Frictional unemployment, bargaining, and agglomeration

Abstract

This paper examines how matching elasticity and labor bargaining power affect industrial agglomeration in an open economy with frictional labor markets. The analysis is based on a footloose capital model of two symmetric regions with a single industry and immobile labor. Unemployment is generated by a Diamond–Mortensen–Pissarides-type search and matching mechanism. We find that the agglomeration force caused by search frictions in the labor market may be strong enough to break the symmetric equilibrium when the matching elasticity with respect to job vacancies is large and/or labor bargaining power is strong. Matching elasticity is crucial for determining the configuration of economic geography.

Appendices

Appendix 1: Calculation of marginal revenue

Equation (1) can be rewritten as

$$\begin{aligned} p_{ii}=d_{ii}^{-\frac{1}{\sigma }} Y_i^{\frac{1}{\sigma }} P_i^{\frac{\sigma -1}{\sigma }},\ p_{ij}=d_{ij}^{-\frac{1}{\sigma }} Y_j^{\frac{1}{\sigma }} P_j^{\frac{\sigma -1}{\sigma }}, \ i\ne j. \end{aligned}$$

The revenue from the local and foreign markets for firm i are expressed as

$$\begin{aligned} R_{ii}&=p_{ii} d_{ii} =d_{ii}^{\frac{\sigma -1}{\sigma }} Y_i^{\frac{1}{\sigma }} P_i^{\frac{\sigma -1}{\sigma }}=\left( \varphi l_{ii}\right) ^{\frac{\sigma -1}{\sigma }} Y_i^{\frac{1}{\sigma }} P_i^{\frac{\sigma -1}{\sigma }},\\ R_{ij}&=p_{ij} d_{ij}= d_{ij}^{\frac{\sigma -1}{\sigma }} Y_j^{\frac{1}{\sigma }} P_j^{\frac{\sigma -1}{\sigma }}=\left( \frac{\varphi l_{ij}}{\tau }\right) ^{\frac{\sigma -1}{\sigma }} Y_j^{\frac{1}{\sigma }} P_j^{\frac{\sigma -1}{\sigma }}. \end{aligned}$$

The marginal revenues of the two markets are calculated as

$$\begin{aligned} \frac{{\rm d} R_{ii}}{{\rm d} l_{ii}}=\frac{\sigma -1}{\sigma }\varphi ^{\frac{\sigma -1}{\sigma }} Y_i^{\frac{1}{\sigma }} P_i^{\frac{\sigma -1}{\sigma }} l_{ii}^{-\frac{1}{\sigma }},\quad \frac{{\rm d} R_{ij}}{{\rm d} l_{ij}}=\frac{\sigma -1}{\sigma }\left( \frac{\varphi }{\tau }\right) ^{\frac{\sigma -1}{\sigma }} Y_j^{\frac{1}{\sigma }} P_j^{\frac{\sigma -1}{\sigma }} l_{ij}^{-\frac{1}{\sigma }}. \end{aligned}$$

Appendix 2: Proof of (3)

Following (2), we can derive that

$$\begin{aligned} R_i=&R_{ii}+R_{ij}\\ =&( \varphi l_i) ^{\frac{\sigma -1}{\sigma }} Y_i^{\frac{1}{\sigma }} P_i^{\frac{\sigma -1}{\sigma }}\Biggr( \frac{P_i^{\sigma -1}Y_i }{P_i^{\sigma -1}Y_i+\phi P_j^{\sigma -1}Y_j} \Biggr) ^{\frac{\sigma -1}{\sigma }}\\&+ \biggr( \frac{\varphi l_i}{\tau }\biggr) ^{\frac{\sigma -1}{\sigma }} Y_j^{\frac{1}{\sigma }} P_j^{\frac{\sigma -1}{\sigma }}\Biggr( \frac{\phi P_j^{\sigma -1}Y_j }{P_i^{\sigma -1}Y_i+\phi P_j^{\sigma -1}Y_j} \Biggr) ^{\frac{\sigma -1}{\sigma }}\\ =&\Biggr[ \frac{P_i^{\sigma -1}Y_i }{\bigr( P_i^{\sigma -1}Y_i+\phi P_j^{\sigma -1}Y_j \bigr) ^{\frac{\sigma -1}{\sigma }}}+\frac{P_j^{\sigma -1}Y_j \phi }{\bigr( P_i^{\sigma -1}Y_i+\phi P_j^{\sigma -1}Y_j \bigr) ^{\frac{\sigma -1}{\sigma }}} \Biggr] ( \varphi l_i) ^{\frac{\sigma -1}{\sigma }}\\ =&\bigr( P_i^{\sigma -1}Y_i+\phi P_j^{\sigma -1}Y_j \bigr) ^{\frac{1}{\sigma }}( \varphi l_i) ^{\frac{\sigma -1}{\sigma }}. \end{aligned}$$

