The Encyclopedia of Economic Principals

Chapter 24

Moral Hazard and Principal-Agent Incentives

Pay for what you can see, and count the rent it leaves behind.

Four of the chapter's worked examples, made interactive: a revenue split between two hidden efforts, the rent a wage floor creates, a deductible that restores care, and a base plus bonus contract. Change one value at a time and watch the figure, the numbers and the hand calculation respond.

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

Splitting revenue between two hidden efforts

Can a revenue split reach the efficient surplus when both partners' efforts are hidden?

Each partner equates its own share of marginal revenue with its full marginal cost, so both efforts fall below the efficient level of 2. Moving the share toward one partner helps one effort and hurts the other more, because cost rises with the square of effort.

Equation, written in LaTeX: R(e,m)=40e+40m,

Equation, written in LaTeX: 30=20e, 10=20m,

Equation, written in LaTeX: 10(1.5^2+0.5^2)=25,

Scroll sideways for the whole equation

e is the developer's effort and m the distributor's. Revenue is 40e + 40m, each effort costs 10 times its square, and the developer keeps share alpha of revenue while the distributor keeps 1 - alpha.

Predict first. Which developer share gives the highest joint surplus under a linear split?

Your prediction

Choose an example

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Figure: Splitting revenue between two hidden efforts. Left: joint surplus against the developer share, peaking at 60 when alpha is 0.50, with the efficient 80 as a dashed line and the current share 0.75 marked at 55.00. Right: bars for developer effort 1.50, distributor effort 0.50 and the efficient level 2.
Developer share alpha: 0.75
Constructed example: the chapter's hypothetical software partnership (shares 0.50 and 0.75); shares 0.60 and 0.90 are added for comparison.

Calculated values

Developer effort e
1.50
Distributor effort m
0.50
Revenue
80.00
Effort cost
25.00
Joint surplus
55.00
Efficient surplus
80.00

With share 0.75 the developer sets e = 1.50 and the distributor m = 0.50. Revenue is 80.00 and effort cost 25.00, so joint surplus is 55.00, 25.00 short of the efficient 80.00. Tilting the share raises one effort and lowers the other by the same amount, so revenue is unchanged, but the convex effort costs rise and joint surplus falls. Hand check: Developer: 0.75 x 40 = 20e, so e = 30.00 / 20 = 1.50; Distributor: 0.25 x 40 = 20m, so m = 10.00 / 20 = 0.50; Revenue = 40 x 1.50 + 40 x 0.50 = 80.00; Cost = 10 x (2.2500 + 0.2500) = 25.00; Surplus = 80.00 - 25.00 = 55.00, 25.00 below the efficient 80.00.

Worked steps

  1. Developer: 0.75 x 40 = 20e, so e = 30.00 / 20 = 1.50
  2. Distributor: 0.25 x 40 = 20m, so m = 10.00 / 20 = 0.50
  3. Revenue = 40 x 1.50 + 40 x 0.50 = 80.00
  4. Cost = 10 x (2.2500 + 0.2500) = 25.00
  5. Surplus = 80.00 - 25.00 = 55.00, 25.00 below the efficient 80.00

Use the idea

When two parties both contribute hidden effort, look for ways to verify at least one effort before tuning a revenue share.

Where the conclusion applies

Linear revenue, quadratic effort costs, risk neutrality and no verifiable effort. With verifiable efforts a contract requiring e = m = 2 restores surplus 80.

Check your understanding: What is joint surplus at alpha = 0.50?
e = m = 1, revenue 80, cost 10 x (1 + 1) = 20, surplus 60, the best any linear split achieves; 0.75 gives 55.

Chapter 24 source: section "Double moral hazard".

Demonstration 2 of 4

The rent from limited liability

Why does a wage floor leave the manager a rent even when the bonus is set as low as possible?

Incentives depend on the spread between the two wages. With a floor at zero, the spread must sit entirely in the success wage, which the manager also collects with probability 0.30 without effort; that payment is the rent.

