Discussion Paper No. 697

DOES PRO-POPULATION POLICY RAISEPER CAPITA CONSUMPTION?

Ken-ichi Hashimoto
andYoshiyasu Ono

July 2007

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

July 2007

Does Pro-population Policy Raise Per Capita Consumption?*by†Ken-ichi Hashimoto and Yoshiyasu Ono‡

Abstract
We theoretically analyze the effects of a child allowance, an improvement in the efficiency
of child rearing and a labor income tax on the fertility rate and per capita consumption. The
effects on per capita consumption are opposite in the absence, and the presence, ofunemployment. For example, a child allowance urges people to have more children andallocate more labor to child rearing, decreasing labor supply for the purpose of commodity
production. Therefore, under full employment, it decreases per capita consumption. In thepresence of unemployment, however, it reduces the deflationary gap and hence stimulates per
capita consumption.
Keywords: child allowance, child rearing, endogenous fertility, wage income tax,
unemployment
JEL Classification Number: E24, J13* Hashimoto’s research is financially supported by the Grants-in-Aid for Young Scientists (B), JSPS. Ono’s
research is financially supported by the Grants-in-Aid for Scientific Research, JSPS.
†

Graduate School of Economics, Kobe University, 2-1 Rokko-dai, Nada, Kobe 657-8501, JAPAN. E-mail:hashimoto@econ.kobe-u.ac.jp‡

Institute of Social and Economic Research, Osaka University, 6-1, Mihogaoka, Ibaraki, Osaka 567-0047,
JAPAN. E-mail: ono@iser.osaka-u.ac.jp

1. Introduction

Many developed countries nowadays face low fertility rates, e.g., France (1.91), Germany(1.36), Italy (1.33), Britain (1.63) Sweden (1.75), and carry out various policies to improvechildcare environment.1 Among these countries Japan’s rate is particularly low (1.29). Japanconsiders a low fertility rate to be a major obstacle for economic growth and subsidizes childrearing (the Parental Leave Act in 1992). However, an increase in the number of children
generates costs as well as benefits. It not only increases total labor supply in the future, but also
requires present child-rearing services. Thus, it decreases labor allocation for commodity
production, which reduces per capita consumption if full employment is realized. In thepresence of unemployment, however, the decrease in the individual labor supply shrinks the
deflationary gap so that per capita consumption may increase.
This paper focuses on such a difference in the employment situation and examines theeffects of a child-rearing subsidy, an improvement in the efficiency of child rearing (nurseryschools, kindergartens, etc.) and a labor income tax on the fertility rate and per capita
consumption. We find that the effects on individual consumption are opposite for the cases offull employment and unemployment, and hence the employment situation must be taken into
account when considering a pro-population policy.
Becker and Lewis (1973), Razin and Ben-Zion (1975), Becker and Barro (1988), Palivos(1995) and Cai (2001) assume that the number of children affects utility and analyze the effects
on fertility and consumption of child-rearing costs, public support for child rearing,productivity and preferences in a static and dynamic setting.2 Yip and Zhang (1996, 1997),
Tamura (1996) and Tabata (2003) extend them to an endogenous growth framework andexamine the impact of demographic transition on growth and welfare. Petrucci (2003)
introduces endogenous fertility to a Sidrauski (1967) type monetary model and finds that theneutrality of money is invalid, i.e., monetary policy changes the fertility rate, labor supply,
consumption and capital. However, all of the abovementioned studies ignore the possibility of

1

The fertility data are those of 2004 from OECD (2006). See Atoh and Akachi (2003) for an international
comparison of family policies.
2

Nerlove et al. (1987, ch. 5), Razin and Sadka (1995) and Nerlove and Raut (1997) provide useful surveys ofvarious approaches to endogenous fertility. For empirical evidence in support of the endogeneity of fertility
choice, see Wang et al. (1994).

1

unemployment. 3 To fill this void, we theoretically analyze the effects of pro-population
policies in the presence of unemployment and compare them with those under full
employment.
The remainder of the paper is organized as follows. Section 2 outlines the basic structureof the model. The dynamics and steady state conditions are shown in section 3. Section 4
focuses on the full-employment steady state and examines the effects of the abovementioned
policies on per capita consumption. Section 5 considers the case where unemployment occurs
in the steady state, which is presented by Ono (1994, 2001), and shows the effects of thepolicies are just opposite to those that arise under full employment. Finally, section 6
concludes the paper.

