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WORKING PAPER

ALFRED P. SLOAN SCHOOL OF MANAGEMENT

IMPLEMENTING ARROW-DEBREU EQUILIBRIA

BY CONTINUOUS TRADING OF FEW LONG-LIVED SECURITIES

by

Darnell Duffie"

and

Chi-fu Huang""

August 1983

MIT Sloan School of Management WP #1501-83

MASSACHUSETTS

INSTITUTE OF TECHNOLOGY

50 MEMORIAL DRIVE

CAMBRIDGE, MASSACHUSETTS 02139

IMPLEMENTING ARROW-DEBREU EQUILIBRIA

BY CONTINUOUS TRADING OF FEW LONG-LIVED SECURITIES

by

Darnel 1 Duffie"

and

Chi-fu Huang""

August 1983

MIT Sloan School of Management WP #1501-83

Graduate student in Engineering-Economic Systems Department,

Stanford University

Assistant Professor of Management and Economics,

Sloan School of Management, Massachusetts Institute of Technology

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The authors would like to thank David Kreps , John Cox, Michael

Harrison, and David Luenberger for helpful comments. They are

also grateful to Larry Jones, Donald Brown, and David Kreps

for pointing out an error in the earlier version of this paper.

Any errors are, of course, those of the authors.

Abstract

A two-period (0 and T) Arrow^Debreu economy is set up with a general

model of uncertainty. We suppose that an equilibrium exists for this

economy. The Arrow-Debreu economy is placed in a Radner [31] setting;

agents may trade claims continuously during [0,T]. Under appropriate

conditions it is possible to implement the original Arrow-Debreu

equilibrium, which may have an infinite dimensional commodity space, in a

Radner economy which has only a finite number of securities. This is

done by opening the "right" set of securities markets, a set which

effectively completes markets for the continuous trading Radner economy.

1.0 Introduction

Event tree A of Figure 1 depicts a simple information structure.

Let's momentarily consider an exchange economy with endowments of and

preferences for (random) time T consumption which depends on the state

U) e J2 chosen by nature from the final nodes of this event tree. A

competitive equilibrium will exist under standard assumptions (Debreu

[3], Chapter 7) including markets for securities whose time T consumption

payoff vectors span R . This entails at least five security markets,

while intuition suggests that, with the ability to learn information and

trade during [0,T], only three securities which are always available for

trading ( long-lived securities [9]) might be enough to effectively

complete markets. (This is the maximum number of branches leaving any

node in the tree. The reasoning is given by Kreps [10], and in an

alternative form later in this paper.) One major purpose of this paper

is to verify this intuition for a general class of information

structures, including those which cannot be represented by

Event Tree A Event Tree B

ta»

f2 ~ oi^f a^> u)j> (i^, ii^

t-o

t=T

Figure 1 Event Trees

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event trees (such as the filtrations generated by continuous-time

stochastic process.) In some cases, where an Arrow-Debreu style

equilibrium would call for an infinite number of securities, we show how

a continuous trading Radner [16] equilibrium of plans, prices and price

expectations can implement the same Arrow-Debreu consumption allocations

with only a finite number of long-lived securities.

A comparision of Event Trees A and B, intended to correspond to the

same Arrow-Debreu economy, obviates the role of the information structure

in determining the number of long-lived securities required to "span" the

consumption space, or the spanning number . (This term is later given a

precise meaning.) Since all uncertainty is resolved at once in Event

Tree B, the spanning number is five (instead of three for Event Tree A).

Roughly speaking, the maximum number of "dimensions of uncertainity"

which could be resolved at any one time is the key determining property.

This vague concept actually takes a precise form as the martingale

multiplicity of the information structure (See Appendix.) A key result

of this paper is that the spanning number is the martingale multiplicity

plus one. The "plus one" is no mystery; in addition to spanning

uncertainty, agents must have the ability to transfer purchasing power

across time.

The notion that certain securities are redundant because their

payoffs can be replicated by trading other securities over time, yielding

arbitrage pricing relationships among securities, was dramatized in the

Black-Scholes [1] option pricing formula. Provided the equilibrium

price process for one security happens to be a geometric Brownian Motion,

and for another is a (deterministic) exponential of time, then any

contingent claim whose payoff depends (measurably) on the path taken by

the underlying Brownian Motion, such as a call option on the risky

-3-

security, is redundant and priced by arbitrage. This discovery curiously

preceded an understanding of its simpler logical antecedents, such as

corresponding results for event tree information structures. Only in the

past few years have the implications of the spanning properties of price

processes (e.g. [10]), the connection between martingale theory and

equilibrium price process (e.g. [5]), and the mathematical machinery for

continuous security trading ([6]) been formalized.

