IOURNAL OF ECONOMIC THEORY 36, 1 lo-1 19 (1985)

Equilibrium in Economies with Incomplete Financial Markets*

JAN WERNER~

University of Bonn, Bonn, West Germany and Polish Academy of Sciences, Warsaw, Poland

Received May 27, 1984; revised December 5, 1984

This paper analyses an exchange economy in the absence of Arrow-Debreu com- plete markets. It is assumed that trading takes place in the sequence of spot markets and futures markets for securities payable in units of account. Unlimited short-sell- ing in securities is allowed. A general equilibrium in such an economy is a set of current and future prices (contingent on uncertain events) and a set of individual plans such that all markets are cleared. The existence of such an equilibrium is proved under usual assumptions. This is in contrast to the case of futures markets for contingent futures commodities where an equilibrium may not exist. The optimality of equilibrium allocations is also discussed. Journaf of Economic Literature Classification Numbers: 021, 313. 0 1985 Academic Press, Inc.

1. INTRODUCTION

In the Arrow-Debreu model of an economy it is assumed that there are markets for all commodities-also for those to be delivered at future dates in uncertain events. In such a situation all economic decisions will be made at one time, markets will open only once and consequently consumers will face a single budget constraint.

If, however, one drops the assumption of complete market structure, then there will be active markets at future dates and agents will face a sequence of budget constraints. A general equilibrium concept for such an economy with incomplete market structure is a Radner equilibrium, see Radner [4]. It is a set of current and future state-contingent prices and a set of individual plans such that all markets, both current and future, are cleared.

* After completing this work I have learned that David Cass has obtained similar results in the paper “Competitive equilibrium with incomplete financial markets,” CARESS Working Paper No. 84-09, University of Pennsylvania.

t I would like to thank Professor Martin Hellwig for helpful discussions. Financial support from Stiftung Volkswagenwerk is gratefully acknowledged.

110 0022-0531/85 $3.00 Copyright 0 1985 by Academic Press, Inc. All rights of reproduction in any form reserved.

EQUILIBRIUM WITH FINANCIAL MARKETS 111

It is well known that a Radner equilibrium may not exist (see Hart [2]). The reason for this is that consumer’s budget correspondence generally need not be continuous. Radner [4] proved the existence of eqnilibri~m imposing a bound on forward transactions.

In contrast to the models of Radner and Hart, where futures markets are markets for contingent futures commodities (in Hart [2] described by securities), we consider in this paper futures markets which are a purely financial phenomenon. They are described by means of securities wit monetary returns. The return of a security may depend on an uncertain event. One may interpret uncertain returns as a source of uncertainty in the model.

The purpose of this paper is to show that, under usual continuity an convexity assumptions, there exists an equilibrium in an exchange economy with financial futures markets. Price-independent monetary returns of securities do not cause any discontinuity in the model and an equilibrium exists even if unrestricted short sales are allowed.

In Section 2 we present a model of an exchange economy with incom- plete financial futures markets. Our economy lasts for two dates but all results can easily be generalized to the case of more dates with securities which cannot be retraded, for example, securities which “live” for only a single period. The equilibrium existence result is proved in Section 3 for a monotone, convex economy. Section 4 discusses some aspects of optimality of equilibrium allocations.

2. THE MODEL

We consider an exchange economy which extends over two dates At date 2 there are K possible states of nature (o,, We,..., oK) = a. There are G consumption commodities available at each date in each state of nature.

At the first date there are spot markets for G current commodities and futures markets for N securities. Each security is described by its state- dependent return at date 2. An agent holding one share of security n receives u,(o) units of account (“money”) if state OIZL~ occurs. Formally, there are N functions v,: Q -+ R + , n = l,..., N. We will denote by v, a K- vector v,(w~),..., v,(oK) and by v(w) an N-vector v~(o),..., vN(o) for @EC!. We will assume that v(w) # 0 for every o E Sz.

At the second date, in each state of the nature all G commodities will be traded on spot markets.

There are m consumers in the economy. For each i= I,..., m, consumer i has an initial endowment of date-l goods ei( 1) and state-dependent endow- ment of date-2 goods e,(2, o) (ef(2, o) denote the endowment of good h at date 2 state w, for some h = l,..., 6). There are no initial endowments of

112 JAN WERNER

securities. Consumption sets are all equal to X, , the nonnegative orthant of X= RGcK+ ‘1. We assume that each agent has a preference relation > i over X, .

Vectors of goods prices and securities prices will be denoted by p = (p(l), ~(2, or),..., ~(2, oK)) E X, and z E RN, , respectively. A con- sumption plan will be typically denoted by x= (x(l), x(2, or),..., x(2, oK)) E X, and a portfolio of N securities by 19 E RN.

