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 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 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 exceptional 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 equilibrium allocations, characterized by the dimension of the equilibrium allocation 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 spanning 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 1x 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 consumption 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)
CT0)
s.t.
po.(x,-wQ+q*8S0,
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$‘= 0.
To simplify the proof of existence of equilibria, we introduce an essentially
equivalent equilibrium concept. Let L denote a linear subspace of R” and
consider the following set of agent problems:
agent 1 solves: maxa,
X
agent iz2
solves: maxni(x)
X
s.t.
p-(x-w’)=O;
s.t.
p*(x-w’)=O,
p&l(x,-W’,)EL.
(2)
A collection ((Xi),p, L) is an effective equilibrium if
(a’) Xi solves (2) given (j, L), 15 ism,
(b’) CiXi=Ciwi,
and
(c’) E is the linear subspace of R” spanned by the columns of V&a).
A final equilibrium concept that is crucial to the analysis is the following. A
collection ((xi), p, L) is a pseudo-equilibrium if it satisfies (a’), (b’), and
(c”) dim(E) = k and L contains the column vectors of V(p, a).
3. Basic results
Let w=(wl,..., W~)E rWy*: denote the list of endowment vectors of the m
agents. We will parameterize an economy by a point (0, a) E Iw!J’:x R”‘. We
are specifically interested in the case 15 k < n. (Generic existence of equilibria
D. Duffie and W Shafer, Equilibrium in incomplete markets
for other cases has been demonstrated
289
in papers cited in the introduction.)
Proposition 1. If ((Xi),p,L) is an effective equilibrium, then there exists some
q~ Rk and 8’ E Rk, 1 s is m, such that ((xi, 8’), (p, 4)) is an equilibrium.
Proposition 2. If ((xi), p, L) is a pseudo-equilibrium and rank( V(p, a)) = k, then
((xi), p, L) is an effective equilibrium.
These two propositions allow us to search for an equilibrium in the guise
of a pseudo-equilibrium
for which V&a) has full rank. That is how we
proceed.
Theorem 1. For every (co, a) E R’$; 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 dimension, 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
there exists a 5~ C and an (n-k)
Fact 1. If LeGk,“,
[Zp-JP,EL.
x k matrix E such that
Proof.
Pick any AE L. Since A has rank (n-k), it has n-k linearly
independent columns. Choose a CJin 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
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)
If rr=a’, then P=Z, so E=E’.
+ E’p’4’]
0
B such
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) cpoo cp; ‘: cp,,( W, n W,,)+rp,( W, n W,,) is smooth for all cr,6’.
(4) Gk,, is compact.
is an atlas for a C” differential StrUCture on Gk,“,
Therefore {K, 4~~)~~~
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 1E]P, = n - k. Thus (pb is onto. To show cpOis bicontinuous, one
uses the calculations in the proof of Fact 1.
(3) From Fact 2, (pb0 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
compact and p(B) = Y/N.
0
set}. Then /? is
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.
(1)
(2)
(3)
(4)
(5)
G’ satisfies:
G’ is C” and homogeneous
of degree 0.
P*G’(~,Y)=_Y.
pTD,Gi(p, 1) = - G’(p, l)T.
hTD,Gi(p, 1)htO for all h #O such that h. G’(p, 1) =O.
and p#O.
lim,.+, \IG’(p,l)/= +co if pears+
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].
293
D. Du@e and W Shafer, Equilibrium in incomplete markets
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)
x
St.
p.(x-W)=O,
[IIcp,JL)]P,p,
0 (X1-Wl)=O].
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,
(4)
If
L, w) = -I.
(P”, L”, on)+@,
(p”, L”, o”)ll = + co.
L, ~5) ~(8R’f
+) x
Gk,, x W7”, and
J?# 0, then
lim,lIZ
D. Duffie and W Shafer, Equilibrium in incomplete markets
294
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
POw, 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 $n- k. This
derivative has the matrix representation
d(s),
.
o’(l),...,aql)
.
,I&),
,
a’(n -
0
s=n-k
L
I...)
0
Pb--k)
0
-
_
where P(s) is the k x kl matrix
r p(s)’
p(s)’
0
P(s)=
L- :
0
.
.
0
0
6. The pseudo-equilibrium
0
0
...
0 -
...
0
...
p(s)’-
0
manifold
Using the construction from the previous section, E = {(p, L, co,a): Z(p, L, co)= 0
and K,(p, L, a) =0 for o such that LE W,}. For each o EC, define H,:
+ R” x R’” - k)k by H,(p, L, CD,a) = (Z(p, L, w), K,(p, L, a)).
G+ x w, x rwyt, x OX”*
D. Duffie and Wl Shafer, Equilibrium in incomplete markets
Fact 8.
Proof.
295
For each 6, 0 is a regular value of H,.
Taking the derivative of H, with respect to w1 and a,
By Fact 6, part (3), and Fact 7, part (2), this matrix has rank I* +(n-k)k.
0
Fact 9. & is a submanifold
dimension ml* + n*.
of
of rW7+ x G+, x rWy’:‘l,
x R”’ without boundary
Proof. Since 0 is a regular value of H,, H; ‘(0) is by the pre-image theorem
a submanifold of R’: + x W, x I!?$: 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
has a subsequence converging to a
belong to rWy+. Thus {(p,, L,,o,,a,)}
- point (p, L, 0, a) with p E rWg+ and (0, a) E X. We pick a permutation CJsuch
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’,@,@,$) is an
equilibrium for (~,a).
0
Proof of Proposition 2.
Proof of Theorem 1.
This is trivial.
Based on the discussion in section 3 and the results of
296
D. Duffie and WI Shafer, Equilibrium in incomplete markets
section 6, it suffices to find some (ii),ci) such that #n-‘(C&C?) = 1 and such
that (C&C?)is a regular value of 7~.We choose an ti and a p such that the last
k rows of I’@, ci) are linearly independent, and define W=(W’, . . . , W”) by
W’= G’(@,1). Then ((W’),~7)is a contingent commodity equilibrium; hence (Wi)
is Pareto efficient. Since V(p, 6) has rank k, there is a unique L E Gk_ spanned
- by the columns of V(p,ii). Since the last k rows of V(p,u) are linearly
independent, there is a representative [ZlE] EL such that [ZIE] V(p, ti) =O;
that is, LE N$. 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),
R(P,
E)= Kdp, CPidl(E),4,
qp, E) = Z(P,(Pii W), a,
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,
i=2
E)h = 0,
h#O,
D. Dufie and W Shafer, Equilibrium in incomplete markets
+D,G’(j&
1)h + f jjro,F’(p,
E)h = 0,
291
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
lJ
relation (4) above. Thus D,Z(p,E) is non-singular.
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
V(o, a) E 0,
and
(i) n-l(o.~, a) = {(cpi(o, a), C&W, a), co,a)?= ,}
(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 Dco..,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.
Then DY’(o,a)
Define Y’: O+R”;+ xR”* by Y’(o,a)=(q~\(o,a),a).
rank I* + n* at every (0, a) E 0, implying that Y’ is a submersion.
(c)
has
(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 lRnkrepresent 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).
‘Then 8 is open in 0 with null complement, and
298
D. Dufjie and W Shafer, Equilibrium in incomplete markets
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.
(co, a) E 0
(f) Making a standard local to global argument, we get an CJc 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:
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:
Example
(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.
D. Dufjie and W Shafer, Equilibrium in incomplete markets
299
(‘4 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 pseudoequilibrium, 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.
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