The Consumption - Based Capital Asset Pricing Model

Arrow-Debreu Existence Result Let K be a normed vector lattice with positive cone K+ (see Schaefer (1974)). We consider an Arrow-Debreu (pure exchange) economy 9'= (?, e') in K, with m agents described by preference relations > i on K+ and initial endowments e' E K+. Throughout, we make the usual assumptions on preferences: each >i is reflexive, transitive, complete, convex, continuous, and strictly monotone. We say that a (Hausdorfl) topology T on K is compatible if it is weaker than the norm topology and if all order intervals [0, x] = { y e K: 0 < y < x } are r-compact. We say that the preference relation >satisfies the forward cone condition at x E K+ if: (FC): There isaveK+, an E> 0 anda p>0suchthatx+Xv-z>-xwheneverx+Xv-zeK+, O < X < p, and llzll < Xe. The forward cone condition was introduced by Yannelis and Zame (1986), and is a variant of the backward cone condition ("properness") introduced by Mas-Colell (1986). The validity of the uniform version of the forward cone condition (that is, that (FC) holds at each x E K+, with v, E, p fixed) is equivalent to uniform properness, which was used by Mas-Colell to obtain the existence of competitive equilibria. However, the pointwise versions of the forward cone condition (which we use) and of properness are incomparable. An allocation (xi,. . . xm) E (K+)m is feasible for d' if E, (e' x') = 0. A competitive equilibrium is a feasible allocation (xl,. . ., xm) and a linear functional v on K such that, for all i, v xi < T * ei and x'>-1yi for any y .E K+ with q r.yi < i7 * ei. A feasible allocation (xl, ... , xm) is in the core CAPITAL ASSET PRICING MODEL 1293 if there exists no other allocation (yl,..., y i) and nonempty subset W of agents such that ? (e' y') = O and y' I xi for all i c-. Our abstract existence result is: THEOREM Al: Let K be a normed lattice, let r be a compatible topology on K, and let 4) be an Arrow-Debreu economy in K. Assume that: (a) each of the preference relations >-, is T-upper semicontinuous; (b) for each core allocation (xi,... xIn) for 4), each >-, satisfies the forward cone condition at x'; and (c) e = 2,e' is strictlypositive. Then 8has a competitive equilibrium (x,...,xxm; r), where i7 is a positive continuous linear functional on K. To prove Theorem Al, we first consider a restricted economy 4. Let K(e) be the order ideal K(e) = {xe K: 3s +R: Ixi ?se}, where I x =x+ + x-, and define a norm 11 Ile on K(e) by lixile=inf{s>O: Ixi sIse}. It is easily checked that, equipped with this norm, K(e) is a normed vector lattice. The economy 4) is simply the restriction of 8) to K(e). That is, 4) has the same agents (1. m}, with the same endowments el,. . ., em, and with the preferences obtained by restricting a, to K(e). The following lemma and its proof are slight variations of those in Zame (1987) as well as Aliprantis, Brown, and Burkinshaw (1987a). The general idea of working on order ideals of the consumption space, allowing one to naturally extend Bewley's (1972) approach, can also be found in Brown (1983) and in Aliprantis, Brown, and Burkinshaw (1987b). LEMMA A2: The economy 4 has a competitive equilibrium (xl..., xm; ii), where X7 is a positive linear functional on K(e) which is continuous with respect to the norm 11 Ile.PROOF: Let f denote the restriction of T to K(e). The feasible consumption sets for &e are the order ideal [0, e], which is r-compact (by assumption). Moreover, the endowment e lies in the interior of the positive cone of K(e). It follows9 from Theorem 2 of Zame (1986) that 4) has a quasi-equilibrium'0 (xl,..., x-; ST), where -6 is continuous with respect to the norm 11 lie. Strict monotonicity of preferences implies as usual that 'i is positive and that (xi,.. I xm; *F) is an equilibrium. Q.E.D. LEMMA A3: If (xi,... I x M; -7) is an equilibrium for 66 and if each of the preference relations >satisfies the forward cone condition at x', then 6 is continuous with respect to the original norm II on K(e). PROOF:11 Let v,, Ii, and pi be as in the forward cone condition for >i at x'. At the cost of replacing ?, by E,/2, we may replace v, with any vector v1' E K+ such that lIv, vI < E,/2. Since K(e) is dense in K, we may without loss of generality assume that vi E K(e) for each i. Set v =E,v, and E = min {e1..,}. We claim that IF V I,ff -YI < -IIYII, Y EK(e)Note first that (since 'i is positive) it suffices to establish this for each y c K(e)+ with IlYll < 1. Suppose that i *y > ST * V/E. For X > 0 small enough, Xy < e, so e-Xy E K(e)+. Since e =?,e' = E,x', the Riesz Decomposition Property (Schaefer (1974)) allows us to find vectors z, E K(e) such 9 Theorem 2 of Zame (1986) assumes that each of the endowments e' lies in the interior of the consumption sets, but this assumption is used only to obtain an equilibrium from a quasi-equilibrium (no monotonicity assumption is used there). 