Discussion Papers In Economics
And Business

A Note on Commodity Taxation and Economic Growth

Kunihiko KonishiDiscussion Paper 13-22

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANA Note on Commodity Taxation and Economic Growth

Kunihiko KonishiDiscussion Paper 13-22

September 2013

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANA Note on Commodity Taxation and Economic Growth∗
Kunihiko Konishi†

Abstract
This note reexamines the growth eﬀects of commodity taxation and a manufacturingsubsidy. By incorporating endogenous labor supply into a variety expansion model following
Grossman and Helpman (1991), we derive new results. First, if households consider leisure
to be important, an increase in the commodity tax rate can decrease the growth rate in the
short run. Second, a small elasticity of substitution and a small manufacturing subsidy halt
economic growth. Third, when the elasticity of substitution is small and sustained growthis possible, a decrease in the subsidy raises the short-run growth rate and decreases thelong-run growth rate.

Keywords: Commodity taxation, Subsidy, Labor supply, Endogenous growth
JEL classiﬁcation: E62, O41.

∗
We would like to thank Koichi Futagami and Tadashi Morita for their helpful comments and suggestions. Of
course, any remaining errors are my responsibility.
†
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043, JAPAN;
E-mail: nge008kk@student.econ.osaka-u.ac.jp

1

Introduction

The eﬀects of consumption taxation have been examined in the endogenous growth literature.
King and Rebelo (1990), Rebelo (1991), Pecorino (1993), Devereux and Love (1994), Stokey
and Rebelo (1995) and Milesi-Ferretti and Roubini (1998) show that taxes on consumption
have negative or no eﬀect on long-run growth. On the other hand, Jorgenson (1998) argues that
implementing consumption taxes can boost the growth rate of the United States. Corresponding
to this statement, Futagami and Doi (2004) considers the eﬀect of speciﬁc consumption taxes(henceforth commodity taxes) in a variety expansion model following Grossman and Helpman(1991). They show that an increase in commodity taxes raises the long-run growth rate. In
their model, an increase in the commodity tax rates reduces the demand for consumption goods
and reallocates labor from production to R&D.
The objective of this note is to reexamine the growth eﬀects of Futagami and Doi (2004) byincorporating endogenous labor supply. By considering labor-leisure choice, the growth eﬀectof commodity taxation can change that of Futagami and Doi (2004); that is, an increase in thecommodity tax rates reduces the growth rate. When the government raises the commodity tax,
households reduce consumption of goods and increase leisure time. This can lead to a decreasein labor supply and the growth rate as well.
In addition to commodity taxation, we reexamine the eﬀects of a manufacturing subsidy.
In Grossman and Helpman (1991), they show that the manufacturing subsidy has no growtheﬀect. On the other hand, Futagami and Doi (2004) shows that a decrease in the manufacturingsubsidy raises the growth rate. In this note, we present some new results diﬀerent from these
existing literature.
The rest of this note is organized as follows. Section 2 establishes the model used in thisnote. Section 3 characterizes equilibrium dynamics and examines the existence of sustained
growth. Section 4 examines the eﬀects of ﬁscal policy. Section 5 states the conclusion.

2

Model

The factor market is perfectly competitive, while the goods market is monopolistically competitive, as explained below. The households have perfect foresight.

2.1

Households

The households maximize the following lifetime utility:()∫ ∞l1−σ − 1
−ρt
U≡
e
D+
dt,
1−σ
0
1

(σ > 0),(1)

where D stands for an instantaneous utility derived from the consumption of a composite good,
ρ > 0 is the households’ rate of time preference, and l is leisure time. D is given by
[∫ n
D=

x(j)

β−1
β

] β

β−1

,

dj

(2)

0

where x(j) denotes the consumption of good j and n denotes the number of available varieties.
We assume that β > 1. β is the elasticity of substitution between any two products. By
denoting the expenditure of the households as E, we obtain the demand function for good j as
follows:x(j) =

q(j)−β E
,
Q1−β

(3)

where q(j) is the consumer price, and Q is the price index deﬁned as
(∫ n

) 1
1−β

Q=

q(i)

1−β

di

.

(4)

0

Then, the households maximize U , subject to Ȧ = rA + W (L̄ − l) − E + T , where A, r,
W , L̄, and T represent asset holdings, nominal rate of interest, wage rate, total time, andlump-sum transfers from the government. Here, capital letters represent nominal variables. By
substituting (3) into (2), we obtain the indirect sub-utility function, D = E
Q . The following
optimal conditions hold1,Q
l−σ = λW,
λ=

r−

Q̇
= ρ,
Q

(5)
(6)
(7)

Q̇
where λ stands for the costate variable attached to the asset holdings. Since Q
is the inﬂation

rate, we obtain r̃ = ρ, where r̃ represents the real rate of interest.

