When Veblen meets Krugman: social network and city dynamics

Abstract

The present paper explores the role of the social structure of suburban areas on city dynamics. We focus on relative concerns in the form of conspicuous consumption and introduce them into a standard economic geography model a la Krugman. We show that the level of social integration within the suburban areas of cities and the level of economic integration across cities are crucial in determining the city sizes. An interesting case arises with moderate trade costs when relatively small shares of income are devoted to the consumption of the differentiated good: if classes of workers are segregated (as in homogenous suburban areas), relative concerns tend to generate dispersed, medium-size cities; when workers of different classes socially interact, relative concerns contribute to foster socially integrated megalopolises. This result shows that keeping-up-with-the-Joneses motives may generate counterintuitive results when agents are able to choose their location.

Appendices

Appendix 1

1.1 Demand side and price index

In this Appendix, we compute the consumer’s demand for the differentiated good $X_{ir}$ and for the agricultural good $A_{ir}$ in region r, the indirect utility function $U(A_{ir},\varPhi \left( X_{ir},X_{-ir},\Lambda _{r}(i)\right) )$, the individual demand in region r for variety v ($X_{ir}(v)$) and we define the price index $p_{X_{r}}$.

Each individual i in region r solves the program

$$\begin{aligned}&\mathrm{Max}_{A_{ir},X_{ir}}U(A_{ir},\varPhi \left( X_{ir},X_{-ir},\Lambda _{r}(i)\right) )=A_{ir}^{1-\mu }\left( \varPhi \left( X_{ir},X_{-ir},\Lambda _{r}(i)\right) \right) ^{\mu }\\&\mathrm{s.t.}\qquad A_{ir}+\int \limits _{s\in N}p_{r}(s)X_{ir}(s)ds=w_{ir} \end{aligned}$$

with

$$\begin{aligned} \varPhi \left( X_{ir},X_{-ir},\Lambda _{r}(i)\right) =X_{ir}+\alpha S(\Omega (\Lambda _{r}(i)))\left[ X_{ir}-\frac{\int _{\Lambda _{r}(i)}X_{jr}\mathrm{d}j}{ \int _{\Lambda _{r}(i)}\mathrm{d}j}\right] \end{aligned}$$

As $\int _{\Lambda _{r}(i)}\mathrm{d}j=\Omega (\Lambda _{r}(i))$ and $X_{ir}=\left( \displaystyle \int \limits _{s\in N}X_{ir}(s)^{\frac{\sigma -1}{\sigma }}ds\right) ^{\frac{ \sigma }{\sigma -1}}$, the Lagrangean function is

$$\begin{aligned} L= & {} A_{ir}^{1-\mu }\left[ \left[ 1+\alpha S(\Omega (\Lambda _{r}(i)))\right] \left( \int \limits _{s\in N}X_{ir}(s)^{\frac{\sigma -1}{\sigma }}ds\right) ^{ \frac{\sigma }{\sigma -1}}\right. \\&\qquad \qquad \left. -\alpha S(\Omega (\Lambda _{r}(i)))\frac{ \int _{\Lambda _{r}(i)}X_{jr}\mathrm{d}j}{\Omega (\Lambda _{r}(i))} \right] ^{\mu } \\&\quad +\,\lambda \left( w_{ir}-A_{ir}-\int \limits _{s\in N}p_{r}(s)X_{ir}(s)ds\right) \end{aligned}$$

with the first-order conditions with respect to $A_{ir}$, the consumption of variety s, $X_{ir}(s)$, and of variety v, $X_{ir}(v)$, and of $\lambda$ , respectively, given by

$$\begin{aligned}&\left( 1-\mu \right) A_{ir}^{-\mu }\varPhi ^{\mu }=\lambda \end{aligned}$$

(29)

$$\begin{aligned}&\mu A_{ir}^{1-\mu }\varPhi ^{\mu -1}\left[ 1+\alpha S(\Omega (\Lambda _{r}(i))) \right] \left( x\right) ^{\frac{\sigma }{\sigma -1}-1}X_{ir}(s)^{\frac{ \sigma -1}{\sigma }-1}=\lambda p_{r}(s) \end{aligned}$$

(30)

$$\begin{aligned}&\mu A_{ir}^{1-\mu }\varPhi ^{\mu -1}\left[ 1+\alpha S(\Omega (\Lambda _{r}(i))) \right] \left( x\right) ^{\frac{\sigma }{\sigma -1}-1}X_{ir}(v)^{\frac{ \sigma -1}{\sigma }-1}=\lambda p_{r}(v) \end{aligned}$$

(31)

$$\begin{aligned}&w_{ir}=A_{ir}+\int \limits _{s\in N}p_{r}(s)X_{ir}(s)ds \end{aligned}$$

(32)

with $x=\int \limits _{s\in N}X_{ir}(s)^{\frac{\sigma -1}{\sigma }}ds$. Considering the ratio of (30) and (31), we obtain:

$$\begin{aligned} \frac{X_{ir}(s)^{-\frac{1}{\sigma }}}{X_{ir}(v)^{-\frac{1}{\sigma }}}=\frac{ p_{r}(s)}{p_{r}(v)} \end{aligned}$$

(33)

Let us notice that from (33), we obtain that

$$\begin{aligned} X_{ir}(s)=\left( \frac{p_{r}(v)}{p_{r}(s)}\right) ^{\sigma }X_{ir}(v) \end{aligned}$$

which can be substituted in the definition of $X_{ir}\equiv \left( \displaystyle \int \limits _{s\in N}X_{ir}(s)^{\frac{\sigma -1}{\sigma }}ds\right) ^{\frac{ \sigma }{\sigma -1}}$ to get

$$\begin{aligned} X_{ir}(v)=\dfrac{p_{r}(v)^{-\sigma }}{\left( \displaystyle \int \limits _{s\in N}\left( p_{r}(s)\right) ^{1-\sigma }ds\right) ^{\frac{\sigma }{\sigma -1}}}X_{ir}\ \ \ \end{aligned}$$

