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JOURNAL OF REGIONAL SCIENCE, VOL. 40, NO. 4, 2000, pp. 671–695 AN EVOLUTIONARY NEW ECONOMIC GEOGRAPHY MODEL* Wei Fan Department of Economics, University of Michigan, Ann Arbor, MI 48109, U.S.A. E-mail: [email protected] Frederick Treyz and George Treyz Regional Economic Models, Inc., 306 Lincoln Ave, Amherst, MA 01002. E-mail: [email protected] and [email protected] ABSTRACT. In this paper we present a general new economic geography model with multiple industries and regions, full labor and capital mobility, land use in production and consumption, and a dynamic adjustment process in which consumers maximize utility and firms respond to nonzero profits. All industries use intermediate inputs as well as land, labor, and capital. Systems of cities form endogenously within this framework, including asymmetrical urban hierarchies and cities of different sizes and industry compositions. Each urban area has a bid-rent gradient and zones with land uses and densities as in the von Thünen model. The equilibrium depends not only on initial conditions but also on speeds of adjustment. The model is a prototype for empirical implementation, as illustrated with a simulation of the effects of transportation cost reductions. 1. INTRODUCTION The emerging area of study known as the new economic geography has greatly enriched spatial economic analysis as evidenced by the burgeoning literature in regional, urban, and international economics (see Krugman, 1998a; Fujita, Krugman, and Venables 1999 for recent reviews). Models developed in this field have led to fresh insights on a wide range of issues such as specialization in a system of cities (Abdel-Rahman, 1996), industry location decisions across countries (Venables, 1996), the effects of congestion (Brakman et al., 1996), and preindustrial agglomeration (Duranton, 1998). However, researchers recognize the need for a more general spatial economic model. Fujita and Krugman (1995) unify von Thünen and Chamberlin models in a way that they suggest will “lead towards the development of a general equilibrium model of urban systems.” Krugman (1998b) calls for development of a “computable *We are grateful to two anonymous referees for their insightful comments and Dr. Omar El-Gayar’s help in many respects. Wei Fan is also grateful to Gordon Hanson, Alan Deardorff, and Michelle White for their constructive discussions, and for financial support from Regional Economic Models, Inc. Received November 1998; revised May 1999 and November 1999; accepted March 2000. © Blackwell Publishers 2000. Blackwell Publishers, 350 Main Street, Malden, MA 02148, USA and 108 Cowley Road, Oxford, OX4 1JF, UK. 671 672 JOURNAL OF REGIONAL SCIENCE, VOL. 40, NO. 4, 2000 geographical equilibrium” model that can answer questions about the effects of technological shocks and policy changes on the economy. In this paper we present a new economic geography model with endogenous land use, labor mobility, interindustry purchases, and N-locations in one- or two-dimensional space. This general structure produces stylized metropolitan forms with rich structures, including an endogenously generated bid-rent and density gradient, and concentric zones specializing in different industries. Additionally, the model presented in this paper has much in common with regional computable general equilibrium models. This suggests uses of the model to evaluate regional economic issues, such as the effect of environmental regulations, economic development initiatives, and productivity changes on regional economic development. As an example, simulation results presented in this paper illustrate the effect of transportation cost changes on the spatial economic structure. The model that we present is not the first that moves towards the goal of a generalized new economic geography model. For example, Puga (1999) presents a model that unifies different strands of work relating to international trade theory. However, his model is not applicable to regional or urban analysis because it does not include land explicitly. A generalized regional economic model is presented in Fujita, Krugman, and Mori (1999) and Fujita, Krugman, and Venables (1999) but the formulation makes constraining assumptions such as the existence of an unlimited amount of land in the economy and the absence of capital stock. The model presented here relaxes some of the most restrictive assumptions of the majority of new economic geography models. In the formulation of this paper we assume the following: workers are mobile between sectors and regions; land is explicit in consumption and production for all sectors, differentiated inputs are used in production; and geographic space is discrete so that numerical solution methods may be applied easily in either one- or two-dimensional space. As in many other models cited in this paper the agglomerative forces in our model are the price effects (both the consumers’ price and producers’ price) and the wage effect. However, unlike models based on immobile production sectors, the spreading force in this model comes solely from demand for a limited supply of land as it is in Helpman (1998). Helpman’s model has only two regions and a single differentiated industry. In addition to land and labor we also have capital as a production factor (Baldwin, 1999a). Despite some common features with existing models, our model is an initiatory model with endogenous land and capital use and labor mobility in a formulation with N-locations and interindustry transactions. In Section 2 we formally present the model. In Section 3 we describe the evolutionary solution method used to solve for the geographical economic equilibrium. We present results of some simulations in Section 4. In particular, the economy in this model organizes itself into cities with different sizes and different industrial structures depending on the initial population and capital stock distributions and adjustment speeds. Simulations show the change in the © Blackwell Publishers 2000. FAN, TREYZ, & TREYZ: NEW ECONOMIC GEOGRAPHY MODEL 673 regional and urban structure caused by reductions in transportation costs. In Section 5 we summarize the paper and discuss possible model extensions. 