Moral Hazard in Principal-Agent Problems

Introduction

In our previous lesson, we investigated adverse selection, where hidden characteristics before a transaction can lead to market failure, as vividly illustrated by the "market for lemons." We now pivot to the second major form of information asymmetry: moral hazard. This issue arises from hidden actions after a contract is in place.

This lesson addresses the classic principal-agent problem. A principal hires an agent to perform a task, but cannot observe the agent's effort, which is costly to the agent. The outcome of the task depends on both the agent's effort and random chance. Your goal for this lesson is to learn how to calculate the optimal contract a principal can offer to incentivize effort and to determine the final payoffs for both parties. This involves a fascinating optimization problem, blending elements of game theory and statistics that should resonate with your analytical background.

The Principal-Agent Problem with Moral Hazard

The core of the problem is a conflict of interest combined with asymmetric information.

The Principal (e.g., a company's owner) wants to maximize profit, which is higher when the agent puts in more effort.
The Agent (e.g., an employee) wants to maximize their own utility, which involves getting paid but avoiding costly effort.
The Asymmetry is that the principal cannot directly see or measure the agent's effort. All the principal observes is the final output (e.g., sales, profit), which is a noisy signal of the agent's effort due to random factors.

How can the principal motivate the agent to work hard? The solution is to design a contract where the agent's pay is linked to the observable outcome. This, however, introduces risk for the agent, and designing the contract requires balancing the need for incentives against the cost of imposing risk on a potentially risk-averse agent.

To get a clear picture of the fundamental setup, let's start with a video that outlines all the necessary components of a typical principal-agent model.

Principal Agent Models Part 1: Moral Hazard with Observability

This video from BurkeyAcademy, though titled "Part 1," provides an excellent setup for the general moral hazard problem we will be tackling.

Please watch from the beginning until the end of the setup. Focus on how the problem is structured: The definition of the principal and agent. The distinction between hidden action (moral hazard) and hidden information (adverse selection). The utility functions for the risk-neutral principal and the risk-averse agent. The agent's reservation utility, which forms the participation constraint. The probabilistic relationship between the agent's effort and the resulting output.

The video introduces a scenario with discrete effort and output levels. While intuitive, we can leverage your background with continuous variables and optimization to analyze a more general and powerful version of this model.

Designing an Optimal Linear Contract

Let's consider a scenario where the principal offers a linear contract of the form $w = F + b p$ . Here, $w$ is the total wage, $F$ is a fixed salary component, $b$ is a bonus rate, and $p$ is a measure of the agent's performance, which is observable. Both output and performance are influenced by the agent's effort, but also by random noise.

Our task is to solve for the principal's optimal choice of $F$ and $b$ . We will do this using backward induction:

First, we solve the agent's problem: For any given contract $(F, b)$ , what is the optimal effort level the agent will choose?
Second, we solve the principal's problem: Knowing how the agent will behave for any given contract, what contract $(F^{*}, b^{*})$ should the principal offer to maximize their own profit?

We will now work through a detailed problem that walks through these steps mathematically.

[PDF] EconS 503 - Microeconomic Theory II Homework #7 - Answer key

This document provides a fully solved example of a moral hazard problem with continuous effort and a linear contract. It's a perfect application of the concepts we're discussing.

Please read the following parts of Question 4, "Moral hazard with multiple tasks": First, carefully read the problem setup on page 2. Understand the agent's cost function, the output and performance functions, and the structure of the linear contract. Next, study part (a), which solves the agent's problem. This involves setting up the agent's expected utility and using first-order conditions to find the optimal efforts, e_1(b) and e_2(b), as a function of the bonus rate b. Then, work through part (b), which tackles the principal's problem. The principal substitutes the agent's optimal effort functions into their own profit function and then maximizes it with respect to b to find the optimal bonus, b^*. Finally, jump to part (g), the numerical example. This section plugs specific values into the formulas you've just seen to calculate the equilibrium bonus, efforts, and expected payoffs for both the agent and the principal.

Unpacking the Solution

Let's break down the key calculations and insights from the document.

