MSU LIBRARIES "3—. RETURNING MATERIALS: RTace in book drop to remove this checkout from your record. FINES wiII be charged if book is returned after the date stamped be10w. TECHNOLOGICAL CHANGE, ECONOMIES OF SCALE, TRADED INTERMEDIATE PRODUCTS, AND SUBSTITUTION BETWEEN ENERGY AND NON-ENERGY INPUTS IN THE U.S. MANUFACTURING SECTOR By Ali Haji Mohamadzadeh A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Economics March 1982 /' ‘ . I . v / I" ' .1 ABSTRACT TECHNOLOGICAL CHANGE, ECONOMIES OF SCALE, TRADED INTERMEDIATE PRODUCTS, AND SUBSTITUTION BETWEEN ENERGY AND NONENERGY INPUTS IN THE U.S. MANUFACTURING SECTOR by Ali Haji Mohamadzadeh In recent multi-input studies of energy demand in U.S. manufacturing, the most frequent model specification has consisted of employing a static profitdmaximization framework defined over inputs of capital, labor, gross energy, gross materials, and gross output ("gross" meaning that these inputs include intra-industry, inter-firm shipments of traded intermediate products). In such models, the prices of the "energy? and Umaterials" aggregate inputs must be treated as endogenous rather than exogenous variables I as has been commonly assumed. Thus, in such Ugross' models, the application of Shepherd's Lemma to obtain Hicksian industry factor demand functions is inappropriate as shown by Samuelson (1953). This study has considered and estimated an alternative model in which cost and factor demand functions for U.S. manufacturing and nonenergy manufacturing sectors (for 1947- 71 period) are conditional upon the level of output of these sectors delivered to final demand (i.e. net" sector output). This "net" model framework provides a proper context for energy policy discussion since we are usually concerned with the energy intensity of a given level of net output. For purposes of estimation (via duality) a translog cost function is specified as a second order Taylor series approximation to the underlying production process. This study, then, presents estimates of two commonly used summary measures of price responsiveness for both sectors, namely, the factor price elasticities and the Allen partial elasticities of substitution among inputs. Our "net" model framework shows considerably mmaller values for factor price elasticities compared to the estimated values obtained in other studies. Regarding other issues, our empirical results indicate that homotheticity, homogeneity, constant returnstxascale, and neutrality of technological change must all be rejected for the manufac- turing sector, while for the nonenergy sector the homogeneous- Hicks-neutral specification is justifiable. The estimation of Hicks biases for the manufacturing sector reveals that over the 1947-71 period technological change has been labor- saving; capital, energy, and material using. This study has also examined returns to scale, and we conclude that in both sectors the source of growth.has been primarily due to utilization of economies of scale. Finally, our data rejects value-added specification for both sectors. DEDICATED TO My wife, Fattaneh, My parents, and my daughter, Sheava. ii ACKNOWLEDGMENTS I wish to express my sincere gratitude to the chairman of my dissertation committee, Professor Anthony Koo, for his guidance and.encouragement. Appreciation is also extended to the other members of the committee, Professor Norman P. Obst, Professor Bruce T. Allen, and Professor Robert L. Gustafson. I would also like to acknowledge my appreciation and indebtedness to Professor Richard G. Anderson of The Ohio State University for his valuable aid and suggestions. I wish to acknowledge the skill with which Terie Snyder typed the final draft. I also wish to thank my good friend John Davis for editorial assistance received from him. My special thanks, along with my love and affection goes to my wife, Fattaneh, for her encouragement, patience, and help which are of a different quality, but equally valuable. Last but not least, I also wish to express my heartfelt gratitude to my parents who supported me morally- and financially during my studies. iii TABLE OF CONTENTS Lxsw 0, TABLEs . . . . . . . . . . . . P85: INTRODUCTION . . . . . . . . . . . . . . . . . . . . . - 1 CHAPTER I. APPLICATION OF DUALITY PRINCIPLE IN THE THEORY OF PRODUCTION AND COST - A REVIEW . . . - - 7 1.0 Introduction - - - - . - - . 7 1.1 Definition of Production Technology . . - - 10 1.2 Production Function - - - . - - - 12 1.3 Duality Between the Input Requirement Set and the Production Function - - - - 14 1.4 Cost Function - - - ~ - - - 14 1.5 Duality Between the Cost Function and the Input Requirement Set - - - - - - - 16 1.6 Profit Function . . . . . . . . 21 1.7 Separability: Definition and Two Related Theorems.. . . . . . 32 1. 7A Separability and the Elasticity of Substitution . . . . . . . 36 1.8 Functional Forms: Choice Criteria . . . . . 44 1.8A Functional Forms Summarized. . . . . . 47 1.9 Translog Production Function and Its 1 Properties . . . . . . . . 51 1. 9A Mbnotonicity and Conve xity Properties . . . . 53 1. 9B Translog Function and Se parability . . 53 1.10 Translog Cost Function . . . . . . . . . 5% 1.11 Translog Profit Function . . . . . . . . . . 6/ II. SPECIFICATION AND ESTIMATION OF INDUSTRIAL FACTOR DEMAND FUNCTIONS WITH EXPLICIT ACCOUNT OF INTERNALLY PRODUCED ENERGY AND MATERIALS INPUTS. 76 2.0 Introduction ... . . . 75 2.1 A Simple Industrial Factor Demand Model. . . 33 2.2 Empirical Specification and Estimation . . . 97 2.3 Empirical Results. - - - - . . . . . . . . 106 III. TOWARDS A MORE GENERAL ECONOMETRIC ESTIMATION OF INDUSTRIAL FACTOR DEMAND FUNCTIONS- - . . . 127 3.0 Introduction - - - - - - . - - ~ - - - - - - 127 3.1 Technical Change - - . - - . - - - - - - . . 130 3.2 Returns to Scale - - - - - - - . - ~ - - 138 3.3 General Empirical Model - . - - - - . . . . 141 3.4 Technical Change and Bias - - . - 145 3.5 Homotheticity, Homogeneity and Return .to Scale . - - - . ° - 147 3.6 Estimation and Hypothesis Testing - - . . . 149 3.7 Empirical Results. . . . . . . . . . . . 153 iv TABLE OF CONTENTS (Cont'd.) CHAPTER III. PAGE 3.7.A Tests of Underlying Assumption, , , , , , 153 3.7.A. 1 Manufacturing , , , , , , , , , 153 3.7.A. 2 Non-Energy Manufacturing , , , _ , . , 162 3.7.3 Elasticity Estimates , , , , , , . , , , 167 3.7.3.1 Manufacturing . . . . . . . . 167 3.7.B. 2 Non- -Energy Manufacturing , . . . 172 3.7.C Economies of Scale and the Rate of Technical Advancement , , , , , _ , _ , , 176 CHAPTER IV. TESTS FOR THE EXISTENCE OF REAL VALUE- ADDED AND OTHER TYPES OF INPUT AGGREGATION IN THE U. 3. MANUFACTURING . . . . . . . . 191 4.0 Introduction . . . . . . . . . . . . 191 4.1 Weak Separability Defined . , , , , , , , _ 191 4.2 Various Specifications Discussed , , , , , , , 193 4.3 Statistical Results. . . . . . . . . . . . . . 204 4.3. A Manufacturing . . . . . . . . . . 204 4.3. B Non— —Energy Manufacturing , , , , , , , , 207 CHAPTER v. SUMMARY AND CONCLUSIONS . . . . . . . . . 211 APPENDICES. . . . . . . . . . . . . . . . . . . . . . 223 BIBLIOGRAPHY. . . . . . . . . . . . . . . . . . . . . 242 Table 2.1 LIST OF TABLES Total Cost of Inputs and Source of Energy and Materials Inputs for Manufacturing; Non- Energy Manufacturing, and Energy-Producing Sectors of the U.S. Economy-Selected Years (Millions of Current $'s).. . . . Maximum Likelihood (IZEF) Parameter Estimates of Translog Cost Function, Net Energy, and Net Materials Data, Without and With First- Order Vector Autocorrelation - U. S. Manu- facturing, 1947- 1971. . - . - - Maximum Likelihood (IZEF) Parameter Estimates of Translog Cost Function, Net Materials Data, Without and With First- Order Vector Autocorrelation - U. S. Non- -Energy Manufac- turing Sector, 1947- 71. - - . Maximum Likelihood (IZEF) Estimates of the Allen Partial Elasticities of Substitution Under First Order Autocorrelation, Selected Years, U.S. Manufacturing.. Maximum.Likelihood (IZEF) Estimates of Price Elasticities of Conditional Factor Demand Under First-Order Autocorrelation (Unres- tricted R) Selected Years, U.S. Manufacturing. Maximum Likelihood (IZEF) Estimates of the Allen Elasticities of Substitution Under First Order Autocorrelation (Unrestricted R) Selected Years, U.S. Non-Energy Manufacturing. Maximum Likelihood (IZEF) Estimates of Price Elasticities of Conditional Factor Demand Under First Order Autocorrelation (Unres- tricted R) Selected Years, U.S. Non-Energy Manufacturing. . . . . . . . . Comparison of Estimated Allen Elasticities of substitution - U.S. Manufacturing Comparison of Input Demand Elasticities - U.S. Manufacturing. . . vi Page 82 110 111 . 112 113 114 115 116 117 Table 2.10 LIST OF TABLES (continued) Alternative Maximum.Likelihood (IZEF) Estimation of the Allen Partial Elasticities of Substi- tution Under Various Specifications of Auto- Correlation: U.S. Non-Energy Manufacturing, 1959. . . . . . . . . . . . . . . . . Alternative Maximum Likelihood (IZEF) Estimation of Demand Price Elasticities Under Various Specifications of Autoregression - U.S. Non-Energy Manufacturing, 1959.. Alternative Model Specifications With Their Corresponding Parameter Restrictions. Maximum Likelihood (IZEF) Parameter Estimates of Translog Cost Function under Various Specifications, U.S. Manufacturing, 1947-71.-. 2 Values of xa n‘ . Maximum Likelihood (IZEF) Parameter Estimates of Translog Cost Function Under Various Specification, U. S. Non- -Energy Manufacturing, 1947- 71. Maximum Likelihood (IZEF) Estimates of the Allen Partial Elasticities of Substitution Under Various Translog Specifications, U S. Manufacturing, 1958. - - - Maximum Likelihood (IZEF) Estimates of Demand Price Elasticities Under Various Translog Specifications, U.S. Manufacturing, 1958.- Maximum Likelihood (IZEF) Estimates of Allen Partial Elasticities of Substitution Under Various Translog Specifications, U.S- Non- Energy Manufacturing, 1959. - - - - Maximum Likelihood (IZEF) Estimates of Demand Price Elasticities Under Various Translog Specifications, U.S. Non-Energy Manufacturing, 1971. . Alternative Estimate of Scale Economies and Rate of Technological Change, U.S. Manufacturing, 1971. . . . . . . . . . . vii Page - 122 123 155 156 159 163 168 169 . 173 174 178 Table 3.10 LIST OF TABLES (continued) Alternative Estimates of Scale Economies and Rate of Technological Change, U.S. Non- Energy Manufacturing, 1971.. - - - - - - Estimated Elasticities of Average Productivity of Inputs With Respect to Factor Prices, Output, and Time; Nonhomothetic-Nonneutral Model, U.S. Manufacturing, 1971. - Estimated Elasticities of Average Productivity of Inputs With Respect to Factor Price, Output, and Time; Homogeneous Hicks-Neutral Mbdel, U.S. Non-Energy Manufacturing, 1971 Alternative weak Separability Tests U.S. Manufacturing, 1947- 71 . . . Alternative Weak Separability Tests, U.S. Non- Energy Manufacturing, 1947-7l.. Total Net Output, Total Cost, and Cost Shares of Capital, Labor, Net Energy, and Net Materials Inputs - U.S. Manufacturing, 1947-71. Divisia Price Indexes of Capital, Labor, Net Energy, and Net Materials Inputs - U. S. Manufacturing, 1947- 71. . . . Total Net Output, Total Cost, and Cost Shares of Capital Labor, Net Energy, and Net Materials Inputs - U.S. Nonenergy Manufacturing,1947-7l Divisia Price Indexes of Capital, Labor, Net Energy, and Net Materials Inputs - U.S. Nonenergy Manufacturing, 1947-71. viii Page 184 185 . 205 206 233 233 234 235 236 INTRODUCTION The oil embargo of 1973 and subsequent enormous and continuous increases in the price of energy since then have accelerated extensive research on energy demand and possi- bilities for substitution among energy and non-energy inputs. Nowadays, the common belief that scarcity of energy will affect output,or at least retard its growth,1ooks so obvious as to require no argument. However, energy consumption in various sectors of the economy is affected differently by energy shortages. Differences in energy consumption are caused both by differences in total output and by differences in the energy intensiveness of production. The manufacturing sector accounts for approximately one- fourth of the aggregate energy consumed for power and heat in the U.S. This sector, therefore, has been broadly classified as an important sector to examine if a wise energy policy is to be pursued. Pursuing such a policy, however, requires empirical estimates of energy demand functionsand elasticities of substitution between energy and non—energy inputs,as the producing units' response to rising energy prices involves substitution. Recently, a large number of econometric studies have focused on possibilities of factor substitution in U.S. manufacturing. Examples of such studies are: Berndt and Jorgenson (1973), Hudson and Jorgenson (1974), Berndt and Wood (1975, and 1979), Griffin and Gregory (1976), Berndt, Fuss and Waverman (1977), Pindyck (1978), Berndt and Khaled (1979), etc. The most frequent model specification in these studies has been consisted of utilizing a flexible functional form for the aggregate cost function defined over four aggregate inputs of capital services, labor services, energy and materials. Then, researchers have employed the well- known Shephard's Lemma of the theory of the firm to obtain the industry's conditional factor demand as first partialv derivatives of the industry's aggregate cost function with respect to input prices. In the theory of the firm, application of Shephard's Lemma to obtain the firm's conditional factor demands is based on the critical assumption of exogenous input prices. Analogous application of Shephard's Lemma in the industry context violates this assumption, since the prices of energy and materials inputs are endogeneous to the manufacturing sector. The aggregate output of the manufacturing sector and purchased.materials and energy inputs have traditionally been ‘measured as gross magnitudes. As such, these gross magnitudes contain the intra-industrx inter-firm shipments of intermed- iate products which move among firms. More specifically, a considerable portion of total materials and energy inputs purchased by the manufacturing sector is produced by other firms within the manufacturing sector; the balance are imported from other sectors of the economy (i.e. are "primary" to the manufacturing sector). While the prices of these primary energy and materials inputs can be considered 3 exogeneous, the prices of the internally produced energy and materials are endogenous, since they respond to any change in the prices of primary factors of production. The subsequent application of Shephard's Lemma to obtain conditional factor demand functions, therefore, must be considered inappropriate in these models. This is precisely the specification error contained in these gross industry-level studies. In this study we consider an alternative specification for the aggregate cost and conditional factor demand functions, and estimate the production structure of the U.S. manufacturing sector as well as the non-energy manufacturing sector for 1947- 71 period separately. In particular, we specify our model, properly, over the level of deliveries of the aggregate manufacturing sector to the balance of the economy, i.e., the "net" output level of the sector. Energy policy discussions are usually, and properly, concerned with the energy intensity of net output. Consequently, we must seek estimates of the price elasticities of factor demand conditional upon the level of net output. In contrast, the "gross" model formula- tion does not provide an appropriate context for a meaningful energy policy discussion, sincean1industry's net output will fluctuate as primary factor prices change. The "net" model formulation also corresponds to the theory of consistent aggregation of factor demand functions of firms with neo- classical production functions (see Green (1964), chapter 9). Once our model is properly specified,this study will present estimates of two commonly used summary measures of price responsiveness; namely, the price elasticity of demand 4 for capital, labor, energy and materials, and the Allen partial elasticities of substitution between energy and non-energy inputs. These estimates, which are considerably different from their estimated values in other studies, constitute a challenge to the existing estimates of these elasticities. Traditionally, a particular assumption frequently employed in econometric studies has been the absence of technical change. In connection with constant returns to scale this implies that all changes in input bundles result from price-induced substitution within a fixed technology. A slightly weaker maintained hypothesis would be that all technical change was of a "Hicks-neutral" character. Again, in such a specification input mix changes are due to factor price changes. However, it is ideally desirable to estimate production structures under weaker assumptions. In particular, we relax both the assumptionkxf”Hicks-neutral'technical change and constant returns to scale which have been maintained in previous studies of manufacturing. This will allow us to examine the effect of biased technical change, namely, input mix changes which occur independently of relative price changes over time. we also can examine the return-to-scale characteristic of the production process. This amounts, therefore, to testing the hypothesis of "Hicks neutrality” and constant returnstxascale rather than to impose them a priori. Our 5 empirical results for the manufacturing sector reveal that technical change has been decidedly non-neutral (labor-saving; and capital, energy, and materials using), and that scale economies are substantial . Finally, this study examines various types of weak functional separability among inputs. The idea originated with Leontief (1947), and provides a criterion for input aggregation. According to Leontief, subsets of inputs which, are weakly separable from others may be formed into consistent aggregates with the property that marginal changes in the level of other input outside the separable subset have no effect on the technical relation among inputs inside the subset. One important application of separability is in the derivation of a value-added function. A large number 0f empirical studies dealing with investment demand and degree of technical substitution between capital and labor have employed a value-added concept instead of output. In this study we put this hypothesis to test along with other weakly separable specifications. To accomplish the objectives of this study we estimate the production relationship via the dual cost function because of its several econometric and theoretical advantages. Obviously it is desirable to employ a specific functional form which does not impose any a priori restriction on the Allen partial elasticity of substitution. The trancendental logarithmic (translog) function is such a candidate. Further, in the empirical implementation of the functional form some authors (e.g. Berndt and Wood (1975), Berndt and Christensen (1973a, 1973b) have taken the translog function as an exact representation of the true underlying production function. But Blackorby, Pirmont, and Russell (1977) and Denny and Fuss (1977) have shown that when the translog function is assumed to be an exact representation of the underlying technology, the separability conditions stated in these studies are too restrictive. This study departs from this restrictive assumption by assuming that the translog cost function is only an approximation to the underlying production technology. CHAPTER I APPLICATION OF DUALITY PRINCIPLE IN THE THEORY OF PRODUCTION AND COST-A REVIEW 1.0 Introduction It is a tradition to start production theory with a set of physica1,technological constraints or possibilities, usually called the production or transformation function or input requirement set, which describe the feasible production activity of the producing unit, also called the "firm", The theory develops,then,by'formulating the decision of the firm which acts to achieve its objectives subject to the limita- tion of its technology and within a certain institutional context. This procedure results in constructed factor demands and output supplies being a function of the technical limitation and the economic environment surrounding the firm. An interesting alternative is to approach the theory of production directly from observed economic data such as demands, supplies, prices, costs, profits,and revenues. This alternative method permits us to formulate the economic theory directly in terms of these functions. This approach is not only as fundamental as the traditional theory, but it is also more tractable. If the production function or input requirement set represents the firm's technological possibilities, the cost and profit functions are concerned with its economic behavior. This procedure gets its power from the so called "duality theorem" between technology and cost functions or, more generally, profit function that establishes that the two approaches are equivalent and equally fundamental. There are two main practical advantages associated with the theory of production duality. First, it enables us to derive, painlessly, the system of demand and supply equations consistent with the optimization behavior of the firm just by direct differentiation of a cost, profit or revenue function,in contrast with solving explicitly a constrained optimization problem by the traditional (Lagrange multiplier) method, where optimization and obtaining the explicit solution involve messy algebraic operation even with objective functions of relatively simple form. Second, duality theory is attractive from the point of View that the "comparative static" results associated with optimizing behavior are very easily derived. Duality theory has its roots in the work of Hotteling (1932), Roy (1942), Hicks (1946) and Samuelson. (1947); but it was the pioneering work of Shephard (1953) which treated the subject comprehensively and provided the proof of basic duality between the technology and cost function. His work was extended and refined later on by a number of authors, among*whom were McFadden (1962), Uzawa (1964), Shephard (19701 Diewert (1971), Lau (1978 and 1976) and others. 