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Holmstrom (1999) - Managerial Incentive Problems (view source)
Revision as of 18:07, 7 April 2010
, 18:07, 7 April 2010→Two Period Model
:<math>h_{2}=h_{1}+h_{\varepsilon }\,</math>
In equilibrium the beliefs are correct and the wage is bid up to the expected output:
:<math>w_{1}=Ey_{1}=E[\eta +a_{1}^{*}+\varepsilon _{1}]=m_{1}+a_{1}^{*}\,</math>:
Likewise:
:<math>w_{2}(y_{1}) &=&m_{2}(z_{1})+a_{2}^{*}\,</math>
where <math>a_{2}^{*}=0\,</math>
From the manager's perspective the second period wage is in expectation:
:<math>\mathbb{E}[w_{2}(y_{1})] = \mathbb{E}[m_{2}(z_{1})] = \frac{h_{1}}{h_{1}+h_{\varepsilon }}\cdot m_{1}+\frac{h_{\varepsilon }}{h_{1}+h_{\varepsilon }}\cdot\underset{\mathbb{E}z_{1}}{\underbrace{(\overset{\mathbb{E}y_{1}}{\overbrace{m_{1}+a_{1}}}-a_{1}^{*})}}\,</math>
Putting this into the utility function:
:<math>\underset{a_{1}}{\max }\;w_{1}-g(a_{1})+\beta w_{2}-g(a_{2})\,</math>
:<math>\therefore \underset{a_{1}}{\max }\;m_{1}+a_{1}^{*}-g(a_{1})+\beta \cdot \left[\underset{E[w_{2}(y_{1})]}{\underbrace{\frac{h_{1}}{h_{1}+h_{\varepsilon }}\cdot m_{1}+\frac{h_{\varepsilon }}{h_{1}+h_{\varepsilon }}\cdot(m_{1}+a_{1}-a_{1}^{*})}}\right] \,</math>
:<math>g^{\prime }(a_{1}^{*})=\beta \cdot \frac{h_{\varepsilon }}{h_{1}+h_{\varepsilon }}\in (0,1)\,</math>