(34)

From (29), (31) and (34), we get that

$$\begin{aligned} A_{ir}=\frac{\left[ \left[ 1+\alpha S(\Omega (\Lambda _{r}(i)))\right] X_{ir}-\alpha S(\Omega (\Lambda _{r}(i)))\frac{\int _{\Lambda _{r}(i)}X_{jr}\mathrm{d}j }{\Omega (\Lambda _{r}(i))}\right] \left( 1-\mu \right) p_{X_{r}}}{\mu \left[ 1+\alpha S(\Omega (\Lambda _{r}(i)))\right] } \end{aligned}$$

(35)

where $p_{X_{r}}\equiv \left( \int \limits _{s\in N}(p_{r}(s))^{1-\sigma }ds\right) ^{\frac{1}{1-\sigma }}$ is the price index of manufactured varieties. Then, we substitute (35) in (32) and we use also (34) to obtain the consumer’s demand for the differentiated good in region r

$$\begin{aligned} X_{ir}=\frac{\mu }{p_{X_{r}}}\left( w_{ir}+p_{X_{r}}\frac{1-\mu }{\mu }\frac{ \alpha S(\Omega (\Lambda _{r}(i)))}{1+\alpha S(\Omega (\Lambda _{r}(i)))} \frac{\int _{\Lambda _{r}(i)}X_{jr}\mathrm{d}j}{\Omega (\Lambda _{r}(i))}\right) \end{aligned}$$

(36)

Previous expression can be substituted in (35) to obtain the consumer’s demand for the agricultural good in region r, that is

$$\begin{aligned} A_{ir}=\left( 1-\mu \right) \left( w_{ir}-p_{X_{r}}\frac{\alpha S(\Omega (\Lambda _{r}(i)))}{1+\alpha S(\Omega (\Lambda _{r}(i)))}\frac{\int _{\Lambda _{r}(i)}X_{jr}\mathrm{d}j}{\Omega (\Lambda _{r}(i))}\right) \end{aligned}$$

(37)

Then, substituting (36) and (37) in

$$\begin{aligned} U(A_{ir},\varPhi \left( X_{ir},X_{-ir},\Lambda _{r}(i)\right) )=A_{ir}^{1-\mu }\left( \varPhi \left( X_{ir},X_{-ir},\Lambda _{r}(i)\right) \right) ^{\mu } \end{aligned}$$

we obtain the indirect utility function:

$$\begin{aligned}&U(A_{ir},\varPhi \left( X_{ir},X_{-ir},\Lambda _{r}(i)\right) )=\frac{\left( 1-\mu \right) ^{1-\mu }\mu ^{\mu }\left( 1+\alpha S(\Omega (\Lambda _{r}(i)))\right) ^{\mu }}{\left( p_{X_{r}}\right) ^{\mu }} \\&\left( w_{ir}-p_{X_{r}}\frac{\alpha S(\Omega (\Lambda _{r}(i)))}{1+\alpha S(\Omega (\Lambda _{r}(i)))}\frac{\int _{\Lambda _{r}(i)}X_{jr}\mathrm{d}j)}{\Omega (\Lambda _{r}(i))}\right) \end{aligned}$$

Expression (34) can be used to derive an expression for the expenditure in manufacturing, $E_{ir}$, that is

$$\begin{aligned} E_{ir}=\int \limits _{s\in N}p_{r}(s)X_{ir}(s)ds=p_{X_{r}}X_{ir} \end{aligned}$$

This expression can be substituted into (34) to get the individual demand in region r for variety v

$$\begin{aligned} X_{ir}(v)=\frac{p_{r}(v)^{-\sigma }}{p_{X_{r}}^{1-\sigma }}E_{ir} \end{aligned}$$

1.2 Individual demand of the differentiated good by workers in region $\mathbf {r}$

In this Appendix, we show how we derive the individual demand by the skilled and by the unskilled consumers in region r. As $\int _{\Lambda _{r}(i)}\mathrm{d}j=\Omega (\Lambda _{r}(i))$, the demands for each skilled individual i in region r as given by (7) and (8) can be rewritten as

$$\begin{aligned} A_{H_{r}}= & {} \left( 1-\mu \right) \left[ w_{H_{r}}-p_{X_{r}}\frac{\alpha S(\Omega _{r}(H))}{1+\alpha S(\Omega _{r}(H))}\frac{\left( h_{H_{r}}X_{H_{r}}+l_{H_{r}}X_{L_{r}}\right) }{\Omega _{r}(H)}\right] , \\ X_{H_{r}}= & {} \tfrac{\mu }{p_{X_{r}}}\left[ w_{H_{r}}+p_{X_{r}}\frac{1-\mu }{ \mu }\frac{\alpha S(\Omega _{r}(H))}{1+\alpha S(\Omega _{r}(H))}\frac{ \left( h_{H_{r}}X_{H_{r}}+l_{H_{r}}X_{L_{r}}\right) }{\Omega _{r}(H)}\right] ; \nonumber \end{aligned}$$

(38)

while for each unskilled individual i in region r the demands are

$$\begin{aligned} A_{L_{r}}= & {} \left( 1-\mu \right) \left[ w_{L_{r}}-p_{X_{r}}\frac{\alpha S(\Omega _{r}(L))}{1+\alpha S(\Omega _{r}(L))}\frac{\left( h_{L_{r}}X_{H_{r}}+l_{L_{r}}X_{L_{r}}\right) }{\Omega _{r}(L)}\right] , \\ X_{L_{r}}= & {} \frac{\mu }{p_{X_{r}}}\left[ w_{L_{r}}+p_{X_{r}}\frac{1-\mu }{ \mu }\frac{\alpha S(\Omega _{r}(L))}{1+\alpha S(\Omega _{r}(L))}\frac{ \left( h_{L_{r}}X_{H_{r}}+l_{L_{r}}X_{L_{r}}\right) }{\Omega _{r}(L)}\right] . \nonumber \end{aligned}$$

