6 [Taken from MWG 19.C.4] Consider an exchange economy with a single consumption good, twostates, and two consumers. Each consumer's preferences are represented byu" (x*) = nx + Txandu" (x*) = 7 log x} + n log x²,where x, is the amount of good consumed in state s by consumer i, and 7 is the subjective prob-ability of consumer i for state s. Since consumer a's (b's) Bernoulli utility function is assumedlinear (concave), a is risk-neutral (risk-averse, respectively). The total endowments of this econ-omy also depends on the state: w = (w,w2). Assume that each consumer is endowed with itshalf: w" = w = }(wr,w2).(a) Suppose that the two consumer share the same belief about the states:nf = n = T1T = n = 12.andShow that at an interior Arrow-Debreu equilibrium consumer b insures completely.(b) Suppose that the two consumers have different beliefs about the states. Show that at aninterior equilibrium consumer b will not insure completely. Which is the direction of thebias in terms of the differences in subjective probabilities? Argue also that consumer a willnot gain from trade.

According to the economic theory, the "Arrow-Debreu equilibrium," all markets in an economy work…

(a) Suppose that the two consumer share the same belief about the states: 7₁ = 7² = 7₁ and 7² = 7² = 7₂. Show that at an interior Arrow-Debreu equilibrium consumer b insures completely. (b) Suppose that the two consumers have different beliefs about the states. Show that at an interior equilibrium consumer b will not insure completely. Which is the direction of the bias in terms of the differences in subjective probabilities? Argue also that consumer a will not gain from trade.