We construct in this paper a dynamic general equilibrium model which displays the central features of the IS-LM model, and notably an income multiplier greater than one, so that crowding out does not occur. It appears that the key to this result is the conjunction of two features of our model: price rigidities (as is usually expected), but also a non-Ricardian economy.IS-LM ; DSGE models ; Keynesian multiplier ; crowding out ; non-Ricardian economies

Jean-Pascal Bénassy

English

Research Papers in Economics

 
PARIS-JOURDAN SCIENCES ECONOMIQUES 
48, BD JOURDAN – E.N.S. – 75014 PARIS 
TEL. : 33(0) 1 43 13 63 00   –   FAX : 33 (0) 1 43 13 63 10 
www.pse.ens.fr 
 
  WORKING PAPER N° 2006 - 14 
 
 
 
IS-LM and the multiplier: 
 
A dynamic general equilibrium model 
 
 
 
Jean-Pascal Bénassy 
 
 
 
 
  JEL Codes : E12, E62, E63 
 
  Keywords : IS-LM, DSGE models, Keynesian multiplier, 
crowding out, non-Ricardian economies. 
 
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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1IS-LM and the Multiplier:
AD y n a m i cG e n e r a l
Equilibrium Model
Jean-Pascal Bénassy∗
October 2005
Revised February 2006
Abstract
We construct in this paper a dynamic general equilibrium model
which displays the central features of the IS-LM model, and notably
an income multiplier greater than one, so that crowding out does not
occur. It appears that the key to this result is the conjunction of two
features of our model: price rigidities (as is usually expected), but also
a non-Ricardian economy.
∗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
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11 Introduction
If one looks at the history of macroeconomics, one sees that two paradigms
have dominated the profession. First, until the early seventies, the IS-LM
model (Keynes, 1936, Hicks, 1937), then the dynamic stochastic general equi-
librium (DSGE) paradigm (Lucas, 1972, Kydland and Prescott, 1982).
For a time many people believed that these two paradigms were totally
irreconcilable, notably due to the fact that the early DSGE models, then
called “real business cycles” (Long and Plosser, 1983), assumed full market
clearing, whereas a central feature of IS-LM models was wage and price
rigidities.
At some point, however, wage and price rigidities were introduced into
DSGE models, which narrowed the gap between the two paradigms. Some
authors further constructed DSGE models displaying some features of the
IS-LM model (see for example McCallum and Nelson, 1999).
The integration is not yet complete, however, as some important features
of the IS-LM model have not made fully their way into DSGE models. For
e x a m p l eo n eo ft h em o s te m b l e m a t i cf e a t u r e so fI S - L M ,a ni n c o m em u l t i p l i e r
greater than 1,h a sb e e nd i ﬃcult to obtain in DSGE models as there is usually
crowding out of consumption by government spending (see for example Fatas
and Mihov, 2001, and Collard and Dellas, 2005).
So the purpose of this paper in to construct a DSGE model which will
display the central features of the IS-LM model, and notably will allow to
obtain a multiplier larger than one.
2T h e m o d e l
It has been shown in Bénassy (2005) that in order to obtain sizeable multiplier
eﬀects it was important not only to have price or wage rigidities, but also to
consider non Ricardian economies. By non Ricardian economies we mean,
as in Barro (1974), economies where, due for example to the birth of new
agents over time, Ricardian equivalence does not hold.
2.1 An OLG economy
So we shall consider a particularly simple non Ricardian economy, a monetary
overlapping generations economy à la Samuelson (1958). In each period t
three markets are open: the goods market at price Pt,t h el a b o rm a r k e ta t
the wage Wt and the bonds market at the interest rate it. The private sector
consists of ﬁrms and households.
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12.2 The agents
The representative ﬁrm in period t has a production function:
Yt = ZtLt (1)
where Lt is employment and Zt a productivity shock.
Households live for two periods. Households born in period t work Lt,
