Implementing Arrow-Debreu Equilibria by Trading Infinitely-Lived Securities

Home , Budget set

Implementing Arrow-Debreu Equilibria

byTrading In nitely-Lived Securities.

Kevin X.D. Huang and Jan Werner

Department of Economics, University of Minnesota

February 2, 2000

1. Intro duction

Equilibrium mo del of an dynamic economy extending over an in nite sequence

of dates plays an imp ortant role in mo dern economic theory. The basic equilibrium

concept in such mo del is the Arrow-Debreu (or Walrasian) comp etitive equilibrium.

In an Arrow-Debreu equilibrium it is assumed that agents can simultaneously trade

arbitrary consumption plans for the entire in nite and state-contingent future.

In most applications of a mo del a dynamic economy a market structure is used

which is di erent from the Arrow-Debreu market structure. Instead of trading

arbitrary consumption plans in simultaneous markets agents trade securities in

sequential markets at every date, in every event. The imp ortance of an Arrow-

Debreu equilibrium rests on the p ossibilityof implementing equilibrium allo cations

by trading suitable securities in sequential markets.

The idea of implementing an Arrow-Debreu equilibrium allo cation by trading

securities takes its origin in the classical pap er by Arrow (1964). Arrowproved

that every Arrow-Debreu equilibrium allo cation in a two-p erio d economy can b e

implementedby trading in sp ot commo ditymarkets at every date and complete

security markets in the rst p erio d. The implementation in the Arrow's mo del is

exact: the sets of equilibrium allo cations in the twomarket structures are exactly

the same. Due and Huang (1985) extended the result of Arrow to an economy

with continuous time (with nite time-horizon).

In this pap er weinvestigate the implementation of Arrow-Debreu equilibrium

allo cations in an in nite-time economyby sequential trading of in nitely-lived se-

curities. Wright (1987) studied a similar question with one-p erio d-lived securities.

The crucial asp ect of implementation in in nite-time securitymarkets is the

choice of feasibility constraints on agents' p ortfolio strategies. On the one hand, a

feasibility constraint is necessary for without it agents would b e able to b orrowin

security markets and roll over the debt without ever repaying it (Ponzi scheme).

On the other hand, a constraint cannot b e to o \tight" for it could prevent agents

from using p ortfolio strategies that generate wealth transfers necessary to achieve 2

a consumption plan of an Arrow-Debreu equilibrium. Wright (1987) employs the

wealth constraint whichsays that a consumer cannot b orrow more than the present

value of her future endowments. He proved that exact implementation holds with

one-p erio d-lived securities: the set of Arrow-Debreu equilibrium allo cations and

the set of equilibrium allo cations in sequential markets are the same.

The diculty in extending the implementation to in nitely-lived securities lies

in the p ossibility of price bubbles in sequential markets. As p ointed out by Huang

and Werner (1999), the wealth constraintgives rise to sequential equilibria with

price bubbles on securities that are in zero supply. (It follows from Santos and

Wo o dford (1997) that there cannot b e price bubbles on short-lived securities or on

long-lived securities in zero supply.) Weprove that if all securities are in strictly

p ositive supply and securitymarkets are complete, then the exact implementation

of Arrow-Debreu equilibria obtains. If some securities are in zero supply, then

Arrow-Debreu equilibria corresp ond to sequential equilibria with no price bubbles.

Usually, there are also sequential equilibria with nonzero price bubbles. We show

that these equilibria corresp ond to Arrow-Debreu equilibria with income transfers,

where the transfers are given by the value of price bubbles on agents initial p ortfolio

holdings.

We consider a second p ortfolio feasibility constraint: the boundedborrowing

constraint. Under this constraint the value of a consumer's p ortfolio normalized

with resp ect to some reference p ortfolio has to b e b ounded from b elow. This

p ortfolio constraint has a remarkable feature that there cannot b e a price bubble

in an sequential equilibrium indep endently of whether the supply of the securities

is strictly p ositive or zero. Weprove that exact implementation of Arrow-Debreu

equilibria holds under the b ounded b orrowing constraint.

