.6) In C(w,y) = a0 + 2 ai In wi + l/2 5 E bij In wi In wj + ay In y + 2 di In wi In y +l/2 ayy (In y)**2 +2'ti T In-wi + t T +I/2 tt (T):':>’:2 where wi and wj are factor prices, y is output level, T is time, and a0, ai, bij, ay, ayy, di, ti, t, tt are parameters. The following constraints are implied by duality theory. First, in order to correspond to a well-behaved production function, a cost function must be linearly homogeneous in factor price. This implies the following relationships among the parameters: S ai-l, 2 bij-O (for all i), S'di-O, and ‘S ti=O. l j I I These restrictions imply that as .input prices rise by a fixed percentage, total cost rises by that same percentage. The bij, di, and ti terms are forced to sum to zero in order to negate any effect they 32 might have on total cost. This leaves ai to exert the only impact on total cost as input prices change, and to maintain the economic meaning of homogeneity. - Second, since the translog function is viewed as a secbnd-order logarithmic approximation, the following symmetry constraint must hold: bij-bji, for all i and j. The symmetry condition is the consequence of‘ the continuity assumption of the parent cost function and Young's Theorem {Footnote 5} from calculus. Combining the symmetry and homogeneity constraints, we have (2.7) 2 aisl, 2 bij= 2 bji= 2 2 bij=0, '21: = o and 2 di=0. i i j i‘j i i- The additional restrictions of ay-I, di-O for all i, and ayy=0 ensure that C(w,y)-y.C(w,l) (where C(w,l) is the unit cost function) so that the corresponding production function . is linearly homogeneous. However, these restrictions are not necessarily always imposed. If ai>0 for all i, Eai-l, and bij-0 for all i and j the translog function collapses to a Cobb-Douglas cost function. Most of the CES-Iike functions may be derived from the translog function as special cases when appropriate restrictions are imposed. The most interesting feature of the translog function is its flexibility. The translog functional form can serve as a local, second-order approximation to an arbitrary cost function. {Footnote 6} {Footnote 5} Young's Theorem: ny and Fyx, are identical to each other, dF/(dx)(dy)-dF/(dY)(dx), as long as the two cross partial derivatives are both continuous. {Footnote 6} See appendix A for a mathematical proof. 33 However, in the econometric model presented in Chapter 3, this translog cost function is assumed as the true data-generating function rather than an approximation to an arbitrary cost function (Berndt and Wood, 1975). This permits additive disturbance terms to be specified for the derived input demand equations and interpreted as random deviations of the endogenous left-hand variables about their cost-minimizing values. The cost minimizing input demand functions xi(w,y), generated via Shephard‘s Lemma, are not linear in the unknown parameters. But it is easy to verify that the factor share equations d In C(w,y) d C wi xi wi ------------ a ---- -- = -—--— = 5] d In wi d wi C C (i.e., via logarithmic differentiation using Shephard Lemma) are linear in the unknown parameters and, hence, are in convenient form for estimation. (2.8) si I ai+ 5 bij ln wj + di In y + ti T i=l,2,...,n By the monotonicity property the cost function must be an increasing function of input prices, i.e., si>0. Constant-output input demand functions showing quantity demanded (xi) as a function of prices and output (w and y) could be obtained from the relationship of xi-si.C/wi, where si is from (2.8) and C is from (2.6). But it is true that the resulting expression would be highly non-linear in w and y, and not convenient for analytical purposes.