2. The Model

We consider a competitive monetary economy populated by infinitely lived households.
At each point in time, they determine consumption c, real money balances m, and the numberof children n that satisfies

N& = nN,
where N represents the population size. Each household’s lifetime utility isU(c, n, m) = ∫ [u (c) + f (n) + v(m)]exp(−ρt )dt ,
∞

0

(1)

where u(⋅), f(⋅) and v(⋅) satisfy the following properties:u'(⋅) > 0, u"(⋅) < 0; f'(⋅) > 0, f"(⋅) < 0; v'(⋅) > 0, v"(⋅) ≤ 0.
He (or she) allocates labor endowment, which is normalized to unity, to child rearing and labormarket activities. Labor input γh(n) is required to bear and raise each child, where γ represents
the inefficiency parameter of child rearing and h(n) is assumed to satisfyh′(n) > 0, h″(n) > 0.
The rest is the labor supply to the market:

3

An exception is Blau and Robins (1989). They empirically investigate the impact of child-care costs on
fertility and employment using a household survey of employment opportunity pilot projects.

2[1 − γh(n)]N.

(2)

The government imposes lump-sum tax z and wage income tax s, and gives childallowance τ to each household. Thus, each household’s flow and stock budget equations area& = ra + (1 − s)w[1 − γh(n)]x − c − Rm + τn − z,(3)

a = m + b,
where a represents real total assets, r and R the real and nominal interest rates respectively, w
the real wage, x the employment ratio and b the real value of equities.4 Given labor demand L,
x is represented by
x = min(1,L).
[1 − γh(n)]N

Maximizing lifetime utility U given by (1) subject to (3) yields the first-order optimal
conditions:u'(c) = δ,v'(m) = Rδ,f′(n) = δ[(1 − s)wγh′(n)x − τ],δ& = (ρ − r)δ,(4)

and the transversality condition:
limt→∞ δaexp(− ρt) = 0,
where δ is the co-state variable of a.
From the conditions given by (4), we derivec&+ π = R = v′(m)/u′(c),c(5)

f ′(n)= (1 − s ) wγh ′(n) x ,u ′(c)

(6)

ρ+σ
τ+

where
σ ≡ − u″(c)c/u′(c).

4

It is later shown that the firm value is zero because a constant-returns-to-scale technology is assumed.

3

We assume σ to satisfy5
σ ≥ 1.

(7)

The firm sector has a production function with constant returns to scale:Y = θL,(8)

where Y represents output and productivity θ is assumed to be constant. Given nominal
commodity price P and nominal wage W, the firm sector determines labor demand L so as tomaximize profits (Pθ − W)L. Under perfect adjustment of P it is always valid thatw = θ,(9)and thus profits are always zero, implying that the firm value is zero.
The government establishes such a social security system that the same level of wealth asheld by an existing household is given to each newly born household. It imposes lump-sum tax
z and wage income tax s to finance the initial setup wealth for newly born households and childallowances.6 Money stock Ms is assumed to expand proportionately to population size N so thatthe effect of monetary policy is neutralized, i.e.,
M& s N&= = n.
Ms N(10)

The government’s budget constraint is thenPzN + Psw[1 − γh(n)]xN = PτnN + AnN − M& s ,which reduces toz + sw[1 − γh(n)]x = (τ + a)n − nMs/(PN).

Given such household, firm and government behavior as mentioned above, the money andequity markets perfectly adjust so that at any point in time7
the money market: mN = Ms/P,

5

(11)

In fact, Hall (1988) and Guvenen (2006) empirically show 1/σ to be much smaller than 1.

6

Developed countries tend to have a mature public redistribution system while developing countries tend to
redistribute wealth to the next generation within each family. For simplicity, we assume that a complete public
redistribution system is established so that each household does not privately care about the initial wealth of thenext generation.
7

Because of Walras’ law for stock variables, (12) is valid if (11) holds.

4the asset market: bN = 0.

