Why Contract Design Matters
A firm may hire a specialist to inspect equipment, a public agency may contract for maintenance, or an insurer may compensate a claims reviewer. In each case, the principal cares about the result but may not be able to observe the agent’s effort continuously. If pay does not depend on anything effort affects, the agent may prefer to conserve effort. If pay depends heavily on uncertain results, however, the agent bears risk that may not be under the agent’s control.
This is the principal-agent problem: the principal chooses contract terms, while the agent chooses an action that may be hidden. The contract must make the desired action privately worthwhile, satisfy the agent’s willingness to participate, and leave enough expected value for the principal after compensation and monitoring costs. The earlier material on moral hazard after contracting introduces the hidden-action problem; here the focus is on how to design incentives and allocate risk.
A Basic Hidden-Effort Model
Consider a one-period model. The agent chooses low effort \(L\) or high effort \(H\). High effort costs the agent \(c\) and changes the probability of success from \(p_L\) to \(p_H\), where \(p_H>p_L\). Success produces revenue \(R\) for the principal. A contract pays \(w_S\) after success and \(w_F\) after failure. Begin with a risk-neutral agent whose utility is expected pay less effort cost, and a risk-neutral principal whose payoff is revenue less pay.
If the agent chooses effort \(e\), expected pay is \(p_e w_S+(1-p_e)w_F\). The agent chooses high effort only if doing so provides at least as much expected utility as low effort. This is the incentive compatibility (IC) condition. If the agent’s outside option has utility zero, the agent must also receive at least zero expected utility from accepting the contract; this is the participation constraint (PC).
The IC condition follows by subtracting the agent’s low-effort utility from high-effort utility. High effort changes the probability of receiving the success payment by \(p_H-p_L\), so its expected-pay advantage is \((p_H-p_L)(w_S-w_F)\). The agent incurs the additional effort cost \(c\). Therefore, high effort is at least as attractive precisely when the expected-pay advantage covers that cost. Equality makes the agent indifferent; a principal who needs high effort to be strictly preferred can offer slightly more than the minimum incentive.
The contract also has to be worthwhile for the principal. Conditional on high effort, expected profit is \(p_HR-[p_Hw_S+(1-p_H)w_F]\), before any monitoring or administrative costs. An incentive-compatible contract need not be profitable: if the effort-induced increase in expected revenue is too small relative to compensation, the principal may prefer not to offer it.
Worked Example: Finding a Bonus That Induces High Effort
A service contract pays a bonus \(b\) if a project is completed successfully and nothing if it fails. Success produces $100 of value for the principal. The agent is risk-neutral, has an outside option of zero, and can choose low effort at no cost or high effort at a cost of $20. Success probabilities are \(0.40\) under low effort and \(0.80\) under high effort. Find the minimum bonus that makes high effort weakly optimal, then check a proposed $55 bonus and calculate the principal’s expected profit.
Method: apply the IC condition to the difference between the two effort choices. Here \(w_S=b\), \(w_F=0\), and \(p_H-p_L=0.80-0.40=0.40\). Thus high effort is weakly preferred when \(0.40b\geq20\), or \(b\geq$50\). At exactly $50, the agent is indifferent, so the principal can use a larger bonus if strict preference is needed.
With the proposed \(b=$55\), expected pay under high effort is \(0.80($55)+0.20($0)=$44\). The agent’s net utility from high effort is \($44-$20=$24\). Under low effort, expected pay is \(0.40($55)+0.60($0)=$22\), with no effort cost, so net utility is $22. High effort is preferred by $2, and its utility of $24 exceeds the outside option of zero.
Expected revenue under high effort is \(0.80($100)=$80\). The principal’s expected profit is \($80-$44=$36\). The calculation checks both implementation and participation, as well as the principal’s payoff. Common mistake avoided: comparing the bonus itself with the effort cost. The agent compares the change in expected pay, which is the bonus multiplied by the change in success probability.
