The degree of indeterminacy of equilibria with incomplete markets

J(KJRNALOF

Mathematical ELSEVIER

Journal of MathematicalEconomics29 (1998) 109-123

The degree of indeterminacy of equilibria with incomplete markets . Ryo Nagata School of Political Science and Economics, Waseda Unit,ersi~', Nishi-Waseda 1-6-1 5hinjuku-ku, Tol,.'vo 169-50. Japan

Received l March 1996; accepted 18 November 1996

Abstract This paper investigates generically the indeterminacy property of a financial equilibrium set with incomplete nominal asset markets in both endowments and utility functions. A permissible utility function is not required to have any convexity, so that the resulting equilibrium set is intuitively expected to become very large. It is, however, proved that generically the degree of indeterminacy of equilibria does not exceed the deficiency in financial asset markets. It is also demonstrated that there is a functional relationship between financial equilibria and an economy parame~,er. JEL classification: C60; D52 Keywords: Nominal assets; Indeterminacyof equilibria; E:~tendedfinancial equilibrium; Deficiency in asset markets

1. Introduction When the number of nominal assets that pay some fixed amounts in units o f accounts as returns is lower than the number of states of nature, then the asset market is called incomplete. It has been shown that in this case competitive equilibria (financial equilibria) are not locally unique and generically contain a smooth manifold with a dimension that is determined by the treatment o f prices o f

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R. Nagata / Journal of Mathematical Economics 29 (1998) 109-123

nominal assets (see, especially, Balasko and Cass, 1989; Geanakoplos and MasColell, 1989; and Werner, 1986, 1990). In earlier works, endowments are taken as a parameter in considering the genericity. We shall prove gene:%ity of a financial equilibrium set in a space of utility functions as well as endowments for economy parameters. Although acceptable utility functions are assumed to be smooth, strictly monotone and satisfy some technical condition, no convexity is required of them. In these circumstances, even demand functions may not exist. Therefore, we shall directly use utility functions and work with the first-order conditions to characterize financial equilibria (the idea of using first-order conditions is attributed to Smale, 1974, and the same approach was adopted by Cass and Citanna to inquire into some welfare aspect in incomplete markets; see Cass and Citanna, 1994). As a result, our equilibrium set becomes an extended equilibrium set that contains the intrinsic financial equilibria. Since the extended equilibria would constitute a very huge set, it may be expected that the degree of their indeterminacy would increase proportionally. It turns out, however, that this is not the case from a generic viewpoint. Given the prices of nominal assets, it is proved that the extended financial equilibrium set is generically a manifold with the same dimension as the one that Balasko and Cass obtained generically for the equilibrium set in endowments on the basis of an economy with given nominal asset prices and strictly quasi-concave uti!fty functions (Balasko and Cass, 1989). They proved that the dimension is equal to the difference between the number of states and the number of assets, called deficiency in financial asset markets, which implies that the degree of indeterminacy is typically equal to the degree of incompleteness of the markets. It follows from our result that as far as the extended equilibrium set is concerned, this fact generically carries over even when preferences are not required to be convex. Our result also claims that the intrinsic equilibrium set has generically the maximum degree of indeterminacy that is equal to the deficiency. We also investigate how the extended set is related to a choice of parameter values and prove that generically in utility functions any extended financial equilibrium point is locally continuous with respect to endowments. The paper is organized as follows. Section 2 describes the basic model. The main feature of the model is that the returns and the prices of nominal assets are exogenously fixed. In the model, each agent's utility function is taken to be a parameter, as alluded to above, and the functional space is accommodated so as to facilitate the analysis of genericity. Then we define the extended financial equilibria by using utility functions, not using demand functions. Section 3 is devoted to ar analysis of the extended financial equilibrium set. There we present our main results on genericity of the degree of indeterminacy for both the extended equilibrium set and the intrinsic equilibrium set. The dependence of the extended equilibrium set on an economy parameter is also investigated and some continuity relation is obtained. Section 4 concludes. The proofs of the propositions in the text are presented in the appendix.

