4. Randomized Wages (20 points) Consider a standard principal-agent model. The set of possible outputs is Y = {y1,y2, . . . ,yn} . The probability of output y,- is 713 (a) where a is the agent's effort. The agent has the following utility u(w)v(a), where u is strictly increasing and strictly concave while 7} is strictly increasing and strictly convex. The agent's outside option is normalized to 0. The risk-neutral prin- cipal wants to design a compensation scheme that maximizes prot. There are two possible schemes. One option is for the principal to define a xed mapping from output to wage, so the set of possible wages is W 2 {01, 102,. . . , wn} and the agent is paid w,- when output y,- occurs. In this case, the principal chooses compensation scheme to maximize M: 715i (a) (312' wi) i=1 subject to the participation and incentive constraints. The other option is for the principal to dene a mapping from output to a lotteryrandomized wages. In essence, if output 3/,- is observed, the wage is being drawn from a lottery with pay outs {w1-1,w,-2,...,w,-m} with probability distribution 1 > IP (wij) 2 pg > 0 and 11-":1 Pij = 1. In this case, the principal chooses compensation scheme to maximize i 7r;- (a) (y.- i Piiwif) i:1 j=1 subject to the participation and incentive constraints. Show that it is never optimal to use randomized wages