Revision Session#
Revenue Function of a Firm#
\[
\Pi = pf (zeH)-wH
\]
1): Diff w.r.t w
\[\begin{split}
\frac{ d\Pi }{dw} = pf'(z E(w) H)*zE'(w)H-H= 0 \\
= pf'(zE(w)H) zE'(w)=1
\end{split}\]
2): Diff w.r.t H
\[\begin{split}
\frac{ d\Pi }{dw} = pf'(zE(w)H)*zE(w)-w = 0 \\
= pf'(zE(w)H)*zE(w)= w
\end{split}\]
Divide 1) by 2)
\[\begin{split}
\frac{ pf'(zE(w)H) zE'(w) }{pf'(zE(w)H) zE(w)} = \frac{ 1 }{w} \\
\frac{ E'(w) }{E(w)} = \frac{ 1 }{w} \\
\frac{ w*E'(w) }{E(w)} = 1 \gets \text{ Solow Condition} \\
\end{split}\]
Ableiten von E(w) mit v=outside option und a=elasticity
\[ \begin{align}\begin{aligned}\begin{split}
E(w) = ({\frac{ w-v }{v}})^\alpha \\
E'(w) = \alpha (\frac{ w-v }{v})^{\alpha-1}
* \frac{ d }{dw}(\frac{ w }{v}-1) = 0 \\\end{split}\\\begin{split}E'(w) = \alpha (\frac{ w-v }{v})^{\alpha-1}
* \bigg( \frac{v(1)-w(0) }{v^2}\bigg) \\\end{split}\\\implies E'(w) = \alpha (\frac{ w-v }{w})^{\alpha-1}
* \frac{ 1 }{v}
\end{aligned}\end{align} \]
Dann
\[ \begin{align}\begin{aligned}\begin{split}
\frac{ E'(w) }{E(w)} = \frac{ \alpha (\frac{ w-v }{v})^{\alpha-1}
* \frac{ 1 }{v} }{({\frac{ w-v }{v}})^\alpha} \\\end{split}\\\begin{split}= \frac{ \alpha (\frac{ 1 }{v}) }{(\frac{ w-v }{w})^{\alpha-\alpha+1}} \\\end{split}\\\implies \frac{ E'(w) }{E(w)}= \frac{ \alpha }{w-v}
\end{aligned}\end{align} \]
optimal wage: place in solow Condition
\[\begin{split}
\frac{E'(w) }{E(w)} *w= 1 \\
\frac{ \alpha }{w-v} * w = 1 \\
\alpha w = w-v \\
v+ \alpha w = w \\
v = w-\alpha w \\
v = w (1-\alpha) \\
w^* = \frac{ v }{(1-\alpha)}
\end{split}\]
Differentation Rule: (important) $\( Diff: \frac{x}{y} \to \frac{ y }{y^2} = \frac{ 1 }{y} \)$
Outside Option#
1): $\( v= w-uw+ucw \\ v = w(1-u)+ucw \\ v = w(1-u+uc) \\ v = w(1-u(1-c)) \)$
place 1) in optimal wage
\[\begin{split}
w = \frac{ v }{1-\alpha} \\
w = \frac{ w(1-u(1-c)) }{1-\alpha} \\
\frac{ w }{w} = \frac{ 1-u(1-c) }{1-\alpha} \\
1-a = 1-u(1-c) \\
u(1-c) = 1-1+\alpha \\
u^* = \frac{ \alpha }{1-c}
\end{split}\]
interpret: unemployment rate depends on
c=replacement ratio (how high unempl. benefit is as fraction of wages)
alpha=elasticity
=> higher unempl. benefits => higher unempl. rate