R. Nagata / Journal of Mathematical Economics 29 (1998) 109-123

i !1

2. The model We consider an exchange economy expanding over two periods (t = 0, 1) with uncertainty over the state of nature in the second period. There are L consumption commodities (1 = 1. . . . . L) all of which have spot markets at time 0 and in any of N states (n = 1. . . . . N) at time 1. Spot markets at time 0 ate labelled n = 0 since time 0 is thought of as a determinate state of nature. There are M types of financial assets (m ~ - I , . . . , M ) that are traded at time 0. We only consider nominal assets, i.e., each asset pays a fixed amount in units of account in each state at time 1. The return that asset m promises at state n is expressed by r,~m. We write I"# = (i-,,l . . . . . r,,u) t, where the superscript t means transpose (n = 1. . . . . N). These vectors amount to an M × N matrix R = ( r t . . . . . r n ) called a return matrix. When M < N, the financial asset market is incomplete. That is the situation we consider throughout in what follows. Spot prices of commodities and assets will be denoted by vectors p = (P0 . . . . . P,~. . . . . PN) ~ [~t+~++1)~ (with Pn = (P,,I . . . . . P,a.)) and q =(qE . . . . . qu) ~ R~+I+, respectively. We make the following assumptions:

Assumption 1. No redundancy: rank R = M. Assumption 2. Non-arbitrage: for the return matrix R, q is a non-arbitrage price vector. We take any R and q fulfilling Assumption l and Assumption 2 and fix them. There are H consumers (h = 1. . . . . H). The consumption vector of each consumer h will be denoted by x h = (Xoh,..., x,,,h.. ., X~) with X,,h_-- (X#I ,h ... , x,a).h Note that x h is taken as a vector in R (N+ l)t.. His (or her) preference is described by a utility function u h defined on consumption vectors. An allocation of consumption vectors of all consumers will be presented by an ( N + I)LH vector x = ( x i . . . . . x u) and an array of utility functions by u = ( u 1. . . . . uU). The portfolio of each consumer, h, will be denoted by 0 I' = ( 0 ~ . . . . . oh), where 0~ > ( < ) 0 implies the purchase (sale) of asset m. Portfolio allocation is presented by 0 - ( 0 ~. . . . . 0 u ) . Each consumer h also has an initial endowment vector eh=(e~ . . . . . • h ) (with e,,h = ( e , lh, . .. , e~L)) an allocation of which will be denoted by e = (e l . . . . . eU), which is also taken as an ( N + I)LH vector. A state of economy is characterized by a tuple (x, p, 0), while an economy itself is parameterized by (e, u). We define these parameters as follows:

Initial endowment space (W). Initial endowments are restricted to be strictly positive so that the admissible set of e (denoted by W) is ~(~'+ ~)Lu Utility function space (~Q). The utility function of every consumer is assumed to (1) be defined on W+~++ I)L (2) be C'-differentiable, (3) be strictly increasing, i.e. Duh( x h) :~ O, V X h E ~(+N++!)/.,

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R. Nagata/Journal of Mathematical Economics 29 (1998) 109-123

(4) be Cr-bounded, i.e., first r(>_ 0) derivatives are all bounded and (5) have non-singular Hessian. The set of utility functions fulfilling these conditions will be denoted by U. The admissible set of u (denoted by O ) is, therefore, the H-product of U, i.e. 1-[ HU.

Remark 1. No convexity is contained in the above conditions. (5) is included for simplicity and can be relaxed (see the proof of Lemma 1). Remark 2. It is known that the set of CLbounded functions forms a Banach space with a canonical norm (see Abraham and Robbin, 1967, p. 24). Since the set of functions fulfilling (3) and (5) in addition to (1), (2) and (4) is obviously open in the Banach space, U is a Banach manifold. Consequently, O is also a Banach manifold. Although the admissible sets of x, p and 0 are WN+ +÷ I)LH, ~(N+÷t)L and R "M respectively, we will denote them by X, P and O for simplicity and call X X P X O the state space. However, W X O will be called the economy space. Next, consider a consumer's behavior and a financial equilibrium. Each consumer h faces the following optimization problem: max

u"( x t')

s.t.

PoX~' + qO" = poe~,

(1)

p.x,l~ =p,,e[~ + r,,O h,

n = l . . . . . N.