Risk Sharing and Performance Pay
A bonus can create an incentive because effort changes the chance of receiving it, but it also makes the agent’s pay uncertain. This matters when the agent is risk-averse. A risk-neutral agent evaluates a payment lottery by its expected value; a risk-averse agent generally values a risky payment less than a sure payment with the same expected amount. If the principal can bear outcome risk more easily, it may be costly to make the agent bear too much of it.
For a risk-averse agent, the principal faces a trade-off. A larger performance differential can strengthen incentives, but it exposes the agent to more outcome risk and may require greater expected compensation to secure participation. A fixed payment shares outcome risk with the principal, but by itself does not reward effort when the agent’s action cannot be observed and the payment is independent of the outcome.
The following example uses a specific utility function to quantify this trade-off. It assumes the agent’s utility from money \(w\geq0\) is \(u(w)=\sqrt{w}\), with effort cost subtracted in utility units. This is a risk-averse preference model; its numerical risk premium should not be treated as a universal value for all agents.
Worked Example: A Risk-Averse Agent and a Success Payment
A risk-averse agent receives \(W\) dollars on success and zero on failure. Success probability is \(0.75\) with high effort and \(0.35\) with low effort. High effort costs 2 utility units, low effort costs zero, and the agent’s outside utility is zero. The agent has utility \(u(w)=\sqrt{w}\). Find a success payment that strictly induces high effort, check participation, and calculate the principal’s profit if success yields $60.
Method: compare expected utility across effort choices. With payment \(W\), high-effort utility is \(0.75\sqrt{W}-2\); low-effort utility is \(0.35\sqrt{W}\). Their difference is \(0.40\sqrt{W}-2\). The incentive condition is \(0.40\sqrt{W}-2>0\), which requires \(\sqrt{W}>5\), or \(W>$25\). Choose \(W=$36, so \(\sqrt{W}=6\).
At \(W=$36\), high-effort utility is \(0.75(6)-2=4.5-2=2.5\). Low-effort utility is \(0.35(6)=2.1\). The agent prefers high effort by \(2.5-2.1=0.4\) utility units, and \(2.5>0\), so participation is satisfied.
Under high effort, expected compensation is \(0.75($36)+0.25($0)=$27\). Expected revenue is \(0.75($60)=$45\), so expected principal profit is \($45-$27=$18\). To measure the payment lottery’s risk cost, its certainty equivalent \(C\) solves \(\sqrt{C}=0.75\sqrt{36}=4.5\). Thus \(C=4.5^2=$20.25\). The expected payment is $27, so the risk premium relative to this certainty equivalent is \($27-$20.25=$6.75\). Common mistake avoided: treating $27 of expected pay as equivalent to a sure $27 for a risk-averse agent.
Monitoring as an Alternative Incentive
Monitoring can make an action more observable, or make shirking more likely to be detected. A contract may then use a fixed payment combined with a penalty for detected low effort. The principal compares the cost of monitoring with the compensation and risk costs of performance pay. Monitoring is not automatically worthwhile: it may be expensive, inaccurate, or unable to verify the action well enough to support a credible penalty.
Suppose high effort is never penalized, while low effort is detected with probability \(q\), and detection leads to a penalty \(F\). For a risk-neutral agent, the expected penalty from choosing low effort is \(qF\). If high effort costs \(c\), and the same fixed wage is paid regardless of effort, the agent prefers high effort when the expected penalty avoided by working is at least the effort cost.
This simple condition depends on the stated assumptions: there are no false detections of high effort, the penalty is enforceable, and the fixed wage is the same under either action. If detection also occurs after high effort, the relevant incentive is the difference in expected penalties between low and high effort. If penalties are constrained by limited liability, the principal may be unable to set \(F\) high enough; a different contract or more effective monitoring may be needed.
Worked Example: Checking Whether an Audit Incentivizes Effort
A public agency pays a contractor a fixed $50. High effort costs the contractor $20, while low effort costs nothing. An audit detects low effort with probability \(0.75\), and a detected contractor forfeits $30. High effort is never falsely penalized. Auditing costs the agency $6. Success produces $100 of value, with probability \(0.80\) under high effort and \(0.40\) under low effort. Determine the contractor’s action and the agency’s expected profit if the contract uses the audit.
