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Cross-Sectional Analysis Of Manufacturing Production Functions In A Partitioned 'Urban-Rural' Sample Space

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Cross-Sectional Analysis Of Manufacturing Production Functions In A Partitioned 'Urban-Rural' Sample Space

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Content 7(1-16,881 NXCKOLSON, Norman Dtright, 1933- CROSS—SECTIONAL ANALYSIS OF MANUFACTURINS PRODUCTION FUNCTIONS IN A PARTITIONED "URBAN-RURAL" SAMPLE SPACE. University of Southern California, Ph.D., 1970 Economics, general : University Microfilms, A XEROX Com pany, Ann Arbor, Michigan THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED CROSS-SECTIONAL ANALYSIS OF MANUFACTURING PRODUCTION FUNCTIONS IN A PARTITIONED "URBAN-RURAL" SAMPLE SPACE by Norman Dwight Nicholson A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Economics) January 1970 UNIVERSITY OF SO UTHERN CALIFORNIA TH E GRADUATE SCHOOL U N IV E R S ITY PARK LOS ANGELES, C A L IFO R N IA 9 0 0 0 7 This dissertation, written by ... under the direction of h%S.... Dissertation Com mittee, and approved by a ll its members, has been presented to and accepted by The Gradu ate School, in partial fulfillm ent of require ments of the degree of D O C T O R O F P H IL O S O P H Y ’’ .'T V Dean D f l * £ L . . J i m n a r x . l 9 . 7 . Q . DISSE^ ^cJTC^CO^ITTEi: ^ ACKNOWLEDGEMENT I wish to express to the members of my dissertation committee— Dr. Gerhard Tintner, Chairman, Dr. John Niedercorn, and Dr. Ira Robinson— my thanks for their patient guidance and helpful comments during the preparation of this dissertation. I owe a special debt of gratitude to Dr. Niedercorn for his suggestions as to suitable research topics. The aforementioned gentlemen are not, of course, to be held responsible for any remaining errors or inadequacies of analysis in this study. I also wish to acknowledge a grant from the University of Southern California for the use of computer facilities. Finally, my greatest appreciation goes to my wife who made this endeavor worthwhile. TABLE OF CONTENTS Page ACKNOWLEDGEMENT.................................... ii LIST OF TABLES.................................... V Chapter I. INTRODUCTION .......................... 1 Chapter by Chapter Development II. SPATIAL LOCATION AND MARKET STRUCTURE . 7 Historical Overview A Short History of Location Theory in Economics Theoretical Approaches to Locational Analysis Classificatory Schemata for Partial Analysis Identification of Agglomerative Factors Land, Labor, and Capital as Factors of Production in Terms of Location Analysis III. THEORETICAL CONSIDERATIONS IN THE ANALYSIS OF PRODUCTION FUNCTIONS WITH RESPECT TO LOCATION OF THE F I R M ..................... 25 Mathematical Development of the Analysis iii Chapter Page A Consideration of the Elasticity of Substitution Theoretical Considerations in Statistical Estimation with Respect to the Analysis of Market Structure The Statistical Estimation Procedure Analysis of the Production Function with Respect to Markets A Digression on the Technological Efficiency of the Factors of Production Measurement of External Economies IV. STATISTICAL FINDINGS .................... 60 Methodology of the Study Industry-by-Industry Analysis Comments Regarding the Industry-by- Industry Analysis V. SUMMARY AND CONCLUSIONS....................... 92 Application of the Analysis to Problems of Regional Economic Development Criticism and Extensions of the Analysis APPENDICES............................................ 102 A. Elasticity of Substitution Estimates . 103 B. Miscellaneous Tables .................... 107 BIBLIOGRAPHY ........................................ 113 iv LIST OF TABLES Table I. Least Squares Estimates of the Cobb-Douglas Production Function (Complete Sample) . II. Least Squares Estimates of the Cobb-Douglas Production Function (Urban Sample) III. Least Squares Estimates of the Cobb-Douglas Production Function (Rural Sample) IV. Restricte4 Least Squares Estimates of the Cobb-Douglas Production Function (Complete Sample) ....................... V. Restricted Least Squares Estimates of the Cobb-Douglas Production Function (Urban Sample) .......................... VI. Restricted Least Squares Estimates of the Cobb-Douglas Production Function (Rural Sample) .......................... VII. Least Squares Estimates of the Cobb-Douglas Production Function with a Dummy Variable to Represent Urbanization Added o c VIII. Capital Intensity Coefficient (3 - ) for Unrestricted Least Squares Estimators pital Intensity Coefficient (J^-) for Restricted Least Squares Estimators v Page 68 69 70 71 72 73 76 79 80 CHAPTER I INTRODUCTION It is a tenet of regional economic theory that industries which are similar in terms of product produced, may differ in terms of production efficiency, scale of operation, and technology employed, depending upon the nature of the market structure in which the industry operates. Yet, very few empirical studies have attempted to estimate differences in production functions for in dustries which operate in more than one market.-*- The usual approach is to estimate an industry-wide production function, which assumes a uniform market for all firms within the industry. For example, two recent studies are concerned with locational aspects of production function analysis; both studies develop elasticity parameters which are con stant for all industries. See: Fred W. Bell, "The Role of Capital-Labor Substitution in the Economic Adjustment of an Industry Across Regions," Southern Economic Journal, XXXI (1964), 123-131; Shin Kyung Kim, "Interregional Dif ferences in Neutral Efficiency for Manufacturing Indus tries: An Empirical Study," Journal of Regional Science CSummer, 1968), pp. 19-27. 1 2 This paper attempts to verify the existence of differences in production functions for firms operating in different markets. Essentially, the technique involves the development of a dichotomized sample space represent ing both "urban" and "rural" markets. Cross-sectional production function estimates are then compared for dif ferences in the estimated parameters. Information regarding external economies, tech nology of the firm, and characteristics of the market structure, in which the industry operates, can be extracted from sample parameter estimates of production functions. All that is needed to accomplish this task is a funda mental understanding of the economic theory of production and distribution and an understanding of the economic effects of distance. Based on the necessary economic theory, a general methodology is developed which utilizes least squares estimates of unrestricted and restricted Cobb-Douglas production functions. With the aid of normal statistical tests, and a comparative analysis of the residuals of the two forms of production function for each market, i.e., "urhan" and "rural," it was found possible to state: (1) the relative factor intensities, in terms of capital or labor, for the average manufacturing industry of the two-digit level of Standard Industrial Classification in each market; (2) in which market the firm enjoys the greatest relative operating efficiency; and (3) whether the industry is operating at its optimum, i.e., least cost point, and, if not, whether there were possible market inelasticities or potential restricted entry of additional firms into the market. Chapter by Chapter Development Chapter II contains a brief development of the locational aspects of economic theory. A short historical overview of the major developments introduced by the major theoreticians in the field, from von Thiinen to present day writers, is provided. The second part of the chapter offers a theoretical development of certain aspects of location theory. This material is presented primarily to differentiate locational factors which affect the shape, or slope, of the firm's cost curve from those which affect the level of the cost curve. Chapter III concerns itself with the theoretical material needed to justify the interpretation of the sample parameter estimates given in Chapter IV. Here again, a short history of economic theory is offered, this time concerning the development of production and distribution theory. The development is then carried out mathematically by considering the production function in terms of its relation to distribution theory and as a technological relationship. Also discussed are the para metric indicators of neutral and non-neutral technologi cal change. The discussion then turns to a development of the elasticity of substitution parameter and its rele vance to the study. In addition, formal theoretical developments for the Constant Elasticity of Substitution and the strictly linear production function are offered. The foregoing material was included to justify the use of a Cobb-Douglas type production function in the empiri cal part of the study. This choice was based on an analysis of elasticity of substitution by the ACMS method.^ Section II of Chapter III details the theoretical considerations involved in choice of statistical methods 2 Kenneth J. Arrow, H. B. Chenery, B. S. Minhas, and R. M. Solow, "Capital Labor Substitution and Economic Efficiency," Review of Economics and Statistics, August, 1961, pp. 225-250. used for the study. A great deal of attention is given to a rigorous theoretical justification of the statistical methodology. Special consideration is also given to the analysis and interpretation of possible inelasticities in the underlying market structure in which the firm operates. In addition, the meaning of technological efficiency is also discussed. Chapter IV is also divided into two sections. The first section discusses the data used in making the statistical estimates and the technical details regarding the construction of the variables from the raw data. This section also presents and explains the tables which make up the empirical portion of the study. The second section gives a detailed analysis of the estimated parameters for each industry in the study. Chapter V presents the major conclusions of the study with respect to observed differences in production functions between the "urban" and "rural" markets. Also, the general validity of the theoretical approach is examined in light of the statistical evidence. Based on the results of the study, some possible criticisms and extensions of the analysis are also presented. 6 Finally, two appendices are included. The first gives a discussion of, and tables for, the elasticity of substitution estimates by the ACMS method. The second offers some miscellaneous tables relating to the technical aspects of the study. CHAPTER XI SPATIAL LOCATION AND MARKET STRUCTURE This chapter provides some prefatory material on industrial location theory and market structure. Its primary purpose is to provide a spatial context in which to view the market structure for factors of input and out put, as described in Chapter III. Historical Overview Although economists have been working for some time to develop a general equilibrium theory of industrial location, no such development seems forthcoming in the near future. Koopmans and Beckmann have reported very pessimistic results regarding a problem of assigning indus trial plants to particular sites.'*' Basically, the plants Tjalling Koopmans, and M. Beckmann, "Assignment Problems and the Location of Economic Activities," Econo- metrica, April, 1957, pp. 53-76. 7 were considered to be competing for a fixed number of sites while minimizing transportation costs for goods flows between plants. It was found that no general solu tion exists for this problem since the movement of one plant to an optimal site made the location of some or all of the plants dependent upon the first plant less than optimal. Perhaps the foregoing illustration explains why modern location analysis, including Beckmann's recent 2 book, has retained a distinctly Marshallian flavor. A Short History of Location Theory in Economics To the German economist, J. H. Von Thunen, must go the credit for the first explicit consideration of the location of economic activities. Von Thunen considered the problem of optimal development of various categories of agricultural productivity with respect to a central town. This town was described as being located on a 2 Martin Beckmann, Location Theory (New York; Random House, Inc., 1968). Johann Heinrich von Thunen, Per Isolierte Stadt in Beziehung auf Nationalokonomie und Landwirtshaft (Stuttgart: Gustav Fischer, 1966). featureless plain, having soil of uniform fertility, and isolated by forest from other towns. All manufacturing was carried out in the central town, which depended on the countryside for raw materials in the form of agricul tural products. A road network of equal efficiency extended from the central town in all directions. Trans portation costs were considered to be proportional to the feed consumed by horses in making the round trip from the place of production in the countryside to the city and back. For a given amount of agricultural product, a certain net revenue was received at the city center. From this revenue the cost of transport had to be deducted. Agricultural producers would bid for the use of a given landsite up to the amount of net revenue available after transport costs were deducted. Since the average revenue curves were related to distance as a function of the energy used in transporting particular commodities, the revenue curves for the relatively heavy or bulky commodities were inelastic with respect to price when compared to the reve nue curves for the lighter, for a given volume, products. Von Thunen's analysis still forms the basis for present day locational theory. The two central aspects of the theory are (1) the costs of overcoming space 10 introduce inelasticities into the producers revenue curves, and (2) the value of a particular parcel of land is related to the revenue curves of thost producers that bid for the use of the parcel. Thus, even though there is an element of supply inelasticity embodied in a parcel of land— in that the parcel has a spatial monopoly— the demand schedule for the parcel remains crucial in deter mining the particular use, and the value in use for a particular parcel of land. Alfred Weber analyzed the location of industry in an analogous manner to von Thunen's work.