DBJ Discussion Paper Series, No.1106

A Keynesian endogenous growth theory
with a rigorous microeconomic foundation

Masayuki Otaki(Institute of Social Science, University of Tokyo)

February2012

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.

A Keynesian endogenous growth theory with a rigorous
microeconomic foundation

Masayuki Otaki
Institute of Social Science, University of Tokyo
7-3-1 Hongo Bunkyo,Tokyo, 113-0033, Japan
E-mail: ohtaki@iss.u-tokyo.ac.jp
Phone: +81-3-5841-4952
Fax: +81-3-5841-4905
Keywords: Microeconomic foundation for the accelerator-principle investment function; Progress of labor productivity; Demand-driven endogenousgrowth; Self-fulﬁlling prophecy
JEL Classiﬁcations: E12, E22, E24, O42

Abstract
Extending the eﬀective demand theory developed by Otaki (2007, 2009), weconstruct a demand-driven endogenous growth theory with a rigorous microeconomic foundation. An accelerator-principle investment function is derived
by the intertemporal maximization behavior of monopolistic competitive employers. Under this investment function, an economy endogenously begins
expanding even if the stability condition for goods markets is satisﬁed.
There are three factors that determine the equilibrium growth rate: thedegree of monopoly (the inverse of the price elasticity of each good) η −1 ;marginal propensity to saving s; Mashallian k that can be manipulated by
the government and is denoted by κ. The higher value of η −1 and s, thelower value of κ, the more rapid the expansion of the economy.

1Introduction

It is important to establish the dynamic microeconomic foundation for theKeynesian endogenous growth theory once we admit that some idling resources, such as labor, possibly exist even when an economy is expanding.
On the basis of the standard two-period OLG model with money developedby Lucas (1972) and Otaki (2007, 2009), we construct a monetary growthmodel possessing such a feature.
Harrod (1939) is the seminal work in this ﬁeld. However, his investmentfunction, which plays a crucial role in his theory, is not compatible with theintertemporal maximization behavior of the ﬁrm.
This paper deﬁnes the equipment investment as the cost for improving
the labor productivity. This is necessary for accomplishing more eﬃcientand lower cost production to accumulate various intangible know-hows besides increasing in capital. Such costs comprise our notion of equipment
investment.
Under the assumption of monopolistic competition in goods markets, real
GDP becomes a shift parameter for each small ﬁrm. If every ﬁrm expects future macroeconomic expansion, the optimal production increases, and thus,
the beneﬁt from cost reduction is also raised. Accordingly, whenever higher
future economic growth is rationally anticipated, current equipment investment is accelerated and the expectations become self-fulﬁlling. This is our
microeconomic foundation for the accelerator-principle investment function
proposed by Hicks (1950).
There are three crucial factors that determine the equilibrium growth
rate: the degree of monopoly (the inverse of the relative price elasticity ofeach good) η −1 ; marginal propensity to saving s; Marshallian k, which isdenoted by κ.
When employers can obtain more marginal monopoly proﬁts η −1 , theyﬁnd the business environment favorable, and thus, increase their equipment
investment. Accordingly, a higher value of η −1 upturns the equilibrium
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growth rate. Second, if the marginal propensity to saving s is high, it meansthat the funds are suﬃcient for equipment investment. This also increases
the growth rate. Third, when the nominal money stock per nominal GDP(i.e., Marshallian k) κ takes a higher value, more resources are allotted tothe older generation’s consumption. As a result, fewer funds are available forinvestment, thereby, dampening economic growth.
The remainder of the paper is organized as follows. In section 2, we build
up the model and solve the equilibrium growth rate, and explore the welfare
economic implications. Section 3 contains the concluding remarks.

