IIIHIIHllIllIIIUIIIILUIMMJMIIIMMM 3 1293 OVERDUE FINES: 25¢ per day per item RETURNING LIBRARY MATERIALS: \. r» ‘ fi‘ifl‘é’? Place In book return to remove " charge from circulation records 6am.“ . ' ' \- . j a ‘ :' }o\\’ '/ n.- v . . ' “.311“, \: @ Copyright by James Howard Grant 1 9 7 9 SUBSTITUTION AMONG LABOR, LABOR AND CAPITAL IN UNITED STATES MANUFACTURING BY James Howard Grant A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Economics 1979 ABSTRACT SUBSTITUTION AMONG LABOR, LABOR AND CAPITAL IN UNITED STATES MANUFACTURING BY James Howard Grant This study addresses questions concerning factor demand, labor-labor substitution, and labor-capital sub- stitution in factor markets for the manufacturing in- dustry across the United States. The objectives are threefold: l) to analyze input demand and labor-labor and labor—capital substitution where labor heterogeneity is captured by disaggregating labor under various schemata, 2) to test for weak separability among the labor sub- aggregates and capital, and 3) to test for differences in the production technology of manufacturers across the United States, specifically to test for regional or area differences in the technical relations among labor and capital inputs. These objectives require estimation of production relations such that there are no a priori constraints on the elasticities of substitution between factors. Thus, where appropriate, either translog cost functions or translog production functions are specified as quadratic approximations to the production process; where the relevant inputs are labor, disaggregated into separate groups by one of four schemes, and capital. Each of the following labor classifications are examined: i) occupation, ii) age, iii) education, and iv) a joint age-education classification. Finally, since Standard Metropolitan Statistical Areas (SMSAs) define the relevant labor markets, the translog functions are estimated across a sample of 84 SMSAs for the year 1969. Estimation assuming technology is similar across the country led to the following conclusions. White collar labor and blue collar labor are substitutes in production. However, when white collar labor is further disaggregated into professional labor and clerical labor, these two groups are very slight complements in production. Labor- labor substitutability increases as the schooling of labor from each group is more similar. As the age groups of labor become more similar, labor-labor substitution is more difficult. Labor of all types are substitutes (or, at least, not complements) with physical capital. As the amount of human capital acquired by laborers increases, they become less substitutable with physical capital. And as laborers acquire more occupation-specific training, their employment is less responsive to changes in the prices paid for their services. Tests of weak separability hypotheses suggest there may not be a consistent aggregate labor index allowing estimation of a two input (labor, capital) production function. However, labor may be disaggregated into white collar—blue collar groups, or white collar pro- fessional-white collar clerical, plus blue collar labor groups. The hypothesis that labor age 25+ form a weakly separable group from younger labor and capital cannot be rejected. Finally, evidence suggests laborers with similar schooling form weakly separable groups. Each model was reformulated to capture first the effect of land, as a factor of production, on the techni- cal relations between labor and capital in production and, second, regional variations which might influence the production technology used. Joint weak separability of labor and capital from land in production cannot be ac- cepted. However, weak separability of labor from both capital and land is more easily accepted. Also, human capital-physical capital substitution increases as land, or plant location, becomes a less important factor in production. Finally, there is strong evidence that the production technology differs across regions. The technology of sunbelt SMSAs differs from that of nonsunbelt SMSAs in that physical capital-human capital substitution is greater in the sunbelt region. Further, the share of total income of either white collar professional labor or highly educated labor is greater in the sunbelt but the income shares of blue collar labor and less educated labor are greater in the nonsunbelt region. TO HOWARD AND LILA WALCOTT ii ACKNOWLEDGEMENTS I wish to acknowledge my thesis committee, Dan Hamermesh, Dan Saks, Anthony Koo, and Ken Boyer, with special thanks to Dan Hamermesh, my chairman, whose aid and encouragement from the beginning were invaluable. There are other members of the faculty, who have con- tinually given me valuable assistance, whom I want to thank. They are Richard Anderson, Robert Rasche, Cynthia Rence, and Lawrence Officer. I appreciate the assistance of Harriet Dhanak and Robert Nevius in the gathering and preparation of data. Noralee Burkhardt has been very helpful in preparing the final draft and earlier drafts of this thesis. I wish to thank my fellow graduate students for their interest and comments concerning this work, and especially for providing a congenial atmosphere for academic pursuits. Finally, Special thanks goes to my family, Bruce, Dorothy, Patty, Todd, little Bruce, and Sue, for their ever present encouragement and support. iii TABLE OF CONTENTS Page List of Tables vi List of Figures ix Introduction 1 chapter I THEORETICAL FRAMEWORK 6 A. Translog Production Functions 6 B. Translog Cost Functions 14 C. Translog Cost Function and Production Function Specifications Compared 19 D. An Extension 21 E. Separability 24 II MODEL SPECIFICATION AND DATA 30 A. Introduction 30 B. Model Specification 31 C. Market Definition 36 D. Statistical Considerations 38 B. Data 42 III MODEL ESTIMATION 46 A. Introduction 45 B. "Goodness" of the Translog Estimates 47 C. Allen Elasticities of Substitution and Derived Input Demand Elasticities 59 D. Summary of Relevant Previous Studies of Substitution Elasticities 70 E. Hicks Elasticities of Complementarity and Derived Factor Price Elasticities 80 F. Separability Tests 87 G. Other Separability Studies 98 H. A Policy Application 103 I. Summary 112 iv IV V TWO ALTERNATE FORMULATIONS A Model of Land Use A Reformulation Including Manufacturing Employment Density Estimation of the Manufacturing Employment Density Model Estimates Using the Density Gradient Reformulation A Geographic Reformulation Estimates Using the Geographic Reformulation Summary CONCLUSIONS APPENDICES A B Proof of Proposition 3 Sufficient Conditions for Weak Separability for Various Translog Functions SMSAs Sampled for Translog Estimates Under Alternative Formulations Estimated Parameters of the Density Gradient Cost Function Estimates When Labor is Disaggregated By Age Under the Alternate Formulations BIBLIOGRAPHY Page 115 115 128 136 140 165 171 184 186 194 194 197 200 207 210 213 Table LIST OF TABLES Models Specified for Cross Section Study of U.S. Manufacturers 1969 Estimated Translog Coefficients Average Values of Dependent and Independent Variables Examination of Regularity Criteria for Each Model Estimated Partial Allen Elasticities of Substitution, 1969 Estimated Input Demand Elasticities, 1969 Studies of Substitution of Production and Nonproduction Workers Studies of Substitution Among Age Groups Studies of Substitution by Education Group Estimated Hicks Elasticities of Complementarity, 1969 Estimated Factor Price Elasticities, 1969 Weak Separability Tests for Labor Disaggregation by Occupation Weak Separability Tests for Labor Disaggregation by Age Weak Separability Tests for Labor Disaggregation by Education Weak Separability Tests for Labor Disaggregation by Age-Education vi Page 32 50 52 56 6O 67 72 77 78 82 84 91 92 93 94 Table 3.15 B1 C1 D1 Studies Testing Weak Separability Hypotheses Elasticities of Substitution and Factor Shares Estimated Translog Coefficients: Under the Density Gradient Reformulation Regularity Criteria for the Density Gradient Reformulation Substitution Elasticities under the Density Gradient Formulation Estimated Change in Factor Share of Input 1, Given a Change in the Density Gradient Changes in Substitution Elasticities, Given Changes in the Density Gradient Total Derivatives of Factor Shares and Substitution Elasticities of the Average SMSA Estimated Translog Coefficient under the Geographic Reformulation Regularity Criteria for Geographic Reformulation Substitution Elasticities under the Geographic Reformulation Sufficient Conditions for Weak Separability SMSAs Sampled Density Gradient Parameter Estimates vii Page 99 110 142 144 149 151 153 160 172 175 179 198 201 207 Table E1 E2 B3 E4 Estimated Translog Coefficients Substitution Elasticities under the Density Gradient Reformulation Changes in Substitution Elasticities, Given Changes in the Density Gradient Total Derivatives of ABS for the Average SMSA Substitution Elasticities under the Geographic Reformulation viii Page 210 211 211 212 212 Figure LIST OF FIGURES .Land Demand and Supply by Location I Land Demand and Supply by Location II Land Demand and Supply by Location III Density Gradient Estimation Alternative Density Gradients of the Representative SMSA ix Page 119 121 124 138 158 INTRODUCTION Substitution among different types of labor and between them and capital has become an important subject among economists interested in labor market policy. Empirical estimates of elasticities of substitution be- tween pairs of inputs are essential for predicting the effect of policy changes. The proposed reduced minimum wage for youth labor is intended to encourage employment of young labor. However, such a policy implies concurrent displacement of adult labor, the extent of which depends on substitution between youth and adult labor. Invest- ment tax credits decrease labor demand, assuming capital and labor are substitutes, but their effects on demand for certain types of labor may be positive if capital is complementary with them in production. Employment and training programs convert low-skilled labor to high- skilled labor; the programs' effectiveness depends on how substitutable these resources are in production, for their substitutability will determine the extent of the change of their relative wages in the post-training equilibrium. The theoretical framework for the analysis of substitution between inputs goes back to Allen (1938), but it is only with relatively recent improvements in the theory of production, empirical techniques, and data 1 2 that Allen's framework has been put to use. Empirical estimation of production relations from a multiple input production framework have only appeared since Griliches' (1969) seminal work considering 3 inputs: low-skilled labor, high-skilled labor, and capital. Currently, estimates of substitution between alternative sets of labor sub-aggregates (except labor classified by occupa- tional groups) and physical capital in production are rare or nonexistent. The advent of estimable multiple input functions and of improved data have recently led to empirical searches for appropriate or consistent aggregates of labor and capital for use in estimating substitution relations. Leontief (1947) introduced the idea of what is now re- ferred to as weak functional separability among inputs, thus providing a criterion for resource aggregation. Groups of inputs which are weakly separable from others may be formed into consistent aggregates in the sense that marginal changes in the level of use of other inputs~ outside the separable group do not alter the technical relations among inputs within the group. There has been very little work exploring weak separability among labor sub-aggregates and capital. This study examines a cross section of manufacturing industry in the United States. Its objectives are: l) to analyze input demand and labor-labor and labor- capital substitution under the assumptions that labor is 3 a set of heterogeneous inputs joined with capital in pro- duction, 2) to test whether sub-aggregates of labor can be considered separable from other labor sub-aggregates and capital, and 3) to test for regional or factor market specific differences in production technology across the United States. Once these objectives are achieved, the results of the study may be put to use examining the influence of public policies such as those mentioned earlier, or of changes in the structure or composition of factors of production, on employment and earnings of various sub- groups of the labor force. To accomplish the objectives of this study it is desirable to estimate production relations via a model which does not impose a priori constraints on elasticities of substitution among factors of production, but which also permits tests of alternative specifications of inputs in the production relation. The transcendental logarithmic (translog) function will provide such a model of production. The translog function is a member of the class of flexible functional forms; these forms are defined by the property that they can provide a second order approximation to any twice differentiable function (see Blackorby, Primont, and Russell, 1977, or Denny and Fuss, 1977, for a demonstration)u Chapter I provides the theoretical framework of analysis' for the following chapters. Translog production functions, translog cost functions, and the concept of weak functional separability among inputs are developed. Their use in de- riving approximations to the relations of substitution and 4 complementarity among inputs in production is also discussed. In Chapter II alternative aggregations of labor and capital as inputs in the production process are developed. The correct model specification, econometric considerations, and data sources are also presented. Chapter III provides estimates of the translog models derived in Chapter II. Assuming the prevailing production technology is the same across the United States, estimates of elasticities of substitution or complementarity among inputs are presented. The results, where appropriate, are compared with those of similar studies. Hypotheses of functional weak separability among inputs are tested. Finally, results are used to estimate the impact of a reduced minimum wage for youth labor on employment and earnings of the remaining segments of the labor force. In Chapter IV two alternate formulations of the translog models are analyzed. The first takes into account factors of production other than labor and capital which may enter the production process in a non-weakly separable fashion and thus alter the relations of substitution among labor and capital. The second tests for geographic differences in production technology which might augment the technical relations among labor and capital. Chapter V summarizes conclusions of the-preceding chapters. Among the more interesting conclusions are that workers from any given labor group are, in general, sub- stitutes for workers from other groups and for physical capital. There is little support for the hypothesis that 5 a consistent aggregate index of labor can be constructed; however, it may be acceptable to disaggregate labor into two or three major sub-groups. Finally, there is some evidence in support of an hypothesis that the technical relations among labor and capital inputs are not the same across the United States. ThereforPa national labor market policy is not likely to have the same impact in each region of the United States. CHAPTER I THEORETICAL FRAMEWORK Throughout the thesis it is assumed that 1) relevant production functions are first-degree homogeneous in the set of all inputs, and relevant cost functions are first— degree homogeneous in the set of all input prices, and 2) markets are competitive. Using these assumptions, translog production function, or cost function analyses are developed. A. Translog Production Functions Consider first the translog production function: finQ = f(£n Xl,£n X2,...,£,1Xn) (1.1) where Q is defined as output; Xi’ i = l,...,n-l, is the ith labor input; and Xn is capital. The specific form of the translog production function is: n Kn Q = ,b1do + 1:1 oi fin Xi n .. £n)L .01X., 1.2) 1—1 jgl Ylj 1 3 ( IML'S l *5 where do represents the state of technological knowledge, mi and Yij are technologically determined production parameters, and a symmetry condition (Yij = in) is imposed (For further discussion see Berndt and Christensen, 1973b.) With the assumption that markets are competitive, the necessary conditions for efficient production are: 6 i = l,..,,n' SQ -——'= P-: 3Xi i where pi is the price of the ith factor service. For each input, Xi, this implies: _3£n Q _ Q , i _ i i _ M 3 X . i 1 0' i=l'ooo,n; '51 I Q) X 10 I (D I p where, given the assumption of first-degree homogeneity in production, Euler's theorem demonstrates that Mi is the relative cost share of the ith input in the total cost of all inputs used to produce Q (i.e., Mi = . . X.. . pi Xi /Zj ij3 for any input l) The following set of share equations are derived by taking the partial logarithmic derivatives 3£nQ/3£nxi and equating them with the cost shares: y..£nX., i = l,2,...,n. (1-3) 13 3 Z 0 Q + "545 j 1 Linear homogeneity in production imposes the following parameter restrictions: 2 a. = 1, E 7.. = 0, f y.. = 0 and i 1 i ij j ij . . (1.4) 2 2 Y1] = 0! 1!] = l,2,...,n. l 3 The equation system 1.3 with the inclusion of the symmetry restrictions, Yij = in’ is more properly expressed: M. = a. + E 3 n y..£nX. + E y..£nX., j = 1,2,...,n.(l.5) 3 i=1 1] l i=j 1 +1 31 With imposition of the linear homogeneity restrictions 8 the system of n equations becomes a singular system. Parameters of the nth equation can be expressed in terms of the parameters in the remaining n-l equations. A non-singular set of share equations would be: n-l M.=a.+ g ..£n(X./X)+ v..£ x.x , 3 3 i=1 Y1] l “ i=§+1 31 n( l/ n) (1 6) i = l,2,...,n-l For example, in the four input case the set of factor share equations, a5), would appear as follows: M1 = 01 + yll£n(Xl/X4) + Y12£n(X2/X4) + Yl3£n(X3/X4) M2 = o2 + y12£n(Xl/X4) + Y22£n(X2/X4) + y23£n(X3/X4) (1.7) :2 I 3 - 03 + Yl3£n(Xl/X4) + y23£n(X2/X4) + y33£n(X3/X4). The remaini ' ng parameters o4, Y14: y24, y34, y44 are determined from the linear homogeneity constraints: 0‘4 = 1 ‘ (“1 + Oi2 + “3" 3 y14 = - .E 713, i = 1,2,3, and j—l Y44 = ’(Y14 + Y24 + Y34" A production function is well behaved if and only if the marginal product of each input is positive, and if it is globally convex. Globally the unrestricted translog function does not satisfy these conditions. However, there are combinations of inputs over which the translog function is well behaved, over which the translog function may provide a good approximation to the underlying produc- tion relation. Therefore, at each observation the mono- tonicity and convexity conditions must be checked. Mono- tonicity is satisfied if aQ/BXi > 0. An equivalent con- dition is that: The condition for strict quasi-convexity is satisfied if the Hessian matrix of partial second derivatives of Q is negative semi-definite. This study examines the transloq function as a quadratic approximation to an arbitrary production func- tion. Denny and Fuss (1977) develop the following de- finitions and propositions relevant to the approximate translog function. Definition 1: A second order approximation to the pro- duction function Q = f(z), where z = (21,...,ZN), is the Taylor series quadratic expansion N A _ * 3f __, * 1=l 1 z N N 2 l 3 f * * + _ Z —————— *IZ. - Z.][Z. - 2.], 2 i=1 j=l aziazj'z 1 1 j j 1: i: a: . . where z = [21,...,ZN] is the p01nt of expansion. Proposition 1: The'translog function of the form(l.2) with symmetry imposed (Yij = in) is a quadratic 10 o I ' 0 * * * approx1mat1on (around the expan51on p01nt x = [Xl,...,X ] n [l,...,1]) to an arbitrary production function of the form fin Q = f[£n Xl,...,£n Xn]° Proof: Define —/z -2 * Q[Z] - D Q! 2.1 - n xi! (10 " f(Z )1 Q =.§_'€._ o-Y =_3__2_f_l __82f -'Y . - *1 -- ,. *-"_—- *— .., 1 321 z 13 3213sz2 azjazi z 31 * Zi = in 1 = 0. Then equation MHZ) vuth‘yij = in satisfies Definition 1. Proposition 2: The translog function of form(l.2)with symmetry (Yij = in ) and the adding up conditions(l.4) imposed is a quadratic approximation (around the expansion point x* = l) to an arbitrary linear homogeneous pro-’ duction function. Proof of Proposition 2 involves writing the Euler equations for linear homogeneity of an arbitrary production function in terms of the definitions of Pro- position 2, then demonstrating that the equation of Definition 1 with the Euler constraints applied is identical to equation(l.2)with symmetry (y.. = Y--) and the adding 1] 31 up conditions(l.4)imposed.l 1See Denny and Fuss (1977),Definition l and Propositions 2 and 3, pp. 406-407. 11 R.G.D. Allen (1938, pp. 503-509) defined the Allen partial elasticity of substitution (AES) between inputs 1 and j as: ’fx 1f_ _ _ k a If..| O1'=k-lxx k '—3‘L’ 3 11' If! where If] is the determinant of the bordered Hessian matrix - _ ” 3f 8f ‘1 f — O 5—: ,.. . . , 8X n 8f 3X1 32f 8X.8X. 1 3 8f 3X L..n .2 and ifij! is the cofactor of the ijth element of f. The AES, Oij’ measures the effect on the quantity of factor i of a change in the price of factor j hold- ing output and other input prices constant. Inputs 1 and j are substitutes in production if Oij > 0, and are complements in production if 013 < 0. Using the share equations(l»5) 0 (< 0), p-complements (substitutes) as Oij < 0 (> 0). The factor price elasticity, or the percent change in price of factor 1 given a change in the quantity of 14 factor j used in production is, (see Sato and Koizumi, 1973a, pp. 48-49): 3 £n Pi a Zn xj = Mjcij. (1.13) B. Translog Cost Functions Now turn to the translog cost function. Shephard (1953, 1970) and Diewert (1971) have demonstrated that input demand and substitution relations can also be in- vestigated by means of cost functions. Assume production of alternative levels of output takes place according to a cost function: c = C(Q, pl, p2,..., pm), (1.14) where C is total cost, Q is output and pi is the price of services of the ith input. Further, assume that C is: i) a positive, real valued function in Q which approaches infinity as Q approaches infinity, ii) first— degree homogeneous in pi, iii) a concave function in pi for all positive levels of output. Given these assumptions Shephard (1953, 1970) and Diewert (1971) have demonstrated that iv) if C is a continuously differentiable function of pi and is minimized with respect to Q > 0, and all input markets are competitive then there exists a well defined production function that is dual in relation to the cost function. This dual production function may not 15 be explicitly expressable in parametric form, but the technological relations among inputs in production can be determined from the cost function alone.3 It is important to point out that the dual to a translog cost function is not necessarily a translog production function. Define the constant returns to scale transloq cost function as: n ZnC = 0180 +£nQ + -E Ei£npi 1—1 (1.15) n n + 6..£n .£n ., 1Z1 jgl 13 pl p3 where 80, Si, éij are technologically determined para- meters and 6 = 6 is the symmetry constraint. ij ji Linear homogeneity in input prices implies the following constraints: (1.16) Z Z Gij = 0; irj = l,2,...,n. 1 3 With the assumption of perfect competition, input prices may be considered fixed. Then, for a given level of out- put, input demand functions subject to cost minimization can be derived from the partial logarithmic derivatives. 3See Humphrey and Moroney (1975) for an extended dis- cussion. 16 . . 3C Sufficient cond1tions for Shephard's lemma,4 -57 = Xi' 1 i = l,2,...,n, are also met. Therefore: we =ac p_i=§i_p_i_=3_‘i_p_i__=s. 82h P. 8p. C C X X-P- 1' l l jjj where total cost C = z ijj. si is therefore defined 1 as the cost share of input i in total cost of producing Q. The set of cost share equations derived from partial logarithmic differentiation is: si = 81 + E aijznpi ; 1 = l,2,...,n. (1.17) With the symmetry and linear homogeneity constraints im- posed the cost share equations become: n-l S.=B..+ E 6..£n(p./p)+ X 6..£n(p-/P) j 3 i=1 1] 1 n i=j+l ji 1 n (l 13) i = l,2,...,n-l; where the parameters of the arbitrary nth equation are determined from the parameters of the remaining n-l equations. For example, for a four-input cost function the cost share equations are: 4See Diewert (1971) for proof of Shephard's lemma. 17 S1 = Eéfl = 81 + 5112M(P1/P4) + 612£n(p2/p4) + 513£n(p3/p4) p2x2 82 = —C—— = 82 + 512£n(p1/p4) + 622£n(p2/p4) + 623£n(p3/p4) p3x3 S3 = c = 83 + 513£”(pl/p4) + 523£“(p2/p4) + 633£“(p3/p4) where the remaining parameters 84, 611, 612, 613, 614, are determined by the linear homogeneity constraints. As with the translog production function the monotonicity conditions i),and the concavity condition iii),do not hold in general for the translog cost function. For any given set of input prices, condition 1) holds when Si > O, i = l,2,...,n, and condition iii) holds when the Hessian matrix of partial second derivatives of C is negative semi-definite. Finally, as with the trans- log production function,the Denny-Fuss Propositions hold for the translog cost function. The function is a quadratic approximation to any arbitrary cost function of the form: £n C = g[£nQ, inpl, £np2,...,£npn]; and the transloq cost function, with the condition of first-degree homogeneity in input prices imposed, is a quadratic approximation around the expansion point [P;,...,P;] = [l,...,l] to any arbitrary cost function which is first-degree homogeneous in input prices. 18 Under the translog cost function specification the partial ABS are easier to determine. Uzawa (1962) has derived the partial AES from cost function 0.14) as: 2 0.. = C a c /(ac ac 1] apiapj api apj ) With the translog specification the partial AES are: 6i1 + S: - Sl Oii = ——- 7. ; 1 = 1,2,. ,n S. 1 611_+ Sisj (1.19) 0.. = , 1 # j ij Sisj As with the production function specification, the price elasticities of demand for inputs are: n.. = 5.0.. . (1.20) 13 J 13 With reSpect to cost function 0.14)the Hicks partial elasticity of complementarity, c.., is defined 13 as where C is the bordered Hessian matrix for C corres— ponding to f of equation(1.8). With the translog specification,the HEC are: cij = [Kijl/IKI (1.21) where K is a symmetric matrix defined in a manner similar to H of equation(l.9). 19 Finally the factor price elasticity is: Eij = Sjcij' (1.22) C. Translog Cost Function and Production Function Specifications Compared On the surface it may appear that results of estimation given a production function specification should be identical with results given a cost function specifica- tion. Both functions are translog in form and empirically they both have the same set of cost shares as dependent variables. However,there exists strong evidence that any comparison should go no further. The first-degree homogeneous cost functions of this study specify input prices as the exogenous vari- ables, while input quantities are exogenous in the pro- duction function specification. If input prices are endogenous then the appropriate cost function is one which is non-homothetic in input prices, C(Q, p(Q)). However available data prohibit its use. When there is no in- dependent quantity measure of output except total revenue, estimation of the non-homothetic function is not possible. Hence the case of endogenous input prices is not easily treated with a translog cost function. Further the (Shephard-Diewert) duality property of cost and production functions and the use of Shephard's Lemma in deriving factor share equations(l.l7)depend on a constant returns to scale technology and exogeneity of input prices. 20 The appropriateness of a cost function or pro- duction function specification, then, depends on the underlying economic characteristics of the problem at hand. For instance if the assumption is that labor supply is relatively price elastic (i.e. factor prices are assumed exogenous), then specifying a cost function is the correct approach. However if labor supply is assumed to be relatively price inelastic (i.e. factor quantities are assumed exogenous), then a production function is the better specified model. While it is true that cost and production func- tions are dual in theory, it is not true that a trans- log approximate cost function satisfying these condi— tions is dual to a translog approximate production func- tion. The translog function does not globally satisfy the (Shephard-Diewert) criteria, but may locally. The dual production function may not even exist in closed form. However,its substitution elasticities can be ob- tained from the translog approximate cost function para- meters. Whereas the production specification assumed i) that the production technology can be approximated by a first-degree homogeneous translog production func- tion, and ii) that producers are profit maximizers. The similarity of estimated elasticities obtained from the models depends, then, on the relative ability of each model to represent the true underlying production 21 relations. For any given problem each model may not be an equally good specification. Finally, the choice of cost function or production function specification affects the computational ease of the elasticity estimates. From a cost function, the partial AES is obtained from estimated parameters of a single equation. Derivation of any partial AES from the production function specification, however, involves in- version of an n x n matrix of estimated coefficients. If any one estimated coefficient in the system has a large standard error, this will be reflected in all the estimates derived from the production function specification. Though this last point is important to keep in mind, the most important criterion for choosing a cost function or production function approach is whether input prices or input quantities can be assumed to be exogenous to the unit under observation. D. An Extension There has been some concern that, with respect to cross section data or data which are constructed in such a way that expansion around the unit vector is not convenient, the translog function may not provide a quadratic approximation to the relevant production func- tion over the range of observations. For instance, the Denny-Fuss propositions demonstrate that the translog 22 function approximates an arbitrary production function at the point x* = [l,1,...,1]. The propositions do not indicate what can be said about the translog function at any other expansion point. In particular, if the exogenous variables are not constructed from an index which can be transformed such that the transformed exogenous variables are clustered around the unit vector, the Denny-Fuss pro- positions do not make it clear that the translog function is still a quadratic approximation to an arbitrary pro- duction function at an expansion point which is more re- presentative of the observed exogenous variables. This problem has not, in general, been of concern in past studies. Nearly all of the previously published studies utilizing translog functions as approximations to production functions have been time-series Studies for which the input measurements have been constructed from Divisia indexes (for example see Berndt and Christensen, 1973b,1974, or Denny and Fuss, 1977). It is possible to construct these indexes such that observations are clustered around the unit vector. In fact in their work Denny and Fuss (1977) are careful to "normalize" the Divisia indices of their study such that x* = [l,1,...,1] at the middle of their time series. However for cross section analysis, such as is presented in this study, where the data are not constructed from an index, in general it is necessary to perform a non-linear transforma- tion of the variables in order for the transformed vari- 23 ables to be clustered about the unit vector. Relative factor prices, relative factor quantities, and factor shares would not remain unchanged after the transformation. The transformed variables are thus no longer representa- tive of underlying economic conditions. The following prOposition extends the Denny-Fuss proposition 2 for any arbitrary expansion point. Proposition 3: The translog function of form(l.2)with linear homogeneity constraints(l.® is a quadratic approxi- mation, around any arbitrary expansion vector x* = [Xl,X;,...,X;], to an arbitrary linear homogeneous production function of the form fin Q = f[£nxl, £nX2,...,£an].5 Specifically,it is desirable to view the translog function as a quadratic expansion around the vector of the means of the observed exogenous variables § = [il' 22,...,Yn] to an arbitrary linear homogeneous production function. The above prOposition justifies this approach. Just as the Denny and Fuss propositions extend to the translog cost function(l.lSL so does the above prOposition extend to the translog cost function a.15)such that with linear homogeneity constraint(1.16)imposed, equation(l.15) is a quadratic approximation around any arbitrary 5For a proof of the proposition see Appendix A. 24 * * t . expansion vector p = [P1,...,Pn] to an arbitrary first-degree homogeneous cost function of the form: Kn C = Kn Q + C(£nP £nP ZnPn). 1! 2100'! E. Separability Separability implies that marginal rates of sub- stitution between pairs of factors in the separated groups are independent of the levels of factors outside the group. Use of aggregate indices of heterogeneous labor and capital inputs has, in the past, required the assump- tion that the production function is separable in these aggregates, and that workers within these aggregates are perfect substitutes. However, with the evolution of less aggregated data on production inputs (especially in manu- facturing)it may not be necessary to assume separability of such heterogeneous aggregates, The specification of separability in many ways alters the characteristics of the production process from those when separability is not assumed. Separability may be viewed as a partitioning of inputs into subsets, if possible, such that efficient production is achieved by a two-stage process. Relative input intensities are optimized within each subset, then optimal factor in- tensities are attained by Optimizing relative intensities among subsets while holding within-subset intensities constant . 25 Leontief (1947) has related functional separa- bility to the existence of sub-aggregates of inputs. The discussion that follows relates the concept of weak func- tional separability to the constant returns to scale translog function as an approximation to the underlying production process. Berndt and Christensen (1973a) extend the Leontief concept through the following definitions and propositions. Weak separability is defined as follows. Definition 2: Consider a twice-differentiable, strictly quasi-concave homothetic production function of n inputs, each with positive marginal product, Q = f(Xl,...,Xn). The set of n inputs is denoted N = [l,...,n] such that a partition, R, of r mutually exclusive and ex- haustive subsets [Nl""'Nr] is constructed. The pro- duction function f(x) is weakly separable with respect to the partition R if the marginal rate of substitution between any two inputs Xi and Xj from any subset NS, 5 = l,...,r, is independent of the quantities of inputs outside of Ns' that is: $2; (sf/axi/af/axj) = o, for all i,j e NS, and k E Ns' A fundamental result of weak functional separa- bility is expressed in the following proposition. 26 Proposition 4: Weak separability with respect to the partition R is necessary and sufficient for the produc- 2 ,...,xr), 6 tion function f(x) to be of the form f(X1,X where XS is a function of the elements of Ns only. Thus a consistent aggregate of a subset of inputs, XS, exists if and only if the subset of inputs, N5, is weakly separable from all other inputs. In the following proposition Berndt and Christensen demonstrated that a group of inputs is separable from others in production if and only if the Allen partial elasticities of substitution between a factor in the separable group and any factor outside the group are equal for all factors within the group. PrOposition 5: Weak separability of f(x) with respect to the partition R at any point in input space is necessary and sufficient for all ABS Oik' Ojk (i,j E Ns' k K Ns) to be equal at that point.7 A weak separability specification then restricts the technology and thus limits the possible functional form of the production function. For example, for a three input first-degree homogeneous production function. Q = f(Xl,X2,X3), weak separability of the type 6See Berndt and Christensen (1973a) pp. 404 for proof. 7For proof see Berndt and Christensen (1973a), pp. 406. 27 Q = f((Xl,X2),X3) implies that a consistent aggregate index of inputs X1 and X2 exists, and is a necessary and sufficient condition for 013 = 023 at any point in the input space. Further, Lau (1972) has demonstrated that weak separability with respect to a partition of input prices of a cost function satisfying the (Shephard-Diewert) duality conditions is necessary and sufficient for weak separability with respect to the same partition of in- put quantities of the dual production function. Denny and Fuss (1977) have derived parameter re- strictions for the translog function, as a quadratic approximation to an arbitrary functional form, which must hold if a set of inputs is separable from others. These restrictions are used to provide empirical tests of separability and thus the possible existence of consistent aggregates of labor and capital. Berndt and Christensen (1973b and 1974) had provided earlier tests of separa- bility: however,their tests implicitly assume the under— lying production process is exactly translog and there- fore more severely restrict the range of possible pro- duction processes. In fact Denny and Fuss demonstrate that a 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 28 function of Cobb-Douglas subaggregates.8 In this study the translog function is assumed to be a quadratic approximation to other functional forms; therefore separability tests employed here are those specific to the specification of the translog as an approximation to other functional forms. The Denny and Fuss (1977) para- meter restrictions for the three-input approximate trans- log production function under the null hypothesis of weak separability are described in the following Denny-Fuss proposition. Proposition 6: The translog function 1.2 is a quadratic approximation to an arbitrary weakly separable production function fin Q = f(b(£nX11£nX2)l£nX3) if 01/02 = Yl3/Y23. The proof consists of: l) specifying an arbitrary pro- duction function with separability imposed, 2) expanding it by a Taylor series, and then 3) restricting the trans- log function to have the same properties as the Taylor series.9 With the imposition of the first-degree homo- geneity constraints(l.4)the above condition is necessary 8See Denny and Fuss (1977), pp. 405-406. The Berndt and Christensen weak separability conditions for a three-in- put translog production function of form 1.2 are _ = 2 91/92 ‘ Y13/Y23' and Y117/22 712' 9For proof see Denny-Fuss (1977) proof of proposition 4, p. 408. 29 and sufficient for: l) the existence of a consistent aggregate of the inputs contained in the weakly separable set, and 2) equality of the appropriate AES. The Denny-Fuss proposition can be expanded to derive weak separability conditions for constant returns to scale translog approximate cost functions and translog approximate production functions with more than three in- puts. Weak separability conditions for translog approxi- mate functions of interest in this thesis are summarized in Appendix B. To summarize, this chapter develops the theory of translog production functions and translog cost functions as second order, or quadratic, approximations to any functional form. Where previously the above statement had been demonstrated to hold only for expansion about the unit vector, it is shown here that for the translog function with first-degree homogeneity imposed, the statement is true for any arbitrary point of expansion. Measures of input demand elasticity, Allen partial elasticities of substitution, Hicks partial elasticities of complementarity, and factor price elasticities are developed for both specifications. Criteria for choos— ing between a cost or production function specification are established, and sufficient conditions for weak separability among inputs are derived. CHAPTER II MODEL SPECIFICATION AND DATA A. Introduction Traditionally, models designed to capture labor force heterogeneity in manufacturing have disaggregated labor by either occupation, education, or age. With respect to the latter two specifications all but a few of these models implicitly maintain the assumption that capital is separable from all labor inputs. Although it may be that any one of the above schemata represents labor force subsets which are more homogeneous than the labor force taken as a unit, the underlying hypothesis for policy purposes many times is that the labor force can be decomposed into sets of relatively high— or low-wage workers. Any classification schema serves as a proxy for the underlying assumption. Labor market policies rarely directly alter the relative prices of workers under the alternative Specifications. Rather, policies affect the relative costs of employing high- or low-wage workers. Therefore an appropriate labor disaggregation is one for which there is little overlap in the earnings distri- butions among the labor force subsets. 30 31 Given any of the above three disaggregation criteria there appears to be substantial overlapping of earnings distributions within each criterion (see Hamermesh and Grant, 1979). The traditional occupation breakdown fares the poorest; with either an age classification scheme or grouping by education the problem seems less severe. In- deed, an alternative classification of labor by occupation, or a joint age-education labor disaggregation may help to ameliorate this problem. B. Model Specification This study examines models of production relations in U.S. manufacturering in which capital and various alternative definitions of labor are the assumed inputs. Four specific models are considered. Three of these de- rive the relevant production relations assuming a four in- put translog function; the other assumes a five input function as the appropriate approximation to the true pro- duction relations. As summarized in Table 2.1 the models are specified as follows. Model A classifies labor by three occupational classifications and also includes capital: labor is disaggregated into the mutually exclusive subsets: 1) professional, technical, and kindred workers plus managers and administrators. Variables involving this group carry a subscript P (for example XP); 2) sales, clerical and kindred workers. Relevant vari- ables are subscripted C (for example XC); and 3) blue 32 Hmanmo Ax + mN age can NH epmnq cco>o£ :oHumooCo "HODMH Av qNIvH mom can NH opted pcoxon :oHumUSUm "noan Am coHumospwlomm + mN mom can mmoH no NH wpmnm LODOMLH :oHumooco “noan AN >3 umoo moncmuu vNIVH mom can mmmH no NH opmem canons» :oHumospo "noan AH a Hmpoz HmuHmmo xx NH wpmum pco>mn :oHumospm "Hoan Am COHuwospm NH momma nmsoucu m wpmum pco>wn :oHumospw "HOQMH H2 xn umoo ooncmuu mmmH no mpmpm ppm NO coHuoHQEoo “HOQMH AH U Hobo: HmuHmmo Ax + me mwmm "noan A: can Sn cvnmN mmmm "noan A: :oHuosconm moncmuu em): 89.. £0an 3 m Eco: Hmuedmo Ax mumxuo3 moH>uwm can anHoo wan "Hoan Am mnmxHOB CQHGCHR pan HMUHumHU .mmHmm "uOQmH AU :oHummsooo muoumnumHCHEpm can mnommcmE an umoo qumcmuu .muwxuo3 omHUCHx can HMUchowu .Hmconmmmoum “noan Am < Hmpoz mumHuomnom mmHQMHnm> mHanum> mme monsuomwscm: .m.D mo xpsum coHuoow mmouu MOM UmHmHowmm mHmUoz H.N OHQMB 33 collar and service workers. Relevant variables are sub- scripted B (for example XB).1 Primarily because the data are convenient, pre- vious studies of labor disaggregated by occupation have assumed labor is either i) nonproduction (white collar) or ii) production (blue collar) labor. An admitted problem with this classification is that white collar workers are defined as all other employees not closely associated with production operations. Included then are nonproduction workers ranging from professionals through clerical and sales workers: this may still be a rather heterogeneous set of workers. Indeed, Hamermesh and Grant (1979) have indicated that with respect to this classification there is a substantial overlap in the wage distributions of production and nonproduction workers.2 The model esti- 1These categories are more fully defined by the 1970 Census of Population Occupational Classification in which labor IE classified by twelve major occupational groups. These groups have been aggregated into the above occupational groupings. The three labor groups by occupation are com- posed of the following Bureau of Census 1970 Major Occupational Groups: P) Professional, technical and kindred workers (1), plus Managers and administrators ex- cept farm (2), C) sales workers (3), plus Clerical and kindred workers (4), B) Craftsmen and kindred workers (5), plus Operatives, except transport (6), plus transport equipment operatives (7), plus laborers, except farm (8), plus service workers, except household (11). plus private household workers (12). Farmers and farm managers (9) and farm laborers and farm foremen (10) were excluded from the occupation aggregation. 2Hamermesh and Grant (1979) report that for the year 1969 39% of production workers earned less than $6000 as did 33% of nonproduction workers. Thirty-four percent of nonproduction workers earned over $10,000 while 20% of production workers did as well. 34 mated in this study would be similar to prior works if labor groups 1 and 2 were combined into one white collar category. Dividing white collar workers into two categories hopefully provides groupings of more homogeneous labor. Model B aggregates labor by age into three mutually exclusive categories: 1) ages 14 through 24 (L), 2) ages 25 through 44 (M), and 3) ages 45 and over (H). Model C groups labor into 3 educational classifications: 1) education through 8th grade or less (L), 2) education through any of grades 9 through 12 (M), and 3) education through 13 or more grades (H). Capital (K) is the fourth input in both models B and C. Finally model D estimates production rela- tions with capital (K) and labor in a joint age-education classification as inputs. The four labor categories are: 1) education up to or including completion of grade 12 and ages 14-24, 2) education up to or including completion of grades 12 and ages 25 and over, 3) education completed beyond grade 12 and ages 14-24, and 4) education completed beyond grade 12 and ages 25 and over (the labor groups are subscripted 1,2,3 and 4 respectively). Given the labor force disaggregation criteria, the next step in estimating input demand relations is to de— cide between cost function or production function specifica- tion of each model. With respect to model A, disaggregation by occupation, specification of a cost function appears to be the better approach. There is evidence that historical 35 changes in occupational wage differentials are consistent with the assumption that labor supply disaggregated by occupation is relatively elastic.3 Thus, wages are the more appropriate exogenous variable. The desired approach for model B, disaggregation by age, appears at first glance to be a production function specification; input quan- tities are then the exogenous variables. It might be argued that changes in relative wages alter the timing or rate of fertility which thus affects the age structure of the work force of future generations. The argument might deserve attention if this were a time series study spanning generations. Since a cross section analysis is undertaken in this study,the argument loses considerable effective- ness. However,in this study only the manufacturing sector is examined. To the extent that manufacturers are not the sole employers of inputs in a region, but may be oligopsonists,if not competitors,in the input markets, the supply of labor facing manufacturers in any given area may be relatively price elastic. It will be assumed that with respect to model B, quantities of labor at a given age 3See Fleisher's (1970, Chapter 13) discussion of "secular changes in skill differentials" for a summary of some empirical work. Reynolds (1978, pp. 256-259) also dis- cusses phenomena which indicate labor supply by occupation is relatively elastic. 36 classification are predominantly exogenous and a produc- tion function approach will be used. The above caution must, however be kept in mind. The appropriate specification of model C, dis— aggregation by education, is less apparent. It is observationally unclear whether labor supply by education is relatively wage elastic or inelastic. Furthermore, the argument that manufacturers may not be the only demanders in the labor market applies for model C also. The choice between a production or cost function approach becomes arbitrary. Therefore, a cost specification is chosen (i.e., exogenous input prices), for, as discussed in Chapter I, computation of the ABS is simpler. Finally for model D, joint age-education labor disaggregation, the appropriate— ness of either a cost or production function approach is equally unclear. Again, input prices are assumed exogenous, and a translog cost function is specified. C. Market Definition In this study Standard Metropolitan Statistical Areas (SMSAs) define the relevant labor markets. Where geographic areas are defined by political boundaries (for example cities, counties, or states), SMSAs are defined by economic boundaries. That is,SMSA boundaries are created such that each area encompasses a center of major economic activity. Politically defined areas may cover more than 37 one such economic center or may arbitrarily exclude neighboring regions important to the center's economy. With SMSAs this result is less likely to occur. Thus SMSAs more correctly define the appropriate factor markets. Mills (1972) points out that of all workers living in SMSAs in 1960 only 4 percent worked outside the SMSA in which they lived. Though SMSAs are the desired labor market defini- tion, there are still inherent difficulties. SMSAs are not identical; each has its own set of specific char— acteristics which may alter the technical relations among inputs in each SMSA. There is no uniform tax structure across SMSAs. This implies that the same changes in relative wages in two SMSAs,identical except for differing tax incentives, will emit slightly differing signals to both factor demanders and suppliers. The resultant bias in estimated variables is unknowable, unless the form of the tax incentives is also known. Further, SMSAs differ with respect to availability and quality of land as a re- source in production. Supportive services and other char~ acteristics of agglomeration differ across SMSAs. These SMSA-specific characteristics may alter the technical relations among inputs across SMSAs as well. Secondly,to the extent that an increase (decrease) in factor prices in an SMSA causes labor migration into (out of)that SMSA, the supply elasticities of labor sub- 38 aggregates are greater than zero. If changes in relative factor prices induce relatively large labor force migra- tion, then a production function specification is in- apprOpriate. D. Statistical Considerations Stochastic versions of equation system(l.6)when production functions are used, and equation system(1.18) when cost functions are assumed, will be estimated. The share equations for the stochastic production function system with symmetry and linear homogeneity constraints imposed become: SZHQ ___ M- = 0.. + g: Y 04(2) + If Y [51(x—i) + u aznxj J 3 i=1 13 Xn i=j+l 31 Xn 3 (2.1) j = l,2,...,n-l; where profit maximization implies equality between the %—%%%- and the factor share M.. ' 3 3 Thus the disturbances (uj, j = l,2,...,n-1) may be output elasticity accounted for because markets may not be perfectly com— petitive, or entrepreneurs may not be able to maximize profits instantaneously. Similarly the estimating equations for the trans- log cost function model with symmetry and linear homo- geneity conditions imposed are: 39 P- n-l p. (SHIN-i) + Z (S- £n(-l) + e. 13 Pm i=j+1 3 pn 1 j = l,2,...,n.—l; where with the cost minimization assumption, Shephard's 8 ZnC * E—IEP; = Sj, the minimum cost share attribut— 1emma implies able to input j. The disturbance terms, Ej' are thus dis- crepancies between s; and the observed cost shares Sj, which may be explained by entrepeneurial misjudgement or other constraints affecting cost minimization. There- fore: Sj = s; + ej, j = l,2,...,n-l. For each system (2.1)or (2.2) the disturbances are likely to be correlated across equations. Random deviations from profit maximization are likely to affect all input quantities. Thus for any i and j, ui is likely to be correlated with uj. Random deviations from cost minimiza- tion ought also to affect all input prices: therefore Bi and Ej will be correlated. This suggests that a Zellner (1962) two-stage estimation will yield efficient parameter estimates. Zellner has shown that when disturbances across equations are correlated,and if the correlation is known, the parameters can be estimated more efficiently by taking this information into account. Furthermore Zellner (1963) 40 has demonstrated that even when the correlation is un- known, it is likely that using an estimate of the cor- relation in the two-stage process will improve estimation efficiency (Zellner Efficient Estimation or ZBF). However, there is a second, computational, prob- lem that exists with respect to either stochastic model. The linear homogeneity constraints imply that with an n equation system, there need be only n-l estimating equa- tions. The parameters of the nth equation are determined by the parameters of the remaining n-l equations. For example in both systems GL1) and (2.2) the nth equation has been arbitrarily deleted. However, the estimators obtained by applying a ZBF estimation procedure depend on which n-l equations are selected from the system. A maximum likelihood procedure would provide para- meter estimates which are independent of the equations chosen. Kmenta and Gilbert (1968),in a series of Monte Carlo experiments,demonstrated that Maximum Likelihood (ML) and Iterated Zellner Efficient Estimation (IZEF) led to identical estimates in all samples.4 William Ruble (1968) demonstrated the computational equivalence of IZEF and ML estimators. The implication is that IZEF and ML estimators have identical asymptotic properties. Therefore 4See Kmenta and Gilbert (1968) for further discussions of the IZEF estimator. 41 in this study parameter estimates will be obtained via the IZEF method; the estimates are invariant to the equa~ tions chosen, and under very general conditions the estimates are consistent and asymptotically efficient.5 Given the parameter estimates it is necessary to verify the regularity conditions for well-behaved cost or production functions. Assuming the conditions are satisfied,tflm:partial AES, input demand elasticities, partial HBC, and factor price elasticities are calculated and various hypotheses concerning the underlying produc- tion technology are tested. Of particular interest is whether in general labor can be assumed weakly separable from capital, and whether certain labor subaggregates can be assumed weakly separable from other labor subaggregates 5Throughout the literature parameters of the translog func- tion have been estimated overwhelmingly from the share equations. Only the very early studies using the translog function have included estimates of the parameters directly from the translog function itself. Presumably direct estimation was difficult either because the requisite data were unavailable or there were no good nonlinear estimation techniques at hand. However, consider the approach taken here: that the translog function with linear homogeneity imposed is only a quadratic approximation to an arbitrary linear homogeneous production function around agy arbitrary expansion point. It seems reasonable to assume that the translog coefficients are estimable directly from equation (l2)with the symmetry and linear homogeneity conditions imposed. The random error term must, however, be assumed to be from a probability density function which is non- zero only for non-negative variables (for example a gamma distribution). The econometric tools then need only be ML estimation. However a problem with such estimation, which does not affect the bias but does affect the standard error, is that multicollinearity among the exogenous variables is very likely. 42 and capita1.(See Appendix B for a summary of sufficient separability conditions.) It is of interest to evaluate which of the alternative schemes of labor disaggregation might be considered "better". Finally, the effect of certain SMSA specific characteristics on the technical relations among input is examined. By comparing para- meter estimates defined as functions of these specific characteristics, it is possible to test for homogeneity of labor-capital production technology under changing economic characteristics of different SMSAs. For example, in Chapterlfilthe translog functions of Chapter I are re- formulated to consider questions concerning similarity of production technology in the sunbelt versus the rest of the U.S., and questions concerning the change in labor- capital production technology as certain SMSA specific characteristics change. B. Data The data for both translog production and cost specifications are constructed for the year 1969. Employ- ment data for manufacturingare taken from the one-in-a- thousand sample file of the County Group Public Use Samples 9f Basic Records From the 1970 Census. Capital and output data are gathered from issues of the Census of Manufactures and the Annual Survey 9; Manufactures. The County Group Public Use Samples identify all of the 125 SMSAs of over 250,000 in population in 1970. Employment statistics are gathered for each of these SMSAs. The 43 Annual Survey and Census of Manufactures provide output and capital data only for selected SMSAs and selected large counties and cities in the United States. Hence the resulting sample size of SMSAs for which there are both employment and capital and output is 84. Also, because the Public Use Sample size is one-in-a-thousand, for each model used there are a few SMSAs for which the observed average price of a labor input or observed amount of a labor service used is zero. These SMSAs were deleted from the sample for the particular model under study. Appendix C lists the 84 SMSAs which make up the overall sample, and gives the sample sizes and the SMSAs deleted for each of the specific models under study. Employment data for 1969 for each SMSA are computed as follows. For the production function specification, the quantity of labor services for each labor subaggregate is total person-hours per year.6 For the cost function Specification, the price of labor services is the average hourly earnings.7 6Total person hours are determined by multiplying [weeks worked in 1969] x [hours worked during the Census survey week] for each individual and then summing the total over observed individuals for each labor subaggregation. 7Average hourly earnings are determined by summing: [wages, salaries, commissions, bonuses or tips from all jobs] + [nonfarm business, professional practice, or partnership income] for each individual, then summing over all individuals in each labor subaggregate, and then dividing the result by total person hours for the given subaggregate. 44 There are no direct capital stock data for SMSAs provided in either the Annual Survey or Census 9f Manufactures. Thus the capital stock for each SMSA is constructed from an investment stream consisting of pur- chases of new capital in each SMSA. For 1969 the capital stock for each SMSA is constructed as follows. A per- petual inventory model is assumed: i.e. Kt = It + (1 - u)Kt_l ; (2.3) where Kt is capital stock at time t, It is investment at time t, and u is the rate of replacement. The Annual Survey and Census g_f_ Manufactures report "New Capital Expenditures" for selected SMSAs and counties for the year 1954 to the present. Data for years 1954 through 1957 are too incomplete for use. Therefore the investment stream, I as represented by "New Capital Expenditures," t’ runs for years 1958 through 1969. The initial year (1957) capital stock is: [Gross book value of depreciable assets on December 31, 1957] minus [accumulated depreciation to 1956] minus [depreciation charged in 1957].8 The replace- ment rate, u, chosen is the average replacement rate implicit in the Office of Business Economics Capital Stock 8Source: Census of Manufacture, 1958, Volume 1, Summary Statistics, pp. 9-24. Data are reported only on a state basis. SMSA estimates were constructed from the state data by multiplying the appropriate state data by the pro— portion of SMSA value added by manufacture to state value added by manufacture for 1957. 45 and Investment series.9'lo u was found to be approximately 0.108. All calculation was in constant 1969 dollars. For the cost and production function specification factor shares, Si and Mi respectively, are identical. (In the remainder of the text the factor share of input 1 from either the production or cost function Specifica- tion is denoted 81') The factor share for each labor sub- aggregate is total earnings ofthe subaggregate (average hourly income multiplied by total person hours) divided by "Value Added by Manufacture" for each SMSA as reported by the 1969 Annual Survey of Manufactures. Both shares of capital (SR) and price of capital services are deter- mined as residuals of value added minus labor income. The share of capital in either total output or total cost is [1969 value added minus total labor income] divided by [1969 value added]. Finally the price of one dollar's worth of capital services for 1969 is computed as [1969 value added minus total labor income] divided by [esti- mated value of capital stock] for each SMSA. 9See Berndt and Christensen (l973b),statistical appendix, pp. 106-107, for a more complete explanation of the appropriateness of the perpetual inventory method and of determining the replacement rate, u. 10The method uses Net Capital Stock, p. 7, and Business Investment, p. 343, over the years 1958 through 1969 to derive the implicit replacement rate. Source: Bureau of Economic Analysis, Fixed Nonresidential Business and Residential Capital in the United States 1925-1975. “—— PB253725, June . CHAPTER I I I MODEL ESTIMATION A. Introduction This chapter presents results of estimates of the translog approximate cost function models and translog approximate production function model for manufacturing. The models summarized in Table 2.1 are model A, trans- log cost function with capital and labor disaggregated by occupation, model B, translog production function with capital and labor disaggregated by age, model C, translog cost function with capital and labor dis- aggregated by education, and model D, translog cost function with capital and a joint age-education dis- aggregation of labor. An additional assumption main- tained throughout the chapter is that the prevailing pro- duction technology is the same across the United States. With this assumption it is appropriate to estimate the function over all observable SMSAs. Among results of interest to be discussed are 1) the "goodness" of the estimated translog functiomsas approximations to the production process, 2) estimated Allen elasticities of substitution and input demand elasticities, 3) review of and comparison with results of other similar studies, 4) estimated Hicks elasticities 46 47 of complementarity and factor price elasticities, 5) tests of selected hypotheses of weak separability among inputs, 6) review of separability hypothesis tests of other studies, and 7) policy implications of empirical findings. B. "Goodness" of the Translog Estimates For each model the appropriate translog function was estimated from the system of derived factor share equations, with symmetry and first degree homogeneity conditions imposed. In each case IZEF estimation was per- formed with the factor share equation for capital eliminated from each system. Therefore for each of models A, C, and D the system of equations(2.2)are estimated,and for model B the equation system(2.l)is estimated.l 1Each model represents a system of seemingly unrelated regressions of the following form: = C Y1 Xlgl + 1 Y2 = x252 + :2 YM = XM + 5M + EM, (m = l,2,...,M) where Ym is an (N x 1) vector of the sample factor shares. Sample size is N, Xm is an (N x Km) matrix of sample values of explanatory variables, gm is a (Km x 1) vector of regression coefficients and em is a (n x 1) vector of sample values of the disturbance such that: E(emn) = 0 where Emn is an element of Em' n=1'2'ooopN E(smem.) = OmmIN 48 In the case of models of four-input translog functions, nine coefficients are estimated directly from the system; the remaining five coefficients are determined from the first-degree homogeneity constraints (equation (1. E(em,sp,) = o mpIN (m.p = 1,2,. .,M) In particular the resultant estimation equations for models A and C are: S1n S2n SBn The M1n 2n MBn And 1n 2n 3n 4n = 81 + all£n(Pl/P4)n + 612£n(P2/P4)n B'2 + 612£“(P1/P4)n + 622£“(P2/P4)n + E3 + 613£n(Pl/P4)n n = 1,2,... + 623£“‘P2/P4)n + + 613£n(P3/P4) 633£D(P3/P4) estimation equations for model B are 0‘1 + Y11£“(X1/x4)n + Y12£“(X2/X4)n + 02 03 + Y12£“‘X1/X4)n + 713£n(xl/x n: 1,2,... + 4)n + Y23 ,N 122€n(X2/X4)n + Yl3£n(x3/x4) 123£n(X3/x4) (“(Xz/X4)n + Y33£n(X3/X4) for model D the set of estimation equations is: 523£“(P3/P4)n + E2 n in n 3n n 1n '7‘,“ n I.“ n -6 .h 81 + 611£“(P1/P5)n + 612£“(P2/P5)n + 513£“‘P3/P5’n + 614£n(P4/P5)n + E:ln + 612£n(Pl/P5)n + 622£n(P2/P5)n + 623£n(P3/P5)n 624£“(P4/P5)n + + 613£n(Pl/P5)n + 623£n(P2/P 634£n(P4/P5)n + E:2n E:3n 5)n + 633£“‘P2/P5) n + 614“”?1/1’5’11 + 624zn‘P2/P5)n + 634£“‘P3/P5)n 544£“‘P4/P5’n + E4n I n=1'2'OOO'No 49 For the five-input function, fourteen coefficients are estimated from the equation system with the remaining six coefficients determined by the linear homogeneity con- straints. Table 3.1 presents translog coefficient esti- mates for models A through D. Throughout the chapter elasticity estimates are reported for the share equations (2.1 or 2.2» represent- ing the average factor shares over all SMSAs. Table 3.2 lists the relevant average and total variables over all SMSAs for each model. Average factor shares range from 0.012 (labor educated beyond grade 12 and between ages 14-24) in model D to 0.48 (capital) in models A, B and C. The average factor shares of Table 3.2 are similar to those obtained in previous studies by others. In their study Freeman and Medoff (1979) estimate average share of non-production workers to be 0.25; the average share of similar labor in model A is 0.24 (Sl + 52). Capital's share appears to be high in all models. Recall that capital is determined as [1969 value added minus total labor income] divided by [1969 value added by manufacturer]. Estimated capital share is thus derived from the residual of value added minus labor income: these estimates are probably biased upward.2 2Total labor income,taken from the Public Use Sampleyis defined as, ”Earnings in 1969: Wages, Salary, Commissions, Bonuses, or Tips from all jobs, Non-Farm Business, Table 3.1. Model A translog cost by occupation 8P .0146 (.0301) 6P? ( 03:3) 6pc (:8109) 5P8 183.13. 6” 28:22. BC .0549 (.0144) écc (:8126) 6C8 (:8125) 6CK ( 8103) BB .219 (.0255) 638 (26:15) 68“ (2316:) (standard errors Model B translog pro- duction by age UL .122 (.0104) YLL 680:9)~ YLM (23632) YL” (I803?) YLK 1233316) GM (:6307) YMM (:0I08) YMH (:8085) YMK (23133) OH .323 (.0136) YHH (:0116) YHK (2313:) 50 in parentheses) Model C translog cost by education BT .0842 H (.0155) 6LL .0180 (.0171) 6 .0051 L” (.0171) 6 -.0101 LE (.0107) 6 -.00223 LK (.0116) B .1757 M (.0248) 6 .0783 A MA (.0275) 6 .0074 “H (.0166) 6 -.0806 ”K (.0186) BB .0289 (.0298) 6 .0800 ”H (.0208) 6HK -.0767 (.0188) Estimated Translog Coefficients Model D translog cost by age-education 81 .0232 (.0079) 0 .0148 11 (.0055) 0 -.0118 12 (.0094) O .0060 3 l (.0036) 6 .0058 14 (.0059) O -.0147 1K (.00552) 82 .243 (.0321) 6 -.0974 22 (.0393) 0 -.0003 23 (.0092) O .0251 24 (.0234) O -.0601 2K (.0239) B3 .0176 (.0082) 6 .0110 33 (.0047) O -.0095 4 3 (.0061) 63K -.00726 (.00571) Table 3.1 (continued) Model A translog cost by occupation B .712 K (.0386) 6 .162 KK (.0263) n = 75 Model B translog pro- duction by age OK .224 (.0560) Y .115 KK (.0242) 78 Model C translog cost by education BK .711 (.0385) 6 .160 KK (.0266) 78 Model D translog cost by age-education B4 .0239 (.0303) 54“ (233(3) 6“ (Z3133) 6“ (23336) GKK (:0382) 61 52 mm.Hoon mH.H onv.o x mm.ow mm.o HoH.o v mm.m mm.m mHo.c m AcOHumosemumoav oo.mmH om.m me.o N o Hmpoz Hmn.mva NN.mm oc.N vmo.o H vn.Noom NH.H omv.o x mN.mm cm.m hmH.o I AcoHumospmv SH.mvH om.m NhN.o z 0 Homo: www.mme om.mv Hv.m Hmo.o H v>.Nomm NH.H omv.o x mm.eHoH m6.e mmm.o : 1064c Ne.vHH mm.v hMN.o z m Hmpoz www.mew Hm.mm mo.N mvo.o A mn.mmmm NH.H omv.o x mo.e6H 66.m mmm.o m Acceummsooov Nm.ov nh.m Hwo.o 0 fl H060: mom.Non om.Nm Hm.©m hmH.o m Aooo.ooo.va Hooo.ooo.Hmv .xx x hence amzm .amzm 90d xooum 6 am Mam .mcHusu00m HmuHQmu wmmuowc HmuHmmU o» cusuwm wumcw (new: CH coped 6cm Hooo.HV ..x momuo>< p:m..m mEoocH osHm> mmmuo>¢ .dmZm mom musom .uonmq ou cusuom mommm>< COmHmm mmmuo>¢ >Huzo: womum>m meanum> ucwpcomoUcH pcm acetcwmoo mo mosHm> omnuw>¢ .N.m oHnt 53 Estimated average hourly income ranges from $2.66 per hour (model D, labor with high school education or less and ages 14-24) to $6.51 per hour (model A, professional, technical and kindred workers). In each model the average hourly earnings range as one would expect a priori. The return is relatively high for labor with greater age, education, or both or for white collar professional workers as compared to clerical or blue collar workers. Estimated average annual return to $1 of capital stock also appears high, for two strong reasons. First estimates of the total return to capital in production may be biased upward (see footnote 1). Second, total capital stock for each SMSA was estimated assuming a perpetual inventory model (see equation 2.3L where the Professional Practice, or Partnership. The Annual Survey of Manufactures gives the following explanation of value added. This measure of manufacturing activity is derived by subtracting the cost of mater- ials, supplies, containers, fuel, pur- chased electricity, and contract work from the value of shipments for products manu- factured plus receipts for services rendered. The result of this calculation is then ad- justed by the addition of valued added by merchandising operations...plus therufi:change in finished goods and work-in-progress in- ventories between the beginning and end of the year. Thus the residual (value added minus labor income) prob- ably overestimates value added attributable to capital. Employers' contributions to Social Security, group insurance, or other group benefits and payments in kind are not included in total labor income. The residual may also include payments to additional unknown factors (for example rental payments for land which are all grouped into a common "capital" category. 54 investment stream was taken to be "New Capital Expendi- tures". There is also an active market for used capital. Yet the data are insufficient to include used capital purchases in capital stock estimates. These two sources of bias lead to an upward bias in the estimated average annual return to $1 of physical capital. Recall, sufficient regularity conditions for a well behaved production function are positive factor shares and negative semi-definiteness of the bordered Hessian matrix. For a well behaved cost function the conditions are positive factor shares and negative semi- definiteness of the Hessian matrix. These conditions were tested for each of the models and were not found to be satisfied over all observations. They were tested for each SMSA using both fitted factor shares and actual factor shares. When fitted factor shares were used the regularity conditions were not satisfied for 25 of 75 SMSAs for model A, 23 of 78 SMSAs for model B, 4 of 78 SMSAs for model C, and 27 of 61 SMSAs for model D. Regularity conditions for actual factor shares were checked with the following results. The conditions were not satisfied for model A in 51 of 75 SMSAs, model B in 45 of 78 SMSAs, model C in 36 of 78 SMSAs and model D in 45 of 61 SMSAs. Failure of the estimated function to meet either regularity conditions for any SMSA leads to some concern. 55 When the regularity conditions are not met,either the marginal product of at least one factor is less than zero, or,if the Hessian matrix conditions are not satisfied,then one or more inputs are not characterized by a diminishing marginal rate of factor substitution. That is,the estimated production function is not well behaved. Elasticity estimates are reported for observa- tions designed to approximate a representative SMSA. Therefore estimates are reported for observations re- presenting the mean of observations over all SMSAs sampled for each model. It is important, then, that regularity conditions hold for the average fitted or average actual factor shares as well. Table 3.3 summarizes results of checks for satisfaction of the regularity conditions for each model evaluated at average fitted and average actual factor shares. All models met regularity conditions when average fitted factor shares were used. When the criteria were tested for average actual factor shares, regularity conditions were satisfied for all models except model D; the Hessian matrix conditions are not satisfied. In summary, as an approximate cost or pro- duction function, model C (education) appears to meet the criteria satisfactorily. Model A (occupation) and model B (age) meet the regularity criteria for fitted shares over at least 2/3 of the SMSAs,and the criteria 56 Table 3.3 Examination of Regularity Criteria for Each Model; First-Order (FOC) and Second Order (SOC) SOC--Sign of Relevant FOC Hessian Determinant 5X5 4x4 3x3 ZXZ 1x1 Model A i ** + - + - (occupation) ii ** *** - + - Model C i ** + - + _ (education) ii ** + - + _ Model D i ** - + - + - (age- education) ii ** + - + + - SOC--Sign of Relevant Bordered Hessian Determinant Model B i ** + - + (age) ii ** + - + * Examination is done for the mean of observations over the sample. ** FOC are satisfied. *** The matrix is algorithmically singular. lFitted factor shares llActual factor Shares. 57 are met for both average fitted and average actual shares. Model D performs poorly, however. Regularity criteria are not met for nearly half of the fitted shares over all sampled SMSAs and are not met as well for the approximate translog function estimated over average actual shares. This result will be important to keep in mind when evaluating elasticity estimates. There remain two additional important possible sources of bias in the parameter'esthmnes.The first possible source arises because the translog function is specified as an approximation to the underlying cost or production function. The translog approximation is based on a truncated and unknown remainder term from the original Taylor series approximation. Estimation of the model with the remainder term excluded amounts to the deletion of an explanatory variable and introduces a specifica- tion bias in an estimation of the translog function co- efficients. In a series of Monte Carlo experiments, Byron (1976 ) checked the reliability of estimation based on translog approximation. A translog function was formulated as an approximation to a two-input CBS function,and trans- log coefficients were estimated both directly from the translog function and from the cost share equations. Both approaches were found to give biased estimates; however the indirect estimation approach fared 58 better than the direct approach. Byron states the follow- ing: What does emerge from the results is that the direct approach is markedly inferior to the indirect approach...and it appears that the poor results of the direct esti- mates are simply due to multicolinearity between the regressor variables resulting from the presence of quadratic terms in the estimated equation. Under the cost share approach, the translog function para- meter estimates are systematically biased toward zero. Estimated Allen elasticities of substitution under the cost shares approach were also evaluated by Byron. Here the results are more encouraging. All elasticity esti- mates were found to be biased only slightly toward zero. Cr,in Byron's words: The results are marginal, and the inter- pretation will depend on one's enthusiasm for the translog position. The results are biased, the biases were not extreme and were found to be due to the exclusion of the remainder term in the Taylor series expansion. The remaining potential bias arises from the nature of the sample observations. The capital stock for each SMSA is estimated with a perpetual inventory model of investment. However, over time the boundaries of various SMSAs have changed through the addition or dele- tion of select counties. If these historical changes could not be corrected for in the investment data, these SMSAs were necessarily deleted from the sample. Therefore, 59 there arises possible bias due to sample truncation. It is reasonable to assume the deleted SMSAs are character- ized by higher population growth rates and/or higher rates of migration into the area. If the rate of growth of capital stock is about the same as the labor force growth rate in these areas,one would expect estimated Allen elasticities of substitution between labor and capital to be unaffected by sample truncation. If not, then there may exist truncation bias. However, it is difficult to tell in which direction, if any, the elasticity estimates are biased.3 Each of these biases must be kept in mind when analyzing the regression re- sults. C. Allen Elasticities of Substitution and Derived Input Demands Elasticities Allen elasticities of substitution are estimated for each model. Table 3.4 presents the estimated AES for the mean of observations over the entire sample. Recall the ABS registers the effect on the quantity demanded of one factor of a change in the price on another factor, with output and other factor prices held constant. A 3Results of Chapter 4 provide some evidence that trunca- tion bias exists but is small. When the SMSAs are divided into nonsunbelt and sunbelt regions (a characteristic of the sunbelt is greater than average growth) labor and capital are in general slightly more substitutable for each other in the sunbelt. 60 Table 3.4. Estimated Partial Allen Elasticities of * Substitution, 1969, Oij (i = row, j = column). Input Input P C B K Model A P -l.15 -0.06 0.62 0.08 (occupation) C -3.10 0.14 0.46 B -l.l3 0.47 K —0.38 L M H K Model B L -213.72 14.38 15.78 5.25 (age) M -ll.50 4.85 1.91 H -lO.42 1.29 K -2.08 L M H K Model C L -8.58 0.77 0.21 0.94 (education) M -l.62 1.16 0.38 H -2.12 0.04 K -0.39 l 2 3 4 Model D 1 -15.57 -0.09 15.85 2.04 0.11 (age- 2 -1.19 0.92 0.51 0.60 education) 3 -3.82 -4.04 -0.30 4 -l.46 0.11 K -O.44 * Estimates are for sample. the mean observations over the entire 61 positive AES implies the two factors are substitutes in production,and a negative AES implies the inputs are complements. It is instructive to examine the results first in a general overview, then examine the elasticities for estimated. translog approximate cost function of models A, C and D and the estimated translog approximate production function of model B, and finally compare and/or reconcile the results with those of previous comparable studies. For nearly all pairs of inputs the estimated ABS, Oij (i # j), are positive. That is,in generaL whether labor is classified by occupation, age, or education it appears that any given labor input is a substitute (or, at least not a complement) for any other labor input as well as for capital. It is also interesting that the order of magnitude of Oij estimates under the produc- tion function specification (model B, Age) is much larger than those under the cost function Specification. This phenomenon need not be disturbing. Remember-ABS esti- mates under the production function specification re- quires constructing the ratio of determinants of n x n and (n+1) x (n+1) matrices which include all estimated translog coefficients. Relatively high standard errors of any of the estimated coefficients may cause all elasticity estimates to have relatively high variances as well. Further such a difference in order of magnitude 62 is not inconsistent with other studies that compare estimates from production vs. cost function specifications.4 Now consider model A (occupation). All factors of production are estimated to be substitutes for each other except professional white collar workers, and clerical and sales white collar workers; they are just slightly complementary (o = -0.06). This is a result which might PC be expected. If much of the clerical work is created by the professionals and managers employed, an increase in the price of managers should not only decrease the number of managers hired (OPP = -l.15) but also decrease the number of supporting clerical workers employed (OPC = -0.06). ( 0.62) Among the: other estimated elasticities OPB is the highest, indicating among all inputs blue collar labor and professional, technical and managerial labor are the most substitutable. It is also interesting that professional white collar labor is the least substitutable of all categories of labor for capital (0P = 0.08, K OCK = 0.46, OBK = 0.47). An increase in the relat1ve price of capital has almost no effect on the quantity of 4In a similar study Anderson (1977) estimated a translog production function of labor and capital with labor dis- aggregated by age over the time period 1947-1972. His estimated 1972 ABS were of the same magnitude as those re- ported in table 3.4. Welch and Cunningham (1978) estimate a Cross Section CES cost function with labor disaggregated by age and excluding capital, by state for 1970. Their estimated AES are closer to the relative magnitude of the AES estimates of models A, C, and D in table 3.4. See Hamermesh and Grant (1979) for a more detailed synopsis of their work. 63 professional white collar labor used, whereas capital and clerical white collar labor or capital and blue collar labor are more evidently substitutes. With respect to Model B (age) the estimated ABS indicate that any given input is a substitute for any other input. WOflmusaged 25-44 and 45+ are nearly equally and rather highly substitutable for labor aged 14-24 = 14.38, = 15.7). Labor aged 25—44 is easily (OLM OLH substitutable (oMH = 4.85) for labor aged 45+,but not as easily as it is for labor aged 14-24. These results are a bit disconcerting. One might expect that the acquired skills of labor in neighboring age groups are more similar to each other than the skills acquired by labor in groups farther separated by age. Thus labor-labor substitution between age groups should increase as the age groups be- come nearer each other. Other studies have borne out this relationship.5 Howeven.if more effort is required to train older labor than younger labor to a new position, or if the institutional structure facilitates substitution of young labor for older labor, then young labor-older labor substitution may be easier than middle labor-older labor substitution. Results of model B support this hypothesis. Turning to the AES of capital for labor of various age groups,more comforting results are attained. Labor 5See discussion p. 76. 64 and capital become less easily substitutable as labor advances in age (oLK = 5.25, OMK = 1.91, CHK = 1.29), with OLK over four times greater than OHK' These re- sults are strikingly similar to those of model A. Model C (education) ABS estimates are, again, positive for all Oij (i # j) and behave as one would ex- pect a priori. Among labor groups the ABS between any two groups are larger the more similar are the groups, as judged by the within-group education level (OLM > OLH ). Further as the level of education in- 1 and OMH > OLH creases, labor becomes less substitutable for capital; in fact,for labor with education beyond high school,a change in the relative price of capital has nearly no effect. No matter how labor is aggregated (model A, B, or C) an interesting result continues to present itself. Regardless of how human capital is approximated, the models predict that as laborers acquire more human capital they become less substitutable for physical capital. That is, each of the three models might be thought of as pro— viding a different proxy for the level of human capital embodied in a labor group. For model A, white collar professionals may represent labor with more human capital than either white collar clerical workers or blue collar workers. For model B, human capital embodied in any age group increases with age. For model C, human capital increases with education. 65 Elasticity estimates of model D (age-education) are interesting, albeit the results of this model are more precarious than those of others. Recall, of the four approximations, the translog function of model D was the least acceptable asaniapproximate cost (or production) function. Pairs of inputs, in this model, are not in general substitutes. Rather, the ABS between young less educated labor and older less educated labor is slightly less than zero (0 =-0.09); young educated labor and 12 older educated labor are complements (0 = -4.04) and 34 educated young labor and capital are slight complements (035 = -0.30). Educated young labor is estimated to be rather highly substitutable for less educated young labor (013 = 15.85). However educated older labor and educated younger labor are not so easily substitutable (024 = 0.51). The general substitution relations between human capital (as approximated by labor sub-aggregates) and physical capital which were identified in models A, B, and C do not hold overall in model D. Holding age con- stant, as education increases young labor and capital move from being substitutes to being complements, while older labor and capital become less easily substitutable. These results are consistent with those of models A, B, and C. But holding education constant, as age increases, the general human capital-physical capital elasticity relation is reversed for both the less educated and the 66 more educated labor groups. As age increases across the less educated group, labor and capital become more easily substi- tutable, and across the more educated group labor and capital move from being complements to being substitutes. A possible explanation for the complementary relations found in model D but not found in models B or C is that model D provides a poor approximation to the production process. The average factor share of young highly educated labor is extremely small and none of the estimated coefficients of this factor are highly significant. The poor quality of coefficients of model D may be reflected in the performance of model D. Turn to Table 3.5, which presents estimated input demand elasticities derived from the eStimated ABS of Table 3.4. Assuming output and all input prices constant, the elasticity of demand for input 1 given a change in the price on input j, nij is: nij = Sjoij' That is Oij is defined as the ABS for inputs i and j weighted by the cost shares of input j. The transformation from oij to nij is sign preserving, therefore the general substitute-or-complement relations between inputs identified from the ABS remain the same. As in Table 3.4 the estimated demand elasticities are relatively larger than the production function specifica- tion (model B) than under a cost function specification.6 6See pp. 75-76 for further discussion. 67 Table 3.5. Estimated Input Demand Elasticities, 1969 I _ o - * nij (1 — row, 3 - column) Input 1 Input P C B K Model A P -0.18 -0.00 0.15 0.04 (occupation) C -0.01 -O.25 0.04 0.22 B 0.08 0.01 -0.32 0.22 K 0.01 0.04 0.13 -0.18 L M H K Model B L -9.68 3.40 3.76 2.52 (age) M 0.65 -2.72 1.15 0.92 H 0.71 1.15 -2.48 0.62 K 0.24 0.45 0.31 -l.00 L M H K Model C L -0.70 0.21 0.04 0.45 (education) M 0.06 -0.44 0.19 0.18 H 0.02 0.32 -0.35 0.02 K 0.08 0.10 0.01 -0.19 1 2 3 4 Model D l -0.54 -0.03 0.19 0.33 .05 (age- 2 -0.00 -0.38 0.01 0.08 .29 education) 3 0.55 0.29 -0.04 -0.65 .14 4 0.07 0.16 -0.47 -0.23 .51 K 0.00 0.19 -0.35 0.17 .21 * Estimates are for the mean of observations over the entire sample. 68 In examining models A, B, and C two striking results appear which are common to all three models. The first is, as indicated in Table 3.4, labor with greater amounts of human capital - whether approximated by occupation, age, or education - becomes less easily substitutable for physical capital. Examine column 4 in Table 3.5 for each of the three models. Cross price elasticities of demand for labor input i given a change in the price of capital, nik’ are lower for white collar professionals than for other workers in model A (nPK = 0.04, ”CK = 0.22, ”BK = 0.22), lower for workers aged 45+ than for others in model B (nLK = 2.52, nMK = 0.92, nHK = 0.62), and lower for the most educated labor-than for others in model C = 0.45, “MK = 0.18 , nHK = 0.02). The second interesting result appears from the (nLK estimated own price elasticities, ”11' (the diagonal elements of the elasticity matrices for models A, B, and C). As labor acquires more human capital, whether approximated by occupation, age, or education, derived demand for labor becomes less elastic. For model A InPPl < Inccl < InBB' and for models B and C lnHHl < InMMI < InLL . It IS a tenet of the theory of human capital that as a worker accumulates human capital specific to his occupation,both the employer and the worker have an increased incentive to continue his employ- ment regardless of possible minor wage fluctuations.7 7 See Becker (1975), pp. 16-37, specifically p. 34. 69 This implies that demand for labor having much specific human capital will be less wage elastic than will demand for labor with less specific human capital. The results cited above bear this hypothesis out. The labor dis- aggregation schemata of models A, B, and C, can be viewed as alternative proxy measures of labor groups with dif- fering combinations of general human capital and specific human capital, with model A (occupation) providing the best approximation for specific human capital ownership. (For example white collar professionals require a great deal of Specific training, compared to white collar clerical labor and blue collar labor), followed by model C (education) and model B (age). Estimated input demand is most price inelastic for white collar professional labor (nPP = -0.18) and across all models estimated input demand elasticity increases as occupation-specific human capital ownership decreases. Now examine the estimated own price elasticities of model D. Holding age constant, the derived labor de- mand becomes less elastic as education increases. And holding education constant, the derived labor demand for the less educated group is less elastic as age increases but is more elastic for the more educated labor group as age increases.€mrmgh this last result is contrary to the expected relation, it may be that formal human capital acquired from schooling depreciates relatively quickly, while experience, acquired through age, does not. 70 D. Summary of Relevant Previous Studies of Substitution Elasticities It is instructive to compare the results dis- cussed above with those of others. In recent years the number of studies of labor-labor or labor-capital sub- stitution in the manufacturing sector has grown. Few of the studies, however, share similar methodologies. They differ, in general, in one or more of the following ways. First, studies differ according to the criterion chosen for labor force disaggregation (occupation, age, or education). Second, studies differ in their-treatment of capital; some either eXplicitly or implicitly choose to assume that capital and labor are separable, thus they do not include a measure of the quantity or price of capital in their estimates. Third, studies differ by choosing to estimate a cost function, treating factor prices as exogenous, or a production function, treating factor quantities as exogenous. Finally, studies are separated by their chosen data: estimates based on time series versus estimates from cross-section data. (See Hamermesh and Grant (1979) for a thorough review of the literature.) First to be examined are studies comparable with model A, workforce disaggregation by occupation. The majority of the recent studies disaggregate labor by occupation, primarily because there are easily accessible 71 data for the workforce disaggregated by production (blue collar) workers, and non-production (white collar) workers, and for capital. Summary results are in Table 3.6. The subscripts B, W, and K denote production workers, nonproduction workers and capital respectively. Model A differs from all previous studies in its treat- ment of white collar workers, by further disaggregating white collar labor into two groups: group P) profes- sional, technical, managerial and kindred labor, and group C) sales, clerical and kindred workers. Nearly all studies find production workers and capital to be substitutes,and production workers and nonproduction workers to be substitutes. The relationship between capital and white collar workers is less clear-cut. The only other study using cross-section data and (appropriately) assuming wages exogenous (Freeman and Medoff, 1979) indicatesthey are substitutes. However most time series estimates using either a cost function or (the inappropriate) production function specification estimate white collar labor and capital to be complements. The results of model A indicate substitutability between capital and white collar labor, as do Freeman and Medoff. However as the level of training of the white collar labor increases (as indicated by the white collar pro- fessional group) capital and labor become much less sub- stitutable. 72 mm. I mm. I Hm. wc.H Hm. coHcscoz Hm. I mN. 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I mm.HI OC.H M3 mN.H VH. om.H OH.N Ho. xm mnocuo .m> mHmconmomoum ammo “mcHusuommscms .ONmH can onH .coHumHsmoa mo momsmcmu .moumum csuImcoH prme HmuHmmov csquuonm «meuHoHumMHm HhmH .moncmuu “HeImmmH meansuommscmz menuHUHummam Hana .moamcmue “Henmoma .mesnsuummzcmz moHuHoHumMHm cmmE .moncmuu unhmH Immma magnsuomescms unencum meuHoHummHm wmmH .moncmuu “onmNmH .mcHusuommscmz mmHuHoHummHm :mmE .moncmuu “mhIommH .mcHnsuomuacmz mmguaunummam Heme .moamcmuu “Heueema .ocausuommscmz vogue: can mumo AmhmHv on3mH£U coHuommImmonu :oHuocsm :oquSpoum Amemac escapees AhhmH. .Hm no cmEHmmmmM AmhmvanuHEmIchcoo Ananv mmsmImccmo HhhmHv :mEowuhIxumHO Amanv wuwsqupcuom mmfiumm mEHB >©5um huomwumo xemscaneoo. 6.m manna 74 .mmmhumsch HmDUH>H©cH now mnouofimuma mo mcmeoE mum mmumEHumm mm.m) em.~- mm.mu oa.m- m.HI m.HI SC mC or. Hm. 3m v m \ON mm.HI vm.HI x3 ow.N Nm.N xm mmeueoeummaw mama .moncmeu “moImmmH .mcausuomuscm: ucofimqum mmusuosnum umeuHoHummHm momH .ooncmnu "wonNmH .oCHusuommscmz moHuHoHummHm momH .moncmnu “wwImNmH .mcHusuommscmz mDCMHQ mo mHHuHmsq umw3oH mmucmHm mo wHHuanq ummcmHs umOHmcmuu .hooH .muwusuommscmz mo msmcou .mwaunmsecH uaoepuv can Im 60:06: new meme .usozcwe you @6660 wnHm> wn poxcmm Q .meHumsccH HnDpH>HCCH now muwumfiwumm mo mamemE mum mmumfiHummm xeemHv mmsmINccmo AnvhmHv :mmcwumHn50Iupcuwm Hmvanv comboumHnsqupcumm mmHumemEHB HmhmHv n.Hm um comcmm Spsum kuomoumu inescaucooc o.m manna 75 Among models which include capital and treat in- put prices as exogenous,the result of model A that demand for labor with relatively more human capital is relatively less elastic is, in generaL.borne out. The only studies with contrary results are from the Freeman-Medoff work which further decomposes labor into union and nonunion groups,and Woodbury's (1978) time series results for the short run. In comparing studies which use a production func- tion approach (factor quantities exogenous) with those using a cost function approach (factor prices exogenous), the production specifications uniformly produce higher ABS estimates than do the cost function specifications. For example, comparing cross-section studies, the Freeman- Medoff (1978) estimates with factor prices exogenous (both excluding capital and including capital) are similar in magnitude to model A estimates; yet elasticity estimates of Chiswick (1978) and Hansen et al. (1975) are consistently higher. More striking are the estimates of Denny and Fuss (1977), who use the same set of time-series data to obtain elasticity estimates from alternately 1) assuming input prices exogenous (cost function), and then ii) assuming input quantities exogenous (production function). Their estimates from the production function approach are higher. There is yet no clear reason for the difference in relative sizes of the elasticity estimates, but these 76 differences have been consistently borne out in previous studies. There are some studies which examine labor dis- aggregated by age summarized in Table 3.7. All of the studies take factor quantities as exogenous variables (as does model B). And all studies, those which include as well as those which exclude capital, suggest that re- gardless of age category any age group of workers is rather easily substitutable with any other age group. How- ever,the study most similar to model B - Anderson (1977) time series analysis - suggests that the ease of sub- stitution between age groups decreases as the number of years between the groups increases. Model B's results suggest the opposite is true. All studies also show that the own price elasticities of derived demand for all groups are greater than one. However, Anderson (1977) does not obtain the expected result that demand for labor becomes less elastic as the amount of human capital em- bodied in the raw labor increases. Three studies reported in Table 3.8 disaggregate labor by education groups. None of them include capital as an input and only one, Johnson (1970), derives elasticity estimates with wages as exogenous variables (as in model C). Results from model C are consistent with the Johnson study results which suggest that workers with college education and high school graduates are fairly easily 77 mm.MI mv.ml VH.hI vm.HI mm.H hm. ov.m +mv VVImN vNIwH om.HH +mv .m> VVImN mw.v +mv .m> VNIOH VH.h vamN .m> VNIQH mosoum xwmImmm mv.H vH MOM wmmnw>¢ nonmq ommcowe HmHIvH soy N¢.H mHImH .m> SHImH Ho.H mHImH .m> mHIVH . .H H 46 noan mo mwm>9 masonw mad mCOE¢ :oHusuHumnsm mo mepsum moHuHoHummHo thH moncmuu “mmnnva .mcwusuomwscmz mmo xenIoemH >Eocoom wHHucm mmu xcan .coHUMHsmom mo wsmcwu .mmumuw posuwz pan mama Hwanv :0mnopcd mHOEmRMHm ICOmcnon Amanv EmnmCHccsu (seam: Npsum .h.m wHQme menwm mEHB msocmmoxm meuHusmsa owesauce Hmuammo moHumm mEHB coHuowm mmouo msocmmoxm moans emesaoxm Hmugmmo whommumo 78 .wcoHumsqm pcmEmc can SHQmsm m>HumHmu mmoH no mnmm> v can Hoocom Hv.H :mHz .w> mmeHou mnmm> m.m Im .w> umm> +m mmumspmum Hoonom 6m.a been .m> mmwaaoo M go A up song; mo mwmwe AcoHuowm muonvv moouo coHumosom >n mo uHmQ m mo uumm msocomOCSo an we pmummnu mum mommsm mmu “mmmH .musuHDoHum¢ mo msmcmo .mmumum AoemHv bonz mmo “oomH .coeumasaom mo msmcou .moumum mmU “oemH .coHnmasmom mo msmcmu .mmumum pozuoz 6cm mama COHDDUHquwsm mo wprsum HthHV >uuonmsoo MAOSmHV :omcnom N63m .w.m mHQmB msocmmoxm mweuHucsmo msocmmoxm mommz emesauxm Hmuammo Naomwumo 79 substitutable. Of the studies using factor quantities as exogenous variables Welch (1970) derives elasticity estimates similar to Johnson and model C, but Dougherty (1972) estimates labor with at least an eighth grade educa~ tion to be very substitutable with other labor. Though model C also finds these inputs to be substitutes, the degree of substitutability is not found to be so great. Finally, in his seminal work, Zvi Griliches (1969) set forth the hypothesis that, "skill or education is more complementary with physical capital than unskilled or raw labor."8 In his work he set out to test the following hypotheses: 0 < o < o and < nk sk' Okn Osn' where Oij is the Allen partial elasticity of substitution between inputs 1 and j, n is raw labor, 5 is skilled labor and k is capital. He tested the hypotheses using two sets of cross-section data. The first were observations on two-digit SIC manufacturing industries for individual states in 1954. The second were based on 1960 Census data on education per worker by industry and 1964 Annual Servey of Manufactures data on capital per worker. Though estimated elasticities of substitution were not reported in Griliches' study, he concluded from 8 Griliches (1969), p. 465. 80 the first set of data: If the model is accepted, it definitely implies a higher elasticity of substitu- tion of 'raw' labor for capital than for 'skilled' labor. and for the second set, Here again the estimated coefficients have the expected signs...Thus, in both sets of data, there is evidence for the hypothesis that 'skill' or 'schooling' is more complementary with capital than unskilled or unschooled labor. The models presented in this chapter, which disaggregate labor by occupation (model A) or by education (model B), derive estimates of the Allen partial elasticities of substitution which are consistent with the Griliches hypothesis. However, in none of the models are the appropriate inputs estimated to be complements. E. Hicks Elasticities of Complementarity and Derived Factor Price Elasticities Recall, the Hicks partial elasticity of comple- mentarity, Cij' registers the effect on the price of one input when there is a change in the quantity of another input used, while holding marginal cost of the input whose quantity is altered, quantities of other inputs, and output price constant. In fact, cij measures the degree to which two inputs jointly contribute to a change in output. Thus when cij is greater than zero, inputs i and j are 9Griliches (1969), pp. 455 and 467. 81 complements in that they work to an increase in output (q-complements). Conversely, when cij is less than zero, the two inputs are substitutes (q-substitutes). The factor price elasticity, e is, then, defined ij' as the percent change in the price of factor 1 given a change in the quantity of factor j used in production. Table 3.9 presents estimates of Hicks partial elasticities of complementarity (BBC) for each of the four models, estimated for the means of observations over the entire sample of SMSAs. Examining elasticity estimates from models A, B, and C, the only general con- clusion is that each labor input, regardless of disag- gregation criterion, tends to be a q-complement with capital. When examined model by model, estimates of Hicks complementarity between labor groups differ considerably. When the workforce is disaggregated by occupation (model A), labor in each category is a q-complement for labor in any other category. Furthermore the q-complementarity between blue collar employees and either group of white collar employees is smaller than the white collar professional- white collar clerical q-complementarity (c = 10.89, PC c = 1.12, = 4.96). Turning to model B (age), PB CCB laborers taken from any pair of age groups are estimated to be q-substitutes. Or, given an increase in the quantity of labor of any type, the marginal productivity of labor Table 3.9. 82 Estimated Hicks Elasticities of Complementarity, 1969, cij (i = row, j = column). Input Input P C B K Model A P -27.71 10.89 1.12 6.56 (occupation C -43.52 4.96 0.90 B -5.91 2.27 K -3.63 L M H K Model B L -2.20 - 0.12 -0.20 0.36 (age) M - 1.03 -0.02 0.51 H -l.l9 0.60 K -O.58 L M H K Model C L -15.33 1.84 3.74 0.25 (education) M -5.17 -l.53 3.15 H -15.04 5.46 K -3.72 l 2 3 4 K Model D . l -22.09 4.24 131.39 -25.06 4.00 (age-education) 2 -4.06 0.71 2.92 1.39 3 464.76 -50.52 -4.34 4 -16.18 6.56 K -3.32 * Estimates are for the mean observations over the entire sample. 83 from any other age group declines. Elasticity estimates from model C (education) Show labor from the least edu- cated group to be q-complementary with labor from either of the other education groups, and labor with education up to and including grades 9 through 12 to be q-substitutes with labor educated beyond grade 12. For reasons stated earlier elasticity estimates from model D (age-education) are questionable. In general model D estimates any labor input to be a q-complement with any other labor input. However, labor ages 14-24 in either education group is an estimated q-substitute with labor educated beyond grade 12 and ages 25+ (cl4 = -25.06, c34 = -50.52). The model also obtains the theoretically impossible result that very young and highly educated labor is a q-complement for itself (c = 464.76). That is, an 33 increase in use of this input would cause output to in- crease in such a manner that the marginal product of the same input increased relatively greatly (assuming, of course, that all other inputs remain the same). From the law of diminishing marginal productivity we know that as the quantity of one input is increased in combination with fixed quantities of all other resources, the marginal product of the one variable resource Egg; decline. Table 3.10 presents estimated factor price elasticities for each of the four models. Consider first models A, B, and C. Factor price elasticity estimates 84 * Table 3.10. Estimated Factor Price Elasticities, 1969, 61' (i = row, j = column).* 3 Input Input P C B K Model A P -4.35 0.89 0.32 3.14 (occupation) C 1.71 -3.54 1.40 0.43 B 0.18 0.40 -l.67 1.09 K 1.03 0.07 0.64 -l.74 L M H K Model B L -0.10 -0.03 -0.05 0.17 (age) M -0.01 -0.24 0.00 0.25 H -0.01 0.00 -0.28 0.29 K 0.02 0.12 0.14 -0.28 L M H K Model C L -l.24 0.50 0.62 0.12 (education) M 0.15 -l.4l -0.26 1.51 H 0.30 -0.42 -2.51 2.62 K 0.02 0.86 0.91 -l.79 l 2 3 4 Model D l -0.76 1.34 1.54 -4.02 9" (age- 2 0.15 -l.28 0.01 0.47 .65 education) 3 4.52 0.22 5.44 -8.11 .07 4 -0.86 0.93 -0.59 -2.60 .13 K 0.14 0.44 -0.05 1.05 .5: Estimates are for the mean observed observations over the entire sample. 85 here compare very favorably with the estimated input demand elasticities of Table 3.7. Compare the estimated own factor price elasticities of the labor inputs in each model (diagonal elements). If occupation, age, or education can be considered proxies for specific human capital embodi— ment, in each model the estimated own factor price elasticity moves farther from zero as labor owns more human BB CC BB and in model C, capital. In model A, e < e and e < app; in model L. HH < Em»: LL‘ CHI-I < am: < ELL' as predicted in the specific human capital literature, B, e < 6 And, the labor group with the greatest amount of specific human capital (white collar professional labor) has the highest estimated factor price elasticity (e = -4.35). Thus a PP given percent change in quantity employed induces a relatively larger price response from labor endowed with occupation-specific human capital than from labor with less human capital. This result is consistent with the previous result that the own price elasticity of demand and specific human capital endowment are inversely related. Examine also the estimated elasticity between capital and labor in models B and C from Table 3.9. In the models, as age increases or as education increases, both the factor price elasticity of labor given a change in the quantity of physical capital used (c ) and the ik factor price elasticity of physical capital given a change 86 in the amount of labor used (ski) increase. That is, these models indicate that, as the amount of human capital em- bodied in labor increases, both 1) a given percent increase in the amount of physical capital used is associated with larger percent increases in the marginal product of labor, ceteris paribus, and 2) a given percent increase in the amount of labor used is associated with larger percent in— creases in the marginal product of physical capital, ceteris paribus. Or, as labor is endowed with more human capital, it becomes more complementary with capital in the sense of working together to an increase in output. (This result is similar to the inverse relation between capital— labor substitutability and human capital endowment found in Section B of this chapter.) The above results are not quite so apparent from model A, however. For a given change in the quantity of physical capital the percent increase in the marginal product of professional white collar labor is greater than that of the remaining two labor groups (a < e n and PK 1\ 6P? < EBY). Conversely, the marginal product of physical capital is most greatly improved from a given percent in- crease in professional white collar labor than from the same percent increase of labor from the remaining groups, ). ceteris paribus (eKP > EKC and EXP > EKB There has not been much empirical work by other authors concerning elasticities of complementarity. 87 However, the work which has been done does substantiate the general relations identified above. Freeman (1979), using time series data over the entire U.S. economy for years 1950-1974, estimated a translog production function with capital and labor disaggregated by age and sex. Among his findings are: Younger men are q-complements with older men, as age increases, Eii (own factor price elasticity of input i) increases, and,as the age of labor increases, the g-complementarity between labor and capital increases. These results found in Freeman's model are consistent with results from model B as well as the general human capital implications arising from models A, B, and C discussed above. F. Separability Tests Historically, empirical analysis of production functions has been undertaken by assuming heterogeneous inputs can be aggregated in some manner into distinct groups or indices of distinct groups (for instance: labor, capital, energy, and/or materials), and that meaningful statements of substitutability relations among members of the groups can be made. Only in recent years have there emerged a growing number of studies which has attempted to disaggregate one or more of the groups of heterogeneous inputs into sub-groups of more homogeneous inputs. This study endeavors to disaggregate labor into more meaningful sub-groups in order to say something not only about the 88 substitutability of labor having a particular set of char- acteristics for another type of labor, but also about the substitutability of different types of labor for capital. The use of aggregate inputs, or indices of aggre— gate inputs, requires the assumption that the underlying production function is weakly separable in these inputs. With the advent of studies which use less aggregate data the assumption of weak separability among inputs in each broad group is testable. The implications of weak separability are numerous. By specifying weak separability among groups of inputs one significantly restricts the available range of technology and therefore the possible functional form of the production function. However, if weak separability exists, there are many very interesting empirical applications. Separability permits the use of aggregate data when disaggregated data are of poor quality or when the analyst is not concerned with within-group substitutability. Separability is con- sistent with a two stage optimization procedure and justifies a two stage estimation of the production relations using consistent aggregates of the inputs in separable groups in the latter stage (see Fuss, 1977). Such a process may pro— vide a feasible estimation method when large numbers of inputs are involved. If the researcher is interested only in substitution relations among inputs in the separable 89 group - because separability implies the marginal rates of substitution between pairs of factors within the group are independent of the levels of those factors outside the group - he need not concern himself with those factors outside the group. Tables 3.11 through 3.14 present weak separability test results for each of the four models in the study. All possible forms of weak separability were tested for models A, B, and C. And for model D all possible forms of weak separability were tested such that all remaining inputs or groups of inputs were at least weakly separable from capital. Since maximum likelihood estimators have been obtained, likelihood ratios may be used to test separability hypotheses. Denoting the value of the likelihood function with separability restrictions imposed as L(R), and that without restrictions imposed as L(UR), the likelihood ratio, A, is then: A = L(R)/L(UR) . Hypotheses may be tested using the fact that -2 ink is asymptotically distributed as a Chi-squared random variable with the degrees of freedom equal to the number of inde- pendent restrictions imposed. In each case the null hypothesis of weak separability of the form described in 90 Tables 3.11 through 3.14 is tested versus the alternative hypothesis that weak separability does not exist. Turn to Tables 3.11 through 3.14 and first con- sider the most commonly assumed form of weak separability: capital from labor, or for each model the null separability hypothesis of the following form: model A (occupation) [(XP,XC,XB),XK], model B (age) [(X XH),XK], model C L'XM' (education) [(XL'XM’XH)’XK]' and model D [(X1,X2,X3,X4),X5]. Ikn:modelA.(occupation) and model C (education) the weak separability hypothesis is rejected at the 10 percent level of significance and 5 percent level of significance respectively. The hypothesis cannot be rejected for model B (age) and model D (age-education). It is interesting that the general labor-capital weak separability hypothesis is rejected for the models which do a better job of approximating labor group disaggregation by occupation- specific human capital accumulation of labor. Model A (occupation) serves to disaggregate labor by the type of occupation-specific human capital acquired as well as the amount acquired. (For example, a white collar professional laborer has probably not only acquired more specific training than a blue collar laborer, but the specific skills required are also different.) With respect to model C (education) both the amount of general human capital and occupation-specific human capital acquired by a worker may be correlated with the workers' schooling. Rejection of the null hypothesis, then, implies that the 91 Table 3.11. Weak Separability Tests for Labor Dis- aggregation by Occupation (Translog Cost Function) Separability Likelihood Ratio Type Parameter Statistic Test among inputs Xi Restrictions -2£n L(R)/L(UR) Results (XP,XC,XB),XK 2 4.638 ** (XP'XC'XK)'XB 2 1.43 (XC,XB,XK),XP 2 0.574 (XP,XB,XK),XC 2 4.214 (xP,xc),xB,xK 2 4.046 (XP,XB),XC,XK 2 5.292 ** (XC,XB),XP,XK 2 0.54 (XP,XK),XC,XB 2 2.704 (xc,xK),xP,xB 2 3.918 (XB,XK),XP,XC 2 0.728 (XP'XC)'(XB’XK) 3 4.362 * (XP'XB)'(XC'XK) 3 10.524 * (XP,XK),(XC,XB) 3 2.99 * Weak separability hypothesis rejected at the 0.05 level. ** Weak separability hypothesis rejected at the 0.10 level. Table 3.12. Weak Separability Tests for Labor Dis- aggregation by Age (Translog Production 92 Function) Separability Parameter Type Restrictions (XL'XM’XH)'XK 2 (XL'XM'XK)'XH 2 (XM,XH,XK),XL 2 (XL’XH'XK)'XM 2 (XL'XM)’XH'XK 2 (XL,XH),XM,XK 2 m >ufiaflnmummom Quanta oo~nccuu 3:722 2.23.8.2... venue-nu 32.3: 6525082.! vogue-nu 6.2-32 .2230825. rod-souu .3342. 63.38.81 coca-l on. 0.0.: xmmz mewumwe mwflpsum .nno—u consouccoa .usou. consoaccoa gavFOuu GOOGOauuusulaiehOI .Ionnav IIOIOUIthUloochII also .mH.m manna .3202 Ou-Iauounns 30‘00316 0.00 coaaucsu Cocaine-sad (Oauocflh uuclu anauusls coduuaooas ocuuou alur 90.38 100 capital, and K is disaggregated in one study (Berndt and Christensen, 1974b) into E, equipment and 8, structures. In each study Specifying a production function every weak separability hypothesis is rejected. Notice the hypotheses are rejected regardless of whether the translog function is seen as an approximate production function, or the stricter assumption of the translog func— tion as the exact function is made. Two studies test separability hypotheses assuming the translog function exactly represents the production process. Berndt and Christrnsen (1974a) when testing for types of (B,W,K) separability conclude "...no consistent aggregate index of BW, KW, or BK exists for our U.S. manufacturing data. The 'natural' aggregate of B and W has the lowest test statistic, but it is still quite decisively rejected."ll In an unpublished work Berndt and Christensen (1974b) test weak separability hypotheses among inputs in the four input production function Q = F(E,S,W,B). They find the following: We conclude that there is a significant loss of explanatory power for the cost shares of E,S,B, and W from imposing the separability conditions necessary to justify any two-stage optimization procedure (weak separability). and 1 . Berndt and Christensen (1974a), pp. 399. 101 It is interesting to note, however, that the grouping of B and W does much more violence to the data than the grouping of E and 5. Though weak separability of any type is rejected by Berndt and Christensen they find less difficulty in using a con- sistent aggregate of capital than using a consistent aggregate of labor (when disaggregated by occupation) in production function estimates. (The measure of capital used in the models estimated in this thesis, remember, is an aggregate of equipment and structures.) Denny and Fuss (1977) use the translog production function approach, but under the assumption that the trans- log function approximates the underlying production pro- cess. The corresponding conditions for weak separability are less restrictive than if an exact representation were assumed.12 However Denny and Fuss still find all forms of weak separability with respect to the production function specification are decisively rejected. Denny and Fuss (1977), using the same set of data, estimate the translog function assuming input prices are exogenous, as an approximation to the underlying cost function. For reasons discussed in chapter 2 this is a more appropriate specification. Under the cost function Specification Denny and Fuss find they cannot reject the 9 1-For an extensive discussion of the characteristics of the restrictions and properties of the resultant functions when weak separability restrictions are imposed under the alternative assumptions see Denny and Fuss (1977) and Blackorby, Primont, and Russel (1977). 