^ Whereas von Thunen viewed demand to be centrally located and supply to be spatially distributed, Weber considered production to be centrally located and demand for manufactures to be spatially distributed. Another contribution of Weber's was to label the composite bundle of forces that caused several producers to congregate in close proximity to each other as "agglomerative factors." To quote Weber: An agglomerative factor, for purposes of our discussion, is an "advantage" or a cheapening of ^Alfred Weber, On the Location of Industries, English translation of Uber den Standort der Industrie (1909) (Chicago: University of Chicago Press, 1929). 11 production or marketing which results from the fact that production is carried on to some con siderable extent at one place, while a degglomer- ative factor is a cheapening of production which results from the decentralization of production (production in more than one place).5 Weber's specific analysis of "agglomerative factors" was carried out in terms of "labor orientation," "transport orientation," et cetera. Modem location analysis is confronted with the same problem; the shift in emphasis lies in the attempt to quantify the "agglomerative factors." After von Thunen and Weber came the "central place" theorists. Christaller and Losch well represent this group. Essentially, their contention is that, given certain basic assumptions, the location of produc tive activities will assume a certain hierarchical order, thus forming a system of production and distribution centers, with the larger centers serving several smaller ^Ibid., p. 126. ^Walter Christaller, Central Places in Southern Germany, trans. by C. W. Baskin (Englewood Cliffs, N.J.: Prentice-Hall, 1966) ; August L'dsch, The Economics of Loca tion, English translation of bie Raumliche Ordnung der Wirtshaft (2nd ed., 1944? New Haven: Yale University Press, 1953). 12 centers more or less in the form of a lattice. Such systems are formed, usually, on the basis of abstract economic reasoning, since structural disturbances— due to historical accident, politics, climate, topography, national boundaries, et cetera— make empirical verifica tion difficult. Present day research into location has tended to be eclectic, with an occasional attempt at general equilibrium analysis.7 But even the attempts at general equilibrium analysis are usually partial in that they assign locations to industrial activities without con sidering final consumption, or other economic activities such as residential or commercial location. Theoretical Approaches to Locational Analysis The theoretical basis for modern locational anal ysis depends upon the division of economic policy planning, whether public or private, into two categories— prescrip 7 For examples of the eclectic approach, see Walter Isard et_al., Methods of Regional Analysis (Cambridge: The MIT Press, 1960). The general equilibrium approach is well represented by Louis Lefeber, Allocation in Space (Amsterdam: North Holland Publishing Co., 1958). 13 tive and adaptive. A prescriptive solution depends upon the ability of an economic planner to order an economic activity (industrial plant) to a particular location, based upon some rational process, while adaptive planning would operate only after the price system had dictated the location of economic activities. Regarding locational models in general, as pre viously stated, there exists no really satisfactory general equilibrium solution to the problem; although attempts have been made at developing such a model. Lefeber has offered a model which he feels would lead to O a general solution. In his model, plant location is influenced by charging a location rent— which is based on minimizing transportation costs for input factors and final goods— such that, in pure competition, the firms net profit after the location rent, or tax, is applied, is zero. Evidently, the location rent forces firms to move to the optimum location point. Lefeber feels that given market prices, net of transportation costs for input factors and final products, a general equilibrium solution exists which can be found by minimizing trans- 8Ibid. 14 portation costs. However, other researchers feel it is too gross a violation of reality to disregard the effect of external economies which can accrue to a firm when it locates in proximity to a supplier of intermediate goods. The external economy, in this case, takes the form of lowered transportation costs for the intermediate goods. Koopmans holds to this point of view: Preliminary exploration of models in the loca- tion of indivisible plants throws serious doubt on the possibility of sustaining an efficient locational distribution of economic activities through a price system. The decisive difficulty is that transportation of intermediate commodities from one point to another makes the relative advantage of a given location dependent on the location of other plants. This dependence of one man's decision criteria on other men's decisions appears to leave no room for efficient price guided allocation.9 Based upon Koopmans' work, other researchers such as Manne, have incorporated the effect of external economies into allocational models; the model offered by Manne, however, is in no sense a general equilibrium solution 9 Tjailing C. Koopmans, Three Essays m the State of Economic Science (New York: McGraw-Hill, 1957), p. 154. 15 to the problem."^ Most locational analysis is conducted in the context of partial equilibrium analysis. The object of the analysis is to consider the locational decision of the firm or industry as part of the profit maximization problem. Detailed attention is given to the market structure in which the firm or industry conducts its operations. The essential program for partial analysis involves finding answers to many empirical questions regarding the firm or industry with respect to its actual location. By studying several firms within an industry, the method of comparative statics can be utilized to identify optimal requirements in the location of a par ticular firm. Classificatory Schemata for Partial Analysis According to the relevant cost considerations, producers and consumers may either cluster together or form various patterns of spatial dispersion. A useful Allan S. Manne, "Plant Location Under Economies of Scale— Decentralization and Computation," Management Science, II, Series A (November, 1964), 213-235. 16 classification system to describe this phenomenon is given by Beckmann, as follows: 1. Producers and consumers are concentrated at point locations and in point markets. la Single point 2a Multiple point 2. Producers are concentrated in points, and consumers are extended throughout an area (market area). la Single point 2b Multiple point 3. Consumers are concentrated, and producers are dispersed (supply area). 3a Single point (central city) 3b Multiple points 4. Both producers and consumers distributed through an area (continuously extended market) .11 It is an empirical question as to which of the situations listed above describe a given industrial location pattern. Any location theory which seeks to explain such a situa tion will have to consider the underlying cost factors, the technology of the industry, and the structure of the market in which the industry operates. (Consideration of these problems forms the subject matter of Chapter III.) ■^Beckmann, Location Theory, p. 26. 17 Identification of Agglomerative Factors Agglomerative factors, as previously stated are the sum total of forces which cause industries to cluster together. It is more useful to define agglomerative factors in terms of economic theory. A useful classifi cation scheme is to consider agglomerative factors as consisting of two types of economies: economies which are external to the firm, and economies which are internal to the firm. External economies. Jacob Viner has classified external economies into two categories: technological and pecuniary.^ Tibor Scitovsky has further elucidated these terms.In Scitovsky's terms, external economies are of two types: (1) those influencing producers through nonmarket interdependence, and (2) those influencing producers through the market. 12 Jacob Viner, "Cost Curves and Supply Curves," Zeitschrift fur Nationalokonomie, III (1931), reprinted in AEA Readings in Price Theory: Volume VI (Chicago: Richard D. Irwin, Inc., 1952), 198-241. ■^Tibor Scitovsky, "Two Concepts of External Economies," Journal of Political Economy, April, 1954, 143-151. 18 External economies of the nonmarket inter dependence type are often referred to as "neighborhood effects" or "spillover effects." Typical examples of such effects with respect to firms are product quality improvements by suppliers of intermediate goods when the improvements are passed on at no increase in cost, and improvements in labor force education, due to public school programs, et cetera. Monetized external economies are most often of two types: (1) external economies realized by reductions in transport costs for intermediate goods, and (2) reduc tions in cost for intermediate goods due to economies of scale realized by intermediate goods producers which are passed on to their customers. Before going on to discuss internal economies to the firm, several important points regarding external economies should be noted. First, locational clustering by firms using large amounts of intermediate goods in their production processes will alter the level of the firm's marginal and average cost curves, but it will not neces sarily change their shape. This can be illustrated mathematically by considering a firm operating in a purely competitive market. The cost function for the i'th good includes a constant term for fixed costs per unit (a^), a variable cost function with respect to quantity [fi(Qi)]f and a transportation cost term proportional to distance with respect to quantity { ^ 3 • Revenue is simply unit price ( P^) times unit quantity (Q^). The solution is straightforward for the profit maximizing Since price is determined by the market, profits are and this term is proportional to the distance from plant to market. The transportation cost term is independent firm: 14 Revenue = P • Q • Cost = a. + f . (Q.) +^.Q.; 1 X I I X a . x directly affected by the constant term (per unit) /S It is just as easily seen that change in the cost 14 James M. Henderson and Richard E. Quandt, Microeconomic Theory (New York: McGraw-Hill Book Co., 1958), pp. 101-102. 20 of intermediate goods, to the firm in question, will affect the variable term fj(Q^), or marginal cost func tion. Therefore, effects due to monetary external econ omies can be classified as to whether they affect the level or the shape of the cost curve. Technological external economies, i.e., nonmonetary economies, are not as easily observed, since their effects are on the effi ciency parameters of the production function, as are the effects of technological internal economies. Internal economies. Internal economies to the firm are normally considered in relation to the quantity of product produced. However, the firm could well adopt a more efficient technology which, in spite of the costs of adopting the technology, would lower the marginal and average costs per unit even with no change in the quantity of product produced. Considering only the efficiency aspect of production, the firm in producing its product has the option of varying (1) the material factors used in production, (2) the capital used, and/or (3) the labor used. As the next chapter will show, the equilibrium theory of production, given no inelasticities in supply or product markets, will insure that the long run operat ing point is at minimum cost. 21 Land, Labor, and Capital as Factors of Production in Terms of Location Analysis Land. Aside from the von Thunen analysis men tioned in the first section, this paper does not treat location rent for the use of a parcel of land; however, there is a justification for not doing so when the analy sis covers an extended market area. Colin Clark has offered a distinction between micro- and macro-location theory which differs in concept from the economist's 15 normal definition of the terms. He defines macro location as the analysis of the location of industry and population between regions and groupings of industrial towns. Micro-location is concerned with the specific location of an economics activity at a particular site within a particular area. The micro-location problem can be thought of as including the bid-rent function. For purposes of ease in explanation, the bid-rent func tion drops out of the macro-location problem. Since von Thunen's analysis assumes a zero rent at the periphery of the central plain servicing the town, another way of looking at the problem would be to think of macro-location -*-^Colin Clark, Population Growth and Land Use (New York: St. Martins Press, 1967), p. 289. analysis as locating an industry at the edge of town, while micro-location theory locates the industry in the town proper. Labor. In nonlocational applications of economic theory, labor supply to the firm has been traditionally treated in terms of supply and demand analysis. More recently, the view of labor as human capital has prevailed In this view the laborer compares the discounted value of alternative income streams available to him and then chooses his occupation accordingly. There are, of course, subjective elements in the discounting of an individual's income stream, such as the "psychic" returns of the job. Two recent studies, one by Lowry and the other by Sjaastad, have given some additional insights into the labor force in terms of migration.- * - 6 The common areas of agreement between the studies can be summarized as follows 1. Labor force migration is a function of employment opportunities being available at a different location, as opposed to migration because of a lack of employment at the present location. I®Ira Lowry, Migration and Metropolitan Growth; Two Analytical Models (San Francisco: Chandler Publishing Co., 1966); Larry A. Sjaastad, "The Costs and Returns of Human Migration," Journal of Political Economy, special supplement, October 1962, pp. 80-93. 2. The young and the educated are more mobile than are older and less educated. This is true for obvious reasons, e.g., older people have a greater investment at their present location than do the young; the costs of migration include the losses incurred in liquidating the present investment in housing, job seniority rights, et cetera; while for the less educated there are fewer higher paying jobs available at alternative locations. 