Chapter 22: Adverse Selection, Lemons, Signaling, and Screening. The Encyclopedia of Economic Principals, interactive companion

Demonstration 1 of 4

The lemons unraveling

When does hidden quality drive the good vans out of the market?

Buyers pay the average value of what is offered. When that average is below what good-quality owners can get by keeping their vans, they leave, the average falls further, and only lemons trade. The loss is the surplus of the good vans that never change hands.

$Equation, written in LaTeX: 0.5(\$6{,}000)+0.5(\$14{,}000)=\$10{,}000.$

$Equation, written in LaTeX: 50(\$14{,}000-\$11{,}000)=\$150{,}000$

Scroll sideways for the whole equation

A market of 100 used vans. Owners value low and high quality at $4,000 and the high value set below; buyers value them at $6,000 and $14,000. Buyers cannot see quality, so they offer the expected value of the vans actually for sale.

Predict first. Will high-quality owners sell at the pooled offer of $10,000?

Your prediction No, they keep their vans Yes, every van trades

High-quality owner value

Share high quality

Figure: The lemons unraveling. Bars for the pooled offer 10,000, the high owners' value 11,000 and the market price 6,000. — High-quality owner value: $11,000, Share high quality: 0.5
Constructed example: the chapter's hypothetical van market (50 and 50 vans, owner values 4,000 and 11,000, buyer values 6,000 and 14,000); high owner values of 9,000 and 13,000 and high-quality shares of 0.3 and 0.7 are added.

Pooled offer: $10,000
Equilibrium price: $6,000
Vans that trade: 50 low-quality vans
Lost surplus: $150,000

With 50 high-quality and 50 low-quality vans, the pooled offer is 0.5(6,000) + 0.5(14,000) = 10,000. The pooled offer 10,000 is below the high owners' 11,000, so they keep their vans. Buyers price the remaining low-quality pool at 6,000 and 50 low vans trade. Lost surplus is 50(14,000 - 11,000) = 150,000.

Worked steps

Pooled offer = 0.5(6,000) + 0.5(14,000) = 10,000
10,000 < 11,000: high owners withdraw
Price falls to the low-quality buyer value 6,000 (above the 4,000 reservation)
Lost surplus = 50(14,000 - 11,000) = 150,000

Use the idea

A market for used goods, loans or insurance works when the pooled price stays above what the best sellers can get elsewhere; certification or warranties can restore it.

Where the conclusion applies

Two quality levels, competitive buyers who know the mix, sellers who know their own quality, and no certification. The price gap between 10,000 and 6,000 is a transfer, not a loss.

Check your understanding: With 70 percent high quality, does the pool survive?

0.3(6,000) + 0.7(14,000) = 11,600 > 11,000, so all vans trade at 11,600.

Chapter 22 source: section "Akerlof lemons equilibrium".

Demonstration 2 of 4

Insurance death spiral

Why can a fair pooled premium drive the low risks out, and what does classifying them cost?

A premium priced on the average claim is a bad deal for the low risks. If their willingness to pay is below it, they exit, and the premium climbs to cover the high risks alone. Classification can bring them back if its cost leaves their premium below what they will pay.

$Equation, written in LaTeX: \pi_0=\frac{50(\$1{,}000)+50(\$3{,}000)}{100}=\$2{,}000.$

$Equation, written in LaTeX: \pi_1=\$3{,}000.$

$Equation, written in LaTeX: 100(\$100)=\$10{,}000$

Scroll sideways for the whole equation

Fifty low risks expect $1,000 of claims and fifty high risks $3,000; high risks pay at most $3,300. pi_0 is the pooled premium and pi_1 the premium once low risks leave.

Predict first. Does the low-risk group buy at the pooled $2,000?

Your prediction No, only high risks stay Yes, both groups buy

Low-risk max willingness to pay

Classification cost per applicant

Figure: Insurance death spiral. Premium bars: pooled 2,000, after exit 3,000, classified 1,100 and 3,100, with the low-risk maximum at 1,300. — Low-risk max willingness to pay: $1,300, Classification cost per applicant: $100
Constructed example: the chapter's hypothetical insurance pool (claims 1,000 and 3,000, maximum payments 1,300 and 3,300, classification at 100 and 400); low-risk maxima of 1,800 and 2,100 and a free classification are added.

Pooled premium: $2,000
Equilibrium premium: $3,000
Low-risk take-up, pooled: 0
Low-risk take-up, classified: 50
Classification cost: $10,000

Pooled premium = (50(1,000) + 50(3,000)) / 100 = 2,000. Low risks pay at most 1,300 < 2,000, so they leave and the premium rises to the high-risk claim, 3,000. Classification sets premiums 1,000 + 100 = 1,100 and 3,000 + 100 = 3,100; low risks enroll (1,100 <= 1,300) and high risks enroll (3,100 <= 3,300). It uses 100(100) = 10,000 of resources.

Worked steps

Pooled premium = (50(1,000) + 50(3,000)) / 100 = 2,000
Low risks: 1,300 < 2,000, equilibrium premium 3,000
Classified low premium = 1,000 + 100 = 1,100
Classification cost = 100(100) = 10,000

Use the idea

Watch the premium, low-risk take-up and average claims together: rising premiums with falling low-risk enrollment is the signature of adverse selection.

