Baby Busts and Baby Booms:
The Fertility Response to Shocks in Dynastic Models
Larry Jones* and Alice Schoonbroodt**
August 2007
Abstract
After the fall in fertility during the demographic transition, many developed countries
experienced a baby bust, followed by the baby boom and subsequently a return to low fertility (BBB event). Demographers have linked these large ‡uctuations in fertility to the series
of ‘economic shocks’that occurred with similar timing –the Great Depression, World War II
(WWII), the economic expansion that followed and then the productivity slow down of the
1970s. This paper formalizes a more general link between ‡uctuations in output and fertility
decisions, in simple versions of stochastic growth models with endogenous fertility. First, we
develop initial tools to address the e¤ects of ‘temporary’shocks to productivity on fertility
choices. Second, we analyze calibrated versions of these models. Qualitatively, results show
that under reasonable parameter values fertility is pro-cyclical. Moreover, following a deviation in fertility (due to a productivity shock, for example) sets o¤ an endogenous cycle even
if productivity is on trend thereafter. Using the U.S. BBB event as a laboratory, we …nd that
the fertility response to a 1 percent increase in productivity lays between 1 and 1.7 percent,
while the response one period later lays between -0.9 and -1.6 percent. Depending on the
nature of child costs, the model captures between 38 and 63 percent of the pre-WWII baby
bust in TFR due to the great depression, while endogenous ‡uctuations in conjunction with
the productivity boom in the 1950s and 1960s captures between 53 and 92 percent of the
post-WWII baby boom in TFR. International evidence on sizes of depressions, baby busts
and booms supports these results.
* University of Minnesota and Federal Reserve Bank of Minneapolis, ** University of Southampton.
The authors thank the National Science Foundation for …nancial support. We also thank Michele Boldrin,
V.V. Chari, Simona Cociuba, Martin Gervais, David Hacker, John Knowles, Ellen McGrattan, Henry Siu,
Mich‘ele Tertilt and participants at Overlapping Generations Days 2010 for helpful comments.
1
1
Introduction
After the fall in fertility during the demographic transition, almost the entire 20th century
appears to constitute, for the United States and for most other developed countries, a period
of unusual deviations in fertility around an otherwise declining trend. First, fertility rates
saw an acceleration of the decrease during 1920-1940: the baby bust. This bust was followed
by a large upswing during 1940-1965: the baby boom. During the period 1965-1985 the total
fertility rate returned to a second slump at even lower levels. Demographers often link these
large ‡uctuations in fertility to the series of economic ‘shocks’ that occurred with similar
timing –the Great Depression, WWII and the economic expansion that followed, and …nally
the productivity slow down of the 1970’s. To economists, this line of argument suggests a
more general link between ‡uctuations in output and fertility decisions, of which the baby
Bust-Boom-Bust event (BBB) is a particularly stark example.
One complementary hypothesis also asserted in the demography literature is that the
baby boom was a ‘catching up’phenomenon from a period of relatively low fertility during
the Great Depression. (See Whelpton (1954), Goldberg, Sharp, and Freedman (1959), and
Whelpton, Campbell, and Patterson (1966)). One di¢ culty with the ‘catching up’hypothesis
that needs to be addressed, according to both demographers and economists, is that, in the
data, this ‘catching up’ clearly does not take place for a given woman (see Greenwood,
Seshadri, and Vandenbroucke (2005) and section 3.1 below for documentation). That is,
completed fertility was low for both the women immediately preceding and immediately
following the baby boom mothers. So, if there is ‘catching up’, it is in a dynastic (aggregate)
sense, not at the individual level. That is, in a dynastic model, it is quite possible that
the dynasty purposefully ‘catches up,’although this need not be observed for any particular
generation of women. This is a distinction relevant in our analysis below.
These two hypotheses can be combined to yield the following relationship: current fertility
is a stable function of productivity levels and the current stock of people in the economy.
That is, unusual drops in income (relative to trend) cause fertility to fall, other things equal;
vice versa, unusual increases in income (relative to trend) cause fertility to rise. If the
increase in income follows a period of below trend growth, the boost to fertility may be even
larger because the current stock of people is low compared to the long run ‘target’ level.
Roughly speaking, in this view fertility is a function of the current income shock and of the
stock of population, with either long run average income or long run ‘trend’ growth rates
determining the target level of population, or of family size. In this paper we formalize this
link by combining simple versions of the stochastic growth model and the dynastic model of
endogenous fertility à la Barro-Becker.
First, we …nd that fertility is indeed pro-cyclical in most cases. Although it seems unlikely
that fertility decisions are a¤ected by quarter to quarter ‡uctuations in productivity (as
addressed in the business cycle literature), longer ‡uctuations, such as the Great Depression
where productivity was below trend for about 10 years, the post-war period where it was
above trend for about the same number of years, and the extended productivity slow-down
at the end of the seventies, are likely to a¤ect current fertility decisions in the interest
of consumption smoothing, both intertemporally and intergenerationally. In the present
1
context, temporary shocks should therefore be understood as extended swings around a very
long term trend. Second, we …nd that if there are (at least) two periods of productive life,
current fertility depends negatively on last period’s fertility.
We then use these models to approximate the elasticity of fertility to productivity shocks,
both contemporaneous and delayed. We …nd that the …rst lays between 1 and 1.7 while the
second lays between 1 and 1.7. After calibrating to U.S. averages, we …nd that, through the
model, deviations in productivity capture between 38 and 63 percent of the pre-WWII baby
bust in Total Fertility Rates (TFR), while endogenous ‡uctuations in response to the earlier
downward deviation and in conjunction with the productivity boom in the 1950s and 1960s
captures between 53 and 92 percent of the post-WWII baby boom in TFR.
