Equilibrium adjustment of disequilibrium prices

ELSEVIER

llCS

Journal of MathematicalEccmomics27 (1997)53-77

Equilibrium adjustmentof disequilibrium prices Jean-Jacques Herings ~'*, Gerard van der ~ I ~ l f Talman ', Richard Venniker b

b,

Department of E¢~mometrics and CentER, Tilb~g Unioer~ity. P,O. Box 90153. 5 ~ LE Tilburg, Netherlands b Dff~r~ment ofEeonometric,~ awd Tinbergen in,~itute. Free University, De Boele~, t 105, 1081 iIV Am.~er~kun, Netherlands

SubmitledAugust 1994; ~'¢ept~ OCtOt~r 1995

Abslrac[ We consider an exchange economy in which price rigidities are present. An always converging price and quantity mtjus~ent process for such an economy is presen~ ~at i8 b~ed on a d i ~ t e algorithmic p r ~ u ~ rather than on more traditional ~justment processes, which ~ b~od on difference or differential ~uati~s. In tr~ shot~ run, all noaonumeraire t~mmodities h+tve a flexible price level with respect to the numemire commodity but their relative price~ me mutually fixed. In the king run, pricem ~r¢ ms~umcd |o !~ complexly flexible, ~ t~dju~tmel~tWt~e~ ~ t ~ with ~ trlv~! ~qutiibflum w!~h

low enou~ price level and complete ~ m ~ d

mtio~in$ on ~II m~e~. Along ~

~th

f o l l o ~ by the ~justmem process, initi+lily ~11 relative price+ of ~ nononUmcmire comm~tti¢~ are kep[ fl~todand t ~ price I~vel i~; incte~d, l~ationing ~heme~ ~ ~ l j u , ~ to k ~ p m~ket~ in equilibrium, In doing so, the pr~ess reaches ~ shasta,run equilibrium with only demand rationing and no rationing on the numemire and at least one of the o|her commodities. In uhe long run, the process allows for a downward price adjustment of unrafionod non-numeraire commodities and eventually reaches a Walrasian equilibrium. JEL cl~m~cal~m: C62; C63; C38; DSl Keyw~ds: Exchlmge economy; Prk~e rigidities; Discqttilibrium; Simplici+tlalgori|hm; Adju,~tmcnt

process; Dr~ze equilibrium;Walm~ianequilibrium

* C~pondtng author.

0304.~)68/97/$t7.00 © 1997 ElsevierScience S.A. All rights re~rvvJ .~$DI 0304-4068(95)0076 ! -X

P J J+ llcrtngs et al+/ Journal ~f Mathematical Economics 27 (1@97)53-77

54

1. Introduction Let us ~ s i d e r an economy where ~ has to take place against a non-Walrao s'taa ~ sys~m. Demand is not equal to supply in the markets for some A Dr~ze equilibrium can ~ be obtained by demand and supply rationing on ~ commodity markets; ~ Dn~ze (1975). However. owing to den*and and supply rationing, prices have a tendency to change. A well-known ~ e e adjustment process is ~ classical Walrasian tatonaement process. ~ i s WOCess ~ s t s ~ prices of ~ c~modities at any point in time ~cording to their n~onal excess demand at that point in lime. Walrasian tatonnement process has a number of ~awbacks+ First+ the adjast~nt of ~ prices acco~Jiag to this process does not g u ~ | ¢ ¢ ~vergeace to a Walr~an equilibrium ~ system. In Scarf (1960) esamples of ~onomics have been given f ~ which ~ Wairasian tatonnement process faih to converge to equilibrium price vector. It has been ~own in Sa~i (1985) that any pro~..ss based on a finile amount of local in_fomlation fails to converge for a s u b ~ t i a l class of economics, ~ i s I~k of convergc~ce hem been ~ v c d in Smale (1976), v~ ~ ~ ~ T a l ~ (1987a+b). ~ Kamiya (1990), where ~vera! ~ ~jastmcnt processes with much ~tter convergence Wopcvties have ~ n pre+ ~nted+ seco~J draw~k of the Walrasi~ tat~neng~nt p~+ess as well as of the ~er ~ s ~ ~ t ~ in ~ ~viou~ p~agra~ is tha~d e ~ d ~ supply ~e not in equilibrium~ long M the ~+mm has ~ ~ : h ~ v ~ a W ~ i m ~ equilibrium price +ys~cm, So, | r ~ must ~ esclu~ until ~ ~uilib~om is re~h~_I+ M ~ ° ~jumtrr~at ~ e + + i. ~ ~ o m y ~ ~ d mq 0~e e+ ~ctive ~mm~ ~ g g i ~ J with a ~ ~ililyflum i ~ + ~ of the m~"k~ml ( ~ m ~ u~d io tl~ ~j~m|.mc~l

is c ~ s ~ d w h ~ ~iows ~ pa~ of D~+¢ equiIib~ia~.I wF~ePepttee~ ad~s+cd ~ in ~ W m l r ~ t ~ m m p m ~ , wi~ a o t ~ e ~ s ~ m ~ Pepl~ by ¢~cfi~ excessdem~o ~ V ~ (1975)( ~ also~ ~:|.ion ia | ~ , I98l) a proof of ~ conve~ge.ce ~ ~hispPocess is given ia a ~ I wi~ thymec o m ~ i t ~ s ~,d two ~ u t r ~ in ~ c ~ where the l~+J e ~ s s dem~¢I h ~ t i o ~ ~ s f ~ s a 8~m gtb~l~bility cc~li~'t. In ge~t~|+ however. mch a ~ s s ~s ~ ~~|y ¢oove~e to a W a ! ~ a n +~+Ioilibfi+m + m my~mm and even ~ i c bebavio~ amy be e ~ ( s ~ D~y ~ Pi++iani. 1991)+~ possibility of chege~c+behaver has b ~ a ~ f ~ in ~ m (1993) h~ I~IIL

In this p ~ r a ~

+h'~ q ~ t i l y Pd~+gt:+mt ~ s s

is I r ~ n ~

wh/~h ~ +

thug ~ the e c ~ y ~s a ~¢ equilibrium ~ in V ~ q ~ (1975), ~ f ~ e + trade is ~ i b k ~ each g~int m f i ~ , F u P I ~ ~ + ~¢~s

Pdd. Heri~s et aL / J ~ r ~ l of Ma~hemcaicat E ~ m ~ i c s 77 ¢L~7t 53-~'~

55

adjusted according to the market situation at this Dr~ze equilibrium ar~l not according to the notional excess demand, i.e. the adjusra-ncnt of prices is based on the relevant market signal. Moreover, the adjustment process does not suffer from lack of convergence, but instead converges to a Walrasian equilibrium in the long run. An ~ f i c i a l function, called the reduced total excess demand function, will be constructed in such a way that the path of zero points of this function yields the ~th of points followed by the desired adjustment process. The adjustment process will be based on discrete algorithmic procedures, as initiated by Scarf (1973), called simplicial algorithms, rather than more traditional adjustment processes, which are based on difference or differential equations. A modification of a simplicial algorithm introduced in van der Laan (1982) will be presented and will applied to the ~ u c e d total excess demand function, The path of points generated by this simplicial algorithm yields an approximation of the desi~d adjustment process. The inaccuracy of the approximation can be made arbiuafily small, The economic interpretation of the adjustment process of this pa~r is inspired by recent experiences in Eastern European countries and the former Soviet Republic, For ge~ral equilibriumotype models of the situation in these countries we refer to Polterovich (1993). In these models markets are cleared by means of demand rationing. Thus far this type of equilibrium has not been used in an adjustment process to obtain a Walrasian equilibrium. We assume that one of ge comm~ities is the numeraire having a fixed price equal to one. The other commodities, called real commodities, in the short term have a flexible price level with respect to this nu~ne~aire commodity, but h~ve n~utu~lly fixed relative prices. When Ihe price level is so low that no consumer wants to ~ll any ~unou~t of the ~ t commodities, a trivhd equilibrium is obtained by complete demand rationing ~n all ihe nononumeraire ~ommoditie~, We in|reduce ~u~ adjustment pr,~:e~ thai ~l~i~ with such ~1|t!vi~! equilibriun~ ~n~d~ubsequently z~d~iu~|sprices ~ d r~tioning ~hen~s in such a way thai at ~ y moment during the ~justment proces~ it holds Sat the markets ~ kept in equilibrium by r~tioni~g the ~lemand for the nonnumeraire commodities, while there is no rationing on the supply side of the markets. At the ~ginning of the process only the price level of the real commodities and the rationing schemes are adjusted simultaneously until at least one of the non~numeraire commodities is no longer rationed in its demand. This part of the process can be seen as the shortoterm adjustment of the ration;'- ~ ~beme given the fixed relative prices of the nononumeraire commodities. St~ing from the trivial equilibrium with complete demand rationing, the rationing on ~mand is reduced unul an equilibrium is found in which the rationing on the ~mand c~umot ~ reduced any further without allowing for more pri~e fi~ibility or supply rationing. From this shorHe~ equilibrium a~ given fixed re!ative pf,ce~, the prices of the unrationed non.numeraire commodities are allowed to decrease relatively with respect to the price level of the real commodities, The pr~z,~ss gerefore continues by simultaneously changing the price level, the prices of ehe

