The Principal-Agent Problem

Definition (The Principal-Agent Problem). The asymmetries of information after the signing of a contract, because of the hidden actions or hidden information.

Moral Hazard

Assumptions The owner of a firm (the principal) wishes to hire a manager (the agent) for a one-time project. The project’s profits are affected, at least in part, by the manager’s actions. Let $\pi$ denote the project’s (observable) profits, and let $e$ denote the manager’s action choice. The set of possible actions is denoted by $E$. We interpret $e$ as measuring managerial effort. Assume that the firm’s profit can take values in $[\underset{―}{\pi },\bar{\pi }]$ and that it is stochastically related to $e$ in a manner described by the conditional density function $f(\pi |e)$, with $f(\pi |e)>0$ for all $e\in E$ and all $\pi \in [\underset{―}{\pi },\bar{\pi }]$. The manager has only two possible effort choices, $e_H$ and $e_L$. $e_H$ is a “high-effort” choice that leads to a higher profit level for the firm but entails greater difficulty for the manager. The distribution functions $F(\pi |e_L)$ and $F(\pi |e_H)$ satisfy $F(\pi |e_H)\le F(\pi |e_L)$ at all $\pi \in [\underset{―}{\pi },\bar{\pi }]$, with strict inequality on some open set $\mathrm{\Pi }\subset [\underset{―}{\pi },\bar{\pi }]$. The manager is an expected utility maximizer with a Bernoulli utility function $u(w,e)$ over his wage $w$ and effort level $e$. This function satisfies $u_w(w,e)>0$ and $u_{ww}(w,e)<0$ at all $(w,e)$ and $u(w,e_H)<u(w,e_L)$ at all $w$. Assume $u(w,e)=v(w)-g(e)$. Assume that the owner is risk neutral and therefore that his objective is to maximize his expected return.

Observable Effort

The optimal problem of the owner is

$$\begin{array}{rl}\underset{e\in \{e_L,e_H\},w(\pi )}{max}& \int (\pi -w(\pi ))f(\pi \mid e)d\pi \\ & \text{s.t.}\int v(w(\pi ))f(\pi \mid e)d\pi -g(e)\ge \overline{u}.\end{array}$$

It is convenient to think of this problem in two stages.

1. First, for each choice of $e$ that might be specified in the contract, what is the best compensation scheme $w(\pi )$ to offer the manager?
2. Second, what is the best choice of $e$?

Conditional compensation scheme

$$\begin{array}{rl}\underset{w(\pi )}{min}& \int w(\pi )f(\pi \mid e)d\pi \\ & \text{s.t.}\int v(w(\pi ))f(\pi \mid e)d\pi -g(e)\ge \overline{u}\end{array}$$

Letting $\gamma$ denote the multiplier on this constraint, FOC

$$\frac{1}{v^{\prime }(w(\pi ))}=\gamma$$

1. If the manager is strictly risk averse, given the contract’s specification of $e$, the owner offers a fixed wage payment $w_e^{\ast }$ such that the manager receives exactly his reservation utility level:$$v(w_e^{\ast })-g(e)=\bar{u}.$$
2. When the manager is risk neutral, any compensation function $w(\pi )$ that gives the manager an expected wage payment equal to $\bar{u}+g(e)$ is also optimal.

Effort choice

The owner optimally specifies the effort level $e\in \{e_L,e_H\}$ that maximizes his expected profits less wage payments,

$$\int \pi f(\pi |e)d\pi -v^{-1}(\bar{u}+g(e)).$$

Proposition 14.B.1. In the principal-agent model with observable managerial effort, an optimal contract specifies that the manager choose the effort $e^{\ast }$ that maximizes $[\int \pi f(\pi |e)d\pi -v^{-1}(\bar{u}+g(e))]$ and pays the manager a fixed wage $w^{\ast }=v^{-1}(\bar{u}+g(e^{\ast }))$. This is the uniquely optimal contract if $v^{″}(w)<0$ at all $w$.

Unobservable Effort

The optimal contract described in Proposition 14.B.1 accomplishes two goals:

• It specifies an efficient effort choice by the manager.
• It fully insures him against income risk.

When effort is not observable, these two goals often come into conflict because the only way to get the manager to work hard is to relate his pay to the realization of profits, which is random.

When these goals come into conflict, the nonobservability of effort leads to inefficiencies.

A risk-neutral manager

Proposition 14.B.2. In the principal-agent model with unobservable managerial effort and a risk-neutral manager, an optimal contract generates the same effort choice and expected utilities for the manager and the owner as when effort is observable.