2.2

Firms

This subsection considers producer behavior. Producers undertake two distinct activities. They
create blueprints for new varieties of diﬀerentiated goods, and manufacture the diﬀerentiatedgoods that have been created by R&D.
We assume that each diﬀerentiated good is produced by a single ﬁrm because the good is
inﬁnitely protected by a patent. We further assume that one unit of labor input produces one
unit of a diﬀerentiated good. The ﬁrm manufacturing good j (ﬁrm j) maximizes its own proﬁt:π(j) = P (j)x(j) − W x(j), where P (j) represents the producer price for ﬁrm j.

2Next, we assume that the government imposes commodity tax on each good. τ represents
the tax when it takes a positive value, or the subsidy when it takes a negative value. We assume
that the commodity tax is constant over time and τ > −1. The consumer price becomes
j = 1, 2, · · · , n.

q(j) = P (j) + τ,

(8)+τFirm j charges the following price: P (j) = βW
β−1 = P . Therefore, all goods are priced equally.

This pricing rule yields per brand operating proﬁts as follows:
π=

E
.
βn

(9)

The wage rate W is normalized to be unity. Thus consumer price, producer price, and demand
for goods become, respectively,
β+τ
,
β−1
β
q=
(1 + τ ),
β−1
β−1
E
x=
.
β (1 + τ )n
P =

(10)

The no-arbitrage condition is given byπ + v̇ = rv,(11)

where v denotes the value of a ﬁrm.
We consider the technology involved in developing a new good.1 The R&D ﬁrms create
blueprints and expand the varieties of goods available for consumption. We assume that one
unit of R&D activity needs na units of labor input. Because the knowledge that has alreadybeen produced includes all that is needed for invention, greater knowledge means faster further
invention. Since the knowledge is non-rival and non-excludable, an expansion in the numberof varieties reduces the labor input. We assume that ﬁrms enter freely into the R&D race.
Therefore, the free entry condition is given by

{
v
2.3

= Wna ,< Wna .

ṅ > 0ṅ = 0.

(12)

Market equilibrium

The households supply L̄ − l units of labor. Labor is used for R&D and production. The labormarket equilibrium becomesag +E= L̄ − l,qwhere g ≡ ṅn . In addition, the government budget constraint is T =1

See Grossman and Helpman (1991) for more details of the R&D process.

3

(13)
∫n

0 τ x(j)dj.

3Equilibrium dynamics

In this section, we characterize the equilibrium dynamics. From (4), (5), (6), and (10), we
obtain111

l = Q σ = q σ n (1−β)σ .

(14)

1
1
Thus, (7) and (14) yields r = ρ + 1−β
g. By using (9) and (11), we obtain vv̇ = ρ + 1−β
g − β1 V ,E. The free entry condition (12) implies that v = na holds under ṅ > 0, and thus,
where V ≡ vnv̇v = −g and E = aV

hold. From these results, we obtain
2−β
1
V =ρ+
g.
β
1−β

(15)

From (10), (13), (14), and (15), we obtain the following diﬀerential equation:
g=

Z
F − 1
− n (β−1)σ ,
Y
Y

(16)
σ+1

1
σ
where Y ≡ β + τ − 1, Z ≡ 1+τ
a L̄ − (β − 1)ρ, and F ≡ a (1 + τ )( β )1σβ−1.

The actual labor time L̄ − l must be positive. From (14) and ṅ ≥ 0, leisure is the highest attime 0. Therefore, we have that L̄ − l > 0 if the initial number of varieties satisﬁes the following
condition (see Appendix A.1):{
n0 > n̂ ≡aF(1 + τ )L̄

}(β−1)σ.

Next, we consider the condition E > 0. From (15) and (16), we obtain{}1aβ− (β−1)σ−(1 + τ )J + (2 − β)F nE=,(β − 1)Y

(17)

(18)

where J ≡ 2−βa L̄ − (β − 1)ρ. As with (17), the initial number of varieties satisﬁes the following
condition so that E > 0:{n0 > ñ ≡(2 − β)F(1 + τ )J}(β−1)σ.

