Moral Hazard in Principal-Agent Relationships

The Principal-Agent Problem: A Concrete Example

The Principal-Agent (P-A) framework models countless real-world relationships: employer-employee, client-contractor, or even insurer-insured. To make this tangible, we will analyze a specific scenario.

Now, let's establish the specific model we will work with for the first part of our lesson, as introduced in the same video.

Scenario: Gold Mining

• Principal (P): A landowner who hires a miner. The Principal is risk-neutral. Their utility is simply profit: $U_P=\text{Output}-\text{Wage}$.
• Agent (A): A miner who can be hired. The Agent is risk-averse. Their utility depends on the wage they receive and the effort they exert: $U_A(w,a)=\sqrt{w}-C(a)$. The square root function for wages mathematically captures risk aversion—the marginal utility of an extra dollar decreases as wealth increases.
• Effort (Action): The Agent can choose high effort ($a=1$) or low effort ($a=0$). The cost of effort is $C(1)=1$ and $C(0)=0$.
• Output (Outcome): The amount of gold found is stochastic. It can be high ($x_H=400$) or low ($x_L=100$).
• Probabilities: Effort influences the probability of finding a lot of gold.
  • If Agent chooses high effort ($a=1$), $P(x=400|a=1)=2/3$.
  • If Agent chooses low effort ($a=0$), $P(x=400|a=0)=1/3$.
• Reservation Utility: The Agent has an outside option (e.g., another job) that gives them a utility of ${\bar{U}}_A=10$. They will only accept a contract if their expected utility is at least 10. This is their participation constraint.

The First-Best Benchmark: What if Effort Were Observable?

Before tackling the moral hazard problem, we must establish a benchmark. What would the optimal contract be if the Principal could perfectly observe the Agent's effort? This is known as the first-best scenario.

In this case, the Principal can write a "forcing contract": "I will pay you wage $w$ if and only if you exert effort $a$." Since the Agent is risk-averse and the Principal is risk-neutral, the most efficient arrangement is for the Principal to bear all the risk. This is achieved by offering the Agent a fixed wage, which provides full insurance against the random output.

The Principal's problem is to figure out which effort level (high or low) is more profitable, and what fixed wage is required for each.

Let's summarize the logic:

1. Cost to induce Low Effort ($a=0$):
   The Principal must offer a wage $w_L$ that satisfies the Agent's participation constraint:
   $U_A(w_L,0)\ge {\bar{U}}_A⟹\sqrt{w_L}-0\ge 10⟹w_L\ge 100$
   The cheapest wage is $w_L=100$.
   The Principal's expected profit is $E[\pi |a=0]=E[\text{Output}|a=0]-w_L$:
   $E[\pi |a=0]=\left(\frac{1}{3}\cdot 400+\frac{2}{3}\cdot 100\right)-100=200-100=100$

2. Cost to induce High Effort ($a=1$):
   The Principal must offer a wage $w_H$ that satisfies the participation constraint:
   $U_A(w_H,1)\ge {\bar{U}}_A⟹\sqrt{w_H}-1\ge 10⟹\sqrt{w_H}\ge 11⟹w_H\ge 121$
   The cheapest wage is $w_H=121$.
   The Principal's expected profit is $E[\pi |a=1]=E[\text{Output}|a=1]-w_H$:
   $E[\pi |a=1]=\left(\frac{2}{3}\cdot 400+\frac{1}{3}\cdot 100\right)-121=300-121=179$

Comparing the two, $179>100$, so the Principal prefers to induce high effort. The first-best contract is to pay a fixed wage of $121$ contingent on observing high effort. This outcome is efficient: the desired action is taken, and risk is optimally allocated to the risk-neutral party.

The Moral Hazard Problem: The Incentive-Insurance Trade-off

Now, let's introduce the information asymmetry: the Principal cannot observe effort. They only see the output (100 or 400).

Consider the first-best contract we just found: pay a fixed wage of 121. If the wage is guaranteed regardless of output, what will the Agent do? Since effort is costly and unobserved, the Agent will choose low effort ($a=0$) to maximize their utility ($U_A=\sqrt{121}-0=11$), saving the cost of high effort. The Principal, expecting an output of 300, will instead get an expected output of only 200, leading to a much lower profit of $200-121=79$. The first-best contract fails.

This brings us to the core dilemma of moral hazard.

The tension is between:

• Optimal Risk-Sharing: The risk-averse Agent wants a stable income (full insurance), while the risk-neutral Principal is the efficient risk-bearer. This pushes towards a fixed wage.
• Providing Incentives: To motivate the Agent to exert costly effort, their payment must be linked to performance. This pushes towards a variable wage that depends on the (stochastic) output.

The optimal contract under moral hazard, known as the second-best contract, must be a compromise. It will expose the Agent to some risk to create incentives, which is inefficient from a risk-sharing perspective but necessary to motivate effort.