The first-order necessary conditions for the solution are

u ~ t - A p o t----O, h u,,! -

#,, P,,t = 0,

I = 1. . . . . L, l = I . . . . . L, n = 1. . . . . N,

- Aq,, + / T'~ ] ~ , r,,,,, = 0,

m=l

(2)

. . . . . M,

~n=! ] h

p o x o + qO

h __ _

-po

e ts

o,

t,,,x'~ =p.e,~ + r,,O",

(3)

n = 1 . . . . . N,

where u,,Ih is short for a u t ' / d x , l ( n = O, I, . . . , N, 1 = 1. . . . , L) and A, /xn(n = 1 . . . . . N) are Lagrange multipliers. The main aim of our approach is to make use of these conditions as a whole. However, market-clearing conditions are H E " x.=

h=l H

H E " e., h=l

T'~ 01' = 0. h=l

n--O.

1. . . . . N,

(4)

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An intrinsic financial equilibrium for a given (e, u) is, of course, a tuple (x, p,O) satisfying (1) for all h and (4). An exact analytical representation of the equilibrium is, however, too involved to work with.

Remark 3. Since no convexity is required for a utility function, demand might not even be a well-defined function. Thus, we are not able to turn the market-clearing condition into a usual equation system on ( p, O) by way of demand fimctions. So we shall adopt a different approach to indeterminacy from the one based on the Walras test, where the Walras test consists of 'counting equilibrium equations and unknowns' (see Magill and Shafer, 1991, pp. 1566-1567, and Cass, 1992, Appendix). Instead, we consider a broader notion of an equilibrium.

Definition (extended financial equilibrium). An extended financial equilibrium for a given (e, u) is a tuple (x, p, 0) satisfying (2), (3) for all h and (4). The set of extended financial equilibria for (e, u) will be denoted by FE~(e, u), which obviously contains intrinsic financial equilibria. We investigate some property of FEc,,(e, u) in what follows. To this end, we exclude Lagrange multipliers from (2) to obtain

u,t',t/pnt - u,'~,/p,,, = 0, n = 0, 1 . . . . . N. ! = i . . . . . L, N

(2')

E (uh,/Pn,)r,,,,,- (U~l/Po,)q,,, = 0 , m = i . . . . . M. n=l

It is obvious that the equation system (2'), (3) for all h and (4) is linearly homogeneous in ( p , 0). So we normalize the variables by taking the first good at the first spot to be the numeraire, i.e. we put Po~ = l. The normalized variables Pnt/Poi, O/pol will be denoted by P'nt, 0', respectively. This normalization requires transformation of the state space X × P × O into X × P ' × O' where P' = { P ~ I~t+S++I)LI Pol = 1}, which is equivalent to I~~N+~L-I++ and t9' is substantially nothing but I~ HM. FEcx(e, u) should also be interpreted to be defined on the altered state space. It is helpful to think of FEex(e, u) as the intersection of two sets, one being the set of (x, p') satisfying (2') for all h and the other the set (x, p', 0') fulfilling (3) for all h and (4). Note that the former set is solely dependent on u (all consumers" utility functions) and the latter on e (all consumers" endowments). We first investigate these two sets generically on e or u, then consider some generic property of their intersection on (e, u).

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3. The structure and generic properties of the extended financial equilibrium set Our first task is to investigate the generic properties of the set of (x, p') satisfying (2') for all h. To this end, we consider the map tr : O --->C r ( X × P', Y), where Y = II~cN÷ ~tt.- I~tt × RuM and the degree of differentiability r is assumed to be large enough: o-(~,)(x ~..... x", p0 ..... Z'N)

[Ch

h

]

u,,t/Pnl -- Unl/P,l ) h= ! ..... tt. ,,= 0..... iV. l= 2..... 1. tt~,/p,,, )

.ii in

-- u~lqm

t

h= I . . . . . H , m = 1. . . . . M ]

tr(u) will be denoted by ~, in what follows. For this map the evaluation map is defined as follows (for the evaluation map see Abraham and Robbin, 1967, Chapter 2).

evtr:O×(X×P')~Y

ev tr(u, (x, p ' ) ) = ~,(x, p'). For the evaluation map we have Lemma 1. 0 is a regular vahte o f ev o'. Proof. See the appendix.

