14.41 Public Economics, 2002 Problem Set # 3 Due: Sunday, October 27, 2002 1.) Consider an economy that is composed of identical individuals who live for two periods. These individuals have preferences over consumption in periods 1 and 2 given by: U = 2log(C1) + 2log(C2)
(1)
They receive an income of 100 in period 1 and an income of 50 in period 2.
They can save as much of their income as they like in bank accounts, earning
an interest rate of 5% per period. They do not care about their kids, so they
spend all their money before the end of period 2. Each individual’s lifetime
budget constraint is given by:
C1 +C2(1+r)=Y1+Y2/(1+r)
(2)
Individuals choose consumption in each period by maximizing lifetime utility
subject to this lifetime budget constraint.
a) What is the individual’s optimal consumption in each period? How much
savings does he or she do in the first period?
b) Now, the government decides to set up a social security system. This
system will take $10 from each individual in the first period, put it in the bank,
and transfer it to them with interest in the second period. Write out the new
lifetime budget constraint. How does the system affect the amount of private
savings? How does the system affect national savings (total savings in society)?
What is the name for this type of social security system?
c) Now suppose that the existence of the new social security system causes
the individual to retire in period 2, so he receives no labor income in period 2.
Solve for the individual’s new optimal consumption in each period in this case.
What is the new level of private and national savings? Does this differ from the
level of savings in part b, and if so, why (explain intuitively)?
d) Now move away from this simple example to our actual US social security
system. Should social security lead to early retirement? Why? What is the
evidence on the relationship between social security and retirement in both the
US and other countries?
2.) You are hired by President Bush to review the Unemployment Insurance
(UI) program, which typically replaces approximately 45% of a worker’s wages
for 26 weeks after he loses his job. In response to the economic downturn,
Congress has made workers eligible for an additional 13 weeks of coverage (this
is in addition to the standard 26 weeks). Answer each part of the question in
1 page or less.
Bush shows you a table comparing the unemployment durations of individuals
who receive UI and do not receive UI. This table reveals that those that
receive UI stay unemployed longer than those that do not receive UI. He claims
that this proves that UI causes longer unemployment durations and the 13 weeks
of additional coverage should be eliminated.
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a) Is he correct? Why or why not? What other evidence could you bring
to bear on this question that might be more useful in gleaning the correct
relationship between UI generosity and unemployment durations?
b) If UI causes longer unemployment durations, does this prove that the
generosity of the program should be reduced? Why or why not?
c) Consider two alternative reforms of the current UI system. The first is to
perfectly experience rate firms, so that the taxes that firms pay are set exactly
equal to the benefits their workers receive (benefits remain at 45% of wages).
The second is a system of individual perfect experience rating - the government
would loan individuals 45% of their wages while unemployed, but they would
have to pay them back when they get a new job.
(i) Contrast the effects of these alternative policies on unemployment durations
and the likelihood of worker layoffs.
(ii) Contrast the extent of insurance for workers provided by each of these
alternatives with the current system.
3.) Consider a worker who has the option to purchase DI (disability insurance)
on the private market. The worker becomes disabled with probability
q. The worker can purchase the DI insurance by paying the premium p. If
the worker is disabled, she will receive a NET payment of b. If the worker is
not disabled, she earns an income of W. If the worker is disabled, she earns no
income. The worker has log utility. Consider this a one-period problem and
assume that the market for DI insurance is perfectly competitive.
a) Write down the expected utility function of the worker if she purchases
insurance.
b) Solve for the worker’s optimal level of insurance (the worker gets to choose
the premium p).
c) What is the worker’s expected utility?
d) Now assume that there are two types of workers in equal number in the
economy. Workers of type i have a greater tendency to become disabled than
workers of type s. qi > qs. Assume that the insurance companies cannot
distinguish between types i and s. As a result, they must offer DI insurance at
a single price. They price the insurance based upon the average probability of
disability qa =(qi+qs)/2.Will the insurance market function? Why or why not?
4.)
An economy produces two goods - sun tan lotion and jackets. Each of these
goods is produced by a separate person. Individual 1 produces sun tan lotion
and receives a profit of $64 if it is hot, $0 if it is cold. Individual 2 produces
jackets and receives a profit of $0 if it is hot, $64 if it is cold.
Individual 1 and 2 maximize expected utility:
E[U1] = π ∗ U(Y1H) + (1 − π) ∗ U(Y1C)
2
(3)
E[U2] = π ∗ U(Y2H) + (1 − π) ∗ U(Y2C)
(4)
where YH and YC are income in state H (hot) and state C (cold), U(Y) is
the utility function, and π is the probability of state H. There is a social welfare
function of the form:
Total Welfare = E[U1] + E[U2]
(a) Suppose that π=1/2, and that the form of the utility function is:
U(X) = (X)1/2
(5)
(6)
(i) What is the initial expected utility of each individual?
(ii) Suppose that the two individuals are considering entering into an arrangement
today which insures their income against this uncertain weather outcome
tomorrow. That is, depending on the weather outcome (state H or C), each person
will either receive some of the other persons income, or give some of their
income to the other person. Can such an insurance arrangement be struck that
makes both parties better off? What arrangement will maximize social welfare?
How does social welfare compare to the level in (i)? Is this arrangement
acceptable to both parties?
(iii) Now, suppose that it is announced that it will be hot tomorrow. How
much insurance will be bought and sold now? What expected utility does this
give each person? Has this increased or decreased social welfare, relative to (ii)?
Why?
(b) Now, suppose that π =1/2 again, but that the form of U changes to
U(X) = ½*X.Answer (i)-(iii) above. Is your answer to (iii) different from (a)?
Why or why not?
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