2. THE MODEL In this model there are N regions (i = 1,2, . . . , N). The population in the economy is L. The population, land area, and capital stock in region i are denoted by Li, Si, and Ki, respectively. Identical individuals each provide one unit of labor and have the same preferences. All consumers consume land and M commodities produced in the economy with an identical Cobb-Douglas utility function M ∏c ui = siθ0 (1) θm mi m=1 where si is the amount of land and cmi is the amount of composite commodity m consumed by a representative consumer in region i. Throughout the paper subscripts i and j are reserved for regions and m is reserved for industries or commodities. A subscript combination mi stands for industry m in region i or is simply called regional industry mi. All m commodities are differentiated. In each region the composite commodity m is composed of different varieties in the Dixit-Stiglitz (1977) form F = G∑ c GH nm cm σ m −1 σm mi, v v= 1 σm σ m −1 I JJ K where nm is the total number of commodity m varieties produced in all N regions, cmi,v is the amount of variety v of commodity m consumed by a representative region i consumer, and σm is the elasticity of substitution between two commodity m varieties. With a Cobb-Douglas utility function a representative consumer spends a constant budget share on land and each of the M commodities. The budget shares for land and the M commodities are θ0 and θm, m = 1, 2, . . . , M, M respectively, and θ 0 + ∑θ m = 1. m= 1 All commodities are produced with increasing returns-to-scale technology using capital, land, labor, and intermediate inputs. A profit-maximizing firm produces only one variety of commodity because of the benefits of increasing returns. Therefore, the same index v is used to represent a firm or a variety. The production function for a representative firm in each industry is assumed to be in the following Cobb-Douglass production function with increasing returns to scale (2) φ mk φ ml φ ms M d k i d s i dl i ∏ d z mi, v mi, v m′ mi, v mi, v m′= 1 © Blackwell Publishers 2000. i φ mm′ = α m + zmi, v 674 JOURNAL OF REGIONAL SCIENCE, VOL. 40, NO. 4, 2000 where zmi,v is the amount of variety v produced by firm v in regional industry mi; φmk, φms, φml are the capital, land, and labor shares in industry m,respectively; and φmm′ is the share of intermediate input m′ in industry m. The fixed cost is αm. Finally, kmi,v, smi,v, and lmi,v are the amount of capital, land, and labor used in the production, and zmm′i,v is the composite intermediate input produced by industry m′ used by firm v in regional industry mi. To simplify the solution method, intermediate inputs are assumed to be of the same form as the composite consumption commodities. The production functions for all industries are in the same form but they can differ in input shares, fixed costs, and marginal costs. However, all regions share the same production technology. Industries can also differ in transportation costs. In order to set up a tractable model, transportation costs are assumed to take the iceberg form. That is, for each unit of commodity m goods shipped from the port of origin region i for a distance of dij, only (1 + γmdij)–1 unit arrives at the port of destination region j (γm > 0). Consequently, for industry m the c.i.f. price will be higher than its f.o.b. price by a factor of (1 + γmdij), that is Pmij = Pmii (1 + γmdij) (3) where dij is the distance between region i and region j. Pmii is the f.o.b. price of commodity m in region i. As is well known, under the assumption that there are a large number of firms in each industry, all firms face the same elasticity of substitution (Krugman, 1980). Assuming that all firms practice mill pricing, each firm will then charge the same f.o.b. price to consumers from all regions. It can be shown that with the production function given in Equation (2) and the CES utility function, the f.o.b. price charged by profit-maximizing firms in regional industry mi is Pmii = (4) σm Ω mi σm − 1 where Ωmi is the marginal cost function of a representative cost-minimizing firm in regional industry mi. Furthermore, it can be shown that M (5) Ω mi = rki ψ mki + rsi ψ msi + wi ψ mli + ∑P m′ i ψ mm′ i m′= 1 where rki, rsi, and wi are the capital rental rate, land rental rate, and wage rate in region i, respectively. Pmi is the price index of commodity m defined as F = G∑ n GH N (6) Pmi j =1 © Blackwell Publishers 2000. 