A p p l . M a t h . J . Chinese U n i v . Ser. B 2001,16(2):203-218
DISCRETE-TIME STOCHASTIC EQUILIBRIUM WITH INFINITE HORIZON INCOMPLETE ASSET MARKETS Zhang Shunming Abstract. This paper examines the existence of general equilibrium in a discrete time economy with the infinite horizon incomplete markets. There is a single good at each node in the event tree. The existence of general equilibrium for the infinite horizon economy is proved by taking limit of equilibria in truncated economies in which trade stops at a sequence of dates.
§1
Introduction
[-1-] proved the existence of general equilibrium w i t h the finite-dimensional commodity space by mathematical technologies, which is the most important work in modern mathematical economics, and they won the Nobel Prize in economics in 1972 and 1983,respectively. The field of mathematical economics develops w i t h general equilibrium theory. G. DebreuE2~surveyed
the finite dimensional theory in V o l u m e II of the Handbook of Mathe-
matical Economics. [3] described t h r e e modelling problems which lead t o quite different infinite dimensional commodity spaces • (A) In intertemporal allocation problems ,the natural commodity bundles are consumption streams; (B) In allocation problems u n d e r uncertainty,the natural commodity bundles are consumption patterns which depend on the s t a t e of the worlds(C) In models of commodity differentiation ,the commodity space is taken as the space M ( K ) of (signed) Borel measures on compact metric space K t o allow for many different commodity characteristics. In this paper ,we study discrete-time stochastic equilibrium from cases (A) and (B).
Received : 1999-06-09. Subject Classification .. 90A09,90A30,90B30,90C05,90C08. Keywords..General equilibrium, infinite horizon incomplete asset markets, infinite horizon economy, truncated economy,associated stochastic economy,purely exchange economy. This research was supported by a project of Financial Mathematics,Financial Engineering and Financial Management,which is one of"Ninth Five-Year Plan'Major Projects of National Natural Science Foundation of China (79790130), Asset pricing theory with frictional security markets, which is a project of National Natural Science Foundation of China (70003002) and 985 Basic Foundation, School of Economics and Management,Tsinghua University.
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I-4-] proved the existence of general equilibrium for two-period economy with incomp l e t e real asset markets by Grassmanian fixed-point theorem. -5~7-] proved it by algebraic topology,game
theory and differentiable
topology,respectively. [-8-] extended the two-peri-
od economy to multiperiod economy and proved the existence of g e n e r a l equilibrium with incomplete real asset markets. 79-] proved the existence of general equilibrium for two-period economy with incomplete nominal asset markets. [-10,11-] extended the two-period economy
to multiperiod
economy and proved the existence of general
equilibrium
for
stochastic economy with incomplete financial markets. T o prove the existence of a finitely effective equilibrium,and the equivalence of a finitely
effective equilibrium with the o t h e r two notions of equilibrium, _c12,13-] found it
convenient to establish the existence of a pseudo-equilibrium relative to some systems of l o o s e , consistent debt constraints firstly. Every suitable finite truncation o f the economy has a
pseudo-equilibrium. T h e limit of these finite horizon
pseudo-equilibrium
pseudo-equilibria
provides a
for the infinite horizon economy ,in w h i c h the debt constraints are tak-
en to be the limit of implicit debt constraints for the finite horizon truncations. [-14-] s t u d ied the role of default when infinite horizon financial markets are incomplete. There is a single good at each state of nature and the single good is used as the numeraire. Thus the existence of general equilibrium with infinite horizon incomplete markets is equivalent to the existence of general equilibrium for a purely exchange economy, c_15-] studied the asymptotic
behavior of infinite horizon incomplete asset markets. [-16-] studied some p r o p -
erties of competitive equilibria for dynamic exchange economies with an infinite horizon and incomplete financial markets. [-17-] studied infinite horizon incomplete markets and the existence of an equilibrium f o r the infinite horizon economy from t a k i n g limit of equilibria in truncated economies in w h i c h t r a d e stops at a sequence of dates. [-18-] studied sequence economies over an infinite horizon with general security structures. W e consider the existence of general equilibrium in a discrete-time economy. Previous authors
(references) have studied the existence of general equilibrium for incomplete mar-
kets having a finite time horizon. In this p a p e r , w e extend these results to an infinite time horizon. W e use an event tree to describe time, uncertainty and revelation o f information over an infinite horizon. There exist eountably infinite nature states. T h e existence of an equilibrium for the infinite horizon economy E~ is obtained by t a k i n g limit of equilibria in truncated economies in w h i c h t r a d e stops at a sequence of dates. T h e associated economy ET is the economy with the same characteristics as E~ in w h i c h agents are constrained to stop trading a t date T c17~. An equilibrium for the associated stochastic economy ET corresponding to the economy ET is one-to-one correspondence with that for the truncated economy Er. W e simplify the economy E r into the purely exchange economy E" and show the existence of its equilibrium by traditional method. From the existence of equilibrium for the economy E" ,we obtain the existence of equilibria for the economy E rE:43,hence f o r the e-
ZhangShunrning
DISCRETE-TIME STOCHASTIC EQUILIBRIUM
205
conomy Er. Then the existence of an equilibrium for the economy E~ is proved by takin~ limit of the equilibria for the economies Er. We prove the existence of general equilibrium as follows. In § 2 the model of the infi nite horizon economy E ~ is built. In § 3 w e study the associated truncated economy Er. h § 4 the associated stochastic economy E-r is analyzed. In § 5 the existence of the equilibri um for the purely exchange economy ~" is presented and in § 6 the existence of the equilib rium for the infinite horizon economy E~. is obtained.