An agent in our economy faces a sequence of budget constraints, one for each date-event, and chooses a portfolio and a consumption plan which he prefers.

We define an equilibrium, called an equilibrium of prices, plans, and price expectations or a Radner equilibrium, for short.

DEFINITION 1. A Radner equilibrium is a price system (p, n) and an allocation (X,, Q,),, 1 ,___, M such that:

(1) for all i = l,..., m, (Zi, Si) belongs to the budget set Bi(p, 7c) = ((x, 0 XEX+ 7 FERN, p(l)x(l)+~@<p(l)ei(l) and p(2,0~)~(2,0~)< ~(2, mj) e,(2, oj) + 8v(oj) for j= l,..., K} and there is no (x, 0) E Bi( p, rc) such that x > i Xi.

(2) CyT, Xi(l)=C~!~ ei(l), Cy=“=, Xi(2, Oj)=CyF”=, fZi(2,Oj) for j= l)...) K, Cy!! 1 si = 0.

3. THE EXISTENCE OF EQUILIBRIUM

THEOREM 1. Assume the following:

(Al) For every i, >i is continuous, convex, and monotonic

(A2) For every i, e,(l) > 0 or e,(2, We) > 0 for j= l,..., K, furthermore x7! 1 ei( 1) % 0 and for every hi = l,..., G there exists a consumer i such that for all j = l,..., K, eF(2, oj) > 0.

Then there exists a Radner equilibrium (p, E, (Xi, S,),, I,...,m) with p%O.

ProoJ: Denote by H the subspace of RN spanned by vectors v(oi) for j= l,..., K, by H + the convex cone spanned by those vectors.

The interior of H, in H (int H, ) is a set of nonarbitrage prices on futures markets. If rc # int H + , then there exists a portfolio 0 E RN such that &c < 0, 8u(oj) B 0 for j = l,..., K and for at least one j,, Bu(wJ > 0. Clearly, if (p, z) is an equilibrium price system then z E int H + . Therefore, in the sequel we will consider only nonarbitrage securities prices.

If 7c is such a price vector, then without loss of generality, we may con- strain consumers to choose portfolios only from the set H. Indeed, if

EQUILIBRIUM WITH FINANCIAL MARKETS 113

(xi, SJ is a consumer’s optimal plan, then (xi, 8,), where gj is an orthogonal projection of fIi on H, is also an optimal plan since preference does not depend on B and gi yields the same returns as 8, at the same cost. Furthermore, if EYE 1 Bi = 0 then X7= I 8, = 0. Consequently, as a budget set of the ith consumer we will now consider

Bj(p,~)=((x,8) such that ~EH, XEX, and 1~(1)~(l)+~e~~(l)e~(l) and ~(2, aj) x(2, ~j) d OU(Wj) +p(2, ~j) e,(2, ~jj) for all j = l,..., K}

LEMMA 1. (i) B, is a closed correspondence

(ii) B,(p, n) is a compact set for p $0 and z E int H +

(iii) B, is lower hen&continuous at every (p, 71) which satisfies one of the following conditions

P(l)41)>0 (*I

p(2,~j)e,(2,~~)>0 forallj=l,..., Kandn.#O. (**I

(The proof of Lemma 1 will be deferred to the end of this section). We define the individual demand correspondence of agent i by

qi(p, n)= ((x7 0)~ Bi(p, n): there is no (x, 6)‘~B~(p, n) with x’>~x). The demand correspondence (pi has the following properties:

LEMMA 2. (i) ‘pi is nonempty-, compact-, convex-valued? and upper hemi- continous at every (p, z) such that p+O, rc E int H +

(ii) ‘pi is a closed correspondence at every (p, z) satisfying (*) or (w)

(iii) if the sequence { p,?, z,, i \ of strictly positive price vectors (i.e., p,PO, n, E int H + ) converges to (p, z) which is not strictly positive and satisfies (*) or (**), then inf {/Ix/I: (x, ~)EP~(P~, nn)for some t3> -+n +oo.

(The proof of Lemma 2 will also be deferred to the end of this section.) Let us now define the total excess-demand correspondence

Z(P, x)= f cpi(P, n)- 2 (e,tW i=l i=i

ff 0 E Z( p, n) then clearly (p, rc) is an equilibrium price system. 2 satisfies the following Walras’ law: for every p % 0, rc E int H + and (z, 0) E Z(p, n),

and

p(1)z(1)+710==0

for all j.