This lemma leads to the following proposition: Proposition 1. There exists an open and dense set ~2 o f .O such that f o r any u ~ ~ the set ( x , p') satisfying (2') f o r all h constitutes an ( N + I X L + H ) HAl - I dimensional submanifold o f X × P'. Proof. By the Transversality Theorem (see Abraham and Robbin, 1967, pp. 46-48) there is an open and dense set .0 of O such that for any u ~ ~0, 0 is a regular value of o-,,. Therefore, from the Preimage Theorem (see Guillemin and Pollack, 1974, p. 21) ~-~(0) is a submanifold of X × P', the dimension of which is equal to dim(X × P ' ) - dimY= ( N + I)(L + H ) - H M - 1. It is obvious that ~:-l(0) represents the set (x, p') satisfying (2') for all h. I:IQED.

For a given u ~ ~ the set (x, p') satisfying (2') for all h will be denoted by ~,, in what follows. Next, consider the generic property of the set of(x, p', 0') satisfying (3) for all h and (4). First, note that (3) for all h and (4) includes redundant equations. There are many ways of excluding them. The way taken here is to omit the market-clear-

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115

ing condition for the first good at every state. For a given e ~ W we define ~b,,: X X

P' X O' ~ R ~'v+ IXH+L- l)+M by , poe

h

, h - qOh' ) h= J..... tt - poXo ( p-,n ~_h -, -h r,,Oh,~l h= 1. . . . . H, n= 1. . . . . n - - p n . ~ n -4r dp,(x, p', 0 ' ) =

h

x" - ~"~e., I

nl

l=2 ..... L,n=O ..... N

h=l H

~ " 0 h, h=!

Lemma 2. For any e ~ W, dp~ is a submersion. Proof. See the appendix.

From the lemma we obtain the following fact. Proposition 2. For any e ~ W the set (x, p', 0') satisfying (3) f o r all h and (4) constitutes an ( N + IX LH + I - H ) + H M - M dimensional submanifoM o f X x P' X 19'. Proof. Since 0 is a regular value of ~b,. by the above lemma, the claim is followed

by the Preimage Theorem.

[]QED

Remark 4. For any e ~e W, 4'~-I(0) ~= O since ~,.(e, p', 0) = 0, Vp' ~ P'.

For a given e ~ W the set (x, p', 0') fulfilling (3) for all h and (4) will be denoted by ~ in what follows. To investigate the generic properties of FEcx(e, u) (the extended financial equilibrium se0 il~ (e, u), it is helpful to express the set by using .Y,, and ~,. as follows: FEcx ( e, u) : ( .Y,, X 0 ' ) N q~..

Thus it turns out that the structure of FE~x(e, u) depends on how ~,. and .Y~XO' intersect each other. To examine the way of intersection, take any u among t'] and fix it. Considering Proposition 1, .Y,,x O' for u is obviously a submanifold of X x P ' X O' with dimension ( N + IXL + H ) - 1. Then it matters how ~, intersects with .~,, X O'. We consider the problem from the generic viewpoint on e. ^

Proposition 3. Given u E O, ~ f o r almost all e o f W intersects transL;ersally with .~. x O' where "almost all" means all except f o r a closed set o f measure O. Proof. See the appendix.

We are now in a position to state our main results.

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R. Nagata/Journal of Mathematical Economics 29 (1998) 109-123

