Inter Temporal Gen Equilib Model Macro 2 Paper

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llWo and there is

Intll'!ril'!mnond

Models

TIMOTHY J. KEHOE

110 introdadion

Hili this we shaU the properties of llWo

oral a model wiRh a finite number of

nnlilDitelv lived consumers, and an overlapping model with an

finfinile number of lived consumers. Both models contain a

cornplete set of markets. 1be Anow-Debreu formulation of the Wal-

Iraman modeJ can. of course, be a UYIIlIillDlIC intpIT.....t"'.

~ i O D in goods are indexed date. The models that we differfrom the standard model in that we allow an infinite

number of As we shall see, models with infinite numbers of

OOissel$S very different from models with finite ofmodels are regarded as however: their

...r<,.....rt i• .., are in so far as provide into models

but finite. nwnbers of

with a finite of lived consumers sbaresUUIPUI win nrnr... r1ti.. ", with the standard Arrow-Debreu modei:

aU equilibria are there is no role to be played

outside mo ney, ulllbacked nominal debt; and, third, there are, in gell le l iU

.ill finite number of uniqne equilibria. In contrast, the O Y 4 ~ 1 1 ~ l p p m g gelllera,tiOllS model may violate each of these three properties: it may

have that are IK)( it may have inwbidt outside money plays an and it may a robust

continuum of equilibria. We see that there is a closebetween the possibility of of and the role for

outside money. The possibility of is a

one

equilibrium models are be.colirUng iIlICll:asi:ngllyparticularly in m a l c r O l ~ 1 1 3 0 n l i c s .

The research in abis paper bas been funded by GTant 00 . SES-8S00484 from ibeNational Fouildatioll.. M, lhinking 00 ,be i.mIer. discussed in this bas been

i l l f t u e ~ i l l ~ l'Iith odIer n:sem::blm same area. Inpartiallar, I woukl to thlIDk David Backus, JOIIlIIthIID Buroell. JotmGellllSlkop'los, Frank Hahll, Patrick Kehoe, Andreu Mas-Coklil.Pot.eman:ludi:is, Paul Romer. Micllad Woodford. WiDiam

Levine.



TIMothy J. K ~ h o ettOOre llIas been to use small general equilibrium models no analyse

macroeconomic issues (see, for example, Lucas 1981, and Kydland and

I?r:escott 1982). Not all macroeconomists have been caught up in this

nrend, however, and the use of explicit general equilibrium models anmacroeconomics has been the subject of much controversy, in which one

side accuses the other of using ad /we and unrealistic models. Manycommentators have interpreted this controversy as an idealogical debate

between monetarists and Keynesians. This interpretation is probably

IlUllfortunate. An explicit general equilibriwn framework impo5CS adiscipline and assures internal consistency. This makes it easy for us to

organize our thinking about economic phenomena and to c o m m u n i c a ~ e dJis thinking to others, mostly because the assumptions of the model have

weU understood implications in this framework. The phrase ad hoc ismuch misused in economics. It bas become a synonym for 'yours' and

'bad' and an antonym for 'mine' and 'good'. Most good economic models

are ad hoc in the strict sense that they are designed for a particular

~ r p o s e and produce resulls that follow closely from a particular set of

assumptions. The advantage of using explicit general equilibrium models

is that they provide a framework. in which sets of assumptions ar e easily

nmderstood and compared.The potential disadvantage is, of course, that the general equilibrium

framework ca D become an intellectual straightjacket. Fortunately, how

ever, this framework is ricb enough to allow a wide variety of results. To

BlDustrate this, we enlploy both of our models to answer Barro's (1974)

question of whether government bonds are net wealth. Different models

a m produce very dilJerent answers this question. 'There is a closerelationship between these answers and the sets of assumptions that

dBstinguisb these models.The models that we study in this paper are both pUJre-excbange models

~ i t b no production or storage. TlDle is discrete and there is no

u:J1Certainty. Furthermore, both models ar e stationary in that the ·strucmre of preferences and endowments is constant over time. These models

are the simplest to analyse. We indicate, however, how ou r results extend

more complicated models. We also oompate the structures of the two

models. On one band, the overlapping generations model has similar

properties to a model with a finite number of infinitely lived consumers

1fJho face borrowing and lending constraints. On the other, a model with

21 finite Dumber of infinitely lived consumers has similar .,.-operties to an

overlapping generations model in which parents leave bequests to their

ciBildreD.

:!'o All lalaitel)' u.etI < C o _ Mecie!J

We begio by analysing an economy with a finite number of agents who

consumer over an infinite number of time-periods. There are FU goods,

ffnlertemporal Equilibrium Models : : o 5

which cannot be stored, in each period and h consumers. Consumelr ,i ns

characterized by a utility function

2: rj-iUj(c:{" .. . • c1",) (1),=1

and an endowment vector wi =(MI., . . .• ~ ) . which he has claim no Dlleach period. Here the discount factor y, satisfies 1> Yj > 0; the mom(m

tary utility function u, is continuously differentiable of the second olTck r

for all positive consumption vectorS, strictly concave, and monotonia Uy

increasing; and the endowment vector wi is strictly positive.

There are two interpretations of this model. 'The first is tbe traditiollaB

Walrasian interpretation in which all trades, including those tbat involve

future delivery of goods, take place in the fust period. In this interpre Ja

tion time plays no explicit role and I can be thought of as merely anotti Ie!'

index on commodities. 1be consumer's budget constraint is

GO

2. p ; ~ E ; 2, p;wJ. ':2}8=1 ,=E

Here p, ,.,. (p'tt . . . • p",) is the vector of futures prices in period 6 and If:<d.is the inner product !:7=t p;,e!,.,.

In the second interpretation trades t a k ~ place over time, bu t there i lperfect capital markets and ratiooal expectations. In this simple m O l ' ~the assumption of perfect capital markets means only that consumers (a n

borrow and lend as much as they want at a competitively determillliW

interest rate, and the assumption of ntional expectations means tl ,at

consumers have perfect foresight. Let 'I, =(ql" . . .• q,.) be the vectorr of

spot prices in period t; let r, be the ioterest rate between I and f + 1; Iud

let be the net lending done by consumer j between t and , + t. We Ulo"ll

iJnterpret m{ as inside money. Consumer j faces a sequence of budl:et

constraints

q ; ~ + m{ q;wi

q + <5i qiwi + (1 + rl)m{ (13)

q : ~ + urt, ~ q : w i + (1 + Y ' - I b m ~ Dividing the budget constraint in period gby (1+ ,.)(1 + ,7) . . . (li + "'8- Y'e= 2, . . . • T. and adding up, we obtain

T T

L p;d,+ m ~ / ( ! +Yt)(l -+ r2) ' •• (U +YT-t) <5i L p;w'' 0 & .-1

where p, = q,/(D + 1"1)(1 + r2) •• . (1 + r, - i)' in the limit dbis produces lle

same budget constraint as does the first interpretation as long as

lim m ~ ( n + V'1)(n+ Y2) • • • (ll + 8'r_ l ) =(]I.T-



=

.J6tj Timothy i. KehOfi

To ~ n s u r e that this condition holds, we need to puQ some constraint OOl

ahe real level of debt that we alloW consumer iO incur. We shalH see thai,

w i i ~ such a constraint, (5) must hold in any equilibrium.

[Let us retum to the tim interpretation of the model. An equilibrium ns

BJ sz:quence of price vectors (p" Pl•...) and a sequence of consumption

vedors <&a, &....) for each consumer. j = 1. . .. , h. such that each

consumer maximizes utility subject SO his budget constraint (2) and

demand is equal to supply:/a /aL ~ = L w I , 1=1,2... . {6}J ~ I I ~ '

N ( j ~ thai any equilibrium must be such that r . ; " ~ I P ; w i converges;

oaherwise the consumer would have infinite income, and his utility

IDwtimization problem would have no solution. Because Uj is monotoni

cally increasing, he would want to consume infinite amounts of at least

on<e good, which would make equilibrium impossible.Since every coosumer has finite income, Rhe yalue of the aggregate

endowment must also be finite:

(1)wi) == ( ~ p ; w i ) 0This impiies that any equilibrium must be Pareto-efticienL The argument

is due to Debreu (J954). Suppose, «0 Ute rootrary, ahlili ihel!"e BS d!I

lP'Etieto-superior allocation plan (c{, c'z, . . . ):

.. ..

y j - l l u , { ~ ; ; . } : yj-U 14J{CI), j = . . . , n, (8)•-1 ,=1

shict inequality for some j. ltiIat is feasible:

" IILi{";;;Lwi, «=1.2• . . . ~ 9 } jm l j ~ 1

l1ilen ihe consumption sequence (c{, &, .. .) must cost at leasa as muchlllS tlhe consumer's income, and strictly more for some consumer;

o ~ ! l l e r w : i s e ( ~ , Ci2> ••• ) would not be utility-maximizing. Consequ ently,

~ t o )(f p:e,) >±( } : p ; w i ) = : ~ ; p ; ( ± wi).I- I ,£1 1: 1 ,ma .=D i=1

SfuItce the Pareto-superior allocation is feasible, however,

±(i p;&,) = i p;(f e.),,;;; ifo;(± wi). Ill)I- I .=1 ,-1 I- I , ~ I i ~ 1

Tillis oontradiction establishes that there can be no allocation that ns

l?i!feto-superior to the competitive allocation and is also feasible.

lniertemporai Equilibrium Models 35J1

That nhe value of the aggregate endowment is finite also implies ttlan

there can be no equilibrium w i t ~ outside money, unbacked de ij ,t.

