Question

1. Starting from a separating equilbirum, the government wants to create a National

Health Insurance Program (NHIP) that would bring full insurance to all its citizens

via a new national pooling contract, while banning any private insurance company

from offering any other contracts. With the aid of an indifference-curve (IC) diagram

for two types of insurees (viz., those with low health-risks and those with high health-

risks), show whether this new policy is a Pareto-improvement. If not, who gains and

who loses?

2. Due to fierce protests from private insurance companies, suppose the government lifts

the ban, and gives people the freedom to purchase insurance from NHIP (a pooling

contract) and/or from the private market (the original separating contracts) as they

see fit. Show diagrammatically whether this .freedom.is welfare-improving. If not,

who gains and who loses?

3. Explain whether there is any alternative policy the government can introduce that

would surely be Pareto-improving and thus favored by everyone.

Answer 1

Insurance markets suffer from adverse selection and moral hazard. The population of individuals subject to the risk of loss (of life, property, health, income, etc.). Wealth if accident does not happen is w > 0. If accident happens loss of wealth is L (0 < L < w). Utility of wealth level w: u(w)

There are two types of individuals: High risk: probability of accident pH. Low risk: probability of accident pL (0 < pL < pH)

An insurance policy is (I, D) where I: insurance premium ,D: deductible. Wealth with insurance policy (I, D).

No accident: w − I Accident: w − D − I.

Expected utility of type i = L, H with insurance (I, D) U_i(D, I) = p_iu(w − D − I) + (1 − p_i)u(w − I)

Separating equilibrium: if complete information policies (FH , FL) were offered.

figure 1

• High risk buys Low risk’s policy
• Since Low does not want to mimic High risk keep offering High risk FH
• But, have to offer Low risk a policy that High doesn’t want to mimic
• Has to be on fair odds line
• Equilibrium candidate is E_L
• Best that can be done for Low risk without attracting High risk
• Note that High risk are fully insured D_H = 0, I_H = p_HL
• Low risk is partially insured at a lower premium D_L > 0, I_L < I_H
• This is the only way to separate Low risk from High
• Low risk is willing to accept higher deductible (deductible is paid in case of an accident, which is a lower probability event for Low risk)
• In exchange for a lower premium Compared to complete information case
• Low risk is worse off
• High risk is indifferent
• Companies still make zero profit
• Asymmetric information has a welfare cost

For equilibrium, Consider policy d. Both types prefer d to their current policies. Does it make a profit?

figure 2

Depends on µ = λpH + (1 − λ)pL

• λ ⁺: average fair odds line with high λ
• λ ⁻: average fair odds line with low λ
• If λ is high (λ +) d does not make a positive profit, We found an equilibrium
• If λ is low (λ −) d makes a positive profit, What we found is not an equilibrium.

In the candidate separating equilibrium Low risk gets low insurance coverage

• they don’t like this because they are risk-averse
• They prefer a pooling policy such as d although they are cross-subsidizing high risk
• If the proportion of High risk (λ) is low, a firm can offer d, attract both types and still make profit
• This destroys the candidate equilibrium
• There is no pooling equilibrium
• There may not be a separating equilibrium either If there is an equilibrium
• High risk gets full insurance at higher premium
• Low risk gets partial coverage at lower premium
• There is a welfare cost of adverse selection
• Entirely borne by Low risk individuals

Pooling Equilibrium: Since both types buy the same policy, expected company profits: w − µL − µw_a − (1 − µ)w_n

where µ is the expected risk µ = λp_H + (1 − λ)p_L

Figure 3

• Suppose there is a pooling equilibrium
• It must be on expected fair odds line
• Low type must prefer it to no insurance: say E
• Then high type also prefers it
• Low risk is worse off and high risk is better off compared to complete information
• Company makes losses on high risk and profit on low risk
• Low risk cross-subsidizes high risk