Among recent conjectures relating cyclical movements in income and fertility, perhaps
the best known one was advanced by Easterlin (2000). He argues that fertility decisions
are based on expected lifetime income relative to material aspirations which are formed in
childhood. When expected income is high relative to individual aspirations, fertility is high,
and vice versa. For the women born in the very early years of the twentieth century who were
making fertility decisions in the late 1920s and during the 1930s, expected lifetime income was
relatively low due to the Great Depression –hence, the baby bust. Vice versa, since the baby
boom mothers grew up in bad times (the 1930s) and therefore had low material aspirations
– while they were making fertility decisions during good times (i.e. lifetime income was
high relative to expectation), they had many children. Operationally, low income for today’s
generation implies high fertility for the next generation, especially if its expected lifetime
income is particularly high.1
It is not hard to see that this version of the relationship between (relative) income and
fertility decisions, as well as the previous one linking above (below) average growth and
high (low) fertility are, in fact, ‘dynamic variations’ of the Malthusian hypothesis, both
in spirit and in their substantive predictions (see Malthus (1798)). Recall that, in the
traditional Malthusian view, the long run population level is determined by a …xed natural
ratio between available economic resources and population size. When income per capita i.e. labor productivity - increases above this natural level, fertility also increases until the
long run ratio is reestablished, and vice versa for periods of economic crisis. The prediction
that periods of unusually high mortality, during which population is depleted while economic
resources remain unchanged, should be followed by years of above average fertility, and vice
versa is not far from the one considered here: periods of very harsh economic conditions, in
which per capita income decreases below the natural level, are also periods in which fertility
decreases. Cipolla (1962), Simon (1977), and Boserup (1981) are some of the best historical
renditions of such a ‘generalized’Malthusian view, and to them we refer for the many details
we must by force omit in our brief historical overview.
While one can see from the above that the theory being considered here dates back
to the very origins of modern economic demography, surprisingly little has been done to
formally address the link between productivity shocks and fertility in a stochastic model of
optimal fertility choice. This paper aims at …lling this gap by investigating the theoretical
1
Formally, one version of this story would be to assume that there is habit formation in consumption, as
in much of the recent literature on asset pricing.
2
and quantitative implications of this link in a stochastic version of the traditional dynastic
model of endogenous fertility (see Becker and Barro (1988) and Barro and Becker (1989)).
Formally, in these models the size of population in period t, Nt , plays a role much like the
capital stock in a standard stochastic growth model. Analogously, fertility plays the role
of investment. This analogy is sometimes imperfect since, for example, in the Barro-Becker
rendition of the dynastic growth model, Nt also enters the utility function of the planner.
Thus, in truth Nt has features that are a mixture of capital and consumption in the stochastic
growth model. Also, if children cost time, there is an opportunity cost component through
wages –absent in the standard stochastic growth model with capital –that makes investment
in children cheap in bad times and expensive in good times.
Given those provisos, recall the simple intuition from the single sector growth model
with productivity shocks. There is a fundamental desire to smooth consumption due to the
concavity of the utility function. Because of this, in a period when productivity is lower
than average (and, as a result, output is correspondingly low), agents lower investment to
smooth consumption. When the shock is high, the opposite occurs. Thus, the growth rate
of K is high when the shock is high and low when it is low. In the case where the analogy to
Nt in the endogenous fertility models holds, this implies that the growth rate of Nt , i.e., the
fertility rate, is high when the shock is high, and low when the shock is low. These …rst order
deviations induced by variations in current productivity, can be either damped or magni…ed
by the particular type of production function one adopts.
In this paper, we take the simplest case by considering a stochastic version of a BarroBecker type model where we abstract from all other inputs besides pure labor (such as
physical and human capital or land). However, people live and are productive for more
than one period. Thus there are young and old workers. Young workers are fertile. Now,
whenever the ratio of young to old workers is low (high) (i.e., last period’s fertility was low
(high)), the "depreciation" into retirement is large. Again, using the analogy to capital in
standard growth models, high depreciation at the end of this period implies high investment
for consumption smoothing purposes. In the case, where the analogy to fertility holds,
the number of births is particularly high (low) if it was particularly low (high) last period.
The fertility rate is the large (small) number of births divided by the small (large) number
of young workers and is therefore particularly large (small). If there is only one period of
working life, the depreciation rate into retirement (or death) is always 100 percent and hence
fertility is independent on last period’s fertility.
Next, we analyze a version of the model in which population does not enter the dynastic
planner’s utility function. This requires a particular con…guration of parameters in the
Barro-Becker type preferences. In this case only total consumption by all members of the
dynasty enters the utility function. This speci…cation simpli…es the model and reduces it
to one which is analogous to a stochastic Ak model (see Jones and Manuelli (1990) and
Rebelo (1991)) with capital vintages. Under the additional assumption that the shocks to
productivity are i:i:d:, we give analytical characterizations of the model for particular cases.
In Section 4 we perform quantitative experiments on this version of the model. To do
this, we …rst calibrate the model parameters to various averages in the data. We then use
actual magnitudes of productivity shocks from 1900 to 2000 and compare the predictions of
3
the model in terms of deviations in fertility rates to the data. We consider two alternative
speci…cations of the model: (1) all costs of raising children are goods costs; (2) all costs are
time costs. Reality probably lies somewhere inbetween these two extreme cases. Indeed, the
time cost may have become more relevant over time as women entered the labor force. Since
women tend to be the parent taking care of children, the relevant opportunity cost is only
procyclical if the latter are actually working whenever they are not busy raising children. In
case (1), the elasticity of fertility to the current shock is 1.04. In the case where all costs are
in terms of time, we …nd that the elasticity is 1.74.