§2
Infinite Horizon Economy
We use an event tree w t o describe t i m e , uncertainty and revelation of informatior over an infinite horizon (see[10,17,193). More precisely,let 3 - = { 0 , 1 , 2 . . . . }denote the set of periods,that is,discrete-time is infinite horizon,and let /2 denote a set of countably infinite possible states of nature. The revelation of information is described by a sequence of partitions o f / 2 , F = ( , ~ 0 , ~ ,,~z . . . . ) , w h e r e the partition ~;~-, is finer than the partition .5v-, i for all t ~ ] . At date t = 0 w e assume that there exists no information so that ,~-c = {Gf , o } . The information available at time t E , ~ - is assumed to be the same for all agents in the economy (symmetric information)and is described by the subset of the partition ~-, in which the s t a t e of nature lies. All agents in our economy are assumed to l e a r n information according to an event tree = The set = consisting of all nodes (vertices) is called the event-tree induced by the power set ,-~=2n,which consists of all subsets of n. 2 = {~,1~, E ~ - , , t = 0,1 . . . . . T}. The set of nodes which succeed a node # E 2 is called the subtree 2(~e) ---
{#'E214:'>J#}
w i t h root ~ E 2 . 2 " ( ~ : ) = {~' E E ( ~ ) l e ' > ~ } = l ~ ' E 2 l ~ : ' > ~ } is the set of strict successors of ~. The subset of nodes of 2 ( ~ ) at date T is denoted by 2 r ( ~ ) a n d the subset of nodes between dates t(~) and T by 2 T ( ~ ) 2-r(~) = {~' E 2 ( ~ ) It(~') = Z},
2 T ( ~ ) = {~' E 2 ( ~ ) ' t ( ~ ) ~ t ( ~ ' ) ~ T}.
When ~ is the initial node,the notation is simplified to 2 " , 2 r , 2T. 2-=
{~E ~lt($) ~=0},
2r=
{~eE 2,t($)
=T},
N r = {~:E 2 1 t ( 8 ) ~ T } .
~+ = { $ ' E 2 ( $ ) i t ( $ ' ) - - - - t ( $ ) + l } is the set of immediate sucessors of ~. Every $ E m w i t h t ( ~ ) ~ l has the unique predecessor node $- of ~. T h e r e is a single good at each node and the single good is used as the numeraire. Any function on ~ is referred t o as a process. A consumption process x:2---,-J~ represents the spot consumption x ( $ ) E . ¢ ? at a typical node $ E =" For simplicity,we assume throughout that all conceivable consumption process are (essentially) bounded E14:,thus the commodity space is l ~ ( 2 ) = {x E . ~ [ s u p e e z l x ( $ ) I < oo}. Real assets are claims t o consumption processes a f t e r date 0 and thus are elements of
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l ~ ( 2 ) . T h e dividend of asset A at node 6: is A(6:). If there are d assets A 1 . . . . . A ' I ; A = (A 1. . . . . A s) E/~(2)<~,y),where
) 7 = {1 . . . . ,d}. A trading strategy 0 is an element of
l ~ ( 2 X ) 7 ) , 0 = (01 . . . . . 0j) E l~ ( 2 X ) z ) , where 0j is the holding of asset j (where 0t'~ 0 m e a n s short-selling asset j). A(6:) • 0(6:-) ----- ~--]sj = Aj(6: ) . 0t(6:_ ) is the n u m b e r of consumption paid in gains at v e r t e x 6: E 2 . Let a°(6:) = A ( 6 : ) • 0 ( 6 : - ) ,
6: E 2 ,
where 0(6:~-)=0,then a ° E / ~ ( 2 ) i s the dividend process on the t r a d i n g strategy O E I ~ ( 2 X ~r). Each asset A t , for j E ,¢¢,is assigned a real-valued price process Ss E l~ ( 2 ) . In o t h e r words,St(6:) is the market value of dr(6:) a t node 6 : E 2 . It will be convenient to t r e a t St (6:) as the market value of dr(6:) a f t e r the dividend dr(6:) has been"declared" and the dividend has been paid. Asset price is the process S E l ~ ( 2 X ) z ) . If O E l ~ ( E X ) Z ) i s a trading strategy,then
S V q O E l ~ ( E ) is the market value of the t r a d i n g strategy at the asset price
processes S defined by ~ S ~ O ~ (6:)----S(6:) " 0(6:)for any 6:E 2 . The dividend process ~°E I ~ ( 2 ) generated by a trading strategy 0 is defined by 3°(6:) = A ( 6 : ) • 0 ( 6 : - ) - - S ( 6 : ) • 0 ( 6 : ) ,
6: E 2 .
