A Continuous Space Model of New Economic Geography with a Quasi-Linear Log Utility Function

Abstract

We consider the extension of a tractable NEG model with a quasi-linear log utility function to continuous space, and investigate the behavior of its solution mathematically. The model is a system of nonlinear integral and differential equations describing the market equilibrium and the time evolution of the spatial distribution of population density. A unique global solution is constructed, and a homogeneous stationary solution with an evenly distributed population is shown to be unstable. Furthermore, it is shown numerically that the destabilized homogeneous stationary solution eventually forms spiky spatial distributions. The number of the spikes decreases as the preference for variety increases or the transport cost decreases.

Appendix

This section is devoted to the proofs omitted in the main text. Recall that $T_1$ and $T_2$ are the lower and upper bounds of T(x, y) in (6), respectively.

1.1 Proof of Theorem 2

Lemma 1

The following inequalities

$$\begin{aligned}&\left( \frac{F}{\Lambda +b}\right) ^{\frac{1}{\sigma -1}}T_1 \le \Vert G\left( \lambda (t)\right) \Vert _\infty \le \left( \frac{F}{\Lambda -b}\right) ^{\frac{1}{\sigma -1}}T_2, \end{aligned}$$

(46)

$$\begin{aligned}&\begin{aligned}\Vert w\left( \lambda (t)\right) \Vert _\infty&\le \frac{\mu }{\sigma }\left( \frac{T_2}{T_1}\right) ^{\sigma -1}\frac{\Phi +\Lambda +b}{\Lambda -b}, \\ \Vert \omega \left( \lambda (t)\right) \Vert _\infty&\le \frac{\mu }{\sigma }\left( \frac{T_2}{T_1}\right) ^{\sigma -1}\frac{\Phi +\Lambda +b}{\Lambda -b} \end{aligned} \end{aligned}$$

(47)

$$\begin{aligned}&+ \mu \max \left\{ \left| \ln \left[ \left( \frac{F}{\Lambda -b}\right) ^{\frac{1}{\sigma -1}}T_1\right] \right| , \left| \ln \left[ \left( \frac{F}{\Lambda -b}\right) ^{\frac{1}{\sigma -1}}T_2\right] \right| \right\} \end{aligned}$$

(48)

hold for the operators (26), (27), and (28).

Proof of Lemma 1

Let us first observe that any $\lambda$ such that $\left\| \lambda (t)-\lambda _0\right\| _{L^1}\le b$ satisfies

$$\begin{aligned} 0<\Lambda -b\le \left\| \lambda (t)\right\| _{L^1}\le \Lambda +b. \end{aligned}$$

(49)

Then, from (26) and (49), it follows that

$$\begin{aligned} \begin{aligned} F^{\frac{1}{\sigma -1}}T_1\left( \Lambda +b\right) ^{\frac{1}{1-\sigma }}\le \left| G(\lambda (t))(x)\right| \le F^{\frac{1}{\sigma -1}}T_2\left( \Lambda -b\right) ^{\frac{1}{1-\sigma }}, \end{aligned} \end{aligned}$$

which immediately yield (46). It follows from (46), (27), and (49) that

$$\begin{aligned} \left| w(\lambda (t))(x)\right|&\le \frac{\mu }{\sigma F}T_1^{1-\sigma }\left[ \left( \frac{F}{\Lambda -b}\right) ^{\frac{1}{\sigma -1}}T_2\right] ^{\sigma -1}\left( \Phi +\Lambda +b\right) , \end{aligned}$$

which immediately yields (47). It follows from (46) and (47) that

$$\begin{aligned} \left| \omega (x)\right|&\le \left| w(x)\right| + \mu \left| \ln G(x)\right| ,\\&\le \frac{\mu }{\sigma }\left( \frac{T_2}{T_1}\right) ^{\sigma -1}\frac{\Phi +\Lambda +b}{\Lambda -b}\\&\quad {5\,mm}+ \mu \max \left\{ \left| \ln \left[ \left( \frac{F}{\Lambda -b}\right) ^{\frac{1}{\sigma -1}}T_1\right] \right| , \left| \ln \left[ \left( \frac{F}{\Lambda -b}\right) ^{\frac{1}{\sigma -1}}T_2\right] \right| \right\} , \end{aligned}$$

which immediately gives (48).

