Moral Hazard After Contracting

Why Actions After an Agreement Matter

A contract can change behavior as well as divide costs. Before an agreement, an insurer may not know how likely a customer is to take precautions. After the agreement, the customer may choose how carefully to maintain equipment, follow a safety routine, or use a covered service. If that action is difficult for the other party to observe or verify, the contract may affect the action the customer chooses.

This is moral hazard: a party’s incentives to take an action change after an agreement because the action or its consequences are not fully observed or borne by that party. The term describes an incentive problem; it does not imply dishonesty. A person can respond to changed financial incentives without misrepresenting anything.

This differs from adverse selection. Adverse selection concerns information about a person or risk type that is hidden when a contract is chosen. Moral hazard concerns an action taken after the agreement. A contract can face both problems, but the analysis here isolates the post-contract action.

Definition: In a simple moral-hazard model, an agent chooses an action $e$ after contracting. The action affects the probability or size of a loss, but the other party cannot observe or verify it perfectly. Moral hazard arises when the contract changes the agent’s private cost-benefit comparison and therefore changes the chosen action.

A Simple Effort-and-Loss Model

Consider two actions: low effort and high effort. High effort costs the individual $c$ dollars and reduces the probability of a loss of fixed size $L$. Let $q_L$ be the loss probability with low effort and $q_H$ the loss probability with high effort, where $q_H<q_L$. Assume a risk-neutral individual, a fixed premium that does not change with the action, and no other effort-related benefits or costs.

With no insurance, the expected private cost of low effort is $q_LL$. The expected private cost of high effort is $c+q_HL$. High effort is privately preferred exactly when its expected cost is lower:

c+q_HL<q_LL \quad\Longleftrightarrow\quad c<(q_L-q_H)L.

The right side is the expected loss reduction from high effort. This same comparison gives a social-efficiency test in this stripped-down model: high effort is socially worthwhile when the expected resource loss it prevents exceeds its resource cost. Insurance payments and premiums transfer money between parties; they do not, by themselves, eliminate the underlying loss or the effort cost.

The private comparison changes under insurance because the insured may pay only part of a loss. For a fixed loss $L$ and a dollar deductible $d$, the insured pays $\min(d,L)$ when the loss occurs. The insurer pays the rest. Assuming the premium is fixed after contracting, the insured’s expected cost of effort $e$ is the effort cost plus the probability of loss times the insured’s out-of-pocket amount. This model assumes one loss size, no policy limit, and no other contract terms; it does not describe every insurance contract.

Worked Example: Finding the Incentive Effect of a Dollar Deductible

A hypothetical equipment policy has a fixed premium and a deductible of $d$ dollars. A covered breakdown costs $L=\$2{,}000$. The breakdown probability is $0.12$ under low effort and $0.07$ under high effort. High effort costs the equipment owner $\$60$. Find the deductible threshold above which high effort is privately preferred, and determine whether high effort is socially worthwhile. Assume the owner is risk-neutral, the loss size is fixed, and there are no other costs.

Method. Compare the expected reduction in the owner’s out-of-pocket loss with the extra effort cost. Because $d\leq \$2{,}000$ at the threshold, the owner pays $d$ if a breakdown occurs.

The decrease in breakdown probability is $0.12-0.07=0.05$. Thus high effort reduces the owner’s expected deductible payment by $0.05d$. High effort is strictly preferred when

0.05d>60 \quad\Longleftrightarrow\quad d>\frac{60}{0.05}=\$1{,}200.

At $d=\$1{,}200$, the owner is indifferent; a deductible greater than $\$1{,}200$ makes high effort strictly preferable. This threshold is below the loss size of $\$2{,}000$, so the assumed out-of-pocket amount $d$ is valid there.

For society, the expected loss reduction is $0.05(\$2{,}000)=\$100$. High effort costs $\$60$, so its net social benefit is $\$100-\$60=\$40$. It is socially worthwhile in this model. The result does not mean that any deductible above $\$1{,}200$ is best: a larger deductible also leaves the owner bearing more financial risk.

How Cost Sharing Changes Incentives

A deductible makes the insured bear the first dollars of a covered loss. With a fixed loss $L$, the out-of-pocket amount is $\min(d,L)$, and the insurer’s payment is $\max(L-d,0)$. If high effort lowers the loss probability, a deductible gives the insured a stronger financial reason to prevent the loss. It also provides less protection when a loss occurs.

