In the setting of Proposition 8.10, show that at equilibrium the optimal demand of every agent does not depend on the price of the risky asset.

Data From Proposition 8.10

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Proposition 8.10 In the context of the model described above (see Grossman [839]), there exists a fully revealing Green-Lucas equilibrium, characterized by the equilibrium price functional * : R R defined by -1 d+By-d) o(1 - p) *(y.....y)= rf rf (8.20) Proof In view of Proposition 8.8, to show that (8.20) defines a Green-Lucas equilibrium price functional of the economy, we first show that it corresponds to the equilibrium price functional in an artificial economy where every agent observes all the private signals y = 1,...,.). As a preliminary, recall that y is a sufficient statistic for the joint distribution of (....) conditioned on d. Hence, the equilibrium of the artificial economy where every agent observes the full set of private signals coincides with the equilibrium of an artificial economy where every agent only observes the sufficient statistic . In this artificial economy, the optimal demand of agent i at equilibrium is given similarly as in (8.17) by Edy-parf ai Var(dly) for every i=1,....I. where p* (V) denotes the equilibrium price corresponding to the artificial economy contingent on the realization = y}. Imposing the market clearing condition, this leads to the price functional Edy pa (y,y) = pa* (V) Var(dy) rf () a