P33. lterings el al. /Journal of M~+hemaficalEconomics 27 (1997~53-77

unrationed commodities at levels below their relative maxima, and the demand constraints of the rationed commodities, in order to keep all markets in equilibrium. It will be shown that this long-term process continues until none of the commodities is rationed and a Walrasian equilibrium has been obtained. This paper is organized as follows. ~n S e c t ~ 2 we introduce the model and define ~ e o ¢ ~ ~ a real denmrd-constrairm'd equilibrium with a given price level. In such an equilibrium the numeraire commodity is not rationed, there may be ~ rationing on ~ other markets, and the price kvel equals a given value. We show the existence of a ~viai equilibrium with comp|ete d e m ~ d rati~ing on the ma~rkets of all noa-numeraire commodities for price |evels low enough. In Sea|on 3 we construct ~ reduced total excess demand function by relati~ to any elemem of '~ (n + I)-dimeasioaal ~ t the total exce~ d e m ~ at some price vector and ~ m ¢ rationing ~ m e and we di~uss the ~haviour of this function. In ~ t i o n 4 we disoass and iilustra|e the ~ u s t m e n t process. Starting from the trivial equilibrium, this process follows a path of real demand=constrained equili~+ ria until a Walra~ian equilibrium is found. In Section 5 we prove by means of

simpticial ~pptoximali=on that the~e indeed exists a path of prices and rationing scl~mes yielding approximate real demaml+ccmstrmned ~uilibna and show that • is path ~ n e e t s a trivial real d e m a n d + c o n s ~ i ~ equilibrium with complete ~ m ~ d rationing on the markets of all non+nu~raice commodities with an a~proximate W a l r ~ i ~ ~uilibrium. where~s ~ i n , e a s y of the approximation can be m ~ ~itratily small To relme Sis path to the set of e ~ t real dema|tdoconstmi~ ~uilibria+ in $ ~ t i ~ 6 the lat~er ~ t i~ considered, It is shown [hat t h ~ e~i~s a c(m~ct~J ~ t of' real ~ m ~ d ~ : ~ s t r ~ r ~ d equilibria ~ | a i n i n g ~ h ~ mviai real dem~t~d+¢~mtrai~ed ~uilihfium ~nd a Wm!t~mi+m~ u !ibnum.

j ~ i+ . . . . n + 1, Fo¢ e a ~ of notation+ in what follo++~ we denote ~ +~| of imiiee~ {I.,+++ k} by I~, g ~ h c o m m i t i ~ Im is c h ~ e r i m e d by ~ c t m ~ m ~ i ~ ~ t X +, a pmfe~ ~ + d e r i n g ++ + tm X ++~ a v ~ k ~ of initial ewJow~n|~ w +, We |~+ke of ~ ~ ~ i t i e ~ . ~ y commodity +++ |, as ~ ~m~l~t~ire eommodi|y+ h~vi~ a ~iee ~ual ~o 1, In thi~ p ~ r ~ ~ ~hal ! ~ eamomy ~ i~ initially f~e+l wi~ completely fl~ed ~htive l m ~ for the ~ - m m ~ m i ~ ~r ~ l ¢ o m ~ b

PJ.L bYerings ~t aL / douenat ¢~'Mathematic¢~t Econtnnics 27 ¢I~'~ ~5.~-77

57

for a given price level o~> 0 the set of admissible prices is given by the set P ( a ) defir~d by

The following assumptions with respect to the economy ~' are made:

AI. For every consumer i ~ !,, the consumption set X ~ belongs to ~ . + ~. is closed and convex, and X ~+ R+ ÷ ~ c X ~. A2. For every consumer i ~ i~, the preference preordering :~ ' on X ~ is complete, continuous, strongly monotonic, and strongly convex. A3. For every consumer i ¢ I,,, the vector of initial endowments w ~ belongs (o the interior of X ~. Notice that the assumption of strong convexity in A2 allows us to work with demand functions instead of demand correspondences. In ge~ral the short-temt fixed relative prices will not be equal to the relative prices in any Walrasian equilibrium of the economy ~ , being a price vector # • ¢~ --~+n*'+~ and consumption vectors x" ~~ X ~. ~'i ~ i,~, such that both ~"o ~x * ~ ' . ~e ~ and x" ~ is a best element for ~ ~ in the budget ~ t {x e ~ X ~ I p" T x ~:g p . T ~t,~} for every i ~ I,,. Hence, there may not exist an (~" > 0 such that the " p " ~ l~(¢~') supports a Walrasian ~lnilibrium. To equilibrate the price Vcolor ~ m ~ d and the supply under price restrictions, we may int~iuce an equilibrium concept involving vectors of quantity constraints on the net detoured. Given a price vet|or # ¢~ ~ " ' ~ and a r~tioning ,theme on demand L ~ I~, ~l~Ined budget ~.et of consnmer ~ ~ I~ is given by

The eo,xes~dir~g constrained demand d~( p, L) of consun~r i is defined ag a ~ s t element for ~ in B~(p, L). Becau~ of the strong convexity and strong monotonieity assumptions this element is unique and lies on the budget h y ~ r . plane, i.e. pXd~(p, L)~pTw~° A real demandoconstrained equilibrium with respect to a g;.ven price level ¢g > 0 is defined as follows.

Defln#ion 2J : A real demandoconstrained equilibrium. For a given a > O. a real demandwonstrained equilibrium with price level ot (RDE~,) for the economy ns a price system p ~ " ~ ' demand L ' ~ R~ '~ ~. and, for every consumer i ~ l,,, a congumption bundle x" ~e X J such that (i) for all i ~ t,., x ' ' ~ d~( p ' . L" );

Sg

PJJ. Herings e¢ aL / J o ~

(iii)

p~ g a ~ , V j e l , , , ~

of M a t ~ t i c a l Economics 27 (1997) 53-77

p~+, = 1;

(iv) for all j e l , , p; < *i i (v) for all i ~ Im, L~+~ > x,,+ ~ - wg+ ~.

xf' - wj for all i ~ Ira;

*

A nat demand-constrained equilibrium with a price level a coincides with the definition of a constrained equilibrium given m Dr~ze (1975) for the set P ( a ) of admissible prices for a given price level a. The rationing scheme on demand is assumed to be uniform, i.e. the ~ for each consumer. This assumption can easily be relaxed. Condition (i) q u i r e s that ~ consumption of each confiner equals his constrained demand, while condition (ii) is the market-clearing condition. Condition (iii) requires that the ~ c e vector p" lies in the set P ( a ) , i.e. for every commodity j ¢ I,, the price is relatively equal to or smaller than the price level a. wher¢~ ihe price of ~ mneraire commodity equals one. Co~itioa (iv) reflec~ ~ natural property that d e m ~ rationing on the market of a commodity will only o c ~ r if its price is m~imal. Condition (v) implies that there is no rationing on the musket of the numeraire commodity. A real demand-constrai~ equilibrium without rationing yields a Walrasian equili~ium. For a given price level ~o there will i n ~ exist such an equilibrium for a large e n n u i . However. it will be shown that for small enough a there exists a uniquely determined trivial RDE~, at which all ~al commodities are completely r a t i ~ ~ d that for ~ chosen large enough any RDE~ yields a WMrasian ¢quili~um. in this paper it is shown by means of a simplicial ~imation that a Walrasian ~ u i l i ~ u m can b¢ reached by generating a path of RDE~'~. t~e, at any point on ~ path all maAe~ ~ in equilibrium and hence t r ~ is posslble, in ~ following way. ~ path ~ s with a irivial RDE~ for ~ eho~n small en~gh~ in |he sho¢lt I¢¢~1, by i~gea$ing ~ and adjusting simul|an¢~ ously !|ie ¢atic-ang ~het~g a p ~ of ~ ' s is general~ whe~ all real t o t a l i t i e s ~ dem~d~rationed and prices ~ relatively fixed, i.e. in e~h ~uillbfium all ~ p¢t¢~ ¢~sttaints in cotgti|ion (iii) of Definition 2,! ate binding, In the long iem~ ~ price co~strain~ will b e c o ~ nonobiwJing ~ a path of RDI~a's h ~ r a t e d where ~ pri~ of a teal t o t a l i t y ~ m e s lower than tbe maximum price as soon as there h no l o ~ r ~mand rationing oct the market for this ~ ~ i t y . To L~ep all ~ in ~ l i b ~ i u m , along ~ path ~e price level, pri~s, ~ ~tioning s c h e ~ s ate ~justed simultaneously. O ~ e Ihe price of a c o c t ~ d i | y without ~ r n ~ ra|ioning again reacbes its maximum pri~, this commodity ~ n e s ~ m a n d o r a t i ~ again ~ ~ price is ke~ equal to its maxiraum. As ~ as no commodity is rationed, a Walrasian equilibrium h ~ ~ reached, It will be shown tha~ ~is will i e ~ ~ ~ c~s¢ eventually. We define t h vector- w by ~' ~ E~/~ w ~, Since at an RDE~ ~ i~ ~ ~ i n g in ~ ~ e ~ f ~ ~ n ~ t ~ r ¢ commodity, i~ is useful consk~t ~ t y ~ r a u ~ i ~ scben~s L satisfying L~, ~~ % . ~ and 0 ~ L, g %, V / ~ t.o The set of ~ s ~ ¢ali~i~g schemes is ~ t e d by ~ ,

P.l.l,, Herings et ~ff,/ , l ~ r n ~ l

{t.e

t

of Mathcnmlieal Et~n~mics 27 ~1997) 53-77

59

L.+, = w...,}.

To show the existence of an RDE~, for any a > 0 the following lemma gives a result about the values of the demand if the price of ~ome commodity is relatively low. The lemma states that for every consumer i ~ I,,, it holds that if the price ratio p j p ~ for any two commodities j, k ~ l,,+t is sufficiently small, then his constrained demand for commodity j exceeds the total initial endowments of this commodity if the demand constraint for it is equal to these total initial endowments. We define the set £¢' by ..~'= {L ~ R~.+ ~ [ L~ < w~, Vj ~ I,,+ t}. Lemma 2.2. Let the economy ~ = ({ X*, :~ ~, "'q'~',~.~ t, g) sati.~y assumpaons A 1- A 3. Then for every i ~ I,~ there exists a number [3 ~> 0 such that for all j ~ 1, + ~ it holds that dj( p, L) > w~ for every ( p, L) ~ ~"+~ ~x ~ satisfying both p / p ~ ~ ~ for some k ~ !, + ~ and Li = ~ .

