Public Economics — Problem Set #2

Due: October 9thth at 2:40pm (submit through the courseworks)

1. Suppose that you have a job paying $2500 per month. With 10 percent probability, you may get sick and your monthly earnings will be reduced by $900. Assume that you spend all of your income on consumption and you have no savings. Your utility from consumption is given by u(C)=√ C and you are interested in maximizing the expected utility.

(a) Suppose that you have no access to insurance. What is your expected income next month? What is your expected consumption?

(b) What is the maximum price that you would be willing to pay for the first dollar of insurance?

(c) Suppose that you can buy insurance that will cover all of your $900 loss. What is the maximum amount that you would be willing to pay for that insurance?

(d) If you divide the amount obtained in part (c) by the amount of coverage purchased ($900), you should get a number that’s smaller than the price that you’d be willing to pay for the first dollar of coverage but larger than the actuarially fair price. Explain why.

(e) Suppose the dollar of coverage costs q=1/9. How much coverage are you going to buy?

2. Consider a person with initial resources of $80,000 that is facing 20% probability of a loss of $60,000. The utility function in each state of the world is given by ln(C) where C is consumption (the derivative of ln(C) is C 1 ). The person maximizes expected utility.

(a) What is the maximum price that the person is willing to pay for the first dollar of coverage?

(b) Suppose that the price of dollar of coverage is equal to 1/3. How much insurance would this person buy?

(c) The answer in the previous part does not involve buying full coverage. At what price would the full coverage be bought? Is it possible that this could be the market price?

3. Suppose that the government considers a new “social insurance” program: a payment to renters who got evicted by their landlords (it is an insurance against an evil landlord). Can you think of a justification for this policy? What are possible costs? What should we know to decide whether introducing this program makes sense?

4. Note: the objective of this problem is to take you through an example of how one might think about consequences and trade-offs involved in social insurance. Think about it as an extension of lectures rather than a problem to be solved. What needs to be done in the problem is mathematically relatively simple, the hard part is interpreting the solution.

Suppose that a person has the utility function given by

√C−D

where C is consumption and D is the dis-utility of work or other effort. A person that is employed has the dis-utility of work given by D. A person that is unemployed and exerts effort of s to find a job experiences disutility of search given by s 2 .

A person that works earns w. She has therefore the utility of √ w−D (we assume that all income is consumed). A person that loses a job searches for a new one and finds it with the probability of s. The utility of a jobless person is given by

s(√w−D)+(1−s) √B−s 2                         (1)

B represents unemployment benefits, the first two terms represent the expected utility from outcomes following the search (with probability s the person finds a job right away, otherwise she receives unemployment benefits).

(a) Find the optimal level of search s. What does it depend on? How does the presence of unemployment insurance affect search? Explain what is the moral hazard here and how the policy induces this behavior.

Now, let’s denote this optimal level of search as s(B) to highlight that it depends on the level of benefits. The objective of the government is to maximize overall utility of the person.

We need two more elements. First, let the probability of losing a job be denoted by 1−p. Thus, the expected utility of the person is given by

p·[ √w−D]+(1−p)[s(B)(√ w−D)+(1−s(B))√B−s(B) 2 ].

Second, benefits have to be financed somehow. We will make a very simple assumption: the cost of a dollar spent by the government is given by some number γ measured in the same units as utility.

The total amount of money that the government needs to spend to finance benefits is given by (1−p)(1−s)B (because (1−p)(1−s) is the probability that unemployment benefits will need to be paid out).

The objective of the government is therefore to maximize

p·[ √w−D]+(1−p)[s(B)(√w−D)+(1−s(B))√B−s(B) 2 ]−γ(1−p)(1−s(B))B.          (2)

with respect to the level of benefits B.

(b) Can you explain using the above expression what are the benefits and costs of increasing unemployment insurance? Can you relate it to the discussion in class about the optimal social insurance?

(c) What do you find unrealistic about this model? Are there examples of moral hazard that we assumed away but that may be important?

(d) (Much harder and boring, only if you are very brave, there is no extra credit for it). See how much progress you can make in actually solving for the optimal level of benefits (do not expect though to get an explicit solution for B, but only to derive a condition that B needs satisfies).

5. Consider the case of unemployment insurance.

(a) Both moral hazard and adverse selection may lead to problems with private provision of insurance products. Both of these are present in the unemployment insurance context. Give an example of each in the context of unemployment insurance and explain why they cause problems for private insurers.

(b) Many governments implement public unemployment insurance programs. Are they immune to the problems you identified above? Explain. If they are still facing one or both of these problems, explain why it still may make sense for them to intervene.

6. Consider an individual that lives for at most two periods. The probability that she survives until period 2 is 0.8. Initial wealth is $10,000 and, if she is alive in period 2, she will then receive a Social Security check for $4,600 and no additional income. Suppose that the price of private annuity is actuarially fair, the interest rate is zero and that the person chooses to have the same level of consumption in the two periods by appropriately annuitizing (so that there is no saving from period one to period two, other than through the annuity that is purchased). What is that level of consumption?