(39)

Considering the second Eqs. in (38) and in (39 ), we obtain a system of two equations in the two unknowns $X_{H_{r}}$ and $X_{L_{r}}$ given by

$$\begin{aligned} \left\{ \begin{array}{l} X_{H_{r}}=w_{H_{r}}\frac{\mu }{p_{X_{r}}}+\frac{\alpha \left( 1-\mu \right) S(\Omega _{r}(H))}{1+\alpha S(\Omega _{r}(H))}\frac{\left( h_{H_{r}}X_{H_{r}}+l_{H_{r}}X_{L_{r}}\right) }{\Omega _{r}(H)} \\ X_{L_{r}}=\frac{\mu }{p_{X_{r}}}+\frac{\alpha \left( 1-\mu \right) S(\Omega _{r}(L))}{1+\alpha S(\Omega _{r}(L))}\frac{\left( h_{L_{r}}X_{H_{r}}+l_{L_{r}}X_{L_{r}}\right) }{\Omega _{r}(L)} \end{array} \right. \end{aligned}$$

(40)

where we used the fact that $w_{L_{r}}=1$ and where $\Omega _{r}(H)=h_{H_{r}}+l_{H_{r}}$, $\Omega _{r}(L)=h_{L_{r}}+l_{L_{r}}$.

Appendix 2

In the case of the additive specification, we know that $S(\Omega (\Lambda _{r}(i)))=\Omega (\Lambda _{r}(i))$. Hence, with $\Omega _{r}(H)=h_{H_{r}}+l_{H_{r}}$ and $\Omega _{r}(L)=h_{L_{r}}+l_{L_{r}}$, the system in (40) becomes

$$\begin{aligned} \left\{ \begin{array}{l} \frac{1+\alpha (h_{H_{r}}+l_{H_{r}})-\alpha \left( 1-\mu \right) h_{H_{r}}}{ 1+\alpha (h_{H_{r}}+l_{H_{r}})}X_{H_{r}}=w_{H_{r}}\frac{\mu }{p_{X_{r}}}+ \frac{\alpha \left( 1-\mu \right) }{1+\alpha (h_{H_{r}}+l_{H_{r}})} l_{H_{r}}X_{L_{r}} \\ \frac{1+\alpha (h_{L_{r}}+l_{L_{r}})-\alpha \left( 1-\mu \right) l_{L_{r}}}{ 1+\alpha (h_{L_{r}}+l_{L_{r}})}X_{L_{r}}=\frac{\mu }{p_{X_{r}}}+\frac{\alpha \left( 1-\mu \right) }{1+\alpha (h_{L_{r}}+l_{L_{r}})}h_{L_{r}}X_{H_{r}} \end{array} \right. \end{aligned}$$

(41)

This system is solved to obtain the individual demands of the differentiated good by the skilled workers and by the unskilled workers in region r, i.e. $X_{H_{r}}$ and $X_{L_{r}}$.

1.1 Complete networks

Let us now consider the complete network and derive the expression for skilled wages as a function of $H_{r}$. Substituting $X_{rr}(s)$ and $X_{rv}(s)$ from (26) and (27) in (19) and making use of (14), the wage paid to skilled workers in region r must satisfy the following equation

$$\begin{aligned} w_{H_{r}}=\frac{\mu }{\sigma }\left[ \frac{\left( L+H_{r}w_{H_{r}}\right) }{ H_{r}+\left( H-H_{r}\right) \phi }\frac{1+\alpha \Omega _{r}}{1+\alpha \mu \Omega _{r}}+\phi \frac{\left( L+H_{v}w_{H_{v}}\right) }{\left( H-H_{r}\right) +H_{r}\phi }\frac{1+\alpha \Omega _{v}}{1+\alpha \mu \Omega _{v}}\right] \end{aligned}$$

(42)

and an analogous expression holds for $w_{H_{v}}$. Hence, we get a system of two linear equations in $w_{H_{r}}$ and $w_{H_{v}}$ that can be solved to obtain the two regional wages for skilled workers as an explicit function of a given distribution of workers, $H_{r}$ and $H_{v}$, between the two regions, and we find that the wage paid in region r to skilled workers is

$$\begin{aligned} w_{H_{r}}=\mu L\tfrac{\sigma A\left( H_{v}+\phi H_{r}\right) +\phi \sigma B\left( H_{r}+\phi H_{v}\right) +H_{v}\mu BA\left( \phi -1\right) \left( \phi +1\right) }{\phi \sigma ^{2}\left( H_{r}^{2}+H_{v}^{2}\right) -\phi \mu \sigma \left( AH_{r}^{2}+BH_{v}^{2}\right) -H_{v}H_{r}\left[ \mu \sigma \left( A+B\right) -\sigma ^{2}\left( \phi ^{2}+1\right) +\mu ^{2}BA\left( \phi -1\right) \left( \phi +1\right) \right] } \end{aligned}$$

(43)

where $A=\frac{\alpha \Omega _{r}+1}{\alpha \mu \Omega _{r}+1}$ and $B=\frac{ \alpha \Omega _{v}+1}{\alpha \mu \Omega _{v}+1}$.³⁶ This expression shows that the wage depends on the value of $\alpha$, and A and B represent a measure of the effects produced by the proximity of neighbours, respectively, in r (where the number of neighbours is given by $\Omega _{r}=H_{r}+L$) and in v (where the number of neighbours is given by $\Omega _{v}=H_{v}+L$).

The wage of skilled workers in (43) evaluated in the case in which they are evenly distributed in the two regions is given by the following expression

$$\begin{aligned} w_{H/2}=\frac{2\mu L}{H}\frac{\alpha \Omega _{s}+1}{\sigma -\mu +\mu \alpha \Omega _{s}\left( \sigma -1\right) } \end{aligned}$$

where $\Omega _{s}\equiv \left( H/2+L\right)$ is the number of neighbours in each region at the symmetric equilibrium.³⁷ It can be readily shown that the wage ($w_{H_{r}}$) increases in the symmetric equilibrium with $\alpha$. Finally, evaluating the wage of skilled workers in (43) when they are all in region r we get that

$$\begin{aligned} w_{H_{r}}=\frac{\mu L}{H}\frac{2+\alpha \left( H+2L\right) \left( \mu +1\right) +2\alpha ^{2}\mu L(H+L)}{\left( \mu \alpha L+1\right) \left[ \sigma -\mu +\alpha \mu \left( H+L\right) \left( \sigma -1\right) \right] } \end{aligned}$$

In what follows, we show that Proposition 3 in Sect. 4.2 holds.

Proof of Proposition 3

First, consider the derivative of $V\left( H_{r},\phi ,\alpha \right)$ evaluated at the symmetric equilibrium. It can be shown using the expressions for skilled wages derived for the symmetric equilibrium in this Appendix that

$$\begin{aligned}&V_{H_{r}}\left( H/2,\phi ,\alpha \right) \end{aligned}$$

(44)

$$\begin{aligned}&\quad =\frac{4\left( f_{2}\phi ^{2}+f_{1}\phi +f_{0}\right) }{H(\phi +1)\left( \sigma -1\right) \left[ \alpha \left( H+2L\right) +2\right] \left\{ 2+\alpha \left[ 2L-H(\sigma -1)\right] \right\} }*\\&\quad \frac{1}{\left\{ 2\left[ \sigma -\mu +\phi \left( \sigma +\mu \right) \right] +\mu \alpha \left( H+2L\right) \left[ \sigma -1+\phi \left( \sigma +1\right) \right] \right\} } \nonumber \end{aligned}$$

(45)

where the coefficients $f_{0}$, $f_{1}$ and $f_{2}$ are functions of $\mu$, $\sigma$, L, H and $\alpha$. We know that all factors in the denominator are positive, given that $\sigma >1>\mu$ and that the term $\left\{ 2+\alpha \left[ 2L-H(\sigma -1)\right] \right\}$ is positive when $W_{\alpha H_{r}}$ evaluated at $H_{r}=H/2$ is positive.³⁸ Thus, the sign of $V_{H_{r}}\left( H/2,\phi ,\alpha \right)$ depends on the sign of the expression $F\equiv f_{2}\phi ^{2}+f_{1}\phi +f_{0}$ in the numerator, and the symmetric equilibrium is stable when $F<0$ and unstable when $F>0$. However, we know that when $\alpha =0$, $F=a_{0}=8(1-\phi )(2\phi \sigma \mu -\mu \phi -\phi \sigma +\phi \sigma ^{2}+\mu ^{2}\phi -\mu ^{2}+\sigma -\sigma ^{2}+2\sigma \mu -\mu )$. In this case, $a_{0}$ is the relevant term in determining the sign of $V_{H_{r}}\left( H/2,\phi ,\alpha \right) ,$ and, as in Forslid and Ottaviano (2003), the symmetry breaking point $\phi _{b}^{FO}=\frac{\left( \sigma -1-\mu \right) \left( \sigma -\mu \right) }{ \left( \mu +\sigma \right) \left( \mu +\sigma -1\right) }<1$ is such that the symmetric equilibrium is stable only for $\phi \in (0,\phi _{b}^{FO})$.³⁹ Note that for $\phi =1$, $F=a_{0}=0.$ As $\alpha$ becomes positive and rises, F becomes negative, that is $F=4\mu H\alpha ^{2}\sigma (\sigma -1)(1-\mu )(2L+H)\left\{ \alpha \left[ H\left( \sigma -1\right) -2L \right] -2\right\} <0$,⁴⁰ and the symmetric equilibrium remains stable for $\phi =1.$ By continuity the result holds for high levels of integration (i.e. large $\phi$) and the symmetric equilibrium is always stable for low trade costs. Moreover, for prohibitively high trade costs, when $\phi =0$, $F=f_{0}$ that is negative for $\alpha =0$ as in this case $f_{0}=8(\mu +1-\sigma ))(\sigma -\mu )=F<0$; by continuity the result holds for positive and sufficiently small values of $\alpha$, with $f_{0}=8(\mu +1-\sigma ))(\sigma -\mu )+a_{1}\alpha +a_{2}\alpha ^{2}+a_{3}\alpha ^{3}=F<0$ where the coefficients $a_{1}$, $a_{2}$ and $a_{3}$ are functions of $\mu$, $\sigma$, L and H. This proves the results in Proposition 3 on the symmetric equilibrium.$\square$

We now focus our attention on equilibria in which all skilled workers move to one region. Note that, provided these full agglomeration equilibria exist, they are stable, and in what follows we show that the final part of Proposition 3 holds.

The value of $V\left( H_{r},\phi ,\alpha \right)$ when all skilled workers are located in region r is given by

$$\begin{aligned} V(H,\phi ,\alpha )=\ln \left\{ \frac{g_{0}\left\{ 2+\alpha \left[ 2L-H(\sigma -1)\right] \right\} \phi ^{1-\frac{\mu }{\sigma -1}}}{d_{2}\phi ^{2}+d_{1}\phi +d_{0}}\right\} \end{aligned}$$

(46)

where

$$\begin{aligned} g_{0}= & {} \sigma \left( 1+\alpha \mu L\right) \left[ \tfrac{\alpha (H+L)+1}{ \alpha L+1}\right] ^{\mu }>1 \\ d_{2}= & {} [1+\alpha \left( H+L\right) ][\mu \alpha L\left( \sigma +1\right) +\sigma +\mu ]>0; \\ d_{1}= & {} -H\alpha \sigma \left[ \mu \alpha \left( H+L\right) \left( \sigma -1\right) +\sigma -\mu \right] <0; \\ d_{0}= & {} (1+\alpha L)\left[ \mu \alpha \left( H+L\right) \left( \sigma -1\right) +\sigma -\mu \right] >0 \end{aligned}$$

When $\alpha =0$, expression (46) becomes

$$\begin{aligned} V(H,\phi )=\ln \left( \frac{2\sigma \phi ^{1+\frac{\mu }{1-\sigma }}}{\sigma (1+\phi ^{2})-\mu (1-\phi ^{2})}\right) \end{aligned}$$

(47)

Equation (47) generates the sustain point $\phi _{s}$, i.e. the value of $\phi$ such that full agglomeration is an equilibrium for $\phi >\phi _{s},$ as in Forslid and Ottaviano (2003).⁴¹ When $\alpha$ is positive, we know that $g_{0}>1$ and that expression $\phi ^{1+ \frac{\mu }{1-\sigma }}$ is increasing in $\phi \in [0,1]$ from 0 (if $\phi =0$) to 1 (if $\phi =1$).⁴² Given the sign of the parameters $d_{2}$, $d_{1}$ and $d_{0}$, expression $d_{2}\phi ^{2}+d_{1}\phi +d_{0}$ in the denominator of (46) is an upward opening parabola in $\phi ~$with positive value $d_{0}$ when $\phi =0$.⁴³ Moreover, we know that, when workers are all agglomerated in r, expression $\left\{ 2+\alpha \left[ 2L-H(\sigma -1)\right] \right\}$ in the numerator must be positive to have $W_{\alpha H_{r}}>0$.⁴⁴ Finally, the parabola in the denominator ($d_{2}\phi ^{2}+d_{1}\phi +d_{0}$) must be positive in order to have $W_{\alpha H_{v}}>0$. Figure 3 presents the two possible scenarios for this parabola by means of the two continuous curves when $\alpha$ is positive and sufficiently small. In Fig. 3, the numerator $N=g_{0}\left\{ 2+\alpha \left[ 2L-H(\sigma -1) \right] \right\} \phi ^{1-\frac{\mu }{\sigma -1}}$ is represented by the dotted line, which characterizes an increasing function in $\phi$ taking the value 0 when $\phi =0$ and $g_{0}\left\{ 2+\alpha \left[ 2L-H(\sigma -1)\right] \right\} >0$ when $\phi =1$.⁴⁵ The lower parabola can be excluded with $\alpha >0$ as it implies that with $\phi =1$ the full agglomeration is an equilibrium, which contradicts direct computation.⁴⁶ The only relevant case is then the other parabola $D=d_{2}\phi ^{2}+d_{1}\phi +d_{0}$ (that intersects the dotted curve in two critical points $\phi _{s}$ and $\phi _{u}$).⁴⁷ In this case, full agglomeration in region r is an equilibrium only for intermediate values of $\phi$ (when the higher parabola lies below the dotted curve) for $\phi \in (\phi _{s},\phi _{u})$ as $V(H,\phi ,\alpha )$ is positive. This proves the results in Proposition 3 on full agglomeration equilibria. $\square$

Fig. 3

Full agglomeration equilibrium

A general overview: the bifurcation diagram.

To explore further the relationship between the strength of relative concerns ($\alpha$), the openness to trade ($\phi$) and the existence and stability of the equilibria we need to focus on numerical simulations of the model. First, let us perform the numerical analysis on a model in which $L=20$, $H=10$, $\sigma =2$ and $\mu =0.4$ (Fig. 4a) or $\mu =0.11$ (Fig. 4b).⁴⁸ We know that the sign of $V_{H_{r}}\left( H/2,\phi ,\alpha \right)$ in (44) depends on the sign of the expression $F\equiv f_{2}\phi ^{2}+f_{1}\phi +f_{0}$, and the symmetric equilibrium is stable when $F<0$ and unstable when $F>0$. The solid curves represent $F=a_{0}$ as a function of $\phi$ when $\alpha =0$, as in Forslid and Ottaviano (2003). They show that in this case the symmetric equilibrium is stable only if $F<0$, that is, only if $\phi <\phi _{b}^{FO}$. Assume now the existence of relative concerns, $\alpha >0$. Figure 4a, b represents the function F for $\alpha =0.02$ , 0.04, 0.06, 0.08. They show that with relative concerns, as $\phi$ rises the symmetric equilibrium becomes stable as soon as $\phi >\phi _{d}$, where $\phi _{d}$ is the “dispersion point” (so-called as the symmetric equilibrium is stable for $\phi >\phi _{d}$). Hence, the symmetric equilibrium is stable both for small values of the freeness of trade $\phi ,$ such that $\phi <\phi _{b}$, and for high levels of integration, such that $\phi$ is above the new critical level $\phi _{d}$.

Then, we can use expression (46) to find the other two critical points $\phi _{s}$ and $\phi _{u}$ for the equilibrium with full agglomeration. Full agglomeration in region r is an equilibrium only for $\phi \in (\phi _{s},\phi _{u})$. As $\phi$ rises from 0 and reaches the sustain point, $\phi _{s}$, full agglomeration becomes a stable equilibrium for $\phi >\phi _{s}$, as in the case $\alpha =0$. However, this equilibrium again disappears with $\alpha >0$ after trade costs have decreased beyond a new critical level, that we name “unsustain point”, $\phi _{u}$.

Fig. 4

Stability of the symmetric equilibrium for the complete network: the plot of F

Table 1 shows the values of $\phi _{s}$, $\phi _{b}$, $\phi _{u}$ and $\phi _{d}$ obtained numerically for the model with $L=4$, $H=10$, $\sigma =2$ and $\mu =0.11$.⁴⁹ The analysis in Table 1 is summarized by the bifurcation diagrams reported in Fig. 1 in the text of the paper.

