Journal of Mathematical Economics 14 (1985) 285-300. North-Holland

EQUILIBRIUM IN INCOMPLETE MARKETS: I

A Basic Model of Generic Existence

Darrell DUFFIE*

Mathematical Sciences Research Institute, Berkeley, CA 94720, USA Stanford University, Stanford, CA 94305, USA

Wayne SHAFER*

University of Southern California, Los Angeles, CA 90089, USA

Final version accepted December 1985

This paper demonstrates the generic existence of general equilibria in incomplete markets. Our economy is a model of two periods, with uncertainty over the state of nature to be revealed in the second period. Securities are claims to commodity bundles in the second period that are contingent on the state of nature, and are insufficient in number to span all state contingent claims to value, regardless of the announced spot commodity prices. Under smooth preference assumptions, equilibria exist except for an exceptional set of endowments and securities, a closed set of measure zero. The paper includes partial results for fixed securities, showing the existence of equilibria except for an exceptional set of endowments.

1. Introduction

This paper demonstrates the generic existence of general equilibria in incomplete markets. Our economy is a model of two periods, with un- certainty over the state of nature to be revealed in the second period. Securities are claims to commodity bundles in the second period that are contingent on the state of nature, and are insufficient in number to span all state contingent claims to value, regardless of the announced spot market prices. As Hart (1975) has shown, equilibria need not exist in this setting, even under the ‘smooth preference’ assumptions of Debreu (1970) that we adopt. Hart’s counterexample is based on a collapse in the span of security markets that occurs on an exceptional set of ‘bad’ spot prices. (An excep- tional set is a closed set of measure zero.) Attempts to resurrect the existence of equilibria have thus concentrated on showing that the exceptional set of ‘bad’ spot prices is only relevant for an exceptional set of economies, a program of generic existence. The generic existence results of McManus

*We are particularly in debt to Andreu Mas-Cole11 for consultations and encouragement, and for conversations with David Cass. Shafer gratefully acknowledges the financial support of the National Science Foundation under Grant SES-851335. Dutlie acknowledges the generous support of the Mathematical Sciences Research Institute.

03044068/85/%3.30 0 1985, Elsevier Science Publishers B.V. (North-Holland)

286 D. Dujie.and W Shafer, Equilibrium in incomplete markets

(1984), Repullo (1984), as well as Magi11 and Shafer (1984,1985), rely on a sufficient number of securities to complete markets, at least for ‘good’ spot prices. An alternative has been to include only purely financial securities, contingent claims to units of account that are independent of spot prices, following Arrow’s early lead of 1953. Although the subspace of claims to value that can be achieved by security trade need not be complete, it is then fixed independently of spot prices. Because of this fact, Werner (1985), Cass (1984), and Duffie (1985) have been able to demonstrate incomplete markets equilibria with purely, financial securities under general conditions, without relying on genericity.

Given the foundational role of the general equilibrium model, an apparent lack of complete markets, and the clear presence of securities whose dividends depend on spot market commodity prices, the existence of equilibrium in our setting had been an open question of some interest to us.

In a comparison paper [DufIie and Shafer (1985)], we extend our results to multiperiod economies under uncertainty with sequential trade, and with mixed real and purely financial securities. Even with this, there are still important open questions for this type of model. Of particular concern is the sense of genericity. Here we prove existence except for an exceptional set of economies parameterized by endowments and securities. A stronger and more appealing result would apply to a fixed set of securities, and prove generic existence parameterizing economies only by endowments. Given restrictions on the set of securities, we are already prepared to do this. In the last section of this paper for example, taking securities that are futures contracts or satisfy other structural restrictions, we sketch out the existence of equilibria except for an exceptional set of endowments. We know of no arguments suggesting that these structural restrictions on securities cannot be weakened or removed entirely.

Related problems are: (i) the question of constrained and full Pareto optimality of incomplete markets equilibrium allocations, pointed out by Hart’s striking examples and further examined by Geanakoplos and Polemarchakis (1985), (ii) the indeterminacy of incomplete markets equilib- rium allocations, characterized by the dimension of the equilibrium alloc- ation manifold by Geanakoplos and Mas-Cole11 (1985) as well as Cass (1985), (iii) the generic optimality of equilibrium allocations with potentially complete markets [Magill and Shafer (1985b)], and (iv) the dynamic span- ning effect of repeated trade of securities, for which sources cited in Duffie and Shafer (1985) may be consulted. Aside from Arrow (1953) all of the papers cited owe a debt to Radner (1972), who demonstrated equilibrium in a general multiperiod model with a lower bound restriction on security portfolios. As Hart (1975) has argued, the short sale of securities is an essential feature of markets, and an equilibrium demonstrated with a lower bound on security trades may depend on the bound chosen by the modeler.