Appendix 3: Proof of Lemma 1

Let $\tilde{k}\equiv k/(1-k)$. For simplicity, we keep the function notations $\mathcal{F}^L(\cdot )$ and $\mathcal{B}_i(\cdot )$ ($i=0,1,2$) even when their variables are $\tilde{k}$ rather than k. Differentiating $\mathcal {F}^L(\tilde{k})$ with respect to $\tilde{k}$ in (27), we obtain

$$\begin{aligned} f_0\equiv&\frac{\partial \mathcal {B}_0(\tilde{k})}{\partial \tilde{k}} +\frac{\partial [\mathcal {B}_2(\tilde{k})\phi ^2]}{\partial \tilde{k}}=\phi ^2 [\beta (\sigma -2)+\sigma ]+ \sigma (1-\beta ) >0, \\ f_1\equiv&\frac{\partial [\mathcal {B}_1(\tilde{k})\phi ]}{\partial \tilde{k}} \\ =&2 \phi (\sigma -\beta ) \tilde{k}^{(\mu -1)(\sigma +1) -1} \Big \{[(1-\mu ) (\sigma -1)-1]\tilde{k}^{3-2\mu } + (1-\mu ) (\sigma -1) \tilde{k}^{2 \sigma (1-\mu ) }\Big \}. \end{aligned}$$

When $(1-\mu ) (\sigma -1)-1>0$, we have $f_1>0$. $\mathcal {F}^L(\tilde{k})$ is an increasing function of $\tilde{k}$. The long-run equation (27) only has one solution, $\tilde{k}=1$.

When $(1-\mu ) (\sigma -1)<1$, we have

$$\begin{aligned} \frac{\partial f_1}{\partial \tilde{k}}=2 (1-\mu ) (\sigma -1) (\sigma -\beta ) [(1-\mu ) (\sigma -1)-1]\tilde{k}^{-2} \Big( \tilde{k}^{(1-\mu )(\sigma -1) }-\tilde{k}^{1-(1-\mu )(\sigma -1)}\Big) \phi . \end{aligned}$$

Since $\mathcal {B}_0+\mathcal {B}_2(\tilde{k})\phi ^2<0$ for $\tilde{k}<1$, $\mathcal {B}_1(\tilde{k})\phi =-\mathcal {B}_0-\mathcal {B}_2(\tilde{k})\phi ^2>0$ holds in the long-run equilibrium. Accordingly, $\tilde{k}^{(1-\mu )(\sigma -1) }-\tilde{k}^{1-(1-\mu )(\sigma -1)}>0$ is true in equilibrium. Therefore, we have

$$\begin{aligned} \frac{\partial ^2 \mathcal {F}^L(\tilde{k})}{\partial \tilde{k}^2}\Biggr |_{\mathcal {F}^L(\tilde{k})=0}=\frac{\partial f_1}{\partial \tilde{k}}\Biggr |_{\mathcal {F}^L(\tilde{k})=0}>0, \quad{\text {for}}\quad (1-\mu ) (\sigma -1)<1 , \end{aligned}$$

where $\mathcal {F}^L(\tilde{k})=0$ has one solution at most in (0, 1). Since the two regions are symmetric, $\mathcal {F}^L(\tilde{k})=0$ has three solutions at most for $k \in (0,1)$.

