Moral Hazard and Incentives

This scene designs a contract when the boss cannot watch the work. A flat wage alone gives no reason to work hard.

Pay must hang on the outcome. Good results need a bonus on top. Two conditions must hold at once. Working hard must beat slacking off. And the job must be worth taking at all. The less the worker fears risk, the gentler the bonus slope. It is a tug-of-war between sharing risk and pushing effort.

For example, you run a 🏪 snack shop and leave a part-timer alone. You cannot see how hard they work. So you do not fix the wage. On days the shop earns more 💰 money, you pay a bonus. Then the part-timer works hard on their own.

(p1−p0)(uH−uL)=c(IC)(p_1 - p_0)(u_H - u_L) = c \qquad \text{(IC)}
p1uH+(1−p1)uL=uˉ+c(IR)p_1 u_H + (1 - p_1) u_L = \bar u + c \qquad \text{(IR)}
wL=uL1/α,wH=uH1/αw_L = u_L^{1/\alpha}, \quad w_H = u_H^{1/\alpha}

Part of these learning paths

Examples & Exercises

Let p1=0.8p_1 = 0.8, p0=0.2p_0 = 0.2, uˉ=20\bar u = 20, u(w)=w1/2u(w) = w^{1/2}, c=10c = 10.

Find the optimal contract (wL,wH)(w_L, w_H) implementing high effort, and compute the expected wage C1C_1, the full-insurance cost, and the risk premium.

uL=uˉ−cp0p1−p0=16.667,uH=uˉ+c(1−p0)p1−p0=33.333u_L = \bar u - \frac{c p_0}{p_1 - p_0} = 16.667, \qquad u_H = \bar u + \frac{c(1-p_0)}{p_1-p_0} = 33.333
wL=uL2=277.78,wH=uH2=1111.11w_L = u_L^2 = 277.78, \qquad w_H = u_H^2 = 1111.11
α=1/2\alpha = 1/2, c=10c = 10 — the optimal contract is where the IR and IC curves cross.
Show proof

  1. Both IC and IR bind, so (0.8−0.2)(uH−uL)=10(0.8 - 0.2)(u_H - u_L) = 10 gives uH−uL=50/3u_H - u_L = 50/3, and 0.8uH+0.2uL=300.8 u_H + 0.2 u_L = 30. Solving, uL=50/3≈16.667u_L = 50/3 \approx 16.667 and uH=100/3≈33.333u_H = 100/3 \approx 33.333.

  2. With u(w)=w1/2u(w) = w^{1/2}, wages are wL=(50/3)2≈277.78w_L = (50/3)^2 \approx 277.78 and wH=(100/3)2≈1111.11w_H = (100/3)^2 \approx 1111.11. The expected wage is C1=0.8×1111.11+0.2×277.78≈944.44C_1 = 0.8 \times 1111.11 + 0.2 \times 277.78 \approx 944.44.

  3. Had effort been observable, full insurance at w=30w = 30 would suffice, costing 302=90030^2 = 900. The risk premium is 944.44−900=44.44944.44 - 900 = 44.44 — the price of loading risk onto the agent through performance pay because effort is unobservable.

    C1−(uˉ+c)2=944.44−900=44.44C_1 - (\bar u + c)^2 = 944.44 - 900 = 44.44

  4. Check that IC really binds: shirking pays the agent 0.2×33.333+0.8×16.667=20=uˉ0.2 \times 33.333 + 0.8 \times 16.667 = 20 = \bar u, and working hard gives expected utility 0.8×33.333+0.2×16.667−10=200.8 \times 33.333 + 0.2 \times 16.667 - 10 = 20 too. IC slack =0= 0 — the agent is exactly indifferent between shirking and working.

With the same parameters but α=1\alpha = 1 (a risk-neutral agent), find the optimal contract and show the risk premium is zero.

Incentive compatibility still demands a wage gap — why does the cost equal full insurance anyway?

wL=uL=16.667,wH=uH=33.333w_L = u_L = 16.667, \qquad w_H = u_H = 33.333
C1=0.8×33.333+0.2×16.667=30=uˉ+cC_1 = 0.8 \times 33.333 + 0.2 \times 16.667 = 30 = \bar u + c
α=1\alpha = 1, c=10c = 10 — the contract sits exactly on the full-insurance isocost.
Show proof

  1. The utility spread (uL,uH)=(16.667,33.333)(u_L, u_H) = (16.667, 33.333) is pinned down by IC and IR regardless of α\alpha. With α=1\alpha = 1, u(w)=wu(w) = w, so wages equal utilities: wL=16.667w_L = 16.667, wH=33.333w_H = 33.333.

  2. The expected wage is C1=0.8×33.333+0.2×16.667=30C_1 = 0.8 \times 33.333 + 0.2 \times 16.667 = 30. The full-insurance cost (uˉ+c)1=30(\bar u + c)^1 = 30 matches it, so the risk premium is zero.

  3. Why: a risk-neutral agent suffers no utility loss from wage variation. Translating the IC-mandated utility spread into a wage spread costs nothing, so the principal buys incentives at the full-insurance expected cost — the pay slope is steep but risk-sharing costs zero.

20. The Principal-Agent Problem