Table 1 Critical points for different values of $\alpha$

1.2 Segregated networks

In the case of the segregated network, the system in (41) becomes

$$\begin{aligned} \left\{ \begin{array}{l} \frac{1+\alpha H_{r}-\alpha \left( 1-\mu \right) H_{r}}{1+\alpha H_{r}} X_{H_{r}}=w_{H_{r}}\frac{\mu }{p_{X_{r}}} \\ \frac{1+\alpha L-\alpha \left( 1-\mu \right) L}{1+\alpha L}X_{L_{r}}=\frac{ \mu }{p_{X_{r}}} \end{array} \right. \end{aligned}$$

(48)

which can be solved for $X_{H_{r}}$ and $X_{L_{r}}$ to find respectively that

$$\begin{aligned} X_{H_{r}}= & {} \mu \frac{w_{H_{r}}}{p_{X_{r}}}\frac{1+\alpha H_{r}}{1+\alpha \mu H_{r}}\,\,\text { and} \\ X_{L_{r}}= & {} \mu \frac{1}{p_{X_{r}}}\frac{1+\alpha L}{1+\alpha \mu L}. \nonumber \end{aligned}$$

(49)

Making use of these solutions, we can rewrite the total demand in region r for variety s produced in region k in (13) as follows

$$\begin{aligned} X_{kr}(s)=\mu \frac{p_{kr}(s)^{-\sigma }}{p_{X_{r}}^{1-\sigma }}\left( w_{H_{r}}\frac{1+\alpha H_{r}}{1+\alpha \mu H_{r}}H_{r}+\frac{1+\alpha L}{ 1+\alpha \mu L}L\right) \end{aligned}$$

(50)

where the price indices and the prices are respectively given by (17), (18) and (16). Making use of (50), the wage in (19) can be rewritten as follows

$$\begin{aligned} w_{H_{r}}=\frac{\mu }{\sigma }\left[ \frac{\left( A_{r}w_{H_{r}}+A_{L}\right) }{H_{r}+\left( H-H_{r}\right) \phi }+\frac{\phi \left( A_{v}w_{H_{v}}+A_{L}\right) }{\left( H-H_{r}\right) +H_{r}\phi } \right] \end{aligned}$$

where $A_{r}\equiv \frac{1+\alpha H_{r}}{1+\alpha \mu H_{r}}H_{r}$, $A_{v}\equiv \frac{1+\alpha \left( H-H_{r}\right) }{1+\alpha \mu \left( H-H_{r}\right) }\left( H-H_{r}\right)$ and $A_{L}\equiv \frac{1+\alpha L}{ 1+\alpha \mu L}L$. Previous expression can be considered together with the analogous expression obtained for $w_{H_{v}}$ to get a system of two equations in two unknowns $w_{H_{r}}$ and $w_{H_{v}}$, that can be solved to find the two skilled regional wages given by

$$\begin{aligned} w_{H_{r}}=\frac{\mu A_{L}\left[ 2\sigma \phi H_{r}+\sigma H_{v}\left( 1+\phi ^{2}\right) -A_{v}\mu \left( 1-\phi \right) \left( \phi +1\right) \right] }{ D_{s}} \end{aligned}$$

(51)

and

$$\begin{aligned} w_{H_{v}}=\frac{\mu A_{L}\left[ 2\sigma \phi H_{v}+\sigma H_{r}\left( \phi ^{2}+1\right) -A_{r}\mu \left( 1-\phi \right) \left( \phi +1\right) \right] }{D_{s}} \end{aligned}$$

(52)

with the denominator of both wages given by $D_{s}\equiv \sigma ^{2}\left( H_{v}+\phi H_{r}\right) \left( H_{r}+\phi H_{v}\right) -A_{r}\sigma \mu \left( H_{v}+\phi H_{r}\right) -\sigma \mu A_{v}\left( H_{r}+\phi H_{v}\right) +\mu ^{2}\left( 1-\phi \right) \left( 1+\phi \right) A_{r}A_{v}$.

Then, $W_{\alpha ir}$ in (9) can be written for skilled workers in region r as

$$\begin{aligned} W_{\alpha H_{r}}=\frac{w_{H_{r}}}{1+\alpha \mu H_{r}}, \end{aligned}$$

while for unskilled it is given by

$$\begin{aligned} W_{\alpha L_{r}}=\frac{1}{1+L\alpha \mu } \end{aligned}$$

Hence, we can rewrite (23) as follows

$$\begin{aligned} V\left( H_{r},\phi ,\alpha \right) =\ln \left\{ \left[ \frac{\left( H-H_{r}\right) +H_{r}\phi }{H_{r}+\left( H-H_{r}\right) \phi }\right] ^{ \frac{\mu }{1-\sigma }}\left[ \frac{1+\alpha H_{r}}{1+\alpha \left( H-H_{r}\right) }\right] ^{\mu }\frac{\frac{w_{H_{r}}}{1+\alpha \mu H_{r}}}{ \frac{w_{H_{v}}}{1+\alpha \mu \left( H-H_{r}\right) }}\right\} \end{aligned}$$

(53)

where the two regional wages for skilled workers $w_{H_{r}}$ and $w_{H_{v}}$ can be substituted respectively from (51) and (52).

We first focus on the equilibrium with full agglomeration, and we show that what stated in Proposition 4 in Sect. 5 for segregated networks holds.

Proof of Proposition 4

Expression (53) evaluated when all skilled workers are located in region r (i.e. $H_{r}=H$ and $H_{v}=0$) is

$$\begin{aligned} V^{s}(H,\phi ,\alpha )=\ln \left\{ \frac{2\sigma \left( 1+\alpha H\right) ^{\mu }\phi ^{1+\frac{\mu }{1-\sigma }}}{\mu H\left[ \sigma -1+\phi ^{2}\left( \sigma +1\right) \right] \alpha +\left[ \sigma -\mu +\phi ^{2}\left( \sigma +\mu \right) \right] }\right\} \end{aligned}$$

