3. (This problem is due to Dave Kreps.)Suppose there are two goods (X=R2+) andthe consumer has a vNM utility index over consumption bundles ofv(x) =f(x1+x2)wheref:R→Ris a strictly increasing function. Letp= (1,3) andp0= (3,1). Letq=.5p+.5p0= (2,2). In Regime 1, there is a 1/2 probability that the realized pricewill bepand a 1/2 probability that the realized price will bep0, so her expected utilityis.5 maxx∈Bp,wf(x1+x2) +.5 maxx∈Bp0,wf(x1+x2).In Regime 2, the price isqwith certainty, so her expected utility ismaxx∈Bq,wf(x1+x2).Prove that, for any strictly increasingf, the consumer is expected to be better off inRegime 1.
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