114 JAN WERNER

In order to prove the existence of (p, z) such that OE Z(p, rc) we will follow the lines similar to the usual proof of the existence of equilibrium in a monotone exchange economy with complete market structure, see, for example, Hildenbrand [ 31.

Consider the following price sets:

P={(p,+RG+ ~H+:n=~;~~++co~) for some A, > 0 such that Xi”= 1 Aj + C,“= 1 ph = 1 }

Q= {PER: :CjL~h=l) P,= ((p, n)EP:Ph> l/n and Aj> l/n for aI1 h=l,..., G,j= l,..., K) Q,= {~~Q:p,>l/n} for n>G+K.

Clearly all those sets are compact and convex, furthermore int P= U, P, and int Q = U, QE.

LEMMA 3. For every n there exist (p,( 1 ), z,) E P,, p,(2, oj) E Q, for j= l,..., K and (z,, 8,) E Z(p,, n,) such that for all (p(l), rc) E P, and ~(29 uj) E Qn 9

P(l)z,(l)+~e,<o (1)

Pt2T uj) zn(2> cL)J.) G env(uj) j= l,..., K. (2)

(The proof will be deferred to the end of this section.) Consider the sequences {p,, z,}, (zn}, and {e,} from Lemma 3.

Without loss of generality we may assume that (p,, z,} is convergent, say ( pnr 71,) -+ (p, 5). Next we show that {z,} is bounded. Let A = ( p = (p( 1 ), P(2, ~I)YYP(& WK)): P>O, c,“=, Ph(l)+x&I c,“=, Phc2, wj)=l}. For

every p E int A, we have (p( 1 ), J$II 1 Ajo( E int P and ~(2, wj)/,Ij E int Q for j= l,..., K, where A,= C,“= 1 ~~(2, oj). Thus for 12 large enough (p(l), cj”=, ;liv(oj)) E P, and ~(2, wj)/Ajc Qn and consequently by (1) and (2),

P(l)zn(l)+ jJ aje,u(Uj),<o

j=l

(3)

PC2, mj) zn(2, Oj) G ajz,env(oj) j= l,..., K. (4)

Summing (3) and (4) over all j, we finally obtain that for every p E int A and n large enough

P(l) zrz(l I+ f Pt2, Oj) 2n(2, Oj) d O* j=l

(5)

Since (zn) is bounded below, (5) implies that {z,, > is bounded. Therefore, without loss of generality we may assume that {zn} converges to some Z.

EQUILIBRIUM WITH FINANCIAL MARKETS 115

Since (z,, 0,) E Z( pn, rc,), we obtain by Walras’ law p,(2, oj) 2,(2, mj) = 6,u(coj) for j= l,.,., K. Taking limits as n + co, we obtain that for every j9 6,u(wj) converges to j42, oj) Z(2, aj). Therefore, since 8, E H= wn(Gw,),..., u(w,)> follows that 8, converges to some BEH.

We claim now that p&O and EEint N,. Since (p(l), %)EP and j(2, w,) E Q for j = l,..., K, there exists at least one consumer for whom (*) or (**) is satisfied. Indeed, if p( 1) # 0 then by assumption (A2), @( 1) ei( 1) > 0 for some i. Otherwise, if p( 1) = 0 then it f0; furthermore since p(2, oj) # 0 for j = l,..., K there exists, by assumption (A2), some con- sumer i with p(2, oj) e,(2, wj) > 0 for all j. Therefore in both cases, (g) or (a*) is satisfied for some consumer. By Lemma 2(iii) follows that (p, E) must be strictly positive, otherwise {zn} would be unbounded.

By closedness of Z we obtain (5, 0) E Z( p, ti). To complete the proof we have to show that (2, 0) =O. From (1) and (2) follows, for all (p(l), n)Eint P and p(2, wj)EintQ,

p(l)Z(l)+7C8<0 (6)

P(2, oj) 2(2, wj) d B"(wj) j = l,..., K. (‘91

By Walras’ law we have

p(l)z(l)+Ene=o (8)

jq2, co,) 2(2, Oj) = fh(Oj) j= l)...) K. (9)

Using (6) and (8), we obtain Z( 1) = 0 and 8 = 0. Indeed, (6) implies T(l)60 and BE(H+)‘-polar cone of H,, i.e., the set (BEH:@<O for every y E N + >. Therefore by (8) p( 1) Z( 1) = 0 and rtf? = 0. Finally, since p(l)>>0 and ?tEintH+, it follows Z( 1) = 0 and 0 = 0.

By (7) and (9), using the fact that 8= 0, we obtain $2, oj) = 0 for j= l,..., K.