The value of $V^{s}(H,\phi ,\alpha )$ in this case does not depend on L. On the one hand, when $\phi \rightarrow 0$ (that is, in autarky), $V^{s}\left( H,0,\alpha \right) \rightarrow -\infty$ and full agglomeration is not an equilibrium. On the other hand, with complete integration, $\phi =1$, we find that $V^{s}\left( H,\phi ,\alpha \right) =\ln \frac{\left( 1+\alpha H\right) ^{\mu }}{\left( 1+H\alpha \mu \right) }$, which is negative, provided $\alpha >0,$ because $\frac{\partial \left( \frac{\left( 1+\alpha H\right) ^{\mu }}{\left( 1+H\alpha \mu \right) }\right) }{\partial \alpha }=-\mu \alpha H^{2}\frac{\left( 1-\mu \right) }{H\alpha +1}\frac{ \left( H\alpha +1\right) ^{\mu }}{\left( H\alpha \mu +1\right) ^{2}}<0$ and full agglomeration is not an equilibrium either with complete integration. Moreover, when $\alpha$ is positive and sufficiently small full agglomeration is an equilibrium only for intermediate values of openness to trade as the numerator, which is a concave function in $\phi$ that increases from 0 when $\phi =0$ to $2\sigma \left( 1+\alpha H\right) ^{\mu }$ when $\phi =1$, and the denominator, which is a convex parabola in $\phi$ that increases from its minimum value $\mu H\left( \sigma -1\right) \alpha +\sigma -\mu >0$ when $\phi =0$ to $2\sigma \left( 1+H\alpha \mu \right)$ when $\phi =1$, intersect twice in $\phi _{s}$ and $\phi _{u}$, both in the range (0,1).⁵⁰ $\square$

We now consider the symmetric equilibrium, and we show that we can derive the results of the numerical analysis mentioned in the final part of Sect. 5 for the symmetric equilibrium. The derivative of $V\left( H_{r},\phi ,\alpha \right)$ in (53) with respect to $H_{r}$ evaluated at the symmetric equilibrium is given by the expression

$$\begin{aligned} V_{H_{r}}^{s}\left( H/2,\phi ,\alpha \right) =\frac{4\left( k_{2}\phi ^{2}+k_{1}\phi +k_{0}\right) }{H(1+\phi )(\sigma -1)(2+\alpha H)\left( h_{1}\phi +h_{0}\right) } \end{aligned}$$

where the coefficients $k_{2}$, $k_{1}$, $k_{0}$, $h_{1}$ and $h_{0}$ are functions of $\mu$, $\sigma$, H and $\alpha$ and they do not depend on L.⁵¹ As the denominator of $V_{H_{r}}^{s}\left( H/2,\phi ,\alpha \right)$ is positive, the sign of $V_{H_{r}}^{s}\left( H/2,\phi ,\alpha \right)$ depends on that of the parabola $G\equiv k_{2}\phi ^{2}+k_{1}\phi +k_{0}$, with the symmetric equilibrium stable (vs. unstable) only when G is negative (vs. positive). We focus our analysis on the simulated model with $H=10$ and $\sigma =2.$ Figure 5a (on the left) represents the value of G as a function of $\phi$ in the case of $\alpha =0$ (solid curve), $\alpha =0.02$ (dash dot), $\alpha =0.04$ (dash), $\alpha =0.06$ (dot) and $\alpha =0.08$ (long dash) when $\mu =0.4$.

Fig. 5

Stability of the symmetric equilibrium for the segregated network: the plot of G

Figure 5b (on the right) represents G for the same values except that now $\mu =0.11$, where $\mu$ is proportional to the share of income, net of the conspicuous effect, devoted to acquire the differentiated good.⁵² The numerical analysis shows that the introduction of weak relative concerns tends to stabilize the symmetric equilibrium for high levels of integration $\phi .$ Stability always holds for $\phi =1$ as then $G=4\alpha ^{2}H^{2}\mu \sigma (\sigma -1)(\mu -1)<0$. The size of the interval of $\phi$ on which stability holds is increasing in $\alpha$ and decreasing with $\mu .$ Figure 5a also shows that for intermediate levels of integration $\phi$ the symmetric equilibrium is destabilized when $\mu$ is large. Indeed, in this case the agglomerative effect they produce is stronger than the dispersion effect. On the other hand, Fig. 5b shows that when $\mu$ is relatively low the symmetric equilibrium can be stabilized for low and intermediate values of economic integration $\phi$ provided $\alpha$ is sufficiently large.

1.3 The mixed case

Let us consider the case of the additive specification. Then when region r has a complete (integrated) network, we know that the aggregate demand in r of variety s produced in r, $X_{rr}(s)$, is given by (26), while the aggregate demand in r of variety s produced in v can be obtained from (27) and it is given by

$$\begin{aligned} X_{vr}(s)=\mu \frac{\left( \sigma -1\right) \tau ^{-\sigma }}{\sigma \beta \left( n_{r}+n_{v}\phi \right) }\frac{\left( L+H_{r}w_{H_{r}}\right) \left( L\alpha +\alpha H_{r}+1\right) }{1+\alpha \mu \left( L+H_{r}\right) } \end{aligned}$$

On the other hand, given that region v has two segregated networks, we know that the aggregate demands in v of variety s produced respectively in v and in r can be obtained from (50). Hence, the wage in (19) in region r in the mixed case can be rewritten as follows

$$\begin{aligned} w_{H_{r}}=\frac{\mu }{\sigma }\left[ \frac{\left( L+H_{r}w_{H_{r}}\right) }{ \left( H_{r}+H_{v}\phi \right) }A+\phi \frac{\left( w_{H_{v}}A_{v}+A_{L}\right) }{\left( H_{v}+H_{r}\phi \right) }\right] \end{aligned}$$

while that obtained in region v is given by

$$\begin{aligned} w_{H_{v}}=\frac{\mu }{\sigma }\left[ \frac{\left( w_{H_{v}}A_{v}+A_{L}\right) }{\left( H_{v}+H_{r}\phi \right) }+\frac{\phi \left( L+H_{r}w_{H_{r}}\right) }{\left( H_{r}+H_{v}\phi \right) }A\right] \end{aligned}$$