Proof. Suppose that there exists a consumer i ~ I,, for which the lemma does not hold. Then without loss of generality there exists a eommtx!ily j ~ 1,+~ and a ~ u e n c e (p', L'),¢~ oi prices and rationing schemes in R~.÷t×.~ ', satisfying, for all r ¢~ [~, lhat L'j = w~, p;/p~. ~ I/r for some k' ~ 1,,. t, and d~( p ~, L') ~ wr Because of the homogeneity of degree zero of the demand function we may assume without loss of generality that for any r ~ M. ~ : [ p~ ~ 1. Hence. there exists a sub~quence (p',, L',, d~( p',, L',))~ ~ ~ converging to some (/3, L. d~) ~ * ~ x ~ x ~ ÷ ~ satisfying ~ = 0, / u v~. and d~ g w; Since ~ . t P~, 1 and there is no supply rationing, the demand function is continuous at (fi. I~) according to the !emma on p. 3IM in Dr~', (!975). Consequently, ~]J ~ dJ( ~L L). Since ~ ~ 0 and lu ~ wj. it folMws from the monotonicity of the preferences thw d~ ~, w~ + w~, which comradic|s ilI ~ w~. OQ, E.D. Given an economy g ~ ({ X*. ~ ~. w~}~"L~, ?), let the numMrs [~ ~, i ~ I,,,. be. so small that ~mma 2.2 holds and define $ by

miDt j ~ I P~

Then a con esponds to a price level in the economy which is so low that under lhe conditions of [gmma 2.2 all consumers ~e demanding net amoums of all real commodities. This gives us the next theorem. ~° ,= ({X s, ~,, ~, vv ~* Theorem 2.3. Let the economy ~.. ~ l, F) satisfy ° assumptions A ! - A 3o Then fiJr any ~ ~ (0, ~ ] there exists an RDE,~. Mvreover, for any RDE~ ( p ' . L*, x* J. . . . , x* "*) with 0 < a ~ ~j it hold,~ that p* =~t~(a), L~ = O, V j ~ 1., and x " ~= w ~, V i G I l . . e

P J J . Heti~gs a al./ Your~l of Mathe~icM E c o ~ i c s 27 U997J53-77

Proof. In Herings (1992) it is shown that for every a > 0 there exists (p" ) L" ) x "~, ' • • , x ' ' ) e ~ + "~' + +x ~ ' × H X ~ such that the first four condii(Z It, tions of Definition 2.1 are satisfied and, moreover, there exists j ~ I. + ~ such that L"s > x y • t - w s,i V i ~ l m. Let any ctE(0, a ] ~.d any ( p = . L * . x * t , . . . , X * m ) with these ~ r u e s be gwen. S u ~ ~ I m and k ~ I. satisfy L~ > +Y~++ w~.

T h ~ x'+=d'(p *, L')=d'(p', L ' )

with

L~

L~, Vyel.+,\{k}, and ['~ =

w+. Since p~ /p~+ ~ < a ~ ~; ~ ~ < 13+ and L + = w~) it follows from L~mma 2.2 that x ~ > w~, wh~h conffadicts equilibrium condition riD. Consequently, L~ =~ x; ~~ w~. Vi ~ I,~. Vj ~ I,. and p" ~ ~ a ) due to equilibrium condition (iv). By e q m h ~ m condition (ii) it follows for ever'./j ~ I, that 0 ~ ~E~'+~(Xy"~ w~) ~, m.L~ ~ d . con~uently+ L~ ~ 0 and xy. i ~ w yt. V i ¢~ 1,m. Since there exists. j ~ !,+ such that L; > x ; ' - w/. V i e l,~. it holds that L'~+ , > x ; ~+, - ,vJ+ ,, ¥ i e tm. Con~quenfly. ( p ' . L ' . x" ~. . . . . x" '~) is an RDE~,. i:]Q.E.D. For a ~ (0. a ]. Theorem 2.3 shows the e x i s t e ~ of a trivial RDE~ in |he sense that the price ratio between the numemi~ and any other commodity becomes h i ~ that nobody ~upplies a ~ o . u ~ r a i r e c o m ~ i t y and therefore an equilibrium is s u s t ~ d by complete demand rationing in ~ m~kets for the nonnumcraire commodi[ie$.

TO { ~ r ! ~ IJ~ l~iee aad qu~lity ~ j u s i ~ a i ~¢~em~, we r~|a|e a price and a ¢ati~iu8 e~tor to aay elemea{ of th~ {a + 1)+di~m+i+mal ~ | C ~+ + 8iv+a by

| ~ e ~l_ C +* ~ t~ i!um~raled+!ia Fig, i {~r # + Z Observe that ~ e t~+unda~ qs + ~ " I d ~ ~¢ ~lon8 to the ~t.

I + q.+

i).+

6,.+

+ I.

())

+mi.().

(,)

++

,,

(s)

PJ.J. Heri~gs et M , / dour~m[ ~'Mether~icat E ~ i e s

-/

27 ¢l ~ 7 ) ~3-77"

6!

7

// // //f //

/_/_d.z

I I i

/

/

/

] 2

Fig. I. "I~e set C ~,

for every j e t . , #(q),~6(q)F~ ~ if q~ ~ I. and j ) j ( q ) < 6(q)7~. if q/>o 1. If q.+ ~ ~ O, then &(q) = a . and if q . . i > O. then &(q) > a. For q ~ C"* ~ we call /~(q) ~, B~(#(q). L(q)) the co-n.-strained" " budget set of consumer i e !., at q, i.e.

{

'--,o ,,:;

vj c l,,, ,}

Le,t ,t~(q) denote the ~ s t elemem for ~ ~ in the constrained budget set ~ ( q ) of t o . s a n e r i G 1~, ~ d~tq) ~ d*( #(q). L(q)), ~ d let us define the total excess ~ m a n d tit q by

The functiol+ +:C "+ + + R "+ i is called the reduced total exce::'+ demand function+ For q ' ~ C "+1 it ho!ds that ( ~ q ° ) , L ( q ' ) , ~ ( q + ) . . . . . d%(q +)) is am RDE,~Iq. ~ if and only if ~(q*)~,Q. Clearly. q" ~ [ ) con'esponds to the trivia! R DE,~ gl L2,. by (/3(~)~(Q ~ * . . i ) . w . . . . . w ). I.|nally we define the ~ t ~,,. e by C"* i ~ {q E C "+ t i n i i n ~ t.q/~ I}. The vet ~ ' * ~ corresponds Io the • ., ° ~n* | A cros~d ~el. in Fig. I for n ~: 2. It a ~ C ° then L(q)~-' w. ,rod h e m e the rationing cons-ra~nts are. ~onobinding~ So, we have that tbr every q ~ C"* t it holds that any RDEa(,~(p(q), ~.(q). ~i(q). . . . . ~ ' ( q ) ) is a Walrasian equilibrium. In Section 5 a cons|ructive proof is given for the exislence of a path of

62

PJ J, Hering,, e~ at. / J t ~ r ~ of Mad~emmic~l Economics 27 ( t~7~ 53-77

approximate ~ of ~ in C "+ 1 corresponding to approximate RDE,/s. This path con~ q = Q. corresponding m ~ trivial RDE,, for a = a, with an approximate zero poim q* of ~ on the boundary ~"+ I of C "+ ! inducing an approxiro~e W~kssian In Secti~ 6 we show ~ existence of a ~ t e d set of RDE,,'s connecting ~ trivial R_DE~ ~ a Wakasian equi!ib~um by ~ i d e r ~ limit of a sequence ~ pa~s of approximate RDE,/s. ~ price and quantity adjustment process foikm, s ~ path of zero points of ~. S ~ n g fi'om ~vial RDE~ ~ the point q = O. it pcoeceds with increasing the price level by h~r¢~ng the v~able q.÷ ,. Since. initially, all consumers ~ ~ ! demanders of $1 ~al commodities, the only way to maintain equilibrium is to ration nit ~mands ~ p ~ e l y . i,e. q , . . , . , q, axe initially ke~ ~ u a l to ~to, At ~ point ~ n ~ con~mcr will s i ~ to ~pply some comn~ity k ~ t,. 11~en it is ~ s i b ~ to w ¢ ~ ~ m m d tmioning in ~ mafl~et for conm~odity k. Le. to i~rease q~. C~tin~ing the adjustmem of Lhe price ~vel. there will also be ~pply by s o ~ confiners ia c~er markets, making it possible to ~ r e a s e ee ~ n t of demand rati~fing in these m ~ L s ~ . Continuing ~is a d j u ~ n t of the price level by a@~gi,g q..~ ~ kce~ng ~ ~ of all c ~ h i e s in equilibrium by ~mg %. Vj ~ I,. it witl be shown Sat ~ ~ point a shorHcm~ equilibrium i~ eeg~hed ~ which there is ~ kruger dema~J r~oning in the market for at least real comn~i~y. ~ay commodity fl ~ 1.. Fcom this point in t i ~ it is ~lIowed $ ~ ~ price ~ com~3i~y fl ~ ~lafive |o ~ ~ice level and is adjusted in ~¢h ~ w~y ~ai the m ~ e l of ~is ¢omm~ihy ig kep! in ~uilibrium. Nest.. at ~ n i in ~ it wili e i ~ h ~ n ~ t there is no d c m ~ rationing in IM m~ket f~r ~nolhcf c~nmodity, ~ y ¢,mmoJity j ~ l.~ Or' in o~e¢ in kCCp iM n~lk¢t fo~ ¢~:~tm~!ity j~ in ~ui!itm~tm, l~ ~ffiee h ~ m M i n c r e ~ l ~ ¢ iM m~imum ~ ¢ ¢ in th~ m~fket f ~ com,t-mMityji m d i c ~ d by |he |~i~ |evel. in the m~simut~ ~ice ~ , . tog¢ll~r wi$ the p~k~ i~v¢l ~ the ra|i~ing ~h~n~s ot" 'he oihe¢ ~~ ~ of ~ c t m ~ i | i e s fi ~ f l ~e ~j~,~.~ simu!lao ~ly ~ h ~a~ the m~kels of ~ ¢ ~otn~gti|~s ~ kept in ~uilg~i,m~ In |he I~tter ease ~ ~ c e of c o ~ o d i i y fl is ke~ equal ~ n m the ~ i m m prk~, wh¢tcas ~ m ~ e t got comrooOi~y fl is equilibrated by ~ n introduci~ d ¢ ~ ra|io~i~. M ~ ~'~lctalty~ ~ ~ poim in t i ~ I~t d c l, M ~ ~ t of ¢ ~ - ~ i t M ~ with r~ ~ d ladoc~ing. ° ~ n the p t ~ p t o c ~ s by Mjustlng s i m u i ~ t ~ s l y ~ price level. ~ ~i,:es a ~ ~mmcMiiics in J. ~ h'~e~mand rationing ~ m e s for ~ ~ I ¢omnmdi~s m the ~ t l. \J~ ~eh th~ a!l markets ~re ~ ; t ia ~uilibtium. Ag ~ ~ fo¢ sonic j ~ J the ~ c ~ ~ h e s its m~imum, prk?e of this ct~m~:l/ty is ~ i n ke~ equal |o 1h¢ ma.~imum ~ ~ the ~ss ~ ~'~'¢ with J\{:~ !he ~l of ~nr~liemed teM c!mm~Jit~es. ~ ~ i ~ ~ ~ ~ ~ r ¢ ~s ~ hmger ~m~mi ra~ionin.g for ~ m e ctmlmc~lity ~ 1,\$. h wiil M shown f i ~ eveni~fl!y ~ wil! M no @m~c] ~ f i ~ n g i~ ~ny m ~ e ~ ~ thetefc,ee a W . ~ : ¢ i ~ equilibrium will M ~ . ~ f ~ we give

