Discussion Paper No. 937

FISCAL POLICYUNDER LONG-RUN STAGNATION:
A NEW INTERPRETATION
OF THE MULTIPLIER EFFECT

Ryu-ichiro Murota
Yoshiyasu Ono

May 2015

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Fiscal Policy under Long-run Stagnation:
A New Interpretation of the Multiplier Eﬀect∗
byRyu-ichiro Murota†
Faculty of Economics, Kinki University
andYoshiyasu Ono‡
Institute of Social and Economic Research, Osaka University
May 20, 2015∗

This is a significantly revised version of ISER Discussion Paper, No. 779, titled “A
Reinterpretation of the Keynesian Consumption Function and Multiplier Eﬀect”. An
earlier version of this paper was presented at the third Graz Schumpeter Summer School
2011 held in the University of Graz, a seminar in Tokyo Institute of Technology andthe conference on Economic Analysis of Dynamics and Preferences. Helpful commentsfrom H. Kurz and the participants of the three seminars are gratefully acknowledged. We
are grateful to financial supports from the Joint Usage/Research Center, the Ministryof Education, Culture, Sports, Science and Technology. In addition, Ono’s research isfinancially supported by the Grants-in-Aid for Scientific Research, the Japan Society for
the Promotion of Science.
†
Corresponding author. Address: Faculty of Economics, Kinki University, 3-4-1
Kowakae, Higashi-Osaka, Osaka 577-8502, Japan. E-mail: murota@eco.kindai.ac.jp. Tel:
+81-6-4307-3213.
‡
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.

1Abstract
We develop a Keynesian cross analysis with a dynamic optimization settingthat explains long-run stagnation caused by aggregate demand deﬁciency.
We show that an increase in government purchases boosts GDP through amultiplier process, but the implication is quite diﬀerent from the conventional Keynesian one. It works not through an increase in disposable income
but through moderation of deﬂation. Thus, countries that have lapsed into
long-run stagnation should expand government spending that directly creates employment in order to reduce the deﬂationary gap.

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

JEL Classiﬁcation Codes: E12, E24, E622

1Introduction

When countries fall into economic depression, their governments tend to increase spending in order to expand aggregate demand and reduce unemployment. In this context policy makers mostly have the Keynesian multiplier
theory in mind. However, this theory has been criticized because the Keynesian consumption function lacks microeconomic foundations. In response,
many economists have analyzed the multiplier eﬀect in various frameworks
with microeconomic foundations. Recently, in particular, it has actively beenstudied because of the Great Recession. For example, using a New Keynesian DSGE model with no unemployment, Christiano et al. (2011) ﬁnd thata large multiplier eﬀect appears under zero interest rates. Eggertsson andKrugman (2012) analyze a short-run deﬁciency of aggregate demand due to aborrowing constraint and show the existence of the multiplier eﬀect. Monacelli et al. (2010) examine the multiplier eﬀect in the presence of unemployment arising because of matching frictions, not aggregate demand deﬁciency.1
In contrast with these studies, we analyze the multiplier eﬀect that appears
when aggregate demand deﬁciency occurs and creates unemployment in thelong run.
Recently, long-run stagnation caused by deﬁciency of aggregate demand
has attracted attention from economists. This is known as “secular stagna1

Empirical studies have also been expanding and various results have been obtained.
For instance, Ilzetzki et al. (2013) ﬁnd that the magnitude of the multiplier depends
on the degree of development, openness to trade and so on. Jha et al. (2014) conclude
that in developing Asia tax cuts may be more eﬀective as a countercyclical policy thangovernment spending increases. According to Hong and Li (2015), the multipliers of public
works investment and consumption vouchers implemented in Taiwan were 1.94 and 1.47,
respectively.