- 3h Since 2 si(w,y)=l, only n-l of the n equations defined by (2.8) can be statistically independent. If all n share equations are included in the estimating system, then the singularity of the residual covariance matrix for the factor share equations becomes unavoidable. {Footnote 7} The singularity problem can be overcome by deleting one of the factor share equations, and consequently, the parameters in the share equations become a subset of those in the cost equation. Now, given data on output (y), input quantity (xi), and input price (wi), all parameters can be statistically determined since 51 can be observed. Although the cost function could be estimated in isolation from the factor share equations, it is more efficient to estimate the parameters jointly with the factor share equations included in the system. A more detailed discussion can be found in Chapter 3.2. The transcendental logarithmic functional form has been discussed by Halter, Carter, and Hocking (1957). Christensen et al. (1971, 1973), Griliches and Ringstad (1971), and Sargan (1971). Empirical applications of the translog profit function have been made by Sidhu and Baanante (1981), Weaver (1983), McKay, Lawrence, and Vlastuin (1983). Empirical applications of the translog cost functional form have been made by Christensen et al. (1973). Berndt and Christensen (1973), Binswanger (197ha, 197hb), Burgess (1975). Christensen and Green (1976), Kako (1978), Nadiri and Schankerman (1979). Ray (1982), {Footnote 7} A singularity problem means the disturbances are linearly dependent, and the covariance matrix cannot be inverted. 35 and Antle and Aitah (1983). 2.3 Derivation of elasticity measures The most natural way of measuring how one input is a substitute for another is the cross-elasticity of factor demand (eij): eij a (dxi/dwj)(wj/xi) However, most researchers prefer a related measure known as the the elasticity of substitution. One such elasticity is given by Varian (1978. p. h6). d(xi/xj) (wi/wj) E sub- -------- - -------- d(wi/wj) (xi/xj) In a two-factor case, this measures the proportionate change in the ratio of the factor quantities per unit change in the ratio of the factor prices when output and other input prices are held constant. This can be pictured as a shift in the input ratio along an isquant as relative input prices change. When xi/xj responds greatly to change in wi/wj, the elasticity will be high and vice versa. The limiting case of E-O occurs when the two inputs must be used in a fixed proportion as complements to each other. The other limiting case, with E infinite, occurs when the two inputs are perfect substitutes for each other. However, an alternative measure is proposed by Allen (1938). The bij parameters in (2.6) and (2.8), where i is not equal to j, can be related to Allen's partial elasticity of substitution (Eij) between 36 inputs i and j. Originally Allen (1938, p. 50A) defined the partial elasticity of substitution in terms of the partial derivatives of the production function. ( 2 x9.fg ) . |Fij| Eij - -------------------- xi . xj . |F| : 0 fl ... fn : : fl fll ...fln : where fg a df/d(xg). IFI ‘ i i I I 1 ° ° I : fn fnl .. fnn : and {FIJI is the determinant of (i,j)th cofactor in F. In order to minimize the cost of producing at a specific level of output, a firm must adjust the input such that the ratio of price to marginal product will be the same for each factor, i.e., fg-wg/k, where k is interpreted as the marginal cost of output. Meanwhile, the rate of change of the independent variables (xl,...,xn) with respect to changes in factor prices is obtained as (see Samuelson (l9A7), pp. 63-9), Substituting these relationships in Eij we have ( 2 wg-xg )[(d xil/(d will Eij - ---------------------------- xi.xj On the other hand, by Shephard's Lemma, xi-dC/dwi, and utilizing the the fact thatE wg.xg-Cost, Eij can be defined in terms of the partial 37 derivatives of the total cost function C(w,y) as follows: If Eij>O, it indicates a substitution relationship between inputs 1 and j. When EijO, and jth factor saving if tj0), since over a long period average farm size has increased, presumably because larger farms are more efficient. 