We show that Keynesian multiplier effects can be obtained in dynamic optimizing models if one combines both price rigidities and a "non Ricardian" framework where, due for example to the birth of new agents, Ricardian equivalence does not hold.multiplier ; Ricardian equivalence ; non Ricardian economies ; price rigidities ; Keynesian multiplier

Jean-Pascal Bénassy

English

Research Papers in Economics

 
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  WORKING PAPER N° 2006 - 15 
 
 
 
Ricardian equivalence and the intertemporal 
 
Keynesian multiplier 
 
 
 
Jean-Pascal Bénassy 
 
 
 
 
  JEL Codes : E62, E63, E12 
 
  Keywords : multiplier, Ricardian equivalence, 
non-Ricardian economies, price rigidities, Keynesian 
multiplier. 
 
CENTRE NATIONAL DE LA  RECHERCHE SCIENTIFIQUE – ÉCOLE DES HAUTES ÉTUDES EN SCIENCES SOCIALES 
ÉCOLE NATIONALE DES PONTS ET CHAUSSÉES – ÉCOLE NORMALE SUPÉRIEURE 
 
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1Ricardian Equivalence
and the Intertemporal
Keynesian Multiplier
Jean-Pascal Bénassy∗†
December 2005
Revised February 2006
Abstract
We show that Keynesian multiplier eﬀects can be obtained in dy-
namic optimizing models if one combines both price rigidities and a
“non Ricardian” framework where, due for example to the birth of
new agents, Ricardian equivalence does not hold.
JEL Codes: E62, E63, E12
Keywords: Multiplier, Ricardian equivalence, Non Ricardian economies,
Price rigidities, Keynesian multiplier.
∗Address: CEPREMAP-ENS, 48 Boulevard Jourdan, Bâtiment E, 75014, Paris, France.
Telephone: 33-1-43136338. Fax: 33-1-43136232. E-mail: benassy@pse.ens.fr
†I wish to thank Fabrice Collard and Harris Dellas, but retain responsibility for all
remaining deﬁciencies.
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11 Introduction
One of the most intriguing features of traditional Keynesian theory (Keynes,
1936, Hicks, 1937) is the so called “multiplier eﬀect” by which an increase
in governement spending can create an increase in consumption, whereas in
Walrasian models it leads to a decrease, via the usual “crowding out” eﬀect.
This multiplier eﬀect is customarily attributed to price or wage rigidities.
Now in the recent evolution of macroeconomic modelling most macroe-
conomic issues are reexamined within the rigorous framework of dynamic
intertemporal maximizing models à la Ramsey (1928). The typical model
depicts consumers as one single dynasty of inﬁnitely lived agents. This model
has notably the property of “Ricardian equivalence” (Barro, 1974), accord-
ing to which, in a nutshell, the distribution of taxes across time is irrelevant
as long as the government balances its budget intertemporally. The model
has been extended to a monetary framework (Sidrauski, 1969, Brock, 1975),
where Ricardian equivalence also holds.
In line with this recent evolution, a natural question to ask is whether
a multiplier eﬀect will arise in these dynamic models. The result has been
actually disappointing: crowding out occurs in the usual DSGE model (see
f o re x a m p l eF a t a sa n dM i h o v ,2 0 0 1 ) ,a n de v e ni nm o d e l sw i t hp r i c er i g i d i t i e s
under standard parameterizations (see for example Collard and Dellas, 2005).
What we want to show in this paper is that, in order to obtain a strong
enough multiplier eﬀect, another ingredient, in addition to price rigidities,
has to be introduced in dynamic models. Namely one should not only have
price rigidities, but also model the economy as “non Ricardian”. By non Ri-
cardian economies we mean, as in Barro (1974), economies like overlapping
generations (OLG) economies à la Samuelson (1958), where Ricardian equiv-
alence does not hold. The non Ricardian economy we shall work with is due
to Weil (1987, 1991). It is a monetary economy where, as in the Ricardian
model, agents have an inﬁnite life but, as in the OLG model, new agents
arrive over time.
We shall call n the rate of growth of the population. In a nutshell the
results are: If n>0, there is a multiplier1, and government spending leads
to an increase in private consumption. This multiplier is larger, the higher
n is, i.e. the more “non Ricardian” the economy is.
1By this we mean more precisely that the income multiplier is greater than one, so that
t h e r ei sn oc r o w d i n go u t .