(12)

The commodity market adjustment is also assumed to be perfect, and hencec = θ[1 − γh(n)]x.

(13)

In contrast, the nominal wage adjustment is sluggish so that we can treat the possibility of
unemployment.8 It is simply represented byW&= α(x − 1),W(14)where α represents the adjustment speed of nominal wage W and x is given by (13).

3. Dynamics and Steady State Conditions
From (6), (9) and (13), n is derived as a function of (c, τ, s, γ):n = n(c; τ, s, γ) ⇔ τ +γh′(n)f ′(n)= (1 − s)c.
u ′(c)
1 − γh( n)(15)Using (7) and (15), we find that as long as τ and s are small enough, the partial derivatives of n,represented by nq for q = c, τ, s and γ, satisfy
nc = (σ − 1)f′(n)/[Φu′(c)c] > 0,nτ = 1/Φ > 0,ns = cγh′(n)/{Φ[1 − γh(n)]} > 0,nγ = − ch′(n)/{Φ[1 − γh(n)]2} < 0,where Φ ≡(16)

− f ′′(n) γh ′′(n)(1 − γh(n)) + ( γh′(n)) 2+
c > 0.
u ′(c)(1 − γh(n)) 2

From (2) and (8), the production capacity per capita isyf(c; τ, s, θ, γ) = θ[1 − γh(n)],(17)

where n is given by (15). From (16) and (17), in the neighborhood where τ = 0,8This assumption is imposed merely to allow disequilibrium to occur in the labor market; otherwise, thepossibility of unemployment is intrinsically avoided. Note that under this assumption the possibility of a full
employment steady state is not eliminated. In fact, section 4 obtains the full employment steady state under it.

5y cf = − θγh′(n)nc < 0, y τf = − θγh′(n)nτ < 0,y sf = − θγh′(n)ns < 0, y θf = 1 − γh(n) > 0,y γf>
1 θh( n) f ′(n) ⎧ h ′(n) h ′′(n) f ′′(n) ⎫= − θ[h(n) + γh′(n)nγ] =
+
−
⎨
⎬=0 ,
Φ
u ′(c) ⎩ h(n) h ′(n) f ′(n) ⎭<(18)

where y qf represents ∂yf/∂q for q = c, τ, s, θ and γ. Note that y γf can be either positive or
negative, depending on the functional forms of h(n) and f(n). For example, ifh(n) = nη where η > 1,

f(n) = en1−ν/(1 − ν) where e > 0,thenh′(n)/h(n) − h″(n)/h′(n) + f″(n)/f′(n) = (1 − ν)/n,i.e., y γf is either positive or negative depending on whether or not ν is smaller than 1.
Since (9) implies that P moves parallel to W, from (13), (14) and (17), the dynamics of P isπ≡P&
= α(x − 1) = α[c/yf(c; τ, s, θ, γ) − 1].
P(19)

By applying this equation into (5) we obtain the dynamics of c:σc&= v′(m)/u′(c) − ρ − α[c/yf(c; τ, s, θ, γ) − 1].
c(20)

From (10), (11) and (19), the dynamics of m ism&
= − α[c/yf(c; τ, s, θ, γ) − 1].
m(21)Equations (20) and (21) form an autonomous dynamic system with respect to c and m.
In the steady state c stays constant and hence (20) yieldsv′(m)/u′(c) = ρ + α[c/yf(c; τ, s, θ, γ) − 1].

Note that this property is valid whether or not full employment is realized.

4. The Case of Full Employment
Under full employment x equals 1 and hence from (13), (17) and (22),6(22)

cf = yf(cf; τ, s, θ, γ) = θ[1 − γh(n)],v′(m)/u′(cf) = ρ,

(23)

where cf represents the per capita consumption under full employment. From the first equationof (23),(1 − y cf )dcf = y τf dτ + y sf ds + y θf dθ + y γf dγ.
Since y cf < 0, as shown in (18), this equation gives
for i = τ, s, θ, γ,

sgn dcf/di = sgn y if

(24)

and thus from (18) we findτ↑, s↑, θ↓ ⇒ cf↓,γ↑ ⇒ cf↑ iffh ′(n) h ′′(n) f ′′(n)> 0.
−+h(n) h ′( n) f ′(n)(25)

Since the total differentiation of the first equation in (23) givesθγh′(n)dn = (cf/θ)dθ − θh(n)dγ − dcf,and (15) implies dcf to satisfy
dn = ncdcf + nτdτ + nsds + nγdγ,(26)

dn satisfies(1 + ncθγh′(n))dn = nc(cf/θ)dθ + (nγ − θh(n)nc)dγ + nτdτ + nsds.
Therefore, from (16)τ↑, s↑, θ↑, γ↓ ⇒ n↑.

