This paper explores the implications of declining population on the economy. A simple model with household childbearing decision with macro-micro feedback rule is introduced to argue that population cannot continue to decrease forever. Then we review the innovation and growth literature to understand the possible implications.

Aoki, Reiko

English

Research Papers in Economics

Hitotsubashi University Repository
Title
Microeconomics of Declining Birthrate: Review of
Existing Literature
Author(s) Aoki, Reiko
Citation
Issue Date 2004-10
Type Technical Report
Text Version publisher
URL http://hdl.handle.net/10086/14289
RightMicroeconomics of Declining Birthrate -
Review of Existing Literature
Reiko Aoki ∗
Institute of Economic Research,
Hitotsubashi University†
October 2004
Abstract
This paper explores the implications of declining population on
the economy. A simple model with household childbearing decision
with macro-micro feedback rule is introduced to argue that population
cannot continue to decrease forever. Then we review the innovation
and growth literature to understand the possible implications.
∗This discussion paper summarizes presentation made by the author at the PIE/A3&A4
Summer Workshop, September 2004 at Institute of Economic Research, Hitotsubashi Uni-
versity. Author is grateful to comments by Noriyuki Takayama and other participants of
the workshop. The research is part of the academic Project on Intergenerational Equity
(PIE), funded by a scientiﬁc grant from Japan’s Ministry of Education, Culture, Sports,
Science and Technology (grant number 603).
†aokirei@ier.hit-u.ac.jp
11 Introduction
There is general agreement that the declining birthrate in Japan is here to
stay, at least in the short run. This is admittedly a belief since there seems to
be no single identiﬁable cause of the decline and there has been no evidence
of eﬀectiveness of any of the policies implemented to address this problem.
While much of the debate has been on the cause and policy to counter the
declining birthrate, perhaps now is the time to think about what will happen
to the economy when the population decline continues. Akihiko Matsutani
has taken this view (Matsutani (2004)). He has made several predictions
about where the economy is going. His policy recommendations have been
not how to stop the decline but what should be done to improve welfare given
the decline continues. As result of population decline Matsutani predicts
that,
• National income will have negative growth
• Per capita national income will not decline
• Firm proﬁtability will not decline
• Changes will occur in
– international specialization
– domestic industrial structure
• Relative increase of imports will decrease the domestic-world price dif-
ferential
• Becomes necessary to change and downsize social capital
In the next section, I introduce a simple framework with household child-
bearing decision. Individual decisions are related to macro variables. Then a
feedback mechanism from labor market to childbearing decision is introduced
1to argue that neither population decrease or increase cannot continue forever.
In the remaining sections, we turn to the innovation and economic growth
literature (Grossman and Helpman (1991)) to understand the economic path
with population decline. There seems to be no literature that analyzes the
economic path when population (labor force) is declining. The economic de-
velopment framework always assumes population is growing. There seems
to be no literature that analyzes the economic path when population (labor
force) is declining. The economic development framework always assumes
population is growing. The innovation growth literature assumes innova-
tion as the sole engine of growth. In particular population (both labor force
and consumer) is assumed constant. We conduct comparative statics on the
population.
2 Economics of Declining Birthrate
In this section we quickly review the economic foundation of birth decisions
and population change.
Household
A consumer or a household lives 2 periods, 0 and 1. In period 0, there is
income (I0), and consumer decides how much to consume in each period
(c0 and c2) and how many children to have, n. We normalize the price of
consumption to 1. Price of a child, p, represents the cost of births and raising
a child. There will be a return of r from each child. This return is pecuniary.
For some households with a family business, this could be quite large. For
instance, in a family of Kabuki actors, a successful child will mean prosperity
of the family business. Cost of raising an heir in the Kabuki family will also
be very costly, meaning a large p. On the other hand, for wage earners, r
may be very low. Although not as expensive as Kabuki training, schooling
costs can be quite high for wage earners too. This return does not include
2any “good feeling” from having a child. Such a factor should be in the utility
function.
A consumer’s utility is,
u(c0,n) + λu(c1),
where λ is the discount factor. We make the usual regularity assumptions
regarding derivatives of u(·).
A consumer’s budget constraint is,
c0 + pn + λc1 = I0 + rn ⇔ c0 + +(p − r)n + λc1 = I0.
Depending on the diﬀerence p − r, a child is a consumption or investment
good. If one can draw substantial income from the family business so that
r > p, then money should be spent on training a successful heir than putting
money in the bank.
Births and Population
In this section we explain the relationship between individual childbearing
decisions and change in population at the macro level.
Assume the diﬀerence p−r is distributed (random variable µ) on [−b,b].
We denote the maximum marginal utility by ¯ un. (This may be an addi-
tional assumption which is consistent with the regularity conditions on the
derivatives.) This occurs at consumption 0 and all marginal utilities are less
than ¯ un. ¯ n denotes the maximum number of children that a household can
physically have.
3Optimal number of children will be,
n
∗(p − r) =

