ISSN (Print) 0473-453X

Discussion Paper No. 1211

ISSN (Online) 2435-0982

THE CONSUMPTION MULTIPLIER
OF GOVERNMENT SPENDING:
THE ROLE OF SUBSTITUTABILITY BETWEEN
GOVERNMENT SPENDING AND LEISURE

Masataka Eguchi
Yuhki Hosoya
Mai Yamada

August 2023

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

The Consumption Multiplier of Government Spending:
The Role of Substitutability between Government
Spending and Leisure 1
Masataka Eguchi2 , Yuhki Hosoya3 and Mai Yamada4
August 28, 2023

1 We thank Masao Ogaki, Masaya Sakuragawa, and Naoyuki Yoshino for their valuable com-ments. The authors gratefully acknowledge the financial support of the Joint Usage/ResearchCenters at the Institute of Social and Economic Research (ISER) and JSPS KAKENHI Grant
Number JP23K01471.
2 Faculty of Economics, Komazawa University; Correspondence to: Faculty of Economics, Komazawa University 1-23-1 Komazawa, Setagaya-ku, Tokyo 154-8525, Japan. E-mail address:
masataka.eguchi@gmail.com
3 Faculty of Economics, Chuo University 742-1 Higashi-Nakano, Hachioji-City, Tokyo 192-0393,
Japan. E-mail address: hosoya@tamacc.chuo-u.ac.jp
4 Faculty of Economics, Meikai University 1 Akemi, Urayasu-city, Chiba 279-8550, Japan. E-mailaddress: myama0157@meikai.ac.jp

Abstract
It has been empirically observed that consumption responds positively to government
spending shock, however, existing models with intertemporally-optimizing households do
not easily reconcile this stylized fact. This paper aims to address this discrepancy between
models and data, focusing on the non-separable preferences with respect to consumption,
leisure, and government spending. We derive conditions for a positive consumption multiplier
under the general utility function and find that consumption can respond positively when
leisure and government spending are substitutes. Examples of government spending thatwould have such an effect include care for children and the elderly, education spending,
highway and public transportation.

JEL Classification: E62; E32; E60
Keywords: Fiscal multiplier, Non-separable preferences, Substitutability

1Introduction

Although many empirical studies indicate that consumption rises in response to increased
government spending (e.g., Blanchard and Perotti (2002); Fatás and Mihov (2001); Mountford and Uhlig (2009); Fisher and Peters (2010); Mertens and Ravn (2010); Ben Zeev andPappa (2017)), a positive consumption multiplier of government spending cannot be easily
reconciled with existing models based on intertemporally-optimizing households. As shownin Baxter and King (1993), increased government spending diminishes the present value ofdisposable income due to higher taxes for financing government spending, resulting in anegative wealth effect that triggers a decline in consumption within a standard real business cycle model. This mechanism also applies to New-Keynesian models with sticky prices.
Previous studies have proposed various modifications to these models in order to establish
a positive consumption multiplier.1
One modification for obtaining a positive consumption multiplier is to assume nonseparable preferences. Linnemann and Schabert (2004), Bouakez and Rebei (2007), Ganelli
and Tervala (2009), Marattin and Palestini (2014) consider a non-separable utility function
between consumption and government spending and show that the consumption multipliercan be positive when consumption and government spending are complements.2 In contrast, Linnemann (2006) considers non-separable utility function between consumption andleisure and shows that the consumption multiplier can be positive when consumption andgovernment spending are substitutes in a frictionless business cycle model for a certain functional form of preferences. However, Bilbiie (2009) shows that, for general utility function,
the parameter restriction under which a positive consumption multiplier occurs in a frictionless business cycle model implies that either the utility function is not concave or that
1

These modifications include productive government spending (Baxter and King (1993); Asimakopoulos,
Lorusso, and Pieroni (2021)), hand-to-mouth households (Galı́, López-Salido, and Vallés (2007)), deep habit(Ravn, Schmitt-Grohé, and Urbe (2006)), and others.
2
In this context, the terms “substitutes” and “complements” are not used in the Hicks’s definition but inthe Edgeworth’s definition. Let the utility function be U (x1 , x2 ). x1 and x2 are “Edgeworth substitutes” if
Ux1 x2 < 0, “Edgeworth complements” if Ux1 x2 > 0, and “Edgeworth independent” if Ux1 x2 = 0. See, Karras(1994); Ni (1995).

1

consumption is inferior. 3
In this paper, we consider a utility function that includes non-separability between leisureand government spending in a frictionless business cycle model. We clarify the conditions
under which the consumption multiplier is positive for general utility function. According toprevious studies, the consumption multiplier can be positive when consumption and government spending are complements and/or consumption and leisure are substitutes. We show
that, in addition to these two conditions, the consumption multiplier can be positive when
leisure and government spending are substitutes; under this condition, consumption neednot be inferior.

2

The model

Our model is similar to a frictionless business cycle model used in Bilbiie (2009), except thatgovernment spending is contained in the utility function in a non-separable form.
Suppose that the representative household aims to maximize the expected present valueof lifetime utility. The momentary utility function of the representative household at time t
takes the general non-separable form:

U (Ct , Lt , Gt ),(1)

where, Ct represents consumption, Lt represents leisure (Lt = T̄ − Nt where T̄ is timeendowment and Nt is hours worked), and Gt represents government spending. Assume thatthe utility function U satisfies UC > 0, UL > 0, UG > 0, UCC < 0, ULL < 0, and the strong
bordered Hessian condition with respect to (C, L), i.e., UCC UL2 + ULL UC2 − 2UCL UC UL < 0.4
The household earns labor wage income and dividends from the firm. The government
3

Bilbiie (2011) extend that framework to an environment with sticky prices, demonstrating that a positive
consumption multiplier can occur under the conditions of concave utility and good normality restrictions
when consumption and leisure are substitutes. Monacelli and Perotti (2008) also reach this conclusion byusing particular functional form of utility function.
4
See Appendix B for the necessity of this assumption.

2

spending is financed through lump-sum taxes on the household and issuing government
bonds. The budget constraint for the representative household can be expressed as:

Bt + Ct = (1 + rt−1 )Bt−1 + Wt Nt + Dt − τt ,(2)

where Bt represents one-period, risk-free government bonds, rt represents the interest rateon government bonds, Wt represents wage, Dt represents dividends received from the firm,and τt represents lump-sum taxes.
The government’s budget constraint is given by:

Bt = (1 + rt−1 )Bt−1 + Gt − τt .

(3)

The government determines a stream of government spending exogenously.
Finally, the production function takes the form of:

Yt = F (Nt ),(4)

where F is non-increasing return to scale, F (0) = 0, FN > 0, FN N ≤ 0.

3

Analysis

Given government spending Gt , the representative household maximizes lifetime utility subject to the budget constraint.
The first-order conditions for the household’s problem are given by

UC (Ct , Lt , Gt ) = β(1 + rt )Et UC (Ct+1 , Lt+1 , Gt+1 ),(5)

UL (Ct , Lt , Gt ) = UC (Ct , Lt , Gt )Wt ,

(6)

and

3

where β is the discount factor.
The optimal condition of the firm for labor is given by

Wt = FN (Nt ).

(7)

Following Bilbiie (2009), we log-linearize the optimality conditions in order to derive
analytical results.5 Let lowercase letters denote deviation from the steady state: that is,
xt ≡ Xt − X where X represents steady state value of Xt .
Log-linearizing Eqs (6) and (7) yields
ULC
ULL
ULG
UCC
UCL
UCG
ct +
lt +
gt =
ct +
lt +
gt + wt ,
UL
UL
UL
UC
UC
UC

(8)

and
wt =

FN N
nt ,
FN

(9)

where the wage is defined as log deviations from steady state: that is, wt ≡ ln(Wt /W ).
Substituting Eq.(9) into Eq.(8), we have


UCC
ULC
−
UL
UC




ct =

ULL UCL
FN N
+
−
FN
UL
UC




nt +

UCG ULG
−
UC
UL


gt .

(10)

In addition, as yt = FN nt and yt = ct + gt ,


5

ULC
UCC
−
UL
UC




ct =

FN N
ULL UCL
+
−
FN
UL
UC



ct + gt+
FN

Analytical results without linearization are shown in Appendix B.

4

UCG ULG
−
UC
UL


gt .

(11)

Eq.(11) can be rearranged as














ULC UCCUCG ULG
















−
−










UL
UC
UC
UL
F
−
1
c
=
F
+
1
gt .

 N

 N
t
















 FN N ULL UCL  FN N ULL UCL 












+
−
+
−
FN
UL
UC
FN
UL
UC

(12)

Solving Eq.(12) for ct yields
ϕFN + 1gt ,(13)
δFN − 1

 


 

ULG
ULL
UCL
UCC
ULL
UCL
FN N
ULC
FN N
where ϕ ≡ UUCG
−
/
+
−
and
δ
≡
−
/
+
−
.
UL
FN
UL
UC
UL
UC
FN
UL
UC
C
ct =

Using the intratemporal optimality condition FN = UL /UC , we obtain the following
proposition.
Proposition 1. In response to increased government spending, consumption increases if andonly ifϕ(UL /UC ) + 1> 0.
δ(UL /UC ) − 1(14)

In Bilbiie (2009), ϕ = 0 as he (implicitly) assumes UCG = ULG = 0. Thus, δUL /UC > 1 isa necessary and sufficient condition for a positive consumption multiplier in Bilbiie (2009)(Theorem 1). In contrast, since we allow for nonzero ϕ, the necessary conditions for a positive
consumption multiplier can be described as follows.
Corollary 1. Under our assumptions for utility function and production function, the necessary condition for satisfying Eq.(14) is,
1. UCG > 0 (consumption and government spending are complements),
2. UCL < 0 (consumption and leisure are substitutes), and/or
3. ULG < 0 (leisure and government spending are substitutes).

5

As the necessary conditions UCG > 0 and UCL < 0 have already been shown in previous
studies,6 we focus on the condition ULG < 0. In this case, the necessary and sufficientcondition for a positive consumption multiplier is obtained as follows.
Proposition 2. If UCG = UCL = 0, a necessary and sufficient condition for a positive
consumption multiplier is,
ULG < UC

FN N
ULL
+
FN
UL


.

(15)

Proof. When UCG = UCL = 0,
− UUCCCδ = FN N+ UULLL

FN− UULGL

ϕ = FN N+ UULLL

FN,.

Since UCC < 0 and ULL < 0, δ < 0 and the denominator of the left-hand side of Eq. (14)is negative. Thus, the numerator of the left-hand side of Eq.(14) must be negative for apositive consumption multiplier.
The condition for the numerator of the left-hand side of Eq. (14) to be negative is,− UULGL!

FN N+ UULLFN
L

UL+ 1 < 0,UC!

− UULGLFN N+ UULL
FN
L

UL
< −1,
UC



ULG UL
FN N
ULL
−
>−
+
,
UL UC
FN
UL


FN N
ULL
ULG < UC+,
FN
UL
where the reversal of the inequality in the third line is due to the fact that the term
6

See, Linnemann and Schabert (2004); Bouakez and Rebei (2007); Ganelli and Tervala (2009) for UCG > 0
and Bilbiie (2009, 2011) for UCL < 0.

6



FN N
+ UULLFNL

is negative.

Bilbiie (2009) shows that the condition for a positive consumption multiplier implies that
consumption is inferior. However, our condition Eq. (15) in proposition 2 guarantees that
consumption is not inferior. See Appendix A for details.

4

Implications and Discussion

In this section, we discuss the intuitive mechanism, economic implications, and empiricalplausibility of the finding that ULG < 0 can lead to a positive consumption multiplier. To
focus on the impact of ULG , we assume UCG = UCL = 0 throughout this section.

4.1

Why is the consumption multiplier positive when ULG < 0?

Let us first give an intuitive explanation that the consumption multiplier can be positive
if ULG < 0. Suppose government spending increases by one dollar. The household will betaxed on the present value of one dollar, which reduces consumption and leisure througha negative wealth effect. Less leisure means more labor, so the decrease in consumption
is canceled out to some extent by the increase in output. As ULL < 0 and FN N ≤ 0,
the marginal utility of leisure is higher, and the marginal productivity of labor is lower
than that prior to the increase in government spending. Thus, output increases by lessthan one dollar when ULG = 0. As a result, consumption must fall because of resource
constraints. This is why the output multiplier of government spending is below one, and theconsumption multiplier is negative in a frictionless business cycle model. The condition fora positive consumption multiplier is equivalent to the output multiplier that is greater thanone. If ULG < 0, government spending reduces the marginal utility of leisure (or marginal“disutility” of labor), it is possible that output would increase by more than one dollar,
allowing for consumption to rise. In other words, if government spending and leisure aresubstitutes, increased government spending enhances the incentive to work and allows for7more room to increase consumption by increasing the output multiplier.

4.2

Is ULG < 0 empirically plausible?

While there are many empirical studies that examine the substitutability between consumption and government spending (Kormendi (1983); Aschauer (1985); Karras (1994); Ni (1995);
Amano and Wirjanto (1998); Bouakez and Rebei (2007); Fève, Matheron, and Sahuc (2013);
Sims and Wolff (2018); Bouakez, Rachedi, and Santoro (2023)), there are few empirical studies for the substitutability between leisure and government spending. Conway (1997), one of
the few exceptions, examines the substitutability between leisure and government spending
using the U.S. data from the University of Michigan’s Panel Study of Income Dynamics(PSID) by estimating a linear labor supply function based on a non-separable utility function with respect to consumption, leisure, and government spending. As a result, Conway(1997) finds a positive substitution effect of government spending on leisure (or complementary effect on labor supply) for men and unmarried women in the U.S.7 Thus, ULG < 0 is
empirically plausible.

4.3

What kind of government spending brings ULG < 0?

Conway (1997) also gives examples of government spending that has substitution effects
on leisure (or complimentary effects on labor). First, public transportation and highways
would have a complementary effect on labor supply by lowering the transportation costs
for working. Second, educational spending may improve opportunities for rewarding workor change preferences for work. Third, care for children and the elderly could substitutehome production and also have a complementary effect on labor. All of these would havesubstitution effects on leisure.
7

Conway (1997) uses Hicks’s definition of substitutes/complements, which differs from the definition of
Edgeworth used in our study. However, we confirm that ULG < 0 is also basically satisfied under theestimated parameters values and utility function in Conway (1997).

8

For clarity, let us consider the following specific form of the utility function:

U = log Ct + γ log(T̄ − Ht − Nt ),(16)

where Ht = H̄ − θGt is hours of home production and/or commuting time, and Lt =T̄ −Ht −Nt . H̄ is home production/commuting time in the absence of government spending. γ
is the scale parameter for the utility of leisure. We assume that the household cannot choose
their own home production/commuting time and that these are determined exogenously
through government spending. When θ > 0, government spending contributes to increasing“disposable time,” defined here as total time T̄ minus Ht . The household can increase either
leisure or labor (in the firm) for the increased amount of disposable time. As the marginalutility of leisure decreases as leisure increases, the relationship between government spending
and leisure will be substitutes when θ > 0.
Here we assume that the production function is constant returns to scale,

Yt = Nt .

(17)

Under these settings, UC = 1/C, UCC = −1/C 2 , UL = γ(1/L), ULL = −γ(1/L2 ), FN = 1,
FN N = 0, UCG = 0, ULC = 0, ULG = −γθ(1/L2 ). As UL = UC from the intratemporal
optimality condition, δ and ϕ in Proposition 1 can be expressed as follows;δ = −γ,ϕ = −θ.

Thus, the necessary and sufficient condition for a positive consumption multiplier is,

θ > 1.

9In summary, government spending that increases disposable time has a substitution effecton leisure, and if the effect is sufficiently large, the consumption multiplier can be positive.

5

Conclusion

This paper examines the conditions for a positive response of consumption to increased
government spending under general non-separable preference among consumption, leisure,and government spending. Consequently, we find that a positive consumption multiplier
can occur when leisure and government spending are substitutes. We do not claim that thiscondition alone can solve the discrepancy between data and existing models for consumption
multipliers, but we believe that it is one factor worth considering.
Although this paper attempts to explain the stylized fact that the consumption multiplieris usually positive, it also has useful implications for the policy debate on the kind of spendingthat can raise the fiscal multiplier. Typically, a public investment that raises the marginal
productivity (or Total Factor Productivity, TFP) of a firm, as shown in Baxter and King
(1993), is thought to yield a large fiscal multiplier. Our findings show that not only policies
that increase a firm’s productivity but also spending that increases a household’s disposable
time can enhance the efficacy of fiscal policy.

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11Fève, Patrick, Julien Matheron, and Jean-Guillaume Sahuc. 2013. “A pitfall with estimated
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Appendix A. Non-inferiority
Bilbiie (2009) shows that a positive consumption multiplier can occur if and only if consumption is inferior when UCG = ULG = 0 and UCL ̸= 0. In this section, we show that apositive consumption multiplier can occur even if consumption is not inferior when ULG < 0.
To focus on the effects of the substitutability between leisure and government spending,we consider the following utility function in which only leisure and government spending are
non-separable:
U (Ct ) + V (Lt , Gt ),(A.1)

where UC > 0, VL > 0, and VG > 0. Here we assume strictly quasi-concavity, UCC < 0 and
VLL < 0. In this expression, VL , VLL , and VLG correspond to UL , ULL , and ULG in the main
text, respectively. The utility function (A.1) implies UCG = UCL = 0.
The budget constraint of the household can be rewritten as

Ct + Wt Lt ≤ Et ,13

where Et represents full income given by Et = Wt T̄ + (1 + rt−1 )Bt−1 + Dt − τt − Bt .
Inferior goods have negative income elasticity at given prices. The demand functions withrespect to consumption and leisure for a given full income level E are derived by solving thefollowing static optimization problem:

max U (C) + V (L, G) s.t. C + W L ≤ E.

Solving this static optimization problem, we obtain as

W UC (C) − VL (L, G) = 0,

(A.2)

C + W L − E = 0.

(A.3)

and

Applying the implicit function theorem, we can examine the effect of changes in full income level E on consumption demand. Given the price of leisure W , the total differentiationof Eqs. (A.2) and (A.3) yields(W UCC )

∂C
∂L
− VLL
= 0,
∂E
∂E

(A.4)

and
∂C
∂L
+W
= 1.
∂E
∂E

(A.5)

Solving Eqs. (A.4) and (A.5),
∂C
VLL
∂L
= 2
> 0,
=
∂E
W UCC + VLL∂Enecessarily hold from UCC < 0 and VLL < 0.

W −1 VLL + W UCC
UCC

∂C
∂E

−1
> 0,

∂L
> 0 and ∂E> 0 imply that neither

consumption nor leisure is inferior.
Bilbiie (2009) shows that the condition for a positive consumption multiplier implies that14

consumption is inferior. However, our condition Eq. (15) in proposition 2 guarantees that
consumption is not inferior.

Appendix B. Analytical Results without Linearization
The strong bordered Hessian condition of U with respect to (C, L):

UCC UL2 + ULL UC2 − 2UCL UC UL < 0,(A.6)

is known as a sufficient condition for strict quasi-concavity of U with respect to (C, L). As
shown in Katzner (1968), the converse relationship does not hold, i.e., there exists a functionthat violates the above inequality, but it is still strictly quasi-concave. Debreu (1972) also
shows that the condition (A.6) is a necessary and sufficient condition for differentiability ofa demand function. To obtain analytical results, we need to differentiate Ct by Gt , so thiscondition must be satisfied. The quasi-concavity of U with respect to G is not necessary
because G is determined exogenously for the household.
Recall the optimal conditions for the household (6) and the firm (7):UL (Ct , Lt , Gt ) = UC (Ct , Lt , Gt )Wt ,Wt = FN (Nt ).

As Yt = F (Nt ) and Nt = T̄ − Lt , we obtain

UL (Ct , Lt , Gt ) = UC (Ct , Lt , Gt )FN (T̄ − Lt ),F (T̄ − Lt ) = Ct + Gt .

15

(A.7)
(A.8)

Define the function H(C, L, G) = (H 1 (C, L, G), H 2 (C, L, G)) as follows:

H 1 (C, L, G) = UL (C, L, G) − UC (C, L, G)FN (T̄ − L),
H 2 (C, L, G) = F (T̄ − L) − C − G.

From equations (A.7) and (A.8), the following relation regarding H 1 and H 2 holds:

H 1 (Ct , Lt , Gt ) = H 2 (Ct , Lt , Gt ) = 0.

To check whether the implicit function theorem is applicable for (C, L), let us calculate theJacobian of H with respect to (C, L):∗H ≡HC1 HL1
HC2= ULL − 2UCL FN + UCC (FN )2 + UC FN N

HL2UCC UCL1

UCC UCL UC= UC FN N − ULC ULL FN = UC FN N − (UC )−2 ULC ULL UL < 0,
1

FN

0

UC

UL

0

where the last inequality comes from the strong bordered Hessian condition (A.6). Therefore, the implicit function theorem is applicable and both Ct and Lt can be represented ascontinuously differentiable functions of Gt . We denote those functions as Ct (Gt ), Lt (Gt ),respectively. Using these functions, H 1 and H 2 can be written as

H 1 (Ct (Gt ), Lt (Gt ), Gt ) = UL (Ct (Gt ), Lt (Gt ), G) − UC (Ct (Gt ), Lt (Gt ), Gt )FN (T̄ − Lt (Gt )) = 0,
H 2 (Ct (Gt ), Lt (Gt ), Gt ) = F (T̄ − Lt (Gt )) − Ct (Gt ) − Gt = 0.

16

Differentiating the above equations with respect to Gt , we obtain






HC2



−ULG + UCG FN .
  = 
HL2
L′t
1





1
HC



HL1  Ct′ 

Thus,


′
C t 



L′t

1
HC

= 

−1 

1
HL  −ULG + UCG FN  
HC2 HL2
1



1
2
1  HL −HL  −ULG + UCG FN 
= ∗

,
H
2
1
−HC HC1which implies that,

Ct′ (Gt ) =−ULL + UCL FN + ULG FN − UCG (FN )2 − UC FN N
.
H∗

(A.9)

Since H ∗ is negative, the necessary conditions of Ct′ > 0 (i.e., Ct increase in response to Gt )are
1. UCG > 0 (consumption and government spending are complements),
2. UCL < 0 (consumption and leisure are substitutes), and/or
3. ULG < 0 (leisure and government spending are substitutes).
Therefore, Corollary 1 is verified.
Suppose that UCG = UCL = 0. (A.9) can be rewritten as follows:
Ct′ (Gt ) =−ULL + ULG FN − UC FN N
.
H∗

17

Since H ∗ < 0, the following relationship holds:

Ct′ (Gt ) > 0 ⇔ ULG <ULL UC FN N
+
.
FN
FN

Using the intratemporal optimality condition FN = UL /UC , we obtainCt′ (Gt ) > 0 ⇔ ULG < UC

Thus, Proposition 2 is also verified.

18



FN N
ULL
+
FN
UL


.