Suppose, to the contrary, that there is an al1ocation tbat is feasible inwhich each consumer satisfies the budget constraint

...

L p;Ci, = L p;w' + ~ , j= l • . . . • Ii (H )l ~ t

where

m= L:,

m'*O. {B)j= n

(U m = 0 but mi =F 0, this is just an equilibrium with transfer paymenb .)

Here m is the stock of outside, or fiat, money, which can be positive l lr

negative. Summing these budget constraints over consumers, we obt.wr

t (i p;&,) =±(i p;wi)+m . Q- l};= i ,=u }= I ,a l

Multiplying the feasibility oonditions (6) by prices and summing, ]hO'lll

ever, we obtain i p ; ( ± ~ ) = f p ; { t w i ) ' . )'=0 J ~ I j t ~ l · ' J - I

Consequently, m =0, which contradicts the assumption t h a ~ mere is a! fU

equilibrium with outside money.

11tis same argument can be used to show that the sequence of budgd

consuaints (3) are equivalent to the single intertemporal budget COil

straint (2). For consumer i to have a weD defined maximization problem•

and for the concept of equilibrium to make any sense, the limit in (5)

would have to exist. Here, unlike the outside money case that we haVI

bust examined, the variables are chosen by the consumers. Since utilit'

is monotonically increasing, every consumer would want to chaos; !

( m { , ~ , . . . ) so that the limit in (5) is negative. 1b e same argument tha (predudes an equilibrium with outside money also precludes Qb i ;

possibility.

For an equilibrium to em. however, we must impose a c o n s t r a i n ~ 06 !

ibe real level of debt that we allow COnsumer j to iRalr.

m{/lIp,1I ~ b , 8 = 1,2.. . . (16

for some b < O. Otherwise. the consumer would try to run a fonz

scheme, rolling over an exponentially increasing amount of debt a n C imaking the limit in (5) as negative as possible. In such a case, as " W ~ argued above, no equilibrium can exist. Any so n of bound on tbe Jreal

level of debt, no matter how large in absolute value, precludes nhh

possibility.



• f":'"::t

J(,'8 Trmothy J. Kehoe

lIn general, this model has a finite number of IocaISy unique equilibria.

1'01 see abis, we transform the equilibrium conditions using an approachdeveloped by Negishi (t96Ob) and applied to inlertemporal models

by Bewley (1982). To simplify the exposition, we ignore the possibility of

oomer solutions to the consumer's utility maximization problem. This can

he justified by imposing an additional restriction on Il } (see Kehoe a n ~ bvine 1985a). 1b e solution to the consumers utility maximizati011l

problem is <:haracterized by the c:onditions

t; - l DU;(d,)=).jP; (li1)

ffJf some Lagrange multiplier it; > 0, and the budget oonstraints (2).(Here DIl/(d,) is the I x II vector of partial derivatives of Ilj.) An

equilibrium is, therefore, characterized by (2) and (17), which are Qhe

utility maximization c:onditions. and (6), wlUch are the market-dearingcooditioos that demand be equal to supply. 'Ibis ns 2 system with an

infinite number of equations and unknowns.Consider now the Pareto problem of maximizing a weighted sum of

ilildividual utility functions subject to feasibility wnstlraints:A ..

max aJ L rl-luAd,)(18)/ .. g , ~ g

Ii Jt

s. t L d,==L wi. i = 1, 2, . . .J ~ I j ~ i

lHere aI , . . . • all are positive utility weighas. fir. solution to iliis problem

os characterized by the conditions

C I l j Y r ' ~ ( d , ) ="" j = 1• . . . • h, 8 = n, 2, . . . (19}

fur some sequence of vectors of Lagrange multipliers Ra =

(nl" . . . , Jr",)> 0, and th e feasibility constraints (6). Notice that, jf wedivi<Be (19) by ai' then it bec:omes the same as (17). This is an alternative','vay of seeing that any c:ompetitive equilibrium is lPareto-efficient, thaa

f:he FlCSt Theorem of Welfare Economics holds.TIle Second Theorem holds as well: any solution io the Pareto problem

i{IS) satisfies all of the conditions for a competitive equilibrium except the

3ndividual budget c:onstraints (2). Such a solution can, therefore, be

':fiewed as competitive equilibrium with transfel" payments. Th e rompetilive prices are, of course, the Lagrange multipliers 11,. We can computeiUle transfer payments needed to implemeot as a competitive equilibrium

,he Pareto-efficient allocation associated with the welfare weights a =la" . , . , a,.):

...

Ij(a) =2: Jr,(a)'{d,(a} - wi!. j == .. . , h. (20),=1

fffflerlemporai Equilibrium Models

Setting these transfers payments equal to 0 produces a characterizatic 'l

equilibria in a finite number of equations and unknowns.

Using the strict concavity of "I ' we can demonstrate that itraRSferr

functions 1/ are c:ontinuous. Also, 'i is homogeneous of degree B JIl\\ CJ1

because 1f, is homogeneous of degree 1 and is homogeneous of degree

0; if we double a, for example, the sequences of consumption veL10nahat solve the problem do not <:hange, but the Lagrange multip liers

double. Furthermore, the transfer functions satisfyPo

L gJ{a} =0 {21]i/=B

because any soiution Qo !the Pareto /problem satisfies the feasii ;i1ity

oonstraints.1lte ronditions that charaaerize the equilibria of this m o d e are

fonnaJly equivalent to those that charaderize the equilibria of a siali<: .pure-exchange model with 13 goods. Indeed, the functions "e,t) =

- IJ(a)1aj have all of the properties of the excess demand functions of fl

p u r e ~ x c h a n g e model: they are rontiouous; they arc homogeneoltS o[

degree 0; and they obey Walras's Law, 1:: ..,«J"(a) - 0 . Debreu ( l 9 7has demonstrated that, if the excess demand functions" are rontinu(tUSlydifferentiable, then aboost all economies have a finite number of Io'.:allyunique equilibria. The phrase 'almost an' is, of coune, given a p r r ~ mathematical meaning. We can use either the transfer functions 'i 01' the

demand functions /; to characterize the equilibria of tbe interteml,orafimodel that we are considering bere. Kehoe and levioe (l985a) :iJaveshown that Debreu's rt:$Oniog extends to this model. FurthermoR . byamposing another. fairly weak. condition on ui t they are abln ft())

demonstrate that IJ is indeed c:ontinuously differentiable.

The proof of Debreu's result relies on fairly oompJex mathemEtk a flmacbinery. The intuition behind it is very simple, however. It is, in faa.

the same intuition as Walras had when he counted equations ancjJ

unknowns: There are h equations, l i(a) = 0, in h unknowns. ai. Bec:luseof homogeneity. one of the weights aj is redundant. Because o fr theadding-up restriction (21), however, one of the equations is alw

redundant. Consequently, the equ.ilibrium c:ondilions can be viewed lM 2l

syetem of II - 1 equations gn Ie - 1 unknowns. Suppose h 8 t i:mese

equations are independent in the sense that 'j(er) =o. j = 1• . . . ,h - n.and the (13 - i) x (h - 1) matrix of partial derivatives

oa l~ ( e r )8a .._a

1 = (22)

01"'_1 (&) 8t,._1 (ci)aerl a«.._ 8

http://slidepdf.com/reader/full/DU;(d,)=).jP

http://slidepdf.com/reader/full/DU;(d,)=).jP



370

,,""

Timothy J. Kehoe

is non·singular. (We have imposed the normalization aA> = i and dropped

filiJe equation 1,.(0') == 0.) Then the inverse function theorem of elementary

calculus says that, in some open neighbourhood of a, it is the only

solution to the equilibrium conditions; that is, ,-8(0) =a. Using ~ h e compactness of the set of possible equilibria and the continuity of the

lequ.ilibriwn c onditions. we c an easily prove that there is a finite number

o j equilibria if J is non-singular at every equilibrium.If J is singular at some equilibrium, then the intuition says that the

sIlightest perturbation in the functions '/ either make it non-singular or

ei!se make it impossible for there to be a solution near a. Figure 16.1

iRlustrates some possibilities in an economy with two consumers.

