Question

1. Ann and Bill have formed a business partnership. They each have to decide how much effort to put into the business. Let x denote the amount of effort Ann puts into the business and let y denote the amount of effort Bill puts into the partnership. The profit of the business partnership depends on the amount of effort Ann and Bill put into the business. It is given by 2x + 2y + 0.5xy hundred thousand dollars. Ann and Bill will split the profit equally however they will separately incur the cost of their own effort. The cost of Ann’s effort is 0.5x². The cost of Bill’s effort is 0.5y². Effort is also measured in hundred thousand dollars.

a. What is Ann and Bill’s payoff function?

b. If Bill puts in an effort level of y = 1, what is Ann’s best response?

c. If they choose their effort levels without knowing their partner’s effort level, what is the Nash equilibrium of this game?

d. What if Ann commits to an effort level first? What are the backward induction effort levels? Does this game have a first mover or a second mover advantage?

Answer 1

a)

A and B's payoff function is given as

b)

The best response for each party given the effort of others is computed from the first-order condition (FOC) of each agent's profit maximization. Therefore, the FOCs are

............ The best response function of A

Given y=1,

c)

The best response of B is

Both players have symmetric best responses, then at equilibrium x=y

d)

If A announces its effort firm A is the first mover, it incorporates B's best response in its profit and maximizes; such that

The FOC of maximization is

In this case, the profit of A is 2.29275 and that of B is 2.7153. Then there is no first mover advantage in this game.