micro theory

3. (20 points) PrincipalAgent The probability of winning a soccer match is determined in part by the level of effort of the players. Assume that the probability of winning a match is given by the function: Pm (e) = 26 62 Where 6 is the level of effort by the average player. 3 is continuous and bounded by the [0, 1] interval. If a team wins a match; the owner of the team will earn $1, 000; if the match ends in a loss or a draw then the owner earns nothing. Then, the expected revenues are equal to R(e) = 1, OOUPw(e). For silnplicity assume that the only cost for the owner is to pay the salary of 11 players. Consider the following wage arrangements: 1. w : $150 (Regardless of the outcome of the game) 1 2. w 2 112(8) 3. w = R(e) 900 The utility of the average player is given by 113(8) 2 w 2508 (a) (10 points) Find the optimal level of effort for each of the proposed wage arrangements. (b) (5 points) Find the prot level for each different contract. (c) (5 points) What's the optimal contract for the owner