Appendix 4: A general form of hiring costs

In this appendix, we show that our results are robust even if the hiring cost is paid by both capital and labor. In real life, there is a human resource department. Now we assume that a firm needs to pay $c \gamma$ units of labor working as human resource sector and $c(1-\gamma )$ units of capital with $\gamma \in [0,1)$ to post one vacancy. The total employment level ($l^t_i$) of a firm located in region i satisfies

$$\begin{aligned} l^t_i=l_i +\frac{c \gamma }{m}\alpha ^{1-\mu }_i l^t_i, \end{aligned}$$

where $l_i$ is the number of workers working for production. Let $A_i\equiv 1-\left( c \gamma /m\right) \alpha ^{1-\mu }_i$ denote the share of workers in the production department. Then, we have $l^t_i=l_i/A_i$. The profit of a firm in region i is

$$\begin{aligned} \pi (l_i^t)=R(l_i^t)\underset{\text{labor}}{\underbrace{- w_il_i-c \gamma \frac{\alpha _i^{1-\mu }}{m}l^t_i w_i}} \underset{\text{capital}}{\underbrace{-c(1-\gamma )\frac{\alpha ^{1-\mu }}{m}l^t_i r_i -r_i}}. \end{aligned}$$

Similar to Sect. 2.4, the wage rate in the Nash bargaining is

$$\begin{aligned} w_i=\mathrm {argmax}\ w_i^{\beta } \cdot \left[ \frac{\partial R(l^t_i)- w_i l^t_i}{\partial l^t_i}\right] . \end{aligned}$$

The solution of the equation above is

$$\begin{aligned} w_i=\frac{\beta (1-\sigma )}{\sigma -\beta } \frac{R_i}{l^t_i}. \end{aligned}$$

Maximizing the profit with respect to $l^t_i$, we have

$$\begin{aligned} \frac{c(1-\gamma )}{m}\alpha ^{1-\mu }r_i=\frac{\left( 1-\beta \right) \left( \sigma -1 \right) }{\sigma -\beta }\frac{R_i}{l_i^t}. \end{aligned}$$

The total capital employed by a firm is $\sigma$. The labor market tightness in region i is solved as

$$\begin{aligned} \alpha _i=\frac{2(\sigma -1)}{c\sigma (1-\gamma )}k_i. \end{aligned}$$

The optimal price is solved as $p_{ii}=w_i/A_i$, $p_{ij}=\tau w_i/A_i$, for $i,j=1,2$, $i \ne j$. Then, we get the price indices in the two regions,

$$\begin{aligned} P_i=\biggr\{ \biggr[ k_i\Big( \frac{w_i}{A_i} \Big) ^{1-\sigma }+\phi k_j \Big( \frac{w_j}{A_j} \Big) ^{1-\sigma }\biggr] \frac{K}{\sigma }\biggr\} ^{\frac{1}{1-\sigma }}. \end{aligned}$$

The labor market clearing condition is given as

$$\begin{aligned} (1-u_1)w_1L_1 A_1=k \frac{K}{\sigma \varphi }(d_{11}+\tau d_{12}). \end{aligned}$$

Following the process in our basic framework, the stability condition at the symmetric equilibrium is written as

$$\begin{aligned} \frac{\mathrm {d}\Delta r}{\mathrm {d}k}\Biggr |_{k=\frac{1}{2}}=&\frac{16 \phi (\sigma -\beta ) \sigma \left\{ \gamma (\sigma -1)- (1-\gamma ) c^2 m \left[ \mu (\sigma -1)+1 \right] \left[ \frac{\sigma -1 }{(1-\gamma ) }\right] ^{\mu } \right\} }{\left\{ \phi ^2 \left[ \beta (\sigma -2)+\sigma \right] +2 (2 \sigma -1) \phi (\sigma -\beta ) +\sigma (1-\beta ) \right\} } \\&\times\frac{1}{\gamma (\sigma -1)+(\gamma -1) c^2 m \sigma \left[ \frac{\sigma -1 }{(1-\gamma ) \sigma }\right] ^{\mu }}-4<0. \end{aligned}$$

The locus of $\left( \mathrm {d}\Delta r/\mathrm {d}k\right) \big |_{k=1/2}$ is an inverted-U shape with respect to $\phi$. The symmetric equilibrium becomes unstable if $\left( \mathrm {d}\Delta r/\mathrm {d}k\right) \big |_{k=1/2}>0$.

