The Encyclopedia of Economic Principals

Chapter 26

Incomplete, Relational, and Dynamic Contracts

When courts cannot see the facts, audits, the future and ownership hold deals together.

Four of the chapter's worked examples, made interactive: debt priced around a costly audit, a self-enforcing bonus, reputation as collateral, and hold-up of a specific investment.

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

Debt priced around a costly audit

How does the cost of verifying a bad outcome feed into the face value of a loan?

Debt with an audit only in default is cheap to enforce: the lender verifies just the outcomes in which it is not repaid. Any audit cost is passed back to the borrower through a higher face value, and the expected audit spending is the deadweight loss.

Equation, written in LaTeX: 0.25(\$40-\$10)+0.75D=\$80.

Equation, written in LaTeX: D=\frac{\$80-0.25(\$30)}{0.75}=\$96.67

Equation, written in LaTeX: 0.25(\$40)+0.75(\$140)=\$115.

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Maya borrows $80. The project returns $40 (low state) or $140 (high state). Only the low state defaults and is audited at cost c. D is the face value of the debt.

Predict first. If the audit cost rises from $10 to $50, does the face value rise by more or less than $40?

Your prediction

Choose an example

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Figure: Debt priced around a costly audit. Stacked bars for the low and high states with audit cost 10 and low-state probability 0.25: face value $96.67, Maya keeps $43.33 in the high state.
Audit cost: $10, Probability of the low state: 0.25
Constructed example: the chapter's hypothetical loan to Maya ($80 loan, returns $40 and $140, audit costs $10 and $50); audit costs of $0 and $30 and low-state probabilities of 0.15 and 0.35 are added for comparison.

Calculated values

Face value D
$96.67
High-state residual
$43.33
Expected audit cost
$2.50
Joint surplus
$32.50
Maya's expected residual
$32.50

In the low state the lender recovers 40 - 10 = 30 after the audit. Break-even needs D = (80 - 0.25 x 30) / 0.75 = 72.50 / 0.75 = $96.67. Maya keeps 140 - 96.67 = 43.33 in the high state. Expected output is 0.25 x 40 + 0.75 x 140 = 115.00, so joint surplus is 115.00 - 80 - 2.50 = $32.50, all of it Maya's because the lender only breaks even. The audit cost destroys value only in the low state.

Worked steps

  1. Low-state recovery = 40 - 10 = 30
  2. D = (80 - 0.25 x 30) / 0.75 = 72.50 / 0.75 = 96.67
  3. High-state residual = 140 - 96.67 = 43.33
  4. Expected audit cost = 0.25 x 10 = 2.50
  5. Surplus = 115.00 - 80 - 2.50 = 32.50

Use the idea

When verification is expensive, prefer contracts that verify only bad outcomes, and keep the probability of those outcomes low.

Where the conclusion applies

Risk neutrality, a lender committed to auditing, perfect audits and a single period. Without commitment the lender may not audit when recovery is negative.

Check your understanding: At an audit cost of $30 with a 0.25 chance of the low state, what face value lets the lender break even?
Recovery 40 - 30 = 10; D = (80 - 0.25 x 10) / 0.75 = 77.50 / 0.75 = $103.33.

Chapter 26 source: section "Costly state verification".

Demonstration 2 of 4

When is a handshake bonus self-enforcing?

How patient must a manufacturer be for an unenforceable bonus promise to be believed?

A relational contract holds when the one-time saving from breaking a promise is smaller than the value of the relationship it destroys. The bonus divides a fixed surplus of $60; it does not add to it.

Equation, written in LaTeX: \$50-\$30=\$20,

Equation, written in LaTeX: \$90-\$50=\$40.

Equation, written in LaTeX: \frac{0.8}{1-0.8}\times\$40=\$160.

Equation, written in LaTeX: \frac{0.2}{1-0.2}\times\$40=\$10.

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High effort costs the supplier $30 and is worth $90 to the manufacturer. b is the promised bonus and delta the continuation factor per quarter. Reneging ends cooperation for good.

Predict first. Does a larger bonus make the promise easier or harder to keep?

Your prediction

Choose an example

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Figure: When is a handshake bonus self-enforcing? Bars comparing the $50 temptation with the $160.00 discounted future loss at continuation factor 0.80: high effort, bonus paid.
Continuation factor: 0.8, Promised bonus: $50
Constructed example: the chapter's hypothetical supply relationship (effort cost $30, value $90, bonus $50, delta 0.8 and 0.2); delta of 0.5 and 0.9 and bonuses of $35 and $70 are added.

Calculated values

Supplier gain if paid
$20.00
Manufacturer gain per quarter
$40.00
Future loss from reneging
$160.00
Temptation
$50.00
Outcome
high effort, bonus paid

Paying the $50 bonus leaves the supplier 50 - 30 = 20 and the manufacturer 90 - 50 = 40 per quarter. Reneging saves 50 now but loses 0.80 / (1 - 0.80) x 40 = 4.00 x 40 = 160.00. Since 160.00 >= 50, the promise is self-enforcing and the supplier exerts high effort.

Worked steps

  1. Supplier gain = 50 - 30 = 20
  2. Manufacturer gain = 90 - 50 = 40
  3. Future loss = 4.00 x 40 = 160.00
  4. Compare 160.00 >= 50: high effort, bonus paid

Use the idea

Keep discretionary rewards small relative to the future value of the relationship, and expect them to fail when the relationship may end soon.

