Question

Consider the following interaction between a single long lived firm, and a sequence of short-lived consumer. In each period t=1,2…, there is a single consumer who only lives for one period. The consumer must choose between the actions BUY and NOT BUY (i.e. B and N ). The firm must choose what quality level to provide, i.e. it must choose between H and L. The payoffs to the two parties are given by the following table, where v H >p>v L and p>c H>c L . a) Solve for the Nash equilibria of the above stage game.

Answer 1

The Nash equilibria of the stage game are (H, B) and (L, N) if vH > p-cH and vL-cL > 0, and (L, B) if vH < p-cH and vL-cL > 0.

Step-by-step explanation:

The Nash equilibria of the stage game can be found by examining the payoffs for each possible combination of actions from the firm and the consumer. In this case, the firm has two options, H (high quality) and L (low quality), while the consumer has two options, B (buy) and N (not buy).

If the firm chooses H and the consumer chooses B, the firm's payoff is vH and the consumer's payoff is p-cH.

If the firm chooses H and the consumer chooses N, the firm's payoff is 0 and the consumer's payoff is 0.

If the firm chooses L and the consumer chooses B, the firm's payoff is p-vL and the consumer's payoff is vL-cL.

If the firm chooses L and the consumer chooses N, the firm's payoff is 0 and the consumer's payoff is 0.

The Nash equilibria occur when no player can increase their payoff by unilaterally changing their strategy. In this case, the firm's best response to the consumer choosing B is to choose H if vH > p-cH, and to choose L if vH < p-cH. Similarly, the consumer's best response to the firm choosing H is to choose B if vL-cL > 0, and to choose N if vL-cL < 0.

Therefore, the Nash equilibria of the stage game are (H, B) and (L, N) if vH > p-cH and vL-cL > 0, and (L, B) if vH < p-cH and vL-cL > 0.