9 These works have built a framework and have paved the way for empirical research where use can be made of flexible functional forms such as the translog, and enabled researchers to use such complex functional forms rather easily, compared with the traditional methods. Empirical works such as Nerlove (1963), McFadden (1964), Diewert (l969a,b), Christensen, Jorgenson, and Lau (1971), Berndt and Christensen (1973a), Berndt and Wood (1975), Humphrey and Moroney (1975), Atkinson and Halversen (1976), and others are examples of works in which the dual cost and profit functions have been used as a basic tool in econo- metric production analysis. In what follows we start with a description of produc- tion technology utilizing the concept of the input require- ment set and then the production function (section 2). Then the duality between the input requirement set and the produc- tion function is explained. In section (4), the cost function and the duality between the cost and production function will be clarified; in particular we explain that, given fixed factor prices and a production function satisfying several properties, a total cost function may be derived under the assumption of cost-minimizing behavior, and conversely, given a cost function meeting some regularity conditions, a produc- tion function can be derived which in turn may be used to derive the original cost function. Finally, in the last section, we study the profit function and demonstrate the duality between the production function and profit function. 10 NOW’We will introduce the following notational conventions which will be utilized: x is an nxl vector with 1 x2 1 2 elements x1, ..., x ° x 3 (with. x and x each being n, n dimensional vector) means that each element of x1 is greater than or equal to the corresponding element of x2; x1 >‘x2 means that each element of vector x1 is greater than or equal to the corresponding element of x2 and additionally at least one element of x1 must be strictly greater than the corresponding element of x2; x1 >> x2 means that each element of x1 is strictly greater than that of x2 correspond- ingly; x' represent the transpose of x; Q is an nxl vector whose elements are all zero; and Rp+ = (x:x 3 Q) is the non- negative orthant in n dimentional Euclidian space. 1.1 Definition of Production Technology The technology of a producing unit, utilizing the service of flows of several inputs to produce a single output, can be described in several ways. One convenient way is to use the notion ofan1"input requirement set”. Suppose that there.are n inputs xi 3 0, i = 1, ..., n. Then the production structure of a producing unit can be characterized by C(y) = {x = (x1 ,..., xn) can produce at least y} (1.1) where x is an nxl vectorcflfinputs,and Q(y)specifies the set of all input combinations which result in a given level of output, y. The familiar isoquant might be seen as the ll boundary of 0(y), i.e. the efficient set of the input requirement set which is associated with a specified level of output y: I(y) = '{x' = (x1 ,..., xn) can produce, exactly, y} (1.2) The input requirement set is assumed to have the following properties: Assumption (1): 9(y) is anonempty subset of the non- negative orthant, RP+A It is possible that some inputs are not put to use; however, a zero level of output will result when no inputs at all are utilized. Then it must be true that 0(0) = Rp+.and y > 0 implies that 0 i 9(y); or stating this differently: if 0 s I(y) =9 y = 0. Assumption (2): 0(y) is closed. Closure means that if a sequence of points xn in 9(y) converges, the limiting point, say x*, also belongs to 9(y) and can produce y. Therefore 9(y) contains all its limiting points. Considering the definition of the isoquant we see that the isoquant of 9(y) belongs to 9(y). Assumption (3): 9(y) is monotonic. If an input bundle x1 can produce a given level of output, then this level of output can also be produced by a larger input bundle, i.e. if xlsQ(y) and x2 3 xE 2 then x CO(y). Similarly, an input bundle capable of producing a given level of output can certainly produce a smaller output level, i.e. if y 3 ya then 9(y) §;Q(y'). This is what is termed "free disposal". 12 Assumption (4): 0(y) is a convex set. Convexity implies that if x1 and x2 can both produce y, then any linear combination of x1 and x2 1 1 can also produce y, i.e. if x and x2e9(y),then 6x + (1-9)x2s9(y), 0 s e s: 1. Convexity ensures that we have a well-behaved technology, i.e., the marginal rate of substitution between inputs is nonincreasing. 1.2 Production Function An alternative way to describe the technology of a firm is to utilize the concept x1 => f(xz) 3 f(xl). iii) Continuity: The production function is continuous 13 from above; i.e., if for every integer N, xN : 0, f(xN) : y, linl xN = x, and y = f(x), then we have lim.f(xN) = y. This N+oo N-yco is, of course, a weaker property than continuity, and.is also consistent with certain discontinuous production processes. (iv) Concavity: the production function is quasi- concave over RP+, because the set {x:f(x) 3 y, xeRP+} is convex for every y 3 0. This property ensures that the marginal rates of substitution are now nonincreasing. Proof: To show (i), f(0) = 0, we must show that: f(Q) A = max {yIOeO(y)} =‘{0}. That is, we must Show that the onlz element of A is zero. By assumption, y > 0 implies that Q C 9(y). Thus the elements of A can not be positive. Therefore the only elements of A are zero. Then 08A and f(0) = max{yIOsQ(Y)} = max {0} ='{0}. Q.E.D. To see (ii),it mast be shown thatyx2 3 x1 implies that f(xz) 3 f(xl). To prove this it is enough to show that A = {y2x189(y)}_c_ B='{y=X2€9(Y)}. Let yoe:A. This implies that x1s9(y0). But x2 3 x1 and monotonicity of 9(y) implies that xsz(yO). Thus yOsB. Therefore A <3 B. ;f(x2) = max B 2 max A = f(x 14 1.3 Duality Between Input Requirement Set and Production Function The production function is derivable from the input requirement set by definition (1.3). As shown by Diewert (1971) it is possible to assume a production function, f(x), with the above properties and derive: 9*(y) = {x f(x) 2 y, xeRP+}. (1.4) It can be shown that 9*(y) posses the four properties men- tioned above about the input requirement set. On the other hand, using 9*(y) in (1.3) results in a production function, f*(x), which is identical to f(x), i.e. f*(x) = f(x). In a similar way, starting with 9(y) to derive f(x), and then using f(x) in (1.4), we obtain n*(y). Then it is true again that 9*(y) = 9(y). This is what we refer to as full duality between the input requirement set and the production function. 1.4 Cost Function One of the main behavioral assumptions, or rules of behavior, in micro analysis of a firm is that of profit maximization. Generally a firm will choose that input bundle that minimizes the cost of producing a given level of output, and then selects that output level, y, which maximizes profit. In what follows we first consider the problem of cost minimization and then that of profit maximization. Suppose that a producing unit, whose technology is given by an n factor production function y==f(xl, ..., xn), 15 or equivalently by input requirement set, O(y), is facing a given vector of factor prices, w' = (w1 ,..., wh), wi > 0, i = 1 ,...,IL and wishes to produce a specified level of output y. Assuming that the producer will minimize the cost of production of that output level, the cost function is defined as C(w,y) = min'{w'x | XEQ(Y): X 2 9} (1.5) or equivalently C(w,y) = min {w'x I f(x) 3 y, x 3 0} (1.5') This simply says that the producing unit takes the factor prices as given,and attempts to minimize the total cost of a specified level of output. Then the total cost function, in general, depends upon the specified level of output, y, the given vector of factor prices,and f(x), the given production function. Theorem (1): C(w,y) has the following properties: (i) C(w,y) is a positive real valued function defined and finite for all finite y > 0. w' = (w ,..., wn),wi>0 and C(w,0) = 0 (ii) (Kw,y) is differtiable in w and 3C(::X) a xi (w,y), i=1, ,n. 1 where the xi(x,y) are the conditional factor demand functions, and depend upon the level of output produced and upon the input prices. This is known as Shephard's Lemma, and for a 16 simple proof see Diewert (1971, pp. 495-496). (iii) C(w,y) is contineous in (w,y). (iv) C(w,y) is a nondecreasing left continuous function in y,and tends to plus infinity as y tends to plus infinity for every w >> 0. (v) C(w,y) is a nondecreasing function in w. (vi) C(w,y) is homogeneous of degree one in w. (vii) c(w,y) is concave in w. (viii) c(w,y) is strictly convex in y. (ix) xi(w,y) is the cost minimizing bundle of input i needed to produce output y > 0 given factor prices w >> 0. (x) xi(w,y) is continuous in (w,y). (xi) xi(w,y) is homogeneous of degree zero in w. 1.5 Duality Between CoSt Function and Input Requirement Set We saw that by using a production function or input requirement set which satisfies certain regularity assumptions (i-iv) we are capable of deriving a cost function which satisfies some desirable properties (i-xi) via definition CL5)or(15'). Now assume that we are given a cost function, C(w,y), satisfying properties (i,iv,v,vi,vii). Then it can be shown that this cost function enables us to derive or generate a family of production possibility set or input requirement sets, L(y), via the following: L(y) ='{x: w'x 3 C(w,y), w >> 0 and x 3 0} (1.6) for y > 0 and L(O) = Rp+ ='{x: x 3 0} for y = 0. 17 The; sets L(y) have properties (i-v) and is identical to My). Enturn we can use L(y) in (1.5) to derive the cost function C*(w,y) which is identical to C(w,y) which has generated L(y). This is what we mean by the duality between the cost function and the input requirement set. But by virtue of the duality between the input requirement set and the production function, there also exists a duality between the cost and production function as f(x) = max'{YIW‘x z C(w,y)} (1.7) y There are two important special cases which impose a certain functional form on the cost function, namely those of constant returnsto scale (CRS)technology and homotheticity of the production function. As Varian (1978) puts it, "There is a convincing argument that all firms should exhibit at least constant return to scale. The reason is that the firm can always duplicate what it has been doing before...thus if the firm is currently producing y, (by doubling) all of its inputs it should be able to build another plant exactly like the first and produce exactly the same amount in each plant. Thus with twice the inputs, the firm can produce twice the output". However, he mentions several counter arguments against this replication argument as follows: First, doubling the inputs may give us twice the output, but this does not necessarily imply that utilizing half the inputs gives us half the output as is required for CRS. Second, the replication argument demands that all the inputs 18 be increased. Considering that, in the short run, some inputs are fixed we would expect that Short run technologies show decreasing rather than CRS, since it cannot be taken for granted that doubling only some of the inputs will result in a doubling of the output. However, even in the long run it might be impossible to increase all inputs. Finally, we might be faced with the possibility of increasing return to scale, since the replication argument for CRS says only that doubling this scale should give at least double the output. However, the long run CRS situation is considered to be a standard situation. The following proposition shows the form of cost function when we have CRS technology (see Diewert (1974)), Proposition (1): If the production function, satisfying properties (i-iv), is, in addition, exhibiting CR8 (or is homogeneous of degree one) the cost function defined by (l.5')may be written as: c(w.y) = yc(.w.l) = y Mw). where C(w,1) = 1(w) is unit cost function. Proof: This proposition may be proved directly from definition (l.5'). By definition (l.5'): C(w,y) = min'IW'leCX) z y. x 3 0} (1.5') x = min {w'xly-1f(x) 31,x 3 0}. x Linear homogeneity of f(x) implies that tf(x) = f(tx). 1 Thus by letting t = y- the above can be written as 19 = min {w'xz f(y-lx) 3 1, x 3‘9} x = min {y W'(y-IX)= f(y'IX) 2 1. y-1 X x 3 Q} = y min {w'uz f(u) 3 1, u 3 0} , where u = y x. u Therefore, C(w,y) = y (w,1) = yl(w) Q.E.D. Another special case is that of homotheticity of the production function. Shephard (1953) introduced the concept of homotheticity and defined a homothetic function as follows: Definition: A function, F(x), is said to be homothetic if it can be written as g(h(x)), where g is a monotonic transformation of h and h is a homogeneous function of degree one; namely §§-> 0 and h(tx) = th(x). Proposition (2): If the production function, F(x), is a homothetic one, then the cost function C(w,y) can be factorized as g-l(y)-l(wx i.e. C(w,y) = g-l(y)A(w), where g-l(-) is the inverse function for g and A(w) is the unit cost function. Shephard has proven this proposition by using the concept ofaidistance function, which is complex and lengthy. We can prove this proposition in a much simplier way directly from the definition. Proof: By definition (5') we write: C(w,y) = miniw'x: F(x) 3 y, x 3 9} x min'{w'x: g(_h(,x)) 3 y, x 3 9} x 20 = min {w'x: h(x) 3 g-1(y), x 3 Q) , by monotonicity x property. min {w'x: (g'1(y>>’1 h(x) z 1. x z 9} X = min {g‘1(y)-w' (g‘1)'1x= h((g'1(y)>’1x> z 1. x (3"1(y))'1°x 2 Q} 8-1(Y) ° min {W'uz h(U) 3 l. u 3 Q} , where u (g’1(y>>'1-x = u . 3-1(y) - CCW.l) = g-l(y)1(w). Q.E.D. This proposition will be utilized later on in testing homotheticity of production function. The reverse of this proposition can also be described and proved. It can easily be shown that a production function, F(x), derived from a cost function of the form g-l(y)°A(W), is homothetic. Starting from the definition of production function, F(x), we may write (assuming that the cost function is separable as above) F(x) = max {yz w'x 3 g-l(y)1(w), for all w >> 0 . From above we obtain h(x) = g‘1> = max {g'1(y>: w'x 2 g’1x. w >> 0}. But h(x), as defined above, is clearly homogeneous of 21 degree one and this proves that F(x) is homothetic. Therefore the proposition may be stated in general as Proposition: The production funCtion, F(x), is homothetic if and only if the associated cost function can be factorized as C(w,y) = g-1(y)l(w). 1.61Profit Function A traditional method of obtaining a profit function is that of maximizing profit, defined as revenue less costs; i.e. Profit = py - Zwixi (1.8) subject to technological constraints the producing unit is faced with, symbolized as production function, f(x). Here we obtain the derived demand and supply,and then substitute back into the formula for profit given by CL$>and obtain profit function as Tr(P.W) = py* - Zwixi* where y* and x* are the optimized values, each being a function of p and w. The difficulty with this procedure is that of tractibility; only those production functions of relatively simple form can be used to solve the profit maximization problem explicitly obtaining the derived demand and supply functions. I An alternative way to study the technology of a 22 producing unit through the profit function is by invoking the duality theorem between the production and profit function. According to this theorem there exists a one-to-one relation between the production function and the profit function under some appropriate regularity conditions. Therefore for the purpose of theoretical or empirical analysis one may be better off to start by using an appropriate profit function. Definition: Assuming that the assumptions (i-iv) are satisfied for the production technology, and assuming further that the production function is bounded1 the profit function is defined as: n(p,w) a max {py - w'x|f(x) 3 y}. X,Y Theorem 2: As in the case of cost function, the profit function possesses several properties as follows: (i) fi(p,W) is nondecreasing in p and nonincreasing in w, i.e.; if pl'3 p and w1 5 w,then h(w1,pl) 3 h(w,p) (ii) n(p,w) is homogeneous of degree one in (w,p). (iii) h(p,w) is differentiable in p and w, and W = Home) 9 3n ,w) awi = -xi(w,p) i=1, ..., n. (iv) n(p,w) is a convex function in (p,w). (v) w(p,w) is a continuous function in (p,w); at least when p>0 and wi>0 (i=1, ..., n). (vi) x(p,w) and y(p,w) are continuous functions in (p w). 23 (vii) x(p,w) and y(p,w) are homogeneous of degree zero in (p.W). 1 Proof: (i):for p we have ply 3 py (i-l) IV 1 for w we have wl-x _< w-x or IA £2 -wl-x 3 -w-x; (i-2) adding (i-l) and (i-2) side by side we have: P]? ' Wl'x Z P? " W’X; then h(p1,wl) = max'{p1y-w1-xlf(x) 3 y} 3 max {py - w-XIf(x) 2 y} = “(p,w) 1 “(p1.wl) z “(p.W) From above it can be seen that for p1 = p but 0 f w1 _< w we have : (a) PY ’ W1"‘ 2 FY ‘ W'Y then 1r(p,w1) 3 n(p,w), i.e. the profit function is non- increasing in w. (b) for p1 3 p but wl = w, we have: ply - w-x 3 py - Wox then «(p1,w) 3 1r(p,w), i.e. the profit function is non- decreasing in p. (11): To show the linear homogeneity of 1r(_p,w) in (.p,w) we multiply p and wbyascalar A>0; then we have: 1r(>.p,).w) =- maxUIpy- lw'x|f(x) 3 y} = l max'{py - w'x|f(x) 3 y} = Ah(p,w) . Q.E.D. 24 Using the above result we can show (vii) as follows: h(Ap, 1w) = Ah(p,w) which, in turn, can be written as Apy (AP.AW)-Aw'x(xp.kw) = Appr,W)-Aw' x(p.W). Therefore it must be true (by the uniqueness property) that y(Ap,Aw) = y(w,p) and x(Ap,lw) = x(p,w); they are homogeneous of degree zero. (iv) To prove the convexity of n(p,w) we must show that: 1r(p*,w*) 5 afl(.Po,wO) + (l-a) 1r (pl,wl) for 0 so: 5 1 where apo + (l-a)pl (iv-l) p* w* awo + (l-o)wl. (iv-2) By definition of profit function we have: h(powo)==p0y(p0,wO)-w0 x(po,WO) 0 3 p y(P*,w*) - wO 'X(.P*.W*) . (.iv'3). 0,w0) and X(p0.w0) where the inequality sign comes because y(p are the maximizing levels of output and inputs at prices (p0,w0). Similarly l 1 l l l ' 1 . h(p .w )=p y(P .w )-w1 X(p :Wl) (iv-4). 3 ply (p*,w*) - wl'x(p*,w*). Multiplying both sides of (iv-3) and (iv-4) by a and (l-a) respectively and adding them up: 25 om + <1-a>w 3 p1)y+<1-a>w 2 P*Y(p*.W*)-W*'X(P*.W*) = n Q.E.D. (vi) To see the continuity of h(p,w) in w,p, we make use of the differentiability of «(p,w) in p and w. Assuming that h(p,w) is differentiable at w=e, we can show that it is continuous at w=e as follows: Since h(p.w) is differentiable at w=e then by definition: 11:: 1r'(pr)-1r(p',6) = 1T'(p.6) W-+ w-e or h(P,W);T:e(P.9) ._. page) + g(w) where lim e(w)=0.' Then w+0 “(p.W) = n(p.e) + (w-e) (n'(p,e) + €(w)). It can be seen from.above that: lim 1r(p,w) = h(p,0). wee 26 Therefore «(p,w) is continuous in w. In a quite similar manner can can show that n(p,w) is continuous in p too. Q.E.D. An interesting approach (due to Lau (1978)) to obtain the profit function and to study its behavior, without actually constructing it, is through the classical Legendre's dual transformation (LT). LT changes a given function of a given set of variables into a new function of a new set of variables. The old and the new variables are related to each other by a point transformation. To clarify this let us consider a given function of n variables 21, ..., zn, F = F(zl, ..., Zn)' A new set of variables may be introduced by the following transformation: t. s ——— , i=1, ..., n . (1.9) Assuming that the determinant of the "Hessian" of F to be different from zero, which guarantees the independence of the n variables ti' the equations<1~9)can be solved for 21 as functions of the ti’ 21 = zi (t1, ..., t ) , i=1, ..., n. We define a new function G as follows: 11 G(t1, ..., tn) = 1:1 tizi(t) - F(zl(t), ..., zn(t)) 27 The function G is known as the Legendre dual transforma- tion function of the primal function F. Consider the following partial differentiation of G with respect to t: n az.