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José MARIO CHIZZOTTI Q1 FIRST WILL SHOW 95 = = V THIS IMPLICATION IS OBVIOUS FIX AND FOR ALL E I = H E = SINCE 9s AND ARE PROBABILITY DISTRIBUTIONS OVER I = = = SINCE WAS ARBITRARY THIS FINISHES THE = NOW, USING BAYES FIND THE EXPRESSION FOR THE POSTERIOR = H = P(s') = V I So, IF 10 FOR ACC WE CAN DIVIDE BOTH SIDES BY = HENCE, GIVEN THE RESULTS ABOVE of = WE HAVE THE FOLLOWING TERIZATION OF THE EQUIVALENT s' = WHICH CLEARLY ONLY DEPENDS ON THE INFORMATION STRUCTURE I 4(s) So WE CAN CONCLUDE THAT THE EQUIVACENT CLASS AS DEFWED DOES NOT DEPEND ON THE PRIOR BELIEF. 1Q2 WE WANT TO SHOW THAT 6 NORE INFOR THAN IN THE SENSE OF a's = SUCH THAT a MAXIMITES THE EXPECTED UTICITY IT OF THE SUPPOSE DECISION MAKER A2 ON A UNDER INFORMATION AND a MAXIMIZES UNDER INFORMATION σ¹. THEN = 2 AND 6 IMPLY MAX ,0) > MAX E a,s THE C.H.S. MAXIMAND ON THE WHOLE SET IT SINCE a IT IN THE SUBSET THE SAME BEING TRUE For THE ALSO IN R.H.S. AND a', HENCE > FOR ALL (A, u, P) (SINCE A WAS WHERE V THE BAYSEAN DECISION MAKER MAXIMUM UTILITY (WHEN IN THE WHOLE SET SINCE THIS ONE OF THE OF 6 BEIN 6 INFORMATIVE THAN IN THE SENSE OF BLACK WE ARE DONE WITH THIS PART OF THE USING THE SAME OF INFORMATIVENESS IN THE SENSE OF BLACKWELL AS ABOVE, WE HAVE: MAX > E ) = WE WANT TO SWOW THAT IF IN ANY OF A} WITH ONLY two ELEMENTS INSTEAD WHOLE SET ,WE GET THE INEQUALITY ABOVE FOR AND LET = THEN la LINEAR FUNCTION OF P so = MAX a P(O) THE MAXIMUM of A FINITE LINGAR FUNCTIONS AND so IT CAN REPRESENTED AS A FINITE SUM WHERE AND IS THE MAXIMUM LINEAR FOR THEN ) 2Q3 a) IF EFFORT THE PRICIPAL CAN CHOOSE BOTH AND e to PROFIT, THE PROBLEM IS GIVEN BY: MAX S.t. w(.) E - J = (*) FIXING e, THE PROBLEM BECOMES MAX - W IT = MIN (*) LET 230 BE THE MULTIPLIER ASSOCIATED to THE K-T CONDITIONS THE THE F.O.C. IS - =0 FOR ) ACC IT SINCE 1/2 >0 AND For ALL IT WE CAN REWRITE THIS AS = 2 15 CONSTANT IS SINCE U' DECREASING. ALSO IMPLIES THE CONSTRAINT (*) IS By THE COMPLEMENDARY SLACKNESS CONDITION. LET = THEN, BY THE RESTRICTION (*) 11 0 (3) WE - 0 = c(e) E - c(e) NOW CETS CALCULATE WHICH LEVEL EFFORT GIVES THE HIGHER EXPECTED UNDER = = 5/3 13 = 15/9 = UNDER =8/5 13 = 64/25 = 4/3 13 16/9 3Q3 6 THIS IN PROFIT FUNCTION WITH THE VALVES for UNDER MAX - = - IS 9 3 + 4 3 2 = 135 27 W(.) UNDER MAX - = -64 25 2 + 186 25 = 122 UNDER MAX w(.) = 2 3 + 74 9 27 42 So CAN SEE THAT C, EXPECTED THEREFORE THE OPTIMAL CONTRACT IF EFFORT IS OBSERVABLE is (e, = (e,, ALL IT b) SUPPOSE SATISFIES MAX - e). FOR SOME e IT L 8 2 - + 5 3 ) UNDER e' 11 ≥ 1/3 4 3 UNDER I WAS GOING to SHOW THAT THE SISTEM IMPOSED BY THIS CANNOT BE D IS NEVER OPTIMAL THE BUT DID NOT HAVE TIME 4Q4 THE EXPECTED UTILITY FROM bi FOR AGENT i MAX SINCE AGENT i RECIVES A PAYOFF OF IF HE WINS THE BID AND 0 WAVE EXPECTED UTILITY BE REWR ITEM AS > FOR ACC bi) by the RESULTS ORDER STATISTICS FOR UNFORM DIST. PEG > = n. THE PROBLEM OF BECOMES MAX n. F.O.C. - = 5SUPPOSE B.W.O.C THAT THERE A MECHANISM THAT IC BUT NOT IC, THEN W.C.O.G. THAT > SUCH - V ASSUME FIRST that RESULT IF - - AND THE MECHANISM IC, THEN > - t PROOF. SINCE THE MECHANISM - > - SUMMING THIS INEQUALITY WITH THE ONE WE ARE ASSUMING IN THE + [ ) WHERE I HAVE SUMMED AND SUBTRACTED FROM THE C.H.S. + it's CLEAR THAT IF THEN - WE ONLY HAVE to SHOW THIS INEQUAL TY. REARRAN GINC WE - SINCE > Oi BY ASSUMPTION THIS INEQUALITY is THEREFORE HAVE SHOWN RESULT BUT PROCEEDING THIS CAN ARRIVE AT SUCH THAT = WHICH > CONTRA DICTION to PROVE THE CASE WHEN Oi WE TAKE THE SAME 6