Arbitrage and equilibrium in economies with infinitely many commodities

Journal

of Mathematical

Economics

8 (1981) 15.-35. North-Holland

Publishing

Company

ARBITRAGE AND EQUILIBRIUM IN ECONOMIES WITH INFINITELY MANY COMMODITIES* David Stanford

Churchill Received

March

M. KREPS

University. Stunford. C.4 94305, USA College,

Cambridge,

1979, final version

received

L’K November

1979

An ‘arbitrage opportunity’ for a class of agents is a commodity bundle that will increase the utility of any of the agents and that has non-positive price. The non-existence of ‘arbitrage opportunities’ is necessary and sufficient for the existence of an economic equilibrium. A bundle is ‘priced by arbitrage’ if there is a unique price that it can command without causing an ‘arbitrage opportunity’ to exist. For economies that have infinitely many commodities, appropriate notions of ‘arbitrage opportunities’ and ‘bundles priced by arbitrage’ depend on the continuity of agents’ preferences. This paper develops these notions, thereby providing a foundation for recent work in financial theory concerning arbitrage in continuous-time models of securities markets.

1. Introduction In the literature of Economics, especially the part that concerns financial markets, one encounters argument ‘by arbitrage’. To argue by arbitrage seems to mean many different things to different people, thus an all encompassing definition cannot be given. But quite often it comes down to something like this. In any economic equilibrium, it should not be possible to purchase at zero cost a bundle of goods that will strictly increase some agent’s utility. For as long as such an ‘arbitrage opportunity’ exists, the agent in question will purchase the bundle, and he will continue to do so until either its price rises or it ceases to increase his utility. The absence of arbitrage opportunities is thus a necessary condition for an economic equilibrium. This paper presents an abstract analysis of ‘arbitrage’ in economies that have infinite dimensional commodity spaces. A major conclusion is that *This research was supported in part by National Science Foundation Grant SOC77-07741A01 to the Institute for Mathematical Studies in the Social Sciences, Stanford University, by the Mellon Foundation, by the Churchill Foundation and by a grant from the Social Sciences Research Council of the U.K. to the Department of Applied Economics, Cambridge. Many if not all of the ideas in this paper were worked out in conversation with J.M. Harrison and P.R. Milgrom, and I am indebted to them for their enormous contributions.

‘arbitrage’ depends on the degree of continuity of agents’ preferences, and a definition of an ‘arbitrage opportunity’ that reflects this dependence is given. In general, the absence of such ‘arbitrage opportunities’ is a necessary but not sufficient condition for economic equilibrium. But under conditions satisfied by many models of interest, it is necessary and sufficient. A brief synopsis of the paper follows. Most technical details have been left to the body of the paper, so all results stated in this synopsis are subject to conditions to be specified. Also, some definitions and theorems given here are reworded in the body of the paper for expositional reasons. The model is formulated in section 2. The commodity space is a real linear space X. Also given are a linear topology r on X and a cone K in X with the origin deleted. These play the following role: An agent in this economy is specified by a complete and transitive binary relation k on X, representing the agent’s preferences for net trades. These preferences are assumed to be convex, T continuous and K strictly increasing. Also given is a subspace M of X and a linear functional 7-con M. Points rnE M represent marketed bundles of goods, bundles that can be bought or sold in (incomplete, if M #X) markets. Prices at which these transactions can be carried out are given by rr. In section 3, the Sability of (M, n) as a model of economic equilibrium is considered. The pair (M,n) is said to be viable if there is some convex, r continuous and K strictly increasing preference relation 2 on X and some m* E M such that 7c(m*)zO

and

m*‘cm

for all

mcM

(1.1)

such that n(m)ZO. If no such 2 and m* exist, then no agent from the class of agents under consideration can find a best net trade from among those which are budget feasible. Thus the existence of such 2 and m* is necessary for (M,n) to be taken seriously as a model of equilibrium. (It is also sufficient in a sense to be explained.) Letting Y be the set of t continuous and K strictly positive linear functionals on X, the following equivalent characterization of viability is obtained: Theorem. (M,n) is ciahle extends 7c to all of X.

if and only

(f there exists

$ E Y such

that

$

In words, to be viable, it must be possible to embed the ‘partial equilibrium’ (M, n) into a general equilibrium (X, $) where the prices $ are r continuous and K strictly positive. In economies with finite dimensional commodity spaces, another condition equivalent to viability is that no ‘arbitrage opportunities’ exist: There exist no rnE M nK with n(m)sO. For economies with infinite dimensional

D.M. Krrps,

Arbitrage

17

and rquilibrrum

commodity spaces, this is still necessary for viability. But in general it is far from sufficient. In se&ion 4, a broader definition of an ‘arbitrage opportunity’ is given: A free lunch is a net ((m,,x,); tx~a} s M x X and a bundle kEK such that m,-x,EK and

u (0)

for all CI,

x,&k,

(1.2)

lim rc(m,)SO.

For (M,n) to be viable, it is necessary that there exist no free lunches. The converse is still not true in general, but under conditions satisfied by many models of interest, the absence of free lunches is equivalent to viability. In section 5 the prices of bundles x$ M are considered. Given such a bundle x, what prices for it are consistent with (M,rc)? If x sold for p (and if markets are frictionless, an assumption made throughout), then any bundle M +ix from span(Mu ix}) could be purchased at price n&m + ;.x): = n(m) +i,p. It is natural to say that the price p for x is consistent with (M,n) if the pair (span(Mu ix)), 7~~)that is so created is viable. It is also natural to say that the price of x is determined bJ urbitrage from (n/r,rr) if there is a single price p for x that is consistent with (M,x). The results of sections 3 and 4 yield fairly sharp characterizations of the set of prices for given s consistent with (M,z) and of the set of bundles whose prices are determined by arbitrage from (M. 7~). The origmal motivation for this paper is some recent work in the Finance literature. Beginning with the seminaI papers of Black and Scho?es (1973) and Merton (1973), this literature has achieved spectacular results from ‘arbitrage’ considerations. These papers employ continuous time diffusion processes as models of security prices, so that the economy of contingent claims that results is one with an infinite dimensional commodity space. In section 6, the application of the general results of this paper to this specific context is sketched. The sketch relies heavily on results derived in a companion paper, Harrison and Kreps (1979), which should be consulted for further details. The concluding section contains miscellaneous remarks about and extensions of the analysis. A final disclaimer must be made. The analysis undertaken here concerns one of the many forms of argument ‘by arbitrage’ and is set in one of the many contexts where ‘arbitrage’ is employed. This is not intended to subsume all the forms that appear, nor (of course) does it.