In all of the above mentioned literature, the takeoff point is a

given set of security price processes (implicity embedded in a Radner

equilibrium). Here we begin more primitively with a given Arrow-Debreu

equilibrium, one in which trading over time is not of concern since

markets are complete at time zero. From that point we construct the

consumption payoffs and price processes for a set of long-lived

securities in such a way that agents may be allocated trading strategies

allowing them to consume their original Arrow-Debreu allocations within a

Radner style equilibrium. In short, we implement a given Arrow-Debreu

equilibrium by continuous trading of a set of long-lived securities which

is typically much smaller in number than the dimension of the consumption

space.

The paper unfolds in the following order. First we describe the

economy (Section 2) and an Arrow-Debreu equilibrium for it (Section 3).

Section 4 provides a constructive proof of a Radner equilibrium which

implements a given Arrow-Debreu equilibrium under stated conditions,

based on a martingale representation technique. Section 5 characterizes

the spanning number in terms of the martingale multiplicity. Section 6

discusses the continuous trading machinery, some generalizations, and two

examples of the model. Section 7 adds concluding remarks.

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2, The Economy

Uncertainty in our economy is modeled as a complete probability space

( n, F, P). The set fl constitutes all possible states of the world

which could exist at a terminal date T > 0. The tribe F is the a

-algebra of measurable subsets of Q, or events which agents can make

probability assessments of based on the probability measure P. Events

are revealed over time according to a filtration F =

Jf , t e [0,T]V , a right-continuous increasing family of sub-tribes

of F, where F = F and F„ is almost trivial (the tribe generated by

fl and all of the P-null sets). The tribe F^ may be interpreted as

2

the set of all events which could occur at or before time t.

Each agent in the economy is characterized by the following

properties:

(i) a known endowment of a perishable consumption good at time zero,

(ii) a random (or state-dependent) endowment of the consumption good at

time T, and

(iii) preferences over consumption pairs (r,x), where r is time zero

consumption and x is a random variable describing time T consumption

(x(uj) in state oj e Ji) .

We will only consider consumption claims with finite variance. The

2 2

consumption space is thus formalized as V = R x L (P), where L (P) is

the space of (equivalence classes) of square-integrable random variables

on ( fl, F, P) with the usual product topology in V given by the

2

Euclidean and L norms.

The agents are indexed by a finite set I = f 1, . . ., Ij. The

preferences of agent i e I are modeled as a complete transitive binary

relation^, on V C V, the i-th agent's consumption set.

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The whole economy can then be summarized by the collection

3

where v = (r. , x . ) e V. is agent i's endowment.

3. Arrow-Debreu Equilibrium

An Arrow-Debreu equilibrium for ^ is a non-zero linear (price)

functional T: V -> R and a set of allocations (v e^.; i eD

satisfying, for all i e I,

V* ^^v' V v' e^^ =|v eV^: -iCv) ^ "^"^i}} >

/* e?^, "Kv) > -Kv*) V V ^ v^

V.

Z V 4 Z V . (3.1)

i=l i=l

We will assume that preferences are strictly monotonically increasing

(in the obvious product ordering on V) so that (3.1) holds with equality

and 4* is a strictly positive linear functional. Since V is a Hilbert

4

lattice, this then implies that 4* is a continuous linear functional

on V, which can therefore be represented as an

element (a, ^ of V itself, or:

>F (r,x) = ar + j{ x(w) I (w)P(dw) ¥(r, x) eV.

Without loss of generality we can normalize f by a constant so that

E( ^ = 1, in order to construct a probability measure Q on ( J^ F) by

the relation

Q(B) = Jg^ (w) P(dw) VB e F.

Equavalently, Q is defined by the Radon-Nikodym derivative ^= dQ/dP.

This leaves the simple representation

iCr, x) = ar + E*(x) ¥(r, x) eV, (3.2)

where E denotes expectation under Q, so the equilibrium price of any

-6-

2. ,

consumption claim x e L (P) is simply its expected consumption payoff

under Q. For this reason we call Q an equilibrium price measure .

For tractability we will want any random variable with finite

variance under P to have finite variance under Q, and vice versa. A

sufficient condition is that Q and P are uniformly absolutely continuous ,

denoted QJ^P (Halmos [4], p. 100) , or equivalently, that the

Radon-Nikodym derivative dQ/dP is bounded above and below away form zero.

A second regularity condition which comes into play is the

separability of F under P. Given Q2;P> it is then easy to show the

separability of F under Q by making use of the upper essential bound on

dQ/dP.

Since uniform absolute continuity of two measures implies their

equivalence (they have the same null sets), we can use the symbols "a.s."

(for "almost surely") indiscriminately in this paper.

4. Radner Equilibrium

A long-lived security is a consumption claim (to some element of

2

L (P)) availaible for trade throughout [0,T]. A price process for

long-lived security is a semi-martingale on ( fl, F, P). In general

the number of units of a long-lived security which are held over time

defines some stochastic process Gon (J2, F, P). We will say is

an admissible trading process for a long-lived security with price

process S if it meets the regularity conditions:

(i) predictability (defined in Appendix), denoted 9 eP,

(ii) 0eLp[S] =((i) ef: E( ^4^ d[S]^) < coj,

where [S] denotes the quadratic variation process for S (Jacod [8]), and

Q

(iii) jOiS is well defined as a stochastic integral.

Th

e stochastic integral L Q(s)dS(s) is a model of the gains (or

-7-

losses) realized up to and including time t by trading a security with price

process S using the trading process 0. Interpreted as a Stieltjes integral,

this model is obvious, but the integral is in general well defined only as a

stochastic integral. This model, formalized by Harrison and Pliska [6], is

discussed further in Section 6, as are the other regularity conditions on 9.

Taking S = (S^^,. . . ,S^) (N ^ ") as the set of all long-lived

security price processes, any corresponding set of trading processes

9= ( Q, , . • •, £U) must meet the accounting identity:

e(t7s(t) = 0(ofs(O) + jj 0(s/dS(s) Vt e[0,T] a.s. (4.1)

meaning the current value of a portfolio must be its initial value plus

any gains or losses from trade incurred. [The shorthand notation in

(4.1) should be obvious.] We'll adopt the notation e(S) for the

space of trading strategies 0= (0,, ... , (^) meeting the

regularity conditions (i)-(ii)-(iii) for each long-lived security and

satisfying (4.1).

A Radner equilibrium for p, is comprised of:

(1) a set of long-lived securities claiming d = (d^^, . . . , d^)

(N ^ oo) with price processes S = (S^, . . . >S„),

(2) a set of trading strategies e6(S), one for each agent i £l

and

(3) a price a e R for time zero consumption, all satisfying:

Budget Constrained Optimality for each i eX •

it '• i''\v

/ r - (0) S(0), X. + (T)d ) is A -maximal in the budget set:

i Cr^ - 0(0)^5(0) , x^ + 0(T)d) £V^ : 0ee(S)] ,

-8-

and Market Clearing:

I

Z 0^(t) = VtE [O.T] a.s.

1=1 2

The space of square-integrable martingales under Q, denoted/T? q' 1*-^

2

multiplicity, denoted M(lf|^); and an orthogonal 2-basis for

2 2

It? , say m = (m^, , . . t^) , N = M(Wf ) ^ a r . + E (x.) ,

and substituting the Radner budget constraint for r and x,

a rj - Q(of SCO) + E* [x. + G^o/sCO) + jj G(t/dS(t)]

ie i:

> a r . + E(x.) ,

or J J

a r , + E (x.) > a r, -I- E (x.). (4.6)

*

* T X r T

The last line uses the fact that E [ J: Q(t) dS(t)] = since J 0(

2

dS

is a Q-martingale for any e 6[S), from the fact that P^

is closed under stochastic integration of this form (see Jacod [8],

Chapter 4). But (4.6) contradicts the Arrow-Debreu budget-constrained

* *

optimality of (r., x.). This establishes the theorem.

Q.E.D.

5. The Spanning Number of Radner Equilibrium

The key idea of the last proof is that an appropriately selected and

priced set of long-lived securities "spans" the entire final-period

2

consumption space in the sense that any x e L (P) can be represented

in the form:

E*(x|F ) = 0(t)^S(t) = 0(O)"'"s(O) + j5 0(S)'^dS(S) VtefO.T], a.s. (5.1)

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where S = (Sq, . . . , S ) is the set of (N + 1) security price

processes constructed in the proof and 6 e 0(S) is an appropriate

trading strategy. As examples in the following section will show,

2

this number N + 1, the multiplicity of Wl ^ plus one , can be

2

considerably smaller than the dimension of L (P). But is this the

"smallest number which will work", or the "spanning number", in some

sense? To be precise, we will prove the following result. (We still

assume Q^P and the separability of F.)

Proposition 5.1 ; Suppose long-lived security prices for f^ are

square-integrable martingales under Q, the equilibrium price measure

for P^ . Then the minimum number of long-lived securities which

2

completes markets in the sense of (5.1), is M(>'1^q) + 1.

2

Proof ; That M(*'1r,) + 1 is a sufficient number is given by

construction in the proof of Theorem 4.2. The remainder of the proof is

devoted to showing that at least this number is required.