1− σ m mj Pmji I JJ K 1 1− σ m FAN, TREYZ, & TREYZ: NEW ECONOMIC GEOGRAPHY MODEL 675 where Pmji is the c.i.f. price in given in Equation (3). The terms ψmki, ψmsi, ψmli, ψmm′i are the standard factor requirement functions for capital, land, and labor, and the intermediate input requirement functions for regional industry mi, respectively. They are functions of the three factor prices and the M intermediate input prices, for example (7) ψ mki LF r I F φ I OP = MG J G MNH r K H φ JK QP si mk ki ms φ ms LMF w I F φ MNGH r JK GH φ φ ml i mk ki ml I OP LMF P I F φ JK PQ ∏ MNGH r JK GH φ M m′= 1 m′i mk ki mm′ I OP JK PQ φ mm′ The profit made by a representative firm in industry m can be shown as (8) π mi = Ω mi Fz GH σ mi,v −1 m − αm I JK The zero-profit output level for a representative firm in industry mi is b g 0 zmi = αm σm − 1 and the equilibrium number of firms after entry in industry mi is (9) 0 nmi = F L GH α σ mi m I bψ g JK −1 mli m Consumers, who are also the owners of the firms, have three income sources: labor income, rental income, and profits. For simplicity, land and capital in all regions are assumed to be owned equally by all consumers in the economy, and each consumer provides one unit of labor. Under these two assumptions, per capita income in region i can be expressed as (10) yi = wi + π i + 1 L N ∑ dr sj S j + rkj K j i j =1 The first term on the right-hand side of Equation (10) is the labor income or the wage rate w. The second term is per capita profit defined as follows πi = 1 Li M ∑n mi π mi m= 1 The last term is rental income, which is simply the total rental spending in the economy divided by the total number of people in the economy. It can be shown that the total spending on capital and land are © Blackwell Publishers 2000. 676 JOURNAL OF REGIONAL SCIENCE, VOL. 40, NO. 4, 2000 L O Fφ I ∑ MMNGH φ JK w L PPQ M rki K i = mk i ml m=1 L O Fφ I ∑ MMNGH φ JK w L PPQ M rsi Si = θ 0 yi Li + mi ms i mi ml m=1 Notice that there is both consumption spending and production spending on land. One can solve per capita income from Equation (10) and express it as r r r r y = A −1 w + π + ∆ e j where LM1 − L θ MM LL θ − A=M L MM LM MN − L θ 1 1 0 0 1 0 L2 θ0 L L 1 − 2 θ0 L M L2 θ0 − L − LN θ0 L L ... − N θ0 L ... M LN θ0 ... 1− L ′ − ... OP PP PP PP Q r y = y1 , y2 ,..., y N b g r ′ w = bw , w ,..., w g r ′ ∆ = b ∆, ∆,..., ∆ g r ′ π = b π , π ,..., π g LF φ + φ I O 1 w L P ∆ = ∑ MG J ∑ L MNH φ K PQ 1 2 1 2 N N M N mk ms j ml m= 1 j j =1 Finally, B and H are defined as the factor share matrix and intermediate input share matrices, respectively LM MN φ 1k B = φ2k φ3k φ 1s φ2s φ 3s OP PQ φ 1l φ 2l , φ 3l and LM MN φ 11 φ 12 H = φ 21 φ 22 φ 31 φ 32 © Blackwell Publishers 2000. φ 13 φ 23 φ 33 OP PQ FAN, TREYZ, & TREYZ: NEW ECONOMIC GEOGRAPHY MODEL 677 Intermediate inputs are produced with the three production factors, so B* is defined as the effective factor share matrix. One can show that B* = (I – H)–1B The effective factor income shares in the economy are M Θf = (11) ∑θ * m B mf f = k, l m= 1 and M Θs = θ + (12) ∑θ * m B ms m=1 Finally, Region 1’s labor is chosen as the numeraire, that is, Region 1’s wage rate is always normalized to one. 3. THE DYNAMIC ALGORITHM Fujita and Mori (1997) formally introduced the evolutionary solution method from complexity theory into the field of economic geography to numerically solve this type of model, which is otherwise very difficult to solve analytically. We adopt their basic methodology to solve the model proposed in Section 2. Essentially, the economy starts with a given initial population distribution and the autarkic equilibrium industrial composition. Then, the economy evolves by itself at each period based on a set of prescribed laws of motion. The evolution process continues until the economy reaches a full equilibrium in which all markets clear and utility levels are the same in all regions. The laws of motion describe how the economy adjusts to its equilibrium from an off-equilibrium position. In this model, there are (M + 3) markets in each region: M commodity markets plus the capital market, the land market, and the labor market. The algorithm assumes that the land rental rate adjusts fast enough to clear the land market in each region in every period. For the capital market, it is assumed that in each period the capital stock is fixed in each region and the capital rental rate adjusts instantaneously so that the capital market is cleared in each region. However, it takes time for capital to flow from one region to the other. Therefore, the capital rental rate can be different among different regions during the adjustment process. Wage rates adjust based on labor market conditions. The number of firms in each regional industry adjusts according to its profit level. Wage rates adjust according to the demand for labor at zero-profit production level. The output level produced by a firm is determined by total expenditure on the regional industry’s output and the number of firms in the regional industry. Price levels evolve by iteration according to Equations (3) to (6). Labor moves across industries based on the difference between the actual employment level © Blackwell Publishers 2000. 678 JOURNAL OF REGIONAL SCIENCE, VOL. 40, NO. 4, 2000 and the hypothetical zero-profits employment level. Finally, the population in each region adjusts through migration to equalize the utility level in all regions. Specifically, the algorithm starts with initial conditions including population distribution in space {Li, t = 0}, capital stock distribution in space {Ki, t = 0}, labor allocation among industries in each region {Lmi,t = 0}, capital stock allocation among industries in each region {Kmi, t = 0}, and initial wage rates {wmi, t = 0}. The capital rental rate and land rental rates are set to clear the capital market and land market, respectively, in each region rki = rsi = 1 Ki M Fφ I ∑ GH φ JK w L mk i m=1 LM MN i ml M F I JK φ ms 1 θ 0 yi Li + wi Li Si φ ml m=1 ∑ GH OP PQ The initial factor and intermediate requirement functions ψmki, ψmsi, ψmli, and ψmm′i are first calculated as if there were no intermediate inputs. Then in each period, commodity prices are calculated using Equations (3) to (6) with previous cost values in Equation (5). Equations represented by Equation (7) update the factor and intermediate input requirement functions and unit cost functions with the newly calculated commodity prices. The initial number of firms in each regional industry is set to the zero-profit number of firms given in Equation (9). The spending by a region on each of the M commodities produced in a particular region is Emij = nmiω mi µ mij ESmj N ∑n mkω mk µ mkj k= 1 where Emij is the spending by consumers and producers in region j on commodity m produced in region i. On the right-hand side, ESmj is total spending in region j on commodity m including both consumption spending and production spending shown as M ESmj = θ m y j L j + Fφ I ∑ GH φ JK w L m′= 1 The ωs and µs are defined as follows ω mi = Ω 1mi− σ m and © Blackwell Publishers 2000. m′m m′l j j FAN, TREYZ, & TREYZ: NEW ECONOMIC GEOGRAPHY MODEL d µ mij = 1 + γ m dij i 679 1− σ m In dollar value, the demand for all commodity-m varieties produced in region i is N EDmi = ∑E mij j The demand for a regional industry’s output is transformed into the hypothetical zero-profit demand for labor by firms in this regional industry, that is D Lmi = φ ml EDmi wi and M LiD = ∑L D mi m= 1 is the total zero-profit demand for labor in region i. The total zero-profit demand is the demand for labor in a regional industry for a given demand for the regional industry’s output if profits for all firms were eliminated by competition. The difference between the zero-profit demand and the actual supply of labor reflects tensions on the labor market. The wage rates adjust according to the zero-profit demand for labor in excess of supply with a damped adjustment factor, that is1 wi,t + 1 = wi,t + λ w LiD,t − Li,t e j where λw is the speed of adjustment for the wage rate. The indirect utility level can be calculated using the income, the rental rate, the commodity price indices and the utility function in Equation (1), as follows (13) Vi = yi rsi− θ 0 M ∏P −θm mi m= 1 The migration decision is based on the indirect utility level for the representative consumer in each region. If the utility level in a region is lower than the average utility level in the economy, residents in that region will move to regions with relatively higher utility levels. For regions with above-average 1 The way zero-profit demand is calculated and the way adjustment process for wage rates is modeled assume no forward-looking behavior in labor contract negotiation. Baldwin (1999b) shows that inclusion of forward-looking expectation leads to absolutely no change in the main results of the standard core-periphery model. © Blackwell Publishers 2000. 680 JOURNAL OF REGIONAL SCIENCE, VOL. 40, NO. 4, 2000 utility level the amount of in-migration into the region is greater with higher utility level. Our model simply assumes that the number of migrants to region i in period t, migi, t, that an above-average region receives is proportional to the deviation of its utility level from the average. Formally Li,t + 1 − Li,t = migi,t R|λ dV − V i L S|λ ρ ∑ dV − V i L T L i,t t i,t L i,t t k, t k,t if Vi,t ≤ Vt if Vi,t > Vt Vk, t < Vt where λL is the migration rate, Vt is the population average of indirect utility, and ρi,t is the proportion of migrants an above-average region i receives. These are defined by 1 Vt = L N ∑V i,t Li,t i=1 and ρ i,t = Vi,t − Vt ∑ dV k,t − Vt i Vk, t > Vt Capital stock moves based on the capital rental rate differences in the N regions. In this model, it follows a similar dynamic process to the movement of people. That is, capital flows into regions with an above-average capital rental rate and out of regions with a below-average capital rental rate. The capital migration rate is λk. In each region and each period labor moves, with friction, from industries with a low zero-profit demand for labor to industries with a high zero-profit demand. If there are any new in-migrants, they are allocated proportionally to each industry’s zero-profit demand for labor because they have already left their old job by out-migration, or in equation F GH Lmi,t + 1 = Lmi,t + λ m Li,t D Lmi ,t LiD,t I JK − Lmi,t + migi,t D Lmi LiD,t where λm is the intersector migration rate. Finally, the output level of a representative firm in regional industry mi is calculated from the material balance equation zmi,v = © Blackwell Publishers 2000. EDmi nmi Pmii FAN, TREYZ, & TREYZ: NEW ECONOMIC GEOGRAPHY MODEL 681 The profit level can then be calculated from Equation (8). At the end of each period the number of firms in each regional industry changes due to the entry and exit decision as follows nmi,t = nmi,t–1 + λnπmi,t–1 (14) where λn is the entry and exit rate. The key to the evolutionary method is that the industrial composition in each region is determined endogenously. A regional industry may disappear and new regional industries may be born as the economic system evolves. A regional industry disappears when the number of firms in that regional industry goes below one following the dynamics specified by Equation (13). A new industry will be born if a tentative firm in the new industry has enough demand for its output to make a positive profit. Specifically, if in each period the number of firms in a regional industry goes down to less than one then the regional industry disappears—both the number of firms and amount of labor in the regional industry are set to zero. On the other hand, for each missing regional industry the profitability of establishing a firm in that industry is determined by calculating the profit level given that there is one firm in the regional industry. If a new firm will be profitable it will then come into existence (a new firm appears); otherwise that regional industry does not form. With the existence of multiple equilibria the economy’s off-equilibrium behavior matters a great deal to the final equilibrium. The path-dependence phenomena in economics is now well known not only in economic geography, but also in other areas as well (for example, Arthur, 1994). Yet, the dynamic process specified in this paper is still more for convenience than for realism. In reality an off-equilibrium economy is characterized by unemployment, temporary profit, inventory, involuntary saving, and so forth. The laws of motion described in this paper are an abstraction of the more complicated adjustment process of the real economy. 4. SIMULATION RESULTS In this section we present results for different model specifications to show the endogenous generation of a spatial economy with urban agglomerations, density and rent gradients within metropolitan areas, and complex systems of cities. An eight-region and a fifty-region “racetrack” model illustrate land uses and rents, as well as change that results from lower transportation costs. Table 1 presents the input-output relationships used throughout the paper unless otherwise specified. Agriculture, manufacturing, and service industries all use intermediate inputs and three factors: capital, land, and labor. Factor use reflects the land intensity of agriculture, capital intensity of manufacturing, and labor intensity of services. Interindustry relationships reflect the use of manufactures in agriculture and the reflexive dependence of manufacturing and © Blackwell Publishers 2000. 682 JOURNAL OF REGIONAL SCIENCE, VOL. 40, NO. 4, 2000 TABLE 1: Factor and Intermediate Input Share in Production Agriculture Manufacture Services 0.01 0.10 0.04 0.10 0.60 0.15 0.05 0.15 0.10 0.15 0.10 0.45 0.02 0.10 0.20 0.10 0.08 0.50 A-Product M-Product S-Product Capital Land Labor TABLE 2: Consumption Share Land A-Product M-Product S-Product 0.20 0.10 0.30 0.40 TABLE 3: Technological Parameters Agriculture Manufacturing Service 1.00 1.00 0.20 6.00 5.00 1.00 0.05 3.00 3.00 1.00 0.50 3.00 Fixed Cost Marginal Cost Transportation Cost Elasticity of Substitution services.2 Table 2 shows consumption shares in land, agricultural and manufactured products, and services. Table 3 shows technological parameters and transportation costs. These reflect a high fixed cost and low transportation cost for manufactured goods, a low fixed cost for agricultural products, and very high transportation costs for services. The elasticity of substitution for production reflects a relatively homogenous agricultural product and differentiated manufactured goods and services. An Eight-Region Model A single-city economy. In this section we present results for an eight-region racetrack economy where the regions are uniformly distributed on a circle and each has the same land area. All goods must be transported along the circumference. The distance between two consecutive regions is one unit of distance. Therefore, there is no first-nature locational advantage (Krugman, 1993). The initial condition is specified such that Region 1 has 93 percent of the population and the other 7 regions each have 1 percent of the population. All regions have the same wage rates and allocate their labor according to their autarkic demand calculated with the effective labor demand as 2 We also ran simulations using more real-world coefficients, such as those in Sonis, Hewings, and Gazel (1995). However, in the experiment the agglomerative force dominated the outcome which located all nonagricultural economic activity in a single location. © Blackwell Publishers 2000. FAN, TREYZ, & TREYZ: NEW ECONOMIC GEOGRAPHY MODEL Lmi = 683 θ m (B*) ml Li Θl The adjustment speeds are λL = 0.1, λw = 0.01, λK = 0.002, λn = 0.1, and λm = 0.25. These speeds remain the same throughout this paper unless otherwise specified. The population distribution in the economic geographic equilibrium is shown in Figure 1. The figure shows the final population distribution and the final labor allocation in each regional industry. It shows that Region 1 develops into service centers, and Regions 3 and 8 specialize in the manufacturing industry. Regions 4, 5, and 6 are occupied by agriculture. Region 2 is a mixture of services and manufacturing and Region 7 is a mixture with mostly manufacturing and agriculture. Regions 1, 2, 3, 7, and 8 could be seen as cities whereas Regions 4, 5, and 6 form a rural area. The spatial distributions of factor prices are shown in Figure 2. As is expected, capital rental rates are the same in all regions and land rental rates are the highest in the service centers and lowest in the rural regions. Moreover, the disparity in wage rates is much smaller than that in land rental rates. This result is consistent with the large variance in rents and the relatively smaller variance in wages observed among U.S. cities. However, the results show higher wage rates in rural regions than in the cities. With labor mobility, at equilibrium, whether workers in the cities receive higher wages depends on whether the disutility of congestion outweighs the benefit of lower price index from the variety effect. It is possible that the benefit of living in a city is sufficiently high that people are willing to accept a lower wage rate for the privilege of city life. In other words, farmers may be highly paid in order to compensate them for living in remote areas with a low variety of goods and services. Smaller wage rate differences are an outcome of the indirect utility function in Equation (13). The effective land income share and labor income share in the economy for the input-output coefficients given in Table 1 can be calculated using Equations (11) and (12). The effective wage share in income is Θl 0.5108 = ≈ 0.8 Θ l + θ0 + Θ s 0.5108 + 0.1271 b g A 1 percent increase in the wage rate increases income and indirect utility by about 0.8 percent. On the other hand, a 1 percent increase in the rental rate results in a Θ0 percent (0.2 percent) decrease in indirect utility. Therefore, other things being equal, a 1 percent difference in regional wage rates must be compensated by a 4 percent difference in the rental rate for the indirect utility to be equal. This explains why the variance in equilibrium land rental rates is much greater than the variance in equilibrium wage rates. Path dependence. It is well known that with increasing returns, economic geographic equilibrium depends not only on the preference and industry © Blackwell Publishers 2000. 684 JOURNAL OF REGIONAL SCIENCE, VOL. 40, NO. 4, 2000 FIGURE 1: The Economic Geographic Equilibrium. (The populations are in thousands) structures in the economy but also on the evolution path leading to the equilibrium. Krugman (1995) is one of several to show the dependence of the final equilibrium on the initial configuration. In this section we reaffirm these earlier results within the context of a generalized new economic geography model. Moreover, the result shows the dependence of the final equilibrium choice on another aspect of the evolution path—the adjustment speed, a phenomenon that has not been observed in previous work in the field. © Blackwell Publishers 2000. FAN, TREYZ, & TREYZ: NEW ECONOMIC GEOGRAPHY MODEL 685 FIGURE 2: Factor Prices in Economic Geographic Equilibrium. The eight-region model is used to generate 1,000 outcomes starting with 1,000 different random initial population and capital stock distributions. A total of 27 different equilibria are found in the 1,000 outcomes.3 Figure 3 shows 3 Two distributions are considered as the same if they differ by a translation or reflection operation because of the symmetry of the system. For example, for a distribution of ABCDEFGH a translation operation to the right by one position changes the distribution to BCDEFGHA. A reflection operation around B changes the original distribution to CBAHGFED © Blackwell Publishers 2000. 686 JOURNAL OF REGIONAL SCIENCE, VOL. 40, NO. 4, 2000 population distributions of the ten most frequent equilibria. The differences in total population distribution between some of the equilibria (e.g., between equilibria #1 and #2, between equilibria #5 and #7) are so small that they are virtually invisible on the bar charts. However, they represent distinct equilibria associated with different industrial composition. FIGURE 3: Effect of Initial Population Distribution on Geographic Equilibrium. (The populations are in thousands) © Blackwell Publishers 2000. FAN, TREYZ, & TREYZ: NEW ECONOMIC GEOGRAPHY MODEL 687 Table 4 shows the 27 equilibria that resulted in an experiment of 1,000 outcomes starting from random initial conditions. The table shows the industry mix for each outcome along with the probability (frequency) of its occurrence. A, M, and S stands for a region specializing in agriculture, manufacturing, or service, respectively. AM stands for a region with a mix of agriculture and manufacturing, MS for a region with a mix of manufacturing and services, and so on. One can see that equilibria #1 and #2 are the most likely outcomes with a combined probability of over 64.5 percent. Although the difference in the population distributions is very small, the industry distributions are clearly different. There are two regions specializing in agriculture in equilibrium #1, but there are three of them in equilibrium #2. In all 27 outcomes there is no region with a mix of agriculture and service. Services and agriculture do not mix because both the forward and the backward linkages are weak. If a region chooses to produce in two industries it is more cost-efficient to produce either in agriculture and manufacturing if the land rental rate is low, or in manufacturing and service if the land rental rate is high. The probability of the TABLE 4: The Equilibrium Industrial Structure Equilibrium Configuration 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 Probability of Occurence Industrial Structure A M A M A MS A S S AM MS S AM S M AM M S S AM M AM S S A MS A A AM A A MS S A AM M S S AM AM MS A S A M M S S AM AM AM M S AM © Blackwell Publishers 2000. AM A A A S M A A A MS M A S M A M MS A MS AM AM S A AM AM AM S M A M A M A MS A A AM A AM M A A AM S A S MS A AM AM MS S A AM S A S M S A S A A A A MS A A AM MS AM AM M AM A A S AM MS A AM MS M MS MS AM A M AM MS M S S AM AM S S AM S A S M A AM S A AM S M MS S S A MS S S S S AM AM S A M M S MS A M MS MS MS AM M S MS AM S M MS A M MS M M AM AM AM MS M S A AM AM AM A S S AM AM S M M 0.335 0.310 0.073 0.053 0.046 0.034 0.027 0.027 0.025 0.017 0.010 0.007 0.006 0.006 0.005 0.003 0.003 0.002 0.002 0.002 0.001 0.001 0.001 0.001 0.001 0.001 0.001 688 JOURNAL OF REGIONAL SCIENCE, VOL. 40, NO. 4, 2000 formation of a two-city equilibrium with at least two pure service regions separated by nonservice regions is very small. From Table 4, this probability can be seen to be 5.6 percent. In fact, among the top-ten list, only the tenth is a two-city equilibrium. As far as we know, there has been no study on the effect of adjustment speed on equilibrium choice. This is not surprising. For most of the models developed so far there are two adjustment