Theoretical Economics Letters, 2013, 3, 288-291
http://dx.doi.org/10.4236/tel.2013.35048 Published Online October 2013 (http://www.scirp.org/journal/tel)
Minimum Wage, Public Investment, Economic Growth
Minoru Watanabe
Graduate School of Economics, Kobe University, Kobe, Japan
Email: mino.watanabe@gmail. com
Received August 8, 2013; revised September 7, 2013; accepted September 15, 2013
Copyright © 2013 Minoru Watanabe. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
This paper considers the relationship between economic growth and minimum wage. Minimum wage helps reduce pov-
erty and maintain a minimum stand ard of living. However, it is also claimed that minimum wage has a negative effect
on employment and GDP. This paper develops a simple two-period overlapping generation model with three economic
policies, minimum wage, unemployment benefit, and public investment that improves labor productivity. The govern-
ment imposes tax on firms to finance public capital and unemployment benefit under a balanced budget. We show that
economic growth is promoted with an increase in minimum wage and the ratio of public investment to tax revenue.
Keywords: Minimum Wage; Public Investment; Economic Growth
1. Introduction
Minimum wage is an economic policy that helps reduce
poverty and maintain a minimum standard of living.
However, it is also claimed that minimum wage has a
negative effect on employment and GDP. Therefore, ear-
lier studies consider the relationship between minimum
wage and economic growth in a dynamic framework.
Cahuc and Michel [1] introduce human capital accumu-
lation in an endogenous growth model. Ravn and Soren-
sen [2] consider skill formation accumulated by school-
ing and training. Askenazy [3] develops endogenous
growth model with an open economy and R&D sector.
Irmen and Wigger [4] consider minimum wage with a
two-country endogenous growth model. Tamai [5] dis-
cusses the interaction between inequality and economic
growth from the view point of political economy. The
above papers assume two sectors or heterogeneous agents.
Fanti and Gori [6] consider the relationship between
economic growth and minimum wage under a simple
one-sector overlapping generation model with 1homoge-
neous agents. It is 2uncertain whether minimum wage
promotes economic growth in earlier papers.
This paper introduces public investment that improves
labor productivity, for example, infrastructure and medi-
cal service. Public investment is an economic po licy with
an important role in macroeconomic performance. Barro
[7], Futagami, Morita and Shibata [8] examine economic
growth with public investment that improves labor pro-
ductivity. Glomm and Ravicumar [9] discuss economic
growth including public investment and human capital
accumulation. Yakita [10] considers public investment in
an aging society.
Following Fanti and Gori [6], this paper introduces
public investment that enhances labor productivity and
considers the relationship between minimum wage and
economic growth. The results obtained in this study are
presented below. First, an increase in minimum wage
always promotes economic growth. Second, an increase
in the ratio of public investment to tax revenue promotes
economic growth.
The remainder of this paper is organized as follows.
Section 2 presents our model. Section 3 describes the
equilibrium. Section 4 summarizes the paper.
2. Model
2.1. Basic Structure
The economy in this paper is based on a basic two-period
overlapping generations framework. There exist three
agents, households, firms, and the government.
2.2. Households
1They introduce heterogeneous agents in extension.
2Whether minimum wage promotes economic growth or not depends
on parameters or assumptions in Cahuc and Michel [1], Ravn and
Sorensen [2], Askenazy [3], Irmen and Wigger [4], and Fanti and Gori
[6].
Households live two periods, young and old and supply
one unit of labor to the labor market. If they are em-
ployed they receive wages, and if they are not employed,
C
opyright © 2013 SciRes. TEL
M. WATANABE 289
they receive unemployment benefit. The utility function
in this paper is:
,
log log
ii i
tyt o
uc c
,1t
(1)
where ,
i
y
t
c
i
is the young-period consumption of house-
hold i, ,1ot is the old-period consumption of household
i; i is the index of both employment and unem-
ployment , and
c
ie

iu
0, 1
is the constant dis-
count factor. The budget constraint of households i is
given as:
,
iii
y
tt
csx
t
x
(2)

,1 1
1
i
ott t
cr

 i
(3)
where i
t
x
is the income in young period, i
t
s
is the sav-
ings, and is the interest rate. If the households are
employed, they receive wags, and if they are not em-
ployed, they receive unemployment benefit. In this
economy, minimum wages exist. Therefore the relation
between minimum wage, and competitive wage,
, is given as:.
1t
r
,mt
w
,ct
w
,mt ct
ww
,
(4)
where 1
is the constant mark-up rate that generates
unemployment in the labor market. Households receive
unemployment benefit when they are not employed,
and is defined as: t
b
t
b
,tm
bw
t
(5)
where is the constant replacement rate.
Therefore unemployment benefit is fraction of minimum
wage. The optimal allocations of household i are given
by:
0,1
,1
1
ii
y
tt
cx
(6)

,1 1
1
1
ii
ott t
c

rx
(7)
2.3. Firms
This paper assumes 3the production function as follows:

1
tttt
YAKGL
(8)
where t
K
is the capital stock, t is the public invest-
ment, is the labor input, A is the constant parameter,
and is the constant parameter. This paper
assumes neither depreciation nor population growth.
Public investment and unemployment benefit are fi-
nanced by tax revenue from firms. The profit maximize-
tion condition s are given as:
G
t
L

0, 1
1
,1
mtt tt
wAKGL


 (9)
 
1
1
1ttt tt
rKGL

 (10)
where t
is the contribution rate for firms to finance
public investment and unemployment benefit. In the
competitive equilibrium, and
1
t
L
*1
,1
cttt
wAKG

hold. Using with Equation (5) and (9), is pre-
sented as:
*,ct
wt
L
1
t
L
(11)
Because t is constant, the unemployment rate, L
t
L1
t
u
, is also constant for any period t.
2.4. Government
The government imposes tax on firms to finance public
capital and unemployment benefit. Assuming a balanced
budget, the budget constraint of the government is given
by:
,
g
ttt
Er
t
K
(12)
,t
GE
gt
(13)
,
1
tt gt
ubE
 (14)
0, 1
where tt t
rK
is the tax revenue from firms, ,
g
t is the
government expenditure, t
G is the public investment,
tt
is the total unemployment benefit,
E
ub
is the con-
stant ratio of public investment to tax revenue, and
1
is the constant ratio of unemployment benefit to
tax revenue. Using Equation (5), (9), (10), (12), (14), and
t
1
t
uL
the budget constraint of the government is
described as follows:



1
1
1
11
1
1
1
ttt t
ttt t
t
uAK Gu
AK Gu




where


1
11
1
ttt t
uAK Gu


is the total expenditure for unemployment benefit, and

1
11
1tttt
t
AK Gu

is the tax revenue. Hence t
is given by:


1
111
t
t
tt
u
u




(15)
3Barro [7] and Futagami and Morita and Shibata [8] assume public
investment in the production functionand public investment increases
labor productivity.
Copyright © 2013 SciRes. TEL
M. WATANABE
290
From Equation (9), (10), (12), (13), and (15), is
given by: t
G


111
11
1
tt
A
Gu
 




tt
uK
(16)
Tax rate t
is an endogenous variable delivered by a
balanced budget and this 4assumption is the same as in
Fanti and Gori [4]. Equation (15) shows that the tax rate
is an 5increasing function of the unemployment rate and
the ratio of public capital to tax revenue. The intuition is
described as follows. If unemployment rate increases,
then minimum wage increases and interest rate decreases
because labor force becomes relatively scarce to capital.
On the other hand, unemployment benefit t also in-
creases with an increase ,mt because unemployment
benefit is fraction of minimum wage, and the total unem-
ployment benefit tt
increases. We assume the ratio of
total unemployment benefit to tax revenue is constant
and balanced budget. From Equation (12) and (14), the
relationship between total expenditure of unemployment
benefit and tax revenue is denoted as tttt .
To satisfy balanced budget when total expenditure of
unemployment benefit, tt
, increases and interest rate,
t, decre ases with an incr ease in unemployment rate, t
b
1
w
ub
ub

ub rK

r
should increase. Hence tax rate is an increasing function
of unemployment rate.
From Equation (16), public investment is also an in-
creasing function of unemployment rate; the reason for
this is described as follows. When the unemployment
rate increases, there are two effects to public investment.
First, an increase in the unemployment rate increases t
from Equation (15), and this enlarges public investment
because the tax revenue increases. Second, an increase in
the unemployment rate decreases t because labor force
becomes relatively scarce to capital, and this decrease
public investment because the tax revenue decreases.
Comparing the two effects, the first effect dominates the
second effect.
r
3. Equilibrium
In a basic overlapping generations model, the capital
stock in period is equal to the savings in period t.
The relationship between ca pi tal and savi ngs is:
1t

11e
ttt
u
tt
K
us us
 (17)
where 1t
K
is the capital in period , is
the total savings of employees, and is the total
savings of unemployment. The dynamics of this econ-
omy is shown as follows:
1tu
tt
us

1e
tt
us

11 1
11
1
tttt
t
KZuuu
K


(18)


1
1
11
ZA


 

where 1tt
K
K
is the growth rate in the economy. To
satisfy sustained growth, a large A is assumed. From
Equation (11), the unemployment rate, t, is constant
and an increasing function of the constant mark-up rate
u
. The derivative of growth rate with respect to
gives:
ddd0
ddd
ttt
t
ggu
u

(19)
1ttt
g
KK
Therefore the following proposition is established.
Proposition 1
An increase in the constant mark-up rate increases the
minimum wage and promotes economic growth.
The total unemployment income tt
increases when ub
increases. Therefore the savings of unemployment
increases. On the other hand, an increase in
has two
effects on the total income of employees tmt
.
First, an increase in
1
,
uw
directly decreases the total in-
come of employees because unemployment increases.
Second, an increase in
increases ,mt
wbecause labor
force becomes scarce and labor productivity is promoted.
Comparing the two effects, the second effect dominates
the first effect. Because the total income increases with
an increase in
, the savings this economy also in-
crease. Therefore economic growth is promoted with an
increase in
in
.
Finally, we focus on the effect of the ratio of public
investment tax revenue on growth rate. Public investment
increases with an increase in the constant ratio of public
capital to tax revenue, from Equation (16). The deriva-
tive of growth rate with respect to
gives:
d0
dt
g
(20)
Therefore the following proposition is established.
Proposition 2
An increase in the ratio of public investment to tax
revenue promotes economic growth.
An increase in the ratio of public investment to tax
revenue,
, promotes public investment, and this in-
creases the minimum wage and unemployment benefit,
from Equation (5). Therefore both the total wage income
and total unemployment benefit increase with an increase
in the ratio of public investment. This means that the
4Fanti and Gori [6] also consider the exogenous replacement rates in
extension.
5From equation (15), the derivative of tax rate with respect to
unemployment rate gives:

 
2
2
11
d0
d11
t
t
tt
uu
 




Copyright © 2013 SciRes. TEL
M. WATANABE
Copyright © 2013 SciRes. TEL
291
total expenditure on unemployment benefit described as
increases even if the ratio of unem-
ployment benefit to tax revenue decreases.

,
1
tt gt
ub E

4. Conclusion
This paper presents a simple endogenous growth model
with minimum wage and public investment that improves
labor productivity. Minimum wage is an economic policy
that helps reduce poverty and maintain a minimum stan-
dard of living. However, it is also claimed that minimum
wage has a negative effect on employment and GDP.
The relationship between minimum wage and economic
growth is uncertain in earlier papers. This paper shows
that a rise in minimum wage always promotes economic
growth. Moreover, this paper shows that a rise in the
ratio of public investment to tax revenue promotes eco-
nomic growth.
5. Acknowledgements
The author would like to thank Tamotsu Nakamura, Ta-
keshi Koba, and Masao Yamaguchi for their helpful
comments and suggestions. The author is responsible for
any remaining errors.
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