10A quasi-equilibrium is the same as an equilibrium, except for the substitution of the condition U(x') > U(x') * * x x' in place of the usual optimality condition. 11 This is the argument of the Price Lemma of Yannelis and Zame (1986). 1294 DARRELL DUFFIE AND WILLIAM ZAME that 0z, 6 x' and AAy =izi. Then e+ Av cAy= (xi+ Avj -z,). Since E2zi = EAy, it follows that Ilzill 6 E,jjAyjj for each i, and hence (by the forward cone condition) that x' + Av, -zi z>x' for each i. On the other hand, iF * (Av EAy) < O so . (x' + As, z1) < I *. xi for some i. This contradicts the equilibrium properties of (x',..., xm; if), so we conclude that f *y 6 (i v/E) jyjj for each y E K(e)+, and hence that * is continuous with respect to the 11 HI-norm on K(e), as asserted. Q.E.D. PROOF (of Theorem Al): By Lemma A2, the economy g has an equilibrium (xi,... xm; if). The equilibrium allocation (xi,... xm) is in the core of o; since all feasible allocations for of are actually in K(e)", the allocation (xi,..., xm) is in the core of 4. Since K(e) is dense in K, the continuity of Xi (Lemma A3) implies that so has a (unique) extension to a positive, continuous linear functional iT on K. Continuity of preferences implies that (xi,. . . xm; iT) is an equilibrium for 4. Q.E.D. An Application to the L2 Case with Additive-Separable Preferences We turn now to the precise result required. From now on, the commodity space L = L2(s2 X [0, T], .9, v). Let 4= (>,, e') denote the corresponding Arrow-Debreu complete markets economy on L, where >-, is represented by a utility function U'. THEOREM A4: Under condition (A.1) of Section 3 and the assumption that the aggregate endowment process e is bounded away from zero, 4 has a competitive equilibrium (xi.... xM; v), where 'r is represented by p E L+ as in relation (6) with p bounded and, for all i, if e' *0, then xi is bounded away from zero. PROOF: Let K denote L equipped with the norm 11 given by lixil = E(j )xt dt). As such, K is a normed vector lattice. It is easily checked that each of the preference relations is continuous in the 11 1l-topology. Let T denote the weak topology on K with respect to the topological dual K* of L. This topology T is compatible.'2 Moreover, since norm-closed convex sets are weakly closed, the preference relations >i are T-upper semicontinuous. Fix a measurable (predictable) subset A of Q2 x [0, T] and a real number a > 0. If y E K+ and y > a on A, then together with the Mean Value Theorem our assumptions imply that (13) U'(y) U'(y z) < E( Tz,dt) sup ui(a2t) Ote[0, T] whenever z E K+ is any vector in K+ supported on A and bounded above by a/2. Similarly, for any set H E. 9, h > 0, and y E K+ with y S h on H, we obtain (14) U'(y+X1A)Ui(y)1>Av(H) inf u,c(3h/2,t) E [O, T ic whenever 0 < A < h. There is no loss of generality in assuming that each e' is nonzero, and hence that U'(ei) > Ui(0) for each i. Suppose (xl,..., xm) is a core allocation for S. Since U'(x')> U'(e') > U'(0), there is a set A, c.9 with v(Ai) > 0 and a number a, > 0 with x' > a, on Ai. If some xi is not bounded away from zero, we can find for each h > 0 some He-9 such that v(H) > 0 and xi < h on H. Since e is bounded away from zero, say e > c > 0 for some c, we can assume without loss that xi > c/m on H for some agent 1. Applying (13) and (14) and our assumption that u,c(k, t) -+ oo as k -0, we conclude that agents j and I could exchange appropriate multiples of the commodity bundles 1H and 1A to effect a Pareto improvement provided h is small enough. Since this would be a contradiction, we conclude that each xi is indeed bounded away from zero. 12 On order intervals, this topology coincides with the weak topology on L with respect to L*; Alaoglu's Theorem implies that order intervals are compact in this topology (Schaefer (1974)). CAPITAL ASSET PRICING MODEL 1295 We now assert that the forward cone conditions hold at the points x. Xm. Since the x"s are all bounded below away from zero and bounded above on some set C E 9 with v(C) > 0, this follows immediately from (13) and (14) (taking