The equlibrium theory in in nite-time economies (see Aliprantis, Brown and

Burkinshaw (1987)) has pro duced several versions of the concept of an Arrow-

Debreu equilibrium distinguished primarily by a sp eci cation of a consumption

space and a space of prices. It is worth p ointing out that Arrow-Debreu equilibria

that, according to our results, can b e implemented bysequential trading of secu- 3

rities have always countably additive prices. The concept of an equilibrium we

employ is closely related to that of Peleg and Yaari (1971).

The pap er is organized as follows: In section 2 weprovide a sp eci cation of time

and uncertainty. In section 3 weintro duce the notion of Arrow-Debreu equilibrium

and in section 4 we de ne a sequential equilibrium in security markets. We assume

that there is a nite numb er of in nitely-lived agents and a a nite number of

in nitely-lived securities available for trade at every date. In section 5 we state

and prove our basic implementation results.

2. Time and Uncertainty

Time is discrete with in nite horizon and indexed by t =0; 1 :::. Uncertainty

is describ ed by a set S of states of the world and an increasing sequence of nite

of S . A state s 2S sp eci es a complete history of the environ- partitions fF g

t=0

ment from date 0 to the in nite future. The partition F sp eci es sets of states

that can b e veri ed by the information available at date t.Anelement s 2F is

called a date-t event. We take F = S so that there is no uncertainty at date 0.

This description of the uncertain environmentcanbeinterpreted as an event

tree. An event s 2F at date t identi es a no de of the event tree. The unique

t t t t

predecessor of s is a date-(t 1) event s 2F suchthats s . Immediate

t t t t t

s . The 2F and s g suchthat s successors of s are date-(t +1)events fs

t+1

+ + +

unique date 0 event s is the root node of the event tree.

The set of all events at all dates is denoted by E . The set of successor events

t + t t

of s is denoted by E (s ), the set of all date- successor events of s is denoted by

t t t + t

F (s ). We also write E (s ) s [E (s ).

3. Arrow-Debreu Equilibrium

There is a single consumption go o d at every date, in every event (see Section

6 for a discussion of an extension to multiple go o ds). A consumption plan is a

scalar-valued pro cess adapted to fF g . Consumption plans are restricted to a a

t=0 4

linear space of adapted pro cesses C . Our primary choice of the consumption space

C is the space of all adapted pro cesses (which can b e identi ed with R ). The

cone of nonnegative pro cesses in C is denoted by C ;atypical elementof C is

denoted by c = fc(s )g.

There are I consumers. Each consumer i has the consumption set C , a strictly

i i

monotone and complete preference on C , and an intial endowment ! 2 C .

+ +

The standard notion of an Arrow-Debreu general equilibrium is extended to

our setting with in nitely many dates as follows: Prices are describ ed by linear

functional P which is p ositiveandwell de ned (and nite) on each consumer's

initial endowment. We call such functional a pricing functional. It follows that a

pricing functional is well de ned on the aggregate endowment! and, since it is

p ositive, also on each consumption plan satisfying 0 c ! .Itmayormay not

be well de ned on the entire space C .

The price of one unit of consumption in event s under pricing functional P is

t t t t

p(s ) P (e(s )), where e(s ) denotes the consumption plan equal to 1 in event s

at date t and zero in all other events and all other dates. We de ne the implicit

t t t t

price of consumption plan c in event s as P (c) P (cjs )=p(s ), where cjs is

a consumption plan equal to c in s and all its successor events, and zero in all

other events. A pricing functional P is countably additive, if and only if P (c)=

t t

p(s )c(s )forevery c for which P (c)iswell de ned. If P is countably additive

t t t

(c)=[1=p(s )] then P p(s )c(s ).

E (s )

An Arrow-Debreu equilibrium is a pricing functional P and a consumption

i i

allo cation fc g such that c maximizes consumer i's preference sub ject to P (c)

P P

i i i

P (! ) and c 2 C , and markets clear, that is c = ! .

i i

We will also need the notion of an equilibrium with wealth transfers. For given

i i

wealth transfers suchthat = 0, a pricing functional P and a consumption

i i

allo cation fc g are an Arrow-Debreu equilibrium with wealth transfers if c maxi-

i i

mizes consumer i's preference sub ject to P (c) P (! )+ and c 2 C , and

markets clear. 5

3. Sequential Equilibrium and Price Bubbles

The notion of sequential equilibrium applies to markets in which trade takes

place at every date. We consider J in nitely-lived securities traded at every date.

