generic regularity of competitive equilibria with restricted participation

Journal of Mathematical Economics 36 (2001) 61–76

Generic regularity of competitive equilibria withrestricted participation�

David Cass a, Paolo Siconolfi b, Antonio Villanacci c,∗a University of Pennsylvania, Philadelphia, PA 19104, USA

b Columbia University, New York, NY 10027, USAc Department of Mathematics for Decision, Universita’ degli Studi di Firenze,

Via Lombroso 6/17, 50134 Firenze, Italy

Received 11 January 2000; received in revised form 2 May 2001; accepted 4 May 2001

Abstract

In this paper, we present a general version of the model of competitive equilibrium with restrictedparticipation on financial markets. Our goal is to accommodate a wide range of portfolio constraintswhile at the same time still permitting (generically) differential analysis of the dependence offinancial equilibria on “fundamental” parameters. © 2001 Elsevier Science B.V. All rights reserved.

JEL classification: D50; D52

Keywords: General equilibrium; Restricted participation; Financial markets; Regularity of equilibria

1. Introduction and motivation

The primary purpose of this paper is to generalize the model of competitive equilibriumwith restricted participation on financial markets (as originally presented in Balasko et al.(1990)). Our goal is to accommodate a wide range of portfolio constraints while at thesame time still permitting (generically) differential analysis of the dependence of financialequilibria on “fundamental” parameters.

There is a large and growing literature in Finance that generalizes the asset pricing modelby allowing for constraints on portfolio holdings. Recently, Broadie et al. (1998) studied the

� This project grew out of an effort to provide a consistent basis for studying the reduced form of simple modelswith both inside financial instruments and outside money, as in Cass (1990, Magill and Quinzii (1988), or Villanacci(1991). Conversations with — and prodding by — Michael Mandler were instrumental in our streamlining the finalproduct.

∗ Corresponding author. Tel.: +39-55-4796-817; fax: +39-55-4796-800.E-mail address: [email protected] (A. Villanacci).

0304-4068/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved.PII: S0 3 0 4 -4 0 68 (01 )00067 -2

62 D. Cass et al. / Journal of Mathematical Economics 36 (2001) 61–76

effect of general convex constraints on portfolio holdings, while Cuoco (1997) and Jouniand Kallal (1999) analyzed the role of convex constraints on the income streams generatedby asset holdings. Most related to our model, Polemarchakis and Siconolfi (1997) provedthe existence of equilibrium for a standard incomplete asset market economy with assetpayoffs denominated in multiple commodities and individuals facing asymmetric linearconstraints on their portfolio incomes.

While Balasko et al. (1990) analyzed only linear homogeneous equality constraints onhouseholds’ portfolio holdings, here, more broadly, we admit any smooth, quasi-concaveinequality constraints. The obvious leading example, non-negativity constraints, representsthe widespread practice of barring short sales. 1 But other, commonly encountered marketrestrictions are also modeled (albeit in somewhat simplified form), for example, obligationsto carry life insurance, or hold compensating balances, or meet margin requirements. Whileit seems likely that our present analysis can be sharpened to incorporate explicit dependenceof the constraints on endogenous variables other than just portfolio holdings, this questionremains to be more thoroughly investigated.

A secondary purpose of the note is to illustrate a fairly general method for encompass-ing various sorts of constraints on individual behavior while still employing differentialtechniques. In particular, it is easily shown that the same sort of argument elaborated herecan be utilized to establish finite, local uniqueness in the standard Walrasian model whenthe consumption set permits zero consumption, or involves joint consumption, or requiresminimum consumption. 2