2. Formulation The formal

pieces of the model

are a real lineur

space

X with

a ~OCU//J

D.M. Kreps, Arbitrage

18

and equilibrium

convex, linear Hausdorff topology T, a cone K in X with the origin deleted, a subspace M of X and a linear functional 71on M. Elements x EX are bundles of goods in some abstract economy. The standard example is where there are N different commodities for some positive integer N. In this case, X =RN. (Throughout, R denotes the set of real numbers.) More exotic examples are where there are infinitely many ‘commodities’, indexed by t E E:. Then X might be some subspace of RE. Or, if Z has a topology, X might be the space of finite signed measures on the Bore1 subsets of E. The fact that X is a real linear space means that bundles of goods are infinitely divisible and that bundles with negative quantities of goods are conceivable. The topology T and cone K are intended to reflect characteristics of the agents who populate this economy. Each agent is described by a complete and transitive binary relation 2 on the space X, which is interpreted as the agent’s preferences over net trade vectors. Agents’ preferences will be assumed to be convex, T continuous and K strictly increasing. Respectively, these three conditions are :

x, x’kx”

and

For all

XEX,

and

%~[0,1]

xgX

2x+(1-i)x’kx”.

(2.1)

the sets {x’~X:xkx’j are closed in T.

(2.2)

and

(2.3)

jx’EX:X’>X)

For all

imply

kEK,

x+k>x.

[Note well that K is a cone with the origin deleted so that (2.3) makes sense.] Conditions (2.1) and (2.2) are standard and require no commentary. AS an example of (2.3), consider the case where X =RN. If preference is strictly increasing in each commodity, then K = (XEX :x,20 for all n = 1,. . ., N and x, >O for some n}. If one gets a strict increase in preference whenever the amounts of all commodities are strictly increased and possibly not when only some are increased, then K=(x:x,>O for all n). If commodity N is noxious once it exceeds a certain level, then K = {x:x, >O for all n
D.M. Kreps,

Arhitruge

ad

19

equilibrium

It is standard in the Economics literature to given agents endowments and consumption sets or, equivalently, net trade sets. This is not done here. 2 are defined on all of X, and agents Instead, preference relations contemplate net trades lying anywhere in this space. As shall be discussed in section 6, this assumption very strongly colors the analysis given here. The subspace M represents the space of bundles that can be constructed out of marketed bundles of goods. Typically, there will be a set M,cX of bundles that be bought and sold in frictionless markets, and M is the span of M,. The linear function n represents the market prices of bundles in M typically there will be numerical function rrO: M, +R, and 7-ris defined by

For this function

TLto be well defined

i$liimi=ijl i;m: i i=l

iLi7c,(mi)=

for

it is necessary

mi, ml E M,

and sufficient

that

implies

2 if7c,(mi).

i=l

This embodies an implicit ‘no arbitrage’ assumption that must hold in any equilibrium where agents are insatiable in directions that are marketed. (That is, where for all XE X there exists rnE M with x+m>x.) Note that because M is a subspace and rc is linear, there is an implicit assumption of frictionless markets (no transaction costs or restrictions on short sales, etc.). A final assumption that is maintained throughout is that M and K have non-empty intersection. In other words, it is possible to purchase a bundle that is sure to increase any agent’s level of satisfaction. The following notation will be used. Binary relations > and 2 on X are defined by x-)x’ if x-~‘EK and x2x’ if .x5x’ or x=x’. Note that 5 is transitive and asymmetric and that 2 is transitive and antisymmetric. Also, if x 5 [resp. 21~’ and 1.> 0, then j.x + ~“5 [resp. %Jx’ + x”. Following

are two examples

of X, z and K.

Exumple 1. Let E be a compact subset of a metric space and field generated from the open sets of 8. Let X be the space measures on (E,B). Let t be the topology of weak convergence z is the weakest topology on X such that x,-+x in r if for functions .f’:X -+ R,

let B be the (Tof finite signed on X. That is, all continuous

20

D.M. Kreps, Arbitrage and equilibrium

Let K be the cone of non-negative,

non-zero

measures,

and

x(B)20

for all

(1975)

analyze

economies

K={xEX:X(E)>O [Hart (1979) and commodity space.]

Mas-Cole11

BEB}. with

this

sort

of

Example

2. Let (B,B,P) be a measure space with non-negative, o-additive, a-finite measure P. Let X = L"(5,B,P) - X is the space of ‘essentially bounded’ measurable functionals on (E,B), where functionals that differ on P-null sets are identified. Let z be the Mackey topology on X relative to L'(Z, B,P). That is, t is the strongest topology on X such that the topological dual of (X,7) is L'. [Bewley (1972) and Kelly and Namioka (1963) discuss this topology.] Let K be the cone K={xEX:P(X
(1972) analyzes

and

economies

P(x>O)>O}.

with this sort of commodity

space.]

N.B. Bewley (1972), Hart (1979) and Mas-Cole11 (1975) take the nonnegative orthant as the agents’ consumption sets, while in this paper the consumption sets are the entire space X. Thus comparisons should be made with care.

