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 ﬁrstorder condition and make sure
that the secondorder 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 ﬁrstorder
condition and assume that the secondorder 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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 Spring '11
 CARRILLO
 Economics, Game Theory, Utility, Juan D. Carrillo

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