Theorem 1. Generically in endowments and utility functions, the set of extended financial equilibria constitutes an N - M dimensional manifoM. Proof. The intersection of two transversal submanifolds is a submanifold, and the codimension of the intersection is the sum of the codimensions of the two submanifolds (see Guillemin and Pollack, 1974, p. 30, Theorem). Thus, from Proposition 3, for any u of .0 and almost all e of W, FE~x(e, u) constitutes a submanifold in X × P' × 61' the dimension of which is equal to dim(X × P' × 61') - (dim(,~,, × 61') + d i m ~ ) = N - M. I:IQED Remark 5. N - M is known as the deficiency in the financial market (Balasko and Cass, 1989; Cass, 1992). Considering the characteristics of the notion of equilibrium under investigation, some implications are given by the theorem. First, note that the equilibrium under consideration consists of three factors, i.e. allocation of consumption vectors (x), price vector ( p ' ) and portfolio allocation (0'), so that the real (resp. nominal) phase of the equilibrium is clarified by focusing only on the first (resp. second) factor. The set of equilibrium allocation of goods is, therefore, represented by the image of the projection of the equilibrium set on X. The theorem implies that the dimension of the image is at most N - M. Otherwise stated, real indeterminacy of extended financial equilibria cannot be above the deficiency in the financial market. The same is true for nominal indeterminacy of equilibria for the same reason. Secondly, the set of extended financial equilibria could grow considerably, for two reasons, one being no requirement of convexity in utility functions and the other the fact that Eq. (2') is only a necessary condition for utility maximization. It is, therefore, intuitively expected that among others the set of equilibrium allocation of goods would be increased to a great extent. Ho~,ever, it is shown from the above argument that the dimension of the set has the maximum N - M. Next, we consider FEex(e, u) to be a correspondence,( e, u) ~ FEex(e, u), and investigate its structure. To this end, take any t-i among ,O and fix it again. Let the set of e ~ W such that ~ × 61' and ~,. are transversal be W(~), which is obviously open and dense in W. Then a particular relatic,n is obtained between e of W(~) and a point of FE~x(e, -~).

Theorem 2. Given any -~ of ~ , for any ~ of W(fi) and anff (To, ~', -O')_of FEex(~, -~) there exist some neighborhoods N(~) of~, N(£, ~', 0') of (~, ~', 0') and a C r map 71" N(~) ~ N(Yc, ~', "0') such that 71(e) is an element of FE~x(e, "~) for all e ~ N(~) and "q(~,) = (~, ~, 0'). Proof. See the appendix. The theorem says that for almost all u, everywhere on the set of e that makes FE~(e, u) an N - M dimensional manifold, any equilibrium point (x, p', 0') of

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117

EFex(e, u) is continuous in e. Roughly speaking, every point included in the extended financial equilibrium set is continuous in initial endowments. 4. Concluding remarks In this paper we examined some generic properties of a financial equilibrium set in utility functions as well as initial endowments. The basic result we established is that the extended financial equilibrium set generically constitutes a manifold with dimension equal to the deficiency in the financial market and is continuous in an economy parameter (initial endowments). Balasko and Cass (1989) investigated the generic properties of the se~s of financial equilibrium price vectors and allocations for a given asset price vector and a given return matrix, which is the same framework as adopted in this paper. They were able to sort out intrinsic financial equilibria to consider since every utility function was required to satisfy strict quasi-concavity. The main result they obtained was that generically in endowments the set of financial equilibrium allocations contains a manifold with dimension equal to the deficiency in the financial market. Comparing their result with ours, a suggestive fact is dra ~n out. That is, even if the consumer's utility function is not strictly quasi-concave, for almost all cases real indeterminancy of financial equilibria is not alcove the deficiency in the financial market. Moreover, the same is true for iaominal indeterminacy. Alternatively put, although the set of financial equilibria would be intuitively expected to grow considerably without strict quasi-concavity on the consumer's utility function, generically dimensional expansion does not take place for both allocation and price system so long as the function is strictly increasing. Indeed, we required C~-boundedness on the utility function, but it is only a technical qualification to facilitate the analysis and not crucial for the result obtained. Without C~-boundedness utility could be infinite only for a consumption vector with an infinite norm as long as C~-differentiability and strict monotonicity are assumed to the utility function. Except for an insignificant case of infinity of e (initial endowments) an allocation including an element with an infinite norm cannot be an equilibrium, so a non-C~-bounded utility function can be replaced by some C~-bounded function without any change in the set of equilibria. Finally, note that our approach possesses general validity. Thus, for the case of variable asset prices (Geanakoplos and Mas-Colell, 1989) generic analysis of the financial equilibria in utility functions and endowments would be successfully accomplished by our approach.

Acknowledgments I am grateful to Professor Kazuo Nishimura for his guidance and comments at the revision process. I am also indebted to an anonymous referee and the editor for

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R. Nagata / Journal of Mathematical Economics 29 (19981 109-123

helpful comments and suggestions. Needless to say, I alone am responsible for any remaining errors.