3. Viability Suppose X, z, K, M and z are specified as above. Could (M,n) be a model of an economic equilibrium if agents are from the class A? A natural criterion for this is the following: Definition.

The pair (M,n) is supported some 2 E A and m* EM such that 7r(m* ) 5 0

and

m*km

for all

by preferences

rnE M

That is, there is some agent from the class economy, is able to find a best net trade vector feasible. The necessity of this if (M,Tc)is to equilibrium is apparent. It is also sufficient in such 2 and m* do exist. Define 2’ by xk’x’

if

x+m*Rx’+m*.

from A if there exist

such that

z(m)sO.

(3.1)

A who, when placed in this among those that are budget be regarded as a model of the following sense. Suppose

D.M. Kreps,

Arbitrage

ud

equilibrium

21

Clearly 2’~ A, and (3.1) is true for k’ in place of 2 if 0 is put in place of m*. So (M,z) is an economic equilibrium for agents from the class A - if all agents have preferences given by k’, then all have 0 in their demand correspondence at prices 7c, thus markets can be trivially cleared at these prices. A criterion equivalent to this support property can be given. Let @ be the set of T continuous and K positive linear function& on X. [To be K positive, 4 must satisfy &k)LO for all k EK.] Let Y be the set of z continuous and K strict/~ positive linear function& on X. [To be K strictly positive, $ must satisfy tj(k)>O for all keK.1 Definition. The pair (A4,n) has the extension property for (X,t,K) if there exists an extension of 7c to all of X that is T continuous and K strictly positive; that is, if there exists $ E Y such that 7c= $ ( M. This criterion has a ‘partial equilibrium/general equilibrium’ flavor, Imagine that all bundles in X are marketed somewhere and that (M,n) is a supposed partial model of the general equilibrium. Then this criterion addresses the ability to embed (M,~c) in a general equilibrium where prices for the general equilibrium are r continuous and K strictly positive. (These two restrictions on the prices for the general equilibrium are necessary if the agents come from the class A. See the remarks following the theorem.) Theorem 1. A pair (M,z) is supported by preferences from it satisfies the extension property for (X,t,K). Proof Define

If (M,n)

x2x

satisfies

if

the extension

property,

A if and only if

let II/ E Y be an extension.

$(x)2$(x’).

Clearly, 2, E A, and (3.1) is satisfied with m* =O. Conversely, suppose that (M,n) is supported by preferences from A. Without loss of generality (see above), it may be assumed that there is some 2 E A satisfying (3.1) with m* =O. For this 2, define B=jx~-$:x>O}

and

C={mEM:n(m)~O}.

The sets B and C are convex, B because preferences are convex, and B is t open because preferences are z continuous. Clearly, B n C = g5. Thus there is a non-trivial continuous linear functional tj such that ll/(b)ZO for b E B and $(c) 20 for c E C. Fix some m, E M n K (which is non-void by assumption). Since m,>O, it is clear that Tc(m,)>O. It is also true that IC/(mo)>O. To see

22

D.M. Kreps, Arbitrage

and equilibrium

this, let XEX be arbitrary such that $(x)>O. (Because $ is non-trivial, such an x must exist.) By continuity of preferences, there exists (small) 2~0 such that m,-Ax>O. Thus r&m,-ix)20 and Il/(mo)~/2$(x)>0. Now as 7+rro) >O and Il/(m,)>O, ij can be renormalized so that $(m,)=n(m,). Repeating the second part of this argument for arbitrary keK shows that for such k, $(k)>O. That is. $ is K strictly positive. This $ extends rc. To see this, pick rnc M and let /I be such that ~(m)+~~(m,)=~(m+~m,)=0. Since m+Am, and -m-Am,, are both in C, t,h(m+im,)=IC/(m)+h+h(m,)=0. Thus $(m)= -h,h(m,,)= -h(m,)=7r(m). Q.E.D. In view of this theorem, (M, TC) will be called (X,t,K)-oiable (or simply &b/e) if it is supported by preferences from A or, equivalently, if it has the extension property. Two comments are in order. First, return to the discussion following the definition of the extension property. It is said there that prices in a general equilibrium for agents from class A must be 7 continuous and K strictly positive. It is now easy to see why this is. If this general equilibrium is to be supported by z from A, then the theorem ensures that it has an ‘extension’ in Y. That is to say, it itself must lie in Y. Second, suppose one considers two topologies 7 and 7’ for X which have the same topological dual. (For example, 7 might be the weak topology relative to some space and 7’ might be the Mackey topology relative to the same space.) Then a pair (M,Tc) is (X,r,K)-viable if and only if it is (X,s’,K)viable.

4. Free lunches In economies with finite dimensional commodity spaces, a third equivalent criterion can be given for viability. This is that (M,z) admits no ‘arbitrage opportunities’. Formally, n(m)>0

if

mEMnK.

(4.1)

Put another way, 7c must be K strictly positive on M. The necessity of this is apparent. Sufficiency is implied by the analysis to follow. In economies with infinite dimensional commodity space, matters are not so simple. Of course, (4.1) is still necessary for viability. But it may be far sufficient. from Suppose for example that there exists a net ((m,,x,); CIE a} c M x X and a bundle kEK such that m,kx,

lim x, = k a

for all and

ccEa,

lim inf 7c(m,) 20. a

(4.2)

D.M. Keeps, Arbitrage

23

and equilibrium

Note that in (4.2), either m,$K or n(m,)>O for all c( is need not be violated. But to an agent of class A, this ‘arbitrage opportunity’ in the following loose sense. At negative cost, the agent can purchase a bundle m, that is a bundle x, that in turn is nearly as good as a unambiguously good. Formally, call a net ((WI,, x,)) EM k E K satisfying (4.2) a f&e lunch. Theorem

2.

possible, thus (4.1) still represents an negligible or even at least as good as bundle k that is x X and a bundle

If (M, 7t) is uiable, then the pair admits no free lunches.