2Model2.1

Structure of the Model

We basically use the same model as Otaki (2007) except for the equipment
investment decision. In every period, individuals are born in the dense of[0, 1] × [0, 1]. This implies that there is no population growth. There are
diﬀerentiated goods z in the dense of [0, 1]. Each good is monopolistically
produced by a single employer z. Fiat money is the only store of value.
Each individual has the identical utility function U :sU ≡ c1−s
1t c2t+1 − δt α,ci,t+j ≡[∫ 10−1]

1−1[cit+j (z)]1−η dz 1−η ,0 < s < 1, (1)where cit+j (z) is the consumption of good z during period t + j at the i-thstage of the life. α denotes the disutility of labor. δ is a deﬁnition functionthat is one when employed and zero when unemployed.
Each employer faces the identical production function y s (z):yts (z) = γt−1 (z)lt (z),γ−1 (z) = γ, ∀z,(2)where γt−1 (z) is the current labor productivity accumulated by equipment
investment. lt (z) denotes the employment level. The real investment cost2

function It (z), which is deﬂated by the price index Pt , is deﬁned as
It (z) = σ[γt (z) − γt−1 (z)],Pt ≡[∫ 10[pt (z)]1−η dz]1−η,(3)where σ denotes the marginal cost for investment.

2.2

Maximization Problem of Economic Agents• Individuals
Since the utility function is (1), we can easily induce the demand function
for good z Dt (z) and saving function S as
pt (z) −η d] yt ,Pt
S(yt ) = syt ,Dt (z) = [(4)(5)

where ytd is the real aggregate eﬀective demand and yts is the real national
income.
Since the expenditure function is Cobb-Douglas form on prices, we caneasily derive the nominal reservation wage WtR ass
WtR = αAPt1−s Pt+1,

A ≡ ss (1 − s)1−s .

(6)In what follows, we assume that equilibrium is interior in the sense that someindividuals are always unemployed. Hence, the equilibrium nominal wage isequal to the nominal reservation wage W R .
• Employers
The optimal behavior of an employer is assumed to be[maxγt (z),pt (z)[ p (z)t

Pt]

R
[]
Wt+1
WtR ]
−Dt (z) − It (z) +Dt+1 (z) .
Pt γt−1 (z)Pt+1 γt (z)(7)

Although the inﬂation rate should be used as the discount rate concerning
future cost reduction, for simplicity, we assume that the gain from the inﬂation is entirely canceled by the proportional corporate tax. It is also assumed
that future wages are actually paid by the employer who will succeed to thebusiness at the next period.
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The solutions of (7) areWtR,[1 − η −1 ]γt−1dIt∗ (z) = [1 − η −1 ][yt+1− ytd ].
p∗t (z) =(8)(9)(9) is our investment function that gives a microeconomic foundation for theacceleration principle. Furthermore, from (3) and (9), we must note that theaggregate employment lt is obtained aslt =σytd=
< 1.
γt−1
1 − η −1

(10)Thus, the unemployment rate is independent of the equilibrium growth rate.
• Government
The government levies a tax that is proportional to the net cash ﬂow of thetﬁrm. The gross tax rate is PPt+1. This tax is entirely and equally transferred

to individuals regardless of whether or not they are employed. Consequently,
the earned income that consists of wages and proﬁts is entirely distributed
to individuals.
Under this set-up, the only government revenue is the seigniorage. We
assume that the monetary-ﬁscal policy of the government keeps the Marshallian k constant, and that accrued seigniorage is entirely spent on wastefulobjects and bears no social utility. That is,
Mt+1 = (1 + g ∗ )πt∗ Mt ,(11)where Mt+j denotes the nominal money stock. g ∗ is the equilibrium growthrate that we shall solve. πt∗ is the equilibrium gross inﬂation rate that is
obtained by (6) and (8). That is,Pt∗ =∗ s]αA[Pt∗ ]1−s [Pt+1−1[1 − η ]γt−1⇒πt∗ =[ [1 − η −1 ]γt−1αA]1s.

(12)

Thus, inﬂation is accelerated by the evolution of labor productivity because
the heightened labor productivity incessantly raises nominal wages. In turn,it implies that if the progress of labor productivity, which corresponds to the4

TFP in our model, stagnates and the growth rate slows down, disinﬂation
becomes prominent. This result is consistent with the empirical research on
Japanese economy by Hayashi and Prescott (2002).

2.3

Market Equilibrium

Since we assume that labor market is located at interior equilibrium, it issuﬃce to analyze the equilibrium condition for aggregate goods market. Thiscondition is obtained by combining (5), (9) and (11):∗syt∗ = [1 − η −1 ][yt+1− yt∗ ] +Mt
Pt⇒

s = [1 − η −1 ]g ∗ + κ.