T h e trading strategy 0 finances a consumption process z if 3 ° ~ z , t h a t is,if 3 ° - - z E l°~ ( ~ ) . There are I agents ( i = 1 . . . . . / ) d e f i n e d by consumption sets l~ (2),endowments d E / ~ ( E ) a n d preference relations ~>~ on l~ ( 2 ) . I-Preference
Assumption l-lEach i = 1 . . . . . I , ~>~ on l~ (E)satisfies the following condi-
tions : (1) Continuous : for each w E l~ ( 2 ) ,the sets {y E l~ ( 2 ) ]y ~>i x }and {y E l~ ( 2 ) ]a: ~.~-.~ y}are closed. (2) Weakly convex:for each x E / 7 ( 2 ) ,the set { y E / ~ ( 2 ) ] y ~>i :c}is convex. (3) Complete :if z I ,wZE l~ ( 2 ) , t h e n e i t h e r zl~>i z2 or w2~>i w1. W e are ready to formulate equilibrium in a simple setting. An dynamic infinitely horizon exchange economy is a collection Eo.= ( ¢ , 2 , A ) consisting of an event tree E,an asset structure
A E l ~ ( 2 X )z) and an exchange economy
~----(1~(2),;~.>~, e ~ ; i ~ - I . . . . . / ) o n
l~ (E). At each node the agents of an economy are imagined to meet at spot commodity markets and security markets to conduct t r a d e at given pairs f o r that node. With a market syst e m ( A , S ) E l ~ ( 2 X d Y ) X l ~ ( 2 > ( a y ) as given,a budget feasible plan (x;,0~)E l ~ ( 2 ) X l ~ ( 2 ) < ~ ) made up of a consumption process x~E l ~ ( 2 ) and a trading strategy 6~'E l ~ ( 2 )( akz) financing the net t r a d e x ~-e ~,that i s , # # - - ( x ~ - - d ) E l~ ( 2 ) , t h e budget constraints of agent i can be written as follows a:~ - d ~ . x'(6:o) - - e'(6:o) + S ( 6 : o ) • 6"(6:0) ~ 0,
t
x ' ( 6 : ) - - e ' ( 6 : ) + S ( 6 : ) .#(6:)~A(6:) .t?(6:-),
6:E2\~0,
then the budget feasible set B ~ ( S ) of agent i who buys 0~E l ° ° ( 2 X ~ 7 ) units of the d assets
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207
is defined by the budget correspondence Bi " IF (~'X J ) - - ~ l ~ ( E ) X 1~ ( E X , ¢¢) as f o l l o w s , Bi(S) = { (x,O) E IT (E) X I~(E X ~iZ) x('~o) -- e'('~o) + S(,~o) " OG%) ~ O }. x($) -- e'($) + S(~) • 0($) ~ A(~) • 0(~-),~ E E\~o
L e m m a 1. T h e budget correspondence Bi satisfies the following: (1) B~ is a closed correspondence, (2) B I ( S ) is a closed,convex and compact set in l ~ ( E ) X I ~ ( E X f l Z ) , (3) If e~E I ~ + ( E ) , B~ is l o w e r hemi-continuous. Proof. (1) and (2) are straightforward. (3) P r o o f m e t h o d is from[20]. Consider the correspondence B,.° : I ~ ( E X ) Z ) - - ~ I ~ ( E : X l ~ ( ~ X flz) defined by Bo(S) = { (x,O) E lT (E) × l~(E X )Z) x(go) -- g(G) + S(G) " O($o) < O x($)
}.
-- ei(~) + S ( , ~ ) • 0($) < A ( 8 ) • 0 ( ~ - ) , ~ E ,~\~o
If eiEl°~+ ( E ) , t h e n ( 0 , 0 ) E B b ( S ) , t h a t is,B°(S) is a nonempty set. Let l i m . - ~ S " = S and ( x , O ) E B ° ( S ) , t h e n I x ( S ) -- e ( $ ) + S ( $ ) • 0($) < A($) • 0 ( $ - ) ,
$ E E\$o.
Then for every {(x",O")}such that lim._~ ( x " , O ' ) = (x,O)and f o r n large enough, x"(6:0) -- e"($o) + S"(~o) • 0"($o) < 0,
I
x " ( $ ) -- e'($) + S"($) • 0"(~) < A ( $ ) • 0 " ( $ - ) ,
~ E ~\~o.
Thus ( x " , 0 " ) E B ° ( S ' ) f o r n large enough,which implies that B° is l o w e r hemi-continuous. Since the closure of l o w e r hemi-continuous correspondence is also l o w e r hemi-continuous, (3) follows. F o r a given market system ( A , S ) E I ~ ( ~ X a Y ) X I ~ ( ~ X , Y ) , l e t -~ denote the set ot dividend processes generated by t r a d i n g strategies , o r - ~ = {c~'eI 0 E l ~ ( ~ X , 7 ) }. Let ~ " denote the set of consumption processes are l ~ ( E ) s u c h that x E ~ . T h a t i s , ~ " is the mark e t e d subspace of l ~ ( E ) , t h a t set of consumption Processes financed by some t r a d i n g strategy w i t h o u t excess dividends. If the marketed subspace ~¢" equals the whole choice space I ~ ( ~ ) , t h e asset markets are called complete. If the marketed subspace ~ is the true subset of the whole choice space l ~ ( E ) ,the asset markets are called incomplete. T h e marketed subspace
~ is a finite-dimensional vector s p a c e , b u t the whole choice space l ~ ( ~ ) i s an in-
finite-dimensional
vector s p a c e , t h u s the marketed subspace ~ d o e s n ' t equal the whole
choice space l ~ ( E ) , t h a t i s , t h e marketed subspace
~ is the true subset o f the whole
choice space l~ ( E ) , therefore the asset markets are incomplete. W e study the n a t u r e o] g e n e r a l equilibrium with incomplete asset markets. Definition 1. An equilibrium for the economy E~ is a three-tuple
(S, (x~,6~)),of an asset
price process S E I ~ ( E X f lz ) ,consumption process x ~ E / 7 ( E ) a n d t r a d i n g strategy 6~E1~ ( E X flZ)such that ( 1) ( x~, ~ ) E Bi ( S ) , i = 1 . . . . , I (budget feasible),
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(2) if ( x , 8 ) E B ~ ( S ) , t h e n x ~ x (optimality), (3) Z I = I (x~--el)=O (commodity markets clearence), (4) Z~=I ~ = 0 (zero net supply assets). The general equilibrium of economy with complete asset markets is equivalent to gene r a l equilibrium of economy without asset markets :2~'22-~. T h e existence o f general equilibriu m of economy w i t h o u t asset markets is summarized i n [ 3 7 . The following four sections prove the main result of this p a p e r : the existence of general equilibrium in a discrete-time economy with the infinite horizon incomplete markets. The significance o f general equilibriu m is from general b o o k s , f o r example [-2] and V23], in the field of mathematical economics.