0n the other hand, for a cost function fitted to aggregate data, the case may be not so clear.. One might argue that the aggregated cost function in some way is an average of, or representative of, or typifies, the individual farm cost functions, and might therefore be expected to exhibit similar characteristics, in particular positive scale economies. Alternatively, one could view the aggregate cost function as being derived form, or related to, an aggregate production function, and argue that, for U.S. agriculture as a whole, there is no reason to suppose anything other than constant returns to scale, that is, zero scale economies. Whichever viewpoint is taken, it was felt that it would be worth investigating what the data, as analyzed through the interpretation of our estimated cost function, seem to imply. Using model 9 (the one 92 which assumed to fit the data best, which allowed for factor-augmenting technological change, and on which our estimated elasticities of demand and substitution are based), the estimated economies of scale parameter turns out to be quite highly negative even when a scale trend term is incorporated (Model 10). (See Table 3.9.) Since this result seems quite unreasonable, I decided to explore the possible effects, on economies of scale measure, of imposing restrictions on the cost function derived from assumptions of homotheticity, homogeneity, and adding a ”scale trend" parameter on the production function. The scale trend parameter, q, is defined as the derivative of the scale measure with respect to time. d d In C d MC 9 = ---- ( -------- 1 = ---- (----1 dT d In y dT AC If the estimate of q is zero, this implies that the ratio of marginal cost and average cost is a constant over time. A positive q indicates that the ratio between these two cost is rising. Table 3.9 displays these results. 93 Table 3.9 Alternative Estimates of Scale Economies, with and without Scale Trend. Model Specification Parameter Log of likelihood SE* restrictions function Group C: Biased technological change and non-homothetic production Sal-1 {2} 2diI22bijI0 bij-bji 9 w/o scale trend qIO 1006.0# -1.9599 10 w/ scale trend 1007.## -1.580# Group D: Homothetic production {Z} and diIO, Vi. ' ' 11 w/o scale trend qIO 992.#6 -.513# 12 w/ scale trend 992.6# .5#83 Group E: Homogeneous production ' {Z} and diIO, Vi, anyO. 13 w/o scale trend qIO 989.61 -.1555 1# w/ scale trend 992.58 1.#215 * These SE measurements are based on the mean value of the series of observed points except in the homogeneity situation where it is a constant. First of all, one may ask whether it is necessary to impose further restrictions on model 9? Do the impositions of homotheticity, homogeneity, and scale trend make any difference? This is subject to the purpose of the analysis. The justification for these restrictions on the estimated parameters is to address and to check the validity of farm production characteristics. Homothetic production models (model 11 and 12) have four additional restrictions (diIO, iI1,2,3,#.) than model 9 and 10, which imply four degrees of freedom. We reject the 9# imposition of the restrictions on group C (homohteticity) models based on the likelihood-ratio test. These two computed chi-square statistics exceed the critical value at the 1% level of significance‘ (i.e., 13.28). For the same reason, the homogeneous production models (i.e., group E) are also rejected on the basis of the likelihood ratio test with the degrees of freedom is five. The critical value of the chi-square statistics at the 12 level of significance and with five degrees of freedom is 15.09. As to the inclusion of the scale parameter, q (which stands for the right hand side cross-product term of time and output), it is easy to see there are almost no differences between models 9 and 10, and between models 11 and 12. The test-statistic is 5.936 between models 13 and 1#. We do not reject the hypothesis that the scale trend parameter is zero at the 12 level since 5.936<6.635. However, we reject the hypothesis if we lower the significance level to the 52 level