Regional mismatch and unemployment: theory and evidence from Italy, 1977–1998

Abstract

We describe the functioning of a two-region economy characterized by asymmetric wage setting. Labour market tightness in the leading-region affects wages in the whole economy. In equilibrium, net labour demand shifts towards the leading region raise unemployment elsewhere and leave regional wages unchanged, causing an increase in aggregate unemployment. Based on SHIW micro-data on earnings, we find strong evidence that wages in Italy only respond to Northern unemployment. We estimate that around 33% of the increase in Italian unemployment during 1977–1998 can be explained by regional mismatch, mainly due to an excess labour supply growth in the South.

Appendix

CES utility function

One of the building blocks of our model is Cobb–Douglas preferences. Below we check how sensitive our results are to this assumption, by adopting (more general) CES preferences.

Suppose that consumers in both regions have CES preferences over regional goods, while the specification of technology in both regions remains unchanged from Eq. (1). Consumers solve the following problem

$$\begin{array}{*{20}c} {{\mathop {\max }\limits_{c_{{1r}} ,c_{{r2}} } }V_{r} {\left( {c_{{1r}} ,c_{{2r}} } \right)} = {\left[ {\alpha c^{\rho }_{{1r}} + {\left( {1 - \alpha } \right)}c^{\rho }_{{2r}} } \right]}^{{1 \mathord{\left/ {\vphantom {1 \rho }} \right. \kern-\nulldelimiterspace} \rho }} ;\quad \rho < 1} \\ {s.to\,p_{1} c_{{1r}} + p_{2} c_{{2r}} \leqslant w_{r} ,\quad r = 1,2} \\ \end{array} $$

(18)

,where σ=1/(1−ρ) represents the elasticity of substitution between the two commodities.

The first-order conditions to the maximization in Eq. (18) are:

$$p_{1} = \alpha {\left( {\frac{{V_{r} }} {{c_{{1r}} }}} \right)}^{{1 \mathord{\left/ {\vphantom {1 \sigma }} \right. \kern-\nulldelimiterspace} \sigma }} $$

(19)

$$p_{2} = {\left( {1 - \alpha } \right)}{\left( {\frac{{V_{r} }} {{c_{{2r}} }}} \right)}^{{1 \mathord{\left/ {\vphantom {1 \sigma }} \right. \kern-\nulldelimiterspace} \sigma }} $$

(20)

$$p_{1} c_{{1r}} + p_{2} c_{{2r}} = w_{r} ,\quad r = 1,2,$$

(21)

Eqs. (19) and (20) can be rewritten as

$$\frac{\alpha } {{1 - \alpha }} = \frac{{p_{1} }} {{p_{2} }}{\left( {\frac{{c_{{11}} }} {{c_{{21}} }}} \right)}^{{1 \mathord{\left/ {\vphantom {1 \sigma }} \right. \kern-\nulldelimiterspace} \sigma }} = \frac{{p_{1} }} {{p_{2} }}{\left( {\frac{{c_{{12}} }} {{c_{{22}} }}} \right)}^{{1 \mathord{\left/ {\vphantom {1 \sigma }} \right. \kern-\nulldelimiterspace} \sigma }} \cdot $$

(22)

Equation (22) implies c ₁₁/c ₂₁=c ₁₂/c ₂₂. Combining this with the market-clearing conditions $Y_{1} = c_{{11}} {\left( {N_{1} + \frac{{c_{{12}} }}{{c_{{11}} }}N_{2} } \right)}$ and $Y_{2} = c_{{21}} {\left( {N_{1} + \frac{{c_{{22}} }} {{c_{{21}} }}N_{2} } \right)}$ gives ${c_{{11}} } \mathord{\left/ {\vphantom {{c_{{11}} } {c_{{21}} = }}} \right. \kern-\nulldelimiterspace} {c_{{21}} = }{c_{{12}} } \mathord{\left/ {\vphantom {{c_{{12}} } {c_{{22}} }}} \right. \kern-\nulldelimiterspace} {c_{{22}} } = {Y_{1} } \mathord{\left/ {\vphantom {{Y_{1} } {Y_{2} }}} \right. \kern-\nulldelimiterspace} {Y_{2} }$ and finally