3

tion”, originally advocated by Alvin Hansen and recently revived by Lawrence
Summers (see, e.g., Eggertsson and Mehrotra (2014) for details). Summers(2014) considers the US economy since the Lehman shock to be in secularstagnation, and suggests that increasing aggregate demand is a way of boosting the economy. Several economists have attempted to theoretically model
secular stagnation. For example, Eggertsson and Mehrotra (2014) develop
an overlapping generations model with a borrowing constraint and show thata persistent deleveraging shock leads to a persistent liquidity trap where aggregate demand deﬁciency and unemployment occur. Moreover, they showthat the borrowing constraint, which makes Ricardian equivalence invalid,
yields a large multiplier eﬀect when government spending is ﬁnanced by issuing bonds. Michaillat and Saez (2014) construct a job search model where
wealth holdings yield direct utility. In their model the marginal utility of
wealth becomes constant, which plays a crucial role in creating persistentstagnation.
However, prior to these studies, Ono (1994, 2001) presents long-run stagnation in a dynamic general equilibrium model with optimizing agents.2 Heshows that if a desire to save money is insatiable (i.e., the marginal utility ofmoney stays positive), aggregate demand deﬁciency and involuntary unem2

Recently, Ono’s model has been extended in various analyses. For example, Matsuzaki(2003) and Hashimoto (2004) consider heterogeneous agents in the model and explore theeﬀects of redistribution. Johdo (2006) combines the model with a spatial model and investigates the relationship between geographical space and stagnation. Johdo (2008a)
introduces monopolistic competition into the model and analyzes the eﬀects of production
subsidies. Johdo (2009) incorporates habit formation into the model and examines therelationship between habit formation and stagnation. Ono (2006, 2014), Johdo (2008b),
Johdo and Hashimoto (2009) and Hashimoto (2011) extend the model to open-economy
models and examine the international spill-over eﬀects of various macro- and microeconomic policies. Using the model, Hashimoto and Ono (2011) study pro-population policies.

4ployment arise in the steady state. Thus, the approach of Michaillat and Saez(2014) is somewhat similar to that of Ono (1994, 2001). Furthermore, Ono(2010) discusses the mechanism of Japan’s long-lasting stagnation since theearly 1990s in his framework. Using a similar model, Murota and Ono (2012)
comprehensively explain various phenomena observed in the Great Depression and Japan’s stagnation, including involuntary unemployment, deﬂation,
zero interest rates and excess bank reserves.
This paper explores what ﬁscal policy is eﬀective for stimulating an economy falling into such long-run stagnation. For this purpose, we examine
the multiplier eﬀect in the framework of Ono (1994, 2001). Whereas Ono(1994, 2001) does not consider the multiplier eﬀect, we derive a consumptionfunction from household optimizing behavior and establish an alternative
Keynesian cross model. In the model, an increase in government purchases
aﬀects consumption and GDP through a multiplier-like process, but a tax
cut (or a transfer increase) has no eﬀect on either of them. The inﬂuences ofvarious parameters such as liquidity preference, potential output and wage
ﬂexibility on the magnitude of the multiplier are also investigated.
The multiplier analysis in this paper is quite diﬀerent from the conventional Keynesian or New Keynesian models in the following respects.
First, it considers persistent unemployment resulting from aggregate demanddeﬁciency. Second, our consumption function represents not the conventional Keynesian relationship between disposable income and consumption
but the eﬀect of an increase in output on consumption through mitigationof deﬂation—i.e., an increase in actual output relative to potential output
narrows the deﬂationary gap and mitigates deﬂation, which makes holding5money more costly and thereby stimulates consumption. Third, in contrast
with Eggertsson and Krugman (2012) and Eggertsson and Mehrotra (2014),
Ricardian equivalence holds in this paper and yet the multiplier eﬀect ofgovernment purchases arises.3 The magnitude of the multiplier eﬀect is independent of the means of ﬁnancing: collecting taxes or issuing bonds. Finally,
since our consumption function is founded on household optimizing behavior, changes in technology and preference parameters aﬀect the form of theconsumption function and vary the magnitude of the multiplier eﬀect.

2

The Model

We start with a brief summary of the model, which is based on Ono (1994,
2001). The government ﬁnances government purchases g and interest payments rt bt , where rt is the real interest rate on government bonds bt , bycollecting a lump-sum tax τt and issuing new bonds ḃt . Thus we haveg + rt bt = τt + ḃt ,where τt denotes a lump-sum transfer if it is negative. Note that bt and τt
are adjusted so that the no-Ponzi-game condition is satisﬁed. The nominal
money supply Mt is kept constant at M , for simplicity, and hence the rateof change in real money balances mt (= M /Pt ), where Pt is the commodity
price, is given byṁt= −πt ,
mt
3

(1)

Using overlapping generations models, Bénassy (2007a, b) argues that non-Ricardianequivalence is important for the appearance of the multiplier eﬀect. Galı́ et al. (2007) develop a New Keynesian model with non-Ricardian consumers, and show that the presenceof such consumer causes the multiplier eﬀect to arise.