P.I.L Hcrings e~ aL / J{mrnal {~fMa~h~rnatical ~\~.m~mi<'s 27 f i ~ 7 ~53~77

~

~ r e ~tails of the process, in the following iemrnas we describe some properties of the reduced total excess demand function ~. Lcmma 3.1. Let the econon~v ~ - ({ X ~, :~ ', w~}~ i, ~) sat/sfy assumptions A 1-A 3. Then the reduced total excess demand function ~ is continuous on C ~+! and ~(q)x ~(q) ~ 0,Vq E C 't+ i. Proof By the iemma in Dr~ze (1975, p. 3(~t) it follows that, for every i ~ I,~. B' is continuous on R~.x{I}×~, using that P,+l = 1 and there is no supply rationing in the market for the numeraire commodity. Using the continuity and the strong convexity of preferences, and the maximum theorem, it follows that, for eve~ i ~ i,,,d ~ is continues on R ~ × {l } × ~ . By the conlinuity of L~refunctions /$ and ~ in q it follows that ~ is continuous on C "+ ~. The strong monotonicity of the preferences yields that ~ q ) + ~(q) = 0,Vq E C n+ s I:IQ.E.D. When q ~ O, e ~ h consumer wants to sell the numeraire commodity in exchange for any of the other commodities. However, as long as % ~ 0 for all j G 1,,, none of the nononumeraire cemmodifies can be bought. So, the consumers must keep their initial e n d o w ~ n t s of the numeraire commodity and we have an equilibrium. When qA > 0 for just one commodity k ~/,,, there is no longer comple~ ~ m a n d ~lioning and the consumers want to buy good k against the numeraire° We ~en have that ~,,+ t ( q ) < 0 and ~k(q)> 0 and therefore the economy is out of ~uilibrium. In the following lemma this reasoning is generalized to the case that qj > 0 for at least o ~ j ~ I,. [.emn~ 3~Z Let the .~concnny $' ~ ({ X ~, ~ J, w~}g ~. ~) satlsfv assumptions A ! oA3o ?hen. for every j e I~, ff q ~ C ~* ~ and qj ~ O. then ~(q) ~ O. t;urthe?more, g q ~ C ~* ~ and q , , ~ ~ O. then ~j(q) ~ 0. Vj ~ 1,,. Moreover..for every k ~ I,. g q ~ C ~ *. q~, ~ ~ O. and q~ > O, ~hen ~ ( q ) > O. Proof, lf q~ ~ 0 for some j ~ I,, ~en L~(q) ~ 0 and hence ~{q) ~ E~% ~(dJ(q) oo w~) g mL~(q) ~ o. ~ t q e c "+ ~ with q,,+ ~ - 0 be given ~ d s u p ~ thal ~ ( q ) g 0 for ~ n ~ k e I, with q~ > Oo Then, for son~ i ~ 1,, d~(q) ~ w~. Since q~ > 0 and hence ~ ( q ) > 0 we have ~at ~ ( q ) is nonobi~ing for this consumer. Therefore, d~(q)*~ d f ( ~ q ) , L) with L ~ S : defined by L~ *~ L~(q). for all j t,+ ~\{k}, and L t ~ w~. Moreover, ~ ( q ) / D , , t ( q ) ~ $ ~ ~ {3~. By Lemma 2.2, d~(~q). L) > w~, a contradiction. Consequently, for every k ~ 1,. if q ~ C "+ ~. q,+ ; ~ 0. and q~ > 0, then ~ ( q ) > 0. From the continuity of ~ it follows that q ~ C ~+ ~ and q,,+ t ~ 0 intplies ~ ( q ) ~ O,Vj e I~. ~Q.E.D, We now need to consider the behaviour of ~ near the boundary of C ~ ~. where q~ ~ 2 for some j ~ 1, or where q,, + ~ ~ I. i.e. when the numeraire commodity is relatively very cheap. To do so, we define the positive nurn~r ~ by

P J J./Icring,~ a ,d. / l o ~

~ M a a ~ t ' w a l Ec~,m~nics 27 ( 1997 # 53- 77

3,3. Let the ecoaomy g~ = ({ X i, ~ ~, wi}~=~, ~) san'sly a s s ~ a o n s A I - A 3 . ~ n , for every j ~ I,, if q ~ C "+ ~ ~ q~ ffi 2, then z~(q) > O. If q ~ C "+ ~ and q,+ t ~ 1 - 8. then ~,+ ~(q) > O. Proof. ~ q~ffi 2, # j ( q ) = O ~ ~(q)ffiws>O. From the monotonicity of Wet~r~e,s it follows thai ~s(q)> 0. By the definition of C ~+ ~ there exists a commodity k ~ I,, such that q~ :g 1. If q. + ~~: 1 - 6, then .....

= .......

s:

~

~ i . # j s; m i n B

~.

Hence, by ~ m a > mw~, t

4. A n

-

2,2, d ~ ÷ ~ ( ~ q ) . ~ q ) ) > w~,~ ~, for all ~ I~,, and so ~,, ~(q) w~, ~ ~ O. ~Q.E.D.

g|~r~

of ~

prk~¢ and qu~tiff adjustment

proce~

In ~is ~ t i o n we con/~der ~ ad~tn~nt p,a~.'ess induced by following p~h of real d c w ~ s t r m - n e d ¢quillbda. ~ path f'wst proceeds from the trivial RDE~ ~ ~ RD~. w ~ ~ l¢~t ~ real ~ i t y is ~ being r a t i O . At this point qj - I lghidS for ~ i , ~ ~ ./~ I~. 1 ~ n ~ proCesS contia~s by keeping the ~|ative pc|cos ~ ~ r a ~ c ~ i ~ m~i~ ~ by allowing a ~ ~ t ~ ~ ~ l a 0 ~ ~ic¢ of ~ u n r ~ commodity by increasing g v ~ of ~ v~i~l¢ %~ Coatinuiag we have tha| in o t ~ r to kccp ~otM espy, ~m~gt ~ual ~o ~ ~ ~ s ~imul~s.ly ~ the ~ s of

rciati~ price ~ ~

~~nt

witt~ut r a ~ i a g , whi~ the reverse hap~ns if

a W ~ | ~ i ~ ¢quili~um ~ ~ obt~ A~ eS~ of ~ pccg~sS is i i | t g ~ i~ Fig., 2 for a - 2 by d~wing ~llon ~ ~ ~ in (q,, q ~ ) - ~ , F~om~ m 2,3 it t o l ~ thU point q - ~ i ~ s ~ ~ v ~ RDE~. Moccove¢. it follows f r ~ Lemm~ 3,2 ,h~, ~ y o ~ r point f with q~, ~- 0 c~u~3t ~ e ~ eq~i|ibrium. ~ f o ~ e , ~ keep ~! ~ , in equilibrium, i~iaiiy ~ have ~o incte~ ~ v ~ of q~. ~us

