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ECN 100B WQ 2024 Problem set 4
DUE MARCH 15TH BY NOON ON GRADESCOPE
1
Rivalry and exclusion
1.1
What are the four types of goods when categorized based on
rivalry and exclusion?
1.2
What is one example of each type of good? Please explain
why.
1.3
For each type of good, how might the government regulate
markets for such goods to ensure the socially optimal level is
produced and consumed?
1
2
Public goods
Three families live at the end of a road outside town. They must pay to install streetlights. Streetlights are a public good for these households—that is if one household buys a a
streetlight, all households will get access to the streetlight. Assume family 1 has an inverse
demand for streetlights of p = 200 − Q, family 2 has an inverse demand for streetlights of
p = 400 − Q, and family 3 has an inverse demand for streetlights of p = 600 − Q. Assume
the supply of streetlights is horizontal at the marginal cost of $400.
2.1
Assuming the families act independently, find the quantity of
streetlights each family will demand. What will be the total
amount of streetlights provided?
2.2
Now find the socially optimal quantity of streetlights for these
three families.
2.3
Please draw this market, including all three demand curves,
the supply curve, and the social demand curve. Indicate the
social optimum.
2.4
What is one way that the households could achieve the socially
optimal quantity of streetlights?
2.5
Explain how this problem illustrates the problem of public
goods and free riding.
2
3
Risk
√
Imagine Alex has the concave utility function U (W ) = 3 W . Imagine Alex is about to
graduate from UC Davis with an economics degree. When she graduates, she will either
get a job paying $2500 per month (being a private economics tutor) or $10000 per month
(coding game theory games in C++). She believes the probability of getting each job is 50%
and she will only get one job.
3.1
What is Alex’s expected utility?
3.2
What is the variance of Alex’s expected utility?
3.3
What risk premium would Alex pay to avoid bearing this risk?
and she has the same job
Now imagine Jamie has the linear utility function U (W ) = 3W
100
prospects as Alex.
3.4
What is Jamie’s expected utility?
3.5
What is the variance of Jamie’s expected utility?
3.6
What risk premium would Jamie pay to avoid bearing this
risk?
3.7
Who is more risk averse—Alex or Jamie?
3
4
Adverse selection
Suppose there is a used car market with lemons and good used cars. Suppose potential
car buyers have asymmetric information—they are unable to tell if a used car is a lemon
or good—and are risk neutral and value lemons at $2000 and good used cars at $10000.
Suppose the reservation price of lemon owners is $1500 and the reservation price of good
used car owners is $8000. The share of current owners who have lemons is 0.1.
4.1
Are the lemons sold in this market?
4.2
Are the good used cars sold in this market?
4.3
What is the price for used cars at the equilibrium in this
market?
Instead of the share of current owners having lemons being 0.1, let the share be θ, a number
between 0 and 1.
4.4
For what values of θ will all potential sellers sell their used
cars?
4.5
Please draw two graphs, one of the market for lemons, and one
of the market for good cars. Each graph should have supply
and demand curves and the demand curve at the equilibrium
price.
4.6
Describe he equilibrium be in this market if there was no
asymmetric information (i.e. if buyers and sellers both knew
whether used cars were lemons or good).
4.7
What are three ways adverse selection could be reduced in
this market? Please explain.
4
5
Bonus
Imagine there are two students who are deciding how to spend their time between consuming a private good, studying alone (Ai ), and a public good, studying together (T ). Each
student has utility function Ui (Ai , T ) = 4 log(Ai ) + 2 log(T ). This utility depends on private
consumption of studying alone and consumption of the total amount of studying together,
which is the sum of how much time each student invests in studying together: T1 + T2 = T .
Each student has 50 hours to allocate (i.e. Ai + Ti = 50).
5.1
What will be the equilibrium levels of Ai and T in this market
if the students make private decisions?
5.2
What is the socially optimal equilibrium in this market?
5.3
Will studying together be under-provided, over-provided, or
provided at the right level in the equilibrium in which the
students make private decisions? Please explain.
5
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