ps2(404)-2

ps2(404)-2 - ECON 404 - Pr. Juan D. Carrillo Assignment 2...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ECON 404 - Pr. Juan D. Carrillo Assignment 2 Problem 1. Job cooperation Two individuals (i ∈ {1, 2}) work independently on a joint project. They each independently decide how much effort ei they put. Effort choice has to be any real number between 0 and 1 (ei ∈ [0, 1] not just 0 or 1). The cost of putting an amount of effort ei is n e2 /2, where n i is a parameter greater or equal than 2. If individual i puts effort ei , then he succeeds with probability ei and fails with probability 1 − ei . The probability of success of the two agents are independent; this means that both succeed with probability e1 × e2 , 1 succeeds and 2 fails with probability e1 × (1 − e2 ), 1 fails and 2 succeeds with probability (1 − e1 ) × e2 , and both fail with probability (1 − e1 ) × (1 − e2 ). If at least one of the individuals succeeds then, independently of who did succeed, both individuals get a payoff of 1. If none of them succeeds, both individuals get 0. Therefore, each individual is affected by the action of the other. However, individuals choose the level of effort that maximizes their own expected utility (benefit minus cost of effort). (a) Write down the expected utility of individuals 1 and 2 (note that the utility of 1 depends on the efforts of 1 and 2 and the utility of 2 depends on the efforts of 1 and 2). [Hint. The expected benefit of 1 is the probability that 1 and/or 2 succeed times the payoff if 1 and/or 2 succeed plus the probability that both 1 and 2 fail times the payoff if both 1 and 2 fail.] (b) Find the Nash equilibrium of this game, that is, the optimal level of effort. Find the expected utility of each individual in equilibrium (use the first-order condition and make sure that the second-order condition is satisfied). Suppose that a benevolent dictator can choose the level of effort that both individuals must exert. He chooses the effort levels that maximize the sum of the expected utilities of both agents (these efforts are also called socially optimal levels). (c) Write down the maximization problem of the benevolent dictator. (d) Find the effort levels that the dictator imposes on each individual (use the first-order condition and assume that the second-order condition is satisfied). Find the expected utility of each individual. (e) Compare the effort level and final utility of each individual in the cases of Nash Equilibrium (selfish individual maximization) and benevolent dictatorship. 1 (f) Interpret the results and conclude (this is the most important part of the exercise). 2. Job competition Consider exactly the same problem as before except for one thing: agents now compete (instead of cooperate). This means that the benefit of each agent is: 1 if he succeeds and the rival fails, 1/2 if he succeeds and the rival also succeeds, and 0 if he fails (no matter what the rival does). The cost of effort for agent i is, just like before, n e2 /2 with n ￿ 2. i Answer questions (a)-(b)-(c)-(d)-(e)-(f) in the new case. (g) Overall conclusion: compare your answer (f) in problem 1 with the answer (f) in problem 2. Interpret and conclude. 2 ...
View Full Document

Ask a homework question - tutors are online