Discussion Paper No. 779

A REINTERPRETATION
OF THE KEYNESIANCONSUMPTION FUNCTION
AND MULTIPLIER EFFECT
Ryu-ichiro Murota
Yoshiyasu Ono

June 2010

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, JapanA Reinterpretation of the KeynesianConsumption Function and Multiplier Eﬀect
Ryu-ichiro Murota∗ and Yoshiyasu Ono†
June 3, 2010

Abstract
We propose a microeconomic foundation of the multiplier eﬀect and that of
the consumption function using a dynamic optimization model that explainsa shortage of aggregate demand and unemployment. We show that government purchases boost aggregate demand through a multiplier-like process
but that the implication is quite diﬀerent. It works through not an increasein disposable income but moderation of deﬂation, which makes money holding costly and stimulates consumption.

Keywords: Keynesian Consumption Function, Multiplier Eﬀect, KeynesianCross, Persistent Unemployment, Aggregate Demand

JEL Classiﬁcation Numbers: E12, E24, E62∗

Address: Faculty of Economics, Kinki University, 3-4-1 Kowakae, Higashi-Osaka, Osaka 577-8502, Japan. E-mail: murota@eco.kindai.ac.jp.
†Address: Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka,Ibaraki, Osaka 567-0047, Japan. E-mail: ono@iser.osaka-u.ac.jp. Ono’s research is ﬁnancially supported by the Grants-in-Aid for Scientiﬁc Research, JSPS.

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1Introduction

The United States, the European Union and Japan have suﬀered from serious
economic depression and unemployment since the recent worldwide ﬁnancial
crisis of 2008. They expand ﬁscal spending so as to stimulate aggregate demand and reduce unemployment.1 When doing so, they mostly have in mind
the Keynesian multiplier theory. However, the theory is criticized for lack ofa microfoundation of the Keynesian consumption function upon which thetheory is founded. In particular, the intertemporal budget equation is ignored
when the consumption function is assumed. New Keynesian economists use
general equilibrium models with imperfect competition and present somemicroeconomic foundations for the multiplier eﬀect. See Costa and Dixon(2009) for a recent survey on those models. However, they consider neithera shortage of aggregate demand nor involuntary unemployment. Full employment always obtains and an increase in labor supply that households
determine by comparing utility of consumption and that of leisure plays akey role in creating the multiplier eﬀect.
This paper reformulates the dynamic optimization model of Ono (1994,2001) and proposes an alternative Keynesian cross model with demand shortage and involuntary unemployment. We obtain consumption as a functionof aggregate demand and obtain the eﬀect of government spending on aggregate demand that looks like the Keynesian multiplier eﬀect. However, thisconsumption function implies not the Keynesian relationship between con1Feldstein (2009) states that the recent revival of interest in ﬁscal stimulus is due todiﬀerences between the current stagnation and previous recessions and to ineﬀectivenessof monetary policy under the current stagnation.

2sumption and disposable income but the eﬀect of an increase in aggregate
demand on consumption that works through a change in the deﬂation rate.
Since the intertemporal budget equation is taken into account in this model,
Ricardian equivalence holds and the multiplier of a tax cut with bond issuance is zero. The multiplier eﬀect works only when ﬁscal spending creates
new employment and the deﬂationary gap shrinks.

2The Model

Let us ﬁrst summarize the model of Ono (1994, 2001). Since it uses theSidrauski-type money-in-the-utility-function model, the ﬁrst-order optimal
condition isρ + η(ct )v 0 (mt )ċt+ π t = Rt = 0,
ctu (ct )(1)

where ρ (> 0) is the subjective discount rate, πt is the inﬂation rate, Rt is thenominal interest rate of bonds, u(ct ) is utility of consumption ct , v(mt ) is utility of real money balances mt and η(ct ) ≡ −[u00 (ct )ct ]/u0 (ct ). u(ct ) and v(mt )satisfy normal properties. The ﬁrst equality implies the Ramsey equationand the second one shows portfolio choice between bonds and money.
The ﬁrm sector is assumed to have linear technology:
yt = θnt ,(2)where yt is output, θ (> 0) the labor productivity and nt labor input. Given
nominal wage Wt and nominal commodity price Pt , the ﬁrm sector chooses
labor demand ndt so as to maximize proﬁts. As far as nt is ﬁnite and positive,
proﬁt maximizing behavior under the linear technology leads to()Wtθ = wt ≡for all t,
Pt3(3)

