Consider the following extensive form game, which ends with an outcome y {0, 1} is...

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Consider the following extensive form game, which ends with an outcome y € {0, 1} is realized: • Player 1 (The Principal) offers a payment to player 2 as a function of y, w(y). • Player 2 (The agent) observes the contract offer and decides whether to accept it, or reject it and obtain a payoff of 0. If the agent accepts it, they exerts effort e € [0, 1], with Ply = 1] = e Both players obtain a payoff of 0 if the agent rejects the offer. If the agent accepts it, payoffs are: 3 Principal: Zy-w(y), Agent: w(y) - e² 1. Calculate the effort level which maximizes the sum of the principal's and the agent's utility. (This is referred to as the first best) 2. Argue that the unique subgame perfect equilibrium involves the effort level from Part (1). What is w(1) and w(0)? Hint: You may want to first give a condition on w(1) and w(0) such that the subgame perfect equilibrium involves this effort level. Then argue further that in fact this condition can be satisfied and that the principal cannot do any better. Another hint: It may be helpful to note that this is a version of the ultimatum game. Consider the following extensive form game, which ends with an outcome y € {0, 1} is realized: • Player 1 (The Principal) offers a payment to player 2 as a function of y, w(y). • Player 2 (The agent) observes the contract offer and decides whether to accept it, or reject it and obtain a payoff of 0. If the agent accepts it, they exerts effort e € [0, 1], with Ply = 1] = e Both players obtain a payoff of 0 if the agent rejects the offer. If the agent accepts it, payoffs are: 3 Principal: Zy-w(y), Agent: w(y) - e² 1. Calculate the effort level which maximizes the sum of the principal's and the agent's utility. (This is referred to as the first best) 2. Argue that the unique subgame perfect equilibrium involves the effort level from Part (1). What is w(1) and w(0)? Hint: You may want to first give a condition on w(1) and w(0) such that the subgame perfect equilibrium involves this effort level. Then argue further that in fact this condition can be satisfied and that the principal cannot do any better. Another hint: It may be helpful to note that this is a version of the ultimatum game.

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1 Calculate the effort level which maximizes the sum of the principals and the agents utility This is referred to as the first best The effort level which maximizes the sum of the principals and the a... View the full answer