Thus 0 E Z( p, 71) and ( p, ii) is a strictly positive equilibrium price vector.

Proof of Lemma 1. (i) Straightforward. (ii) By assumption, there are Aj > 0 for j = I,..., K such that

5 /2iU(Oj) = 71

j=l

Let p’ = (p(l), 1, ~(2, ol) ,..., A,p(2, ox-)). Define

Bg,(p’) = {x E x, : p’x <p’qj

116

and

JAN WERNER

F,(p,~)=(e~H:~B~p(l)e,(l)and

-p(2, oj) e,(2, coj) < 8u(oj), j= l,..., K}.

Both sets are compact for p $0 and 7c E int H + . It is easy to show that

NP? n) = &i(P’) x F,(P, n).

Hence Bi(p, n) is compact, as a closed subset of a compact set. (iii) Consider the correspondence ji defined by

&(p,n)={(x,8):xEX+, eEH,p(l)x(l)+nO<p(l)ei(l)

andp(2, oi) x(2, oj) < Bu(coj) +p(2, oj) e,(2, oj),j= l,..., K};

di(p, n) is nonempty for every (p, n) satisfying (*) or (**). Indeed, if p( 1) ei( 1) > 0 then x = 0 together with some 8 such that n0 <p( 1) e,( 1) and e2++) > 0, j = I,..., K (recall that we assumed v(mj) > 0 for all j) belongs to di(p, rc). If p(2, wi) e,(2, oi) > 0 for j = l,..., K and rc # 0 then x = 0 together with some 0 such that nti < 0 and 8v(oj) > -p(2, oj) e,(2, oj) belongs to ji(P7 n).

Let x,EX+ and 8,EH with X,-+X and 8,-t& where (x,~)E&(~,Tc)

for some (p, rc) satisfying (*) or (**). Then for every { p,, rc,} such that pn +p, n, + rc, and for II large enough, pn( 1) x,( 1) + rc,e, <p,(l) ei( 1) and p,(2, wi) x,(2, wj) < 8,u(oj) +p,(2, oi) ei(2, oi), j= l,..., K. Thus (x,, 0,) E Bi( pn, 71,) for n large enough, which implies that Bi is 1.h.c. at (p, n). Since the closure of 1.h.c. correspondence is also l.h.c., (iii) follows.

Q.E.D.

Proof of Lemma 2. (i), (ii) Consider the relation >j defined on X, x RN by (xi, 0,) >,! (xz, 0,) if xi >i x2. This is a continuous and con- vex preference relation. In the usual way (see Hildenbrand [3, pp. 99-1031) one concludes using Lemma 1, that (i) and (ii) holds. It should be noted that, by assumption (A2), (*) or (**) is satisfied for every i at every p%O and zEint H,.

(iii) Assume the contrary, then there is a subsequence, say x, such that x, +x and (x,, 0,) E qi(pn, 71,) for some 8,. Since (x,, 0,) E qi(pn, q) and preference relation is monotonic, it follows p,(2, oi) x,(2, oi) = 8,u(oj) +p,(2, mj) e,(2, mj) forj= l,..., K. Taking limits as IZ + co, we obtain that 8,v(oj) converges for every j to ~(2, oj) x(2, oj) -p(2, oi) e,(2, oi). Since 8, E H and H = span { o(ol),..., A} it follows that 8, converges to some 8 E H. By (ii) cpi is closed at (p, z), therefore (x, 0) E qpi(p, TC). On the

EQUILIBRIUM WITH FINANCIAL MARKETS 117

other hand, since (p, n) is not strictly positive (i.e., some spot prices are zero or there is an arbitrage opportunity) and every commodity is desired, it follows that qi( p, rr) = 0, a contradiction. QED.

Proof of Lemma 3. Let B, be a convex, compact set such that the image Z( P, x (QJK) c B,. For every (z, 0) E B, we consider a set

andp(2, wj) ~(2, mj) =t;“~“e” qz(2, c+) forj= I,...) K}. n

We apply the Kakutanis’ theorem to the correspondence ,u~ x Z o B, x P, x (QJK into itself. Thus we obtain a fixed point (p,, A,, z,, pcl, x Z,,. It is now easy to show, using Walras’ law, that (p,, E,, z,, 8,) satisfy conditions of Lemma 3. Q.E.D.