(t) n 32 (t) i j=l J i j=1 azj(t) ati < > I a—lz' < 5—” > = z. t + t0 - 1:1, .00, no By substituting %§—" tj we obtain: 3 i=1, ..., n. a) C) 2., ti 1 Q) This is a remarkable result which expresses the duality of TUT. The following scheme summarizes this duality: Old System New System Variables: zl, ..., zn t1, ..., tn Function: F=F(zl, ..., zn) G=G(t1, ..., tn) Transformation 8F EC .— I: t. z I: __ 3Z1 1 i ati a = , ..., z G G(t1, ...tn) F F(zl n) As this scheme reveals,the new variables are the Partial derivativescfifthe old function with respect to the old variables,and the old variables are the partial derivatives ofthe new function with respect to the new Variables. 28 The remarkable property of this transformation is that of its symmetry in both systams, i.e., the same transforma- tion that leads from the old to the new system leads back from the new to the old system. This means that the LT of G, G*, is: 6*(21, ...zn) = Eti(z) zi - G(t1(z), ..., tn(z)) = F. Therefore the two functions F and G are related to each other by the following set of dual relations: F(zl, , z ) + G(t1, , tn) = 22 1’ BF = 3G g ‘7: t "E z By using this transformation we are able to construct the profit function and study its behavior. Substituting the production function in the profit equation (1.8) results in Profit = pf(x) - Zwixi Assuming profit maximization as a behavioral assumption, a price-taker firm will maximize profit with respect to x, taking p and w as given. This resultsiJIthe profit function h as a function of p and w, which gives the maximized level of profit for each set of values of p and was Tr(P.W) = Pf(X*) — Zwixi* 29 Before going further it might be more convenient to work with normalized profit function given as TrNO!) = f(x*) - Erixi* , Wi th where r1 = If is the normalized price of 1 input; while the one-to-one correspondence between «(p,w) and nN(r) must be clear, because: n(p,w) = max {pf(x)-Zwixi} = x p max {f(x) - Erixi} = ponN(r). x In terms of our problem we are faced with the production function f(x) which may be identified as F(z), a normalized profit function nN(r) as G(t), and x as 2. The partial derivative of f(x) with respect to x is set equal to t, the new variable according to LT: 020) ”H: ll ('1 But profit maximization implies that: n [I 020) xlm I. ,1 Therefore r may be identified as t. The LT function can be constructed, by definition, and by recognizing that r=t, as G(r) = Zrixi(r) - f(x1(r), ..., xn(r)) , which is precisely —nN(r), i.e., the negative normalized profit function. Moreover, from LT we obtain = -x, (1.10) 30 which is the derived factor demand function. Furthermore, the relation between the "Hessian" matrices of the production function and the normalized profit function can be obtained by differentiating Q2 HI l H r; 09 N with respect to r and by treating x as a function of r, which results in -—1-32f [3-35]= I arar 3r (1 11) where I is the matrix of unit. But form (1.10) 3x BZWN —E a -§E§ET =[‘U ij]. (1-12) Therefore, by substituting (1.12) into (1.11) we obtain N l [U ijJ = ’[fij] where [“Nij] and [fij] are the ”Hessian" matrices of the normalized profit function and production function respectively. The concept of the profit function may easily be extended to a case when some inputs are fixed. If u represents the vector of fixed factors of production, then the production function may be written as Y = f(X.u) Then the profit function may be defined as 31 n(p,w,u) = max {py-Zwixilf(x,u) 3 y}. va This may be called the "variable profit function” due to Samuelson (1953-1954, p. 20). The dual corresponding to this is obtained as follows: f(x,u) E'{(x,u)lw(p,w,u) 3 py-Zwixi for p>0, w >>0, u > 0}. Alternatively, Legendre's transformation may be used to obtain the restricted or short run normalized profit function as hN(t,u) with the following dual transformation relations: (1) f(x,u) - wN(r,u) = Zrixi (ii) 8f(-) = r SEESELEL = -x 3x 3r N ... _ a (a) _ 3f(0) (111) x — - "3r r _ 3x N N - 3f(°) 3 (-) a -) _ af(.) (IV) 311 = “Bu ‘ITau _ Bu N . (V) f") = “NW - 2r: if: ”NM =af<-> - wig—1% (Vi) u u From.this LT relationship,which represents a system of partial differential equations,one may either construct the normalized profit function or study its behavior, given the production function and the first order necessary condition for max (and vice versa). 32 Moreover, asis shown by the fact that the production function and the normalized profit function are LT of each other, a partial differential equation for f(x,u) in x and u transforms to a partial differential equation for nN(r,u) in r and u; therefore the equivalent properties of the production function and the normalized profit function may be deduced immediately. This technique has been extensively used by Lau (1978) to study the conse- quences of assuming several differential properties regarding the production function on the normalized profit function and vice versa. Furthermore, the LT has the advantage of being quite useful in deriving the solution of certain partial differential equations,being intractible otherwise. 1,7 Separability: Definition and Two Related Theorems The concept of separability is of quite an importance in some areas of economic theory. Sono (1945) and Leontief (1947) are pioneers. 3 g (xk+l""’xn-1)’ xn) Here one might think of g1 (i=1,2,3) as the intermediate output of sector i (iron mining industry, coal industry,...) which lateris combined with xn’ say labor, to produce f(x), steel. In fact gi acts as a new factor which is combined with the original input, xn, to produce steel. One might ask, as Leontief (1947b) puts it, "given a quantitative description of the overall relationship such as f(x),can one without any additional outside information, i.e., solely through examination of the mathematical properties of the function, f(x), establish the possible existence of such 34 subsidiary groups of variables (such as (x, ...,xr), (xr+1,...xk)...) and describe the properties of the corresponding intermediate functions (g1( ). 82( )...)?” Different theorems on separability will provide a general answer to this question. The concept of separability has been studied under two conditions which have been named weak and strong separability, coined by Strotz (1957) and (1959). Let y = f(x) represent the production function, where X = (X1.....Xn) is a vector of inputs. Partitioning this vector into s mutually exclusive and exhaustive subsets N* = (N1,...,NS), the weak separability of f(x) may be defined as follow: Definition: The production function, f(x), is weakly separable with respect to the above position, if the marginal rate of substitution (MRS) between any two inputs 1 and j from any subset Nm,msl,...,s is independent of the inputs outside Nm’ namely, ;3—.(fi) = o v i j CN and k ¢ N axk f; ’ ’ m m Definition: Letting s > 2,the production function f(x) is said to be strongly separable if the following condition is fulfilled: f 5%; (I?) = o, v ist, jeNh. k t NmU’Nh h # m J 35 This means that MRS between any two inputs from sub- sets Nm and Nh is independent of the quantities of third inputs which are not in Nm or Nh’ These conditions may also be written as fjfik - fifjk = 0 V i,jeNm and k e Nm for weak separability and V ist, jeNh and k é NmUNh for strong separability, where f. - —££§l , partial derivative of the production function 1 2 th = 3 1? input and f i j _—3x13xj Note that if s=2, then ieN1 implies that kELNz and hence with respect to i jENl; then the condition for strong separability reduces to the condition for weak separability. The following two theorems are of fundamental importance regarding functional separability: Theorem (1): The production function f(x) is weakly separable with respect to the partition N*==(N,... ,Ns) if and only if f(x) is of the form G(g1(x1),...,gs(xs)),where gm(xm) is a function of the subvector xIn alone which are the elements of Nm only. Theorem (2): The production function f(x) is strongly separable with respect to the partition N* = (Nl"°"Ns) if and only if f(x) is of the form G(gl(x1)+...+gs(xs) where gm(xm) and xm are defined as before. These theorems have been proved by Goldman and Uzawa (1964). 36 1.7A Separability and Elasticity of Substitution To understand the nature of the relation between functional separability and the elasticity of substitution it is necessary to interpret functional separability in economic language. Keeping the argument simple, let us assume a production function with three inputs, x1, x2, and x3; and let inputs x1 and x2 be functionally separable from the third input, x3. This means, by the definition of weak separability, that the marginal rate of substitution between inputs 1 and 2 is independent of the level of input 3 used. In other words, if the usage of x1 and x2 is held constant and the usage of x3 increases, the increased flow of x3 makes x1 and x2 more effective at the margin, and raises their effective- ness by exactly the same relative amount. This, as we saw above, was stated as 3%; (:%)==0. Here we see that the augmented usage of x3 shifts the marginal products of x1 and x2 by the same proportion (which is observationally the same as the familiar Rick's-neutral technological change); there- fore x3 must have an equally close substitution or complemen- tary relationship to both inputs, namely, the partial elasticity of substitution between x1 and x3 is the same as the partial elasticity of substitution between x2 and x3, one appropriate measure of the elasticity of substitution being the Allen partial elasticity of substitution in cases where more than two inputs are involved. This inequality of the partial elasticities of substi- tution, in the case of functional separability, can be 37 demonstrated mathematically. Berndt and Christensen (1973b) have shown this equality in the case of the homothetic production function, but in what follows it can be shown that the homotheticity of the production function is not required. Before showing the relation between the separability of the production function and the equality of the elasticities of substitution, we shall first transform the Allen partial elasticity of substitution into a definition in terms of the cost function. This transformation has been shown by Uzawa (1962) for a production function which is homogeneous of degree one and subject to a diminishing marginal rate of substitution, and in terms of the unit cost function; while this may be shown for a twice differentiable, strictly quasi- concave production function f(x) with strictly positive marginal products. Let y = f(xl,...,xn) be a twice differentiable, strictly quasi-concave production function with a finite number, n, of inputs, each having a strictly positive marginal product. Also assume that the vector x = (x1....,xn) is partitioned into s mutually exclusive and exhaustive subsets N* = (Nl”"’Ns)' The Allen partial elasticity of substitution, Oij’ between two factors 1 and j (i#j) is defined n hi1 thh'Fij' °ij ‘ xileFl where 2 fh - %§;" fhg - giiiig , h,g 1, , n, ro £1 . ..fn ‘ |F| = 51 £11. .fln LfP°'°'f“1°°"fRnJ and [Fiji is the determinant of (ij)th cofactor in F. Using this formula for two particular inputs, 1 and k, such that ist and k¢Nm (mel, ..., s), the partial elasticity of substitutiog, oik' between them can be written by: X f lFol 01k ' hgi.:: IFI 1k 1 for ist and k¢Nm- It is a well-known fact that the firm, in order to minimize the cost of producing of a specific level of output, must adjust the factor inputs such that the ratio of price to 'marginal product will be the same for each factor or w .f = --E ; where A is interpreted as being the h A marginal cost of output. Also the rate of change of the independent variables (x1, ..., xn),with respect to changes in factor input prices, is obtained (see Samuelson (1947), pp. 63-69) as 8xi = lFikl FE; XTFT Substituting these relations in Gik we obtain On the other hand by Shephard's Lemma we have: x = acgw, ) = C. and W1.. 1 :4- 3x 2 3 Z 3w1 3 awpéx' ) = Cik ‘ k i k By substituting these in Oik and utilizing the fact that thxh = C(w,y) we get: CC ik . Oik 3 EEC; , IENm, kiNm (1.13) Similarly CC. k . o k 35%; . JeNm. k¢Nm Now setting oik and ojk equal to each other we get: .CC CC. 11‘ - TUE; i,jeN , keN tick jk m m Or ClClJ - CjC1k = 0, i,jeNm, keNm (1.14) This means that Oik = ojk (i,jeNm, k¢Nm)if and only if (1.14) holds. But(l.l3):h3nothing but the condition for weak separability of the cost function C(w,y). On the other hand these conditions, under a suitable assumption such as 40 homotheticity of the production function or homotheticity of the micro production functions gm(xm), are also the conditions for weak separability of f(x). Thus, it was shown how weak separability of the production function with respect to the partition N* implied that Oik = Ojk (i,jeNm, kéNm). But still we need to know under what conditions weak separability of the production function implies weak separability of the cost function in the same partition. To see this, let y = f(x) = G(gl(x1), ..., gs(xs)) where gm(xm) is a micro production function and xm is a subvector with its elements being that of NIn only. The cost function associated with G( ) may be defined by: 1 . S ' C(w , ..., wS,y) = min { Z w mmeG(gl(x1),... , 1 l s = x,...,x m gs(x3)) 3 y} = min x}...,xs;y1,..., ys . S l m { E W In): |G(Y1.~. Y3) 3 Y; m=1 ym = gm 1}. m - x s In the second step we must minimize Z ¢m(ym)-Am(wm) mal with respect to ym and subject to G(yl, ..., ys), i.e., 1 s . - S m. m m m C(w, ....w;y)=m1n {Z¢(y)-A(w)= l 3 mal y ,...,y G(yl. ..., VS) 2 y} = C(y; 11(w1 , ..., As(ws)). Q.E.D. In this proposition it should be noted that G is a homo- thetically separable function.2 Next we might consider the case in which the G S function itself is homothetic in x1, ..., x . In this case it can easily be shown that a weakly separable and homothetic function implies the homotheticity of each subfunction,3 and so the problem becomes like the one we had before. Another case is the one in which the G function and the subfunctions gm(xm) exhibit constant return to scale, i.eu they are homogeneous of degree one. The cost function in this case is obtained as: s C(wl, ..., ws;y) = min '{ Z w'mxm:G(gl(xl),..., 1 s m=l x , O I I ’x ss(xs> 3 y} 42 = min '{Xw'mxm:G(Y1,...Ys) Z Y; x1,...xs;yl,..-.ys Since each gm(xm) is homogeneous of degree one,it has a dual cost function of the form ymlm(wm), where 1m(wm) = min{wrmxm:gm(xm) 3 l}; m=l,...,s. m X Therefore we have: C(') = min '{Zymkm(wm):G(yl.....ys) z y} y ,...,y5 = Y‘C(Al(Wl)..... Xs(w§))since G is homogeneous of degree one. As Diewert (1974) has pointed out,the elasticity of substitution will have a special feature in this case as will be described below. Evaluating the elasticity of substitution between the ith jth input from group two, x?, we have, by definition (1.13) input from group one, xi,and.the C 32C 1 2 12 awi awi oi.- 3 3C ac l 2 8wi 23wj l 2 3A 8A 0C 0C __2. (y ) y 12 3w; 3w. = 1 _] =CCl2 = 012 y“: 311 y“: 312 Clcz 1 a??? 2 5:? 43 where Ci is the first order partial derivative of C with respect to its 1th argument, and 012 is the second order partial derivative of C with respect to its first and second arguments. The special feature about this elasticity is that oi? does not depend on the subgroup indices 1 and j or generally this means that 0;?1= Gmn. Therefore, the elasticity of substitution between the two primary inputs 1 and j from subgroups (intermediate inputs) m andtn is the same as the elasticity of substitution between intermediate inputs m and n. Interestingly,the relation between a weakly separable production function and the profit function may be established. 1, ...ws) be the profit function associated.with Let n(p,w G(g1(x1),...,gs(xs)); then we can show that the profit function is also weakly separable in input price aggregate as follows. 8 ' w(p,wl,...,ws) = max '{py- Z W mimlG(gl(Xl). y;x1,...,xs m=1 ., gS(xS) 3 y} = max y;y1,...,ys;xl,...,xs ' 8 .mm 1 S {py - 1w x No .....y) 23'. gm (XIII) = ym} 44 This consists of two steps: in the first step,we maximize S I py - Z1 w mxm subject to gm(xm) = ym and with respect to m: x1,...,x% which results in a cost function of the form ymxm(wm) provided that gm(xm) is homogeneous of degree one. In the second step,we maximize py - Zym1m(wm) subject to G(yl,...,y8) = y and with respect to y and yl,...,ys, i.e., maxl '{py - Xymlmwm): G(y1,....ys) z y}. s y;y ,...,y which results in n(p;Al(wl),..., As(ws)), i.e., if the produc- tion function is weakly separable and the subfunctions are homogeneous, then the profit function will be weakly separable. 1.8 Enngfiigné; Forms: Choice Criteria In studying a producing unit, it is a common procedure to assume an objective function and then a behavioral assumption which enables the producing unit to optimize its objective subject to a side relation, given the conditions surrounding the problem such as market conditions which determines the prices of inputs and outputs, etc. However.to answer questions such as the elasticity of demand for fuel, elasticity or ease of substitution between capital and labor,auu1economies of scale, which are the most concrete applicationscfifeconomic analysis to policy questions, it is necessary to know the specific parameters describing the producing unit's behavior. 45 The answers to such questions which might have serious policy implications can be found only by examining the data and estimating the parameters of the agent's objective function, using some statistical method. In doing so, it is necessary to choose a functional form for the objective function, say, the production function, and try to estimate its unknown parameters. The task of choosing a functional form is not an easy one, since the researcher must compromise between different and often unreconcilable properties a functional form has. The main two desirable properties a function must have are: first, the capability of representing a wide range of techno- logies, in order to minimize the prior assumptions imposed on estimating the equations, and second, the tractability, i.e., the ease of computation, estimation, and interpretation. Flexible functions such as the translog (TL), the generalized Leontief (G.L.), the generalized Cobb-Douglas (GCD), etc., are the ones with the first property. These functions may be considered as a second order approximation to any arbitrary function; their elasticities of substitution between different pairs of factors are variable, they permit the existence of uneconomic regions, and, as far as estimation is concerned, they are linear in parameters. However, these functions have a complex form.and often involve too many parameters, which makes estimation not an easy job. On the other hand, there are simple functional forms such as the Cobb-Douglas (CD), the constant elasticity of substitution (CBS), and the 46 Leontief functions which are tractable, while they have restrictions on the elasticity of substitution, form of separability, constancy of factor shares (in CD case); still the choice between these forms needs careful consideration and depends upon the use to which they are to be put. It is a well known fact that the CD and the Leontief production functions have an Allen elasticity of substitution (AES) of unity and zero for all input pairs respectively, although there is nothing in theory which suggests that such a restriction will be universally met. The degree of factor substitution is of great importance, since it has a number of repercussions. A change in an input price ratio will affect the cost of production, product prices, and income distribution according to factor ownership, and consequently consumption and savings patterns will be affected. In this sequence of events, a proper measure of the degree of factor substitution plays an important role. This shortcoming decreases to some extent in the case of the CES function, since it allows the ABS to deviate from unity, although it is doomed to be constant by construction. Although in the case of the two factor production function it might be justified, on the ground that the elasticity of substitution is constant, in the case of a multi-factor tech- nology it necessitates that all inputs be equally substituta- ble, which is not realistic unless it is tested empirically. In fact, in the multi—factor case the CES model stands in sharp 47 contradiction to economic common sense, as well as to the very purpose of such studies. This theoretically and empirically unjustifiable restriction has led to several studies which have indicated that the elasticity of substitution between capital and labor varies, usually inversely and sometimes significantly, as capital deepening occurs. This evidence casts doubt on the empirical uSefulness of any CES function, and has led several authors4 to develop functions with the property of making the common elasticities of substitution a function of some variables such as output level of factor ratio, etc. However, the CD function has been widely used in empirical work, regardless of its shortcomings apparently because, first, the direct estimation of the CD function, using aggregate time series data, is not inappropriate, due to the fact that substitution effects are not well identified by highly colinear data, and second, as Fisher (1969) argues, the constancy of the factor shares of labor and capital in aggregate data fits the CD hypothesis. With this general background in mind, we will now review several functional forms, and will study their properties in brief. 