consume C1t and hold a quantity of money M1t in period t.I np e r i o dt +1
they consume C2t+1 a n dh o l daq u a n t i t yo fm o n e yM2t+1. They maximize
the expected value of their utility Ut:
Ut = αLogV1t +( 1− α)Log V2t+1 −
Lt
a
(2)
with:
Vit = V
µ
Cit,
Mit
Pt
¶
i =1 ,2 (3)
where the function V is homogeneous of degree one in its two arguments and
Mit is money held at the end of the period. As an example we may have the
following C.E.S. speciﬁcation:
V
µ
Cit,
Mit
Pt
¶
=
"
C
(σ−1)/σ
it +
µ
Mit
θPt
¶(σ−1)/σ#σ/(σ−1)
(4)
where σ is the elasticity of substitution between consumption and money bal-
ances. Since we want to model the idea that money facilitates transactions,
and is thus somehow “complementary” to consumption, we shall assume that
σ<1. As an example we may note that for σ =0we obtain:
V
µ
Cit,
Mit
Pt
¶
= Min
µ
Cit,
Mit
θPt
¶
(5)
which gives rise to a demand for money Mit = θPtCit very akin to the tradi-
tional “cash in advance” speciﬁcation. For positive values of σ the demand
for money will be sensitive to the interest rate, as we shall see below.
2.3 Government policy
The government uses monetary and ﬁscal policy. Monetary policy consists
in choosing the interest rate it on the bonds market. We may note that,
although the interest rate is the policy tool, it may be manipulated so as to
stabilize a monetary aggregate, such as the quantity of money.
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1The second tool, ﬁscal policy, consists in choosing a level of government
spending Gt and taxes Tt in real terms. We shall assume that taxes are levied
on the young households, proportionately to their total income1 at the rate
τt.T h et a xr a t eτt is chosen so as to yield a real amount of taxes Tt,s ot h a t
we have:
τt (WtLt + Πt)=τtPtYt = PtTt (6)
2.4 Budget constraints
The old household starts period t with total ﬁnancial assets Ωt.H es p e n d s
PtC2t, and splits the rest between bonds B2t and money M2t:
M2t + B2t = Ωt − PtC2t (7)
If he lived through another period his wealth next period would be:
(1 + it)B2t + M2t =( 1+it)
µ
Ωt − PtC2t −
it
1+it
M2t
¶
(8)
But since he is not going to be alive next period, he wants this wealth to
be zero, so that his budget constraint is:
PtC2t +
it
1+it
M2t = Ωt (9)
Let us now consider the young household born in t. H es t a r t sl i f ew i t h
no assets, but receives wage income WtLt,p r o ﬁti n c o m eΠt = PtYt − WtLt,
both being taxed at the rate τt.H e a l s o s p e n d s PtC1t,s ot h a tt h es u mo f
money M1t and bonds B1t is:
M1t + B1t =( 1− τt)(WtLt + Πt) − PtC1t (10)
And his ﬁnancial holdings at the beginning of next period Ωt+1 will be:
Ωt+1 =( 1+it)B1t + M1t
=( 1+it)
·
(1 − τt)(WtLt + Πt) − PtC1t −
it
1+it
M1t
¸
(11)
1Proportional taxes are used because lump sum taxes create income eﬀects on labor
supply which make some equations look less "natural". But the central results are the
same.
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13 The demand for money and the LM curve
Budget constraints and maximization programs become particularly easy to
study if we use the intermediate variable “total spending” Sit deﬁned as:
Sit = PtCit +
it
1+it
Mit (12)
The second term represents the value of interest foregone because of hold-
ing money instead of bonds. From the preceding budget constraints we see
that the optimal choices of the households can be somehow broken in two:
T h ec h o i c eo fSit (i =1 ,2) and the allocation of this “total spending” Sit
between consumption and money. Let us start with this second problem: For
a given value of Sit the household, whether young or old, will maximize the
value of Vit, i.e. he will solve the following problem:
Maximize Vit = V
µ
Cit,
Mit
Pt
¶
s.t.
PtCit +
it
1+it
Mit = Sit
The ﬁrst order conditions are:
V2
V1
=
it
1+it
(13)
where V1 and V2 are the ﬁrst and second partial derivatives of the function
V . In view of the homogeneity of degree one of the function V , equation (13)
deﬁnes a “demand for money” for the agent, which has the following form:
Mit
PtCit
= Φ
µ
it
1+it
¶
Φ
0 < 0 (14)
As an example with the C.E.S. subutility function above (4) we obtain
the following demand for money:
Mit
PtCit
= θ
µ
θit
1+it