because the test statistic 5.936 is greater than the critical value of 3.8#1. The estimates of q in models 10, 12, and 1# are 0.0059, 0.0235, and 0.0#05, respectively. Although the imposition of these restrictions had been rejected on statistical grounds, the results of models 11, 12, 13, and 1# (in Table 3.9) still indicate negative scale economies when the scale trend term is not included, but positive scale economies when it is included. The derived estimates of scale economies between models 12 and l# (with scale trend) and models 11 and 13 (without scale trend) are opposite. One cannot conclude much of anything from this mixed bag of results. They are. inconcluded here because they might be of some interest or use to future investigators. One possible hypothesis is that there might be more confounding between our estimates of technological change and scale effect. Our estimate of technological progress on average is more or less higher than those investigtor's have obtianed. If this estimate is in fact biased upward, it may be accompanied by some downward bias in the scale effect estimate. 3.6 Summary The empirical study of U.S. farm input demand was designed and reported in this chapter with five sections. In the first section, five input subgroups were categorized as follows: (I) hired labor, (2) capital, (3) feed, seed, and livestock purchased, (#) fertilizer, and (5) miscenllaneous items. Annual data (1910-81) of input prices, farm production expenses, and output levels are collected from USDA publications. Econometric estimation procedure of a translog cost function was discussed in section 3.2. Zellner's seemingly unrelated regression technique, an interative version, was used to estimate the demand system and the cost function. In section 3.3, a series of hypothesis testing of the validity of duality theory implied regularity conditions -- symmetry, linear homogeneity, monotonicity, and concavity -- were carried out. The specified biased technological change model successfully passed the linear homogeneity and monotonicity conditions. 96 However, the symmetry condition has failed and the local concavity condition was indefinite. To complete the rest of the analyses, the symmetry restriction was imposed as the theory suggested. The investigation of technological change showed that agricultural industry was characterized by labor-saving, capital-using technology. The annual rate of technological advancement was 2.9 percent. In section 3.#, the own-price demand elasticity and Allen's partial elasticity of substitution were derived based on the selected factor-augementing technological change model. Each own-price elasticity of demand has the anticipated negative sign except for the miscellaneous items. Farm capital appeares to be a substitute for all other farm inputs. Finally, the investigation of economies of scale for the U.S. farming industry as implied by our estimated cost functions led to inconclusive. 97 CHAPTER # Summary and Conclusion This chapter contains a summary of the study, conclusions drawn from the analysis, discussion of the results, and suggestions for further research. #.1 Summgry and Implication of Results In this study I attempted to analyze the structure of U.S. farm input demand and the technology of farm production. By employing the flexible translog cost functional form and the duality theory, a set of results on elasticity of farm input demand, elasticity of substitution, technOIOgical change, and economies of scale was derived. In view of model specifications, a translog cost function which exhibits factor-augmenting technological change and linear homogeneity with respect to factor prices seems appropriate. The derived Allen's partial elasticity of substitution between input subgroups and the 98 price elasticity of input demand, are the main areas of investigation in this study. Capital was found to be a substitute for all other inputs. Labor is a substitute for capital, feed, seed, livestock