2. Restricted participation

For our objective here we simply modify the specific version of the Balasko–Cass–Siconolfi model of restricted participation described (in fairly mnemonic terms) by Cass(1990). Brief discussion of several of the most important simplifying assumptions can befound there, as well as in the overview by Cass (1992). There are C types of physicalcommodities (labelled by the superscript c = 1, 2, . . . , C, and referred to as goods) , andI types of credit or financial instruments (labelled by the superscript i = 1, 2, . . . , I, andreferred to as bonds). Both goods and bonds are traded on a spot market today, while onlygoods will be traded on a spot market in one of S possible states of the world tomorrow(these markets are labelled by the superscript s = 1, 2, . . . , S, so that s = 0 representstoday and s > 0 the possible states tomorrow, and are referred to as spots). Thus, altogetherthere are G = (S + 1)C goods, whose quantities and (spot) prices are represented by thevectors

x = (x0, . . . , xs, . . . , xS) with xs = (xs,1, . . . xs,c, . . . , xs,C),

and

p = (p0, . . . , ps, . . . , pS) with ps = (ps,1, . . . ps,c, . . . , ps,C),

1 An alternative approach for dealing with this particular form of restriction is developed by Geanakoplos et al.(1990) in the course of their analysis of constrained suboptimality in a model with an incomplete stock market.

2 For works in this area, see Bonisseau and Cayupi (1995), Shannon (1994) and Villanacci (1993).

D. Cass et al. / Journal of Mathematical Economics 36 (2001) 61–76 63

respectively. The quantities and prices of bonds are represented by the vectors

b = (b1, . . . , bi, . . . , bI )

and

q = (q1, . . . , qi, . . . , qI )

respectively. 3 All prices are measured in units of account, referred to as dollars. Assetsare nominal, i.e. the typical bond, which costs qi dollars at spot s = 0, promises to returna yield of ys,i dollars at spot s > 0.

Since we will be treating bond yields as potentially variable, there is no loss of generalityin normalizing goods prices spot-by-spot; under this convention, variation in ps,1 is simplytranslated into (scalar) variation in yS .

By introducing the above price normalization, we are indeed transforming the modelwith nominal assets in a model with numeraire assets, i.e. assets which pay in units of anumeraire good in each spot. Therefore, under suitable interpretation, our results apply toboth the case of nominal and numeraire assets. Let

y =

y1,1

. . .

ys,i

. . .

yS,I

=

y1

...

ys

...

yS

= (S × I )-dimensional matrix of bond yields.

It greatly facilitates our analysis to assume that

Assumption 1. Rank Y = I (which implies that I ≤ S).

Finally, there are H households (labelled by the subscript h = 1, 2, . . . , H ) who arespecified by (i) consumption sets Xh = RG++, (ii) utility functions uh : Xh → R,(iii)goods endowments eh ∈ Xh and portfolio sets Bh ∈ RI . As in most of the literature on“smooth economies” we will assume throughout that

Assumption 2. uh is C2, differentiably strictly increasing (i.e. Duh(xh) � 0) and differen-tiably strictly quasi-concave (i.e.�x = 0 and Duh(xh)�x = 0 ⇒ �xTD2uh(xh)�x < 0),and has indifference surfaces with closure in Xh.

What distinguishes this model from its predecessor is that now

Bh = {bh ∈ RI : ah(bh) � 0},where the portfolio constraints ah : RI → R

#Jh exhibit the following properties. 4

3 It will be convenient, for example, in representing dollar values of spot market transactions , to treat every priceor price-like (say, for instance, marginal utility) vector as a row. Otherwise we maintain the standard convention.

4 It will be convenient to index the households’ portfolio constraints by j ∈ Jh, where Jh is simply some finiteset of dimension #Jh. Otherwise, as we have indicated earlier, the typical index k simply runs over the first Knatural numbers.


Assumption 3. ah is C2, differentiably quasi-concave, i.e.

Dah(bh)�b = 0 ⇒ �bTD2ah(bh)�b ≤ 0,

does not require participation on the bond markets, i.e. ah(0) � 0, and satisfies the rankcondition

ajh(bh) = 0, j ∈ J ′

h ⊂ Jh ⇒ rank[Dajh(bh), j ∈ J ′

h

]= #Jh

(which implies that #J ′h ≤ I ).

