Intermediate microeconomics
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1. Using the Conjectural Variations model, answer the following:
(See handout for details of this model.)
a) Cooperative Conjectural Variations: If both firms match one another’s quantity
restrictions (dqz/dqi = dqi/dqz = +1), what will be the industry output (qi + qz)?
If the firms colluded to form a. monopoly, what will be the industry output (qm)?
How does q] + qz compare with qm?
b) Competitive Conjectural Variations: If each firm assumes that if it reduced output
by 1 unit, the other firm will increase output by 1, (dqz/dqi = dqi/dqz = -1), what will
be the industry output (q1 + qz)? If the firms were in perfect competition with each
other, what will be the industry output (qpc)? How does q1 + q: compare with qpc?
c) Cournot Conjectural Variations: Between these two extremes, if each firm takes
the other’s output to be given, (dqz/dqi = dqi/dqz = 0), what will be the industry
output (q1 + qz)? If the firms were Cournot competitors, what will be the industry
output (qc)? How does q1 + qz compare with qc?
d) Consistent Conjectural Variations: If we don’t know how films conjecture about
each other’s decisions, what is an equilibrium set of conjectural variations (dqz/dql
dqi/dqz =?). Does this match with any of the cases above (a, b, c)? Discuss the result.
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Two identical firms each have TC = 2Q and face a market demand of Q = 8 – P. They
compete on the basis of price.
a) If this game is played once, and the firms cooperate (collude), what is the maximum
profit they can make together, and how much will each firm make? What is the Nash
Equilibrium of this game, and why?
b) If this game is repeated for a very large but finite number of rounds, what is the Nash
Equilibrium, and why?
c) If the game is indefinitely repeated, for what values of the discount factor will the firms
be able to sustain a cooperative Nash Equilibrium, and why (give the conditions)?
i. A risk averse person with a von-Neumann-Morgenstem utility index of: U = ln(Y)
has a 20% chance that a disaster will reduce her regular income of $100,000 to zero.
She can buy insurance at a rate of $0.40 per dollar of coverage.
a) Will she fully, under, or over-insure against this risk, and why?
b) What is her optimal bundle of contingent claims?
c) How much insurance will she buy and at what cost?
The Acme Auto Insurance Company is risk neutral and seeks to offer insurance at its
actuarially fair price. All drivers have the same initial income, $400, and if they are involved
in an auto accident, their income falls to $0. All have the same utility index: U(Y) = Y”
There are two groups of drivers: half are safe drivers with a 25% chance of being in the
accident, and the others are risky drivers with a 75% chance of being in an accident. Each
individual knows whether she is a safe or risky driver.
a) If the Company can identify a safe driver from a risky driver, what premium will it
charge a driver in each group? What amount of insurance will each type of driver buy?
How much income will each type of driver have in the two states of the world?
b) If the Company cannot identify a safe driver from a risky driver, it only knows that there
is an overall 50% chance of an individual being in an accident, so it charges a premium of
50% to everyone. Solve for the optimal decision for each type of driver (assume there is
no limit on the amount of insurance an individual can buy). What is Acme’s expected
profit per driver? Is it in long run equilibrium? Explain precisely what is going on here.
l. The gasoline market can be represented by these functions: 03 = SOP & Q1) = 120 – 3oP
a) What is the equilibrium price and quantity in the market?
b) If gas stations are required to pay a tax of $0.50/ gallon, what is the new equilibrium price
and quantity? How much will stations receive and how much will consumers pay/gallon?
0) Instead, if consumers have to purchase government coupons for $0.50/ gallon before
buying gas, what is the new equilibrium price and quantity? How much will stations
receive and how much will consumers pay/ gallon? How does this compare with (b)?
d) What are the levels of consumer and producer surpl us before (a) and after taxes (b), (c)?
e) If burning gasoline causes pollution costing $l/gallon, what is the optimal level (price
quantity) of gasoline consumption, taking this extemality into account?
0 Describe three policies which would achieve the optimal level of consumption, and find
the levels of consumer surplus, producer surplus, government revenue (if any) and total
social surplus associated with each policy.
g) “If there is an extemality, there is a unique efficient level of production, but there is no
unique efficient price.” What is “efficiency” in this context? Is this statement correct?
5. Tarzan and Jane live alone in the jungle, and Cheetah does all the work for them: patrol the
perimeter of their clearing and harvest tropical fruits. Cheetah can collect 3 pounds of fruit an
hour, and currently spends 6 hours patrolling, 8 hours picking, and 10 hours sleeping.
a) What (who) is the production process? What are the public and private goods?
b) If Tarzan and Jane are each willing to give up the consumption of one pound of fruit for
one hour of patrol, is the allocation of Cheetah’s time Pareto efficient? Why?
0) Should Cheetah patrol more or less? Why?
6. Widgetco Inc. has opened a new factory next to Lake Littauer. In the production of widgets,
Widgetco dumps 5 thousand gallons of a pollutant, red dye #10103, into the lake. The
pollutant reduces the fish population and results in a loss to fishermen. The table below
shows the total cost of reducing the volume of discharge and the total loss to fishermen at
each level of reduction. Assume that it is only possible to reduce the volume of red dye
101oa by these fixed amounts.
Reductions in Annual Cost to Widgetco Annual Loss to
Red Dye #10103 of Reducing Discharge Fishermen
{1000 gallons! {millions ofSQ (millions ofS)
0 0 20
l 1 l6
2 4 l2
3 9 8
4 16 4
5 25 o
a) What is the eflicient reduction in the discharge of red dye #101oa?
b) As a specialist with the EPA, you must choose between imposing a tax on the discharge
and doing nothing. Under what conditions, if any, is each of the proposals appropriate?
c) Suppose you determine that the tax is appropriate. What size tax per 1000 gallons of
discharge would you choose?
(I) Suppose you determine that it is better to do nothing. Will the amount of pollution
depend on whether Widgetco has the legal right to dump red dye #10103 into the lake or
the fishermen have the right to fish in unpolluted water?