This is trivially proven using the extension property characterization of viability. A slightly more difficult proof is given below for expositional purposes. Note that this argument formalizes the loose intuition above. Also note that this argument does not employ the convexity of preferences in A. Proof: Suppose that (M,n) is supported by 2 E A at m*. Further suppose that {(m,,x,)j and ke:K are such that m,kx,+k. Then m*+k>m*, so by the continuity of 2, there exists some A>0 such that m* -E.m, + k>m* (where m, E M nK as before). Using the continuity of 2 again, for all sufficiently large ci, m* - Am, +x,>m*. Now using the fact that 2 is K strictly increasing, for all sufficiently large r, m* --iurn, +m,>m*. Thus z(m* -~~m,+m,)=3r(m*)-~~(m0)+~(m,)~0 is necessary for all large SI. Of course, n(m*) = 0, and n(m,) >O is necessary as m, + m* >m*. Thus lim inf,rr(m,) 2 in(m,) > 0. There exist no free lunches. Q.E.D. Even absence example

with this broadened definition of an ‘arbitrage opportunity’, the of such opportunities does not imply viability. The following illustrates this:

Example 3. Let X = I” (the space of bounded be the product topology (topology of pointwise K={x=(x~,x~,...)E/~:x”~~ and (Note

x”>O

the use of superscripts

sequences of real numbers), convergence) and

T

for all n

for some n) to denote

coordinates.)

Let M and 7-cbe given

by M=;(x’,x’

,... ):x1=x2=...)

and

n(m)=m’

Then (M, n) admits no free lunches. To see this, suppose {(m,,x,);a E a} E M xX and kEK are such that m,hx,+k. Since ke:K, for some n, k”>O.

As X,-+/C, for sufficiently large x it follows that s; > k”/2>0. And as m, 2x, for all a, rn: > k”/2 > 0 for all large M. Thus lim inf,rr( m,) = lim inf,mi 2 k”/2 > 0. But (M,rr) is not viable. This can be seen from the fact that Y is empty! Suppose $ is a K strictly positive linear functional on X. Let e, be the unit vector in direction n in X. That is, e, = (ef , ei,. .) where 2; = 1 if j=n and ej, =0 if jjn. Then e,EK, thus $(e,)>O. Let y,=e,-[$(e,)/$(e,)]e,. By linearity, $(y,)=O. But yn-+el in z, so if $ is r continuous it follows that $(e, ) = 0, a contradiction. Since the absence of free lunches does not in general imply viability of the model, one might look for a still broader definition of ‘arbitrage opportunity’, the absence of which is equivalent to viability. Results of this sort are not reported here. Rather. conditions on (X, z, K) will be given under which the current definition of a free lunch is adequate. (The reader’s patience is begged, as a bit of spadework is necessary before these results can be given.)

Lemma 1. cj,I M=z

Zf (M, 7r) admits no free lunches, then there exists

Q,E @ such that

(Recall that @ is the set of z continuous and K positive linear functionals on X. This lemma is nearly the desired result, lacking only strict positivity.)

Proof

Because

(M,Tc) admits

sup l limsupz(mz);

no free lunches,

m,Lx,+O

1 SO,

(4.3)

where the outer supremum is over all nets {(mu, x,); CIEu} E M x X such that m,sx, for all CI and x,+0. To see this, suppose (4.3) is violated. Then there and lim, z(m,)>O. Take m, E M n K and is a net {(m,, x,)> with m, 2x,+0 let &=min(l,rc(m,)/~(m,)}. Define m;=i,m,-m,, x:=&m,-xx, and k = lim,&m,. Then for all CC,mj, &xh and n(A,m, - m,)sO. Also x;+k, and keK. Thus there would be a free lunch. From (4.3) there is a convex open neighborhood N of the origin such that supfn(m):m~N-(Ku{O})}sl. Let f be the gauge of N - (K u {O}). Then f homogeneous, f(x) =0 for XE -K, and rrsf on M. theorem, there exists a linear extension C/I of rc such 5f(x)=O for XE -K, c$ is positive. Since C#I 5 1 on an Q, is continuous. Q.E.D.

(4.4) is convex, positively By the Hahn-Banach that $sf. Since b(x) open neighborhood N,

D.M. Kreps, Arbitrage

25

and equilibrium

For XEX define

77(x) = inf

c(x)=

lim inf 7c(m,); J

sup i

lim sup rc(m,); a

m,Zx,+x

,

(4.5)

I m,sx,+x

,

(4.6)

I

In (4.5) the outer infimum is over all nets {(m,,x,)j c M x X such that m,kx, for all a and x,+x. If no such nets exist, 71(x)= + cl3. Analogous conventions apply to (4.6). The functions 71 and 71 have a natural economic interpretation - they give upper and lower bounds on the price of a bundle x ‘by reason of arbitrage’. (This interpretation is developed in section 5.) Lemmu

2.

If Q,E @ is such that qb 1M = 7c, then 715 4 s %.

Proof.

Suppose d, E @ is such that Q,1M = n. Then for all {(m,, x,)} such that m,~x,+x, liminfZ~(m,)==liminf,~(m,)>=lim,&x,)=Q)(x). Thus %(x)24(x). The other inequality is proven analogously. Q.E.D.

Proqf:

2.

If there are no free

1.

CorollaI?

By Lemma Q.E.D.

Lemma

3.

Ifk EK

Proof:

1, there

and there

If C(k)zO, a standard such that m,kx,+k free lunch. Q.E.D. [(m,,x,))

The following

lunches,

exists

then 7~2 TJ, I? > - s

d, E @ such that

are no free

lunches,

and z < x.

