Two individuals (i ∈ {1, 2}) work independently on a joint project. They each independently decide how much eort ei they put. Eort choice has to be any real number between 0 and 1 (e_{i} ∈ [0, 1] not just 0 or 1). The cost of putting an amount of effort e_{i} is n e^{2}_{i}/2, where n is a parameter greater or equal than 2. If individual i puts effort e_{i}, then he succeeds with probability e_{i} and fails with probability 1 - e_{i}. The probability of success of the two agents are independent; this means that both succeed with probability e_{1}x e_{2}, 1 succeeds and 2 fails with probability e_{1} x(1 - e_{2}), 1 fails and 2 succeeds with probability (1 - e_{1})e_{2}, and both fail with probability (1 - e_{1}) (1 - e_{2}).
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 aected by the action of the other. However, individuals choose the level of eort that maximizes their own expected utility (benet minus cost of eort).
(a) Write down the expected utility of individuals 1 and 2 (note that the utility of 1 depends on the eorts of 1 and 2 and the utility of 2 depends on the eorts of 1 and 2). [Hint. The expected benet 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 eort. Find the expected utility of each individual in equilibrium (use the rst-order condition and make sure that the second-order condition is satised). Suppose that a benevolent dictator can choose the level of eort that both individuals must exert. He chooses the eort levels that maximize the sum of the expected utilities of both agents (these eorts are also called socially optimal levels).
(c) Write down the maximization problem of the benevolent dictator.
(d) Find the eort levels that the dictator imposes on each individual (use the rst-order condition and assume that the second-order condition is satised). Find the expected utility of each individual.
(e) Compare the eort level and nal utility of each individual in the cases of Nash Equilibrium (selsh individual maximization) and benevolent dictatorship.