By (29) and (49), we have that

$$\begin{aligned} \left\| \Psi \left( \lambda (t)\right) \right\| _{L^1}&= \gamma \int _{\mathcal {M}}\left| \left[ \omega (\lambda (t))(x)-\frac{1}{\Lambda }\int _{\mathcal {M}}\omega (\lambda (t))(y)\lambda (y) dy\right] \lambda (x)\right| dx\\&\le \gamma \left\| \omega (\lambda (t))\right\| _\infty \left( \Lambda +b\right) \left( 1+\frac{\Lambda +b}{\Lambda }\right) . \end{aligned}$$

Together with (48), this completes the proof.$\square$

1.2 Proof of Theorem 3

Firstly, we show that G is Lipschitz continuous. For $\lambda _1,\lambda _2\in Q$, let us define

$$g_i(x):= \frac{1}{F}\int _{\mathcal {M}}\left| \lambda _i(y)\right| T(x,y)^{1-\sigma }dy,~i=1,2$$

so that $G\left( \lambda _i\right) (x) = g_i(x)^{\frac{1}{1-\sigma }}$. Then, based on the mean-value theorem in a Banach space, we obtain

$$\begin{aligned} \left\| G\left( \lambda _1\right) -G\left( \lambda _2\right) \right\| _\infty \le \mathcal {C}\left\| g_1-g_2\right\| _\infty , \end{aligned}$$

(50)

where $\mathcal {C}=\frac{F^{\frac{\sigma }{1-\sigma }}T_2^{\sigma }(\Lambda -b)^{\frac{\sigma }{1-\sigma }}}{\sigma -1}$. Then, we see that

$$\begin{aligned} \begin{aligned} \left\| g_1-g_2\right\| _\infty&=\max _{x\in \mathcal {M}}\left| \frac{1}{F}\int _{\mathcal {M}}\left( |\lambda _1(y)|-|\lambda _2(y)|\right) T(x,y)^{1-\sigma }dy\right| \\&\le \frac{T_1^{1-\sigma }}{F}\int _{\mathcal {M}}\left| \left| \lambda _1(y)\right| -\left| \lambda _2(y)\right| \right| dy\\&\le \frac{T_1^{1-\sigma }}{F}\int _{\mathcal {M}}\left| \lambda _1(y)-\lambda _2(y)\right| dy= \frac{T_1^{1-\sigma }}{F} \left\| \lambda _1-\lambda _2\right\| _{L^1}. \end{aligned} \end{aligned}$$

(51)

From (50) and (51), we have

$$\begin{aligned} \left\| G\left( \lambda _1\right) -G\left( \lambda _2\right) \right\| _\infty \le \mathcal {L}_G \left\| \lambda _1-\lambda _2\right\| _\infty , \end{aligned}$$

(52)

where $\mathcal {L}_G>0$ is a constant.

Secondly, we show that w is Lipschitz continuous.

$$\begin{aligned} \begin{aligned} \Vert w(\lambda _1)-w(\lambda _2)\Vert _\infty&= \max _{x\in \mathcal {M}}\left| w(\lambda _1)(x)-w(\lambda _2)(x)\right| \\&= \max _{x\in \mathcal {M}}\left| \frac{\mu }{\sigma F}\int _{\mathcal {M}}\left( \phi (y)+\left| \lambda _1(y)\right| \right) G(\lambda _1)(y)^{\sigma -1}T(x,y)^{1-\sigma }dy\right. \\&\quad\left. -\frac{\mu }{\sigma F}\int _{\mathcal {M}}\left( \phi (y)+\left| \lambda _2(y)\right| \right) G(\lambda _2)(y)^{\sigma -1}T(x,y)^{1-\sigma }dy\right| \\&\le \frac{\mu }{\sigma F}T_1^{1-\sigma }\left( \Phi +\left\| \lambda _1\right\| _{L^1}\right) \left\| G(\lambda _1)^{\sigma -1}-G(\lambda _2)^{\sigma -1}\right\| _\infty \\&\quad+ \frac{\mu }{\sigma F}T_1^{1-\sigma }\left\| G^{\sigma -1}\right\| _\infty \left\| \lambda _1-\lambda _2\right\| _{L^1} \\&\le \frac{\mu }{\sigma F}T_1^{1-\sigma }\left( \Phi +\Lambda +b\right) \left\| G(\lambda _1)^{\sigma -1}-G(\lambda _2)^{\sigma -1}\right\| _\infty \\&\quad+ \frac{\mu }{\sigma }\left( \frac{T_2}{T_1}\right) ^{\sigma -1}\frac{1}{\Lambda -b}\left\| \lambda _1-\lambda _2\right\| _{L^1} \end{aligned} \end{aligned}$$

