JOURNAL OF ECONOMIC THEORY 49, 379-382 (1989)

uilibrium with Incomplete Ordered Preferences

JAN WERNER

Department of Economics, University of Mirmesota, Minneapolis, Minnesota SS455

Received October 12, 1988; revised December 13, i988

III this note we prove the existence of Radner equilibrium ~6 plans, prices, and price expectations in an exchange economy with incomplete financial markets without monotonic or ordered preferences. Journal of Economic Literature Classification Numbers: 021, 313. cj 1989 Academic I’xss. Inc.

The purpose of this note is to prove the existence of Radner equiIibri~m of plans, prices, and price expectations (Radner [6]) in an exchange economy with incomplete financial markets under weak conditions on consumers’ preference relations. All the previous existence results in the case of markets for tinancial assets (i.e., assets with exogenously specific money returns) require monotonicity of consumers’ preferences (see Werner [9], Cass [ 11, Duffie [2]). §trict monotonicity was used proving generic existence of equilibrium with markets for comma futures (see Duftie and Shafer [3], Magi11 and Shafer [5]), since the results rely on the regular economy arguments. Evidently? mouotonicity is a restrictive assumption in the context of uncertainty.

The result of this note allows preferences to be non-monotonic and even non-complete and non-transitive. Assumptions are essentially the same as in the case of an economy with complete markets (see §hafer [X] which is also the main reference for the idea of the proof). 0nfy the non-satiatio assumption is strengthened to non-satiation in every state of natures also prove (Remark 2) a variant of the existence result which allows “zero-probability” states, i.e., states such that consumer’s satisfaction does not depend on consumption conditional on this state.

An economy extends over two time periods f = 0, I7 and there are s possible states of nature s = I, . . . . s in period I. There are I. commodities for consumption at date 0 and in each state at date I. There are iV ~~a~c~a~ assets for trade at date 0. Asset n, B = 1, . . . . N$ yields ?Jn units of account in states at date I. There are m consumers m the economy. Each consumer

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Copyright c 1989 by Academic Press. lx AI1 rights of reproductmn in my form reserved

380 JAN WERNER

has a consumption set Xi c R’, where I = L(S + 1 ), an initial endowment vector e’~ Xi, and a preference relation P’ c Xix Xi. (xi, z’) E Pi means “xi is preferred to zi,” and is denoted by xi E P’(z’). Each xi E Xi is of the form xi = (x6, Xi) . ..) xi), where xbe RL (xt E RL) denotes date 0 consumption (date 1 state s consumption, respectively). Vectors of commodity prices and asset prices will be denoted by p = (pO, pl, . . . . pS) E R’ and qE RN, respectively. y = (yl, . . . . yN) denotes a portfolio of N assets. Let Bi(p, q) denote consumer’s i budget set in the incomplete markets economy for given prices (p, q), i.e., B&J, q) = {(xi, vi) E Xi x RN: pox; + qyi 6 poei, psxf d r, y’+ pseL, s = 1, . . . . S}. Let A be the set of attainable consumption- portfolio allocations, i.e., A = {(x, y) : x = (x1, . . . . xm), y = (y’, . . . . y”), xi E Xi, yi E RN, Cy= I xi = X7= 1 ei, Cy= 1 yi = O}. Let 2i be the coordinate projection of A on Xi.

Radner equilibrium is an attainable allocation (X, jj)~A and a price system - -

(p, q) E R’x RN such that (Xi, j’)~B,(p, q) and there is no (xi, y’) E Bj(p, 4) with xi~ Pi(?), i= 1, ,.., m.

The following theorem is the main result of this note:

THEOREM 1. Let an economy satisfy the following conditions for each i = 1, ,.., m:

(1) Xi is bounded below, convex, and closed,

(2) ei E int Xi, (3) P’ is open in Xix Xi,

(4) for every X’E Ti, xiE bd Pf(x’) for every s = 0, 1, . . . . S, where Pi(xi) = {z E R’: z E P’(x’) and z,, = xf, for s’ #s, s’ = 0, 1, . . . . S},

(5) for every xi E Xi, xi $ co P’(x’).

Then there exists an equilibrium.

ProoJ: We shall convert the economy into an abstract economy and apply the existence result in Shafer and Sonnenschein [7]. Let B,= {pscRL: IIp,ll<~}, s=l, . . . . S,~~~B~={(P,,~)~R~XR~:II(P~,~)II~~}. Let r be an abstract economy with m + S+ 1 individuals with the following characteristics: For i= 1, . . . . m, individual i has a non-empty, convex, and compact choice set Kix Yi (to be defined later), a constraint correspondence Cj: fl,“=, B, + Ki x Yi (where n,“= ,, B, denotes Cartesian product) defined by Ci(p, q) = {(x’, yi) E Ki x Yj: pox; + qyi < Poe; + l- Ilh 4)/l, ~~~f~r,~‘+p,ef+ 1 - IIPJ, s = 1, . . . . S}, and a preference rela- tion P’ defined by pi(x’, yi) = (( zi, vi) E Ki x Yi: zip P’(x’)} for (xi, yi) E Ki x Yi. For i = m + s, s = 1, . . . . S, individual i (“state s market player”) has a choice set B,, a constraint correspondence C,,, = B,, and a preference relation P, + s defined by P, + S(x, p,) = (pi E B,: pi(C’“, 1 xf - Cy= 1 et) >

EQUILIBRIUM WITH INCOMPLETE MARKETS 381

p,(Cy= I x:- Cy= I ef)>. Individual i = m + S + 1 (“date 0 market player”) has a choice set B,, a constraint correspondence G, c s + 1 = B,, an preference relation P, + s + 1 defined by P,, + s + ,(x, y, po, q) = (( &, q’) E B,: p&(Cy!= 1 x; -cyt, e;, f q’ c;7, yi > p&C?= 1 xb - cr=“=, eb> + q c/= l y’).