To mate some of the concepts that we have discussed in this sectioo

more concrete, let us consider a simple model with one good in each

period and two oonsumers. Suppose that ",(c,) -= uz(c,) = log c, and t h a ~ W I = Ml2 = 1. lbe only difference between the awo consumers is t h a ~ Yo < 12· A. solution to the utility maximization 85 characterized by the

<c()ooitions-1/ ' iri C:=A;P, (23}

....

LP,d,= ~ p , . (24)' ~ I g

l ~ 1 1 l equilibrium satisfies these conditions and the condition ihai demand

t,ta,.i}

/,- I I

I I1

I

-- - 11,~ - - - - - - . . . . . . .

n!!i.ll

lnlertemporai Equilibrium Models nn

equals supply:

C: + 11::+ 2, 8 = 1, 2, . . .

1rhe Pareto pli'Oblem is

GO ..

max at LA" ...-1110ge t0+ "" '

'-IIog c,O «2 La 1'22

(:!6),I:: t r=- R

s.L 11:: + -= 2, 0= n, 2, . . .

A. solution to this problem is characterized by the oonditions

O'jy j- ! lei, = JIl" j = n, 2,

and (25). These equations can easily be solved to yiekB

-1 2 a j Y r ~ '- n 2(;!8)

" t P j-"0'11'1 + C'U2

X, = (m.t.- I -+ a2rl-2)12. { ~ 9 )

The transfer payments needed to implement as a competitive equilibril·m

ahe allocation associated with the weights a. and a2 are therefore

.. I ~ _ ~ )Ig(1lr1> Ilrz) = 2: Jf,(c, - 1) _ 'l'y. n_ Y I'=.

.=1 I 2

2 =t a2 a.

(30)

'z(a., (2 ) = .1r,(c, -1 ) -.--)u=1 - Jl - Yt . .2

Notice that these functions are continuously differentiable, are bomog.e

neous of degree 1, and sum to O.

The llllique equilibrium of this model is found by setting these transi\!u'

payments equal to O. It is at =(1 - Yt)/(1 - yJ. 0'1 = 1. Notice thai tVevalue of the aggregate endowment is finite since

i n,(l + i) = 25 : (1 - Y. ia- I + y ~ - I ) / 2 , e 8 ,cD l -yz

2(3 1)= - 12 .

There is, of course, no outside money in this model. There is. boweve r,inside money: consumer 1, who is more impatient than consumer

spends more than his endowment early in his life. later he c o n s u ~ ! S less, paying back his debt. In the limit, his mnsumption in each peri()rll

approaches 0 and consumer 2'5 consumption approaches 2.



313

'""

; )12 TlIYIOlhy J. Kehoe

; ~ . A:IJ On....... Geueratioas Mood

1n this section we consider an overlapping generations model in which'lhere is a single good in each period and a single consumer, who lives for

iJWO periods, in each ~ o e r a t i o n . This is tbe model originaUy developed

!by Samuelson (1958) and analyzed extensively by Gale .(1973).B.n

the'DCxt section we discuss more general models.

The consumer born in period 8, t = li. 2 , . . . . solves the utility

~ m i v t t i o n pcoblemunax u(G. c ~ + » }

(32}s.t. p,c!+ P ' + B C ~ + l =P,li'l +PH IWz·

We make the same sort of assumptions on u and (WI> W2) as in the

!)JI'evious secliOil. As in the previous model. we can also think of this

r:onsumer as facing two budget constraints:

q.c:+nr=q,w, } «33)q,+lc!+I = q,+J W:z + R + r,)m'.

l1f we normalize the spot prices so that q,...1 :::: q, = 1, divide tlJe second

oonstraint by (1 + r,). and add both together. we l!:aIl produce a single

!!Judget constraint in which P,/P'+I = (1 +r,).'The solution to this pcoblem is cbara<:terized by (be conditions

aU lD 8 }"C .. CJoj·I P= ~ I P , (34)

au f I

"'.... (C,. C,+l}= A,P,"'1. ~ t + i

and tbe budget constraint in (32). Given the strict concavity of i i, this

oonsumer has continuous excess demand functions y(p" P.+i) = c! - w,

wheD young and z(p.. PHI) =C : ~ t - Wz wben old. TIle form of the budgeaooustraint impUes that these functions are homogeneous of degree 0 in

(p " P,+!) and obey Walras's Law:

P,y(p,. P,+I) +p,z(P . 1',+,) o. (35)

Consider. for example, the case where fI(e!. e!+l) = loge;+ y log C;+I-

The excess demand functions can easily be computed using (32) and (34).

They zre

p,w, +P,+!"'2 - YP,WB +P,...1W2 }y(P"P,.l) (1 + y)p, w, = (1 + y)p,

(36}y(p,w, +p,+ ,"'2) yp,w.- P , + , ~

2(p" PHI) -W z = .(1 + y)PH O (1 + y)P, ... i

InleTtemporal Equilibrium Models

Notice that these functions do indeed satisfy continuity, bomogclI eity •

and Walras's Law.in addition to the consumers born in periods 1, 2 , . . . . there us Ml

ronsumer who is alive only in period 1 an d who solves the problem

m a x u o ( c ~ (31)

s.u. P R C ~ =PIwg +m.

Here m, which can be positive. negative, or zero, is the stock of oolside

money held by generation O. Xf m is non-negative, then it is tasil,

interpreted as fiat money. Even if it is negative, however, tben are

institutional stories to go with it. Think of an institution that makes Cuans

to consumers when they are young. The institution oollects the repay

ments of these loans when the coosumers are old and uses them to nateloans to the young oonsumers in tile next generation. TItere U f '. off

course, many other interpretations.S this consumer bas preferences for, and endowment of , o n l I ~ th(C

IIirst good, we need oot be careful about specifying Uo or w ~ . The el cess

demand function for this consumer ism

Zo(Ph m}= Pi '

An equilibrium of this model is a stock of outside money mand a r i c e sequence (flh P2, .. .) that satisfies the conditions that excess deman·j be

equal to 0 in every period:

zJ,.PI, m) +yWu. P2) =0

[n period! 1 and

Z(PH. p,)+ y(p" P, ... ,)=0 .(40)

UR period 8, 8 = 2. 3, . . . .One way to compute the equilibria of this model, developed by iaie

(1973) and Cass. Okuno, and Zilcha (1919), is to use the offer curve, abe

wage of (y(p" P, ... ') , z(p" PH'»)' This curve passes tbrough the oi'lgin.stays always m the second and fourth quadrants. and intersects rrays

tbrough the origin only once (except at the origin itself). In fact, Wld,:as's

!Law (35) tells us that

z(p,. P,+8)/Y(P" PHI» = - P,/P,+I; (4ll}

that is, the point where it intersects th e ray with slope - p,!p,+. b a s ~ : s itscoordinates excess demands at (Pit P,+1)· In addition, the oller c .uv e

always s a t ~ 6 e s y> -W I and z> -Wz.

For example, in ou r simple log-linear example, we can use the fOf1l!tui8)

11'or yep,. 1"+1) in (36) to solve for p,lp,+O in terms of)l and substitut<r 1lhe



D

3?.IJ Timothy J. Kehoe

rresoRt in&o the fonnula for z(p,. P"'i) obtain the oilIer CURVe:

YK'ltIl'2 W2z= -- (412)

(I + y)2y + (1 + y)yw! 11 + y.

TilDe result is pictured in FIgUre 16.2.

