An Inquiry into the Same-Movement Pattern
in Two Cases of Labor Supply

Fujio Takata
February 2019
Discussion Paper No.1904

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANAn Inquiry into the Same-Movement Pattern in
Two Cases of Labor Supply
Fujio Takata
Graduate School of Economics, Kobe University,
1-1, Rokkodai-cho, Nada-ku, Kobe, 657-8501, Japan,
jikairf@clock.ocn.ne.jp.
January 30, 2019

Abstract
Let us consider movement patterns in an economy based on a Ramsey type model, as do many papers in the R.B.C. school.We focus on
how labor supply is determined and on how this determination aﬀects
the movement patterns. We discuss two kinds of labor supply, namely,
an intensive arrangement and an extensive one. We conclude that thelabor supply in both arrangements is similar from the viewpoint of amacro economy. This leads us to conjecture that the two forms of labor
supply do not aﬀect the movement pattern.
JEL classification: E20; E24; E32
Keywords; Divisible labor ; Indivisible labor; Movement patterns

11

Introduction

The R.B.C. theory has been dealing with fluctuations, especially businesscycles, based on Ramsey (1928). The original Ramsey model has been transformed into a general equilibrium theory by introducing perfect foresight.1
This means that Ramsey type models can endogenously determine several
economic variables, such as savings, consumption, wage rates, rental rates
of capital, and labor supply, in every period assumed. Moreover, as savings
lead to investments, those models can explain how an economy moves. In
particular, the question of whether business cycles can be produced or nothas been a big issue. Along this line, we are interested in indeterminacy, inwhich many upward and downward paths in an economy can be detected.
This indeterminacy has been found in many economic circumstances, suchas increasing returns to scale, externality and monopoly. We can list Benhabib and Farmer (1994) as a representative and classical study. But theseassumptions all depend on economic outcomes, which leads to an inevitablediﬃculty in managing economic policies.
However, some papers maintain that only fiscal policy can produce thisindeterminacy. Fiscal policy here specifically means income taxation and abalanced budget rule. Schmitt and Uribe (1997) is a significant and controversial essay based on this line of reasoning and on an important assumption
in relation to labor supply, that is, an indivisible labor supply. Thus the paper discusses what conditions cause indeterminacy in an an economy withan indivisible labor supply. On the other hand, Takata (2017) oﬀers a mathematical proof concerning the conditions for indeterminacy in an economy
with a divisible labor supply. But both papers derive similar conclusions,because in the former model, the disutility of a household is expressed interms of an expected disutility function with two arguments, one of which isinevitably null, while in the latter model, the disutility of a household is not
an expected disutility, and the disutility function consists of an argument.
This is why the same conclusion is derived on appearance. Additionally, thereason one argument in the disutility function is null is derived from theassumption that some parts of households do not work. Naturally, this does
not cause any disutility.
In this paper, we assume that an indivisible labor supply shows an extensive arrangement of the labor force, that is, some parts in the fixed working
population work and others do not in every period, while a divisible labor
supply shows an intensive arrangement, that is, the whole working popu1

See Attanasio (2015) for subsequent contributions and extensions to Ramsey (1928).

2

lation works in every period, but changes its working hours, which is its
utilization.
Why do both economies follow the same movement pattern? As mentioned above, the similarity seems to depend on the normalization of disutility, which is supposed to be null when the potential labor force is out of
the labor market. Is this absolutely true? The two papers cited above apparently share a similar ambiguity concerning how they deal with a macroeconomy from the viewpoint of the two types of labor supply. Both models
assume a representative agency. Does this not entail any problems?
To this question, it is appropriate to investigate Mulligan (1999), .whichanalyzes the relationship between the labor supply determined by the two
arrangements, focusing on the relationship between the micro and macrosides of an economy.
Mulligan (1999) concludes that the labor supply in both arrangements issimilar from the macroeconomic viewpoint. But this demonstration is only
about the labor supply; it doesn’t analyze how the economy moves.
In this paper, based on Mulligan (1999), we derive a conjecture that thesame movement pattern in an economy occurs in an economy in the twocases of labor supply, i.e. of an intensive arrangement and an extensive one.
In the next section we investigate Mulligan (1999) in detail, in the casesof both divisible and indivisible labor. In what follows, in section 3, we
explore an aggregate labor supply in terms of an extensive arrangement.
Subsequently, in section 4, we demonstrate the similarity between the twolabor supply arrangements. We demonstrate that when the Mulligan modelis extended to an analysis of economic movements, its conclusions are thesame as those in Takata (2017). Finally, section 5 is devoted to concluding
remarks, in which we mention that the diﬀerence between the two arrangements does not aﬀect the movement pattern in an economy.