Equation, written in LaTeX: (0.70-0.30)w_H\geq12,

Equation, written in LaTeX: 0.70(21)+0.30(-9)=\$12,

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High effort costs the manager c and raises success probability from 0.30 to 0.70. Success is worth 100 to the company. w_H and w_L are the wages after success and failure.

Predict first. If effort cost rises from 12 to 16 under limited liability, does the rent rise by the same 4?

Your prediction

Choose an example

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Figure: The rent from limited liability. Bars for expected wage $21.00, manager rent $9.00 and company return $49.00 with effort cost $12 under limited liability.
Effort cost: 12, Wage floor: Limited (w_L = 0)
Constructed example: the chapter's hypothetical launch (cost 12, probabilities 0.30 and 0.70, value 100, both liability regimes); costs 8 and 16 are added for comparison.

Calculated values

Success wage w_H
$30.00
Failure wage w_L
$0.00
Expected wage
$21.00
Manager rent
$9.00
Company return
$49.00

High effort needs a pay spread of $30.00. The manager is then indifferent between efforts, and the chapter's convention assigns high effort. Expected wage is $21.00 and the company keeps $49.00. The wage floor forces the whole spread into the success wage, so the manager keeps a rent of $9.00, equal to 12 x 0.30 / 0.40. Hand check: Spread: 0.40 x (w_H - w_L) = 12, so w_H - w_L = 12 / 0.40 = 30.00; With w_L = 0, w_H = 30.00; Expected wage = 0.70 x 30.00 = 21.00; Rent = 21.00 - 12 = 9.00; Company = 0.70 x 100 - 21.00 = 49.00.

Worked steps

  1. Spread: 0.40 x (w_H - w_L) = 12, so w_H - w_L = 12 / 0.40 = 30.00
  2. With w_L = 0, w_H = 30.00
  3. Expected wage = 0.70 x 30.00 = 21.00
  4. Rent = 21.00 - 12 = 9.00
  5. Company = 0.70 x 100 - 21.00 = 49.00

Use the idea

Before raising a bonus, ask how much of it is paid in states the agent would reach anyway.

Where the conclusion applies

Risk neutrality, a zero outside option, two effort levels and the high-effort tie-break the chapter states. Removing liability protection is not presented as desirable.

Check your understanding: With cost 16 under limited liability, what are the rent and the company return?
w_H = 16 / 0.4 = 40, expected wage 0.70 x 40 = 28, rent 28 - 16 = 12, company 70 - 28 = 42.

Chapter 24 source: section "Limited-liability information rents".

Demonstration 3 of 4

A deductible restores care

How large must a deductible be before an insured driver pays for care again?

Insurance removes part of the loss from the driver's own account, so the private return to care shrinks. A deductible puts enough of the loss back on the driver to make care worth its cost.

Equation, written in LaTeX: (0.10-0.02)(\$2{,}000)=\$160,

Equation, written in LaTeX: \$80+0.02(\$2{,}000)=\$120.

Equation, written in LaTeX: (0.10-0.02)(\$1{,}500)=\$120,

Scroll sideways for the whole equation

Care costs $80 and lowers the chance of a $2,000 accident from 0.10 to 0.02. The deductible D is the part of the loss the driver pays; 0 is full insurance and 2,000 is no insurance.

Predict first. What is the smallest deductible that restores care?

Your prediction

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Figure: A deductible restores care. Bars of the driver's private saving from care at deductibles 0, 500, 1,500 and 2,000 (0, 40, 120 and 160 dollars), with the current $1,500 highlighted and a line at the $80 care cost.
Deductible ($): 1,500
Constructed example: the chapter's hypothetical driver (care 80, accident 2,000, probabilities 0.10 and 0.02, deductible 1,500, full and no insurance); the 500 deductible is added.