2

If M(*l1„) = 00, we are done. Otherwise, suppose S =(S, , . . . , S„)

K < "i is a set of square-integrable Q-martingale security price

processes with the representation property (5.1). By the definition of

2

multiplicity, it follows that K ^M(W ). It remains to show that K

2

= M(iy? ) implies a contradiction, which we now pursue.

T 2

Let X = k + 1 S(T) e L (P), where k is any real constant and 1 is

a K-dimensional vector of ones. If S has the property (5.1) there exists

some 0e 6(5) satisfying (5.1) for this particular x. Furthermore,

since S is a vector of Q-martingales,

(xiF^) = k + l's(t) = k + 1 S(0) + j^ l'dS(s) V t e IO,T] a.s.

T * T

Since 0(0) S(0) = E (x) = k + 1 S(0), equating the right hand sides

of (5.1) and (5.2) yields

E

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jj0(s/dS(s) = jj l"^dS(s) Vt e [0,T] a.s.

Lemma A. 2, which is relegated to the appendix due to its technical

proof, then implies

Q !at e [0,T] : 0(t) = l] > 0.

Since QJ^P, the same event has strictly positive P-probability , and

equating the second members of (5.1) and (5.2) yields

P [at e [0,T] : l^S(t) = l'^S(t) + kj > 0,

an obvious absurdity if k ^ 0. Q.E.D.

The reader will likely have raised two points by now. First, having

2

shown that the "spanning number" is M(W|q) + 1 when long-lived

security prices are square-integrable martingales under Q, what do we

know about the "spanning number" in general? From the work of Harrison

and Kreps [5], we see that a "viable" Radner equilibrium must be of the

form of security price processes which are martingles under some

probability measure. Their framework, somewhat less general than this,

was extended in Huang [7] to a setting much like our own. We leave it to

readers to convince themselves that the same conclusions can easily be

drawn here. We have chosen to announce prices as martingales under Q,

rather than some other probability measure, as this follows the natural

selection of a numeraire claiming one unit of consumption in every state

(dQ in the proof of Theorem 4.2). Other numeraires could be chosen; if

a random numeraire is selected then in equilibrium security prices will

be martingales under some other probability, say P, and the "spanning

number" would be M(Ol a) + 1 (if the proper regularity conditions are

2

adhered to). Does this number differ from M(WIq) + 1; that is, can

the martingale multiplicity for the same information structure change

under substitution of probability measures? Within the class of

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equlvalent probability measures (those assigning zero probability to the

same events), this seems unlikely. It is certainly not true for event

trees. We put off a direct assault on this question to a subsequent

paper. We will show later, however, that if the information is generated

2 2

by a Standard Brownian Motion, then MClp) = M(Bl^).

The second point which ought to have been raised is the number of

securities required to implement an Arrow-Debreu equilibrium in a Radner

style model (dropping the requirement for complete markets). For

example, with only two agents, a single security which pays the

differences between the endowment and the Arrow-Debreu allocation of one

of the agents will obviously allow the two to trade to equilibrium at

time zero. This is not a very robust regime of markets, of course. By

fixing such agent-specific securities, any perturbation of agents'

endowments or preferences which preserves Arrow-Debreu prices may

preclude an efficient Radner equilibrium. Agents will generally be

unable to reach their perturbed Arrow-Debreu allocations without a new

set of long-lived securities. A set of long-lived securities which

completes markets (in the sense of (5.1)) is contrastingly robust,

although our selection still depends endogenously on Arrow-Debreu

prices. It remains a formidable challenge to show how markets can be

completed by selecting the claims of long-lived securities entirely on

the basis of the (exogenous ) information structure. There are no

economic grounds, of course, precluding the selection of security markets

from being an endogenous part of the equilibrium. This would indeed be

an interesting problem for future theoretical and empirical research.

6.0 Discussion

In this section we discuss some definitional issues, generalizations

of the model, and some specific examples.

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6.1 The gains process and admissible trading strategies

2

Why Is Lq[S] the "right" restriction on trading strategies

against a security with price process S? Why is the stochastic integral

2

J 9iS, for e e Lq[S] then the appropriate definition of

12

gains from such a stragety? The trading strategies required to

represent certain claims In the general case could never be carried out

in an actual securities market. No broker or floor trader could move

quickly enough.

Following Harrison and Pliska [6], we will say that a predictable

trading strategy Is simple , denoted 9 e A, if there is a

partition =| t. , t, , . . . ,t ,, t = TJof [0,T] and bounded

K. V 1 n-1 n J

random variables jh.V , h. e F^^ , satisfying

e(t) = h^, t e (t^t^ + 1].

A simple trading strategy 9, In words, is one which is piecewise