vi = lc for each i). We now conclude from Theorem Al that d' has an equilibrium (xi I ., xm; f7). Since (xl,... I xm) is a core allocation, xi is bounded away from zero for each i. Moreover, since v is 111ll-continuous, the Riesz representation theorem implies that ST is represented by a positive bounded function p; since v is a finite measure, p is in L. Q.E.D. APPENDIX B: ITO'S LEMMA We have used Ito's Lemma repeatedly, and it may be best to state it formally for the record in the setting of Ito processes. If dXt = ,ux(t) dt + ux(t) dBt is an Rn-valued Ito process for some n, and f: R"n x [0, T] -D R is C2 (when extended to an open set), then the process Ydefined by Yt =f(Xt, t) is also an Ito process with stochastic differential dYt =.uy(t) dt + ay(t) dBt defined by ay(t)= fx(Xt, t)ax(t) and ALy(t) =fx(X , t)*x (t) +ft(Xt, t) + trace [x(t)Tfxx(Xt, t)Ox(t)], where the vector fx( Xt, t) and matrix fxx( Xt, t) denote the first and second partial derivatives of f with respect to the X arguments. (The same formula applies under weaker conditions.) We actually use two special cases of this result in the paper. One is the case of n = 1, Xt = et, and Yt =f ( Xt, t) = U.c(et, t). In this case, ay(t) = uxcc(et, t)oe(t). The second application of Ito's Lemma in the body of the paper is the case of n = 2, XtM) -P,, = pt, and Y, =f(Xt, t) = Xt1lXt2). In the latter case, application of Ito's Lemma leads to Ay (t) =PtAv (t) + VtJp (t) + UV(t) * op(t). Details and extensions can be found, for example, in Lipster and Shiryayev (1977). APPENDIX C: SPANNING WITH REAL PRIMITIVE SECURITY DIVIDENDS Just as with Arrow's (1953) model of security markets, we have found it convenient in this paper to define the primitive, exogenously given, security dividends in (nominal) unit of account, not in (real) consumption numeraire terms. This allows an easy development of spanning in continuous-time, based on primitive assumptions on the nominal dividend process D. In a general multi-commodity model, there is little alternative, as shown by Hart (1975), since equilibria need not exist in general. (Although Hart worked in a discrete-time setting, the fact that the span of markets may collapse discontinuously at certain endogenous pot price processes does not disappear in continuous-time; it only becomes more difficult o deal with.) In the single-commodity setting of this paper, however, one can define spanning assumptions directly on a real risky security dividend process y = (Yy.yN), provided there is also a nominal numeraire security DO as defined in Section 3. In this case, the primitives of an economy are J%= ((Q, -F, F, P),(Ub, e'),(Y, D?)). Suppose, for N > K, that Y is an RN_valued Ito process with stochastic differential representation dYt = uy (t) dt +ay (t) dB, and that Mt = Jjtry(s) dBs is a martingale generator (as defined in (A.3)). For this, it is basically enough that the essential infimum of the rank of ay is maximal, and therefore qual to the dimension K of the Brownian motion B. As an alternative to (A.3), this is in principle enough to demonstrate the existence of an equilibrium satisfying all of the results of the paper. To guarantee this, we need only complete the proof of Step (C) of Theorem 1, as follows. We take static equilibrium (xl,..., xm; in) as given by Appendix A, Theorem A4, where if has the representation ff . c = E(f JTptc dt), for p bounded and bounded away from zero. The corresponding nominal dividend process D' is defined by dDt' =pt dYt'. By Girsanov's Theorem (under technical regularity conditions on .&y and ay(t) given in Lipster and Shiryayev (1977)), there exists a new probability measure Q uniformly equivalent to P and a Brownian motion B relative to Q under which dDt =pAay(t) dBt. Since p is bounded above and below away from zero, the spanning assumption (A.3) is therefore satisfied under the new measure Q. Let S, = EQ[DT Dt [JFt define the nominal security price processes, where EQ denotes expectation under Q. In order to define a suitable 1296 DARRELL DUFFIE AND WILLIAM ZAME new consumption spot price process pl, first define the density process Zt= [dP t ], t[O,T]. Now let Pt =Pt/Zt for all t. By Lemma 4.1 and Proposition 4.1 of Duffie (1986), there exist trading strategies (61. f) such that ((S,p),(xl,61). (xm,Gm)) is an equilibrium for e'. In order to recover the pricing formula (6) on which the rest of the paper is based, we use the fact that, for any integrable random variable W,