Doshisha University Life Risk Research Center
Discussion Paper Series No。2010・03

A Note on Fertility and Revenue Variance
Hidenobu Hirata(Graduate School ofEconomics,

Doshisha UniversityJandTadashi Yagi
Ooshisha universityp〃 ″れDiscussion Paper Senes

Abstract
This paper investigates the relationship between fertility and revenue
variance. Evidence shows that there is positive correlatiOn between incomeand fertility.We theoretically study this relationship and explttn why.
Imphcation from this result is that it is better to change the social system
to have more performancebased pay than to have subsidy for higher birthrate.

Introduction
This paper investigates the relationship between fertility and revenue variance.
The importance of income variance to fertility is that the va」

liance raises thevalue Of children who would be born in the future. One's variance of revenue
increases the expectation of income that one and one's children trnight get in
the future and thus one tries to bear more children. We reach to this result byusing both HalFniltOnian and Bellman equation as optilnal control theorェ

This

paper seems to be the flrst theoretical study regarding the association between
fertility and revenue variance. There exsists econometric evidence that incomevariance and fertility would have positi℃relationship(PerOtti,1996).ThiS iS

because, by his paper, in a society where there is l鑓ge income distribution,

government tries to amend it by nscal pOlicy and thus,low economic growth,whereas fertility has a large and highly signincarlt negative coemcient in thegrowth regression.There is also a study that investigates theoretically therelationship between fertility rate and income density. If the population densityand thus income density is concentrated around discontinuity of human capital,a negative aggregate relationship is likely to be observed and Otherwise it would
be positive,where discorltinuity of human capitalis the disindivibilites of human
xed lentth Of diferent opportunities(primarycapital investments such as in■2school,secOndary school,and etc)(Docq■lier,2004).In thiS Study only thedensity is concerned and nOt the variance of income distribution. Croix andDoepke(2003)includes nOt Only total fertility but also diferential fertility intheir llnodel. They assume an unequal initial distribution of human capital andshow that endogenOus fertiliむ r rate and schoohng tilne are functiOns Of it. In
our model we show that it is rather uncertainty that in■uence the long― run

dynalnic behavior Of fertility lt is quite natural to assme that the incolFne
distribution is enlarged by each agent's revenue rather than diSerentialヽ

Ⅳagewith no uncertaillty about their income in the future. Besides,if their income
have nO uncertainty in the htllre and just the disparity between them,incOme
distribution、 vOuld not be enlarged

Further we later shoM″in Section 2 thatdynamically optimal fertility rate is not a inction of wage,υOr its difFerence.

BarrO and Beckel・ (1986),BarrO and Beckel・ (1988),and Barro and Becb(1989)
use Lagrange muLilplier fOr evew period thtt dOes not anow for tt savintt ineach periOd.Dahan and Tsiddon(1998)showed theoretically that the fertility
and GDP gravth fO1low an inverted‐
U shaped relatiollship.They examine thenumber Of O■ spring fOr the unskilled,the skilled number Of ofFspring of skilledparents,and the skilled number of ofFspring of unskilled parents and show thatthe fertility of the unskilled is higher than thtt of the skilled which is also highαthan that of the la.st One. Agellts choose the higher leve1 0f incOme incurring
the cost tO become skilled wOrker along、/ith the econonlic grclwth and thus

the fertility rate alters.[rhat is,along、 Ⅳith the econOlrnic groM″th and to someGDP,agel■ ts have high level of incOme and thus highおrtiliサbut hereatter,the price of bearing a child becomes expensive and consequently it brings low
fertih年 ヽlorand(1999)alSO re∝h a similar conclusbn that theねrtnity andGDP grOwth follow an inverted― U shaped relationshipセッ considering old―age
supporじ fOr child bearing. Al1 0f the theoretical models above and most Of the3ぼ (1986;1988)andlitσ
atllre in this ndd are along the lines of Barro and Beは石e pursue for more complete model with regards toBarro and Becker(1989).ヽ ヽeconolllic transition in capital accumulation through savings,levels of fertili可ぅ石e appけthe model of Barro andcomsumption,allld more than one parents.ⅥSala―卜Martin(1995).