Kn general, there are two steady Siaaes, inOation ~ a c t o r s fJ > 0, suchtltat the price sequence p, =fJ' satisfies

z(fJl-I, 11) +y(fJ', 11'+1) =: z(ll. fj ) +y(li. P) =@. ~ . s 3 )

These are given by the two intersections of aile offer curve with the tiiDe

\lRurougb the origin with slope - n. z = - y. There is only one steady stalefilm the degenerate case where the slope of the offer curve is -! a« t!he

origin.The steady state where fJ =1 Pareto-dominates the steady state the

origin. One way to see this is to show that the consumption plan found bysolving the representative consumer's maximization HKoblem when P, =

PH R also sotves the problem of maximiziog the utiliay of steady-state

oonsumption plan:max 14(C1, C 2 ~ ( ~ ) lS.t. CR YCa= Wi -+ MIa·

2

,i • Y

A. - -

~ . 1 l ' ~

lnleJ1emportal Equilibrium Models 3" 5'

Alternatively, notice ihai, since DO trade ns always feasible, tbe consum!fcan only be better of f if be chooses to tnde. Indeed, a simple reveall':d

preference argument implies that the consumer prefers the net trade[y(l. 1), z(l. 1)] to any point that lies on or to the left of the line wi,1b

slope -1 . (Look at Figure 16.2 again.)

To compute equilibria besides the two steady stales, we start m i lu

= mP and read horizontally to me line with slope -1 to find the Vai l Ie

of y for which y(fi .. pJ = - ZG. We then read vertically to the offer culve

to find the point [Y(PI> Pz). z(fi" 1'2»). We now continue by readir lghorizontally to the ray with slope -! to find the value of y for willi( h

y(P2. fi3) = - z(P .. P2) . This process is illustrated in Figure 16.3. neoffer curve in the figure corresponds to the case where )'W, > "2. Noti( e

that, for auy value of sucb that =,;,Ip,< z(l, 1), there is anequilibrium that converges to tbe autarkic steady state in which tbere s

no trade. (There is a natural lower bound on ' ; '/PI provided by - w ~ bt \!

ahis is indepeodent of the offer curve of (y , z).) The price sequence 5

computed by nonnalizingpl = I, dieD using the slope of tbe line througb

abe origin passing through the offer curve at [y(fiJ, P2), z(fh, P2>J to finj

P2, using the slope of the line through the origin passing througb the offer

curve at ["(P2, P3). z(pz. ;;3») to find p,. and so on. Notice that e'ler lequilibrium of this model, except for the one that starts at ' ; '{PI '""

z(1. !) , involves nnJIatiolli. At the autarkic steady slate p, which is

\ z

J l U



31 f§ Tunothy 1. Kehoe

negative of the reciprocal of the slope of the offer culrVe at the origin, is

rw.lwz> U.

!Not only is there a oootinuum of equilibria in this example, but outside

money plays a crucial role and equilibria are DOt necessarily Pareto

efficient. Observe that any equilibria that starts with 0< mlfh < z(l, 1) is

l?areto-dominated by the equilibrium with ,;,Ip, = z(l . 1): the firstgeilleration prefers the highest Zo possible, and subseqllent generations are

worse off the further they are from [y(l, 1), z(l , 1)1 an d the closer they

ar@ Uo autarky. In fact, equilibria with higher m/p, P a r e t ~ o m i n a t e those with lower starting-points. In the next section we shall see that the

equilibria with ii i IPI < 0, altbough not necessarily Pareto-dominated by

equilibria with "'lpI:= z(1, 1), are not Pareto-eflicient. As Shell (1911)

haJS indicated, this failure of the FlTSt Welfare Theorem depends on the

double infinity of consumers and goods. Althougb it is possible to mimic

llB'I ils failure of the Farst Welfare Theorem in a model with aDcomplete

markets, as done, for example, by Cass and Yaari (1966), it should be

stressed tbat it occurs even if all markets are complete.

!Figure 16.4 depicts the offer curve for the log-lioear model where

ytY, < "2. Notice that, for any values of m/P1 such that ,nIp I <0, there ism equilibrium that C O I l v e r ~ to the steady state ·where {J =1. There is

aBoo an equilibrium that starts with ,nlpi = 0 and stays at the autarkic

si,!8dy state. Here jJ = Y W I / ~ < 1. This equilibriUDll is Pareto-efficient

Jns. ll(!).(l

intertemporal Equilibrium Models :),17

since the value of the aggregate endowment is finite:

+ + i: jJ'- Wa + M.Iz) == 1 l V ~ + 191, +L . (w. + Wz). (A5),=2 A fJ

As we shall see in the nexl section, aU of the equilibria of ~ h i s modleB weI?areto-eflicient.

These two examples suggest ihree hypotheses. FlCSt. any indetennim lCY

of equilibrium is connected to inflation if there is positive outside m O l l <Second, all equilibrium price paths converge to some steady statc. Thi ."d,

any indeterminacy of equilibrium is associated with a non-zero stad, of

outside money. We now study counter.examples to the tirsi ' l f O propositions. In the next section we shan see that the third, although iliJle

in any model with one good in each period and consumers who live f1JI1ltwo periods, fails in more general models.

The log-linerar exampl es that we have analysed have the property t h ; jas the price ratio P,/P,.-I increases, ,(P.. PHI) decreases and z(p,. P'd )

increases. This means that the demand functions , and z exhibit gf( 35

substitutability. Consider the offer cllrve depicted in Figure 16.5. HEre

gross substitutability fails in ihe backwards-bending scdion of the o 6 ~ J r curve. Notice that, for any value of ';'/P1 sufficiently close to z ( ~ , I).

there is an equilibrium that converges to the steady state where fJ = R.

The crucial feature of this offer curve is that the slope of the offer cut' "e

z

y

1F!tz. M.5



318Timothy J. Kehoe

aa (y ill, 1), z(l, 1)1 is positive and !ess than one. There are also equilibriathat start with m/PI near, or even equaJ to . z(1, 1) and converge ao the

autarkic steady state: whenever there are two vaiues of z that correspond

to a single y, we have a choice of two ways to read from the line with

siope -1 to the offer curve.'Jl'"hls example also has equilibria that do no t converge to any steady

s ~ t e . Consider the offer curves in Figure 16.6. Here there is a two-period

qcle Zo, ZIP Zo, Z" • • • • Th e second offer curve is the reflection of the

fiBt across the line with slope -1 . Cyc8es are points where these two

z. -y

)'. -z

fig. RUi

cwves intersect, where

(y(p" p,), z(p" p,)) = - Z(P2 ,PI)' - Y(P2, PI)}' (46)

This implies that mand (PI' P2. Pa, h, .. .) at e an equilibrium of this\!D00Ie1. The possibility cycles in this sort of model was first pointed out by

Gale (1973). Benhabib and Day (1982) have shown that there are .also

examples with equilibria that do no t converge to any steady state or 10 en

cycle of any length. Th e possibility of such strange behaviour, often

rrefcmd to as chaotic dynamics. has been a n a l y ~ d extensively byf,. ( j ~ ~ d ~ 1 (1985).

intertemporal Equilibrium Models :·19

4. GeaenU O"erbppiag Geaendioas Models

We now lum ou r attention to overlapping generations models with mllll1lY

goods in each period and many consumen; in each generation. If we aU'iW

many goods and many consumers, the assumption of two periods of life is

completely general: BaJasko, Cass, and SheD (1980) present a s i m ~ i e procedure for redefining periods and generations that converts a model mwhich consumers live for any finite Dumber of periods into one in wbk:h

they live for only two. Suppose that consumelS live for k periods. ThmlJ

redefine generations so that generations - k +2, - k + 3, . . . ,0 becon :e

generation 0, generations 1,2, . . . • " -) become generation 1. and : 0

on. Redefine periods in the same way. Figure 16.7 illustrates th is

procedure for the case k =4. Notice that each generation lives for ju 'lt

iWO redefined periods. If there are n ,0005 in each origioal period, th eue

are (k -1)n goods. indexed by date. in each redefined period. If t h e ~ ~ are h consumers in each original generation, there are (k - U)'1

consumers in each redefined generation.