D. Dufjie and WI Shafer, Equilibrium in incomplete markets 287

The remainder of the paper is ordered as follows. Section 2 presents the economic setting and the delinition of equilibrium. Section 3 states the main results. Section 4 briefly outlines the theory of Grassmannians, a class of manifolds of linear subspaces that plays a key role in our analysis. Section 5 presents a number of facts on the incomplete markets excess demand function. In section 6 we characterize the relevant set of parameters forming an equilibrium as a manifold. The proofs of the main results are found in section 7. We rely heavily on differential topology, introduced to the study of general economic equilibrium by Debreu (1970,1972). The final section contains preliminary results on generic existence of equilibrium for fixed securities satisfying structural restrictions.

2. The basic equilibrium problem

This section outlines the economic setting, stating the definition of an equilibrium with incomplete markets in a two period model with uncertainty over the state of nature in the second period.

In period 0 there are spot markets for commodities, and security markets for assets that pay bundles of commodities in period 1, the bundle paid depending on the state of nature. In period 1 agents cash in their portfolios of assets and their endowments, trading the proceeds on spot markets for commodities. There are 1 commodity types, n possible states of nature in period 1, and k assets. Asset j, for 14 js k, is characterized by a collection (u’(s)):,, of vectors in R’. An agent holding one unit of asset j after trading in period 0 is entitled to the vector uj(s) of commodities in period 1 if state s occurs. A portfolio 13 = (0,). . . , 13,) E Rk of assets thus represents a claim to the vector USE Iw’ of commodities in state s, where a(s) is the 1 x k matrix with jth column u’(s). Let n* = Ink and a = (a( l), . . . , u(n)) E KY” denote a given asset structure. Let 4 E Rk denote a vector of asset prices, giving a portfolio 8 the market value 4.8 in period 0.

Let x,E!?++ denote a period 0 consumption vector of a typical agent, and let xi(s) E rW$ + denote a consumption vector of an agent in period 1, state s. For 1* = l(n + l), we can thus write x=(x0, (x1(s)):= 1) E R’f + as a consump- tion vector. Similarly, p = (pO, (pi(s)):, 1) E WY + represents a collection of spot market price vectors. Each agent i, for 15 i S m, is characterized by an initial endowment vector wi E rWz: + and a utility function ui: RI:+ +R satisfying:

(i) Ui is C” (partial derivatives of every order exist and are continuous), (ii) DUi(X) E IF@: + for all x in R’: + (strict monotonicity), (iii) hTD2ui(x)h < 0 for all h # 0:h . Du,(x) = 0 (differentiably strictly convex

preferences), and (iv) {xER~+: ui(x)zuI(X)} is closed in R” for all X in rWy + (a boundary

condition).

288 D. Duffie and WI Shafer, Equilibrium in incomplete markets

Given a spot price system PE R’f + and a portfolio 0~ Rk, the market value of the portfolio in state s is p,(s)‘a(s)8. Denote by @,a) the n x k ‘returns’ matrix whose sth row is pr(s)ra(s), so that V(p,a)8~R” is the vector of dividends across the n states in period 1 generated by a portfolio 8. For any x=(x,, (xi(s)):, 1) E RI*, let p1 0 x1 E denote the vector (pi(s) * x1(s)):, 1 of units of account required to purchase R” in the n states. In order to consume XEIWiI+, agent i thus needs the vector p1 0 (x1 -w\) of units of account. Given prices (p, q) E R’f + x Rk, agent i is therefore faced with the problem

maxni(x) s.t. po.(x,-wQ+q*8S0, CT 0)

p1 q (X,-W;w(p,4~. (1)

An equilibrium is thus a collection ((Xi, 8’), (p, 4)) satisfying

(a) (Xi,@) solves (1) - - given (p,q) for all i, and (b) xi9 =Ci,’ and c$‘= 0.

To simplify the proof of existence of equilibria, we introduce an essentially equivalent equilibrium concept. Let L denote a linear subspace of R” and consider the following set of agent problems:

agent 1 solves: maxa, s.t. p-(x-w’)=O; X

agent iz2 solves: maxni(x) s.t. p*(x-w’)=O, X

p&l(x,-W’,)EL. (2)

A collection ((Xi), p, L) is an effective equilibrium if

(a’) Xi solves (2) given (j, L), 15 ism, (b’) CiXi=Ciwi, and (c’) E is the linear subspace of R” spanned by the columns of V&a).