For J ′h ⊂ Jh′ , all h, let

J ′ = ×hJ ′h

and

J′ = {J ′ : there are bh ∈ RI such that ajh(bh) = 0, j ∈ J ′h, all h, and

∑h

bh = 0}.

Also, hereafter simply denote a′h(bh) = (a

jh(bh), j ∈ J ′

h).

Assumption 4. For every J ′ ∈ J′ and every i there is some h such that

a′h(bh) = 0 ⇒ a′

h(bh + (0, . . . , �bi, . . . , 0)) = 0 for �bi ∈ R.

Remark 1. In this context, Assumption 1 is not at all innocuous. When their portfolioholdings are constrained, households may very well benefit from the opportunities affordedby the availability of additional bonds whose yields are not linearly independent.

Remark 2. Assumption 3, like Assumption 2, is designed to facilitate using differentialtechniques. At the cost of fairly moderate complication, the last rank condition in thisassumption can be weakened. Such weakening would be necessary, for instance, in orderto encompass having some linear equality constraints.

Remark 3. In effect, Assumption 4 states that, on each bond market, there is some house-hold who is unrestricted. Permitting this latitude is quite credible; otherwise it is hard tosee why one would want to differentiate a particular bond as a distinct economic object.However, now at the cost of fairly considerable complication, it too can be weakened. 5

3. Financial equilibrium

In order to describe and analyze competitive equilibrium in this setting, it is useful (andalmost unavoidable) to introduce a fair bit of more abstract notation. So, to begin with, let

P = {p ∈ RG++ : ps,1 = 1, all s} = set of possible(normalized) spot good prices,

5 Just for the purposes of exposition we will, when it is convenient, presume that it is household h = i who isunrestricted in transacting on bond market i.


Q = {q ∈ RI } = set of possible bond prices,

Y = {Y ∈ RSI : rank Y = I } = set of possible bond yields, and

E = {e = (e1, . . . , eh, . . . , eH ) ∈ (RG++)

H }= set of possible goods endowments(and allocations).

An economy is described by the utility function vector (uh)Hh=1, the return matrix Y andthe endowment vector e. In our analysis, u and Y are kept fixed.

Now also let

Ψ =

p0

. . . 0

ps

0. . .

pS

= the spot good matrix,R

=[−q

y

]= the overall return matrix � = (p, q) ∈ Π = P × Q

Then, given e ∈ E,� ∈ Π is a financial equilibrium if, when households optimize, i.e.given � (and Y ), (xh, bh) is the optimal solution to

maximize uh(xh) with associated multipliers

subject to −Ψ (xh − eh) + Rbh � 0, λh = (λ0h, . . . , λ

sh, . . . , λ

Sh)

ah(bh) � 0 µh = (µjh, j ∈ Jh)

and xh � 0, all h,

(1)

both spot goods and bond markets clear, i.e.∑h

(xh − eh) = 0 (2)

and ∑h

(bh) = 0. (3)

Recall that, by the Kuhn–Tucker theorem for quasi-concave programming, the optimalsolution to (1) is characterized by the Kuhn–Tucker conditions 6

6 For this particular problem, sufficiency of the Kuhn–Tucker conditions is related to the strict quasi-concavityof uh, and necessity to the full rank of Da′

h(bh) (where now J ′h = {j ∈ Jh : ajh(bh) = 0} denotes the subset of

binding constraints) — given that all the constraints are themselves quasi-concave.


Duh(xh) − λhΨ = 0

λhR + µhDah(bh) = 0

−Ψ (xh − eh) + Rbh = 0

ah(bh) ≥ 0

µah(bh) = 0

xh � 0, λh � 0 and µh ≥ 0, all h.

(4)

Thus, now taking

N = {(�, e) ∈ Π × E : Eq.(4) has a solution},the principal focus of our investigation is on the structure of the equilibrium set

M = {(�, e) ∈ N : Eqs.(2)–(4) have a solution}.Define also

M ⊂ N ⊂ Π × E, andΦ = restriction toM of the projection of Π × E ontoE.