Each security j is sp eci ed by a dividend pro cess d which is adapted to fF g and

j t

t=0

nonnegative. For at least one security j the dividend pro cess d is not eventually

zero, that is, for each s there exists a succesor event s suchthatd (s ) > 0 The

t t

ex-dividend price of security j in event s is denoted by q (s )andq is the price

j j

pro cess of security j . A p ortfolio strategy sp eci es a p ortfolio of J securities (s )

t t J

held after trade in event s for each s .Thus is a IR -valued pro cess adapted to

fF g .

t=0

Each consumer i has an initial p ortfolio at date 0. The supply of securities is

.We assume that 0. When comparing sequential equilibria therefore =

with Arrow-Debreu equilibria wewanttoallow for nonzero supply of the securities

and at the same time have the total endowment of go o ds unchanged. Therefore

we sp ecify consumption endowments in sequential markets by y 2 C and assume

that

i i i

! = y + d: (1)

Consumers face feasibility constraints when cho osing their p ortfolio strategies.

Such constraints are necessary to prevent consumers from using Ponzi , see Huang

and Werner (1999). In the de nition of sequential equilibrium b elow the set of

feasible p ortfolio strategies of consumer i is . Sp eci c feasibility constraints will

be intro duced in later in this section.

A sequential equilibrium is a price pro cess q and consumption-p ortfolio allo ca-

i i

tion fc ; g such that:

i i i

(i) for each i, consumtion plan c and p ortfolio strategy maximize sub ject to

0 0 0 i 0 0 i

c(s )+q (s ) (s ) y (s )+q (s ) ;

t t t i t t t t t 0

c(s )+q (s ) (s ) y (s )+[q (s )+d(s )] (s ) 8s 6= s ;

c 2 C ;2 ;

+ 6

(ii) markets clear, that is

X X

i t t t i t t

c (s )=y (s )+d(s ); (s )=; 8s

h h

Security markets are one-periodcomplete in event s at prices q if the one-p erio d

t+1 t+1

payo matrix fq (s )+d(s )g t+1 has rank equal to the numb er of immediate

s 2fs g

successors of s . Securitymarkets are complete at q if they are one-p erio d complete

at every event. Of course, for markets to b e complete it is neccessary that the

numb er of securities is at least as large as the numb er of immediate successors of

anyevent.

If markets are complete, then for eachevent s there exists a p ortfolio strategy

that has payo equal to one in s ,zeroatevery other eventandinvolves no holding

after date t. The date-0 price of that p ortfolio strategy is the event price (present

t t

value) of one unit of consumption in event s , denoted p(s ).

The present value at s of security j is de ned using event prices as

p(s )d (s ); (2)

p(s )

s 2E

t t

where we assumed that p(s ) 6= 0 for every s .To see that this in nite sum is well

de ned, we rst observe the following relation b etween event prices and security

prices

t t t+1 t+1 t+1 t

p(s )q (s )= p(s ) q (s )+d (s ) 8s ;j: (3)

j j j

t+1

s 2fs g

Using (3) recursively we obtain

X X X

p(s ) p(s )

r t

d (s ) + q (s ) (4) q (s )=

j j j

t t

p(s ) p(s )

r t t

=t+1

s 2F (s ) s 2F (s )

t t

for each s , and for any r>t. Since the price q (s ) is nonnegative, (4) implies

that

X X

p(s )

q (s ) d (s ) (5)

j j

p(s )

=t+1

s 2F (s )

for every s and r>t. Since dividends d (s ) are nonnegative, we can take the

limit on the right hand side of (5) as r go es to in nityandwe obtain that the

presentvalue (2) is well de ned and do es not exceed the the price of the security.

The p ositive di erence b etween the price and the presentvalue of security j is

the price bubble on that security.We denote the price bubble by (s )sothat

t t

p(s )d (s ): (6) (s ) q (s )

j j j

p(s )

nfs g s 2E

Wehavethat (s ) 0.