(27)

The properties given by (25) and (27) are summarized as follows:

PROPOSITION 1. If full employment is realized in the steady state, an increase in child allowanceτ or income tax s increases the fertility rate and decreases per capita consumption. An increasein productivity θ increases both the fertility rate and per capita consumption. An increase in
the efficiency of child rearing (viz. a decrease in γ) always increases the fertility rate, but theeffect on per capita consumption is ambiguous. More precisely,7>>h ′(n) h ′′(n) f ′′(n)−+dc /dγ = 0 iff= 0.
h(n) h ′(n) f ′(n)<<fIntuitively, a rise in child allowance τ or wage tax s reduces the opportunity cost of childrearing, causing people to raise more children and reduce labor supply to the market. Thus, nincreases and c declines. A change in γ has two opposite effects on cf. A decrease in γ (viz. an
improvement in the efficiency of child rearing) directly increases labor supply and thus raises
cf. However, it increases fertility rate n, causing labor supply to fall and thus cf to decrease.

Therefore, the total effect of a change in γ is ambiguous.

5. The Case of Stagnation under a Liquidity Trap
We next consider the case where a liquidity trap arises and unemployment persistently
remains, as presented by Ono (1994, 2001), and compare the effects of the abovementioned
policies and parameter changes with those in the case of full employment.
Money demand is given by the second equality of (5):R = v′(m)/u′(c).

Thus, a liquidity trap arises if the representative household has an insatiable desire to holdmoney:
limm →∞ v′(m) = β > 0,
where β is an exogenous and positive constant.9 Figure 1 illustrates the money demand curve inthis case.
In the presence of the liquidity trap the second equation of (23) may be invalid for any m. It
is indeed the case if
ρ < β/u′(cf) (< v′(m)/u′(cf) for any m),

9

(28)

See Ono (1994: 4-8) for discussions on the insatiable utility of money in the history of economic thought(e.g., Veblen, Marx, Simmel, Keynes) and its economic implications. The validity of this property is empiricallysupported by Ono (1994: 34-8) using the GMM (generalized method of moments), and more extensively by Ono,
Ogawa and Yoshida (2004) using both parametric and non parametric methods.

8

i.e., the full-employment steady state does not exist. In this case a stagnation steady state is the
only steady state, as proved by Ono (1994, 2001). Because of persistent unemployment P
continues to decline and hence v'(m) converges to β. Thus, the steady state level of c thatsatisfies (22) is given by the intersection point of the following two curves:the l curve: R = β/u′(c),the π curve: R = ρ + α[c/yf(c; τ, s, θ, γ) − 1].

(29)

Since u″(⋅) < 0 and y cf < 0 from the first property given in (18), both curves of (29) are
increasing functions with respect to c, as illustrated in Figure 2. Intersection point E denotes
the steady state in which c is smaller than cf, the full-employment level of consumption percapita. Since under (28) the l curve is located above the π curve, as shown by the Figure 2, inorder for this c to exist, the following must be satisfied:β/u′(cf) > ρ > α.

(30)

As is also evident from Figure 2, under (30) the l curve is more inclined than the π curve at theintersection point, i.e.,10
Λ = −βu″(c)/[u′(c)]2 − α[1/yf(c; τ, s, θ, γ) − c y cf /(yf)2] > 0.

(31)

From (29) we obtain
Λdc = − [αc/(yf)2] y if di(for i = τ, s, θ, γ).

(32)

By comparing (32) with (24) we find

LEMMA 1. If unemployment arises in the steady state, the effect of an increase in each (policy or

technology) parameter (i.e., τ, s, θ, γ) on consumption per capita is just opposite to what
obtains when full employment is realized in the steady state.