where wt denotes the real wage.
The government ﬁnances government purchases g and interest payments
rt bt , where rt is the real interest rate on government bonds bt , by imposing
lump-sum tax-cum-subsidy τt and issuing new bonds ḃt . Thus,g + rt bt = τt + ḃt .
It adjusts bt and τt so that the non-Ponzi game condition is satisﬁed. Nominal
money supply Mt is kept constant at M and hence real money balances
mt (= M /Pt ) followṁt= −πt ,mt(4)where πt (≡ Ṗt /Pt ) is the inﬂation rate.
Since we take into account the possibility of unemployment, employment
nt is determined by the short side of labor demand ndt and inelastic labor
supply n:}{nt = min ndt , n .
Nominal wage Wt is assumed to adjust in a sluggish manner:Ẇt=αWt( d)nt−1 ,nwhere α (> 0) is exogenous and constant.2 Meanwhile, commodity price Pt
instantaneously adjusts so as to satisfy (3) and realize the commodity market
2

Although how nominal or real wages adjust is an important issue that many economists
have long addressed, we assume a simple adjustment process because our purpose is not to
examine why wages are rigid but to analyze how government spending aﬀects GDP understagnation. Using the idea of the fair wage, Ono and Ishida (2009) give a microfoundation
to such an adjustment process, and show that under the microfounded adjustment process
an economy also reaches a steady state of which properties are similar to those of theunemployment steady state of the present paper.

4equilibrium:
ct + g = yt .

(5)

Since πt = Ẇt /Wt from (3) and yt = θnt from (2), we have
( d)
()
nt
yt
πt = α−1 =α−1 ,ny(6)where y denotes full-employment output:y ≡ θn.

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The Consumption Function and the Multiplier Eﬀect

This section derives a relationship between aggregate demand y and consumption c that looks like the Keynesian consumption function. Using it we
propose an analytical framework similar to the Keynesian cross. A multiplierlike eﬀect of government purchases on aggregate demand arises but the economic implication is quite diﬀerent.
In the full employment steady state (n = nd = n), if it exists, ċ = 0 and
y = y. Thus, from (1), (4), (5) and (6) we ﬁnd
π = 0, ṁ = 0, c + g = y, ρ =v 0 (m).
u0 (c)(7)

However, if the marginal utility of money has a positive lower bound β:3
lim v 0 (m) = β > 0m→∞
3Ono (1994, chapter 1) mentions the validity of the assumption of insatiable liquidity
preference, quoting the statements by Karl Marx and Georg Simmel. Based on recent
ﬁndings in neuroscience, Ono and Ishida (2009) also discuss the validity. Using bothparametric and non-parametric methods, Ono, Ogawa and Yoshida (2004) empiricallysupport the assumption. Murota and Ono (2008) show that the marginal utility of money
stays positive in the presence of status preference, and Murota and Ono (2009) demonstratethat it reaches a positive lower bound under zero nominal interest rates if liquidity ofdeposits is considered.

5and β is high enough to satisfy
ρ<βu0 (y − g),(8)then there is no value of m that satisﬁes (7) and hence the full employment
steady state does not exist.4 The second equality of (1), R = v 0 (m)/u0 (c),implies that this is the case of the Keynesian liquidity trap, where money
demand m is inﬁnite when the nominal interest rate R takes a positive lower
bound β/u0 (c).
Under (8) the marginal beneﬁt of money (the liquidity premium) exceedsthat of consumption (the time preference rate ρ) if consumption c takes thefull-employment level y − g. In order for the ﬁrst order condition (1) to besatisﬁed, c is set to be lower than y − g and then unemployment (n < n)and deﬂation (π < 0) occur. Hence, the economy reaches an unemployment
steady state where ċ = 0, y < y, and ṁ/m > 0, which causes v 0 (m) = β.
Then, (1) and (6) lead to(ρ+α)
yβ−1 = 0 ,yu (c)(9)

which gives the consumption function in the present model:c = u0−1 α  ≡ c(y).
ρ−α+ yy(10)[u0 (c)]2 α> 0.
u00 (c)βy(11)βFrom (10) we obtain5c0 (y) = −4This condition for the non-existence of the full-employment steady state is the sameas in Ono (1994, 2001).
5If u(c) is a logarithmic function, c0 (y) is constant.