2

iw 1

B

Fig,

2, Illustralion of the adjustment path;

n ~ 2,

R DE,~tqfs by relaxing the constraint on the demand for conunodity 1 according to the value of q~ and changing the price level accusing to q~. At point A also the value of q~ b e c o ~ s positive, inducing a non-zero demand constraint in the m ~ e t for commodity 2. At point B the path reaches an RDE,~(q~ with no rationing in the market for commodity I, Then the path continues with values of ql above o n . This par[ of the path induces RDE,~(#~'s in which for commodity I a si|uation c o ~ s ~ n d i n g to condition (iv) of Definition 2,1 occurs, i,e. no rationing in the demand for comm~ity I, while the price of this commodity is relatively ~low the price level ~(q), which is de|ermined by the value of q:~. From l,~mma :~.3 we ~ o w that ~,(q) > 0 if ql ~ 2 ~ld that $~(q) > 0 if q:~ ~= I =~ & Since on Ihi~ path all markets ate in equilibrium, the path can neither reach values of q:~ above I o~o5 not the boundary of C ~ where q~ ~ 2. Moreover, if q~ ~come,~ equal m ~ m , the path is continued along the bound~y of C :~ by ~dju~ting qa ~ld q:~ ~ d comp~te ~ m ~ ralioning in the m~ket for commodity 2. 'I'he~tbre, by continuing the pr~ess in the ~gion where q~ > 1, the path has to re~h either a point on ~e boundary of C ~ where q2 ~ 1, or qi must again hecome equal to one, The former case is similar to point W and is discus~d below. The later ca~ is illuseated in Fig. 2 where the pa$ reaches point C. At this point a ~cond RDE~tv~ with qe ~ 1 is reached. From this point on, the path again induces RDEa{e)'s with rationing on both commodities. By continuing the process the path must ~aln ~ h a point q with q~ ~ 1 for either j ~ 1 or j ~ 2~ This hapI~ns at point D. where q~ ~ !. From ~his point on, the path induces RDEa(q~'s with no rationing in the market for commodity 2. Similarly to the reasoning given above, the path must re~h either a point where q~ ~ 1 or a point where q~ ~ I, In the former c ~ the path continues ~s in point C The latter ca~e is illuswated in the figure by point W, where the process re~hes a Walr~ian equilibrium. Notice that Song the path the value of q~ initially inc~a~s. However, in general ['~ is not g u ~ a n ~ d that this value incre~y~es monotonically. Along some p ~ of the path it

P J J . Hce~gs et al./ Jemnm~ of M a 1 ~ * ~ ' o ~ ~c,momR:~ 27 ~1997J 53-77

p~

Fill 3. ~

piii~ti(m of t!~ p ~ c ~

in di~iluili~ilm

re g i me s; n ~ 2.

i,s p~sib~ ~ t the v ~ of ~ v~ab~ q~ that determi~s ~ price level will t~c~a~ ~ l ~ the price ~vel or(q) will decrease in ~ r to keep ~ |oral ¢~ces~ ~ m ~ equal to zero. Using the ~ f i n i l : ~ of ~ q ) ~ c ~ translale ~ piciui~ of Fig. 2 in (qt+ q~):qme¢ ~ a #ctme in (p,. p~)~spe~e. Recall ~at p~ ~ 1 i~ fixed. To do ~0 we firs¢ ¢~i,i.a~r Fig. 3. Assuming thai tl:.~re i~ no rationing ~ ihc m ~ l for nun!era~ t o t a l i t y , in Rgo 3 we have ~itwn ~ diffe~nt a~itioniltg t ~ g i ~ s ~ ¢ ~ i n g m ~ v l i l ~ of p~ ~ t #~. The point W' ~not¢~i the Wlilri~ian ,quilibdom v a ~ a tP~ ~ ¢ ~ 0 ~ ¢ ~ ¢ ~ t!~Mngi h ~ g h thi~ point ~lmrlite the r~tMr high ~ ~plfly i~i~min~ in ~ m~kel.~ is nc~ded t~ ~tliiibritle ill~¢i,i, in ~ g i ~ 11 (lid |M vth~ of p~ (p~) i,~ ¢iti~r M~, ~ d the vMue of p~

!n ~ ml t i~ t ~ I o I I ~ illtrseCli~ of two tgiont we ~ only taliOiting in ~ of ~ ltltilll,~, for hl~tallce ~tfe isi i~ rltlioe~g in |he mtlrkel f ~ ¢ o m ~ i t y I whel~ ~ g l ~ 1 ~ !I t ~ L At ~ h a p o l l ~ e l l I swii¢ites flora demllild r ~ e . g in reili¢~t I 1 ~ p # y , l ~ i n g in ~ g i 11. ~ clmr~, ~i poini Fig, 4, ~ ~ ~ i g h i I i ~ kt'lvillg ~ ~ g i n ~ t e n l t ~ iiliii!ly filed relalive peic~es of i ~ nimomimertlire ¢ o m ~ i ~ s . At rely point on this Iil~ we M ~ thai p - ~ ~i) for ~ p d ~ 1~!¢1 ~ > O, P o l l 0 ~ l ~ ~ l~iCe t¢¥¢1 7, At this ~ n i ii ~iviil ~uilil~i~m is ot~tth~,~l wii,~ ~ p t ~ t d e ~ rationing of

~ q~

io ~

in ~

li~e !~vei ~

henc~ in Fig. 4 ~

pat

P J J . tlering,~ e~ e L / J o u r ~ i ~'Ma~h¢~eic~+t /fco*~mk's 27 {t997) 53-77

67

Fig. 4. lllusUation of the adjustment path in the price space; n ~ 2.

consumer starts to supply commodity 1. This point still corresponds to the point Q in Fig. 2, beean~ this latter point is the projection of the pact of the path along which only q:~ i n c ~ s . At the point O' complete demand rationing is relaxea and q~ becomes positive. Going from 0 to A in Fig. 2 corresponds to going from O' to A' in Fig. 4. The path from 0 to A shows that demand rationing on commodity I is relaxed from rJero, while the path from O' to A' shows that the price level incre~s simul~eously. At point A also q2 becomes positive. Continuing aloP~g the path in Fig. 2 from A to B, Fig. 4 shows thai simultaneously the price level (i.e. c~(q)) increases unlil at point B'. corresponding to poh~t B in Fig. 2, the h<~undary t~twcen regions I and l! is reached, at which the market regime fo~ commodity 1 switches from demand raritaaing into supply rationing. At this point the p~th in Fig. 2 continues with values of q! above I and hence with ~Ice p~ ~low |he masimum ~co~ing to the price level while the m~kees are kept in ~utlibrium wit+hoot rationing in the market for commodity I. In Fig. 4 this i~ illustta~d by the fact that the Imth leaves the ray Ihr~gh O in the up+yard direction, inducing a price ratio Pl/P2 < Vt/F2 by following the curve between ~gions i and !1. At point C', corresponding to point C in Fig. 2, this curve again ~ t s the ray of fixed relative prices. Observe that going along this curve from B' to C ~ absolu~ value of P2 is first increasing and then decreasing, showing that the price level and hence q3 does not increase monotonically. Continuing at poinl C the path in Fig. 2 again induces ~ equilibrium with fixed ~lative prices and ~mand rationing in both markets, and hence the correspoMing path in Fig. 4 continues along the ray through O going fu~.her upwards in region 1. At this p ~ of the ~th the price level increases again. At point D'. con+esponding to point D in Fig. 2, the borer between regions I ~ d Ill is re,bed+ Now the path continue~ along the curve ~tween ~ e ~ regions, keeping the m~ketg in equilibrium by allowing !he price of commodity 2 to v~y ~:~low the allowed maximum value

68

PJ J. Heriag s e~ al, ~Journal ~ Ma~bemagc~d Eomomic s 27 ( 1 ~ 7 ~M - 7 7

(q2 > 1) and imposi~ a ~max~d constraint in the market for commodity l (qt < 1), until at poim W', cotrespcmding to p~nt W in Fig. 2, the Walrasian equilibrium value of the prices is reached.

5. A p p r e ~

~

dema~-cen~.r~|ned equilibria

In this section attention is focused ~ approximate real d e m a n d ~ t r a i n e d ~uflibria. By using fimplicial tech~aes we give a co~tructive ~ f of the e~stence of a ~ of points in C" + ~ inducing a path of approximate equilibria ~ , ~ ~ mv!~ R~++ i ~ d by q - ~ wire m ~ x i m a t e Wakasi~ ~H~um (WE) reduced by some ~ i n t q ~ C "++ such Sat min+mp+qy~ I. Apply~g ~ s + ~ i ~ ~iq~ ~ v i d e s m with t ~ ~sibility to foliow ~he path ~ b e d in ~ previous s e c t . . By taking ~ mesh m+m of the ~ r l y i n g triangulation small enough+the excess demand at ~ ~ x i m a t e equilibria being ~ t e ~ e d can be ~ arbiu'ariiy close to ~ero. In ~ following definition an ++pproximate RDE+o for any price level a > O, is introduced. De]+ni+om 5./: ~.RD£+ ~ ++WE. For m given price level ~ > 0 ~ a real l~mber e ~ O, an ~-RDE~ (~WE) for ~ ~ y ~ ~ ([X'. ~+'. we}?. ,. P) is a I ~ ¢ + r a m p, ++P~ionimg mch+~ L and ccmmmp~i~ !m~les x+,..., +'+ such

$ $ all co~ifio~ of an RD~+ (WE) ~

~isfied. e ~ p ~ lhmt the condid(m of

(~:~ly+ an OgDE+ is an RDE+ ~ an O°WE ~+~ WE+ To +how the e~islem~¢ of a ~th of +P~RDF+++s~ t l n g ~ ~r6vial RD[+~+~ an ~+W~ for ~im+ry + > 0+ we u+~ ~ ~hm+q~+ of mimpli+cifl~ M m m | i c m of fm++ti+~mm.~im a~om:h is me+ t+~ in vm ~ ~ (|91+2) ~ Hell+++ 0993)+ ~ +veto tm N, 0 + ++ k, a t~i~~ ~m~ ~ vsi~ it ~ f l ~ ' d ~+ the ~ v e x hull of ++1 ~+~ly v~,~ in R~+ q~....+q+++., and it+ ~ m ~ d by + ( q t .... icily+ by o+ ~ ~ q+~,+.+, q+++~ ~ c a l ~ the ve~+ices +. A(+eL+ l)++Jmp~m++P~ m g ~ v e x hull of t vettimP++ +(q+. . . . . q++ +') im e a l ~ a fm+m of +++ ~ 4 m +P+++++~ fmet +P(++.+++.+,+*° +,, q*+ '+..., q++ +) is c + l ~ ~ Pm+t of + + + m m ~ vem~ q+. P + O m / + +, m j-sim~ex ~ing ¢~v¢~ hull of ) + I vet!ices of m ++~m~x o + is c a l ~ a fmPe of ¢, A finite 0~1¢¢ti¢~ ~ of k°simolices is a ~fiangula~ ~ a ¢ o m ~ subset S of some ~ t k ~ a n ~ a + if'+ (~) ~ h+~P+~t~t++,~ two +[mplk-em+m,~ imei~+ empty ~ a + : o r e + f e e of It ¢ m + + w m ~ + $ i~++ + ~ $ i c m m~ v e ~ m|+ then e+h f~et r of m k+sim~x + ++~ eRhcr l~g m ~ ~|~i+e b c m ~ of S + ~ is ~ | y a fuel of