(13)Thus, we obtain the equilibrium growth rate as+∞
∑s−κ
g =≡
[s−κ]
η −j .
−11−η
j=0
∗(14)

Equations (13) and (14) have interesting economic implications. First, economic growth begins endogenously by the increase in equipment investment.
The initial investment is caused by the rational animal spirits (Keynes (1936,ch. 12)): the employers believe in the future expansion of the economy.
If employers consider that the economy will forever stagnate, the equipment investment will become zero, and so will the economic growth rate. Ina such case, as proved by Otaki (2007), traditional Hicks-Samuelson’s 45◦
analysis is valid. Hence, we must note that the mechanism of endogenousgrowth never depends on the instability of the goods-market equilibrium.
Second, the equilibrium growth rate g ∗ is an increasing function of η −1and s, and is also a decreasing function of κ. s − κ indicates the surplusof the economy normalized by current real GDP. Accordingly, the economy
has abundant funds for the economic growth, thereby increasing the growthrate. Accordingly, the expansionary monetary policy under the credibilityof money (Otaki (2011)) can stimulate the economy in the short run, but
shortens the loanable funds surplus and lowers the growth rate in the longrun.
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Finally, the degree of monopoly η −1 enhances the economic growth sincea higher η −1 implies that an improvement of in the income distribution to
proﬁts and stimulates equipment investment. It is also worthy to note that
this eﬀect is persistent, and hence, powerful, because the increment in laborproductivity never depreciates.

2.4

Welfare Implication

Since, mainly for simplicity, we assume that the equilibrium nominal wageequals the nominal reservation wage, there is no welfare gain from the increasein employment. Accordingly, the source of welfare gain is conﬁned to the real
net cash ﬂow N CFt∗ :N CFt∗ = [η −1 − [s − κ]]yt∗ .

(15)

The indirect utility function IU is Cobb-Douglas:
IU ∝N CFt∗.
[π ∗ ]sSubstituting (10), (12) and (15) into the above equation, we obtain
IU ∝[η −1 − [s − κ]]yt∗[η −1 − [s − κ]] ∗=l = αAσ[η −1 − [s − κ]]. (16)∗[αA]−1 [1 − η −1 ]γt−1[αA]−1 [1 − η −1 ] tThus, the economic growth never provides any additional beneﬁts as long as
economy is bothered by an unemployment problem. The fruits from economic
growth are entirely consumed by the acceleration of inﬂation.
Since IU is an increasing function of κ, the most urgent problem in anational economy is to reduce the number of unemployed individuals as Otaki(2009) proves, even if the equilibrium growth rate becomes zero. Economic
growth is the second priority in the economy as a whole.

3Concluding Remarks

We succeeded in constructing a Keynesian endogenous growth model basedon neoclassical microeconomics that is as faithful as possible. Results ob6

tained are as follows.
First, the equipment investment is the driving force of economic growth.
Second, a higher degree of monopoly η −1 (product diﬀerentiation) stimulates
the economic growth because it bears more proﬁts, and thus, invokes the cost
reduction investment. a high ratio of surplus funds s − κ also contributes tothe economic expansion since it eases employers to invest much resources forthe equipment investment.
Third, although the economy grows autonomously, as long as there isa serious unemployment problem, economic welfare cannot be improved byexpansion. In this sense, before attaining the full-employment equilibrium,
we should adopt an active aggregate demand policy even if the velocity ofeconomic expansion is lowered.

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[2] Hayashi, Fumio and Prescott, Edward C., 2002, The 1990s in Japan: A
lost decade, Rev. Econ. Dyn. 5, 206-235.
[3] Hicks, John R., 1950, A Contribution to the Theory of Trade Cycle,Oxford University Press, Oxford.
[4] Keynes, John M., 1936, The General Theory of Employment, Interestand Money, Macmillan, London.
[5] Lucas, Jr. Robert, E., 1972, Expectations and the neutrality of money,J. Econ. Theory 4, 103-124.
[6] Otaki, Masayuki, 2007, The dynamically extended Keynesian cross andthe welfare-improving ﬁscal policy, Econ. Lett. 96, 23-29.
[7] Otaki, Masayuki, 2009, A welfare economics foundation for the fullemployment policy, Econ. Lett. 102, 1-3.
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[8] Otaki, Masayuki, 2011, A pure theory of aggregate price determination,
Theor. Econ. Lett, 1, 122-128.

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