In addition, it is also necessary for E > 0 that J takes a diﬀerent sign from Y (see Appendix
A.2). As a result, we can sum up the condition that labor time and expenditure take positive
values as follows:{}n0 > max n̂, ñ
and JY < 0.
To illustrate equation (16) in an (n, g) space, we have to take account of the signs of J,
Y , and Z. Figure 1-1 illustrates the case wherein J < 0, Y > 0, and Z > 0. The number ofvarieties when g = 0 in equation (16), is given as follows:[ ](β−1)σ
F
∗
.
n ≡
Z
4

From Figure 1-1, n0 > n∗ holds so that sustained growth is possible.2 Similarly, Figure 1-2
illustrates the case wherein J > 0, Y < 0, and Z < 0. This implies that sustained growth ispossible as long as n0 > ñ holds.3 Along the same lines, Figure 1-3 illustrates the case wherein
J > 0, Y < 0, and Z > 0. As shown in Figure 1-3, economic growth stops at n∗ : the economy
does not grow under n0 ≥ n∗ .4[Figure 1]
To characterize these three cases, it is worth focusing on the relation between β and τ (SeeFigure 2 where the three lines represent J = 0, Y = 0, and Z = 0, respectively). Table 1
presents the characteristics of regions (A), (B), and (C). Region (A), where J < 0, Y > 0, and
Z > 0, corresponds to Figure 1-1. Region (B), where J > 0, Y < 0, and Z < 0, corresponds toFigure 1-2. Region (C), where J > 0, Y < 0, and Z > 0, corresponds to Figure 1-3.5[Figure 2 and Table 1]
The following proposition can be stated.

Proposition 1• For an economy where (τ, β) is in region (A) and n0 > n∗ , sustained growth is possible(see Figure 1-1).
• For an economy where (τ, β) is in region (B) and n0 > ñ, sustained growth is possible (seeFigure 1-2).
• For an economy where (τ, β) is in region (C) and ñ < n0 < n∗ , the number of varieties
converges to n∗ (see Figure 1-3).
A suﬃciently small β implies that the economy is in region (B) or (C); that is, Y is negative.
Recall that β is larger than one. In these regions, τ is negative, which represents the subsidy.
An intuitive explanation of (B) and (C) can be given as follows. When the government pays
a subsidy to manufacture diﬀerentiated products, the labor demand for production is large
enough. In region (C), this concentrates labor on production, and as a result, economic growthceases. However, the larger the subsidy, the lower is the consumer price for goods. Households
then reduce leisure time and consume more goods. This increases labor supply. In region (B),labor can be allocated to R&D activities and thus sustained growth becomes possible.
2
1+τ
When β > 2, 2−β
J > 1+τL̄ > Z holds. Therefore, n∗ > n̂ > ñ holds. In contrast, 1 < β < 2 implies thatan∗ > n̂ and E > 0. Thus, there is no need to consider the value of ñ.
3
1+τ
Since Y < 0 implies that 1 < β < 2, 1+τ
L̄ > 2−βJ holds. Hence, we have ñ > n̂.
a
4
1+τ
1+τ
Because Y < 0, 2−β < 1 holds. This implies that 2−β
J > Z and n∗ > ñ hold.
5
We omit the case where J < 0, Y > 0, and Z < 0 because g = 0 holds for all n.

5The result in region (C) is diﬀerent from that of Futagami and Doi (2004), where eachhousehold supplies one unit of labor inelastically and the tax/subsidy aﬀect only the labordemand for production and not the labor supply. Futagami and Doi (2004) exhibits scaleeﬀects, that is, a large labor supply delivers a high growth rate. In this model, in contrast, even
if households’ total time is quite large, a small subsidy can halt economic growth.

4Eﬀects of commodity taxation

In this section, we investigate the eﬀects of commodity taxation on economic growth. By
diﬀerentiating equation (16) with respect to τ , we obtain
FK
∂g
J
− 1
n (β−1)σ ,=− 2 −2∂τY
σ(1 + τ )Y

(19)

∂g
where K ≡ (1 + τ ) − (σ + 1)(2 − β). The sign of ∂τ
depends on the signs of J and K, andthe value of n. To study the eﬀects of commodity taxation on economic growth, we consider
regions (A), (B), and (C).

∂g
Region (A): J < 0, Y > 0, and Z > 0. When K > 0, the sign of ∂τ
depends on the value of∂gn. Let n′ be the number of varieties at which ∂τ= 0. From (19), n′ is deﬁned as follows:[FKn′ ≡ −σ(1 + τ )J](β−1)σ.