The lnodel
We think of assets per erective labor,た as the cumulation ofindependent iden‐
tically normally distributed increments of savings and a function of asset prices.
Thus,we can write

αた。=μ んをdιttσれdB,
where B is a standaFdiZed Brownian mdion(Wiener Process)whOSe increment
aた has zer。lnean and variance at. This is because thatたdepends on savings per

erective labor and savings depends on the cost to raise children and consump―tion,c,and price of Anancial assets that alter stochasticdly7.ThuS,increments
ofた per time depends on savings p∝

お or.Because of(1),time by the efFective l〔a competitive llnarket,open economyO and arms'pront maDamization b濾(鳥)](1+g̀)・(鳥), υを=Iノ(鳥)一鳥ノ′
rt=ノ′ngs,(2)and鳥 =乙 /1(1+θ)jZ』aswhere tt is the expected value of tt for generationづin Barro and Becker(1989).Heretter we omit t and use represelIItttive agentsand writeたinttead ofれ.
We assume that卜θy=JTtt・―一Ⅳ肱
気πの匂(3)4

where c stands for consumption and its price is norΠ
equation is iom Barro and Sala―

lalized to one The abovei―
Martin(1995)and nOtations are same asthem.The diference with Barro and Sala‐

V isi‐
L〔額tin is that we suggest that」

artin,Ⅳ is the scale of a fanlily where n isa function πo .f I n B a r r o a n d S a l■aM‐
the fertility rate. And also we use a diferent cost function for the education ofchildren. WVe take into our consideration of the quality ofthe child and the costto raise the child to the quality lVe maximizeび

applying Halniltonian function.

Fo1lowing Becker and Lewis(1973)we notate

π the price of 9,c their qualityt,n),N=N(0)and N(t)=N(0)eれ( a S S u m e d t o b e t h e s a m e f o r t t o fitlldlreeぬand c quantity of other consumptions,where the price of c is normalized to l.

er andSubstituting these notations illto a simple budget constraint as in BeよLewis(1973),i.e・ ,T〆
λ= 颯① 十Ц→五(4)綺 υ 十 戒 一 η τ一 の Ⅳ″ ≧ Qt,.Thus
c→=c―Цρ釘we assume thttr=ρτ.H∝e砒where )=fr(7)αRけand we have,here

力(3)=鳥 (υ+rた 一π9π―c).
Wejust assume here that力

is a function of aたand dた is a function of savings per

eSective labor s,at a tilne. The reason of this is that,as in Beder and Barro(1988)and Barro and Becker(1989),we include human capital inた.Becallseん

includes human capital,the transfer of it beyond generations lnight take morethan the integal of aた . Rather it is natural to assume that the transfer ofた,human captial,to ones'children take several tilnes.It inight depend on abilitiesof children or hOw parents teach them.It is thus difncult to denne and therefore
bたa
5 7 n d w e j u s t m e n t i o n t h aitsた aw e d o n o t i n s i s t h cwlowuたl d b e a t t a i n e d α

5ows lvhen the asset is bought

function of αた。Because the asset yields no cash■by s,the value of holding it is its capital appreciation and there is IIo immediatepayout or dividend iom the asset. Thus a no‐arbitrage or equllibriun■collditionbril瑶s,ι=EIαたrttα1・Following the model of D破it and Pindyck(1994,pp.140‐(6)142),we therefore get

た=4た β,(7)whereた =O whenた =0,

ス =響

β=:一 多 +(8),̲:)2+『q争》

and r is the tangible cost of investment or buying an(9)asset, た。In derivingequation(7),we uSe

ぽ = λ ・― I(10)*′ん (j)=1・(11)and*are the critical value where the net gain from the t りIn equation(10),鳥 ・ムndん
equals I.Equation(11)iS the Sm。
expression becomes as below.

。th pasting condition.Thus,Hamiltonia雌Ц→―″=鶴 ‖硫―πⅣⅣ%し「一九η
なのの旬■十レ十十 λ2 ( ηανV ・(12)The second term in RHS isλl multiplied by diferentiated budget constraint,
where λl and λ2鉗 e the shadow prices regarding with the two state variables,■
01 Variables to maximize I1/in equation(3).
たand Ⅳ,whereas c and n are co■when solving Ha■ liltonian.

We put T=cЮ

1=0
θβψ
一ψ
=Ⅳ―
ス
けλたlβけ納くⅣ鋤
等β1=λAたlβ0⇒=等 =0,andⅣ←)=Ⅳω)ent
Because of鮮θψ̈0‑のφ句―
Cし酬ⅣⅣψ:ι霧 =〆ρθlιⅣψC(2‑α)旬 [ψⅣψ ⅣC(2‑a)φ十 θ t〔β19π― λlle RO(βスλⅣ十η9πιⅣ)l十Ⅳl = 0 ,λ2 1 Ⅳ十 ( η一 d ) ι(14)where combining the nlst and second term on the right hand side we get,

