Location in a Disk City with Consumer Concentration Around the Center

Abstract

The paper studies a two-stage location-price duopoly game in a disk city with consumer concentration around the city center. When consumers are uniformly distributed over the plane, unconstrained firms locate outside of the city. Consumer concentration, however, induces firms to locate nearer to each other and, when the degree of concentration is sufficiently high, inside of the city. Prices and firm profits decrease in the degree of consumer concentration. We explicitly solve the model for classes of cone-shaped, dome-shaped, and bell-shaped consumer densities. In all cases we identify a loss of welfare due to the strategic effect which causes the firms’ spatial differentiation being too large.

Appendix

As described in the main text, the demand function of firm 1 reads

$$D(\tilde{x})=2\int_{\tilde{x}}^{1}f(r)r\arccos(\tilde{x}/r)dr$$

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and the demand of firm 2 is $1-D(\tilde{x})$. By using Leibniz’s rule and integrating by parts, we obtain the derivatives

$$D^{\prime}(\tilde{x})=-2\int_{\tilde{x}}^{1}\frac{f(r)}{\sqrt{1-(\tilde{x}/r)^{2}}}dr =-2\left[f(1)\sqrt{1-\tilde{x}^{2}}-\int_{\tilde{x}}^{1}f^{\prime}(r)\sqrt{r^{2}-\tilde{x}^{2}}dr\right]$$

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and

$$D^{\prime\prime}(\tilde{x})=2\tilde{x}\left[\frac{f(1)}{\sqrt{1-\tilde{x}^{2}}}-\int_{\tilde{x}}^{1}\frac{f^{\prime}(r)}{\sqrt{r^{2}-\tilde{x}^{2}}}dr\right].$$

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In the case of $\tilde{x}=0$ (symmetric firm location around the city center), one obtains¹⁰

$$D(\tilde{x}=0)=1/2.$$

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The first two derivatives amount to

$$D^{\prime}(\tilde{x}=0)=-2\int_{0}^{1}f(r)dr<0$$

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and

$$D^{\prime\prime}(\tilde{x}=0)=0.$$

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