3

2

The Mulligan models

2.1

Optimal divisible laborFirst, Mulligan (1999) introduces a divisible labor environment as
[
]
T
∑
max
C λ−
e−ρ(t−1) gt ν(Nt ) , λ > 0, gt > 0,
C{Nt }T
t=1

(1)

t=1

s.t, Nt ∈ [0, nmax ] , C = a +T∑e−ρ(t−1) Qt wt Nt .

t=1

The divisible environment shows that representative individual works range
anywhere from 0 to nmax units of time. During time interval t, a wage is paid
at a rate wt for time worked Nt . The present value of lifetime consumption
C is financed out of lifetime earnings and an initial asset stock a.
Here, the definitions of notations are as follows: U (C) = λ C is theconsumption value function and λ constant; ρ is the rate of time preference;
gt is a preference parameter, which shows diﬀerent degrees of disutility in a
lifetime; Qt is the period interest rate factor; ν(N ) is the disutility of work.
Furthermore, Mulligan (1999) makes assumptions concerning disutility
such as′ν >0′′∫ nmaxν >0In ν ′ (N ) dN = 0.

(2)
(3)
(4)

0

We need to analyze (4) in detail, utilizing a specific disutility function suchasN 1−χν(N ) =.
(5)1−χHere, χ < 0. Based on the above function, (4) is transformed as∫ nmaxlog(N ) dN = 0.
0Solving the above integration, we obtain (−1 + log(nmax ))nmax = 0.
′
Therefore, nmax = E 2 and ν (1) = 1 prevail. The latter relationship implies
that in 0 < N ≤ 1, the marginal disutility in relation to N is less than or2

E means a base logarithm.

4

is equal to unity, while in 1 < N < E, it is more than unity. Thus, inthis model, the disutility in relation to N increases as N does, and reaches
its maximum ν(E). So Mulligan (1999) assumes that the function ν(N )is constant over time, but the marginal disutility schedule varies as does gt
over time. Based on these assumptions, Mulligan (1999) deduces the optimallabor supply as′

gt ν (Nt ) = λ Qt wt . t = 1, 2, · · · , T.

(6)

In short, the labor supply is determined to satisfy the condition that thedisutility in relation to labor is equal to the discounted wage multiplied by
λ.

2.2

Indivisible labor

Second, Mulligan discusses indivisible labor.
Individuals choose whether to work or not, during a time interval. Wagewt is payed to labor during time interval t. The labor supply is determinedto satisfy the following relationships:[]T∑max λ C −e−ρ(t−1) γt nt ,(7)
C{nt }Tt=1

t=1−s.t, nt ∈ {0, n}, C ⩽ a +T∑e−ρ(t−1) Qt wt nt .

t=1

Here, γt is a preference parameter for labor and nt is labor input in timeinterval t, respectively.
Moreover, the disutility function of labor ν(n) is defined as−−

ν(n) = γt n

and is normalized as νt (0) = 0. This shows that when an individual doesn’t
work, which means the least disutility, disutility is normalized as 0. Ad−
ditionally, an individual is assumed to oﬀer fixed time n. Although theindividual is engaged in a fixed time of work, disutility here is aﬀected by
the preference for labor in a lifetime.3 Based on (7), we can obtain some3This postulation is supposed to depend on the fact that an individual decides whether
to participate in the labor market or not. However, Hansen (1985) and Rogerson (1988)
oﬀer a model in which this decision is determined by lotteries which firms oﬀer. According
their postulations, this risk appears as a fixed cost for the firms, because the allowancesfrom the lotteries are payed even to unemployed workers.

5

conditions regarding the way decisions are made, as follows: Consider amaximization problem such as
(
)
T
T
∑
∑
−ρ(t−1)ξ =λ a+e
Q t wt n t −e−ρ(t−1) γt nt .
t=1

t=1

Here, the following prevails:∂ξ= e−ρ(t−1) (λQt wt − γt )∂ntBased on this, we can see that an individual chooses whether to work or notaccording to the following conditions:−nt = n as λ Qt wt ⩾ γt , t = 1, 2, · · · , T.
nt = 0as λ Qt wt ⩽ γt .

The above conditions imply that an individual chooses to work when themarginal utility of consumption is larger than the disutility of labor, andvice versa.