In the background, it is important for the applicability of our analysis that a financialequilibrium exists for every specification of the “fundamental” parameters e). (Note thatnow Φ represents the projection of the equilibrium set M into the “fundamental” parameterspace E.)

Theorem 1 (Existence; (Siconolfi, 1989)). Φ(M) = E.

4. Generic regularity

We are going to establish that, generically in the “fundamental” parameters, financialequilibria are finite in number and locally unique. The essential idea underlying the argu-ment is to imbed the financial equilibria of the original model among those of a related(but artificial) model in which only binding portfolio constraints are accounted for — andtherefore to which standard techniques from differential topology can be directly applied.We will develop this argument through a sequence of steps formalized in four lemmas. Sincethe proofs of the lemmas themselves are rather routine, their details are merely sketched inan Appendix A.

We begin by observing that the concept of a financial equilibrium can always be ex-panded to explicitly include various of the (normally redundant) endogenous variablesxh, bh, λh and µh. In the present case, we find it especially convenient to incorporate justthe households’ spot goods consumption bundles,

x = (x1, . . . , xh, . . . , xH )

as follows. Identify the dummy variable ξ together with x and let

� = (p, q, ξ) ∈ Π = P × Q × E.


Then simply add the variable ξ together with the definitional equation

ξ − x = 0 (5)

to the specification of the equilibrium set,

N = {(�, e) ∈ Π × E : Eq.(4) has a solution}M = {(�, e) ∈ N : Eqs.(2)–(5) have a solution}, andΦ = restriction to M of the projection of Π × E ontoE.

Our goal is to show that for a generic set of economies there is a finite number of equilibriaand that equilibrium prices and allocations are locally a smooth function of the parameters.These results are stated in Theorem 2. The argument is broken in four Lemmas.

Lemma 1. The smooth mapping α : M → M such that (�, e) �→ α(�, e) = (�, e) is ahomeomorphism.

By virtue of Lemma 1 we can, in principal, concentrate on this nominally broader re-formulation. To simplify notation, we will suppress the tildes until the very end of theargument. Now observe that, for every J ′ ∈ J ′, if

ajh(bh)

{>

=

}0 according as j

{/∈∈

}J ′h, (6)

then a solution to Eq. (4) also yields a solution to the system


λhR + µ′hDa′

h(bh) = 0

−Ψ (xh − eh) + Rbh = 0

a′h(bh) = 0

xh � 0, λh � 0 and µ′h ≥ 0, all h,

(4′′)

where µ′h = (µ

jh, j ∈ J ′

h). So, in the first instance, replacing Eq. (4) by Eq. (4′′), define

N ′+ = {(�, e) ∈ Π × E : Eq.(4′′) has a solution},M ′+ = {(�, e) ∈ N ′+ : Eqs.(2), (3), (4′′) and (5) have a solution}, and

Φ ′+ = restriction toM ′+of the projection ofΠ × E ontoE.

(The subscript “+” is intended to underline the importance of the non-negativity con-straints µ′

h ≥ 0 in Eq. (4′′)).

Lemma 2. M ⊂ ∪J ′∈J ′M ′+ and, for every J ′ ∈ J′, Φ ′+ is proper.

Unfortunately, M ′+ itself is not a smooth, boundaryless manifold. In order to overcome

this difficulty, we simply enlarge the set to permit µjh < 0 for some h and some j ∈ J ′

h.


Toward this end, let

z′h = (xh, bh, λh, µ

′h, p, q, Y, eh)∈Z′

h=Xh×RI×RS+1++ ×R#J ′

h×P×Q×Y × Xh

and define a mapping

F ′h : Z′

h �→ RG × RI × RS+1

++ × R#J ′h

such that

z′h �→ (Duh(xh) − λhΨ, λhR + µ′

hDa′h(bh),−Ψ (xh − eh) + Rbh, a

′h(bh).