Remark 1. Proving Theorem 1 we have shown the existence of a nor- malized equilibrium price system, i.e., such a (p, n) that (p(l), TL) E P and ~(2, oj) E Q for j = l,..., K. Clearly individual demand is homogeneous of degree zero in (p(l), rr), hence there is no loss of generality in the nor- malization of date-l prices. There is no such homogeneity in date-2 prices but it should be noted that individual demand at (p(l), n,p(2, or),..., ~(2, wK)) in the economy with securities v is e the demand at prices (p(l), rc, ,I, ~(2, al),..., AKp(2, oK)) in the economy with securities Au, where IJJ(w,) = ,Ijv(wj) for a, > 0. Therefore looking for all equilibria with securities v one has to compute all normalized equilibria with securities dv for i % 0.

Remark 2. Discussion of assumption (A2). What we assume about date- 2 endowments is, of course, weaker than e,(2, wj) 9 0 for all i and all j; it is even weaker than e,(2, wj) +O for only one consumer i, for allj. The follow- ing simple example shows that it can not be weakened ts the assumption C?!! 1 e,(2, oj) $0 for all j. There are two states of nature, one good, two consumers and one security u(wr) = v(w2) = 1. Consumer i, i= I, 2, maximizes the expected value of a utility u~(x(~)) + 4x(2,0)). We assume that lim,,,, u:(v) = +co. Consumer l’s endowment is zero in state aa3; and positive in state w2 and at date 1; consumer 2’s endowment is zero in state oa and positive in or and at date 1. Clearly in equilibrium, the market for security would be inactive. On the other hand, there are no prices at which xi = ej, Bi = 0 would be an optimal decision of consumer i.

ence, there is no equilibrium.

118 JAN WERNER

4. THE OPTIMALITY OF EQUILIBRIUM

In this section we discuss optimality properties of equilibrium allocations in an economy described in Section 2.

There is no reason to expect a Radner equilibrium allocation to be Pareto optimal. In fact it is Pareto optimal, in general, only if the market structure is “essentially” complete, i.e., if the space of monetary returns span(v,,..., oN} is equal to the whole space RK.

Furthermore, even a constrained Pareto optimality cannot, in general, be expected. The reason for this is, loosely speaking, that the market struc- ture involves constraints on net trades and only via initial endowments on consumption plans. Consequently the constraint on consumption plans depends on endowments and is different for different consumers. Therefore it seems to us that the more appropriate efficiency concept for an equilibrium in incomplete markets is V-efficiency. We say that a com- modity allocation (&= I,...,m is V-efficient for some subset V of RGcK+l) if it cannot be (weakly) Pareto dominated by any attainable allocation (x;)~= r ,__,, m such that xi --x~E I’ for all i.

Let us consider the set of admissible net trades for the given prices (p, n). There are K+ 1 types of such net trades: date-l admissible net trades

M,(p, n) = {(z(l), ~(2, wi) ,..., ~(2, wK)) E RGcK+l): there

exists 0 such that ~(2, oj) ~(2, wi) = Bv(oj),

j = l,..., K);

i.e., a set of trade plans which involve only date-l income transfer, and date-2 state mj admissible net trades

Mj(p, n) = {(z(l), ~(2, or),..., ~(2, oK)) E RGcK+l): there

exists 0 such that p( 1) z( 1) + ~8 = 0 and

~(2, co,+) 42, wk) = Ov(o,), k = I,..., K k #j}

for j = l,..., K; i.e., a set of trade plans which involve only date-2 state oi income transfer.

It can be easily shown under nonsatiation assumption that a Radner equilibrium allocation is not Pareto dominated by any allocation which can be attained by retrading admissible at equilibrium prices (p, TC). Hence, V( p, z)-efficiency for V( p, rt) = UF!= 0 MJ p, rr ) follows.

An evident disadvantage of the above optimality description is its (equilibrium) price-dependence. It is, however, a feature of incomplete markets that the set of admissible net trades depends on prices. In the case of commodity futures markets, a set of net trades admissible for all prices

EQUILIBRIUM WITH FINANCIAL MARKETS 119

(which is a subset of V(p, K)) provides a complete description (i.e., a characterization) of Radner equilibrium allocations (see Gale [ 1, pp. X3-219] ). In our case, this subset of V( p, 7~) is, in general, too small to give a meaningful optimality description. Indeed, it may happen that there are no contracts for futures commodities admissible for all prices.

REFERENCES

1. D. GALE, “Money: In Equilibrium,” Cambridge Univ. Press, Cambridge, Mass., 1982. 2. 0. D. HART, On the optimality of equilibrium when the market structure is incompiete, J.

Econ: Theory 11 (1975), 418443. 3. W. HILDENBRAND, “Core and Equilibria of a Large Economy,” Princeton Univ.

Princeton, NJ., 1974. 4. R. RADNER, Existence of equilibrium of plans, prices, and price expectations in a sequence

of markets, Economekica 40 (1972), 289-303.