The sets K, c R’ and Yic RN are constructed in the following way: assumption (1 ), pi is compact. Ki is taken to be a compact, convex set s that 8, c int Kc. Define a correspondence Fi: ‘=lB,+RN by F<(p)= I ,~E:H: min,=, .., s inf,i.. p,(x: - ei) - (1 ,ll) 6 v d ~-xs=1.....s Sup,ItK,~s(~s-kb)+l-~ld,l~,~=l~ . . . . S}, where H=span(v,:s=l,..., S). F, is a compact-valued, u.h.c. correspondence on in Hildenbrand 14, p. 241 the image of F, is co to be a convex, compact subset of H such that the i is contained in the relative interior of Y, (r.i. Y,).

The abstract economy r we have constructed satisfies t Theorem 1 in Shafer and Sonnenschein [7] so it has (2, 7, p, S) such that, for i= 1, . . . . m,

We show that (X, 7)~ A, II@,11 = 1, s= 1, . . . . S, ii(&, q)ll = 1, and th restriction of the consumption-portfolio choice to Kj x Yi is not b in equilibrium: Summing up consumers’ budget constraint at dat get j$(C~T, Zb-C~z”=l e~)4-~~~~l jidm(l - ll(po, qo)l/j. Suppo x7= l 2; f CyT 1 eb or Cy* I yi # 0, then by (iv) we must have /I (PO, @\I = 1 and pO(Cy= i 2; - Cr= i ek) + S Cyz I y’ > 0, which is a contradiction. Using the fact that x7= I y’=O, we prove using the same argument that C~z”=lZ~=C~z,e~, s==i ,..., S. Therefore 2”~ 2j for each i. Assumption (4) and condition (v) imply that all budget constraints in Ci(p, 4) are fu~~~Ie~ with equality. Hence y’ E r.i. Yi for each i. Note also that the restriction of portfolio choices to the subspace H of RN is without loss of generality. Indeed, if H # RN, i.e., if some assets are redundant, then for every portfolio YERN its orthogonal projection on H has the same payoff in every state and, by a non-arbitrage argument, the same value in equilibrium. Summing up all budget constraints (state-by-state), it easily follo lI(j$,, ijo) = 1 and iip,II = 1, s= 1, ,.., S. This completes the proofs

382 JAN WERNER

The following two remarks deal with relaxing the assumption (4) of local non-satiation in every state:

Remark 1. Assumption (4) can be replaced by the assumption of non-satiation in every state, i.e., that Pi(x’) # @ for every s = 0, 1, . . . . S and xi E yi. This technique is standard.

Remark 2. Assumption (4) can be relaxed to allow for “zero-probabil- ity” states. If for some s = 1, . . . . S, z E P’(x’) implies z’ E P’(x’) for every z’EX~ such that z$ =zs, for s’ fs, s’= 0, 1, . . . . S, and every xi, ZEX~, then we call state s a “zero-probability” state for consumer i. For such s, we have Pi(xi) = 0, xie Xi. Let S,c (1, . . . . S} be the set of “zero-probability” states for consumer i. We assume that for each s there is at least one consumer i with s # Si. Assumption (4) can be relaxed to hold only for each s $ Si. Then there exists an equilibrium. The proof requires a minor modification: We have to replace the correspondence Ci by ci defined by ei(p, q) = {(xi, yi) E Ki x Yi: pOxA + qy’ ,< p&+ 1 - lIpo, q/l, p,xf d psef.+r,yi+ 1- IIpJ for s$Si, psef+r,yi-(l/m)(l - IIpsll)~psxf~pseS+r,yi+ l- (Ips(l for s E Si> for (p, 4) E nfsi=, B,. The reader can easily chetk that Ci is continuous, and that all the arguments of the proof hold for Cf.

Remark 3. The obviously strong assumption (2) can be relaxed at the cost of strengthening assumptions on preference relations. The standard methods of the complete markets model can be applied here.

REFERENCES

1. D. CASS, “Competitive Equilibrium with Incomplete Financial Markets”, CARESS Working Paper 84-09, University of Pennsylvania, 1984.

2. D. DUFFIE, Stochastic equilibria with incomplete financial markets, J. Econ. Theory 41 (1987), 405416.

3. D. DUFFIE AND W. SHAFER, Equilibrium in incomplete markets. I. A basic model of generic existence, J. Math. Bon. 14 (1985), 285-300.

4. W. HILDENBRAND, “Core and Equilibria of a Large Economy,” Princeton Univ. Press, Princeton, NJ, 1974.

5. M. MAGILL AND W. SHAFER, “Equilibrium and Efficiency in a Canonical Asset Market Model,” Working Paper, University of Southern California, 1985.

6. R. RADNER, Existence of equilibria of plans, prices and price expectations in a sequence of markets, Econometrica 40 (1972), 289-303.

7. W. SHAFER AND H. SONNENSCHEIN, Equilibrium in abstract economies without ordered preferences, J. Math. Econ. 2 (1975), 345-348.

8. W. SHAFER, Equilibrium in economies without ordered preferences or free disposal, J. Math. Econ. 3 (1976), 135-137.

9. J. WERNER, Equilibrium in economies with incomplete financial markets, J. Econ. Theory 36 (1985), 110-119.