(F ′h is nothing other than the left hand side of the equations in Eq. (4′′)). Furthermore,

observe that a solution to Eq. (4′′) also yields a solution to the system


λhR + µ′hDa′

h(bh) = 0

−Ψ (xh − eh) + Rbh = 0

a′h(bh) = 0

xh � 0 and λh � 0, all h,

(4′)

(The only difference is that we have dropped the non-negativity constraints µ′h ≥ 0 from

Eq. (4′′)). So, in the second instance, replacing Eq. (4′′) by Eq. (4′), define

N ′ = {(�, e) ∈ Π × E : Eq.(4′) has a solution such that rank D(x,b,λ,µ′)F′h(z

′h)

= G + I + (S + 1) + #J ′h, all h},

M ′ = {(�, e) ∈ N ′ : Eqs.(2), (3), (4′) and (5) have a solution such that rank

D(x,b,λ,µ′)F′h(z

′h) = G + I + (S + 1) + #J ′

h, all h},and

Φ ′ = restriction toM ′ of the projection of Π × E ontoE.

Lemma 3. For every J ′ ∈ J′,M ′+ ⊂ M ′,M ′ = ∅ or M ′ is a smooth, GH-dimensionalmanifold and, for M ′c = {(�, e) ∈ M ′ : (�, e) is a critical point of Φ ′},

Φ ′+(M

′c ∩ M ′+) = Φ ′(M ′c ∩ M ′

+) is closed and null.

At this point, consider the set

E∗ = E\{ ∪J ′∈J′Φ

′(M ′c ∩ M ′+)}.

From the last part of Lemma 3, we know that E∗ is an open, full measure subset of E. Butby combining Theorem 1, Lemma 2 and the first two parts of Lemma 3 we can concludemuch more. In particular, we also know that for every e ∈ E∗,


1. #Φ−1(e) < #∪J ′∈J ′Φ ′−1+ (e) < ∞ (since, for every J ′ ∈ J ′, Φ ′+ is proper while

Φ ′−1+ (e)), 7 and that

2. for every (�, e) ∈ Φ−1(e), there is some J ′ ∈ J ′ and some open neighborhood of (�, e),say, U ′ ⊂ Π ×E, such that Φ ′ restricted to U ′ ∩M ′ is a local diffeomorphism (since M ′is a smooth manifold while e is a regular value of the smooth mapping Φ ′ : M ′ → E).

The only remaining problem, then, is that we need somehow to guarantee first, that such aJ ′ is unique, and second, that for some open neighborhood of (�, e), say,U ⊂ U ′, U∩M ′ =U ∩ M(so that Φ ′ restricted to U ∩ M ′ is identical to Φ restricted to U ∩ M). In short, weneed somehow to further limit e so that the only solution to the system (4) at an admissiblefinancial equilibrium (�, e) ∈ Φ−1(e) ∩ Φ ′−1(e) satisfies both Eqs. (6) and (7)

µ′h � 0. (7)

This last step can be achieved in the following way. Let

µ′′h = (µ

jh, j ∈ J ′′

h ), for J ′′h ⊂ J ′

h,

and contemplate further restricting M ′ by appending the additional constraint µ′′h = 0.

Thus, define

M ′′h = {(�, e) ∈ M ′ : µ′′

h = 0}, and

Φ ′′h = restriction toM ′′

h of the projection of Π × E ontoE.

Lemma 4. For every J ′ ∈ J′, h and J ′′h ⊂ J ′

h,M′′h = ∅ or M ′′

h is a smooth, GH − #J ′′h

-dimensional submanifold of M ′ and, for

M ′′ch = {(�, e) ∈ M ′′

h : (�, e)is a critical point ofΦ ′′h},

4′+(M ′′ch ∩ M ′+) = Φ ′′

h(M′′ch ∩ M ′+) = Φ ′(M ′′c

h ∩ M ′+) is closed and null. 8

So now take

E∗∗ = E∗\{ ∪J ′∈J ′,h,J ′′

h ⊂J ′h

Φ ′(M ′′ch ∩ M ′

+)}.

