Homework #5
1. First and Second Degree Price Discrimination
Consider that, in the Chocolate market, there is a monopolist, Charlie Factory. The demand curve for Chocolate is P = 48 - 4Q, and MC = Q + 3.
a. If this market is a competitive market (P=MC), how much is the producer surplus for Charlie Factory?
b. If this market is served by a single-price monopolist, what is the monopolistic price? Under this monopolistic price, how much is the producer's surplus?
c. Charlie can find each customer's valuation for Chocolate using his magic, so he can use 1st Degree Price Discrimination! How much is the producer surplus under 1st Degree Price Discrimination?
d. Charlie lost his power to find each customer's valuation of chocolate, so he decides to use 2nd Degree Price discrimination. His plan is as follows. Until Q=4, he sets the price as X. From Q=4 to Q=8, he sets the price as Y. From Q=8 to Q=9, he sets the price as Z. How much are X, Y and Z?
e. How much is the producer surplus under his plan described in part (d)?
2. Third Degree Price Discrimination
Consider that, in the market, there is a monopolist who can distinguish two types of customers, Group A and Group B (such as men and women, or young and adult). Group A's demand curve is P = 10 - 2Q, and Group B's demand curve is P = 7 - Q. In both markets, MC=ATC=4.
Note: If MR for the total market (the Z curve in our class notes) =MC, the profit maximizing quantity is a total production of 3 units.
a. What is the profit-maximizing quantity in each market under 3rd degree price discrimination? What are the prices in each market?
b. Derive the total profit for this monopolist who practices third degree price discrimination.
c. Which group of customers has the more elastic demand curve?
d. Based on your answer, choose the appropriate word in the following bracket.
"Group A which has the more ( elastic, inelastic ) demand curve faces the higher price under 3rd degree price discrimination."
e. Explain the statement in part (d).
3. Monopolistic Competition
The Digital Camera Market is monopolistically competitive. Each firms faces a demand curve of P = 200 - 0.5Q. There is a new firm, SOYO, who would like to enter this market. The Total Cost for this firm is given as TC= 20 -11Q+Q^2 and marginal cost for this firm is MC= -11+2Q.
a. Draw a picture, including the demand curve, MR curve, TC, MC.
b. Write down the equation for MR.
c. What is the optimal quantity for this firm to produce in the short run? What is the optimal price in the short run? What is the firm's profit in the short run? In your graph in part (a) indicate the optimal price, optimal quantity, and the area of profits for this firm.
d. What is going to happen in the long-run in this industry? Illustrate this long-run outcome in a new graph.
4. Game Theory
Mr. Crane is considering whether or not he should evade his taxes. If he doesn't evade paying his taxes (strategy "N"), he will need to pay taxes of $10. On the other hand, if he evades his taxes (D), there is a possibility he could be caught by the tax authorities. Once he is caught, he will be forced to pay the fine for tax evasion. The tax authorities (the IRS) also have two potential strategies: they may do regular detection (RD) or extra detection (ED). Extra detection costs more, but it can catch tax evaders far more frequently. To sum up the choices facing Mr. Crane and the IRS we have the following payoff matrix where the first number in the cell refers to the IRS payoff while the second number refers to Mr. Crane's payoff:
|
|
Mr. Crane
|
|
IRS
|
|
D
|
N
|
|
ED
|
1, -7
|
7, -10
|
|
RD
|
2, -1
|
8, -10
|
a. What is Mr. Crane's dominate strategy? What is the IRS's dominate strategy. (Note: it is possible that they don't have a dominate strategy)
b. What is the equilibrium outcome?
Now the IRS decides to increase the punishment fines for taxpayers caught engaging in tax evasion. As a result, the payoff matrix becomes:
|
|
Mr. Crane
|
|
IRS
|
|
D
|
N
|
|
ED
|
2, -7-0.8X
|
7, -10
|
|
RD
|
3, -1-0.5X
|
8, -10
|
c. The IRS sets the value of X at 10. What is Mr. Crane's dominate strategy? What is the IRS's dominate strategy? Is there an equilibrium to this situation, and if so, what is that equilibrium? (Note: it is possible that they don't have dominate strategy)
d. If the IRS wants to guarantee Mr. Crane will never engage in tax evasion, what is the minimal value of X that will achieve this outcome?
5. Oligopoly
There are only two companies, Bigsoft and Smallsoft, in the computer-software market. The demand function for software in this market is P=10-Q, so the MR is MR=10-2Q (why?). The MC is constant and can be expressed as MC(=ATC) is 2.
a. Two companies try to coordinate to set quantity and price like a single-price monopoly market. After they set this profit maximizing price and quantity the two firms plan to share the resultant profit fifty-fifty. What is the profit for each company if they both successfully adhere to this plan?
b. However, one company, say Smallsoft, deviates from this coordination, and sets its price equal to $4. What is the profit for Smallsoft? What is the profit for Bigsoft? (Hint: No one wants to buy higher price goods.)
c. Both Smallsoft and Bigsoft set price equal to $4, and then share the total market profit fifty-fifty. What is the profit for each company?
d. Both firms has two choices they face: they may either charge the joint monopoly price (found in part (a) or they can set their price equal to $4. Create a payoff matrix representing these choices for Smallsoft and Bigsoft. What is the dominant strategy for this game? What is the outcome of this game?
e. Explain the intuition for your answer in part (d).