A Simple City Equilibrium Model with an Application to Teleworking

Abstract

We propose a simple semi-discrete spatial model where rents, wages and the density of population in a city can be deduced from free-mobility and equilibrium conditions on the labour and residential housing markets. We prove existence and (under stronger assumptions) uniqueness of the equilibrium. We extend our model to the case where teleworking is introduced. We present numerical simulations which shed light on the effect of teleworking on the structure of the city at equilibrium.

Appendix

1.1 A. Softmax as the Expectation of a Max

Let us first recall that a random variable $\varepsilon$ has a centered standard Gumbel distribution if its cdf has the following double exponential form:

$$\begin{aligned}\mathbb {P}(\varepsilon \le t)=\exp (-\exp (-t-\gamma )), \; \forall t\in \mathbb {R}, \text{ with } \gamma :=-\int _0^{+\infty } \log (s) \exp (-s) \text{ d }s.\end{aligned}$$

Consider now $\varepsilon _0, \cdots , \varepsilon _N$, $N+1$ i.i.d. distributed with a centered standard Gumbel and for $\beta =\beta _0, \ldots , \beta _N \in \mathbb {R}^{N+1}$ set

$$\begin{aligned}V(\beta ):=\mathbb {E}(\max _{i=0, \ldots , N} (\beta _i + \varepsilon _i))\end{aligned}$$

since

$$\begin{aligned}\mathbb {P}( (\max _{i=0, \ldots , N} (\beta _i + \varepsilon _i) \le t)=\exp \Big (-\Lambda (\beta ) \exp (-t-\gamma )\Big ), \text{ with } \Lambda (\beta )=\sum _{i=0}^N e^{\beta _i}\end{aligned}$$

we have

$$\begin{aligned}\begin{aligned} V(\beta )&=\int _{\mathbb {R}} t \exp \Big (- \exp (-t +\log \Lambda (\beta )-\gamma )\Big ) \exp (-t +\log \Lambda (\beta )-\gamma ) \text{ d }t \\&= \int _{\mathbb {R}} (s+ \log (\Lambda (\beta )) \exp \Big (- \exp (-s -\gamma )\Big ) \exp (-s -\gamma ) \text{ d }s \\&= \mathbb {E}( \varepsilon + \log (\Lambda (\beta )))= \log (\Lambda (\beta ))=\log \Big (\sum _{i=0}^N e^{\beta _i} \Big ). \end{aligned}\end{aligned}$$

Recalling the expression (2.13) for $R_\sigma$, we thus have

$$\begin{aligned}\mathbb {E}(\max _{i=0, \ldots , N} (w_i-c_i(x)+\sigma \varepsilon _i))= \sigma \log \Big (\sum _{i=0}^N e^{\frac{w_i-c_i(x)}{\sigma }} \Big ) = R_\sigma (x,w)\end{aligned}$$

which shows (2.15). Moreover, it follows from Lebesgue’s dominated convergence theorem that V is differentiable with

$$\begin{aligned}\frac{\partial V}{\partial \beta _i} (\beta )= \mathbb {P}( \beta _i+\varepsilon _i \ge \beta _j +\varepsilon _j, \; \forall j=0, \ldots , N)=\frac{e^{\beta _i}}{\sum _{j=0}^N e^{\beta _j}}\end{aligned}$$

which shows formula (2.16).

1.2 B. Existence and Uniqueness of Equilibria in the Teleworking Model

1.2.1 Existence

Under the assumptions of Sect. 4, we claim that there exists equilibrium wages i.e. a vector $\widetilde{w}\in \mathbb {R}_+^{2N}$ solving the fixed-point equation (4.14). To see this, we first argue as in Lemma 3.1 and find that for every $\mu \in {{\mathscr {P}}}(X)$, the functional $\widetilde{J}_\mu$ defined in (4.13) admits a unique minimizer $\widetilde{w}=(w_i^k)_{i=1, \ldots ,N, k=1,2}$ which satisfies

$$\begin{aligned}\max _{i,k} w_i^k + \sum _{i=1}^N \widetilde{\pi }_i(w_i^1, w_i^2) \le \overline{w}:= 2M + \sum _{i=1}^N \widetilde{\pi }_i(w_0, w_0) +\sigma \log (2N+1)+ w_0\end{aligned}$$

where $M:=\max _i \Vert c_i\Vert _{\infty }$. Thanks to the nonnegativity of $\widetilde{\pi }_i$ and (4.5), we find that the minimizer of $\widetilde{J}_\mu$ belongs to $[\underline{w}, \overline{w}]^{2N}$ where $0<\underline{w}\le \overline{w}$ are bounds that do not depend on $\mu$. The conclusion of Lemma 3.1 still holds for $\widetilde{J}_\mu$ so that the existence of an equilibrium follows from Brouwer’s theorem exactly as in the proof of theorem 3.2.

