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**Topic: risk-neutral insurance companies**- Consider an insurance market in which risk-neutral insurance companies offer full health

insurance coverage to consumers. Consumers differ in their demand for insurance and

their expected medical costs. As in lecture, the market is perfectly competitive, but there

is asymmetric information: each consumer knows their own expected medical costs and

demand for insurance, but insurance companies only know the distribution of these values

in the population. There is no moral hazard.

Let P

D(Q) denote the inverse demand curve, where Q ∈ [0, 1] is the fraction of consumers

who purchase insurance. P

D(Q) describes the price at which Q consumers buy.

The marginal cost curve is

MC(Q) = 50 × (1 − Q)

If Q consumers demand insurance, MC(Q) describes the expected health costs of the

marginal consumers. The marginal cost curve remains the same throughout this problem.

(a) Given P

D(Q) and MC(Q), write an equation that describes the socially efficient

quantity of insurance, Qe

. (You cannot solve for Qe yet; just provide the general

expression)

(b) Derive the average cost curve, AC(Q). Explain (in words) why the intersection of

the average cost curve and the demand curve describes the equilibrium quantity and

price of insurance. Denote this quantity by Q∗

. (You cannot solve for Q∗ yet)

(c) Suppose the market demand curve is

P

D(Q) = 100 × (1 − Q)

Find Qe and Q∗

, and compare them. What is the deadweight loss due to adverse

selection?

(d) Find a different demand curve for which everyone buys insurance in equilibrium.1

Write this demand curve algebraically and illustrate the new market equilibrium

graphically. Is this equilibrium efficient?

(e) Find a different demand curve for which it is efficient to insure everyone, yet no one

buys insurance in equilibrium. Define the demand curve algebraically, and illustrate

the market equilibrium graphically. What is the deadweight loss due to adverse

selection?

(f) Find a different demand curve for which it is socially efficient to insure 90% of the

population, but where only 50% buy insurance in equilibrium. Define the demand

curve algebraically, show that it satisfies the required properties, and graphically illustrate the market equilibrium and efficient quantity.

1Many demand curves satisfy this condition.

1 - An individual values her health as well as money. She has income y, can purchase medical

services m. Her utility function is given by

u(m; λ, ω) = h(m − λ; ω) + y(m)

where λ and ω are parameters, m is the market value of the medical services she purchases,

and y(m) is her remaining income after purchasing medical services m.

We’ll consider health insurance contracts that take the following form: the individual pays

a premium p to the insurance company regardless of her medical spending. Then, for each

dollar of medical spending, she pays c ∈ [0, 1] out of her own pocket, and the insurance

company covers the remaining expenses. Therefore,

y(m) = y − p − c × m

We call c the co-insurance rate, p the premium. A higher c denotes less generous

coverage. The agent’s utility function takes the following form:

u(m; λ, ω) = (m − λ) −

1

2ω

(m − λ)

2

| {z }

h(m−λ;ω)

- y − p − c × m | {z }

y(m)

Throughout the problem you may assume that both λ and ω are positive numbers, and

that they are known to the individual (there is no uncertainty).

(a) Suppose the individual has already purchased an insurance contract (p, c), and knows

her health parameters (λ, ω).

i. What is her utility-maximizing level of medical spending m∗

?

ii. Compare her optimal medical spending under full insurance (c = 0) and no

insurance (c = 1). How does the difference depend on λ and ω?

iii. Briefly, provide an economic interpretation for each of the parameters λ, ω.

(b) Let u

∗

(p, c) denote the indirect utility the individual obtains from a contract (p, c)

given her optimal medical spending m∗

. This will of course depend on λ, ω as well as

the contract.

i. Derive an expression for u

∗

(p, c) given (λ, ω).

ii. Calculate u

∗

(0, 1), i.e. the individual’s indirect utility from no insurance.

iii. What is the highest premium p that the individual would be willing to pay for

an insurance plan with co-insurance rate c, instead of having no insurance?

(c) What is the insurance company’s profit from offering (p, c) to the individual? Remember that the insurance company always receives the premium p, but has to pay

the fraction (1 − c) of her medical costs.

(d) Is there any contract (p, c) that the insurer would be willing to sell to this individual

and which the individual would be willing to buy?

(e) In lecture, we discussed how the value of insurance depends on (i) risk type and (ii)

risk preferences. With that discussion in mind, explain (in words) why the setup of

this problem implies that there are no gains from trade through insurance.

**Type of Service: Math/Physic/Economic/Statistic ProblemsType of Assignment: CalculationSubject: Not definedPages/words: 1/550Academic level: Master’sPaper format: MLALine spacing: SingleLanguage style: US English**