GROUP A
problem1)
An agent has a utility function over goods 1 and 2 of the form U= X_{1}^{c }X_{2}^{d}, where c is your individual number and d is your minimum number. The agent’s income is equal to your 2-digit number. The price of good 1 is your maximum number and the price of good 2 is your median number. Derive the agent’s demand functions for good 1 and good 2. find out the quantities of good 1 and good 2 in the agent’s optimum bundle.
problem2)
An agent has a utility function over goods 1 and 2 of the form U= X_{1}^{c }X_{2}^{d}, where c is your 1- digit number and d is your minimum number. The agent’s income is equal to your 2-digit number. Initially, the price of good 1 is your median number and the price of good 2 is your individual number.
Let the price of good 1 change to your maximum number. For good 1, determine for this price change the
a) total price effect
b) the substitution effect
c) the income effect
problem 3)
For the same problem you analysed in problem 2, find for that price change the
a) Laspeyres measure of the welfare change
b) Paasche measure of the welfare change
c) compensating variation
d) equivalent variation
GROUP B
problem 4)
a) Consider the problem you analysed in problem 2. Instead of the income value you used there, allow the agent to have an endowment of good 1 equal to the first digit of your 2- digit number, and an endowment of good 2 equal to the second digit of your 2-digit number. Derive expressions for the ordinary demands for both goods and find out the gross and net demands for each good.
b) An agent has a utility function over wealth given by U= W^{.5/c} where c is your 1-digit number. Their wealth if not robbed is equal to your 2-digit number multiplied by 1000. Should they be robbed, their wealth will be your maximum number multiplied by 1000. They assess the probability of being robbed as 1/(median number x 10). How much would this agent be prepared to pay for full insurance? How much would they have to pay for full (actuarially) fair insurance?
problem 5)
Consider the two agents, A and B.
• Agent A has the utility function U_{A}= X_{1}^{c }X_{2}^{d} where c is your minimum number and d is your median number. A’s endowment of good 1 is the first digit of your 2-digit number, and A’s endowment of good 2 is the second digit of your 2-digit number.
• Agent B has the utility function U_{A}= X_{1}^{c }X_{2}^{d }where e is your maximum number and f is your 1-digit number. B’s endowment of good 1 is the second digit of your 2-digit number, and B’s endowment of good 2 is the first digit of your 2-digit number.
• The price of good 2 is your 1-digit number.
a) Find the equilibrium price for good 1 and the gross and net demands of both agents for goods 1 and 2.
b) Repeat the analysis for the cases where
i. the values of c and d are swapped for A, and e and f are swapped for B.
ii. the endowments of goods 1 and 2 are swapped for A, and the endowments of goods 1 and 2 are swapped for B [with c, d, e, f at their original – i.e part a) values].
problem6) Consider the two agents, A and B. Each can choose one of two strategies, 1 and 2. The payoffs for the various outcomes are illustrated below (A’s payoffs listed first in each cell):
Player B
Strategy 1 Strategy 2
Player A Strategy 1 3.5, b c, 2.5
Strategy 2 e, f g, 1.5
where:
• b is your individual number
• c is your 1-digit number
• e is your median number
• f is the first digit of your 2-digit number
• g is the second digit of your 2-digit number
a) Assume that A and B act simultaneously. Find all equilibrium strategy combinations of this game, including, where appropriate, mixed-strategy equilibria. Show A and B’s equilibrium payoffs.
b) Reprepare this game in extensive form. Determine the equilibria and payoffs for the case in which A moves first, and the case in which B moves first.
GROUP C
problem7)
Consider a market in which all output is produced by two firms, A and B. The market inverse demand curve is given by P= a-bQ where a is your two-digit number x 10 and b is your individual number. Both firms have a constant marginal cost equal to your median number.
a) Find the Cournot equilibrium outputs for firms A and B, the equilibrium market price and the equilibrium profit for each firm.
b) Repeat for
i. the case where the marginal cost of firm B is constant and equal to your maximum number.
ii. The case where there are n firms with marginal cost equal to your median number. Find the output of each firm, the market price and each firm’s profit, where n is the sum of your individual number and your median number. [Hint: with identical costs each firm’s output will be the same].
iii. The case where there are two firms A and B and the marginal cost for firm A is m_{A}Q_{A} (where m_{A} is your minimum number) and the marginal cost for firm B is m_{B}Q_{B} (where m_{B} is your 1-digit number).
problem8). Consider a market in which all output is produced by two firms, A and B. The market inverse demand curve is given by P= a-bQ where a is your two-digit number x 10 and b is your individual number. Both firms have a constant marginal cost equal to your median number.
a) Find the Stackelberg equilibrium outputs for firms A and B, the equilibrium market price and the equilibrium profit for each firm, on the assumption that firm A is the leader and firm B is the follower.
b) Repeat for
i. the case where the marginal cost of firm B is constant and equal to your maximum number.
ii. The case where there are two firms A and B and the marginal cost for firm A is m_{A}Q_{A} (where m_{A} is your minimum number) and the marginal cost for firm B is m_{B}Q_{B} (where m_{B} is your 1-digit number).
iii. The above two cases on the assumption that B is the leader and A the follower.
problem9)
Consider a market a market for used cars in which cars can be either high-quality or lowquality. The demand for both types of car is perfectly elastic. The price buyers are willing to pay for a car known to be of low quality is your individual number x $2000 and the price they are willing to pay for a car known to be of high quality is your maximum number x $4000. Sellers are willing to accept a price equal to your minimum number x $1000 for a car known to be of low quality, and to accept a price equal to your median number x $3750 for a car known to be of high quality. The number of cars available for potential sale is equal to your 2-digit number x 200. The number of high-quality cars in that group is equal to your maximum number x 100. The supply of both cars is perfectly elastic up to the quantity of cars available.
What will be the outcome in the market in terms of the prices and quantities of cars of each type sold, the welfare gains from trade, and how those gains are distributed, is each of the following cases:
a) Information on quality is complete and symmetric.
b) Information on quality is zero and symmetric, and both buyers and sellers have the utility function U=V, where V is wealth.
c) Information on quality is complete for sellers but zero for buyers, and buyers have the utility function U=V.
d) Information on quality is complete for sellers but zero for buyers, and buyers have the utility function U=c ln v, where c is your 1-digit number.
For cases c and d above, find the maximum value of the sellers’ valuation of good-quality cars (given your original value of θ) that would allow a market for good-quality cars to exist. For the original sellers’ valuation of good cars find the minimum value of θ that would allow a market for good-quality cars to exist.
If the sellers of good quality cars in cases c and d were able to spend $18000 on a certification process that buyers regarded as 100% credible, would they do so? If not, what would be the maximum amount they would be willing to pay?
problem10) Consider a good for which production generates external costs. Let the marginal external cost function be MEC=aE, where a is your 1-digit number, and E the quantity of emissions. The pollution can be abated at a cost. Let the marginal cost of abatement function be MCA=B-cE, where B is your 2-digit number and c is your median number.
a) Find the socially optimal level of emissions, and the optimal value of abatement costs and external cots.
b) If an emission fee were levied on producers, what would be the deadweight social loss associated with setting the fee at 90% of the correct value?
c) If an emission standard were enforced, what would be the deadweight social loss associated with setting the standard at 110% of the correct value?