3Aggregation

We have discussed an individual’s behavior based on (7) in an economy
with many heterogeneous individuals. They naturally have heterogeneous
preferences for labor in their own lives, and the diﬀerences are revealed as
the life-cycle disutility of the work profile.
In this section, we intend to aggregate the results of the labor supply insystem (7). In this situation, Mulligan (1999) assumes that the geometricmean preference in period t, gt varies over time, but the distribution F ofpreferences around that mean is the same in every period. In short, theseassumptions can be approximately expressed as
( )
γt
log
= F (x),gt∫
x dF (x) = 0.
We can know the primitive function of F (x) as f (x) = 1/x, because F (x) =log(γt /gt ) = log(x). Moreover, let us investigate the second relationshipintensively. We can easily deduce the following:∫
xf (x) dx = 0,
6which implies that the expectation of x is null.
Mulligan (1999) assumes that the continuum of agents equals population
frequencies in terms of probability. As a result, if (γt /gt ) is small, we cansay that the participation rate in the labor market is large, and vice versa.
In other words,((γt /g(t ) can))be an indication in relation to the labor supply.
γtThen, since F log
can be a criterion concerning the fraction ofgt
consumers working at time t, Πt , can be described as( ( ))
γtΠt = F log.
gt
Additionally, based on λ Qt wt ⩾ γt or λ Qt wt ⩽ γt , consumers determine
whether to enter the labor market or not. In light of this, we can replace γtwith λ Qt wt . Therefore, since the above criterion can be written as( ())λ Q t wtΠt = F log,
gt
the average labor supply at time t, Nt , is expressed as( ())λ Q t wt
Nt = F log
n̄.
gt

(8)

On the other hand, the average budget constraint is obtained by averaging
the budget constraint in (7) across consumers, as follows:
C =a+

T
∑

e−ρ(t−1) Qt wt Nt .

t=1

In addition, C means the present value of lifetime consumption.

4

The similarity between the divisible and indivisible models

Here we face an interesting outcome, that is, the two types of labor supply
produce a similar supply on the whole. This can be summarized by thefollowing propositions.
The basic idea is as follows: a general equilibrium system consists ofendogenous and exogenous variables.. Obviously, the endogenous variables(solutions) are expressed in terms of exogenous variables, that is, parameters. Assume that we investigate whether system 1 and system 2 are identical7

or not. Then, assume that the parameters in system 1 are replaced by thosein the system 2. In this situation, if the solutions in system 1 are identicalto those in system 2, and vice versa, we can say that the two systems are
identical.
Proposition 1 The whole labor supply determined by extensive arrangement becomes similar to that determined by intensive arrangement.
Here we begin to prove Proposition 1.
Proof 1 (i) Choose an arbitrary set of parameters ( a, ρ, λ, n̄, {wt gt Qt }) ∈+2
R2 × R3T
for the extensive arrangement.
+(ii) In a whole economy, the average labor supply Nt satisfies (8).
(iii) Choose the same arbitrary set of parameters in the case of the in+2
tensive arrangement, (a, ρ, λ, nmax , {wt gt Qt }) ∈ R2 × R3T, and choose+
the disutility function ν(N ) satisfying(′ν (N ) ≡ e

F −1

)

N
n̄ .

(9)

′

′′

Additionally, ν has properties such as ν > 0, ν > 0, and these satisfy(4). These properties are completely coincident with those defined in (2) ∼(4).
(iv) In light of (6),(′ν (N ) =λ Q t wt
gt

(

)

F −1

=e

N
n̄

)

prevails. The above relationship shows
( (
))
(
)
N
λ Q t wt′.
= F log(ν (N )) = F logn̄gt′Moreover, noticing λ = U (C), we can obtain a conclusion that eventually
the average labor supply equals (8). ■Next, let us consider the reverse process. Then, the second proposition
prevails:

8

Proposition 2 The aggregate labor supply expressed in terms of the parameters in the economy with an intensive arrangement is equal to that expressedin terms of the same parameters in the economy with an extensive arrangement.
Proof 2 (i) Choose an arbitrary set of parameters for the economy with an+2
intensive arrangement: (a, ρ, λ, nmax , {wt gt Qt }) ∈ R2 × R3T.
+(ii) Based on the arguments in subsection 2.1, the labor supply is denotedas (6).
(iii) Choose the same parameters for the economy with an extensive ar+2
rangement: ( a, ρ, λ, n̄, {wt gt Qt }) ∈ R2 × R3T. Furthermore, define a+
distribution function such as′0
if x < log(ν (0)),
ϕ(ex )′′
F (x) ≡
if x ∈ [log(ν (0)), log(ν (nmax ))],

n

 max
′
1
if x ⩾ log(ν (nmax )).