1.8A Functional Forms Summarized There is an interesting way, due to Mundlack (1973), to describe several kinds of functional forms by using a quadratic form and by imposing some appropriate conStraints on 48 that. The form is yo = (1. 1') = ao+z'e+x'Bz *9 B Z (1.15) To derive various functions, the following transformations will be utilized: T1 yr 3 xi ' pi I 0 T2 y1 = 1n xi ; p1 = 0, 1=0,l, ,n, where x0 is output and x1, ..., xn are inputs. (1) Cobb-Douglas (CD) The CD function is derivable from (1.15) by imposing: B E 0, and by obtaining variables by applying the logarithmic transformation T2. The following notation is used to state this as (1.15) ()(B E 0) n T2 The result is n 1n x = a + Z a. 1n x. 0 0 i=1 1 1 (ii) CES-like This function may be obtained by (l. 15) 0(a0=0)O(B E 0) n T1 (1 {oi=o} for i=0.l. - . . .n; which results in 1 pp I x = ( a. x. ) 0 i=1 1 1 49 (iii) CRES (Constant Ratio of Elasticity of Substitution) The CRES function is developed by Mukerji (1963) and Forman (1965), and can be derived from (1.15) by imposing: (1.15) n(o0=0) n (B s o>(\ Tl which results in o = (121 aixi ) Unlike the CES, this function does not have the identical and constant partial elasticities of substitution. However, the ratios of the elasticities of substitution are constant for this function, and are not necessarily the same. The CRES is a homogeneous function only when the 0's are all equal, and it obviously reduces to the CES-like in this case. (iv) CRESH (Homogeneous or Homothetic CRES) (Hanoch 1971). The CRES functions are not homogeneous or homothetic, and this causes the Allen-Uzawa elasticities of substitution to vary with output, as well as with the factor combination. This makes the expansion path (for given factor prices) curved in a predetermined and often undesirable way, dictated by the form of the function. In the CRESH, this problem is removed due to homogeneity and homotheticity of the function, so that the elasticities of substitution vary along the isoquants and differ between pairs of factors, although the elasticities of substitution stand in fixed ratios everywhere. 50 The CES and its limiting forms (the CD(o-l),the Leontief (o=0), and linear (08m) functions) are special cases of CRESH. Its functional form is p xi pi a (——) - l E 0 i=1 1 x0 The parameters of this function are estimatable from a system of log-linear equations, given data on factor prices, quantities, and output, and assuming cost minimization. (v) Generalized Leontief (Diewert 1971) This function is obtained from 0115)as: (l.15)n(a020)()(a50)nTln{pO=l and pj=%} j=l, ..., n . The functional form for this function is written as: x = X 28.. x x. 0 i j 13 i 3 (vi) Generalized Diewert (Generalization of the Generalized Leontief) This function is derived as (v) by setting the parameters as CO = p and . = ‘=l ..., . ojtoJ. n The resulting functional form is also known as the quadratic mean of order p, and is written as 2 2 1/0 x0 (2 §Bij xi xj ) 51 (vii) Translog Function (Christensen, Jorgenson and Lau (1971) (1973)). The translog (TL) function developedby Christensen, Jorgensen and Lau may be obtained from (1.15) by imposing: (1.15) 0T2 which results: n I = + . l . + .. . . n x0 o0 til a1 n x1 g jle 1n x1 1n xJ (viii) Quadratic form; This form is obtained by imposing (L15){)Tln{oj=l} j=0,l,...,n . which results in x0 = a0 + Zoixi + XZBij xixj 1.9 Translog Production Function and Its Properties This function is defined in Christensen et al (1970) as: ‘2‘ n a n g ( y . ln x.) y = no ( H x 1)( Hlxi j=l 13 J ) Taking logarithm of both sides we obtain: n 1n y = 1n a0 + 1:1 oi ln xi + g g g Yij 1n xi 1n xj , (1.16) where y is the quantity of output and the x's are inputs; a0 is a parameter which represents the state of technical knowledge; oi and Yij are technologically determined parameters Equality of Yij and in (i # j) is necessary for the 52 applicability of Young's theorem to integrable functions, while this equality could in principle be tested. In this function all input quantities must be strictly positive, because 1n}(+ -e as x + 0 and output would be ill-defined. The translog function has several interesting properties in both its theoretical and empirical application, especially if all the log quadratic terms are omitted, when it reduces to a Cobb-Douglas function; and as Christensen et al (1970) have shown, most of the CES-like functions may be derived from it as special cases when appropriate restrictions are imposed. The interesting feature of the translog function, which has entitled this function, is its flexibility. It can be considered a second order approximation to any functional form for values of factors near unit. To see this, let us start, in general, with the following function: y = ¢(x1, ..., xn) where y denotes output, and the x's are services of inputs. This function may be rewritten as: In x1, e1n x 1n y 1n ¢(e n) . or 1n y f(ln x1, ..., 1n xn) . (1.17) Applying Taylor's method (see Allen (1938)), we expand 017) around point x_ = (1) or 1n 5 = (Q), (where x is a n dimentional vector of input), obtaining: 53 “ 3f 1n y = f(ln 1, ..., 1n 1) + Z 3 In x. i=1 n x 1 1n §=9 . 2 a f + i Z Z aln x. aln x. 1“ xi 1“ xj + R 1 J 1 3 1n =0 —'- (1.18) where R represents the higher order terms. Note that 2 3f 3 f and are constant fo 3In xi lnx=g 81n xialn xj 1n §=Q r i,j=l, ..., n. Therefore we let the following: r 3f mi Edi 1:1, ...,n. In ETC = 32f = 32f J Yij ' alnxialnx. 81nx.31nxi J In §= Q 3 1n i=9. = = (1.19) - in . 1.3 l. . n L f(ln 1, ..., 1n 1) = 1n a0 . Therefore, by substituting (1.19) in (1.18) and omitting the higher order terms, R, we get (1.16) which is the translog function.Q.E.D. ‘l-9A Monotonicity and Convexity Properties An important neoclassical assumption on production function is that of increasing marginal productivity of all factors, namely: 1 3x. - , O. O ’ 54 This implies that X 3 ln _ i 31 ._ a'l'n'xi - 3’ 3x1 2 0» 1‘1: “- Because xi and y are positive therefore a 1n n . = 0+ .01 O>O =1, 0.0, ’ TTIE_§; oi jél 713 n xJ _ 1 n (1.20) It is obvious that in general these monotonicity conditions cannot be globally satisfied for all quantity configurations. Notice that the translog function possesses an uneconomic region over ranges of input space if (i) xj;+ 0 (jsl, ..., n) and Yij > 0 , or (11) xj + m (j=l, ..., n) and Yij < 0 . Both cases imply negativity of fi, the marginal product of the 1th input. This indicates that the translog function exhibits much more flexibility than either the Cobb-Douglas or the C.E.S. function which do not allow the existence of an uneconomic region. Should this case (negativity of marginal product) happen, a profit maximizing or cost minimizing producer will not operate in that region as long as there exists a non- negative price associated with that input. In general, however, a local satisfaction of the monotonicity conditions, especially at the point of approximation, x = (1), is expected. This implies that Q IV 0 i=1, ..., n . 55 Having Ci 3 0, i=1, ..., n, and an arbitrary set of Yij's’ it is always possible to find 1n xj's, not all zeros, in order to have (1.20.),satisfied. Another property that a neoclassical production function must also have, in addition to the monotonicity property, is that of the concavity, i.e.,it exhibits decreasing returns to scale. To satisfy this property,1fijj must be negative semidefinite. Hence a necessary condition is that fii 3'0.or 2 ,3 y a a a}; a a alny fii 8x. (axi) 8x. (x. a lei) 8x. 1 1 1 1 32 xi 8x. ' y galny( 1 )+_y_3(81ny) aInxi x2 x13x1 Blnx. i 1 :12 <<§—}§‘T,Xq-1>§-§%§i—+yfi>go 1 This inequality must be satisfied in particular at the point of approximation. Therefore ,2 L12 = y ((ai-l) ozi + yii) 5 0 (1.21). 3x 1 1n r=9. To have (1.21), satisfied, given monotonicity, i.e. oi 3 0, the following must be true: 0 f oi f l Yiifo‘ 56 Moreover, we compute cross partial derivatives as 322 ="y 3'lny alny + y. .) axiaxj xixj 3 In xi 3 In xj 1j Evaluating this at the point of approximation we obtain 2 LL. axiax. y (aid. + vi.) J J Concavity of production function at 1n xég implies, by a continuity argument, that.f(-) is locally concave in.the neighborhood of 1n ETQF and this is all we need, because, as has been shown in standard microeconomic texts, the economic region is the concave region, even though the function might have uneconomic or convex regions which are locally convex and might exhibit increasing return to scale in certain ranges of the inputs. Therefore a necessary and sufficient conditions for f(-) to be locally concave at ln x=Q is that the matrix ‘ r Carnal "’ 111 O‘1‘”‘2"'1‘12 °‘1°‘n+11n [f..] 1‘3 1n §=O {anal + Ynl . . . . . . . (an-1)on+ynn J be negative semidefinite which in turn requires that all the principal minors be negative semidefinite. Therefore to check concavity we must compute these principal minors to see whether the concavity conditions are met. 57 It might seem desirable to actually test for the concavity assumption either in general or at some predeter- ‘mined level of inputs. In this regard, the translog function is appropriate and ideal since global concavity is not assumed a priori. In addition to the above test, it is very convenient to test statistically the homogeneity of the production function. If the production function exhibits constant returns to scale (CRS), first degree of homogeneity, it must be true that: 1n f(lxl, ...,lixn) = 1n A°f(x1, ..., xn) = 1n f(xl, ..., xn) + 1n 1 This implies the following set of linear parametric restrictions: g Yij = g yji = zzyij = o v 1,J=1, ..., n . which can be tested easily, given the required data. Thus we notice that how flexible the translog production function is, since it allows increasing, decreasing, and CR8, as well as completely variable returns to scale over the range of inputs. The translog function will also provide a suitable framework for empirical work since the function and its corresponding marginal productivity conditions are linear 58 in their parameters and therefore may be estimated, taking into account the linear equality restrictions across equations, by applying multiple regression techniques. 1-93 Translog Function and Separability When we were discussing separability we saw that different aggregate indices of heterogeneous capital, material, and labor inputs could be used in the production function, based on the assumption that the production function was separable in those aggregate indices. Separability allows us to use aggregate data when disaggre- gated data do not exist or have poor quality. Another advantage of separability is that of its consistency with decentralization in decision making or equivalent optimization by stages (multistage). In some cases the appropriate disaggregated data exist,but even in these cases only multistage optimization and estimation is feasible, due to the large number of inputs involved. On the other hand,the separability specification will cause severe restrictions on the structure of technology,and hence on the possible form of the production function. In spite of this shortcoming, separability is a pivotal concept in production function estimation; although in most of the production function studies separability and the existence of aggregate inputs have been assumed a priori. However, there are two recent studies, done by Berndt and Christensen (1973 and l974),in which empirical tests of separability 59 and the possible existence of consistent aggregates of labor and capital have been furnished for the first time, using a translog function. Instead of using the translog function as a second order approximation to some unknown arbitrary production function, they have implicitly assumed the translog function as a £323 representation of the underlying technology. This procedure results in a different and more restrictive test for separability, and is not accepted as a general test of the separability hypothesis. In fact, the real problem with tests of separability, based on an exact interpretation of the translog function, is that they are not only tests of the null hypothesis of separability. Instead they result in tests of the joint null hypothesis of separability and a particular inflexible functional form for either the aggregate functions or the production function as a function of the aggregate inputs. The following proposition is due to Denny and Fuss (1977) and summarizes the whole argument. Proposition (3): The separable form of a translog function,interpreted as an exact production function,must be either a Cobb-Douglas function of translog subaggregates or a translog function of Cobb-Douglas subaggregates. To clarify this proposition,assume that the three input translog function(116), interpreted as being exact, is weakly separable as 1n y = f(ln g (1n x1, ln x2), 1n x3) , where g is an input aggregator function. Since f is weakly 60 separable then it must be true that: f.f. - f.f. 1 3k J 1k = O i,j=l,2; and k=3 Substituting for first and cross partial derivatives in the above we obtain: Si ij " Sj Yik = 0 (o:i + g Yij 1n xj)yjk - (aj + g in 1n xi)yik = 0 or 3 (“ink‘o‘j Yik)+ Z (Yimyjk-Yikyjm) 1“ "m = O m'1 (1.22) To have G»22)equal to zero we need both parentheses equal to zero. However, a sufficient condition for(L22) to hold is: Yik = ij = 0 ° which are termed "linear" separability constraints. For nonzero Yik and ij a necessary and sufficient condition for Ou22) to hold are aink ‘ “j Yik = O Yiijk ‘ ijYik = 0 ’ which may be written all together as sq“? y. y. —£E.= vigi= 6. m=l, ..., 3 J J'k J'm 61 Now there are two possibilities: (a) Substituting the linear separability constraints (y13 = y23 = 0 in 3 inputs caSe) in translog production function (interpreted as'gxggg) results in a production of the form (see Denny and Fuss (1977)) 1n y = 1n a0 + 6g 1n g + 5h 1n h (1.23) where g is a translog funCtion of x1 and x and h is a translog function of x3, while 6 and 5h are the correspond- 8 ing parameters. Obviously(1.23)is a Cobb-Douglas function of translog input aggregates. Here separability implies a unitary elasticity of substitution between aggregate inputs g and h along the y isoquant as well as 023 = 013. (b) Now if we substitute the non-linear constraints 0‘ Y Y Y . (—l = —l2 = -ll = —12 = 6) in translog function, again “2 Y23 Y12 Y22 interpreted as exact, we obtain a production function of the form5 : 1n y = 1n a0 + 8g 1n g + 8h In h +1/2-H _ 3,3. lngln h, (1.24) 1.1-gm ' where g here represents a Cobb—Douglas function of x1 and x2 and do, Bg,and Bh are parameters. The function(l.24)is a translog function of Cobb-Douglas input aggregates. In this case,separability implies a unitary elasticity of substitution between x1 and x2 (i.e. along a g isoquant) as well as 62 013 a 023, ij being the Allen partial elasticity of substi- tution between factor i and j along a y isoquant. Then a rejection of the maintained (separability) hypothesis might be due to rejecting a unitary elasticity of substitution for the subaggregate function instead of the correct maintained hypothesis 013 = 023. To avoid this problem one must consider the translog function not as a true function but rather as a second order approximation. The following proposition about the approximate weak separability, which has been proven by Denny and Fuss (1977), will illustrate the point. Proposition (4)6: The translog function (16) is a quadratic approximation to an arbitrary weakly separable production function 1n y = f(ln g (1n x1, ln x2), 1n x3) if 0‘ Y .1 = .12 (1.25) “2 As we see the constraint (1.25) in this case is identical to the first set of constraints needed for separability in the exact case ( ). For inputs 1 and 2 being “2 Y23 Y21 Y22 separable from.3,it is shown by Berndt and Christensen (1973a) that the remainder of the constraints reduce to one independent constraint of the form Y11Y22 = (Y12)2. But this constraint is the one which forces 1n g (1n x1, 1n x2) to become a Cobb-Douglas sub-function, as can be seen from the proof of the proposition (3). Therefore, approximate weak separability involves imposing the constraint 63 a1y23 - a2y13 = 0 while for testing exact weak separability the imposition of the additional constraint Y11Y22 - (y12)2==0 is needed. 1.10 Translog Cost Function We saw that a producing unit was capable of producing alternative rates of output according to a cost function C - C(y; w1.~..‘wn). A translog cost function may be represented as: 2 ln y + h 62 (In y) + 2a. In wi ln C = a0 + 6 1 1 (1.26) + g ZXYij 1n wi 1n wj + 281 In wi 1n y . One of the properties of the cost function for the cost mini- mizing firm is that of linear homogeneity in factor prices (see section 4). For this property to hold, it must obey the following restrictions on the parameters of (1.26): ZY°’=§in=§§Yij=O £8. = 0. One may wish to test the validity of these restrictions as a test of the cost minimization hypothesis; or estimate the cost with these restrictions imposed g priori. As was shown (section 4), invoking the well known Shephard's Lemma, the derived factor demand function can be 64 obtained by partial differentiation of the cost function with respect to the factor prices; namely, x. a _ i=1, ..., n; which can appropriately be written in logarithmic form for the translog cost function as alnC =3 "“’i="i"'i= 3 i=1 n a InRH- a i 77 C i. ’ "" ' where Si represents the relative share of the 1th input in total cost, which is computed from translog as 51 = oi + Bi 1n y + 1 Yij 1n wj 1=l,..., n . J By the monotonicity property the cost function must be an increasing function of input prices, i.e., Si 3 0. Unlike the derived factor demand function, xi, these shares are linear in parameters and may conveniently be estimated. Note that since these shares add up to one, i.e., 2 Si = 1, only (n-l) of these equations are independent; therefore, i the (n-l) equations may be estimated in conjunction with the cost function itself. Later, the estimation procedure will be discussed in detail. Another property of the cost function is that of concavity in input prices, which implies that the matrix 2 (gfisfifi—J must be negative semidefinite for a specified 1 5 range of input prices. There are other economically interesting hypotheses 65 which may be suitably be tested within the framework of the translog cost function. These are homogeneity and homo- theticity of the production function in inputs. Recall that homogeneity of the production function implies that C(y,w) = Y'1 (W) To have this satisfied the following parametric restrictions must hold for (1.26). ll 0 62 II C B. 1 i=1, ..., n . For a homothetic production function the cost function is factorized as C(y.W) = h(.y)'A(w) . which implies the following restriction on (1 26) B. = 0 i=1, ..., n . As was shown, the Allen partial elasticity of substitution between two inputs 1 and j could be calculated from the cost function as CC . i o.. a C_Cl 13 i j where the indices on C represent partial differentiation of the cost function with respect to factor prices. Computing this for the translog cost function we obtain,7 66 .y. “ij‘STSl.+1 iii 1 J o.. = v.1. + Si (Si-1) 11 2 Si Similarly the price elasticity of demand for a factor of production Ei' = 311'xi is computed as J 3 In wj E.. = o..S. 1J 1J J th and the own-price elasticity of demand for i input is (see Allen (1938, p. 519, problem number 12)), E.. = 0.. . . 11 1181 Finally economiescflfscale are widely defined as the relative increase in output resulting from a proportional increase in all inputs along a ray through the origin. However, Hanoch (1975) has discussed a more relevant concept for microeconomic analysis as "the increase in output relative to costs for variations along the expansion path where input prices are constant and costs are minimized at every output." Thus the extent of scale economies can be expressed as the elasticitywmftotal cost with respect to output, i.e., a 1n C/ a 1n y. An alternative way to arrive at this elasticity is the following argument: positive economies of scale are associated with decreasing average cost (AC), i.e.,é§ (g)< 0 or yC' - C < 0, where C' = g; is the marginal cost (MC). This implies that %% < l, which 67 in turn can be written as {73% = (g—g)/(%) '3 g—i—E—g— < l . Thus we can measure economies of scale by d ln C/d 1n y; and in order to make positive scale economies associated with positive numbers and negative scale economies (scale diseconomies),we subtract it from unity i.e.,scale economies = l-a 1n C/a 1n y. lmll Translog Profit Function The translog profit function is given as ln1r(p, wl, ...wn)==ao + 61 1n p + k 62 (lnp)2 -+ Kai ln:wi + g XXYij ln-wi 1n wj + 2 Bi 1n wi Int). where p and W1 are the money pricescxfthe output and the ith factor of production respectively. It was shown that a valid profit function was characterized by certain conditions such as linear homogeneity in p andww,monotonicity, and concavity. Linear homogeneity in p and w implies following restrictions on the parameters: 61 +- f £1. = 1 i=1 1 n Xv i=1 ..