P,Jd. Heriegs ¢t a L / Jourraff of Machrmati(:.t ECoe~t~:s 27 {i997~ M-77

69

tr or it is a facet of exactly one other k-simplex in 3 . The ~ s h of a triangulation .~ is defir.ed by ~ s h ( . ~ ) ~ max, ~ max{il q-/~ll~ l q. # E o,}. In this section the set C~* i = {q ~ C,,+ ~! q,,+ ~ ~: I -/~J will be triangulated. where B is as deC'met in Eq. (6) of Section 3. It is also useful to define the set ~ + i ~={q E C~ + i ! min~a l.qj = 1}. An example of a triangulation of C~ + l with arbitrarily small mesh size is obtained by using the K-triangulation described in Freudenthal (1942). Tbe K-triangulation of C~ +l is obtained as follows. For k ~ i , , , let ¢~ denote the vector in R "+. with e ~ = 1 and e~= 0, for all jEl,,+~\{k}, and let e "+1 denote the vector in R "+I wml -'' % .~+t +t=l--~ and e~ +l = 0 , for all j E I n. Let r E N be given. Then the K-triangulation of C~ +l with grid size r -t is the collection of all simplices o~q,a,) with vertices q~,...,q"+~ in C~ +i such that q~ is a multiple of r ~1 if j E l , , , q ~ + l is a multiple of (I - 8)r" t, ~r~ ('n"I, . . . . ~r.+ ~) is a permutation of the elements of 1,4 I, and for every h ~ 1,4 i, qh÷ i ~ qh + r-le~'~. The me~ size of the K-~angulation of C~ 41 with grid size r- I is r -I, Let the labelling function ~b: C~ 41-~ I,,+a be defined by ~(q) max[arg min{~(q)j j ~ !,+~}], i.e. the last component for which the total exee~ demand at q is minimal. Let some triangulation ~ of C~ +~ be given. Now a procedure is used which starts at q ~ _0 and generates a sequence of simplices of varying dimension being faoes of simplices in 3 r. For a simplex o'(q~,..., qt+ ~) in this ~ u e n c e it holds for every ) ~ 1.4, that q~ ~ 0 for every q ~ (r or j ~ ~{q~, .... q'+ ~}). In the first ca.~ ~j(q) < 0 for every q ~ cr by Lemma 3.t and Lemma 3.2, and in the second case ~(q~)~ 0 for a vertex q~ of o" with ~(q~) ~ j by the definition of the labelling function ~ and the fact that ~ ~ti~fies Walras' law. It will be shown below ~at them properties guar~t~ that for a ~ i m q in such a simplex ~(q) i~ approximately ~ro. Two sub.quest si|nplices in the ~4ucn~ either s h ~ a common f~et. or one simple~ i~ a facet of the other° Such ~implice~ ~ said to be adj~ent. The pr~edure umd is clo~ly related to ~e one given in van tier ~ (1982) and is demribed below, [:or J c l.+ ~, we dcfim t ~

where I J I ¬es the number of elements of the set J. It can be shown that ~ ( J ) is a triangulation of A(J). We denote by C~+ ~ the part of the boundary of C~ + ~ where some component of q is

maximal, so

The ~ t C~+ ~\¢2~ ~'~ correslxmd~ ~o the ~ i ~ d a~a in Fig. 1. In the de~riptio~ of the procedure given below, ~ will ~nole a simplex and q~ a vertex generated by the pr~edure. J~ is a sub~t of labels of 1.~ ~ generated by the procedure a~d

70

P J J. tle¢,ings e~ aL /Jo~rna~l t~ Mathematical Et~nomics 27 ~1997) 53-77

induces a set A(J ~) and a triangulation 5r(J ~) in which the procedure generates simplices. Given a set S c ~ t, co(S) denotes the convex hull of the set S The procedure operates as follows. P~e S~O.

~

t = 0 , qt = 0 , a ~ = ~r(q*), Jt =~, i f j f k =

l. Go to Step 1. S t ~ 1. If ~(qt) ~y~ then go to S~p 3. Otherwise there is a unique vertex ~ of ~ such that # ~ q~ and ~ # ) ~ ~(q~). Go to S~p 2. Step :2. Let ~ ~ ~ facet of o" ~ ~ i ~ ~° If ~ exists h ~ J~ such that

~CA(Yt\{h}), then go to S ~

4. If ~ c C ~ + ~. then stop. ~ i s e

there

is a undue point q ' ~ l ~ A ( J t ) ~tch that o ".~+~ ~co(~U{q~+~}) is a t.simplex of ,~(J~) and o-~+ ~ ~ ¢r ~. |ncre~se the values of i and j by !. Go to Step ~. S ~ 3. ~ f i n e J~+~ ~J~u{~(q~)}. ~ r e is a undue point q~+~ ~ A ( J t+~) such that o'J~i~co(o'~U{q~+~}) is a ( t + l)-simplex of ~(J~+~). Increase the val~s of i, j0 k, and t by 1. Go to Step 1. Step 4. I~t ~ ~ the ~ n ~ vertex of o'~ ~ch th~ ~ ) ~ h and ~ ~ ~. Define j t • ~ , j t \ { h}. [~fine o ~~+ ~ ~ ~-. Increase ~

~re~

the value of t by |. ~ t ~ be ~

values of j and k by I

element ~. Go to Step 2.

~ e pro~dure is illustra'~t in Fig. 5 for n ~ I and r ~ 3. The procedure starts wi$ the 0~dimensio~l ~imp~ ~ ~ {[~] in A(~) awl terminates with a simplex having tt f~et ia ~'~*IN A((2}). A ~ r ~ st~ing simplex {Q} ~ pn3cedure

~rl~s

a I o~mp~x in A({2)), ~ n

two adj~rtt 2-$impliccs in A({l. 2})

~aerated, S u ~ n f l y , two ~ j ~ r t | l osimplk~ in A{(I}) ~ ~taint~, fol~ Iow~j by ~ight ~ j ~ e n t 2osimpli~ i~l ~4({i, 2})0 Finally. two ~jacem l o~implices in A({2}) ~ ~,er~ted, wi~ | ~ 1~! ~impte~ h~ving the f~e! ~|ermir~d in Step

2

/

.... 7 2

ii

/ //

PJJ. Herings ¢t at,/Journat of Math~m~ticat E~m~n~cs 27 ~19¢~7~$3-77

71

2 in the set C~+ ~. It is easily verified that the properties of a triangulation guarantee that each s~:~t~in the procedure is feasible and unique. Definition 5.2: J-completeness. Let Jcl~+~ be given with [ J t = t. A ( t - l)simplex ,'(qt, . . . . q~) in C~ ÷ t is J-complete if ~b({q~. . . . . q t }) = J. A J-complete simplex 'r in ,4(J) and a .l-complete simplex ~ in ,4(]) are said to be adjacent complete simplices if either J = J and r and ~ are both facets of a same simplex or in .9"(J), or if r is a facet of ~ and ~- is a simplex in a(d), or if ~r is a facet of r and ~r is a simplex in A(,I). It is easily verified that if two simplices o r l ~ . ~ ( J ) and o"~÷t ¢~,9"(,/) are subsequently generat~ by p ~ d u r e , then r~ ~ or'~ n orJ*~ is a ( J U J)-complete simplex in a(J rid). Let or t, or ~,... be the sequence of simplices generated by the procedure and c o n s i s t the sequence r t, ~'~,... given by ~'~* or~nor i*|, for j ~ 1. ~ e s u b ~ u e n t simpliees in the latter sequence are adjacent complete simplices. It will be shown that by generating a finite number of simplices in U .~c t,. , ~ ( J ) the procedure terminates in Step 2 with a simplex having, for some J c I,,+ t, a J-complete facet in ~ + t. To prove this, we first give the ~ x t lemma. Lemma 5.3. Let a triangulation .~° of Cg + t and a labelling #nca'on ~ : C~* ~ -~ I,, ~ be given. Let r be a J.comp/ete simplex in A(J ) fi~r some J c !, + ~. Then r has exactly one adjacent complete simplex if T ~ {~} or ff • lies in Cg÷ ~. Otherwise, r has two adjacent complete simplices. Pr~mf. Fill, consider the simplex or i . ~.I ~ {Q}°'finis is a J°complete simplex in