processes at most: the wage-rate adjustment and migration. Migration is assumed to occur after wage rates adjust to clear the markets. Under these assumptions, changing the adjustment speeds is equivalent to changing the unit of time in two separated processes. Therefore, there is no effect on the equilibrium the economy eventually reaches. In this model, all adjustment processes occur in the same time so a change in relative adjustment speed can have real effect on the economy. For example, a change in capital migration rate may result in a higher per capita capital stock in some regions, and a lower per capita capital stock in others. Figure 4 illustrates this property of the model with two population migration rates. For the eight-region model starting with 93 percent of the population in Region 1, a migration rate λL of 0.1 produces a configuration with two large equally populated regions. One of Region 1’s neighbors ends up in exactly the same position as Region 1, although they started with different initial conditions. On the other hand, the outcome of a lower migration rate of 0.001 is a different equilibrium with the first region remaining as the biggest center for the economy. It is interesting to notice that the reflection symmetry about Region 1 of the initial distribution is broken in the first equilibrium, but is maintained in the second equilibrium. The dependence on the adjustment speed reemphasizes the importance of the off-equilibrium behavior in economies characterized by increasing returns. Urban systems. Elaborate metropolitan areas emerge within a regional system as more regions are added to the model. This is illustrated with a fifty-region economy. Figure 5 shows the equilibrium distributions of total population, labor forces employed in each of the three industries, capital stock, capital rent, land rent, and wage rate in each region. Characteristics of the von Thünen isolated state begin to emerge, including patterns of population density, industry composition, and land rents that are found in a spatial economy with urban, suburban, and rural regions. Figure 5 illustrates a system of cities and rural areas in which there are multiple metropolitan centers, each with increasing population densities and land rents moving from the hinterland to the urban core. There are five separate cities with the largest one centered on Region 37. The two medium-sized cities centered on Regions 19 and 50 are about the same size. The two smallest cities are located at Regions 10 and 26, respectively. There are five small towns scattered in rural regions specializing in manufacturing at Regions 6, 29, 32, 42, and 44. Within each metropolitan center, the service industry is located at the © Blackwell Publishers 2000. FAN, TREYZ, & TREYZ: NEW ECONOMIC GEOGRAPHY MODEL 689 FIGURE 4: Effect of Migration Rate on Geographic Equilibrium. (The populations are in thousands) center and is surrounded by the manufacturing industry. This is qualitatively similar to observed patterns of urban development in which a central business district is occupied by banking, insurance, and other service firms, and is ringed © Blackwell Publishers 2000. 690 JOURNAL OF REGIONAL SCIENCE, VOL. 40, NO. 4, 2000 FIGURE 5: The Economic Geographic Equilibrium. (The populations are in thousands) by zones of manufacturing and agriculture. There are 12 pure-service regions, 14 pure-manufacturing regions, 12 pure-agriculture regions, 10 regions with a mix of agriculture and manufacturing, and 2 regions with a mix of manufacturing and service. Not surprisingly, capital stock is heavily concentrated in the © Blackwell Publishers 2000. FAN, TREYZ, & TREYZ: NEW ECONOMIC GEOGRAPHY MODEL 691 metropolitan centers. The land rental rate is the highest in Region 37, the center of the largest city, with a value of 68.30. The lowest rental rate is in Region 6 with a value of 14.26. The highest land rental rate is about 4.8 times as high as the lowest land rental rate. Again, the difference in wage rates is very small, and the wage rates in the rural region are slightly higher than the wage rates in the metropolitan centers. The highest wage rate is in Region 31, a pureagriculture region, with a value of 1.105, while the lowest wage rate of 0.997 is in Region 18, a suburban region of one of the medium-sized cities with a mix of manufacturing and service. The wage difference is only about 10 percent. The effect of reductions in transport costs on urban systems. The model presented may be used to show the equilibrium effects of various exogenous shocks. Figure 6 shows the equilibrium distributions when the transportation costs for all three industries are reduced to 30 percent of the original costs after the economy reaches the equilibrium presented in Figure 5. One can see that reduction in transportation costs changes the equilibrium configuration significantly. The most substantial change is that population becomes more concentrated around a large metropolitan center in Regions 21 and 22. The agriculture production that existed in Regions 23 and 24 is squeezed out from the large metropolitan center. The largest metropolitan center that used to be in Region 37 shrinks to the second largest metropolitan center. The other four metropolitan centers are all significantly smaller than previously. Furthermore, the five small towns are all gone. A dramatic change also occurs with the land rental rate. The distribution for the land rental rate peaks in Region 21 and again in Region 36. The land rental curve is flattened in the sense that the highest rental rate (55.7 in Region 21) is only about 3.2 times higher