A final equilibrium concept that is crucial to the analysis is the following. A collection ((xi), p, L) is a pseudo-equilibrium if it satisfies (a’), (b’), and

(c”) dim(E) = k and L contains the column vectors of V(p, a).

3. Basic results

Let w=(wl,..., W~)E rWy*: denote the list of endowment vectors of the m agents. We will parameterize an economy by a point (0, a) E Iw!J’: x R”‘. We are specifically interested in the case 15 k < n. (Generic existence of equilibria

D. Duffie and W Shafer, Equilibrium in incomplete markets 289

for other cases has been demonstrated in papers cited in the introduction.)

Proposition 1. If ((Xi),p,L) is an effective equilibrium, then there exists some q~ Rk and 8’ E Rk, 1 s is m, such that ((xi, 8’), (p, 4)) is an equilibrium.

Proposition 2. If ((xi), p, L) is a pseudo-equilibrium and rank( V(p, a)) = k, then ((xi), p, L) is an effective equilibrium.

These two propositions allow us to search for an equilibrium in the guise of a pseudo-equilibrium for which V&a) has full rank. That is how we proceed.

Theorem 1. For every (co, a) E R’$; x R”‘, there exists a pseudo-equilibrium for the economy (co, a).

Theorem 2. There is an open set Qc [WY’: x R”’ with null complement such that an equilibrium exists for every economy (co, a) in 52.

Theorem 2 states that equilibria exist generically, that is, except for an exceptional set of economies. The gap between Theorems 1 and 2, basically that a pseudo-equilibrium generically corresponds to an equilibrium, was tilled earlier by Mas-Cole11 (1985b), using a somewhat different proof.

The basic idea of the proofs is as follows. Let G,*. denote the collection of k-dimensional subspaces of R”. For convenience, we refer to (p,L) as a pseudo-equilibrium if there exist xi E rWy +, 1 I ilm, such that ((xi), p, L) is a _ _ pseudo-equilibrium. Define the pseudo-equilibrium manifold E by

E = {(p, L, w, a) E R’f + x G,,, x [WY’: x IF!“*:

(p, L) is a pseudo-equilibrium for economy (0, a), with p. w1 = l}.

Define rc: &+ Ryr> x R”’ by n(p, L, co, a) =(~,a). We note that n-‘(W,ti) is the - -

set of (p, L, co, a) such that (p, L) is a pseudo-equilibrium for (0, a). We will apply mod 2 degree theory to rc along the following lines: [We

refer readers unfamiliar with differential topology to Guillemin and Pollack (1974) for the basic definitions.] Suppose f: X+ Y is a smooth proper map between two boundaryless smooth manifolds X and Y of the same dimen- sion, and suppose Y is connected. For any regular value y of f, let #f-‘(y) denote the number of points in f-‘(y), the set of x in X such that f(x)=y. Then #f-‘(y) mod 2 is the same for every regular value y of f. In particular, if there exists a regular value j of f such that #f-‘(j) is odd, then f-‘(y) cannot be empty for any y in I: for y is by definition a regular value if f-‘(y) is -empty. Thus to prove Theorem 1, we can show the following:

290 D. Duffie and W! Shafer, Equilibrium in incomplete markets

(a) E is a smooth manifold without boundary of dimension ml* + n*, (b) rc is proper, and (c) there is a regular value (0, a) of a such that # rc-l (W, a) = 1.

- - At a regular value (~,a) of rr, the points in n-l(W,Cr) will be locally

smooth functions of (0, a), and the (p,L) co-ordinate functions are in fact submersions (have derivatives that are onto). This allows us to prove Theorem 2 by showing that a small perturbation of a regular value of rr will yield a pseudo-equilibrium (p, L) with rank (V(p, a)) = k. (The basic idea here is Sard’s Theorem: If f: X+Y is a smooth map of manifolds, then almost every point in Y is a regular value of J) Such an equilibrium (j&E) is by definition an effective equilibrium, which, by Proposition 1, corresponds to an equilibrium.

4. Grassmannians

In order to show that E is a manifold, we will explicitly describe an atlas for a differentiable structure on Gk,n. This structure shows G,,, to be a smooth compact manifold without boundary of dimension k(n - k), called the Grassmannian of k-planes in R”.

Let Y c [W”(n-k) denote the manifold of (n-k) x n matrices of rank n-k, an open subset of Iw ‘(n-k) Any A in Y naturally induces some L in Gk,” by . L= {y E IF: Ay = O}. If A induces L in this manner, then A’ E Y also induces L if and only if there exists a non-singular (n-k) x (n- k) matrix B such that BA = A’. We thus define an equivalence relation - on Y by

A - A’o3 non-singular (n-k) x (n-k)

matrix B such that BA = A’. (3)

We identify G,,. with Yl- endowed with the quotient topology: U is open in Yf- if and only if p-‘(U) is open in Y; where p: Y+ Y/N is the identification map. Thus, from now on, L will denote an element of

Y/- 5 Gk,,,, and ‘A EL) means that A is an element of the equivalence class L, or ‘A induces L’.