The Agent's Effort Choice

The agent's problem is to choose effort levels $e_{1}$ and $e_{2}$ to maximize their expected utility:

e_{1}, e_{2} max E [U] = E [F + b (g_{1} e_{1} + g_{2} e_{2} + ϕ) - \frac{1}{2} e_{1}^{2} - \frac{1}{2} e_{2}^{2}]

Since the expected value of the random shock $ϕ$ is zero, this simplifies to:

e_{1}, e_{2} max F + b g_{1} e_{1} + b g_{2} e_{2} - \frac{1}{2} e_{1}^{2} - \frac{1}{2} e_{2}^{2}

Taking the first-order conditions with respect to $e_{1}$ and $e_{2}$ gives us the agent's optimal effort levels in response to the bonus $b$ :

\frac{\partial E [ U ]}{\partial e _{1}} = b g_{1} - e_{1} = 0 ⟹ e_{1}^{*} (b) = b g_{1}

\frac{\partial E [ U ]}{\partial e _{2}} = b g_{2} - e_{2} = 0 ⟹ e_{2}^{*} (b) = b g_{2}

This is the incentive compatibility condition in action. It shows precisely how the agent's unobservable effort responds to the observable incentive $b$ . A higher bonus rate directly induces higher effort.

The Principal's Optimal Contract

The principal anticipates the agent's reaction and substitutes $e_{1}^{*} (b)$ and $e_{2}^{*} (b)$ into their own expected profit function. The principal's expected profit is $E [π] = E [y - w]$ . After substituting everything in, the principal's problem becomes:

b, F max b g_{1} (f_{1} - b g_{1}) + b g_{2} (f_{2} - b g_{2}) - F

Notice that profit is strictly decreasing in the fixed payment $F$ . To maximize profit, the principal will set $F$ at the lowest possible level that ensures the agent is willing to work. In this simplified model, that level is $F^{*} = 0$ (assuming a reservation utility of zero). The main decision is the bonus rate $b$ . Taking the first-order condition with respect to $b$ and solving yields the optimal bonus rate:

b^{*} = \frac{f _{1} g _{1} + f _{2} g _{2}}{2 ( g _{1}^{2} + g _{2}^{2} )}

Calculating Equilibrium Payoffs

Once we have the optimal contract $(F^{*}, b^{*})$ , we can determine the equilibrium outcomes. The numerical example in part (g) of the reading demonstrates this perfectly. With $f_{1} = 1/2, f_{2} = 1/3, g_{1} = 2/3, g_{2} = 1/4$ , we find:

Optimal Bonus: $b^{*} = \frac{30}{73}$
Agent's Effort: $e_{1}^{*} = \frac{20}{73}$ and $e_{2}^{*} = \frac{15}{146}$
Principal's Expected Profit: $π^{*} = \frac{25}{292} \approx 0.08$
Agent's Expected Utility: $U^{*} = \frac{25}{584} \approx 0.04$

This completes the exercise: we have used the parameters of the problem to calculate the specific effort the agent will exert and the resulting expected payoff for both the principal and the agent under the optimally designed contract. The difference between the wage required in a world of perfect information and this incentive-based wage represents the agency cost—the price the principal pays to overcome the moral hazard problem.

Conclusion

In this lesson, you have moved from the pre-contractual problem of adverse selection to the post-contractual problem of moral hazard. You have learned how to model the principal-agent relationship and solve for the optimal incentive contract when the agent's actions are unobservable.

Key Takeaways:

Moral hazard in a principal-agent context arises when an agent's costly actions are unobservable to the principal after a contract is signed.
The principal's challenge is to design an incentive contract, typically linking the agent's pay to a noisy but observable performance measure, to motivate effort.
The problem can be solved using backward induction: first determine the agent's optimal effort as a function of the contract, then the principal chooses the optimal contract anticipating that response.
This requires satisfying two implicit constraints: the Participation Constraint (the agent must be willing to accept the contract) and the Incentive Compatibility Constraint (the agent must find it optimal to choose the effort level the principal desires).
By setting up and solving the parties' respective optimization problems, we can calculate the optimal contract parameters, the agent's resulting effort, and the expected payoffs for both principal and agent.

In our next lesson, we will shift our focus back to a more direct form of government intervention by analyzing the welfare effects of imposing a price cap on a monopoly.

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