purchased, and miscellaneous inputs, but there is a weak complemetary relationship between labor and fertilizer. The results show a decline in the substitutablity between labor and capital. Another important conclusion is that biased technological change in U.S. agriculture is mainly labor-saving and capital-using technology. This confirms previous empirical studies in the same direction. A 2.92 annual growth rate of agricultural productivity sustained over the last seven decades is quite impressive. The estimated translog cost function passed the test of linear homogenity and monotonicity regularity conditions implied by the duality theory. However, it fails the test of symmetry condition and the concavity condition is indefinite.' This once again implies the difficulty of working with aggregate data. This is an unfortunate limitation imposed by considerations of computational manageability within the scope of this econometric study. #.2 Future research needs Another important limitation in this study is the exclusion of farm family labor as an input. This is primarily due to the unavailability of data on the farm family labor component of production expense and the wage rate of family labor on the farm. If we assume that farm family labor were paid as the hired labors, we still have to 99 approximate the farm employment, either in hours or as a portion of the entire p0pulation. It is possible to include the farm family labor at least in an approximate way. Selecting and introducing new flexible functional forms requires the application of many mathematical formulations, statistical tests, and large computer accounts. Berndt and Khaled (1979). Chalfant (1983) and other researchers have put out tremendous amounts of effort to search for a better function. As mentioned in chapter 2.2, pursuing a "perfect" functional form is a never-ending job. However, if a super flexible functional form (e.g., Fourier functional form proposed by Gallant (1981, 1982)) is tested repeatedly and shows its strength in most cases, (such as the translog dominating the past decade) then the form is justified for further investigation. Is output really exogenous? Sometimes it may not be. Endogenous output leads to profit maximization and other explicit behavioral assumptions. The profit function approach, which assumes both exogenous product and factor prices, is an alternative way to investigate U.S. farm input demand structures. However, one must be very careful to explain those so-called "external shocks” to a producer. A better way to deal with the exogeneity is to incorporate risk and uncertainty into the profit function specification. Pope (1982a) demonstrates that risk aversion biases the certainty (i.e., risk-neutrality) results regarding factor demands and output supplies derived from the profit function. The derivatives of expected profit with respect to input prices are not necessarily negative. Ignoring risk might result in analytical biases, but estimation biases from 100 econometric analyses still remain unsolved. Finally, neoclassical duality theory was developed to describe a firm's behavior. Ideally, this study should use farm level data and investigate the producer's behavior. Use of aggregate data introduces a measure of aggregation bias. Therefore, my estimates should be regarded only as broad indicators and Interpreted with care. APPENDICES 101 APPENDIX A Translog Function as a Second-Order Approximation of any Function The flexibility property of the translog functional form as a second-order Taylor's approximation is developed as follow: (A.1) Cost function: C I g(wl, w2, ..., wn, y) (A.2) Any function: y = f(xl, x2, ..., xn) By taking the log of the arbitrary functional form (A.2), we get (In x1) (1n xn) 1n y I In { f [ e ... e ] } (A.3) 1n y I f ( In x1, ..., In xn) Applying Taylor's expansion method, we expand (A.3) around point XII, or equivalently, (1n x)IO, and obtain n df I (11.11) In y - f(ln 1, In 1) +2 --------- I In xi iIl d(ln xi) Iln XIO 2 1 df I +- 22 ------------------- I (lnxi)(lnxj)+R 2 I j d(ln xi) d(ln xj) Iln XIO where R represents the higher order terms. Since df : df I -------- I and ------------------- I d(ln xi) Iln x=0 d(ln xi) d(ln xj) Iln XIO are constant for i,j=1,...n, therefore, we assume the following 102 identities: df I (A.5) """"" 1 I ai d(ID Xi) Iln XIO 2 df : ------------------ I Ibij 1, d(ln xi) d(ln xj) Iln XIO I l I 2 I----> where biijji df I I ------------------ I iji J and f(ln l, ..., In 1) I a0. Then, by substituting (A.5) into (A.