Some information problems only bite after a contract is signed. Your manager cannot observe how hard you actually work, so your pay must be built so that working hard is your own best move. That is moral hazard, and the principal–agent model is how economists tame it — a staple of intermediate micro, and one of the topics microeconomics tutors Yale students raise most. This page builds it from the primitives to the optimal contract.
1 · Hidden action, not hidden type
Moral hazard is a hidden action: the agent takes a decision the principal cannot see or verify. The classic case is effort — a worker knows how hard they tried, an employer sees only the result. Because effort is costly and unobservable, the agent has a private incentive to do less than the principal wants.
Keep this separate from its famous cousin. Under adverse selection the hidden thing is a type — a used car’s quality, a borrower’s riskiness — known before the deal. Under moral hazard the hidden thing is an action, chosen after it. Action versus type, after versus before: examiners use exactly that pairing to sort the two halves of asymmetric information, and mislabelling a question loses easy marks.
2 · Why a fixed wage fails
Strip the problem to its bones. The agent chooses effort — high or low. High effort costs c; low effort costs nothing. Effort does not fix the outcome, it shifts the odds: the project succeeds with probability pH under high effort and only pL under low, where pH > pL. The principal sees only success or failure, so pay can depend on that outcome but never on effort directly.
Now try a flat wage — the same amount whatever happens. High effort still costs c, low effort is free, and the wage is identical either way. So the agent pockets it and shirks. Every time. Pay that ignores the outcome cannot buy effort; it has to vary with what the principal can actually see.
3 · The two constraints every contract must clear
Write the contract as a base wage w plus a bonus b paid only on success. Two conditions decide whether it works.
Incentive compatibility (IC). The agent must prefer high effort. High effort raises the chance of the bonus by pH − pL, so its expected extra pay is (pH − pL)b. That must cover the effort cost:
![\[(p_H - p_L)\,b \ge c \quad\Longrightarrow\quad b \ge \frac{c}{\,p_H - p_L\,}\]](https://econtutor.com/wp-content/ql-cache/quicklatex.com-077325c5462b7ebd4ea217876aa312f8_l3.png "Rendered by QuickLaTeX.com")
A small gap pH − pL — outcomes that barely reveal effort — forces a large bonus.
Participation (IR). The agent can walk away to an outside option worth ū, so the contract must deliver at least that in expected terms, net of effort:
![\[w + p_H\,b - c \ge \bar{u}\]](https://econtutor.com/wp-content/ql-cache/quicklatex.com-f6ce89a56f90d52de20438babcfb97bd_l3.png "Rendered by QuickLaTeX.com")
IC fixes the smallest workable bonus; IR then fixes the base wage that just keeps the agent on board. Solve both as equalities and you have the cheapest contract that buys high effort.
4 · Insurance versus incentives
There is a cost hiding in that bonus: it makes pay risky, moving with an outcome the agent cannot fully control. A flat wage gives perfect insurance but zero incentive; a high-powered bonus gives sharp incentive but dumps risk on the agent. That is the trade-off at the heart of the model. When the agent is risk-neutral, loading on risk is costless and moral hazard costs nothing. When the agent is risk-averse, the principal must pay a risk premium to keep participation intact — so the agency cost is really the price of risk.
Worked example — paying a sales rep
A rep puts in high or low effort. High effort costs c = £100. It shifts the odds of closing a big account: pH = 0.75 versus pL = 0.25. A closed account is worth £1,000 to the firm, a miss £0. The rep’s outside option is ū = £100.
Step 1 — The pieces. Contract = base wage w + bonus b on a closed account. The effort gap is pH − pL = 0.5.
Step 2 — A flat wage fails. Set b = 0. High effort nets w − 100; low effort nets w. Low wins, so the rep coasts.
Step 3 — The minimum bonus (IC). High effort must pay for itself: b ≥ c/(pH − pL) = 100 / 0.5 = £200. Any smaller and the rep shirks. So b* = 200.
Step 4 — The base wage (IR). With the bonus set, participation binds: w + 0.75 × 200 − 100 = 100, so w* = 100 + 100 − 150 = £50. The optimal contract is (w*, b*) = (£50, £200).
Step 5 — The first-best benchmark. If effort were observable, the firm would just demand high effort and pay a flat wage covering cost plus outside option: ū + c = £200, with no bonus. But that contract, (£200, £0), has a zero bonus — it fails IC. Under hidden action the firm cannot use it.