3. Labor force migration from one location to another is an inverse function of the distance involved. The foregoing factors will affect the supply curves of labor for firms in various ways depending on the nature of the firm's labor requirements and the firm's location. All that may be said in general is that the elasticity of labor supply will intrinsically vary for different industries and different locations. The exact nature of this variation can only be determined empiri cally. Capital. Investment theory requires that the profit maximizing entrepreneur equate the expected margi nal internal rate of return from a given investment to the market rate of return on alternative income streams, 24 usually a "safe” bond. From the lender's point of view the expected rate of return on loaned capital is to be maximized, subject to a risk or uncertainty premium deter mined from knowledge, or lack of knowledge, regarding the probability distribution underlying the prospective investment. When locational considerations are introduced, it would be normal to expect the risk or uncertainty premium to increase; thus, the cost of capital at a new location may well be greater than the present market rate. Additionally, if the firm in question is considering a move from an established site to a new location, the costs of disposing of the fixed investment at the present loca tion must be included in the decision making process. With capital then, as well as with labor, the elasticity of supply will vary from place to place and from firm to firm. CHAPTER III THEORETICAL CONSIDERATIONS IN THE ANALYSIS OF PRODUCTION FUNCTIONS WITH RESPECT TO LOCATION OF THE FIRM The preceding chapter sought to delineate the varied aspects of the general location problem for a firm. In this chapter the scope of the problem is narrowed to a consideration of the information that can be obtained from an analysis of production functions regarding loca tional optimality with respect to the market structure in which the firm operates. The production function specifies a technical relationship regarding the transformation of the factors of input into output.^ In this view, engineers maximize output for a given set of inputs through their choice of technology. While the engineer is concerned with the particular technological processes by which a firm produces 1 Lawrence R. Klein, An Introduction to Econo metrics (Englewood Cliffs, N.J.; Prentice-Hall, Inc., 1962), pp. 83ff. 25 26 its output, the economist is often concerned with the production function at a higher level of aggregation. This is because of the economist's concern with the market structure in which the firm conducts its business. In fact, as Stigler makes apparent in his book, Production and Distribution Theories, the form of production func tions utilized by economists has been largely determined by the requirement that each of the factors of production be paid in proportion to their marginal productivity. If a firm pays each factor of input the value of that factor's marginal physical product, then it can be shown by Euler's therom that total output is just exhausted when the production function is homogeneous of degree one. Wicksell first put forth a production function of the form P = A01 B^9 (where OC + ^3 = 1) as a suitable theoretical production function which met the Euler 3 conditions. Later, this form of production function became popularized as the Cobb-Douglas production function 2 George J. Stigler, Production and Distribution Theories (New York: The Macmillan Co., 1941), pp. 320ff. 3Ibid., p. 377. 27 through the empirical work of Professor Paul H. Douglas.^ Professor Douglas, however, came to use this particular function at the suggestion of his colleague C. W. Cobb,^ who chose the function because it satisfied the Euler Theorem. Thus, on the one hand the production function specifies a technical relationship regarding the trans formation of factor inputs into output, while on the other hand, the production function, as used by economists, takes into account certain restrictive requirements regarding the value of the product which the factor shares receive for their contribution to marginal productivity. Apparently, then, economic as well as technological con siderations are involved in the production functions normally used by economists. In fact, the assumption of a homogeneous production function is not necessary to fulfill the postulates of 4Gerhard Tintner, Econometrics (New York: John Wiley and Sons, Inc., Science Editions, 1965), pp. 51ff. 5 Paul H. Douglas, "Comments on the Cobb-Douglas Production Function," The Theory and Empirical Analysis of Production, Murray Brown, ed. (National Bureau of Economic Research; New York: Columbia University Press, 1967) , p. 16. 28 marginal productivity theory. The conditions can be fulfilled given an underlying market structure in which the firm maximizes profit and the finn's maximum long run profit is zero. These are characteristics of the purely competitive model where there is free exit and entry of competing firms. Moreover, it can be shown that the assumption of an invariable linearly homogeneous produc tion function, without consideration of the underlying / r market mechanism, leads to inconsistent results. Interesting economic interpretations can be made, however, about the underlying market structure by assuming a model of a profit maximizing firm which is constrained by a Cobb-Douglas production function, wherein the param eters of the function are allowed to vary, i .e., (X+ yS j| 1. The siam of the exponents indicates the degree of "returns to scale," where OC+/^ <1 indicates decreasing returns to scale; 0(+ = 1 indicates constant returns to scale, and OC + > 1 indicates increasing returns to scale. Since the price of the factors of input and the price of 6 James M. Henderson and Richard E. Quandt, Micro- economic Theory (New York: McGraw-Hill Book Co., 1958), pp. 54-67, and chap. iv. output are taken into account by the profit maximizing firm, differences in the observed returns to scale for the firm imply differing market conditions. The fore going analysis can perhaps be better illustrated with the aid of mathematics. Mathematical Development of the Analysis The following mathematical development assumes the firm's production function to be of the form Q = AK0' ’ , where Q = output, A = an efficiency constant, K = capital, N = labor, OC = elasticity of output with respect to capital, and yS = elasticity of output with respect to labor. The Euler conditions. Marginal productivity theory has been developed to a large extent from a con sideration of the Euler Theorem. Given a homogeneous function, f (x,y), of degree k, then xfx (x,y) + yfy(x,y) kf(x,y), To prove this the homogeneous function f(tx,ty) = t^f(x,y) is partially differentiated with respect to t this results in xftx(tx,ty) + yfty(tx,ty) = kt^-^f(x,y). The desired result follows directly, by setting t = 1. Properties of the Cobb-Douglas production func- 30 tion. The Cobb-Douglas function is homogeneous, i.e., AftK)^ (tN)-^ = (AK^N^) = t0*- ^ Q. When CC + JS = 1, the function is homogeneous of degree one, as speci fied by Cobb-Douglas. Thus, the function obeys the Euler conditions. It can also be shown that the average and marginal products of the factors are homogeneous in degree zero; therefore, equal increments of the factors yield constant returns to scale: Let £ = K* also ^ = K* N tN The average and marginal physical products of capital and labor are, APPn = $ = A (K*)^ ,oc—/ APPk = 9. E = = A (K*)( * N K K* MPPk = AOCk ^"1 N'^“° = AOC(K*)0C‘ '/ MPPn = A (1-0C)N = A(1-0C ) (K*^ r* Given Euler's Theorem it can be shown that, Q = ^ K + S Q n = KACXCK*^"' + NA(l-dC) (K*)00 9K <9N = NACK*)** K0C + 1-dC NA NK* {K*f^ [oc + l-oc] = NA(K*)P^ 31 Therefore, the relative shares of total product accruing to labor and capital are K(^Q/^K) = K c f \ (K*)0^"1 = QC Q NA (K*)^ N(g>Q/3N) = NA(l-0( ) (K*)* = 1-a- Q NA(K*p: Factor rewards, in terms of purely competitive markets for inputs and output for a profit maximizing firm, are found by solving a constrained maximization problem. The additional variables needed are 7C- = profit, Pg = price of output, Pn = price of labor, Pk = price of capital. Profit is to be maximized subject to the side condition of a Cobb-Douglas production function. The solution is as follows: Maximize -7c. = PgQ - PkK - PnN subject to Q = AK^N &. (Since pure competition is assumed in the factor and product markets, all prices are taken as given con stants .) The augmented profit function is 7C* = PgQ - P^K = PnN — /\ (Q — AK^N^) . The first order conditions for a maximum are i f - p - A = 0 32 = -p +/\<x _ 9K ^k K S7C* _ „ . IJK^ll^ = -p ) = 0 *1 vr o»N 1 1 N d%* £>A = -Q + AKa N^ = The foregoing equations imply pq° V The second order conditions for a maximum imply the following results. OC(cC-l) P ^ < 0, ^ 6 h )-£ < „ and K N 2 K N K2N2 For the parameters <2C and the second order conditions imply thatoC 1 o r ^ > 1 is inconsistent with maximum profits. Also (X + <, 1 is implied. If OC + ^ ^ 1 then the assumptions of perfect competition and profit maximization are inconsistent.7 (In this case 1 may be 7Marc Nerlove, Estimation and Identification of 33 taken as a limiting value.) It should be repeated that the equilibrium con ditions outlined above are long run equilibrium conditions for the firm and not the industry. They are long run conditions for the firm, since prices are predetermined for output and input in the short run. Also, the quan tity of product produced is determined by a technological relationship, i.e., the production function, which is fixed in the short run. In terms of the theory of pure competition, the firm maximizes profit by equating two constants— an impossible task. 7C = pq - CQ dTL d 2- f C dQ = P - C = 0, dQ2 < ° Three different results are possible: (1) P = C, (2) P >C, or (3) P<C.® In the short run, then, the price of output equals the cost of output only by chance. Equilib rium in the market occurs through the entry of additional Cobb-Douglas Production Functions (Chicago: Rand-McNally and Co., 1965), chap. i. O Henderson and Quandt, Microeconomic Theory, pp. 62-67. 34 firms in the market, in the case of P >C. When P< C firms either leave the market or become more efficient by changing their method of production. There is no real dichotomy between the Euler con ditions and the possibility of a profit residual (negative or positive) accruing to the firm in the short run. The Euler conditions for linearly homogeneous production functions hold only if each input factor is paid the value of its marginal product, which, of course, requires under lying market equilibrium.^ Returns to scale. As previously stated, the Cobb- Douglas production function can indicate other than unitary returns to scale. For a percentage increase in labor and capital inputs, output will charge as follows: Q = AK^N^, r = percentage Q (1 + r)a+/S = A [K (1 + r)]^ [N(l + r) Y* CK + equals the degree of homogeneity of the production g Joan Robinson, "Euler's Theorem and the Problem of Distribution," Economic Journal, September 1934, pp. 398-414. 35 function. If the production function is a truly inde pendent technical relationship, then information regarding returns to scale in the production process may be used as an aid in making inferences regarding the underlying market structure. . Measuring efficiency with the Cobb-Douglas func tion. Efficiency is a relative criterion, at least in so far as it is measured through the use of the Cobb-Douglas function. Since the production function is strictly a technological relationship, only technological efficiency can be measured. Two types of technological efficiency can be distinguished, "neutral," and "non-neutral." Neutral technological efficiency is represented by the constant parameter (A) of the Cobb-Douglas func tion. Since^. = kP^n^, a change in (A) does not affect wA the factor inputs or their elasticities. A change in (A) can only alter the level of output; therefore, a change in the (A) parameter is said to represent a neutral technological change. Non-neutral technological change can be measured by shifts in the elasticity parameters ( OC and $ ). As previously stated, for purely competitive markets factor 36 rewards are proportioned by the relationships OC = p Q ^ PnH Oc V = ---- . Considerina the ratio = , if OC PqQ PnN increases, factor prices remaining constant, then pro portionality is maintained by a decrease in the use of labor (N). Relative to labor, the use of capital (K) has increased; therefore, the ratio of the two elasticities is said to determine the capital intensity of the produc tion process.10 A Consideration of the Elasticity of Substitution Since the production function is a purely tech nical relationship, it is desirable to have some measure to indicate the effect of a price change in the factors of production upon the proportions of the input factors used in producing the product. Such a measure can be thought of as indicating the ease with which capital can be substituted for labor. The elasticity of substitution was first defined by Hicks. His measure relates a Murray Brown, On the Theory and Measurement of Technical Change (Cambridge, Eng.: Cambridge University Press, 1966), pp; 38-41. 