(53)

Here, (46) and (49) are used in the last deformation. Then, by the mean-value theorem and (46), we obtain

$$\begin{aligned} \left\| G(\lambda _1)^{\sigma -1}-G(\lambda _2)^{\sigma -1}\right\| _\infty \le \mathcal {C}\left\| G(\lambda _1)-G(\lambda _2)\right\| _\infty , \end{aligned}$$

(54)

where

$$\mathcal {C} = \left\{ \begin{aligned}&(\sigma -1)\left( \frac{F}{\Lambda -b}\right) ^{\frac{\sigma -2}{\sigma -1}}T_2^{\sigma -2},~~~\textrm{if}~\sigma -2\ge 0\\&(\sigma -1)\left( \frac{F}{\Lambda -b}\right) ^{\frac{\sigma -2}{\sigma -1}}T_1^{\sigma -2},~~~\textrm{if}~\sigma -2 <0. \end{aligned} \right.$$

From (53) and (54), we obtain

$$\begin{aligned} \left\| w(\lambda _1)-w(\lambda _2)\right\| _\infty \le \mathcal {L}_w \left\| \lambda _1-\lambda _2\right\| _\infty , \end{aligned}$$

(55)

where $\mathcal {L}_w>0$ is a constant.

Thirdly, we show that $\omega$ is Lipschitz continuous. By the mean-value theorem and (46), we obtain

$$\begin{aligned} \left\| \ln G(\lambda _1)-\ln G(\lambda _2)\right\| _\infty \le \left( \frac{\Lambda -b}{F}\right) ^{\frac{1}{\sigma -1}}T_1\left\| G(\lambda _1) - G(\lambda _2)\right\| _\infty \end{aligned}$$

(56)

It follows from (55) and (56) that

$$\begin{aligned} \begin{aligned} \left\| \omega (\lambda _1)-\omega (\lambda _2)\right\| _\infty&\le \left\| w(\lambda _1)-w(\lambda _2)\right\| _\infty + \mu \left\| \ln G(\lambda _1)- \ln G(\lambda _2)\right\| _\infty \\&\le \mathcal {L}_{\omega } \Vert \lambda _1-\lambda _2\Vert _{L^1} \end{aligned} \end{aligned}$$

(57)

where $\mathcal {L}_\omega >0$ is a constant.

We are now able to show the Lipschitz continuity of $\Psi (\lambda )$. By (48) and (49), we see that

$$\begin{aligned} \begin{aligned} \left\| \Psi (\lambda _1)-\Psi (\lambda _2)\right\| _{L^1}&= \gamma \int _{\mathcal {M}}\left| \omega (\lambda _1)(x)\lambda _1(x) - \frac{1}{\Lambda }\int _\mathcal {M}\omega (\lambda _1)(y)\lambda _1(y)dy\cdot \lambda _1(x)\right. \\&\quad \left. -\omega (\lambda _2)(x)\lambda _2(x) + \frac{1}{\Lambda }\int _\mathcal {M}\omega (\lambda _2)(y)\lambda _2(y)dy\cdot \lambda _2(x)\right| dx \\&\le 2\gamma \left\| \omega \right\| _\infty \left\| \lambda _1-\lambda _2\right\| _{L^1} \\&\quad + \gamma (\Lambda +b)^2\left\| \omega (\lambda _1)-\omega (\lambda _2)\right\| _\infty \end{aligned} \end{aligned}$$

(58)

Thus, (57) and (58) complete the proof.$\square$

1.3 Proof of Theorem 6

For any $\lambda \in L^1_\Lambda$, discussion similar to that in the proof of Theorem 2 but now with $\left\| \lambda \right\| _{L^1}=\Lambda$ instead of (49) completes the proof.$\square$