For use later, we note that (4) and (6) imply that

t+1

) p(s

t t+1

(s )= (s ) (7)

j j

p(s )

t+1

s 2s

for each s .

Whether nonzero price bubbles can exist in a sequential equilibrium dep ends

crucially on the form of p ortfolio feasibility constraints (see Huang and Werner

(1999)). There are two feasibility constraints that are imp ortant for our imple-

mentation results. The rst applies to complete securitymarkets. It prohibits a

consumer from b orrowing more than the presentvalue of his future endowment,

that is

t i t i t

q (s ) (s ) p(s )y (s ) 8s : (8)

p(s )

+ t

s 2E (s )

We refere to constraint (8) as the wealth constraint.

If the supply of securities is strictly p ositive, that is >>0, then there cannot

b e nonzero price bubbles in a sequential equilibrium (Santos and Wo o dford (1997)).

However, if some securities are in zero supply, then nonzero price bubbles are

p ossible.

Tointro duce the second p ortfolio feasibility constraintwe rst de ne a nor-

malization of security prices. Whenever security price vector q (s ) is p ositive and

nonzero we can de ne the normalizedsecurity pricevector q^(s )by

q (s )

(9) q^(s )

q (s )

j 8

Under normalization (9) security prices are measured relative to the price of p ortfo-

lio consisting of one share of each security.We could have used any other p ortfolio

with strictly p ositiveweights as a reference p ortfolio in the de nition of price

normalization.

Portfolio strategy satis es the boundedborrowing constraint if

t t

inf q^(s ) (s ) > 1: (10)

s 2E

We refer to a sequential equilibrium in which each consumer's set of feasible p ort-

folio strategies consists of all p ortfolio strategies satisfying (10) as a sequential

equilibrium with boundedborrowing.

There cannot b e a nonzero price bubble in sequential equilibrium with b ounded

b orrowing.

Theorem 2.1. If q is a sequential equilibrium priceprocess with bounded

borrowing and if security markets arecomplete at q , then (s )=0 for every j

and every s .

Pro of: Weprovethat (s )=0 forevery j . The same argument will apply to

anyevent s for t>0.

Since securitymarkets are complete at q , for eachevent s and each security j

j t

there exists a p ortfolio (s )suchthat

t+1 t+1 j t r r t+1 t

[q (s )+d(s )] (s )= p(s )d (s ) 8s 2 s ; (11)

t+1

p(s )

(

s 2E (s t+1)

t+1 t+1 t

If wemultiply b oth sides of (11) by p(s ), sum over all s 2 s and use (), then

we obtain

t j t r r

q (s ) (s )= p(s )d (s ): (12)

p(s )

r (

s 2E (s t)

Equations (11) and (12) imply that

t t jt t r jt t t t

[q (s )+d(s )] (s ) q (s ) (s )=d (s ) 8s :

Thus is a p ortfolio strategy with payo that replicates the dividend d of security

j and with date-0 price equal to the presentvalue of d at date 0.

0 j

Supp ose that (s ) > 0forsomej . Consider a p ortfolio strategy that results

j j

j t t

from holding and selling one share of security j at date 0. Thus (s )= (s ) 1

j j

t t t

and (s )= (s )fork 6= j and for every s .Wehave

k k

t t j t t j t t 0

[q (s )+d(s )] (s ) q (s ) (s )=0 8s 6= s :

Using (12) and (6) wealsohave

t j t t t

q (s ) (s )= (s ) 8s : (13)

Further, since

(s )

1 8s ;

q (s )

strategy is feasible under the b ounded b orrowing constraint.

Let b e consumer i equilibrium p ortfolio strategy. Consider p ortfolio strategy

j j

+ . This strategy is budget feasible, satis es the b ounded b orrowing constraint

and generates higher consumption at date 0 without decreasing consumption in any

i i

other event. This is contradiction to the assumption that c and are equilibrium

t t

consumption and p ortfolio choices. Therefore (s )=0 forevery s . 2

5. Implementation.

Theorem 5.1. If consumption al location fc g and a countably additive pric-

ing functional P areanArrow-Debreu equilibrium such that security markets are

complete at prices q given by

(d ); (14) q (s )=P

j j s

i i i

then there exists a portfolio al location f g such that q and the al location fc ; g

areasequential equilibrium under the wealth constraint.