a ( J ) if and only if J ~ {d,(~)}. Since .~t({~(9)))is a triangutalton of A({~/~(@) and T t is a facel ta the relative ~undary of A({~(Q)}), there is a unique i~simplex ~r ~ ~ a ( D, q) in A({~(~)}) such that ~ is a facet of (r ~, Either ~(q) ~ 4~{~) ~ d f~ ~ {q} is a {~(~)}ocomplete simplex in A({~{O)})0 or q~,(q) * ~b(~) and , ~ q z is a {~(~), ¢h(q)}ocomplete simplex in A({~(~), ~(q)}), Hence, ~r~ has exactly one adjacent complele simplex, Secondly, let r " ~ .dq t, .... qa) ~ Jocomplete in A(J) with I J t ~ t, whik,. ~r" is a subset of C,~+ ~, so ~'" lies in the relative ~ u n d ~ of A(J). It is e~ily shown that r cannot lie in A(J') for a proper sub~t J' of J. Since b~(J) is a triangulation of A(J) there is a unique simplex or" ~ tr(q t .... ,q'* *) in ~ ( J ) containing r " as a facet. Either ~ b ( q ' * l ) ~ J and a " h ~ a unique J.eomplete facet in A ( J ) not equal to r ' , or ~(q'+t)q~J and rr* is a J U { 6 ( q ' * l ) } o complete simplex in A(J U {¢t~(q'* ~)}). Since T" d ~ s not lie in A(J') for any prop~zr sub~t J' of J this ~ows that T* has exactly one adjacent complete simplex. Now let "r(q ~.... ,q*) be a docomple~c simple~ in A(J) wi~ [ J t ~ t. ~r¢~{0}, a n d , no~ being a sub~t of ~~"*~ . 'H~ere are two ~ssibilities, e i t h e r , lies in A(J') for some uniquely detenuined proof sub~t J' of .] or ~" ¢|~s not lie in the ,j

o

*

72

PJ J. fte,rlngs e¢ aL / Jtm,rttal ~ M a t ~ i c a t

Econamics 27 f1997) 53-77

relative boundary of A(J). In the first case, by the properties of a triangulation, ~ is a unique t-simplex tr(q ~. . . . . q'+~) in ~q'(J) having r as a facet. As in the previous paragr~, either o" is J ~ {~b(q'÷ ~)}-complete in A(J U {~b(q'÷ t)}), or tr has a J-complete facet r ' ~ r in A(J). This yields exactly one adjacent compt¢~ s:m~plex to ~'. The other adjacent complete simplex is given by the unique / ' - c ~ p l e t e facet of r. Hence, r has exactly two adjacent complete simplices. In ~ c ~ when r ~ s not lie in the relative boundary of ,4(J), then by the properties of a triangulation there are e'~actly two different simplices in 6~(J) containing r as a common facet, and ~ before this yields exactly two adja~nt complete simpliees to r. It is easily verified that there cannot be any ~ e r adjacent comple~ simplex no r. t:2Q.E,D. Theorem 5.4. ~ t a triangulaKon ~ of C~ + ' and a labelling.~nction ~ : C~* ~ -~ I,, ~ be given. Then the procedure terminates, after generating a finite number of si~lices in U s c t,. ,if( J)...in Step 2 : ( the procedure with a simplex having a J.complete ~ c e t in A(J) ~ C~ + ~ for some J c !~ + ~. Proof. ~ t ~ . ~ ~.,.. be the ~ u e n c e of a d j ~ n t simplices generated by the pr~edure. E i ~ r the procedure terminates, after ~ r a t i n g a finite number of simplices, in S~p 2 with a tosimplex in A ( I ) having a J-comple~ facet in A(J ) ¢~ C~* ~, or due to the finiteness of the num~r of simplices in U ,~c ~.. ,if( 1 ), after a finite number of steps a Jocomplete simp~x in A(J) is gestated which ha~ already ~ n ~ r a t o J ~ f ~ . However, applying the well.known door4nd~roOUt ptincip~ of l~mke ~ d How~m (1964)(~e also Scarf, 1973) it follows from l~mma 5.3 ~a~ each J~¢on~plete si|npl¢~ in A(/) c ~ be visited at mo~t em,ee, Henc:e, the p~'edure mu~| terminale~ OQ,E.D,

$o given any triagulali{m of (:~' ~ ge prc¢~u~ ge~ra{es a finite number, say M, of s i m i l e s ~,~,.~., (~ u ~ a ¢o~tespo~ding seqt~n~ of adjacen| comple~ ~"* ~, The sim~|e~ ~ induces the trivial RD~e with eomple~ dema~M rationing ~m ~1 n~:monume~ai~ ctmm~,odities, h~ It~e following theorem it is shown that the m~imal ab~lu~ val~ ~ the total e~tcess d e ~ , tl ~q)U .~. at ~ y point q in any simple~ ~ne~aled by ~ ~ ~ can ~ m ~ a~bitrarily ~mall by t~ ~ ~ si~ of tl,e ~r~gulati~ small enough. '~ ~ ~) ,~as~ assumptions AI-A3, Thea f i x et~ery ~ > O, thet'¢ e~srs ? > 0 such ~ l fi~r r~¢O~ ~iaagutaao~ ;~ ~ t h ~ ' / J ) ~ ~, for et~ery l ~ n t q ~n ~ny sim#tc~ generated by the

P~,~j: I~. {r ~ any s~m~e~ ger~ra{¢d by the p ~ r d u r e ~ld t~e any pt~int q~ in ~r, F ~ ~ YC/~+ ~, {r c{~t~ia~ ~ J,comp~e simp~x r in A(J) ~ith vertices

PJJ. Herings e~ a~./ df~urnal of Marhemath'~ff Ecene~mics 27 ~t997) 53-77

73

q~ . . . . ,ql.~l. It will be shown that n + I ~ J . Suppose not, then q,,+ t = O, for all q ~ , . By L~mma 3.2 it holds then for any vertex q~ of v that ~.~(q~) ~ O, for all j~l,,. By Lemma 3.1, #(qh)r~q~,)= 0 and hence ~,,+ z(qa) < O. So ~ ( q ~ ) = n + l. a contradiction with n + l ~ J. Moreover, for every k ~ d there exists some vertex q~ of ¢ such that ~,~(qh) ~: O. If k ~ I,,+ ~ \ J = l,\g. then for every q ~ r, q~=O. and by l.emma 3.1, ~i(q) ~;0. Consequently. for every j~-l,+t there exists a point q ~ r with t~(q) a 0. Let us define m i n ~ ~.., ~ ( _ a )

p,( _la ) Since ~ is a cominuous function on a compact set C~* ~ there exists -,/> 0 such that for every 4. ~ ~ C~* t it holds that II ~ - ~ It ~ ~ implies I1 ~(q) - ~(q)iI < ~. Hence, mesh(.5r ) g ~ implies *,~(q') < Z < e, for all k ~ ! , . ~. Since by Lemma 3.1, #(q,)T~(q.) = 0, it holds for every k ~ 1,. ~ that

E,¢,..,,,,,;3A q') PJ q') Hence, il~(q')U,, <

~.

q')

[:IQ.E.D.

The next corollary follows immediately from the fact that £(Q) =~0. The corollas implies that initially only the price leve~ is increase&

(~m)#ary 5.6. Let tl~e ee,momy ~" oo(1X', ~ '. W'}~?~~. ~) satis:~ assumptions AlooA3. Then ¢b({)) ~° n + 1, if I[ ~(q)[I,~, < ~:, then it is easily verified that (~(q). L(q), ,~t~(q). . . . . d%'(q)) ~atisfi¢~ all prot~rli¢~ of an (;:.R,DE~{q), excepl possibly the ~'equireenenl that ~ m a a d rationing on comm~ities with a price below the maximum pine or demand r a t t l i n g on the numcraire c~)mmodity is nonobindin8. However, r¢.c~ll lhat we defined L~(q)~ ~5 if q ~ C"* R and q ~ 1, and £.+ ~(q)oo **~,~~. for eve~ q ~ C . So, ff ~ < n n n ~ t n u n ~ z w;, then for every consumer < w~, and an t:-RDE,ttq ) is obtained. Let us define ~ = min~ut m i n ~ t,. %~. Since q,~ i implies that Lj(q) = %. we now have that for every ~ < ~. li~2(q)ll < a and q ~ ~,~+ t implies (#(q), L(q), ,~t(q) . . . . . d%(q)) is an e>WE. We ,ire now able to prove the next theorem, saying that there indeed exists a path of approximate equilibria.

77~eorem 5.7, Let the ecol~omy 2;"~ ({X~, ~ ', w*}~ 1, ~) satio~fy a.ssumptions A!,-43. Then jbr every t: > 0 there exists a piecewise linear, omtinuous ]~nction 'rr :[0, 1] ,~, C~ * I sa6s~dng

74

PJJ. He~g~ et aL / d ~ n a l of Motbemaric~d Eco~,~ic,," 27 f ~ 7 ~ 53-77

(i) ( ~ ( 0 ) ) . ~(~(0)). dq(w(O)) . . . . . ~(~'(0))) is the trivial RDE~; (ii) (/K,r(1)). L,(~'(I)). d'~(ar(l)) . . . . . d ' ( , r ( | ) ) ) is an e-WE; a~d (iii) (#(~(t)). l,(,r(t)). ~(~-(r)). . . . . d~(ar(t))) is an ,-RDE~(.,~o P f o r all t~[O. l]. Proof. Without loss of geaerali~ ~ ~ e < ~. We choose y as in ~ m 5.5 and consider ~ s~ueace r ~,..., r ~ of adjacent complete simplices obtmned by u~ing ~ ~ . ¢ d u r e . F ~ h simplex in this ~ u e n c e is /-complete in A(J) for ~ n ~ $c:I..,. ~ j~l~. let b ~ dehorn ~ bmycen~ of eJ. Clearly. b ~ ~0. Since for every j ~ IM- ~ ~ convex hull of the union of ¢ s and Ts÷ ~ equals o"s+* and a simOex is convex, it holds tha~ convex combinations of the ba~centres of cs ~ cs+~ ~ elements of c~~+~. |J.*t N ~ M ,,~ ~ arid let US

.o ( 1 -

+( t , - LN,

where [rJ ~ s , for ~ y ~al number r. ~ g r e ~ s t in~ger less than or equal to r. Notice that in d~e c ~ t ~ I. b N, : ~ b ~ ~`* can ~ taken equal to an arbitrary v~. C~afly, ¢r is ~ cominuous, piecewi~ linegr ~ t i o n , ~r(O) yields the trivial R D ~ . and for all r ~ [0. 1]. ~r(t) i n d u s ~/,'-RDE~t,m ), It ~mains to verified ~ n(l) i~d~ ~ ~ , or ~uivalentiy. *r(I)~#"*t.--.a Clearly & |~! q* .... .#~ M &~ ve~tice~ of , m S u p ~

¢~,,,~(I)~ I ~ & "I~n since

~o 8 ~ 1 by ~ m m a 3,3 th~ ~,,, ~(q~)> 0, m $(qO ¢ n + t, But tMn ¢~ is J~comp|¢te io A(J ) f ~ ~}O~e $ not ¢~minit~g n + i, implying q~. ~ - O. lot all j *~/~. a con~Micti~,

far the ¢ x i ~ of a ¢~tinu~s p ~ w i ~ | i , ~ p~h of ~RDE~,'s ha~ ~ n shown for eve~ e > 0, M ~ v c r , this path h ~ been c'o~¢g~d by applying tech~ue of simpl~i~ ~¢~ima|ion. |n this fh~1 ,~tion ~ c ~ ~ ~ 0 wi|l consi~d, We ~ t t ~ Ot~ U ~ ¢ ~ i t ~ , di~n|J~bi|i~y ~ i l b ~ s

P.}3. t t ¢ h ~ s ct al,/ l ~ !

