Part 1 - Problems and calculations

1. You are given the following information for a good.

Demand: Qd = 280 - 70P

Supply: Qs = -40 + 30P

A) What is the equilibrium price of this good?

B) What is the price elasticity of demand at the equilibrium price?

C) What is the price elasticity of supply at the equilibrium price?

D) If a $1 per-unit excise tax is levied on the buyers of the good, how much will buyers pay after tax?

E) What quantity of the good will be sold after the tax?

2. Suppose there are two consumers, A and B, and two goods X and Y. Consumer A is given an initial endowment of 3 units of good X and 9 units of good Y. Consumer B is given an initial endowment 7 units of good X and 6 units of good Y. Consumer A's utility function is given by:

U_A(X, Y) = X²*Y,

And consumer B's utility function is given by:

U_B(X, Y) = X*Y,

Therefore, consumer A's marginal utilities for each good are given by:

MU_X = 2X*Y

MU_Y = X²

Also, consumer B's marginal utilities for each good are given by:

MU_X = Y

MU_Y = X

a) Suppose the price of good Y is equal to one. Calculate the price of good X that will lead to a competitive equilibrium.

b) How much of each good does consumer A demand in equilibrium?

Good X:

Good Y:

c) How much of each good does consumer B demand in equilibrium?

Good X:

Good Y:

3. Suppose a firm's production function is given by Q = AL^αK^β. Thus the marginal product of labor is given by:

MP_L = αAL^α-1K^β

And the marginal product of capital is given by:

MP_K = βAL^αK^β-1

Suppose A = 1, α = ½, and β = ½.

Also, the price of labor, w = 1 and the price of capital, r = 9.

a) What is the total cost of producing Q units of output?  

b) What is the marginal cost of production?

c) Suppose the demand curve facing the monopolist is P = 78 - 6Q. How much output should the monopolist produce in order to maximize profits?

d) What price should the monopolist charge?

e) What is the deadweight loss due to the monopoly?

f) What value does the Lerner Index take in this situation?

4. Suppose there are two types of consumer: Type A and Type B. The demands for a monopolist's product for each type of consumer are given by:

Type A: Q = 100 - 4P

Type B: Q = 60 - P

Assume the marginal cost of production is constant and MC = 4, and there are no fixed costs.

a) If the monopolist is able to charge different prices to each type of consumer, what price will she charge to type A consumers?

b) If the monopolist is able to charge different prices to each type of consumer, what price will she charge to type B consumers?

c) How much profit will the monopolist make?

For parts d) - e), suppose the monopolist cannot successfully price discriminate between the two groups, and therefore is forced to charge a single price for both types of consumers.

d) What price will the monopolist charge?

e) How much profit will the monopolist make?

5. Suppose there are two firms in a market that each simultaneously choose a quantity of output to produce. Firm 1's quantity is q₁, and firm 2's quantity is q₂. Therefore the market quantity is Q = q₁ + q₂. The market demand curve is given by P = 70 - 5Q, Also, each firm has constant marginal cost equal to 10. There are no fixed costs.

The marginal revenue of the two firms are given by:

MR₁ = 70 - 10q₁ - 5q₂

MR₂ = 70 - 5q₁ - 10q₂             

A) How much output will each firm produce in the Current equilibrium?

B) What will be the market price of the good?

C) What is the deadweight loss that results from this duopoly?

D) How much profit does each firm make?

E) Suppose Firm 1 chooses their output first, and then firm 2 chooses their output. What is the Nash equilibrium in quantities?

6. Suppose that a perfectly competitive firm has Fixed Cost = 392 and Variable Cost = 2q², so that MC = 4q.

a) In the short run, what is the maximum profit this firm could make if the market price of the good is $60?

b) What is the Break Even price?

c) Suppose the market demand is given by P = 532 - 2Q. How many firms will there be in long run equilibrium?     

7. Suppose a consumer's utility function is given by U(X, Y) = X*Y. Also, the consumer has $288 to spend, and the price of X P_X = 9, and the price of Y, P_Y = 4.

a) How much X and Y should the consumer purchase in order to maximize her utility?

b) How much total utility does the consumer receive?

c) Now suppose P_X decreases to 4. What is the new bundle of X and Y that the consumer will demand?

d) How much money would the consumer need in order to have the same utility level after the price change as before the price change?

e) Of the total change in the quantity demanded of X, how much is due to the substitution effect and how much is due to the income effect?

Part 2 - Short Answer Definitions

Please give concise accurate responses to each of the following.

1. Suppose Consumer A has a utility function U_A(X, Y) = X + 3Y. Consumer B's utility function is U_B(X, Y) = X*Y. Also, there are a total of 18 units of Good X and 9 total units of Good Y available. Using an Edgeworth Box, illustrate the contract curve. Your graph should be neat and accurate. Make sure your graph is carefully and accurately labeled.

2. Suppose a consumer's preferences are represented by the utility function, U(X, Y) = MIN(3X, Y). The price of X is P_X = 2, and the price of Y is P_Y = 4. Sketch the graph of the consumer's Engel Curve for Good X.

3. Suppose a consumer has utility function U = MIN(X, 2Y). The consumer has $27 to spend and the price of god Z is P_X = 4 and P_Y = 1. If the price of good X falls to P_X = 1, how much of the total change in demand for X is due to the substitution effect? (Hint: Sketch the graph.)

4. Suppose Ann's marginal rate of substitution between good Y and good X is -2, Bill's marginal rate of substitution between good Y and good X is -3. Each consumer has 10 units of each good. Propose a trade between Ann and Bill that will make each consumer better off.

5. Suppose a consumer has utility function U(X, Y) = X + 2Y. Also, the consumer has $30 to spend, and the prices are given by P_X = 3 and P_Y = 1. Using a graph, sketch the consumer's budget constraint. Also, identify the bundle of budget constraint which maximizes the consumer's utility, and draw the indifference curve which runs through this bundle. Be sure to clearly label graph.

6. Suppose that two players are playing the following game. Player A can either choose Top or Bottom, and Player B can either choose Left or Right. The payoffs are given in the following table:

Where the number on the left is the payoff to player A, and the number on the right is the payoff to player B.

a) Does player A have a dominant strategy, and if so what is it?

b) Does player B have a dominant strategy and if so what is it?

c) For each of the following strategy combinations, write TRUE if it is a Nash Equilibrium and FALSE if it is not:

i) Top/Left

ii) Top Right

iii) Bottom Left

iv) Bottom Right

d) If the game is played where Player A moves first and Player B moves second, using the backward induction method discussed in class, what will be the outcome of the game?