¶−σ
(15)
Now we obtain the following relation between total spending and con-
sumption:
Sit = PtCit +
it
1+it
Mit = ζtPtCit (16)
where ζt is deﬁned as:
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1ζt =1+
it
1+it
Φ
µ
it
1+it
¶
> 1 (17)
Let us now move to the aggregate values:
Ct = C1t + C2t Mt = M1t + M2t (18)
Aggregating the above relation (14) for young and old (i =1and 2), we
obtain the total demand for money:
Mt
PtCt
= Φ
µ
it
1+it
¶
Φ
0 < 0 (19)
In which, since it/(1 + it) is srictly increasing in it, we recognize a demand
for money very close to the traditional LM curve.
4 Walrasian equilibrium
Let us start with the old households, who enter period t with a ﬁnancial
wealth Ωt. Combining (9) and (16) we ﬁnd the consumption of the old:
C2t =
Ωt
ζtPt
(20)
The young household is subject to the intertemporal budget constraint
(11), which, in view of (16), becomes:
Ωt+1 =( 1− τt)(WtLt + Πt) − ζtPtC1t (21)
In the ﬁrst period of his life the household maximizes the expectation of
his utility (2) subject to the budget constraint (21). We may note, from (20),
that C2t+1 is proportional to Ωt+1. So the young household will solve:
Maximize αLogC1t +( 1− α)LogΩt+1 −
Lt
a
s.t.
Ωt+1 =( 1− τt)(WtLt + Πt) − ζtPtC1t
The ﬁrst order conditions for this program yield:
ζtPtC1t = α(1 − τt)(WtLt + Πt)=α(1 − τt)PtYt = α(PtYt − PtTt) (22)
Wt =
PtYt
a
(23)
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1Turning now to ﬁr m s ,t h er e a lw a g em u s tb ee q u a lt om a r g i n a lp r o d u c -
tivity:
Wt
Pt
= Zt (24)
Finally the equation of equilibrium on the goods market is:
Yt = Ct + Gt = C1t + C2t + Gt (25)
Combining equations (20), (22), (23), (24) and (25) we ﬁnd the values of
Walrasian output, employment, consumption and price:
Yt = aZt Lt = a (26)
Ct = aZt − Gt (27)
P
∗
t =
Ωt
(ζt − α)aZt − ζtGt + αTt
(28)
We note that an increase in government spending Gt decreases Ct by an
equal amount (equation 27), the traditional “crowding out” eﬀect.
5T h e I S - L M s y s t e m
Traditionally the IS-LM system is associated with price rigidities, wage rigidi-
ties, or both. So we shall look for equations that are valid in all these cases.
If wages are sticky equation (23) is not valid anymore. If prices are sticky
equation (24) is not valid. But equations (20), (22) and (25) are valid in all
circumstances. Combining them we obtain:
Yt =
1
ζt − α
µ
Ωt
Pt
+ ζtGt − αTt
¶
(29)
which looks very much like an IS curve. This has to be supplemented by the
LM curve, which we saw above (equation 19):
Mt
PtCt
= Φ
µ
it
1+it
¶
Φ
0 < 0 (30)
Equations (29) and (30) form a system that is highly similar to the IS−
LM system. We shall now investigate whether, as IS−LM,t h i ss y s t e mc a n
display a strong multiplier eﬀect.
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16 Preset prices and the multiplier
We shall now introduce a traditional price rigidity, preset prices in the tra-
dition of Gray (1976):
1
Pt
= Et−1
1
P∗
t
(31)
We still have to specify how monetary policy is operated. There are two
traditional polar ways, setting the interest rate or setting the money supply,
and we study them in turn.
6.1 Setting the interest rate
We ﬁrst assume that the government sets a given interest rate (it = i), letting
the quantity of money adjust endogenously. As a consequence ζt is constant
and equal to ζ, and the IS curve is rewritten:
Yt =
1
ζ − α
µ
Ωt
Pt
+ ζGt − αTt
¶
(32)
Combining (28), (31) and ζt = ζ we ﬁnd:
1
Pt
= Et−1
(ζ − α)aZt − ζGt + αTt
Ωt
(33)
Inserting this into (32) we obtain:
Yt = aEt−1Zt +
ζ
ζ − α
(Gt − Et−1Gt) −
α
ζ − α
(Tt − Et−1Tt) (34)
We see that the multiplier, ζ/(ζ − α),i sa l w a y sg r e a t e rt h a n1.
6.2 Setting the money supply
We now study how the analysis is modiﬁed when the government controls
the money supply instead of the interest rate. Let us recall the two IS−LM
equations:
Yt =
1
ζt − α
µ
Ωt
Pt
+ ζtGt − αTt
¶
(35)
Mt
PtCt
= Φ(jt) Φ
0 < 0 (36)
with:
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1ζt =1+jtΦ(jt) jt =
it
1+it
(37)
Since we are actually interested in the sign of the consumption response,
let us rewrite, deducting Gt from both sides, the IS curve as:
Ct =
1
ζt − α
µ
Ωt
Pt
+ αGt − αTt
¶