Method: calculate the expected penalty avoided by high effort. If the contractor shirks, the expected penalty is \(0.75($30)=$22.50\). The contractor’s net payoff under high effort is \($50-$20=$30\). Under low effort it is \($50-$22.50=$27.50\). High effort is preferred by \($30-$27.50=$2.50\), so the incentive condition is satisfied. The high-effort payoff of $30 also exceeds the zero outside option.
With high effort, expected agency revenue is \(0.80($100)=$80\). The fixed wage costs $50 and the audit costs $6, giving expected profit of \($80-$50-$6=$24\). For comparison, if the same fixed wage were paid with no effective audit, the agent would choose low effort because the wage would not vary with effort and low effort costs nothing. Expected revenue would then be \(0.40($100)=$40\), and profit before audit cost would be \($40-$50=-$10\).
The audit makes high effort privately worthwhile and changes the agency’s expected outcome, but it is worthwhile only if the value of inducing effort justifies its cost. Common mistake avoided: comparing the $30 penalty directly with the $20 effort cost. The agent expects to pay the penalty only when low effort is detected, so the relevant amount is \(qF=$22.50\).
A Practical Contract-Design Sequence
The model separates three questions that can otherwise be confused: can the contract induce the desired action, will the agent accept it, and does the principal benefit from offering it? A contract that passes only one or two of these tests may not solve the principal’s problem.
State the available effort choices, their costs, and how each choice changes the probability or value of outcomes.
Compare expected utility from the intended action with expected utility from each alternative. This produces the incentive compatibility condition.
Confirm that the agent’s utility is at least the outside option. If the agent is risk-averse, account for how uncertain compensation affects the amount needed to secure participation.
Subtract expected compensation and monitoring or enforcement costs from expected revenue under the action the contract induces.
This sequence is useful in settings such as outsourced maintenance, public-service delivery, and claims review. In a hypothetical insurance arrangement, for example, an insurer might pay a reviewer partly according to the speed or accuracy of completed reviews. A speed bonus could increase throughput, but an excessively strong target could shift risk to the reviewer or encourage rushed decisions. The contract must reward the outcome the principal values without making payment depend on a measure that fails to represent it.
Check Your Understanding
For each problem, identify the relevant incentive or risk calculation before checking the answer.
- A risk-neutral agent chooses between low effort, which has no cost and success probability \(0.30\), and high effort, which costs $12 and has success probability \(0.70\). A contract pays a bonus only on success. What is the minimum bonus that makes high effort weakly optimal? (Answer: $30.)
- A risk-neutral agent receives a fixed $15 plus a $40 success bonus. Success probabilities are \(0.25\) under low effort and \(0.65\) under high effort; high effort costs $14. Check the incentive condition and calculate the agent’s net expected payoff under high effort. (Answer: High effort is preferred by $2; its net expected payoff is $27.)
- An agent with utility \(u(w)=\sqrt{w}\) receives \(W\) dollars on success and zero otherwise. Success probabilities are \(0.20\) under low effort and \(0.60\) under high effort; high effort costs 1.5 utility units. What is the minimum success payment that makes the agent indifferent between the two efforts? (Answer: $14.0625.)
- An audit detects low effort with probability \(0.80\), and high effort costs $16. What is the minimum enforceable penalty that makes high effort weakly optimal, assuming high effort is never falsely penalized and the same fixed wage is paid either way? (Answer: $20.)
- Under \(u(w)=\sqrt{w}\), an agent faces a payment of $64 on success and $0 on failure, with success probability \(0.50\). Calculate expected payment, the certainty equivalent of the payment lottery, and the risk premium. (Answer: Expected payment $32; certainty equivalent $16; risk premium $16.)
The contract and incentive calculations in this tutorial are aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.