If the manager is risk neutral, the problem of risk sharing disappears.

A risk-averse manager

To characterize the optimal contract in these circumstances, we again consider the contract design problem in two steps:

1. Wwe characterize the optimal incentive scheme for each effort level that the owner might want the manager to select;
2. We consider which effort level the owner should induce

First Step

$$\begin{array}{rl}\underset{w(\pi )}{min}& \int w(\pi )f(\pi |e)d\pi \end{array}$$$$\text{s.t.}\begin{array}{rl}& \text{(i)}\int v(w(\pi ))f(\pi |e)d\pi -g(e)\ge \bar{u}\\ & \text{(ii)}e\text{ solves }\underset{\tilde{e}}{max}\int v(w(\pi ))f(\pi |\tilde{e})d\pi -g(\tilde{e}).\end{array}$$

Definition (Incentive Constraint). Constraint (ii) is known as the incentive constraint: it insures that under compensation scheme $w(\pi )$ the manager’s optimal effort choice is $e$.

Implementing $e_L$

• In this case, the owner optimally offers the manager the fixed wage payment $w_e^{\ast }=v^{-1}(\bar{u}+g(e_L))$, the same payment he would offer if contractually specifying effort $e_L$ when effort is observable.

Implementing $e_H$

• $$({\text{ii}}_H)\int v(w(\pi ))f(\pi |e_H)d\pi -g(e_H)\ge \int v(w(\pi ))f(\pi |e_L)d\pi -g(e_L).$$Letting $\gamma \ge 0$ and $\mu \ge 0$ denote the multipliers on constraints $(\text{i})$ and $({\text{ii}}_H)$, respectively, $w(\pi )$ must satisfy the following Kuhn-Tucker first-order condition at every $\pi \in [\underset{―}{\pi },\bar{\pi }]$:$$\begin{array}{}\text{(14.B.10)}& \frac{1}{v^{\prime }(w(\pi ))}=\gamma +\mu [1-\frac{f(\pi |e_L)}{f(\pi |e_H)}]\end{array}$$

Lemma 14.B.1. In any solution to problem (14.B.9) with $e=e_H$, both $\gamma >0$ and $\mu >0$.

Consider, for example, the fixed wage payment $\hat{w}$ such that $(1/v^{\prime }(w))=\gamma$. According to condition (14.B.10),

$$w(\pi )>\hat{w}\text{if}\frac{f(\pi |e_L)}{f(\pi |e_H)}<1$$

and

$$w(\pi )<\hat{w}\text{if}\frac{f(\pi |e_L)}{f(\pi |e_H)}>1.$$

The optimal compensation scheme pays more than $\hat{w}$ for outcomes that are statistically relatively more likely to occur under $e_H$ than under $e_L$ in the sense of having a likelihood ratio $[f(\pi |e_L)/f(\pi |e_H)]$ less than $1$.

In an optimal incentive scheme, compensation is not necessarily monotonically increasing in profits.

• Requires monotone likelihood ratio property.
  • Definition (monotone likelihood ratio property). The likelihood ratio $[f(\pi |e_L)/f(\pi |e_H)]$ is decreasing in $\pi$.

Condition (14.B.10) also implies that the optimal contract is not likely to take a simple (e.g., linear) form.

The expected value of the manager’s wage payment must be strictly greater than his (fixed) wage payment in the observable case, $w_{e_H}^{\ast }=v^{-1}(\bar{u}+g(e_H))$.

Second Step

The owner compares the incremental change in expected profits from the two effort levels with the difference in expected wage payments in the contracts that optimally implement each of them.

Conclusion

We know that the wage payment when implementing $e_L$ is exactly the same as when effort is observable, whereas the expected wage payment when the owner implements $e_H$ under nonobservability is strictly larger than his payment in the observable case.

Thus, in this model, nonobservability raises the cost of implementing $e_H$ and does not change the cost of implementing $e_L$. The implication of this fact is that nonobservability of effort can lead to an inefficiently low level of effort being implemented.

Proposition 14.B.3. In the principal-agent model with unobservable manager effort, a risk-averse manager, and two possible effort choices, the optimal compensation scheme for implementing $e_H$ satisfies condition (14.B.10), gives the manager expected utility $\bar{u}$, and involves a larger expected wage payment than is required when effort is observable. The optimal compensation scheme for implementing $e_L$ involves the same fixed wage payment as if effort were observable. Whenever the optimal effort level with observable effort would be $e_H$, nonobservability causes a welfare loss.

