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On spatial systems of cities

Gianandrea Lanzara, Matteo Santacesaria

TL;DR

This paper develops a unified spatial framework that merges systems of cities with regional models to address whether spatial economies admit multiple equilibria. It endogenizes commuting areas, representing them as additively weighted Voronoi tessellations, and embeds an Armington gravity structure with spillovers in productivity and amenities. The main contributions are analytical conditions for the existence and uniqueness of equilibria, including a regime with multiple equilibria in urban locations but a unique labor distribution across active centers, which helps reconcile conflicting empirical findings on geography and scale economies. The framework thereby explains how historical contingencies can persist in the urban system while city sizes respond in a determinate way, and provides a tractable approach to studying geographic policy and infrastructure effects in spatial economies.

Abstract

Are there multiple equilibria in the spatial economy? This paper develops a unified framework that integrates systems of cities and regional models to address this question within a general geographic space. A key feature is the endogenous formation of commuting areas linking a continuum of residential locations to a finite set of potential business districts. Using tools from computational geometry and shape optimization, we derive sufficient conditions for the existence and uniqueness of spatial equilibria. For plausible parameter values, urban location is indeterminate, but, conditional on an urban system, city sizes are uniquely determined. The framework reconciles seemingly conflicting empirical findings on the role of geography and scale economies in shaping the spatial economy.

On spatial systems of cities

TL;DR

This paper develops a unified spatial framework that merges systems of cities with regional models to address whether spatial economies admit multiple equilibria. It endogenizes commuting areas, representing them as additively weighted Voronoi tessellations, and embeds an Armington gravity structure with spillovers in productivity and amenities. The main contributions are analytical conditions for the existence and uniqueness of equilibria, including a regime with multiple equilibria in urban locations but a unique labor distribution across active centers, which helps reconcile conflicting empirical findings on geography and scale economies. The framework thereby explains how historical contingencies can persist in the urban system while city sizes respond in a determinate way, and provides a tractable approach to studying geographic policy and infrastructure effects in spatial economies.

Abstract

Are there multiple equilibria in the spatial economy? This paper develops a unified framework that integrates systems of cities and regional models to address this question within a general geographic space. A key feature is the endogenous formation of commuting areas linking a continuum of residential locations to a finite set of potential business districts. Using tools from computational geometry and shape optimization, we derive sufficient conditions for the existence and uniqueness of spatial equilibria. For plausible parameter values, urban location is indeterminate, but, conditional on an urban system, city sizes are uniquely determined. The framework reconciles seemingly conflicting empirical findings on the role of geography and scale economies in shaping the spatial economy.
Paper Structure (33 sections, 16 theorems, 167 equations, 1 figure)

This paper contains 33 sections, 16 theorems, 167 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A spatial model of urban systems
  3. Setup
  4. Consumer's problem.
  5. Firm's problem.
  6. Spillovers.
  7. Market clearing.
  8. Welfare.
  9. Equilibrium.
  10. Restrictions on the geography.
  11. Endogenous commuting patterns.
  12. Existence and uniqueness of spatial equilibria
  13. Using the Voronoi weights to describe the equilibria
  14. Y*-centric equilibrium
  15. From Y*-centric equilibria to spatial equilibria
  16. ...and 18 more sections

Key Result

Proposition 1

Suppose that Assumptions ass:Ab-ass:admiss hold, and fix $\mathcal{Y}^* \subseteq \mathcal{Y}$. Then, the vector of Voronoi weights $\lambda$ and the level of welfare $V$ that are consistent with a $\mathcal{Y}^*$-centric equilibrium coincide with the solutions of the system of equations $i = 1, \dots, n^*,$ together with the aggregate population constraint eq:aggpop.

Figures (1)

  • Figure 1: Multiple spatial equilibria and model parameters

Theorems & Definitions (41)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • Theorem 2
  • proof
  • Lemma 3.1
  • ...and 31 more