DBJ Discussion Paper Series, No.1105

The Aggregation problem in employment theory:
The representative individual or indivisible employees models?

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

January

2012

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,
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not reflect the views of the Institute.

The Aggregation problem in employment theory:
The representative individual or indivisible employees models?

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

Abstract
Employment theory does not agree with whether variation in employment
should be expressed by a change in the hours worked by a representative
individual or by a change in the population of employed individuals.
Thus, the present article describes how this distinction brings about serious theoretical consequence by using an OLG model. The crucial factor
that separates the property of employment theories is the existence of theintertemporal substitution eﬀect.
Monetary expansion increases the rate of return for money if it is crediblein the sense of Otaki (2011). It enhances the hours worked in the representative individual model, and thus, aggregate supply causes demand.
In the indivisible employees model, by contrast, such an intertemporal
substitution eﬀect does not exist. Monetary expansion directly improves thepurchasing power of money, and thereby increases the aggregate demand forgoods by the older generation. Thus, demand pulls supply.
Keywords: Representative Individual, Indivisible Labor Supply, Intertemporal Substitution, Credibility of Money, Fiscal Multiplier.
JEL Classiﬁcations: E24, E31, E32, E42, J22.

1Introduction

Independent of whether scholars adopt the neoclassicals or new Keynesian
economic models, recent employment theories have rested on the assumptionof a representative individual. However, it is important that the variation ofthe hours worked by a representative individual essentially diﬀers from thatof the number of employed indivisible workers. In the present paper, we showthat how this distinction can bring about serious theoretical consequences.
For a representative individual, monetary expansion increases the rateof return as long as money is credible, and stimulates labor supply. Hence,apart from certain spurious diﬀerences, both neoclassicals and new Keynesian
economics seek to ﬁnd the causes of employment variation due to supply-sideincentives.
By contrast, no such substitution eﬀect exists in the indivisible employees model.1 Monetary expansion directly heightens the purchasing power ofmoney even if the money-supply rule obeys that proposed by Lucas (1972)
as long as money is credible. It also implies that monetary expansion leads toan upturn in aggregate demand, which increases real GDP. In other words,
demand causes the corresponding supply as Keynes (1936) points out.
The remainder of this paper is organized as follows. Section 2 constructs
alternative models of employment theory, and section 3 concludes.

1

Although we can principally separate the adjustment of hours worked from that ofemployment level (see Fukao and Otaki (1993)), this requires far more complex dynamics,
which is not essential to the present discussion. Furthermore, if there is no ﬁxed sunkcost for being employed, it is clear that every ﬁrm uniformly oﬀers minimal hours worked
because the increasing marginal disutility of labor requires higher wages for compensation. In such a case, the production adjustment by working hours never incurs a diﬃcultproblem.

12The Model

2.1

The Structure of the Model

We consider a standard two-period deterministic OLG model operating ina production economy. In every period, a unit of individual is born. Eachindividual can work only when they are young. Unit working hour produces
unit goods.
Money supply obeys Lucas’s (1972) rule; that is,
mt = mt−1 x,(1)where mt−1 is the nominal money stock per capita that is carried over fromthe previous period. x is the gross growth rate of money. In this sense, newmoney is supplied at its own nominal interest rate.
We make the following alternative assumptions about labor supply: (i)in the representative individual model, he or she can choose working hours
and there is nounemployment; (ii) each individual faces a discrete choice overwhether to work in the indivisible employees model.

2.2

The Representative Individual Model

2.2.1

The Deﬁnition of Equilibrium

For simplicity, we assume that a representative individual has the following
utility function UR :UR ≡ u(c1t , c2t+1 ) − v(ht ),v ′ , v ′′ > 0,(2)where u is a well-behaved strictly concave and homothetic function. c1t , c2t+1
denote the consumption level of generation t during the young and old stagesof life, respectively. h is the hours worked. Following Lucas (1972), further
we assume that leisure and current consumption are mutually normal goods;that is,
u11 −

pt+1
u12 < 0,
pt x
2

(3)

where uij is the second derivative concerning i and j arguments of u. The
lifetime budget constraint is
mt−1≤ ht , pt+1 c2t ≤ mt−1 x
pt
pt+1
c1t +
c2t+1 ≤ ht .
pt x
c1t +
⇔(4)
(5)

The optimality conditions areu2
pt+1=,
u1
pt xv′= 1,u1
h∗t = c∗1t +(6)(7)
pt+1 ∗c,
pt x 2t+1(8)where ui is the partial derivative of the i-th argument of u.
In addition to these three optimality conditions, there is an independent
market equilibrium condition. Here, we consider the condition for the money
market equilibrium; that is,
mt−1 x = p∗t+1 c∗2t+1 .

(9)Furthermore, we assume the credibility of money in the sense proposed byOtaki (2011):2

dp∗t+1= 0.
(10)dx
There are ﬁve endogenous variables (c∗1t , c∗2t+1 , h∗t , p∗t , p∗t+1 ) and ﬁve independent equations (6), (7), (8), (9), and (10). Hence, the model is closed, andthe solution consists of temporary rational expectation equilibrium.

2The concept of credibility of money is a device to select a unique rational expectation

equilibrium (REE) among multiple REEs that are generic to the OLG model of the monetary economy. Credibility economically means that people rationally believes its intrinsic
value is kept intact even if the velocity of monetary acceleration is changed.

32.2.2

Comparative Statics

By combining Equation (8) with (6), we obtainh∗t = c∗1t +u2 mt−1 x.
u1 p∗t+1(11)From Equations (6) and (10), the right-hand side of Equation (11) is solely
a monotonously increasing function of c∗1t , which contains x as an exogenousparameter.
The curves U U and V V in Figure 1 are loci of Equations (7) and (11),respectively. The intersection E0 is the temporally rational expectation equilibrium that presumes the credibility of money.
When monetary expansion occurs (i.e., x increases), the curve V V shifts
in the south-east direction because of the growth in future consumption stimulates labor supply and economizes leisure. Thus, the equilibrium moves frompoint E0 to point E1 .
From the concavity of u and Equation (6), it is clear that the eﬀectivep∗
inﬂation rate (the inverse of the real interest rate) t+1
decreases with anp∗t x
acceleration in monetary growth (i.e., an increase in x).
To summarize up, as long as money is credible, easy monetary policy
increases the real rate of interest, thereby a representative individual works
longer hours in order to enjoy a greater degree of future consumption.
Accordingly, monetary expansion advances the intertemporal substitution
from current consumption and leisure to future consumption by increasing
the real rate of interest. Consequently, the expansionary eﬀect of monetary
policy is entirely based on labor supply incentives not on the expansion ofaggregate demand. In this sense, the representative individual model is can
be classiﬁed as a neoclassical macroeconomic model.
2.2.3

The Time-Independency of the Model

Assume that a representative individual rationally expects that the real efp∗
fective inﬂation rate t+1is kept intact after period t + 1, since no economicp∗t x
4environment is changed after period t. This implies thatdp∗
dp∗
dx= ∗t+j+1 − ∗t+j , ∀ j ≥ 1.
x
pt+j+1pt+j(12)In addition mt+j−1 x = p∗t+j+1 c∗2t+j+1 and mt+j−1 = p∗t+j c∗2t+j holds from themoney market equilibrium condition (9). Thus, we obtaindc∗t+j+1 dc∗t+j
dp∗t+j+1 dp∗t+jdx−=−[− ∗ ] = 0.
c∗t+j+1c∗t+jx
p∗t+j+1pt+j(13)In other words, future consumption c∗2 becomes time-independent. Hence,from Equations (6), (7), and (8), (c∗1 , h∗ ) are also time-independent. Consequently, the rational expectation equilibrium characterized by the initialcondition x and the expectation formations (10) and (12) is stationary.

2.3

The Indivisible Employees ModelHere, we assume that labor supply is indivisible, and that each individual
has the identical utility function UI :UI ≡ u(c1t , c2t+1 ) − δt v,(14)where u is the same consumption utility function as that described in Equation (2). δt denotes a deﬁnition function that takes value unity when he/sheworks unit time and becomes zero when he or she does not work. v is thedisutility of labor.
According to Otaki (2011), the nominal minimal revenue that individuals
decide to work N R is represented aspt+1)f (v), ψ1 > 0,
N R = ψ(pt ,xwhere ψ is a linear homogeneous function.

ψ2 > 0,(15)

At any interior equilibrium, all individuals are indiﬀerent about whetherto work, and thus, we obtain the following diﬀerence equation concerning theevolution of price sequence:3p∗p∗t+j = ψ(p∗t+j , t+j+1 )f (v)x3⇒

1 = ψ(1,p∗t+j+1)f (v), 0 ≤ j.
p∗t+j xSee Otaki (2011), for the case that the equilibrium is located at the boundaries.

5(16)Hence, the equilibrium real interest rate r∗ ≡p∗t+j xis independent of x andp∗t+j+1takes a constant value.
The equilibrium condition for the money market is∗s(r∗ )yt+j=mt+j−1 x≡ m̃∗t+j ,p∗t+j+10 ≤ j,(17)

where s( · ) is the marginal propensity to save. Assuming the credibility ofdp∗
dx
dp∗
money (i.e., t+1 = 0), according to Equation (16), ∗t = −holds.
dx
ptx
That is, an increase in the monetary growth rate x increases the current
value of money and empowers the purchasing power of older individuals aslong as money is considered a credible asset. As such, monetary expansion
stimulates the economy through the multiplier eﬀect developed by Otaki(2007).
It is facile to show the time-independency of the equilibrium. Assume
people rationally believe that the price level p∗t+j+1 grows proportionately
with the monetary expansion rate x, that is,
dp∗t+j+1dx=,∗
pt+j+1x1≤j(18)then the equilibrium real cash balance m̃∗ becomes time-independent. So is
the real equilibrium GDP y ∗ .
In addition, as we precisely aforementioned, the individual employeesmodel belongs with Keynes’ (1936) type model in the sense that a monetary
expansion stimulates the aggregate demand unlike the representative individual model that crucially relies on the supply-side intertemporal substitutionincentives.

3Concluding Remarks

This article analyzed how the aggregation problem aﬀects employment theory
and drew two main conclusions.

6First, owing to the intertemporal substitution between goods and leisure,a change in working hours in the representative individual model is supplyside oriented even if money is credible and non-neutral. Acceleration in monetary growth increases the real interest rate of money, and thus, intertemporal
substitution occurs from leisure and current consumption to future consumption.
Second, the individual employees model possesses the demand-driven
property proposed by Keynes (1936). Although the real rate of interest on
money is endogenously ﬁxed, whenever money is credible, it becomes to behighly valued and this increases the purchasing power of the older generationby the acceleration of monetary growth. As such, eﬀective demand expands
and real GDP increases because of the multiplier eﬀect.
To summarize, Keyes’ (1936) economics can be characterized by two attributes: credibility of ﬁat money of which its intrinsic value is basicallyindeterminate and the speciﬁcity of labor as a commodity, namely, the indivisibility of labor.

References[1] Fukao, K., M. Otaki, 1993, Accumulation of human capital and thebusiness cycle, J. Polit. Econ. 101, 72-99.
[2] Keynes, John M., 1936, The General Theory of Employment, Interest,and Money, Macmillan, London.
[3] Lucas, Jr. Robert, E., 1972, Expectations and the neutrality of money,J. Econ. Theory 4, 103-124.
[4] Otaki, Masayuki, 2007, The dynamically extended Keynesian cross andthe welfare-improving ﬁscal policy, Econ. Lett. 96, 23-29.
[5] Otaki, Masayuki, 2011, A pure theory of aggregate price determination,Theor. Econ. Lett. 1, 122-128.
7

c1UUVVE0V ′V ′E1hOFigure 1

18

The title of Figure 1
Figure 1: The determination of (c∗1 , h∗ )29