37 proportional change in the relative factor inputs to a proportional change in the marginal rate of substitution between labor and capital. (This measure allows substitu tion of a proportional change in the relative factor price ratio in the denominator, as shown below.) Mathematical development of the elasticity of substitution. The elasticity of substitution is, by definition: d(N/K) ^ = d(Qk/Qn) WhSre Qk = 9K ' and Qk/Qn The index can be extended to factor prices by a consideration of minimum cost optimization for the firm operating in a purely competitive market. Production costs are to be minimized as follows: Min C = KPk + NPn subject to Q(K,N) = Q0. By utilizing Lagrangian func tions, the solution is found to be: Min Z = KPk + NPn + A. [Q0 " Q (K,N) ] 0K = Pk " ^ Qk 0 9Z _ 3n “ pn " ?l Qn 0 38 Q0 " QCK,N) = 0 and Pk Qk Given the equilibrium solution that = Qk f factor prices may be substituted directly into the elasticity of substitution formula. this study. Since the production function expresses a technical relationship, it is an important empirical question as to whether factor substitution is possible, given a shift in factor prices. For example, a strictly linear production function exhibits many characteristics which are similar to the Cobb-Douglas function. Given a linear function: The function is linearly homogeneous, i.e., tQ = tOCK + t(l-0C )N tQ = t[0CK + (1-0C)N] It is similar to the Cobb-Douglas function, when that function has parameters such that OC + /3 = 1 >0) , in that both functions exhibit a straight line expansion Relevance of the elasticity of substitution to Q = <XK + (1-0C)N [0C+ (1-0C)=1] 39 path, give constant returns to scale, and exhibit average and marginal products which are homogeneous to degree zero. However, factor substitution is ruled out, as are increasing or diminishing returns to scale, when the pro duction function is strictly linear. (It should be noted that the firm can still optimize, by changing its tech nology, if the factor prices shift. But this implies a shift in the capital composition of the firm, and thus, a time dimension is introduced into the analysis, while a response via the elasticity of substitution mechanism is thought to be instantaneously carried out by the firm. Such an assumption, of course, violates reality to a degree.) For the Cobb-Douglas production function the elasticity of substitution is found to be: Pk Qk AOCK^N OcN Pn Qn A T&/S N^-1 d (N/K) = /3 d(W V Pn or N/K _ yS 40 d(N/K) CJ— = d(Pk/Pn) N/K V Pn Thus, the solution dictates that, for the Cobb-Douglas production function, the elasticity of substitution is invariable and equal to one. Homohyphallagic or Constant Elasticity of Sub stitution production function. Because there is no reason a priori why the firm's production function should exhibit any particular elasticity of substitution, a production function having greater generality has been popularized by Arrow, Chenery, Minhas, and Solow. These authors characterize their function as a Constant Elasticity of Substitution (C.E.S.) production function. However, the elasticity is empirically determined and may take values greater or less than one. This function is also variable in capital and labor, but has three 11 Kenneth J. Arrow, H. B. Chenery, B. S. Minhas, and R. M. Solow, "Capital Labor Substitution and Economic Efficiency," Review of Economics and Statistics, August 1961, pp. 225-250. oc >£ oc 41 parameters: (A) a neutral efficiency parameter, (6 ) a distribution or relative factor shares parameter, and {/*) a parameter which determines the elasticity of substitu tion. Both the Cobb-Douglas and the strictly linear production functions are special cases of the C.E.S. func tion. The equation for the C.E.S. function is: Q = A[S K** + (1 -S (A > 0, 0 < ‘ S , /> > -1) The C.E.S. function is also linearly homogeneous: A [ S (tK)_/° + (1-6) (tN)-^ ]-1^° A £ t"'* [ 6 K-^ + (1-6) N_/3 ]J “V 0 (t-^ )~1//> Q = tQ The elasticity of substitution for the C.E.S. function can be derived after obtaining the marginal physical products of capital and labor: First, by partial differentiation and rearranging it is found that Qn = A(^) [$Tf/°+ (i-6)r/r ( 1//’)-1(i-S)(-/)N_ / -1 = (1-6 )A[ S k" - / ° + (1-S )N'/>]"(1+/’ ) ^ N-(1+'° } = (i-S)^[Sk^+ (i-6 )N-^]-a+^)//> N-a+/°) A' " 3 42 = (l-c£ ) / Q V-+/7 "> 0, and by a similar derivation, £ fQ)1+/ Qk = A'* \kJ > o . The isoquant for output is given by dQ = QkdK + QndN = 0 then, dK = -Qn _ -(1-5 ) /K\1+^ . dN Qk “ £ (^Ny / U‘ Marginal productivity theory requires the least cost com bination of factor inputs to satisfy the equality — P n ( 1 - & ) / K V - +/ * Qk pk thus pk = / -1 . Solving this equation for the equilibrium factor input ratio gives k 7 _ & _ Y ' W ) K Y / u + / ’> p . Finally, the marginal N yi-S and average relationships necessary to solve the elastic ity equation are found to be a<vM> - / 6 y/ii+f) / 1 i /P \i/d+/o)-i ( l * J K/H ^ V pk pk J 43 Substitution into the elasticity equation gives (J~ = as the elasticity of substitution for the C.E.S. function. The elasticity value depends upon rho. If /°— $ > o, then the elasticity equals one (Cobb-Douglas case). If -> < = o , then the elasticity approaches zero, in the limit. This is the case of nonsubstitutable factor proportions. This study proceeds by estimation of Cobb-Douglas type functions. However, elasticity of substitution estimates, by the ACMS method, were first obtained for the samples used in the study. Based upon the statistical findings, it appeared that little was to be gained by estimating a C.E.S. type function. (A discussion of the elasticity of substitution estimates is to be found in Appendix A.) Theoretical Considerations in Statistical Estimation with Respect to the Analysis of Market Structure The preceding paragraphs have developed some of the theoretical aspects of the production function as it is related to the firm operating in a purely competitive market. The paragraphs of this section will detail the theoretical considerations involved in statistical esti mation of the production function with special regard for 44 the underlying market structure. The Statistical Estimation Procedure The method of estimation used in this paper is that of "naive least squares," as opposed to a model which attempts to achieve identification of the production func tion by the method of simultaneous equations. The rationale for the use of a single equation model for examining underlying market structure was well stated in Nerlove: A surface fitted to observations on XQ£, X-^ and X2f (^of=®i' ^lf=^i* ^2f=^if i=l#2, ..., n firms) would tell us what production a firm might be expected to have if the characteristics of its markets and its possession of managerial ability are such as to make it use certain quantities of labor and capital. Such a surface would tell us virtually nothing about the output a typical firm could produce if it were given certain quantities of labor.12 Nerlove goes on to express models offering to determine the production function with simultaneous equations in corporating market prices into their s t r u c t u r e . 1 3 12 Nerlove, Estimation and Identification, p. 24. 13 Ibid., chap. ii. 45 The approach offered in this paper is to estimate different production functions within the same industry, thus allowing for statistical parameters to change with the hoped for market difference. To be more specific, the production function to be estimated is of the form: log V = log A + a log K + b log N + u, where a and b are sample parameter statis tics, and u is a log normally distributed disturbance term with zero expectation. The sample available consists of 462 observations on twenty industries (two-digit level of S.I.C.) ranging over forty-six states and the District of Columbia. There are empty cells in the sample at the state level, i.e., a particular state may or may not yield observations for a particular industry. Additionally, an autonomous calculation which was based on the ratio between employment in Standard Metropolitan Statistical Areas within a state to the total employment for that state was used to divide the sample state into so-called "urban" and "rural" areas.This dichotomization of the ■^A detailed discussion of the sample data and the estimation procedure is presented in chapter iv. The material presented here is only given as a basis for the 46 sample develops abstract, i.e., not truly geographic marketing areas. Thus, the sample parameter variance is constrained into an "urban" and "rural" market. Thus, two production functions for each industry are estimated. Mathematical development of the statistical considerations. The Cobb-Douglas function Q = AK N can be transformed into a linear relation among its logarithms: Let q = log Q, k = log K, n = log N, and r = log A. Given a set of observations on several firms (i = 1,2,..., n: firms), average sample parameter esti mates (a,b) of the population parameters (OC fJS) can be determined by a multiple regression of q onto k and n. Errors in measurement and errors in specification are assumed to be independent for each variable; they are expressed by a log-normally distributed disturbance term (u). However, factor quantities are not truly exogeneous variables. Given an optimizing firm operating in a par ticular market situation, the firm will choose quantities theoretical considerations underlying the estimation procedure. of capital and labor based on its markets. Once the quantities of capital and labor are chosen, the quantity of output is determined. For a purely competitive market situation the firm will equate: Thus, underlying the equation for estimating the produc tion function q = r + ak + bn + uc, are the implied These two relationships must be further modified to express the possibility of market imperfections. This can be done by modifying the factor shares relationship to include elasticity parameters for factor and product markets which will vary with differing market situations. PqQ relationships: (q - k) + log a = log + u^ P (q - n) + log b = log + u2 15 For notational simplicity, subscripts for firm, industry, and market have been omitted. 48 The factor shares relations become: fl*&\ ^ 1-£ d / PnN 1+- 1- ea where £ k, £n, £d are the elasticities of supply for capital, labor, and the elasticity of demand, respec- 16 tively. When the related elasticities are incorporated, the equilibrium relationships become: Pk(l+ 1) (q - k) + log a = log -------^-- P (1- _1 ) ea + u, (q - n) + log a = log Pk(1+ v 1- ^ + u- The resulting set of simultaneous equations is: q - ak - bn = r + uQ 16 The relation of factor shares to markets and the introduction of elasticities is fully discussed in H. H. Liebhavsky, The Nature of Price Theory (Homewood, 111.: The Dorsey Press Inc., 1963), pp. 304-318. 49 q - k q - n Based on the above system of equations, Marschak and Andrews have concluded that the residual components are interrelated, including variations in the firm's technical efficiency, or ability to optimize, in addition to the random component. They assume prices as fixed for the firm as well as a similar production function for all firms in the industry. From this point of view, 1 7 estimation results are biased by the residual components. taneous Equations and the Theory of Production," Economet- rica, July-October, 1944, pp. 143-205. See also, Nerlove, chap. iii; A. A. Walters, "Production and Cost Functions: An Econometric Survey," Econometrica, January-April, 1963, pp. 1-66; A. Zellner, J. Kmenta, and J. Dreze, "Specifica tion and Estimation of Cobb-Douglas Production Function Models," Econometrica, October, 1966, pp. 784-795. There are many additional source articles on this topic, many of which are listed in the bibliography of the Walters article. 17 Jacob Marschak and W. H. Andrews, "Random Simul- 50 The approach used in this paper, as previously stated, is to attempt to achieve at least partial identification of the production function with respect to the underlying markets by estimating a separate cross-section for each market. In this way, the estimated value of the produc tion function parameters ( OC and ), and the neutral efficiency parameter (A) are allowed to vary between markets. However, the parameters are assumed constant, for a given industry, within a particular market. To the extent that nuisance variables still exist as residuals, the estimated production functions include a degree of bias. Analysis of the Production Function with Respect to Markets Since the estimation process develops an "average" production function for each industry, the industry may be considered to consist of n identical firms within a given market. Each industry operates in two markets, i.e., "urban" and "rural." If certain basic assumptions are allowed, then the estimated industry production func tions can be said to be consistent with certain markets and inconsistent with others. The necessary assumptions 51 are as follows: 1. The elasticity of demand for output is independent of the factor supply elasticities. 2. The slopes of the demand curve for output and the factor supply curves have the usual shapes. 3. Firms in the industry attempt to maximize profits and the industry moves toward long-run equilib rium. 4. The input factors are not perfect substitutes for each other. Whether a market is consistent with the estimated factor input elasticities can be determined by an examina tion of the factor market equilibrium conditions. Given the factor input market equilibrium conditions: It is immediately apparent that the numerator and denomi nator of the right hand term in the two equations shown directly above can be reinterpreted in terms of marginal revenue and partial marginal costs.18 Thus, it is possible to state that, in equilibrium, dc ac M — ±— 3N. If the markets for dR dQ factor inputs and for output are initially perfectly elastic, then any change in elasticity, in either market, would serve to constrain production, either by increasing costs of factors or decreasing revenue from output. ( CK i /3 ) represent the technology employed in the produc tion process, it is possible to deduce, given the assump tions stated in the first paragraph of this section, the nature of the market in which the industry operates: 18Given R = P^Q, then -577 = P + QdPq . q dQ dQ Additionally, 25 = P_ (1 + 2. dPq) = P (1+—1) . Since the dQ Pg--- ' 6 d elasticity of demand normally is negative, = (1- — ) . dQ £d By analogous reasoning, the elasticities of supply can be determined; the supply elasticities are partial since each factor influences the cost function. Since the partial elasticities of factor input 1. If the estimated parameters of the production function show factor elasticities to be greater than unity (a + b 1) , then there is an implication that the markets for fcLCtor supply or demand or or both are less than perfectly elastic. If this were not the case, then the profit maximizing firms would increase their scale of operations— thus taking advantage of greater marginal physical productivity, while prices remain constant. 2. If returns to scale are constant (a + b = 1), then the firm size, strictly speaking, is indeterminate. However, the profit maximizing conditions equate revenue is greater than marginal cost, the firm would continue expanding until marginal cost was equal to marginal revenue. If, given constant returns to scale, marginal cost was greater than marginal revenue, the firm would either have to become more efficient or go out of business. If the firm has no influence upon market prices, it is left with the sole recourse of becoming more efficient. dR dC dQ ~ dQ 0, and d2c dq Therefore, if marginal 54 3. If returns to scale are less than unity (a + b< 1) , then an incremental increase in output can only be pro duced at a higher marginal cost. Additionally, since the elasticity parameters are bi-directional at the margin, the present costs to the firm are already pegged on the rising portion of the firm's marginal cost curve. Such a result implies that elements of monopoly are present in the market structure in which the firm operates. At the very least entry into the market is restricted in some fashion, or other firms having more efficient factor utilization would enter the market.