A contract can instead require the insured to pay a fraction of the covered cost. That fraction is coinsurance. A fixed-dollar copayment, by contrast, is an amount paid per service. These terms are not interchangeable: a payment equal to 50% of a service’s cost is coinsurance, not a copayment. Cost sharing can discourage unnecessary use, but a fixed copayment does not vary with the resource cost of each service.

Worked Example: Interpreting a Retained Share Correctly

A hypothetical contract leaves the insured responsible for a fraction $r$ of a fixed $\$1{,}000$ loss; the insurer pays the remainder. The premium is fixed. The loss probability is $0.10$ with low effort and $0.06$ with high effort, which costs $\$20$. Find the boundary value of $r$ at which the insured is indifferent, and state when high effort is strictly preferred. Assume risk neutrality and no other costs.

Method. The insured’s out-of-pocket amount when the loss occurs is $r(\$1{,}000)$. The reduction in expected out-of-pocket loss from high effort is the probability reduction times that amount:

(0.10-0.06)r(\$1{,}000)=0.04r(\$1{,}000)=\$40r.

High effort is strictly preferred when its expected private saving exceeds its $\$20$ cost:

\$40r>\$20 \quad\Longleftrightarrow\quad r>0.5.

The boundary is $r=0.5$, or 50% of the loss. At that boundary the insured bears $\$500$ if the loss occurs and is indifferent between effort levels. A retained share greater than 0.5 makes high effort strictly preferred. The boundary value 0.5 is a fraction, not a dollar deductible.

The total expected loss reduction from high effort is $0.04(\$1{,}000)=\$40$, greater than the $\$20$ effort cost. Thus high effort is socially worthwhile in this model. At the boundary $r=0.5$, the insured receives only half of the expected loss-reduction incentive, illustrating why a privately chosen action can depend on the contract even when the underlying social comparison is unchanged.

Monitoring and Performance Conditions

Cost sharing is not the only way to address hidden actions. Monitoring can make an action more observable. A contract may also condition a payment on a verifiable result, such as completing a documented safety check. Such terms can strengthen incentives, but monitoring uses resources, and an outcome is not always a reliable measure of effort: it may depend on luck or factors outside the individual’s control.

To assess monitoring, separate three questions: Does it change the individual’s incentives? What real resources does monitoring use? Does the resulting action reduce expected losses by more than the effort and monitoring costs? A penalty or withheld payment is generally a transfer between parties in a simple social-cost calculation; the monitoring work itself is a resource cost.

Worked Example: Checking Private Incentives Against Monitoring Costs

A hypothetical property manager chooses low or high preventive effort. A fire loss of $\$1{,}000$ occurs with probability $0.08$ under low effort and $0.03$ under high effort. High effort costs the manager $\$35$. An audit detects low effort with probability $0.20$; if detected, a contractually valid condition withholds $\$300$ from the manager. Assume high effort is never wrongly identified as low effort, the audit has no other effect, and the manager is risk-neutral. Monitoring costs the insurer $\$12$ per contract. Is high effort privately induced, and is the monitored arrangement socially worthwhile?

Method. First compare the manager’s expected contract consequences. Under low effort, the expected withheld amount is $0.20(\$300)=\$60$. Under high effort, the expected withholding is zero, but the manager pays the $\$35$ effort cost. Since $\$60>\$35$, the manager prefers high effort under this condition.

Next compare social resource costs. High effort reduces the expected fire loss by $(0.08-0.03)(\$1{,}000)=\$50$. Subtract high-effort cost and monitoring cost: $\$50-\$35-\$12=\$3$. The monitored arrangement has a positive net social benefit of $\$3$ under the stated assumptions. The $\$300$ withholding is not subtracted as a social resource cost because it is a transfer; the audit’s $\$12$ cost is a resource cost.

The conclusion depends on the assumptions. If monitoring cost exceeded $\$15$, the avoided expected loss would no longer cover the combined effort and monitoring costs. If audits sometimes wrongly penalized high-effort managers, that risk could also change their incentives.

