Monitoring, moral hazard, and turnover

Abstract

I studied the effects of monitoring on political turnover, when the politicians’ early actions affect future economic outcomes. I considered an infinite-horizon environment, where the expectation about the potential successor’s policy is endogenous. As a result, the incentive to replace the incumbent is endogenous. In a stationary Markov equilibrium, the relationship between monitoring and turnover is non-monotone. The model sheds light on dynamic agency problems when the agent’s initial effort has persistent effects, and on the role of reputation in models with endogenous turnover.

Appendix: Derivations

1.1 Distribution of $\log R(q)$

The log likelihood of $q$ is $\log R(q)$ given by:

$$\begin{aligned} \log R(q) = \log \phi \left( \frac{\log q - \mu }{\sigma } \right) - \log \phi \left( \frac{\log q + \mu }{\sigma } \right) \end{aligned}$$

where $\phi$ is the standard normal pdf. Thus, we get

$$\begin{aligned} \log R(q)&= \frac{- (\log q - \mu )^2}{2 \sigma ^2} - \frac{- (\log q + \mu )^2}{2 \sigma ^2} \\&= \frac{ - ({\log q}^2 - 2 \mu \log q + \mu ^2 ) + ({\log q}^2 + 2 \mu \log q + \mu ^2 ) }{2 \sigma ^2} \end{aligned}$$

Hence,

$$\begin{aligned} \log R(q) = \frac{ 4 \mu \log q }{2 \sigma ^2} = 2 \frac{\mu }{\sigma ^2} \cdot \log q \end{aligned}$$

Since

$$\begin{aligned} \log q \sim \left\{ \begin{array}{ll} N( \mu ,\sigma ^2), &{} \hbox {if}\quad a^P = \mathtt{good }; \\ N(-\mu ,\sigma ^2), &{} \hbox {if}\quad a^P = \mathtt{bad }. \end{array} \right. \end{aligned}$$

we get:

$$\begin{aligned} \log R(q) \sim \left\{ \begin{array}{ll} N( 2 \frac{\mu ^2}{\sigma ^2},4 \frac{\mu ^2}{\sigma ^2}), &{} \hbox {if}\quad a^P = \mathtt{good }; \\ N(-2 \frac{\mu ^2}{\sigma ^2},4 \frac{\mu ^2}{\sigma ^2}), &{} \hbox {if}\quad a^P = \mathtt{bad }. \end{array} \right. \end{aligned}$$

1.2 Conditional probabilities of being removed

Since under a good policy we have $\log R(q) \sim N\left( 2 \frac{\mu ^2}{\sigma ^2}, 4 \frac{\mu ^2}{\sigma ^2}\right)$, we get the following:

$$\begin{aligned}&\Pr \{q: R(q) < R^* | a^P = \mathtt{good } \} \\&= \Pr \left\{ q: \frac{\log R(q) - 2 \mu ^2/\sigma ^2 }{2 \mu /\sigma } < \frac{\log R^* - 2 \mu ^2/\sigma ^2}{2 \mu /\sigma } \bigg | a^P = \mathtt{good } \right\} = \\&= \Pr \left\{ q: \frac{\log R(q) - 2 \mu ^2/\sigma ^2 }{2 \mu /\sigma } < \frac{\sigma \log R^*}{2 \mu } - \frac{\mu }{\sigma } \bigg | a^P = \mathtt{good } \right\} \end{aligned}$$

But $\frac{\log R(q) - 2 \mu ^2/\sigma ^2 }{2 \mu /\sigma } \sim N(0,1)$, so

$$\begin{aligned} \Pr \{q: R(q) < R^* | a^P = \mathtt{good }\} = \varPhi \left( \frac{\sigma \log R^*}{2 \mu } - \frac{\mu }{\sigma } \right) \end{aligned}$$

where $\varPhi$ is the standard normal CDF. In a similar way we obtain that

$$\begin{aligned} \Pr \{q: R(q) < R^* | a^P = \mathtt{bad }\} = \varPhi \left( \frac{\sigma \log R^*}{2 \mu } + \frac{\mu }{\sigma } \right) \end{aligned}$$