$I 1M =7c. Apply

Lemma

then ii(k)>O.

diagonalization procedure and lim,~(m,)=7?(k)~O,

would yield a net which would be a

result is the heart of the argument.

Lemma 4. If(M, n) admits no free lunches, #big@ such that &( M=zz and &(k)>O.

then for

every

keK

there

exists

This sharpens Lemma 1: rr can be extended to some continuous and positive linear functional which is strictly positive in a given direction kEK. Proof

Fix some k EK. By Lemma 1, there exists 4 E @ such that 4 1M = 71. If there is nothing to do, so suppose that 4(k) =O. Clearly, k # M is By Lemma 2, 7S(k)L4(k)=Ozg(k), and by Lemma 3 %(k)>O. Thus %(k)>F(k). Pick some real number p such that E(k)>p>z(k) and p>O.

& k)>O implied.

26

D.M. Kreps,

Arbitrage

and equilibrium

(Note that this is always possible.) Let M’ denote the span of M and (kJ, and let 7~‘denote the linear functional on M’ given by rc’(m + Ak) = rc(m) + Ap. (M’,Tc’) admits no free lunches. Suppose ((r&x;)} is a net in M’ xX and k' EK are such that rn:& x:-+k’. Write rn: =m2 +&k where m,s M. It suffices to consider three cases, namely (i) lim,i,=O, (ii) there exists i>O such that i,LJ. for all LY,and (iii) there exists ;1O. But then liminf,rr’(m~)=liminf, [7c(m,)+j.,p] =liminf,7r(m,)>O. Suppose there exists i. >O such that A,?/1 for all M. Then

and by the definition of z(k), limsup,(rr(-mJi.,))j_n(k) 0, which, since 2, ZE, ~0, implies lim inf, [n(m,) + i.,p] > 0. That is, lim inf, n’(mk) >O. By a similar argument, if there exists E.O, completing the demonstration that (M’, 71’) admits no free lunches. Finally, apply Lemma 1 to (M’, z’). This yields 4’ E @ such that 4’ 1M’ =7c’. Since 4’1 M=Tc’) M= 7c and 4’(k)=rc’(k)=p>O, this 4’ will serve for Q.E.D. @P The following

Lemma

5.

strengthening

If (X,t)

is normable

exists an equicontinuous &I M=n and &(k)>Ofor

Consider

of Lemmas

1 and 4 is left to the reader:

and (M,z)

admits

no free

family of linear functionals all k.

the following

four properties

that (X, t,K)

lunches,

then there

{&; k EK} E @ such that

might possess:

There exists a countable subset (k,, k,,. . .> of K such every k EK there exist ;1>O and an n such that k& Ak,.

that

Every subset of Cpis weak* separable. For $$;i”

every

countable subset {41,42,. . .} of @ there exist that A,,>0 for all n, I;& A,= 1, s;ch

l

(X, z) is normable.

for (4.7)

(4.8) real and (4.9) (4.10)

D.M. Kreps, Arbitrage

Theorem

3.

sufficient lunches.

condition

Zf either for

and equilibrium

21

(4.7) or (4.8) and either (4.9) or (4.10) hold, then a the viability of (M,x) is that (M,rc) admit no free

Proof. If (M,JT) admits no free lunches, then by Lemma 4 there exists a collection 14,; kEK} G Q, such that & ( M = z and &(k)>O for all k. Moreover, if (4.10) holds then this can be an equicontinuous family of functionals by Lemma 5. Suppose (4.7) holds. For n = 1,2,. . ., define 4, -&,. Then [4i, +2,. . .> s @ satisfies (i) 4,I M = n for all n and (ii) for all kEK there exists n such that 4,(k)>O. For (ii), note that if k&ilk,, then ~,(k)~~,(~.k,)=~~~,(k,)>O. And if (4.10) holds, the $,, 42r.. . are equicontinuous. Suppose (4.8) holds. Let ($,,I$~, . . .} be a weak* dense subset of {&; kEK}. Then for any kEK, if &-+& weak*. it follows that 4,,,(k)+&(k) >O and so 4,.(k) is eventually strictly positive. Thus {4,,42,...) satisfies (i) &,I M =n for all n, and (ii) for all k EK there exists n with 4,(k)>O. And if (4.10) holds, 4i, c#I~,. . . are equicontinuous. Thus if either (4.7) or (4.8) holds, {bi, 42,. . .} can be produced as above. And if either (4.9) or (4.10) hold, one can find iL,zO such that ~~=I i,,,= 1 and $ =I:= 1 j.,4 E @. [Condition (4.9) is that such 1, exist, while if (4.10) holds, equicontinuity of the 4, assures that $ E !P for any i,, with cz: 1R, as 4,(k)>O for some n. = 1.1 Clearly, II/ ( M = n. And for all kEK, $(k)>O Q.E.D. Thus $E Y, proving that (M,~c) is viable. (1) Condition (4.7) is tailor-made for economies with a countable Remarks. number of commodities, so that X is a subspace of R”, and K=[k=(k1,k2,...)~X:k”z0 and because

k”>O

for all n

for some n}.

in this case the ‘unit vectors’ will serve for the k,.

(2) Recall Example 1. There X is the space of finite signed measures on the Bore1 o-field of a compact subset E of a metric space, T is the topology of weak convergence, and K is the cone of non-negative, non-zero measures. Let F denote the space of continuous real valued functions on Z. Then 8: X +R is a r continuous linear functional if and only if e(x)=j,f (t)x(dt) for some f EF. This linear functional is K positive iff f is non-negative and is K strictly positive iff f is strictly positive. Let (I.)) denote the usual sup norm on F. Then f,-+f weak* if Il,f,--f Il+O. Fd ,ifE is separable, (F, )I .)I) 1s a second countable topological space. Thus z IS separable, (4.8) holds. In any case, (4.9) holds: given a sequence {,f,}

28

D.M. Kreps,

from @, take i., proportional 3 applies.