Hidden Information

Assumptions The manager’s effort level, denoted by $e$, is fully observable. Effort can be measured by a one-dimensional variable $e\in [0,oo)$. What is not observable after the contract is signed is the random realization of the manager’s disutility from effort. Gross profits (excluding any wage payments to the manager) are a simple deterministic function of effort, $\pi (e)$, with $\pi (0)=0$, ${\pi }^{\prime }(e)>0$, and ${\pi }^{″}(e)<0$ for all $e$. The manager is an expected utility maximizer whose Bernoulli utility function over wages and effort, $u(w,e,\theta )$. $u(w,e,\theta )=v(w-g(e,\theta ))$. $g(0,\theta )=0$ $$\begin{array}{l}{g}_e(e,\theta )\{\begin{array}{ll}>0& \text{ for }e>0\\ =0& \text{ for }e=0\end{array}\\ g_{ee}(e,\theta )>0\text{ for all }e\\ g_{\theta }(e,\theta )<0\text{ for all }e\\ g_{e\theta }(e,\theta )\{\begin{array}{ll}<0& \text{ for }e>0\\ =0& \text{ for }e=0\end{array}\end{array}$$ Single-crossing property. The manager is strictly risk averse, with $v^{″}(\cdot )<0$. The manager’s reservation utility level is denoted by $\bar{u}$. $\theta$ can take only one of two values, with ${\theta }_H>{\theta }_L$ and $\mathrm{Prob}({\theta }_H)=\lambda \in (0,1)$.

The techniques we develop here can also be applied to models of monopolistic screening.

• Definition (Monopolistic Screening). In a setting characterized by precontractual informational asymmetries, a single uninformed individual offers a menu of contracts in order to distinguish, or screen, informed agents who have differing information at the time of contracting.

A contract must try to accomplish two objectives here:

1. The risk-neutral owner should insure the manager against fluctuations in his income;
2. A contract that maximizes the surplus available in the relationship must make the level of managerial effort responsive to the disutility incurred by the manager, that is, to the state $\theta$.

Observable State

Step 1

A complete information contract consists of two wage effort pairs: $(w_H,e_H)\in \mathbb{R}\times {\mathbb{R}}^{+}$ and $(w_L,e_L)\in \mathbb{R}\times {\mathbb{R}}^{+}$.

$$\begin{gathered} \begin{array}{rl}\underset{w_L,e_L\ge 0 \\ {w}_H,e_H\ge 0}{max}& \lambda [\pi (e_H)-w_H]+(1-\lambda )[\pi (e_L)-w_L]\\ & \text{s.t.}\lambda v(w_H-g(e_H,{\theta }_H))+(1-\lambda )v(w_L-g(e_L,{\theta }_L))\ge \bar{u}.\end{array} \end{gathered}$$

The reservation utility constraint must bind.

In addition, letting $\gamma \ge 0$ denote the multiplier on this constraint, the solution must satisfy the following first-order conditions:

$$\begin{array}{}\text{(12.C.2)}& -\lambda +\gamma \lambda v^{\prime }(w_H^{\ast }-g(e_H^{\ast },{\theta }_H))=0\end{array}$$$$\begin{array}{}\text{(12.C.3)}& -(1-\lambda )+\gamma (1-\lambda )v^{\prime }(w_L^{\ast }-g(e_L^{\ast },{\theta }_L))=0\end{array}$$$$\begin{array}{}\text{(12.C.4)}& \lambda {\pi }^{\prime }\left(e_H^{\ast }\right)-\gamma \lambda v^{\prime }(w_H^{\ast }-g(e_H^{\ast },{\theta }_H))g_e(e_H^{\ast },{\theta }_H)\{\begin{array}{l}\le 0\\ =0\text{ if }{e}_H^{\ast }>0\end{array}\end{array}$$$$\begin{array}{}\text{(12.C.5)}& (1-\lambda ){\pi }^{\prime }\left(e_L^{\ast }\right)-\gamma (1-\lambda )v^{\prime }(w_L^{\ast }-g(e_L^{\ast },{\theta }_L))g_e(e_L^{\ast },{\theta }_L)\{\begin{array}{l}\le 0,\\ =0\text{ if }{e}_L^{\ast }>0\end{array}\end{array}$$

First, rearranging and combining conditions (14.C.2) and (14.C.3), we see that

$$\begin{array}{}\text{(14.C.6)}& v^{\prime }(w_H^{\ast }-g(e_H^{\ast },{\theta }_H))=v^{\prime }(w_L^{\ast }-g(e_L^{\ast },{\theta }_L))\end{array}$$

• So the manager’s marginal utility of income is equalized across states.
  • The manager therefore has utility level $\bar{u}$ in each state.