Then it is easily seen that

e ∈ E∗∗ and (�, e) ∈ Φ−1(e) ∩ Φ ′−1(e) ⇒ Eqs.(6) and(7), all h, (8)

and the reasoning we sketched after the statement of Lemma 3 yields the desired result(since the conclusion in (8) will also obtain in some open neighborhood of (�, e)).

Theorem 2 (Finite, local uniqueness). There is an open, full measure subset ofE,E∗∗ ⊂ E,

such that, for every e ∈ E∗∗,

7 In particular, note that these two properties entail that, for every γ /∈ Φ ′(M ′c ∩ M ′+), Φ ′ can be employed to

construct a finite open cover of the set Φ ′−1+ (γ ), say, Uk, k ∈ K, such that Uk ∩ Φ ′−1

+ = {(�k, γk)}k ∈ K.8 Here it may be useful to be reminded that, when J ′′

h = ∅ and M ′′h = ∅, dimM ′′

h = (SI+GH−#J ′′h ) < dimΓ =

(SI + GH) and therefore, M ′′ch = M ′′

h .


(i) #Φ−1(e) < ∞, and(ii) for every (�, e) ∈ Φ−1(e), there is an open neighborhood of (�, e), U ⊂ Π × E,

such that Φ restricted to U ∩ M is a local diffeomorphism.

Finally, returning to the distinction between M and M introduced at the beginning of thissection, we emphasize that, for (�, e) ∈ U ∩ M , the solution to Eq. (4) is also a solution to


λhR + µhDah(bh) = 0

−Ψ (xh − eh) + Rbh = 0

ajh(bh) = 0, j ∈ J ′

h,> 0, otherwise

xh � 0, λh � 0, and µjh > 0, j ∈ J ′

h = 0, otherwise, all h.

(9)

But from Eq. (9) it follows immediately that α restricted to U ∩ M is a local diffeomor-phism, so that Theorem 2 actually applies to the original model as well.

Appendix A

Here we merely sketch proofs of the four lemmas which are instrumental in establishingTheorem 2.

Proof of Lemma 1. Given the maintained Assumptions 1–4 it can be easily shown that,for every (�, e) ∈ N , the solution to Eq. (4) is unique. So, now represent these solutions bythe mappings

Gh : N → Xh × RI × RS+1++ × R#J ′

h

such that

(�, e) �→ Gh(�, e) = (fh(�, e), . . . ) = (xh, . . . ),

and consider as a candidate for α−1 the mapping

β : M → M

such that

(�, e) �→ β(�, e) = ((f1(�, e), . . . , fh(�, e), . . . , fH (�, e)),�, e) = (�, e).

Then one can readily verify that (i) β◦α = idM,(ii) α◦β = idM and (iii) β is continuous.In particular, the proof of the last property is essentially the same as the proof that Φ ′+ isproper (though focusing on the system (4) rather than (4′′), outlined next. �

Proof of Lemma 2. As we noted earlier, for J ′h = {j ∈ Jh : ajh(bh) = 0}, all h, a solution

to the system (4) also yields a solution to Eq. (4′′). Hence M ⊂ ∪J ′∈J ′M ′+. To see that Φ ′+is proper, consider a sequence eν ∈ E, ν = 1, 2, . . . , with eν → e ∈ E and any associated


sequence (�ν, eν) ∈ Φ ′−1+ (ev), ν = 1, 2, . . . Then on the one hand, from xν

h � 0, all h,and Eq. (2), xν

h also satisfies

xνh =

∑h′

eνh′ −∑h′ =h

xνh′ �

∑h′

eνh′ .

On the other hand, a solution to Eq. (4′′) satisfies the Kuhn–Tucker conditions for Eq. (1)with ah replaced by a′

h and (since ah(0) � 0 ⇒ a′h(0) � 0) the optimal solution to this

problem also satisfies

uh(xνh) ≥ uh(e

νh).

Hence (since cl{xh ∈ Xh : uh(xh) � uh(eh)} ⊂ Xh), without any loss of generality wecan take

xνh → xh � 0, all h.

