Theoretical Economics Letters, 2013, 3, 216-219
http://dx.doi.org/10.4236/tel.[phone] Published Online August 2013 (http://www.scirp.org/journal/tel)
Monetary Growth Theory under Perfect and
Monopolistic Competitions
Masayuki Otaki1, Masaoki Tamura2
1Institute of Social Science, University of Tokyo, Tokyo, Japan
2Institute of Innovation Research, Hitotsubashi University, Tokyo, Japan
Received May 13, 2013; revised June 13, 2013; accepted July 13, 2013
Copyright © 2013 Masayuki Otaki, Masaoki Tamura. This is an open access article distributed under the Creative Commons Attribu-
tion License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
ABSTRACT
This article analyzes the difference of properties of economic growth theory between perfect and monopolistic competi-
tion. Whether or not capital investment is constrained by effective demand is the crucial factor which characterizes
economic growth theories in different degree of competition. Whenever each firm faces a downward sloping demand
curve the location of which is determined by the strength of effective demand (i.e., the real GDP), its capital accumula-
tion is inevitably constrained by effective demand. Thus, as far as business environment is kept unchanged, so is capital
investment. However, when the good market is perfectly competitive, firms never perceive such demand constraint,
thereby capital investment advancing autonomously independent of the phase of business cycle. An important macro-
economic implication of such a difference of the attitude toward capital investment is as follows. When an economy is
in perfect competition, capital investment becomes an independent driving force of economic growth as Keynes con-
siders, although it is subject to other independent expenditure (e.g., the government expenditure) and falls into a sub-
sidiary component of effective demand otherwise.
Keywords: Sustainability of Fiscal Expenditure; Monetary Economic Growth Theory; Investment Theory under
General Equilibrium
1. Introduction
It is almost unknown how the market structure of goods
markets affects economic growth in a monetary economy.
Although Dixit and Pindyck [1] and Smets [2] built
models of investment function under uncertainty and
pointed out that the function depends on the level of ef-
fective demand, it is not their concern with how such
investment relating to economic growth as a whole each
other. Otaki [3] developed a general equilibrium growth
model under monopolistic competition, and also found
that there exists no endogenous economic force for sus-
tainable growth in a monetary economy.
In his seminal work, Uzawa [4] analyzed properties of
the investment function under perfect competition in the
context of general equilibrium model. Although his the-
ory entirely excludes the existence of money, he found
that the optimal investment ratio to capital is free from
the level of effective demand. The optimal ratio is de-
pendent on the profit rate which is endogenously deter-
mined only by relative prices. Such a prominent property
of the investment function implies that, differing from
the monopolistic competition case analyzed by Otaki [3],
capital investment enables an economy to sustain its
growth. This is because the accumulated past capital in-
vestments (i.e., existing capital stock itself) empowers
current investment without referring to the condition of
effective demand1, and the resources are rewarded by
whole earned quasi rents within the firm.
The main theoretical issue addressed in this article is
to check the validity of Uzawa’s [4] assertion that capital
investment calls forth future investment expansion under
1It is quite ambiguous why more capitals enhance more investment in
Uzawa [4]. He attributes such a property to the existence of manageria
l
resources. However, it seems difficult to exhibit the substance o
f
managerial resources. Otaki [5], instead, introduces the concept o
f
dexterity of labor forces, which provides physical capital with positive
externalities such as process innovations. A firm is regarded as an in-
genious device for the internalization of such externalities, and thus,
since there is no limit to sale under perfect competition, a firm attains
sustainable growth together with the accumulation of dexterity.
C
opyright © 2013 SciRes. TEL

M. OTAKI, M. TAMURA 217
perfect competition and that economic growth is sus-
tainable even in a monetary economy. The result is as
follows. Since the effective demand principle works be-
cause of the indeterminacy of equilibrium price sequence
(see, for more detail Otaki [6]), an economy is not nec-
essarily able to attain GDP which guarantees the full re-
source utilization even though goods markets are com-
petitive. However, since capital investment becomes an
autonomously expanding independent expenditure of
effective demand, the suspending power to economic
growth becomes fortified compared with the case of
monopolistic competition as analyzed in Otaki [3]. Con-
sequently, government deficits necessary for attaining
full resource utilization grows only at a constant rate,
which is equalized to the GDP growth rate, although
such a rate is accelerated together with capital accumula-
tion in monopolistic competition case (see Theorem 2 in
Otaki [3]).
The remainder of this paper is organized as follows.
We construct and analyze the two-period overlapping-
generations monetary growth model with impure altru-
ism2 in Section 2. In Section 3, we analyze the relation-
ship between the competitiveness of markets and the
fiscal sustainability. Section 4 contains concluding re-
marks.
2. The Model
2.1. Structure of the Model
There are two strata in this economy: employers and em-
ployees. Each employee provides his unit labor when he
is young at his discretion. The disutility is denoted as