Figure 3 illustrates the effect of an increase in τ, an increase in s or a decrease in θ. From(18), each of them exogenously reduces yf and thus shifts the π curve upward toward the π′
curve, moving the intersection point of the two curves from E to E′. Consequently, per capita

10

In the appendix it is shown that (30) must hold for the present dynamics to be saddle-path stable.

9consumption c increases from c1 to c2 while full-employment consumption per capita cf
decreases from c1f to c2f.
From (19), at the intersection point given by (29)β/u′(c) = ρ + α(x − 1),from which we obtain{−βu″(c)/[u′(c)]2}dc = αdx.

(33)

Therefore, employment rate x moves in the same direction as c does.
Since (29) yields c as a function of τ, s, θ and γ, totally differentiating (15) leads to
dn = (ncdc/dτ + nτ)dτ + (ncdc/ds + ns)ds + (ncdc/dθ)dθ + (ncdc/dγ + nγ)dγ.

(34)

From (16), (18), (32) and (34), we obtaindn/dτ > 0, dn/ds > 0, dn/dθ < 0.

(35)

The effect of an increase in γ on n is calculated in the appendix. It satisfies
dn/dγ < 0.

(36)

Thus, from lemma 1, (33), (35) and (36), the following proposition obtains.
PROPOSITION 2. If unemployment arises in the steady state, an increase in child allowance τ or

income tax s raises the fertility rate, consumption per capita and the employment rate. An
increase in productivity θ decreases the fertility rate, consumption per capita and theemployment rate. An increase in the efficiency of child rearing (viz. a decrease in γ) always
increases the fertility rate, but the effects on per capita consumption and the employment rate
are ambiguous. More precisely,>><h ′(n) h ′′(n) f ′′(n)−+= 0.
dc/dγ = 0 and dx/dγ = 0 iff
h(n) h ′(n) f ′(n)<<>Intuitively, the child allowance policy urges people to have more children and thus
decreases labor supply. In the presence of unemployment it reduces the deflationary gap,
causing consumption to increase. Wage income tax s reduces net income from labor supply andhence decreases the opportunity cost of child rearing. It urges people to have more children and10decreases labor supply, which generates the same positive effect on per capita consumption as
the child allowance policy does. An improvement in productivity θ widens the deflationary gap
and increases deflation, which causes people to decrease consumption. It then reduces wage
income and hence people decrease the number of children.
A decrease in γ (viz. an improvement in the efficiency of child rearing) increases thenumber of children, but the amount of labor that is required to rear each child decreases.
Therefore, the effect on the labor allocation between child rearing and labor supply isambiguous. It also implies that the effect on household consumption is ambiguous.

6. Conclusion

Various policies that promote fertility increase total labor endowment, but reduce labor
supply per capita because parents have to take care of children. Thus, if full employment isalways realized, they reduce consumption per capita. For example, a rise in the child allowanceor the wage tax reduces the opportunity cost of child rearing, causing people to have morechildren and lower labor supply to the market. Thus, consumption per capita decreases.
In the presence of unemployment, however, the decrease in the labor supply caused by apro-population policy reduces the deflationary gap, lowers deflation and thus stimulates
consumption per capita. The effects of such policies in the presence of unemployment are just
opposite to those under full employment.
An improvement in the efficiency of child rearing (e.g., nursery schools andkindergartens) generates two opposite effects on consumption. It decreases the volume of labor
required to rear each child and hence increases labor supply as long as the number of children isinvariant. However, it increases the fertility rate, which requires parents to allocate more laborto take care of children and thus lowers labor supply to the labor market. Therefore, the labor
supply per capita can either increase or decrease, depending on the shape of the utility functionof having children and that of the cost function of child rearing. If it increases, consumption percapita increases as long as full employment is realized, whereas in the presence of
unemployment it decreases, and vice versa. The effects of the policies that affect the fertility
rate in the presence of unemployment are just opposite to those under full employment.

11

Appendix
Derivation of equation (36)

From (17), (18), (32) and (34),Λdn/dγ = ncαch(n)/[θ(1 − γh(n))2] + {Λ + ncαcγh′(n)/[θ(1 − γh(n))2]}nγ.