~fMa~licai E c ~ i c s 27 f1997)/$3-77

75

a WE. To substantiate this conjecture, notice that the first n components of the function ~ map an (n + l)-din~nsional set into an n-dimensional set. thereby leaving one degree of freedom. Clearly. a zero point of the first n components of the function ~ yields a zero point of ~ by Walras' law. In this ~ t i o n we take another approach. We do not make any differentiability assumptions, instead we only make assum~ons A I-A3. The result, being that the set of points q ' E C~ + t satisfying ~(q" ) = 0 contains a c o m ~ n t con~ning both the point q =. 0 and a point in C~ + t. holds for every economy ~tisfying the previously mentioned assumptions. The proof of the result follows the approach of Herings (1993). Given an economy ~ .~ ({X d, :~ ~..~.~" ~). we define the set Q as

Q~{q"

¢ C £ +'

i~(q') ~Oi.

A topological space is connected if it is not the union of two non~mpty, disjoint, closed sets, A subset of a topological space is connected if it becomes connected when given the induced topology. The component of a point in a topological space equals the union of all c o ~ c l ~ subsets of the topological space containing point, It is not difficult to show that the component of a ~ i n t is fl~e largest connoted sub~t of the topological space confining the point. The c o l ~ t i o n of c o m ~ n t s of a set partitions the set. For a non-empty compact ~ t S c ~ ~ we define the distance function gs : ~ ~ "* ~ by s~S

It iS easily ~OWn that the function gs is continuous, l~t S t and S ~ ~e non.empty. c o m e t s u b , I s of ~ . We define e(S ~. S ~) by e( S ~. S "~) ~

rain

I1 s ~ "~ s ~ ]J..

~ViOtJ~ly, ,~ and S" ~ i n g disjoint implie~ ~ S ~. 8 ~) > O. The¢~em 6.1. Let the economy ~ ( { X ~, ~ . w q ~ t . ~) satirize assumptions AI~o.A3. Then Q has a comp~ent ¢~mmining ~ and an element in ~"* ~. i.e. there exists a connected set cbCpoints in C~* i inducing a set of RDE..'s comaining both the trivial RDEg and s ~ e WE. Proof. Let n", r ~ N, denote a function ,r as defined in Theorem 5.7 satisfying [[ ~(~r'(t))[[ ~. < l / r , for all t 6! [0, !]. l~t us consider an accumulation point of the ~ n c e {~r'(l)},¢~, say q ' . Clearly, q" ~ (~"+ ~ and R q ' ) ~ Q , ~o q" i ~ t e e s a WE. So, 0 ~ Q and q" ~ Q. Exerci~ 4c of section 5.1 in Munkre~ (1975. p. 235) atates thatthe component of a point in a compact Hausdofff space equals the in~rsecticm of all ~ts containing the point which ~ bo~ o ~ n ~ d c ! o ~ in the compact Hau~lorff spaee~ Support q* is not an element of the comp~'ment of Q. From the fact that (~g+l is a compact Hausdorff spau,'e when

PJJ. Heri~.~ a ~, / J o u ~ l ~ M ~ ~

E~i~s

27 f 1~97J 5 3 - ~

given ~ indtmed topoMgy, it follows that there exist ~mpact disjoint sets Q~ and ~: s,~h that ~ E ~ . q" ~ ~ z ~ ~ U O~ = ~. Hence. ~ exists ~ > 0 such that ~Qt, Q~)> e. consider a s u b s ~ ( ~ r ' , ) , ~ with l l * r " ( l ) - q" I1= < ~/2 for ~1~ s ~ ~. For s E N we define the function f~ : [0, i ] ---*~ by

/°(0 By ~ continuity ~ ~ ftmctions gO', g~=, arid ~r', it follows thaL for any s ~ N, ~ fimction f~ is co~nmms. Moreover, f~(O) < - e and f~(l) > O. Let ~' ~ [o, i] ~ s f y / ' (: t ~f) ~ go~(~",(tO) = ~#~',(t')) > e / 2 , Let ~s con~,ider ~ t the compact set C~ L Without loss of generality, lira, ~..~'
Her,ce+ go(~r ) ~ O. Since

A IoA 3. ?7~en theee eo~ist;~'a c~mn.eefed ~et ~¢ RDE,~'s ¢~f ~ e ~ m m' m' ~ , the rffeMI l¢

°

P . m f C~B~,~k~ the ~i of RDE,~./~

((

!3(q),g.(q)

d*~(q)

°

d~(q))e~*'×N:"×t~N~*'lqee'~

)

with Q0 ~e c o m | ~ n t ol the mt Q c ~ i n i ~ g ~+ By Theorem +.l ~ ~ l ~ove Cotlt~tm~ rite ~viM RDE+ and ~ ~ , ~ since the imam. of ii eonnec|ed m| by c o n | i n ~ fu~|itm i~ c~rn~¢t~ed~ the ¢ ~ 1 t . ~ fotlow~. OQ.E.Do

~ulh~ w~s,~ ~ ~ k

~

~NWO),

two ~yn'~.'~ts ~fe.~_s fo~ ~ i r va,l u ~

fln~i~ly by ~

~fi~

c o m ~ t , ~ oa

(~ai~Ikm f ~ Sdemific Re~ar_¢h

PJ~ e Hermgs ef ~d~/ $om'~t qf Mathema:~¢'atEco~'~ics 27 ¢~997~33-77

References Bi~hm, V., |993, Rc~mc~ce in Keynesian m~roeconomic models, in: F. Gori. L. G©ron~zo and M. Galeo~, ¢ds.. Nonlinear dyuamics in eecc~mics and the social sciences(Spdnger-Verlag. Berlin) 69-94. Day, R.H. and G. Pianigili, 1991, Slalistical dynamics and economics. Journal of Economic Behavior and Olgaaizalkm 16, 37-83. ~, J2L, 1975, E x ~ of an exchange equilibrium under price rigidities, inPmalional Economic Review 16. 3OI-320. Fmudealhal, H°, 1942, Simplizialzedegungen yon beschr~ler Flachheit, Annals of Mathematics 43. 580-582. Hedngs, P J J , 1992, On Ihe structure of constrained equilibria, FEW Research Memorandum 587, Tilburg University, Till~rg. Hedngs, PJJ., 1993, On the co,neclednesa of the ~l of constrained equilibria. CentER D i s c , | o n Paper 9363, Ttlburg Unive~ty, TUburg. Kamiya, K., 1990, A globally stable price adjustment process, ~onomelm'a 58, 1481-1485. van der Laan, G, 1982, Simplicial appcoximalion of unemploymcn| equilibria° Journal of Mahhematica| Economics 9. 83-97. van der I~tan, G. and AJJ. Ta|man, lgBVa. Adjusunent pr~csscs for tie,ling ecofloNlicequilibria, in: AJJ. Talman trod G, v&-t der La~, eds. Computatioll and modelling of economic equilibria (NoHh~Hollm~d,Amsteldam) 85~ 123. v ~ dee L~m~%G, and AJ J, Talman. 19~7b, A convergent i~ce ~Jjus|ment prt~ess, ~,onomics Letters ~3, | 19~123. L ~ u ¢ , (L. 1961, A ¢ommenl on 'Stable spillover~ amtmg sublli~s', Review of ~momi¢ Studies ~mk~.~ C0~, and J~T, Howson, 1964, Equi|ibrium points of bimalxix garaes, SlaoM $oumal on Applied Mathcmattc~ 12, 413~423. Mtl.k~a. J,Ro, 197~. Topoh~y, a f~r~tcourse (~nfic~oHall, E n g | e w ~ Cliff~. NJ). ~d~ovt('h, V,, 1~3, I~I|(~|ng, queues, ~ d b|~k ro~kel F~tmomewlca61, l~B, S ~ D.O°. 1985, I~r~|ive price mecha,tsm~, Ec~om~l~it:a ~3. i l 1% 1131~ 9 ~ ; !t, !~)~ S ~ ¢ e x ~ p ~ of ~!ob~l in~bil!|y of the cooq~cttllvc cquliibrium. |oicmalio~al Scarf. H.. 19~/:LThe comim|al|~ o| ~¢:~,~omlc~uilibritt (Yal~ Ua~ve~ity P~e~. New Haven. CT). M~tl~m~t~ ~ o m l c s 3, 10%o120o V ~ o t ] ~ B~CH., 197~i, $1~b!¢. spillovf~s ltmo~g ~ilb~|~t.l~.RevlfW Of ~onomie Slttdlc~ 4~,