In Exercises 4.23 and 5.1 you are being asked to make up examples that have certain characteristics. In each case you just need to make up a payoff table for a two-player strategic form game that has these charactertics (i.e., you need to make up the numbers in the table). You don't need to make up a story on which the game is based.

Exercises 4.12 - 4.15 refer to the model of Bertrand price competition described prior to Exercise 4.7 and prior to 4.12: Assume there are two firms; the firm that charges the lower price serves the entire demand (the other firm makes no sales); if the firms charge the same price they divide the market equally; only whole dollar prices can be charged; production costs are zero; and the market demand curve is downward-sloping.

Two additional exercises on Bertrand competition that build on 4.12-4.15:

1. Suppose production costs are positive, and that the average per-unit cost of production is the same for each firm and does not increase or decrease with increased production ("constant returns to scale"). Retrace the argument in Exercises 4.12 to 4.15 to determine the IEDS prediction.

2. What if prices need not be whole dollar amounts, but can be in cents instead? Then what is the IEDS prediction?

1. There are two identical firms in a market, each with constant per- unit production costs of $20 per unit. The market demand is given by the equation P = 80 - 4Q, where P is the price per unit, in dollars, and Q is the number of units sold in the market. Determine the Cournot equilibrium quantities the firms will sell and the resulting market price.

2. Now suppose there are n firms in the market of Problem #1, each with the same $20 per-unit cost. Determine the Cournot equilibrium (firms' output levels, market quantity and price) as a function of n. For n = 2, 4, and 14 determine the consumer surplus, total profits (producer surplus), and deadweight loss. How do Q and P compare with the competitive values of Q and P if n is very large? (Hint: Since all firms are identical, you might guess that in a Nash equilibrium each would produce the same quantity, say q. A firm's reaction function contains the total production of the other firms, which would be (n-1)q if the all firms produce the same amount.)

3. Three firms produce an identical product, but at different costs: it costs Firm A $20 for every unit it produces, it costs Firm B $40 per unit, and it costs Firm C $80 per unit. The market demand is given by the equation P = 300 - Q, where P is the unit price, in dollars, and Q is the total number of units sold. (Since the firms' output is identical, they all sell at the same price.) These are the only three firms in this market. Determine the Cournot equilibrium production levels of the three firms and the resulting market price. (Hint: Any firm's reaction function is an equation containing each firm's output level. In firm 1's and firm 2's reaction functions try replacing q₃ with the right-hand side of firm 3's reaction function, which has only q ₁ and q₂ in it. This gives you just two equations with the two variables q₁ and q₂.)

4. There are only two firms producing natural spring water, Airhead and Bubbles. The firms draw their water from different springs (Airhead's water is "still" and Bubbles' is carbonated), so each firm has some "market power." Specifically, the demands for the two firms' waters are given by

where p_i denotes the price per gallon charged by Firm i and q_i denotes the resulting number of gallons that will be purchased from Firm i. Each firm can produce as much water as it chooses, costlessly. The firms compete with one another via price: each firm chooses a price to charge, taking as given the price the other firm is charging. Draw the two firms' reaction functions in a diagram with p₁ and p₂ on the axes. Determine the Bertrand equilibrium prices and quantities -- i.e., the Nash equilibrium when each firm, in making its own pricing decision, takes the other firm's price as given.

Two men share access to a common grazing area. Each man can choose to own either no cows, one cow, or two cows. Each cow's daily yield of milk, in quarts, depends on how many cows in total are grazing, as follows:

               Total cows grazing:          1   2   3   4

               Each cow's daily yield:      8   5   3   2

(a) What are the Pareto efficient (i.e., socially optimal) allocations of milk? What are all the patterns of cow ownership and transfer payments that will support these allocations?

(b) How many cows do you predict each man will own? Explain your prediction. Indicate, in particular, whether your prediction is some sort of equilibrium. If your prediction is not some sort of equilibrium, explain why you have predicted as you have.

Now suppose that 100 men have access to a common grazing area. Again, each man can choose to own either no cows, one cow, or two cows to provide milk for his family. The more cows the grazing land is required to support, the lower is each cow's yield of milk; specifically, each man obtains

(c) How many cows do you predict each man will own? Explain your prediction. Indicate, in particular, whether your prediction is some sort of equilibrium, and if so whether it is a unique equilibrium, and whether this equilibrium is one that would be likely to be reached quickly or only after a long period of time during which the men learn how one another behaves. If your prediction is not some sort of equilibrium, explain why you have predicted as you have.

(d) Assume that the men can make transfers of milk among themselves (in particular, that men with more cows can give milk to those with fewer cows to compensate them for agreeing to own fewer cows). Is your prediction in (c) Pareto efficient for the 100 men? If so, verify it. If not, find a Pareto efficient allocation of milk to the men that makes everyone strictly better off, and a pattern of cow ownership and transfer payments (in quarts of milk) that will support that allocation.