§3
Associated Truncated Economy
The existence of an equilibrium for the infinite horizon economy E~ is proved by taking limit of equilibria in truncated economies in w h i c h t r a d e stops a t a sequence of dates. T h e truncated economy Er is the economy with the same characteristics as E~ in w h i c h agents are constrained to stop trading at date T. T h e truncated economy E r = (er,E,A)consists of an event tree 2 , a n asset structure A E / ~ ( E × , 7 ) and an purely exchange economy e r = ( l Y ( E ) , > ~ i , e i ; i = l . . . . . I ) on l~ ( 2 ) . T h e budget constraints o f agent i can be written as f o l l o w s : x'~(~o)
t
- e'(~o) -'- S~(~o) • ~ ( ~ o ) <~ O,
X~r(~) -- e'(~) + S r ( ~ ) " 6~T(8) ~ A(~) " 6~r($- ),
xiT(S) -- e'(S) ~- A ( S ) • 6~r(S-),
X~r(S) - e'(S) = O,
~r(¢) = 0,
~ E 2r-'\$o
S E 2r
S E 2\Er
S E ~\E=~-I
then the budget set Br(ST)Of agent i who buys g T E / ~ ( 2 X , J ) units of the J assets is defined by the budget correspondence B,T : lT (HXflz)--~l'~(E)XI~(HX~. ¢) as f o l l o w s , BT(Sr) = Xr(So) -- e'(So) + S t ( S o ) • St(So) <~ O. XT(S) -- e'(S) q- ST(S) • Or(S)
A(S) • 8 r ( S - ) , ( x r , S r ) E l~ ( 2 ) X l ~ ( 2 X o~¢)
S 6 ~'~-"r-l\S0
x r ( S ) -- e'(S) = A ( S ) • O r ( S - ) , X T ( S ) - - el(S) = O,
8~(S) = o,
S E
S E 2r
>.
S E ~~'~"\2 T
=._,\~,r-,
Definition 2. An equilibrium for the economy E r is a three-tuple
( S t , (x~-,8'r))of a n asset
price process S r E I ~ ( H X ¢ .¢) ,consumption process x ~ E l : ( 2 ) and t r a d i n g strategy ~rEl ~ (H X ¢-¢) such that
DISCRETE-TIME STOCHASTIC EQUILIBRIUM
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209
(1) (X~r,6~T) E BT(Sr) , i = 1 . . . . . I (budget feasible), (2) if (xT,Or) E Br~(Sr) , t h e n x~-~,xr(optimality), (3) ~-']i=~ ( x ~ - - - g ) = 0 (commodity markets clearence), (4) ~ = ~ 0'r----0 (zero net supply a s s e t s ) , (5) S r ( ~ ) = 0 ,
~ E ~ \ Er - '
§4
AssoCiated Stochastic Economy
The existence of general equilibrium for the truncated economy E r depends on the existence of general equilibrium for a stochastic economy /~r= (~T,Er,Ar)consisting of an event tree E r,an asset structure A-z" E lT ( E r N ,7) ,where fA(~),
~E=-r,
"N
~r(~)
10,
¢ E ~\~r,
and an exchange economy i t = (/~ ( E r ) , ~ , g r ; i = 1 . . . . . / ) o n l ~ ( E r ) , where e~-(¢)=e' (¢) , ¢ E ~ r a n d preference relation ~ r on i r (~r) as follow: --I
- - ' / ' - -2 --1 XT - ~ I XT{;=~X ~ i X-- z ,
where x* (~)=x]-(~),~EET,k----1,2
and x~( ~ ) = x 2 ( ~ ) , ~ E ~ \ H r.
If ~ , satisfies Preference Assumption 1 , t h e n --~T also satisfies the following Preference Assumption 2. [Preference Assumption 2]For each i--1 . . . . . I,~_~fon l~ ( E r) satisfies the following conditions • (1) Continuous . f o r each x r E l ~ . ( E r) , t h e s e t s
{YT E
l~ (..~r) [YT ~-,rXr }and {YT E l~ ( E r) ]xr ~TyT}are closed,
(2) Weakly convex:for each XTEI~(,.~~T) ,the sets { y r E l 7 ( ~r) l.yr~,ryrr }is convex, (3) Complete :if xJ-,x~-E 17 ( ~ r ) , t h e n e i t h e r xlr~-,rx~ o r x~-~-,TxJ-. The budget constraints o f a g e n t i can be written as follows : ZC'r(~o) - e'r(~o) + Sr(~o) • 0'r (,~0) ~< 0,
J
2.~(~)
[~-(~)
-
--
e'r(~) + S r ( ~ ) • ~r(~) <~ A r ( ~ ) • ~ r ( ~ - ) , e~(~) = A r ( ~ ) • ~ r ( ~ - ) ,
~ E ~r-,\~o,
~ E ~r
then the budget feasible set ~ r ( S r ) o f agent i who buys ~ ; E l ~ ( ~r-~ X,,,Y)units of the J assets is defined by the budget correspondence B,.r : l°2(~r-~ X y ) - ~ l ~ ( E r) N F ° ( Nr-' × , 7 ) as follows : ~ (~0) -- e.~.(~0) + Sr(~0) • ~r(~0) ~< 0 K"(S.:)
=
) /
(.7-~.,~-) ~ t 7 ( ~ ' ) x ~ * ( s " - , x f ) ] ~r('~) - ~ ( ~ ' ) + 3',.(,f) • ~,.(~) ~<
A-r(~') " 0~'(,~-),
•
E
~-r-~\e0
2rr(,e ) - e-~(~) = ~.(~) • ~r(~-), Definition 3. An equilibrium for the e c o n o m y / ~ r is a three-tuple
} ~ E ~J
(~r, (x~-,~'r))of an asset
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price p r o c e s s ST E I ~ ( ~r-~ XoY),consumption process x ~ E l ~ ( ~ r ) a n d trading strategy ~'r
E I ~ ( Er-~ × J ) such that (1) (x}, ~r) E ~ r ( S t ) , i = 1 . . . . , I (budget feasible), (2) If ( 2 r , 0 r ) EB,r(ST) , t h e n x}~,rxT(optimality), ( a ) ~,~=t ( x ( ~ - & ) = o (commodity markets clearence), (4) ~--]~=, g - - 0 ( z e r o net supply assets). Proposition 1. If (Sr,(x~r,O~))is a n equilibrium for the economy E r , t h e n ( ~ r , ( x } , ~ r ) ) i s an equilibrium for the economy ET, where ~r ($) = ST ( $ ) , $ E N r - i x} (~e) = x} ($), $ E ~Tand i~r ($) = ~r ($), $ E E T - , . Conversely, (ST, (x}, 0'r) )is an equilibrium for the economy E r , t h e n ( S t , (x},gr))is an equilibrium f o r the truncated economy E r , w h e r e
= ~Sr(~),
x}(~) = ~ x } ( $ ) ,
and
8 E E,r-'
O~r(~) =
§ 5 Purely Exchange Economy F o r i E ~.~,let u i E l ~ ( ~ T ) , w h e r e ui(~)=-0 f o r all ~ E ~ T , a n d x~-=e~+ui+a~',then
x~(~o) = ~ ( ~ o ) + , u ' ( ~ o ) , x~(~) = e~(~) + u ( ~ ) + ~ ( ~ ) " ~ ' ( ~ - ) ,
f
~ E -~-'r\~o.