+Z s.=0 j=l,...,n. 31 i 1 1'1 +28 i=1 8 =0. 2 i Monotonicity requires that n(p,w) be an increasing function in p and a decreasing one in w. Thus the following must be met: 68 3'17: laln‘fl =1 - ...,) p'flrT'p" p(91+921n p+ZiBiani)20. (1.27) an n aln n w =_-————_=—(a.+ zY.. ano+B° lnp)fo 3W1 Wi alnwi wi 1' j 1] J 1 It is obvious that these conditions can not be globally met for all price combinations. But.as long as there exists the actual observed ranges of price for which the monotonicity conditions are met,we should not be concerned so much about global monotonicity. These conditions must be particularly satisfied at the point of approximation, 1n p 8 0, ln‘w = 0, This implies, since u, w,and p are positive, that: IV 0 E’1 mi 5 0 i=1, ..., n . These restrictions are easily testable. For the convexity condition it is required that the matrix (nij) be positive semidefinite - a necessary condition is that "ii 3 0, or 2 3 "JL aln n aln u _ :2 2<-ainp-31np ””930 p P 2 3 n N aln'n 3 1n n —2-= ( ( '1)+Y--)>0- awi wi2 aln wi a 1n wi 11 - Evaluating these at the point of approximation, we obtain: 69 2 3 1r -—7 = «(e (e -l) + e ) 3 0 3p ln P=0, 1n w_=_0_ 1' l 2 32'“ a ( ( -l)+ )>0 '=I aw Troioi Yii , ,1 ,... i n p=0, 1n w=(_)_ From these conditions, given monotonicity, we derive a sufficient set of conditions as 6121, 6220, Y..>O. In order to form the matrix (fiij) we compute the cross partial derivatives as: an N (311117 alnw apawi pwi a 1np a In wi + Bi) 32w 3 1r (8 In 1r 3 In 1r + ) awiwj wiwj a In wi 3 In Wj Yij At the point of approximation these are equal to: 2 3 1r _ apawi ’ "(elai + Bi) 1n P=0, 1n w_=Q 2 3 1r ._ awiaw. - “(aiaj + YIJ) 3 1n p=0, ln w=0_ Hence the matrix (1.13.)l O 1 0 becomes: n w=_, n p= "70 I 61a1+ BI “1(“1"1y+fi1.“I’2+712""“1“n+11n elOIn-Han analfinl O'no' 2+Yn2' ‘ ' ' o‘n(mn.l)+"nn To see that convexity conditions are satisfied locally we must have all the principal minors positive semidefinite. One of the main advantages of using the profit function is that, by invoking the duality theorem as we saw, we are able to derive the supply and demand functions conveniently by differentiating the profit funCtion with respect to the prices of output and inputs respectively. For the translog profit function the supply function of output and the demand functions for factors of production are given by the linear forms (1.27). Note that only n of these equations are independent and that linearity of the equations is an additional advantage for econometric estimation. 7 The own and cross-price elasticities of demand for the factors of production are derivable from the demand functions, and are defined as: ax * w. Eii .. a: 53* and i i * E = 3X1 W 13 awj xii: Utilizing: 71 - a" a * 3wi x1 and * 3x1 3 _ 32w 3 W1 awi2 Eii can be written as 32 w-—'—'—: 2 l E 3 3W1 = Si ' Si ' Yii (1.28) 11 “§?"" 31 SW].- where 31’ as before, is the ratio of expenditure on ith input to profit, i.e.. * s = _ a 1n n = _ an ‘3; = wixi (1.29) i 3 In W1 awi N N Similarly 2 Wj 3 n 3.3. - .-.. E = wl VJ 2 $183 Y13 (1.30) ij 3n Si 3wi Partial elasticities of substitution are obtained by the following definitions: 2 E11 ’(31 + Si + Yii) a a = 11 Si $2 1 g Ei - ’(Sisi+ Yi'L) 1 8.8 J J 1 J An interesting formula for the partial elasticities of substitution is derived by substituting for Si and Sj’ E.. and EJ.- 11 . from (1.28) - (1.30) which results in J 2 w. 3xi* a “ 1 n -——2 = 1L; 8W. 8W. 1r1r.. Oii 1 1 = 1 = 11 31: w 2 ‘ “"7 (1.31) - aw. -i'- (3") ("1) 1 W 3w. 1 and similarly TITI'.. 0,, = ___Ll_ 1‘] 1T. 1T. 1 J This formula, (1-31), for the elasticity of substitu- tion is Similar to Uzawa's formulation of the elasticity of substitution which was in terms of the cost function and its first and second partial derivatives. Here we have shown that it is possible to formulate the Allen partial elasticity of substitution in terms of the profit function and its first and second partial derivatives with respect to factor prices. This may be considered a generalization of Uzawa's. To summarize, the purpose of this chapter is to proe vide a description of the theoretical issues which form the, basis of empirical research. We have reviewed the principle practical application of duality theory between production and cost function, which establishes that the two approaches are equivalent and equally fundamental. In particular, we have studied the two main advantages of the duality approach to the theory of production. First, the advantage that enables us to derive, painlessly, the system 73 of demand and supply equations consistent with the optimi- zation behavior of the firm, just by direct differentiation of cost or profit functions. Second, the advantage that the "comperative static" results (elasticity of substitution, etc.) associated with optimizing behavior are very easily derived. we have then studied the concept of weak separa- bility among inputs and have seen that the assumption of weak separability on the cost function leads to severe restrictions on the partial elasticities of substitution. In this chapter we also have surveyed various functional forms;i11 particular, the many features of the translog functions have been described. More specifically, we have seen that (i) the translog function provides a second order Taylor's series approximation to any twice differentiable function, and therefore, there are no a priori assumptions on functional form to be estimated in empirical investigations; (ii) many economically meaningful hypotheses appear; as linear restrictions on the parameters of the translog function and thus may be readily tested; (iii) the translog function and the marginal conditions are linear in parameters and hence may be easily estimated by standard linear regression methods; and (iv) the translog cost or production function, unlike the CES, or CD functions, imposes no §_priori constraints on the partial elasticities of substitution and,therefore, is a powerful vehicle for the testing of specific functional forms“ CHAPTER I FOOTNOTES 1The production function f(x) is said to be bounded if lim ————f(:x)-- o A” This ensures that an attainable solution exists for the normalized profit maximization problem. 2A function is defined as being "homothetically separ- able" if it is weakly separable and each subfunction is homb- thetic; however, note that this does not necessarily imply the homotheticity of the function itself in x1, ..., x3. 3 4One obvious approach is to make the common ABS, 0, a function of some variable such as the level of output or the factor ratio or factor share, etc. Such generalizations have been called Variable Elasticity of Substitution (VES) functions, and have been discussed by Ravankar (1971), Lu ??g7§%etcher (1968), Sato and Beckmann (1968), and Lovell 5 6Instead of repeating their proof of this proposition a simpler proof is given as follows. SinceY the translog of the form (1.16) with symmetry imposed (Yi%j=Y311) is a quadratic approximation (around the expansion point (1, ..., 1)) to an arbitrary production function of thex form lny= f(ln x1, ..., ln xn), by evaluating the Leontief conditions for weak separability of the translog function (i. e. , equation (1.22)) at the point of approximation (by substituting ln x=0 in equation (1.22) one obtains aiYk - anik = O. For the three input case, where the third inpat is weakly separated from the other two, this condition can be written as For proof, see Lau (1969) p. 385. For the details, see Denny and Fuss (1977). “1 13 . . -—— = ——— which 18 equation (1 25) Q.E.D. a Y 2 23 7 To see this we substitute in Oi; for Ci Cj, and Cij in terms of Si S , and the parameters of the translog cost function. We 1obtain C1 and C13 as follows. 74 75 3C w’ S.C - a In C a. i g . 8 1 Si. W W; '6‘, therefore Ci ~31- , and S C 3C. S.C . . - 1 - 3 '1. Similarly C. —%— . Then Cij 333— “aw (—w.) . j j j 1 38 S 35 Y- C i i 3C i 1 =- — + — . B Th C. . = Y wi ij wi 5wj “t 5wj wj us 13 c . . . +815j 5:53; Substituting for Ci’ Cj' and cij 1n 0. we obtain the above formulas. CHAPTER II SPECIFICATION AND ESTIMATION OF INDUSTRIAL FACTOR DEMAND FUNCTIONS WITH EXPLICIT ACCOUNT OF INTERNALLY PRODUCED ENERGY AND MATERIALS INPUTS 2.0 Introduction Tremendous increases in energy prices and interruption in energy supplies associated with the oil crises of 1973 have led to a rising interest in extensive research on the characteristics of energy demand and factor substitution. The fact that the manufacturing sector accounts for more than one quarter of annual energy consumption in the United States has made this sector a potential source of reduction in energy demand. The growing number of econometric studies, which have appeared in the literature in recent years with the objective of examining the possibilities of dealing with continued increases in the real cost of energy is an indication. Examples 0f such studies are: Berndt and Jorgenson (1973), Hudson and Jorgenson (1974, 1976, 1978a,l978b), Berndt and Wood (1975, 1979), Griffin and Gregory (1976), Atkinson and Halvorsen (1976), Berndt, Fuss and Waverman (1977), Brock and Nesbit (1977), Fuss (1977), Halvorsen (1977), Pindyck (1978), Berndt and Khaled (1979), Berndt and White (1977), and Magnus (1979). Since energy inputs along with other inputs (capital, labor, and material) enter a producing unit's production process, a cost minimizing producing unit responds to a continued 76 77 energy price increase obviously through substitution. These studies have sought to deal with the degree of energy substitution with other inputs, which becomes a crucial factor in driving policy implications(xfincreasingly scarce and higher priced energy inputs. If one finds that energy and labor are substitutable, then ceteris paribus, an increase in the price of energy leads to an increase in the demand for labor,and therefore employment rises. On the other hand,if it is found that energy and labor are comple- ments, then ceteris paribus, higher priced energy will restrain the demand for energy.and the demand for labor and therefore employment will fall. As another example, one may consider the relation between capital and energy inputs. If energy and capital inputs complement each other in the production process, then any restraint on energy prices (e.g. energy price control) will increase the demand for capital goods.and therefore favor capital formation, while an increase in energy cost reduces the demand for new plant and equipment and consequently discourages capital formation. 'On the other hand, if energy and capital inputs turn out to be substitutable in the production process, then rising energy prices facilitate capital formation by increasing the demand for capital goods. Aside from its attractiveness, the knowledge of technical substitutability of energy and non-energy inputs is very important in the choosing of an appropriate policy. If nonreplenishable,cnrslowly growing energy such as oil and 78 natural gas must be technically employed in fixed or almost fixed proportion with other inputs such as labor or capital, then the growth of real national product in resource-scarce economies may become severely restricted in the near future. Accordingly then, some knowledge of technical substitution is essential for rational planning of private and govern: ment policy concerning foreign trade and allocation of revenues for energy resource development. In general,it can be argued that the limited degree of substitution between energy and non-energy inputs makes the adjustment process by an industry to higher priced energy somewhat difficult,and might cause a substantial rise in the unit cost. The industry in question might even shift the composition of the product away from an existing energy intensive to a non energy-intensive production process. The estimation of Hicks Allen elasticity of substitu- tion between energy and non-energy inputs in manufacturing have, therefore, been the center of attention in these recent studies. Review of the literature indicates that these estimates have not always been consistent. In fact, we have witnessed apparently contradictory results, the most interesting being the substitution possibilitieSr between energy and capital,which may be summarized briefly in two main categories,according to energy-capital comple- mentarity and energy-capital substitutability. Among these studies Berndt and Wood (1975), Berndt and White (1979), Berndt and Khaled (1979), Berndt and 7.9 Jorgenson (1973), using time series data on capital (K), labor (L), energy (E), and materials (M) for the United States manufacturing, have found energy and capital inputs as complements. Similarly Fuss (1977) and Magnus (1979) employing the above four (KLEM) time series data for Canadian manufacturing pooled by different regions and KLE time series data for Dutch manufacturing respectively found the same K-E complementarity result. In contrast to the above results, in studies by Griffin and Gregory (1976), as well as Pindyck (l979),based on KLE time series data pooled by organization for Economic Cooperation and Development (OECD) countries, both found K-E substitutability. Also Wills (1979), based on KLEM time series data on the United States primary metal industry, found capital and energy as being substitutable. Similar results obtained in a study by Halvorsen and Ford (1970), where employing cross section data on capital, two types of labor and three types of energy by state for eight two-digit SIC manufacturing industries, found either significant (E-K) substitutability or insignificant (E-K) complementarity. In contrast to the conflicting evidence from econometric studies regarding E-K substitution possibilities, the substitution possibilities regarding (K-L), (E-L), (E-M), and (L-M) have been relatively consistent throughout these studies with few exceptions. It is not, however, the purpose of this study to reconcile these apparent contradic- tions regarding factor substitution possibilities,since this 80 has been done elsewhere.1 This study rather deals with another subtle issue raised by Anderson (1980) in his estimation of the united States non-energy manufacturing sector. In the large number of the empirical studies on the possibilities of energy reduction in the manufacturing sector which has been conducted in the recent years - which were mentioned above — the most frequent model specification has been consisted of choosing a flexible aggregate cost function defined over four aggregate inputs of capital services (K), labor services (L), energy (E), and materials (M). Then, utilizing the well-known Shephard Lemma Of the theory Of the £3393 researchers have, similarly, obtained the industry's conditional factor demand functions, as first partial derivatives of the industry's aggregate cost function with respect to factor prices. In the theory of the firm the application of Shephard's Lemma for obtaining the firm's conditional factor demand, as first partial derivativeszof the firm's cost function with respect to factor prices, is based on the crucial assumption of exogenous input prices. The analogous application of Shephard's Lemma in the context of industry violates this crucial assumption,and leads to an erroneous result as we will see below. The source of this error rests in the measure of "output" in the industry-level studies due to the lack of sufficiently detailed data. .Aggregate "output" and 81 purchased "material" and "energy" have traditionally been measured as "gross" magnitudes. As such, these gross magnitudes contain the inter-industry shipment of inter- mediate products which move between firms. Therefore, this method measures industry-level "output" as firm's total value-of—shipments; and simdlarly "energy” and ”materials" as total amount of "energy" and "materials" purchased by all the firms which comprise emu industry, regardless of source. Thus, given these measures of gross output, energy, and materials inputs, the application of the well-known Shephard Lemma of the theory of the firm to obtain the conditional factor demand function for industry fails. The utilization of Shephard's Lemma, as was discussed above, is based on the assumption that the prices of inputs are exogeneous, while at industry level studies this assumption obviously does not hold. A glance at table (2.1)2 reveals that of the total "materials purchased by all firms in the non-energy manufac- turing sector a considerable portion have been produced by' the firms within the game non—energy manufacturing sector. These "intra-industry inter-firm.shipments" of traded intermediate products (materials) constitute slightly more than 60 percent of total "materials" purchased by firms in the non—energy United States manufacturing sector. Similarly, in the manufacturing sector (energy and non-energy sectors combined) the intra-industry inter-firm shipment of traded intermediate products ("materials" and "energy") 82 .AMNmAV mouauoomn< uuoosmm cw nonpau souu vmuodauuwu "auusom .u0uoom o;u ovmmuao manna scum new A:u0uoomlcao:v acuuoo wsu c—zuqa mauuu scum pmoasuuaa muwuuouma use Anuocm + .mu:a=« Ham mo monocuuaa .uauuu ago we umou awDOH « ocomu «a «NHN ~oom~ mmmon finned hm oqommu -~can Gammoo N~ -mo «coon on wecs- «assoc momnON mnou cmsm ma Oahu mnem~ wcoon Nmooa no memoNN nwamn concoo NN nNNn mmhnu No cemnmn smnmnn «macaw acmu sods - fiend wm-~ comcn doom cc anoeom «Omamw emoqme «a Game omama No cameoa aaqomm barman meow aNNo Nd dang nucOu cmnuu omen no eemoeu numnnw mafiooc Hm damn «word so Muenca ~o~ccw manmwc mead amen «a eqmfi c~oo~ nmomd 0500 cc ommsnu «macaw common an anon nosed No nwcenu «comma uocnmm omo~ nmcc mu ~x- dqma ooosu meco no cocnnm oncmON ommcnn an nonn manna co naccna soaaow cnwemn enmu co—n ca now. moon named «man no cosmwn «comma HomOCn oa coca m~oMu we wuocma «enema wmmemn nmaa odcm Cg Nan cmco unwed some do naenm mooncg cummNN ma coma sceca we n-oa uoqmcn mesona cmma mama cm ado wmqq «nmm «men an Hawao oMNoua noosmn NN mwnn cam~ mm “memo mencun nemeam “emu .!:!alsl N noncomuczo dquH N uouuomuszo ”much N uOuommncso Haupu N acuoomucao annoy _o_uoua: +mwuocm «umou Houoh anuocm +m~uuuoumz «uuoo Hench +>m-=u +muauuoum: sumou Hmumh use» Uz~zahoU¢mzm mo NUKSOm 02¢ mfiamzH ho Hmoo Azozoom .m.= uzh no machoum OZuosaozmlwozmzm 92¢ H.N maaa change in primary factor price by these "net" and "gross" models. Differentiating the conditional factor demand function (2.6) - for the "gross” model - and (2.8) for the "net" model - with respect to the prices of primary factor 5 = l, ..., r, respectively, we obtain from.(2.6): 2 i M: _§__ * . '= . = awka Ws aws xik(. )9 l l,...,n, k,S l,...,r (2.9) and from (2 . 8) : O 2 2 1 r+n a w —— = — x. (.) + }' x‘é’.(-) awk aws aws 1k j=r+l awkaws 13 r+n 3w 3 ___. * +.’1=:Z:+1 aws Wk xiJ' ) (2.10) 96 Comparison of (2.9) and (2.10) obviously reveals the different responses of these two models to a change in a primary factor price. The first term on the right hand side of (2.10) is equivalent to (2.9), which according to the law of demand for factors is negative or equal to zero for k a s = l,...,r; i.en the conditional demand curves for each factor are downward sloping. This follows from the fact that the cost function is concave in input prices, and that the second derivative of a concave function is non-positive. This term can be regarded as the direct effect of a change in primary factor prices. The other terms on the right hand side of (2.10), representing the terms relating to the effects on total unit cost of substitution among intermediate products within the industry, can be regarded as indirect substitution among primary factor inputs. The different responses of these two models may also imply different factor demand elasticities as can be seen from equations (2.9) and (2.10). One must, however, note that under two special circum- stances both the "net" and the "gross" model will be equivalent: first, if the firms comprising the industry employ a fixed coefficient Leontier technology with, obviously, no induced factor substitution among the traded intermediate products; second, if there are no traded intermediate products. In this case the traded intermediate products must have been netted out in terms of their primary factor content.‘5 One also must realize from equations (2.9) and (2.10) that the 97 "gross" model can be considered a special case of the more general "net" model. In the next section we briefly describe the translog unit cost function and the system of share equations derived from it, then describing the estimation problems associated with the system of share equations, finally presenting the empirical results based upon the U.S. manufacturing and the U.S. non-energy manufacturing data,and comparing these results with other studies. 