(A1)

From (17), y cf in (18), and (29), Λ given by (31) satisfies
Λ + ncαcγh′(n)/[θ(1 − γh(n))2] = (ρ − α)σ/c + α(σ − 1)/[θ(1 − γh(n))].

(A2)

Using (15) in which s = 0 and τ = 0, and nc and nγ given by (16), we obtainnc = − nγ[1 − γh(n)](σ − 1)γ/c.

(A3)

Substituting (A2) and (A3) into (A1) and rearranging the result gives
Λdn/dγ = [α(σ − 1)/θ + (ρ − α)σ/c]nγ.

Since σ ≥ 1 from (5), nγ < 0 from (16), ρ > α from (30), and Λ > 0 from (31), dn/dγ is negative,
as described in (36).

Dynamic stability

We examine the stability of the dynamics described by (20) and (21). For simplicity, we
treat the case where child allowance τ and wage income tax s are both zero. Let k = 1/m,
substituting (17) into (20) and (21) yields
c& =

⎛ c
⎞⎤
c ⎡ v ′(1 / k )− ρ − α⎜⎜ f− 1⎟⎟⎥ ,⎢σ ⎣ u ′(c)⎝ y (c ) ⎠ ⎦⎡ ⎛ c⎞⎤k& = kπ = k ⎢α⎜⎜ f− 1⎟⎟⎥ .
⎣ ⎝ y (c) ⎠⎦Linearizing them in the neighborhood of the steady state values (c*, k*) gives⎛ c& ⎞ ⎡ A1⎜⎜ & ⎟⎟ = ⎢⎝ k ⎠ ⎣ A3

A2 ⎤⎛ c − c* ⎞⎟,⎜A4 ⎥⎦⎜⎝ k − k * ⎟⎠

where

12

(A4)

A1 (≡ ∂ c& /∂c) =⎤c ⎡ − u ′′(c)v ′(1 / k )− α[1 / y f − cy cf /( y f ) 2 ]⎥ ,⎢2σ ⎣ [u ′(c)]⎦A2 (≡ ∂ c& /∂k) = −c ⎡ v ′′(1 / k ) ⎤⎢⎥,σ ⎣ u ′(c)k 2 ⎦A3 (≡ ∂ k& /∂c) = kα[1/yf − c y cf /(yf)2],A4 (≡ ∂ k& /∂k) = π = α(c/yf − 1).
Since the characteristic equation isΨ(λ) = λ2 − (A1 + A4)λ + (A1A4 − A2A3),
eigenvalues λ1 and λ2 satisfy
λ1λ2 = A1A4 − A2A3 =+⎤c ⎡ − u ′′(c)v ′(1 / k )− α[1 / y f − cy cf /( y f ) 2 ]⎥ π⎢2σ ⎣ [u ′(c)]⎦c ⎡ v ′′(1 / k ) ⎤α[1/yf − c y cf /(yf)2].
⎢⎥
σ ⎣ u ′(c)k ⎦(A5)

If full employment is reached in the steady state, π = 0 and hence the first term of (A5)becomes to zero. Since y cf < 0 from (18) and v″(.) < 0, the second term is negative. Thus, λ1λ2< 0, implying that one of the eigenvalues is positive and the other is negative. Since c is
jumpable and k (≡ 1/m) is non-jumpable, the dynamic path is saddle-point stable.
If there is no full-employment steady state and persistent stagnation arises, x < 1 and π < 0in the steady state. Therefore, k (≡ 1/m) approaches zero and the value given by the square
brackets of the first term of (A5) equals Λ, which is positive from (31). Thus, the first term of(A5) is negative. The second term is negative since y cf < 0 from (18) and v″(.) < 0. Therefore,(A5) is negative (λ1λ2 < 0), implying that the dynamic path is saddle-point stable.

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15

R

β
u ′(c)

Om

Figure 1: Money Demand with a Liquidity Trap

16

Rl curveβu′(c f )π curve

ρ
E

ρ−α

O

cf

c*

Figure 2: Stagnation Steady State

17

c

R
l curveπ′ curve
ρ−α

π curve

E′
E

O

c1

c2

c2f

c1f

Figure 3: Pro-population Policies

18c