6

where πt (≡ Ṗt /Pt ) is the inﬂation rate.
The household sector maximizes the following lifetime utility:∫ ∞[u(ct ) + v(mt )] exp(−ρt)dt,
0

subject toȧt = rt at + wt nt − ct − Rt mt − τt ,where u(ct ) is the utility of consumption ct , v(mt ) is the utility of real money
holdings mt , ρ is the subjective discount rate, at (= bt + mt ) is real total
assets, wt is the real wage and Rt (= rt + πt ) is the nominal interest rate.
As usual, we assume that the ﬁrst derivatives of u(ct ) and v(mt ) are positive
and that the second derivatives are negative. The household inelasticallysupplies its labor endowment n. However, as shown below, it may not be
fully employed. Therefore, employment nt is given by the short side of labordemand ndt and labor supply n:{}
nt = min ndt , n .

(2)The optimality condition for this utility-maximization problem isρ + η(ct )ċtv ′ (mt )+ πt = Rt = ′,ct
u (ct )

(3)

where η(ct ) ≡ −[u′′ (ct )ct ]/u′ (ct ). The ﬁrst equality in (3) indicates the Ramsey equation and the second implies portfolio choice between bonds andmoney.
While the commodity price Pt is perfectly ﬂexible, the adjustment of thenominal wage Wt is assumed to be sluggish as follows:
( d
)
nt
Ẇt
=α
−1 ,
Wt
n
7

(4)

where α (> 0) denotes ﬂexibility of the adjustment, in order to take into
account the possibility of unemployment due to demand deﬁciency. See Ono
and Ishida (2014) for a microeconomic foundation of this wage adjustment.4
It is noteworthy that recently studied Phillips curves, such as the New Classical Phillips curve, the New Keynesian Phillips curve and the hybrid of
the forward- and backward-looking Phillips curves, are not appropriate forthe analysis of persistent stagnation due to aggregate demand deﬁciency
because the possibility of market disequilibrium is not allowed from the beginning or because the inﬂation–deﬂation rate cumulatively expands as longas market disequilibrium exists.5 Thus, the possibility of unemployment ina steady state, which we focus on, is intrinsically eliminated under these
Phillips curves.
The ﬁrm sector has linear technology:
yt = θnt ,(5)

where yt is output, θ is labor productivity, which is constant, and nt is labor
input. Since the production function is linear in labor, the ﬁrm sector decides

4

They apply various fairness concepts to the mechanism of nominal wage setting andobtain nominal wage movements that depend on the unemployment rate if unemployment
exists. They ﬁrst obtain the dynamics of fair wages and ﬁnd that with unemployment, ﬁrms
set wages to be the same as the fair wages so as to urge their employees to work eﬃciently.
In this setting 1/α is the average duration of employment because wage adjustments are
due to alternation of incumbent workers, whose fair wages depend on their past and theirrivals’ wages, by new recruits who have no preconceptions about fair wages.
5
See Woodford (2003) for the properties of these Phillips curves.

8

labor demand ndt and commodity supply yt to bendt = ∞, yt = ∞if

θ > Wt /Pt ,0 < ndt < ∞, 0 < yt < ∞if

θ = Wt /Pt ,ndt = 0, yt = 0

if

θ < Wt /Pt .

