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Gibbons Murphy (1992) - Optimal Incentive Contracts In The Presence Of Career Concerns (view source)
Revision as of 21:01, 27 April 2010
, 21:01, 27 April 2010no edit summary
Substituting in our last result we have:
:<math>\begin{eqnarray}-\mathbb{E}\left[ e^{-r\left[ c_{1}+b_{1}(\eta +a_{1}+\varepsilon_varepsilon_{1})-g(a_{1})+\beta \left( (1-b_{2}^{*})\left[ \frac{h_{1}}{%h_{1}+h_{\varepsilon }}\cdot m_{1}+\frac{h_{\varepsilon }}{%h_{1}+h_{\varepsilon }}\cdot (\eta +a_{1}+\varepsilon _{1}-a_{1}^{*})\right]+b_{2}^{*}(\eta +a_{2}^{*}(b_{2}^{*})+\varepsilon_varepsilon_{2})-g(a_{2}^{*}(b_{2}^{*}))\right) \right] }\right]\end{eqnarray}\,</math>
Which can be written in certainty equivalent form as:
:<math>\begin{eqnarray}&&e^{-r\left[ c_{1}+b_{1}(m_{1}+a_{1})-g(a_{1})+\beta \left( (1-b_{2}^{*})%\left[ \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]+b_{2}^{*}(m_{1}+a_{2}^{*}(b_{2}^{*}))-g(a_{2}^{*}(b_{2}^{*}))\right) \right]} \\&&\cdot e^{\frac{1}{2}r^{2}\left[ \left( b_{1}+\beta \cdot(1-b_{2}^{*})\cdot \frac{h_{\varepsilon }}{h_{1}+h_{\varepsilon }}\,+\betab_{2}^{*}\right) ^{2}\left( \frac{1}{h_{1}}+\frac{1}{h_{\varepsilon }}%\right) -2\left( b_{1}+\beta \cdot (1-b_{2}^{*})\cdot \frac{h_{\varepsilon }%}{h_{1}+h_{\varepsilon }}\right) \beta b_{2}^{*}\frac{1}{h_{e}}\right] }\end{eqnarray}\,</math>
and so:
:<math>\begin{eqnarray}b_{1} &=&\frac{1}{1+r(\frac{1}{h_{1}}+\frac{1}{h_{\varepsilon }})g^{\prime\prime }(a_{1}^{*}(b_{2}^{*}))} \label{b1} \\&&-\beta (1-b_{2}^{*})\frac{h_{\varepsilon }}{h_{1}+h_{\varepsilon }}-\frac{%r\beta b_{2}^{*}\frac{1}{h_{1}}g^{\prime \prime }(a_{1}^{*}(b_{2}^{*}))}{1+r(%\frac{1}{h_{1}}+\frac{1}{h_{\varepsilon }})g^{\prime \prime}(a_{1}^{*}(b_{2}^{*}))} \end{eqnarray}\,</math>