ps2(404)-2

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

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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 ...
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