2.2 Empirical Specification and Estimation As was discussed in the previous chapter, the fundamental result of the duality theorem states that, given certain regularity conditions, the specification of the production function implies a particular cost function and vice versa. Therefore, the structure of technology can be studied empirically employing either cost or production function - the choice being a statistical matter. Direct estimation of the production function is attractive when the output level is endogenous, while direct estimation of the cost function becomes more attractive whenever the level of output is exogenous. Suppose the technology for the manufacturing sector can be represented by a positive, finite, continuously twice differentiable, strictly monotone increasing, and strictly concave aggregate production function as: 98 Q = F(K,L,E,M) (2.11) where (K,L,E,M) are the input quantities of capital services, labor services, primary energy,and primary materials. We further assume that any technological change affecting the aggregate inputs of K,L,E, and M is Hicks neutral. Now given the production function (2.11) and the vector of input prices, (WK, WL, WE, WM), the corresponding dual cost function of the manufacturing sector can be obtained as the solution to the following constrained minimization problem: cch,wL,w M;Q) E LmEnM {WKK+WLL+WEEWMM|F(K,L,E,M) : q}. (2.12) Further, if the production function is assumed to be a first degree homogeneous function in input space,then the cost function (2.12) factors into the following expression: C(WK, WL’WE’WM'Q) = Q-Kc(W ,WL ’EW ,WM) (2.13) where c(-) is the manufacturing sector's unit cost function. For the purpose of estimating the unit cost function. c(°), a specific functional form.must be employed. Until very recently the estimation of the parameters of the cost or production function has been based upon highly restrictive functional forms, the most popular being the Cobb-Douglas and C.E.S. The shortcomings of these functional forms - as we have seen - have led researchers to employ a relatively new generation of flexible functions such as the translog, 99 generalized Leontief, etc . These functional forms permit the technology to exhibit an arbitrary set of partial elasticities of substitution between pairs of inputs at a given point in input price or quantity space. These functional forms, therefore, offer substantial gain of flexibility and create a great opportunity for the investi- gator to test important maintained hypotheses of previous works. With these points in mind the functional form we choose to work with is necessarily of this family. In particular we choose to work with the translog functional form proposed by Christensen, Jorgenson, and Lau (1971). This function provides a second order local approximation to an arbitrary underlying function about a point. Therefore, we assume that the unit cost function can be approximated up to the second order by the following translog cost function: 1n C(W) = (10+: aiani+g Z Zyij 1nWi1nW., i,j=K,L,E,M; 1 1 J 3 (2.14) where (yij) is a symmetric matrix. The equality of yij yji is a necessary for the applicability of Young's theorem to integrable functions, and as Denny and Fuss (1977) have stated in proposition (1) of their paper,the symmetry constraints must hold when one views the translog function as a quadratic approximation to an arbitrary cost function. Recalling the Samuelson (1947) - Shephard (1953) duality theorem of chapter one, if we differentiate the 100 unit cost function with respect to the price of an input we obtain the derived demand for that input. Therefore, assuming cost minimizing behavior and industry-level exogenous factor prices, we apply Shephard's Lemma which yields a set of estimable factor share equations linear in logarithm.of factor prices, = a 1n c = - - _ Si W mi '1‘; Yij 1n Wj, 1,1 - K,L,E,M (2.15) where Si is the cost share for ith factor. To correspond to a well-behaved production function, a cost function must be homogeneous of degree one in factor prices. This yields the following set of parametric restrictions: 201:]. [‘4 2y.. - 2y.. 3 X y.. = 0 (2.16) i 13 j 13 i ' 1.] U These restrictions will be imposed throughout the chapter. Following Uzawa (1962) the Allen partial elasticity‘ of substitution (AES) can be computed from the cost function by the following formula: (’13- = my» -’cij(W_.Q)/ci(fl.Q)-cj(_1~1.Q) (2.17) and since c(W,Q) = Q-c(W), equation(2.17) can be reformulated in terms of the unit cost function: Oij = c(W)- cij(W)/ci(W)cj(W), i,j = K,L,E,M (2.18) 101 2 = 3c 3 3 c where ci SWI" and cij SHEEN; . Clearly, as has been seen from the above formulas, AES's are variable. For the trans- log cost function,we earlier saw that the own and cross AES could be derived as: 7.. = 1' O O (Y .-S-) - i 1 on — 1+ % (2.19) S i The corresponding own and cross~price elasticities of demand for factors were obtained as: .. 0.. . 1J 1J J ”ii = oii Si 1,3 = K,L,E,M; (2.20) where nij (cross-price elasticity of demand) measures the percentage change in derived demand for input i for an exogeneous change in the price of input, given.that all other input prices and output quantity remain constant. Note that while cij = 0. ji by definition “1' # "ji in general. J To characterize the structure of technology of the manufacturing and non-energy manufacturing sectors, we estimate separately the parameters of the unit cost function, using the stochastic version of the cost share equations (2.15) as a multivariate regression system. To do this we specify an additive disturbance pi for each of the share 102 equations on the assumption that entrepreneurs make random errors in adjusting to their exact cost-minimizing input levels. The disturbance pi'S are likely to be correlated, because random deviation from cost minimization should affect all of the market for the inputs, and hence the estimation procedure suggested by Zellner (1962) is expected to yield more efficient estimates. Because of adding-up constraints on the cost shares-which imply a zero sum of disturbances across the four equations at each observation-togethe ‘with the symmetry restrictions across equations, one of the cost shares must be deleted to avoid a singular estimated disturbance var-co 'matrix. By deleting one of the share equations from the system the Zellner procedure can be made operational; however, the estimate so obtained will not be invariant to which equation is deleted. Barten (1969) has shown that the maximum likeli- hood estimates of the system of share equations with one equation deleted would be invariant to which equation is dropped. Kmenta and Gilbert (1968) and Dhrymes (1970) have shown in a series of Monte Carlo experiments that the iteration of the Zellner (IZEF) estimation procedure until convergence results in estimates which are identical to those of maximum likelihood estimates. Accordingly the application of the iterative Zellner estimation procedure will yield, computationally, maximum likelihood and therefore 103 consistent and asymptotically efficient estimates for the parameters of (2.15). This is the procedure we employ here. In preliminary application of IZEF procedure to our data we have found a substantial degree of serial correlation in the postwar U.S. manufacturing and non—energy producing manufacturing sectors. To increase the efficiency for this case we assume that the additive disturbance term, ui, in each input share equation is both serially and contemporan— eously correlated. As was discussed above, the adding-up constraint on the dependent variables (2.151 together with the symmetry restrictions across equations imply the parametric restriction of (2.16) which in turn requires that one cost share equation be omitted. Therefore, imposing parametric restriction (2.16) on the system of equations (2.15), and adding an additive disturbance term “i to each of the equations we find: Sit = “i + Z Yij 1“ (th/WMt) + “it 1’j='K'L'E; J t=1,,,,,1~ (2.21) writing (2.21) in matrix notation we have: ( 1 f j (11 X r W SKt “K YKK YKL YKE “(Wm/Wm) “Kt SLt = °‘L YK1. YLL YLE ln ousnomn< + .oosaouumcmwc no ~o>o~ mc. an open Iouu acoumuuuc >~ucuowwucmnu cu ucowuuuuooo 4 ans.msn m~a.~mn can.emn sno.ssn "coasoasa 32:33-3; s~m.~ ano.a sea.e ns~.~ ”m ms0._ msa.a Hea.fl ~oe. an ass.” ~a~.~ sog.~ new. a ":omuszncucuaa ans. ems. use. has. mm ans. ass. «as. «as. em ass. now. see. see. a "N1 at.s v sewn. mw as.~ . we”. as.“ V «as. - . a as. e coo. “m1. 3. e Sc. - SQ .o.~ s «so. Ao.n v anco.n - ass Aa.s V .aNo. Re.“ C same. use Ae.~ C can. - use As.s . soc. - as. v sec. 42.. 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I o Am.AI cu n.Iv An.aI ou o.cIv 4M oc.HI cm.m I n¢.c I mn.w I oo.~I mm.~I ~H.NI oo.mI . o easesaV AnV AssoHV AeV AseaeV as maanmuv concowuon naonmav buomouu Ammmav Amhouv mugs: county Amsouv woo: a sun dug a mad“: can comps: soavawm van aqumwuo enema: can accuse paw autumn van autumn uwuuoaszm accomaan vououuummuas oux mm< AuV oszDPQ 0, the output has moved from one to another isoquant of the ggme production function (scale effect), or there has been a shift from one production technology to another (technical change). This suggests that it is useful to try to measure economies of scale and technical change, and separate the effect of one from the other. First we start with the definition of technical change. Assume a neoclassical aggregate production function Q=F<§.T>. g; > o. -§—§_§ _>_ o; (3.1) where Q and E represent output and a vector of inputs respectively while the variable T represents time to allow for technical change. Technological change by its very nature allows a producing unit to produce more output with the same level of input quantities. 0r equivalently, the existing level of output, after technological advancement, can be produced with smaller quantities of at least one input while the quantities of other inputs remain the same. Therefore, it seems quite natural to measure the rate at which technology advances as ' = a 1n % a If factor input prices and output quantities are (3.2) X exogenously determined, the duality theory between cost and production implies that, assuming cost-minimizing behavior, the production technology given by'(3.1) can be uniquely represented by the cost function C = C(H- Q. T) (3.3) where C is total cost and‘w;is the vector of factor input prices. Technological change, therefore, can be measured by utilizing the cost function. A producing unit at internal equilibrium, ceteris paribus, can produce a given level of output at a lower cost after technological advancement. Thus, the rate of technological change can 132 directly be measured by a 1n C O - = 3.4 C 31‘ E: Q ( ) on the cost side. To undertake empirical work, we need to be more specific about the nature and character of technological change. Technological change may be biased with respect to one factor or the other, or it may be neutral with regard to all inputs involved. According to Hicks (1932, p. 121), technological changes are classified as labor- saving, neutral, or capital-saving respectively, "as their initial effects are to increase, leave unchanged, or diminish the ratio of the marginal product of capital to that of labor". If the producing unit is to attain a position of internal equilibrium, then one must examine the effect ofatechnological change along the firm's expansion path where the firm minimizes the cost of producing any given level of output. In this sense Hicks neutral technological change can be interpreted as "expansion path ' saving". Technological change is said to be Hicks neutral, in this sense, if marginal rates of technical substitution between each pair of factor inputs are independent of technological change. Therefore, representing technical change by the production function (3.1) the following can ‘ be defined (see Lau (1978)). Definition: A production function exhibit Hicks neutrality, in the above sense, if it can be written in the 133 form F(zs. T) f(gcg) .T) (3.5) An alternative interpretation of Hicks' classification, which is widely accepted, is the one which seeks the effect of technical advancement along a ray from.the origin where factor proportions remain constant at their pre- technical-change level. However, if one considers the effect of technological change along a ray,as suggested by this interpretation, then we are faced with a quite different notion of neutrality, i.e., implicit Hicks neutrality the term used by Blackorby et a1 (1976). Technological change, therefore, is defined to be implicitly Hicks neutral if marginal rates of technical substitution between each pair of factors, at constant factor ratios, are independent of technological change. Constructing the implicit representation of the production function(3.l) in the implicit form of H(Q, X, T) = 0, Blackorby et a1 (1976) have proved that technical change is implicitly Hicks neutral if and only if H(;) can be written as HQ, 32. T) E G(Q. T. g(Q. 29) (3.5) According to this interpretation, technical change is classified as labor-saving, neutral, or capital- saving depending on whether, at a constant capital-labor ratio, the marginal rate of technical substitution increases, stays unchanged, or decreases. This classification can 134 immediately be interpreted in terms of factor input shares. If at a constant value of a factor ratio, the marginal rate of substitution (or the ratio of capital price to labor price) is rising, then the labor share is declining. Similarly, if the marginal ratio of substitution is declining, the capital share will decrease, and if technical change is classified as neutral, factor shares remains constant.1 Another type of technological change, different from the former two types, but often confused with them, is one which can be written in the following decomposable form: F(g, T) a A(T) . f(_)_() (3.7) In this case the isoquant map remains unchanged,but the output number attached to each isoquant is multiplied by A(t). This type of neutrality is what Blackorby et al (1976) call "extended Hicks neutrality". The practical implication of Hicks neutrality, as is evident from all three types of Hicks neutrality, is that the ratio of the marginal productSIxfany two factors of production is independent of time.2 However, it is clear that these three types of Hicksian neutrality are not equivalent in general. Besides the fact that extended Hicks neutrality (3-7)implies Hick neutrality (35), none of the three types of Hicks neutrality implies either of the other two, unless additional assumptions are imposed. One such assumption is that of homotheticity in inputs which serves as a necessary and sufficient condition for simultaneous Hicks and implicit Hicks neutrality. The reason is that 135 the expansion path through any arbitrary point coincides with a ray through that point under input homotheticity. Therefore homotheticity is a necessary condition for the equivalence of all three types of Hicks neutrality. The second assumption is that of input homogeneity,which is a sufficient condition for the equivalence of all three types. (See Blackorby et a1 1978). Corresponding to Hicks neutrality in the production function F(-), we may define indirect Hicks neutrality for the dual cost function C - C(w, Q, T), as Lau (1978) did for the normalized profit function. Definition: A cost function is said to be indirectly Hicks neutral if it can be written in the following form: c = Ema-m). Q. T). (3.8) Practically, indirect Hicks neutrality implies that the ratio of the derived demands of each pair of inputs is independent of technical change. In general Hicks neutrality does not imply indirect Hicks neutrality or vice-versa, unless additional assumptions are made. Under the homotheticity assumption, Hicks neutral- ity implies and is implied by indirect Hicks neutrality. Also, as Lau (1978) has shown in the context of the normalized profit function, "a technology is both directly and indirectly Hicksian neutral only if either it is homothetic or it is additive in T." (See Lau (1978)) An alternative classification of technical change, 136 which has played a more central role in growth literature, is the Harrod classification of technical change. According to Harrod, technical change is defined to be neutral if at any constant value of the capital-output ratio the marginal product of capital remains unchanged. Here we compare points with constant capital-output ratios, while in Hicks' classi- fication points with a constant capital-labor ratio are compared. The Harrod classification can also be stated in terms of the effect upon factor input shares as technological progress proceeds. Technological change is defined to be labor-saving (capital-saving) in Harrod's sense if, at any constant level of the capital-output ratio, the capital share is increasing (decreasing) relative to the labor share. Technical change is classified as Harrod neutral if, at any constant value of the capital-labor ratio, the capital share and labor share increase at the same rate. The importance of Harrod neutrality stems from the fact that only technical change of this form can be consistent with balanced growth in the usual growth model. In parti- . cular, it has been demonstrated by Joan Robinson (1938) and Uzawa (1961) that Harrod neutral technical change is exactly equivalent to pure labor-augmenting technological pro- gress, and can easily be incorporated in the usual growth model. Harrod neutral technical change, therefore, may be defined as labor-augmenting technical change as follows: Definition: Production function (3.1) exhibits Harrod neutral technological change if one can write the production 137 function in the following separable form: F(zgff) s f(h(L,T),)_9 (3.9) where L is labor, the primary factor of production. The practical implication of Harrod neutrality, for- mulatedirIthe weakly separable form above, is that the ratio of the marginal product of labor to the rate of technological change measured in terms of output,is independent of K. For the dual side of the problem,and corresponding to the dual cost function (3.3),indirect Harrod neutrality can be defined as: Definition: Cost function (3.3) exhibits indirect Harrod neutral technological change if it can be written in the following weakly separable form: c = 6(g(wL.T). WES-Q) (3.10), The practical implication of indirect Harrod neutrality is that the ratio of the demand for labor to the rate of technical changepmeasured in terms of cost,is independent of wx. Here again direct Harrod neutrality does not imply indirect Harrod neutrality or vice-versa. Only under the two following conditions is a production function both directly and indirectly Harrod neutral (See Lau (1978) for a proof in the case of the normalized profit function): 1) h(L.T) = h(A(T)L) 2) Q = mg) +h(L.T) (3.11) 138 One interesting form of technical change is that of factor and/or output-augmenting technical change. Under general specification of factor and output-augmenting, the production function can be written as follows: F(X,T) = c(T) f(gl(T)Xl, ...., an(T)Xn) (3.12) where c(T) represents output-augmenting technical change and ai(T) is "factor-augmenting" technical change correspond- ing to input 1. Hicks and Harrod neutrality may be considered two special cases associated with (3.12). The former occurs when ai(T) = 8(T) for all i and f is a homothetic function. In this case it can be shown that: Q = c(T)-g(;(T)'lh(xl, ,..., Xn)) (3.13) which clearly exhibits Hicks neutrality,3 while Harrod neutrality occurs when the c(T) and ai(T)'s are all constant, except for the ai(T) corresponding to labor, the primary factor of production. 