than the lowest one (17.6 in Region 6) compared to 4.8 times before the reduction. Population densities also become more evenly distributed. The largest population decreases from 63.8 thousand to 49.7 thousand, while the lowest population increases from less than 1.7 thousand to 1.9 thousand. The more uniform metropolitan population densities and rents are not surprising in the context of the underlying role of transportation costs in the determination of rent and density gradients in urban economic theory (e.g., Fujita, 1989). In a regional context, the simulation is consistent with the two-region result of Kilkenney (1998), which shows that very low transportation costs lead to the development of rural areas. If transportation costs are reduced to one-tenth of the original level, rather than 30 percent, the new equilibrium configuration comes very close to a monocentric core-periphery structure. The new equilibrium shown in Figure 7 has only one large metropolitan center in Region 25. The land rental rate decreases almost monotonously as one moves away from the center, except for a couple of small deviations at Region 9 and 37. Agriculture is all located in the periphery. The major reason for this is that the agricultural sector uses the largest proportionate land and transportation inputs. Thus, as transportation costs decrease it pays farmers to move to the periphery as they increase the isolation of the city centered at the two ends so that it ultimately disappears. © Blackwell Publishers 2000. 692 JOURNAL OF REGIONAL SCIENCE, VOL. 40, NO. 4, 2000 FIGURE 6: Equilibrium with Reduced Transportation Cost. (Populations are in thousands) The other cities then combine with services (the heaviest user of its own and nonfarm intermediate inputs and the smallest user of land) and next with manufacturing that uses both land and the intermediate inputs surrounding it. © Blackwell Publishers 2000. FAN, TREYZ, & TREYZ: NEW ECONOMIC GEOGRAPHY MODEL 693 FIGURE 7: Equilibrium with Further Reduced Transport Cost. (Populations are in thousands) 5. CONCLUSION This paper presents an N-region, N-industry, three-factor evolutionary economic geography model with monopolistic competition. The distinguishing features include: (1) multiple industries and regions, (2) complete mobility of labor between industries and regions, (3) land use in both consumption and production, (4) intermediate inputs used in all industries, (5) concurrent adjustment processes including firm location changes based on nonnegative profits, and (6) discrete formulation for easy numerical implementation. The prototype model presented in this paper can generate urban systems with cities of different sizes and industry compositions. The simulation results show that the © Blackwell Publishers 2000. 694 JOURNAL OF REGIONAL SCIENCE, VOL. 40, NO. 4, 2000 economic-geographic equilibrium reached by the economy depends not only on initial conditions, but also on the speeds of adjustment. In contrast to the models presented in Fujita, Krugman, and Venables (1999), which separately include land rental markets, intermediate inputs, and urban hierarchies, this paper brings together these separate lines of inquiry into a single, unified model. The model generates an economy with multiple urban centers, each of which has a von Thünen-type land use pattern and rent gradient. This outcome is due to all industries and households using land and being able to substitute between land, labor, and capital. The result is that households and each of the industries concentrate in separate zones and compete with each other in the market for land and in all other markets. The model structure produces complex, asymmetrical equilibrium outcomes. Most new economic geography models produce only two spatial configurations (economic activity is either concentrated in one location or is equally dispersed); only Kilkenney (1998) and Helpman (1998) have obtained asymmetrical results. Furthermore, we show that although many equilibrium configurations are possible, some are more likely to occur. A simulation based on transportation cost reductions illustrates how the model can be used to evaluate the effects of external shocks on the economy. The simulation shows changes in the land use by industry, population densities, and the number and location of cities. In particular, as transportation costs decrease industries with a high use of intermediate inputs and relatively low land intensity tend to agglomerate. On the other hand, industries with fewer backward linkages and higher land use are more dispersed. These particular results depend on the model parameters. The basic structure may be extended to achieve its potential as a practically useful model. For example, it can be expanded to incorporate less than instantaneous speeds of adjustment in land and capital markets, and to allocate workers across sectors based on wage mechanisms during the transitory period. Heterogeneous labor within and among skill or industry categories can be incorporated into the model. To fully capture income and product relationships the model can be extended to account for government activity and for the savings and capital formation process. The model also serves as a prototype for an empirical new economic geography model. Although initial conditions can be observed in a real economy, estimates of parameters are also required to develop a model that can be used for realistic policy analysis. Key parameters include the speed of response with which economic actors respond to market signals and elasticities. 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