Since Wt can only adjust sluggishly according to (4) while Pt is ﬂexible, Pt
instantaneously rises if θ < Wt /Pt because commodity supply is zero, anddrops if θ > Wt /Pt because ﬁrms try to expand their shares by undercutting
the price. Consequently, Pt takes the following value:6
θ=Wt(≡ wt ) ,
Pt

(6)

Ẇt
.
Wt

(7)

which straightforwardly implies
πt =When (6) holds, yt satisﬁes
ct + g = yt = θnt ,(8)

where nt is a positive ﬁnite value satisfying (2): nt = min{ndt , n}.
In the following analysis we focus on the case where unemployment occurs,i.e., nt = ndt < n. In this case, (4), (5) and (7) yield
(
)
(n
)
Ẇt
yt
t
πt =
=α
−1 =α
−1 ,
Wt
n
y

(9)

where y denotes full-employment (or potential) output:y ≡ θn.
6

From (6), the proﬁts and the ﬁrm value are zero, which implies that the household
sector’s total assets at consist of only mt and bt , as mentioned about household behavior.

9Note that the dynamic behavior of the economy is characterized by diﬀerential equations for consumption and real money balances.7

3

The Consumption Function and the Multiplier Eﬀect

In this section, we ﬁrst show long-run stagnation with aggregate demand deﬁciency and unemployment and then propose a new analysis of the multipliereﬀect.

3.1

The Stagnation Steady State

Following Ono (1994, 2001), we assume that the marginal utility of money
has a positive lower bound β:
lim v ′ (m) = β > 0,

m→∞

(10)

which creates the Keynesian liquidity trap. In fact, from the second equalityof (3):
R = v ′ (m)/u′ (c),we ﬁnd that R approaches a positive lower bound β/u′ (c) even when money
demand m diverges to inﬁnity.
7

From (1), (3), (8) and (9), we obtain ċt and ṁt as functions of ct and mt , respectively:[()]
ct + gv ′ (mt )ct−α−1 + ′−ρ ,ċt =η(ct )yu (ct )()
ct + gṁt = −αmt
−1 ,
y

∞
which yields {ct }∞t=0 and {mt }t=0 and consequently the sequences of all variables, including()
ct + gv ′ (mt ), πt = α− 1 , rt = Rt − πt .
Rt = ′u (ct )y

See in detail Ono (1994, 2001) for the present dynamics.

10

When a money-in-the-utility-function model is adopted,8 it is almost always assumed that the marginal utility of money eventually decreases to zero
as money holdings increase.9 However, Ono et al. (2004) empirically ﬁndthat the assumption (10) is better supported than the zero lower bound. Theoretically, Murota and Ono (2011) show that it remains positive if money isa status symbol, and Murota and Ono (2012) show that it reaches a positive
lower bound when nominal interest rates are zero in a model that incorporates both money and deposits into a utility function. Camerer et al. (2005)
mention the possibility that the utility of money has little association with
consumption. If this is true, it may be possible that the marginal utility ofmoney, in contrast to that of consumption, does not decline to zero. Furthermore, Michaillat and Saez (2014) consider the case where the marginalutility of wealth consisting of money and bonds always becomes positively
constant in a model where accumulating wealth is an end in itself or a wayof gaining social status.
We further assume that y is so large, ρ is so small or β is so large as tosatisfyρ<βu′ (y − g)

,

(11)

which implies that from (3) the marginal beneﬁt of money (the liquidity
premium) exceeds that of consumption (the time preference rate ρ) if steady8

The dominant view of money in contemporary economics is that people do not derive
utility directly from money. However, Camerer et al. (2004, 2005) argue that money may
directly provide utility on the ground of neuroscientiﬁc evidence that money and various
reinforcers, i.e., attractive faces, funny cartoons, cultural objects and drugs, activate thesame dopaminergic reward circuitry of the brain.
9
Devoe et al. (2013) present evidence that may conﬂict with the assumption of thedecreasing marginal utility of money. They ﬁnd that individuals who earn more moneyfrom labor view money as more important.

11

state consumption c is at the full-employment level y − g. Therefore, in orderfor the optimality condition (3) to be satisﬁed, c must be set to be lower thany − g, which leads to n = nd < n and π < 0 from (2), (8) and (9). Intuitively,
the household prefers saving money rather than consuming enough to attainfull employment, which gives rise to aggregate demand deﬁciency, involuntary
unemployment and deﬂation. In this case, from (1), (2), (3), (8) and (9), we
obtain the following stagnation steady state:(c + g = y < y, n = nd < n, π = α)y− 1 < 0,yβṁ= −π > 0, ρ + π = R = ′ , r = ρ.
m
u (c)

(12)

From the ﬁrst, third and ﬁfth properties of (12), steady-state consumption
c satisﬁes(
Φ(c) ≡ ρ + α)c+gβ−1 − ′
= 0,
y
u (c)

as shown by Ono (1994, 2001). From (11), one obtains
Φ(y − g) = ρ −βu′ (y − g)

< 0.

Therefore, in order that c uniquely exists within (0, y − g), it must be valid
that

(

)
g
Φ(0) = ρ + α− 1 ≥ ρ − α > 0,yα βu′′ (c)Φ′ (c) = +< 0,y[u′ (c)]2(13)

where the condition that ρ − α > 0 is required so that Φ(0) > 0 even wheng = 0. Given c, we uniquely obtain y and all the other endogenous variables.
In what follows we present a new interpretation of the multiplier eﬀect,
which is not examined by Ono (1994, 2001). Consumption c is given as a12

function of output y, as in the Keynesian consumption function, and thedetermination of GDP (output y) is expressed in an alternative way, as in
the 45-degree diagram. An increase in government purchases boosts GDP
through a new multiplier process, and the magnitude of the multiplier isinﬂuenced by various parameters.

3.2

The Consumption Function and Keynesian Cross

We begin by expressing c as a function of y. From the third and ﬁfth properties of (12), we obtain
(
ρ+α

)
y
β
−1 = ′ ,
y
u (c)

(14)

which gives the consumption function in the present model:
c = u′−1 
α  ≡ c(y; α, y, β).
ρ−α+ yyβ(15)In (15) c does not depend on m, implying that the Pigou eﬀect does notwork when the economy is caught in the above-mentioned liquidity trap.
This is the cause of the persistent aggregate demand deﬁciency. In contrast,in the usual money-in-the-utility-function model that does not consider theliquidity trap, from the second equality of (3) consumption is represented asa function of real money holdings and the nominal interest rate:c = ϕ(m, R).
Therefore, the Pigou eﬀect works:∂cu′ (c)v ′′ (m)
= ′′
> 0,
∂m
u (c)v ′ (m)13and c eventually reaches the full-employment level as deﬂation continues andm expands.
Diﬀerentiating (15) with respect to y yields∂c(y; α, y, β)α [u′ (c)]2
cy ≡
=−
> 0.
∂y
βyu′′ (c)(16)

From (13), (15) and (16), one obtains′ −1(c(0; α, y, β) = u

β
ρ−α

)
> 0, 0 < cy < 1.

(17)

These properties are mathematically the same as those of the Keynesian
consumption function. If u(c) = log c, for example, the consumption function
represented by (15) becomes linear:c = c0 + cy y, where c0 =αρ−αand cy =
,
β
βy

(18)

which indeed looks like the textbook consumption function.
The implications of the properties given in (17) are, however, quite different from the conventional ones. c(0; α, y, β) does not imply autonomous
consumption, nor does cy represent the marginal propensity to consume.
c(0; α, y, β) is simply the magnitude of consumption when y = 0 (and thenπ = −α from the third property of (12)). cy indicates the eﬀect on consumption c of an increase in output y through mitigation of deﬂation. An increasein output requires an increase in employment, which mitigates deﬂations in
the nominal wage and price, as is clear from (9). This makes holding money
more costly and thereby stimulates consumption (dc/dπ = −(u′ )2 /(βu′′ ) > 0
from the ﬁfth property of (12)). Thus, consumption c depends not on disposable income y − τ but rather on output y.
14

The consumption function (15), which satisﬁes the properties in (17), isvalid only under long-run stagnation in which aggregate demand deﬁciency
and involuntary unemployment persist. In fact, the Keynesian consumptionfunction was observed during the 1930s (the Great Depression) in the US(see, e.g., Davis, 1952; Shapiro, 1988; Emerson, 2011). Meanwhile, whenfull-employment output is realized (y = y), we have the following linear
relationship:c = y − g,where its slope equals one (dc/dy = 1) and its intercept takes a negativevalue. Shapiro (1988) and Emerson (2011) ﬁnd these properties from the US
data in the period after World War II.
Using the consumption function (15), we present an analysis similar to
the Keynesian cross. By substituting (15) into the ﬁrst property of (12), we
obtainc(y; α, y, β) + g = y,(19)where the left-hand side (LHS) denotes aggregate demand and the right-handside (RHS) denotes aggregate supply. (11), (15) and (17) imply thatc(0; α, y, β) + g > 0, c(y; α, y, β) + g < y,and that the LHS of (19) has a positive slope less than one. Therefore,
steady-state GDP is uniquely determined. This is illustrated in ﬁgure 1,
where it is given by y ∗ . Note that the ﬁgure presents the case of such a linear
consumption function as (18)As shown in the ﬁgure, GDP is determined mathematically in the samemanner as the conventional Keynesian cross. However, the positively sloped
15

consumption function does not imply the Keynesian income eﬀect on consumption. In the present framework y is output rather than disposable income. As output y increases, the deﬂationary gap shrinks and deﬂation
declines. This decline in deﬂation stimulates household consumption, whichleads to an increase in aggregate demand. We will discuss the multipliereﬀect generated by this process in the next subsection.

3.3

The Multiplier Eﬀect

From (19), we obtain seemingly the same multiplier eﬀect as that of theconventional Keynesian model:
dy
1
dc
cy
=
> 1,
=
> 0.
dg
1 − cydg
1 − cy(20)

However, the multiplier process substantially diﬀers from the conventional
one.10 An increase in government purchases g by dg initially boosts output yby dg. It reduces the deﬂationary gap and moderates deﬂation, which urges
households to increase consumption c by cy dg. The increase in c additionally boosts y by cy dg, which again moderates deﬂation and increases c by(cy )2 dg.11 Such interactions between the moderation of deﬂation and theincrease in consumption repeatedly occur, cumulatively increasing consumption and output, and eventually leading to (20).12
10

Ono (2011) discusses the implication of the conventional multiplier eﬀect and argues
that the multiplier eﬀect of a ﬁscal expansion may be seriously misunderstood. He showsthat even in the conventional Keynesian framework, the true eﬀect of ﬁscal spending
depends not on the amount of spending but on the beneﬁt directly generated by the
spending.
11
Note that this is not the actual adjustment process over time but the conceptual process, as is the case of the conventional multiplier eﬀect. The economy, in fact, immediately
jumps to a new steady state when g unexpectedly changes in the stagnation steady state.
12
Mankiw (1988) also obtains a multiplier eﬀect that is mathematically similar to theconventional Keynesian one and explicitly shows the multiplier process in a general equi-16In the present model, Ricardian equivalence holds and hence the magnitude of the multiplier eﬀect does not depend on the means of ﬁnancing:issuing government bonds or collecting the lump-sum tax τ . This is clear
from the consumption function (15), where c does not depend on τ . Moreover, from (15), a change in τ aﬀects neither consumption nor GDP:13
dc
dy
= 0,
= 0.
dτ
dτ
Therefore, in order to stimulate the economy by ﬁscal expansions, the government has to allocate the budget not to direct transfers or tax cuts but tocommodity or service purchases that create new employment, because thisincrease in employment moderates deﬂations in the nominal wage and price.
Since the present multiplier eﬀect works through moderation of deﬂation,
it disappears in the typical Keynesian case where nominal wages and prices
are ﬁxed. In fact, from (15), if α = 0 then c is constant:c = u′−1(β/ρ) ,and hence neither g nor τ aﬀects c. This result is the same as the conventional
Keynesian case with a balanced budget. It is because Ricardian equivalenceholds, which essentially leads to the same situation as that where the government adopts a balanced budget in the conventional Keynesian model.

librium model. However, his model is static, and neither aggregate demand deﬁciency nor
unemployment exists. Moreover, imperfect competition among ﬁrms is crucial for creating
the multiplier eﬀect.
13
Feldstein (2009) and Shapiro and Slemrod (2009) ﬁnd that the 2008 tax rebate in the
US was not very eﬀective in increasing private consumption. According to Shapiro andSlemrod (2009, table 1), only one-ﬁfth of households receiving the rebate planned to spend
most of it while the remaining four-ﬁfths planned to use it mostly to save or to pay oﬀ
debt.

17

We have so far considered the case where g is wasteful. Let us brieﬂy
discuss the case where g increases productivity or utility. When the laborproductivity θ is a function of g:θ = θ(g), θ′ (g) > 0,
an increase in g raises full employment output y (= θ(g)n), which expands
the deﬂationary gap and worsens deﬂation. Thus, the eﬀect of g on y via achange in θ is negative, making the multiplier eﬀect lower (or even negative).
When the utility is given by u(c, g), (14) is rewritten as
(
ρ+α

y
−1
y

)
=

β
uc (c, g)=⇒ c = c(y; α, y, β, g),from which we obtain
cg =

ucg
.
−ucc

(19) is replaced byc(y; α, y, β, g) + g = y,and the multiplier eﬀect on output given in (20) turns to bedy
1 + cg≡=dg
1 − cyucg−ucc,
1 − cy1+i.e., it depends on ucg . If c and g are complementary (i.e., ucg (c, g) > 0), an
increase in g encourages private consumption, which enhances the multipliereﬀect. If they are substitutes (i.e., ucg (c, g) < 0), the multiplier eﬀect issmaller (or may even be negative) because g is substituted for c. In particular,
if c and g are perfect substitutes (i.e., u(c, g) = u(c + g)), ucg = ucc and hence
the multiplier is zero, implying that an increase in such government spending18

completely crowds out private consumption. If ucg (c, g) = 0, the multipliereﬀect is of the same magnitude as that in the case where g is wasteful, while
the utility increases.

3.4

The Comparative Statics

Because the consumption function is derived from household optimizing behavior, we can analyze the eﬀects on the consumption function of changes
in various preference and technology parameters such as wage ﬂexibility α,potential output y and liquidity preference β. From (15), they are()[u′ (c)]2 y∂c(y; α, y, β)= − ′′− 1 < 0,cα ≡∂αβu (c) y∂c(y; α, y, β)αy[u′ (c)]2
=
< 0,
∂y
βy 2 u′′ (c)∂c(y; α, y, β)u′ (c)
cβ ≡
=
< 0.
∂β
βu′′ (c)
cy ≡

(21)

When y is given, an increase in α accelerates the decline in the nominal
wage whereas an increase in y expands the deﬂationary gap, both of which
aggravate deﬂation and thus urge the household to save money and reduce
consumption. An increase in β straightforwardly induces the household tosave more and consume less.
From (19) and (21), the eﬀects of α, y and β on GDP are
cα
dy
cy
dy
cβ
dy
=
< 0,
=
< 0,
=
< 0.
dα
1 − cydy
1 − cydβ
1 − cy(22)

The ﬁrst and second properties, respectively, show that more ﬂexible wage
adjustments lower GDP, which implies the “paradox of ﬂexibility”, and thatan increase in potential output decreases GDP, which implies the “paradox19

of toil”, both of which are discussed by Eggertsson and Krugman (2012).14
The third is the “evil of thrift” mentioned by Keynes (1936, p. 358): an
increase in the household’s desire to hold money reduces GDP.
We next explore the inﬂuences of α, y and β on the magnitude of themultiplier eﬀect dy/dg. Because the multiplier eﬀect is given by the ﬁrst
equation of (20) and from (15), (16) and (19) cy is expressed as a function ofα, y and β, one obtains
d
di

(

dy
dg

)

1
=
(1 − cy )2(∂cy ∂cy dy
+
∂i
∂y di

)
, where i = α, y, β.

(23)

The ﬁrst term in parentheses on the RHS implies the direct eﬀect of eachparameter on the multiplier eﬀect, whereas the second term represents theindirect eﬀect through a change in the output level due to the parameterchange. Therefore, we ignore the latter by assuming a logarithmic utility
function (u(c) = log c) and focus on the former.15 In this case ∂cy /∂y = 0,
as is clear from (18), and hence the second term disappears.
To examine the sign and implication of the ﬁrst term ∂cy /∂i in (23), using
the third and ﬁfth equations of (12), we decompose cy into the moderationeﬀect of an increase in y on deﬂation, ∂π/∂y, and the stimulative eﬀect of
this moderation on consumption, ∂c/∂π. With logarithmic utility, they are
cy =

∂c ∂π
∂c
1
∂π
α
·
, where
=
and
= ,
∂π ∂y
∂π
β
∂y
y

14

In Eggertsson and Krugman (2012), the paradox of ﬂexibility means the case wherea negative shock decreases output more if price ﬂexibility increases. Eggertsson (2010)
originally uses the paradox of toil to describe the case where the desire of everyone towork more results in decreasing aggregate employment, whereas Eggertsson and Krugman(2012) use it in the sense that an increase in potential output leads to a decrease in actual
output.
15
If u(c) has a general form, the indirect eﬀect depends on the third derivative of u(c),as is clear from (16), and thus is ambiguous.

20

which yields
1
∂cy
α
α
∂cy
∂cy
=
> 0,
= − 2 < 0,
= − 2 < 0.
∂α
βy
∂y
βy
∂β
β y
A rise in α makes π more sensitive to a change in the output gap, which
increases ∂π/∂y and hence cy . As y is larger, an increase in y becomesless eﬀective in narrowing the deﬂationary gap, causing ∂π/∂y to decline
and cy to be lower. As the liquidity preference is stronger (as β rises), thedesire for money compared with that for consumption becomes the dominantfactor in the consumption decision, making the role of deﬂation in deciding
consumption less important. Therefore, a rise in β reduces ∂c/∂π, while∂π/∂y is intact because β has nothing to do with the wage–price adjustment.
Consequently, cy decreases. From (23), the magnitude of the multiplier eﬀect
varies in the same direction as cy , namely,
d
dα

(

dy
dg

)

d
> 0,
dy

(

dy
dg

)

d
< 0,
dβ

(

dy
dg

)
< 0.

(24)

Let us summarize the properties in (22) and (24). Increases in liquidity
preference and potential output are deﬁnitely harmful to the stagnant economy. They not only decrease the level of GDP but also weaken the multipliereﬀect. This result may explain why Japan’s stagnation since the early 1990s
has seriously persisted and why an increase in government purchases was not
as eﬀective as expected (see Kameda (2014) and therein references for this ineﬀectiveness). In fact, the following phenomena associated with increases in
β and y were observed during this stagnation. Money demand motivated byfactors other than the transaction motive increased (Otani and Suzuki, 2008),
and the government of Japan repeatedly implemented policies intended to21

increase potential output, such as deregulation and privatization, despite thepresence of the deﬂationary gap.16 Meanwhile, an increase in wage ﬂexibility
strengthens the multiplier eﬀect but reduces the level of GDP.17 This result
suggests that improving imperfection in the labor market may not necessarily
be beneﬁcial to the economy.

4

Conclusion

Long-run stagnation with aggregate demand deﬁciency occurs if intertemporally optimizing households have insatiable preferences for holding money.
In this long-run stagnation, consumption is expressed as a function of output, as is the Keynesian consumption function, and an increase in government purchases boosts GDP through a multiplier process. However, thisconsumption function represents not the Keynesian relationship between income and consumption but the eﬀect on consumption of an increase in output
through moderation of deﬂation. Therefore, the multiplier eﬀect of government purchases results from the repetition of the interactive process between
moderation of deﬂation and an increase in consumption.
This multiplier of government purchases is larger than one although Ricardian equivalence holds. This is because an increase in government purchasesof goods and services directly creates new employment, which mitigates deﬂations in the nominal wage and price. Meanwhile, a tax cut (or a transfer
increase) has no eﬀect on GDP because a tax cut in itself does not create
new employment. Thus, direct creation of new employment is essential for
16

See, e.g., Nishizaki et al. (2014) for Japan’s deﬂationary gap.
Using a DSGE model, Christiano et al. (2011) obtain a similar result in a short-run
slump.
1722

stimulating the economy. Moreover, we ﬁnd that an increase in potential
output reduces GDP and weakens the multiplier eﬀect. These results lead us
to the conclusion that expanding government purchases is eﬀective, cutting
a tax is ineﬀective and increasing potential output is harmful for economies
falling into long-run stagnation with aggregate demand deﬁciency, probably
such as the USA and Japan in recent years.

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RHS of (19)

LHS of (19)45 degree∗O

Figure 1: The determination of GDP

30y