Th e model with many consumers and many goods has the sam ·;

potential for equilibria that are Pareto-inefficient and equilibria witl Qunbacked nominal debt as does the simple model of the previous section ,

234 567 8 S

It 0 0 0 0 0'0 0 0 IIt C 0 co: 0 000

X It I 0 Q 0 : 0 ij 0--------r-------r------x x It I X 00 1 0 DO)

I Io X X I X X 0 I 0 0 0

G x 'I

A X X II

ij u 0 ;_________L _______

~ - - - - - - - -o DO:X x X IX CO l

I ,

000 ( 0 ) ( lt lX X C 2, ,0 1 )( 0 IL G ______It ,________)( ___ ._ _ _ _ _ _

o 0 0 I 0 c 0 l x x )( I0 0:0 0 0:0 X)( 3

I Io 0 0:0 0 0 10 0 )(

o I

2 3Period

1fiI. nlu

-2

-1

o

2c:

..g 3

4

5

6

7

8



300 Trmothy J. KehGtE

Hi hali e'Ven more potentiaJ for indeterminacy of equilibria. Consumer j nDgenerntioo , solves the problem

max ",(yj + IN " + wz}

u. ,,:1.+,,;.. z! =O.(47)

lHere )Ii> Z{, "'I. Pt, and PHI are aU n-dimensionaB vectors. i f bis0p Jexce5§ demaDd functions ar e Y IJ Pt+ ;) and z (p" P,+I)' then th e

aggregate excess demand functions for generation 8 are y(p" P,+I) andl

z( DIl" 81Hol) where, for ~ a m p l er - h

y(p" P.+.} .. Ly/(p" PHI}· {48)i - I

We ~ m e that y and z are conlinuousiy difJerenoabJe fOfr ail strictly

positive ~ r i c e vectors (P,. P.+I). are homogeneous of degree 0 iii

.. f !+i)' and obey Walras'slaw,

P;y(p" P,+n) +P:+IZ(P" P,+l} !!!!ij. (49)

JlKn arllditioo, there is an old generation. alive only in the tina period. abaa

!bas tl11e aggregate excess demand function ~ .. m). We assume that Zo isoontifJlUOusly differentiable for aJl strictly positive'price vectoq PD and an

open interval of moaey stocks m that includes 0, is homogeneous of

ciiegree 0 in (Pit m). and obeys Walras'$IaVi.

p:Zo(p., m) iE m. (SO)

Am equilibrium of this model again is a stock of outside money mand asequence of price vectors (PI' P2•. . .) that satisfies (39) and (40) where

Ilhe 'Yariables are reinterpreted as vectors. To see the possibility of

indeterminacy. let us count equ ations and unknowns in the equilibriumronditioDS. The equilibrium amdition in the first period,

z.{J . efr)+ y(.Pi.Pa) =0, {Sn

rontains $I equations in 2n +1 unknowns. Since the equations are anhomogeneous, we can impose a normalization to · reduce this to 1n

IlUlkIilOWDS. The equiJibriam conditions in subsequent periods,

Z(p,-i, p,) +y(Pl> P.+I) ::: 0, g = 2, 3, . . . , (52)

each add n equations and n unknowns. The entire system therefore has ndegrees of freedom. I f we set ", =0 II priori, there is one fewer unknown.

and this reduces the degrees of freedom io n - 1. Th e idea is tbat we

choose m, Ph and h to satisfy (51) and then use (52) as II nonlineardifference equatio n to determine Pl , P• . .. •

'Jrilte problem with simply counting equations and unknowns is tbat we

do 00 i aJways know wbether we can use {52) to continue an equilibriumprire sequence for arbitrary (Pit P2). lin /Figure 16.3, fO Ir example, fif we

inlertemporai Equilibrium MOlkls 2':H

start with any value of m/PI above z(l . I), we can continue li lle

equilibrium for a few periods but eventually we reach a situation whf re

we cannot continue because z exceeds "'I and there is no offer ClU"Ve

read to vertically! In general. we want to a ~ o i d situations where \!Ie

cannot use (52) &0 compute a positive value of P'+I as a function of P,_I

and P, . One way to do this is w require that the equilibriumsequence converge to a steady state at which the matrix of part ial

derivatives of y e p " ~ P,+I) with respect to P'+I is non-singular. This implies

that in some open neighbourhood of the steady state, for fixed (P,- I,pll}'tbe function Z(P'_I. Pt) +Y(P., .) is invertible. 1be implicit functiullI

theorem tells us that in this neighbourhood PHI can be compuir"!d

uniquely as a function of ( p ~ , P,). Restricting ou r attention to tHsneighbourhood of the steady slate, we can avoid the problem iUustTatuCi

in figure 16.5. where tOOre may be more titan one P.+. that satisfies Rile

equilibrium conditions. 1bis restriction may force us. however, to i g n C lsome equilibria.

A steady state of this model QS lR vector of relative prices p and! <lBi

inflation factor fJ such that P' == , 8 ~ ! s a t i s f i e s (52). There are two types :)JI

steady states: nominal steady states, in whidt there is a DOo-uro arnolD II I

of nominal debt transferred from generation to generation, and reaH

steady states, in which there is no such transfer. Notice that in iili l y

equilibrium the amount of nominal debt transferred from generatiu lIl

to generation stays constant over time: (SO) and (51) imply tltaQ

-P;Y(Pe. pz} == P ~ ~ P .. Ifr); Walras's Law implies that piz(Plt P2) =-p;y(p., pz); (51) impJies that - P21(P2, Pl ) = j J ~ ( p " P2); and 50 OiD.

"The steady-state condition is

zur-ip, fJ'p) +y(fJ'p, f J ' ~ i p ) =z(p, fJp) + yep, flp) =o.

This implies that p'z(p, (Jp) +p'y(p, (Jp) ... O. Walras's Law implies tlHa

p'y(p,(Jp)+!Jp'z(p,!Jp)=O. Subtracting on e from another, we obta illl

(1-,8)p'z(p, fJp) ... o. This says lbat fJ =1 at any nominal steady stat.! .Kehoe and Levine (1984b) prove that almost aU economies are such dunfJ :1= 1 at every real steady state.

Balasko and Shell (1980) and Burke (1987) have shown that ZI

necessary and sufficient condition for Pareto efficiency of ao equilibril! 'l!iis abat ..

L I/p,,,-I =00. (5l j).=1

Here IIp,11 =(p;p,)lf2. the standard Euclidean norm. They impose a!

nniform curvature condition 00 indifference surfaces that is natural instationary environment. Notice that any equilibrium that converges !to

steady state where fJ> K, an inflationary steady state, is P a r e t ( ) - i n e f f i c i esince the

sumin (54) converges. Any eqqjJjbrium that converges Ro

steady ~ t e where fJ l!ii: n howcve!l. is Pareto-efficient since ihe SUlllll i Illl

http://slidepdf.com/reader/full/y(.Pi.Pa





382 Timothy .11. Kehoe

(54) cJIiverges. In fact it is easy to show that if fJ = 1 ahe equilibriumallocaoon maximizes a weighted sum of utilities of the consumers in III

representative generation subject to steady-state consumption

constraints.When there are many goods in each period and many consumers in

each generation, there is no need for there to be a unique nominal sleadystate and a unique real steady state as there are in the example of the

previous section. Even with one good in each period, but more than one

consumer in each generation, there can be multiple real steady states,

a1thoogh there is a unique nominal steady state. Consider, for example. astatic w o - p e n o n exchange model witb mUltiple equilibria. Such a model

is ea5l' Uo construct in an Edgeworth box; see Shapley and Shubik (1m )

~ o r aIlI example. Now convert this into an overlapping generations model

in which there are two consumers in each generation with the same

preferences for and endowments of the two goods in the two periods of

«heir lives. The multiple equilibria of the static model are real steady

states of the overlapping ~ c n e r a t i o n s model in which each consumer

trades only with the other consumer in the same generation. This

iilustllltes the point that real steady states ar e not, in general. autarkic, asilieyare in the simple model. With many goods in each period. not even

nominal steady states need be unique. Kehoe and Levine (l984b) prove,

Iliowe"lfcr . that in general eNery economy bas an odd-in particular, anon-zero-number of oominaJ steady states a n ~ an odd number of real

Sieady states. Furthennore, the matrix: of partial derivatives of y with

respect to its second vector of arguments is almost always non-singular at

every steady state.To analyse the behaviour of equilibrium price sequences that converge

eo steady state, Kehoe and Levine (l985a) linearize the equilibrium

oonditjons (51) and (52). The local stable manifold theorem of dynamical

systems theory says that the behaviour of the nonlinear system near the

steady slate is qualitatively the same as that of tbe linear system (see

[£Win 1980). They consider the set of price pairs (PI. P2} that satisfy the

equmbrium condition in the first period and lead to convergence to thesteady state when employed as starting conditions for the nonlineardifference equation (52). This set is a manifold. a set of points that is

Bcaliy equivalent to an open subset of a Euclidean space of dimension

s m a l ~ e r than 211. (The prototypical manifold is a linear subspace.) Kehoe

and !Levine demonstrate that this manifold can bave dimension as large as

Fe if h e r e is outside money and as large as n - i if there is no money.