102 hypothesis of weak separability of the form: [(W,K),B]; however, they do reject the two remaining forms of weak separability. Unfortunately there appear to be no existing studies which test separability hypotheses from models using cross-section data. In addition there is no evidence of separability tests from models which disaggregate labor by a criterion other than occupation. Therefore there is little ground for comparison of separability tests for models in this study with those which have gone on before. With respect to labor disaggregation by occupation, model A results concur with others in that the hypothesis of the existence of a consistent aggregate index of labor over occupational categories is rejected. Model A also rejectes the hypothesis of [(B,K),W] separability. How- ever when W is disaggregated into two subgroups: X P professional, technical, and managerial labor, and X C sales and clerical labor, the weak separability hypothesis that XP and K as well as XC and B can be re- aggregated into consistent groups, [(XP,XK),(XC,XB)], is not rejected, indicating that it is reasonable to: i) aggregate over labor with less human capital;and: ii) aggregate over physical capital and labor with more human capital. To conclude this section, the evidence is sub- stantial that one cannot construct a consistent aggregate index of blue collar and white collar workers in the U.S. 103 There is more favorable evidence for constructing labor sub-aggregates of the following form with capital: [(XP,XC),XB,XK], or [(XC,XB),XP,XK]. With respect to labor disaggregated by age the hypothesis of a consistent aggregate index of labor cannot be rejected. However the separability test of labor sub-aggregates with a far more "favorably impressive" test statistic is [(X ’XH)'XL'XK]' that is separability of labor aged 25+ (XM'XH) from younger labor (XL) and from capital (XK). With respect to labor disaggregated by education,a consistent index of aggregate labor over all levels of education cannot be constructed; however,consistent aggregates of either labor with education up through grade 12 or labor educated at all beyond grade 8 might be found. Finally, recall with reSpect to models A, B, or C, in each case when capital is included in a group with other labor inputs the hypo- thesis of weak separability of that group from each of the remaining inputs cannot be rejected. H. A Policy Application The previous sections of this chapter dealt with evaluating the estimated translog function for each model as an "appropriate" approximation of the underlying pro- duction process and interpreting Allen partial elasticity of substitution estimates, Hicks partial elasticity of complementarity estimates, and tests of weak separability hypotheses. This section focuses on the use of the 104 elasticity estimates of preceding sections to evaluate policy issues which involve inputs or groups of inputs in production. Policy actions alter factor markets in one or more of the following ways: 1) by changing factor prices or 2) by changing the supply of factors of production. In- vestment tax credits alter the price of capital services. A youth sub-minimum wage law would reduce the cost of employing teenage labor. Programs which upgrade the skills of a portion of the unskilled labor force increase the supply of more highly skilled workers. Three dif- ferent effects of any labor policy action may be identified: i) the own price or quantity employed of the factor at which the policy is aimed is altered, ii) prices or quantities of other factors which are substitutes or com- plements of the factor are altered, and iii) income shares of each of the factors may increase or decrease due to the policy action. It is the purpose of this section to examine policies which are implemented by altering factor prices or quantities. Consider first the impact of a policy designed to alter directly the price or quantity employed of a specific factor. Tables 3.5 and 3.10 presented the estimated price elasticity of demand for factors (assuming constant output and constant prices of other inputs), and estimated 105 factor price elasticities (assuming constant output prices i.e. variable output and constant quantities of other in- puts). Recall the general relations which held in each of models A (occupation), B (age), and C (education). As the human capital embodied ix: labor increases, as approximated by age, occupation or education, 1) the quantity of labor demanded is less responsive to a change in its own price, ceteris paribus, and 2) thecnniprice of labor is more responsive to a change in the quantity of labor used, ceteris paribus. The immediate implications are: as the amount of human capital owned by workers increases (de- creases), 1) policy actions affecting workers' own wages have a smaller (larger) impact on the quantity employed and 2) policies aimed at increasing employment of workers in any group have a larger (smaller) impact on the workers' own wages. Turn to the second question, the impact on other resources,and examine the off-diagonal elements of Table 3.5. In each of the models A, B, and C estimated AES is positive, i.e. each input is an estimated substitute or any other input. Further models A and C estimate all of the cross-price input demand elasticities to be less than one. However when labor is disaggregated by age the esti- mated demand for youth labor is highly responsive to a change in the price of any other input = 3.40, (nLM ”Ln = 3.67, nLK = 2.52). Estimated factor price 106 elasticities are much less similar across models (Table 3.10). Only in model A is each input an estimated q- complement of any other.13 The relative sizes of estimated factor price elasticities also vary considerably across models. Finally, to analyze the effect of policy actions on cost or income shares of various factors requires further familiarity with concepts of elasticities of sub- stitution. Three specific elasticity measures will be discussed: Samuelson elasticities, Allen-Hicks elasticities, and composite elasticities. Paul Samuelson (1968) was the first to relate an elasticity of substitution to distributive factor shares in a multifactor setting. Samuelson's elasticity of sub- stitution considers the substitutability of one factor against all remaining inputs taken together. The Samuelson elasticity of substitution of input Xi is defined as: -(1 - Si)fi 00 = I i f..X. ll 1 where fi and fii are the respective first and second derivatives of production function f with respect to input i. What Sato and Koizumi (1973) have called the dual Samuelson elasticity of substitution of factor Xi lRecall, input i and j are q-complement if, as output changes to maintain constant output price an increase in the quantity of input i used in production is associated with an increase in price of input j. 107 is defined as: o“? = -(1 - Si)gi 1 911x: where 91 and gii are the first and second derivatives respectively of the unit cost function g which is dual to production function f.14 The effect of an increase in the quantity or price of one factor on the relative share of that factor is indicated by Oi and 0:. That is,the Samuelson result states that the relative share of factor i increases or decreases as the quantity of that factor increases, de- pending on whether oi is greater or less than one; or: BS. sgn(§§%) E O as o. A|V H l Sato and Koizumi (1973) derive a relation from of, with respect to a change in the price of factor i: 381 > d sgn(§§f) ? 0 as Oi A|V l-‘ That is the relative share of factor i increases (de- creases) as the price of factor i increases,depending on whether of is greater (less) than one. 4It should be noted that when production and cost functions are homogeneous of degree one the Samuelson elasticities of substitution, the Allen Partial elasticity of sub- stitution and the Hicks partial elasticity of complementarity are related in the following manner: 1 1 1 1/0. =—:—- 2 so. and ‘3: _ Z s. . 1 1 Si j¢i 3 13 0 1 81 j#i 3 13 108 Now consider the effect of a change in the price (quantity) of one factor upon the relative share of an- other factor. Rather well known results concerning the Allen partial elasticity of substitution and the Hicks partial elasticity of complementarity shed some light here. These results are: i > . . sgn(§E;) < 0 as 0ij ? 1, (1 ¢ 3): and BS. sgn<§§%) E 0 as cij E 1, (i g j). 3 To analyze the effect of a change in the price (quantity) of factor i on the composite shares of a group of remaining inputs, say j and k, one final set of substitution measures is needed. Sato and Koizumi (1973) have derived the composite elasticity of substitu- tion of input Xi, wi' and the dual composite elasticity of substitution of input Xi’ ¢i' They are defined as follows: w. = (l - S-)l— + S c i '1 Ci 1 i and 0 = (l - S )l— + s d i i Cd 1 i i where S S Thus as the price (quantity) of factor i changes the income share of factor i changes relative to the com- posite share of factors j and k together as: AIV o D) (n e Alv H and The relationship between the various elasticity concepts and the factor shares are summarized in table 3.16. The Samuelson elasticities relate changes in the own factor share of inputs to changes in its own price or quantity. The Allen and Hicks elasticities relate changes in the factor share of input i to changes in the price or quantity of input j. The composite elasticities relate changes in the factor share of input i relative to a set of other factor shares to changes in its own price or quantity. It is instructive at this point to examine a policy issue concerning manufacturing industry in light of elasticity estimates presented in this chapter. For example, an issue of current interest is the 15See Sato and Koizumi (1973). PP. 487-488. 110 a a .6 .9 muflmomeoo HMSp muwmoaeou ha ha ..o ..O cofiupuflumnpm xuflumucwfimamfioo flo mo mo mufloflummHm mo muflOHummam p Hmwuumm cwHH< ammuumm meflm comawpsmm Hmsp comHmsEmm n n a .m .x Hm .x :H mmmcmno mmumcm uouomm ppm coausuflumnsm mo mmflufloflpmwam .mH.m manna :0 muommmm 111 legalization of a below-minimum wage for youth as a means of increasing employment. A youth sub-minimum wage pro- gram would reduce the legal minimum wage for teenage or near teenage labor. The labor group immediately affected by such a program is most closely represented by labor aged 14-24 from model B. Estimates from Table 3.5 show demand for youth labor to be more elastic than demand for any other input (nLL = -9.69). Thus a decrease in the wages of youth labor causes a rather large increase in number of youths employed. Furthermore, (from the AES estimates of Table 3.4) though young labor is an estimated substitute for labor of both remaining groups, the estimated of cross price elasticities of demand and nML ”HL' Table 3.5, for labor in these groups are relatively small. Remaining questions concern the effect on income distribu— tion among labor inputs due to a decrease in the price of youth labor. The estimated dual Samuelson elasticity of sub- stitution (at the mean of all observations), 0:, is 0.16 implying a decrease in youth wages is associated with an increase in youth's income share relative to all other in~ puts. All the relevant estimated Allen partial elasticities of substitution are greater than one, implying a new youth sub-minimum wage decreases each share of total output received by the remaining inputs. Finally the estimated dual composite elasticity of substitution of youth labor for all remaining labor inputs, = 6.61, indicates the ¢L 112 income share of young labor increases relative to the re- maining shares of labor income combined (i.e. (S2 + S3)/S1 decreases as Pl decreases). Thus estimates of model B suggest a youth sub- minimum wage would bring about relatively large increases in youth employment, some but not large declines in employment of older labor. The share of total income received by youth is expected to increase both absolutely and relative to the income shares of the remaining labor groups. I. Summary The four models developed in Chapter II — A (occupation), B (age), C (education) and D (age-education) - were estimated under the assumption that production technology is the same across the United States. When the estimated translog functions were checked as quadratic approximations to well defined cost or production func- tions, only model D faired poorly. Various elasticity measures of substitutability between inputs were constructed from the coefficient estimates of the estimated translog functions. Among the technical relations among inputs revealed by the elasticity estimates are the following. Contrary to previous studies, white collar labor and physical capital are not complements; rather,white collar clerical labor and physical capital are substitutes and white collar professional labor and 113 physical capital appear to be neither substitutes or comple- ments. As labor obtains more human capital it becomes less substitutable with physical capital. As labor obtains more occupation-specific human capital, the quantity of labor employed is less responsive to minor changes in its own price. Finally labor-labor substitution is easier as the level of education of labor from each group becomes more similar, but labor-labor substitution is more difficult as the age of labor from each group becomes more similar. Hypotheses of weak separability among inputs were also tested for each model. There is evidence to support aggregation of labor into two groups, white collar and blue collar, However, a "better" labor disaggregation might be to divide labor into white collar professional labor and remaining white collar labor plus blue collar labor. There is little support for an aggregation of all labor into one group. However other plausible labor subaggregates are: i) by age, labor aged 14-24 and labor aged 25+, or ii) by education, either labor with education up to grade 8 and labor educated beyond grade 8, or labor educated up to grade 12 and labor educated beyond grade 12. Finally,one policy issue, sub-minimum wage legisla- tion for young labor, was analyzed. The conclusions are that such legislation would increase employment of young labor with little decrease in employment of older labor. At the same time the share of total income received by young 114 labor would increase both absolutely and relative to in- comes received by the remaining labor groups. CHAPTER IV TWO ALTERNATE FORMULATIONS A. A Model of Land Use The work of the previous chapter was undertaken under the assumptions: 1) the underlying production function for manufacturing is the same across SMSAs, and 2) though there are other inputs in the production process, these other inputs are weakly separable from labor and capital and there- fore do not alter the technical relations among the labor and capital inputs. Thus it was not necessary to specify other factors as relevant inputs in the production or cost functions of Chapter III. The intent of this chapter is to present two alternate formulations of the underlying produc- tion process. The first formulation attempts to take into account the presence of other factors of production which may enter the production process in a non-weakly separable manner with labor and capital, and therefore alter the rela- tions of substitution among labor and capital. The second reformulation tests for geographic differences in production technology which might augment the technical relations among labor and capital inputs. 115 116 One of the purposes of this study is to examine labor and physical capital substitution across relevant labor markets in the United States, where relavent labor markets are Standard MetrOpolitan Statistical Areas. Though SMSAs are constructed such that they provide very good definitions of regional labor markets, no SMSA has exactly the same set of characteristics. If other im- portant inputs are not weakly separable from labor and capital, then their use at varying levels across the dif- ferent SMSAs will alter the technical relations among labor and capital. One cfi’ the most important additional resources is land. There are locational advantages to establishing a manufacturing plant in or near an urban area. Costs of transporting materials to the plant and distributing the final product are reduced as plants are established nearer to terminals of major transportation networks such as rail, water, or air. Urban areas contain large stocks of labor. By locating near large pools of labor, the cost of trans- porting labor to or from the workplace is reduced. There may be easier access to intermediate materials for pro- ducers in urban areas. Large urban areas also offer finan- cial, legal and other ancillary services to the production of manufactures. Therefore, there are cost advantages to locating manufactures in close proximity to these factors of production. 117 The productivity of land, or rather location, as a factor of production, depends largely on its location relative to the various factors of production and distri- bution. As land is closer to the source of other inputs, the more valuable is the land in production because the cost of transporting resources, or the cost of acquiring ancillary services, is reduced. Manufacturers are, there- fore, willing to pay a higher rental price of land the nearer it is to the remaining desired resources. Consider a: single SMSA and the land available for use in production within the SMSA. For simplicity, assume that all land is of uniform quality, and the desired re- sources and major transportation terminals (major water ways, railroad terminals, highest concentration of the labor force, etc.) are located at the center of the SMSA's central city. The supply of land within any ring concen- tric with the city center, with width of one mile, and radius of k miles, is fixed. Assuming a world with com- petitive input markets and competitive output markets, the equilibrium rental price of land is a function of the quantity of land available within the ring and the dis- tance of the ring from the city center, other things being equal. A profit maximizing firm will employ labor (X), capital (K), and Land (L) until the value of the marginal product (VMPi) of any group i is equal to the price of an input from that group (Pi). Specifically, with respect to 118 land, each firm will employ land until VMPL = PL; that is, until the output price (PQ) multiplied by the marginal product of land (MPL) is equal to the rental price of land (P That is, L). PQ * MPL = PL' or MPL = PL/PQ. The marginal product of land equals the ratio of price of land to output price. However, for any amount of land employed, the pro- ductivity of land is a functioncxfits distance from the city center. Figure 4.1 illustrates this relation. Consider two concentric rings of radius i miles and j miles from the city center respectively, i less than j. The land of ring j is further from the city center than the land from ring i. The marginal product of land, attributed to location in ring j, is everywhere smaller than the marginal product of land atrributed to location in ring i. Because land is assumed to be uniform in all things except location, VMPL(i) must be everywhere greater than VMPL(j). Furthermore, because firms are profit maximizers, the following conditions must hold: P ° MPL(i) Q PL(i), and P - MPL(j) Q PL(j). 119 Figure 4.1 Land Demand and Supply by Location I PL SL(i) PL SL(j) ‘\\\\YMPL(j) 120 Therefore, MPL(i) > MPL(j) implies PL(i) > PL(j)° That is, as land is located closer to the city center, its equilibrium rental price would be expected to increase in such a way that manufacturing is equally profitable re- gardless of location. Manufacturing is not the only productive use which can be made of the land. Land near the city center may be more productive for manufacturing relative to other indus- tries because of its location advantages. Moving farther from the city center, as location advantages to manufac- turing decrease, alternative uses of the land become more productive relative to its productivity in manufacturing. Figure 4.2 illustrates demand and supply for land by manu- facturers when there are alternative uses of the land. With increased distance from the city center the marginal product of land used in other industries rises relative to that for manufacturing (more land intensive industries such as agriculture or mining, for extreme examples, may provide more productive uses of the land). As there are more alter- native productive uses of the land from a ring farther from the city center, the supply of land within the rings to manufacturing becomes more price elastic at each quantity of land supplied. (For the same quantity of land in rings 1 and j, SL (i) is less price elastic than SL (j).) The proportion of land used for non-manufacturing purposes 121 Figure 4.2 Land Demand and Supply by Location II SMSA Center 1 3' VMPL CH 122 increases with distance from the city center. From Figure 4.2, as each supply schedule of land from each ring becomes everywhere more price elastic for a given quantity of land with increased distance from the city center, the prOportion of land within the ring used for manufacturing decreases; or L:/Li > Lg/Lj. That is, once alternative productive uses of land are taken into account, the density of manufactureres per square mile is expected to decrease with distance from the city center. The foregoing simple models were presented to pro- vide a theoretical justification of two observations predom- inant in the urban economics literature. First, the rental price of land decreases with distance from the city center; and second, the employment density of manufactures decreases with distance from the city center. Both of these facts were seen to be the result of differential productivity of land due to its location relative to the city center. The interesting question is what does this all imply for the production relations among land, labor, and capital in the manufacturing industry. Does location play an important role in determining the relations between labor and capital? Or might land, or location, be considered weakly separable from the remaining inputs? The above simple models were constructed for a single SMSA. Each SMSA has its own unique geographic shape and its own set of locational incentives for production. 123 Therefore, for each SMSA the equilibrium distribution of the rental prices of land as a function of distance from the city center is unique, and the manufacturing density dis- tribution is unique. A third common observation, identified by Mills (1972), Muth (1969) and Kain (1975), is that over the last few decades, manufacturing density, or the density of em- ployment in manufactures, has declined in the central cities of SMSAs. Furthermore, manufacturing has become more evenly distributed throughout the SMSA. It is not difficult to find justification for this fact from our simple model. Over recent decades transportation and communications net— works have advanced greatly. Improved highways have made transportation of materials, labor, and the final output to and from manufacturing plants less costly. The argument has also been made that as firms become larger, they in- corporate within the firm many of the ancillary services previously performed by individuals outside the firm.1 All these changes over time have decreased the importance of location relative to the city center in pro- duction. Figure 4.3 illustrates this. As plant location becomes less important in production, the value of the marginal product of land from a ring nearer the city center «nun (i))decreases relative to the value of the marginal lSee Chinitz (1961). PP. 98. 124 Figure 4.3 Land Demand and Supply by Location III SMSA Center (j) (j) 125 product of land from a ring farther from the city center (VMPL Qj)). At the same time, the equilibrium prices of land from either ring begin to converge (P;(i) > P£(i) > P£(j) > P;(j)). The proportion of land from the nearer ring used in production decreases and the proportion of land from the farther ring used in production increases, (LE/Li:>L;/Li:>L5/Lj:>L3/Lj). That is, manufacturing density is more evenly distributed across the SMSA. The interesting question, again, is whether this historic change in the importance of location in production alters the technical relations between labor and capital and between labor and labor. First, consider the effect of plant location re- lative to the city center on relative factor intensities in production. For manufacturers on the periphery of the SMSA, land is cheap. As production moves closer to the city center, the marginal product of land increases, and land becomes more expensive. Assuming land and other re- sources are substitutes in production, labor and capital will be substituted for land and production becomes less land intensive. Is labor-labor substitution or labor- capital substitution altered by this? If labor and capital are weakly separable from land, then marginal changes in the use of land, or marginal changes in location, do not alter substitution among these remaining inputs. If this is not the case, then location of production does alter 126 substitution. With the available data, it is not possible to test directly the hypothesis that labor and capital are weakly separable from land. However, representative SMSAs with high central city employment densities (indicating low overall land intensity of production) can be compared with representative SMSAs with low central city employment densities (indicating higher overall land intensity of production), to examine the hypothesis that land intensity of production does not alter the production relations among labor and capital inputs. Second, it is interesting in its own right to examine the direction of change in factor substitution as manufacturing density flattens out in each SMSA. There are also some Specific areas which should be discussed. Estimates of white collar labor-capital substitution of previous studies have not consistently found the inputs to be either substitutes or complements. The time-series studies of Denny and Fuss (1977), and Berndt and Christensen (1974a and 1974b) have estimated a complementary relation between labor and capital. In these studies the time-series run from 1929-1968. Substitution elasticity estimates of time-series Spanning fewer and more recent years are mixed. Clark and Freeman (1977) and Kesselman et a1. (1977) estimates indicate these factors are com- plements; however, Berndt and White (1978) and Dennis and Smith (1978) estimate them to be substitutes. Finally, 127 results of cross section studies of manufacturing in recent years estimate white collar labor and capital to be sub— stitutes in production. Freeman and Medoff (1979) in a cross section study over 1972, and the results of this study discussed in Chapter III, estimate positive white collar labor-capital substitution. The estimates from these studies are not inconsistent if the decline in manufac- turing density observed over recent decades implies production relations have been altered such that white collar labor and capital are becoming increasingly sub- stitutable (or less complementary). A model encorporating manufacturing density as an independent variable can be used to examine this hypothesis. The effect of decreased manufacturing density on labor-labor substitution has implications with respect to labor market segmentation hypotheses and with respect to implications concerning the internalization on ancillary services. A decrease in substitutability of labor from different groups as manufacturing density decreases lends support to the hypothesis of increasing labor market seg- mentation. Internalization of ancillary services may also alter labor-labor substitution. In a production function model including manufacturing density, these two influences probably cannot be separated. 