Then the consumption sets of a g e n t i is changed as follows.
Proposition
2. F o r each i E ~¢,the consumption set X' is a closed,convex and compact set in
l°; (ET-1) X F ° ( E T - 1 x ~Z) ,and (O,O) E X i. Thus the budget constraints for agent./is
u~-t-NTVlgT~O,that i s ,
u ' ( ~ ) + S r ( ~ ) • ~ r ( ~ ) <~ O,
~ E ~r-i
and the budget feasible set br(Sr)of a g e n t i who buys ~ r E l ~ ( E r-1 × J ) units of the J assets is defined by the budget correspondence bri : 17(E r-1X6P')--~Xi as f o l l o w s :
bT(S~) = { ( u , g ) E x'lu + S~mg<~ 0}. L e m m a 2. T h e budget correspondence bT satisfies the following : (1) b,-r is a closed correspondence, (2) bri(Sr) is a closed,convex and compact set in X~,and ( 0 , 0 ) E b r ( ~ r ) , (3) If e r - 1 E l T + ( E r - l ) , t h e n &r is l o w e r hemi-eontinuous. Proof. (1)and (2) are straightforward. (3)Proof m e t h o d is from E20~. Consider the correspondence bY • l g ( E r-i Xfla)--~X~ defined by
DISCRETE-TIME STOCHASTIC EQUILIBRIUM
Zhang Shunming
If er_lElT+(~T-1),then -i
where
211
(-- 1 e~-_~,0)Eb°(~T).Let S~ffl~(~T-1x,ce)with llm.--~ST--'~T • -, _
(u,Or)Eb°(ST)for
( u , O r ) E X i , t h e n u+STv10T<0. Then for every
{(u",~T)}~___X i
with lim,-o~(u",~r)= (u ,0T) and f o r n large enough ,u"q-S~-I-]~T<0. Thus (u" , ~ r ) Eb7(S}) for n large enough,which
implies that by is l o w e r hemi-continuous. Since the closure of
l o w e r hemi-continuous correspondence is also l o w e r hemi-continuous, (3) follows. Agent i E . Y is defined by consumption sets Xi ,endowments ( 0 , 0 ) E Xi and preference relations >-; on Xi as f o l l o w s .
If ~ r satisfies Preference Assumption 2 , t h e n ~ 2 also satisfies the following Preference Assumption 3.
EPreferenee Assumption 3-] F o r each i = 1
. . . . . I , >2 over X~ satisfies the following condi-
tions : (1) continuous.for each ( u , 0 r ) E X ' , t h e sets { ( u ' , 0 ~ - ) f X i](u',OrT)>" / (u,OT) }and {( u ' ,~-) 6 X ' I (u ,0T) _>7 ( u ' ,0~) }are closed, (2) weakly convex :for each (U,OT) E X~ ,the set { (u' ,O~r) E Xi I (u' ,O'T)>__; (U,OT) }is convex, (3) complete., if (u I ,01T), (U2 ,0~) E X ~,then e i t h e r (u I ,@~-) ;~i" (u 2,0~) o r (u 2,0~) ~ 7 (u ~
,~).
Definition
4. An equilibrium for the exchange economy e" = ( X~, 2>~~ , (0, O) ~i = I . . . . . I )
on I ~ ( H T - I ) X I ~ ( Er - I X ~ . c ) i s a three-tuple
( ~ r , ( u ' , ~ r ) ) , o f asset price process ~ T E
/~(H r-1Xo,g ) and consumption processes (ui,~'T)EX i such that (1) (U¢,~") EbT(ST) , i = 1 . . . . . I (budget feasible), (2) if (U,OT) ~ b ~ ( S r ) , t h e n ( u ~ , g r ) ~ ; (U,OT) (optimality), (3) E [ = l
(W'01T)= (0'0)(markets clearence).
From Definitions 3 and 4 we obtain the following proposition.
Proposition 3. If (ST, (W,~r))is an equilibrium for the exchange economy e* , t h e n (ST, (e~-k-ui+a~')',gr))is
an equilibrium f o r the economy Er.