Step 2

Now consider the optimal effort levels in the two states.

Combining condition (14.C.2) with (14.C.4), and condition (14.C.3) with (14.C.5), we see that the optimal level of effort in state ${\theta }_i$, $e_i^{\ast }$, satisfies

$$\begin{array}{}\text{(14.C.7)}& {\pi }^{\prime }(e_i^{\ast })=g_e(e_i^{\ast },{\theta }_i)\text{ for }i=L,H.\end{array}$$

This profit is exactly equal to the distance from the origin to the point at which the owner’s isoprofit curve through point $(w_i^{\ast },e_i^{\ast })$ hits the vertical axis.

Proposition 14.C.1. In the principal-agent model with an observable state variable $\theta$, the optimal contract involves an effort level $e_i^{\ast }$ in state ${\theta }_i$, such that ${\pi }^{\prime }(e_i^{\ast })=g_e(e_i^{\ast },{\theta }_i)$ and fully insures the manager, setting his wage in each state ${\theta }_i$, at the level $w_i^{\ast }$ such that $v(w_i^{\ast }-g(e_i^{\ast },{\theta }_i))=\bar{u}$.

With a strictly risk-averse manager, the first-best contract is characterized by two basic features:

1. The owner fully insures the manager against risk.
2. It requires the manager to work to the point at which the marginal benefit of effort exactly equals its marginal cost.
   • Because the marginal cost of effort is lower in state ${\theta }_H$ than in state ${\theta }_L$, the contract calls for more effort in state ${\theta }_H$.

Unobservable State

Proposition 14.C.2: (The Revelation Principle). Denote the set of possible states by $\mathrm{\Theta }$. In searching for an optimal contract, the owner can without loss restrict himself to contracts of the following form:

1. After the state $\theta$ is realized, the manager is required to announce which state has occurred.
2. The contract specifies an outcome $[w(\hat{\theta }),e(\hat{\theta })]$ for each possible announcement $\hat{\theta }\in \mathrm{\Theta }$.
3. In every state $\theta \in \mathrm{\Theta }$, the manager finds it optimal to report the state truthfully.

Definition (revelation mechanism). A contract that asks the manager to announce the state $\theta$ and associates outcomes with the various possible announcements is known as a revelation mechanism.

• Definition (incentive compatible revelation mechanism). Revelation mechanisms with truthfulness property that the manager always responds truthfully are known as incentive compatible (or truthful) revelation mechanisms.

Assumptions infinite risk aversion: Take the expected utility of the manager to equal the manager’s lowest utility level across the two states.

So we can ignore the choice of workers.

$$\begin{array}{rl}\underset{w_H,e_H\ge 0,w_{L,},e_L\ge 0}{max}& \lambda [\pi \left(e_H\right)-w_H]+(1-\lambda )[\pi \left(e_L\right)-w_L]\\ & \text{s.t.}(i)w_L-g(e_L,{\theta }_L)\ge v^{-1}(\overline{u})\\ & (ii)w_H-g(e_H,{\theta }_H)\ge v^{-1}(\overline{u})\\ & (iii)w_H-g(e_H,{\theta }_H)\ge w_L-g(e_L,{\theta }_H)\\ & (iv)w_L-g(e_L,{\theta }_L)\ge w_H-g(e_H,{\theta }_L)\end{array}$$

The pairs $(w_H,e_H)$ and $(w_L,e_L)$ that the contract specifies are now the wage and effort levels that result from different announcements of the state by the manager.

We show that the optimal level of $e_L$ satisfies the following first-order condition:

$$[{\pi }^{\prime }\left(e_L\right)-g_e(e_L,{\theta }_L)]+\frac{\lambda }{1-\lambda }[g_e(e_L,{\theta }_H)-g_e(e_L,{\theta }_L)]=0$$

Proposition 14.C.3. In the hidden information principal-agent model with an infinitely risk-averse manager the optimal contract sets the level of effort in state ${\theta }_H$ at its first-best (full observability) level $e_H^{\ast }$. The effort level in state ${\theta }_L$ is distorted downward from its first-best level $e_L^{\ast }$. In addition, the manager is inefficiently insured, receiving a utility greater than $\bar{u}$ in state ${\theta }_H$ and a utility equal to $\bar{u}$ in state ${\theta }_L$. The owner’s expected payoff is strictly lower than the expected payoff he receives when $\theta$ is observable, while the infinitely risk-averse manager's expected utility is the same as when $\theta$ is observable (it equals $\bar{u}$).