128 B. A Reformulation Including Manufacturing Employment Density To include land, or location, as an input in pro- duction, the translog models developed in the first two chapters are reformulated. A conclusion in the previous section was that manufacturing density in any SMSA depends on the productivity of land as a function of its location relative to the city center. Land nearer the city center is more productive than land farther from the city center. The more sensitive the productivity of land is to distance from the city center, the more densely will manufacturing be distributed near the city center and the more rapidly will manufacturing density fall with distance from the city center. Because data on the price and quantity of land used in manufacturing are unavailable, a distribution function of manufacturing employment density is estimated as a proxy for land, or location, for each SMSA. Mills (l972a)argues that his own studies and studies of others have Shown that an exponential function provides as good an approximation to SMSA population density, as well as employment density, as any other function.2 Define the exponential density function as follows: -§k D(k) = De (4.1) 2Mills qualifies this position by stating: 129 where D(k) is the manufacturing employment density k miles from the center of the SMSA, and D and E are parameters estimated from employment data. D is the measure of employment density at the center of the SMSA's central city (in this chapter D is in terms of thousands of employees) and E is a measure of the rate at which density declines with distance from the center. An SMSA where the productivity of land is highly sensitive to its distance from the city center would have a large cen- tral employment density parameter D and a large rate of decline of density parameter 5. If distance from the city center has little influence on land productivity, then both D and E would be smaller. In order to incorporate these characteristics into the models developed in the previous chapters, consider a multiple input cost function where land, as approximated (continued) Employment density functions are, however, another matter. There is practically no evidence that the exponential density func- tion provides an accurate description of the variation of employment density with distance from the city center...At best the exponential function may provide a broad summary statistic to measure the concen- tration of employment near the center of metropolitan areas. There are two justifications for proceeding boldly to calculate exponential employment density functions from central city suburb data. First there is no other way to do it, since more detailed data hardly exist. Second, exponential employment density functions provide extremely interesting comparisons between the suburbanization of population and that of employment. (Mills, 1972a, p. 36). 130 by the urban setting of production (U), is also an input.3 The cost function for each SMSA becomes: C = C(Q’U'Pl'PZOOO'Pn) (402) Because direct inclusion of land as an input in the model is not possible, the urbanness of production (or location) can be seen to alter the equilibrium prices of the remain- ing inputs. For instance, wages in plants located nearer the residences of labor should be lower than wages of the same labor in more distant plants, because labor is willing to take a pay cut in return to decreased transportation cost and transportation time to and from work. Similarly, firms nearer the major transportation terminals will be willing to pay higher prices for materials and capital because transportation costs of these inputs to the plants as well as transportation costs of ancillary services to the plants are decreased. In particular, U is an input such that it augments each of the remaining input prices in a specific fashion. That is: 3Only the cost function will be derived here. The . production function derivation is similar and the final form will be shown. 131 Pi = Pi(U) thus: C = C(QIP1(U)IP2(U)I-o-IPn(U))I where urbanness, U, is itself a function of the density gradient parameters D and 5. Or: U = U(Dr€)° Thus Pi = Pi(U) implies Pi = Pi(D,£). Assume Pi(D,§) takes the following functional form: Pi(D,€) = Pie 1 (4.3) where A: and A? are parameters to be estimated from the cost function 4.2. Each input price is now augmented by city center employment density, D, and the rate of decline of employment density from the center, 5. This specification is chosen so that in alternative ex— treme cases, D approaching either zero or infinity and E approaching either zero or infinity, the separate in— fluences of D and C on production relations can be identified. The cost function with input prices augmented for urbanness becomes: (lie—n+lie-g) (Aie'D+A:e'€) C = C(Q,P1e ,...,Pne . 132 The translog function is then: n (A: C-D+Aie-£) 1n C = 2n 80 + in Q + .2 Bi £n(Pie i-l n n (11e-D+A?e-g) + 1/2 2 y aij12n(Pie 1 1 ) (4.4) i=1 jél (11e-D+A?e-£) £n(Pje 3 3 )] Rearranging the terms, equation(4.4)becomes: n n n in c = B + y a. in P. + 1/2 2 X 6.. in P. En P. =1 1 1 i=1 j=1 13 1 3 -D 1 , 1 + 1/2 E g 5.. e (lj in P1 + xi Rn Pj) -g .2 2 + 1/2 E g 5.. e (Aj 2n Pi + 1i 2n Pj), 1] where: B0 = 2n 80 + in Q l -D 2 -£ + Z Bi(lie + lie ) i -2D 2 2 -2§ + 1/2 E z 6ij(AiAj e + lilj e 3 + (111? + A111)e'(D+€)). i j i j The symmetry property, 5.. = 6.., implies: 1] j]. 133 n n n in C = B + Z 8. 1n P. + 1/2 5 0.. 2n P. in P 0 i=1 1 1 151 jgl 13 1 j n n l -D + Z X 6.. A. kn P. e (4 5) i=1 j=1 13 3 1 n n 2 g + Z 1 6.. A. in P. e . i=1 j=l 13 J 1 Let: 1 n 1 2 n 2 1. = ) 0..1. and T. = 6.. l. ( 4.6 ) l j=l 1] J '1 jZl '13 3 then: Ii 3 )1 in C = B + 8. in P. + 1/2 , 0.. in P. in P. 0 i=1 1 1 i=1 j=1 13 1 3 (4.7) n n + Z 1 in P. e D + 2 12 in P. e g . 1 . 1 1:1 1:1 Thus the translog cost function takes the form of equation (4.7.) The first degree homogeneity conditions imply the following constraints exist for the T and A coefficients: (4.8 ) 134 and As in Chapter I, with the assumption of perfect competi- tion and with symmetry and first degree homogeneity con- ditions imposed, a set of n-l share equations is derived from equation (4.7): ' n-l a in c E , . = 6.j £n(Pi/Pn) + ; éji £n(Pi/Pn) 1 1 i=j+l + T. e + Ti e ( 4.9 ) Cost function coefficients are estimated from this set of equations. The effect on the ith factor Share of a marginal change in employment density at the city center, D, or the rate of decline of pOpulation density,€, is computed as follows: 3 Si = -T e-D 3 D (4.10) a s. _ l = -T e g. 135 Finally changes in the Allen partial elasticities of sub- stitution, Oij' as defined in Chapter I given a marginal change in either D or g are:4 -D l l 30.. ..e T. S. + . S. _;1= 13 (J_1 T14) 17y. 90 (s s )2 ' 3' i j l -D Boii = Ti e (20ii - Si) . 3D 53 i -g 2 :01. = 6ij e (13 512+ T. S ) ' i # j. 5 (5.5.) 1 3 2 '5 35 s? i 4For all four models, A,B,C, and D, a cost function is estimated. However for model B a production function is estimated as well, assuming input quantities are augmented by the urbanness of production. Under the production funca tion Specification the production function is: Q = f(X1(U)IX2(U)I°°°IXn(U)) where Xi = Yi(D,g) and (lie-0+Age-g) Xi(D,g) = Yie The translog function to be estimated is then: in Q = A0 + E “i in xi + 1/2 E g Yij 2n Xi 2n Xj 3 -D 4 - + E Ti in Xi e + g Ti 2n Xi e 5 and aSsuming first degree homogeneity among the inputs the system of n-l Share equations is: 136 The system of equations estimated is a stochastic version of the n-l equation system (4.9). For reasons discussed Chapter II, each share equation is assumed to have a linear disturbance term (Si), and the disturbances are assumed to be correlated across equations. An Iterated Zellner Efficient Estimation procedure (IZEF) is used to obtain cost function coefficient estimates. (See Chapter II, pp. 39 to pp. 43 for further discussion.) C. Estimation of the Manufacturing Employment Density Model The purpose of this section is to describe how density gradient parameters D and E from equation (4.1) are estimated for each SMSA. The procedure used to estimate parameters D and g is the same as that used by Mills (1972). Assume equation (4.1) correctly represents the employment density in an SMSA. Not all SMSAS are approxi- mately circular in shape, but are bounded by lakes, oceans, national borders, or have other geographic irregularities. Therefore, assume that the SMSA is circular except that a Slice of arc length 2n - 0 radians has been removed. Then the number of employees n(k) within a ring of width dk, concentric with the central city center is: n-l £n(X./X ) + f 7.. £n(X./X ) i n i=j+1 ij i n 137 n(k)dk = D(k)¢k dk (4.11) The total number of employees within q miles of the city center, N(q), is: N(q) = fq D(k)dk. (4 12) 0 . Substituting (4.11) into (4.12) and integrating, N(q) be- come 5 : N(q) = 9% (1 - (1 + zq)e'€q) (4.13) € Finally, letting q approach infinity, from (4.13): lim N(q) = 9% , (4.14) q-mo g which is the total number of employees in the entire SMSA. The 1967 Census of Manufactures publishes employ- ment data for SMSAs. The 1970 Census of Population reports land areas of each central city. Maps of SMSA central cities are published in 1970 Census of Population census tract data. These data were used to obtain estimates of D and g for each SMSA. For the following discussion consider Figure 4.4. Let g be the radius of the SMSA's central city; q is estimated by constructing on a census tract map a circle centered at the center of the central city and whose boundary closely approximates the city's boundary. The radius of this circle is then q: and 0 is estimated as the value which equates the area of an arc of radius q 138 Figure 4.4 Density Gradient Estimation Central City Boundary /’ 0 radians / / ‘\ q’ \ / l < (22*) - 6) radians \ I Natural Boundary , 139 to the land area of the central city in question. That is, the value of 0 is obtained from the following equation: , (4.15) where A is the land area of the central city. With an estimate of 0 in hand, equation (4.13) and (4.14)are solved simultaneouslyfbr D and E. Central city manufacturing employment is N(q) of (4.13), and SMSA manufacturing employement is lim N(q) of (4.14). Sub- stitute(4.l4)into (4.13)and E i: the only remaining unknown; 5 has no analytic solution and is solved for iteratively via the Newton-Raphson method (see Scarborough, 1955). Once E is found, then D is obtained from (4.15). Not all SMSAS lend themselves to employment density gradient estimation Via this method. Those SMSAS whose central cities have irregularly shaped boundaries which include large amounts of sparsely populated areas are excluded. Those SMSAS with multiple central cities which are not contiguous are excluded. Within the sample of SMSAS, estimates of D, employment density at the center of the city, range from 14911 to 135. Similarly estimates of E, rate of decline of employment density, range from 1.93 to --.007.5 Over 5For all but two SMSAS the estimated 5 is greater than zero. However for the Chicago and Detroit SMSAS the estimated 5 are -.073 and -.O77 respectively, indicating for these two SMSAS manufacturing employment density increases (does not diminish) with distance from the city center. 140 the set of SMSAS sampled the mean estimated D is 2587 and the mean estimated E is 0.469. These estimates are consistent with density gradient parameter estimates of Mills (l972a)and those of Muth (1969) as reported by Mills (1972a). The complete set of density gradient parameters for each SMSA is listed in Appendix D. D. Estimates Using the Density Gradient Reformulation The task of this section is to evaluate the esti— mated cost or production functions of the density gradient formulation as approximations to well behaved cost or production functions, and to test for the significance of land, or location, in the determination of production relations among the various labor inputs and capital. For each formulation the four models developed in Chapter II are estimated. The models are, again: A, with capital and labor disaggregated by occupation: B, with capital and labor disaggregated by age: C, with capital and labor disaggregated by education: and D, with capital and labor disaggregated jointly by age and education. For reasons stated in Chapter II, cost functions are estimated for models A, C, and D and a production function is estimated for model B.6 6For both alternate formulations a cost function was also estimated for model B and summary results will be discussed in footnotes where appropriate. 141 Coefficient estimates of the translog cost func- tions of models A, C, and D, and the translog production function of model B (equation system (4.9)) are presented in Table 4.1. Specific coefficient estimates will be dis- cussed later, however it appears that across the models the Ti, coefficients of e-D, the city center density variable, are more significantly different from zero than are the Ti coefficients of e-g, the rate of decline of employment density variable.7 Remember that as approximations the estimated trans- log cost and translog production functions should satisfy the regularity conditions for well behaved cost or produc— tion functions. The conditions are that the first deriva- tives of the production function with respect to each input quantity be greater than zero. And the Hessian matrix of the relevant cost function, or the bordered Hessian matrix of the relevant production function, should be negative semi- definite (See chapter I, pp.6-7 and p.15 for further dis- cussion). These conditions are examined for each model. Table 4.2 presents summary results for the density gradient reformulation. The appropriate derivatives were taken first at a point representing the mean of observations of the in- dependent variables including the mean values of D and E over all ovservations and the mean of actual factor shares 7 . . . The estimated coeffiCients of the translog cost function approximation of model B under each alternate formulation are presented in Appendix E. 142 Table 4.1. Estimated Translog Coefficients: Under the Density Gradient Reformulation (Standard Errors in Parentheses) Model A Model B Model C Model D translog cost translog produc- translog cost translog cost by occupation tion by age by education by age-education Bp -.0245 GL .107 8L .104 81 .0119 (.0343) (.0177) (.0241) (.0114) 6 .0796 6 .0372 6 .0199 6 .0124 PP (.0192) LL (.00353) LL (.0185) 11 (.00549) 6 -.0213 y -.0122 6 —.00781 6 -.0112 PC (.0106) L” (.00437) L” (.0188) 12 (.00982) 6 -.00614 y -.0113 6 -.00821 6 .00770 PB (.0210) L“ (.00417) L” (.0116) 13 (.00352) 6 , -.0566 y -.0136 6 -.00389 6 .00500 P1 (.0160) LK (.00586) LK (.0131) 14 (.00609) 6 -.0139 1K (.00557) B .0470 .304 B .163 8 .211 C (.0215) GM (.0478) M (.0350) 2 (.0503) 6 .0561 y .120 6 .100 6 .131 CC (.0123) M“ (.0130) ”M (.0297) 22 (.0406) 6 .0133 y -.0355 6 .00677 6 -.00455 CB (.0153) M“ (.00991) “H (.0166) 23 (.00902) 6 -.0215 y -.0725 6 -.0994 6 -.0359 CK (.0111) "K (.0164) “K (.0193) 24 (.0236) 6 -.0801 2K (.0258) BB .205 aH .357 an -.0474 53 .0225 (.0368) (.0534) (.0376) (.0114) 6 .0973 y .127 6 .0609 6 .0121 BB (.0329) H” (.0150) ”H (.0185) 33 (.00440) 6 -.0823 y -.0807 6 -.0595 6 -.0129 BK (.0190) ”K (.0182) ”K (.0171) 34 (.00599) 63K -.00233 (.00558) Table 4.1 (continued) 6 .772 y .232 K (.0912) K (.0870) 6 .160 y .167 xx (.0262) xx (.0304) 3 I: .144 TL .00896 (.0258) (.00821) I: .0198 I: .0371 (.0163) (.0236) I; -.0503 I: .0597 (.0281) (.0278) r: -.0837 I: -.106 (.0385) (.0415) I: .0470 I: .0151 (.0498) (.0140) I: .00969 I: .0763 (.0316) (.0412) I: .00689 T4 .0415 (.0542) H (.0485) 2 4 TX -.0636 1K -.133 (.0751) (.0726) N = 61 64 143 6K H k‘ M 3:63 2 k) b N 7:»: m H le .781 (.0489) .163 (.0262) .00248 (.0194) -.0340 (.0284) .129 (.0309) -.0952 (.0393) -.0320 (.0345) -.00238 (.0507 .101 (.0549) -.0666 (.0701) 64 44 4K -.00904 (.0417) .101 -.0572 (.0179) .764 (.0603) .154 (.0297) .00444 (.00805) -.O435 (.0344) .0379 (.00802) .0999 (.0289) -.0987 (.0428) .0142 (.0156) .0411 (.0697) -.00949 (.0158) .0148 (.586) -.0606 (.0876) 51 144 Table 4.2. Regularity Criteria for the Density Gradient Reformulation: Order (SOC) Criteria SOC--Sign of Relevant Hessian Determinant First Order (FCC) and Second FOC 4X4 3X3 282 1x1 Model A i ** - - + - (occupation) ii ** - - + - Model C i ** + - + - (education) ii ** + - + - Model D i ** - + + - (age-educa- tion) ii ** - + + - SOC--Sign of Relevant Bordered Hessian Determinant Model B i ** - + (age) ii ** - + * Examination is done for the mean of observations over the sample. ** FOC are satisfied. lfitted factor shares 1 1actual factor shares 145 for each model. Second, appropriate derivatives were taken after replacing the mean actual shares with the fitted factor a shares calculated from the mean of all observations. The results from using either fitted shares or actual w~-w . . shares are the same. The estimated production function of model B (age) and the estimated cost function of model C (education) meet the regularity criteria. However models A (occupation) and D (age-education) fail to satisfy the regularity criterion of alternating signs of the relevant bordered Hessians, implying that one or more of the inputs is not characterized by a diminishing marginal rate of factor substitution.8 That the regularity conditions are not always met need not be disconcerting. Regularity conditions are tested with the estimated parameter coefficients of each model and they are, therefore, stochastic variables them- selves. When regularity conditions are not met it is always because one of the Hessian matrix determinant conditions is not satisfied. Due to the nature of the determinant it is not possible to estimate the variance of the estimated value oftflmadeterminant, and therefore, it is not possible to assign a statistical significance to the test. In fact, because of high variances of ‘the 8The estimated translog cost function of model B under the density gradient reformulation fails to meet the Hessian matrix regularity criterion when either average fitted shares or average actual shares are used. 146 coefficient estimates of some models, the significance level of tests of the regularity condition of these models may also be very low. From the coefficient estimates of Table 4.1, produc- tion relations among inputs are derived in the form of Allen partial elasticities of substitution and Hicks partial elas- ticities of complementarity. If labor and capital are weakly separable from land, then marginal changes in an SMSA's density gradient will not alter the estimated substitution relations among the labor and capital inputs. Therefore, the interesting question is whether labor-labor substitution or labor-capital substitution is sensitive to changes in D or g , or to changes in combinations or D and £1. In particular, two areas of interest were dis- cussed in the first section of this chapter. The first is that no two SMSAS have the same set of attributes: thus in each SMSA the relation between land produc- tivity and distance from the city center is different. An SMSA with land whose productivity increases greatly as its location is nearer the city center, relative to other SMSAS, was demonstrated to have large coefficients D and E, and was demonstrated to use a relatively less land intensive production process. For an SMSA whose land productivity is not so much altered by distance from the city center, relative to other SMSAS,‘the employment density gradient will be relatively flat. Thus both co- efficients D and 6 will be relatively small, and the 147 land intensity of production will be greater. To examine the hypothesis that land intensity of production does not alter substitution among various labor groups and capital, the partial derivatives of the estimated Allen partial . elasticities of substitution (AES) or Hicks partial elasticities of complementarity (HEC), between paris of these inputs with respect to D and with respect to 5 will be examined. Second, among the observations discussed in the first section of this chapter is that manufacturing employment density, overtimelast few decades, has been flattening out. Land productivity has become less sensitive to distance from the city center. This implies that for the representative SMSA, central city employment density is decreasing, and the rate of decline of density is also decreasing. To test for the effect of this trend on labor-labor and labor-capital substitution, a repre- sentative SMSA will be examined. Flattening of the density gradient will be simulated by imposing a reduction in D of a fixed amount and simultaneously decreasing g by an amount sufficient to maintain the level of manufacturing employment in the representative SMSA. To test the hypo— thesis that factor substitution is not altered by this trend, changes of the estimated AES or HEC for these specific simultaneous changes in D and 5 will be examined. 148 Before discussing these two sets of hypotheses, recall from Table 4.1 that the 12 i’ estimated coefficients of the rate of decline of employment density e-g. are somewhat less reliable than the Ti, estimated co- efficients of the city center density e-D. The standard errors of estimation of the estimated 1: are all rather large; though many are greater than one standard error from zero, all of the estimated Ti are within two standard errors of zero. Table 4.3 presents the estimated AES of the cost function models A (occupation), C (education), and D (age- education), and the estimated HEC of the production func- tion model B (age). The elasticities are estimated at the mean of observations over the entire sample for each model. These elasticities are not directly comparable with those of Tables 3.4 and 3.9 because the sample sizes are different. However, the general relations identified inChapterlfl] still hold under this density gradient re- formulation (See Chapter III for further discussion.)9 Examine now the first question: the influence of differing land productivity across SMSA's, as approximated 9The estimated AES of the cost function specification of model B (age) under the density gradient reformulation are reported in Appendix E. Again, the elastiCity estimates follow the general relations identified in Chapter III. Except that under the cost function specification young and middle aged labor are estimated complements and the relation between young labor and capital is neutral. 149 Table 4.3. Substitution Elasticities under the Density Gradient Formulation* Input Input P C B K Model A P -2.l32 -.886 .963 .220 (occupation) C -2.354 .396 .404 Oi' B -1.279 .416 3 K -.386 L M H K Model B L -3.234 -.116 -.0389 .387 (age) M -l.079 .364 .362 Ci. H -.949 .291 3 K -.358 L M H K Model C L -8.271 .655 .268 .901 (education) M “1.298 1.153 .257 0.. H -2.875 .224 13 K -.375 l 2 3 4 K Model D l -l7.181 .0375 20.924 1.90 .13; (a e-educa— 2 -0833 -0318 .278 ° 9tion) 3 13.256 -6.82 .542 0.. 