We defer the pro of of Theorem 5.1 until later in this section when we state and

prove a more general theorem. 10

Theorem 5.1 says that an Arrow-Debreu equilibrium in which the implicit se-

curity prices with zero price bubbles make the security markets complete can b e

implemented as a sequential equilibrium under the wealth constraint. Our next

result says that the implementation is exact when sequential equilibria with zero

price bubbles are considered.

i i

Theorem 5.2 If security prices q and consumption-portfolio al location fc ; g

areasequential equilibrium under the wealth constraint, security markets arecom-

plete at q and price bubbles arezero, then consumption al location fc g and the

t t t

pricing functional P given by P (c)= p(s )c(s ), where p(s ) is the event

s 2E

priceofs ,areanArrow-Debreu equilibrium.

Again, the pro of of Theorem 5.2 is deferred until later in this section.

If the supply of the securities is strictly p ositive, then, as mentioned in Section

4, there are cannot b e a nonzero price bubble in a sequential equilibrium. Then

Theorems 5.1 and 5.2 provide a one-to-one relation b etween Arrow-Debreu equi-

libria and sequential equilibria. However, if the supply of some securities is zero,

sequential equilibria with nonzero price bubbles are p ossible. These equilibria do

not corresp ond to Arrow-Debreu equilibria. Weprovenow that the equilibria with

nonzero price bubbles are Arrow-Debreu equilibria with wealth transfers.

0 J 0

Theorem 5.3. Let (s ) 2 IR and satisfy (s ) =0. If consumption

al location fc g and countably additive pricing functional P areanArrow-Debreu

0 i

equilibrium with wealth transfers f (s ) g and security markets arecomplete at

prices q given by

t t

q (s )= (s )+P (d ) (15)

j j s j

t J i

for some (s ) 2 IR satisfying (7), then there exists a portfolio al location f g

i i

such that q and al location fc ; g areasequential equilibrium under the wealth

constraint.

Conversely 11

i i

Theorem 5.4. If security prices q and consumption-portfolio al location fc ; g

area sequential equilibrium under the wealth constraint and if security markets are

complete at q , then consumption al location fc g and the pricing functional P given

t t t t

by P (c)= p(s )c(s ), where p(s ) is the event priceof s ,areanArrow-

s 2E

0 i 0

Debreu equilibrium with with wealth transfers f (s ) g, where (s ) is the vector

of price bubbles at date 0.

Theorems 5.3 and 5.4 indicate that there is a multiplicity of sequential equilibria

under the wealth constraint when some securities are in zero supply. Unless the

initial p ortfolios of securities are all zero, the multiplicity is re ected in di erent

consumption allo cations.

We turn now our attention to sequential equilibria under the b ounded b orrow-

ing constraint. It follows from Theorem 2.1 that there cannot b e a nonzero price

bubble in a sequential equilibrium with b ounded b orrowing. Wehave

Theorem 5.5. If consumption al location fc g and a countably additive pric-

ing functional P areanArrow-Debreu equilibrium such that security markets are

complete at prices q given by

q (s )=P (d ); (16)

j j

and there exists a portfolio strategy which is boundedand

t t t t t t t

y(s ) [q (s )+d(s )] (s ) q (s ) (s ) 8s (17)

i i i

then there exists a portfolio al location f g such that q and the al location fc ; g

areasequential equilibrium with boundedborrowing.

Conversely,

i i

Theorem 5.6. If security prices q and consumption-portfolio al location fc ; g

areasequential equilibrium with boundedborrowing, security markets arecom-

plete at q ,thenconsumption al location fc g and the pricing functional P given by

t t t t

p(s )c(s ), where p(s ) is the event priceofs ,areanArrow-Debreu P (c)=

2E s

equilibrium. 12

A sucient condition for there b eing a b ounded p ortfolio strategy satisfying

P P

t i t

(17) is thaty is eventually b ounded relativeto d , that is,y (s ) d (s )

j j

t i

for all s where t , for some > 0;>0.