.
His lifetime utility which comes from the consumption
stream is a Cobb-Douglas function (note that
such a function is common with employers). Thus, the
lifetime utility is defined as

121
,
tt
cc

U


 
1
121 ,0 1,
ss
tt t
Uc cs 


 (1)
where t

is a definition function the value of which
takes unity when employed and zero when unemployed.
Without loss of generality and mainly for simplifying the
calculation, we assume that the economy is located at the
full employment equilibrium (The full-employment level
is fixed to unity).
On the other hand, each employer hires employees to
produce goods and gain profit when he is young. In addi-
tion, he makes investments for the next generation, be-
cause we assume that every employer holds impure al-
truism; he not only concerns with his own lifetime utility
but also descendant’s utility. His inheritances are seeds
of the dexterity of future employees via investments on
education (we assume that it takes a considerable length
of time for such education being effective). His invest-
ments enable young employers in the next period to hire
employees. There is no disutility of labor in this stratum.
The government newly issues fiat money to finance its
fiscal expenditures which is, for simplicity, bear no addi-
tional utility in the private sectors. It is also assumed that
the government pays real dole in proportion to his
dexterity t, which is null since our reference point is
the full-employment equilibrium. The arbitrage condition
within the labor market requires
d
L
1,
R
ttt
pdW L  (2)
where
R
t
W is the nominal reservation wage, which is
endogenously determined as below.
The budget constraint of the government becomes
 
1
1
11
1
1π1
tt tt
tt t
tt tttt
t
tt
t
MM pG
1
M
MG
pL pLpL
m
gm











, (3)
where
11
11
,,1,
tttt
ttt
tt tttt
MGp L
mg
pL pLpL


  2
1.


t is current price of the good which is produced in
the economy. t
p
g
is the real government expenditure per
an efficient unit labor force. t is the labor force per an
employee measured by the efficiency unit. θ denotes the
degree of progress in dexterity nurtured by the em-
ployer’s capital investment.
L
2.2. Agent’s Maximization Problems
2.2.1. Employers
An employer is assumed to be impure altruistic. His
marginal substitution rate between his own consumption
and his descendant’s income is fixed to unity. Further-
more, since the marginal substitution rate between cur-
rent and future consumption is equalized to the gross
inflation rate 1πt

, which is common in both capital
investment and money hoarding decisions, the optimal
capital investment decision problem becomes equivalent
to the maximization problem on the discounted net cash
flow obtained from capital. Accordingly, the optimal
economic behavior of an employer can be expressed by
the following equations3.
3Although, for simplicity, we henceforth assume that the nominal wage
is equal to the nominal reservation wage, it is natural to consider that
the nominal wage is determined through bargaining process since labor
forces are regarded as quasi fixed production factor. However, even
though we introduce such negotiation process into the model, obtained
results are ke
p
t intact. For detail, see Otaki [8].
2See, for example, Acemoglu [7] on the detail of impure altruism.

M. OTAKI, M. TAMURA 218
1,
ER R
ttt
SrLWL




1



(4)

*,1 πr





 (5)
where
E
R
S
r
is the aggregate savings of employer stra-
tum. is the rate of return from skilled labor force.

denotes the average adjustment cost for educating and
nurturing dexterity, which is defined as
 
11
,,0,
tt t
LL L  


 0,
where is the total adjustment cost.

2.2.2. Employees
Since the lifetime utility function of consumption is
Cobb-Douglas form, the aggregate savings of employees
E
E
S is
1.
EE R
tt
SsWL

 (6)
In addition, the indirect lifetime utility t
I
U becomes
  
1
1
1
1
1
.
R
1
s
s
R
tt
tttt
ss
tt
WL
IUW Lpp
pp






t 
(7)
Combining (7) with (2), we obtain the following fun-
damental equation concerning the dynamic motion of
equilibrium price sequence.
 
