A Theory of the Cross-Sectional Fertility Differential:
Jobs’ Heterogeneity Approach
Daishin Yasui∗February 19, 2014

Abstract
This paper presents a theory of the cross-sectional fertility differential, which produces
the negative wage-fertility relationship based on jobs’ heterogeneity. Compared to the existing literature, the theory not only captures the realistic situation where productivity andworking conditions differ across jobs, but also requires only standard conditions on preferences to generate the negative relationship. Moreover, the result is robust to changes
in economic environments (e.g., public policy and technology). The theory reconciles thenegative cross-sectional wage-fertility relationship with various time-series variation in theaggregate fertility. Furthermore, this study adds an important viewpoint to the empiricalliterature.

∗ Kobe University. yasui@econ.kobe-u.ac.jp

1

1 Introduction
This paper presents a tractable theory of the cross-sectional fertility differential, which produces
the negative relationship between wages and fertility based on the heterogeneity of job productivity. Since the seminal work of Gary Becker (Becker, 1960), a large number of studies havedeveloped fertility theories by using microeconomic techniques. This paper introduces new insights into the literature. In contrast to the existing literature, which focuses on heterogeneity of
the individual ability, we focus on heterogeneity of job (or firm) productivity.1 To introduce thejobs’ heterogeneity, we construct a labor market model where firms offer jobs specifying wages
and working hours, individuals apply to preferable jobs, and one job and one individual form a
production unit. This is in contrast with the rest of the literature, which conventionally assumes
that individuals facing given per-time wages (or given wages per unit of human capital) allocate
their time between labor supply and child-rearing. In our model, the cross-sectional fertility
differential arises as an equilibrium outcome in the competitive labor market, where jobs withdifferent productivity levels offer different labor contracts.2 Compared to existing theories, ourtheory captures the realistic situation where productivity and working conditions differ acrossjobs and obtains the result that fertility negatively correlates with wages under moderate restrictions on preferences. Moreover, the result is robust to changes in economic environments (e.g.,
public policy and technology).
A widespread finding in the economics of fertility is the negative cross-sectional relationship between wages and fertility: wealthier parents tend to have fewer children.3 A large partof theoretical literature has attempted to explain this negative relationship without assumingthat children are an inferior good. The assumption commonly used in the literature is that the

1 Jones, Schoonbroodt and Tertilt (2011) provide an excellent overview of existing theories in the literature.
2 Since the structure of our model considerably differs from that employed by other studies, one may argue thatthe results obtained in this paper are attributable to the special structure, but it is incorrect. The structure is just an
assumption for introducing jobs’ heterogeneity. It is the introduction of jobs’ heterogeneity that departs our resultsfrom those obtained in the literature so far. See Section 5 for details.
3 The negative relationship between wages and fertility is widespread across time and regions, but some exceptional findings are reported. See Section 5 for details.

2cost of children is largely parents’ time, and because of this, parents with higher wages face ahigher price of children. On the other hand, parents with higher wages have more wealth. The
usual substitution and wealth effects coexist: if the substitution effect dominates the wealth effect, wealthier parents have fewer children, and vice versa. Thus, the literature conventionallyimposes restrictive assumptions on preferences that diminish the wealth effect relative to thesubstitution effect in order to explain the negative effect of wages on fertility. Alternatively, nota few studies in the literature add the parental choice on child’s quality to derive the negative
relationship: wealthier parents want more quality and thus less quantity. Adding the quality
choice by itself, however, does not generate the negative relationship, and additional restrictive
assumptions on preferences and/or the child-quality production function are needed (see, forinstance, Becker and Tomes (1976) and Moav (2005)).
By using the US data on 30 birth cohorts between 1830 and 1960, Jones and Tertilt (2008)
find that the negative cross-sectional relationship has been surprisingly stable. Over the same
period, technological progress has led to a rise in real wages, the educational level of the population has risen, the government has conducted various policies (e.g., tax reform, education
reform, and social-security reform), and there has been a long secular decline in the average
fertility rate, interrupted by a temporary rise (i.e., a baby boom). In other words, in the lasttwo centuries of the United States, the cross-sectional relationship between wages and fertility remained negative despite considerable changes in economic environments and time-series
variations in the aggregate fertility. However, with few exceptions, the negative relationshipderived from existing theories are not robust to changes in assumptions, for example, changes
in the form of preferences, the form of education function, and the type and the size of policy
variables. This paper is intended to construct a model in which the negative cross-sectional relationship is produced without imposing restrictive assumptions on preferences and is robust tochanges in economic environments.
The only crucial assumption in our model is that it takes time to raise children. Individuals
prefer higher consumption (i.e., higher wages) and higher fertility. By the assumption that child3

rearing takes time, longer working hours mean lower fertility. Thus, to induce individuals towork longer hours, firms must pay higher wages to compensate lower fertility. Firms are facedwith this tradeoff in offering labor contracts, and firms with higher-productivity job want theiremployees to work longer hours even if they must pay higher rewards because those jobs can
produce the larger quantity of output per unit time. In contrast to the conventional fertility
model based on individuals’ heterogeneity, our model does not entail welfare differential amongindividuals because individuals with different wages merely choose different contracts on thesame indifference curve: there is only the substitution effect, not the wealth effect. Therefore,
our model needs no restriction on the relative size between the substitution and wealth effects togenerate the negative cross-sectional wage-fertility relationship.
Jones, Schoonbroodt and Tertilt (2011) share with us the motivation to produce a robust
negative cross-sectional relationship between wages and fertility, and they present a prospective theory. In contrast with our theory, which focuses on the heterogeneity of job productivity,
theirs sheds light on the heterogeneity of individual preferences. They attribute the negative
relationship to unobserved heterogeneity in parental preferences for fertility. Persons who havea preference for children receive less education in anticipation of having more children and supplying less labor to the market in the future, and as a result, they actually earn lower wages andhave more children.4 The causality is opposite to the conventional fertility theory, from fertility
to parental wages. The negative relationship derived from their theory is quite robust to changes
in the form of preferences: it does not depend on specific functional forms or parameter restrictions. Our theory is as robust as theirs with respect to changes in the form of preferences. An
advantage of our theory is that we do not even need to introduce education into the model. Evenin the United States, in the 19th century, there was limited scope for human capital investments
based on one’s own choice, but the negative relationship was observed (Jones and Tertilt, 2008).
Our theory is applied to the economy where the free choice of employment is guaranteed, even
4 Kimura and Yasui (2007) also introduce parents’ own educational choice to derive the cross-sectional negative
wage-fertility relationship. In Kimura and Yasui (2007), the source of education differential is not taste heterogeneity,
but the arbitrage between becoming skilled or remaining unskilled.

4

if the education system is underdeveloped there.
Mookherjee, Prina and Ray (2012) have a similar motivation. They try to produce a robust
negative cross-sectional relationship between wages and fertility based on a dynamic generalequilibrium approach. In their model, choice between skilled and unskilled occupations by parents plays an essential role in producing the negative relationship.5 Their breakthrough is torestrict individuals’ fertility behavior by using the steady-state condition of the dynamic generalequilibrium, not imposing restrictions on preferences itself. Based on the dynamic general equilibrium model, however, they inevitably rely on the specification of preferences to some degree:
although they impose no restriction on the elasticity of substitution between consumption andfertility, they impose a restrictive assumption on the elasticity of substitution between fertility
and child quality.6
There has been a recent increase in economic theories endogenizing fertility choice.7 One
broader aim of this paper is to offer an “off-the-shelf” fertility model used as one of the building
blocks in applied theories. In addition to the tractability, our theory has a useful characteristic as a building block: it can reconcile the negative cross-sectional wage-fertility relationshipwith various time-series variation in the aggregate fertility. Based on our model, the driving
force behind the aggregate fertility variation across time can be separated from that behind thecross-sectional fertility differential. It is possible, for instance, that the dominant wealth effectassociated with changes in overall productivity and public policy raises the aggregate fertility
whereas the cross-sectional relationship remains negative. Especially in the macroeconomics
literature, the impacts of various policy changes are analyzed (e.g., Greenwood, Guner and
5 Doepke (2004) uses a similar mechanism. The result of Mookherjee, Prina and Ray (2012) is a generalizationof Doepke (2004) to the case in which restrictions on preferences concerning the relative magnitude of wealth andsubstitution effects are relaxed.
6 Another merit of the theory of Mookherjee, Prina and Ray (2012) is that it can explain the widespread negative
correlation between wages and fertility, while allowing for exceptions that arise within occupations. Our theory alsopresents a new insight on this point. See Section 5 for details.
7 For instance, there are studies on the demographic transition and the long-run growth (e.g., Galor and Weil
(1996), Galor and Weil (2000), Galor and Moav (2002), and Hansen and Prescott (2002)), those on the international
differences in fertility (e.g., Adsera (2004) and Manuelli and Seshadri (2009)), and those on the baby boom following
World War II (e.g., Greenwood, Seshadri and Vandenbroucke (2005), Doepke, Hazan and Maoz (2007), Jones andSchoonbroodt (2010), and Kimura and Yasui (2010)).