#) and omitting the higher order term, R, we get equation (2.6) which is the translog cost function without time variable. The translog function can be expressed in natural exponential form n 1/2 (EbIj In xi) n ai y I a0 (111 xi ) (il1 xi ) Taking the natural log of both side, we have 1 In y I In a0 +12 ai 1n xi + - 2 2 bij In xi 1n xj 2 i j 103 APPENDIX B The Second Order Partial Derivatives of the Translog Cost Function The first partial derivative of the translog cost fucntion (2.6) (2.6) In C(w,y) I a0 + 2 ai In wi + 1/2 2 2 bij In wi 1n wj + ay 1n y + 2 di In wi In y +1/2 ayy (In y)**2 + 2 ti T In wi + t T +1/2 tt (T)**2 with respect to factor price, d [In C(w,y)J/d [In wi], is the factor share equation (2.8). (2.8) $1 - a1+2 bij In wj + di In y + ti T 1-1,2,...,n By Shephard's Lemma, the first partial derivative of the cost itself with respect to the factor price is the input demand equation in quantity-dependent form, i.e., indC/dwi. Therefore, d In wi (l/wi) d wi C or equivalently, xi I si . ---- wi The second order derivative of the translog cost function with respect to factor prices can be stated as the partial derivative of the input demand, xi. There are two situations: (1) d xi/d wi and (2) d xi/d wj, 10# where i'is not equal to j. First let us derive d xi/d wi. 2 . d C(w,y) d xi d [si(wi) . C/wi] d wI d wi d wI d 51(wI) C d (C/wi) I --------- . --- + ---------- . 51 [by the product d wi wi d wi rule.] Since d 51 d si d 51 bii ------- I bii, hence ---------- I bii, and ------ I ----- . d In wi (1/wi)d wi . d wi wi Therefore, the complete derivation is 2 d C(w,y) bii C wi.xi - C ---------- I ---- --- + (-----------).si [by the quotient rule] d wiz' wi wi wi.wi bII . C si.C - C = """""" + I """"" ) (w1)**2 (wi)**2 105 The second case of the partial derivative can be derived as follows: 2 d C(w,y) d xi d [ si(wi) . C/wi ] dwj dwi dwj dwj d (si(wi)) C d [ C(wJ1/wi ] a ----------- --- + ---------------- . sI d wj wi d wj bIj C WI.XJ . ----- .--- +( ------- - 0) $1 wj wi w1 w1 le C XJ I ----- . --- + (----) . si wj wi wi bIj C (xj.wj) . SI . ----- . --— + -------------- wj wi wI WJ . --------------------------- [since xj.wj I SJ-C] These second order partial derivatives are employed in calculating the Allen's partial elasticity of substitution and the own-price elasticity of demand as well as checking the concavity condition. 106 APPENDIX C Divisia Index The Divisia index is named after Francois Divisia (1889-196#). Originally the index was related to a general equation of exchange. Using this equation, Divisia, by differentiation, distinquishes two indexes whose product is always proportional to the total payment of the period to which the equation applies. Instead of comparing two discrete price situations, the constructed Divisia index can analyze the continuous effects of price changes. The Divisia index is defined in terms of a weighted sum of growth rates. By denoting the proportional rate of change of price level by ”d In P", which is equivalent to .P/PI(dP/dt)/P (the dot over variable P denotes derivative with respect to time). then, for any fixed production level the Divisia price index, in its continuous form, is defined as: n n P1 (8.1) Pindex =1n P - f 2 si (d In Pi) -f 2 si (----) t i-l t iIl Pi where siIPi*xi/2 Pikxi is the relative share of the value of ith input in total expenditure. However, in practice, neither the quantities nor the prices are continuously observable. Most economic data take the form of observations at discrete points in time. Tornquist (1936) and Theil (1965, 1967) constructed an index, 107 Mt) 1 . (8.2) In I """" 1 ' - Z [si(t)+si(t-1)] In ( -------- ) w(t-1) 2 win-1) 01' In w(t)-ln w(t-l)= 1/22 [51 (t)+si (t-1)] [In wi.