DBJ Discussion Paper Series, No.1302

Monetary Economic Growth Theory
under
Perfect and Monopolistic Competitions

Masayuki Otaki(Institute of Social Science, University of Tokyo)
Masaoki Tamura(Innovation Research Center, Hitotsubashi University)

May

2013

Discussion Papers are a series of preliminary materials in their draft form. No quotations,reproductions or circulations should be made without the written consent of the authors inorder to protect the tentative characters of these papers. Any opinions, findings,
conclusions or recommendations expressed in these papers are those of the authors and do
not reflect the views of the Institute.

Monetary Economic Growth Theory
under
Perfect and Monopolistic Competitions
Masayuki Otaki∗ and Masaoki Tamura
Institute of Social Science, University of Tokyo,
Innovation Research Center, Hitotsubashi University
Abstract
This article analyzes the diﬀerence in the properties of economic growththeory between perfect and monopolistic competition. Whether or not capital investment is constrained by eﬀective demand is the crucial factor thatcharacterizes economic growth theories in diﬀerent degrees of competition.
Whenever a ﬁrm faces a downward sloping demand curve the location ofwhich is determined by the strength of eﬀective demand (i.e., the real GDP),
its capital accumulation is inevitably constrained by eﬀective demand. Thus,
as far as the business environment is kept unchanged, capital investment is
as well. However, when the goods market is perfectly competitive, ﬁrms
never perceive such a demand constraint, and thus, capital investment autonomously advances independent of the phase of the business cycle.
An important macroeconomic implication of such a diﬀerence in the attitude toward capital investment is as follows. When an economy is in perfectcompetition, capital investment becomes an independent driving force ofeconomic growth as Keynes considers, although it is subject to other independent expenditure (e.g., government expenditure) and falls into a subsidiary component of eﬀective demand otherwise.

∗Corresponding author. ohtaki@iss.u-tokyo.ac.jp

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Introduction

Little is known how the market structure of goods markets aﬀects economic
growth in a monetary economy. Although Dutta and Rustichini [1] andSmets [2] built models of investment function under uncertainty and pointed
out that the function depends on the level of eﬀective demand, they were notconcerned with how such investment related to economic growth as a whole.
Otaki [3] developed a general equilibrium growth model under monopolisticcompetition and also found that there exists no endogenous economic force
for sustainable growth in a monetary economy.
In his seminal work, Uzawa [4] analyzed the properties of the investment
function under perfect competition in the context of a general equilibrium
model. Although his theory excludes the existence of money, he found thatthe optimal ratio of investment to capital is free from the level of eﬀectivedemand. The optimal ratio is dependent on the proﬁt rate which is endogenously determined only by relative prices. Such a prominent property of
the investment function implies that, in contrast of the monopolistic competition case analyzed by Otaki [3], capital investment enables an economy
to sustain its growth. This is because the accumulated past capital investments (i.e., the existing capital stock itself) empowers current investment
regardless of the condition of eﬀective demand1
The main theoretical issue addressed in this article is to check the validity of Uzawa’s [4] assertions that capital investment stimulates future
investment expansion under perfect competition and that economic growthis sustainable even in a monetary economy. The result is as follows. Since
the eﬀective demand principle works because of the indeterminacy of the1

It is quite ambiguous why more capitals enhance investment in Uzawa [4]. He at-tributes this property to the existence ofmanagerial resources, which are rewarded bywhole earned quasi-rents within the ﬁrm. However, it seems diﬃcult to identify the substance of managerial resources. Otaki [5] instead introduces the concept of the dexterityof labor forces, which provides physical capital with positive externalities such as process
innovations. A ﬁrm is regarded as an ingenious device for the internalization of suchexternalities, and thus, since there is no limit to sale under perfect competition, a ﬁrm
attains sustainable growth together with the accumulation of dexterity.

1equilibrium price sequence (for more detail Otaki [6]), an economy is notnecessarily able to attain a GDP that guarantees full resource utilizationeven though goods markets are competitive. However, since capital investment becomes an autonomously expanding independent expenditure of effective demand, the suspending power to economic growth becomes stronger
compared with the case of monopolistic competition as analyzed in Otaki[3]. Consequently, the government deﬁcits necessary for attaining full resource utilization grow only at a constant rate, which is equalized to theGDP growth rate, although such a rate is accelerated together with capital
accumulation in the monopolistic competition case (see Theorem 2 in Otaki[3]).
The remainder of this paper is organized as follows. In Section 2, we construct and analyze a two-period overlapping-generations monetary growthmodel with impure altruism2 . Section 3 contains concluding remarks.