PROOFS:

Pro of of 5.4: It suces to show that for each consumer i the sets of budget

feasible consumption plans are the same in sequential markets with the wealth

constraint under security prices q andinArrow-Debreu markets under the sp eci ed

0 i

countably additive pricing functional P and wealth transfers f (s ) g.

If c is budget feasible in sequential markets with the wealth constraint under

q , then there exists a p ortfolio strategy such that

i 0 0 i 0 i 0 0 i

c (s )+q (s ) (s ) y (s )+q (s ) ;

i t t i t i t t t i t t 0

c (s )+q (s ) (s ) y (s )+[q (s )+d(s )] (s ) 8s 6= s ;

t i t t i t

q (s ) (s ) [1=p(s )] p(s )y (s ) 8s :

+ t

s 2E (s )

t t

Multiplying b oth sides of the budget constraints by p(s ) and summing over all s

for t ranging from 0 to some arbitrary , and using (3), we obtain

X X X X X

t i t 0 i t i t i

p(s )y (s )+q (s ) : (18) p(s )c (s )+ p(s )q (s ) (s )

t t

t=0 t=0

s 2F

s 2F s 2F

t t

P P

t i t

Adding p(s )y (s ) to b oth sides of (18) and using the wealth con-

t= +1 s 2F

straint lead to

t i t 0 i

p(s )y (s )+q (s )

s 2E

2 3

X X X X

t i t i t i t

4 5

p(s )c (s )+ p(s )q (s ) (s )+ p(s )y (s )

t t +

t=0

s 2F

s 2E (s )

X X

t i t

p(s )c (s ): (19)

t=0

s 2F

t 13

Taking limits in (19) as go es to in nity results in

X X

t i t t i t 0 i

p(s )c (s ) p(s )y (s )+q (s ) : (20)

t t

s 2E s 2E

t i t

Since p(s )y (s ) is nite in sequential equilibrium under the wealth con-

s 2E

strain, P (y ) is nite in Arrow-Debreu markets under the countably additive pric-

ing functional P . In light of endowment relation (1) and inequality (20), P (! )

i 0

and P (c ) are also nite under P . Using (6) for s and (1), (20) simpli es to

i i 0 i i

P (c ) P (! )+ (s ) . Thus c satis es the Arrow-Debreu budget constraint

0 i

under P with wealth transfers f (s ) g.

Consider now a consumption plan c that is budget feasible in Arrow-Debreu

0 i

markets under P with wealth transfers f (s ) g, that is

i i 0 i

P (c ) P (! )+ (s ) : (21)

i i

That P (! ) is nite and (21) imply that P (c ) is nite. Since securitymarkets are

i i t i t

complete at q and P (c )and P (y ) are nite, for each s there exists p ortfolio (s )

suchthat

t+1 t+1 i t i i t+1 t

t t

[q (s )+d(s )] (s )=P (c ) P (y ) 8s 2 s ; (22)

s s

Since P is countably additive, equation (22) can b e written as

p(s )

i i t+1 t t+1 t+1 i t

: (23) (c (s ) y (s )) 8s 2 s [q (s )+d(s )] (s )=

t+1

p(s )

t+1

s 2E (s )

t+1 t+1 t

Multiplying b oth sides of (23) by p(s ), summing over all s 2 s , and using

(3), we obtain

p(s )

i i t t i t

(c (s ) y (s )) 8s : (24) q (s ) (s )=

p(s )

+ t

s 2E (s )

It follows from (23) and (24) that

i t t i t i t t t i t t 0

c (s )+q (s ) (s )=y (s )+[q (s )+d(s )] (s ) 8s 6= s : (25)