We now describe a C” atlas for C&. Readers may refer to Hirsch (1973) for the definition of an atlas, charts, and related concepts. Let Z = {o:a is a permutation of { 1,. . . , n}), and let 0-l denote the inverse permutation of any 0 in C. For any o in Z, let P, denote the n x n permutation matrix corresponding to 0. We will have occasion to write certain matrices in partitioned form, as follows. If A is (n-k) x n, then

D. Duffie and I+! Shafer, Equilibrium in incomplete markets 291

If P is n x n, then

n-k

P= k

Fact 1. If LeGk,“, there exists a 5~ C and an (n-k) x k matrix E such that [Zp-JP,EL.

Proof. Pick any AE L. Since A has rank (n-k), it has n-k linearly independent columns. Choose a CJ in C such that the first n-k columns of AP, are linearly independent. Then

where o=a-’ and E=([AP,](l))-l[AP,](z). 0

Fact 2. If [ZIE]P,EL and [ZIE’]P,EL, then for P=P,,P,-I, we have E= [P(l) + E’PC3)] - ’ [PC’) + E’PC4’]. In particular, if o = rr’, then E = E’.

Proof. We have [Z(E]P,-[Z(E’]P,., so there exists a non-singular B such that

[Z(E]P,=B[Z(E’]P,..

Let P= P,,P,-,; then this becomes

[ZIEJ=B[ZjE’]P.

In partitioned form,

z = B[P”’ + E’P’3’], E = B[P”’ + E’Pt4’].

Thus [PC’) + E’ Pc3)] is invertible, and

E = [p(l) + E’p’3’] - 1 [p(2) + E’p’4’]

If rr=a’, then P=Z, so E=E’. 0

292 D. Duffie and W Shafer, Equilibrium in incomplete markets

For each 0 E C, let

W,=(LEG,,,:~EEIW(“-~)~ such that [Z(E]P,EL)

and let qrr:W,-,R(“-k)k be defined by [Z\C~,,(L)]P,EL.

Fact 3

(1) oKAT,, is an open cover of G,,,. (2) qb is a homeomorphism of W, onto I!@~-~)~. (3) cpo o cp; ‘: cp,,( W, n W,,)+rp,( W, n W,,) is smooth for all cr, 6’.

(4) Gk,, is compact.

Therefore {K, 4~~)~~~ is an atlas for a C” differential StrUCture on Gk,“, making G,,, a compact C” manifold without boundary of dimension k(n - k).

Proof

(I) Let p:Y-+Y/- be the identification map. Then p-‘( W,) = {AE Y: the first n-k columns of AZ’,-1 are linearly independent). This is clearly open in Y so W, is open in Y/N. By Fact 1, Gk,nt u, W,. (2) qp, is one-to-one by Fact 2. For any EE R(n-k)k, [ZlE]P, induces some L since rank [Z 1 E]P, = n - k. Thus (pb is onto. To show cpO is bicontinuous, one uses the calculations in the proof of Fact 1. (3) From Fact 2, (pb 0 cp; ‘(E’) = [P(l) + E’P@)] - ‘[P(‘) + E’Pt4)], where P = P,,P,-1. (4) Let /I= {BE Y: the rows of B form an orthonormal set}. Then /? is compact and p(B) = Y/N. 0

5. Demand functions

For each agent i, define G’: IRf + x R, + W’: + by G’(p, y) = arg [max, U&X) s.t. p.x=y].

Fact 4. G’ satisfies:

(1) G’ is C” and homogeneous of degree 0.

(2) P*G’(~,Y)=_Y. (3) pTD,Gi(p, 1) = - G’(p, l)T. (4) hTD,Gi(p, 1)htO for all h #O such that h. G’(p, 1) =O. (5) lim,.+, \IG’(p,l)/= +co if pears+ and p#O.

Proof. This is well known. 0

For each agent i, define F’: WY + x Gk,, x R’: +-+Rlf + by F’(p, L, w) = arg[max,z+(x) s.t. p.(x-w)=O, p1 0(x, -wl)eL].

D. Du@e and W Shafer, Equilibrium in incomplete markets 293

Fact 5. Each F’ satisfies

(1) F’ is C” and homogeneous of degree zero in p. (2) pF’(p,L,w)=p*w. (3) If w = G’(& l), then

(i) F’(p, L, w) = w VL, (ii) DpF’(p, L, w) is negative semi-definite VL, and (iii) p’D,F’(p, L, w) = 0.

Proof

(I) Pick a p, L, w, and cr such that L E W,. Then

F’(p,L,w)=arg[maxUi(X) St. p.(x-W)=O, [IIcp,JL)]P,p, 0 (X1-Wl)=O]. x

For F’ to be C”, it is sufficient that the bordered Hessian for the above maximization problem is non-singular, which the reader is invited to write out and check. (2) This is obvious. (3) Suppose w = G’(p, 1). Then ai 2 ui(x) Vx: p. (x - w) 5 0. For any L, w is affordable at the constraints corresponding to F’, so w=F’(p, L, w) for all L. We consider p #P_ Since w is affordable at any (p, L), Ui(F’(p, L, w)) > ui(w) if F’(p, L, w) # w. Thus from above we have ~7. F’(p, L, w) zp* w =p. F’(p, L, w). In addition, p. F’(p, L, w) = p. w = p. F’(p, L, w). Combining these two relations, (p -p). (F’(p, L, w) - F’(p, L, w)) 5 0 Vp, L. Thus F’ is a monotone decreasing operation in p, and D,F’(p, L, w) is negative semi-definite. The fact that pTD,Fi(p, L, w)=O can be obtained by differentiating both sides of the relation p. F’(p, L, w) =p * F’(p, L, w) and evaluating at p = p. 0

We now define an excess demand function Z: rW? + x Gk,, x [WY’: +R’* by

Z(p, L, o) = G’(p, 1) +i% F’(p, L, w’) - .f wi. i=l

Fact 6

(1) Z is C”. (2) Z(p, L, co) = 0 if and only if p. w1 = 1 and

G’(p, p . w’) + iz2 F’(p, L, w’) = F wi. i=l

(3) D,lZ(P, L, w) = -I.

(4) If (P”, L”, on)+@, L, ~5) ~(8R’f +) x Gk,, x W7”, (p”, L”, o”)ll = + co.

and J?# 0, then lim,lIZ

294 D. Duffie and W Shafer, Equilibrium in incomplete markets

Proof. Parts (1) and (2) follow from Facts 4 and 5. Part (3) is easy. For part (4) one applies Fact 4 to G’ and uses the fact that F’(p, L, w’) 20. 0

Finally, for each cr in Z, define K,: rWy + x W, x Rn*+R(n-k)k by K,(p, L, a) =

CI I %(-w PO w, 4

Fact 7. Each K, satisfies

(1) K, is C”, (2) D,K,(p, L, a) has rank (n - k)k.

Proof

(I) This is obvious. (2) For simplicity, let a=id, the identity permutation. To verify (2), we take the derivative of K, with respect to a’(s), 1 g j j k, 15s $n- k. This derivative has the matrix representation

o’(l),...,aql) . d(s), . ,I&), , a’(n -

0 -

s=n-k L 0 I...) 0 Pb--k) _

where P(s) is the k x kl matrix

r p(s)’ 0 0 . . . 0 -

0 P(s)= .

p(s)’ . . . 0 . L- : 0 0 0 . . . p(s)’ - 0

6. The pseudo-equilibrium manifold

Using the construction from the previous section, E = {(p, L, co, a): Z(p, L, co) = 0 and K,(p, L, a) =0 for o such that LE W,}. For each o EC, define H,:

G+ x w, x rwyt, x OX”* + R” x R’” - k)k by H,(p, L, CD, a) = (Z(p, L, w), K,(p, L, a)).

D. Duffie and Wl Shafer, Equilibrium in incomplete markets 295

Fact 8. For each 6, 0 is a regular value of H,.

Proof. Taking the derivative of H, with respect to w1 and a,

By Fact 6, part (3), and Fact 7, part (2), this matrix has rank I* +(n-k)k. 0

Fact 9. & is a submanifold of rW7 + x G+, x rWy’:‘l, x R”’ without boundary of dimension ml* + n*.