5

Knowles (2003), Boldrin, De Nardi and Jones (2005), Zhao (2009), Erosa, Fuster and Restuccia(2010)). In such studies, a careful decision about which one to use as a building block is needed
because for the results to be consistent with data, at least on the last two centuries of the UnitedStates, the negative wage-fertility relationship must be preserved in response to various policychanges. Our theory satisfies this criterion as a building block: without imposing restrictive
assumptions on preferences, various types of policies can be introduced and their sizes can bechanged flexibly.
The remainder of this paper is organized as follows. Section 2 presents the basic model andconducts some comparative-static analyses. Section 3 provides an example, where functional
forms are specified and a closed-form solution is derived. Section 4 extends the model to include
quantity-quality tradeoff and market childcare. In Section 5, we discuss the relationship between
the conventional fertility theory and our theory. Section 6 concludes this paper.

2

Model

2.1

Structure

Individuals The economy is populated by a continuum of measure N of ex ante homogeneous
individuals. All individuals are assumed to have the same strictly increasing, strictly quasiconcave, twice continuously differentiable utility function u (n, c) over the number of children,n, and consumption, c. We omit leisure from the model for simplification, but the introductionof leisure does not undermine our advantage in the robustness of the negative wage-fertility
relationship over existing theories (see Footnote 10 for details) The indifference curve in the(n, c) plane corresponding to utility level U can be written asc = ΨU (n) ,6(1)

which satisfies∂ ΨU (n)∂ 2 ΨU (n)∂ ΨU (n)< 0,> 0.
> 0, and2∂n∂n∂U
Throughout this paper, functions and variables with the subscript “U” represent those associatedwith the utility level U. We further add an Inada-type condition to guarantee that all the time
available is not devoted to work:
lim

n→0

∂ ΨU (n)= −∞.
∂nThis assumption is only for simplification, not essential to the model. These are all the restrictions imposed on preferences to derive the negative cross-sectional wage-fertility relationship.
We impose no further restriction on the curvature of utility function, as well as the elasticity of
substitution between fertility and consumption.
The total time endowment of an individual is normalized to 1. Producing a child takes τ > 0
units of time. We let l ∈ [0, 1] and w ≥ 0 denote the time spent working and the labor income,respectively. The time constraint and the budget constraint are respectively given byτn + l = 1(2)(1 + tc ) c = (1 − tw ) w + θ + sn n,

(3)

and

where tc ≥ 0, tw ∈ [0, 1), sn ≥ 0, and θ ≥ 0 are the consumption tax rate, the labor-income
tax rate, the child-rearing subsidy, and the non-labor income, respectively. To demonstrate therobustness of our theory, we include various variables, some of which are often omitted in theliterature. It should be especially noted that a negative child-rearing subsidy is the same as the
goods costs of childcare for parents. As will become apparent, our main result (Proposition 1)
does not depend on the value of sn , whether positive or negative, implying that our theory isrobust to the inclusion of goods costs of childcare.8
8 A negative non-labor income is the same as a lump-sum tax for parents.

7

The value of θ , whether positive

Jobs There is a large pool of prospective entrants into the market. Prior to entry, firms are identical. To open a vacant job, firms must incur an initial investment cost κ > 0, which is thereafter
sunk. Vacant jobs then draw their productivity parameter α from a common distribution F (·).
It is assumed that the support of F (·) is R+ and that F (·) is strictly increasing and twice continuously differentiable. One job and one individual form a production unit. When an individual
works in a job with productivity α , the match produces α units of good per unit time. If a job
with productivity α hires a worker on the condition that working hours is l and compensation isw, the (after-tax) profit isΠ (α ) = (1 − tΠ ) (α l − w) ,(4)

where tΠ ∈ [0, 1) is the corporate-profit tax rate. Note that after the entry, the initial investment
cost was sunk.9

Labor market

Individuals and vacancies come together in the competitive labor market: eachvacancy offers a contract specifying working hours, l, and compensation, w, and each individual
freely applies to a vacancy to work. There are no frictions in the matching process.

Timing of events The timing of events is as follows.
1. Each firm decides whether or not to open a vacancy with a fixed cost κ .
2. All vacancies draw their productivity α from a common distribution F (·).
3. Each vacancy offers a contract specifying working hours, l, and compensation, w, andeach individual observes all posted contracts and decides on which of these to apply freely.

or negative, does not affect our main result (Proposition 1), implying that our theory is robust with respect to theinclusion of lump-sum tax. Furthermore, we assume the linear child-rearing cost, τ n, but we do not need thisassumption to derive the main result, which does not depend on whether child production is decreasing, increasing,or constant returns to scale.
9 In reality, investment cost is at least partially deductible in many countries. Even if we assume that investment
spending is tax deductible, our results do not change qualitatively except for the comparative-static analysis on thecorporate-profit tax rate.

84. All matches implement their contracts. Individuals work a promised schedule and production takes place. Firms pay promised earnings to employees. Further, each individual
raises children during out-of-work hours.

2.2

Competitive equilibrium

In this subsection, we derive a competitive equilibrium of this economy. An allocation is a tuple∞∞{α , {w (α )}∞α =α , {l (α )}α =α , U, {Π (α )}α =α , p, M}, where α is the threshold productivity

below which jobs do not operate, w (α ) is compensation in jobs with productivity α , l (α ) is
working hours in jobs with productivity α , U is the individuals’ utility level, Π (α ) is the profit
of jobs with productivity α , p is the proportion of individuals employed, and M is the measureof entrants. We now define a competitive equilibrium.
∞∗∗
Definition 1. A competitive equilibrium is {α ∗ , {w∗ (α )}∞α =α ∗ , {l (α )}α =α ∗ , U ,∗∗{Π∗ (α )}∞α =α ∗ , p , M }, which satisfies the following conditions.

1. All individuals optimally choose their jobs (or not to be employed) and have the same
utility level.
2. Each job chooses working hours and compensation (or not to post any labor contract) tomaximize the profit.
3. The labor market clears: the measure of employed individuals is equal to the measure ofactive jobs.
4. Firms freely open vacancies: firms continue to open vacancies until the expected profit isdriven to the initial investment cost.
We solve the model in two steps. In the first step, given the utility level of individuals, U,∞∞
we derive the equilibrium allocation {α , {w (α )}∞α =α , {l (α )}α =α , {Π (α )}α =α } as a functionof U. In the second step, given the result of the first step, we derive the equilibrium allocation

9

{U, p, M} by using the labor-market clearing and free-entry conditions. Then, combining them,
we arrive at the competitive equilibrium allocation.

First step
Let us start with the first step. Since each firm is small relative to the market size and has nomarket power, it takes the utility level of individuals as given and must post a contract guaranteeing the utility level to attract them. Properties of individuals’ induced indifference curves inthe (l, w) plane are central in considering labor contracts posted by firms. In Lemma 1 below,
we demonstrate that an indifference curve mapping in the (l, w) plane is as shown in Figure 1.
Lemma 1. (a) The induced indifference curve in the (l, w) plane corresponding to utility levelU, w = ΩU (l), satisfies∂ ΩU (l)∂ 2 ΩU (l)∂ ΩU (l)∂ ΩU (l)> 0,> 0, and lim
= ∞.
> 0,
2
l→1
∂l
∂l
∂U
∂l
(b) The bottom curve passes through the origin.
Proof. (a) Substituting (1) and (2) into (3), we obtain
w=

[
)
]
(
1
1−l
1−l
(1 + tc ) ΨU
− θ − sn
≡ ΩU (l) ,1 − twττ(5)which is the induced indifference curve in the (l, w) plane corresponding to utility level U. Using
the property of ΨU (n), we establish that[]
∂ ΩU (l)1∂ ΨU (n)=+ sn > 0,− (1 + tc )∂l(1 − tw ) τ∂n∂ 2 ΩU (l)1 + tc ∂ 2 ΨU (n)
=
> 0,
∂ l2
(1 − tw ) τ 2 ∂ n2
1 + tc ∂ ΨU (n)∂ ΩU (l)=> 0,∂U
1 − tw ∂ U
10

(6)

w
ΩU0

Higher U

01l

Figure 1: Induced indifference curves in (l, w) plane.
and[]∂ ΩU (l)1∂ ΨU (n)− (1 + tc )= lim
+ sn = ∞.
n→0 (1 − tw ) τ
l→1
∂l
∂n
lim

(b) Since individuals can choose not to be employed, their utility level is bounded below by U0 ,which is the utility level associated with l = 0 and w = 0. Thus, U0 is given byΩU0 (0) = 0,

(7)

and w = ΩU0 (l) passes through the origin.

The induced indifference curve in the (l, w) plane is upward sloping and convex to the l axis.
The former property is derived from the fact that u is increasing in both arguments: to be indifferent, lower fertility (i.e., longer working hours) must be compensated by higher consumption(i.e., higher pay). The latter property is derived from quasi concavity of u: since the marginal
rate of substitution of c for n increases as (n, c) moves toward the northwest along an indifference curve, c = ΨU (n), the increment of consumption (i.e., compensation) required to induce11individuals to reduce fertility (i.e., work extra hours) increases as working hours and compensation increase. The induced indifference curve associated with higher U is located above theone associated with lower U. Obviously, shorter working hours and higher compensation areassociated with higher utility.
Given U, vacancies post labor contracts maximizing their profits within the constraint ofw ≥ ΩU (l). A vacancy with productivity α solves the following maximization problem:max
l∈[0,1],w≥0

Π (α ) = (1 − tΠ ) (α l − w) ,s.t. w ≥ ΩU (l) .

Solving this problem leads us to the following proposition.
Proposition 1. A job with higher productivity offers a labor contract with longer working hours
and higher compensation. An employee in a higher-productivity job has fewer children.
Proof. Since the constraint w ≥ ΩU (l) holds with equality, we can rewrite the job’s problem asmax ΠU (α ) = (1 − tΠ ) [α l − ΩU (l)] .

l∈[0,1]The solution to this problem can either be interior or at a corner; there is a threshold level ofproductivity, α̃U , below which jobs maximize their profits by setting l = 0. The threshold level
is

α̃U =
The solution is∂ ΩU (0)
> 0.
∂l




α − ∂ Ω∂Ul(l) = 0 if α > α̃U ,



(8)





 l=0

if α ≤ α̃U .

The case of l = 1 is excluded because liml→1 [∂ ΩU (l) /∂ l] = ∞. Totally differentiating the first-

12

w
ΩU

wU (α2 )wU (α1 )0α1α2lU (α1 ) lU (α2 )Π̃U (α1 )1lΠ̃U (α2 )ΠFigure 2: Labor contracts offered by jobs with α1 and α2 (> α1 ).
order condition of the interior solution, we obtain[lU′ (α ) =∂ 2 ΩU (lU (α ))∂ lU (α )2

]−1
> 0.

(9)

The sign comes from Lemma 1. It follows from (2) and (5) that this result implies nU′ (α ) < 0
and wU′ (α ) > 0.

This proposition is the main result of this paper. If jobs with different productivity levels coexist, then we observe a negative cross-sectional relationship between labor income and fertility.
The intuition is simple. For individuals to be indifferent, longer working hours must be compensated by higher rewards. Firms are faced with this tradeoff in posting labor contracts, and firms
13with higher-productivity jobs want their employees to work longer hours even if they must pay
higher rewards because those firms can produce a larger quantity of output per unit time. Figure 2graphically illustrates this argument. Individuals’ induced indifference curves in the (l, w) plane
are as shown in Figure 1. Equation (4) implies that jobs’ iso-profit curves are upward sloping
and the slope, α , is steeper for higher-productivity jobs. If a higher-productivity job decreases
labor inputs, l, then the output, α l, decreases by the larger amount. Thus, to keep the profit ofsuch a job constant, compensation, w, must decrease by the larger amount. A posted contract
is a point of tangency between these two curves, which is unique. It follows that a higherproductivity job chooses longer working hours (implying fewer children) and higher compensation (wU (α1 ) < wU (α2 ) and lU (α1 ) < lU (α2 ), where α1 < α2 ).10 Figure 2 also depicts the pretax profits associated with each labor contract, Π̃U (α ) ≡ ΠU (α ) / (1 − tΠ ) = α lU (α ) − wU (α ).
Although the above analysis focuses on the relationship between total pay and fertility, it
is customary in the literature to analyze the relationship between per-time pay and fertility. We
check whether our argument about the relationship between total pay and fertility also holds forthe relationship between per-time pay and fertility. Differentiating the per-time pay with respect
to α and using (4), (8), and (9), we obtain[][]∂ wU (α )∂ ΩU (lU (α ))l ′ (α ) ΠU∗ (α )
=
= U
> 0,
∂ α lU (α )∂αlU (α )
(1 − tΠ ) lU (α )2where ΠU∗ (α ) is the maximized profit of a job with productivity α for given U. The maximized
profit is positive because jobs whose profit is negative do not operate. It follows that the statement of Proposition 1 regarding the relationship between total pay and fertility also holds for the
relationship between per-time pay and fertility: a job with higher productivity offers a labor contract with longer working hours and higher per-time pay. This comes from strict quasi concavity
10 The negative wage-fertility relationship does not crucially depend on the assumption that child-rearing time is
the mirror image of labor supply. The negative relationship derived from our model is more robust to the introductionof the third time use (e.g., leisure) than that derived from individuals’ heterogeneity approach. Even if we introduce
leisure into the model, the result that longer working hours must be compensated by higer pay does not change. Bothchild-rearing time and leisure decrease as working hours increases unless the complementarity between consumption
and leisure is sufficiently large compared to the complementarity between consumption and children.

14

of u. The increment of earnings required to induce individuals to work an extra hour increases
as working hours and pay increase. Since the marginal compensation required increases withworking hours, so does the average compensation.
The productivity level α̃U separates jobs whose profit is maximized at l > 0 from those
whose profit is maximized at l = 0. However, all jobs with α > α̃U do not necessarily operatebecause the maximized profit might be negative even for such jobs. Since firms can freely choose
whether to operate or not, only jobs with positive maximized profit offer labor contracts. Taking
this into consideration, we can derive the condition for jobs to operate.
Lemma 2. There exists a threshold productivity level α U , below which jobs do not operate:(a) α U ≥ α̃U for all U ≥ U0 , with strict inequality if U > U0 , and(b) α U increases with U.
Proof. Given the utility level of individuals, U, the maximized profit of a job with productivity

α isΠU∗ (α ) = (1 − tΠ ) [α lU (α ) − ΩU (lU (α ))] ,(10)

where lU (α ) is given by (8). The productivity level that makes jobs indifferent between operating or not, α U , is given by ΠU∗ (α U ) = 0. It follows from this and (8) thatαU =ΩU (lU (α U )) ∂ ΩU (lU (α U )).
=lU (α U )∂l

(11)

For the threshold job, the per-time compensation, Ω/l, and the marginal rate of substitution
between l and w, ∂ Ω/∂ l, are equalized to the productivity level. Since Ω is strictly convex tothe l axis,∂ ΩU (l) ∂ ΩU (0)>for all l > 0.
∂l∂l
If U > U0 , thenαU =ΩU (lU (α U )) ∂ ΩU (lU (α U )) ∂ ΩU (0)=>= α̃U .
lU (α U )∂l∂l
15

If U = U0 , then w = ΩU (l) passes through the origin, and thus

α U0 =limlU (α U )→0ΩU0 (lU (α U )) ∂ ΩU0 (0)== α̃U0 .
lU (α U )∂l

By the envelope theorem, we obtain∂ ΠU∗ (α )= (1 − tΠ ) lU∗ (α ) > 0,∂αand

∂ 2 ΠU∗ (α )= (1 − tΠ ) lU∗′ (α ) > 0.
∂ α2Therefore, the threshold productivity level α U is unique and exists in [α̃U , ∞). Differentiating
ΠU∗ (α ) with respect to U, we obtain∂ ΠU∗ (α )∂ ΩU (lU (α ))= − (1 − tΠ )
< 0.
∂U
∂U
The maximized profit of a job with productivity α , ΠU∗ (α ), decreases with U for all α . Therefore, the threshold productivity level increases with U.
Since the maximized profit increases with α , there exists a threshold productivity, below
which jobs do not operate. We can graphically confirm this by investigating Figure 2. The(pre-tax) profit of jobs whose iso-profit curve tangent to the relevant induced indifference curve
passes the origin is zero, and those jobs are the exit thresholds, that is, Π̃U (α U ) = 0. Thislemma also states that as the utility level of individuals increases, the ratio of active jobs to all
the entrants decreases. Higher utility means that higher compensation is required; thus, the jobs’
profits are reduced, driving low-productivity jobs out of the market.

16

Second step
It follows from (5) and (7) that the value of U0 is exclusively determined by parameters, notaffected by other endogenous variables. Since all individuals must have the same utility level inequilibrium, the utility level is fixed to the outside-option value, U0 , as long as unemployment
exists, that is, p < 1. Taking this into consideration, we can write the market-clearing conditionof the labor market as
pN =




[1 − F(α U0 )]M if p < 1,







 [1 − F(α )]MU(12)
if p = 1.

The LHS is the measure of employed individuals and the RHS is the measure of active jobs.
Using this market-clearing condition leads us to the following lemma.
Lemma 3. The utility level of individuals is a nondecreasing function of the measure of entrants:

U ′ (M)
= 0 if M ≤ M̃,







 > 0 if M > M̃.

where M̃ ≡N
1 − F(α U0 )Proof. In the case of p < 1, the values of p and M are not uniquely determined. Since α U0 is
independent of p and M, all the pairs (p, M) satisfying pN = [1 − F(α U0 )]M are possible. In thisrange, an increase in the measure of entrants, M, only results in a proportionate increase in theratio of employed individuals, p, but does not affect the individuals’ utility level and the firms’
profits at all. Once p reaches 1, on the other hand, since the ratio of employed individuals cannot
increase further, in response to an increase in M, α U must increase to keep the measure of active
firms constant and balance supply and demand. Therefore, U (M) is constant over the interval[]()
0, N/[1 − F(α U0 )] and increasing over N/[1 − F(α U0 )], ∞ .

17

As the measure of entrants increases, the measure of high-productivity jobs (as well as thatof low-productivity jobs) increases because the productivity distribution is exogenously given.
Owing to the competition among those jobs, the utility level of an individual increases. The
utility level continues to increase until the exit threshold becomes high enough for the supply
and demand to balance.
We close the model by considering the firms’ free entry. Since firms continue to open vacancies until the expected profit is driven to the initial investment cost, the following condition
holds:∫ ∞αU

ΠU∗ (α ) dF (α ) ≤ κ ,with equality if there are positive entrants. Combining this free-entry condition with Lemmas 2
and 3, we establish the existence of the competitive equilibrium.
Proposition 2. There exists the competitive equilibrium, where

M∗




= 0 if




∫∞





 > 0 if

¯
∫∞ ∗
¯
Π
(
α
)
dF
(
α
)
¯
αU U

¯

¯
∗
α U ΠU (α ) dF (α )¯

M=0

≤ κ,
(13)

M=0

> κ.

Proof. From Lemma 3, U (M) is constant over M ∈ [0, M̃] and increasing over M ∈ (M̃, ∞).
From Lemma 2, aU increases with U and ΠU∗ (α ) decreases with U for all α . Therefore,∫∞∗α U ΠU (α ) dF (α ) is constant over the interval M ∈ [0, M̃] and decreasing over M ∈ (M̃, ∞). If

∫∞

∫∞ ∗
∗
∗
α U ΠU (α ) dF (α ) ≤ κ when M = 0, then there is no entry, and thus M = 0. If α U ΠU (α ) dF (α ) >κ when M = 0, then firms continue to open vacancies until∫∞∗
α U ΠU (α ) dF (α ) = κ , and thus

M ∗ > 0.
Figure 3 graphically illustrates this proposition (E (·) represents the expectation over α ).11
∞
∞
∗
∗
∗
∗
∗
The competitive equilibrium is {α ∗ , {w∗ (α )}∞α =α ∗ , {l (α )}α =α ∗ , U , {Π (α )}α =α ∗ , p , M },11 Strictly speaking, if the expected profit evaluated at M = 0 is equal to κ , then multiple equilibria arise. However,
we do not emphasize the multiplicity because it arises only when the parameters happen to satisfy such a condition,
which is practically impossible. Furthermore, it is not robust to changes in assumptions. For instance, if the initial

18

κ

κ
E(Π)
E(Π)

0 = M∗

M

M̃

0

M̃

M∗

M

(b) M ∗ > 0(a) M ∗ = 0

Figure 3: Equilibrium measure of entrants.
which satisfies (4), (5), (8), (11), (12), and (13). The equilibrium utility level, U ∗ , depends onfunctional forms and parameters, but the negative wage-fertility relationship holds for any levelof U (Proposition 1), and thus we always observe the negative cross-sectional relationship inequilibrium, independent of functional forms and parameters.

2.3

Comparative statics

In this subsection, we conduct comparative-static analyses on the effects of changes in policy variables (consumption tax rate, tc , labor-income tax rate, tw , child-rearing subsidy, sn , andcorporate-profit tax rate, tΠ ) and the effect of technological change. For the analysis, this subsection adds the following assumption:∂ 2 ΨU (n)
< 0.
∂U∂ n

(14)

For any given n, the indifference curve associated with higher U has the steeper (negative) slope
than that associated with lower U. This assumption is not too restrictive: if preferences areinvestment cost is a strict convex function of the measure of entrants or if the output produced in this economy faces
a downward-sloping demand curve, then the multiplicity vanishes.

19homothetic, (14) is satisfied. The assumption implies that[]1
∂ 2 ΩU (l)∂ 2 ΨU (n)=− (1 + tc )
> 0.
∂U∂ l
(1 − tw ) τ
∂U∂ n

(15)

For any given l, the induced indifference curve associated with higher U has the steeper (positive) slope than that associated with lower U.
Changes in policy variables affect firms’ and individuals’ behavior in equilibrium throughtwo channels. One is the substitution effect: changes in policy variables vary the form of theinduced indifference curves in the (l, w) plane (i.e., the marginal rate of substitution between l
and w). The other is the wealth effect: policy changes affect the firms’ profits and their entry
decision, resulting in changes in the utility level of individuals and thus changes in their behavior.
Lemma 4. The wealth effect on fertility is positive: if U < U ′ , then nU (α ) < nU ′ (α ) for all α .
Proof. Totally differentiating (8) and using Lemma 1 and (15), we obtain∂ 2 Ω (l)UdlU (α )l< 0.
= − ∂ ∂2 ΩU ∂(l)
U
dU
2

∂l

This immediately leads us to the result.
Equation (15) alone represents the wealth effect: the induced indifference curve associatedwith higher U has a steeper (positive) slope than that associated with lower U. Thus, jobs withproductivity α decrease working hours (i.e., increase fertility) and increase compensation in
response to an increase in U. The intuition is simple. For guaranteeing a higher level of utility, it
is more profitable for jobs to increase a bit of both fertility and compensation rather than increase
only one of these by a large margin. Thus, the wealth effect on fertility is positive. The effects
of changes in policy variables on fertility choice are summarized below.
Proposition 3. The effects of policy changes on fertility of employees in firms with productivity

α are as follows:
20

(a) If tc increases, fertility increases through the substitution effect but decreases
through the wealth effect. The total effect is ambiguous.
(b) If tw increases, fertility increases through the substitution effect but decreases through thewealth effect. The total effect is ambiguous.
(c) If sn increases, fertility increases through both the substitution and wealth effects.
(d) If tΠ increases, fertility decreases through the wealth effect.
Proof. (a) Totally differentiating (8), we obtainτ ∂ Ψ∂Un(n)∂ lU (α )
=
< 0.
2
∂ tc
(1 + tc ) ∂ ΨU2(n)

(16)

∂n

This implies ∂ nU (α ) /∂ tc > 0, and thus fertility increases through the substitution effect. Differentiating (10) with respect to tc and using the envelope theorem, we obtain∂ ΠU∗ (α )
1
=−
ΨU
∂ tc
1 − tw

(

1−l
τ

)
< 0.

(17)

Since an increase in tc decreases the profit for all α , the expected profit of entry falls and themeasure of entrants decreases in response to an increase in tc . The utility level must fall torestore the balance of supply and demand in the labor market. It follows from Lemma 4 that
fertility decreases through the wealth effect. (b), (c), and (d): In the same way, we establish that

∂ lU (α ) /∂ tw < 0, ∂ ΠU∗ (α ) /∂ tw < 0, ∂ lU (α ) /∂ sn < 0, ∂ ΠU∗ (α ) /∂ sn > 0, ∂ lU (α ) /∂ tΠ = 0,and ∂ ΠU∗ (α ) /∂ tΠ < 0.
We can understand the substitution effect by investigating how the first-order condition, (8),varies. The first-order condition represents the allocation when U is exogenously given. Thus,(16) indicates that if the utility level remains unchanged, an increase in consumption tax rate
reduces working hours (i.e., increases child-rearing time) in jobs with productivity α . The slopeof the induced indifference curve is the increment of wages required to induce individuals to21

work an extra hour. An increase in consumption tax rate makes the required increment larger, andas a result, firms become reluctant to offer labor contracts with longer working hours (i.e., shorterchild-rearing time). This acts to increase the number of children. That is not the only effect,
however, because policy changes also affect the utility level of individuals. We can understandthe wealth effect by investigating how the firms’ profit, (10), varies. If the expected profit ofentry decreases as a result of policy changes, then the measure of entrants decreases. Since theexcess supply of labor arises, the utility level declines, the threshold level of productivity falls,
and then, the balance of supply and demand is restored. Thus, (17) implies that an increase in
consumption tax rate lowers the profit, curbing the jobs’ entry and decreasing the utility level.
Since the wealth effect on fertility is positive, a decrease in U lowers fertility. It is ambiguouswhether employees in jobs with productivity α increase or decrease the number of children in
response to an increase in tc because the substitution and wealth effects have opposing effects
on fertility. It follows from the analogous arguments that we can understand the effects ofchanges in other policy variables: the effects of increases in tw , sn , and tΠ on fertility choice areambiguous, positive, and negative, respectively.
The focus of the above analysis is on the fertility choice of each employee. The effects ofpolicy changes on the aggregate fertility are more complex. The average fertility rate of thiseconomy is1n̄ ≡1 − F (α ∗ )
∗

∫ ∞
α∗

n∗ (α ) dF (α ) .

To examine the effects of policy changes on n̄∗ , we must consider not only the behavior ofeach agent but also the composition effect because the threshold level α ∗ is endogenous andpolicy changes might affect the productivity distribution among active jobs by affecting α ∗ . For

22

instance, the effect of an increase in consumption tax rate on n̄∗ is

∂ n̄∗
=
∂ tc

[∫ ∞

]

∗
α ∗ n (α ) dF (α )− n∗ (α ∗ )1 − F (α ∗ )+f (α ∗ ) [1 − F (α ∗ )]

∫ ∞

∂ α∗
∂ tc

∂ n∗ (α )f (α ) d α ,
∂ tc
α∗

(18)

where f (α ) is the density. The first and second terms represent the composition effect and theeffect discussed in Proposition 3, respectively.12 Since individuals in higher-productivity jobs∫

have fewer children, α∞∗ n∗ (α ) dF (α ) /[1−F (α ∗ )] < n∗ (α ∗ ). Thus, the sign of the composition
effect depends on the sign of ∂ α ∗ /∂ tc : through the composition effect, increase (resp. decrease)in the threshold productivity decreases (resp. increases) the average fertility rate.
Let us elaborate the composition effect. In the competitive equilibrium where there arepositive entrants, the following two conditions must hold: (i) the threshold job’s profit is 0 (i.e.,Π∗ (α ∗ ) = 0) and (ii) the expected profit at the time of entry is 0 (i.e., E (Π∗ (α )) − κ = 0).
Investigating these two conditions leads us to the following proposition.
Proposition 4. The effects of policy changes on the exit threshold α are as follows:(a) If the utility elasticity of consumption increases with fertility (i.e.,∂ [(∂ ΨU /∂ U) (U/ΨU )] /∂ n > 0), then an increase in tc decreases the threshold, and viceversa.
(b) It is ambiguous whether an increase in tw increases or decreases the threshold.
(c) An increase in sn decreases the threshold.
(d) An increase in tΠ decreases the threshold.
Proof. See Appendix.

12 Note that ∂ n∗ (α ) /∂ t differs from ∂ n∗ (α ) /∂ t : ∂ n∗ (α ) /∂ t is just the substitution effect (i.e., the effect when
c
c
c
U
U
keeping U constant), while ∂ n∗ (α ) /∂ tc includes the wealth effect as well as the substitution effect.

23

As shown in Proposition 3, when tc increases, the equilibrium is restored by a reduction in
U. Since an increase in tc directly curbs the entry whereas a reduction in U promotes the entry,
whether the measure of entrants in the new equilibrium is larger or smaller than the measureof entrants in the initial equilibrium depends on which effect is dominant. Proposition 4 statesthat the effect of an increase in tc depends on the form of preferences, the effect of increase in
tw depends not only on the form of preferences but also on other parameters, and the effects ofincreases in sn and tΠ on the threshold are negative.
Combining the results of Propositions 3 and 4 enables us to understand the effects of policychanges on the average fertility rate: Propositions 3 and 4 respectively indicate the signs of thesecond and first terms in (18). It is only for the child-rearing subsidy, sn , that whether its change
raises the average fertility rate or not is unambiguously determined: increase in sn necessarilyraises the average fertility rate.
Next, we examine the effect of technological progress. Although various formulations might
be possible as a representation of technological progress, here, we capture the technologicalprogress as a change in the productivity distribution from F (·) to G (·) such that G (·) first-orderstochastically dominates F (·), i.e., F (α ) ≥ G (α ) for all α .
Proposition 5. A change in the productivity distribution from F (·) to G (·), where G (·) firstorder stochastically dominates F (·), raises the average fertility through the wealth effect but
lowers it through the composition effect. The total effect is ambiguous.
Proof. Since Π∗ (α ) is nondecreasing function, it follows from the first-order stochastic dominance that the following inequality holds:∫ ∞α∗Π∗ (α ) dG (α ) ≥∫ ∞α∗Π∗ (α ) dF (α ) = κ .

Thus, such a change in the productivity distribution promotes entry of jobs, increases the exit
threshold, and raises the utility level. By the wealth effect, employees in jobs with productivity

α increase children for all α . By the composition effect, on the other hand, the proportion24

of employees in higher-productivity jobs, who have fewer children, increases. It is ambiguouswhether average fertility rate increases or decreases because the wealth and composition effects
act in opposite directions.
Cross-sectionally, fertility falls as wages rise. However, an increase in the overall wage levelcaused by technological progress does not necessarily result in a fall in the overall fertility rate,
because of the wealth effect: an increase in the utility level triggered by technological progressinduces jobs with productivity α to offer shorter working hours (i.e., higher fertility) for all

α . If the wealth effect dominates the composition effect, technological progress leads to anincrease in the average fertility rate, and vice versa. This property enables our model to reconcile
the negative cross-sectional wage-fertility relationship with various time-series variation in theaggregate fertility over the course of economic growth and business cycle.
The effect of the technological progress considered here is similar to the effect of a fallin the corporate-profit tax rate. Both have no substitution effect and have the positive wealthand negative composition effects on the average fertility rate. However, note that the effects oftechnological progress and a fall in the corporate-profit tax rate on the average fertility rate are
not identical. A change in the corporate-profit tax rate affects the average fertility rate only byaffecting the measure of entrants. On the other hand, a change in the productivity distribution
varies the average fertility rate even if it does not entail a change in the measure of entrants.
Suppose that a change in the expected profit is offset by a proportionate change in the initialinvestment cost and the measure of entrants does not change. In this case, a change in thecorporate-profit tax rate has no effect on the equilibrium allocation of this economy. On theother hand, a change in the productivity distribution varies the average fertility rate. If theequilibrium measures of entrants under F (·) and G (·) are the same, then the following must
hold for the balance of supply and demand in the labor market:( )F (α ) = G α ′ ,25where α and α ′ are the equilibrium exit thresholds under F (·) and G (·), respectively. It follows
from the first-order stochastic dominance that α < α ′ . The higher threshold level is associatedwith the higher utility level. Even if technological progress is accompanied by an increase in
the entry cost and the resultant measure of entrants does not change, the positive wealth andnegative composition effects coexist, and it is ambiguous whether the resultant average fertility
rate increases or not.

3

Example: Constant wage elasticity of fertility

Thus far, we have tried to keep the model as general as possible to illustrate the robustness of
our theory. Since one broad aim of this paper is to offer an “off-the-shelf” fertility model as abuilding block in applied research, this section shows that specifying functional forms enables
our model to derive a closed-form solution, illustrating the tractability of our theory. This example predicts a constant wage elasticity of fertility, which is consistent with a finding reported
in Jones and Tertilt (2008). Based on the parameterized model, furthermore, we examine theeffects of technological change, illustrating that our theory can reconcile the negative crosssectional wage-fertility relationship with the various time-series variation in aggregate fertility
over the course of economic growth and business cycle.
To parameterize our model, we need to specify the utility function and the productivity distribution. We assume a Cobb-Douglas utility function, which is commonly used in the literature:u (n, c) = nγ c1−γ ,where γ ∈ (0, 1) represents the relative weight for children. Assume that the productivity distri-26

bution, F (α ), is a uniform distribution over [0, A]:

F (α ) =



α



 A

if α ∈ [0, A] ,





 1

if α > A.

The parameter A > 0 represents the maximum technology level available in this economy. Here,technological progress is captured by an increase in A (the distribution with higher A first-orderstochastically dominates the one with lower A). We assume that increase in A might be also
associated with increase in costs: the jobs’ initial investment costs and the individuals’ per-childtime costs are respectively given by κ (A) and τ (A). This formulation allows us to include
the case that the required investment and education costs increase with the rise of technology.
Especially, the initial investment cost has to depend on A for balanced growth.
For simplification, we ignore tax, subsidy, and non-labor income here. Then, the time constraint and the budget constraint are, respectively, τ (A) n + l = 1 and c = w. Before proceeding,we would like to emphasize that based on the conventional fertility model, where individuals facing given per-time wages allocate their time between labor supply and child rearing, wecannot derive the negative relationship between wages and fertility under the assumption that
preferences are given by a Cobb-Douglas utility function and child rearing requires only thetime costs: since the wealth and substitution effects cancel out each other, fertility does not varywith wages. As will become apparent, in contrast, this model produces the negative relationship.
A firm with productivity α solves the following problem:max
l∈[0,1],w≥0

αl − ws.t. nγ c1−γ = U, τ (A) n + l = 1, and c = w

27

The first-order conditions imply[lU (α ) = 1 −(1 − γ ) α
γ

]γ −1

[τ (A)]γ U,

[

and

]
(1 − γ ) ατ (A) γ −1nU (α ) =
U,
γ

(19)

]
(1 − γ ) ατ (A) γwU (α ) =
U.
γ

(20)

[

These establish that higher-productivity vacancies offer labor contracts with longer working
hours (i.e., lower fertility) and higher wages. It should also be noted that an increase in the utility
level, U, increases both nU (α ) and wU (α ). As shown above, the wealth effects on fertility andwages are positive (the utility function assumed here satisfies (14)). Using (19) and (20), we
obtain1− 1−γ

nU = U γ wU γ ,which indicates the negative wage-fertility relationship: the wage elasticity of fertility is − (1 − γ ) /γ .
The maximized profit of a job with productivity α isΠU∗ (α ) = α −[ατ (A)]γ U

γ γ (1 − γ )1−γ

.

(21)

It follows from ΠU∗ (α U ) = 0 that the threshold productivity is{αU =[τ (A)]γ U

γ γ (1 − γ )1−γ

} 1−1 γ.

(22)

Suppose that U > U0 . Then, the market-clearing condition of the labor market, N = [1 −F(α U )]M, is[] 1−1 γ γ[τ (A)] U1
N = 1−M.

A γ γ (1 − γ )1−γ

28

Using this, we can write the individuals’ utility level as a function of the measure of entrants:U=()
M − N 1−γA.
Mγ γ (1 − γ )1−γ
[τ (A)]γ(23)

It follows from (21) that the expected profit of entry is∫ A
αUΠU∗ (α ) dF (α ) =

∫ A
αU

{

α−

[ατ (A)]γ U

γ γ (1 − γ )1−γ}
dF (α )

Using (22) and (23), we can write the expected profit of entry as a function of the measure ofentrants:∫ A
α U(M)[∗
ΠU(M)(α ) dF (α ) = A

)1−γ
)2 ]
(
(
1 + γ − 2 M−N+ (1 − γ ) M−N
M
M
2 (1 + γ )Equalizing this to the initial investment cost, κ (A), we obtain the free-entry condition:
(
(
)1−γ
)2
1 + γ − 2 M−N+ (1 − γ ) M−N
MMA= κ (A) .
2 (1 + γ )(24)Whether an increase in A promotes entry or not depends on the form of κ (·). If κ (A) /A decreases (resp. increases) with A, then the measure of entrants increases (resp. decreases) withA.
Now all the equilibrium conditions are set. Using them, we can compute the average fertility
rate:
1
n̄ =
1 − F (aU )∫ A
αUn (α ) dF (α ) =1
1 − M−N
M( M−N )1−γ
M

− M−N
M
.
τ (A)

A change in A affects the average fertility rate through two channels: one directly affects childrearing costs, τ (A), and the other affects the measure of entrants (i.e., through equation (24)).

We conclude this section by presenting a numerical example. We consider three cases: (i)κ (A) = κ and τ (A) = τ , (ii) κ (A) = Aκ and τ (A) = τ , and (iii) κ (A) = Aκ and τ (A) = Aτ . The
29

Figure 4: Wage-fertility relationship.
measure of individuals, N, is normalized to 1. We assume that γ = 0.75 (the relative weight forchildren in the utility function),13 κ = 0.25, τ = 0.4, and A = 1. It follows from the free-entry
condition, (24), that the associated measure of entrants, utility level, and threshold productivity
are, respectively, M ∗ = 1.038, U ∗ = 0.496, and α ∗ = 0.301. The lowest and highest wages arew (α ∗ ) = 0.045 and w (1) = 0.109, respectively. Figure 4 depicts the wage-fertility relationship.
Figure 5 exhibits how the average fertility rate changes as A changes in the cases of (i), (ii), and(iii): in case (i), increase in A entail the dominant wealth effect, raising the aggregate fertility(the upward dotted line); in case (ii), the wealth and composition effects associated with increase
in A cancel out each other, and thus, the aggregate fertility does not change (the horizontal
dotted line); in case (iii), in addition to the effects observed in the case (ii), increase in childrearing costs directly decrease the aggregate fertility (the downward dotted line). The crosssectional wage-fertility relationship is negative at any point in Figure 5. Our model reconciles
the negative cross-sectional wage-fertility relationship with various time-series variation in the
13 This value of γ corresponds to the wage elasticity of fertility of −1/3. Jones and Tertilt (2008) find that for 30
birth cohorts between 1830 and 1960 in the United States, the income elasticity of fertility remained roughly constantat about −0.30.

30Figure 5: Changes in average fertility in response to changes in A. Case (i): upward line. Case(ii): horizontal dotted line. Case (iii): downward dotted line.
average fertility rate.

4

Extension

4.1

Quantity-quality tradeoff

It is rather standard in the literature to include in parents’ preferences not only the number ofchildren, but also some measure of the quality of children. In this subsection, we extend themodel so that parents care about the educational attainment of their children, and we explore therobustness of our model to such an extension. The utility function is modified as u (n, c, h), where
h is the educational attainment of children. As in the basic model, the utility function is strictlyincreasing, strictly quasi-concave, and twice continuously differentiable and satisfies an Inadatype condition. We assume that child rearing takes time while education requires purchased
inputs as in de la Croix and Doepke (2003) and Moav (2005).14
14 Although this assumption is often used in the literature, we do not need it to derive the negative wage-fertility
relationship. However, with this assumption, we can derive the positive relationship between parental wages and31

In this subsection, we introduce the leisure time into the model, despite not generating utility.
Producing a child not only takes time but also increases education costs, which are the essenceof the quantity-quality model that the product of quality and quantity of children appears in thebudget constraint (see Equation (25)). It might be preferable for individuals to discard a partof the time available for reducing education expenditure, then they might wish to work longer
hours without increases in compensation. To avoid such an implausible situation, we introduce
the disposable time as “leisure” and denote it by z. The time constraint, the budget constraint,
and the education function are respectively given byτ n + l + z = 1,
(1 + tc ) c + en = (1 − tw ) w + θ + sn n,

(25)

and
h = m (e) ,where e is the education spending per child and m (·) is the education production function. We
assume that m (·) is strictly increasing, strictly concave, and twice continuously differentiable.
An important departure from the basic model is that once offered labor contract (l, w), individuals further allocate their time between child rearing, τ n, and leisure, z, and allocate theirincome between education for children, en, and own consumption, c, to maximize their utility.
Given this individual maximization behavior, each firm chooses a contract to post. To solve themodel backward, we first consider the individuals’ problem. Given l and w, an individual solves
the following maximization problem:
max u (n, c, h) ,s.t. τ n + l + z = 1, (1 + tc ) c + en = (1 − tw ) w + θ + sn n, and h = m (e) .
educational level, which is widely observed in the reality, under moderate restrictions on preferences as well.

32

The first-order conditions of this problem are given by−u1 − u2sn − e≤ 0,
1 + tc(26)

with equality if z > 0, andu2

1 1−l −z
= u3 m′ (e) ,1 + tcτ(27)

where ui represents the partial derivative of u with respect to the i-th argument. It follows fromstrict quasi concavity of u that the second-order condition is satisfied.
Therefore, the maximization problem faced by a job with productivity α is the following:max
l∈[0,1],w≥0(s.t. uΠ (α ) = (1 − tΠ ) (α l − w) ,)1 − l − z̃ (1 − tw ) w + θ + (sn − ẽ) 1−l−z̃τ,, m (ẽ) = U,τ
1 + tc(28)

where z̃ and ẽ are given by (26) and (27). Solving this problem leads us to the following proposition.
Proposition 6. (a) A job with higher productivity offers a labor contract with longer working
hours and higher compensation. An employee in a higher-productivity job has fewer children.
(b) An employee in a higher-productivity job educates children more if and only if(u22 u3 − u32 u2 ) u1 + (u31 u2 − u3 u21 ) u2 < 0.

(29)

Proof. (a) Showing that induced indifference curves in the (l, w) plane, (28), are upward sloping
and convex to the l axis proves part (a) of this proposition. Totally differentiating (28), we obtaindw 1 1 + tc=
dl
τ 1 − tw(u1 sn − ẽ+u2 1 + tc

33

)
≥ 0.

(30)

This sign comes from (26). Consider the case where (30) holds with strict inequality.15 Differentiating (30) with respect to l and noting that the relationship between l and w on an induced
indifference curve is given by (30), we obtaind2w
1 + tc u22 u11 − 2u1 u2 u12 + u21 u22
=−
> 0.
2
dl
(1 − tw ) τ 2
u32This sign comes from strict quasi concavity of u. Induced indifference curves in the (l, w)
plane are upward sloping and convex to the l axis, as in the basic model (Figure 1). Thus,
part (a) of this proposition follows from the analogous argument of Proposition 1. (b) Totallydifferentiating (27) with respect to w, l, and e, and using (27) and (30), we obtain(u22 u3 − u32 u2 ) u1 + (u31 u2 − u3 u21 ) u2 1 ndl
u2 u3
τ 1 + tc
[
]
2
(
)
n
1
= u22 u23 − 2u23 u3 u2 + u33 u22 2+ u3 m′′ (e) de.
u3 (1 + tc )2
It follows from strict quasi concavity of u that u22 u23 − 2u23 u3 u2 + u33 u22 < 0. Therefore, de/dl <0 if and only if the inequality (29) holds.
Even if the quantity-quality tradeoff is introduced, the model can generate the negative wagefertility relationship using only standard conditions on preferences (quasi concavity) and thequality-production function (concavity). A widespread finding in the literature is that wealthier parents tend to have fewer children, as well as educate them more. To explain this finding based on our model, we need an additional assumption on preferences because the strict
quasi concavity of u alone does not guarantee (29). The assumption of homothetic preferences
qualifies as the additional assumption required. Adding the homotheticity of u establishes that(u22 u3 − u32 u2 ) u1 + (u31 u2 − u3 u21 ) u2 = (u22 u3 − u32 u2 ) u1 < 0. The assumptions required here
to obtain the desired result are still considerably moderate compared to most of the existing lit15 There might be regions where dw/dl = 0 (i.e., z > 0). However, points (l, w) belonging to such regions are notoffered because they are inconsistent with firms’ profit maximization: if firms do not need to raise wages, they want
their employees to work as long as possible. Thus, we can focus on the case where dw/dl > 0.

34

erature.

4.2

Nannies

What happens in our model if parents are allowed to outsource childcare? Although it is fairly
obvious that childcare is a time-intensive activity, childcare can be, at least partly, bought in themarket (e.g., nannies, nurseries, and wet nurses) and outsourcing childcare has been growing
in developed countries. In this subsection, we extend the model so that parents are allowed tooutsource childcare, and we explore the robustness of our model to such an extension. The time
constraint and the budget constraint are respectively given by(1 − q) τ n + l = 1

and(1 + tc ) [c + v (q) qτ n] = (1 − tw ) w + θ + sn n,(31)

where q ∈ [0, 1] is the proportion of the nanny’s time and v (q) is the cost of a nanny per unitof time. It is assumed that v (·) is strictly increasing, twice continuously differentiable, andlimq→1 v (q) = ∞. Furthermore, we assume thatv′′ (q) q + 2v′ (q) > 0,(32)

which is the condition for ∂ 2 [v (q) qτ n] /∂ q2 > 0 to be satisfied. These assumptions on v (·)
capture the idea that there are various activities in childcare, they differ in cost, and individuals
outsource childcare in ascending order of cost.
Given an offered labor contract (l, w), individuals further allocate their income between the
nanny’s cost, v (q) qτ n, and own consumption, c, to maximize their utility. Given this individual
maximization behavior, each firm chooses a contract to post. To solve the model backward,
we first consider the individuals’ problem. Given l and w, an individual solves the following

35

maximization problem:
max u (n, c) ,s.t. (1 − q) τ n + l = 1 and (1 + tc ) [c + v (q) qτ n] = (1 − tw ) w + θ + sn n.
The first-order condition of this problem is given by
[
]
u1
sn
+
− v′ (q) q (1 − q) + v (q) τ ≤ 0,
u2 1 + tc(33)

with equality if q > 0. Note that q = 0 if u1 /u2 + sn / (1 + tc ) − v (0) τ ≤ 0 and that the case ofq = 1 is excluded by the assumption of limq→1 v (q) = ∞.
Therefore, the maximization problem faced by a job with productivity α is the following:max
l∈[0,1],w≥0(s.t. uΠ (α ) = (1 − tΠ ) (α l − w) ,1 − l (1 − tw ) w + θ + sn (1−q̃)τ
1−l,− v (q̃) q̃
(1 − q̃) τ
1 + tc(1 − q̃)
1−l

)
= U,

(34)

where q̃ is given by (33). Solving this problem leads us to the following proposition.
Proposition 7. (a) A job with higher productivity offers a labor contract with longer working
hours and higher compensation. (b) An employee in a higher-productivity job has fewer children.
Proof. (a) Showing that induced indifference curves in the (l, w) plane, (34), are upward sloping
and convex to the l axis proves the part (a) of this proposition. Totally differentiating (34), we
obtain[]dw
1 + tcsnu1=+− v (q̃) q̃τ > 0.
dl
(1 − tw ) (1 − q̃) τ u2 1 + tc

(35)

This sign obviously holds in the case of q̃ = 0. In the case of q̃ > 0, the sign holds because(33) holds with equality. Differentiating (35) with respect to l and noting that the relationship36

between l and w on an induced indifference curve is given by (35), we obtaind2w
u22 u11 − 2u1 u2 u12 + u21 u22
1 + tc
=−
> 0.
2
2
dl
u32
(1 − tw ) (1 − q̃) τ 2
This sign comes from strict quasi concavity of u. Induced indifference curves in the (l, w) plane
are upward sloping and convex to the l axis, as in the basic model (Figure 1). Thus, the part (a)of this proposition follows from the analogous argument of Proposition 1. (b) Since childcare
can be outsourced, longer working hours do not necessarily mean fewer children. The numberof children is given by n = (1 − l) / [(1 − q) τ ]. Thus, noting that the value of q chosen by an
employee depends on l, the relationship between working hours and fertility is given bydq∂ n 1 − (1 − q) + (1 − l) dl=.
∂lτ
(1 − q)2(36)

Totally differentiating (33), we obtain
u2 u11 −2u1 u2 u12 +u1 u22
1(1−q)τ
u32
2

dq
=
dl

2

1−l u2 u11 −2u1 u2 u12 +u1 u22− [v′′ (q) q + 2v′ (q)] (1 − q) τu32(1−q)2 τ
22.

Substituting this into (36), we obtainτ[v′′ (q) q + 2v′ (q)] (1−q)
∂n
(1−l)
= 2
3 2 < 0.
u2 u11 −2u1 u2 u12 +u21 u22∂l− [v′′ (q) q + 2v′ (q)] (1−q) τ
2

(37)

(1−l)

u32

This sign comes from (32) and strict quasi concavity of u.
Even if parents are allowed to outsource childcare, the model can generate the negative
wage-fertility relationship. The intuition is simple. Owing to the assumption that the cost of ananny per unit of time increases with the proportion of the nanny’s time, individuals with longer
working hours (i.e., shorter own childcare time) face a higher price of childcare. Thus, theychoose lower fertility and higher consumption to obtain the same utility level as individuals with37

shorter working hours obtain. Equation (37) indicates that though the negative wage-fertility
relationship is still preserved, the improvement in the availability of market childcare reduces
the fertility differential. Let us interpret the size of v′′ (q) q + 2v′ (q) as the difficulty in using
market childcare. The absolute value of ∂ n/∂ l decreases as v′′ (q) q + 2v′ (q) decreases andconverges to 0 as v′′ (q) q + 2v′ (q) approaches 0. The linear cost (i.e., v (q) is constant) meansthat v′′ (q) q + 2v′ (q) = 0; then, ∂ n/∂ l < 0 for q = 0, while ∂ n/∂ l = 0 for q > 0. This implies
that if all types of childcare service can be purchased at the same price, individuals who use
nannies’ time have the same number of children: the difference in own childcare time is offset
by the use of market childcare.

5 Discussion
5.1 Jobs’ heterogeneity vs individuals’ heterogeneity
This subsection organizes the relationship between the jobs’ heterogeneity approach (ours) andthe individuals’ heterogeneity approach (the conventional one). We introduce individuals’ ability heterogeneity into our framework, illustrating that in contrast with the negative wage-fertility
relationship stemming from the jobs’ heterogeneity, the one stemming from the individuals’ heterogeneity depends on the form of preferences. It establishes that the robustness of our theory
comes not from the model’s structure (wages and fertility are simultaneously determined in acontract-posting market) but from jobs’ heterogeneity. The discussion here proposes an important viewpoint for the empirical literature.

For simplicity, suppose that there is no heterogeneity in jobs’ productivity. There are two
types in individuals’ ability, β1 and β2 (> β1 ). When a job hires an individual with ability βi ,the match produces βi units of good per unit time. Individuals with different ability might havedifferent utility levels, while homogeneous jobs must earn the same profit. As in the case where
there is heterogeneity in jobs’ productivity, we can graphically depict the offered contracts.
38wΩU2 ΩU1wU2 (β2 )
wU1 (β1 )0β2 β1lU2 (β2 ) lU1 (β1 ) 1lΠ̃U1 (β1 )= Π̃U2 (β2 )ΠFigure 6: Labor contracts offered to individuals with β1 and β2 (> β1 ).
Figure 6 exhibits a possible case. Since both the substitution and wealth effects on consumption(i.e., wages) are positive, a higher-ability individual receives higher wages. On the other hand,the substitution effect on fertility is negative while the wealth effect on fertility is positive; thus,
it is ambiguous whether a higher-ability individual has more or less children. Figure 6 depicts thecase where the wealth effect is dominant, and thus, a higher-ability individual has more children(lU1 (β1 ) > lU2 (β2 ) implies nU1 (β1 ) < nU2 (β2 )). This illustrates that the conventional fertility
theory based on individuals’ heterogeneity must impose restrictive assumptions on preferences
to explain the negative relationship between wages and fertility. Furthermore, the wage-fertility
relationship is not robust to changes in economic environments (e.g., policy changes) because
they generally vary the form of the induced indifference curves. Even if the negative relationship39

holds under a parameter configuration, it does not necessarily hold under others.
The above description suggests that the cross-sectional wage-fertility relationship depends
on the source of wage differential. Several studies in the literature argue that in the reality, thewealth effect is typically dominant and that the negative wage-fertility relationship is mainly astatistical fluke due to a problem of missing variables (e.g., Becker (1960), Becker and Lewis(1973), and Becker and Tomes (1976)). The argument here introduces a new insight into thisstrand of the literature: the source of wage differential, whether jobs’ or individuals’ heterogeneity, is a candidate for one of the variables missed in the past empirical studies.16 The literature
has often pointed out the importance of distinguishing sources of family income, for example,
labor and non-labor income and husband and wife income.17 The discussion here suggests thatin addition to these, the source of wage differential also matters. Attempting to produce thenegative wage-fertility relationship based solely on individuals’ ability might be misleading anda careful examination on whether fertility differential stems from jobs’ heterogeneity or individuals’ heterogeneity is needed for calibrating and estimating the relevant parameters.

5.2

Exceptional findings

The negative relationship between wages and fertility is widespread across time and regions,but not universal. Several studies have reported exceptional findings, and one may argue thatthe robustness of our theory is inconsistent with these exceptions. However, it is inappropriate
as a criticism toward the jobs’ heterogeneity approach. The reality might be a hybrid of jobs’
heterogeneity and individuals’ heterogeneity. The jobs’ heterogeneity approach is more likely
to complement rather than contradict the individuals’ heterogeneity approach. Combining both
approaches might open the way for solving seemingly puzzling problems related to fertility
16 There are a large number of empirical studies on fertility choice. For instance, they are surveyed in Hotz,
Klerman and Willis (1997) and Schultz (1997).
17 It is sometimes argued that an increase in family income due to an increase in the return on nonhuman assets islikely to raise fertility through the wealth effect, while that by an increase in wages is likely to reduce fertility throughthe substitution effect (e.g., Schultz (1981, 1994)), and an increase in the husband’s wages is likely to raise fertility
through the wealth effect, while an increase in the wife’s wages is likely to reduce fertility through the substitution
effect (e.g., Schultz (1986) and Blau and van der Klaauw (2007)).

40

choice. In what follows, we discuss two exceptional findings well known in the literature by
combining both approaches.
Positive relationship in the early stage of development

It is sometimes argued that in theearly stage of the development process, there exists a positive income-fertility relationship (e.g.,
Wrigley (1961), Weir (1995), Clark and Hamilton (2006), Clark (2007), and Boberg-Fazlic,
Sharp and Weisdorf (2011)). For instance, the following might be possible explanations tothis finding. Jobs’ productivity differential is large in advanced economies, while it is not in
less-advanced ones; thus, wage differential in advanced economies largely reflects the jobs’ productivity heterogeneity, while that in less-advanced economies largely reflects the individuals’
ability heterogeneity. If that is the reality and an increase in wages associated with individuals’ ability has a dominant wealth effect on fertility, the wage-fertility relationships in advanced
and less-advanced economies are reversed, negative and positive, respectively. Alternatively, inagrarian economies, the mechanism proposed in this paper may not work well simply because
the free choice of employment is not guaranteed. If individuals face a limited occupational
choice, they are forced to accept their immediate environments, say, the farming villages they
were born into, as their workplaces. Then, the utility levels among individuals are not equalized,
and thus the individuals’ heterogeneity approach fits well with the analysis on such economies.
Positive relationship within the same occupation

Empirical findings in some earlier literaturesuch as Freedman (1963) and Simon (1969) report that the overall cross-sectional relationshipbetween wages and fertility is negative, whereas within the same occupation, higher-income
households tend to have more children. Combining the jobs’ heterogeneity approach with theindividuals’ heterogeneity approach might provide a theory reconciling the overall negative relationship with the within-occupation positive relationship. In many countries, the freedom
to choose own occupation is guaranteed, but the individuals’ performance in the occupation
depends on their ability. This leads us to the supposition that inter-occupation wage differential largely reflects productivity differential between occupations, not inherent in individuals,
41whereas within-occupation wage differential largely reflects ability differential between individuals. Then, if the wealth effect associated with the individuals’ ability heterogeneity dominates
the substitution effect, the observed pattern is obtained.
Needless to say, the hypotheses presented above are just theoretical possibilities and shouldbe empirically tested in a rigorous way. What we would like to emphasize here, however, isthat the difference between implications drawn from the jobs’ and individuals’ heterogeneity
approaches can broaden our options available to analyze fertility choice. We hope that the discussion presented here facilitates the testing of theories on fertility choice.

6

Conclusion

This paper has constructed a tractable theory of the cross-sectional fertility differential. The
theory sheds light on heterogeneity of the jobs’ productivity, one that has largely been ignored in
the literature. By using only standard conditions on preferences, we show that the negative wagefertility relationship can be produced and it is quite robust to changes in economic environments(e.g., public policy and technology). Our theory reconciles the negative cross-sectional wagefertility relationship with various time-series variations in aggregate fertility over the course of
economic growth and business cycle.
This paper is not intended to say that existing theories based on individuals’ heterogeneity
are wrong because they are not robust to changes in assumptions. On the contrary, we believethat our theory based on jobs’ heterogeneity complements the existing literature to improve ourunderstanding of fertility choice. The reality might be a hybrid of jobs’ heterogeneity and individuals’ heterogeneity. Making explicit the difference between implications drawn from thejobs’ and individuals’ heterogeneity approaches adds an important viewpoint to the literature.
For instance, this paper shows that the cross-sectional wage-fertility relationship depends on thesource of wage differential. Without the investigation on the source of wage differential, estimating the form of preferences from observed wage and fertility differentials might be misleading.
42

Since our model is simple and tractable, it can be theoretically extended in several ways.
Let us propose some potential directions for future research. Explicitly distinguishing between
the time of husband and wife would be an interesting extension. It shows what types of labor
contracts are chosen by males and females and what types of couples are formed in equilibrium. Furthermore, extending our model to dynamic models or incorporating our theory into
overlapping-generations frameworks is useful to analyze the cross-sectional relationship and thetime-series variation simultaneously.
Although this paper is motivated by empirical patterns in fertility and the main aim is todevelop a fertility-choice theory producing a robust negative wage-fertility relationship, ourframework can also be used to analyze labor-leisure choice. If we modify the proposed modelso that individuals receive utility from leisure time, instead of children, it becomes the model for
analyzing labor-leisure choice. Modern-growth and business-cycle theories accept the long-run
stability of leisure per capita as a stylized fact (Ramey and Francis (2009) provide a thorough evidence). Our model reconciles the long-run stability of leisure per capita with the cross-sectional
upward-sloping labor supply curve and the time-series upward trends in real wages. Furthermore, almost all the arguments concerning fertility choice presented in this paper, for example,
different effects between firms’ and individuals’ heterogeneity, are applied to the discussion onlabor-leisure choice.

7

Appendix

7.1

Proof of Proposition 4(a) Suppose that M is fixed. Since M is fixed, the exit threshold must remain unchanged to keep
the balance of supply and demand in the labor market. Thus, the profit of the threshold firm mustremain 0. For this, the utility level U must decline to offset the negative effect of an increase in43

tc on the profit. Totally differentiating ΠU∗ (α U ) = 0, we obtaindU
ΨU (n (α U ))=−,∂ Ψ (n(α U ))dtc
(1 + tc ) U∂Uwhich represents the iso-profit condition of the threshold job. To keep the profit of the thresholdjob 0, one unit of increase in tc must be accompanied by
ΨU (n (α U )) / [(1 + tc ) ∂ ΨU (n (α U )) /∂ U] units of decrease in U. It follows that given M, thevariation in the profit of jobs with productivity α , Π∗ (α ), caused by such a simultaneous changeof tc and U is∂ Π∗ (α )∂ Π∗ (α )
dtc +
dU
∂ tc
∂U
∂ Π∗ (α )∂ Π∗ (α ) ΨU (n (α U ))
dtc −
=
dtc
∂ tc
∂ U (1 + tc ) ∂ ΨU (n(α U ))dΠ∗ (α ) =∂UΨU (n (α )) ΨU (n (α ))1 − tΠ
[εU (n (α U )) − εU (n (α ))] ∂ Ψ (n(α U))
dtc .
=−
U
U
1 − tw
U∂U

where εU (n (α )) ≡ [∂ ΨU (n (α )) /∂ U] · [U/ΨU (n (α ))] is the utility elasticity of consumption
evaluated at n (α ). This implies that the profit of jobs facing the higher (resp. lower) utility
elasticity of consumption than that the threshold job faces increases (resp. decreases). If theutility elasticity of consumption increases with fertility (i.e., ∂ [(∂ ΨU /∂ U) (U/ΨU )] /∂ n > 0),then the profits of all jobs with productivity α > α fall because n (α ) < n (α ) for all α > α .
Then, the expected profit of entry for given M falls, the measure of entrants decreases, andthe exit threshold declines. Conversely, if the utility elasticity of consumption decreases withfertility, then the exit threshold rises.
(b) We follow a procedure similar to the proof of (a). The iso-profit condition of the thresholdjob isdUwU (α U ).
=−∂ Ψ (n(α U ))dtw
(1 − tw ) (1 + tc ) U∂UGiven M, the variation in Π∗ (α ) caused by a simultaneous change of tw and U such that Π∗ (α )44remains 0 is∂ Π∗ (α )∂ Π∗ (α )
dtw +
dU
∂ tw
∂U
∂ Π∗ (α )∂ Π∗ (α )wU (α U )
=
dtw −
dtw
∂ tw
∂ U (1 − tw ) (1 + tc ) ∂ ΨU (n(α U ))∂U][wU (α )wU (α U ) ∂ ΨU (n (α ))1 − tΠ
=−dtw−∂U(1 − tw )2 ∂ ΨU (n(α )) ∂ ΨU (n(α U ))dΠ∗ (α ) =∂U∂USince wU (α ) > wU (α U ) and ∂ ΨU (n (α )) /∂ U > ∂ ΨU (n (α U )) /∂ U for all α > α U , it is ambiguous whether it is positive or negative without further specification. Since wU (α ) depends
not only on preferences, but also on policy variables, in contrast to the proof of (a), we cannot
determine the sign of dΠ∗ (α ) only by imposing a restriction on preferences.
(c) We follow a procedure similar to the proof of (a). The iso-profit condition of the thresholdjob isdUn (α U )=.
dsn (1 + tc ) ∂ ΨU (n(α U ))∂UGiven M, the variation in Π∗ (α ) caused by a simultaneous change of sn and U such that Π∗ (α )remains 0 is∂ Π∗ (α )∂ Π∗ (α )
dsn +
dU
∂ sn
∂U
∂ Π∗ (α )∂ Π∗ (α )n (α U )
dsn +
dsn
=
∂ tc
∂ U (1 + tc ) ∂ ΨU (n(α U ))∂U][
1
1 − tΠ ∂ ΨU (n (α U ))∂ ΨU (n (α ))=n (α ) −n (α U ) ∂ Ψ (n(α )) dsn
UU
1 − tw∂U∂UdΠ∗ (α ) =∂UThis is negative for all α > α U because n (α ) < n (α U ) and ∂ ΨU (n (α U )) /∂ U <∂ ΨU (n (α )) /∂ U for all α > α U .Thus, the expected profit of entry for given M falls, the measureof entrants decreases, and the exit threshold declines.
(d) Even if tΠ changes, the profit of the initial exit threshold remains 0((1 − tΠ ) [α ∗ l (α ∗ ) − w (α ∗ )] = 0). On the other hand, an increase in tΠ lowers the expected profit

45

of entry, E ((1 − tΠ ) [α l (α ) − w (α )]). The measure of entrants decreases and the exit threshold
declines.

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