σ甲町Ⅳ% いの午J[島十 期 = 「 ″札 βA ♂ ・ Ⅳd 島

壼om equ就lon(13).Thll%器
beCOmes

十期0つ

4Ⅳ
Юttβ
ご
4♂
d∴+朝

∂fr
∂π

β19πλlレーR(t)(βスんⅣ十η9πtⅣ)]
十λ2 1 Ⅳ十 ( η― α) t Ⅳl = 0 ・Because of transversality condition,when t=Oo,λ(16)r=0. Thereforelυ =λ 2ハthe above equation becomes,‐ス♂
Ⅳ為βd∴+切 λlβスたβ19πⅣ 十λ129πtⅣld∴十効 β労影妥=+9π(η― d)静and deviding both sides by t and substituting t=∞″′(π一a)
Cφtt Cψι+9πwe have,一αバ■の= い一のf Z=農百η* =型 β五たβ‑1.

(18)

9π

This is the steady―state dynamically optimal fertility rate with uncertainty
regarding the return Of human capital exanlined.

ヽヽb see here in equation(18)that η*iS not a function ofυ or its diference,because even thollgh wehave included wage direrences,it would be deleted by transversality condition,λlυ =λ 2ハ「=0・ This result has ilnplicatiOn that although revenue diference
has no efFect on fertility, one's hture income variance has l証
eSect on fel・titlity.That is,richness per sθ

ge and positive

is not encouraged for fertility but,

8emciency Wage is encollraged,because this expands the uncertainty about one's
futllre income and thus its expectation. The policy implication iom this result
is that,though there is some diect policy thtt has some impact on fertilitly,itmtht be better tO change the social systeln to have higher b士

th rate.恥 re see

how it is encouraged in Section 3. The expectation of higher return in the future
σ lead parents to have more children.Besides,to have thoseexpressed by larger
high return with lttge

σ,i.e. uncertaintL assoCiated with their children,oneinvOlves human capital also,expectationneeds numerous children. Becauseた

towards their Own children becomes incentive to give bむ

th to mOre children.

This implication could also be applied to those who get tax exemption. lrhe
fact that tax expemption and subsidy to child in poor families have positive ef―

fect on fertilitァ
COuld be understood as the future income that the children in the.l But fOr those who are not poor the credit pr(vided
futtre bring to their famiけand expanded,the gO℃ rnment such as the earned inCOme tax credit(ETIC)O n けp r O d u c e d e x t r e m e l y s m a l l r e d u c t i o n s i n h i g h e r O r d e r f e r t i l i t y a m o n gwomen(Baughman and Dick=t―Conlin,2009).The reason of this,understoodiom our study, is that because ETIC is phased out with additional incomee the uncertainty
a b o v e a c e r t a i n a m o u n t o v e r t h e p h a s o o u t r a n g e , i t moandけor valriance of revenuc including the credit and income smaller. And thus,thevalue of children in the futllre are lnade lower and therefore reduced the fertility.
This endogenous fertility decislons involve some exogenous factors involving
the transfer of assets including human capital.Equation(2)is giVen by themean ofたand each agent's wage and therefore it is endogenous,but becauseたis the cumulation of independent identically normally distributed increments ofsavings,dynamically optimal fertility rate involves uncertailllty regarding humancapital transfer.

lSee Whittington(1992)and ttLittington,ct

α:(1990)

3

Silnulation

W e c a t n u n d e r s t t t d i o m e q u a t i ocn (a1l8l)dtぽh a¨v e p o s t t i v e a n d l i n e a r
relatiolllship.
ヽb sinlllate connection between
σand η*beneath. In Figure l,ψ =0.2,r=0.05,μ=0.01,た=5,alldw e s u b s t i t u t e c = 1 0 , 9 = 1=02,,π
I=10.

I︲動
Figllre l:Relationship between optimalお武iliサand σ.

It is evident that the variance in agents' future income have l証
ge e3ect onf e r t n iたT
ちhiS iS because uncertainty regarding human capital transfer towards
their children enhance the value of their Ofspring.

Concluding remarks
We study in this paper the positive correlation between revenue variance andfertility lt is the expectation regttding theむfuture income that raises thevalue of their Orspring and thus it becomes the incentive to have rnOre children.

10The expectation that parents and their o、

vn children llnight yield higher returnin the futllre lead parelllts to bear more children.Subsidlr is only efFective topoor people that could be seen ioIIn econometric evidence and the reason that
we understand from our study is that subsidy raipes the expectation that theymight be rewarded by their coming children,but for those who are not poor therevenue variance caused by the subsidy is not large as much as for those who arespring.The policy implication from thispoor and thus also the value of their o■so that performancebased
paper is that it is better to change the social systen■wage is more accepted.

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Altruism and the economic theory ofOl・
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図 Barro,R.J.and BeおA reformulation of the economic,G.S.(1988)

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