Two individuals (i ∈ {1, 2}) work independently on a joint project. They each independently decide how much e ort ei they put. E ort 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 e ffort ei is n e2i/2, where n is a parameter greater or equal than 2. If individual i puts e ffort 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 e1x e2, 1 succeeds and 2 fails with probability e1 x(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 payo of 1. If none of them succeeds, both individuals get 0. Therefore, each individual is a ected by the action of the other. However, individuals choose the level of e ort that maximizes their own expected utility (bene t minus cost of e ort).
(a) Write down the expected utility of individuals 1 and 2 (note that the utility of 1 depends on the e orts of 1 and 2 and the utility of 2 depends on the e orts of 1 and 2). [Hint. The expected bene t of 1 is the probability that 1 and/or 2 succeed times the payo if 1 and/or 2 succeed plus the probability that both 1 and 2 fail times the payo if both 1 and 2 fail.]
(b) Find the Nash equilibrium of this game, that is, the optimal level of e ort. Find the expected utility of each individual in equilibrium (use the rst-order condition and make sure that the second-order condition is satis ed). Suppose that a benevolent dictator can choose the level of e ort that both individuals must exert. He chooses the e ort levels that maximize the sum of the expected utilities of both agents (these e orts are also called socially optimal levels).
(c) Write down the maximization problem of the benevolent dictator.
(d) Find the e ort levels that the dictator imposes on each individual (use the rst-order condition and assume that the second-order condition is satis ed). Find the expected utility of each individual.
(e) Compare the e ort level and nal utility of each individual in the cases of Nash Equilibrium (sel sh individual maximization) and benevolent dictatorship.