Arrow-Debreu budget constraint (21), equation (6), and endowment relation (1)

i i 0 i i 0 i

imply that P (c ) P (! )+ (s ) = P (y )+q (s ) ,which together with (24) for

s lead to

i 0 0 i 0 i 0 0 i

c (s )+q (s ) (s ) y (s )+q (s ) : (26)

i i

(25) and (26) imply that consumption-p ortfolio plan (c ; ) satis es the sequential

i i

budget constraints under q . Moreover, since c 0, equality (24) implies that

satis es the wealth constraint. This completes the pro of.2

Pro of of 5.3: Similarly as in the pro of of Theorem 5:4wecanshowthatfor

each consumer i the sets of budget feasible consumption plans are the same in

Arrow-Debreu markets under countably additive pricing functional P with wealth

0 i

transfers f (s ) g and in sequential markets under the wealth constraint and

the security prices q sp eci ed by (15). Sp eci cally, since sequential markets are

i i

complete at q and P (! )and P (y ) are nite in Arrow-Debreu equilibrium, for

i i

each budget feasible consumption plan c in Arrow-Debreu markets P (c ) is nite

i i i

and there exists a p ortfolio plan such that (22)-(23) hold. Plan (c ; ) satis es

the sequential budget constraints and satis es the wealth constraint under q .

It is worth noting that for each consumer i there generally exist more than

one such p ortfolio plan if the numb er of securities is larger than the number

t t

of immediate successors of some event. For each s ,letJ (s ) b e a collection of

as many securities with linearly indep endent one-p erio d payo s as the immediate

t i t

successors of s .We consider such p ortfolio plans f g for whichineachevent s

t t i

(s )= 8j=2 J (s ): (27)

Wenow show that these p ortfolio plans clear security markets if fc g are Arrow-

Debreu equilibrium consumption allo cations under P .To pro ceed, summing (23)

P P

i i

over all i, and using the market clearing condition for consumption c = !

i i

and the endowment relation (1), weobtain

X X

p(s )

t t i t t 0

)= [q (s )+d(s )] (s d(s ) 8s 6= s : (28)

p(s )

s 2E (s ) 15

Using (15) and the fact that P is countably additive, we can rewrite (28) as

# "

t t 0 t t i t

) = (s ) 8s 6= s : (29) [q (s )+d(s )] (s

Since preferences are nonsatiated, the Arrow-Debreu budget constraint holds with

i i 0 i 0

equality in equilibrium,thatis, P (c )=P (! )+ (s ) (s ). This observation

combined with the market clearing condition for consumption implies that

(s ) =0: (30)

In light of (7) and (30), wehave

p(s )

t 0

(s ) = (s ) =0 8t 0: (31)

p(s )

s 2F

J t J t

Since 2 IR and (s ) 2 IR for all s , (31) implies that

+ +

t t

(s ) =0 8s : (32)

Given (32), equation (29) reduces to

" #

t t i t t 0

[q (s )+d(s )] (s ) =0 8s 6= s : (33)

Since sequential markets are complete at q , equations (27) and (33) imply that

i t t i

(s )= for all s ,which means that f g clear security markets. This com-

pletes the pro of.2

Proofof5.6: Itsucestoshow that for each consumer i the sequential budget

set under the b ounded b orrowing constraint at security prices q coincides with

the set of budget feasible consumptions in Arrow-Debreu markets at the sp eci ed

countably additive pricing functional P .

Let c b e budget feasible for consumer i in sequential markets at q with b ounded

b orrowing. Then there exists a p ortfolio strategy suchthat

i 0 0 i 0 i 0 0 i

c (s )+q (s ) (s ) y (s )+q (s ) ;

t 0 i t t i t i t t t i t

) 8s 6= s ; c (s )+q (s ) (s ) y (s )+[q (s )+d(s )] (s

t i t

inf q^(s ) (s ) > 1:

s 2E 16

These budget constraints imply (18).

We claim that

lim inf p(s )q (s ) (s ) 0: (34)

s 2F

Toprove (34) note that since involves b ounded b orrowing at q , (9) and (10)

imply that q (s ) (s ) B q (s ) for all s , and for some B . It follows that

X X X

p(s )q (s ) (s ) B p(s )q (s ) 8s : (35)

s 2F j s 2F

Taking !1 in (35) and using (?? ) result in

X X

i 0

p(s )q (s ) (s ) B (s ): lim inf

s 2F

By Theorem 2:1 there is no price bubble in sequential equilibrium with b ounded

b orrowing; particularly, (s )=0 forevery j . The desired result then follows.