Thus we study the exchange economy ¢* and its equilibrium. W e define the individual d e m a n d correspondence of agent i by ~ " I~(BT-] X~c)__~X; as follows •
(~t(ST) = {(U,0T) E
bri(-~T)[there
is no ( u ' ,0)) E
bT(-ST)
with ( u ' ,0!r) ~ ; (U,0T)},
T h e d e m a n d correspondence N has the following properties. Lemma 3. U n d e r Preference Assumption 3,if e~--] C/Y+ (ur-a),then (1)N is nonempty,compact and c o n v e x valued and u p p e r hemi-continuous, (2)N is a closed correspondence.
212
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Vol. 16,No. 2
Proof. (1) is from [-24]. (2) Let (u",~T)E~,(S.~) with lim(u",~T) = (U,~T) and l i m ~ = ST, then (u,O,)Eb,T(Sr). Next we claim that ( u , g r ) is a maximal element for ~ ; in bT(Sr). T o see this ,let (u' , ~ ' r ) E b ] ' ( S r ) , t h e n u' + SrVqS'r ~ 0. If e~_, ElT+ ( ~ r - ~ ) , t h e n for each 0 < A
- ,1~_~ + S ~ D ~ r < 0
for all n > n ~ . Thus (n",~.) >,." (u'--,k;~--z ,~'r)for all n > n o , t h e n (u,Or)>_2 (u'--2gr-~ ,~r) from the continuity of >'~'. Letting 2 ~ 0 , t h e n (u,Or)_>~" ( u ' , ~ ) f r o m the continuity of > , ' . Therefore (u ,0r) E p , ( S r ) . Let us now define the t o t a l excess-demand
correspondence r p : l~ ( ~ r - ~ × of)--~
F " ( ~ r - l ) Xl'~'(Hr-: x of) by I i=1
Corollary. U n d e r Preference Assumption 3 ,if e-~-_: E IV+ ( ~ r - ~ ) ,i = ] . . . . . I , t h e n (1) ~0is nonempty,compact and convex valued and u p p e r hemi-continuous, (2) ~0is a closed correspondence. If
( 0 , 0) E ~ ( S T ) , then clearly
Sr is an
equilibrium
price process, f satisfies
Walras'I.aw :for every S - r E F 2 ( E r- ~Xof) and (z,Or) E f(Sw) , t h e n Z-Jf-ST[-']0T-~- O. T h e o r e m 1. U n d e r Preference Assumption 3 , l e t e~--~ E 1~_ ( ~ r - ~ ) , i = 1 . . . . . I , then there exists an equilibrium (ST, (u',~r))for the exchange economy ¢'. Proof. Let Z be a compact and convex set such that the image qfflT(~ r-: Xof))~__Z. F o r every ( z , ~ r ) E Z , w e consider a correspondence p
:
Z--'~I~(~TIxof) by
(z,Or)ffZ,
,u(z,~r) = {ST E IT (~._~T-: X f ) [ Z -~ STSZ,~T -~- max.~i.e,~rsr-~×1)z -4- ~'[-q0r}. Claim 1. F o r any ( z , ( J r ) E Z , p ( z , O r ) is a non-empty convex subset of l~(~.r-1X J ) . Claim 2./z is a closed correspondence. Proof. Consider a sequence { ( z~ , ~ ) }~_~_Z converging to (z, ~) ~___Z,and a sequence {S~-}~___ 12 ( ~ 7 - ' X ~')converging to S T E 17 ( ~ r -, X of) such that , for every n , S~ E ,u(z~, ~T), then z" -+- S~-~l~r = maxx~.e~(er-~x..~)z" + S~v[-l~v, that i s , for all S}-E lv (~r-1X,)Z) , t h e n Z ~, ~[~OT ~ Z + STF]OT
f o r all ~ E 17 ( E r - i X ¢1z) , t h a t i s , z + SrV10r = maxr+¢ 17(~r-,xt)Z + ~rE3Or.
Zhang Shunming
DISCRETE-TIME STOCHASTIC EQUILIBRIUM
213
Thus S r E /l(z,Or). Then the correspondence , u × f : Z×12"(Er-~×a~z)-*ZXl°~(Er-~ Xa.¢) defined by
(/~ × 9)( ( z , O r ) , S t ) = (~S~-) ,,,(z,Or) ) satisfies the conditions of Kakutani's Fixed-Point T h e o r e m of Fan :as1 and Glicksberg :z~3, then there exists a fixed point ( S t , (z, 0r)) o f / a × 9, that i s , (z, ~r) • 9(St)and ST • /~ (Z,
OT). From (Z,OT) • f ( S r ) ,z-l-Srl~Or = 0by Walras'Law. From ST • P(z,Or) ,z-q-Nr[]Or = maxx).e~7~zr
,×.,~z~S~lZ]0r,then for any S'.rEI~(E r-* >(af),zq-S'rg3Or~zq-Sr~O.r= 0 , t h u s
z ~ 0 and Or=0,and z = 0 from z-+-Sr~0r=0,that i s , (z,Or) = ( 0 , 0 ) . Corollary. U n d e r Preference Assumption 1, let e-~.-, • IS+ ( E "r-* ) , i = 1 . . . . . 1, then there exists an equilibrium ( S t , (x-~,gr))for the stochastic economy Er.
§6
Main Result
(Sr, (x~-,g)-))is an equilibrium for the truncated economy Er for all T E , ~ , w h e r e ST E /~(EXo5z) ,x~-E I F ( E ) and
0'TE/~(EX,.~).
If the security price process Nr is strictly arbitrage--free for the asset structure A t , t h e n there exists a strictly postive process tiE l't+ ( E ) such that [zr? 1 ST(S) - tiff) ~ ti(r/)Ar(r/),
$ E E r-I,
,/C. $+
thus - S r E l ~ 2 ( E T - " × j ) i s b o u n d e d , S r • l T ( E × j ) i s
also bounded:
{St(S0) }is bounded in ~.l and has a converging subsequence {Sr0($0)}with lira Sr0($o) = S($0).
