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Separating Equilibrium. Imagine the usual setup: perfect competition, no government intervention, no transaction costs,

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1. Separating Equilibrium. Imagine the usual setup: perfect competition, no government intervention, no transaction costs, rational risk-neutral profit-maximizing firms, and rational risk-averse expected utility maximizing individuals (utility function only of wealth).

 

All individuals have the same initial wealth, $200,000.

All people face the possibility of a loss that may or may not occur. If it occurs, everyone has the same loss size, $150,000.

Individuals differ in their probability of suffering that loss (but are otherwise identical). Let’s keep it simple and assume only two types of individuals – H and L types – with high and low probability of loss, respectively. Type H people have a nine-tenths probability (90%) of suffering the loss; Type L people have a one- tenth probability (10%) of suffering the loss.

Each individual knows his or her own probability of suffering a loss but insurers do not; individuals appear identical to insurance companies. Insurers know only that the population has equal numbers of L and H types.

This setup is common knowledge. There are no other risks.

a) Please clearly draw a large diagram showing a SEPARATING EQUILIBRIUM. Be sure to include (and label with the roman numerals below, putting relevant dollar values on the axes and labelling your axes):

i. the point representing no insurance

ii. the set of contracts that provide full insurance (at a variety of prices)

iii. the set of contracts that would be break-even if sold only to H types;

iv. the set of contracts that would be break-even if sold only to L types;

v. the set of contracts that would be break-even if sold to BOTH types;

vi. the contract that would be sold to H types in a separating equilibrium;

vii. the indifference curve of H types going through (vi);

viii. the contract that would be sold to L types in a separating equilibrium; and,

ix. the indifference curve of L types going through (viii).

x. the contract or contracts we should expect to see with a government mandate (see (c) describing this below.

 

b) Compare the separating equilibrium with asymmetric information (that you showed in (a)) to the equilibrium you would expect to see in an otherwise identical setup but with symmetric information (where everyone knows who is a Type L and who is a Type H). Are outcomes different under asymmetric information and symmetric information? If so, how are they different? Answer for three types of market participants:

i. For H types?

ii. For L types?

iii. For insurers?

 

c) Government intervention. Now assume that the government mandates that everyone must buy full insurance. The rest of the problem setup with asymmetric information outlined above is unchanged. What equilibrium should we expect?

a. Label the contract or contracts you would expect to see offered (as x) on the figure described above.

b. Refer to the diagram in explaining how this government mandate benefits or harms various groups (e.g., H types, L types, insurers).

c. Refer to the diagram in explaining how the government mandate corrects a problem associated with the equilibrium without the mandate.

 

2. Signaling and screening.  Please explain briefly how signaling and screening are similar and also how they are different (e.g., how are the problems they solve similar/different and how the solutions to those problems are similar/different). Briefly give an example of signaling and an example of screening that was discussed in class.  For each, be sure to include:

a. the behavior involved;

b. who does this behavior;

c. the costs and benefits of doing this behavior, and;

d. how signaling/screening solves a problem.

 

3. Safety Investment.

An individual starts with $100 in wealth.

There is a potential loss of $80 that may or may not occur.

Imagine that without any investment in safety there is a 20% chance that the loss will occur.

There is a safety investment that costs $5; this investment will lower the probability of a loss to 10%.

The individual is rational, aims to maximize expected utility and is risk-neutral.

a) If the person DID NOT buy insurance, will she choose to make the safety investment? (Show your work.)

b) If that individual DID NOT buy insurance and makes the optimal safety investment in (a), what will her wealth be i) if the loss does not happen, ii) if the loss does happen, and iii) on average?

c) Imagine that full insurance coverage were available. No other insurance is available. If the person DID BUY insurance for some reason, will she choose to make the safety investment? (Assuming the safety investment were unobservable to the insurer.)

d) What price would a rational insurance company in a competitive market (with no transaction or other costs) charge for this full insurance? (Explain in 1 sentence.)

e) If that individual DID BUY full insurance for some reason, makes the optimal safety investment in (c), and if she faced the price of insurance in (d), what will her wealth be i) if the loss does not happen, ii) if the loss does happen, and iii) on average?

f) Explain how comparing your answers in (b) and (e) shows the risk vs expected wealth trade-off you face when buying insurance. (Explain in 2-3 sentences)

 

 

 

 

4. A continuous safety problem. Consider a risk neutral individual who faces a potential loss of size L with probability p. The individual can invest in safety to affect p, so that p depends on safety expenditure as p=1/(1+S) (S cannot be negative); so P(S=0)=1 for S=0.

a. Please find the optimal level of safety investment for the individual, with no insurance.

b. Now imagine that the individual has purchased I units of coverage at a premium P(I), and then has to choose the level of safety investment. What is the optimal level of safety investment as a function of I?

c. Now imagine the insurer is smart and knows the person will choose safety as in (b). They set actuarially fair prices taking optimal safety behavior into account. What actuarially fair premiums will they set as a function of I, P(I)?

d. Given the premium structure in (c), what amount of insurance I will the individual choose? Please show your work.

 

5. Game. Consider the game we played after the exam.

a) What product was typically sold in the 5th round (just before we changed the rules of the game)? Why?

b) What product was typically sold in the last round?  Why?

c) Please explain how reputation might be used in this game. (What products would be offered at which prices in which rounds?)

 

 

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