∂g
Q 0 holds. Next, we examine whether n′ is larger than n∗ . We calculate
Hence, when n Q n′ , ∂τthe following diﬀerence:
′

(n )

1
(β−1)σ∗− (n )1(β−1)σ1
1= − (1 + τ ) σ
a

(

β
β−1

)1 (
σ

)
YR
,
σJZ

(20)

′
∗
where R ≡ 1+τ
a L̄ − (σ + 1)(β − 1)ρ. Therefore, R R 0 implies that n R n .

Figure 3-1 shows how the economy responds to an increase in the commodity tax rate whenR > 0. The locus of g, which represents equation (16), shifts down if n∗ < n < n′ and shifts up
if n > n′ . Figure 3-2 shows how the economy responds to an increase in the commodity tax rate∂g
> 0; thatwhen R < 0. The locus of g shifts up if n ≥ n∗ . Furthermore, K < 0 implies that ∂τ

is, the locus of g shifts up. This case also corresponds to Figure 3-2. From the above discussion,
region (A) can be divided into regions (A-1) and (A-2) (see Figure 4 in which K = 0 and R = 0
are added to Figure 2). Besides, Table 2 presents the characteristics of regions (A-1) and (A-2).
Regions (A-1) and (A-2) correspond to Figures 3-1 and 3-2, respectively.
[Figures 3, 4 and Table 2]
6

Region (B): J > 0, Y < 0, and Z < 0. From Figure 4 and Table 2, the region (B) implies∂g
R 0. This case can be illustrated as
K < 0 and R < 0. Therefore, from (19), n Q n′ implies ∂τin Figure 5 which shows how the economy responses to a reduction in the subsidy. The locus
of g shifts up if ñ < n < n′ and shifts down if n > n′ .
[Figure 5]
Region (C): J > 0, Y < 0, and Z > 0. From Figure 4 and Table 2, region (C) implies that∂g
K < 0. From (19), ∂τ
R 0 holds when n Q n′ . Next, we reexamine whether n′ is larger thann∗ . From (20), R R 0 implies that n′ R n∗ .
Figure 6-1 shows how the economy responds to a decrease in the subsidy when R > 0. The
locus of g shifts up if ñ < n < n∗ ; a fall in the subsidy raises the growth rate and the number ofvarieties in the steady state. Figure 6-2 shows how the economy responds to a decrease in thesubsidy when R < 0. The locus of g shifts up if ñ < n < n′ and shifts down if n′ < n < n∗ . If n
is suﬃciently small, a decrease in the subsidy increases the short-run growth rate and decreases
the long-run growth rate. In addition, the number of varieties in the steady state becomes
small. From the above discussion, region (C) can be divided into regions (C-1) and (C-2) (SeeFigure 4). Regions (C-1) and (C-2) correspond to Figures 6-1 and 6-2, respectively.
[Figure 6]
The following proposition can be stated.

Proposition 2
Suppose that n0 is suﬃciently small and satisﬁes ṅ > 0.
• For an economy where (τ, β) is in region (A-1), when the government increases the commodity tax rate, the short-run growth rate decreases and the long-run growth rate increases(see Figure 3-1). If (τ, β) is in region (A-2), in contrast, an increase in the commodity
tax rate raises the growth rate (see Figure 3-2).
• For an economy where (τ, β) is in region (B), a decrease in the subsidy increases theshort-run growth rate and decreases the long-run growth rate (see Figure 5).
• For an economy where (τ, β) is in region (C-1), when the government lowers the subsidy,
the growth rate increases and the number of varieties in the steady state becomes high (seeFigure 6-1). If (τ, β) is in region (C-2), in contrast, a decrease in the subsidy increases theshort-run growth rate and decreases the long-run growth rate. Furthermore, the numberof varieties in the steady state becomes low (see Figure 6-2).
7