Since during the Great Depression productivity was about 12.5 percent below trend, we
get about one-half of the 21 percent downward deviation in fertility (TFR) in the …rst case
and about two-thirds in the second. For the baby boom, about 35 to 55 percent of 30 percent
upward deviation during late 1950s are accounted for by the 10 percent productivity boom
alone, while the response to low productivity in the 1930s— and its ensuing low fertility—
accounts for another 20 to 40 percent.
All of these statements are contingent on a particular way of identifying the ‘shocks to
productivity’ in the data. And the results will certainly be a¤ected by how this issue is
treated. Also, the inclusion of physical capital may a¤ect the results but adds an additional
state variable. We perform these and other sensitivity in Section 5.
Finally, in Section 6, we show that countries with a smaller Depression in the 1930s also
tend to have smaller baby busts and that those with smaller baby busts tend to have smaller
baby booms in the 1950s, conditional on the economic circumstances in the 1950s. These
observations further support our theory.
2
U.S. Data
In this section, we lay out the basic facts about the time paths of productivity and fertility
in the U.S. over the 20th century. We begin with the facts pertaining to the growth in
productivity using a consistent measure for Total Factor Productivity (TFP) and Labor
Productivity (LP) from Chari, Kehoe and McGrattan (2006). As most economists know,
this period is one of more or less continued growth in productivity with a few interruptions.
The most signi…cant of these is the great depression. Figure 1 shows the natural logarithms
of TFP and LP over the period from 1901 to 2000.
The facts about productivity over this period can be described as follows:
1. the continual upward trend;
2. the marked decline below trend that took place in the 1930s and early 1940s;
3. the return to trend in the early 1950s;
4. the signi…cant increase above trend that took place in the 1950s and 1960s;
5. the productivity slowdown since the 1970s.
4
Figure 1: Total Factor Productivity and Labor Productivity, 1901-2000 (1929=100)
6.5
Log TFP
Log LP
Productivity
6
5.5
5
4.5
4
1900
1920
1940
1960
1980
2000
Year
This timing of the movements of productivity around trend …ts well with the movements
in fertility seen in the data. Figure 2 shows the time path of the Total Fertility Rate (TFR)
and Cohort Total Fertility Rates (CTFR) (by birthyear of mother +23 years) over the period
from 1850 to 2000. We have two time series for TFR, which calculates how many children a
woman would have over her lifetime if current age-speci…c fertility rates were to prevail in the
future. The …rst series is the one prepared by Haines (1994) using Census data and hence is
available only every 10 years. The second comes from the Natality Statistics Analysis from
National Center for Health Statistics. It is available at annual frequencies, but only since
1917. The CTFR series comes from Jones and Tertilt (2008) and counts how many children
were born to a particular cohort of women at the end of their fertile period. Implicitly, it
is equivalent to adding age-speci…c fertility rates pertaining to a particular cohort of women
over time. Its frequency is …ve-year birth cohorts.
At the beginning of the period, fertility is still in the midst of what is known to demographers as the demographic transition, the marked fall in fertility (and mortality) that has
occurred in all developed countries. This fall accelerates from the late 1920s to the mid
1930s. Fertility then increases to reach its peak in the baby boom of the 1950s and 1960s.
It appears that a good description would be:
1. high, and fairly constantly decreasing fertility from 1850 until 1925, when it reaches a
TFR of about 3.5 children per woman;
2. an acceleration of the rate at which fertility is falling between 1925 and 1933 (from
TFR=3.5 to TFR=2.1);
5
Figure 2: (Cohort) Total Fertility Rate, 1850-2000
6
TFR
CTFR (+23)
5.5
5
Fertility
4.5
4
3.5
3
2.5
2
1.5
1850
1900
1950
2000
Year
3. constant, but low, fertility over the period from 1933 to 1940, with the level at about
TFR=2.2;
4. rapidly rising fertility from 1940 to 1957, with TFR going from 2.2 up to 3.7;
5. high, stable fertility from 1957 to 1961 at about TFR=3.6;
6. a rapid decrease from 1961 to 1976, with TFR going from 3.6 down to 1.7;
7. a slight increase and then stable low fertility over the remainder of the period, with
the level at about TFR=2.
We will refer to 2 and 3 as the pre-WWII baby bust, 4 and 5 as the post-WWII baby
boom and 6 as the baby bust of the 1970s. The exact sizes of these features of the data
depend on how one treats the trend growth in productivity and trend decrease in fertility over
the period. For example, was there a common, exogenous growth rate in productivity over
the entire period with higher frequency (albeit highly autocorrelated) ‡uctuations around
this trend? Or, were there several regimes of growth? For fertility, one can see that while
TFR decreases smoothly over time, the early CTFR data shows that the largest decrease
happens for cohorts of women born between 1858 (4.9 children per women) and 1878 (3.25
children per woman). The ‡uctuations thereafter, however, look very similar in both series,
though somewhat larger in TFR than CTFR. This pattern also suggests that if the baby
boom is a catching up of fertility from the baby bust, it is at the aggregate (dynasty) level,
6
not at the level of the individual woman.2
These detrending considerations will have an impact on the analysis we present below
and because of this, we try several alternatives. For now, we …t a linear trend to the (ln)
TFP and LP series from 1901 to 2000 (ln Pt = 0 + 1 t + t , where P = fT F P; LP g), and
detrend TFR using an HP …lter (smoothing parameter, w = 20; 000). We obtain annual
percent deviations over this period plotted in Figure 3.