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12T h e m o d e l
We shall thus use the model of Weil (1987, 1991), which has the great ad-
vantage of having the Ricardian model as a particular case. Each household
lives forever, but new “generations” are born every period. Denote as Nt
the number of households alive at time t. We will work below with a con-
s t a n tr a t eo fg r o w t ho ft h ep o p u l a t i o nn ≥ 0,s ot h a tNt =( 1+n)
t N0.T h e
Ricardian model corresponds to the special limit case n =0 .
2.1 Households
C o n s i d e rah o u s e h o l db o r ni np e r i o dj. W ed e n o t eb ycjt, yjt and mjt his
consumption, endowment and money holdings at time t ≥ j. This household
maximizes the following utility function:
Ujt =
∞ X
s=t
β
s−tLogcjs (1)
and is submitted in period t to a “cash in advance” constraint:
Ptcjt ≤ mjt (2)
Household j begins period t with a ﬁnancial wealth ωjt.F i r s tt h eb o n d
market opens, and the household lends an amount bjt at the nominal interest
rate it. The rest is kept under the form of money mjt,s ot h a t :
ωjt = mjt + bjt (3)
Then the goods market opens, and the household sells his endowment yjt,
pays taxes τjt in real terms and consumes cjt, subject to the cash constraint
(2). Consequently, the budget constraint for household j is:
ωjt+1 =( 1+it)ωjt − itmjt + Ptyjt − Ptτjt − Ptcjt (4)
2.2 Aggregation, endowments and taxes
Aggregate quantities are obtained by summing the various individual vari-
ables. There are Nj −Nj−1 agents in generation j, so for example aggregate
taxes Tt are given by:
Tt =
X
j≤t
(Nj − Nj−1)τjt (5)
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1The other aggregate quantities, Yt,C t,Ωt,M t and Bt, are deduced through
similar formulas from the individual variables, yjt,c jt,ωjt,m jt and bjt.
We now have to describe the distribution of endowments and taxes among
households. We assume that all households have the same income and taxes:
yjt = yt =
Yt
Nt
τjt = τt =
Tt
Nt
(6)
3 The dynamic equations
3.1 Taxes and government budget constraint
The dynamics of government liabilities Ωt is:
Ωt+1 =( 1+it)Ωt − itMt + PtGt − PtTt (7)
Government budget balance corresponds to Ωt+1 = Ωt, i.e. since Ωt =
Mt + Bt:
PtGt = PtTt − itBt (8)
W ew o u l dl i k et oh a v eat a xi n d e xTt such that budget balance is achieved
under the traditional condition:
Gt = Tt (9)
This will be the case if we deﬁne Tt through:
PtTt = PtTt − itBt (10)
Budget balance now corresponds to (9), and the government budget con-
straint (7) is rewritten as:
Ωt+1 = Ωt + PtGt − PtTt (11)
3.2 Consumption dynamics
The dynamics of consumption is given by (see the appendix):
Pt+1Ct+1 = β (1 + n)(1+it)PtCt − (1 − β)nΩt+1 (12)
The dynamic system consists of equations (11) and (12).
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14 The multiplier
We shall assume that nominal prices are fully rigid, now and in the future.
Besides simplifying the algebra this assumption, which is clearly the most
favourable situation for a multiplier eﬀect to arise, will allow to make crys-
talclear that a non Ricardian environment plays a central role in addition
to price rigidities. We also assume that the nominal interest rate is pegged
(it = i). Now equation (12) is rewritten:
PtCt = γPt+1Ct+1 + γ (1 − β)nΩt+1
= γPt+1Ct+1 + γ (1 − β)n(Ωt + PtGt − PtTt) (13)
with:
γ =
1
β (1 + n)(1+i)
(14)
We shall assume γ<12. Integrating forward equation (13) we obtain:
PtCt = γ (1 − β)n
∞ X
j=0
γ
j (Ωt+j + Pt+jGt+j − Pt+jTt+j) (15)
Now from (11):
Ωt+j = Ωt +
j−1 X
k=0
Pt+k (Gt+k − Tt+k) (16)
so that:
∞ X
j=0
γ
jΩt+j =
∞ X
j=0
γ
j
"
Ωt +
j−1 X
i=0
Pt+k (Gt+k − Tt+k)
#
=
Ωt
1 − γ
+
∞ X
k=0
Pt+k (Gt+k − Tt+k)
∞ X
j=k+1
γ
j
=
Ωt
1 − γ
+
∞ X
k=0
γk+1
1 − γ
Pt+k (Gt+k − Tt+k) (17)
and ﬁnally, inserting (17) into (15):
2This condition also plays a role for price determinacy in models with market clearing.
See Bénassy (2005) for a formal analysis and economic discussion.
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1PtCt =
γ (1 − β)n
1 − γ
"
Ωt +
∞ X
j=0
γ
jPt+j (Gt+j − Tt+j)
#
(18)
So with n>0 consumption reacts positively to government spending,
and from formula (18) the multiplier is greater, the higher n is.
5C o n c l u s i o n
We have thus seen that in order to obtain a “multiplier”, price rigidities