The result in region (A-2), which corresponds to Figure 3-2, is the same as that in Futagami
and Doi (2004). However, Figure 3-1 shows that the result in region (A-1) is diﬀerent. In region(A-1), the commodity tax is larger than in region (A-2) (see Figure 4). When the commodity
tax is larger, households prefer leisure to consumption. An intuitive explanation can be given as
follows. When the government raises the commodity tax, households’ demand for consumption
goods decreases, and leisure time increases. Because the latter eﬀect is larger, the short-run
growth rate declines. From (14), however, the larger the number of varieties, the lower the priceindex. Households reduce the leisure time and consume more goods. Thus, the eﬀect of labor
supply becomes negligibly small. An increase in the commodity tax rate decreases the demandfor consumption goods and reallocates labor from production to R&D. Consequently, this leads
to the higher long-run growth rate. In addition, region (A-1) is relatively large if households
put weight on leisure. This is captured by the following fact. From the leisure utility functionof (1), a small σ makes the households’ leisure utility high. When σ decreases, R = 0 shifts up
and K = 0 shifts down as shown in Figure 4.6 Therefore, a smaller σ implies a larger region(A-1).
An intuitive explanation of the result in region (B), which corresponds to Figure 5, is asfollows. A fall in the manufacturing subsidy raises leisure time and reduces the labor demandfor consumption goods. Since the latter eﬀect is larger, the reallocation of labor increases thegrowth rate in the short run. As argued above, the large number of varieties implies a low leisuretime, and as such, the eﬀect of labor supply becomes negligibly small. However, a decrease in
the manufacturing subsidy lowers the growth rate in the long run (this is contrary to Futagami
and Doi (2004)). The underlying reasoning is that when the government decreases the subsidy,
households expand their expenditure to avoid a reduction in their utility from the consumptionof goods. Hence, the labor demand for production increases and R&D declines. This results inthe lower long-run growth rate. The mechanism behind the result in region (C-2), corresponding
to Figure 6-2, is the same as that behind (B). In contrast, the mechanism behind the result
in region (C-1), which corresponds to Figure 6-1, applies to only the short-run mechanism inregion (B).

5Conclusion

In this note, we have analyzed the eﬀects of commodity taxation and a manufacturing subsidy
in a variety expansion model with endogenous labor supply. The main results are as follows.
First, when the government raises the commodity tax, households reduce labor supply. This
6

From the definition of J, Y and Z, these values are independent of σ.

8can lower the growth rate in the short run if households prefer leisure. Second, if the elasticity
of substitution and the subsidy are suﬃciently small, economic growth ceases. Third, when thegovernment reduces the subsidy, the short-run growth rate increases and the long-run growthrate decreases.

Appendix
A.1

Condition L̄ − l > 0

From (10), (14), and the deﬁnition of F , condition L̄ − l > 0 is given by1+τ
− 1L̄ − F n (β−1)σ > 0.
aWhen ṅ > 0, this equation must be concluded at time 0,{}(β−1)σan0 > n̂ ≡F.
(1 + τ )L̄

A.2

Condition E > 0

β > 2 implies that J < 0 and Y > 0. From (18), condition E > 0 is given by−(1 + τ )J + (2 − β)F n1− (β−1)σ> 0.

When ṅ > 0, this equation must be concluded at time 0,}{(2 − β)F (β−1)σn0 > ñ ≡.
(1 + τ )J
If 1 < β < 2, the signs of J and Y are ambiguous. When J < 0 and Y > 0, obviously, E > 0.
When J > 0 and Y < 0, condition E > 0 can also be given as n0 > ñ. However, when J andY have the same sign, E > 0 must not hold under ṅ > 0.

References[1] Devereux.M and Love.D.R.F (1994) The eﬀects of factor taxation in a two-sector model ofendogenous growth, Canadian Journal of Economics, Vol.27, pp.509-536.
[2] King.R.E and Rebelo.S (1990) Public policy and economic growth: developing neoclassicalimplications, Journal of Political Economy, Vol.98, pp.S126-S150.
[3] Futagami.K and Doi.J (2004) Commodity taxation and economic growth, Japanese Economic Review, Vol.55, pp.46-55.
[4] Grossman.G.M. and Helpman.E (1991) Innovation and growth in the global economy. Cambridge, Mass.: MIT Press.
9[5] Jorgenson.D.W (1998) The growth-boosting power of a consumption tax, in J.Jasinowski(ed.), The rising tide, New York: John Wiley, pp.223-229.
[6] Milesi-Ferretti.G.M and Roubini.N (1998) Growth eﬀects of income and consumption taxes,
Journal of Money, Credit and Banking, Vol.30, pp.721-744.
[7] Pecorino.P (1993) Tax structure and growth in a model with human capital, Journal of
Public Economics, Vol.52, pp.251-271.
[8] Rebelo.S (1991) Long-run policy analysis and long-run growth, Journal of Political Economy,
Vol.99, pp.500-521.
[9] Stokey.N.L Rebelo.S (1995) Growth eﬀects of ﬂat-rate taxes, Journal of Political Economy,
Vol.103, pp.519-550.

10

Figure 1-2

Figure 1-1

Figure 1-3

11Table 1

Figure 2

Figure 3-1

Figure 3-2

12

Table 2

Figure 4 : the shift represents lowering σ

Figure 513

Figure 6-2

Figure 6-1

14