This manifold can also have dimension as small as 0, un which case il

consists of isolated points. (The best linear approximation to this

manifold near the steady state is the intersection of the stable subspace of

the unearized version of (52) with the sel of vectors that satisfy the

linearized version of (51).) Almost all economies are such that any small

lmertemporm Equilibrium Models ~ 8 3 perturbation produces an economy with the same qualitative propertil$.Kehoe and Levine also prove that there are robust examples of stea lystales with no equilibria at all that aH\verge to them. This cannot h a p pwith only one good in each period because Walras's Law implies that mand p can be chosen so that the sleady-state price vector (P. /Jp) satisfies

Zo(p,m)+y(p,/Jp)=O. Consequently, the steady slate itself i§ ;91l

equilibrium.

Notice Ihat we can use a similar irick to that used to convert economieswith consumers who live for k periods inro economies in which they Ii'/e

fo r two to convert the study of equilibria that converge to cydes of aLYfinite length into the study of equilibria that converge to steady states .Suppose that an economy has a k-period cycle in the sense tin t(PH .. •. • •P,+Ic) = (fJ'PIo .• . , {J'Plc) satisfies (52). Redefine generatiolilsso that, for example, generations i , 2, . . . •k become generation

Similarly, redefine goods. A k-period cycle is now a steady state of ni eredefined model.

De RbnIiu EqlliYaIeKe 1IbeOl'elll .

In 18!7 Ricardo asked the question. Does it make any difference whetht r

a government finances an increase in expenditure by raising taxes or h,\,

selling bonds? (See Ricardo 1951: 244-9.) The simple answer that i n ~ came up Wilh, although he realized that there were complications, W " $

that it makes DO difference. because consumers anticipate that they hav

to pay more taxes in the future if there is a bond sale SO that

government can malce interest payments. This is at odds to Keynes';

answer to the &arne question; that a bond-financed increase in govern

ment expenditure has the full multiplier effect, but that a tax-financec l

increase has a much smaller balanced multiplier effect. 1b e crucia ldistinction between the two analyses is that in one consumers' saving. ;

behaviour is altered by the bond issue and in the other it is not. H reduces to , as Barro (1974) puts it, Are government bonds net wealth?

!Let us first answer this question using our model with infinitely live"consumers. We introduce into that model a government that p u r c h a s egoods g, = (gl" . . . ,gN) in period I, t = 1, 2, . ... We require that thitexpenditure pattern be feasible in tbat

III

0". g, " . } : wi. 6= I , 2. _" (55);=1

Suppose first thaQ these purchases are financed by lump-sum-g:•...• r,o so that the government budget balances in every period:

h

L -r{=p;g" g = 2•.. . .

J=!



185J8.e. Timothy J. KehO€

Then the budget consuaini faced by coosumer j as.. ..

L p;d, =L (p;wi - ~ ) . (57)1'*1 t . : . ~

§uppose, on the other hand, that the governmenq issue bonds b"

8

=U.

2, • •.• that pay interest at the competitively detennined interesi

rate. it finances these interest payments by lump-sum taxes w,. Th e

government must balance its budget in tbe sense tbat the presen!

dis.oounted value of its expenditures is equal to the present discounlecl)

value of its revenues:

.. "(" )~ p ; g t + h,= ~ + b , tID _ (58)

LP;8,·L L Wr,=1 '=1 j= 1

1:;'=1 h, shows up on both sides of the budget constraint since tbe presenft

disoounted value of Ii bond is equal to sum of the interest payments on iu.

The consumer's budget constraint becomes

- GO ..

L (P:c, +bO = L (p;wI w,)+ }: II.11-1 , - I 1- 1 (59).. ..

L P;c, =L (P;wi - 11,).,_1 ,=1

Here is the net purthase of bonds by consumer j in IJ)eriod g and!

" (OO)L ~ = b " I ~ I

Notice that, if

(61)L -G= 2 fY",= 1 ,.:::1

~ h e s e two models ar e identical in their essentials. in particular, tbe agenis

face the same budget constraints. This is the Ricardian Equivalence

Theorem. There are a number of important maintained bypotheses.

First, there ar e perfect capital markets. .This implies that each consumelT

faces a single budget constraint. Second, all taxes are lump-sum;

1'\,\,<,.1\otherwise, relative prices would be distorted in different ways by different

. . _" taxation scbemes. taxes are not redistributional. In other words,

consumers face the same total tax bill under the two taxation scheme;

oaherwise, relative prices would change because of income effects.

We are not claiming that the equilibrium is the same as ir g, = 0, 1=1,

2, . . . Since the government is consuming some of the goods tbat would

oiliet'wise have gone to consumers, this cannot be the case. Governmeni

fiscal policy always bas real effects. II is the way it ns financed that Ds

arreievant.

lmenemporm Equilibrium Models

Hi is difficult to give the Ricardian Equivalence Theorem an ioterpn tao

tion in an overlapping generations model: alternative tax schemes t\.'la t

!ime tax collections differently necessarily have redistributional effe,cts

because consumers are alive at different times. There are very :ial

situations in wbich differeut tax schemes do no t affect the equilibricn HQ

does not matter, for example, in which period of life ronsumers pay ta'Ies

as long as each consumer faces a single budget constraint, all taxes ;u-e

lump-sum, and each consumer faces the same total tax biD under ·'.he

different schemes. Rather than say that tbe Ricardian EquivalcliICe

1beorem does not hold in an overlapping generations model. we sho llM

say that the range of tax schemes that do no t affect the equilibria is much

more limited in an overlapping generations model than it is B!ll 2lJ1

infinitely lived consumer model.

Another problem with interpreting the Ricardian EquivaleMCtl

Theorem in a model with infinite numbers of consumers and goods is, as

we have seen, that there is no reason for !;;'_I p;g, to converge. Thegovernment can therefore issue bonds that it need never pay back. Th!Se

bonds act like injections of outside money and are, tberefore, net.W8fl&: 'l9leA.

figure 16.8 depicts an example with a steady state in which 8, = g :> 0every period. Here inflation erodes the value of the initial stock

outside money al the same rate as that at which the value of the t c ~

lFig,lIM



386 Tunothy J. Kehoe

stode of government bonds increases. The total real stock of nominal

debt. outside money and bonds, remains constant at z +g. 1l1e steady

interest rate is r = liP -1 <0 . Notice that, even though the

government is consuming g> 0 of the single good in every period, this

equilibrium Pareto dominates the autarkic equilibrium where it consumes

notiling. Examples of this sort are discussed by Sargent (1987: 0. 7).

Barro (1974) has argued that the Ricardian Equivalence Theoremholds for overlapping generations models in which consumers include

their otrspring's utility in their own utility functions. Since their offspring

similarly value the utility of their own offspring, this can make 1Il

consumer's utility maximization problem into the problem of maximizing

utility of an infinitely lived family. 1be problem with this story is that,

in general. we have to allow some consumers io pass on debts. as well as

bequests, to their offspring. In tbis case we would want consumers io

include their progenitors' utility in their own utility functions. Think of

ou r example of the two infinitely lived consumers with log-linear utility

fUlllctions as a model of such families. On e family of consumers

asymptotically consumes notbing. They use almost aD of their income to

service their family debt. which they inherit (com their progenitors and

pass on to their offspring. This sort of problem always occurs if different

families have different discount factors in ~ h e i r reduced-form utility

fu iilctions. Institutional arrangements in modem societies make this

feature of the bequest story very unrealistic. As Barro himself points out,

nf a ~ a m i l y is at a comer solution because of a non-negativity oonstraint

00 . bequests. the family faces a sequence oR" budget coostrainls that

cannot be aggregated into one.

~ m j j a r l y , a model with infinitely lived consumers who face liquidity

OO!JlsQraints can have similar chnacteristics to an overlapping generationsmodel (see Woodford 19863. io r example.) I f we cannot reduce the

consumer's utility maximization pt"oblem to one with a finite number or

budget constraints. then we cannot prove that the value of the aggregate

emoowment is finite. Consequently, equilibria Deed not be Paretoefficient, and there may be equilibria in which outside money plays 2

rroIle. Even ou r argument that there is a finile nwnbu of equilibria faUs

apart. The essential feature of that argument is that each consumer

characterized by a single Lagrange multiplier AJ = 1/aj . Rf the consumer

caHUlOt equate his marginal utility of income in diJlerent periods, t hen he

aci>:S, to some extent, like a sequence of dillerent consumers. There may

be iJ robust continuum of eqUilibria, and the Ricardian Equivalence

11'!eorem need not hold.

