'% EXERCISE* 3.18 (Completeness via dimension). In the single-period model we have been considering, assume that there is no arbitrage, that P{w} > 0 holds for all m E Q, and that excess returns are non-degenerate in the following sense: if 2161' gt (R'I (w) r) = 0 holds for some 3; 6 RI and all u) E Q, then y = 0 for all i E I. Establish the following: (1) IfXD(1+ 'r) + 2,301, (saw) 3(1 + r)) = 0 holds for some (X0, 6') E R X R1 and all a; E Q, then X3 : 0 and Eli : 0 for all i E I. (Hint: Use no arbitrage to show that X0 = 0; then use non-degeneracy or excess returns to show 6 E 0.) (2) If the number of states in .Q is equal to the number of traded (riskless and risky) assets, i.e., if |.Q| = 1 + |I|, show that the market is complete. (Hint: Use part (1) and linear algebra.) (3) Conversely, if the market is complete, show that |Q| = 1 + |I |. (Hint: This is again linear algebra, you will have to match the dimension ll of the space of all random variables and the dimension 1 + |I| that is spanned by the assets.)