1. to incorporate heterogeneous state price densities across investors. Good market clearing (21) together with investors' optimality (18)-(19), imply (26)-(27), where we have substituted ?(t) = y 1 ? 1 (t)/y 2 ? 2 (t);Appendix: Proofs Proof of Proposition 1. This is a variation on Karatzas et,1990
2. A(t)/A i (t). Then, using ?c 1 (t)/??(t) + ?c 2 (t)/??(t) = 1 the last equality in (33), differentiate c i in (29) twice with respect to ? and use ?c i (t)/??(t) = A(t)/A i (t) to obtain ? 2 c i (t)/?? 2 (t) = (A(t)/A i (t)) 2 P i (t) ? (A(t)/A i (t))P (t). Then, using ? 2 c 1 (t)/?? 2 (t) + ? 2 c 2 (t)/?? 2 (t) = 0 (implied by good clearing);differentiate c i in (29) with respect to ? to obtain ?c i (t)/??(t) =