1
1
11π.
s
ss
ttt
d
pdp p 





 (8)
2.3. Market Equilibrium: The Relationship
between Market Competitiveness and
Autonomy of Capital Investment
We have two markets in the model: the good market and
money market. By Walras’ law, we can concentrate the
equilibrium condition for the good market. By adding up
(4) and (6), the saving function of the economy as a
whole is
t
S
1. tt
SsrL

 (9)
To avoid the unessential non-linearity in the invest-
ment function, we assume that the average adjustment
cost function

is a power function. That is,

,1

  .
Then, from the optimality condition for the optimal
capital investment (5), the investment function t
I
is
derived as

1
1
1π,
tt
r
1t
I
L


 









L

(10)
where 1


 is the equilibrium inflation rate in (8).
Furthermore we assume that, differing from Otaki [3],
the growth rate of fiscal deficits per labor force measured
in efficiency unit is set at zero and 1tt
mm m

 holds.
The government budget constraint (3) is transformed into
1
1.
1π1
g
m














(11)
Equations (9), (10) and (11) lead us to the following
equilibrium condition for the good market normalized by
existing labor force in terms of efficiency unit 1t L

as
1
1
1
111
1,
r
s
rg
r
sr m















 

 





 



m
(12)
where the third term of the right-hand side of above
equation in (12) is the expenditure of the old generation
per efficiency unit of labor force. As far as the real cash
balance in terms of efficiency unit of labor force is
determined so that both sides of (12) are equalized, the
economy can sustain the full capacity utilization4. Since
the first term of the right-hand side of (12) is the contri-
bution of the capital investment to the growth rate of
GDP per capita, this implies, in contrast with Otaki [3],
that the capital investment autonomously and steadily
grows free from the level of effective demand.
m
3. The Analysis: Market Competitiveness
and the Fiscal Sustainability
Since it is apparent that the monetary growth rate under
the full capacity utilization equilibrium is equal to that of
nominal GDP, 11





 


 , we finally obtain the
following theorem.
Theorem
The growth of the monetary economy under perfect
competition is sustainable in the sense that the ratio of
public debts to nominal GDP is kept constant over time.
The above theory is quite contrastive with properties
of the monetary growth model under monopolistic com-
petition in Otaki [3], in which the public debts-nominal
GDP ratio t
t
G
Y is explosive. The decisive economic rea-
son whether such a ratio is explosive or not is whether
employers are subject to the effective demand constraint.
Employers’ capital investment is substantively affected
by whether or not they face the demand constraint.
4As Otaki [9] argues, if individuals rationally believe that the future
p
urchasing power of money is unaffected by the change of current
nominal money supply Mt (i.e., money is credible), current price pt
also becomes insensitive to Mt (see (8), and thus, the government can
control the real cash balance.

M. OTAKI, M. TAMURA 219
In perfect competition case, capital investment enables
every employer to expand the production without the
demand constraint in the long run. It immediately implies
that the aggregate capital investment is autonomously
expanded, and hence it consists of an endogenous force
of economic growth as incessant effective demand stim-
uli. The above perfect competitiveness case is a limit
case of Otaki [3] in which the price elasticity of each
firm’s good

.
Meanwhile, capital investment is constrained by the
effective demand in case of monopolistic competition
. In other words, capital investment does
not have a power enough to create new additional de-
mand per se. Capital investment enables every employer
only to reduce production cost. Thus, other exogenous
expansionary shocks, such as acceleration of fiscal ex-
penditure, are indispensable with sustaining economic
growth. Accordingly, fiscal deficits and the public debts-
nominal GDP ratio become explosive as proved by Otaki
[3].

0  

To summarize, as goods are standardized, markets are
more competitive and

becomes large, the constraint
of aggregate demand to which capital investment is sub-
ject fades away. Thus, capital investment is empowered
enough to create new demand by itself and the fiscal
sustainability is heightened. The limit case, where mar-
kets are under perfect competition, is precisely analyzed
in this article because this is the most prominent case for
exhibiting the relationship mentioned above.
4. Conclusions
This paper analyzed how market competitiveness relates
to the sustainability of economic growth. The obtained
result is as follows. Because there is no demand con-
straint whenever market is competitive, capital invest-
ment creates additional effective demand in the future by
itself. Such fact implies that an economy steadily grows
without unsustainable help from its government.
In turn, as Otaki [3] shows, goods provided in the
economy are differentiated even narrowly and markets
become less competitive, every employer perceives that
he faces a downward-sloping demand function the loca-
tion of which is determined by effective demand. There-
fore, capital investment is subject to effective demand,
thus, loses the driving force for economic growth. The
progress of labor productivity by capital investment
needs an explosive fiscal expenditure to maintain the full
resource utilization equilibrium.
In this sense, competitiveness plays a key role on sus-
taining stable fiscal balance with moderate economic
growth.
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