1.2.2 Uniqueness

Let us now further assume that the production functions $\widetilde{f}_i$ are of class $C^2$ on $(0,+\infty )^2$ and that (4.15) holds. Since $-\nabla \widetilde{f}_i$ is the inverse of $\nabla \widetilde{\pi }_i$, this implies that for every $(w^1, w^2)\in [\underline{w}, \overline{w}]^2$, the $2\times 2$ matrix $D^2 \widetilde{\pi }_i(w^1, w^2)$ is positive definite so that its smallest eigenvalue $\lambda _{\min }(D^2 \widetilde{\pi }_i(w^1, w^2))$ is positive (and depends on $(w^1, w^2)\in [\underline{w}, \overline{w}]^2$ in a continuous way). Setting $\alpha :=\frac{\theta }{1-\theta }$, and

$$\begin{aligned}\widetilde{\mu }(x, \widetilde{w}, \alpha ):=\frac{ \widetilde{R}_\sigma (x, \widetilde{w})^{\alpha }}{\int _X \widetilde{R}_\sigma (y, \widetilde{w})^{\alpha } \text{ d } y }, \; \forall (x, \widetilde{w}, \alpha ) \in X\times \mathbb {R}_+^{2N} \times \mathbb {R}_+\end{aligned}$$

we write the system of equilibrium conditions as the system of 2N equations in the 2N unknown $\widetilde{w}:=(w_1^1, w_1^2, \ldots w_N^1, w_N^2)$

$$\begin{aligned}\widetilde{G}(\widetilde{w}, \alpha )=0, \text{ where } \widetilde{G}_i^k(\widetilde{w}, \alpha )=\frac{\partial \widetilde{\pi }_i}{\partial w_i^k}(w_i^1, w_i^2) + \int _X \frac{\partial \widetilde{R}_{\sigma }}{\partial w_i^k} \widetilde{\mu }(x, \widetilde{w}, \alpha ) \text{ d }x \end{aligned}$$

so that the Jacobian matrix (with respect to $\widetilde{w}$) $\widetilde{A}$ of $\widetilde{G}$ reads

$$\begin{aligned}\begin{aligned} \widetilde{A}=&\left( \begin{array}{ccccc} D^2 \widetilde{\pi }_1(w_1^1, w_1^2) &{} O &{} O &{} \cdots &{} O \\ O &{} D^2 \widetilde{\pi }_2(w_2^1, w_2^2) &{} O &{} \cdots &{} O\\ \cdot &{} \cdot &{} \cdot &{} \cdots &{} \cdot \\ \cdot &{} \cdot &{} \cdot &{} \cdots &{} \cdot \\ O &{} O &{} O &{} \cdots &{} D^2 \widetilde{\pi }_N(w_N^1, w_N^2) \\ \end{array}\right) \\&+ \int _X D^2_{\widetilde{w}\widetilde{w}} \widetilde{R}_{\sigma } \widetilde{\mu }\\&+ \int _X \nabla _{\widetilde{w}} \widetilde{R}_\sigma \nabla _{\widetilde{w}} \widetilde{\mu }^{\top } \end{aligned}\end{aligned}$$

where the matrix in the first line is written in $2\times 2$ block-diagonal form (and O denotes the zero $2\times 2$ matrix) and the matrix on the second line is positive definite. Since all wages $w_i^k$ belong to $[\underline{w}, \overline{w}] \subset (0,+\infty )$, setting

$$\begin{aligned}\nu :=\min _{i=1, \ldots , N} \min _{(w^1, w^2) \in [\underline{w}, \overline{w}]^2} \lambda _{\min }(D^2 \widetilde{\pi }_i(w^1, w^2)) >0,\end{aligned}$$

we have, for every $\widetilde{\xi }\in \mathbb {R}^{2N}\setminus \{0\}$

$$\begin{aligned}\widetilde{A}\widetilde{\xi }\cdot \widetilde{\xi }> \Big ( \nu - \int _X \vert \nabla _{\widetilde{w}} \widetilde{R}_\sigma \vert \; \vert \nabla _{\widetilde{w}} \widetilde{\mu }\vert \Big ) \vert \widetilde{\xi }\vert ^2. \end{aligned}$$

Obviously $\vert \nabla _{\widetilde{w}} \widetilde{R}_\sigma \vert \le \sqrt{2N}$ and assuming that $\alpha \le 1$, arguing as in the proof of theorem 3.3, we get

$$\begin{aligned} \int _X \vert \nabla _{\widetilde{w}} \widetilde{\mu }\vert \le \frac{ \alpha }{w_0} \sqrt{2N}\end{aligned}$$

hence, we deduce that $\widetilde{A}$ is invertible as soon as

$$\begin{aligned} \alpha \le \frac{ w_0 \nu }{ 2N} \end{aligned}$$

For such a choice of $\alpha$, uniqueness of the equilibrium can be shown exactly as in the proof of theorem 3.3.