Figure 3: TFR, TFP and LP Percent Deviations From Trend, 1901-2000
0.4
0.3
TFP
LP
TFR
Percent Deviations
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
1910 1920 1930 1940 1950 1960 1970 1980 1990 2000
Year
Although it is not perfect, there is an impressive coincidence in timing. The coe¢ cient
of correlation between annual TFP and TFR deviations for the years 1901 to 2000 is 0.4,
with a coe¢ cient of 0.71 from 1901 to 1940 and a correlation of 0.2 from 1941 to 2000. This
suggests that the U.S. TFR is procyclical during the early time period but the correlation is
much weaker thereafter (see also Butz and Ward (1979)). As suggested by the model below,
one reason for the decrease in the correlation may be the increase in female labor supply
which made the opportunity cost of children, women’s wages, procyclical.
What our model simulation is also going to capture is the large downward deviation in the
1930s due to the large negative shock during the great depression and a baby boom following
endogenously as a response to the baby bust itself, one generation later. We therefore also
bt denote the percent deviation from trend in variable X
run the following regression. Let X
2
See also Greenwood, Seshadri, and Vandenbroucke (2005), Doepke, Hazan, and Maoz (2007) and Jones
and Tertilt (2008) for more on the make-up, across birth cohorts, of the fertility pattern during the baby
boom.
7
in period t:
T[
F Rt =
0
+
b +
1 Pt
b
2 Pt l
+ "t
where P = fT F P; LP g and l = f20; 25g: The results are given in Table 1.
Table 1: U.S. TFR and Productivity: Regression results
Indep. Var.
Coe¢ cient
Std. error
t-statistic
p-value
Constant
T[
FPt
T[
F P t 20
-0.0051
0.8363
-0.8401
0.0155
0.1851
0.1951
-0.3284
4.5183
-4.3068
0.7435
0.0000
0.0000
Constant
T[
FPt
[
T F P t 25
-0.0112
0.7636
-0.6348
0.0174
0.2045
0.2154
-0.6468
3.7329
-2.9471
0.5198
0.0004
0.0043
Constant
ct
LP
c t 20
LP
-0.0058
0.6004
-0.6218
0.0169
0.1885
0.2009
-0.3395
3.1845
-3.0952
0.7351
0.0021
0.0027
-0.0139
0.4774
-0.4796
0.0187
0.2252
0.2411
-0.7441
2.1194
-1.9892
0.4592
0.0375
0.0505
Constant
ct
LP
c t 25
LP
These regressions show that the coe¢ cients on contemporaneous productivity ( 1 ) are
all positive and signi…cantly di¤erent from zero, while the coe¢ cients on productivity a
generation ago ( 2 ) are all negative and signi…cantly di¤erent from zero. Also, 1 and 2 are
of similar magnitude in absolute value, while the constant ( 1 ) is likely to be zero.
3
The Model
In this section, we lay out a model of the response of fertility to period by period stochastic
movements in productivity. To do this, we use a model of fertility based on that developed
in Becker and Barro (1988) and Barro and Becker (1989) (Barro-Becker henceforth). The
simpli…cation that we make is to assume that there is no physical (or human) capital in the
model. Thus, the ‡ow of income is solely due to wage income. On the other hand, we add a
stochastic component as well as an explicit age-structure to the basic Barro-Becker model.
8
3.1
Model setup
A period is 20 years. Every person lives for four periods, one as a child and T = 3 as an
adult.3 There is an initial age distribution of the population given by (N03 ; N02 ; N01 ) where N03
is the number of initial old (i.e., their age in period t = 0 is a = 3), etc. We will normalize
by assuming that N03 = 1.
No decisions are made in childhood (age a = 0). At age a = 1, young workers make
fertility and consumption decisions and supply one unit of labor inelastically to earn a wage,
wt1 . At age a = 2, old workers make consumption decisions and supply one unit of labor
inelastically to earn a wage, wt2 , but are no longer fertile: In the last period of their lives,
age a = 3, they are retired and only consume.
Adults care about consumption, the number of children and their children’s future utility.
Following the original Barro-Becker formulation, we assume that the utility of a person who
was born in period t 1 and whose …rst period as an adult is in period t is given by:
Ut1 = Vt1 +
1
g(nt )Ut+1
where Ut1 represents the full value of utility of an age a = 1 adult in period t looking from
that point forward, Vt1 is the utility this person gets from his own path of consumption, nt is
1
the number of children that he has and Ut+1
is the utility that his typical child will receive.
We distinguish between time preference as measured by the discount factor, and the
degree of altruism between generations within a period, : I.e., = 1 means that a person
cares as much about the utility of his children as he cares about his own (see, Manuelli and
Seshadri (Forthcoming)).
Let cas be the amount of consumption for the typical person in period s that is age a.
We will assume that utility from the time path of own consumption ((c1t ; c2t+1 ; c3t+2 ) –young
worker, old worker, retirement) is of the form:
Vt1
=
3
X
a 1
u(cat+a 1 ):
a=1
Substituting gives:
Ut1 =
3
X
a 1
u(cat+a 1 ) +
1
g(nt ) Vt+1
+
1
g(nt+1 )Ut+2
a=1
Under the assumption that g(n) = n simpli…cation occurs because g(nt )g(nt+1 ) =
g(nt nt+1 ). Thus:
3
The extension to a larger number of (shorter) periods is straightforward, but drastically increases the
number of state variables in the model. This complicates the numerical procedures studied below while
adding little to the intuitions we want to focus on in this paper.
