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 H, (a) where a is the agent's effort. The agent has the following utility \"(HO0(a), where u is strictly increasing and strictly concave while '0 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 dene a xed mapping from output to wage, so the set of possible wages is W = {tub wz, . . ., wn} and the agent is paid :0,- when output y,- occurs. In this case, the principal chooses compensation scheme to maximize M: \"i (a) (y:- \"'0 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 y,- is observed, the wage is being drawn from a lottery with pay- outs {2011, wig, . . .,w,-m} Wlth probability distribution 1 > IP (wif) = Pfj > 0 and :1 mi = 1. In this case, the principal chooses compensation scheme to maximize in: \"a (a) (yr f: Paws) i=1 j=1 subject to the participation and incentive constraints. Show that it is never optimal to use randomized wages