2The Model

2.1

Structure of the Model

There are two strata in this economy: employers and employees. Each
employee holds impure altruism; he is not only concerned with his own
lifetime utility but also his descendant’s young-time utility. The inheritanceshe leaves are the seeds of the dexterity of future employees via the educationof current incumbent employees (we assume that it takes a considerable
length of time for such education to be eﬀective). There is no disutility oflabor in this stratum.
Each employee provides his unit of labor when he is young at his discretion. The disutility is denoted as α. His lifetime utility which comes fromthe consumption stream (c1t , c2t+1 ) is a Cobb-Douglas function (note thatsuch a function is common with employers). Thus, the lifetime utility U isdeﬁned as
U ≡ [c1t ]1−s [c2t ]s − δt · α,
20 < s < 1,See, for example, Acemoglu [7] on the details of impure altruism.

2(1)where δt is a deﬁnition function the value of which takes unity when employed and zero when unemployed. Without loss of generality to simplify thecalculation, we assume that the economy is located at the full employment
equilibrium (the full-employment level is ﬁxed to unity.).
The government newly issues ﬁat money to ﬁnance its ﬁscal expenditureswhich is, for simplicity, bear no additional utility in the private sectors. Itis also assumed that the government pays real dole d in proportion to hisdexterity Lt , which is null since our reference point is the full-employment
equilibrium. The arbitrage condition within the labor market requires
pt d = WtR Lt−1 ,(2)where WtR is the nominal reservation wage, which is endogenously determined as below.
The budget constraint of the government becomesMt − Mt−1 = pt Gt⇔gt = mt −mt ≡Mt,
pt Lt−1⇔MtMt−1
Gt−=
pt Lt−1 pt Lt−1
pt Lt−1

mt−1, (3)
[1 + πt ][1 + θ]

wheregt ≡Gt,
pt Lt−1πt ≡pt+1− 1,ptθ≡

Lt−1− 1.
Lt−2

pt is the current price of the good produced in the economy. gt is the real
government expenditure per eﬃcient unit of the labor force. Lt is the laborforce per employee measured by the eﬃciency unit. θ denotes the degree ofprogress in dexterity nurtured by the employer’s capital investment.

2.2
2.2.1

Agents’ Maximization Problems
Employers

Since an employer is assumed to be impure altruistic, his marginal substitution rate between his own future come and his descendant’s income is ﬁxed
to unity. Hence, the consumption/saving decision is independent of whether
savings are put into capital stock or money. Furthermore, since the marginal
substitution rate between current and future consumption is equalized to the3gross inﬂation rate 1 + πt , which is common in both capital investment andmoney hoarding decisions, the optimal capital investment decision problembecomes equivalent to the maximization problem on the discounted net cash
ﬂow obtained from capital. Accordingly, the optimal economic behavior ofan employer can be expressed by the following equations3 .
S ER = s[rLt−1 − WtR Lt−1 ],(4)ϕ′ (θ∗ ) = [1 + πt ][r − ϕ(θ∗ )],(5)

where S ER is the aggregate savings of employer stratum. r is the rate of
return from skilled labor force. ϕ denotes the average adjustment cost for
educating and nurturing dexterity, which is deﬁned as
Φ(Lt , Lt−1 ) ≡ ϕ(θ)Lt−1 ,ϕ′ ϕ′′ > 0,

where Φ is the total adjustment cost.
2.2.2

Employees

Since the lifetime utility function of consumption is the Cobb-Douglas form,
the aggregate savings of employees S EE is
S EE = s · WtR Lt−1 .

(6)

In addition, the indirect lifetime utility IUt becomes
IUt =WtR Lt−1
[pt ]1−s [pt+1 ]s⇒

WtR Lt−1 = α[pt ]1−s [pt+1 ]s .

(7)

Combining (7) with (2), we obtain the following fundamental equation concerning the dynamic motion of the equilibrium price sequence.
⇒pt d = α[pt ]1−s [pt+1 ]s3d 1
1 + π∗ = [ ] s .
α(8)Although, for simplicity, we henceforth assume that the nominal wage is equal to thenominal reservation wage, it is natural to consider the nominal wage to be determinedthrough a bargaining process since labor forces are regarded as quasi-ﬁxed production
factors. Even though we introduce such a negotiation process into the model, the obtained
results intact. For details, see Otaki [8].

42.3Market Equilibrium

We have two markets in the model: the goods market and the money market.
By Walras’s law, we can concentrate the equilibrium condition for the goodsmarket. By adding up (4) and (6), the saving function of the economy as awhole St is
St ≡ s · rLt−1 .

(9)

To avoid unessential non-linearity in the investment function, we assume
that the average adjustment cost function ϕ is a power function. That is,ϕ(θ) ≡ θβ ,β > 1.