9
Ut1
=
=
=
=
3
X
a=1
3
X
a 1
a=1
u(c1t )
u(c1t )
u(cat+a 1 ) +
a 1
u(cat+a 1 ) +
+
u(c2t+1 )
1
+(
)
+ ( )2
+ :::
+
1
)2 g(nt nt+1 )Ut+2
1
+(
g(nt )Vt+1
g(nt )
"
3
X
a=1
1
g(nt )u(ct+1 )
a 1
#
u(cat+1+a 1 ) + (
+
2
1
)2 g(nt nt+1 )Ut+2
u(c3t+2 ) + g(nt )u(c2t+2 ) +
2
g(nt nt+1 )u(c1t+2 ) + :::
u(c2t+1 ) + g(nt )u(c1t+1 )
2
u(c3t+2 ) +
1
g(nt )u(c2t+2 ) + g(nt nt+1 )u(c1t+2 )
Similarly, under the same assumptions, we can de…ne the continuation utility for a person
who was a young adult in period t (i.e., he was born in period t 1) looking forward from
the point when he reaches age a.
Of particular interest is the continuation utility of an adult of age T in period 0 – i.e.,
the initial old. These are the only agents in the model who care about all agents of all ages
in all periods. Utility for these agents, U0T , is given by:
U0T =
1
X
t=0
(
)t
"
T
X
1 a
#
g(Nta )u(cat )
a=1
where Nta is the number of descendants of the initial age T agent that are of age a in
period t. I.e.,
Nt1 = nt 1 Nt1 1 is the number of births in period t 1;
Nta = Nta 11 for a = 2; :::; T ;
Nta = 0 for a > T ;
N0T = 1.
To ensure the existence of a balanced growth path, we assume that the cost of children
born in period t is in terms of period t consumption but allow this cost to depend on the wage
of young workers – t (wt1 ). Thus, this allows for the two most common ways of modeling
child costs: goods costs – t (wt1 ) = t –and time costs – t (wt1 ) = bwt1 . Thus, feasibility in
period t is given by:
3
2
X
X
a a
1
b
Nt ct + t (wt )Nt
wta Nta Wt
a=1
Ntb
= nt Nt1
1
t = bwt ;
a=1
1
Nt+1
where
=
is the total number of births in period t. If the costs are in terms
of time,
there is an additional constraint, namely that 0 bNtb Nt1 :
We will assume that wages grow at a constant rate, , on average. Moreover, there is
an aggregate shock so that the entire age speci…c wage pro…le shifts up and down by st in
period t. Thus, in period t, wages are given by:
(wt1 ; wt2 ) = ( t st w1 ;
10
t
st w2 ):
We assume that st is a …rst order Markov Process (i.i.d. soon).
We will also assume that the costs of children also grow at rate : That is, we will assume
that, (wt1 ) = t in the goods cost case and (wt1 ) = t bw1 in the time cost case.
Given this, the Planner’s Problem is:
P( ;
; fN0a g ; s0 )
max
U0T
=E
subject to:
hP
1
t=0
(
t
)
hP
T
a=1
1 a
i
g(Nta (st a ))u(cat (st ))
P3
P
a t a a t
t
)ct (s ) + (st )Ntb (st )
st 2a=1 wa Nta (st a );
a=1 Nt (s
Nt1 (st 1 ) = Ntb 1 (st 1 ) is the number of births in period t
Nta (st a ) = Nta 11 (st 1 (a 1) ) for a = 2; :::; T ;
Nta (st a ) = 0 for a > T ;
N0a given, a = 1; :::; T .
js0
1;
where st = (s0 ; s1 ; ::::; st ) is the history of shocks up to and including period t:
Nb
Let us introduce one additional piece of notation. Let nbt = 0:5Nt 1 be the number of
t
births per woman. This is the model quantity that we will identify with the Cohort Total
Fertility Rate (CTFR) in the Data, while TFR will be a weighted average over several
cohort’s fertility.
3.2
Consumption Across the Age Distribution
First, we analyze how consumption within a given period, t, is distributed across the di¤erent
ages of agents alive at the time. The relevant term in a typical period, t, is:
P3
a=1
1 a
g(Nta )u(cat ) =
P3
a=1
1 a
g(Nta )u
Nta ca
t
Nta
=
P3
a=1
1 a
g(Nta )u
Cta
Nta
;
where Cta = Nta cat is the total consumption of the age a cohort in period t.
Given any level of aggregate consumption in a period,P
Ct , the planner will choose a
distribution across ages to maximize the above subject to 3a=1 Cta = Ct . It follows that
this is done by equating the marginal utility of a unit of aggregate consumption across the
di¤erent ages.
1
To gain more intuition restrict attention to the case where, u(c) = c1 , and, as above,
g(N ) = N . For this choice of functional forms, there are two sets of parameter restrictions
that satisfy the natural monotonicity and concavity restrictions, both in terms of the aggreC
gate, or dynasty variables, (C; N ), and in terms of per capita values, (N; c) = (N; N
). These
11
i
are: i) 0 < < 1, 0 < < 1 and 0
+
Planner’s objective within a period is:
P3
1 a
a=1
(Nta )
+
a 1
1 (Ct )
1
1 < 1, and ii)
> 1 and
+
1
0.4 The
.
As can be seen, the marginal utility of Cta is a¤ected by three things –First, is g increasing
(case i) or decreasing (case ii) in N ? Second, is N growing over time? Third, what is ?
Ca
For example, suppose = 1. In this case, the age a term is g(Nta )u Nta . If population
t
is growing (the case typically of empirical interest), then Nta is decreasing in a –there are less
people in older generations. Thus, if g is increasing (decreasing) in N , g(Nta ) is decreasing
(increasing) in a. Thus, other things equal, the marginal value of an increase in per capita
consumption within a period is decreasing (increasing) in age. On the other hand, in this
case a given level of aggregate consumption in a cohort is split across fewer people in older
groups increasing per capita consumption. This leads to a lower value of u0 .
Thus if g is increasing in N , whether the Planner will want consumption to be increasing
or decreasing in age within a particular period will depend on which of these two e¤ects
is larger. If g is decreasing in N , then if population is growing, per capita consumption is
increasing in a for sure. For example, specializing further to the case where = 1
, and
= 1, period t utility becomes:
P3
a=1
Cta1
1
.
0
Thus, aggregate consumption of all age groups within a period will be equalized, C a = C a
for all a; a0 , and hence, larger age groups (younger cohorts if N is increasing) will have smaller
0
per capita consumption –cat < cat for a < a0 .
3.3
Procyclical fertility and catching up
In this section we study the properties of the solution to the Planner’s Problem outlined
above. In particular, we will characterize how the policy functions from this problem depend
on both the current shock and the initial state.
To gain some intuition about the working of the model, notice that if +
1 = 0, then
N does not enter the period utility function except in aggregate consumption, and hence,
if wa = 0, a 6= 1; N plays exactly the same role in this model as k does in a stochastic Ak
model.5 There is one twist however. This is that, at least in the case where child-rearing
is modeled as a time cost, t –the cost of the investment good –is also stochastic. In that
case, since t = bwt1 , periods when productivity is high are also those when children – the
analog of the investment good in the Ak model –are expensive. Also, in this case aggregate
consumption, C; grows at the same rate as N (if = 1), but per capita consumption is
constant (without shocks). Other than that, the analogy is very close.
4
The knife-edge case where ! 1 and = (1
) can also be analyzed. See Jones and Schoonbroodt
(2010b) or Schoonbroodt and Tertilt (2010) for details.
5
See Jones and Manuelli (1990) and Rebelo (1991), seminal papers on this model.
12
As is usually true in models with exogenous, trend growth, solutions can be obtained by
solving a related model with no growth and a di¤erent discount factor. Thus, the solution
to P ( ; ; fN0a g ; s0 ) can be obtained directly from the solution to P (1; ^; fN0a g ; s0 ) – i.e.,
= 1 (no growth) and the discount factor, ^, depends on ; ; ; and . Because of this
result, we will abstract from trend growth through most of the remainder of the paper. In
those cases where the solution to the model depends on the discount factor, we will use this
result to calibrate to the appropriate discount factor in the detrended model.
Given these standard results, we now derive comparative statics of current fertility with
respect to productivity shocks and last period’s fertility. To do this, we …rst simplify the
problem to one with only one state variable. We then take …rst order conditions and analyse
comparative statics across steady states/balanced growth paths therein.
Denote by V (N 1 ; N 2 ; N 3 ; s) the maximized value obtained in the problem P (1; b; fN0a g ; s0 )
when the initial conditions are:
(N01 ; N02 ; N03 ) = (N 1 ; N 2 ; N 3 ) and s0 = s.
Because of our assumptions on the functional forms for the utility function, it is straightforward to show that the value function is homogeneous of degree in (N 1 ; N 2 ; N 3 ); i.e.,
V ( N 1 ; N 2 ; N 3 ; s) = V (N 1 ; N 2 ; N 3 ; s). The problem P (1; b; fN0a g ; s0 ) is therefore a
standard, stationary dynamic program as long as s is …rst-order Markov. Because of this
result, we can characterize the solution through Bellman’s Equation. That is V satis…es:
P
a 1
V (N 1 ; N 2 ; N 3 ; s) maxN a0 ;C a ;N b 3a=1 1 a (N a ) + 1 (C1 ) + E [V (N 10 ; N 20 ; N 30 ; s0 )js]
s.t.
P3
a=1
Ca + N b
s
P2
a=1
N a0 = N a 1 , for a = 2; 3;
N 10 = N b .
Under the assumptions that = 1
and this problem simpli…es to:
V (N 1 ; N 2 ; N 3 ; s)
max
T
0
Na
s.t.
TC + Nb
;C;N b
s
P2
C1
1
a=1
wa N a ;
, and
+
= 1, as discussed in Section 3.2, C a = C a0
E V (N 10 ; N 20 ; N 30 ; s0 )js
(1)
wa N a ;
N a0 = N a 1 , for a = 2; 3;
N 10 = N b .
As can be seen from this, since w3 = 0 by assumption, we have that V (N 1 ; N 2 ; N 3 ; s) =
V (N 1 ; N 2 ; N 3 ; s) for any N 3 ; N 3 . Because of this, we will write V as depending only on
(N 1 ; N 2 ; s).
13
Assuming further that shocks are i.i.d., using the homogeneity property of V , the above
0
BE is equivalent to the following one in terms of fertility per household, n = N b =N 1 ; with
1
:6
Vb (n; s) = V (N 1 =N 2 ; 1; s)=T , where n = N
N2
Vb (n; s)
maxn0
"
s[
w1 n+w2
]
(s)n0 n
1
) + n1
=(1
T
#
i
E Vb (n0 ; s0 ) ;
h
Taking the …rst order condition with respect to n0 and rearranging gives
(F OC)
LHS(n0 ) =
(s)
T E V^1 (n0 ;s0 )
=
i
h
2
s w1 + wn
(s)n0
= RHS(n0 ).