are not the only decisive ingredient, but it is also important to have a “non
Ricardian” economy, corresponding here to n>0. We shall now give a brief
intuition for this.
Let us start with the limit case n =0 .A sw es h a l ln o ws e et h ea r g u m e n t
is quite reminiscent of the famous “balanced budget multiplier” of Haavelmo
(1945). Consider indeed an increase in government spending, say Gt,a l l
other spending variables being held constant. Because of the government’s
intertemporal budget constraint, Gt will have to be compensated by taxes,
either now or in the future, and the total discounted value of these taxes must
be exactly equal to Gt. As a result we have a “balanced budget” formula, as
in Haavelmo.
Now if n>0, new generations arrive over time, and the future, but yet
unborn, generations will bear some of the future taxes. So there is part of
government spending that will not be paid by currently alive generations,
and as a result the traditional multiplier mechanism comes into play.
6 Appendix: derivation of equation (12)
We shall derive in this appendix the dynamic equation (12). Consider the
household’s budget equation (4). We assume that it is strictly positive, so
households always want to satisfy the “cash in advance” equation (2) exactly.
We thus have mjt = Ptcjt and equation (4) is rewritten:
ωjt+1 =( 1+it)ωjt + Ptyt − Ptτt − (1 + it)Ptcjt (19)
Deﬁne the intertemporal discount factors:
Rt =
t−1 Y
s=0
1
1+is
R0 =1 (20)
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1A p p l y i n gt h ed i s c o u n tf a c t o r s( 2 0 )t ot h eb u d g e tc o n s t r a i n t( 1 9 ) ,i tb e -
comes:
Rs+1ωjs+1 = Rsωjs + Rs+1Ps (ys − τs) − RsPscjs (21)
If we aggregate all discounted budget constraints (21) from time t to
inﬁnity, and assume that Rsωjs goes to zero as s goes to inﬁnity (the usual
transversality condition), we obtain the intertemporal budget constraint of
the household:
∞ X
s=t
RsPscjs = Rtωjt +
∞ X
s=t
Rs+1Ps (ys − τs) (22)
Now maximizing utility function (1) subject to the intertemporal budget
constraint (22) yields the following consumption function for a household j:
RtPtcjt =( 1− β)
"
Rtωjt +
∞ X
s=t
Rs+1Ps (ys − τs)
#
(23)
Summing this across the Nt agents alive in period t, we obtain the aggre-
gate consumption Ct:
RtPtCt =( 1− β)
"
RtΩt + Nt
∞ X
s=t
Rs+1Ps (ys − τs)
#
(24)
In equilibrium Yt = Ct + Gt, so the equilibrium equation is:
RtPt (Yt − Gt)=( 1− β)
"
RtΩt + Nt
∞ X
s=t
Rs+1Ps (ys − τs)
#
(25)
Divide both sides by Nt and use Yt = Ntyt, Gt = Ntgt:
RtPt (yt − gt)=( 1− β)
"
RtΩt
Nt
+
∞ X
s=t
Rs+1Ps (ys − τs)
#
(26)
Let us rewrite this equation for t+1and subtract it from (26). We obtain:
RtPt (yt − gt) − Rt+1Pt+1 (yt+1 − gt+1)=
(1 − β)
·
RtΩt
Nt
−
Rt+1Ωt+1
Nt+1
+ Rt+1Pt (yt − τt)
¸
(27)
Now multiply the government’s budget equation (7) by Rt+1/Nt:
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1RtΩt
Nt
=
Rt+1Ωt+1
Nt
+( Rt − Rt+1)Pt (yt − gt) − Rt+1Ptgt + Rt+1Ptτt (28)
Insert this into equation (27):
Rt+1Pt+1 (yt+1 − gt+1)=βRtPt (yt − gt) − (1 − β)
µ
1
Nt
−
1
Nt+1
¶
Rt+1Ωt+1
(29)
and multiply by Nt+1/Rt+1:
Pt+1 (Yt+1 − Gt+1)=β
Nt+1
Nt
(1 + it)Pt (Yt − Gt) − (1 − β)
µ
Nt+1
Nt
− 1
¶
Ωt+1
(30)
Taking ﬁnally Nt+1/Nt =1+n,a n du s i n gCt = Yt − Gt we obtain:
Pt+1Ct+1 = β (1 + n)(1+it)PtCt − (1 − β)nΩt+1 (31)
which is equation (12).
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A mathematical theory of saving”,

A simple perfect foresight monetary model”,

An exact consumption-loan model of interest with or without the social contrivance of money”,

Are government bonds net wealth?”,

Fabrice and Harris Dellas (2005), “Poole in the new Keynesian model”,

Interest rate rules, price determinacy and the value of money in a non-Ricardian world”,

Is money net wealth?”,

Multiplier eﬀects of a balanced budget”,

Permanent budget deﬁcits and inﬂation”,

Rational choice and patterns of growth in a monetary economy”,

The eﬀects of ﬁscal policy on consumption and employment: theory and evidence”, working paper,

The General Theory of Employment, Interest and Money.

Ricardian equivalence and the intertemporal Keynesian multiplier

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Ricardian equivalence and the intertemporal Keynesian multiplier

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