Ta~m
F o r $~ E E l , (ST° (St)}is a bounded sequence in Euclidean space ,-~". El is e i t h e r finite ( a ) E*~l = { # i , . . . ,~e~'}or countably infinite ( b ) E , = {$I,$~,... }. W e consider the two cases as follows. ( a ) ~ : = {$] . . . . . ~f}.
{Sro(~l) } is a bounded sequence in Euclidean space .~" and has
a converging subsequence {Sr~(~l)} with lira Sr~(~]) = S ( $ I ) and lim Sr~(~o) = S($o). {ST~($~)}is a bounded sequence in Euclidean space a~s and has a converging subsequence {ST~($~)} with lira S r ~ ( ~ ) ---- S($~), T~m
limSr~($o) = S($0) and l i m S ~ ( ~ ) = S ( $ I ) ,
A p p l . M a t h . d. Chinese Univ. Ser. B
214
Vol. 16,No. 2
{Sro_~(~) }is a bounded sequence in Euclidean space , ~ and has a converging subsequence {Sro ( ~ ) }with lim S r o ( ~ ) = S ( ~ f ) , T ~
lira Sr~,(~o) = S(go), lim S o ~
S(~o)
k = 1
K -
1
TOK ~ , ,o ° . . . . ° °
that i s , lim Sr~,(~o) = S(~o), T ° ~
lim Sr~,(#~) = S ( ~ ) ,
k = 1 . . . . K.
T ~
Let T a = T ° , t h e n {Sro(~l)}has a converging subsequence {St1 (gl) }with limSrl(~,) = S ( ~ , ) ,
t = 0,1.
T I ~
( b ) ~ l = {~ ,~e~ . . . . }. {Sr0($l)} is a bounded sequence in Euclidean space ~.I and has a converging subsequence {ST~ ( ~ ) }with lira Sr?($]) = S($~) and lim Sty(S0) = S($o). {ST~($~)}is a bounded sequence in Euclidean space ~ s and has a converging subsequence {ST~($~)}with lim Sr~($~) = S($~), T ~
limSr0($0) = S($0) and l i m S ~ ( $ l ) = S($~). . . . . . , . , . .
{SrO_l ($f)}is a bounded sequence in Euclidean space ~.1 and has a converging subsequence {Sro($f) }with lim S r ~ ( $ f ) = S ( $ f ) , T ° ~
lim Sr~:($0) = S(~0), limSro($~) = S ( $ ~ ) ,
k = 1 . . . . . K - - 1.
T ~ °..
o . . . . . .
F o r any n a t u r a l n u m b e r K , lim S ~ ( £ o ) = S(~o), lim Sr~,(~l) = S(£~),
k = 1 .... K,
T ~
thus {Sro(~1)}has a converging subsequence {St1 (~1)} with limSrl(~,) T I ~
= S(~,),
t = 0,1.
Zhang Shunming
DISCRETE-TIME STOCHASTIC EQUILIBRIUM
215
U s i n g the same m e t h o d , w e can obtain the following claims• F o r ~ezE •z, {Sr~ ($2)}has a converging subsequence {Sr2(4:z) } with limSr2(4:,) = S(4:,),
t = 0,1,2.
T 2 ~ • . . . . . . •.
F o r 4:r-t E E r - t , {STr-2 (4:r-1 ) } has a converging subsequence {STI-~ (4:r-1) } with lim T'I'-I
STT--I(4:¢)
~---
*.~(4:L) ,
t -= 0,1 . . . . , T -- 1.
~
° , . • , . • • •
Therefore {ST} has a converging subsequence,say {S/.},with limS~ ---- S. i
I
i
Consider the sequence {(x~,O~) }. x:~ ~ ~]~= a:4.
~
I
i
then
i
~
I
i
for all 4 : E E . If A E l ~ + ( N X J ) , t h e n 6¢~(4:) is bounded f o r all 4:EE from the budget constraints. {(x~(4:o) ,0~(4:o))} is a bounded sequence in Euclidean space ~ r ) < ~ u and has a converging subsequence
(x~(4:o),0'~0(4:o))} with lim (x~o(4:o) ,0~o(4:0)) =
(x(4:o),0(4:o)).
F o r 4:1E E~, {(x~o (4:~), 0'¢o (4:1)) }is a bounded sequence in Euclidean space ~ I × ~l.J. E~ is e i t h e r finite (a)E~ = {4:1, . . . . 4:f} o r countably infinite (b)E~ = {4:~ ,4:~ . . . . }. W e consider the two cases as follows. ( a ) ~-,~1 = {4:1 . . . . . 4:IK } •
{(xfo (4:1),0~o (4:1))}is bounded sequence in Euclidean space ~ z X ~.Z'Zand has a converging subsequence { (xf~ (4:1) ,0:~ (4:~) ) }with lim (x~7(4:1),0~.~(4:1)) = (x(4:1),0(4:1)) and lim (x:~(4:o),0~(4:0)) = (x(4:o),0(4:0)). { (xf~ (4:~),0f~(4:~))}is a bounded sequence in Euclidean space ~ z × ~ H a n d has a converging subsequence { (xi-o (4:~), 0~ (4:~)) }with lim (x~(4:~),0f~(4:~))
= (x(4:~),0(4:~)),
lira (x¢~ (4:0) ,0~(4:0)) = (x(4:o),0(4:0)) and lim (xf~ (4:1) ,O~ (4:1)) = (x(4:l) ,0(4:1))•
{ ( x f o ( 4 : f ) , O f o (4:f))} is a bounded sequence in Euclidean space ~.~zX~Z'Zand has a converging subsequence { (xfo (4:f), 0~, (4:f)) }with
lira (xei (4:f) ,0cO (4:f)) = (x(4:f) ,0(4:f)), lim (x~ (4:o) ,0~ (4:o)) ----- (x(4:0) ,0(4:o)), lim (xe~(4:~),0~(4:~)) = (x(4:~),0(4:~)), j-~-oo that i s ,
k = 1 ....