^ However, an alternative long run possibility also exists for the profit maximizing firm. If market in elasticities are dominant, then the firm can adapt its technology over time, with respect to the market situation in which the firm conducts its business. If this were the case, the firm would eventually be operating at its least cost point, i.e., constant returns to scale, but the employment of the.' factors would differ in different locations. An indication of this would be given by widely 19 Liebhavsky, The Nature of Price Theory, pp. 319- 320. 55 diverging capital intensity coefficients between "urban" and "rural" markets for a particular industry. Finally, if both "urban" and "rural" production functions exhibit constant returns to scale and similar capital intensity coefficients the interpretation would have to be that markets were perfectly elastic. Even in this case it would be expected that the neutral effi ciency parameter, which represents differences in nonpecu- niary external economies which have been effectively internalized by the industry, would vary between markets. A Digression on the Technological Efficiency of the Factors of Production The technological efficiency of the factors of production had, perhaps, its earliest expression in the famous dictum of Adam Smith— "the division of labor is limited by the extent of the market"— and in his also famous illustration of the efficiencies of specialization with increases in scale in the pin factory. Joan Robinson has expressed the matter succinctly: In every case increasing returns arise from improvements in productive technique. As output increases the efficiency of the factors can be increased by the fuller utilization of indivisible 56 units of the factors, or by the adoption of more specialized methods of production.2* The foregoing analysis has in all likelihood served as an antecedent for the more recent development of process analysis. In the view of process analysis, the change in the employment of factor proportions with an increase in the quantity of output would be defined as a change in process. In this case, the change in process would be specified by a different activity vector. How ever, at a higher level of data aggregation, such as for the entire firm, the change in process is submerged below the level of perception. This information is expressed, in a gross aggregation, by the partial elasticities of factor inputs with respect to the quantity of product produced. 20 Joan Robinson, The Economics of Imperfect Com petition (New York: St. Martins Press, 1965), p. 343. See also, George Stigler, "The Division of Labor is Limited by the Extent of the Market," Journal of Politi cal Economy, June 1951, pp. 185-193. 57 Measurement of External Economies The close relationship between external economies and economies of location has been discussed in Chapter II. Analysis of estimated production functions yields only limited information without explicit consideration of the cost structure of the markets in which the firm operates. (In order to elucidate the foregoing state ment, external economies to the firm will be categorized as technological or monetary.) An often cited example of an external economy which affects a firm, but which is not monetized, is the case of beneficial schooling of the labor force by another firm in the same industry. In the ordinary course of labor turnover, the firm in question may receive the benefits of such schooling embodied in the labor it hires •2- L Of course, labor may attempt to monetize the schooling received. See G. S. Becker, "Investment in Human Capital: A Theoretical Analysis," Journal of Political Economy, Supplement, October 1962, pp. 9-79. 58 A monetized external economy would occur if, for instance, an intermediate goods supplier, to the firm in question, was able to lower the unit selling price of the factor inputs it supplied. This might occur if the intermediate goods supplier had a production function exhibiting returns to scale greater than unity, and, based on an increase in demand for its product, the firm expanded production. Any price reductions passed on would be external economies to the firm under considera tion . The class of nonpecuniary external economies will affect the estimates of the efficiency parameters, but monetized external economies are not estimated, since they are represented on the cost side of the equations. In an attempt to take into consideration, at least partially, the effects of monetized external economies upon the loca tion of industries, a relation was developed to measure the effect of urban size upon the quantity of output. (A discussion of the construction of this variable, and the results of the analysis are presented in Chapter IV.) Finally, before bringing this chapter to a close, it should be noted that the strict separation between production technology and costs— upon which the analysis contained in this chapter has been based— is not, unfortunately, rigidly maintained in the empirical por tion of this study. This is because of the nature of the data involved. Pure technological data for a large sample of several industries at a single point in time are unavailable. The data utilized in this study for value added in production, which represents final goods output, and for capital stock are money aggregates. Using money aggregate data allows the effects of cost savings due to vertical integration of the firm to affect O p the neutral efficiency parameter. 22 If a firm decides to make an intermediate good, which it had previously purchased from an outside sup plier, the transfer cost savings will reduce the costs of intermediate product deducted from gross revenues. CHAPTER IV STATISTICAL FINDINGS This chapter is presented in two sections. Section one details the methodology of the study, and section two contains a detailed breakdown of the statistical findings reported on an industry-by-industry basis. Methodology of the Study This study reports on an analysis of sample parameter estimates of restricted and unrestricted Cobb- Douglas production functions which have been developed from data gathered from the 1958 Census of Manufactures, and from the 1957 Annual Survey of Manufactures.^ The data consist of observations on twenty industries (two- digit level of S.I.C.) reported on a state-by-state basis. United States Bureau of the Census, Census of Manufactures: 1958 (Washington, D.C.: U.S. Government Printing Office, 1961), Vol. I, chap. ix; idem., Annual Survey of Manufactures: 1957 (Washingtpn, D.C.: U.S. Government Printing Office, 1959), chap. i, Table 9, and chap. ii, Table 4. 60 61 The data yielded a total of 462 observations. Sample sizes for individual industries ranged from a high of forty-one observations to a low of six observations. A total of forty-six states plus the District of Columbia O are represented m the observations. The following are the main variables developed from the data: 2 A cross partition of the observation set arrayed across industries and states is given in Appendix B. The following sources all used the capital data from the Census of Manufactures, 1958 in estimating manu facturing production functions. None of the studies cited has the thrust of this study, in terms of the use of the sample parameters in making inferences regarding market structure. Two of the studies show certain similarities in some of the estimates, although there are explicit differences. The study by Bell uses restricted least squares to develop a capital efficiency parameter which is used to make inferences regarding capital-labor substi tution across regions. The study by Kim develops a produc tion relation which utilizes a diversification of industry variable to test for external economies effects. The studies are Fred W. Bell, "The Role of Capital-Labor Sub stitution in the Economic Adjustment of an Industry Across Regions," Southern Economic Journal, XXXI (1964), 123-131; George H. Hildebrand and Ta-Chung Liu, Manufacturing Pro duction Functions in the United States, 1957 (Ithaca, N.Y.: Cornell University, The New York State School of Industrial and Labor Relations, 1965); Shinkyung Kim, "Interregional Differences in Neutral Efficiency for Manufacturing Indus tries: An Empirical Study," Journal of Regional Science, Summer, 1968, pp. 19-27; Zvi Griliches, "Production Func tions in Manufacturing," The Theory and Empirical Analysis of Production, Murray Brown (ed.) (New York: Columbia University Press, 1967), pp. 285-340. 62 V = Value added, measured in monetary terms, is given as the value of final product produced during the year adjusted by the Bureau of the Census to remove inconsistencies in bookkeeping practices between firms. K = Capital, measured as the gross capital stock, in money terms, plus the value of leased plant and equipment. Both net and gross concepts were tested; the gross concept produced, in general, a slightly better statistical fit. The use of capital stock, rather than capital services, in troduces problems of dimensionality. However, the capital stock concept produced a better fit.^ In economic terms, the appropriate hypothesis would be that the entire capital stock was avail able for the use of labor during the year. Regard ing the use of gross instead of net capital stock, depreciation practices vary widely between firms; additionally, depreciation is taken largely for The capital stock concept is most often used; however, Griliches, "Production Functions," pp. 280-281, uses a flow of capital services concept. 63 tax purposes, often with accelerated write-offs for new investment. N = Labor, measured in production man hours per year, is developed as a variable by dividing the total payroll by the average wage rate per hour for production workers. This was done to include nonproduction workers into the accounting. Skill differences between production and nonproduction workers are lost in the aggregation, but separate wage and man-hour data for nonproduction workers are not given in the available data. W = Wage, measured as the hourly average wage rate for production workers. This is obtained by dividing the wage bill for production workers by the number of production worker man-hours worked per year. X = Proxy variable, representing the degree to which the work force is located in urban areas for the several states included in the study.5 This vari able is constructed as the proportion of the labor 5 This variable is tabled m Appendix B. 64 force living in Standard Metropolitan Statistical Areas within a particular state to the total labor force within the state. In 1957, there were seventy-four SMSA's, of which some states had several and some had none. Where the SMSA straddled state boundaries, ancillary data from the 1960 United States Census or 1957 County Yearbook were used to proportion employment. After compilation the data were key punched case- wise for use in machine computation. Most of the computa tions were run with the aid of a multivariate regression program (BMD-03R).^ This program is especially useful in that it allows for transformation of the data variables and partitioning of the sample data into subsamples. The statistical analysis of the data was designed after consideration of the theoretical material presented in Chapter III. In order to highlight differences in production functions between "urban" and "rural" markets, an arbitrary decision rule was used to assign the sample University of California, BMP Biomedical Computer Programs, W. J. Dixon (ed.) (Publications in Automatic Computation No. 2; Berkeley and Los Angeles: University of California Press, 1968). 65 observations into an abstract sample space. (The space is abstract in the sense that no real geographical boundaries can be assigned to it.) If the industries within a particular state had 25 per cent or less of their employment in a SMSA, the sample observations were put into the "rural" category. If more than 25 per cent of the state's employment was in a SMSA, then the observa tions were categorized as "urban." The 25th percentile cut-off point was chosen in an effort to preserve a significant number of sample observations in both cate gories, since the procedure of dividing the sample space certainly is not parsimonious in terms of using precious degrees of freedom. Hopefully, the procedure can be justified as a legitimate attempt to improve the group homogeneity of the estimated production functions. The major hypothesis to be proved or disproved, is that the production functions examined are linearly homogeneous. One way to test for linear relations is to analyze the sums of squared deviations obtained from regressions consisting of restricted and unrestricted least squares estimates of the Cobb-Douglas production 66 7 function. Useful information can also be gained by computing confidence intervals on the elasticity parameters (OC r $ ) • Additionally, the elasticity parameters can also be used to develop capital intensity coefficients, as explained in Chapter III. These coefficients can then be used in making comparisons between the technological mix of capi tal and labor in the "urban" and "rural" markets. Also, a comparison between the neutral efficiency parameters (log "&) estimates for the "urban" and "rural" markets gives an indication of the relative efficiency of an industry operating in a particular market. (As pre viously explained the neutral efficiency parameter estimates give an indication of nonpecuniary external economies.) Since these estimates are in terms of a ^Gerhard Tintner, Econometrics (New York: John Wiley and Sons, Inc., Science Editions, 1965), pp. 81-91. This test is directly applicable to the analysis of restricted versus unrestricted production functions. It develops an "F" statistic as follows: (SSDr - SSDU)(d.f.) F =--------------------- SSDU This empirical statistic is tested against Pq,95(l,d.f.) 67 geometric mean, it is probably best to consider them only in an ordinal sense, i.e., efficiency for an industry in one market is greater, equal to, or less than that for another market. (This information is tabled in Appendix B.) Tables I, II, and III are unrestricted Cobb- Douglas production estimates of the form log = log + a log + b log (i = 1,2,...,20) for twenty industries (two-digit level of SIC). Table I is a com posite production function developed from the total sample available for each particular industry. Table II is developed from a subsample of each industry, consisting of those industries thought to be located in "urban" areas. Table III is also developed from a subsample and consists of those industries which are "rural," i.e. nonurban. Tables IV, V, and VI are similar to the first three tables, except that they report on restricted Cobb- Douglas production functions of the functional form log = log Ajl + a log / + Uj_ (i = 1,2,...,20). W A casual inspection of the reported multiple determination coefficient (R^) from Table I and from 68 TABLE I LEAST SQUARES ESTIMATES OF THE COBB-DOUGLAS PRODUCTION FUNCTION (Complete Sample) SIC log A a b a+b R2 d.f 20 -.05204 .636 (. 