1. Two identical firms have MC = $2 and face a market demand function of QD = 8 - P. The firms compete on the basis of (continuous) price; if their prices are equal they split the market but if one has a lower price it gains the entire market. If the game is repeated, a discount factor δ applies to each firm, and the discounted value of profits is maximized.
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 a very large finite number of times, what is the Nash Equilibrium, and why?
c) If the game is indefinitely repeated, for what values of δ will the firms be able to sustain a cooperative Nash Equilibrium, and why?
2. A risk averse person with a von-Neumann-Morgenstern 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?
3. The Acme Auto Insurance Company is risk neutral and seeks to offer insurance at its actuarially fair price. All drivers in the community have the same initial income, $400, and
know that if they are involved in an auto accident, their income falls to $0. All drivers have the same von Neumann-Morgenstern utility index: U(Y) = Y½
There are two groups of drivers in the community. One half are safe drivers and face a 25% chance of being in the accident. The others are risky drivers and face a 75% chance of being in an accident.
a) If the insurance company can identify who is a safe driver and who is a risky driver and charge them different prices, what premium will members of each group pay for insurance? How much insurance will a representative driver in each group buy? What level of income will each type of driver enjoy in the two states of the world?
b) Now assume that while each individual knows whether she is a safe or risky driver, the insurance company does not. Acme only knows that overall there is a 50% chance of an individual being in an accident, so it charges $0.50 per $1 of coverage to everyone. Solve for the optimal decision for each type of driver. You may assume that there is no limit on the amount of insurance an individual can buy. Is the insurance company in long run equilibrium at this price? [Hint: what is Acme's expected profit per driver?] Explain briefly what is going on in this case.
4. The gasoline market can be represented by these functions: QS = 50P & QD = 120 - 30P
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?
c) 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? How does this compare with (b)?
d) What are the levels of consumer and producer surplus before (a) and after taxes (b), (c)?
e) If burning gasoline causes pollution costing $1/gallon, what is the optimal level (price & quantity) of gasoline consumption, taking this externality into account?
f) 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 externality, 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?
c) 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 S thousand gallons of a pollutant. red dye #1010a. 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 #1010a by these fixed amounts.
Reductions in Annual Cost to Widgetco Annual Loss to
Red Dye #1010a of Reducing Discharge Fishermen
(1000 gallons) (millions of S) (millions of S)
|
0
|
0
|
|
|
1
|
1
|
16
|
|
|
4
|
12
|
|
3
|
9
|
8
|
|
4
|
16
|
4
|
|
5
|
25
|
0
|
a) What is the efficient reduction in the discharge of red dye #1010a?
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?
d) Suppose you determine that it is better to do nothing. Will the amount of pollution depend on %%tether Widgetco has the legal right to dump red dye #1010a into the lake or the fishermen have the right to fish in unpolluted water?