Where the conclusion applies

Fair premiums, two risk groups, buyers who know their own risk and truthful classification. Risk aversion enters only through the willingness to pay figures.

Check your understanding: If classification costs $400, does the low-risk group enroll?

The premium is 1,000 + 400 = 1,400 > 1,300, so no.

Chapter 22 source: section "Asymmetric Information and Adverse Selection".

Demonstration 3 of 4

When does a credential separate?

How much more costly must a credential be for low types before it reveals who is who?

A signal works only if it is cheaper for the type it is meant to reveal. The high type must gain from it and the low type must not.

$Equation, written in LaTeX: \$100{,}000-\$10{,}000=\$90{,}000,$

$Equation, written in LaTeX: \$100{,}000-\$50{,}000=\$50{,}000$

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Employers pay $100,000 with the credential and $60,000 without. The credential costs the high type and the low type the amounts set below, and adds nothing to output.

Predict first. Will the low type imitate at a $50,000 cost?

Your prediction No, 50,000 is below 60,000 Yes, the wage gain is worth it

Low type credential cost

High type credential cost

Figure: When does a credential separate? Payoff bars: high type 90,000 with the credential, low type 50,000 if it imitates, both against 60,000 without it. — Low type credential cost: $50,000, High type credential cost: $10,000
Constructed example: the chapter's hypothetical workers (output 100,000 and 60,000, credential costs 10,000 and 50,000, and 30,000); a low-type cost of 40,000 and a high-type cost of 45,000 are added.

High type net with signal: $90,000
Low type payoff from imitation: $50,000
Separating: yes

High type: 100,000 - 10,000 = 90,000 > 60,000. Low type imitating: 100,000 - 50,000 = 50,000 < 60,000. The credential separates the types. Imitation breaks even at a low-type cost of 100,000 - 60,000 = 40,000.

Worked steps

High: 100,000 - 10,000 = 90,000 > 60,000
Low: 100,000 - 50,000 = 50,000 < 60,000
Separating: yes

Use the idea

Ask whether a credential is harder to get for weaker candidates; if not, it cannot carry information, however expensive it is.

Where the conclusion applies

Competitive employers, two types, a credential with no productivity effect and wages that follow beliefs. The credential's cost is a real resource cost; the wage gap is a transfer.

Check your understanding: At what low-type cost does imitation just break even?

100,000 - c = 60,000 gives c = 40,000.

Chapter 22 source: section "Costly Signaling".

Demonstration 4 of 4

Deductible menu that sorts

Can a deductible plan make buyers reveal their own risk?

A deductible costs more to the buyer who expects to use it. Set right, the low risk takes the cheap plan with the deductible and the high risk prefers full cover, so choices reveal type.

$Equation, written in LaTeX: \$225+0.10(\$500)=\$275,$

$Equation, written in LaTeX: \$225+0.40(\$500)=\$425$

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A $1,000 loss happens with probability 0.10 for low risks and 0.40 for high risks. Plan F gives full cover for $420. Plan D has the premium and deductible set below.

Predict first. Which plan does each risk type choose?

Your prediction Low risk D, high risk F Both choose D Both choose F

Plan D premium

Plan D deductible

Figure: Deductible menu that sorts. Expected cost bars: low risk 275 under Plan D and 420 under Plan F; high risk 425 under Plan D and 420 under Plan F. — Plan D premium: $225, Plan D deductible: $500
Constructed example: the chapter's hypothetical plans (loss 1,000, probabilities 0.10 and 0.40, Plan F 420, Plan D 225 or 175 with a 500 deductible); a 250 premium and a 700 deductible are added.

Low risk cost under D: $275
High risk cost under D: $425
Low risk chooses: Plan D
High risk chooses: Plan F
Separating: yes

Low risk under D: 225 + 0.10(500) = 275 against 420, so it chooses Plan D. High risk under D: 225 + 0.40(500) = 425 against 420, so it chooses Plan F. The menu separates: each type picks a different plan.

Worked steps

Low: 225 + 0.10(500) = 275 < 420, Plan D
High: 225 + 0.40(500) = 425 > 420, Plan F
Separating: yes

Use the idea

Design plan menus so that each one is the best choice for the group it targets; a lower premium that attracts everyone ends the sorting.

Where the conclusion applies

Risk neutral buyers, coverage required, known loss probabilities and no change in care after buying.

Check your understanding: With a $175 premium and a $700 deductible, does the menu separate?

Low: 175 + 70 = 245 < 420 chooses D; high: 175 + 280 = 455 > 420 chooses F; it separates.

Chapter 22 source: section "Screening and Self-Selection".

The lemons unraveling

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Calculated values

Worked steps

Use the idea

Where the conclusion applies

Insurance death spiral

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Calculated values

Worked steps

Use the idea

Where the conclusion applies

When does a credential separate?

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Calculated values

Worked steps

Use the idea

Where the conclusion applies

Deductible menu that sorts

Choose an example

Calculated values

Worked steps

Use the idea

Where the conclusion applies