089) .436(.091) 1.072 .988 38 21 -.04350 .896(.042) .231(.052) 1.127 .999 3 22 .13156 .840(.097) .154(.091) .994 .968 18 23 .43023 .603(.127) .422(.112) 1.025 .937 21 24 .16111 .428(.083) .596(.102) 1.024 .952 20 25 .04145 .185(.148) .902(.159) 1.087 .970 19 26 .39701 .380(.034) .614(.043) .994 .991 27 27 .17436 .834(.243) .214(.255) 1.048 .782 15 28 .49939 .183(.105) .854(.110) 1.037 .954 29 29 .64836 .39 8(.175) .539(.210) .937 .971 15 30 .64833 .573(.141) .359(.152) .932 .980 13 31 .37998 . 178 (. 074) .836 (.077) 1.014 .985 14 32 .18322 .183 (.095) . 893 ( . 094) 1.076 .952 24 33 .57260 .378(.109) . 584 (.117) .962 .962 26 34 .43075 .241(.076) .772(.083) 1.013 .994 30 35 .27466 .197(.138) .853(.154) 1.050 .984 27 36 .34637 .390 (.118) .640 (.118) 1.030 .982 22 37 .91359 .481(.083) .443(.086) .924 .940 26 38 .30082 .305(.137) .743(.150) 1.050 .994 11 39 -.39448 . 291 ( . 151) . 883 (. 141) 1.170 .977 4 NOTES: 1. The functional form is log V = log fa + a log K + b log N. 2. The standard error of the estimated coeffi cient is given in parentheses. 69 TABLE II LEAST SQUARES ESTIMATES OF THE COBB-DOUGLAS PRODUCTION FUNCTION (Urban Sample) SIC log k a b a+b R2 d.f. 20 - 21 .05560 .767(.151) insufficient sample .293(.159) 1.060 .988 22 22 .34376 .838 (.103) .124(.081) .962 .955 8 23 .53547 .685(.186) .328(.156) 1.013 .940 12 24 .92724 .601(.201) .248(.311) .849 .966 10 25 .23154 .687(.157) .353(.162) 1.040 .984 10 26 .31437 .470 (.035) .530(.039) 1.000 .996 18 27 .21997 .835 (.265) .204 (.272) 1.039 .773 13 28 1.00344 .335 (.098) .575(.113) .910 .962 19 29 .71419 .382 (.255) .545(.311) .927 .978 10 30 .60274 .492(.144) .457(.158) .949 .986 10 31 .34795 .181(.082) .843(.077) 1.024 .989 9 32 .04708 .066(.112) 1.059(.114) 1.125 .960 14 33 .38686 .307 (.089) .705(.097) 1.012 .983 18 34 .40931 .206(.097) .813(.103) 1.019 .993 19 35 .26618 .411(.100) .625(.106) 1.036 .995 19 36 .04624 .374(.178) .716(.187) 1.090 .994 12 37 .93872 .217 (.068) .716(.079) .933 .978 17 38 .34484 .189(.180) .856(.187) 1.045 .994 9 39 -.39448 .291(.151) .883(.141) 1.174 .977 4 NOTES : 1. The functional + b log N. form is log V = log k + a log K 2. The standard error of the estimated coeffi cient is given in parentheses. 70 TABLE III LEAST SQUARES ESTIMATES OF THE COBB-DOUGLAS PRODUCTION FUNCTION (Rural Sample) SIC log a b a+b R2 d.f 20 .08290 .525(.148) .526(.131) 1.051 .976 13 21 -.06950 .861(.069) .276(.084) 1.137 .999 1 22 .23156 .159(.101) .855(.103) 1.014 .999 7 23 .47771 .182(.072) .788(.062) .970 .989 6 24 -.35108 .541(.142) .580(.104) 1.121 .956 7 25 .17431 -.229(.087) 1.278(.093) 1.049 .994 6 26 1.05786 .394(.213) .444(.291) .838 .934 6 27 insufficient sample 28 -.13984 .303(.165) .841(.164) 1.144 .982 7 29 .31063 .410(.421) .616(.573) 1.026 .908 2 30 insufficient sample 31 .49313 -.125(.352) 1.083 (.406) .958 .977 2 32 .55793 .428(.083) .516(.070) .944 .986 7 33 1.16188 .584(.313) .195(.391) .779 .876 5 34 .34352 .310(.145) .722(.182) 1.032 .979 8 35 -.16850 -.199(.467) 1.377(.617) 1.178 .896 5 36 1.11599 .304(.184) .544(.214) .848 .801 7 37 -.64395 .873 (.250) .407(.180) 1.280 .757 6 38 insufficient sample 39 insufficient sample NOTES: 1. The functional form is log V = log A + a log K + b log N. 2. The standard error of the estimated coeffi cient is given in parentheses. 71 TABLE IV RESTRICTED LEAST SQUARES ESTIMATES OF THE COBB-DOUGLAS PRODUCTION FUNCTION (Complete Sample) SIC log & a R2 d.f. 20 .28746 .642(.102) .502 39 21 .48675 .932(.114) .944 4 22 .09923 .845(.087) .827 19 23 .53999 .578(.110) .557 22 24 .25144 .430(.081) .570 21 25 .42310 .193(.133) .095 20 26 .35111 .409 (.040) .787 28 27 .40057 .816(.230) .440 16 28 .67638 .177(.104) .087 30 29 .53541 .284(.146) .190 16 30 .38291 .537(.149) .480 14 31 .43865 .173(.071) .281 15 32 .53160 .141(.093) .084 25 33 .40043 .375(.109) .303 27 34 .48200 .265(.069) .321 31 35 .47482 .307(.127) .174 28 36 .48812 .377(.117) .311 23 37 .50015 .49 8(.085) .563 27 38 .48767 . 404 (. 137) .421 12 39 .47817 .174(.168) .177 5 NOTES: 1. The functional form is log a log K N log & + 2. The standard error of the estimated coeffi cient is given in parentheses. TABLE V RESTRICTED LEAST SQUARES ESTIMATES OF THE COBB-DOUGLAS PRODUCTION FUNCTION (Urban Sample) SIC log a R2 d.f. 20 .20201 .848(.159) .552 23 21 insufficient sample 22 .15687 .87K.077) .933 9 23 .59394 .667 (.147) .612 13 24 .29024 .396(.116) .512 11 25 .41271 .686(.156) .636 11 26 .31615 . 470 (. 034) .911 19 27 .40708 .818(.247) .439 14 28 .61194 .304(.105) .295 20 29 .64604 .189(.173) .095 11 30 .41028 .444(.144) .465 11 31 .44956 .162(.075) .321 10 32 .62940 .013(.119) .001 15 33 .44154 . 309 (.087) .397 19 34 .49419 .226(.094) .225 20 35 .42429 .462(.105) .490 20 36 .45028 .554(.232) .305 13 37 .60275 .205(.073) .304 18 38 .52544 .243(.189) .141 10 39 .47817 .174(I168) .177 5 NOTES: 1. The functional form is log 2 . lo g 1 1 - |= log £ + N ' The standard error of the estimated coeffi cient is given in parentheses. 73 TABLE VI RESTRICTED LEAST SQUARES ESTIMATES OF THE COBB-DOUGLAS PRODUCTION FUNCTION (Rural Sample) SIC log & a R2 d.f. 20 .34925 .448(.130) .460 14 21 insufficient sample 22 .29675 .164(.102) .244 8 23 .35051 .206(.059) .633 7 24 .21525 .423(.106) .663 8 25 .37832 -.225(.094) .447 7 26 .47942 .252(.196) .191 7 27 insufficient sample 28 .55274 .223 (. 206) .128 8 29 insufficient sample 30 insufficient sample 31 insufficient sample 32 .29620 .483(.072) .848 8 33 .35462 .459(.322) .253 6 34 .46411 .341(.125) .453 9 35 .52022 .034(.366) .001 6 36 .49777 .344(.178) .319 8 37 .56009 .685(.151) .747 7 38 insufficient sample 39 insufficient sample NOTES: 1. The functional form is 'K a log N log V N = log A + The standard error of the estimated coeffi cient is given in parentheses. 74 Table IV will show that it is above .9 for all but one industry in the unrestricted least squares estimates, while it is below .5 for the majority of industries estimated by restricted least squares. An examination of the correlation matrix for the regressions listed in Table I shows K and N to be highly correlated which no doubt accounts for a great deal of the deterioration in between the two sets of estimates. Analysis of the composite production function (Tables I and IV) residuals showed that only in one industry, SIC 20— Food Products, could the hypothesis of linear homogeneity be rejected. The implication drawn is that there are market constraints the size on the food products industry. Analysis of the residuals of the regressions developed from the "urban/rural" dichotomization yields the following results: (1) For the urban sample the hypothesis of linear homogeneity is rejected for: SIC 20— Food Products, SIC 2 8— Chemicals and Allied Products, SIC 32— Stone, Clay, and Glass Products, SIC 35— Machinery, except electrical, and SIC 36— Electrical Machinery; (2) with respect to the "rural" sample, the linear homogeneity hypothesis is now accepted for SIC 20— 75 Food Products. Only for one industry, SIC 28— Chemical and Allied Products, is the linear homogeneity hypothesis rejected. Based on the foregoing results of the analysis, the general conclusion reached is that, with several exceptions, the production functions of the firms analyzed do exhibit constant returns to scale. This result, coupled with an analysis of the capital intensity coeffi cients (Tables VIII and IX), which tend to vary a great deal between markets, leads to a general confirmation of the hypothesis that the technology of an industry is highly adaptable to inelasticities in the underlying market structure. (More will be said about this when Tables VIII and IX are discussed.) Table VII presents the results of an analysis of a production relationship of the form log V = log & + a log K + b log N + cD, where D is a dummy variable representing the industries thought to be in "urbanized" areas for a given sample observation. (This variable is obtained by the decision rule previously explained.) The "t" statistic for the parameter (c) of the D variable was tested at the 10 per cent level of significance. The null hypothesis is that this parameter is not SIC 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 TABLE VII LEAST SQUARES ESTIMATES OF THE COBB-DOUGLAS PRODUCTION FUNCTION WITH A DUMMY VARIABLE TO REPRESENT URBANIZATION ADDED log A a b c tc R2 .03342 . 631(.091) .436(.092) .010(.020) 0.508 .994 .06030 .929(.021) .179(.027) -.044(.012) -3.650 .999 .00093 .838 (.079) .165(.075) .140(.044) 3.163* .980 .40872 .570(.122) .442(.107) .107(.059) 1.795* .945 .22839 .427 (.079) .569(.097) .064 (.036) 1.801* .959 .22554 .093(.131) .938(.137) .107(.038) 2.809* .979 .36223 .441(.047) .546(.054) .030 (.023) 1.230 .989 .17301 .818(.256) .219(.263) . 063 (.202) 0.311 .784 .65249 .244(.090) . 724 (.101) .154(.043) 3.574* .968 .64962 .400(.187) .535 (.231) .003 (.063) 0.043 .971 .62936 .479(.143) . 469(.485) -.083 (.049) -1.697 .984 .37904 .174(.085) .839(.085) .004(.044) 0.092 .985 .20031 .179(.098) .891(.096) .015(.058) 0.053 .952 .53052 .358(.110) .629(.122) -.072(.061) -1.190 .964 .40361 .253(.077) .768(.084) - .022 (.022) -0.968 .994 .27296 .199(.141) .852(.156) -.006 (.045) -0.123 .984 .37832 .391(.120) .628(.123) .023 (.053) 0.433 .982 .98898 .428(.098) .470(.093) .087 (.084) 1.033 .942 TABLE VII (Continued) SIC A log A a b c tc R2 d.f. 38 .29579 .243(.169) .801(.177) .039(.058 0.664 .995 10 39 -.39448 .291(.175) .883(.163) .000(.000) 0.000 .977 3 NOTES: 1. The functional form is log V = log \ + a log K + b log N + c.D where D is a dummy variable representing urbanization. 2. The standard error of the estimated coefficient is given in parentheses. 3. Column 6 lists the computed "t" value of the parameter. An asterisk indicates that is different from zero at the 10% level of significance. 78 significantly different from zero. This measure is designed to be a test of the degree of association of particular industries with urban areas. As such, it is a measure of the factors of agglomeration. It should be noted that this variable also measures disassociation with urbanization— or deglomeration. The following industries were found to enjoy significant agglomerative economies: SIC 22— Textile Mill Products, SIC 23— Apparel and Related Products, SIC 24— Lumber and Wood Products, SIC 25— Furniture and Fixtures, and SIC 2 8— Chemicals and Allied Products. Tables VIII and IX give the capital intensity coefficients for the three types of production functions, i.e., composite, "urban," and "rural." Table VIII gives the coefficients for unrestricted least squares, and Table IX lists the coefficients for restricted least squares estimates. In general, the coefficients vary considerably between "urban" and "rural" areas, indicating the employment of different technologies within the same industry, depending upon the market in which the industry is located. At the level of data aggregation used in this study, it would also be possible to assume that the composition of products produced by the industry in 79 TABLE VIII CAPITAL INTENSITY COEFFICIENT (^) FOR UNRESTRICTED LEAST SQUARES ESTIMATORS SIC Production Function Composite Urban Rural 20 1.46 2.62 0.99 21 3.88 see note 2 3.12 22 5.45 6.76 0.19 23 1.43 2.09 0.23 24 0.72 2.42 0.93 25 0.20 1.95 see note 3 26 0.62 0.89 0.89 27 3.90 4.09 see note 2 28 0.21 0.58 0.36 29 0.74 0.70 0.66 30 1.60 1.08 see note 2 31 0.21 0.21 see note 3 32 0.20 0.06 0.83 33 0.65 0.43 2.99 34 0.31 0.21 0.43 35 0.23 0.66 see note 3 36 0.61 0.52 0.56 37 1.09 0.30 2.14 38 0.41 0.22 see note 2 39 0.33 0.33 see note 2 NOTES:1.See Chapter III for a discussion of the capital intensity coefficient. 2. Insufficient sample. 3. Negative capital coefficient. 80 TABLE IX CAPITAL INTENSITY COEFFICIENT {Of ) FOR RESTRICTED LEAST SQUARES ESTIMATORS Production Function SIC Composite Urban Rural 20 1.42 5.58 0.81 21 13.70 see note 2 see note 2 22 5.45 4.54 0.20 23 1.37 2.00 0.26 24 0.75 0.65 0.73 25 0.24 2.18 see note 3 26 0.69 0. 89 0.34 27 4.43 4.49 see note 2 28 0.21 0.44 0.29 29 0.40 0.23 see note 2 30 1.16 0.80 see note 2 31 0.21 0.19 see note 2 32 0.16 0.01 0.93 33 0.60 0.45 0.85 34 0.49 0.29 0.52 35 0.44 0.86 0.03 36 0.60 1.24 0.52 37 1.00 0.26 2.17 38 0.68 0.32 see note 2 39 0.21 0.19 see note 2 NOTES: 1. See Chapter III for a discussion of the capital intensity coefficient. 2. Insufficient sample. 3. Negative capital coefficient. 