   
   
0 p − r > ¯ un
ˆ n(p − r) 0 < p − r ≤ ¯ u
¯ n p − r ≤ 0
,
where ˆ n(p−r) is the interior solution to the optimization problem. In other
words, it solve the ﬁrst-order condition,
∂u
∂n
(c0,n) = p − r. (1)
For the society as a whole, the total number of children born is
n
∗ =
Z
p−r<0
¯ ndµ +
Z
0≤p−r<¯ un
ˆ n(p − r)dµ.
The new number of people, n∗, can be greater or less than the number of
parents’ generation. It depends on both cost of children (p) and return from
children (r). Since children can be form of savings, generous pension could
contribute to less children. This eﬀect is the income eﬀect which would be
reﬂected in (1).
Feedback Relationship
The change in labor force follows
Lt+1 − Lt = sL (rt − pt)Lt,
where rt and pt are return and cost of having a child at time t. sL is a
constant that measure eﬃciency of child rearing, reﬂecting social capital.
4We can make the following approximation regarding return from a child,
rt = Et

∂f
∂Lt+1

, pt =
∂f
∂Lt
.
Return is future marginal product of labor and cost is the forgone wage , i.e.,
current marginal product of labor. When population declines over a period
and labor supply declines, wages should become higher. This will increase
the return of a child. If the population increases over a period, the wage
and and in turn the return of a child will decline. Thus labor force should
not continue to decrease or increase over long periods of time unless there is
signiﬁcant change in capital.
Labor productivity also depends on level of capital. Capital accumulation
proceeds according to
Kt+1 − Kt = sKYt, Yt = f(Kt,Lt).
If Kt+1 > Kt, it is reasonable to assume,
∂f
∂L
(Kt+1,Lt+1) >
∂f
∂L
(Kt,Lt)
with Lt+1 − Lt > 0 . In this case, labor will increase for a prolonged length
of time. If return to labor is perceived to decline in the future, it will be,
∂f
∂L
(Kt+1,Lt+1) <
∂f
∂L
(Kt,Lt),
and labor will start to decrease. This will raise the marginal product of labor,
and labor decline should not last forever.
This simple analysis suggests
• Population decline cannot last forever.
• Policy variable other than p should be considered, such as social capital
(sL).
5The proceeding simple model is a skeleton model and other variables
should be considered. There are also likely to be a other relationships be-
tween the existing and yet to be added variable. As the ﬁrst step towards
a more comprehensive model, we review the innovation growth literature in
the following sections.
3 Solow Model
We start with the classic Solow Model. The economy is,
Z = F(K,AL),
where Z is output, K is capital, A is productivity of labor and L is labor. It
is assumed that the function F(·) is concave and linearly homogeneous.
We can write output per eﬀective labor, z as function of capital per
eﬀective labor, k,
z =
Z
AL
= F(
K
AL
,1) = f(k), where k =
K
AL
.
The ﬁrst equality can be interpreted to mean population decline results in
increased per capita output (actually per eﬀective capita). However Z also
depends eﬀective labor as well as capital. Denoting saving rate by s, capital
accumulates according to
˙ K = sZ = sALf(k).
Capital per eﬀective labor grows according to,
˙ k = sf(k) − (
˙ A
A
+
˙ L
L
)k.
In traditional growth theory, technical advance
˙ A
A and population growth
˙ L
L
6are both assumed positive. On the other hand,
˙ L
L < 0 will increase capital
per eﬀective labor ceteris paribus. The relationship also implies that decline
in savings (which is often associated with aging society) will reduce capital
accumulation ceteris paribus. Overall if per labor capital K
L = Ak and
˙ A
A are
not very large, then ˙ k < 0. Since z = f(k), this means output per eﬀective
labor will decline.
Getting back to outputs,
˙ Z = ˙ (AL)z + (AL)˙ z.
Growth may be negative even even if ˙ z > 0. Per capita (not per eﬀective
capita),
˙ (Az) = ˙ Az + A˙ z,
may not be positive if ˙ z < 0.
Last but not least, we note that without homogeneity of the function,
F(K,AL),
˙ Z =
∂F
∂K
˙ K +
∂F
∂L