Taking limits in (18) as go es to in nity leads to

X X X

i t i t 0 i t i t

p(s )q (s ) (s ) p(s )y (s )+q (s ) : (36) p(s )c (s ) + lim inf

t t

s 2F

s 2E s 2E

Substituting (34) into (36) yields

X X

t i t t i t 0 i

p(s )c (s ) p(s )y (s )+q (s ) : (37)

t t

s 2E s 2E

t i t

Note that p(s )y (s ) is nite in sequential equilibrium since there exists a

s 2E

p ortfolio plan which is b ounded and satis es (17). Thus P (y ) is nite in Arrow-

Debreu markets under the countably additive pricing functional P . By virtue of

i i

(1) and (37), P (! )andP (c ) are also nite under P . The non-existence of price

bubble and (6) imply that

t t t

q (s )= [1=p(s )] p(s )d (s ) 8s 8j: (38)

j j

+ t

s 2E (s )

i i i

In light of (1) and (38), inequality (37) can b e written as P (c ) P (! ). Thus c

satis es the Arrow-Debreu budget constraint under P . 17

Supp ose now that a consumption plan c is budget feasible in Arrow-Debreu

markets under P ,thatis

i i

P (c ) P (! ): (39)

i i

Since P (! ) is nite, (39) implies that P (c ) is nite. Since security markets are

i i i

complete at q and P (c )and P (y ) are nite, there exists a p ortfolio plan such

that (22)-(25) hold. Equation (38) also holds due to Theorem 2:1. (1), (38) and

i i i 0 i 0

(39) imply that P (c ) P (! )=P (y )+q (s ) ,which together with (24) for s

i i

give (26). (25) and (26) say that consumption-p ortfolio plan (c ; ) satis es the

sequential budget constraints under q .

It remains to showthat satis es the b ounded brrowing constraint. Since

c 0, (24) implies that

t i t t i t

q (s ) (s ) [1=p(s )] p(s )y (s ) 8s : (40)

+ t

s 2E (s )

Since there exists a p ortfolio plan which is b ounded and satis es (17), y is

d . Therefore, there exists some suchthat eventually b ounded relativeto

X X

t i t t t

q (s ) (s ) [ =p(s )] p(s ) d (s ) 8s : (41)

+ t

s 2E (s )

Substituting (38) into (41) yields

t i t t t

q (s ) (s ) q (s ) 8s : (42)

t i t t i

It follows from (9) and (42) thatq ^(s ) (s ) for all s .Thus satis es the

b ounded b orrowing constraint under q .2

Pro of of 5.5: Similarly as in the pro of of Theorem 5:6wecanshowthatfor

each consumer i the Arrow-Debreu budget set under countably additive pricing

functional P coincides with the set of budget feasible consumptions in sequential

markets at the security prices q sp eci ed by (16) under the b ounded b orrowing

constraint. Particularly, since sequential markets are complete at q and P (! ) and 18

P (y ) are nite in Arrow-Debreu equilibrium, for each budget feasible consumption

i i

plan c in Arrow-Debreu markets P (c ) is nite and there exists a p ortfolio plan

i i i

such that (22)-(25) hold. The Arrow-Debreu budget constraint P (c ) P (! ),

i i i

price relation (16), and endowment relation (1) imply that P (c ) P (! )=P (y )+

0 i 0

q (s ) . This together with (24) for s and the fact that P is countably additive

i i

result in (26). (25) and (26) saythatplan(c ; ) satis es the sequential budget

constraints under q .To showthat satis es the b ounded b orrowing constraint,

note that (40)-(41) hold for the same reason stated in the pro of of Theorem 5:6.

Since P is countably additive, substituting (16) into (41) yields (42). The said

result then follows.

Among all suchportfolioplansf g consider those that satisfy (27). We claim

that these plans clear securitymarkets. Toprove our claim, summing (23) over

P P

i i

! and c = all i, and using the market clearing condition for consumption

i i

endowment relation (1), we obtain (28). Substituting (16) into (28) and using the

fact that P is countably additive lead to (33). Since securitymarkets are complete

i t t i

at q , equations (27) and (33) imply that (s )= for all s . Therefore, f g

clear security markets. This completes the pro of.2 19

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