81 question varies between "urban" and "rural" markets. However, at least some indication of within group homo geneity is available from an examination of the respective coefficients of determination. A low coefficient would indicate a highly diversified industry, in terms of products and technology. The restricted least squares coefficients are the proper ones to examine in this case since they are not affected by the multicollinearity problem. Industry-by-Industry Analysis This section proceeds to a detailed discussion of the statistical implications of the study with respect to each industry in the study. The analysis is meant to be illustrative rather than definitive. SIC 20— Food and Kindred Products. The analysis of residuals rejects the linear homogeneity hypothesis for the "urban" sample; there are evidently potential economies of scale which imply the existence inelasticities in the market structure. Examination of the capital intensity coefficients shows capital to be more intensely applied in the "urban" market. The neutral efficiency parameter 82 is greater in the rural market. Thus, in spite of technological adaptation, apparently the locational advantage for the food products industry is in the rural market. SIC 21— Tobacco Products. Examination of the raw data for the tobacco products industry shows the industry is concentrated in the "rural" market. Thus, a de facto case for "rural" advantage exists. The production process for tobacco has a capital intensity on the order of three to one. Since the "urban" sample is too small, there is no estimate of the "urban" production function to compare with the "rural" estimate. SIC 22— Textile Mill Products. The most interest ing aspect of the elasticities developed for the textile mill industry was the almost exact reversal of proportions between "urban" and "rural" markets. In terms of capital intensity the "urban" market technology waa highly capital intensive, while the "rural" technology wa.s highly labor intensive. Neutral efficiency was found to be greater in the urban market. Additionally, Table IX indicates a significant agglomerative influence operating in the 83 industry. SIC 23— Apparel and Related Products. Agglomera tion economies are significant for the apparel industry. The industry also shows greater neutral efficiency in the "urban" market. In terms of the capital intensity co efficients, the industry is capital intensive in the "urban" market and labor intensive in the "rural" market. SIC 24— Lumber and Wood Products. The wood products industry shows greater neutral efficiency in "urban" areas. Also, the industry is capital intensive in "urban" markets and labor intensive in rural markets. There are, additionally, statistically significant agglomerative factors in the wood products industry. SIC 25— Furniture and Fixtures. Evidently, sig nificant agglomerative economies are also in operation for the furniture and fixtures industry. An examination of the neutral efficiency parameter shows it to be greater in the "urban" market. The capital intensity coefficient is on the order of 2:1 in the urban market; in the "rural" market capital had a negative elasticity coefficient, meaning that an increase in capital would lead to a 84 decrease in output. The elasticity coefficient for labor was above unity. This result may imply a shortage of appropriately skilled labor for the furniture industry in the rural market. (Although it is only fair to note that the low coefficient of determination given in the restricted least squares estimates for the "rural" sample suggests that no correlation exists in the estimated equation. This may also imply nonhomogeneity in produc tion, i.e., diversified technology for the "rural" sample. SIC 26— Paper and Allied Products. For the paper products industry neutral efficiency is found to be greater in the "rural" sample. The two technologies in the "urban" and "rural" markets are found to be equal (0.89) and to be labor intensive. SIC 27— Printing and Publishing. The printing and publishing industry has a high capital intensity coeffi cient in the "urban" market. (No comparison with the rural market is possible because of the small "rural" sample.) SIC 28— Chemicals and Allied Products. This industry is an interesting one to examine, since the 85 analysis of residuals test rejected the hypothesis of linear homogeneity for both the "urban" and "rural" samples. Even more interesting is the fact that the "rural" sample showed potential economies of scale (a + b > 1) , while the "urban" sample gave evidence of decreasing returns to scale (a + b<l); the implication being market inelasticities in the "rural" market and restricted entry in the "urban" market. In terms of neutral efficiency, the advantage is to the "urban" market. The capital intensity coefficients are lower than unity in both markets, implying somewhat similar technologies. Additionally, Table VII shows that agglom eration factors are significant for the chemical industry. However, in all fairness, it should be pointed out that the coefficients of determination for the restricted least squares estimates are quite low in both markets which vitiates the analysis to a degree. SIC 29— Petroleum and Coal Products. The petrol eum industry shows evidence of a similar technology in both markets. The capital intensity coefficient in both markets is less than unity. An examination of the neutral effi ciency parameter shows it to be greater in the "urban" 86 market. SIC 30— Rubber and Plastics Products. This indus try is almost entirely concentrated in the "urban" sample; thus, there is a de facto case for agglomeration econo mies. The capital intensity coefficient exhibits a ratio slightly greater than unity. SIC 31— Leather and Leather Products. For the leather products industry neutral efficiency is greater in the "rural" market. A direct comparison of the capital intensity coefficients cannot be made because of a negative sign attached to the capital coefficient for the "rural" market. SIC 32— Stone, Clay, and Glass Products. Analysis of the stone, clay, and glass products industry yields quite interesting results. To begin with, the linear n homogeneity hypothesis is rejected for the "urban" sample, but it is accepted in the "rural" market. Examination of the urban sample shows economies of scale to be greater than unity (a + b >1), implying the existence of market inelasticities. The neutral efficiency parameter shows the industry to be more efficient in the "rural" market. 37 Additionally, the coefficient of determination for the restricted least squares estimates gives a relatively good fit for the "rural" market, while the "urban" market estimate is uncorrelated. This fact suggests a diversity of processes for the industry in the "urban" market. (In addition to the fact that the analysis given for the urban market cannot be maintained in a literal sense.) Finally, a comparison of the capital intensity coefficients between the two markets shows that the coefficients are less than unity in both markets, but much more so for the "urban" coefficient. SIC 33— Primary Metal Industries. In terms of capital intensity coefficients, technology is similar for the two markets, being labor intensive in both. The neutral efficiency parameter is greater in the rural market. SIC 34— Fabricated Metal Products. In the fabri cated metal products industry, efficiency is greater in the "urban" market. Technology, as evidenced by the capital intensity coefficients, is somewhat similar, being labor intensive in both markets. 88 SIC 35— Machinery, except electrical. The analysis of residuals test rejects the linear homogeneity hypothesis for the "urban" sample; however, the suggested interpretation of existing market inelasticities (because a + b > 1) would have to be quite weak, in view of the standard errors for the estimated parameters. The capital intensity coefficients cannot be directly compared since the capital estimate for the "rural" market has a negative sign. However, the intimation is that technology is capital intensive in the "urban" market and labor inten sive in the "rural" market. Neutral efficiency is greater in the "urban" market. SIC 36— Electrical Machinery. The linear homo geneity hypothesis is statistically rejected by the analysis of residuals test for the "urban" market of the electrical machinery industry; the indicated economies of scale (a + b > 1) lead to the conclusion that market inelasticities exist in the "urban" market. However, computation of a confidence interval on the elasticity parameters leads to the conflicting result that the linear homogeneity hypothesis should be accepted. The capital intensity coefficients are quite similar for both markets, 89 being labor intensive. The neutral efficiency parameter is greater in the "rural" market. SIC 37— Transportation Equipment. Neutral effi ciency is greater in the "urban" market for the trans portation equipment industry. Additionally, the capital intensity coefficients give evidence of considerable differences between the two markets, being capital inten sive in the "rural" market and labor intensive in the "urban" market. SIC 38— Instruments and Related Products. The instruments and related products industry is located almost entirely within the urban sample. Thus, no com parison between the two markets is possible. Regarding the capital intensity coefficient for the "urban" sample, it is found to be labor intensive. SIC 39— Miscellaneous Manufacturing. No compari son between "urban" and "rural" samples is possible since most of the sample observations fall into the "urban" category. Additionally, not much within group homogeneity for the industry is to be expected, since the miscellaneous manufacturing group includes such diverse industries as 90 the precious jewelry group and the morticians' goods group. Therefore, for all practical purposes, this industry is exempt from formal analysis. Comments Regarding the Industry- by-Industry Analysis Before bringing this chapter to a close, it should be restated that the analysis is predicated on the accept ance of the theoretical basis, including the ceteris paribus assumptions, contained in Chapter III. Also, the analysis has abstracted completely from level changes in efficiency that might be obtained from possible locational changes with respect to transport costs. Also, it is a - normal practice to examine the coefficient of multiple determination when assessing the overall worth of an estimated function. Because of the multicollinearity problem with the unrestricted least squares estimates, the of the restricted estimates is probably the best test statistic to use. However, even here, a word of caution is offered. When comparing the between the "urban" and "rural" sample, attention must be given to the degrees of freedom involved since the fewer the degrees of freedom there are, the higher the R^ or degree of explanation of the estimate is likely to be. 91 As a final comment, it should be noted that the neutral efficiency parameter (log and the dummy vari able estimator (Table VII), which represent nonpecuniary external economies and agglomeration factors, respec tively, are in exact agreement with each other. A com parison of the neutral efficiency parameter between the two cross-sections shows neutral efficiency to be greater in "urban" markets in nine out of the fifteen cases where a comparison was possible. CHAPTER V SUMMARY AND CONCLUSIONS In terms of a strict statistical interpretation of the findings of the analysis, the major conclusion of the study is that technology is adaptable, to a great extent, between "urban" and "rural" markets for most of the industries in the study. However, this interpretation is subject to qualification because of the highly aggre gated nature of the data. If it were possible to examine the nature of the products produced by each industry in both the "urban" and "rural" markets, it might be found that they were producing essentially different products which required differing technologies. The second major conclusion is that, while urban agglomerative factors predominate for a majority of the industries in the study, some industries are more effi cient when located in rural areas. An examination of the respective industries involved shows a good degree of conformity with, what might be called, reasonable a priori theoretical expectations. For instance, resource oriented 92 93 industries (such as food products, and stone, clay, and glass products) are more efficient in the "rural" market. Industries that use large amounts of capital, in absolute terms, were generally found to be more efficient in "urban" markets. However, two industries (electrical machinery and primary metal products), which might be expected to be more efficient in "urban" markets, were found to be more efficient in the "rural" market. In view of the seeming reasonableness of the results for the other industries in the study, the apparently anoma lous results for the two industries in question may, in fact, state the true situation. But it could also be the case that the industries are producing different products, and therefore using different technologies, in the two markets, or the data could simply be so lacking in precision as to produce a false result. It should also be remembered that the neutral efficiency parameter measures only nonpecuniary external economies. On a strictly statistical interpretation, one aspect of the study gave slightly disappointing results, since a goodly portion of the theoretical development was keyed to the interpretation of Cobb-Douglas production functions that were other than linearly homogeneous. 94 Based on the statistical tests performed, the estimated production functions gave results that generally supported the Cobb-Douglas hypothesis, i.e., (0C+/5’ = 1). How ever, the validity of the theoretical approach for inter preting the underlying market structure is not brought into question by the results. Indeed, based on the weaker criterion of reason able agreement with a priori expectations, the results are quite interesting. An examination of the unrestricted least squares estimates, Tables II and III, for the "urban" and "rural" samples respectively, yields quite interesting results. For example, SIC 37— Transportation Equipment, shows decreasing returns to scale in the "urban" market and increasing returns to scale in the "rural" market. Again, an examination of SIC 32— Stone, Clay, and Glass products, an industry known to be resource oriented, shows increasing returns to scale in the "urban" market and decreasing returns to scale in the "rural" market. The implication of decreasing returns to scale being restricted entry into the market, while increasing returns to scale implies inelasticities in the market structure. (It has to be noted again that these interpretations are not based on statistically significant parameters, but rather, 95 only on reasonableness.) Turning to an examination of Table I which gives the results for the unrestricted least squares estimates for the composite sample, it is found to be the case that industries which employ large absolute amounts of capital tend to have estimates for the elasticity parameters which indicate decreasing returns to scale. The four industries showing greatest negative deviation from linear homo geneity are, in order, SIC 37— Transportation Equipment, SIC 30— Rubber and Plastic Products, SIC 29— Petroleum and Coal Products, and SIC 33— Primary Metal Industries. Although these findings are not statistically significant, they are interesting since the four industries cited are oligopolistic to a degree, and therefore, tend to confirm the validity of both the theoretical and statistical approach of this paper. Application of the Analysis to Problems of Regional Economic Development Application of the foregoing analysis to problems of regional economic development yields some interesting implications. Proper interpretation requires that both the technological elasticity parameters and the neutral efficiency parameter be considered mutually. 