Fig. 8

Loci of $\left( \mathrm {d}\Delta r/\mathrm {d}k\right) \big |_{k=1/2}$ with different $\beta$

Fig. 9

Loci of $\left( \mathrm {d}\Delta r/\mathrm {d}k\right) \big |_{k=1/2}$ with different $\mu$

Figure 8 ($\sigma =4$, $\mu =0.95$, $\gamma =0.4$, $c^2 m=0.8$) shows that the locus of $\left( \mathrm {d}\Delta r/\mathrm {d}k\right) \big |_{k=1/2}$ crosses the horizontal axis twice for a large $\beta$. That is, the symmetry breaks when the bargaining power is large and trade costs are intermediate. In Fig. 9 ($\sigma =5$, $\beta =0.98$, $\gamma =0.4$, $c^2 m=0.8$), we observe that the symmetric equilibrium becomes unstable when $\mu$ is sufficiently large. We can calculate that $\left( \mathrm {d}\Delta r/\mathrm {d}k\right) \big |_{k=1/2, \phi =1}=0$ when $\mu =1$. Hence, for $\mu =1$, the dispersion pattern moves from symmetry to asymmetric agglomeration when trade costs are small and re-dispersion does not occur.

Simulations show that (i) the symmetry breaks for a large bargaining power and/or a large matching elasticity and (ii) re-dispersion emerges for $\mu <1$ and disappears for $\mu =1$. In conclusion, our main results are robust when vacancy costs are paid in a general form.

Appendix 5: Proof of the properties for real incomes

(i) According to (28), the wage rate is not affected by trade costs in the symmetric equilibrium. According to (18), the price indices increase with trade costs in the symmetric equilibrium path. Both employed workers and unemployed workers are better off in the symmetric equilibrium with small trade costs.

(ii) According to Corollary 1, $(1-\mu )(1-\sigma )+1>0$ is a necessary condition for the symmetry to break. Plugging (28) into (18), the price index in region i is rewritten as

$$\begin{aligned} P_i=\frac{2^{1-\mu }c^{\mu }(\sigma -1)^{1-\mu } \beta }{ m(1-\beta ) \sigma ^{1-\mu } }\Big \{[ k_i^{(1-\mu )(1-\sigma )+1 }+\phi (1-k_i)^{(1-\mu )(1-\sigma )+1}] \frac{K}{\sigma }\Big \}^{\frac{1}{1-\sigma }}. \end{aligned}$$

If $(1-\mu )(1-\sigma )+1>0$, we have

$$\begin{aligned} \frac{\partial }{\partial k}\mathrm {Log} \Big (\frac{P_1}{P_2}\Big )=&- \frac{(1-\phi ^2)[(1-\mu ) (1-\sigma )+1](1-k)^{-2}}{(\sigma -1) \Big [\Big ( \frac{k}{1-k} \Big )^{\mu -1}+\phi \Big (\frac{k}{1-k} \Big )^{\mu \sigma }\Big ] \Big [\phi \Big ( \frac{k}{1-k} \Big )^{\mu -1}+\Big ( \frac{k}{1-k} \Big )^{1-(1-\mu )\sigma } \Big ]} \\ <&0. \end{aligned}$$

In the long-run equilibrium, the more agglomerated region always has a lower price index. In Proposition 1, we have shown that the nominal wage is always higher in the more agglomerated region for $\mu <1$ and the relative nominal wage equals 1 for $\mu =1$. Hence, real incomes are always higher in the agglomerated region for both employed and unemployed workers.