˙ AL + A ˙ L

.
The relationship between population change and output is eve more complex.
4 Grossman-Helpman Product Variety Model
In this section we consider an economy where consumers have preference for
variety and innovation increases variety. Speciﬁcally, utility of representative
household
Ut =
Z ∞
t
exp
ρ(τ−t) log
Z n
0
(τ)x(j)
αdj
 1
α
dτ, 0 < α < 1,
where elasticity of substitution is  = 1
1−α > 1. Parameter n is measurement
of product variety available. Additional variety is achieved by innovation
7which requires ` units of labor,
dn =
`
a
dt, a > 0.
Each ﬁrm produces one variety and products are perfectly product diﬀeren-
tiated. Thus manufacturer of brand j is a monopolist, and gets proﬁt
π =
1 − α
n
,
with price p = w
α where w is the wage.
Equilibrium in capital markets requires ,
π + ˙ v = rv,
where v is the value of ﬁrm and r is the interest rate. The exogenous supply of
labor is a constant L and demands are from the innovation and manufacturing
sectors. The equilibrium condition in the labor market is,
a˙ n +
1
p
= L.
Denoting discount rate by ρ, we have ρ = r. Incorporating the capital
markets equilibrium we have the no arbitrage condition,
˙ v = ρv −
1 − α
n
.
This is the downward sloping curve in Figure 1a. Free entry and resource
constraint implies,
˙ n =



L
a − α
v v > ¯ v = αa
L
0 v ≤ ¯ v.
(2)
This is the horizontal line in Figure 1a. The steady state, (¯ n, ¯ v), is the
intersection of the two curves, point E. More speciﬁcally, ¯ v = αa
L and ¯ n =
8(1−α)L
αρa .
Now we consider what happens if labor decreases from L to L0. Only
equation (2) is aﬀected and it shifts upward. The steady state variety de-
creases to ¯ n0 < ¯ n, and ﬁrm value increases to ¯ v0 > ¯ v (Figure 1b). However,
this is only comparison of steady states. In the short run, as result of sud-
den reduction of labor implies economy is below equation (2). There is no
innovation and v decreases. This suggests that the speed of adjustment to
steady state after reduction is very important.
5 Product Variety Model with Public Knowl-
edge Capital
In this model, innovation results in accumulation of knowledge capital which
can be shared by everyone in society. Knowledge capital in turn increases
product variety, each product again comprising a single market (ﬁrms are
monopolists). Knowledge capital Kn (n varieties) is a public good and con-
tributes in the following way to increasing product variety,
˙ n =
LKn
a
,
where L is the total labor of the economy. We can normalize so that Kn = n.
Labor is used for manufacturing and innovation. The labor market equilib-
rium is.
a˙ n
n
+
1
p
= L. (3)
The following substitutions are made for convenience,
V =
1
nv
, g =
˙ n
n
.
9Then the free entry condition with (3) implies,
g =