Calculated values

Private saving from care
$120.00
Care taken
yes
Expected resource cost
$120.00
Smallest deductible for care
above $1,000

With a $1,500 deductible, care cuts the driver's expected out-of-pocket loss by $120.00, more than the $80 care cost, so the driver takes care and the expected resource cost is $80 + $40 = $120. Any deductible above $1,000 restores care. Hand check: Saving = (0.10 - 0.02) x 1,500 = 120.00; Compare with care cost 80: 120.00 > 80, so care is taken; Resource cost = 80 + 0.02 x 2,000 = 120.00; Care needs D > 80 / 0.08 = 1,000.

Worked steps

  1. Saving = (0.10 - 0.02) x 1,500 = 120.00
  2. Compare with care cost 80: 120.00 > 80, so care is taken
  3. Resource cost = 80 + 0.02 x 2,000 = 120.00
  4. Care needs D > 80 / 0.08 = 1,000

Use the idea

Size a deductible from the care cost and the change in accident probability, not from the size of the loss alone.

Where the conclusion applies

A risk-neutral driver, a premium fixed before the care decision and no experience rating. A deductible of exactly 1,000 is a tie and is left out of the control.

Check your understanding: At a 500 deductible, does the driver take care, and what is the resource cost?
0.08 x 500 = 40 < 80, so no care, and the resource cost is 0.10 x 2,000 = 200. Care needs D > 80 / 0.08 = 1,000.

Chapter 24 source: section "Moral hazard".

Demonstration 4 of 4

Base plus bonus

How large must a success bonus be for a sales representative to work hard and still accept the job?

The bonus must make high effort pay at least as well as low effort, and the total must clear the outside option. The book's 22,000 bonus does both; smaller bonuses fail one test or both.

Equation, written in LaTeX: \$15{,}000+0.20(\$22{,}000)=\$19{,}400.

Equation, written in LaTeX: 0.50(\$100{,}000)-\$26{,}000=\$24{,}000.

Scroll sideways for the whole equation

The representative's outside option is $20,000. High effort costs $6,000 and raises the chance of a $100,000 sale from 0.20 to 0.50. Pay is a base salary plus a bonus paid on a sale.

Predict first. Will a $20,000 bonus with a $15,000 base induce high effort and acceptance?

Your prediction

Choose an example

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Figure: Base plus bonus. Left: agent net payoff $19,400 under low effort and $20,000 under high effort, with the $20,000 outside option as a line. Right: principal profit $24,000.
Success bonus ($): 22,000, Base salary ($): 15,000
Constructed example: the chapter's hypothetical sales representative (base 15,000 with bonus 22,000, and the 20,000 flat salary); bonuses 15,000 and 20,000 and their pairings are added.

Calculated values

Low effort pay
$19,400
High effort pay
$26,000
High effort net
$20,000
Effort chosen
High
Contract accepted
yes
Principal profit
$24,000

Low effort nets $19,400; high effort pays $26,000 and nets $20,000 after the $6,000 effort cost. The contract induces high effort and the representative accepts, leaving the principal $24,000. Hand check: Low effort: 15,000 + 0.20 x 22,000 = 19,400; High effort: 15,000 + 0.50 x 22,000 - 6,000 = 20,000; Agent picks high effort and nets 20,000 against the 20,000 outside option; Profit = 0.50 x 100,000 - 26,000 = 24,000.

Worked steps

  1. Low effort: 15,000 + 0.20 x 22,000 = 19,400
  2. High effort: 15,000 + 0.50 x 22,000 - 6,000 = 20,000
  3. Agent picks high effort and nets 20,000 against the 20,000 outside option
  4. Profit = 0.50 x 100,000 - 26,000 = 24,000

Use the idea

Check incentive compatibility and participation separately when setting a bonus.

Where the conclusion applies

Risk neutrality and two effort levels. An exact effort tie is resolved toward high effort, and an agent exactly at the outside option accepts, as in the chapter's example.

Check your understanding: With a 15,000 base and a 20,000 bonus, does the contract work?
Low effort pays 15,000 + 0.20 x 20,000 = 19,000; high effort nets 15,000 + 10,000 - 6,000 = 19,000. Effort ties, and both fall short of the 20,000 outside option, so the contract fails.

Chapter 24 source: section "Principal-agent problem".