Gibbons Murphy (1992) - Optimal Incentive Contracts In The Presence Of Career Concerns

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Gibbons, R. and K.J. Murphy (1992), "Optimal Incentive Contracts in the Presence of Career Concerns: Theory and Evidence," Journal of Political Economy, 100(3): 468 505. pdf

Abstract

This paper studies optimal incentive contracts when workers have career concerns-concerns about the effects of current performance on future compensation. We show that the optimal compensation contract optimizes total incentives: the combination of the implicit incentives from career concerns and the explicit incentives from the compensation contract. Thus the explicit incentives from the optimal compensation contract should be strongest for workers close to retirement because career concerns are weakest for these workers. We find empirical support for this prediction in the relation between chief executive compensation and stock market performance.

Summary

Gibbons and Murphy (1992) is much like Holmstrom (1982) with two important differences:

• Firms can now write explicit contracts
• Managers are now risk averse

The model uses a rational expectations equilibria, and competition amongst firms (a zero profit condition).

The Model

The model is a two period game. As in Holmstrom, production is given by:

[math]y_{t}=\eta +a_{t}+\varepsilon _{t}\,[/math]

However, firms can write explicit contracts (but only on a single period):

[math]w_{t}(y_{t})=c_{t}+b_{t}(y_{t})\,[/math]

and the manager's utility function is:

[math]U(w,a)=-e^{-r\sum_{t=1}^{\infty }\beta ^{t-1}[w_{t}-g(a_{t})]}\,[/math]

Wages in period 2 take the form:

[math]w_{2}(y_{2}|y_{1})=c_{2}(y_{1})+b_{2}(y_{1})y_{2}\,[/math]

In equilibrium effort is correctly anticipated so:

[math]z_{1}\equiv \eta +\varepsilon _{1}=y_{1}-a_{1}^{*}\,[/math]

and

[math][\eta|z_{1}]\sim N(m_{2},\frac{1}{h_{2}})\,[/math]

Giving:

[math]m_{2}(z_{1})=\frac{h_{1}}{h_{1}+h_{\varepsilon }}\cdot m_{1}+\frac{h_{\varepsilon }}{h_{1}+h_{\varepsilon }}\cdot z_{1}\,[/math]

and

[math]h_{2}=h_{1}+h_{\varepsilon}\,[/math]

In the second period the agent maximizes:

[math]\max_{a_{2}}\;-\overset{\text{constant}}{\overbrace{e^{-\frac{r}{\beta }[c_{1}+b_{1}y_{1}-g(a_{1})]}}}\cdot \mathbb{E}\left[ e^{-r[c_{2}+b_{2}(\eta+a_{2}+\varepsilon _{2})-g(a_{2})]}|z_{1}\right]\,[/math]

which, given that period 1 is past and we are using exponential utility, is equivalent to:

[math]\mathbb{E}\left[ e^{-rx}\right] =e^{-rE[x]+\frac{1}{2}r^{2}var(x)}\,[/math]

yielding a first order condition of:

[math]g^{\prime }(a_{2})=b_{2}\,[/math]

The zero profit condition on the firms then gives us:

[math]c_{2}(b_{2}) = (1-b_{2})\mathbb{E}[y_{2}|y_{1}] = (1-b_{2})[m_{2}+a_{2}^{*}(b_{2})] \,[/math]

So for any [math]b_{2}\,[/math] we can calculate [math]c_{2}(b_{2})\,[/math].

However, the zero profit condition on the firms imply that all rents are transfer to the managers, so we can maximize the manager's expected utility subject to the contraints that:

• [math]g^{\prime }(a_{2}^*)=b_{2}\,[/math]
• [math]c_{2}(b_{2}) = (1-b_{2})[m_{2}+a_{2}^{*}(b_{2})]\,[/math]

Therefore we take:

[math]\max_{b_{2}}\;-e^{-r[m_{2}+a_{2}^{*}(b_{2})-g(a_{2}^{*}(b_{2}))]+\frac{1}{2}(rb_{2})^{2}(\frac{1}{h_{2}}+\frac{1}{h_{\varepsilon }})}\,[/math]

which gives:

[math]\frac{da_{2}^{*}(b_{2})}{db}-g^{\prime }(a_{2}^{*}(b_{2}))\cdot\frac{da_{2}^{*}(b_{2})}{db}-r(\frac{1}{h_{2}}+\frac{1}{h_{\varepsilon }})b_{2}=0\,[/math]

Implicitly differentiating [math]g^{\prime }(a_{2}^*)=b_{2}\,[/math] with respect to [math]b_{2}\,[/math] gives:

[math]\frac{da_{2}^{*}(b_{2})}{db}=\frac{1}{g^{\prime \prime }(a_{2}^{*}(b_{2}))}\,[/math]

Which can be substituted in to solve:

[math]b_{2}^{*}=\frac{1}{1+r(\frac{1}{h_{2}}+\frac{1}{h_{\varepsilon }})g^{\prime\prime }(a_{2}^{*}(b_{2}^{*}))}\,[/math]

This does not depend on [math]y_1\,[/math] (because of the assumption of CARA utility there are no wealth effects). So career concerns will be on [math]c_2\,[/math] and not [math]b_2\,[/math], which gives us:

[math]c_{2}^{*}(b_{2}^{*})=(1-b_{2}^{*})\left[ \frac{h_{1}}{h_{1}+h_{\varepsilon }}\cdot m_{1}+\frac{h_{\varepsilon }}{h_{1}+h_{\varepsilon }}\cdot \underset{z_{1}}{\underbrace{(\overset{y_{1}}{\overbrace{\eta +a_{1}+\varepsilon _{1}}}-a_{1}^{*})}}\right]\,[/math]

We can now solve the first period problem. The manager solves:

[math]\max_{a_{1}}\;-\mathbb{E}\left[ e^{-r[c_{1}+b_{1}(\eta +a_{1}+\varepsilon_{1})-g(a_{1})]}\cdot e^{-r\beta [c_{2}^{*}(b_{2}^{*})+b_{2}^{*}(\eta+a_{2}^{*}(b_{2}^{*})+\varepsilon _{2})-g(a_{2}^{*}(b_{2}^{*}))]}\right]\,[/math]

Substituting in our last result we have:

[math]-E\left[ e^{-r\left[ c_{1}+b_{1}(\eta +a_{1}+\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_{2})-g(a_{2}^{*}(b_{2}^{*}))\right) \right] }\right],[/math]

Which can be written in certainty equivalent form as:

[math]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 }}\,+\beta b_{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] }\,[/math]

Taking a first order condition yields:

[math]g^{\prime }(a_{1})=\underset{\text{incentive}}{\underset{\text{explicit}}{\underbrace{\underset{}{b_{1}}}}}+\underset{\text{implicit incentive}}{\underbrace{\beta \cdot (1-b_{2}^{*})\cdot \frac{h_{\varepsilon }}{h_{1}+h_{\varepsilon }}}}\,[/math]

Recalling that [math]b_{2}^{*}\in (0,1)\,[/math] we can see that the implicit incentive is positive. Furthermore the model has many of the same characteristics as Holmstrom (1982) - effort in increasing in patience and the precision of the output process, but decreasing in the precision of prior ability.

To complete the model we need to solve for [math]b_1\,[/math], which we do using the zero profit condition implictly in the manager's maximization as before:

[math]c_{1}(b_{1}) =(1-b_{1})E[y_{1}] =(1-b_{1})[m_{1}+a_{1}^{*}(b_{1})]\,[/math]

and so:

[math]b_{1} =\frac{1}{1+r(\frac{1}{h_{1}}+\frac{1}{h_{\varepsilon }})g^{\prime\prime }(a_{1}^{*}(b_{2}^{*}))} -\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}^{*}))}\,[/math]

Note that [math]b_{1}^{*}\lt b_{2}^{*}\,[/math] for three reasons:

1. [math]\frac{1}{h_{1}}+\frac{1}{h_{\varepsilon }})\gt (\frac{1}{h2}+\frac{1}{h_{\varepsilon }})\,[/math] because learning occurs about the manager's ability. There are less explicit incentives in the first period because there is more risk.
2. There are incentives provided by career concerns, so they need not be provided by explicitly
3. There is an insurance effect. The manager is uncertain about his own human capital and needs insuring in case he is 'bad'