From here on the rest of the argument is pretty much standard (using various of the prop-erties enumerated in the maintained Assumptions 1–4; we simply verify that the sequenceof solutions to Eq. (4′′) have a limit which is also a solution to Eq. (4′′) (as well as Eqs. (2)and (3)):

• xνh → xh � 0 together with the first equation in Eq. (4′′) ⇒ λνh → λh � 0 ⇒ pν →

p ∈ P such that Duh(xh) − λhΨ = 0;• Y ν → Y ∈ Y and λνh → λh � 0 together with the second equation in Eq. (4′′) (for

h = i) 9 ⇒ qiν → qi, all i;• pν → p ∈ P,Rν =

[−qν

Y ν

]→ R =

[−q

Y

]with Y ∈ Y, eνh → eh and xν

h →xh together with the third and fourth equations in Eq. (4′′) ⇒ bνh → bh such that−Ψ (xh − eh) + Rbh = 0 and a′

h(bh) = 0; and• Rν → R, bνh → bh and λνh → λh together with the second equation in Eq. (4′′) ⇒

µ′νh → µ′

h ≥ 0 such that λhR + µ′hDa′

h(bh) = 0. �

Proof of Lemma 3. As we noted earlier, a solution to the system (4′′) also yields a solutionto Eq. (4′). It follows from application of a well-known argument (see, for example, Theorem1 in (Balasko and Cass, 1991)) that if z′

h is a solution to Eq. (4′′), then — for F ′h as defined

in the text (with DF′h as displayed in Eq. (A.1) below) — rank D(x,b,λ,µ′)F

′h(z

′h) = G +

I + (S + 1) + #J ′h. Hence, M ′+ ⊂ M ′.

9 Bear in mind here and later on that, for simplicity, we are specializing Assumption 4 so that, for h = i,

a′h(bh) = 0 ⇒ a′

h(bh + (0, . . . , �bi , . . . , 0) = 0 for �bi ∈ R⇒ Dbi a′h(bh) = 0.


To see that M ′ itself is a smooth, GH-dimensional manifold, consider the Jacobian of F ′h

DF′h(z

′h) =

D2uh(xh) 0 −Ψ T 0 0 0 0

0∑

j∈J ′hµj

hD2a

j

h(bh) RT Da′h(bh)

T

. . .

−λsH I

. . .

[ · · · λshI · · · ] 0

−Ψ R 0 0

. . .

−(xsH − esH )T

. . .

. . .

bTh

. . .

Ψ

0 Da′h(bh) 0 0 0 0 0

(A.1)

By definition, for (�, e) ∈ N ′,D(x,b,λ,µ′)F′h(z

′h) = G + I + (S + 1) + #J ′

h, so we canemploy the Implicit Function Theorem to represent the relevant solutions to Eq. (4′) by thesmooth mappings

G′h : N ′ → Xh × RI × RS+1

++ × R#Jh

such that

(�, e) �→ G′h(�, e) = (f ′

h(�, e), g′h(�, e), . . . ) = (xh, bh, . . . );

the derivatives of G′h (with respect to the directions (�p,−�q, . . . ,�yS,�eh)) are

described by the solutions to the linear system

DF′h(z

′h)�z′

h = 0. (A.2)

It follows that we can rewrite Eqs. (2), (3), (4′) and (5) more compactly as∑h

(f ′h(�, e) − eh) = 0, (A.3)

∑h

(g′h(�, e)) = 0 (A.4)

and

ξ − (f ′1(�, e), . . . , f ′

h(�, e), . . . , f ′H (�, e)) = 0, (A.5)

that is, so that

M ′ = {(�, e) ∈ N ′ : Eqs.(A3)–(A5) have a solution}.This is a system of G + I + HG equations in the [G − (S + 1)] + I + HG + HG

variables (�, e). However, by virtue of the budget constraints in Eq (4′), S + 1 of theseequations — say, for specificity, those describing market clearing for the first type of good


at each spot — are redundant. Moreover, by virtue of the definitional character of Eq. (A.5),the variables ξ can be perturbed independently of all the others. So, in order to employ thePreimage Theorem to establish that M ′ has the desired smooth structure, it suffices to findsolutions to Eq. (A.2), say,