T
Then, LHS is increasing in n0 , with LHS(0) = 0 if E V^1 (0; s) = 1. Also, RHS(n0 ) is
decreasing in n0 with a positive intercept at n0 = 0. Thus, there is a unique solution.
To see the behavior of n0 as a function of the shock, we consider the two extreme cases,
(s) = , a goods cost, and a time cost, (s) = bsw1 .
In the …rst case, (s) = , the FOC is:
(F OC) Goods cost
T E V^1 (n0 ;s0 )
=
h
i
2
s w1 + wn
n0
T
.
In this case, RHS shifts up when s goes up while LHS is unchanged. Thus, n0 is
increasing in s – fertility is procyclical. This e¤ect is larger, the larger . Also, it follows
that n0 is linear in s.
On the other hand, when (s) = bsw1 ; a time cost, the FOC becomes
(F OC) Time cost
bw1 s1
T E V^1 (n0 ;s0 )
2
=
w1 (1 bn0 )+ wn
T
In this case, if > 1 LHS shifts down when s goes up while RHS is unchanged. Thus,
again n0 is increasing in s –fertility is procyclical. If, < 1 the opposite occurs and fertility
is countercyclical.
Moreover, in both cases, RHS shifts down when the current state, last period’s fertility
per capita, n, increases while LHS is independent of n. Thus n0 decreases when n increases,
generating cycles.
Finally, note that if there is only one period of productive life, w2 = 0, F OC becomes:
(F OC)
bAw1 s1
T E V^1 (n0 ;s0 )
=
w1 (1 bn0 )
T
:
Hence, in this case, current fertility is independent of last period’s fertility.
This leads to the following proposition.
6
See Appendix A.1 for details.
14
Proposition 1 Current fertility, n0 (n; s) is
1. a.
b.
2. a.
b.
procyclical if (w1 ) = , or if (w1 ) = bsw1 and > 1;
countercyclical if (w1 ) = bsw1 and < 1;
independent of last period’s fertility, n if w2 = 0;
decreasing in last period’s fertility, n if w2 > 0.
Thus, if w2 > 0 the model generates endogenous cycles, triggered by productivity shocks.
The intuition for why fertility is procyclical in this model is similar to that in many growth
models. Here, fertility plays the role of an investment good and the usual consumption
smoothing logic implies that when the shock is high, investment should be high so as to
o¤set future negative shocks e¤ects on consumption. This argument is direct when the cost
of children is a goods cost. It is tempered when the cost is a time cost by a second e¤ect.
This is that the cost of the investment good is also higher than average when the shock is
high. Thus, whether or not it is a good idea to invest in those periods depends on how
strong the desire is to smooth consumption. When this force is strong – is large – the
consumption smoothing e¤ect is large relative to the cost e¤ect and fertility is procyclical.
Thus, the more important the time cost in raising children, the more procyclical is the cost
of children itself. That is, whenever productivity is high, the cost of children is also high
and vice versa. This dampens the procyclicality of fertility and, indeed, when the desire to
smooth consumption is very low, fertility actually is countercyclical.
Some intuition for the fertility cycles in point 2 of the proposition can also be obtained
by analogy with growth models. Here, what we …nd is a source of endogenous cycles or
‘catching-up.’ The relevant analogy from capital theory here is to consider a model in which
depreciation is not constant. Here, we have an extreme version. Capital (i.e., bodies) that
are built in period t 1 have full productive capacity in period t and period t + 1 – there
is no depreciation between periods t and t + 1 –but has zero productive capacity in period
t + 2. Thus, age speci…c depreciation rates here would be 1 = 0 and 2 = 1 –no depreciation
after one period, full depreciation after two. In a situation like this, when n = N 1 =N 2 is
higher than usual, the planner expects next periods depreciation rate to be lower than usual
(because N 2 is relatively low). Because of this, to smooth consumption, current investment
–n0 = N b =N 1 –will be relatively low. Thus, n0 is low when n is high. When there is only one
period of working life, the depreciation rate is always 100 percent and hence this e¤ect is not
present. Because of this, fertility is independent of last period’s fertility. The example with
one period working life, wt2 = 0, is interesting because it corresponds most closely to that of
the original Barro-Becker model while the model with more than one period productive life
is more realistic.
Thus, according to our theory, while the baby bust in the 1930s may be explained by
the Great Depression, the baby boom in the 1950s is …rst and foremost a response to low
fertility in the past: ‘catching up.’
15
4
Quantitative Results
In this section, we use the facts about the time paths of productivity and fertility in the
U.S. over the 20th century laid out in Section 2 to perform quantitative experiments on
the model. To do this, we calibrate parameters to selected moments of our data. We then
use this model to explore two kinds of questions. First, what does the calibrated model
say about the size of responses to shocks to productivity – e.g., what is the elasticity of
fertility with respect to a productivity shock? In keeping with the theoretical results of the
previous section we study both the current response to a shock and also the lagged response
one generation later due to the misalignment of the age structure of the workforce. Second,
based on the estimated policy functions from the calibrated model we study the predicted
response to a productivity shock like those seen in US history –e.g., the Great Depression.