4 -1.236 -221 *Estimates are for the mean observations over the entire sample for each model. Estimated Allen Partial Elasticities of Substitution oij for Cost Function Models A, C, and D; and Estimated Hicks partial Elasticity of Complementarity for Production Function Model B. (i = row, j = column) 150 by the Mills density gradient, on labor and capital pro- duction relations. The influence on the marginal product of an input, given a change in D or g, is reflected in. the partial derivative of the input's Share in production with respect to either D or E (BSi/BD and 851/85). Iftflmemarginal product of input i increases (decreases) with increases in either D or 5, so will the factor share of input 1 increase (decrease). Table 4.4 pre- sents estimated partial derivatives of input shares with reSpect to D and g for each model. The marginal product of input 1 is unambiguously increased in SMSAS where production is concentrated near the city center (steep density gradient) if both BSi/BD and 881/86 are positive. Similarly, the marginal product of input i is unambiguously increased in an SMSA where production is less concentrated around the city center (flat density gradient) if both 851/80 and 881/85 are negative. Over all models, the marginal product of capital unambiguously increases as the manufacturing employment density gradient becomes steeper. Thus, there is support for the hypothesis that land and capital are substitutes. The marginal product of labor endowed with the greatest amounts of human capital (professional labor of model A. labor age 45+ of model B, and labor with education at least beyond grade 12 of model C) increases as the density 0 gradient becomes flatter. The impact on blue collar workers is both small and ambiguous (SSE/3D >.0 and 151 Table 4.4. Estimated Change in Factor Share of Input i, Given a change in the Density Gradient: Either D or 5 (standard errors in parentheses) Model A Model B Model C Model D (occupation) (age) (education) (age-education) i BSi/BD i BSi/BD i BSi/BD i BSi/BD P -.01267 L -.000674 L -.0000187 1 -.000446 (.00227) (.000616) (.00146) (.000809) C -.00174 M -.00279 M .00256 2 .00437 (.00143) (.00178) (.00214) (.00346) B .00443 H -.00449 H -.00870 3 -.00381 (.00247) (.00209) (.00232) (.000806) K .00737 K .00797 K .00716 4 -.01004 (.00339) (.00312) (.00296) (.00290) K .00992 (.00460) Q “r r ' 0: 8-1/65 851/3, BSi/Bg asi/o, P -.0302 L -.00945 L .0200 1 -.00949 (.0320) (.00876) (.0216) (.0104) C —.00622 M -.0477 M .00149 2 -.0275 (.0203) (.0258) (.0317) (.0466) B -.00442 H -.0260 H -.0632 3 .00634 (.0348) (.0303) (.0344) (.0106) K .0408 K .0832 K .0417 4 -.00989 (.0482) (.0454) (.0439) (.3916) K .0405 (.0585) * Estimates are for the mean observations over the entire sample for each model. 152 SSE/BE < 0). While increases in employment density in— crease blue collar productivity, increases in the rate of decline of employment density decrease blue collar pro- ductivity. However, in general, labor productivity and labor's share of income increase as production is less concentrated around the city center. For labor and capital to be weakly separable from land, marginal differences in the employment density gradient should not affect the Allen partial elasticities of substitution, or the Hicks partial elasticities of complementarity, between pairs of inputs. Table 4.5 pre— sents the partial derivatives of estimated ABS with respect to D and g for cost function models A (occupa— tion), C (education) and D (age-education), and the partial derivatives of estimated HEC with respect to D and 5 for the production function model B (age). The hypothesis that labor and capital are weakly separable from land implies that the partial derivatives of the relevant elasticities in Table 4.5 would be small relative to the actual elasticity estimates of Table 4.3 In particular, since a general increase (decrease) in land productivity, induced by improved (eroded) loca- tional advantages to operating nearer the city center, causes an increase (decrease) in both D and a, weak separability should be examined with respect to joint changes in these density gradient parameters. Using this criterion, the hypothesis that labor and capital are Table 4.5 . Model A occupation 301. .___1 (3D ) Model B age 3c.. 11 (SB—’7 Model C education 30.. i ___1. (3D ) Model D age-educa- 30.. (41) 3D Model A occupation 30.. (5.11) 153 Changes in Substitution the Density Gradient? X m O m i/j KEEP." X2231“ i \ 3 waNH 7:!110'17 i \ j ?<:I:3t" P C .0306 -.2018 .1537 L M .1927 -.0295 Elasticities Given Changes in B -.00255 .0049 .0173 H -.0348 .000783 -.0196 L M -.00144 .00238 .00928 1 2 -.1071 .00114 .00790 P C .0729 -.532 .548 L M 2.702 -.454 .0134 .00602 -.0386 .00790 -.0901 3 7.340 -.451 41.962 B -.00797 .0302 -.Ol73 -.3254 -.1983 .0348 -.0536 -.0047 .0178 .0107 .0012 .0042 -.0017 .0105 .00145 .0179 -.0357 .00997 4 K .0153 .171 -.153 -.0345 .0154 .0696 -.0372 ~3.291 .131 -.0897 .00118 .0407 .0569 -.0193 -.0186 .0330 .1159 Table 4.5 Model C education 30.. (-—11) 3 5 Model B age-educa- tion 30.. (~11) 35.. * Estimates are for the mean observations over Estimated partial derivatives of (Ji- Models A, C, and Bient paramenters each model. . of Model B with respect to density gra D and ci D and 5. (continued) i/j L L 1.542 M H K '/j 1 1 -2.279 2 3 4 K 154 M .0829 .00541 2 -.3345 -.0497 H -.O953 .0598 -.587 3 -6.576 .6687 -69.893 .0330 .0684 -.2404 .058 .2137 -.1076 4.133 .1290 -.1438 .00013 .3100 -.0163 .0631 the entire sample for 155 weakly separable from land is not supported in any of the four models. There is evidence that human capital-physical capital substitution decreases as the employment density gradient becomes steeper: in model A the AES between white collar professional labor and physical capital decreases unambiguously, and in model C the AES between educated labor and physical capital decreases unambiguously. Other labor-capital substitution elasticities are not greatly altered by changes in the density gradient. With respect to labor-labor substitution, pro- fessional white collar labor and professional clerical labor are increasingly complementary as the employment density gradient becomes steeper. However, blue collar labor-white collar labor substitution is not much affected by changes in land productivity. Thus, it may be reason— able to include blue collar and white collar labor in a group weakly separable from the individual inputs capital and land. Decreased human capital-physical capital sub- stitution, and increased white collar professional labor- white collar clerical labor complementarity, in SMSAS with steeper density gradients may be a reflection of the characteristics of employment near the centers of these SMSAS. That is, firms'headquarters may be more pre- dominantly located in these areas, and the relations among white collar labor and physical capital in these nonpro- duction tasks may be more complementary. 156 Substitution between labor from any pair of groups disaggregated by age (model B) increases as density gradients become steeper. With respect to the education disaggregation (model C), substitution between any pair' of labor groups is not much affected; however; workers from groups with more similar schooling (OLM and OMH) are slightly more easily substitutable, but‘workers from the most dissimilar groups (OLH) are less substitutable, as the employment density gradient becomes steeper. These results do not support the hypothesis that labor market segmentation is increasing. Further, the implica- tion of this exercise is that the hypothesis that labor, as a group, is separable from capital and land probably cannot be accepted. This result is consistent with the finding of Chapter III that, in general, labor-capital separability is not accepted. Turn, now, to the second area of interest: labor and capital production relations as the manufacturing density gradient becomes flatter. A slightly different approach is taken. For a representative SMSA, given a fixed amount of manufacturing employment, an increase in employment density on the fringes of the SMSA and a corres- ponding decrease in employment density nearer the city 10 . . When model B (age) is estimated from a cost function specification, there is little evidence of appreciable change in labor-labor substitution or labor-capital sub- stitution as the density gradient changes: except that as the density gradient becomes steeper the corresponding Table in appendix E indicates young and middle aged workers become more complementary and older workers and capital be- come more substitutable. 157 center is simulated by imposing a flatter density gradient on the SMSA. The functional relation between density gradient parameters D and 5 described in equation (4.2) implies that given a fixed level of employment, N, as city center density D decreases, so must the rate of decline of density E decrease. In fact, from equation (4.14) g% = % 4(ND)’1/2 Assume the representative SMSA is the average SMSA. That is the amount of inputs used, input prices, value of out- put, and manufacturing employment density parameters are the means of the observations over all SMSAS. Further, assume the average SMSA is circular in shape, implying 6 = Zn. Figure 4.5 depicts changes in D and g as the density gradient becomes flatter. The natural logarithm of the exponential density function(4.1)is plotted as a function of distance in miles (k) from the city center. That is, from (4.1) in D(k) = in D - 5k For a fixed level of employment and a given change in D, 6 also changes according to equation (4.14). Both the factor shares of inputs and the substitution elasticities are altered by this joint change in D and g. The effect of this joint change on the representative SMSA is approximated by solving for the total derivatives of these variables. In particular: 158 Figure 4.5 Alternative Density Gradients of the Representative SMSA in D(k) (3) g k in D(k) = in D - 5k Gradient D E (1) 2587 .469 (2) 2328 .449 (3) 1294 .332 aSi 8S1 dSl ’ 5'5— dD + T (15, 36. 33 = __1_.l _1_l d013 3D dD + 35 d5, and _ 351 361 d 13 - 3D dD + 35 d5, i = l, ..,n, i # j where 8., 0 ., and 5.. are the estimated variables of i ij ij the representative SMSA. Table 4.6 presents the total derivatives of these variables, first for a 10% decrease in city center density D, and the corresponding decrease in g, and second for a 50% decrease in D, and the corresponding decrease in g. The fifty percent change is probably a closer approximation to what occurred between 1945 and 1970.11 The first conclusion from this exercise is that a small decrease in city center density has very little effect on the productivity of labor and capital, or on substitution between pairs of labor or capital inputs. However, large changes, or continuing gradual changes, llMills (l972a) estimates manufacturing employment density gradients, using U.S. Census data over the years 1920 through 1963, for the following six SMSAS: Baltimore, Denver, Milwaukee, Philadelphia, Rochester, and Toledo. Over the postwar years 1948-1963, the percent decline in D and E ranged from 63.6 percent to 37.5 percent and from 57.6 percent to 21.2 percent, respectively; the respective median declines in D and 5 were 51.1 percent and 32.8 percent. Over the entire span of years 1920-1963. the respective percent declines of D and E ranged from 67.7 to 24.0 and 66.3 to 18.8: the respective median declines were 55.5 percent and 47.1 percent. 160 whmco.l eves. ammo. cocoa.) vanes.) emmoo.- cease.) meao. amass.) mono.) x z 2 A “a as memos.) mmooo.u mmmco.u mesooo.- vomoo. assess.) mmsoooo. mmeo. ammo. 60H.) x x z a ma 06 Lamas.) memos.) mecca.) maaco. amass.) same.) sumo. mamooo. ammo. amass.) x m o a md ..66 mo.- u mt .amm.u r as .Q Ca wmmouowp wcH new mm>wum>wu¢p ammo» émzm womum>d ecu mo mofiufloflummam c0wusuflumnsw pcm wwnmgm nobocm mo memoc.l hemoo. mmoooo.l mmmooo.l H mp mnm00.) moaoo. moHoo. momooo.. a mo thoo.l ®OHOO.I memooo. mmmoo. .mp wo>wum>wuoo AZIX ca (42254 -H (140th -r-( usmcw fiance .o.v Acofiumoopmv U Hmpoz Ammo. m H060: Acoflummsouo. ¢ Home: 632. 161 mwmo.| mmmoo.l mvfloo. mOHOO. x wHNO.I cwmo.l Namoo. meo. mmmoo.( mNmoo. vmmo. vmmo. mom. cho.| mHHo. mOHoo.1 mmmo.l Hmvoc. mmmc. names.) come. cod. 2 2 ma ..oo Demo.) mOHo.I vnm.( mmvoo. vmm. m 0 ma ..op nma.) u p Q a“ mmmmuowp www.ml MOH. hmh.al moHoo.I ovwoo. N '7" "'4 pp FANG.) hmmoo. HOMO. 0N0.) hHNoo. w mp HmHo.( mamoo. Ohmoo. wove. memo. ~H mp .va.HI u no mom no“ mm>flum>flump Hooch mmmoo.l ommoo. ammooo. Ammooo.l mmno. momooo. H .mp “DIEM :v-I AZIX 0H HNMQ‘M '0'! .mmmv m H000: Acoflummpuuov d Home: 186.. (moppmummmv 6 Hope: AmeCwucoov.®.¢ wanna 162 came.) mwvo. wmma. HNN.) wbmoooo.) m th.I Nm@.m ammo. OMH.I meme.) Hobo. mmmO.) vaco.l mmb.vvn Nov. Hemoo.l hmm.wl cvvo. m N .3 pp boa. vmao.( nNHc.I omoo. vvao.u I 2 ma ..cp vac.I mvHo. movoo. mmaoo.l Hmv. mmHoo. 1:648 (moppmummmv 0 H0002 HNMVX 4 .mp omHo.I NHNO. Nmmoo.( omom.l NhNoo.I ACOwumozpmv 0 Hope: 422% .mp prscflucoov.w.v manna 163 in the distribution of manufacturing within an SMSA might be associated with changing production relations among labor and capital inputs. In general, the partial AES between labor endowed with the greatest amounts of human capital (both white collar professional labor and labor with schooling beyond high school) and physical capital are more positive as the density gradient becomes flatter. However, substitution between capital and the remaining labor inputs is not much affected. The estimates of white collar labor-capital com- plementarity (0ij < 0) from time-series studies which span the earlier years (1929-1968) are reconcilable with the positive white collar labor-capital substitution estimates (0ij > 0) of more recent time-series and cross section studies. There is little evidence to support the labor market segmentation hypothesis that, as the density gradi- ent becomes flatter, laborers from different groups become less easily substitutable for each other. Neither does the internalization of ancillary labor services appear to alter greatly labor substitution. With respect to the most releVant labor force disaggregation, by occupation (model A), estimated blue collar labor-white collar labor substitution remains virtually unaltered for both small and large changes in the employment density gradient. However, within the white collar labor group, white collar professional labor becomes increasingly complementary with white collar clerical 164 labor. With respect to labor disaggregated by education (model C), there is evidence that the elasticity of sub— stitution between labor from the more similar education groups (0LM and OMH) moves closer to zero. However, these changes are negligible when compared with the initial estimated elasticities of substitution between these inputs. Furthermore, the best evidence of labor mar- ket segmentation would be that in this model substitution between least educated labor and most educated labor (i.e., d0LH ought to be negative. In fact, the partial AES between these inputs increases as the density gradient be- comes flatter (i.e., is positive), indicating that 12 d0LH labor from these groups becomes more substitutable. To summarize, with land indirectly included into the production function as a resource which influences the productivity of labor and capital, hypotheses of weak separability among inputs and the influence of location on the production relations among the remaining labor and capital inputs were examined. Empirical evidence indicates that land and capital are substitutes in production, but in general, the relation between labor and land in production 12Total derivatives of the estimated AES of the cost func- tion specification of model B (age) for a 10% decrease in D and a 50% decrease in D are computed in Appendix A. As with the production function specification, estimated changes in factor substitution under this model, as the density gradient becomes flatter, are negligible. 165 of manufactured goods is neutral. These is a strong indi- cation that a hypothesis of weak separability of labor and capital from land cannot be accepted. However, weak separability of labor from capital and land is somewhat more easily acceptable. The effect of less concentration of manufacturing near the city center on substitution between labor and capital is such that human capital-physical capital substitution increases. In particular, white collar labor-physical capital substitution increases. However, labor-labor substitution is generally unaffected by this trend of decentralization of manufactures. The major exception being increased complementarity between white collar professional labor and white collar clerical labor. E. A Geographic Reformulation Consider the second alternate formulation of the underlying production process: accounting for possible geographic differences in production. In particular, the point of interest is whether the production process of the sunbelt SMSAS is the same as the production process used in the rest of the United States. Over the most recent decades the southern states have experienced rapid growth in nearly all industries. With respect to manu- facturing, growth in the number of establishments, employ- ment, capital accumulation, and value added are all above the national average}:3 This change in the importance of BOver the years 1954 through 1972 manufacturing in southern states has experienced growth in total number of establish- ments, total employment, annual expenditures on new capital, 166 the sunbelt States as manufacturing centers relative to other regions of the U.S. may be for a number of reasons. There are obvious climatic differences among regions. Different state and local governments provide different investment and tax incentives for the establishment of plants within their jurisdictions. Air conditioning has increased worker comfort in the South. The quality or availability of resources demanded by manufacturers may differ across regions. Certainly because of the greater growth of new plants in the sunbelt the age structure of capital may be younger than in other areas. New highway systems have facilitated locations of plants in the South Whatever the reason for the differences across regions, it is interesting to test whether the technical relations among labor and capital inputs are different in the sun- belt than they are in the rest of the U.S. This formulation does not concern itself with reasons for the growth of the sunbelt as a manufacturing region but rather with: 1) whether the underlying production process itself can be assumed to be homogeneous across the United States, and if it cannot, 2) how the technical relations among labor and capital differ. The manufacturing employment density gradients of SMSAS from southern and southwestern states tend to be (continued) and value added by manufacture of 31%, 47%, and 124%, respectively: while the U.S. average growth has been 28%, 17%, 98%, and 100%, respectively. 167 flatter than those of SMSAS from other regions. If the hypothesis that white collar labor-capital Substitution increases as manufacturing densities flatter is true, then these inputs Should be more substitutable in the southern regions. To the extent that racial prejudice may still be greater in the South, or the quality of education poorer, labor-labor substitution may be more difficult in the South, and the shares of income received by more educated or more highly skilled workers may be higher relative to their counterparts in other regions. For this second alternate formulation consider a multiple input production function or cost function where regional location (L) is also a determinant of production relations. In what follows a tranSIOg cost function is derived. The derivation of the appropriate translog production function is summarized in footnote 15. The generalized cost function now takes the follow- ing form: C = C(Q,L,P1,P2,...,Pn). In particular regional location, L, augments each of the input prices in the following Specific fashion: Pi = Pi(L) such that 8.L P.(L) = Re 1 (4.16) l l 168 That is, there exists a set of region Specific determinants of the equilibrium price of each input. If region has no influence on the price or marginal product of input i, then 1; equals zero. The cost function can be written: 8 L 8 L 8 L l 2 n C = C(Q,Ple , Pze ,...,Pne ), Where L is a binary variable taking the value 1 for an SMSA located in the sunbelt and 0 for any other SMSA. Therefore 8i represents the degree to which regional location augments the price of input i.l4 Inclusion of the binary variable L allows the hypothesis test that there is no regional difference in production technology (81 = 0, i = l,...,n). Second, possible regional differences in labor-labor and labor- capital substitution may also be tested. The translog cost function is then: n 81L in c = 2n 8 + in Q + Z 8 £n(P.e ) 0 . 1 i=1 n n 8iL 81L 4' 2 ) Oij £D «I a: «I . 53 :5 .nusu . ~53: coda-luau. I: 5 :30". 08:1. 13 I. c «It: I II no. noun-9r. can... of. an. - n»\: I 3.3» I 2» .przic I 7 z . 2 ~ .338 3:893! . u... . 9|» I ..II . .2]. I “In 1...: .32: an: 3.2:... I... III 31.: .8: III I I I I -29.. as; 38... III... on. «I ......~.. I a; fl~l I ml u «N an an an Ha lrlklll: p. pInIIJTIIIL» .I II? In a 2 ~ .I. ..HF . H”, . “d’vr . ' O “flh . ‘fl'ou. o .4” . mm QIOIwIH ' I‘ll” —.'C~ .AH..~ ON-‘U o:.-.-U dn‘Ud. M O .8~ . 0 .~ 003-5230 0..-annual 0.33.390 0.3.3qu. I 05. .03: 53.3.3300 uoos .3395. 932.93 2.....— upon muflawnmummmm xmoz MOM wcofluwccou ucwfioflwwsm Hm wanna 199 pI~I I nap I -p I «apwtn II .III I III I In» I ..p.~I n~ In» I I I.» 'llllllll ..n» I nap..~» nu» Illllllllllzr .«n» I -I.n.p Inupsn~II~I I I.I .napxnupnI I .I 8‘? I AN» 0 an? on”? O flu>onur .anpxnupcap I I.» .na»\n.»~I I «I In U I! . .nucu...nnu.nue~.~uu~..ucu.a.u .Iueg...I....I~..~I~.-I..a.a .nuaa...u:~.nuau.-..«uc~.—ucu.a_. .nucu..uc~.nuc~ ..«un~.—uc~.a.u Aconcflucoov .0598-Pa scan-luau. cc» 3 coda-9'0 .3030 0.3 a. n 0.8:. no. .3315. can I: £03.00 3.3-0.3! m 2.9: vs- 3.83 no... ace-Ina: oInInIu :32: 05!... nIIIinIu I «I. an...u ..I m w * I aucuom IQI~IOI~ l3»!!- !o‘uu is 9:“ Hm magma APPENDIX C SMSAS SAMPLED FOR TRANSLOG ESTIMATES UNDER ALTERNATIVE FORMULATIONS (/ indicates inclusion in the sample of observations) 200 201 Table Cl SMSAS Sampled mmflmwaI-J .b w w w w u w M N N N H I—- I—a I—l I—a ,_. o \l m U1 J:- u N ox U1 to N \o co \1 ow m N O O C O O O O O Q 41. 42. 44. 46. 48. SMSA Akron, OH Albany-Schenectady-Troy, N.Y. Albuquerque, NM Allentown-Bethlehem-Easton, PA-NJ Anaheim-Santa Ana-Garden Grove, CA1 Atlanta, GA Augusta, GA-SC Austin, TX Baton Rouge, LA Beaumont-Port Arthur-Orange, TX Boston, MA Bridgeport, CT Buffalo, NY Canton, OH Chattanooqa, TN-GA Chicago, IL2 Cleveland, OH Columbia, SC Denver-Boulder, CO Des Moines, IA Detroit, MI Duluth-Superior, MN-WI El Paso, TX Erie, PA Fort Wayne, IN Fort Worth, TX Fresno, CA Grand Rapids, MI Greenville, NC Hartford, CT Overall Translog Formulation \.<\ \.<\ \.<\ \.<\ \.<\ \.<\ \.<\ \.<\ \.<\ \.<\'\ > \'\'\\'\\"\ Model: B \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ <\ \.<\ \.<\ \.<\ \.<\ \.<\ \.<\ \.<\ \.<\ \.<\ x <\'\ 0 '\ \"\"\\\\\ <\ \.<\ \.<\ \.<\ \.<\ \.<\ \.<\ \.<\ \.<\ \.<\ D f \"\'\\'\\ 202 Sunbelt-Nonsunbelt Density Gradient Reformulation Reformulation Model: Model: 12 13 16 17 18 19 22 23 25 26 32 33 34 35 36 37 4O 41 42 44 46 48 51. 54. 55. 56. 57. 58. 59. 60. 63. 65. 66. 67. 68. 70. 73. 77. 78. 79. 80. 81. 85. 86. 87. 88. 90. 95. 99. 100. 101. 102. 103. 104. 105. 203 Huntington-Ashland, WV-KY-OH Jacksonville, FL Jersey City, NY-NJ Johnstown, PA Kansas City, MO-KS Knoxville, TN Lancaster, PA 3 Lansing, MI Lorain-Elyria, OH Louisville, KY-IN Madison, WI Memphis, TN-AR-MS Miami, FL Minneapolis-St. Paul, MN-WI New Haven, CT Newport News-Hampton, VA Norfolk-Portsmouth, VA-NC Oklahoma City, OK Omaha, NB Orlando, FL Philadelphia, PA Phoenix, AZ Pittsburgh, PA Portland, OR Reading, PA St. Louis, MO-IL Riverside-San Bernardino- Ontario, CA San Diego, CA San Francisco-Oakland, CA San Jose, CA Santa Barbara, CA Seattle-Everett, WA Shreveport, LA \"\"\3’ \\\\\\\\\\\\\\\\\\\\\\ \\'\"\'\"\\ \\\\'\\\\\\'\\\\'\\\\\\\\\\\\(I! \\'\\\'\\ \\\\\\\\\\\\\\\\\\\\\\\\\0 v \ \"\"\\