Arbitrage

to (llfnll ‘2”)

and equilibrium

‘. Thus if E is separable,

Theorem

(3) Recall Example 2. The space X is L”(E”,B,P) where (E,B) is a measure space and P is a non-negative, a-additive, o-finite measure on (E,B). The topology r is the Mackey topology on X relative to L’ (5, B, P). And the cone K is the set of non-negative, non-zero functions x EX. Of course, O:X+R is a r continuous linear functional if and only if 0(x)=jsx(<)f(t)P(dt) for some f~ L’ (E, B, P). Moreover, this t) is K positive iff f is non-negative, and 0 is K strictly positive ifff is strictly positive P-a.s. 1f (E’,B,P) has a countable base, then L’(E, B,P) is a separable Banach space. See Kolmogorov and Fomin (1970, p. 382). For (E, B,P) to have a countable base, there must exist a sequence {B,,B,,.. .}sB such that for E > 0 and BE B there exists some n with P(B A B,) 0 for all n and I,& = 1, the linear functional $=x$,4, is not r continuous. (5) Suppose it was not known that agents have convex preferences. Let A’ be the set of complete and transitive binary relations on X that are r continuous and K strictly positive. Say that (M,rr) has the weak support property if there exists k E A’ and m* EM such that n(m*)sO

and

m*km

for all

mEM

such that n(m)lO.

As the proof of Theorem 2 did not use the convexity of preference, it serves to prove that if (M,n) has the weak support property, then (M,n) admits no free lunches. Thus in any context (such as those ennumerated in Theorem 3) where not admitting free lunches implies the extension property, the weak support property is equivalent to viability. (6) In general, the emptiness of Y is equivalent to the emptiness of A. (Proof: Given 2 E A, separate {x:x>O) from the origin.) So in Example 3, A as well as Y is empty. Clearly, a necessary condition for the absence of free

D.M. Kreps,

Arbitrage

29

and equilibrium

lunches to imply viability is that Y and A are conjecture is that this condition is also sufficient.

non-empty.

An enticing

5. Arbitrage Fix X, T, K and viable (M, n). Suppose x E X \ M can be bought and sold in some other market. What can be said about the price of x? Suppose x sold at price p. Then any bundle in the space M’ =span (M u (x}) can be purchased at price x’(m+ ix)=n(m)+ Q. Given what is known about agents (that they come from the class A), it is natural to say that price p for bundle x is consistent with (M,~c) if the extended model (M’,z’) so created is (X,z,K) viable. It is convenient to recast this definition in terms of GE Y. Definition. Given a viable (M, 7c), price p for bundle x is consistent (M, z) if there exists $ E Y such that $1 M = n and e(x) = p.

with

Fix viable (M, 71) and a bundle x E X. It is clear that the set of prices for x that are consistent with (M,n) is non-empty. Moreover, suppose $. $‘CSY are such that $ ) M = I,!/ / M = 7~. Then for all lj E [IO,11, p$ + (1 - fl)$’ E Y and ([j$ + (1 -p)$‘) ( M = n. Thus the set of prices for x consistent with (M, 7~) is an interval on the real line. To pin down this interval, reconsider the functions C and 7~defined in (4.5) and (4.6). Repeating those definitions,

TC(x)=inf l liminf7r(m,); z(x)=sup i

limsupn(m,); 3

w1,kx,-+x

I ,

mz$x,+x

(5.1) .

(5.2)

I

Note that these functions give upper and lower limits on the price that x might sell for ‘by reason of arbitrage’. For if x sold for a price outside of this range, a free lunch would exist in the extended model. Exactly this is stated in Lemma 2; that if Q,E @ is such that C#J 1M = X, then ~5 4 5 71. Thus the set of prices for x consistent with (M, n) is a subinterval of [z(x), f(x)]. A partial converse is the following. Theorem 4. is u price for

If (M, n) is viable and %(x)>~(x), x consistent with (M, 7~).

then any price

PE (z(x),

71(x))

Proof. As (M,x) is viable, there exists $ E Y with $ ( M =x. Of course, $(-u) E [z(x), n(x)] and $(x) is finite. Thus for any p E (z(x), E(X)), there exists

30

D.M. Kreps, Arbitrage

arid equilibrium

P’E (z(x), n(x)) and /?E(O, 1) such that /?p’+ (1 -p)$(x)=p. proof of Lemma 4. With trivial modifications this proof exists $E@ such that C$/ M = ?I and 4(x)=$. But then satisfies $’ E Y, $’ / M = 7c and $‘(x) =p. Q.E.D.

Reconsider the shows that there $‘=[j4+(1--p)$

This leaves the question: If n(x) >z(.x), are 5(x) and c(x) prices for x that are consistent with (M,rr)? In economies with finite dimensional commodity spaces the answer is always no. But in general, the answer may be yes or no in either or both cases. (Examples are easy to construct.) A special case is where the price of x is entirely

determined

Definition.

The price of x is determined by arbitruge ,from there is a single price p for x that is consistent with (M,n). Corollary

2.

arbitrage

from

If (M,n)

is

uiable,

then

the

price

oJ’ x

by (M, rr): (viable)

(M, II) if

is determined

b)

(M, 7~) if und onl!, $” C(x) = z(x).