(t) - 1n wi (t-1)] as a discrete approximation to the Divisia index in logarithms. It approaches the continuous form as the change of t approaches zero. This composite Divisia price index uses the arithematic averages of the relative shares in two periods as the weights. Obviously, the discrete and continuous index numbers are equal if, and only if, relative shares are constant. The prevailing usefulness of the Divisia index is due to the fact that Diewert (1976) shows that this index, in view of the second-order approximation property, is consistent with a homogeneous translog aggregator function. Diewert introduces the term "aggregator function” as a neutral term for the underlying production or utility function. Since the index is a line integral, the index is dependent, in general, upon the path on which the integral is taken. Hulten (1973) shows that if the aggregate exists, is homogeneous of degree one in its components, and there exists a corresponding price normal at each point unique up to a scalar multiple, then the Divisia index is path independent and can retrieve the actual values of the aggregated function, subject to an arbitrary normalization at some base period. Given these desirable properties, the Divisia index is the best choice among other index numbers. Constructed Divisia Price Index Table C.l Constructed Divisia Price Index for Feed, Seed, and Livestock Purchased, 1910-I981. (1977-1.0) Price Index of Divisia ------------------------------------- Price Year Feed Seed Livestock Purchased Index 1910 .25 .15 .15 .205779 1911 .25 .17 .I# .203927 1912 .26 .19 .16 .2200#2 1913 .2# .15 .18 .213#62 191# .26 .1# .19 .226289 1915 .25 .l# .19 .221030 1916 .27 .20 .20 .2#2861 1917 .## .26 .26 .358935 1918 .#7 .28 .30 .393595 1919 .52 .32 .30 .#23112 1920 .52 .38 .26 .#12729 1921 .26 .20 .1# .210901 1922 .28 .19. .18 .23#315 1923 .39 .20 .18 .266067 192# .35 .20 .19 .27#8## 1925 .36 .23 .20 .286896 1926 .31 I .27 .21 .268689 1927 .32 .2# .2# .282096 1928 .35 .21 .30 .31672# 1929 .3# .21 .29 .307839 1930 .31 .21 .22 .266762 1931 .22 .17 .15 .189906 1932 .16 .11 .12 .1#0222 1933 .18 .12 .11 .1#75#1 193# .26 .17 .11 .1930## 1935 .27 .20 .19 .23#9#3 1936 .27 .18 .18 .228#01 1937 .31 .25 .21 .270073 1938 .23 .18 .20 .21#609 1939 .23 .15 .23 .218952 19#0 .25 .16 .23 .231111 19#1 .27 .16 .26 .25108# 19#2 .33 .21 .31 .30676# 19#3 -39 -26 -35 -359677 19## .#3 .30 .33 .383896 108 (continue of Table C.l) 19#5 19u6 19#7 1998 19#9 1950 1951 1952 I953 195# 1955 1956 1957 1958 1959 1960 1961 1962 1963 196# 1965 1966 1967 1968 1969 1970 1971 1972 I973 197# 1975 1976 1977 1978 1979 1980 1981 dddd .... .#3 .50 -59 .63 .52 .31 .32 .36 .#2 .38 -37 .33 .#2 dd -393135 .##9#97 .535063 -597516 .5081u9 .5#0396 .608090 .610752 .5216h3 .520938 .998899 .#757b9 .#89209 .513821 .512895 .999287 .523618 .511369 .513713 .#96899 .519658 .5#9265 .5#37h7 .52930h .557867 .5868#0 .608661 .661975 .926710 .985021 .951872 .00086 .00000 .1113# .31505 .38631 .##963 109 nnnnnnnnnn nnnnnnnnnnnn 110 Appendix D The Computer Program for Calculating the Eigenvalues This program is written in FORTRAN. *JOBCARD* FTN5. HAL. L60. *EOS PROGRAM HSU This program calculate the eigenvalues of Hessian matrix. If eigenvalues are negative, then the Hessian is negative semidefinite. "IORDER" is a parameter which stands for the order of the input Hessian matrix. ”NVAR" is another parameter which shows the total number of the upper-triangular elements of a symmetric matrix. PARAMETER (10R0ER-5) PARAMETER (NVARI(IORDER+1)*IORDER/Z) Variable descriptions: A: Input real matrix of order N, an array. N: Input order of the matrix A. JOBN: Input optional