Proof. Since 0 is a regular value of H,, H; ‘(0) is by the pre-image theorem a submanifold of R’: + x W, x I!?$: x R”‘, and hence of R’” x G,_ x rWY$ x R”‘, of dimension [1* + k(n - k) + ml*+ n’] - [l* + k(n - k)] = ml* + n*. Clearly, H,‘(O)=(IW~+xW,xIW”:*,xIW”*)nE. Since {IW~+xW,xIW”::x[W”*:~EC) is an open cover of R’: + x G,,, x R’;‘?,. x R”‘, this shows that & is a submanifold of the latter set of dimension ml* + n*. 0

Fact 10. The projection map rc:&+R$ x R”’ is proper,

Proof. Let XcRY!$ x KY” be compact. Suppose (p,, L,, w,, a,) E K- ‘(X), for n=l,2,... . We will show that this sequence has a convergent subsequence in n-‘(X). Since (L,, w,, a,) E Gk, n xX, we can assume without loss of generality that (L,, o,, a,)+@, 0, ii) EX. Since pn. w,’ = 1 for all n and {wi} is bounded, we know {p,> is bounded. By Fact 6, part (4), any limit point of {p,] must belong to rWy +. Thus {(p,, L,,o,,a,)} has a subsequence converging to a point (p, L, 0, a) with p E rWg +

- - and (0, a) E X. We pick a permutation CJ such

that LE W,. Then by continuity, Z(p,L,O) =0 and K,(p, L, 6) =0, implying that (p, L, W, 5) err-‘(X). 0

7. Proofs of the theorems

Proof of Proposition 2. Define 4= eT V(p, a), where e = (1, 1, . . . , 1) E R”. By (c’) of the definition of effective equilibrium, for each i>= 2 there is a 8’ E Rk such that p1 q (Zi,-wi)= V(p,a)8’. Let B’= --cEz8’. Then ((X’,@,@,$) is an equilibrium for (~,a). 0

Proof of Proposition 2. This is trivial.

Proof of Theorem 1. Based on the discussion in section 3 and the results of

296 D. Duffie and WI Shafer, Equilibrium in incomplete markets

section 6, it suffices to find some (ii), ci) such that #n-‘(C&C?) = 1 and such that (C&C?) is a regular value of 7~. We choose an ti and a p such that the last k rows of I’@, ci) are linearly independent, and define W=(W’, . . . , W”) by W’= G’(@, 1). Then ((W’), ~7) is a contingent commodity equilibrium; hence (Wi) is Pareto efficient. Since V(p, 6) has rank k, there is a unique L E Gk_ spanned

- - by the columns of V(p,ii). Since the last k rows of V(p,u) are linearly independent, there is a representative [ZlE] EL such that [ZIE] V(p, ti) =O; that is, LE N$. By Fact 5, ti’=F’(Z& L, W’), so that ((W’),p, L) is an effective equilibrium. By Pareto efficiency and the fact that each W’ is affordable to agent i at any (p,L), this is the only pseudo-equilibrium. Hence ((p,E,~&a)} = ?+(W,Li).

We need to show that (O,ti) is a regular value of 71, which is equivalent to showing that D (p,L~Hid(P, L, Us, E) has full rank Z* +(n- k)k. In order to ease the computation of derivatives with respect to L, we write E = Cp,(L) for any L E W,, and define

F’(p, E) = F’(p, q;‘(E), \iii), qp, E) = Z(P, (Pii W), a,

R(P, E) = Kdp, CP id l(E), 4, R(P, El = @(P, ~3, If(p, J-3.

Since (pi,,, Wid) is a chart on Gk,“, it suffices to show that DR(p,E) has full rank. We have

By Fact 5, part (3)(i), D&p, E) =0, so we need only show that D,Z(p,E) has rank I* and that Z&R@, E) has rank (n- k)k.

D&p, E) has rank (n- k)k: Let V,(& ii) denote the k x k matrix consisting - -

of the last k rows of V(p,a). Then V&7,6) is non-singular by construction. The derivative of R at (Z&E) with respect to any row vector of E is just V,(p, a). Thus Z&Z@, E) can be represented as a block diagonal matrix with I’,(& @) in each diagonal block. Since there are (n-k) diagonal blocks, DE@& E) has rank (n - k)k.

D,Z(p,E) is non-singular: Suppose, to get a contradiction, that there exists h E R” such that

D,Z(j& E)h = 0, h#O

e-Z&G’@, 1) h + 2 D,F(p, E)h = 0, h#O, i=2

D. Dufie and W Shafer, Equilibrium in incomplete markets 291

+D,G’(j& 1)h + f jjro,F’(p, E)h = 0, h#O, i=2

=>G’@, l)*h=O, h#O.

