Homework 5
Price discriminating monopolist
1. Suppose the market for a health drink has two classes of buyers. The demand function for the two classes are:
Class 1: Q = 10 - P/2
Class 2: Q = 32 - 2P
Suppose that the cost function for a monopolist operating in the market is TC = Q2.
a. Calculate the profit that the monopolist would make if a single price were set for both the classes. Assume that the monopolistic price is below $16. (You will need to carefully figure out the aggregate demand function.)
b. Suppose the monopolist acts as a third degree price discriminator charging different prices for the different classes of buyers represented in the above information. Find the price and quantity pairs that a price discriminating monopolist would set for the two classes.
c. Compare the total profits from price discrimination with the profits without price discrimination.
d. Find the price elasticity of demand for the two classes at the optimal price and quantity pairs that you found for each of them in part (b). How are the prices in the two market classes related to the corresponding elasticities of demand for these two classes? Is this relationship between prices and elasticity of demand intuitive?
Now suppose the cost function for the monopolist changes to TC = 4Q and the demands for the two classes remain the same.
e. Calculate the new price and quantity pairs for the two classes when the monopolist acts as a price discriminator. Compare the profits generated here with the profits generated in part (b).
f. Show that the relationship between price elasticity of demand and price for the two market classes that was observed in part (d) also holds true here.
Game Theory
2. Al and Ben are two bickering siblings. They detest each other and one has exactly the opposite preferences as the other. However having been left alone at home they decide to go to a restaurant where they will share a single meal. At the restaurant Al gets to choose the main dish and Ben the starter or appetizer. For simplicity assume that there are two main dishes, a chicken based one and another that is ham based. There are two starters, noodle soup and popcorn shrimp.
Al gets a utility of 2 utils from consuming chicken and soup; 1 util from chicken and shrimp, 0 utils from ham and soup and -2 utils from ham and shrimp.
Ben's utilities from the four pairs are exactly the opposite (negative) of Al's.
a. Write down a matrix showing the four pairs of choices and the utility pairs of both Al and Ben for each choice of a main dish and a starter.
b. What are the dominant strategies for Al and Ben?
c. Which combination of main dish and starter would Al and Ben agree on if they were to follow only their dominant strategies? Is there more than one combination that is a possible solution to this problem?
d. Is there a main dish and starter combination that makes both Al and Ben better off than their common dominant strategy pair? If yes name it, if no then show that none of the other pairs make both of them simultaneously better off.
e. Now suppose the utilities of Al and Ben are just as before except for Ham and Soup. Al now gets a utility of 3 from this pair. Ben still detests Al and has utilities exactly opposite of Al's. Does there exist any dominant strategy for Al and Ben?
Monopolistic Competition
3. Often monopolistic competition is viewed as a market structure similar to perfect competition with one of the assumptions of perfect competition being relaxed. Out of the following assumptions about perfect competition which one is altered for monopolistic competition?
a. There are lots of buyers and sellers
b. There is perfect information
c. Products are homogeneous or identical across all producers
d. There is free entry and exit of firms
4. The demand for a good is given by P = 100 - 2Q and the cost function is given by TC = 20 + 12Q.
a. Suppose there is a monopolist operating in this market. What is the profit realized by the monopolist?
b. Now suppose that the market is characterized by monopolistic competition. Describe what will happen in the long run to the profits of the firms and the demand function. In particular draw two separate graphs, one showing the monopolistic price and quantity and the other showing long run equilibrium if the market is monopolistically competitive and is in long run equilibrium (You need not compute exact numbers, just show how the ATC will look like and how a possible demand function will look like in the long run for monopolistic competition.)