A model of intraurban employment location : an application to Bogota, Colombia

World Bank Reprint Series: Number 249 Kyu Sik Lee A Model of Intraurban Employment Location An Application to Bogota, Colombia Reprinted with permission from Journal of Urban Economics, vol. 12 (November 1982), pp. 263-79. Copyrighted by Academic Press.  World Bank Reprints No. 215. Michael Cernea, "Modernization and Development Potential of Traditional Grass Roots Peasant Organizations," Directions of Change: Modernization. Theory, Research, anid Realities No. 216. Avishay Braverman and T. N. Srinivasan, "Credit and Sharecropping in Agrarian Societies," Journal of Development Economics No. 217. Carl J. Dahlman and Larry E. Westphal, "The Meaning of Technologi- cal Mastery in Relation to Transfer of Technology," Annals of the American Academty of Political and Social Science No. 218. Marcelo Selowsky, "Nutrition, Health, and Education: The Economic Significance of Complementarities at Early Age," Journal of Develop- menit Economics No. 219. 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George Psacharopoulos, "The Ecoromics of Higher Education in Developing Countries," Comparative Education Review No. 226. Katrine Anderson Saito and Delano P. Villanueva, "Transaction Costs of Credit to the Small-scale Sector in the Philippines," Economic Development and Cultural Chanige No. 227. Johannes F. Linn, "The Costs of Urbanization in Developing Coun- tries," Economic Development and Cultural Change No. 228. Guy P. Pfeffermann, "Latin America and the Caribbean: Economic Performance and Policies," Soluthw-t7estern Review of Management and EconiOmnics No. 229. Avishay Braverman and Joseph E. Stiglitz, "Sharecropping and the Interlinking of Agrarian Markets," American Economic Review No. 230. Abdun Noor, "Managing Adult Literacy Training," Prospects No. 231. Bela Balassa, "Shifting Patterns of World Trade and Competition," Grozjth and Entrepreneurship: Opportunities and Challenges inZ a Changinlg World No. 232. Johannes Bisschop, Wilfred Candler, John H. Duloy, and Gerald T. O'Mara, "The Indus Basin Model: A Special Application of Two-Level Linear Programming," Mathlematical Programnmintg Study  JOURNAL OF URBAN ECONOMICS 12, 263-279 (1982) A Model of Intraurban Employment Location: An Application to Bogota, Colombia' KYiU SIK LEE Urban Development Department, The World Bank, 1818 H Street NW, Washington, D,C. 20433 Received March 6, 1981; revised May 18, 1981 A micro model is formulated to study the location behavior of manufacturing firms in urban areas. A bid-rent function is derived from the profit function and captures the firms' locational equilibrium situations. The theoretical model is ex- tended to a multinomial logit specification and estimated using establishment survey results for Bogota, Colombia. The survey included information on (1) attributes of the establishment such as plant space, and (2) attributes of the plant site such as access to markets. The estimated model is capable of predicting the location choices of different types of firms. 1. INTRODUCTION The work reported here is part of a World Bank urban study project. In this paper a theoretical model of employment location is formulated and extended to an empirical specification in the multinomial logit framework. In the descriptive phase of the study, the employment location patterns of Bogota, Colombia, and their changes were extensively analyzed using in- dustrial directory data. The analysis, performed in terms of births, deaths, and relocation of firms, revealed a high degree of employment location dynamics: both the birth and relocation rates were high and evidence of spatial decentralization of manufacturing employment was strong (Lee [9]). Although researchers have drawn attention to the need for modeling employment location behavior, the gap in this area remains unattended in the literature. The analytical work reported in the present paper is an 'Presented at the Econometric Society Annual Meetings, Denver, Colorado, September 5-7, 1980. Support for the work reported in this paper was provided by the City Study research project (RPO 671-47) funded by the World Bank. The views reported here are those of the author and should not be interpreted as reflecting the views of the World Bank or its affiliated organizations. The author thanks Maria Clara de Posada who conducted the survey of establishments and Jose Fernando Pineda who supervised it, M. Wilhelm Wagner and Leslie Kramer for research assistance, and members of the World Bank research staff for comments with particular appreciation for Gregory K. Ingram and Douglas H. Keare. Discussions with Professor Marc Nerlove and comments received from Professor Edwin Mills' seminar at Princeton University were helpful at the early stage of this work. Roger Schmenner provided valuable suggestions for designing the survey instrument. 263 0094-1190/82/060263- 17$02.00/0 Copyright 4 1982 by Academic Press, Inc. All rights of reproduction in any form reserved.  264 KYU SIK LEE attempt to model the location behavior of the firm and to explain observed patterns of employment location. For this purpose, a survey of manufactur- ing establishments was conducted in Bogota, a rapidly growing city com- parable to such United States cities as Phoenix and Houston. This paper presents estimation results based on the survey. The model is presented in the next section, the survey is then briefly described, and finally, the estimated results are reported. 2. A MODEL OF EMPLOYMENT LOCATION Consider T types of manufacturing firms in an urban area. The firm mnaximizes profits as a price taker in both product and factor markets. The firm uses a set of variable and fixed inputs to produce an output. The problem is to determine the optimum combinationi of inputs, including the lot size and the plant location, to attain locational equilibrium profits in an urban area. Consider a production function in the general form Q = f(L, X; Z) () where Q is the output, L the lot size, X a vector of other inputs such as labor, and plant and equipment; Z a vector of site characteristics that are independent of lot size and can be considered as "local public goods"2 such as the quality of public utility services, accessibility to markets, and ameni- ties of the zone of plant location. The profit of the firm is II =pf(L, X; Z)-RL- wX (2) where II is the profit, p the output price, R land rent per unit, w other input prices, such as wage rate, and price of capital input. From the first-order conditions for profit maximization, one obtains the following demand equations for variable inputs: af R (3) aL p af _W (4) ax p( Solving (3) and (4) for the optimal input quantities L* and X*, and substituting them into (2), the "profit function," based on the duality 2Burstein [I] included this variable in the household utility function of her housing demand study.  