Here, F (x) has properties such as F ∈ [0, 1], F (x) > 0, and the expectation of the primitive function of F (x) is null.
At this stage, we need to specify F . We assumed a disutility function
(5),
N 1−χ
.
ν(N ) =1−χ′Based on the above assumption, since ν (N ) = N −χ , log (ν ′ (N )) = log(N −χ ) =x prevail. Moreover, since we denote log (ν ′ (N )) = x, it is obvious thatx
N = e −χ . On the other hand,( ())
N≡ F log ν ′ (N ) = F (x)n̄
can be defined. Thus, function F (x) can be specified as
x

e −χ
N
=
,
F (x) =
n̄
n̄

(10)

which shows that the distribution F (x) is the product of exponential functions concerning x. Before proceeding, we need to confirm some properties9

in relation to F (x). Since Nmax is E, it is obvious that −∞ < x < 1.
−1
Therefore, F ∈ [0, 1] if n̄ = e χ , and F (x) > 0.
Moreover, ϕ shows a reciprocal of the disutility of labor, which can be′easily proved, as follows: Based on our definition, ν (N ) = ex , we candeduce′−1
N = ν (ex ) ≡ ϕ(ex ).
In other words, by definition log (ν ′ (N )) = x, so we can deduce labor supply
as a mapping of the marginal disutility of labor, and this mapping is a reciprocal of the marginal disutility of labor.
(iv) In total, the average labor supply satisfies (8), and in light of thedefinition of F , this results in equality with (6). We can confirm this logic
as
))
( (
(
)
λ Q t wt′n̄ = (x) n̄.
Nt = F log ν (N ) n̄ = F log
gt■(
In the above proof, notice that F(log

λ Q t wt
gt))
shows the partici-

pation rate in the labor market.

5

Concluding remarks

Mulligan (1999) demonstrates that the indivisible model is equal to thedivisible model, if in the two models the marginal utility of income acrossthe population is equal. This equality is proved only in relation to aggregatedlabor.
However, in this paper, we implicitly refer to Hansen (1985) and Rogerson (1988) as providing an indivisible model, respectively established on thebasis of a linear disutility function. Their postulations imply an extensive
arrangement in the labor supply, and therefore an expected disutility in relation to the aggregated labor supply. In their papers, it is assumed thatpeople choose whether to work or not in their lifetime. In this sense, in theirmodel some parts of a labor population are necessarily unemployed.. In thissituation, the work force arrangement comprises a group newly entering or
quitting the labor market. Furthermore, when a person chooses not to work,
this person is compensated by firms in terms of allowances in lotteries. Additionally, this mechanism is repeated over the person’s lifetime. If a person
opts to be idle, this person’s disutility concerning labor is assumed to be10

at a maximum and null. This normalization is a way to produce a linear
expected disutility function.
On the other hand, Mulligan introduces a divisible model, and discusses
how the labor supply is determined. In the divisible economy, the labor
supply arrangement is determined by intensity in terms of labor hours, thatis, in terms of utilization. All members of a labor population are supposed
to participate in the labor market, changing utilization over a lifetime.
Mulligan concludes that there is no diﬀerence in the labor supply between the two economies. His scope in this demonstration is limited to thelabor supply, but we can conjecture a scenario in which, in a general equilibrium, the two economies can move in a parallel pattern. In short, we candemonstrate that the diﬀerence between the two labor supply arrangements,
one intensive and one extensive, does not aﬀect the movement pattern of
the two economies.
We are especially interested in indeterminacy in an economy based ona balanced budget rule. Schmitt and Uribe (1997) discusses this significant
issue based on Hansen (1985). On the other hand, Takata (2017) oﬀers a
proof regarding an economy with an intensive arrangement. Both papers
were written without referring to Mulligan (1999). Our conjecture is true.

Acknowledgements
I was advised about my English by Prof. Jeﬀrey Herrick. Naturally, I accept
full responsibility for possible errors.

References[1] Attanasio, P. O.,. (2015), “FRANK RAMSEY’S MATHEMATICAL
THEORY OF SAVING,” The Economic Journal, Vol. 125, No. 583,
269-277.
[2] Benhabib, J. and Farmer, R. F., (1994), “Indeterminacy and Increasing
Returns,” Journal of Economic Theory, 63, 19-41.
[3] Hansen, G. D., (1985), “Indivisible Labor and the Business Cycle,”
Journal of Monetary Economics, 16, 309-327.
[4] Mino, K., (2017). Growth and Business Cycles with Equilibrium Indeterminacy (Tokyo: Springer) Advances in Japanese Business andEconomics, 13.
11

[5] Mulligan C., (1999), “Microfoundations and Macro Implications of Indivisible Labor,” NBER Working Paper, No. 7116.
[6] Ramsey, F. P., (1928), “A Mathematical Theory of Saving,” The Economic Journal, 38, No. 152 (Dec.), 543-559.
[7] Rogerson R., (1988), “Indivisible Labor, Lotteries and Equilibrium,”
Journal of Monetary Economics, 21, 3-16.
[8] Schmitt-Grohé, S. and Uribe, M., (1997), “Balanced-Budget Rules,
Distortionary Taxes and Aggregation Instability,“Journal of Political
Economy, Vol. 105, No. 5, 976-1000.
[9] Takata, F., (2017),“Two Steady States and Two Movement Patterns
under the Balanced Budget Rule-An Economy with Divisible Labor,”
Discussion Paper No.1702, January 2017, Kobe University.

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