Step 6 — The agency cost. Compare the wage bills. First-best: £200 flat. Second-best: expected pay = 50 + 0.75 × 200 = £200 — identical. Because the rep is risk-neutral, shifting £150 of pay into a risky bonus costs nothing extra, so the agency cost is zero. The firm earns 0.75 × 1,000 − 200 = £550 either way.
Step 7 — Where the cost would come from. Make the rep risk-averse and Step 6 changes: the risky bonus now needs a premium to satisfy IR, the wage bill climbs above £200, and that excess is the true cost of moral hazard.
Could you derive the optimal contract — the smallest bonus from IC, then the base wage from IR — rather than just describe it? That derivation, and the first-best-versus-second-best distinction, is exactly what principal–agent questions reward. Building it from the primitives on your own past papers is what a one-on-one microeconomics tutor does with you. Book a trial session.
Practice
Q1. A driver-app project has pH = 0.8, pL = 0.4, effort cost c = £60 and outside option ū = £80. (a) Find the minimum bonus that satisfies IC. (b) Find the base wage from binding IR. (c) What flat wage would work if effort were observable, and why does it fail under hidden action?
Q2. With pL = 0.5 and c = £40, a firm can raise pH from 0.7 to 0.9 by training its staff. Compute the minimum IC bonus before and after. What does the change say about how informative outcomes are?
Q3. A closed account is now worth £2,000. Using the worked contract (w = £50, b = £200) with pH = 0.75, pL = 0.25: (a) find the firm’s expected profit from inducing high effort; (b) compare it with inducing low effort under a flat wage of £100.
Answers. Q1: (a) b ≥ 60 / (0.8 − 0.4) = £150. (b) w = 80 + 60 − 0.8 × 150 = £20, so the contract is (£20, £150). (c) The first-best flat wage is ū + c = £140; with no bonus it fails IC (0 < 150). Q2: before, b ≥ 40 / 0.2 = £200; after, b ≥ 40 / 0.4 = £100. A wider gap makes outcomes more informative about effort, so a smaller bonus buys it — the bonus halves. Q3: (a) 0.75 × 2,000 − (50 + 0.75 × 200) = 1,500 − 200 = £1,300. (b) low effort pays 0.25 × 2,000 − 100 = £400; high effort dominates.
Key takeaways
- Moral hazard is a hidden action taken after the contract — unobservable effort, distinct from adverse selection’s hidden type known before it.
- A flat wage cannot buy effort; pay must depend on the observable outcome, so the contract needs a bonus.
- Two constraints pin the contract: IC sets the smallest bonus, b ≥ c/(pH − pL); IR sets the base wage. Both bind at the optimum.
- The agency cost is the price of risk — zero with a risk-neutral agent, positive once risk aversion forces a premium.
Why Yale students choose our microeconomics tutoring
- Built around the contract-theory core: sessions derive IC and IR from the primitives, so you can build a contract under exam pressure rather than half-remember a formula.
- The distinctions that carry marks, drilled: hidden action versus type, first-best versus second-best, risk-neutral versus risk-averse — rehearsed until you label them cleanly.
- One-on-one and matched to your course: tutors work from your own problem sets and past papers, in your lecturer’s notation.
FAQ
Q: What is the difference between moral hazard and adverse selection?
A: Moral hazard is a hidden action taken after a contract is signed — unobservable effort is the standard case. Adverse selection is a hidden type known before the deal, like a used car’s quality. Action versus type, after versus before.
Q: Why can’t the firm just pay a fixed wage?
A: Because effort is costly and a fixed wage is the same whatever the outcome. The agent gains nothing from working hard, so they shirk. Pay has to vary with the observable outcome to reward effort.
Q: What are the IR and IC constraints?
A: IR (participation) says the agent’s expected payoff must beat their outside option, or they walk away. IC (incentive compatibility) says the agent must prefer the effort the principal wants. A workable contract satisfies both.
Q: How do you find the minimum bonus?
A: From IC. High effort raises the success probability by pH − pL, so its extra expected pay is (pH − pL)b. Set that at least equal to the effort cost c and solve: b ≥ c/(pH − pL).
Q: What is the agency cost of moral hazard?
A: The extra the principal pays because effort is hidden. With a risk-neutral agent it is zero — the bonus aligns incentives for free. With a risk-averse agent the principal must add a risk premium, and that premium is the cost.
Book a microeconomics tutor for Yale coursework and exams
Principal–agent questions reward students who can derive a contract, not just describe one. One-on-one sessions build that from the primitives — IC, IR, the first-best benchmark, and the risk trade-off — on your own past papers. Tell us your course and exam date, and we will match you with the right tutor this week.