Deriving the Deductible Incentive Condition

The deductible rule follows by comparing expected private costs under the two actions. Let $c$ be the additional cost of high effort, $q_L$ and $q_H$ the loss probabilities, and $L$ the fixed loss. With a dollar deductible $d$, the insured pays $\min(d,L)$ when a loss occurs. A fixed premium is the same under both actions, so it cancels in the comparison.

\begin{aligned} C_L&=q_L\min(d,L),\\ C_H&=c+q_H\min(d,L),\\ C_H<C_L &\Longleftrightarrow c+q_H\min(d,L)<q_L\min(d,L)\\ &\Longleftrightarrow c<(q_L-q_H)\min(d,L). \end{aligned}

This derivation explains why the deductible affects behavior: it increases the amount of a loss the insured expects to bear, making prevention more valuable to that person. If $d\geq L$, further increases in the deductible do not change the out-of-pocket payment for this fixed-size loss. The formula also does not apply unchanged when loss sizes vary, premiums depend on effort, or the individual is risk-averse; those features require a model that includes the relevant probabilities, contract terms, or risk preferences.

Common Mistake: Mixing Fractions and Dollar Amounts

A frequent error is to calculate a threshold as $20/(0.10-0.06)=500$ and then treat 500 as the required retained fraction. The probability difference alone is not the expected saving per dollar of a $\$1{,}000$ loss. The correct comparison for a retained fraction $r$ is $20=(0.10-0.06)(\$1{,}000)r$, giving $r=0.5$ at indifference. That fraction corresponds to $\$500$ retained on the $\$1{,}000$ loss.

If the contract instead specifies a dollar deductible $d$, write the dollar comparison directly: $20=(0.10-0.06)d$, so the indifference deductible is $\$500$, provided the deductible does not exceed the loss. Here 0.5 is the retained fraction; $\$500$ is the equivalent dollar amount. Do not attach dollar units to a fraction or treat a dollar amount as a rate.

Another error is to conclude that stronger incentives always improve welfare. A deductible, copayment, monitoring system, or performance condition may reduce overuse or encourage prevention, but it can also impose risk on the insured, discourage valuable actions, or consume monitoring resources. A sound analysis states whose incentives change and compares avoided resource costs with all real resources used.

Key takeaway: Moral hazard is a post-contract incentive problem. To analyze it, identify the hidden action, compare the action’s private expected costs under the contract, and then separately assess whether its real benefits exceed its real costs. This treatment is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Check Your Understanding

For each problem, assume risk neutrality and the stated contract terms. Distinguish a fraction from a dollar amount when reporting the answer.

A hypothetical policy has a fixed premium, a $\$1{,}500$ loss, and a dollar deductible $d$. Loss probability falls from $0.20$ to $0.12$ with high effort, which costs $\$45$. Find the deductible boundary for indifference and say when high effort is strictly preferred. (Answer: $d=\$562.50$ is the boundary; high effort is strictly preferred for $d>\$562.50$.)
A contract leaves the insured a fraction $r$ of an $\$800$ loss. Loss probability falls from $0.15$ to $0.10$ with high effort, which costs $\$24$. Find the retained-share boundary. Is high effort strictly preferred at the boundary? (Answer: $r=0.60$; no, the insured is indifferent at 60% and strictly prefers high effort only for $r>0.60$.)
A $\$1{,}200$ loss occurs with probability $0.09$ under low effort and $0.04$ under high effort. High effort costs $\$48$, and monitoring costs $\$10$. Calculate the net social benefit of high effort with monitoring. (Answer: avoided expected loss is $0.05(\$1{,}200)=\$60$; net social benefit is $\$60-\$48-\$10=\$2$.)
A verified maintenance condition withholds $\$250$ from a manager if an audit identifies low effort. Audits occur with probability $0.30$, and high effort costs $\$65$. Assume high effort is never wrongly identified as low effort. Does the condition privately induce high effort? (Answer: expected withholding under low effort is $0.30(\$250)=\$75$, which exceeds $\$65$; yes, high effort is privately preferred.)
A health plan charges a fixed $\$15$ per visit for a service that has a resource cost of either $\$60$ or $\$180$. Is the member’s payment a copayment or coinsurance, and what share of the resource cost does it represent in each case? (Answer: it is a fixed copayment; it is 25% of the $\$60$ cost and $8.33\%$ of the $\$180$ cost, so it is not a constant coinsurance rate.)

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