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Appendix

The partial derivatives with respect to $w^1$ have the indicated sign if

\begin{displaymath}
n^1 K \tau - s M \beta + n^1 K \beta >0,
\end{displaymath}

i.e.
\begin{displaymath}
\frac{\beta +\tau}{\beta}> \frac{s M}{n^1 K}.
\end{displaymath} (20)

The budget constraint of the UI (9) or (14) can be transformed into

\begin{displaymath}
n^1 K w^1 (\beta +\tau) + n^2 K w^2 (\beta + \tau)=
s M \beta w^1 + (1-s) M \beta w^2
\end{displaymath}

or

\begin{displaymath}
\frac{\beta +\tau}{\beta}=\frac{s M w^1 + (1-s) M w^2}{n^1 K w^1 + n^2 K w^2}.
\end{displaymath}

Insertion of the right-hand side of the equation in place of the left-hand side of inequality (20), and multiplication with the denominators of the fractions yields

\begin{displaymath}
(1-s) M n^1 K > s M n^2 K.
\end{displaymath}

Multiplication with $1/(s (1-s) M)$ then gives


$
\hfill \displaystyle \frac{n^1 K}{s M}>\frac{n^2 K}{(1-s) M}. \hfill \framebox[2.5mm]{} \vspace{2mm}
$
The left-hand side is the rate of employment in region 1, the right-hand side is the rate of employment in region 2. For the derivative with respect to $w^2$, a parallel consideration applies.



Helge Sanner: Regional Unemployment Insurance, Potsdam 2001