Appendix Proof of Lemma 1. Let the set of C'-bounded functions in C'(Rt+~+ ~)~ 1~) be B. Then note that I-I ttB is the tangen space of O at each u o f / 2 . First, consider the derivative of the evaluation map o- at (u, (x, p')). By the construction of g, we obtain Dev or(u, ( x , p ' ) ) ( r ' , ( y , z))

= D~ev or( u, ( x , p ' ) ) v + D.,ev o ' ( u , ( x , p ' ) ) ( y, z) = or,(x, p ' ) + D ( , . , , ) g , ( y ,

z),

where v is an element of HUB and (y, z) of ~(N+ j)t.u X Te, P'; that is, the tangen space of X × P' at (x, p'). Note that an element of Tp, P' is represented by a vector of the form (0, a 2 . . . . . aCN+ ~)~). TO prove the claim of the lemma we need to show that Dev or(~i, (.~, ~')) is surjective at any (~, (£. ~')) ~ ev o'-z(0). To this end, we shall demonstrate that for any element (o~, /3) ofl~ (N+ I×L-I)H × I~HM which is the tangen space of Y at 0 ~ Y, there exist v ~ I I n B and (y, z ) ~ R(N+l)t'n × Tp, P' such that or,.(~, ~') + D(:~.~,)~(y, z ) = ( a , /3). o~ will be described e l e m e n t w i s e as ( Olnl)h h = i . . . . . H , n = 0 . . . . . N . I = 2..... L and /3 as (/3,1',)r,= i..... n.m = z..... m in what follows. Step 1. First consider (y, z). It is obtained by calculating that the matrix representation of D(x.p,)g , (the ( ( N + IXL - I ) H + HM) × ( ( N + I ) L H + ( N + I)L) matrix) is of the form:

Al

AH [ r . - q ' ] o . , [u ° .....

[ r . - q ' ] o . , [u ° ..... u

l.

R. Nagata / J~urnal of Mathematical Economics 29 (1998) 109-123

119

where --h

r

t

--h

t

".,.o,/p., ) - (,i~, .o,lp-., ) ..... (".,.oLIP.,)

An = ((

(U.tm,-h/ p , a )

• .. -

,

I

_ (ff.,.lv,/p,,,) . . . . ,

1XL-

(~,.oLlp.,).r

(U,,t.Nt/p.,)t --h

,

p,,,)).=o. that is. an ( N +

-

.....

I)X(N+

I ) L matrix (h = 1. . . . . H ) and

l /P'll

l/'p'2~

o

Dp. =

..

o

(diagonal matrix)

1/p'N, I

and U" (n = 0, ! . . . . . N) of [U ° . . . . . UN]h is an ( N + i) × L matrix of the form

!

..... ~hl,nl

.....

'Ol,nL /

From Assumption 1 and qualification (5) of ,12, each M × ( N + I)L submatfix [R, -qt]D#[U° . . . . . UN]dh has a maximal rank M. (Note that the latter condition (qualification (5)) is only a sufficient condition for [U ° . . . . . uN]h tO have maximal rank and it is unnecessarily tight as such.) Take z = 0, then there exists an appropriate y = (yl ..... yn) with yh ~ I R ( N + I ) L ( h = ! . . . . . H) such that ([R, - q t ] D p , [ U 0. . . . . u N ] ! ( y l ) t . . . . . [ R, - q t ] D p , [ U 0 . . . . . U N ] H ( y H ) t ) _ 1~" ( y , 01 is the desired tangen vector. Step 2. Our task now is to choose some v so that ~(.~, ~') can make up the deficiency a - (A~(yl) t. . . . . A , ( y n ) t ) . To this end, pick up and fix any C" function f : R+---. R satisfying (i) D f x 4: 0, Vx ~ R+. and (ii) C~-boundedness. We define a C r function v h" Rt+s++ t)L...,, I~ (h = 1. . . . . H ) by

h

h

)

4" ~,~,f(x,2 + ' "

h

h

)

h

h

+PNLF(XNL),

+ ~2f(X~v2) +"" r t

h

+ I,t,Lf( X,L

~

-h

t

--h

,

r

-

where lX,~t = p,ttot,j - ~,~=o~,~=l y j k ( ( u . L j J p , a -- ( U , l . j ~ / p , , i ) ) ) / f (X,t). n = 0, 1 . . . . . N, l = 2 . . . . . L with p ~ ffi 1. Note that v t' is an element of B since it is

R. Nagata / Journal o f Mathematical Economics 29 (1998) 109-123

120

obviously Cr-bounded ( h = 1. . . . . H ) . B y using these v h we obtain the desired v = {t,'l . . . . . v H) for it is easily checked that

¢,.(~, F)

[

+ (

),,= ,..... ,

.... o

.....,,.,o .....,I

|

--I ("Etu,,,/P,,,) -h

~n=

t

1

-h r,.... -(Uot/1)q,,,

] h = ! .....