Where the conclusion applies

Both parties observe effort, breach is punished by permanent reversion to low effort, and the continuation factor is common knowledge. Renegotiation after breach weakens the threat.

Check your understanding: At delta = 0.5 and a $70 bonus, is the bonus self-enforcing?
Manufacturer gain 90 - 70 = 20; future loss 0.5 / 0.5 x 20 = $20 < $70, so it reneges and the supplier exerts low effort.

Chapter 26 source: section "Relational contracts".

Demonstration 3 of 4

Reputation as collateral

How much future business must be at stake for a repairer to keep doing honest work?

A reputation is worth keeping when the premium business it protects, discounted, exceeds the one-time saving from cheating. A separate asset lost on detection adds to the penalty without adding output.

Equation, written in LaTeX: \pi^C=\$90-\$70=\$20.

Equation, written in LaTeX: G=\$70-\$40=\$30.

Equation, written in LaTeX: \frac{0.75}{1-0.75}\times\$20=\$60.

Equation, written in LaTeX: \$30\leq\frac{\delta}{1-\delta}\times\$20,

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The trusted price is $90, honest work costs $70 and hidden low quality $40. delta is the continuation factor; B is a separate reputation asset destroyed by detection.

Predict first. Below which delta does the repairer cheat when there is no separate reputation asset?

Your prediction

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Figure: Reputation as collateral. Rent at risk rising with the continuation factor, against the $30 cheating gain. At delta 0.75 with asset 0 the loss is $60.00: honest quality is sustained.
Continuation factor: 0.75, Separate reputation asset: none
Constructed example: the chapter's hypothetical repairer (price $90, costs $70 and $40, delta 0.75 and 0.5, threshold 0.6, asset $15). All values are the book's.

Calculated values

Honest profit per job
$20.00
Cheating gain
$30.00
Continuation rent
$60.00
Total loss from cheating
$60.00
Threshold delta
0.600
Outcome
honest

Honest profit is 90 - 70 = 20 and cheating saves 70 - 40 = 30. The rent at risk is 0.75 / (1 - 0.75) x 20 = 60.00. So honest quality is sustained. The threshold solves 30 - 0 = delta / (1 - delta) x 20, so delta = 30 / 50 = 0.600.

Worked steps

  1. pi = 90 - 70 = 20
  2. G = 70 - 40 = 30
  3. Rent = 3.00 x 20 = 60.00
  4. Total loss = 60.00 + 0 = 60.00
  5. Threshold = 30 / (30 + 20) = 0.600

Use the idea

Trust sellers more when they have a long future of repeat business or a visible asset to lose.

Where the conclusion applies

Certain detection before the next job, permanent customer withdrawal and a constant price. Slow or noisy detection lowers the rent at risk.

Check your understanding: At delta = 0.5 with B = $15, is honest quality sustained?
Rent 0.5 / 0.5 x 20 = $20, plus 15 gives $35 > $30, so yes; with B = 0 it is $20 < $30 and cheating wins.

Chapter 26 source: section "Reputation as Contract-Enforcement Capital".

Demonstration 4 of 4

Hold-up and the bargaining share

Why can an efficient relationship-specific investment fail to happen?

Once the line is built its cost is sunk, and bargaining hands part of the new surplus to the other side. The investor weighs only its own share against the whole cost.

Equation, written in LaTeX: \$100-\$40=\$60\text{ million}.

Equation, written in LaTeX: \$60-\$35=\$25\text{ million},

Equation, written in LaTeX: \$50-\$20=\$30\text{ million}.

Equation, written in LaTeX: 0.70(\$60)=\$42\text{ million}.

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Surplus is $40 million without the specialized line and $100 million with it. k is the supplier's sunk cost and s its bargaining share of the increment after the line is built.

Predict first. What minimum bargaining share makes the supplier invest at the book's $35 million cost?

Your prediction

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Figure: Hold-up and the bargaining share. Bars for the supplier's return of 30 million, the 35 million cost and the 25 million joint net surplus at share 0.50: the supplier will decline.
Supplier bargaining share: 0.5, Investment cost: $35m
Constructed example: the chapter's hypothetical supplier and assembler (surplus $40 and $100 million, cost $35 million, shares 0.50 and 0.70); a share of 0.60 and costs of $25 and $45 million are added.

Calculated values

Incremental surplus
$60m
Joint net surplus
$25m
Supplier's incremental return
$30m
Decision
decline
Supplier's private gain
-$5m
Threshold share
0.583

Investing raises surplus by 100 - 40 = 60 at a cost of 35, a joint gain of 60 - 35 = 25 million. The supplier keeps 0.50 x 60 = 30 million, which is < 35, so it will decline (private change 30 - 35 = -5 million). It needs a share of at least 35 / 60 = 0.583.

Worked steps

  1. Increment = 100 - 40 = 60
  2. Joint net = 60 - 35 = 25
  3. Supplier return = 0.50 x 60 = 30
  4. Private gain = 30 - 35 = -5: decline

Use the idea

Protect specific investments with ownership, long-term contracts or a larger bargaining share for the party that invests.

Where the conclusion applies

Investment cannot be contracted on, has no outside value, outside payoffs are zero and bargaining splits the increment in fixed shares.

Check your understanding: At s = 0.60 and k = $35 million, does the supplier invest?
0.60 x 60 = $36 million > $35 million, so yes, by $1 million.

Chapter 26 source: section "Hold-up problem".