A potential house buyer (principal) hires a real estate broker (agent) to collect information about a house. The house has quality (1 E {15,11}. A high quality house delivers utility 1 to the principal, a low quality house delivers utility 1 (this is net of the price paid]. The prior is Pr(q = H ) = 1r. Both the agent's and principal's utility are quasilinear. The agent invests effort r: at cost (3(6) into observing a signal 3 E {QB}. The signal is informative with probability Pr(s qu H) Pr(s qu L) g + e =: 19(6) The signal provides 'hard' information, so the agent cannot lie about the value of the signal. The cost function (:(e) is increasing and convex, and obeys (1\"!(6) > D. To obtain internal Optima asslnne that (5(0) = U, (:"(U) = U and limenl (3(6) = 00. After observing the signal, the principal can choose to buy the house or not. If her decision to buy is independent of the signal, there is no reasOn to have the agent exert effort. Hence we assume the principal buys if s = G and does not bu},r if s = B. (a) Suppose the principal can observe the agent's effort choice. Show the welfare maximising e'ort satises the rst order condition c"(e) = 1. [Here, welfare is the sum of the agents and principal's utility]. Now, consider the second best contract, where r: is not observed by the principal. A contract consists of a wage ma 2 0 when the good signal is observed, and a wage 11.13 2 0 when the bad signal is observed [Note the limited liability constraint; there is no other (IR) constraint]. (b) Write down the agent's utility. Show the agent's Optinlal level of effert satises the rst order condition (mg m3)(27 1] = (5(6)