96 Considering first the elasticity parameters for capital and labor, it is apparently possible for many industries to adapt the technology employed depending upon the nature of the market in which the industry operates. The apparent range of choice in manufacturing processes may enhance the feasibility of locating indus try in "poverty pockets," thus taking advantage of relative wage differentials, and at the same time employ ing previously unemployed individuals. In addition, since most of the twenty manufacturing industries sampled were found to be producing at constant returns to scale, plant size may be more readily adaptable to potential markets for produced goods than previously thought. However, when the estimates of the neutral effi ciency parameter and the urbanization parameter are con sidered together the situation becomes more complex. In nine out of the fifteen industries where estimates were possible, it was found that the industry was more effi cient when located in the urban market. The implication being that urbanization promotes efficiency for certain industries. This conclusion agrees completely with the economic view of the production process as a continual flow of goods and services between the producing and 97 consuming sectors of the economy. Urbanization simply reduces the sum total of costs involved in transmitting goods and services between producing and consuming sectors of the economy. The full implication for regional economic development policy would be that while economic develop ment on a relatively small scale is perhaps feasible in rural areas, greater economic efficiency could be gained by creating entirely new urban areas. Criticism and Extensions of the Analysis Certain criticisms regarding the nature of the data have already been mentioned, but being forced to use the data at hand leads naturally to a consideration of extensions to the analysis made possible by removing the restrictions. The most obvious faults of the present analysis are the small quantity of sample observations available and the aggregated nature of the data. Objec tion could also be made to the methodology used in con structing the variables utilized in the study. Addition ally, no account has been taken of intermediate goods as a factor of supply; this is certainly a legitimate criticism since supply inelasticity for intermediate 98 goods could have a direct effect on the production function. Still another class of criticisms could concern the method of estimation utilized for the study. It might also be argued that costs could be incorporated into the model, and therefore, conceivably take into account pecuniary external economies. Upon examination it will be seen that all of the foregoing criticisms involve the use of more and better data. If more observations on industries at the three- and four-level digit of SIC classification were available, it should be possible to reduce the variance of the estimates. Also, the ambiguity of interpretation due to the homogeneity of industry problem should be reduced or eliminated. Regarding the construction of variables, this study has followed the more or less accepted practice used by most researchers when working with cross-section data. Among the difficulties in constructing variables are the following: (1) book value of capital is given in historical rather than current value which allows the actual quantity of physical capital to vary between firms showing the same book value, depending upon when the capital goods were bought; (2=) no account is taken of the 99 vintage of capital, i.e., new capital is supposedly more efficient capital; (3) the capital data are not adjusted to reflect the actual amount of capital in use by the firm during the year; (4) the labor variable is not adjusted to reflect skill differences in the labor force utilized by different firms. The foregoing problems are serious, but their solution is involved with con structing the proper weighted indexes given adequate data. The conceptual problem, therefore, is with the data, not with the theoretical production function. The incorporation of intermediate goods as a supply factor does, however, involve a theoretical change in the model. Since value added is a representation of final product, some adjustments to the production function are necessary. The function used in this paper has implicitly assumed intermediate goods input to be a constant (G). Thus, the true estimated function is V = G + AK^N^. Intermediate goods can be simply put into the model by considering G to be a weighted index variable (G = P-.G-,) . The new function is simply 1=1 Q = AK^ N^GV . 100 This study has sought to develop and compare elasticity estimates for a number of industries. In order to accomplish this, there had to be data compati bility between industries, which led to the choice of data actually utilized. Other researchers have developed very good techniques for developing production function estimates from cost data for individual firms within an industry. This approach would be a worthwhile extension to the analysis presented here, but the task of gathering such data would most certainly require a large and well- funded, research effort. Finally, the validity of the results rests upon a fundamentally correct approach in terms of economic theory, and a proper interpretation of the statistical analysis. Hopefully, the conclusions offered in Chapter IV, and the first section of Chapter V, will serve to Two such studies are Marc Nerlove, "Returns to Scale in Electricity Supply," Measurement in Economics: Studies in Mathematical Economics in Memory of Yehuda Grunfeld, by Carl F. Christ et al. (Palo Alto: Stanford University Press, 1963); George H. Borts, "The Estimation of Rail Cost Functions," Econometrica (January 1960), pp. 108—131. 101 indicate that useful information regarding locational effects can be extracted from cross-sectional estimates of manufacturing production functions which are segre gated as to markets. APPENDICES 102 APPENDIX A ELASTICITY OP SUBSTITUTION ESTIMATES Elasticity of substitution estimates for the twenty industries involved in the study were computed by the ACMS method in order to determine whether estimation of production functions of the Cobb-Douglas type could be theoretically justified. By computing confidence intervals on the cT parameter, to test the hypothesis that the true value of ct~ is one, it was found that for nineteen out of the twenty industries in the composite sample, it could not be maintained that the elasticity of substitu tion was other than unity. Similar results were also obtained for the "urban" and "rural" samples. Based on the foregoing results the Cobb-Douglas production function was chosen for the study. 103 104 APPENDIX A-I ESTIMATED ELASTICITY OF SUBSTITUTION (Complete Sample) SIC log “ A ^ R2 CJ— K d.f. 20 -1.33723 . 860 ( .121) .566 39 21 -9.31878 4.577 ( .516) .952 4* 22 -2.71650 1.450(1.350) .057 19 23 .13452 .117( .703) .001 22 24 -1.74750 . 977( .082) .871 21 25 -2.09251 1.151( .109) .848 20 26 -3.59416 1. 831( .317) .544 28 27 -3.53643 1.795(1.246) .115 16 28 -1.31901 .916( .260) .293 30 29 - .90510 . 736( .453) .373 16 30 -2.57839 1.367( .296) .604 14 31 -1.29699 .771{ .301) .308 15 32 -1.94150 1.123( .272) .406 25 33 -1.61331 . 962 ( .625) .081 27 34 -1.00706 .6 81( .170) .342 31 35 -1.63335 .942 ( .457) .132 28 36 -1.67425 . 984( .348) .258 23 37 -4.01374 1.959(1.170) .094 27 38 -2.04208 1.119( .291) .552 12 39 -1.21889 .756( .145) .844 5 NOTES: 1. The functional form is log log W. 2. The standard error of the estimated coefficient is given in parentheses. ♦Significant at 5%. . . 1 1 J 105 APPENDIX A-II ESTIMATED ELASTICITY OF SUBSTITUTION (Urban Sample) SIC log A Cf R2 d.f. 20 -1.58682 .972 ( .251) .391 23 21 insufficient sample 22 - .23495 .352 (3.004) .001 9 23 4.19956 -1.742 (1.089) .164 13 24 -1.49360 .866 ( .127) .808 11 25 -2.30350 1.244( .110) .921 11 26 -3.66498 1.852 ( .293) .677 19* 27 -3.79229 1.901(1 .622) .089 14 28 -1.51675 1.004 ( .467) .188 20 29 - .64931 .626 ( .430) .161 11 30 -1.90293 1.074 ( .273) .584 11 31 -1.39662 . 826 ( .330) .385 10 32 -2.81349 1.496( .520) .355 15 33 - .74523 . 592 ( .612) .047 19 34 -1.82555 1.033 ( .150) .704 20 35 - .79858 . 591 ( .428) .087 20 36 -2.67948 1.404( .402) .483 13 37 2.10050 - .606 ( .800) .031 18 38 -1.40306 . 848 ( .386) .326 10 39 -1.21889 .756 ( .145) .844 5 NOTES: 1. The functional form is log ( — ] = log A + $ log W. / 2. The standard error of the estimated coefficient is given in parentheses. *Significant at 5%. 106 APPENDIX A-III ESTIMATED ELASTICITY OF SUBSTITUTION (Rural Sample) SIC log % R2 d. f. 20 -1.08192 .741( .134) .687 14 21 -10.76109 5.221( .564) .977. 2* 22 .32968 .015( .429) .0002 8 23 -1.21927 .716( .343) .383 7 24 -1.88037 1.033 ( .102) .927 8 25 -1.20762 .738( .692) .140 7 26 -2.68751 1.462 ( .831) .306 7 27 - .33315 .461( .432) .276 7 28 - .55713 .574( .403) .203 8 29 -2.97170 1.605 ( .957) .234 3 30 insufficient sample 31 2.08023 -.801( .946) .193 3 32 -1.78148 1.059 ( .356) .526 8 33 -3.73910 1.879(1.484) .211 6 34 - .00125 .243( .396) .040 9 35 -2.82851 1.453(2 .504) .053 6 36 -3.00576 1.599 ( .645) .434 8 37 4.05695 4.057(2 .638) .252 7 38 insufficient sample 39 insufficient sample NOTES: 1. The functional form log W. is log >1 z ^ = log A A + 2. The standard error of the estimated coefficient is given in parentheses. *Significant at 5%. V \* APPENDIX B MISCELLANEOUS TABLES 107 108 APPENDIX B—I COEFFICIENT OF URBANIZATION State % State % 1. Maine .01 25. West Virginia .20 2. New Hampshire .01 26. North Carolina .01 3. Vermont .01 27. South Carolina .01 4. Massachusetts .60 28. Georgia .01 5. Rhode Island .99 29. Florida .01 6. Connecticut .65 30. Kentucky .33 7. New York .88 31. Tennessee .32 8. New Jersey o 00 • 32. Alabama .01 9. Pennsylvania .76 33. Mississippi .01 10. Ohio .66 34. Arkansas . . .01 11. Indiana .27 35. Louisiana .36 12. Illinois .72 36. Oklahoma .01 13. Michigan . 66 37. Texas .48 14. Wisconsin .42 38. Montana .01 15. Minnesota .66 39. Idaho .01 16. Iowa .01 40. Wyoming .01 17. Missouri .80 41. Colorado .67 18. South Dakota .01 42. Arizona .01 19. Nebraska .01 43. Utah .01 20. Kansas .83 44. Nevada .01 21. Delaware .98 45. Washington .44 22. Maryland .76 46. Oregon .45 23. Washington, D.C. .99 47. California .83 24. Virginia .25 Source: U.S. Bureau of the Census, 195 7 Annual Survey of Manufactures (Washington, D.C.: Government Printing Office, 1959). NOTES: 1. The coefficient of urbanization is computed as the ratio of employment in SMSA's within a state to the total employment in the state. Where SMSA's overlap a state's boundary, ancillary data were used to make cor rections . 2. A value of .01 was assigned to states having no SMSA. This was done to satisfy requirements for machine computation. 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 109 APPENDIX B—II INDUSTRIAL CLASSIFICATION Market Having Greater Neu- Industry tral Efficiency Food and Kindred Products Tobacco Products Textile Mill Products Apparel and Related Products Lumber and Wood Products Furniture and Fixtures Paper and Allied Products Printing and Publishing Chemicals and Allied Products Petroleum and Coal Products Rubber and Plastics Products Leather and Leather Products Stone, Clay, and Glass Products Primary Metal Industries Fabricated Metal Products Machinery, except Electrical Electrical Machinery Transportation Equipment Instruments and Related Products Miscellaneous Manufacturing rural urban urban urban urban rural urban urban rural rural rural urban urban rural urban APPENDIX B—III INDUSTRY OBSERVATIONS WITHIN STATES State 20 21 22 23 24 25 26 27 SIC 28 29 30 31 32 33 34 35 36 37 38 39 1. Maine X X X X X X X 2. New Hampshire X X X X X X X 3. Vermont X X X 4. Massachusetts X X X X X X X X X X X X X X X X X X 5. Rhode Island X X X X X X X X 6. Connecticut X X X X X X X X X X X X X X 7. New York X X X X X X X X X X X X X X X X X X 8. New Jersey X X X X X X X X X X X X X X 9. Pennsylvania X X X X X X X X X X X X X X X X X X 10. Ohio X X X X X X X X X X X X X X X X 11. Indiana X X X X X X X X X X X X X X X X 12. Illinois X X X X X X X X X X X X X X X X X 13. Michigan X X X X X X X X X X X X X 14. Wisconsin X X X X X X X X X X X X X X 15. Minnesota X X X X X X X X X X X X 16. Iowa X X X X X X X X X X 17. Missouri X X X X X X X X X X X X X X 18. South Dakota X 19. Nebraska X X X APPENDIX B-III (Continued) State 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 20. Kansas X X X X X 21. Delaware X X X X X X X X 22. Maryland X X X X X X X X X X X X X X 23. Washington, D.C. X X 24. Virginia X X X X X X X X X X X X X X 25. West Virginia X X X X X X X X X X X 26. North Carolina X X X X X X X X X X X X X 27. South Carolina X X X X X X 28. Georgia X X X X X X X X X X X X 29. Florida X X X X X X X X X X X 30. Kentucky X X X X X X X X X X X X X 31. Tennessee X X X X X X X X X X X X X X 32. Alabama X X X X X X X X X X X X 33. Mississippi X X X X X X X 34. Arkansas X X X X X X X X 35. Louisiana X X X X X X X X X 36. Oklahoma X X X X X X X X X X 37. Texas X X X X X X X X X X X X X 38. Montana X X 39. Idaho X X 40. Wyoming X APPENDIX B—III (Continued) State 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 41. Colorado X X X X X X X X X X 42. Arizona X X X X 43. Utah X X X X 44. Nevada X X 45. Washington X X X X X X X X X X 46. Oregon X X X X 47. California X X X X X X X X X X X X X X X X Total Sample Size 41 6 21 24 23 22 30 18 32 18 16 17 27 29 33 30 25 29 14 7 462 BIBLIOGRAPHY BIBLIOGRAPHY A. Books Allen, R. G. D. Macro-Economic Theory: A Mathematical Treatment. New York: The Macmillan, Co., 1968. Alonso, William. Location and Land Use. 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Measurement in Economics: Studies in Mathematical Economics and Econometrics in Memory of Yehuda Grunfeld, Stanford. Ed. by Carl F. Christ et al. Palo Alto: Stanford University Press, 1963. Nerlove, Marc. "Returns to Scale in Electricity Supply." Measurement in Economics: Studies in Mathematical Economics in Memory of Yehuda Grunfeld, Stanford. Ed. by Carl F. Christ et al. Palo Alto: Stanford University Press, 1963. Pp. 108-131. 121 Solow, Robert M. "Some Recent Developments in the Theory of Production." The Theory and Empirical Analysis of Production. Ed. by Murray Brown. New York: Columbia University Press, 1967. Pp. 25-53. Viner, Jacob. "Cost Curves and Supply Curves." American Economic Association. Readings in Price Theory, VI. Chicago: Richard D. Irwin, Inc., 1952, 198-241. Zellner, A.; J. Kmenta; and J. Dreze. "Specification and Estimation of Cobb-Douglas Production Function Models." Readings in Economic Statistics and Econometrics. Ed. by Arnold Zellner. Boston: Little, Brown and Co., 1968. Pp. 323-335.

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Creator Nicholson, Norman Dwight (author)

Core Title Cross-Sectional Analysis Of Manufacturing Production Functions In A Partitioned 'Urban-Rural' Sample Space

Degree Doctor of Philosophy

Degree Program Economics

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Advisor Tintner, Gerhard (committee chair), Niedercorn, John H. (committee member), Robinson, Ira M. (committee member)

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