3.2 Return to Scale A common assumption in theoretical and empirical research is the assumption of linear homogeneity of the production function implying the existence of constant returns to scale. The assumption of constant returns to scale is of some importance, because the justification of this assumption at the industry level is that in the long run, a perfectly competitive industry with price taking firms 139 and free entry and exit will be able to duplicate what it has been producing before. Thus, by doubling all of their inputs the firms should be able to build another plant identical to the first and produce twice the output. In this case, then, the average cost remains the same for all levels of output. Scale effecna however, at the industry level may be present due to externalities or lack of freedom for entry and exit, or it may be the case that the firm's production functions are not homogenous of degree one. It might also be the case that the production function exhibits decreasing returnsto scale,because the scale effects are genuinely decreasing or because some factor remains fixed in the long run. Similarly, we might be faced with increasing scale effects; for example one factor explaining the rapid post- WOrld War II economic growth in the U.S. might be due to the presence of increasing scale effects. In these circumstances it may be desirable to allow for non-constant return to scale,and not to impose constant returns to scale a priori on the estimating model. In the theory of production,we recognize two different concepts of returnstx>scale. The first concept, which has widely been used as a measure and definition of returnsto scale, is the proportional change in output relative to the prOportional change in the inputs for move- 'ments along a ray through the origin. This is, in other words, the elasticity of output with respect to an 140 equiproportional variation of all inputs. This concept is known as the function coefficient, and using the production function (3.1), it may be given more precisely by - d , dk = d 1n E’UQ'F sun—1% where k > 0 is a scalar. ~ According to Hanoch (1975) the second concept, which is more relevant as a measure of scale economies, is obtained from the relationship between total cost and output along the expansion path where the producing unit must remain if it is to minimize the cost of producing any given level of output. For this purpose,elasticity of cost with respect to output appears to be a natural measure to express returnstx>scale. This elasticity measures the proportional increase in cost relative to the proportional increase in output for movement along the expansion path where input prices are constant and costs are minimized at every level of output. This can be written symbolically as: E = 3 ln‘C a In 0 w ' where E S l,depending on whether the cost function exhibits a o o . 4 Increa31ng, constant, or decrea31ng returnstx>scale. By subtracting this elasticity from unity, one can associate positive numbers with economies of scale and negative numbers with scale diseconomies, i.e.,we have SE = l - E E 0 as there is increasing, constant, or decreasing returns to scale. 141 For purposes of empirical implementation, it is neces- sary to adopt an explicit function form for the cost function (3.3). Our choice is the translog function. In this flexible framework we are able to explicitly deal with these issues and further examine various other specifications as we proceed. 343 The General Empirical Model we assume that the production technology in U.S. manufacturing can be presented by the production function Q = F(K,L,E,M3T) (3.14) where the flow of output Q is related to the service flow of four aggregate inputs; capital (K), labor (1), energy (E), and intermediate materials GM). T is an indicator of technical change and is measured in years. we further assume that the production function F is positive, finite, contineously twice differentiable, strictly monotone, strongly quasi concave, and nondecreasing in technical change, T. If factor prices and output levels are exogenously determined, and assuming costaminimizing behavior, the production structure implied by (3-14) can uniquely be described by a cost function of the form: = ' - 3.15 c C(wK_,WL,wE,wM,Q,T)_ , ( ) where C represents total cost and the Wi’ i = K,K,E,M, are factor prices. 142 For purposes of estimation, it is necessary to employ a specific functional form for the cost function C. The highly general functional form we have chosen for this study is the translog cost function proposed by Christensen,JOrgenson and Lau (1971, 1973). It places no apriori restrictions on the Allen partial elasticities of substitution (AES) among the factors of production. An important property of the translog function is that it can be interpreted as a second order approximation to an arbitrary cost function. It also allows scale economies to vary with the level of output. Also, technical change can be incorporated into translog cost function conveniently. The production structure is represented by a non- homothetic translog cost function which can be written by the following approximation to an arbitrary cost function: 1 1n C = a0 + g a. 1n w. + 2 Z X yij 1n wi 1n wj 1. l l J + 5 In W 1n Q + 1n Q +'1 (1n Q)2 1 YiQ i “Q I YQQ - . + Z T ln W + T + L T2 + T ln Q Yit i 0‘I: 2 Ytt YtQ ' i, j = K,L,E,M. (3.16) The derived demand functions for each factor of production are conveniently obtained by partially differentiating the cost function with respect to factor prices, namely, 143 3C _ = X. 3wi 1 where X. is the cost-minimizing quantity demanded of ith input. 1Using this result, known as Shephard's lemma (Shephard (1953, 1970)),one can express the input demand functions in terms of cost shares simply by logarithmic differentiation of the translog cost function as ‘a In C i i a In Wi C i where Si denotes the cost share of the ith factor input. Then the translog cost function produces cost share equations of the form Si = oi + g Yij K,L,E,M. 1n Wi + YiQ 1n Q + Yit T: (3.17) where i,j Since the systems of demand equations (3-17) must satisfy the adding-up restriction (X Si = l) the following parameter restrictions hold: X a. = 1 1 gym, = g,“ = £32,113, = 0 (3.18) 2Y1: = 0 i,j = K,L,E,M 1 In addition to these parameter restrictions (3.18) the following symmetry constraints, implied by equality of the cross partial derivatives, are required. 144 y.. = y.. ia‘j (3.19) These restrictions, (3.18) and (3.19), will be imposed through- out the study. The translog cost function must satisfy the following conditions: (1) Linear homogeneity in prices; that is, for a given level of output, total cost must increase proportionally with a proportional increase in all factor prices. This implies the same parameter restrictions as (118)- (ii) MOnotonicity: that is,the cost function must be an increasing function of input prices. This implies that the cost shares be strictly positive. (iii) Concavity 32C , awi VJ. in input prices: this means that the Hessian matrix, must be negative semidefinite. The Allen-Uzawa partial elasticity of substitution between input 1 and j, as we sawrbefore, can be obtained from the cost function by the formula CC - i' Gij ‘ Ctr} (3‘20) 1 J For the translog cost function these elasticities are: Yi' . . Oij = l + —-g—Sij l # J (3.21) 7.. -S. o. = 1+_1-i___£ (3.22) 11 52 1 Obviously these elasticities entail no apriori restrictions with respect to their value of constancy. The own and cross-price elasticities of demand for factors of production 145 can be obtained as: a 1n Xi (3 23) “ij "' Trix—V3 = Gijsj ' 3 1n X = i a - - = 3.24 "ii S‘Ifi‘fi; °iiSi 1'3 K'L'E'M ( > where the nij (demand price elasticity) measures the percentage change in demand for input i for an exogenous change in the price of input j, given all other input prices and output quantity remain constant. Note that while 0.. - 0.. by l] 31 definition, "ij # ”ji in general. 3.4 Technical Change and Bias Inclusion of T, as an input, in the translog cost function (16) will facilitate the study of technical change biases. The Hicksian concept, which is the most common concept of the biases of technical change, can be handled conveniently in the cost function framework in terms of input cost shares. Letting Si represent the share of the ith input of total cost, as in(3J7). a technical change is said to be i-using, idsaving, or neutral if the ith cost share (Si), at constant input prices, increases, decreases, or remains the same. Therefore the Yit's are the estimates of factor using or factor saving Hicks biases of technological change. These parameters represent, in fact, the rate of change in the cost shares not attributable to prices, i.e., 1 = . i = K,L,E,M. (3.25) 146‘ Here, a zero value for the Yit (for all i==K,L,E,M) implies Hicks neutral technical change. If the production structure is non-homothetic, technological change may be biased with respect to the returns to scale. A biased technological advancement of this kind will increase the scale level at which decreasing returns set in, and thus may change the output level at which minimum average cost could be attained. The scale bias (SB) therefore, can be defined as a time (technical change) derivative of the scale measure, i.e. _ a 3 1n C SB ' 331* (m)- With respect to our translog cost function this,derivative is equal to YtQ’ and thus we can measure the scale bias after estimation of the parameters of the translog cost function. Other alternative models regarding technical change may be tested. For example a parametric restriction of Ytt = O, in addition to Hicks neutrality constraints (git = 0), implies scale bias technical change along with a linear Hicks neutral technical change. While a further imposition of th = O, in addition to Yit = O and Ytt = 0, implies the usual type of exponential Hicks neutral technical change. Finally, we may have a production structure with no overall technological effect which implies the following parameter restrictions; (3.26) = 0 O i = K,L,E,M 147 3J5 Homotheticity, Homogeneity, and Return to Scale A common assumption in theoretical and empirical research is that the production function is linear homogenous, meaning that there are constant returns to scale. This is the kind of assumption which one may wish to test, rather than imposeaipriori. The cost function specified in.(3.15) is assumed to be non-homothetic with the corresponding trans- log approximation represented.by (3.16). If the production technology exhibits homotheticity, the cost function (315) can be written as a separable function of output and factor prices; that is, C = h(Q,T)oC(w,T) , (3.27) where C(W,T) is the unit cost function corresponding to F(§,T). To apply the translog approximation to (3.27) we take logarithms of both sides and obtain In C = ln h(Q,T) + ln c(flqT) (3.28) The non-homothetic translog cost function (3.16) will correspond to a homothetic production technology, represented by the cost function (3.27) , if we impose the restrictions: yiQ = O 1 = K,L,E,M. (3.29) Two other versions of (27) which are also implied by the homotheticity of the production structure are C = k(Q)-C(W,T) (3.30) 148 C = g(Q,T):CCfl). (3.30) For these cases the additional parameter restrictions are: th =- 0 for (3.30) and Yit = 0 for (3.31) respectively. These two specifications will exhibit, in addition to homotheticity, scale unbiasedness (YtQ = O), and Hicks neutrality (Yit = 0) respectively. A homothetic production technology is further restricted to be homogeneous (of degree %) if and only if the elasticities of cost with regard to output are constant. The cost function (3.15) can be written in the form _ k C - Q -cQ_J_,T) (3.31) The translog cost function (3.16) can serve as a second order approximation to (332) if we impose the following restrictions on (3.16). = 0, = 0, = 0 i = K,L,E,M (3.32) ‘YQQ YtQ *iQ Finally, if the underlying production function exhibits constant returns to scale (linear homogeneous), then the correspOnding cost function can be written as C = Q-c(fl,T) (3.33) This imposes an additional parameter restriction on the translog cost function (3.16), i.e., in addition to (3.33) we 5 have = 149 Economies of scale are among the factors , such as technical change and relatively cheap material resource inputs, which have been mentioned as contributing phenomena in the growth of industrial output after WWII. Earlier, scale economies (SE)were defined in terms of the cost elasticities of output. With the translog cost function.(3l6)this elasticity is obtained as: g l: C = aQ + YQQ 1n Q + zYiQ 1n Wi + YtQ T ~(3.35) Underwidifferent specification and with different parameter restrictions imposed, this elasticity can be derived accordingly. For example, if the homotheticity restriction (JiQ = O) is imposed,then, under this restriction, one can rewrite this formula as: a In C §_IE—Q aQ + YQQ 1n Q + YtQ T; (3.36) and so on. 3.6 Estimation and Hypothesis Testing The parameters to be estimated are contained in the derived factor share equations(3 LT)and the translog cost function itself, which form our estimable equations. It is of course feasible to estimate the parameters of the translog cost function alone, using ordinary least squares. This procedure, however, may result in a high degree of multi- collinarity and therefore highly imprecise coefficient estimates, due to the large number of terms involved in the 150 translog cost function. In addition, this method neglects the additional information contained in the factor share equations. An alternative estimation method used by many (e.g. Berndt and Wood (1975)) is to estimate only the share equations as a multi-variate regression system. This method is satisfactory when the factor share equations contain all the parameters of the translog cost function (for example, in case we assume constant returns to scale - Hicks neutral technical change). Since we have adopted a nonhomotheticé nonneutral specification for the translog cost function, many parameters do not appear in the factor share equations. Consequently, the estimation of factor share equations is not an appropriate approach,and must be abandoned in this case. An optimal approach, practiced by many authors, has consisted of joint estimation of the translog cost function and the factor share equations as a multivariate regression system. Therefore the model to be estimated consists of the translog function itself and the three share equations for capital, labor, and primary energy, after deleting one share equation - arbitrarily the materials factor share - to avoid singularity of the disturbance var-cov matrix. With constraints for linear homogeneity in factor prices and symmetry constraints imposed (i.e., restrictions (3.18) and (3.19)), the estimating equations are written as: 151 si = oi +32 aij 1n (Wj/WM) + aiQ 1n Q + yit T + pi ln(C/WM) = a0 + E oi magi/WM) + 35 E J; Yij 1n (wi/Wm) 1n (,wj/wM) + g YiQ 1n (“i/WM) 1n Q + E yit 1n (Vii/WM) T + aQ 1n Q + g YQQ(1n Q)2 + atT + 35 Ytt T2 + th T 1n Q + no; i,j = K,L,E. where yij = and "O and pi are random disturbances. in’ Assuming that the random error vector u = (“0’ “K’ “L’ “E) is independently and identically distributed as multivariate normal with mean vector zero and nonsingular covariance matrix, we have estimated the parameters of the model employing the fiterative'Zellner" estimation method. This estimation procedure is well known to yield coefficient estimates identical to maximum—likelihood estimates (see Kmenta and Gilbert (1968)), and therefore the estimates are consistent and asymptotically efficient. Since the parameter estimates so obtained are maximumrlikelihood estimates,we can test the validity of various hypotheses (specifications) such as homotheticity, homogeneity, etc. Our statistical tests for various specifications are based on the likelihood ratio method. The likelihood ratio is A = Lmax(m)/Lmax(a), where 152 Lmax(m) and Lmax(a) stand for the maximum of the likelihood function under restricted and unrestricted models respectively. It is W811 known (Wilks, (1938)) that the test statistic -2 In A has an asymptotic distribution that is chi-square with degrees of freedom equal to the number of independently imposed restrictions. 153 3.7 Empirical Results 3.7.A Tests of Underlying Assumptions 3.7.A.1 Manufacturing We now proceed to a discussion of our various specifications estimated by employing the Zellner iterative estimation method. The parameter estimates for these speci- fications are reported in Table (3.2) along with their corresponding sample log-likelihood values. The reported sample log-likelihood values indicate that casting Medel 1 as the unconstrained model, the homothetic specification (Model 3, YiQ = 0) must be rejected. This can be seen by computing the likelihood ratio test statistic which is twice the difference of the sample log-likelihood values. The test statistic for the homothetic model (Model 3) is 19.854, while the .01 chi-square critical value with three (parameter restrictions) degrees of freedom (.01 x23) is 11.345. The two other versions of the homothetic production function, specified in equations (3.30) and (3.31), and presented as Model 4 and Model 5 in table (3.2), are also rejected. The likelihood ratio test statistics are 20.842 and 23.128 for Model 4 and Model 5 respectively, while the .01 x24 and .01 X26 critical values are 13.277 and 16.812 respectively. Model 4 assumes homotheticity but no scale bias (th = 0), while Model 5 assumes homotheticity along with Hicks neutral technical change. Since the homotheticity hypothesis is rejected, the technology is not homogeneous of any degree. As 154. indicated in table (3.2) the homogenous and constant returns to scale (CRS) hypotheses are rejected, where test statistics for these models (Model 6 and 10) are 21.590 and 47.628, compared with the corresponding.01x2 critical values of 11.070 and 12.592 respectively. However, if the (rejected) homothe- tic model is maintained, the homogeneith restrictions would not be rejected, while the CRS model would be rejected. As for the effect of technological advancement, the model with 4"no technical change" (Model 12: at- Ytta Yit- th = 0) has decisively been rejected against the unrestricted model (Model 1). For our nonhomothetic,unrestrained model (Medel 1), "Hicks neutral technical change" (Model 13; yit-IO) is also rejected; the test statistics is 14.322 while the .01 X23 is equal to 11.345. However, once the (rejected) homothetic, homogeneous, or CRS specifications are maintained as out unconstrained models, the imposition of Hicks neutrality restrictions on each causes very little additional loss of fit respectively; and thus "Hicks neutrality" will not be rejected for these models (Model 5, 8, and 11). For example, if homotheticity is maintained, then the imposition of Hicks neutrality will result in a test statistics of 3.274, while the .01 x23 is ll.345,and hence "homothetic-Hicks neutral technical change" can not be rejected. Model 11 (CRS-Hidks neutral technical change) is the same model chosen by Berndt and Wood (1975) in their study of the U.S. manufacturing sector. Berndt and.Wood, however, did not estimate the rate of technical change,as they only 155 TABLE 3.1 ALTERNATIVE MODEL SPECIFICATIONS WITH THEIR CORRESPONDING PARAMETER RESTRICTIONS Model Specification Parameter Restrictions l Unrestricted: None 2 No scale bias: YtQ - 0 3 Homothetic: Y,Q = 0 4 Homothetic Y = Y = O - no scale bias: 1Q tQ 5 Homothetic Y1Q = Yit = 0 - Hicks neutral technical change 6 Homo eneous: . = = 0 7 Homogeneous/conditional on Y1Q - YQQ . 0 no scale bias: 8 Homogeneous Y = Y = Y 8 Y = 0 - Hicks neutral technical 1Q QQ tQ it change: 9 Homogeneous Y, = Y = Y; = O - Hicks neutral/conditionalcn1 1Q QQ Lt no scale bias 10 Constant return to scale: GQ = 1, YiQ = YQQ a YtQ = 0 ll Constant return to scale a - 1, Y = Y I Y - Y = 0 - Hick neutral technical Q 1Q QQ tQ it change: 12 No technical change: at = Yit = Ytt = YtQ = 0 13 Hicks neutral technical change: Yit - 0 14 Hicks neutral technical change y, - Y = 0 it tQ - no scale bias: lS Hicks neutral linear technical Y1t ' Ytt = YtQ = 0 change: 16 Hicks neutral linear technical Yit = Ytt = 0 change/conditional on no scale bias: l7 Unitary elasticity of Yij = 0 substitution: 18 Homogeneous Y Y Y a Y - Y - 0 - no technical change: 1Q QQ tQ t it Ct 19 Homogeneous = 0 - Hicks neutral linear technical change: .ouooUnuucmmo we ~o>ou we. on ooou eouu acououuwv aducoouuucwuo no: on acouuquuooo « co; co coo.o~ c-.n~ ~oc.c~ ooc.co on“. s c =-.. uuooaooooo coon: ooocooosoo n o c n a use: "acouuoquuoox co~.ocn coo.ocn noo.oon anc.~cn aco.ocn ooc.ocm "couscous coogcooxoo oo moo coo. coo. coc. coc. coo. coo. om moo. 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Wtwwm.IoIoo oo oovoz co ooooz w ooooz o Homo: m omooz m omvoz mo ooooz mo oovoz mo ooooz o ooooz oomomoozmmooHH0 indicates that higher factor prices have a depressing effect on scale economies and cause Q/Xi to decline, while YiQ<0 indicatestfluaopposite. Therefore, as long as economies of scale are present (SE>0) a negative value of YiQ implies a positive value for the elasticity of average factor product with respect to output. For the manufacturing sector (unrestricted model-1971) these elasticities are positive for K,E, and M since YiQ To have this satisfied we must have: (Q-gg - C) E O, which implies 38 E g . Or further, dQ ' Q d In Q > ' 188 189 ‘sThe parameter restrictions in this case and.the homogeneous case involve the restriction YtQ - 0, due to the requirement that the elasticity of cost With respect to output must be constant or equal to one,depending on whether the production structure is homogeneous or linear homo: geneous respectively. The impos1tion of this restriction implies, at the same time, the absence of scale bias. 6Thismodel has been rejected at-the .05 level of significance when tested against the unrestricted model, but accepted at the .025 level. 