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Agglomeration, congestion, and regional unemployment disparities

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Abstract

Regional labor markets are characterized by huge disparities between unemployment rates. Models of the New Economic Geography explain how disparities between regional goods markets endogenously arise but usually assume full employment. This paper discusses regional unemployment disparities by introducing a wage curve based on efficiency wages into the New Economic Geography. The model shows how disparities between regional goods and labor markets endogenously arise through the interplay of increasing returns to scale, transport costs, congestion costs, and migration. The level and stability of regional labor market disparities depends on the extend of labor market frictions.

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Notes

  1. Blien (2001) delivers results for Germany, Nijkamp and Poot (2005) present a meta-analysis, Blanchflower and Oswald (2005) and Montuenga-Gomez and Ramos-Parreno (2005) survey empirical results.

    Fig. 1
    figure 1

    Regional labor market disparities. Sources Official Statistics Federal Republic of Germany (2010); Official Statistics Federal Republic of Germany (2011), own calculations

  2. Blanchard and Katz (1992) focus on the role of shocks for regional labor market dynamics rather than the wage curve per se. However, according to Elhorst (2003), they present the probably most encompassing model of regional labor markets.

  3. Examples are Chen and Zhao (2009), Helpman and Itskhoki (2010), Helpman et al. (2011), Méjean and Patureau (2010), Monford and Ottaviano (2002), Picard and Toulemonde (2006), or Strauss-Kahn (2005).

  4. Examples are Peeters and Garretsen (2004) and Pflüger (2004).

  5. Peach and Stanley (2009) present a meta-analysis of existing evidence.

  6. By replacing Eqs. 2 and 3 through housing supply, a housing-market can be explicitly modeled within the same framework, see e.g. Helpman (1998) and Pflüger and Südekum (2008).

  7. See for example Fujita and Thisse (2002).

  8. Our wage curve hence relates real wages to the unemployment rate. Empirical studies from the wage curve literature usually rely on nominal wages due to lack of regional price data, whereas theoretical models for the wage curve are foremost based on real wages. However, empirical analyses using real wages show that the wage curve results also appear when regional price levels are available, see for example Eckey et al. (2008).

  9. Baldwin (2001) further shows that the “history vs. expectations”-discussion of the core–periphery model needs a reevaluation when using forward-looking expectations, though this is not relevant in our case since we do not address this discussion. Further results regarding forward-looking expectations in the core–periphery model are presented in Baldwin et al. (2003).

  10. The basic function of the agricultural sector is to serve as a reference for prices and wages. We do not need the sector as a centrifugal force due to the presence of congestion costs. However, alternative approaches to fix the price levels or to gain a reference for prices are more complex without qualitatively changing the results. Hence we decide to keep the agricultural sector in our model to keep the analysis as simple as possible.

  11. If one unit of the manufacturing good is transferred to the neighboring region, only 1/\(\tau \) units arrive. Therefore \(\tau \) units have to be sent when one unit shall arrive.

  12. The real wage in region \(r\) is \(w_r K P_r^{-\mu }\).

  13. The chance to find employment depends on the endogenous job creation rate which is directly linked to the unemployment rate.

  14. Additional results for these parameters are available upon request from the author.

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Acknowledgments

Helpful comments and suggestions from Uwe Blien, Johannes Bröcker, Jochen Michaelis, Michael Pflüger, and the participants of the HWWI PhD Seminar, GfR Winterseminar, MAGKS PhD Seminar, IAB-Colloquium, IfR Brown Bag Seminar, ERSA and EEA/ESEM Conferences are gratefully acknowledged. The usual disclaimer applies.

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Correspondence to Ulrich Theodor Zierahn.

Appendices

Appendix: Definition of the break-point

To calculate the break-point point, \(\lambda \) is set to \(0.5\) and we can utilize the fact that all endogenous variables are equal in both regions. Furthermore, the change of one variable in a region is equal to the negative change of the same variable in the other region. Therefore, the system can be expressed in units of region \(r\) and the index for regions is dropped. Consequently, the break-point is expressed by the system of Eqs. 1830:

$$\begin{aligned} P&= \left[ \frac{1+\tau ^{1-\theta }}{\mu } \left( L_M w^{1-\theta } \right) \right]^{1/(1-\theta )}, \end{aligned}$$
(18)
$$\begin{aligned} w&= \left[ (1+\tau ^{1-\theta })YP^{\theta -1} \right]^{1/\theta },\end{aligned}$$
(19)
$$\begin{aligned} Y&= w L_M + L_A,\end{aligned}$$
(20)
$$\begin{aligned} w&= \frac{P^{\mu }}{K} e \frac{\rho + \psi + \delta + 1 - \gamma }{1-\gamma },\end{aligned}$$
(21)
$$\begin{aligned} \delta&= \frac{\psi L_M}{\mu \lambda -L_M},\end{aligned}$$
(22)
$$\begin{aligned} H&= h^{1-\frac{\lambda }{1-\lambda }},\end{aligned}$$
(23)
$$\begin{aligned} \text{ d}P&= \frac{P^\theta (1-\tau ^{1-\theta })}{\mu } \left[ \frac{w^{1-\theta }}{1-\theta }\text{ d}L_M + L_M w^{-\theta } \text{ d}w \right],\end{aligned}$$
(24)
$$\begin{aligned} \text{ d}w&= \frac{w^{1-\theta }P^{\theta -1}(1-\tau ^{1-\theta })}{\theta } \left[\text{ d}Y + (\theta -1)Y\frac{\text{ d}P}{P} \right],\end{aligned}$$
(25)
$$\begin{aligned} \text{ d}Y&= w \text{ d}L_M + L_M \text{ d}w,\end{aligned}$$
(26)
$$\begin{aligned} \text{ d}w&= \mu \frac{w}{P} \text{ d}P + \frac{P^{\mu }}{K} \frac{e}{1-\gamma } \text{ d}\delta ,\end{aligned}$$
(27)
$$\begin{aligned} \text{ d}\delta&= \frac{\mu \psi }{(\mu \lambda - L_M)^2} \left( \lambda \text{ d}L_M - L_M \text{ d}\lambda \right), \end{aligned}$$
(28)
$$\begin{aligned} \frac{dM}{2}&= 0 = \frac{\delta }{\rho +\psi +\delta } \left[ \Big [ w \frac{K}{P^{\mu }} - e \Big ]\text{ d}H + \frac{H K}{P^{\mu }}\text{ d}w - \frac{\mu HwK}{P^{1+\mu }}\text{ d}P \right] \nonumber \\&+ \frac{\rho + \psi }{(\rho +\psi +\delta )^2} H \left[\frac{wK}{P^{\mu }}-e \right]\text{ d}\delta ,\end{aligned}$$
(29)
$$\begin{aligned} \text{ d}H&= -\frac{1}{(1-\lambda )^2} H \ln (h) \text{ d}\lambda . \end{aligned}$$
(30)

Any solution to this system is a break-point.

Appendix: Definition of the sustain-point

The sustain-point is defined by the equilibrium conditions 613, 16 and the definition of the income \(Y_r = w_r L_{Mr} + (1-\mu )/2\) for region \(r\) and for region \(s\) accordingly, the definition of the congestion costs 2 and 3, the no-migration condition 17, and the derivative of the system against \(\lambda \) (only the derivative for region \(r\) is presented here, the equations for region \(s\) are calculated in the same way):

$$\begin{aligned} \text{ d}P_r&= \frac{P_r^{\mu }}{\mu } \left[\frac{w_r^{1-\theta }}{1-\theta } \text{ d}L_{Mr} + \frac{L_{Mr}}{w_r^{\theta }} \text{ d}w_r + \frac{(w_s \tau )^{1-\theta }}{1-\theta } \text{ d}L_{Ms} + \frac{L_{Ms} \tau ^{1-\theta }}{w_s^{\theta }} \text{ d}w_s \right]\qquad \end{aligned}$$
(31)
$$\begin{aligned} \text{ d}w_r&= \left(\frac{w_r}{P_r} \right)^{1-\theta } \frac{1}{\theta } \left[\text{ d}Y_r + (\theta -1)\frac{Y_r}{P_r} \text{ d}P_r \right] \nonumber \\&+\left( \frac{w_r}{P_s} \right)^{1-\theta } \frac{\tau ^{1-\theta }}{\theta } \left[ \text{ d}Y_s + (\theta -1) \frac{Y_s}{P_s} \text{ d}P_s \right]\end{aligned}$$
(32)
$$\begin{aligned} \text{ d}Y_r&= w_r \text{ d}L_{Mr} + L_{Mr} \text{ d}w_r\end{aligned}$$
(33)
$$\begin{aligned} \text{ d}w_r&= \mu \frac{w_r}{P_r} \text{ d}P_r + \frac{P_r^{\mu }}{K} \frac{e}{1-\gamma } \text{ d}\delta _r\end{aligned}$$
(34)
$$\begin{aligned} \text{ d}H_r&= -\frac{1}{(1-\lambda )^2} \ln (h) H_r \text{ d}\lambda \end{aligned}$$
(35)
$$\begin{aligned} \text{ d}\delta _r&= \frac{\mu \psi }{(\mu \lambda - L_{Mr})^2} \left( \lambda \text{ d}L_{Mr} - L_{Mr} \text{ d}\lambda \right)\end{aligned}$$
(36)
$$\begin{aligned} \text{ d}M&= \frac{\delta _r}{\rho +\psi +\delta _r} \left( \left[ w_r \frac{K}{P_r^{\mu }} - e \right] \text{ d}H_r + H_r \frac{K}{P_r^{\mu }} \text{ d}w_r - \mu H_r \frac{w_r K}{P_r^{1+\mu }} \text{ d}P_r \right) \nonumber \\&+ \frac{\rho \psi }{(\rho + \psi + \delta _r)^2} H_r \left[w_r \frac{K}{P_r^{\mu }} - e \right] \text{ d} \delta _r \nonumber \\&- \frac{\delta _s}{\rho +\psi +\delta _s} \left( \left[ w_s \frac{K}{P_s^{\mu }} - e \right] \text{ d}H_s + H_s \frac{K}{P_s^{\mu }} \text{ d}w_s - \mu H_s \frac{w_s K}{P_s^{1+\mu }} \text{ d}P_s \right) \nonumber \\&- \frac{\rho \psi }{(\rho + \psi + \delta _s)^2} H_s \left[w_s \frac{K}{P_s^{\mu }} - e \right] \text{ d} \delta _s \end{aligned}$$
(37)

Appendix: Frictions and disparities

Labor market frictions are characterized by the parameters \(e, \rho , \gamma \) and \(\psi \). When these parameters increase, frictions increase. These parameters are only included in the wage curve, whereas they do not appear in the equations for the goods market equilibrium; hence, they influence the goods market only indirectly through their effects on the wage curve. From the goods market the wage results, at which firms reach their break-even point. Until this point is reached, new firms enter the market. Therefore, unemployment adjusts, until the break-even point (and its corresponding wage) is reached. Therefore, the primary influence of labor market frictions is only on the unemployment rate, whereas the break-even point and its corresponding wage changes in reaction to the changes of the unemployment rate. Thus, the real wage is fixed if we only look at the wage curve. The same is true for migration. Thus, we reformulate the wage curve:

$$\begin{aligned} \frac{v(w_r)}{H_r} = e \left(1+ \frac{\rho +\psi /u_r}{1-\gamma } \right) \end{aligned}$$
(38)

The same holds true for region \(s\). We further know from the arguments above that:

$$\begin{aligned} \text{ d}\left( \frac{v(w_r)}{H_r} \right) = 0 \quad \text{ and}\quad \text{ d}\left( \frac{v(w_s)}{H_s} \right) = 0 \end{aligned}$$
(39)

Then, we can show for changes in the labor market frictions that changes in the unemployment rate are higher for regions, where the unemployment rate is higher:

$$\begin{aligned}&\text{ d}u_r = \left( \frac{u_r^2 \rho }{e \psi } + \frac{u_r}{e} + \frac{u_r^2 (1-\gamma )}{e \psi } \right) \text{ d}e\end{aligned}$$
(40)
$$\begin{aligned}&\text{ d}u_r = \frac{\text{ d}\rho }{\psi } u_r^2\end{aligned}$$
(41)
$$\begin{aligned}&\text{ d}u_r = \frac{\text{ d}\psi }{\psi } u_r\end{aligned}$$
(42)
$$\begin{aligned}&\text{ d}u_r = \text{ d}\gamma \left( \frac{\rho +\frac{\psi }{u_r}}{(1-\gamma )\psi } u_r^2 \right) \end{aligned}$$
(43)

Hence, the primary effect of increasing frictions (which occur through the wage curve) is that they enforce disparities. The increasing disparities then spur migration from the periphery to the agglomeration which further enforces disparities through its effect on the goods market.

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Zierahn, U.T. Agglomeration, congestion, and regional unemployment disparities. Ann Reg Sci 51, 435–457 (2013). https://doi.org/10.1007/s00168-013-0555-3

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