next up previous
Next: Endogenous benefit rate Up: Endogenous tax rate Previous: Central UI, endogenous contribution

Regional UI, endogenous contribution rate (model TR)

In the case of regionally balanced UI budgets, equilibrium conditions (8) and (9) must be modified. In model TR, there are two regional contribution rates, whereas the benefit rate remains uniform and exogenous. The condition for a migration equilibrium then reads:
$\displaystyle F^{mig}$ $\textstyle =$ $\displaystyle \frac{n^1}{s}\frac{K}{M} u[(1-\tau^1) w^1-k] +\left(1-\frac{n^1}{s} \frac{K}{M} \right) u(\beta w^1 -k)$  
    $\displaystyle - \frac{n^2}{1-s} \frac{K}{M} u[(1-\tau^2) w^2]-\left(1-\frac{n^2}{1-s} \frac{K}{M} \right) u(\beta w^2) =0,$ (10)

which differs from (8) only with respect to the superscrips of $\tau$. More differences arise concerning the condition of an equilibrated UI budget. Here, two equations, one for each region, express the requirement of self-financing UI:
\begin{displaymath}
F^{UI1}=\tau^1 n^1 - \beta \left(s \frac{M}{K}-n^1 \right)=0
\end{displaymath} (11)

and
\begin{displaymath}
F^{UI2}=\tau^2 n^2 - \beta \left[ (1-s) \frac{M}{K} -n^2\right]=0.
\end{displaymath} (12)

Differentiating implicitely equation (10) yields the partial derivatives of $s$. The derivatives with respect to $n^i$ and $w^i$ remain unchanged with the exception of the definitions of $u^{m1,j}$ and $u^{2,j}$. Therefore, only the derivatives with respect to $\tau^1$ and $\tau^2$ are calculated. They read

\begin{displaymath}\frac{\partial s}{\partial \tau^1}=\frac{-s (1-s)^2 n^1 w^1 u...
...u^{m1,u} \right) + n^2 (s)^2 \left( u^{2,e}-u^{2,u} \right)}<0 \end{displaymath}

and

\begin{displaymath}\frac{\partial s}{\partial \tau^2}=\frac{(s)^2 (1-s) n^2 w^2 ...
...u^{m1,u} \right) + n^2 (s)^2 \left( u^{2,e}-u^{2,u} \right)}>0.\end{displaymath}

Equations (11) and (12) show that the contribution rates only depend on variables related to the respective region. Wages have no impact because both, revenues and expenditures, depend linearly on the respective wage. Solving for $\tau^i$ and differentiating partially yield:

$\displaystyle \frac{\partial \tau^1}{\partial s}=\frac{\beta M}{n^1 K}>0, $ $\displaystyle \frac{\partial \tau^2}{\partial s}=-\frac{\beta M}{n^2 K}<0, $
   
$\displaystyle \frac{\partial \tau^1}{\partial n^1}=-\frac{\beta s M}{{(n^1)}^2 K}<0, $ $\displaystyle \frac{\partial \tau^2}{\partial n^2}=-\frac{\beta (1-s) M}{{(n^2)}^2 K}<0. $
The derivatives hold the expected signs. When the number of employed workers is given, an increase of the population of one region is accompanied by a rise of the number of unemployed. If the budget of the regional UI is to be balanced, the contribution rate has to increase, too. If, in contrast, the number of employed workers increases, the revenue of the UI rises and the expenditures are lower. The contribution rate, which corresponds to an equilibrated budget is lower.

Model TR consists of the characteristic submodel described above, and of the submodels determining employment and wages in both regions. The relevant equilibrium conditions are thus equations (1) and (6) respectively for region 1 and region 2, and equations (10), (11) and (12). Figure 2 summarises the partial effects of the endogenous variables on one another. As the formal analysis shows, under the assumptions set above, all effects can be derived unambiguously. The only link between the regions is migration, symbolised by the variable $s$. If the situation of workers in region 1 improves by lower UI contributions, higher gross wages, or higher employment, immigration from region 2 increases. This lowers the equilibrium UI contribution rate in region 2, which has an impact on the wage rate and consequently on employment. As can be seen, all variables mutually depend on each other. The complexity of the simultanous equations brings about that the total effects of variations of exogenous variables cannot be determined in general. Therefore, a comparison of the models TC and TR is undertaken only in the calibrated form of the models (section 5).

Figure 2: Partial effects in model TR
\begin{figure}\begin{center}
\special{em:linewidth 0.4pt}
\unitlength 1.00mm\l...
...3}} \put(62.5,20){\makebox(0,0){$-$}}
%
%
\end{picture}
\end{center}\end{figure}


next up previous
Next: Endogenous benefit rate Up: Endogenous tax rate Previous: Central UI, endogenous contribution
Helge Sanner: Regional Unemployment Insurance, Potsdam 2001