Question

Suppose there are two firms that sell bubble gum: Bazooka Joe and Gatling Gordon. Each firm faces a marginal cost of 1 per unit. There are 100 customers, who will buy one piece of gum each at the lowest price, (provided that price is no more than $1). That is, if Bazooka Joe’s price is greater than Gatling Gordon’s, Bazooka Joe gets no customers, if Bazooka Joe’s price is less than Gatling Gordon’s, Bazooka Joe gets all of the customers (provided Bazooka Joe’s price is less than or equal to $1). If they have the same price less than or equal to $1, they each get 50 customers. Bazooka Joe and Gatling Gordon cannot charge any price they want. They can only charge prices 0 cents, 1 cent, 2 cents, 3 cents, etc. That is, they cannot charge prices with decimals (e.g. they cannot charge 1.3 cents or 2.4 cents). Suppose the two firms set price once and simultaneously. (a) Is it a Nash equilibrium for each firm to set a price equal to marginal cost? Explain. (b) Are there any other Nash equilibria in pure strategies? If so, identify them. If not, explain why not. (c) Now suppose instead that Bazooka Joe has a marginal cost of 1 cent per unit, but Gatling Gordon has a marginal cost of zero per unit. Identify all pure strategy Nash Equilibria, and explain your answer.

Answer 1

a) Yes, it is a Nash equilibrium for each firm to set the prices equal to marginal cost. Because if one of the firm sets the price greater than marginal cost, then that firm will lose all the customers because another firm is selling at a lower price. On the other hand, no firm would set a price lesser than marginal cost beacuse setting a price lower than 1 would mean incurring losses on the sale of bubble gum. Hence, no firm moves to any other price given that the other firm chooses marginal cost. Therefore, choosing marginal cost by both firms is a nash equilibrium.

b) This is an example of bertrand price competition. In this case, no other point than the marginal cost is a nash equilibrium. Price above marginal cost is not an optimal move for any firm because the other firm can attract all the customer base by choosing a lower price and this price war among the two firms will lead them to choosing marginal cost. Any price less than marginal cost say 2 cents will let the firm get the customer base but will earn losses on the sale. Hence no firm will move below marginal cost. Therefore, marginal cost is the only pure strategy nash equilibrium in this case.

c) If Bazooka Joe has a marginal cost of 1 per unit and Gatling Gordon has a marginal cost of 0 per unit. Then setting a price above 1 would give profits to both the firms, but any of the firms can reduce the price slightly and earn all the customer base. If Bazooka Joe chooses price equal to marginal cost i.e 1 per unit, Gatling Gordon can reduce the price slightly and still earn profits as the marginal cost is 0 for them. Hence choosing marginal cost for Bazooka Joe is no more optimal. Any price less than 1 will induce losses for Bazooka joe and hence they wouldn't want to choose that. In his case then, there is no pure strategy equilibrium and only mixed strategy nash equilibrium exists for this game