I

P',,t/P,,l)f ( x,,t)),,=,

J

H.m= I . . . . M

..... n .... o ..... N.,=2 ..... L

(o) so that o;.(.~, F ) + D(.v.~')o;-~(.~, O) = ( a , f l ) .

EIQED

Proof of Lemma 2. Direct calculation reveals that for any given e of W, the derivative of 4~ at any (x, p', 0 ' ) ~ X × P ' × O' (matrix representation) has partially those ( N + I)H + ( N + IXL - !) + M columns that lead to the following square matrix that has no operating part on Tp,P': 0

-"1 "Po~' ""-POE' -Ptl

- q l .... qt~

0

.... PtL

0

r l ! " " r lid

0 0

• piJ I" """ Pt~,"

I'i'| I "" "I'NM

i .......................................

~Plt' ... "~'IJ I"

0

0

0

-I "Pll' .,,

OI

.....

0 "'"

0 I)

"PNI'

(AI)

0

1

0 1 0

0 "'"

O0

I OI

0

0 ,.,

0 0 l)

1

I 0

0 "'"

It is easily seen that the square matrix is non-singular. However, obviously Tx~(N+++I ) L l l _ ~ ~(N+ I)LH and T¢ 0 ' - ~ H M . So Dqb~,~,.,,¢) (the derivative of ~b~ at

R. N a g a t a / J o u r n a l o f M a t h e m a t i c a l E c o n o m i c s 2 9 ( 1 9 9 8 ) 1 0 9 - 1 2 3

121

(x, p', 0')) is surjective. Since (x, p', 0') is taken arbitrarily, the claim follows. EIQED

Proof of Proposition 3. For any e of W we define a map F: @,. X .++ll~tN+t)Ln~ X X P' × (9' by F((x, p', 0'), e ) = ( x + e, p', 0'). Step 1. Consider a composite map f --- H. F : ~e × I ~ + OLU_, X × P' consisting of a projection map H : X X P ' × O ' - - , X × P ' and F. We show that f is a submersion. It is easily checked by calculation that the matrix representation of the 0erivative of f at any ((x, p', 0'), e ) ~ tbe × {~+N++l)Ln is as follows: (N+I)LH

(N+I)L

(N+I)LH

HM

1 I

',o 1

0

(N+I)LH 0

0---

o I o -.. I

!

.

(N+I)L

0

I

0

.

.

.

.

.

.

I!

°l °

0 "-i

J

We have to show that the image of (T~x,f.o,)@e) × R oN+ I)Ln by the above matrix as a linear transformation coincides with [~( N + I ) L H × Tf P ,t which is the tangent space of X × P' at (x + e, p'). It is obvious from the construction of the matrix that the image covers R ('v+ ~)Ln. To see that TvP' is also covered, investigate the construction of Tc~.p, 0~qb~. Obviously T~x.f.0,~e c T~X× Tp, P'× To, O', on the one hand. On the other hand, T~x.f,o,)~e = KerDck~,~x,p,,o,j (Ker means kernel). Recall that D~be,
•

=

=

l(x, p', o') xxP'x

0'[

v,

(_,_h Po¢o _,_h

_,J,

-PoZo

_,_h

PnCn --~nJ~n

X

~,hfl

= q 0t~

) h:- t . . . . .

--r n Val~'1]h~-

~

H,

1 . . . . . H. n = i . . . . . N "

OJ~,ffi0

h

enl

I ~-

hfl

n=0

.....

N , I = 2 . . . . . L. h = !

122

R. Nagata / J o u r n a l o f Mathematical Economics 29 (1998) 109-123

we have 1

~(¢,~) = {(.', p'. 0') ~ x × P ' x h

-PoXo = q0~)~,= ~..... tt,

, h + e,',') (P,,(e,, ~.h, "~nt ~--" ~,h=l

O't

-, __,~-~- - r

~Pn'~n

n

0~

,lh~l

..... H.n=l

h h en~ + enl

h=|

..... N'

O h,

, n=O .....