L
a − αV V < L
αa
0 V ≥ L
αa
. (4)
Higher rate of innovation leaves less labor for manufacturing. This leads
to higher higher price and higher proﬁt for those ﬁrms that can produce.
Graphically, this the downward sloping line and the vertical axis above its
vertical intercept in Figure 2a. Finally the no arbitrage condition is,
˙ V
V
= (1 − α)V − g − ρ.
This is the upward sloping line in Figure 2a.
The steady state equilibrium is the intersection E. In steady state, va-
riety continues to increase (g > 0) but ﬁrm value is constant. Division of
labor between innovation and manufacturing also remains constant in steady
equilibrium.
Now we consider what happens if labor decreases from L to L0. This
means equation (4) moves downward (Figure 1b). The new steady state
equilibrium must be E0.1 The rate of variety increase declines. The value of
the ﬁrm remains constant at a lower level.
6 Human Capital Investment
We consider a model where there is both product and labor heterogeneity.
High-technology products require high-skilled labor but resources can be used
1Other expectations on return on ﬁrm value are not consistent. If expectation on
return on v is greater than at E0, then variety stops at ﬁnite number (g = 0) and V goes
to inﬁnity or v goes to 0. But this is not possible since ﬁrm proﬁt remains strictly positive
with ﬁnite number of variety. If expectation on return is less than E0, then v goes to
inﬁnity while variety grows at maximum rate. Increasing number of variety means there
is an upperbound of v. Again, not possible.
10to improve quality of labor. Speciﬁcally, there is unskilled labor (L) and
high skilled labor (H). Individuals invest in human capital (take time for
schooling) to become high skilled labor. Innovation requires high skilled labor
but manufacturing uses low skilled labor. The rate of innovation is γ.
There are three types of manufactured goods: traditional (Z), high-tech
(Y ) and intermediate input (X). Manufacturing of the traditional manu-
factured good requires only low-skilled labor. The high-tech product uses
high-skilled labor and the intermediate good. The intermediate good is di-
rectly inﬂuenced by innovation. Each good has ﬁxed coeﬃcients production
process.
Steady state equilibrium can be characterized by the following three re-
lationships. First, the no-arbitrage condition is,
(1 − δ)pXX
cγ(wL,wH)
= ρ + γ
⇔ BXcX − BZcZ =
(1 − δ)σ
ρ + γ
, (Π)
where cZ, cY, and cX are unit costs of production, cγ = cγ(wL,wH) is the unit
cost of R&D, wH and wL are wages, and pX is the price of the traditional
good.
Using the notation convention aLX as input coeﬃcient of L labor for
producing X, the labor market clearing condition, one for each type, are,
aLγγ +
aLXσδ
cX
+
aLZ(1 − σ)
cZ
= L, (L)
aHγγ +
aHXσδ
cX
+
aHZ(1 − σ)
cZ
= H. (H)
The three equations are depicted in Figure 3a. The common intersection of
all three lines, E, is the steady state equilibrium.
First let us consider the reduction of both types of labor. This shifts
both labor curves (H) and (L) downward. Then the rate of innovation γ
11must be reduced to move the (Π) curve downward. (The new intersection A
in Figure 3b.) This makes the labor curves shift upward until all three lines
again intersect at the same point. (Intersection E in the same Figure.) This
is the new steady state with a lower γ.
Now let us consider the reduction of L only. This time only L moves
downward. Accordingly now Π must move upward to the new intersection
A in Figure 3c. This increases γ and both labor curves move downward.
(The new equilibrium E). As result of reduction of low skilled labor, rate of
innovation increases.
The result of reduction on only high-skilled labor is summarized in Fig-
ure 3d. The rate of innovation is reduced as result. The proceeding analysis
shows that there are diﬀerent eﬀect of labor reduction on rate of innova-
tion. We have shown that it is not possible to categorically claim eﬀect of
population reduction on pace of innovation.
7 Innovation and Trade
Japan is an open economy and trade is an essential part of the economy. We
consider a model with two countries, A and B. Each country produces two
goods (innovative and traditional). There is only one type of labor and both
goods require it. Labor also is used to improve the the innovative good.
Each country has labor market and no-arbitrage conditions as in the
previous section. In addition, there is a market clearing condition.
In equilibrium, only Country A innovates if A has suﬃciently large stock
of innovation. The production of the traditional good may be specialized.
Speciﬁcally, both countries produce traditional good if,
LB
LA + ρ
≤
1 − σ
σ
. (5)
12Only Country B produces traditional good if,
LB
LA + ρ
≥
1 − σ
σ
. (6)
In equilibrium, initially wages will be equal in both countries, meaning
consumers in both countries are equally well-oﬀ. But as only country A
continues to innovate, it must provide increasingly many high-technology
products. Eventually, labor becomes very scarce in country A and wA > wB.
A decrease in population of country A may change the world regime from
(5) to (6). Then Country A will only produce high-tech goods which Coun-
try B will only produce traditional good. As result of population decrease in
one country, trade increases and specialization becomes more extreme. Even-
tually, the high demand for labor in one country leads to wage diﬀerentials.
References
Gene M. Grossman and Elhanan Helpman (1991): Innovation and
Growth in the Global Economy, The MIT Press.
Akihiko Matsutani (2004): “Declining Births and Population Decline
(Shyoushika to Jinkou Genshyou) (1),” Nihon Keizai Shinbun, August 3,
25.
13v
n
0 = v &
0 = n &
E
v
n
Figure 1a: Innovation in Product Varietyv
n
0 = v &
0 = n &
E
E′
' v
v
n n′
Figure 1b: Innovation in Product VarietyE
a
L
V
0 = V &
a
L
α
α
ρ
− 1
g
Figure 2a: Innovation  with Public KnowledgeE
a
L
a
L′ E′
V
0 = V &
a
L
α
a
L
α
′
α
ρ
− 1
g
Figure 2b: Innovation  with Public KnowledgeX c
1
Π
Figure 3a: Human Capital
E
L
H
Z c
1X c
1
Figure 3b: Human Capital
A
L
H
Π
E
Z c
1Figure 3c: Human Capital
E
L
H
A
X c
1
Π
Z c
1X c
1
Figure 3d: Human Capital
E
L
H
Π
Z c
1

Declining Births and Population Decline (Shyoushika to Jinkou Genshyou) (1),” Nihon Keizai Shinbun,

Grossman and Elhanan Helpman

Microeconomics of Declining Birthrate: Review of Existing Literature

http://hermes-ir.lib.hit-u.ac.jp/rs/bitstream/10086/14289/1/pie_dp228.pdf

Microeconomics of Declining Birthrate: Review of Existing Literature

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics