Theoretical Economics Letters, 2012, 2, 365-368
http://dx.doi.org/10.4236/tel.[phone] Published Online October 2012 (http://www.SciRP.org/journal/tel)
A Keynesian Endogenous Growth Theory with a Rigorous
Microeconomic Foundation
Masayuki Otaki
Institute of Social Science, University of Tokyo, Tokyo, Japan
Received June 2, 2012; revised July 2, 2012; accepted July 31, 2012
ABSTRACT
Extending the effective demand theory developed by Otaki [1,2], we construct a demand-driven endogenous growth
theory with a rigorous microeconomic foundation. An accelerator-principle investment function is derived by the in-
tertemporal maximization behavior of monopolistic competitive employers. Under this investment function, an econ-
omy endogenously begins to expand even if the stability condition for goods markets is satisfied. Three factors deter-
mine the equilibrium growth rate: the degree of monopoly (the inverse of the price elasticity of each good) 1


, the
marginal propensity to saving
s
, and the Mashallian that can be manipulated by the government and is denoted by
. The higher values of
k

1

 and
s
, and the lower value of

, the more rapidly the economy expands.
Keywords: Microeconomic Foundation for the Accelerator-Principle Investment Function; Progress of Labor
Productivity; Demand-Driven Endogenous Growth; Self-Fulfilling Prophecy
1. Introduction
It is important to establish the dynamic microeconomic
foundation of the Keynesian endogenous growth theory
once we admit that some idle resources, such as labor,
possibly exist even when an economy is expanding. On
the basis of the standard two-period OLG model with
money developed by Lucas [3] and Otaki [1,2], we con-
struct a monetary growth model possessing such a fea-
ture.
Harrod [4] is the seminal work in this field. However,
his investment function, which plays a crucial role in his
theory, is not compatible with the intertemporal maximi-
zation behavior of the firm.
This paper defines the equipment investment as the
cost of improving labor productivity. This is necessary
for accomplishing more efficient and lower cost produc-
tion to accumulate various intangible know-hows besides
increasing in capital. Such costs constitute 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 macroeco-
nomic expansion, the optimal production increases, and
thus, the benefit from cost reduction also increases. Ac-
cordingly, 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 accelera-
tor-principle investment function proposed by Hicks [5].
There are three crucial factors that determine the equi-
librium growth rate: the degree of monopoly (the inverse
of the relative price elasticity of each good) 1


, the
marginal propensity to saving
s
, and the Marshallian ,
which is denoted by
k

.
When employers can obtain more marginal monopoly
profits 1


, they find the business environment favor-
able, and thus, increase their equipment investment. Ac-
cordingly, a higher value of 1

 increases the equilib-
rium growth rate. Second, high marginal propensity to
saving
s
implies that the funds are sufficient for equip-
ment investment. This also increases the growth rate.
Third, when the nominal money stock per nominal GDP
(i.e., the Marshallian )
k

takes a higher value, more
resources are allotted to the older generation’s consump-
tion. As a result, fewer funds are available for investment,
thus, dampening economic growth.
The remainder of the paper is organized as follows. In
Section 2, we construct the model and solve the equilib-
rium growth rate, and explore the welfare economic im-
plications. Section 3 contains our concluding remarks.
2. Model
2.1. Structure of the Model
We use essentially the same model as that of Otaki [1]
except for the equipment investment decision. In every
C
opyright © 2012 SciRes. TEL

M. OTAKI 366
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 . Each good
is monopolistically produced by a single employer z. Fiat
money is the only store of value.
[0,1]
Each individual has the identical utility function : U

1
121
1
11
1
1 1
,0
,
d,0<<
ss
tt t
it jit j
Ucc
cczz s
















1,
(1)
where is the consumption of good z during
period at the ith stage of the life.

it j
cz

tj

denotes the
disutility of labor.

is a definition function that is one
when employed and zero when unemployed.
Each employer faces the production function


s
yz:
 
11
=,=
s
ttt
yzzlzz z 

,,  (2)
where is the current labor productivity accu-
mulated by equipment investment. denotes the
employment level. The real investment cost function

1tz



t
lz

t
I
z, which is deflated by the price index , is de-
fined as
t
P
 
1
1 1
10
=,
ttttt
Izzz Ppzzd,


 



 






(3)
where

denotes the marginal cost for investment.
2.2. Maximization Problem of Economic Agents
2.2.1. Individuals
Since the utility function is (1), we can easily induce the
demand function for good z, and the saving func-
tion S as

t
Dz
 
=
td
t
t
pz
Dz y
P
 




,
t
,
t
(4)

=
t
Sy sy (5)
where d
t
y
is the real aggregate effective demand and
s
t
y
is the real national income.
Since the expenditure function is of the Cobb-Douglas
form on prices, we can easily derive the nominal reserva-
tion wage
R
t
W as

1
1
1
=,1.
s
Rsss
ttt
WAPPAss 


 (6)
In what follows, we assume that the equilibrium is in-
terior in the sense that some individuals are always un-
employed. Hence, the equilibrium nominal wage is equal
to the nominal reservation wage
R
W.
2.2.2. Employers
Every employer consists of a dynasty. Employers con-
sider not only their own interests but also their descen-
dants. The optimal behavior of an employer is assumed
to be
 

  
 
,1
11
1
1
max
.
R
tttt
zpz
tt ttt
R
tt
tt
ttt
pz WDz Iz
PPz
PW
Dz
PP z








 







 


(7)
The first term of (7) represents the benefit of the cur-
rent employer. The second term corresponds to the net
benefit of their descendant (The labor-productivity im-
provement investment causes more efficient production
(i.e., cost reduction).)
Although the inflation rate is the discount rate con-
cerning future cost reduction, for simplicity, we assume
that the gain from the inflation is entirely canceled by the
corporate tax levied on the net benefit from the cost re-
duction. We denote the rate as t

. That is, we assume
that
1
=.
t
tt
P
P


It is also assumed that future wages are actually paid
by the employer who will assume control of the business
in the next period1.
The solutions of (7) are
*
1
1
()= ,
1
R
t
t
t
W
pz






(8)

*1
1
=1 .
dd
ttt
I
zy 

y

 


 (9)
Equation (9) is our investment function that gives a
microeconomic foundation for the acceleration principle.
Furthermore, from (3) and (9), we must note that the ag-
gregate employment is obtained as
t
l
1
1
== <
1
d
t
tt
y
l



1. (10)
Thus, the unemployment rate is independent of the
equilibrium growth rate whenever effective demand grows
at the same velocity as does labor productivity improve-
ment2.
2.2.3. Govern men t
The government levies a tax that is proportional to the
net additional cash flow of the firm. The gross tax rate is
1From the envelop theorem, it is clear that the maximization problem
(7) is invariant even if future total sales 
11tt
p
Dz
 is introduced to
the objective function.
2Note that the unemployment level temporally diverges from (10) when
an unexpected shock, such as monetary expansion, occurs in the eco-
nomy.

M. OTAKI 367
1
=t
tt
P
P


.
This tax is entirely and equally transferred to individuals
regardless of whether 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 wasteful
objects and bears no social utility. That is,

**
1=1 π,
ttt
M
gM
 (11)
where tj denotes the nominal money stock. M
*
g
is
the equilibrium growth rate for which we solve. is
the equilibrium gross inflation rate that is obtained from
(6) and (8). That is,
*
πt
1
1
** 1
1
**
1
1
1
=π=
1
ss
1
s
tt t
tt
t
AP P
PA








 

 




 






(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 the TFP in our model, stagnates and the growth rate
slows, disinflation becomes prominent. This result is
consistent not only with the empirical research on the
Japanese economy by Hayashi and Prescott [6] but also
with the downward-sloping long-run Phillips curve.
2.3. Market Equilibrium
Since we assume that the labor market is located at inte-
rior equilibrium, it is sufficient to analyze the equilibrium
condition for the aggregate goods market. This condition
is obtained by combining (5), (9), and (11):
*1** 1*
1
=1=1 .
t
ttt
t
M
syy ysg
P




 

 


(13)
Thus, we obtain the equilibrium growth rate as

*
1
=0
=.
1
j
j
s
g


s



 


 (14)
Equations (13) and (14) have interesting economic im-
plications. First, economic growth begins endogenously
via the increase in equipment investment. The initial in-
vestment is caused by the rational animal spirits (Keynes
[7], Ch. 12): the employers believe in the future expan-
sion of the economy.
If employers assume that the economy will forever
stagnate, the equipment investment will become zero,
and so will the economic growth rate. In such a case, as
proved by Otaki [1], the traditional Hicks-Samuelson’s
45˚ analysis is valid. Hence, we must note that the me-
chanism of endogenous growth never depends on the in-
stability of the goods-market equilibrium.
Second, the equilibrium growth rate *
g
is an increase-
ing function of 1


and s, and is also a decreasing func-
tion of

.
s


denotes the surplus of the economy
normalized by the current real GDP. Accordingly, the
economy has abundant funds for the economic growth,
thereby increasing the growth rate. Accordingly, the ex-
pansionary monetary policy under the credibility of money
(Otaki [8]) can stimulate the economy in the short run,
but shortens the loanable funds surplus and lowers the
growth rate in the long run.
Finally, the degree of monopoly 1

 enhances eco-
nomic growth since a higher 1

 implies an improve-
ment in the income distribution to profits and stimulates
equipment investment. It is also worth noting that this
effect is cumulative. That is, at the first stage, future real
income grows by
s


per cent3. In the next stage,
s


bears the additional monopoly profit that amounts
to


1
s
m 


because of the rightward shift of each
good demand curve; thus, each firm expands its supply in
response to the increment of the demand. Similarly, this
income again bears income of


11
s.

 


 
Such a
cumulative process continues infinitely and converges to
1
1
s




.
2.4. Welfare Implication
Since, mainly for simplicity, we assume that the equilib-
rium nominal wage equals the nominal reservation wage,
there is no welfare gain from the increase in employment.
Accordingly, the source of welfare gain is confined to the
real net cash flow :
*
t
NCF


*1
=
tt
NCFs y 


*
.



 (15)
The indirect utility function
I
U is Cobb-Douglas:
*
*
.
π
t
s
NCF
IU 

Substituting (10), (12) and (15) into the above equa-
tion, we obtain




 
1*
11*
1
1
*1
11
1
==
1
t
t
t
sy
IU A
slA s
A



.

 






















(16)
3Note that the employers rationally confirm that future effective de-
mand proportionately increases with the progress of labor productivity.

M. OTAKI 368
Thus, the economic growth never provides any addi-
tional benefits as long as economy suffers from an un-
employment problem. The fruits of economic g
entirely consumed by the acceleration of inflatio
Since
se
[1] M. Otaki, “The Dynamically Extended Keynesian Cross
and the Welfolicy,” Economics
Letters, Vol. 9.
icks, “A Contribution to the Theory of Trade Cy-
in Japan: A
ry of Aggregate Price Determina-
nse, before attaining the full-employment equilibrium,
we should adopt an active aggregate demand policy even
if the velocity of economic expansion is lowered.
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We succeeded in constructing a Keynesian endogenous
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First, t
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