We use these last two equations to find the wages of skilled workers in the two regions, which are respectively

$$\begin{aligned}&w_{H_{r}}=\mu \\&\dfrac{\sigma \phi ^{2}A_{L}H_{v}-AL\mu A_{v}+AL\sigma H_{v}+\sigma \phi A_{L}H_{r}+AL\mu \phi ^{2}A_{v}+AL\sigma \phi H_{r}}{ \sigma ^{2}H_{r}H_{v}{+}\sigma ^{2}\phi H_{r}^{2}{+}\sigma ^{2}\phi H_{v}^{2}{+}A\mu ^{2}A_{v}H_{r}{+}\sigma ^{2}\phi ^{2}H_{r}H_{v}{-}\sigma \mu A_{v}H_{r}{-}\sigma \mu \phi A_{v}H_{v}{-}A\sigma \mu \phi H_{r}^{2}{-}A\mu ^{2}\phi ^{2}A_{v}H_{r}{-}A\sigma \mu H_{r}H_{v}} \end{aligned}$$

and

$$\begin{aligned}&w_{H_{v}}=\mu \\&\frac{\sigma A_{L}H_{r}-A\mu A_{L}H_{r}+\sigma \phi A_{L}H_{v}+AL\sigma \phi ^{2}H_{r}+A\mu \phi ^{2}A_{L}H_{r}+AL\sigma \phi H_{v}}{\sigma ^{2}H_{r}H_{v}{+}\sigma ^{2}\phi H_{r}^{2}{+}\sigma ^{2}\phi H_{v}^{2}{+}A\mu ^{2}A_{v}H_{r}{+}\sigma ^{2}\phi ^{2}H_{r}H_{v}{-}\sigma \mu A_{v}H_{r}{-}\sigma \mu \phi A_{v}H_{v}{-}A\sigma \mu \phi H_{r}^{2}{-}A\mu ^{2}\phi ^{2}A_{v}H_{r}{-}A\sigma \mu H_{r}H_{v}} \end{aligned}$$

where the two denominators in the two equations are equal.

Then, we find that with $W_{\alpha Hr}$ for skilled workers in the integrated region r given by

$$\begin{aligned} W_{\alpha Hr}\equiv w_{H_{r}}-\alpha \mu \frac{L+H_{r}w_{H_{r}}}{L\alpha \mu +\alpha \mu H_{r}+1} \end{aligned}$$

and $W_{\alpha Hv}$ for skilled workers in the segregated region v

$$\begin{aligned} W_{\alpha Hv}=\frac{w_{H_{v}}}{\alpha \mu H_{v}+1}, \end{aligned}$$

the log of the indirect utility levels $V\left( H_{r},\phi ,\alpha \right)$ in the mixed case is given by

$$\begin{aligned} V^{mix}\left( H_{r},\phi ,\alpha \right) =\ln \left[ \left( \tfrac{p_{X_{v}} }{p_{X_{r}}}\right) ^{\mu }\left( \tfrac{1+\alpha \left( L+H_{r}\right) }{ 1+\alpha H_{v}}\right) ^{\mu }\tfrac{w_{H_{r}}-\alpha \mu \frac{ L+H_{r}w_{H_{r}}}{L\alpha \mu +\alpha \mu H_{r}+1}}{\frac{w_{H_{v}}}{\alpha \mu H_{v}+1}}\right] \end{aligned}$$

(54)

Evaluating $V^{mix}\left( H_{r},\phi ,\alpha \right)$ in (54) when all skilled workers are located in the integrated region r we obtain the following expression⁵³

$$\begin{aligned} V^{mix}\left( H,\phi ,\alpha \right) =\ln \tfrac{\left( 1+\alpha \mu L\right) \left\{ 2+\alpha \left[ 2L-H(\sigma -1)\right] \right\} \left[ \alpha (H+L)+1\right] ^{\mu }\sigma \phi ^{1-\frac{\mu }{\sigma -1}}}{ \left\{ [L\alpha \mu \left( \sigma +1\right) +\mu +\sigma ][1+\alpha (H+L)]\right\} \phi ^{2}+(L\alpha +1)[\alpha \mu \left( \sigma -1\right) \left( H+L\right) +\sigma -\mu ]} \end{aligned}$$

From the inspection of $V^{mix}\left( H,\phi ,\alpha \right)$, we know that agglomeration in r can be an equilibrium only for high or intermediate $\phi$. When $\phi =1$, the argument of the logarithm in $V^{mix}\left( H,1,\alpha \right)$ is equal to 1 when $\alpha =0$. With a positive value of $\alpha$, we can show that the argument of the logarithm in $V^{mix}\left( H,1,\alpha \right)$ is smaller than 1 and, thus, agglomeration is an equilibrium when $\phi =1$, only for relatively large value of $\mu$ provided that the number of unskilled workers is relatively large with respect to that of skilled workers, that is $H<2L/(\sigma -1)$.⁵⁴ More generally, we know that in this last case full agglomeration in r is an equilibrium also for all values of $\phi \in \left( \phi _{s},1\right)$.

Instead, evaluating $V^{mix}\left( H_{r},\phi ,\alpha \right)$ in (54) when all skilled workers are located in the segregated region v, we find that

$$\begin{aligned}&V^\mathrm{{mix}}\left( 0,\phi ,\alpha \right) \\&\quad =\ln \tfrac{\left\{ (L\alpha +1)[H\alpha \mu \left( \sigma +1\right) +\sigma +\mu ]\phi ^{2}-H\sigma \alpha [\left( \sigma -1\right) \alpha \mu H+\sigma -\mu ]\phi +(L\alpha +1)\left[ H\alpha \mu \left( \sigma -1\right) +\sigma -\mu \right] \right\} }{2\sigma (L\alpha +1)^{1-\mu }\left( \alpha H+1\right) ^{\mu }\phi ^{1-\frac{\mu }{\sigma -1}}} \end{aligned}$$

From the inspection of $V^{mix}\left( 0,\phi ,\alpha \right)$, we know that agglomeration in v can be an equilibrium only for high or intermediate $\phi$. When $\phi =1$, the argument of the logarithm in $V^{mix}\left( 0,1,\alpha \right)$ is equal to 1 when $\alpha =0$.