We have also benefited from the suggestions of James Foster, Rob Reed, and participants of the Midwest Mathematical Economics Conference at the University of Michigan and the Regional Science Association. This diminishing geographic effect is measured both by the Euclidean distance to the mean of the distribution of firms and by an overall index of firm distribution. The main determinants of the underlying urban configuration are: (i) the cost of consumer travel per unit distance to firms, (ii) the extent to which knowledge spillovers become less effective as a result of overall firm distribution, and (iii) population of firms.
Regardless of the different types of urban configurations, the potential location function is always (strictly) concave in the distance over business areas, thus implying a (strictly) convex penalty cost for the robust distribution. 2 As argued in Section 4.4 below, multicentric equilibria can never be generated in models such as ours, regardless of the convexity or concavity properties of the penalty or transaction cost plane. The key to generating the multicentric equilibria in Fujita and Ogawa (1982) is the fixed factor proportions and the separable potential location function that compares pairwise firm locations to obtain the overall penalty for a firm from integration.
4 For an extensive discussion and mathematical characterization of degradability properties (poverty indices), the reader is referred to Foster and Shorrocks (1991). The central feature of the model is the impact of uncompensated inter-firm knowledge spillovers in production. Both the locational potential function approach in Fujita and Ogawa (1982) and our consideration of the Q function treat agglomeration of firms as an external economy.
Taking the constant capital rent r and the labor wage schedule {w(z)} as given, firms make their choice of production location by facing the trade-off between land rent and labor cost and the external benefit from knowledge.
Equilibrium
Under this condition, a representative household (x, z) has no incentive to change employment or residential location, because the increased benefit from changing location exactly offsets the increased cost. If this condition is not met, there will be a positive measure of unoccupied land in equilibrium. 9 The employment choice function parallels the “driving pattern” function in Fujita and Ogawa (1982).
As we will see in Section 4 below, the symmetric nature of the model implies that it is not necessary to solve the job choice function in the characterization of the equilibrium. Part (ii) defines the equilibrium prime rate as the upper bound of the two bid rate functions RF and RC. Since both the demand for and the supply of land at the frontier are completely inelastic, the equilibrium land rent is indeterminate.
Part (iii) specifies no profit under perfect competition, and (iv) implies no vacant land in the city. It is therefore sufficient to examine the equilibrium conditions on the right side of the city where z, x $ 0.
Endogenous Determination of Urban Configuration
- The Completely Mixed Urban Configuration
- The Monocentric Urban Configuration
- The Incompletely Mixed Urban Configuration
- Can the Duocentric Urban Configuration be an Equilibrium Outcome?
A fully mixed urban configuration (denoted by J = C) is one in which firms and workers occupy all locations along a linear city (see Figure 1). Figure 2 shows the graph of equilibrium land rent and wage schedules in a linear city. This, together with (6), ensures that for each household the equilibrium choice of residence is equal to its workplace, resulting in an urban configuration in which firms and households are mixed in all locations, and the equilibrium cost of commuting is reduced to zero.
As a result, companies are more willing to pay higher wages to attract workers from the suburbs to commute to work. Simple comparative statistics show that a larger penalty on firm proliferation due to less effective knowledge spillovers (a larger g) results in an increase in the equilibrium price, the equilibrium. Therefore, in the remainder of the article we will limit our attention to the analysis of the equilibrium basic rent, ignoring the equilibrium wage for the sake of brevity.
We next turn to examine the case of a monocentric city configuration (denoted J = M), where all firms locate towards the city center within an interval [-q, q] (0 < q < 1), while households live on the outskirts [ -1, -q] and [q, 1] (see Figure 3). Thus, for different values of the unit commuting cost, t, we can determine the corresponding equilibrium city configuration as illustrated in Figure 5. Intuitively, when the unit commuting cost is sufficiently high, the fully mixed city configuration emerges as the unique equilibrium outcome; when unit commuting costs are sufficiently low, the unique urban equilibrium configuration turns out to be monocentric.
A question remains which urban configuration emerges in equilibrium and whether the urban equilibrium configuration is unique when unit commuting costs are moderate [i.e. B2(g,N) < t < B1(g,N). Only in the area around the city center are companies and households completely mixed. Proposition 3: Under condition I there is a competitive spatial equilibrium with an incompletely mixed symmetric urban configuration, in the sense that firms locate in the area [-f2, f2], while households move both towards the city center [-f1, f1] if living nearby. suburbs [-1, -f2] and [f2, 1] for 0 < f1 < f2 < 1.
Since firms and households in the central cluster [-f1 , f1 ] are fully mixed, their bid rates must be identical, as given by (15). The first is similar to the fully mixed case, which guarantees that no household in the central cluster wants to work outside the home. This theorem states that when the unit commuting costs and the penalty for business dispersal are moderate, the urban configuration is incompletely mixed; it is neither fully dispersed (as in the fully mixed case) nor fully concentrated (as in the monocentric case).
Thus, there is a key difference between an imperfectly mixed and a duocentric urban configuration: in the latter case, the bid rent for firms in (-2qD, 2qD) is strictly lower than that for households, so the equilibrium land rent is equal to the household bid rent within this central cluster, where no company found. By similar arguments, any multicentric urban configuration can be ruled out as a configuration of spatial balance.
Further Discussion
Importantly, under our knowledge spillover setup, a firm's output penalty for distance from the average location of firms is strictly increasing and strictly convex, implying that it is unfavorable for firms to be spatially separated into different clusters. Given such a duocentric configuration, a firm will always move to the average location of firms; penalties, ground rents and wages are all lower there. It is important to note that the proof (by contradiction) of Proposition 4 relies on mutually contradictory slope conditions.
Specifically, the properties of the function that pairwise compares fixed locations rather than its integral seem important.
Existence) For any commuting cost and dispersion penalty parameters, there is a competitive spatial equilibrium
Uniqueness) Almost surely in commuting cost and dispersion penalty parameters, there is a unique competitive spatial equilibrium associated with a symmetric urban configuration which is
Concluding Remarks
Based on Romer-type production externality, our paper has developed a general equilibrium framework in which the unique urban equilibrium configuration (completely mixed, monocentric or incompletely mixed) is determined analytically depending on the population of firms, the residential cost work traffic and company size. ' knowledge spillover parameters. We show that the integration of distance-dependent production externalities is sufficient to ensure that firms are always clustered together in a competitive spatial equilibrium, ruling out the possibility of multicentric urban configurations, in contrast to findings in Fujita and Ogawa (1982 ) and in more recent sequels using models of product differentiation or spatial competition. First, we can relax the assumptions of fixed labor supply and fixed demand for land.
The main purpose of these exercises is to check the robustness of the absence of multi-centered city. Of course, they are achieved at the cost of increased complexity, which makes analytical results less likely. Furthermore, it may be interesting to investigate the welfare properties of competitive spatial equilibrium.
In particular, the presence of uncompensated knowledge spillovers can lead to a suboptimal equilibrium – in equilibrium, firms do not take into account the positive externalities of production and thus underinvest compared to the optimal one. An interesting question is whether such inefficiencies are smaller in one urban configuration compared to others. In addition, we can add an externality to the consumer's utility through local congestion or neighborhood effects.
Finally, our findings provide empirically testable hypotheses regarding (i) the shape of the rental density, (ii) the locations of firms and consumers, and (iii) comparative statics that account for the dependence of the urban configuration (measured as a discrete variable) on commuting, population and production overflow parameters. In order to support the fully mixed urban setting, it is required that no household has an incentive to work away from home, because by working for a firm closer to the city center, the incremental profit from a higher wage dominated by the induced cost of commuting. . In order to support this urban configuration, it is required that in spatial equilibrium, (i) no household in the central cluster desires to work away from home (a condition similar to that in the completely mixed case).
In condition I, (A20) holds for every 0 < f1 < f2 < 1, thus ensuring the non-mixed urban configuration in spatial equilibrium. Fujita, 1992, "Alonso's Discrete Land Use Population Model: Efficient Allocations and Competitive Equilibria", International Economic Review. Chatterji (ed.), Regional Science: Perspectives for the Future, (St. Martin Press, Inc., New York, NY).
Ogawa, 1982, "Multiple Equilibria and Structural Transition of Non-Monocentric Urban Configurations," Regional Science and Urban Economics. Henderson, 1993, "Geographic Localization of Knowledge Spillovers as Evidenced by Patent Citations," Quarterly Journal of Economics.