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As is proven by Ono (2001), the unemployment steady state uniquely exists
and satisﬁes saddle-path stability under the following conditions:ρ > α, −βu00 (c) α− > 0,y[u0 (c)]2

and then from (10) and (11) c(y) satisﬁes()
β0 −1c(0) = u> 0, 1 > c0 (y) > 0.
ρ−α(12)

These properties look the same as those of the Keynesian consumption function but the implications are quite diﬀerent. Neither does c(0) mean autonomous consumption nor does c0 (y) mean the marginal propensity to consume. It means an eﬀect that works through a change in deﬂation, and hence
consumption c depends on not disposable income y−τ but aggregate demandy.
Substituting (10) into (5) yields
c(y) + g = y.

(13)

Note that (13) gives a unique value of y satisfying 0 < y < y because (8)implies c(y) + g < y and (12) is valid. This leads to seemingly the same
multiplier eﬀect as the conventional Keynesian one but the present eﬀect
works very diﬀerently. An increase in g by the magnitude of dg initiallyexpands y by dg, which moderates deﬂation and causes the household toincrease c by c0 (y)dg. It further expands y by c0 (y)dg, which again moderates
deﬂation and increases c and y by [c0 (y)]2 dg. This process continues andeventually6
dcc0 (y)
dy
1=,=.
dg
1 − c0 (y) dg
1 − c0 (y)6Blanchard and Perotti (2002) ﬁnd that an increase in government purchases leads toan increase in consumption. See Galı́ et al. (2007) for a similar ﬁnding.

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Since the intertemporal budget equation is taken into account, Ricardianequivalence holds in the present model.7 Therefore, there is no diﬀerence inthe multiplier eﬀect between under a balanced budget and a deﬁcit budget.
Government purchases g increase consumption c even under a balanced budget. Lump-sum tax-cum-subsidy τ has any eﬀect on neither consumption
nor aggregate demand even under a deﬁcit budget.8
If wage adjustment speed α is low, the deﬂation rate is less sensitive tothe output gap and therefore the eﬀect on c of an increase in g is small.
Moreover, if α = 0 and hence prices and wages are ﬁxed (i.e., π = 0), from(10) and (13) c is independent of y and is not aﬀected by g. Thus, in thetypical Keynesian case with ﬁxed prices and wages, ﬁscal expansion does not
aﬀect consumption. This property holds true whether a liquidity trap occurs
or not since from (1) where π = 0 and P is constant at P̄ one ﬁnds
ρ=v 0 (M/P̄ ),u0 (c)

implying that c is constant. Although the stimulative eﬀect of government
purchases g decreases as α declines, a lower α leads to larger consumption
and aggregate demand for given g. This property is derived from (10) and7

Our multiplier eﬀect is also very diﬀerent from those discussed by Bénassy (2007a,2007b) and Galı́ et al. (2007). They assume non-Ricardian frameworks with price rigidities. Bénassy uses overlapping generations models, and Galı́ et al. assume rule-of-thumbconsumers, who do not take intertemporal decisions.
8This may be consistent with the consequence of the tax rebate implemented by the US
government in 2008. Feldstein (2009) mentions that consumption in the second quarter of2008 increased by only 12 billion dollar although 80 billion dollar went back to taxpayers
in May and June 2008, and mentions that the estimated marginal propensity to consume
from the corresponding rebate variable is only 0.13 whereas that out of real per capita
disposable income is 0.70. Similarly, Shapiro and Slemrod (2009) report that only one-ﬁfth
of the recipients of the tax rebate planed to mostly spend the rebate and the rest planed
to mostly save it or to mostly pay oﬀ debt with it.

8(13):dcdy[u0 (c)]2 (y − y)
== 00
< 0.
dα
dα
u (c)βy[1 − c0 (y)]Note that the present multiplier analysis holds only in the unemployment
steady state given by (9). If (8) is not valid and the economy is not in theliquidity trap, the second equality of (1) yields consumption c as a functionof m and R:
c = φ(m, R),
and the real balance eﬀect works. A decrease in P expands m and hence
increases c until full employment is reached and (7) is satisﬁed.

4Conclusion

From a dynamic optimization monetary model with a liquidity trap we derive a consumption function and a multiplier eﬀect of ﬁscal expansion on
aggregate demand. In the steady state with demand shortage and deﬂation consumption is a function of aggregate demand and aggregate demandis determined in a way similar to the Keynesian cross. However, workingof the eﬀect is quite diﬀerent since the consumption function does not represent the relationship between disposable income and consumption, as in
the conventional Keynesian model. An increase in government purchases expands consumption by mitigating deﬂation. This eﬀect arises whether undera deﬁcit budget or a balanced budget since households take into account theintertemporal budget equation.

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