ECON 425: Economics of Public Expenditures
Homework Assignment 4
You need to explain your answers. The answers are due on Thursday, April 2nd at
11:59 EST. Please upload your submission directly in Canvas.
Problem 1: Adverse Selection, continued. Every person has $1,000 to spend. There
are two groups of people, 100 sick people and 300 healthy people. A sick person visits the
hospital with probability p = 0.25. A healthy person visits the hospital with probability
p = 0.05. Any hospital visit costs $400. All individuals have the following expected utility
function:
EU = (1− p)√Cg + p√Cb
a) If the insurance company charges the price x = 40 to everyone for full insurance, who
will buy insurance? Who will not buy insurance?
b) Consider the insurance company offers full insurance at a high price of $100 and partial
insurance (only covers $20 in case of hospital visit) at a low price of $1. Will a sick
person buy full, partial, or no insurance? Will a healthy person buy full, partial, or no
insurance?
c) Which outcome is more efficient, a) or b)?
d) Describe why health insurance providers typically offer “high deductible, low premium”
and “low deductible, high premium” insurance packages. Consider the results from this
problem.
Problem 2: Moral Hazard (25 points). Kennedy has some non-wage income $30,000
and a dangerous job that pays $60,000. If Kennedy chooses a high level caution, she will
get injured with probability p = 0.1. If Kennedy chooses a low level of caution, she will get
injured with probability p = 0.5. If she gets injured, she can no longer work. Low caution is
costless, but high caution costs Kennedy $10,000. Kennedy’s expected utility follows
EU = (1− p)√Cg + p√Cb
where Cg represents Kennedy’s consumption if she does not get injured and Cb represents
Kennedy’s consumption if she gets injured.
a) Suppose there is no insurance. What is Kennedy’s expected utility from choosing low
caution? What about from choosing high caution? Will Kennedy choose low or high
caution?
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b) Suppose that the government provides workers’ compensation, so that Kennedy gets
$40,000 if she gets injured and can’t work. Will Kennedy choose low or high caution
now?
Problem 3: Calculating Social Security Benefits. Beth has earnings from age 18 to
age 43 (25 years in total) and from age 47 to age 67 (20 years in total). From age 18 to age
43, Beth’s real earnings are $700 every month. From age 47 to 67, Beth’s real earnings are
$1,400 every month. Use the procedure described in the lecture notes to answer the following
questions.
a) What is Beth’s AIME? (Take the 35-year average of monthly earnings.)
b) What is Beth’s PIA? (Use the PIA calculation rules in the lecture slides.)
c) Beth retires exactly one year after qualifying for the full benefits amount. What are
her monthly Social Security benefits in retirement?
d) Repeat parts a) – c), but now calculate the AIME as the 40-year average of monthly
earnings. If the results change, briefly explain why.
Problem 4: Baby Boom. Suppose that all workers live for two periods (young and old).
All workers earn income of $300 when young and $0 when old. Suppose there is an unfunded
Social Security system. In every period, the government taxes each young agent $50 and
immediately pays each old agent an even share of the tax revenue.
In addition, suppose there is a baby boom. So where all generations were the same size, now
one and only one generation is twice as large as the others.
a) When the baby boom generation is young, what are the social security benefits of the
older generation as compared to the usual benefits?
b) Once the baby boom generation is old, what social security benefits does it receive
compared to the usual social security benefits?
c) What could the government do to smooth out the effect of the baby boom? Assume
that the government has no other funds that it can use to solve the problem.
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