K - 1,
216
A p p l . M a t h . J . Chinese U n i v . Ser. B
Vol. 16,No. 2
lim (x~($o),0~o ($o)) = (X(~o),6($o)), "o
TK~
lim ( x ~ ( ~ ) , 6 i ~ , ( $ ~ ) ) = ( x ( $ ~ ) , 8 ( ~ ) ) ,
k = 1 . . . . K.
Let ~1 = ~-~, then {(x~-o ( $ 1 ) , 8~o ($1))} has a converging subsequence { (x~-, ( ~ 1 ) , 0~, ($~) ) }with lim ( x f , ( $ , ) , ~ - ~ ( $ , ) ) = ( x ( $ , ) , 9 ( $ , ) ) ,
t = 0,1.
(b)& = {~I , ~ . . . . }. {(xfo (~i),6¢o (~I))}is a bounded sequence in Euclidean space ~ * X :9~.rl and has a converging subsequence { (x¢~(~]) , 0 ~ ( ~ I ) ) }with lira (:ci,~(~l) ,8r~(~i)) = (x(~;1) ,6(~1)) and lira (x~(~o) ,6~(~o)) = (X(~o),8(~o)). { (x~-~ ($~),0f~ ( ~ ) ) } i s a bounded sequence in Euclidean space . ~ t X ~-~*aand has a converging subsequence { (x:~ ($~), 8~-~($~)) } with lim (x,~($~),0:~(~)) = (x($~),0($~)), lim (xf~($o),0~($o)) = (X($o),0($o)) and lim ( x ~ ( $ 1 ) , 8 ~ ( ~ ] ) ) = ( x ( $ ] ) , 0 ( $ I ) ) .
rx ~ ~ ~)}is a bounded sequence in Euclidean space : ~ X~.*Jand has a {(x~,_ ($~) , 8-o_ converging subsequence { (x~ ($~) , 0 ~ ( ~ ) ) }with lim (x~ ($~) ,0~ ($~)) = ( x ( $ ~ ) , 0 ( $ ~ ) ) , lim (x~ ($o) , 0 ~ ($o)) = (X(~o) ,0($o)), ~o lim (xf~,($~),0f~($~)) = (x($~),0($~)), .
.
.
.
.
.
k = 1 ....
K - 1.
° .
For any n a t u r a l n u m b e r K , lira ( x ~ ( $ o ) ,g~o($o)) = (x($o) ,0($o)), ' o
TK~
lim ( x ~ ( $ ~ ) , 0 ~ ( $ ~ ) ) = (x($~),0($~)),
k = 1 .... K,
thus {(X~o ($~),0~-o ($~) ) } has a converging subsequence { (x~-, ($~), 0~, ($~)) t with lim ( x f , ( $ , ) , 0 ~ ( $ , ) ) = ( x ( $ , ) , 9 ( $ , ) ) ,
t = 0,1.
U s i n g the same m e t h o d , w e can obtain the following claims. F o r $~ ~ H~, { (x~, ($~), 0:~ ($~)) } has a converging subsequence { (x~-* ( ~ ) ,0~-~ ($~)) } with lim (x~*($,),~-~($,)) = ( x ( $ , ) , O ( ~ , ) ) , ° .
° . . .
,
t = 0,1,2.
°
F o r $~_~ ~ ~-_ ~, { ( x ~ ' - , ($~-~), 0~-* ($~--i) ) }has a converging subsequence { (x~ ~-, ($~_~) , 0 ~ - ~ ($~_~)) } with
Z h a n g Shunming lim ,
.
°
o
°
.
°
DISCRETE-TIME STOCHASTIC EQUILIBRIUM
(x##-,($,),#~-,($,))
= (x($,),8($,)),
217
t = 0,1 . . . . . T - - 1.
.
Therefore { (x~- ,8~) }has a converging subsequence { (x~- ,0F) }with lira ( x 2 , 0 ~ ) = ( x , 0 ) and l i m S ~ = S. Let T~ = T , t h e n the sequence { (St, (X~,0'T))}has a converging subsequence { (ST ( x - ~ , 0'T~, ) ) }w i t h limv ~ { ( S t , , ( x ~ , , ~r~ ) ) = ( S , ( x ; , i f ) ) E 1~- ( ~ X ~,Y) X l'~ ( H ) X l ~ ( ~ "
f). Theorem 2. U n d e r Preference Assumption 1,1et e~E l~+ ( E ) , i = 1 . . . . . I and A E l : + ( ~ " ~ ) , t h e n (S, (x~,0~) )is an equilibrium for the economy E~. Proof. Conditions (1), (3) and (4) of Definition 1 are obvious,we now want t o show cor dition (2) only :if ( x , O ) E B ~ ( S ) , t h e n x ~ x ( o p t i m a l i t y ) . If ( x , O ) E B~(S)with l i m r ~ S r ~ = S , t h e n there exists (Xr~,Or~) E B~(Sr~)such the l i m r a ~ (XT,,0r a) = ( x , O ) from the lower hemi-continuity of ~ , ( L e m m a 1 ( 3 ) ) . From (XTa
O T ~ ) E B ~ ( S T , ) w e have x ~ . , ~ i Xw~. Thus x ' ~ x f r o m the continuity of ~>,. R e m a r k . If I ~ ( ~ , # ~ - , P ) takes the place of 1 7 ( E ) , t h e n the conclusions hold as before,an, w e need not c h a n g e the proof in this paper.
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School of Economics and Management ,Tsinghua Univ. ,Beijing 100084. Dept. of Economics, Univ. of Western Ontario, London O N , Canada N6A 5C2.