1    Review questions

Indicate whether each statement is true, false, or uncertain, and give a brief explanation.

1. Feldman and Slemrod (2007) find that different evasion rates across different types of income not subject to third-party reporting.  True. Feldman and Slemrod (2007) find “on average, reported positive self- employment , nonfarm small- business, and farm income must be multiplied by a factor of 1.54, 4.54, and 3.87, respectively. ”

2. Imputation eliminates the corporate income tax in Australia.   Uncertain:  if im- putation credits are fully valued by shareholders then the corporate income tax is eliminated.   However, some investors, such as foreigners,  cannot use imputation credits.

3.  Government provision of social insurance overcomes the moral hazard problem. False:  moral hazard occurs when people change their behaviour in response to in- surance, e.g. by searching less intensively for work when unemployment benefits are provided or by making riskier health choices when health insurance is provided. Both

government and private provision of insurance are constrained by moral hazard.

4. In contrast to social insurance, private insurance can provide perfect insurance be- cause customers pay the full cost of expected insurance claims through the premi- ums. False. Private insurance is also subject to moral hazard (actions taken by the person insured that increase the likelihood of insurance payment).  Therefore, private insurance typically offers only partial insurance (e.g., auto insurance premiums go up after you have an accident).

2    Problems

2.1    Social Insurance

Consider an economy of individuals who each have income W and increasing and concave utility u(·). These consumers may or may not develop a health condition, which requires money to fix. Without insurance, their utility is u(W) if they do not develop the health condition and u(W − d) if they develop the health condition (and therefore have to pay to get it fixed).

These individuals have the option of buying health insurance.   This health insurance requires them to pay a premium regardless of whether they develop the health condition, but completely covers the cost of fixing the condition if they develop it. With insurance, their utility is u(W − k) if they are insured, where k is the insurance premium. They are fully insured, so their utility is u(W − k) regardless of whether they develop the health

condition. We will assume that W − d < W − k < W .

There are three types of consumers:

i high-risk individuals who develop the condition with probability pH ii medium-risk individuals who get it with probability pM

iii low-risk people who develop it with probability pL

with pH  > pM  > pL . One third of consumers are of each type.

Finally, the cost to the insurance company of fixing the condition for someone who has purchased their insurance is also d (that is, it is equal to the cost to the consumer to fix the condition).

(a)  Suppose that the insurance company can observe the risk characteristics of each individual.  Assuming that the company offers actuarially fair insurance, explain why each individual will get insurance if and only if:

pj u(W − d) + (1 − pj )u(W) < u(W − pj d)                           (1)

where pj  ∈ {pH ,pM ,pL } is the probability that the individual develops the condition, given their type.

The left-hand side of this equation is the expected utility of the individual if they do not purchase insurance.   The right-hand side is the fixed utility the individual gets if she has insurance.     The individual will purchase the insurance if her utility with insurance is greater than her expected utility without it.

(b) Assume that equation (1) holds for individuals of all risk types.  But now suppose that the insurance company cannot price its insurance based on the risk character- istics of individuals.  If the company offers a single insurance product with a fixed premium and assumes that everyone will buy it, explain why their expected cost is pd, where:

pd = (pH + pM + pL )d

If you take an individual at random, the probability they are high-risk is 1/3.  Conditional on their being high risk, the chance that they develop the health condition is pH . Similar logic applies to those with medium and low risk.  Taking into account all these possibilities, the overall chance that the insurance company encounters an individual who later develops the health condition is given by p .  Their expected cost is this probability times the cost if they do have to pay out for such an individual, which is d for everyone.

(c) Under what conditions would low risk individuals be willing to buy the insurance offered by the company in part b above?  What about medium risk?  What about high risk?   Hint:  Assume the company charges everyone a premium equal to the expected cost from part 2 above.  Combine that with equation (1), which describes

under what conditions each individual would be willing to purchase insurance. For the three types, they will buy the insurance if the following conditions hold:

pH u(W − d) + (1 − pH )u(W) < u (W −  (pH + pM + pL )d)  if j = H  pM u(W − d) + (1 − pM )u(W) < u (W −  (pH + pM + pL ))d)  if j = M

pLu(W − d) + (1 − pL )u(W) < u (W −  (pH + pM + pL ))d)  if j = L

(d)  Continuing from part c, is it possible for low-risk people to be willing to purchase the insurance offered by the company in part b, while high risk individuals are not? Show it mathematically using your results from part c.

It is not possible for low-risk people to be willing to purchase the insurance, but not high- risk individuals.  This would require the following to be true: u(W) < u(W − d) which is ruled out by the assumption of increasing utility.

(e)  Suppose that low-risk individuals are unwilling to buy the insurance but that other individuals still are. Explain why the expected cost of the insurance is now:

pd =  (pH + pM ) d

If you take an individual at random from those who buy the insurance, the probability they are high-risk is 1/2.  Conditional on their being high risk, the chance that they develop the health condition is pH .  Similar logic applies to those with medium risk.  Taking into account all these possibilities, the overall chance that the insurance company encounters an individual who later develops the health condition is given by p =  (pH + pM ) .  Their expected cost is this probability times the  cost if they do have to pay out for such an individual, which is d for everyone.

(f) Under what conditions will a high risk individual buy the insurance offered by the

company in part e above? What about medium risk?

For the two types, they will buy the insurance if the following conditions hold:

pH u(W − d) + (1 − pH )u(W) < u (W −  (pH + pM )d)  if j = H  pM u(W − d) + (1 − pM )u(W) < u ( (W −  (pH + pM )d)  if j = M

(g)  Suppose that medium-risk individuals are actually unwilling to buy the insurance at this cost, but that high-risk individuals still are. If medium-risk individuals drop out of the market, what is the expected cost now?  Under what conditions will an individual of type H indeed be willing to buy this insurance?  Will they purchase it?

Now only high risk individuals  are  left,  and the  expected cost is just pH d .   High risk individuals will purchase the insurance if:

pH u(W − d) + (1 − pH )u(W) < u(W − pH d)

Since this individual has concave utility, the high-risk individual will always be willing to purchase this insurance.

(h) We are left in a situation in which only high-risk individuals will purchase insurance. However, low-risk and medium-risk individuals would have been willing to pay for actuarially fair insurance that covered their own expected costs. What caused this problem? What policy could help fix this problem?

This problem was caused by adverse selection:  only those with unobervably high risk of developing a medical problem are willing to purchase insurance. An insurance mandate or government-provided provision are two possible solutions.  These ensure that everyone gets insurance, but do lead to some redistribution from low-risk to high-risk individuals.  An alternative, if possible, would be to allow insurance companies to use information about individuals’ risk characteristics.   This will be worse  than the mandate  or government provision for high-risk individuals, but better for low-risk individuals.

2.2    Insurance and Redistribution

Explain the why, during the lecture on social insurance, I stated that the motivation for insurance is similar to the mathematical logic for redistribution.

Consider a utilitarian social planner.  Their goal is to maximize the sum of everyone’s utility.  If Person A has marginal utility = 2 and Person B has marginal utility = 10, I can take one unit of consumption away from Person A (costing the sum of everyone’s utility 2 units) and redistribute that unit of consumption to person B (which adds 10 to the sum of everyone’s utility) for a net gain of 8 to the sum of everyone’s utility.  This is exactly the same logic for the basic insurance problem, except rather than redistributing utility between two different people, I am redistributing utility for one person between two possible states of the world, resulting in increased expected utility.  In particular, I am shifting utility from a state of the world where I have low marginal utility to a state where I have high marginal utility.