Ii. rOl" FWIe Models

What does ou r analysis of the overlapping generations model tell usabout the properties of models with large. bu t iiinite. number or

//nlerlemporlll /Equilibrium Models 381'

consumers and! goods? Suppose that we truncate the modei aa 5 )lIII1e

period T using a terminal young generation Yr(PT, m) analogous to the

initial old generation Zo(P .. m) . Outside money now corresponds ' 0 III

transfer from the terminal young generation to the initial old. TItefe isnow a finite number of equilibrium conditions:

zJ..Ph if!) +Y(Pt. P2) = 0

z(p" P2 ) +Y(h, Pl ) = 0 62}

z(Pr-I' PT) + '!T(ftn ria) =O.

All of the equilibria of this model are Pareto-efficient. In general. t iller: isa one-dimensional continuum of equilibria indexed by the real tran,.ferpayment tn/\lftdl.

This method of truncating this model is often equivalent to specil1il!llg

expectations of prices in periods after the model ends. For example , 'We

could specify YT(Pr, m) as YT(Pr, IIPTII (Jp) where (P, P) is a st eAdy

state. Here, of course. m=: IIPrll {JP'Z(PT7 IIPTII (Jp). (See Aue lrbac!1l.

KotJikoff, and Skinner 1983 for an application of this approach.)

Consider a situation where one equilibrium Pareto-dominates anolber

in the infinite horizon model. Each of these can be made an equilibr ium

of tbe truncated model with a suitable choice of Yr. Since both of !the

equilibria of the truncated model are Pareto-efficieot, the equilibr:ulIIIll

that dominates in the infinite borizon model must assign some memhers

of the terminal generation lower utility tban does the inferior eq ui

librium. Notice tbat the functions Yr do no t necessarily bear l i l ly

relationship to utility maximization by generation T illl the infin ite

borizon model. I f T is large enougb. the model is clear: by sacrificing the

welfare of one generation, all otbers are made beUer off, and socie1ly . \5 awhole is made better off from a utilitarian viewpoint.

In an infinite horizon model there can be n dimensions of indeler

minacy if there is outside money and n - 1 dimensions i f there is not. '(hesingle dimension of indeterminacy that shows up because of fiat mO l ley

conesponds to the indeterminacy parunetrized by the real tnuder

payment. What about the otber dimensions? To answer this question, let

US suppose tha t we have two equilibrium price sequences. (Pt.P2, . ' .)and (P h P2, .• . ), which both converge to tbe same steady state. Supp>setoo thaI both involve tbe same real stock of outside money, .

WI m--1:1-lIP111 lIP III"

If we truncate using a tennioai young genention

1TWr.m) = YT(PT, fin-I)' {1i3)

m = ii i, b e D (Pn.P2 . , •• [ ly } is an eql!ilibrium. I f we truncate wiRb lhe



388

-

Tunothy J. / 1 ( ~

Pl •• fp,

pi I

T

lFII. n631

analogous choice of YT. then (PI, PZ•.•.• fir) is an equilibrium. Figure

»6.9 depicts this sort of situation. For large enough T. ({Jr. PT+') is going

to be arbitrarily close to (PT. (IT+I) no matter bow far apart are PI andPn indeterminacy of equilibrium therefore corresponds to sensitivity to

terminal conditions. sensitivity of initial prices that becomes more acute

as the time horizon T becomes larger. See Kehoe and Levine (1986) for

nwneuical simulations of an eumple with this propery.

We should point out one other way of reducing an overlapping

generations model to a model with a finite time horizon. Suppose that in

eve!)' period the probability that the world ends before the next period isP. (]I<P < 1. It is then natund to assume that consumer j in generation g

solves the expected utility maximization problem

m a x ~ U J ( y : ; WI! + (1 - p)vlr.+ Wa. z ~ + WZ)} (64)

S.t. p,1, +P,+lz, = O.

Here is his utility function if the world ends before the second period

of his tife and Vj is his utility function if it does not. Even though theworld ends in finite time with probability 1. this model is identical to anoverlapping with an infinite horizon. It may have equilibria that are not

rareto-efficient, equilibria in which outside money plays a role. and

equmlJ)ria with one or more dimeosions of indeterminacy.

lntertemporllK /Equilibrium Models 3 i ~

1. EueasioDS aaci CoIIdasioIls

The results presented in this paper can be extended ao more g e n e rmodels. Kehoe et aI. (1988) have exteoded this analysis of tbe model m(ll2 finite number of infinitely liyed agents to similar models that a l l oproduction and capital accumulation. The only difficulty is in ensuwlg

tbat the transfer functions used in the equilibrium conditions ale

continuously differentiable. Muller and Woodford (1985) have extended!ihis analysis of the overlapping generations model to models that indue! einfinitely lived ( ; ( ) n , u m e ~ , assets. and production. They find thatpresence of infinitely lived consumers or infinitely lived assets can force

the value of the aggregate endowment to be finite. This rules out Paren!)

inefficiency of equilibria and outside money_ It does not rule o t , lindeterminacy of equilibria. however.

Do Pareto inefficiency of equilibria and outside money depend! ()i1l

there being an infinite number of consumers or on some c o n s u m e ~ s baving finite life-spans? Kehoe (1986) considers a simple p u ~ x c b a n gmodel in which there is an infinite number of consumers woo all live fC lTever. This model has equilibria that are Pareto-inefficient and equiJibni'lwith outside money. It also has equilibria with several dimensions () findetenninacy

As we have seen. indeterminacy is a ·U'elatively separate issue f!rolia

Pareto inefficiency and outside money. Kehoe et Ill. (1986b) consider

abstract model with infinite numbers of c o n s u m e ~ and goods. ]be onU,r

prices that are allowed assign finite value to the aggregate endowmen€,This rules out Pareto inefficiency and outside money. Even so, there ar(;

Il'obust exarnple& with any dimension of indeterminacy. The reason

this indetermioacy is tbat we cannot reduce the equilibrium conditions

a finite number of equations and unknowns. These authors also find tha&here is a finite number of locally unique equilibria if consumers am

similar enough. This generalizes our results on economies with a f i n i tnumber of consumers.

Santos and Bona (1986) and Geanakoplos an d Brown (1985) b a , ,extended the results of Kehoe and Levine (l985a) for stationary ,

pure-exchange, overlapping generations models to models with 00IiJ '

stationary structures. Like Kehoe and Levine, these authors need

restrict their attention to equilibria that remain close to each other I

some sense. They find that, even in a noo-statioaary environment, there

are n dimensions of potential indeterminacy if there is outside money aM.1'1 - 1 dimensions if t here is

One disturbing aspect to the potential indetennioacy of equilibria ir ,

ahat it occurs for some values of the parameters of a model but not fOl ·

others. We would like to somehow classify the parameter values rot



390 Timothy./J. !Kehoe

whiclhl indetenninacy does no t occur. A first step in this direction has

!been tak.en by Balasto and Shell (1981), who consider a model withmany goods in each period but a single two-period-lived consumer with a

Cobb-lDouglas utility function in each generation. TIley prove that there

Us HUG indeterminacy without outside money and only on e dimension of

ondererminacy with it. Geanakoplos and Polemarchakis (1984) have

shoWlII that the essential feature of this analysis is that tbe singleuwo-period-lived consumer bas intertemporally separable preferences.

[{eooe and !Levine (l984a) have shown further that any small perturbation

m model with a single two-pe riod-lived consumer with intertemporally

sepauable preferences, even if it introduces small heterogeneities among

consumers or small interdependencies in consumption ove/!' time, results

ma IlVIOdeI with these same features.A more significant finding is that of Kehoe el m. (l986a), who cousideJr

geneiJ'ai pure-exchange, overlapping generations economies with many

goods in each period and many consumers in each generation, ill which

all demand functioos edlibit gross substitutability. TItey pro ve that there

n5 a i1lnique equilibrium if there is no money; although there may be 3l

one-dimensional indeterminacy with outside money, there is at most oneequiiibrium for each level of real outside money in the first period.

!Furthermore, their analysis is global rather than local. I f the economy is

stationary, then there is a unique nominal steady state and there is i lwniqK real steady state, and every equilibrium converges to one of them.