N,/=2 .....

L

= 0

h=!

From Lemma 2, ~ + ~ is a manifold. So it is transversal to ,~,, × O' for almost all ~. Since e is arbitrarily taken from W, the claim follows. [2QED

Proof of Theorem 2. ~t,v+nXn+L-~)+M will be denoted by Z in what follows. Consider a C" map ~:" W(~) x (X x P' × t9') ~ Y× Z defined by ~(e, (x, p', O ' ) ) : ( ~ , ( x ,

p'), ~(x,

p', 0 ' ) ) .

First, note that the component (.~, ~', 0') of any (~, (.~, if', 0')) satisfying ~:(e, (x, p', 0 ' ) ) = 0 is an element of FEex(~, ~). Then a C r map ~(~, -): X x P ' × 0 ' 4 Y × Z is submersive at (.~, if', 5) so that by the Local Submersion Theorem there exist local coordinates around (~, ~', 0') and 0 ( ~ Y × Z) such that ~(~, (a n. . . . . a r ) ) = a n. . . . . a r, where T = d i m ( X × P ' x O ' ) = ( N + I ) ( L H + L ) + HM - 1 and K = dim(Y x Z) = (N + I)(LH + L - l) + HAl + M (Guillemin and Pollack, 1974, p. 20). Since the preimage of 0 by ~(~, • ) is nothing but FE+x(~, "~), around (~, ~', 0') FEex(~, "~) is locally represented by the set of those points that have coordinates (0 . . . . . 0, at+ n. . . . . a r) (note that T - K = N - M). So we restrict ~(~,, • ) on the complement space of FE~(~, -~) near (~, ~', 0') to obtain the identity map. Then consider the same restriction of s~( ., • ) with respect to the second argument (x, p', 0') near (~, (.~, ~', 0')). It is obvious that for the restricted ~:, D2~(~, (~, ~', 0')) (the partial derivative in the second argument at (~, (~, ~', 0'))) is an isomorphism. Therefore, from the Implicit Function Theorem there exist (i) an open neighborhood of ~,, (ii) a relatively open neighborhood of (.~, if', 0') in X X P' x O' that does not intersect FEex(~, ~) except (.~, ~', 0'), and (iii) a unique C ~ map r/such that r/(~) = (.~, ~', 0') and ~(e, r/(e)) = 0. It is obvious by the definition that FE~(e, ~) includes ~ (e). DQED

References Abraham, R. and I. Robbin, 1967, Transversal mapping and flows (W.A. Benjamin, New Y~rk). Balasko, Y. and D. Cass, 1989, The structure of financial equilibria with exogenous yields: The case of incomplete markets, Econometrica 57, i 35-162.

R. Nagata / Journal of Mathematical Economics 29 (1998) 109-123

123

Cass, D,, 1992, Incomplete financial markets and indeterminacy of competitive equilibrium, in: J-J Laffont, ed., Advances in economic theory - Sixth World Congress, vol II (Cambridge University Press, Cambridge). Cass, D. and g. C!~anna, 1994, Welfare-improving financial innovation, CARESS Working Paper, University of Penns: i vania. Geanakoplos, J. and A, Mas-Coleli, I989, Real indeterminacy with financial assets, Journal of Economic Theory 47, 22-38. Guillemin, V. and A. Pollack, 1974, Differential topology (Prentice-Hall, Engeiwood Cliffs, NJ). Magill, M. and W. Shafer, 1991, Incomplete markets, in: W. Hildenbrand and H. Sonnenschein, eds., Handbook of mathematical economics, Vol IV (North-Holland, Amsterdam). Smale. S., 1974, Global analysis and economics II A; Extension of a theorem of Debreu, Journal of Mathematical Economics I, 1-14. Werner, J., 1986, Asset prices and real indeterminacy in equilibrium with financial markets, Discussion Paper No. B-50, Universit~it Bonn. Werner, J., 1990, Structure of financial markets and real indeterminacy of equilibria, Journal of Mathematical Economics 19, 217-232.