parameter. If joanO, the subroutine compute eigenvalues only. 0: Output vector of A. The length is N. Z: Output NxN matrix. If joanO, Z is not used. WK: Work area. If joanO, WK is at least N. IER: Error parameter (for output). NOBS: Number of observations. I 111 (continue of program HSU) OPEN (UNIT-1.FILE-'TAPE1') OPEN (UNIT-6,FILEI'OUTPUT') IZIIORDER NOBSIO C----Begin the loop ----------------------------------------------------- 100 CONTINUE ’ READ (1,900,EN0I999) (A(I),II1,NVAR) NOBS-NOBS+1 WRITE (6,910) NOBS 00 110 II1.NVAR WRITE (6,920) A(I) 110 CONTINUE C C The following subroutine is called from IMSL, Vol. 2, 9th ed., C June, 1982. C C The subroutine "EIGRS'I compute the eigenvalues of a symmetric C matrix. C CALL EIGRS(A.IORDER,O,D,Z,WK,IER) IF(IER .NE. 0) THEN PRINT 20. IER ELSE WRITE (6.80) 00 120 II1.IOR0ER WRITE (6.30) 0(1) 120 CONTINUE ENDIF GO TO 100 C----End of loop -------------------------------------------------------- 10 FORMAT (13,2x,15(F7.3,1x)) 20 FORMAT ( 'ERROR NUMBER IS ',I3.' (SEE THE IMSL DOCUMENTATION FOR A 8DESCRIPTION OF THE ERROE CODES. ') 30 FORMAT (F12.2) #0 FORMAT ( 'THE EIGEN VALUES ARE: ') 900 FORMAT (15(F8.3,1X)) 910 FORMAT ( 'OBSERVATION: ',I3,' INPUT DATA ARE: ') 920 FORMAT (Ix,F9.3) 930 FORMAT ( 'NUMBER OF OBSERVATIONS - ', I3) 999 PRINT 930,NOBS STOP END 112 (Table 0.1) The Computed Characteristic Roots of the Hessian Matrix Based on Model 9. year root 1 root 2 root 3 root # root 5 1910 -#,502 78 -#7.57 -.01 61 02 #.551 16 1911 -#,78# 35 -#9.02 -.01 62 39 #,830 78 1912 -212 31 -.05 9.16 10 85 26# 35 1913 -#35 7# -.02 15.12 17 38 #82 72 191# -309 59 -.03 12.89 31 33 358 61 1915 -329 29 .03 13.28 36 10 379 30 1916 -273 59 .01 12.00 29 #7 305 72 1917 -583 7# -17.2# -.01 25 01 608 16 1918 -289 5# -12.17 - 03 15 32 30# 92 1919 -215 28 -IO #6 - 05 11 39 22# #5 1920 -99.59 -7 13 -.O7 7 9# 106 #0 1921 -2.80 - #8 2.#6 12 82 28 36 1922 -2.03 -1 06 1.66 13 67 23 80 1923 -6.#2 59 2.20 1# 0# 21 79 192# -30.77 02 #.10 9.05 39 60 1925 -22.20 11 3.66 13 91 35 10 1926 -31.78 - 02 #.17 6.89 #0 77 1927 -16.00 - 18 3.03 3.29 ' 2# 76 1928 -10#.09 -7 3# -.02 11 55 113 68 1929 -128.89 -8 l# -.02 ll 78 138 55 1930 -23.12 - 02 3.65 #.73 3# 09 1931 -60.03 07 5.81 25 ## 78 66 1932 -133.56 -8 58 .00 37 85 16# 81 1933 - -5.92 1.57 5-70 35 06 59 05 193# -2#6.56 .03 11.#7 #9 70 282 92 1935 -530.12 -l6.#9 -.01 27 01 560 20 1936 -1,237.53 -2#.99 -.01 32 56 1,262 30 1937 -#3#.72 -1#.93 -.0# 16 85 #59 62 1938 -9##.76 -21.85 -.06 18 39 968 #3 1939 -802.77 -20.17 -.07 16 5# 827 0# 19#0 -577.09 -17.1# -.11 13.36 601.77 19#1 -#72.55 -15.50 -.29 7.82 #91.32 19#2 ~62.80 -l#.89 -5.53 #.71 77.#0 19#3 -l60.61 -1#.79 -8.90 1.00 165.39 19## -#57.9# -15.08 -1#.17 .#2 #57.05 19#5 -2,182.12 -32.95 -18.22 .18 2,163.32 19#6 -3.577.7# -#2.20 -10.05 .22 3.5#7.60 19#7 -7,991.68 -63.06 -.07 15.05 7,919.60 19#8 -805.55 -28.79 -19.98 .23 797.25 113 19#9 -6#7.23 -17.92 -13.61 .31 6#1.96 1950 -1,100.#2 -23.38 -9.02 .30 1,089.#6 1951 -561.29 -16.67 -5.85 .## 553.86 1952 '20.319 -17.9# .73 #.5# 20.66 1953 -16.51 -12.#1 -1.3# 3.35 16.79 195# -15.25 -13.76 1.38 #.#7 15.58 1955' '37.58 -11.01 .68 #.31 #1.19 1956 ‘9-95 ”7.26 '-57 2.75 9-7‘0 1957 -9.50 -5.90 -1.#1 2.21 9.#7 1958 -61.89 -6.69 .12 5.58 6#.9# 1959 -37.16 -5.56 .18 #.30 39.20 1960 -27.8# -3.59 .13 3.68 28.68 1961 -37.38 -1.98 .06 #.26 37.37 1962 -17.30 -2.57 .16 2.85 17.31 1963 -22.97 -.73 .07 3.29 22.35 196# -5.9# -1.81 .33 1.#7 #.85 1965 -5.35 -1.26 .21 1.36 #.01 1966 -2.10 -1.28 .35 .57 1.0# 1967 -2.56 -1.02 .23 .71 1.07 1968 -2.0# -1.10 -.12 .#5 .59 1969 -1.30 -.99 -.31 .2# .#9 1970 -1.07 -.9# -.#6 .15 .#7 1971 -1.65 -.51 .07 .#8 .52 1972 -1.38 -.#0 .05 .## .56 1973 -1.20 -.50 “.28 .01 1.21 197# -.76 -.26 .01 .08 .18 1975 -.6# -.26 -.15 .09 .12 1976 -.#7 -.18 -.02 .02 .10 1977 -.#O -.16 .00 .01 .09 1978 -.26 -.12 .00 .08 .13 1979 -.19 -.09 -.02 .00 .07 1980 1# -.07 00 0# 07 BIBLIOGRAPHY llh Bibliography Abkin, Michael H. 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