EMPLOYMENT LOCATION MODEL 265 z FIG. 1. The firm's bid-rent function. theorem,3 is obtained as I* =pf(L*, X*; Z) - RL* - wX* = I*(p, R, w; Z). (5) Let t be the unit transport cost for shipment of output; then p - t is the factory price of output. Using p as the numeraire and introducing the location subscript (u), (5) becomes Il*(u)=g[l - (u), K(u), w(u); Z(u)] (6) where II, ti, R, and iw7 are values normalized by p; u refers to the distance to the product market. In locationial equilibrium, for a given u every firm should have the same profit, and there is no incentive for any firm to relocate. An equilibrium rent profile must satisfy fl*(u) =g[ -t T(u), R (u), w~(u); Z(u)] = const.4 (7) As with residential location, a useful interpretation of this formulation of firm location choice is in terms of the bid-rent function of the firm, giving the price for site with characteristics Z that yields profit 17*, Let R*(u) denote the bid rent, then (as in Fig. 1) k"(u)-h[l -T(u), uF(u); Z(u); lI*(u)]. (8) For convenience, suppose the unit transport cost is site invariant within an urban area and include it as an element in the constant term. Also 3For the duality relations between the production function and the profit function, see Diewert [2] and Lau and Yotopoulos [6]. 4Solow [121 shows an equilibrium rent profile of households in an urban area.  266 KYU SIK LEE suppress JI*(u) which is constant. Hence (8) can be written *(u)- h[iw(u); Z(u)] (9) where -= <0; ->0. (10) 3w az For illustration, consider the case of labor input. As the labor-land ratio increases the marginal product of land increases relative to that of labor, and the relative price of land with respect to labor also rises. This argument supports the empirically observed rent gradient in an urban area in the sense that as the distance to the CBD becomes shorter, the intensity of a variable input such as labor increases and the land rent rises.5 In other words, producers respond to input price differentials over space to obtain optimal input combinations including lot size. Also the value of land increases as desirable site characteristics, such as public service provision and accessibility, are improved. Since w;7 is the input price vector normalized by output price, (4) can be rewritten as af (u) = i(U). (11) Substituting (11) into (9), we have the bid-rent function expressed in terms of firm characteristics af/aX and site characteristics Z. For expository reasons, rewrite (9) as K*(u) h[x(u), Z(u)], (12) where x(u)[= (fA/8X)(u)] now represents a vector of firm characteristics, namely input combinations, which in turn depend on technology char- acterized, for example, by type of production process and building struc- ture. As mentioned earlier Z(u) is a vector of site characteristics. Now suppose that there are T types of firms defined by x and S types of sites defined by Z. Let N, be the numiber of type t firms in the market. Then using (12), the bid rent for a site with characteristics Z by the nth firm of type t is given by R*, = h,n(ZZ) n E N, (13) 5A measure of the land price gradient using the survey data used in this study resulted in the following: In land price 8.029 - 0.1126 distance, R2 = 0. 1093, which can be written as land price = 3069e - 0.1126 distance.  EMPLOYMENT LOCATION MODEL 267 Note that we have now suppressed the vector x(u) that is used to define the firm type t. For example, all firms of type t are siniilar in terms of output, input combination and technology, that is, they have an identical production function. Following Ellickson's [3, 41 work on residential location, we can interpret this model in terms of predicting the probability of a certain type of firm t to locate at a site with a specified set of characteristics Z. The stochastic version of (13) is R*,= h,,(Z - ) I e,,, ,E N (14) where e,,n is a random disturbance term reflecting unaccounted variations of firm characteristics of type t. Since a given site is occupied by the firm with the highest bid, the relevant variable for determining the probability that a given site is occupied by a firm type t is the maximum bid given by firms of type t. R, = max(K1 ) = h,(Z) + e,, t E- T (15) n where el = max (e,n), n E N,. n If the et are identically and independently distributed Weibull,6 the specification of a logit model follows, namely, the probability that a firm of type t occupies a site with characteristics Z takes the logit specification7 p(t Z) exp[h,(Z)] (16) t'ET The above discussion shows that the basic theoretical approach used in the study of residential location can provide a useful analytical framework for the study of employment location.8 The optimnizing behavior of the firm is postulated as location specific, that is, the choice by the firm of a specific site is part of the production decision; furthermore, the location specific 6For example, the maximum value of an identically and independently distributed normal variate has the Weibull distribution. 7Ellickson [3, 4] derived this variation of the logit model in his residential location study. 8Theoretical and empirical work is rare in this area; Mills [10] and Solow [12] offer basic micro foundations; the work by Hoover and Vernon [5], Struyk and James [13], and Schmenner [11], although descriptive, serves as the empirical bases in the field.  268 KYU SIK LEE equilibrium position of individual firrns is extended to the "locational equilibrium" situation of all firms in an urban area. The theoretical model is easily extended to the stochastic specification of the model in an estimable form. 3. THE DATA i he sample of 126 establishments was drawn for the survey from DANE's 2629 distinct firm records in the industrial directory files covering 1970- 1975,9 stratified by the following four categories: (1) location history, that is, stationary firms, movers, and births'