7The concavity condition is violated, though not severely at some point for models with no restriction on.the Yit(i=K,L,E,M) coefficient. The noncavities disappear for these models when neutrality (Yitgo) is imposed. This violation of regularity conditions, as Wales (1977) explains, does not necessarily imply the absence of the underlying cost- minimization process, rather it may indicate the inability of the flexible functional form to approximate the true cost function over the range of the data. Here, in Tables (3.5) and (3.6) we have reported the estimated substitution and price elasticities for the manufacturing sector for the year 1958 (which is very close to the mid-point of the sample year, 1959L since the concavity condition is satisfied for all specifications at that point. ' 8Berndt and Khaled computed the dual rate of returns to scale as one over the elasticity of cost with respect to out ut, i.eo (3 1n C/3 1n Q)‘ . To compare with our estimates, we ave inverted their estimate and subtracted it from one . 9Although the point estimate of ”EE is positive (”WEE < 0) for the year 1971, this curvature violation does not seem to be statistically significant when judged by its standard error. The standard error for ”EE’ under the assumption that the share SE is constant and equal to the mean of its estimated valueI turned out to be about .15I which is quite large. See also footnote 7 above. 10The elasticity estimates reported‘by Berndt and Khaled are: 'nEj = .293, -.434, .546, and -.405 for j = KILIEIM respectively. 190 11'The reasons for choosing the homogeneous Hicks neutral specification are twofold: (1) this specification appeared to be justifiable for representing the non-energy manufacturing sector, and (2) for the unrestricted model the estimates of YiQ and Yit turned out to be insignificant. Since for the homogeneous Hicks neutral specifi- cation YiQ == Yit = 0 holds, we only have reported -n. . in 13 Table (3. 12) . CHAPTER IV TESTS FOR THE EXISTENCE OF REAL VALUE-ADDED AND OTHER TYPES OF INPUT AGGREGATION IN U.S. MANUFACTURING 4.0 Introduction The estimation of the production relation and factor demand functions requires some measure of the output or acti- vity level. Real value-added has been used as such a measure in virtually all empirical studies of labor-capital substi- tubility and investment demand, where in the existence of real value-added has been assumed a priori. It has been shown by some authors (e.g. Sims (1969), Gordon (1969), and Arrow (1972)) that when material inputs are used in the produc- tion process, as they are in the manufacturing sector, then the existence of real value-added rests upon the existence of weak separability between the primary input and material inputs. This allows us to write the production function, F, as G(g(K,L),E,M) where g(-) has been identified as real value-added. In this chapter we will test the value- added specification and other types of separability among inputs; but first we summarize the discussion of weak separability in Chapter I. 4.1 Weak Separability Defined The notion of weak separability, which we are mainly concerned with, is defined as follows: If we partition therp- tuple vector of the input x= {XII XZI u-I Kn} into r mutually exclusive and exhaustive subsets as N* = {N1, ..., Nr}, a function, f(x), is said to be weakly separable with respect 191 192 to the partition N* if the marginal rate of substitution between a pair of inputs i, jeNm (mel, ..., r) is indepen— dent of changes in the level of inputs outside Nm’ i.e., f. 3 1 ___ ( ) = 0 a IT xk J or f.f. -f.f. = 0 Vi,j cN , andkéN (4.1) m m Goldman and Uzawa (1964) have proved that f(x) is 'weakly separable with respect to the partition N* (r > 2) if and only if it is of the form _ 1 1 r r (4 2) f(X) - G68 (1:). g (X)) - where gm(xm) is a function of the subvector x!11 e Nm alone. Berndt and Christensen (1973a) have shown that if f(x) is homothetically separable,then the dual cost function C(Q,W) is also separable and therefore we must have: C.C. - C.C. = 0 (4.3) Berndt and Christensen (1973a) have also proved, in the context of production theory, that a strictly quasi-concave homothetic production function, f(x), is locally weakly separable with respect to the partition N* if and only if Oik = ojk’ i.e.,the Allen partial elasticity of substitution between input i and k is the same as that between input j and k. 193 4.2 Various Specifications Discussed In several recent studies of manufacturing a restric- tive assumption about the structure of production has been the weak separability of capital, labor, and energy inputs, as a group from the materials input. Examples are Humphrey and Moroney (1975), Griffin and Gregory (1976), Pindyck (1979), and Magnus (1979). This separability assumption has been necessarily adopted in these studies due to unavailability of reliable data from which to construct price or quantityindices of the materials input. However, one might ideally wish to test this hypothesis rather to impose it apriori. But it is feasible to do so only when price and quantity indices for the materials input are available. In this section we attempt to test this and other similar restrictions on the production structure. Specifically we assume that the general non-homothetic production function (3.14) is weakly separable as: F(KILIE.M;T) = f(g(KILIEIT)IMIT) (4.4) where g(g) is an input aggregator function or microfunction. If g(}) is homothetic in primary inputs, then the dual non- homothetic cost function (3.15) will be weakly separable as: C(wKIW .wE,wM.Q.T) = H.(h.(.WKIWLIWEIT)..WMIQ.T) (4.5) where h(-) is aggregate input price. 194 The translog cost function (_3,lo)._will be approximately1 weakly separable as in.Qi5)if the following parameter restrictions hold: “inM ‘ “j YiM " 0 and ainQ - oj YiQ C) i,j = K,L,E (4.6) These constraints follow directly by applying condition (4.3) to the translog function (3.16),. The price aggregator function h(°) is homothetic, and thus independent of the level of output. Since homotheticity of the aggregator flmction is a necessary and sufficient condition for the validity of the two-stage optimization,one may estimate equation (4.5) in stages. Excellent examples of this two-stage optimization procedure are Fuss (1977), and Pindyck (1979). In the first stage, by choosing an appropriate price aggregator function for energy, they have optimized the mix of fuels that make up the energy input; and then they have optimally chosen quantities of capital, labor, and energy. While the price aggregator functions chosen by Fuss (1977) and Pindyck (1970) are linear homogeneous and homothetic translog functions respectively, we may observe a less restrictive alternative. The less restrictive alternative we may consider is the following formulation which may be quite useful in many practical situations. More specifically, we assume that 195 the non-homothetic cost function (3.15) is weakly separable as: C = H,.gwL.wM.Q.T> ,Q,T) (4.16) where h(x) and g(°) are homothetic in input prices and independent of the level of output. For our non-homothetic translog cost function, inputs K and B will be approximately weakly- separable from.L and M if the following parameter restrictions hold: “'ij ' a-Yik = 0 i,j = K,E, and k = L.M.Q; and (4.17) = 0 i,j = L,M and k = K,E,Q 202 Another version of ((K,E),(L,M)) weak separability is c = H(h(WK,WE,Q,T),g(WL,WM,Q,T)) (4.18) where h(-) and g(-) are non-homothetic functions in input pricesB. The non-homothetic translog cost function (3.15) is approximately weakly separable as specified in equation (4.18) if the parameter restrictions implied by (4.18) are satisfied, i.e., Qink-GjYik = 0, i,j K,E, and k = L,M; and oiyjk-ajyik = 0, i,j L,M, and k = K,E . (4.19) There is of course no reason to restrict our analysis to various possibilities of weak separability among factor inputs. The separability specification also permits us to analyze technological change. We earlier saw that a production function is defined to exhibit Hicks neutrality if it can be written in the following form c = H(h(!/J_.Q).Q.T) (4.20) The parameter restrictions implied by (4.20) on our non- homothetic translog cost function (3.16) are oinT - aniT = 0 1,3 = K,L,E,M . (4.21) An unfortunate limitation of this specification is that the parameter restrictions (4.21) cannot be imposed simultaneously with adding up constraints (281 = 1) without 203 imposing extended Hicks neutrality. For example, if there are only two inputs,the condition for indirect Hicks neutrality is alYZT - azle = 0. This constraint, however, cannot hold simultaneously with the adding-up constraints o1 + a2 - l and le + YZT - 0 unless le = Y2T = O, which only holds for extended Hicks neutrality. Therefore, we adopt and test indirect Hicks neutrality for a subset of the inputs; namely we write (4.20) as C = HChCWK.W .WE.WM7Q).WM.Q,T) (4-208) and the implied parameter restrictions as oinT - aniT = O i,j = K,L,E. (4.21a) Finally, we check the condition for the validity of Harrod neutral technical change with our data. We saw earlier that a cost function exhibits indirect Harrod neutral technological change if it can be written in the following weakly separable form; c = H(h(WK.WE.WM.Q).WL.Q.T) (4.22) The non-homothetic translog cost function (3-16) Will be approximately weakly separable if the following parameter restrictions are satisfied: “inL ‘ “jYiL ‘ 0 - = °°= 4.2 aiij aniT 0 1,3 K,E,M. ( 3) 204 4.3 Statistical Results 4.3.A Manufacturing We have statistically checked the conditions for the validity ofthe imposed parametric restrictions implied by the various types of separability specifications,which were discussed above. To perform these statistical tests we have estimated all eleven specifications,and have computed the likelihood ratio test statistics as twice the difference of the sample log-likelihood values of the unrestricted and restricted models. As is generally well known,the asymptotic distribution of this test statistic is x2 with degrees of freedom equal to the number of restrictions. Of the eleven specifications we have tested, five specifications could not be rejected as the likelihood ratio test statistic for these models turned out to be small compared to the .05 (.01) x2 critical values. These specifications are the ((K,L,E),M) separability specified in equation (4.7) , the value-added specification of equation (4.18), and the two versions of the ((K,L,E),M) weak separability specifications (see footnote 3), and finally one of the ((K,E),(L,M)) weak separability specifications specified in equation (4.18), i.e., H(h(WK,WE.Q.T) . gCWL.WM.Q.T)) . The likelihood ratio test statistics we have obtained for these five models (in above order) are 0.282, 3.732, 4.076, 2.818, and 7.346; while the 0.05 x2 critical values (with two and three degrees of freedom) are 5.991 and 7.815 205 TABLE 4.1 ALTERNATIVE WEAK SEPARABILITY TESTS U.S. MANUFACTURING, 1947-71 Type of Separability Independent Likelihood Ratio Test Parameter Statistics: Result Restrictions - 2 1n L(w)/L(Q) [(W ! :L,:E,T) ,WM’Q,T] 4 26.290 * [(WK EQ T) W M’Q’T] 2 .282 [(WK WE.W T)W M’Q' T] 2 19.506 * [(W K WL’ T).W E’ WM.Q,T] 3 42.370 * [(W K,W L’Q’T)’WE’WM’Q'T] 2 3.732 [(W K WE,T)’(WL :WM ,T):Q9T] 5 15.986 * [(WK,WE ,T)W L’ WM’Q’ T] 3 4.076 [(W K’ WE,1»IQ,'JL‘),(L WM,Q.T)1 3 7.346 [(WK ,WE ,Q,T),WL WM,Q,TJ 2 2.818 [(W, Q), WM,Q, T] 2 14.168 * [(WK ,WE ,W M) L,Q T] 4 25.504 * [(WK ,WL ,W E), WM L] 2 12.536 * W( W) WE ,W M] 2 40.426 * [(w:, E) WL ,W M] 2 14.748 * 3 24.176 * [(wK. We) .04 L.wM>J * Weak separability hypothesis is rejected at .01 level of significance. 206 TABLE 4.2 ALTERNATIVE WEAK SEPARABILITY TESTS U.S. NON-ENERGY MANUFACTURING,.I947v7l Type of Separability Independent Likelihood Test Parameter Ratio Results Restrictions Statistics ~21nL(m)/L(9) [(WK .WL. T)W MQ, T] 4 7.644 [(WK ,WL W,w:,Q, TLW M,Q, T] 2 6.522 [(WK ME’W ,T)W M,Q, T] 2 3.184 [(W K’ W:,T),WE, WM,Q,T] 3 1.310 [(WK,WL,Q,T),WE;WM,Q,T] 2 .018 [(WK,WE,TL(WL,WM,TLQ,TJ 5 11.920 [(WK,WE,T),WL,WM,Q,TJ 3 10.498 [(WK»WE.Q.T).(Wt,WM.Q.T)] 3 10.744 [(W KL,wE,Q,T),w ,WM,Q,T] 2 10.498 * [(W, Q),W M,,Q T] 2 3.616 [(W’ ’WE ,W N) L,.Q T1 4 4.722 [(WK ,LW ,W EL WM] 2 15.138 7" [(WK W) WE ,W M] 2 15.772 *, [(w:, WEL WL ,W M] 2 22.968 * [(Wk, WE),(WL,WM)J 3 41.432 * *Weak separability hypothesis is rejected at .01 level of significance. 207 respectively. These five specifications, therefore, cannot be rejected based on these test statistics. However, we are inclined to reject the first four specifications on the grounds that none of them will satisfy the concavity condition required for well-behavedness of the cost function. The ((K,E),(L,M)) weak separability specification of equation (418)13 the only specification that our net output data and model specification (nonhomothetic-nonneutral) cannot reject, while at the same time the concavity conditions are satisfied for this model. However, the other version of this separability specification,i.e.,H(h(Wk,WE,T), g(Wi.WM.T),Q,T) is rejected at .01 level of significance. It is interesting to note that the ((K,E),(L,M)) separability specification was the only specification that Berndt and WOod (1975) could not reject with their data and model specification (CRS - Hicks neutral technical change), and that the cost function was alsowell-behaved.4 One important result is the "rejection" of the value- added specifications. Many empirical studies of investment‘ demand and capital-labor substitutability in the U.S. 'manufacturing sector have in fact assumedaapriori the value« added specification. One, therefore, is inclined to view the results of such studies as unreliable due to the rejection of value-added specification. 4.3.B Non—manufacturing Energy As for the non-energy manufacturing sector, we have 208 repeated the same number of statistical tests to check whether any of the parametric separability conditions imposed on the translog cost function (16) are satisfied. 0f the eleven separability specifications we have tested for the non-energy manufacturing sector, we could reject only one at the 0.01 level of significance, where the general "nonhomothetic-nonneutral" specification was chosen as the unrestricted model. The rejected model is the weak separ- ability specification of H(h(Wk,WE,Q,T),WL,WM,Q,T)5. This "nonrejection" situation here is quite similar to what we experienced in the previous chapter when we were estimating and testing for the validity of various model specifications for the nonenergy manufacturing sector. There we concluded that, forthe nonenergy manufacturing sector, the characterization of the production structure by a nonhomo- thetic-nonneutral cost function was unnecessary, and that the adoption of a homogeneous-Hicks neutral specification was justifiable. Maintaining the "homogeneous-Hicks neutral" specification as the unrestricted specification, we have tested for the validity of three well-known separability specifications; i.e., ((K,L,E),M) weak separability; the value-added specification, (-(K,L) ,E,M); and ((K,E),(L,M)) weak separ— ability. These specifications have frequently been utilized by many researchers in a number of empirical studies. For example, ((K,L,E),M) weak separability has been assumed by Griffin and Gregory (1976) and Magnus (1979) in their 20.9 studies, due to lack of reliable data on.the materials input price. The value-added specification has also been employed in many empirical studies of investment demand and capital- labor substitutability in U.S. manufacturing. And finally, the ((K,E),(L,M)) weak separability is the separability that Berndt and Wood (1975) were not able to reject in their study of U.S. manufacturing. All these specifications, however, are rejected at the 0.01 level of significance with our data for'the nonenergy manufacturing sector. The likelihood ratio test statistics for these specifications (in the same order mentioned above) are 15.138, 22.968, and 41.432, while the 0.01 x2 critical values with two and three degrees of freedom are 9.210 and 11.345 respectively. CHAPTER IV FOOTNOTES 1There is a distinction between the translog as an exact representation of a functional form.and as an approximation to a functional form. Tests ofthe separability hypothesis based on an exact interpretation of the translog are restrictive. Specifically, the separability constraints imply that the separable form of a translog function must be either a Cobb-Douglas function of translog subaggragates or a translog function of Cobb-Douglas subaggragates. To avoid this restriction in carrying out our tests, we have chosen the more general interpretation oftfluatransloe as a second-order approximation to some unknown arbitrary cost function. 2See footnote 1 above. 3Two less restrictive versions of (4-16) and (4.18) are C = H(h(WK,WE.T). WLWM.Q,T) and c = H(h(WK,wE.Q»T),wL,wM,Q.T) respectively. The parameter restrictions for the former spec1f1cation are: aink-anik=0’ and aiij-ajyiQ=0; i,j=K,E, and k=L,M. While for the latter specification the following implied parametric restrictions must hold: aiyjk-ajyik=0; i,j=K,E, and k=L,M. 4For the record, however, we have also checked the validity of the separability conditions for three commonly utilized specifications using the same model specification A (CRS-Hicks neutral technical change) employed by Berndt and Wood (1975). The specifications tested are: ((K,L,E),M) weak separability, the value-added specification, and ((K,E),(L,M)) weak separability. All these three specifi- cations have been rejected at the .01 level of significance when each was tested against the maintained "CRS-Hicks neutral technical change" specification. 5In addition to this specification, there are four other separability specifications which our data can reject, although at a lower level of significance. Specifically, the specification number312,6, 7, and 8 in Table (4.2) are rejected (the two former ones at the .05, and the two latter ones at the .025 level of significance). 210 CHAPTER V SUMMARY AND CONCLUSIONS The oil crisis of 1973 and the subsequent continuing increases in the price of energy have led to increased interest in the characteristics of energy demand and substi- tution elasticities between energy and nonenergy inputs. The U.S. manufacturing industries that account for approximately one-fourth of aggregate energy consumption have attracted a- great deal of attention as a potential sourse of reductions in energy demand. In recent years, a growing number of econometric studies have focused on the estimation of the Hicks-Allen substitution elasticities among energy and nonenergy inputs imithe manufacturing process. The information regarding these elasticities provided by empirical studies is very essential in deriving policy implication of increasingly scarce and higher priced energy inputs. A review of the literature, however, indicates that these estimates have not always been consistent. One interesting result is the contradictory evidence concerning substitution possibilities between capital and energy. It has not been, however, the purpose of this study to reconcile these differences concerning factor substitution. This study rather has dealt with another subtle issue ignored in past studies of 211 212 manufacturing. The primary objective of this study has been to examine past studies of manufacturing energy demand and to employ an alternative model for the estimation of industrial conditional factor demand functions. More specifically, in past econometric studies of manufacturing sectors the researchers have specified a flexible cost function over four aggregate inputs of capital services (K), labor services (L), energy (E), and materials (M). In these studies, output, materials and energy have been measured as "gross" magnitudes in the sense that they contained intra-industry shipments/pi, . . (B.2) x(t>/x(t) = 2 s1 xi(t)/xi(t) respectively. 238 239 However, due to the nature of economic data, which take the form of observations at discrete points in time, the following discrete form of the Divisia index is used in practice: 1°3 Pt‘ 1°55 pt-l ”X g (~Si,t+si,t-1)(1°g Pi,t' logpi,t-l) (B.3) where sit = Pitxit/E p. The discrete Divisia index (B.3) itxit' is in fact the discrete approximation to the continuous-time Divisia index (3.1). This is Tornqvistt's discrete approxima- tion to the continuous formula. It approaches the continuous form as At+0. The Divisia index has many desirable properties such as an aggregation procedure discussed by Richter (1966), Theil (1967), Hulten (1973% and Diewert (1976). It is also a fact that this index suffers from one extremely serious problem. Since the index is a line integral, it is dependent, in general, upon the path on which the integral is taken. Hulten (1973), however, has shown that if the aggregate (x)‘ exists, is homogeneous of degree one in its components (xi) and there exists a corresponding price (p) normal at each point unique up to a scalar multiple, then the Divisia index is path independent,and retrieves the actual values of the aggregating function, subject to an arbitrary normalization in some base period. Therefore, the Divisia index preserves, up to the normalization, all the information in the problem; and it is at least as good as any other 240 index. In other words, the Divisia index is the best choice among index numbers, given the above conditions. 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