Then, from the optimality condition for the optimal capital investment (5),the investment function It is derived asIt ≡ ϕ(θ∗ )Lt−1 =ϕ′ (θ∗ )r[1 + π ∗ ]Lt−1 =
Lt−1 ,ββ[2 + π ∗ ](10)where 1 + π ∗ is the equilibrium inﬂation rate in (8).
Furthermore, we assume that, in contrast to from Otaki [3], the growthrate of the ﬁscal deﬁcit in terms of the labor forces is set to zero and m =mt = mt−1 holds. The government budget constraint (3) is transformed into
[g = 1−]1m.
[1 + π ∗ ][1 + θ∗ ](11)Equations (9), (10) and (11) lead us to the following equilibrium condition for the good markets normalized by the existing labor force in terms ofeﬃciency units Lt−1 asr[1 + π ∗ ]1+g+m
β[2 + π ∗ ][1 + π ∗ ][1 + θ∗ ]r[1 + π ∗ ]s·r =+ m,
β[2 + π ∗ ]

s·r =
⇒(12)

where the third term of the right-hand side of the above equation in (12)is the expenditure of the old generation in terms of the eﬃciency units oflabor force. As far as the real cash balance in terms of eﬃciency units of5the labor force m is determined so that both sides of (12) are equalized, theeconomy can sustain full capacity utilization4 .
Since it is apparent that the monetary growth rate under the full capacity
utilization equilibrium is equal to that of nominal GDP [1 + π ∗ ][1 + θ∗ ], we
ﬁnally obtain the following theorem.
Theorem 1 The growth of the monetary economy under perfect competitionis sustainable in the sense that the ratio of public debts to nominal GDP iskept constant over time.
The theory contrasts sharply with the properties of the monetary growthmodel under monopolistic competition in Otaki [3], in which the publicGt
debts-nominal GDP ratio ( ) is explosive. The decisive economic reason
Yt
that determines whether such a ratio is explosive or not is whether employers
are subject to the eﬀective demand constraint. In the perfect competition
case, every employer can expand his ﬁrm without any constraint in the longrun. This implies that the aggregate capital investment is autonomously
expanded, and hence it consists of an endogenous force of economic growth
as constant eﬀective demand stimulus.
On the other hand, capital investment is constrained by eﬀective demandin the case of monopolistic competition. Thus, other exogenous expansionary shocks such as acceleration of ﬁscal expenditure are indispensable tosustaining economic growth because capital investment does not have thepower to create new additional demand per se. Accordingly, ﬁscal deﬁcits
Gtand the public debts-nominal GDP ratio ( ) becomes explosive as proved
Ytby Otaki [3].

3Concluding Remarks

This paper analyzed how market competitiveness relates to the sustainability of economic growth. The obtained result is as follows. Because there is4

As Otaki [9] argues, if individuals rationally believe that the future purchasing powerof money is unaﬀected by a change in the current nominal money supply Mt (i.e., moneyis credible), the current price pt also becomes insensitive to Mt (see (8)), and thus, thegovernment can control the real cash balance.

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no demand constraint whenever the market is competitive, capital investment creates additional eﬀective demand in the future by itself. This fact
implies that an economy steadily grows without unsustainable help from itsgovernment.
In turn, as Otaki [3] shows, if 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 location ofwhich is determined by eﬀective demand. Therefore, capital investment issubject to eﬀective demand, and thus, it loses the driving force for economicgrowth. The progress of labor productivity by capital investment needs anexplosive ﬁscal expenditure to maintain the full resource utilization equilibrium.
In this sense, competitiveness plays a key role in sustaining a stable ﬁscal
balance with moderate economic growth.

References[1] P. K. Dutta and A Rustichini, “A Theory of Stopping Time Games
with Applications to Product Innovation and Asset Sales,” Discussion
Paper No.523, Department of Economics, 1991, Columbia University.
[2] F. Smets, “Exporting versus FDI: The Eﬀect of Uncertainty, Irreversibilities, and Strategic Interactions,” Working Paper, 1991, YaleUniversity.
[3] M. Otaki, “On the Endogenous Sustainability of Economic Growth:
Why is the Scale of Government Enlarged?” Theoretical Economics
Letters, 2013, forthcoming.
[4] H. Uzawa,

Time Preference and the Penrose Eﬀect in a Two-Class

Model of Economic Growth,

Journal of Political Economy Vol. 77,

1969, pp. 628-52.
[5] M. Otaki, “ The Evaluation of Dexterity and a Theory of the Growthof a Firm,” Modern Economy 2013, forthcoming.

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[6] M. Otaki, “The Dynamically Extended Keynesian Cross and theWelfare-improving Fiscal Policy,” Economics Letters Vol.96, 2007,
pp.23-29.
[7] D. Acemoglu, Introduction to Modern Economic Growth, Princeton
University Press 2009, Princeton, New Jersey.
[8] M. Otaki, “A Welfare Economics Foundation for the Full-employmentPolicy,” Economics Letters Vol. 102, 2009, pp.1-3.
[9] M. Otaki, “A Pure Theory of Aggregate Price Determination,” Theoretical Economics Letters Vol.1, 2011, pp.122-128.

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