To design such a contract, the Principal must ensure two conditions are met:

1. Participation Constraint (PC): The agent must be willing to accept the contract. Their expected utility from the contract (assuming they exert the desired effort) must be at least their reservation utility.
2. Incentive Compatibility Constraint (IC): The agent must voluntarily choose the desired effort level. Their expected utility from choosing the high-effort action must be greater than or equal to their expected utility from choosing the low-effort action.

A Solvable Model: Moral Hazard with Limited Liability

Solving the gold-mining model with a risk-averse agent involves complex calculations. To clearly see the strategic response to moral hazard, let's analyze a simpler but equally important model where both parties are risk-neutral, but the agent is protected by limited liability—their wage cannot be negative. This constraint prevents the principal from imposing extreme punishments for bad outcomes, making it harder to provide incentives.

This model is representative of many corporate finance and employment situations.

Let's formalize and solve the example from the notes.

Model Setup:

• Parties: Risk-neutral Principal and Agent.
• Effort: High ($a_H$) or Low ($a_L$). Costs: $c(a_H)=c_H$, $c(a_L)=0$.
• Output: Success ($x=1$) or Failure ($x=0$).
• Probabilities:
  • $P(x=1|a_H)=p_H$
  • $P(x=1|a_L)=p_L$, where $p_H>p_L$.
• Contract: A pair of wages $(s_1,s_0)$, where $s_1$ is the wage for success and $s_0$ for failure.
• Constraints:
  • Limited Liability (LL): $s_1\ge 0,s_0\ge 0$.
  • Reservation Utility $\bar{U}=0$.

The Principal wants to implement high effort ($a_H$) at the minimum possible cost. The cost is the agent's expected wage: $E[s|a_H]=p_H{s}_1+(1-p_H)s_0$.

The contract must satisfy two constraints:

1. Incentive Compatibility (IC): The Agent must prefer high effort.
   $\underset{\text{Expected payoff from high effort}}{\underbrace{p_H{s}_1+(1-p_H)s_0-c_H}}\ge \underset{\text{Expected payoff from low effort}}{\underbrace{p_L{s}_1+(1-p_L)s_0-0}}$

2. Participation Constraint (PC): The Agent must prefer accepting the contract to their outside option.
   $p_H{s}_1+(1-p_H)s_0-c_H\ge 0$

Solving for the Optimal Contract:
To minimize the expected wage cost, the Principal should provide the weakest possible incentives. This means making the IC and PC constraints hold with equality. The Principal will also want to set the wage for failure, $s_0$, as low as possible. The limited liability constraint means the lowest it can go is $s_0=0$.

Let's set $s_0=0$ and see if we can satisfy the constraints. The IC constraint becomes:
$p_H{s}_1-c_H\ge p_L{s}_1⟹(p_H-p_L)s_1\ge c_H⟹s_1\ge \frac{c_H}{p_H-p_L}$
To minimize cost, the Principal sets $s_1$ to the lowest possible value: $s_1^{\ast }=\frac{c_H}{p_H-p_L}$.

The optimal contract to induce high effort is $(s_1^{\ast },s_0^{\ast })=(\frac{c_H}{p_H-p_L},0)$.

The Strategic Consequence: Information Rent
Now, let's check the Agent's actual payoff under this contract. Does the PC bind?
$E[\text{Payoff}]=p_H{s}_1^{\ast }+(1-p_H)s_0^{\ast }-c_H=p_H\left(\frac{c_H}{p_H-p_L}\right)-c_H$
$E[\text{Payoff}]=c_H\left(\frac{p_H}{p_H-p_L}-1\right)=c_H\left(\frac{p_H-(p_H-p_L)}{p_H-p_L}\right)=\frac{p_L{c}_H}{p_H-p_L}$
Since $p_L>0$ and $c_H>0$, the Agent's expected payoff is strictly greater than 0 (their reservation utility). This extra payoff is called an information rent. It is the premium the Principal must pay the Agent because their action is hidden. This is a direct cost of moral hazard.

Conclusion

In this lesson, we have explored the strategic problem of moral hazard. By moving from a world of observable actions to one of hidden actions, we uncovered a fundamental conflict between providing incentives and insuring against risk.

Key Takeaways:

• Moral Hazard arises from hidden actions, where one party (the Principal) cannot observe the effort of another (the Agent).
• The first-best contract, which is optimal when actions are observable, typically involves the risk-neutral Principal fully insuring the risk-averse Agent with a fixed wage. This contract fails under moral hazard as it eliminates the incentive to exert effort.
• The strategic response to moral hazard is a second-best contract that links pay to a performance measure (output). This contract forces the Agent to bear some risk in order to incentivize effort.
• This incentive-risk trade-off is costly. To induce the desired action, the Principal often must design a contract that leaves the Agent with an information rent—an expected payoff greater than their reservation utility. This rent is a core consequence of the information asymmetry.

In the next module, we will transition to a new topic: Bargaining and Negotiation. We will leave behind the world of information asymmetry for a moment and instead focus on how two fully-informed rational players divide a surplus between them. Our main tool will be the Rubinstein alternating-offers model, which provides a powerful and elegant solution to this fundamental strategic interaction.