Throughout this section, the interpretation has been that x is marketed somewhere and that 71 gives the prices of bundles in M that are simultaneously being marketed. Another useful interpretation would be that bundles in A4 are marketed at prices rc prior to the opening of a market in x, and that prices for x consistent with (M, n) are prices for which x might sell in the newly opened market. Of course, this second interpretation cannot be made in general, as the introduction of a market for x creates new economic opportunities for agents and may thereby change the prices of bundles in M. But suppose the price of x is determined by arbitrage from (M,n). In economies with finite dimensional commodity spaces, this is true if and only if XE M, thus the introduction of a market for x does not change economic opportunities and the second interpretation may be made. (More precisely, the prices rr remain equilibrium prices in the new setting. If there are multiple equilibria, it is possible that prices jump from n.) In economies with infinite dimensional commodity spaces matters are not quite as simple. The price of x might be determined by arbitrage and still x$M. For example, if M is not r closed, the price of every bundle in closure(M)‘,M is determined by arbitrage. [In general, bundles outside of closure(M) may have their prices determined by arbitrage.] Still. the second interpretation can be made, even for preferences in A’. Theorem

5.

Let

(M,x)

x(nz*)zO

be ciuble, and

such that 7c(m)50.

m*zm

und let 2 E A’ and m* E M be such that

for all

mEM

(5.3)

Suppose

the price

the economy

qf’x

is determined

M’=span m*km’

(Mu for

all

{x))

by arbitrage with prices

m’E M’

from

(M, z) to he p. Then

in

z’(m+~x)=z(m)+i,p,

such that

~‘(m’)~O.

(5.4)

Proof.

Suppose m’ E M’ is such that z’(m’)zO and m’+m*. Write m’ as m where rnE M. Clearly j_#O, for otherwise (5.3) is violated. Suppose i >O. Let m, E M nK. By continuity of 2 there exists p>O such that &-pm, =m+i,x--pm,>m*. Of course, z(m,)>O. so yc(m,)/i.>O. As rC(x)=~), there exists i(m,,x,)) GM x X such that m,kx,+x and n(m,)~p+p(m,)/i for all U. Because 2 is continuous and K increasing, for sufficiently large a, m + Am, -pm,>m*. But 71(m+~m,-~m0)=~(m)+1.7c(m,)-~7c(m,)~z(m)+~(p +~71(m,)/i.)--~(m,)=7c(m)+i.p=n’(m+i.x)~O, contradicting (5.3). A similar argument works for ,4.< 0. Q.E.D. +Xx,

Thus, if there is an economy in which bundles in M are traded at equilibrium prices x, and if a market in x opens wherein the price of x is p, all agents are content with their original net trades, and markets clear.

6. An application In this section, an application of the analysis above is sketched. The interested reader should consult Harrison and Kreps (1979) for further details. Black and Scholes (1973) posit a pure trade economy in the spirit of Radner (1972). Given is a probability space (a, F, P) on which lives a standard Brownian motion {B(t);Osts I}. Let {I;,} be the Brownian filtration, and assume that F= F,. The filtration (F,j gives the information held be agents at dates t E [0, l] in the usual fashion. Agents in this economy consume a single good at dates zero and one. Consumption at date zero is non-stochastic. Consumption at date one is state contingent. The net trade space is taken to be X = R x L2(Q, F, P). Consumption can be traded between dates and states by trading in two financial securities. The first security pays a dividend of d, =e’ units of the consumption good at date one. The second pays a dividend of d2(a) =exp (oB(1, o)+pj units at date one in state cc). (Here, r, ~1 and a>0 are given constants.) The two securities are traded for each other and for the consumption good on date zero at relative prices one apiece. At dates t > 0, the two securities are traded for each other at prices pi(t)=exp respectively.

{rt)

and

p2(t,W)=expjoB(t,o)+~tS,

(6.1)

32

D.M.

Kw~s,

Arbitrage

and equilibrium

Agents can employ srljlfinl~kng simple trudilzg strategies when trading in these securities. A simple trading strategy is a two-dimensional stochastic process O= {d,(t, a); k= 1,2, tE [0, 11) where (i) t-+Q(t, w) is piecewise constant on intervals of the form [t,, t,, 1) for some.fixed finite collection of dates O=t,O at which 0 may change, d(t,_ ,). p(t,) =O(t,) . p(t,). That is, the value of the portfolio prior to date t, trading is equal to the post-trade value. This is the budget constraint on agents: After date zero, any purchases must be paid for by proceeds from simultaneous sales. (Equality is assumed w.l.o.g., as preferences will be strictly increasing.) There are a finite number of agents in this economy, indexed by i = 1,. . ., I. Agent i has net trade space X and is characterized by a preference relation ki on X. Letting 5 be the product of the usual topology on R and the L2norm topology on L2(Q,F,P), and K={(~,~)EX:I~O,~~O P-as., and either r>O or P(y>O)>O), it is assumed that agents’ preferences are z continuous and K strictly increasing. If agent i adopts the simple self-financing trading strategy 8, he obtains the net trade vector (--Q(O). p(O), 0(l) .d). Following Radner (1972), prices p are equilibrium prices if for each i there exists a simple self-financing trading strategy 0’ such that :

(-@(O).p(O),

@(l).d)

is ki’maxini;ll

{xEX:x=(-o(O).p(O), self-financing

for some simple

19};

c 0’=0. Call such equilibria

e(l).d)

in the set

(6.2b) Radner equilibria.

D.M. Kreps,

Arbitrage

33

and equilibrium

It is not difficult to recast this as follows. Define M={xE~:)c=(Y-_(O).p(O), 8(1).d) for rER and 0 a simple self-financing trading strategy),

(6.3)

and define rt : A4+ R by rr(r-G(O).p(O),

Ql).d)=r.

(6.4)

Suppose (f?)f=, together with the prices p constitute Then for xi= (-@(O).p(O), B’(l).d) it follows that rc(xi)sO

and

.y’ is ki maximal

x’~hl in

a Radner

equilibrium.

for each i,

(6Sa)

{m~M:rr(m)~Oj,

(6Sb) (6.5~)

T:@=O.