(38)
Let us diﬀerentiate (36) and (38), calling φ the absolute value of the
elasticity of Φ:
dCt
Ct
− φ
djt
jt
=0 (39)
dCt
Ct
=
αdGt
(ζt − α)Ct
−
dζt
ζt − α
(40)
Now we have from (37):
dζt = Φ(jt)(1− φ)djt (41)
Combining (37), (39), (40) and (41) we obtain:
dCt
dGt
=
αφ
φ(1 − α)+( ζt − 1)
(42)
We see that consumption is againa ni n c r e a s i n gf u n c t i o no fGt.T h em u l -
tiplier is an increasing function of φ.I n t h e l i m i t c a s e w h e r e φ =0(the
“cash in advance” case), the consumption does not move in response to a
government spending shock.
7 Partial price ﬂexibility
Let us now assume that the current price is given by:
1
Pt
= γEt−1
1
P∗
t
+( 1− γ)
1
P∗
t
(43)
For γ =0the price is fully ﬂexible, for γ =1we obtain the “Gray con-
tract” above. We shall study here the interest rate setting case. Combining
(28) and (43) we ﬁnd:
1
Pt
=( 1− γ)
(ζ − α)aZt − ζGt + αTt
Ωt
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1+γEt−1
(ζ − α)aZt − ζGt + αTt
Ωt
(44)
Inserting this into (32) we ﬁnd:
Yt =( 1− γ)aZt + γaEt−1Zt
+
γζ
ζ − α
(Gt − Et−1Gt) −
γα
ζ − α
(Tt − Et−1Tt) (45)
We see that the multiplier is greater than 1 if:
α>ζ(1 − γ) (46)
As should be expected from traditional Keynesian analysis, price ﬂexibil-
ity decreases the multiplier.
8C o n c l u s i o n
We saw in this article that by adequately combining price rigidities and a
non Ricardian framework we could obtain an income multiplier greater than
one, or in other words that an increase in government spending could lead to
an increase in private consumption. This was done in a dynamic stochastic
general equilibrium model with maximizing agents.
Although it can be greater than one, the multiplier is not always so, and
using a rigorous model allowed to specify the range of parameters for which
no crowding out occurs. We studied notably the degree of price rigidity, and
whether monetary policy is implemented by controlling interest rates or the
supply of money.
It is thus comforting to know that the two paradigms that dominated
macroeconomics for so long are, after all, not contradictory.
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1References
[1] Barro, Robert J. (1974), “Are government bonds net wealth?”, Journal
of Political Economy, vol. 82, pp 1095-1117.
[2] Bénassy, Jean-Pascal (2005), “Ricardian equivalence and the intertem-
poral Keynesian multiplier”, PSE and CEPREMAP, Paris.
[3] Collard, Fabrice and Harris Dellas (2005), “Poole in the new Keynesian
model”, European Economic Review, vol. 49, pp 887-907.
[4] Fatas, Antonio and Ilian Mihov (2001), “The eﬀects of ﬁscal policy on
consumption and employment: theory and evidence”, working paper,
INSEAD.
[5] Gray, Jo-Anna (1976), “Wage indexation: a macroeconomic approach”,
Journal of Monetary Economics, vol. 2, pp 221-235.
[6] Haavelmo, Trygve (1945), “Multiplier eﬀects of a balanced budget”,
Econometrica, vol. 11, pp 1-12.
[7] Hicks, John R. (1937), “Mr. Keynes and the “classics”: a suggested
interpretation”, Econometrica, vol. 5, pp 147-159.
[8] Keynes, John Maynard (1936), The General Theory of Employment,
Interest and Money. New York: Harcourt Brace.
[9] Kydland, Finn E. and Prescott, Edward (1982), “Time to build and
aggregate ﬂuctuations”, Econometrica, vol. 50, pp 1345-1370.
[10] Long, John B. and Charles I. Plosser (1983), “Real business cycles”,
Journal of Political Economy, vol. 91, pp 39-69.
[11] Lucas, Robert E. Jr (1972), “Expectations and the neutrality of money”,
Journal of Economic Theory,v o l .4 ,p p1 0 3 - 1 2 4 .
[12] McCallum, Bennett T. and Edward Nelson (1999), “An optimizing IS-
LM speciﬁcation for monetary policy and business cycle analysis”, Jour-
nal of Money, Credit and Banking, vol. 31, pp 296-316.
[13] Samuelson, Paul A. (1958), “An exact consumption-loan model of inter-
est with or without the social contrivance of money”, Journal of Political
Economy, vol. 66, pp 467-482.
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IS-LM and the multiplier: A dynamic general equilibrium model

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IS-LM and the multiplier: A dynamic general equilibrium model

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