Monetary Growth Theory under Perfect and Monopolistic Competitions

doi:10.4236/tel.2013.34036

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Theoretical Economics Letters, 2013, 3, 216-219

http://dx.doi.org/10.4236/tel.2013.34036 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

Email: [email protected], [email protected]

Received May 13, 2013; revised June 13, 2013; accepted July 13, 2013

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

resources. However, it seems difficult to exhibit the substance o

managerial resources. Otaki [5], instead, introduces the concept o

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.

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







 

121 ,0 1,

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

ttt

pdW L

 (2)

where

W is the nominal reservation wage, which is

endogenously determined as below.

The budget constraint of the government becomes



1π1

tt tt

tt t

tt tttt

MM pG

pL pLpL

















, (3)

where

,,1,

tttt

ttt

tt tttt

MGp L

pL pLpL





 





t is current price of the good which is produced in

the economy. t

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.

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

t intact. For detail, see Otaki [8].

2See, for example, Acemoglu [7] on the detail of impure altruism.

M. OTAKI, M. TAMURA

218

ER R

ttt

SrLWL















(4)



*,1 πr











 (5)

where

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



,,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

S is

EE R

SsWL



 (6)

In addition, the indirect lifetime utility t

U becomes

  

tttt

IUW Lpp















(7)

Combining (7) with (2), we obtain the following fun-

damental equation concerning the dynamic motion of

equilibrium price sequence.

 

11π.

ttt

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

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,





 

.

Then, from the optimality condition for the optimal

capital investment (5), the investment function t

derived as



1π,



 

















(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

1π1





























(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



111

sr 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.

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

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

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].









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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