(�z′h(s, c), all h), all (s, c) with c > 1,

and

(�z′h(i), all h), all i,

such that

∑h

(�xs′,c′h (s, c) − �e

s′,c′h (s, c)) =

{1, (s′, c′) = (s, c)

0, (s′, c′) = (s, c) with c′ > 1

and ∑h

�bi′h (s, c) = 0, all i

and ∑h

(�xs,ch (i) − �e

s,ch (i)) = 0, all (s, c) with c > 1

and

∑h

�bi′h (i) =

{1, i′ = i

0, i′ = i

One can readily verify (from direct inspection of Eq. (A.2)) that choosing, say,

�es,11 (s, c) − ps,c and �e

s,c1 (s, c) = −1, all (s, c) with c > 1

everything else zero, and

�bii (i) = 1

and

�es,1i (i) =

{1/qi, s = 0

−ys,i , s > 0, all i,

everything else zero, works for this purpose. 10

10 One reason for being so pedantic at this point is that the same kind of construction, but involving much moreintricate perturbations, will be required in the proof of Lemma 3. Another is that we want to underscore howgreatly Assumption 4 simplifies this particular argument; a weaker assumption would necessitate considerablymore complicated construction of (�z′

h(i), all h).


Finally, we observe that, by Sards Theorem, Φ ′(M ′) is null, so that Φ ′(M ′c ∩ M ′+) ⊂Φ ′(M ′c) is also null. Furthermore, since Φ ′ is proper while M ′c ∩ M ′+ is closed (becauseM ′+ is closed and det DΦ ′ is continuous — so that the image of 0 by its inverse is closed inM ′+), Φ ′(M ′c ∩ M ′+) is closed as well. �

Proof of Lemma 4. Aside from changes in notation, this basically repeats the proof ofLemma 3. The only point which really requires further elaboration is the demonstrationthat, for given h, we can find solutions to Eq. (A.2), say,

(�z′h(j), all h′), j ∈ J ′′

h ,

such that∑h′

(�xs,ch′ (j) − �e

s,ch′ (j)) = 0, all (s, c) with c > 1,

∑h′

(�bih′(j) = 0, all i,

and

�µj ′h (j) =

{1, j ′ = j

0, j ′ = j

For simplicity, assume that H > I and take h = H (a similar, but messier calculationworks in general), and suppress the argument “j”. Then, choose �z′

h, all h, as follows: fix�q = �ys = 0, all s, and pick �p such that, for h = H ,

(�xh,�bh) = 0,

−Ψ T�λh +

. . .

−λsH I

. . .

�p = 0,

RT �λH + Da′H (bH )T �µ′

H = 0,

. . .

−(xsH − esH )T

. . .

�p + Ψ�eH = 0

�µj ′h (j) =

{1, j ′ = j

0, j ′ = j

and

�es,cH = 0, all (s, c) with c > 1,


and for h < H,

D2uh(xh) 0 −Ψ T 0

0∑

j∈J ′hµjhD

2ajh(bh) RT Da′

h(bh)T

−Ψ R 0 0

0 Da′h(bh) 0 0

�xh

�bh

�λh

�µh

+

. . .

−λsH I

. . .

0

. . .

−(xsH − esH )T

. . .

0

�p = 0

R

(−∑h′

(�bhh′) + Ψ (. . . ,�es,1h ,

∑h′

�xs,2h′ , . . . ,

∑h′

�xs,ch′ ), . . .

)= 0, h = 1,

R

(−∑h′

(�bhh′) + Ψ (. . . ,�es,1h , 0, . . . , 0), . . .

)= 0, 1 < h � I,

�es,ch = 0, all (s, c) with c > 1, 1 < h ≤ I,

and

�eh = 0, I < h < H. �

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