Unfortunately, there are plausible examples that violate gross sub-

stitullability. Kehoe and Levine (1986) consider a model with a single

,oociI in each period and a single three-period-lived consumer in each

generation. 1bey give tbis consumer a constant elasticity of substitutiollJ

u t i l i ~ y function and show that, for plausible parameter values, this model

~ " h i b i t indeterminacy without outside money and more than one

dimension of indeterminacy with it. Moreover, they choose the crucial

parameter. the elasticity of substitution in consumption over time. to

agree with empirical evidence.nne present analysis has f o c ~ on the differences between models

witlu a finite number of infinitely lived consumers and overlapping

generations models. Yet these two types of models have important

properties in COmmon. In both, for example, equilibria always exist.

Sinoe the equilibriwn conditions for a model with a finite number of

nnfiniteJy lived consumers can be transformed using Negishi's (196Ob)

approach into those of a model with a finite nwnber of goods, it isstf'anghtforward to prove the existence of equilibria in such models. This

is done, for example, by Kehoe a al. (1988). Proving the existence of

equilibria in overlapping generations models involves more subtle issues.

Considering the limit of a sequence of truncated economies. Halasko et

!lit. :J.1980) prove the existence of an equilibrium with riB = 0 in :m

Kmertempoyai EqlUiJibrium Models 39)'

pure-exchange, overlapping generatiQns model. For general models witlh

countably many consumers and goods, Burke (1986,1988) and Wilsoll

(1981) have proven the existence of equilibria. The presence Qf outsfide

money may be necessary, however, for an equilibrium tQ exist.

Another property that these two iypeS of models have in commolll i t ~ tbat the Second Welfare Theorem holds: any Pareto-efficient allocation

can be supported as a competitive equilibrium witb transfers. Thisproven fQr tbe overlapping generations model by Balasko and S h e ~ [ (1980). The role that outside money plays in supporting a P a r e t o - e f f i c i e i l l ~ allocation can be interpreted as such a transfer. Unfortunately, Cass et (61.

(1979) and Millan (1981) have examples in which no P a r e t o - e f f i c i e l 1 lallocation can be supported as a competitive equilibrium by giving aJ

transfer ooly to the first generation. Burke (1987). however, shows thaQ a

transfer to the first generation does support efficiency if t is foUowed by :at

sequence of taxes on subsequent generations. Furthennore, the sum ( I J Ireal tax payments can be made arbitrarily smaU.

How much of this analysis extends to intertemporal models with

uncertainty? If all markets are complete, titen the analysis of the modlefi

with a finite number of infinitely lived consumers remains the same. Hillparticular. all equilibria are Pareto-efficient, there is no role for outside

mouey, and there is generically a finite number of equilibria. Goods are

indexed by histories of states of nature as weD by date. (See Kehoe anell

Levine 1985b for an analysis of a model of this sort.) In a stoch.asftic

overlapping generations model the assumption of complete markets iis

unnatural. however. In a deterministic setting we have argued thai on

makes no difference whether aU trade takes place in the first period or

takes place sequentially; in a stochastic setting setting this is no longer tbe

case. Consumers would want to make trades in periods before they are

born tQ insure themselves againsa being born into UDfavourable clilfo

cumslances. Dutla and Polemarchakis (1985) present an analysis of aJ

simple stochastic overlapping generations model and show the differerK:e

between equilibria with complete markets and equilibria where COlll

sumers are allowed to trade only during the ir own lifetimes.

As we have seen, models with infinitely lived consumers who fu e

incomplete markets have similar properties to overlapping generations

models. Bewley (1980, 1983), Scheinkman and Weiss (1986), and Levine

(1986) analyse simple stochastic models in which there are infinitely livecll

consumers who are constrained in their borrowing and lending decisions.

Not surprisingly. they find that such models have equilibria that arre

Pareto-inefficient and equilibria in wbicb outside money plays an impor

tant rolc. Presumably, these models also have indeterminate equilibrim.

but this property has not received much attention.

The most worrying property of the overlapping generations modeD 8s

probably its potential for indeterminate equilibria even if there ns 1l1\i!J)



39 .1

• " i

3 9 Timothy J .Kehoe

ou;side money; There are two reasons for this. First, indeterminacy

nwJces the model unsuitable for comparative statics analysis. Second, it

makes the concept of perfect foresight problematical. Multiplicity of

equihoria of any sort presents difficulties for an economist interested in

usnng a model to do a comparative-statics analysis of the impact of s

change in parameters. Suppose, however, that a model has a finitelIIumber of locally unique equilibria that vary continuously with its

parameter values. (Almost all static general equilibrium models possessthese properties.) Then the economist could hope that, by appealing to

history to justify focusing on one particular equilibrium. and to a (usuallyunspecified) dynamic adjustment process to justify focusing on .thedisplacement of that equilibrium after a change in parameter values,

comparative statics Still makes some sense. Even these hopes vanish if

there is a continuum of equilibria.The idea underlying perfect foresight in a model with no uncertainty is

the same as . hat underlying the rational expectations hypothesis in amooel with uncenainty: the agents know the structure of the model and

use it to predict the relevant values of future variables. I f the model does

not make determinate prediction, then hypothesis of perfect foresight

becomes less attractive. I f there is a continuum of perfect foresight paths,

the theory is incomplete. Geanakoplos and Polemarchakis (1986) arguethat indeterminacy leaves room for factors like fixed nominal wages and

animal spirits of investors. AF. we have seen in our discussion of theRicardian Equivalence Theorem, if there is a continuum of equilibria,

some may bave Keynesian features and some may not.

One way to tr y to make the theory complete would be to fix the values

0): some variableS in the first period, for example the real money stock or

a relative price. Even this approach tails if y(p, , PHI) is not always 'an

mvertible function of p,+ 1. With the backwards-bending offer curve in

Figure 16.5, for example, there is an infinite number of equilibria even if

we fix the value of mlpl: at every point where there are two values of

P+l such that z(p,-t. p,)+ y(p" Pt+I) = 0, we have a choice of a different

price path to foUow.

Modelling expectations has long been a difficulty in economic theory.Keynes (1973/1936), for example, realized the importance of expecta

tions formation, but claimed to work with a model in which the timeperiod was short enough so that expectations could be taken as

exogenous. The simplest way of making expectations endogenous is to

nl8ke them adaptive as done by, for example, Friedman (1968) andPhelps (1967). The equilibria of the overlapping generations .models

would be generically determinate if we specified expectations as eitherexogenous or adaptive: since values of past variables can be taken as

exogenous in any period, the equilibrium conditions reduce to a system

of a finite number of equations in the same finite number of unknowns.

lnlertemporal Equilibrium Models

Computing the equilibria of such a model would reduce to computing the

equilibria of a sequence of models that look like static models.

The indetennmacy of equilibria in the overlapping generations models

is aU the more wonying because it can be aSsociated with the existence of

self-fulfilling prophecies. Even though the prefererices, endowments, and

technology of an economy are detenninistic. a random variable can affect

the equilibria merely because agents expect it to. This phenomenon is

referred to as a 'SUDSpot', although actual sunspots may actuaUy affect thetechnology of an economy (see, Mirowski 1984), and may not be

themselves stochastic (see, Weiss 1985). There is a large and growing

literature on sunspots. A very incomplete list of references includes:Azariadis (1981). Azariadis and Guesnerie (986), Cass and Shell (1983),

and Farmer and Woodford (1984). Woodford (1986b) presents an

example in which agents employ a simple leaming rule and the economyconverges to a perfect-foresight sunspot equilibrium.

Just as worrying as iodetenninacy of equilibria is the possibility that aJJ)

economy may have no equilibrium that converges to a steady state. I f thepath followed by equilibrium prices is chaotic or periodic of a very long

length, the perfect-foresight hypothesis is unattractive for a different

reason: it requires too much computational power of the agents of the

model. Any theory of expectations fonnation that is designed to copewith the problem of indetenninacy of equilibrium must also be able to

relax the requirement of perfect foresight when equilibrium pricedynamics are chaotic or periodic of very long length. Unfonunately, as

Benhabib and Nishimura (1985) and Boldrin and Montrucchio (1986)have shown, even the model with a finite number of infinitely lived agenks

can have equilibria that exhibit periodic or chaotic dynamics.The above analysis of intertemporal general equilibrium models has

provided us with a clear understanding of why Pareto inefficiency and

outisde money occur in the overlapping generations model but not in the

model with a finite number of infinitely lived consumers. It is also clear

how these properties manifest themselves in a truncated version of the

model. Although we have attained some understanding of the possibility

of indeterminacy, we are stili faced with the dilemma that indetenninacyis symptomatic of an incompleteness of the modeL What is needed is a

serious theory of expectations formation . .

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