That is, the Radner equilibrium at prices p corresponds to an ‘ArrowDebreu’ equilibrium where bundles in M are sold at prices given by 7~. Moreover, if (xi) is any allocation satisfying (6.5), then there exist 8’ with xi = (-Q’(O). p(O), Q’(l) ‘d) such that these 8’ together with p constitute a Radner equilibrium. Does the Black-&holes price process admit a Radner equilibrium for agents as above? The answer is yes if and only if (M,Tc) is viable. If the answer is yes, what other bundles in X are priced by arbitrage? Exactly those bundles that are priced by arbitrage for (M,n). It turns out [cf. Harrison and Kreps (1979, theorem 3)] that the Black-Scholes model is viable and eaery claim in X is priced by arbitrage, because there is a unique extension of 7c in Y. Following Theorem 5 above, in any economy where the Black-Scholes prices are equilibrium prices and agents are as above, the equilibrium allocation is (unconstrained) Pareto efficient. Moreover, the characterization of bundles whose price is determined by arbitrage that is developed above (especially Corollary 2) is useful in obtaining ‘asymptotic efficiency’ results for economies that approximate the Black-Scholes economy [cf. Kreps (1979)].

7. Remarks Two assumptions comment.

maintained

throughout

this

paper

are

worthy

of

It has been assumed that all agents in the economy are known to come from a single class A corresponding to a given triple (X,t,K). Writing A(X, T,K) to denote the dependence of A on (X,z,K), suppose instead that one knows that there are some agents from class A(X,, z,,Kl), some from A(X,,t,,K,), some from A(X,,z,,K,), and (possibly) some others about which nothing is known. In the spirit of this paper, (M,n) would be called viable if it is supported by some &E A(X,,s,,K,), by some other k2 E A(X,,z,,K,), and by some third k3 E A(X,,z,,K,). Equivalently, for n = 1,2,3 there must exist a rn continuous and K,, strictly positive linear functional $, on X, such that $, / M=z Note that if there is a class of agents whose consumption spaces X, are largest and whose preferences are continuous in a weakest topology and strictly increasing on a largest cone, then it is that class of agents that ‘calls the tune’ concerning viability and arbitrage. Throughout it has been assumed that agents’ consumption sets are an entire linear space X rather than a subset of that space. This is quite different from other treatments of economies with infinite dimensional commodity spaces. For example, Bewley (1972) takes X = L”(Z,B,P) and agents’ consumption sets as the positive orthant of X. This difference has enormous consequences for the analysis. Consider X = L” (Z, B, P) where Z is countable, B is discrete and P(t)>0 for all j’ EE. Imagine that agents are ‘von Neumann-Morgenstern expected utility maximizers’. That is, every agent is characterized by a preference relation 2 that is represented

x2x’

if and only if

2 U(x(<))P(S)Zc r

U(x’(i;))P(S), :

for Cl :R+R that is strictly increasing and concave on [0, a). Moreover imagine that every such U is bounded. Then preferences are L’ Mackey continuous, so if the consumption set is the positive orthant in X, Bewley (1972) guarantees the existence of an economic equilibrium. But if the consumption set is all of X, no equilibrium exists (for M=X). It is easy to see why ~ if preferences are as above, then they are continuous in the product topology as in Example 3. It is easy to show that for any price system $ and any bundle x E X with $(x)=0, there exists keK and {xZ> CX such that Ii/(x,) =0 and x, approaches x + k in the product topology. Thus as x + k>x, for large enough CI it follows that x,>x. This says that for every budget feasible x, there is a ‘better’ budget feasible x, (that is, for any agent whose preferences are continuous in the product topology). In an economy with such agents (expected utility maximizers with bounded utility functions and net trade sets equal to X), therefore, no equilibrium could ever be attained. [This can also be seen as a consequence of Remark (5) following Theorem 3.1

It is therefore of interest to ask how the results given here must be modified if agents are allowed to have general consumption or net trade sets. One way to proceed is to say that agents are characterized by t and K as before and by a set TcX, where it is known that every agent’s net trade set is at least T. It seems natural to suppose that such T satisfy OE T and JET

kEK

and

imply

t+k~T.

Then (M, 7~) is called viable if there exists k E A(X, s,K, T) (the set of convex, r continuous, K strictly increasing preference relations on T) and in* EM such that m*EMnT. such that

n(m*)zO

and

m*km

forall

m~Mn7

rr(m)~O.

(7.1)

This paper has analyzed one extreme case, T=X. The results modification if T has non-empty interior and if one adds to requirement that WI*E irlt (T). But consider the other extreme =K u [O). Then any (M, 7-c)satisfying n(nz)>O

if

need no (6.1) the case, T

mEMnK

is viable. It would be interesting to see the analysis cases that lie between these two extremes.

(7.2) given

here adapted

to

References Bewley. I-., 1972, Existence of equilibt-ia 111economres with infinitely many commodities, Journal of Economic Theory 4, 514-540. Black, F. and M. Scholes, 1973. The pricing of options and corporate liabilities, Journal of Political Economy 81, 6377659. Harrison, J.M. and D. Kreps, 1979, Martingales and arbitrage in multiperiod securities markets, Journal of Economic Theory 20, 381-408. Hart. O., 1979, Monopolistic competition in a large economy with differentiated commodities, Review of Economic Studies 46, l-30. Kelley, J. and I. Namioda, 1963, Linear topological spaces (Van Nostrand, New York). Kolmogorov, A. and S. Fomin, 1970, Introductory real analysis (Prentice-Hail, Englewood Cliffs, NJ). Kreps, D., 1979, Multiperrod securities and the efficient allocation of risk: A comment on the BlackkScholes option pricing model, in: J. McCall, ed., The economics of uncertainty and information, forthcoming. Mas-Colell, A., 1975, A model of equilibrium with differentiated commodities, Journal of Mathematical Economics 2, 2633296. Merton, R., 1973, The theory of rational option pricing, Bell Journal of Economics and Management Science 4, 141-183.