The last implication is a consequence of pTo,G’(p, 1) = - G’(& l)‘, and the fact that pro,F’(j& E) =0 for iz 2. (See Facts 4 and 5.) Since G’(p, 1). h = 0 and h#O, Fact 4 implies that hTD,G’(& 1)h -co. By Fact 5, part (3), O,F’(p, E) is negative semi-definite. Hence hTD,Z(p, E)h -CO, contradicting relation (4) above. Thus D,Z(p,E) is non-singular. lJ

Proof of Theorem 2

(a) Define Q2, to be the set of regular values of rc. By Sard’s Theorem, Q’, is null. Since n is proper, 52, is open. In fact, we can write the ‘stack of records’ theorem as follows:

V(Qti) E 52,, 3 a neighborhood 0 of (@a) in Sz,, and smooth junctions 9;: rf+R’f+ and ~0:: t7+Gk+ i=l,..., ‘I: where T is the number of pseudo-

- - equilibria for the economy (0, a), such that

(i) n-l(o.~, a) = {(cpi(o, a), C&W, a), co, a)?= ,} V(o, a) E 0, and (ii) for each i there is a ai EC such that cp\(w, a) E W,iV(o, a) E 0.

(b) Since HJcp’,(o,a), &(~,a), ~,a)-0 for (O,U)E 0, we can differentiate to

get

(One can identify L with E via the diffeomorphism (PJ to take derivations.) The above equation says that every row of Dco,@,i is a linear combination of the rows of D(rpi,cp\). Since the rows of D co. .,H,i are linearly independent and there are I* +(n - k)k rows, the rows of D(cpi, 40:) must also be linearly independent. In particular, rank (Drpi(o, a)) = l*V(o, a) E 0.

(c) Define Y’: O+R”;+ xR”* by Y’(o,a)=(q~\(o,a),a). Then DY’(o,a) has rank I* + n* at every (0, a) E 0, implying that Y’ is a submersion.

(d) Define P={(p,a)~&!~+ x Rn’: rank(V(p,a))=k}. We view I/ as a’map v: IV; + x R”*-+Rnk, and let -Y c lRnk represent all n x k matrices of rank k. Clearly 9’” is open and has null complement. We note that P = V-‘(r), and that D,V(p,a) always has maximum rank nk. (See the calculation in Fact 7.) Thus V is a submersion, implying P= V/-‘(v) is open in I%; + x UP* with null complement.

(e) Since Yi is a submersion, (Y’)-l(P) is open in U with null complement. Let I?= ni(Yi)-‘(P). ‘Th en 8 is open in 0 with null complement, and

298 D. Dufjie and W Shafer, Equilibrium in incomplete markets

(co, a) E 0 implies that rank( V(cp’, (0, a), a)) = k, for i = 1,. . . , T Thus, for every (W,U)E 0, every pseudo-equilibrium for the economy (~,a) is an effective equilibrium.

(f) Making a standard local to global argument, we get an CJ c s2, with the following properties. First, 52 is open and has null complement in !2,, and hence in Ry’: x R”*. Second, for every (~,a) in Q, there is an odd number of equilibria for the economy (~,a), each of which is locally a smooth function of (%a). 0

8. Generic existence for fixed asset structures

We now indicate some of the problems involved with demonstrating generic existence with a fixed asset structure. In this case the basic difficulty is the absence of parameters with which to perturb the functions K,. The following two examples illustrate, however, two different approaches to overcoming this problem:

Example 1 (commodity forward contracts). Let a be the asset structure for which u(s) = I for each state s, where I is the I x 1 identity matrix. In this case V(p, a) is the n x 1 matrix whose sth row is pi(s)r. One can show generic existence of equilibria for the fixed asset structure a as follows:

(a) In this case D,K, always has rank (n-01, implying that 0 is a regular value of H, with a=& We thus have the pseudo-equilibrium manifold E,= {(p,L,o): (p, L, w,a) EC}. As before, we can examine the projection map 71: &,+ rW:‘t, defined by z(p, L, w) = o.

(b) As before, for most o the pseudo-equilibria will be locally smooth functions of o. The equation K,(p, L, 5) = 0 can be solved for n-Z of the spot price vectors in terms of the remaining 1 spot price vectors. Some standard calculations then show that the functions assigning each o the pseudo-equilibrium values of these 1 spot price vectors are submerions. Then, for most w, these 2 spot price vectors will be linearly independent, implying that V(p,C) has full rank 1. This approach works for any 3 such that rank (C(s)) = k for all s.

Example 2 (commodity forward contracts contingent on an event). Again we suppose k = 1. We also assume that E is given by a(s) =0 for s In- 1 and Z(s) = I for s 2 n - 1 + I. (By reordering the states, one can select any such ‘event’.) It follows that V(p, 5) is the n x 1 matrix whose first n - 1 rows are identically zero, and row s, for szn-Z+l, is just Pan.

(a) In this case D (P, ,,K,(p, L, 5) fails to have full rank for C= id, q&L) =O, and linearly dependent (pl(s)):,,_r+l. Thus, in contrast to Example 1, we cannot guarantee that E, is as manifold.