$$\frac{\alpha } {{1 - \alpha }} = \frac{{p_{1} }} {{p_{2} }}{\left( {\frac{{Y_{1} }} {{Y_{2} }}} \right)}^{{1 \mathord{\left/ {\vphantom {1 \sigma }} \right. \kern-\nulldelimiterspace} \sigma }} = \frac{{p_{1} }} {{p_{2} }}{\left( {\frac{{A_{1} N_{1} }} {{A_{2} N_{2} }}} \right)}^{{1 \mathord{\left/ {\vphantom {1 \sigma }} \right. \kern-\nulldelimiterspace} \sigma }} \cdot $$

(23)

The profit maximization condition for firms implies w _r =p _r A _r , r=1, 2. Therefore Eq. (23) can be rewritten as

$$\ln {\left[ {\frac{{\alpha A^{\rho }_{1} }} {{{\left( {1 - \alpha } \right)}A^{\rho }_{2} }}} \right]} = \frac{1} {\alpha }\ln {\left( {\frac{{N_{1} }} {{N_{2} }}} \right)} + \ln {\left( {\frac{{w_{1} }} {{w_{2} }}} \right)}$$

(24)

or, alternatively,

$$\ln {\left[ {\frac{{A^{\rho }_{1} \alpha l^{{\rho - 1}} }}{{A^{\rho }_{2} {\left( {1 - \alpha } \right)}{\left( {1 - l} \right)}^{{\rho - 1}} }}} \right]}^{\sigma } = {\left[ {\ln {\left( {\frac{{N_{1} }}{{N_{2} }}} \right)} - \ln {\left( {\frac{l}{{1 - l}}} \right)}} \right]} + \sigma \ln {\left( {\frac{{w_{1} }}{{w_{2} }}} \right)}$$

(25)

$$\cong {\left( {u_{2} - u_{1} } \right)} + \sigma \ln {\left( {\frac{{w_{1} }} {{w_{2} }}} \right)} \cdot $$

(26)

According to Eq. (26), the regional mismatch index under CES preferences is

$$D^{\sigma }_{{12}} \cong d{\left( {u_{2} - u_{1} } \right)} + \sigma d\ln {\left( {\frac{{w_{1} }} {{w_{2} }}} \right)},$$

(27)

with $D^{1}_{{12}} = D_{{12}}$.

In Table 5 we estimate the trend in $D^{\sigma }_{{12}}$ for values of σ in the range [0,2]. The first column reports the annual change in the South–North unemployment differential. The second column reports estimates of $D^{\sigma }_{{12}}$, as the sum of the annual average change in u ₂−u ₁ and the proportional annual average change in w ₁/w ₂, multiplied by σ. Note that, given Eq. (27), $D^{\sigma }_{{12}}$ represents the part of the change in the South–North unemployment differential that can be explained by regional mismatch.

Table 5 The impact of regional mismatch on North–South unemployment differentials for alternative values of σ

As σ increases, the estimated mismatch index is reduced due to the higher weight on relative wage changes. Since wage differentials evolved in favor of the South, it turns out that for high-enough values of σ, relative wage changes overweight changes in the unemployment differential and the demand index switches sign. In any case, the change in net relative demand is not significantly different from zero for values of σ above 1.5. For σ=0, corresponding to Leontieff preferences, the demand shift is exactly equal to the change in unemployment differentials: relative wage changes do not induce any substitution between the two labour inputs. For σ=0.5, observed demand shifts account for approximately 60% of the total change in unemployment differentials. When σ=1, which is the Cobb–Douglas case, this accounts for approximately 40% of the rise in the unemployment rate differential. Note that such predicted change in u ₂−u ₁ is simply equal to the rise in u ₂, given that Northern unemployment is not affected by regional mismatch when preferences are of Cobb–Douglas type, as illustrated in Sect. 2.