Interurban population distribution and commute modes

Abstract

This paper explores the interplay between trade costs and urban costs within a new economic geography model in which workers are mobile. As in former research, we show that workers tend at the same time to agglomerate in order to limit trade costs of manufactured goods and to scatter in order to alleviate the burden of urban costs due to large urban areas. In this paper, special attention is paid to the role of congestion, which acts as a dispersion force and hampers workers from agglomerating in the same urban area. We show that the development of public transport, or the construction of road infrastructure, modifies the spatial organization of the economy and fosters agglomeration, as it reduces congestion.

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Notes

  1. 1.

    The city border is defined as the location \(X_r\) where a worker can exactly make a round trip within the time constraint T, namely \(a_{r}(X_r)=2\).

  2. 2.

    The city border is then equal to \(X_r=\displaystyle \frac{v_{0}T}{2\left( 1+\left( \frac{L_{r}}{\kappa _{r}}\right) ^\alpha \right) }\).

  3. 3.

    These assumption is straightforward and may seem far from the facts. Indeed, the speed of public modes, such as buses, may be subject to congestion (this is less true for subways, trams and bus rapid transits). However, the assumption made allows us to represent the main features of a public transport network and to study its implication on spatial organization.

  4. 4.

    This equilibrium is reached at the short term, long before the equilibrium of workers’ migrations between the two regions is reached (namely \(\omega ^v\ll \omega ^U\)).

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Correspondence to Cédric Allio.

Appendix

Appendix

Proof 1 The symmetric configuration is a stable equilibrium if, and only if, \(\frac{dU_r}{U_r}\) has not the same sign than \(\frac{dL_r}{L_r}\). From Eq. (16), we have:

$$\begin{aligned} \left. \begin{array}{lll} \frac{dU_r}{U_r}\frac{dL_r}{L_r}<0 &{} \Leftrightarrow \; \mu \displaystyle \frac{(2\epsilon -1)Z}{(\epsilon -1)(\epsilon (Z+1)-Z)}<1\\ &{} \Leftrightarrow \; \left[ (2\epsilon -1)\mu -(\epsilon -1)^2\right] Z<\epsilon (\epsilon -1)\\ &{} \Leftrightarrow \;\left\{ \begin{array}{ll}Z<\displaystyle \frac{\epsilon (\epsilon -1)}{(2\epsilon -1)\mu -(\epsilon -1)^2} &{} \quad \mathrm{if} \;(2\epsilon -1)\mu -(\epsilon -1)^2>0\\ Z>\displaystyle \frac{\epsilon (\epsilon -1)}{(2\epsilon -1)\mu -(\epsilon -1)^2} &{} \quad \mathrm{if} \;(2\epsilon -1)\mu -(\epsilon -1)^2<0 \end{array}\right. \end{array}\right. \end{aligned}$$

Note that \(Z\in [0,1]\). We obtain:

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \frac{\epsilon (\epsilon -1)}{(2\epsilon -1)\mu -(\epsilon -1)^2}<0 &{} \Leftrightarrow \; \mu<\displaystyle \frac{(\epsilon -1)^2}{2\epsilon -1}\\ \displaystyle \frac{\epsilon (\epsilon -1)}{(2\epsilon -1)\mu -(\epsilon -1)^2}>1 &{} \Leftrightarrow \; \mu <\epsilon -1 \end{array}\right. \end{aligned}$$

Thus, when \(\mu <\displaystyle \frac{(\epsilon -1)^2}{2\epsilon -1}\) or \(\mu <\epsilon -1\), we have \(\forall Z, \; \displaystyle \frac{dU_r}{U_r}\frac{dL_r}{L_r}<0\). And, the symmetric configuration is always a stable equilibrium. However,

$$\begin{aligned} \forall \epsilon>1, \quad \epsilon -1>\displaystyle \frac{(\epsilon -1)^2}{2\epsilon -1} \end{aligned}$$

Let

$$\begin{aligned} Z^*=\frac{\epsilon (\epsilon -1)}{(2\epsilon -1)\mu -(\epsilon -1)^2} \end{aligned}$$

Eventually, when \(\mu <\epsilon -1\), the symmetric configuration is always a stable equilibrium. Otherwise, the symmetric configuration is stable if, and only if, \(Z<Z^*\).

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Allio, C. Interurban population distribution and commute modes. Ann Reg Sci 57, 125–144 (2016). https://doi.org/10.1007/s00168-016-0766-5

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JEL Classification

  • R120 Size and Spatial Distributions of Regional Economic Activity
  • R410 Transportation: Demand, Supply, and Congestion; Travel Time; Safety and Accidents; Transportation Noise
  • R300 Real Estate Markets, Spatial Production Analysis, and Firm Location: General