(‘4

D. Dufjie and W Shafer, Equilibrium in incomplete markets 299

Nevertheless, equilibria generically exists in this case. We note that [IlO] V(p, Cz) = 0 always holds. We take L E G1,, to be the linear subspace induced by [IlO]. Then Z(p,L,w) =0 implies that (p,L) is a pseudo- equilibrium, and that 0 is a regular value of Z. Thus we can take E,t = {p, L, 0): L = L and (p, L, w, ti) E &} to be the equilibrium manifold As before, for most o the pseudo-equilibrium prices are smooth functions of o, and are also submerions. It follows that, for most CD, (pi(s)):,,_ 1 +t

are linearly independent. We note that this approach applies to any asset structure a such that aj(s) ~0 for 15 j5 k and ss n - k, provided the vectors ((aj(s))5= l)i=n-k+ r are in general position.

References

Arrow, K.J., 1953, Le role des valeurs boursieres pour la repartition la meillure des risque% Econometric, 4147, discussion 4748, Colloq. Internat. Centre National de la Recherche Scientilique, no. 40 (Paris, 1952) (C.N.R.S., Paris). Translated in Review of Economic Studies 31 (1964), 91-96.

Balasko, Y., 1975, The graph of the Walras correspondence, Econometrica 43,907-912. Balasko, Y. and D. Cass, 1985, Regular demand with several, general budget constraints,

CARESS workine oaner number 85-20. Universitv of Pennsvlvania. Philadelnhia. PA. Cass, D., 1984, Competitive equilibria with incomplete financial markets, CARESS working

paper number 84-09, University of Pennsylvania, Philadelphia, PA, April. Cass, D., 1985, On the ‘number’ of equilibrium allocations with incomplete financial markets,

CARESS working paper 85-16, University of Pennsylvania, Philadelphia, PA. Debreu, G., 1970, Economies with a finite set of equilibria, Econometrica 38, 387-392. Debreu, G., 1972, Smooth preferences, Econometrica 40, 603-615. Debreu, G., 1976, Smooth preferences, a corrigendum, Econometrica 44, 831-832. Dierker, E., 1974, Topological methods in Walrasian economica, Lecture notes in economics and

mathematical systems number 92 (Springer, Berlin). D&e, D., 1985, Stochastic equilibria with incomplete financial markets, Research paper number

813, Graduate School of Business, Stanford University, Stanford, CA, Jan. Duffie, D., and W. Shafer, 1985, Equilibrium in incomplete markets: II. Generic existence in

stochastic economies, Mathematical Sciences Research Institute working paper, Dec. Geanakoplos, J. and A. Mar.-Colell, 1985, Real indeterminacy with financial assets, Preliminary

draft, May. Geanakoplos, J. and H. Polemarchakis, 1985, Existence, regularity and constrained suboptim-

ality of competitive portfolio allocations when the asset market is incomplete, Yale University, New Haven, CT, Feb.

Golubitsky, M. and V. Guillemin, 1973, Stable mappings and their singularities (Springer, New York).

Guillemin, V. and A. Pollack, 1974, Differential topology (Prentice-Hall, Englewood Cliffs, NJ). Hart, O., 1975, On the optimality of equilibrium when the market structure is incomplete,

Journal of Economic Theory 11, 418-443. Hirsch, M., 1976, Differential topology (Springer, New York). McManus, D., 1984, Incomplete markets: Generic existence of equilibrium and optimality

properties in an economy with futures markets, University of Pennsylvania, Philadelphia, PA, Aug.

Magill, M. and W. Shafer, 1984, Allocation of aggregate and individual risks through futures and insurance markets, Department of Econom&, University of Southern California, Dec.

Magill, M. and W. Shafer, 1985, Equilibrium and efficiency in a canonical asset market model, Department of Economics, University of Southern California.

300 D. D@e and W Shafer, Equilibrium in incomplete markets

Mas-Colell, A., 1985a, The theory of general economic equilibrium, a differentiable approach, Econometrica Society Publication no. 9 (Cambridge University Press, Cambridge).

Mas-Colell, A., 1985b, Written communication: Pseudo-equilibria are generically equilibria, Oct. Kadner, K., 1972, Existence of equilibrium of plans, prices and price expectations in a sequence

of markets, Econometrica 40,289303. Repullo, R., 1984, Equilibrium and efficiency in economies with a sequence of markets, Ph.D.

dissertation, University of London. Werner, J., 1985, Equilibrium in economies with incomplete financial markets, Journal of

Economic Theory 36, 1 l&l 19.