We …nd that the answers to these quantitative questions also depends on the nature of
the costs of children. Because of this, we give results for two alternative speci…cations. These
are: (1) all costs are time costs ( (s) = bw1 s); (2) all costs of raising children are goods costs
( (s) = ). As noted above, these two speci…cations are qualitatively di¤erent in that with
a time cost, the costs of children are higher in productivity booms than in busts.7
We …nd that the quantitative responses in the model are economically signi…cant in all
cases. For example, the elasticity of fertility to a contemporaneous productivity shocks lies
between 1 and 1.7,8 while the elasticity one period later lies between 0.94 and 1.5. Standard
recessions will have rather modest e¤ects on fertility, however. Large recessions, such as the
Great Depression have important and long lasting e¤ects, however.
We therefore turn to the historical record of the United States and study the predicted
response of fertility to the productivity booms and busts over the 20th century. To do this,
we must …rst construct a series of shocks to feed into the model. It is not obvious how to do
this realistically. On the one hand, in the model, it is assumed that the shock for the current
period is realized at the beginning of the period. E¤ectively, this means that individuals
know, at the beginning of the period, what the sequence of annual shocks will be over the
next 20 years. Even with highly correlated shocks at annual frequency, this assumption
seems extreme at best. Another alternative is to decrease the length of a period. This
necessarily increases the size of the state space. For example, even decreasing the period
length to 10 years and maintaining the i.i.d. assumption increases the size of the state space
from 2 dimensional to 5.9 Relaxing the i:i:d: assumption which might be required for 10as opposed to 20-year periods, adds another state variable. In addition, one would have to
7
Reality probably lies somewhere inbetween these two extreme cases. Indeed, the time cost may have
become more relevant over time as women entered the labor force. Since women tend to be the parent taking
care of children, the relevant opportunity cost is only procyclical if the latter are actually working whenever
they are not busy raising children.
8
As will be seen below, the model response to a shock is smaller when the cost is in terms of time than
when it is in terms of goods. Thus, ’between 1 and 1.7’ corresponds to 1 with a time cost and 1.7 with a
goods cost.
9
The state space is three-dimensional in our problem: N 1 ; N 2 and s. Using the homogeneity results,
this can be reduced to two state variables, N = N 1 =N 2 and s. With 10-year periods (and age-groups) the
problem becomes six-dimensional: Nb2 ; N 1;1 ; N 1;2 ; N 2;1 ; N 2;2 and s, which using the same methods can be
reduced to a …ve-dimensional problem.
16
address timing of births (between age 20-30 versus 30-40) more seriously. This is not an easy
problem (see Doepke, Hazan, and Maoz (2007) *** Sommer (2010) *** for examples). Thus,
there are technical di¢ culties with following the strategy of decreasing the period length.
Because of this, we present results for several alternative methods for constructing the
relevant series of productivity shocks for TFP and LP. First, since the data shows that over
the period 1940 to 1980 at least 60 percent of all births are to women age 20 to 30,10 we assume
that women have all their children between age 21 and 30. As our baseline experiment, we
therefore assume that the relevant productivity shock for them is the average one in the data
for that 10 year period. We conduct sensitivity analysis on this speci…cation in Section 5.
We also have to choose which productivity measure to use, Total Factor Productivity
(TFP) or Labor Productivity (LP). Second, we use TFP in our baseline case and discuss LP
in Section 5.
In our baseline experiment, a shock to productivity (or labor income) the size of the Great
Depression gives rise to a contemporary baby bust that accounts for 64 to 103 percent of the
reduction in CTFR and 39 to 63 percent of the reduction in TFR seen in the data. Further,
the model prediction of the lagged response to such a shock is a baby boom. Combined
with the productivity boom in the 1950s and 1960s, the predicted size of the baby boom lies
between 84 and 150 percent of the actual size of the baby boom observed in the CTFR data
and 53 to 92 percent of the one observed in the TFR data.
4.1
Parameterization
Preference Parameters: Throughout, we assume that = 1
and set = 3 following
Jones and Schoonbroodt (2010b) among others.11 The discount factor is set to = 0:9620
to match an annual interest rate of about 4 percent.
Wage and Productivity Parameters: From the model, wages are given by wta = st t wa ,
where wa is the base wage (in period 0) for workers of age a = 1; 2 , is the trend productivity
growth rate for a 20-year period and st is the productivity shock which we assume to be
i:i:d: over time.12 Further, for computational reasons, it is convenient to assume a functional
form for the distribution of productivity shocks, st . We assume that ln s^t N (0; s2 ) where
2
s^t = st e s =2 so E(st ) = 1. Thus, the parameter values to be determined are w1 ; w2 ; and s .
We normalize w1 = 1 and choose w2 = 1:25. This is in line with life-cycle earnings pro…les
from Hansen (1993) and Huggett (1996). For example, using these pro…les, Andolfatto and
Gervais (2008, p.3750) report that the wages of 40 to 60 year old workers are 25 percent
higher than those of 20 to 40 year old workers.
The growth rate of productivity, , and the standard deviation of productivity shocks,
,
s are calculated from the TFP series plotted in Figure 1. First, we compute the linear
10
See Vital Statistics of the United States, Table 1-7.
Mateos-Planas (2002), Andolfatto and Gervais (2008) and Scholz and Seshadri (2009) also use a value
of = 3: In the Appendix we perform sensitivity analysis with respect to all parameters.
12
This is a reasonable approximation for long movements in labor productivity across generations, which
within a dynasty are 20 years apart.
11
17
trend in productivity by running the following ordinary least-squares regression:
ln T F Pt =
0
+
1t
+ t:
(2)
We …nd that 1 = 0:0159 and therefore set = (e 1 )20 = 1:016120 . That is, productivity
grows at an average of 1.61 percent per year over the 20th century.13
To pin down the value of s several steps are required. First, in our baseline experiment,
we use the 10 most fertile years of each cohort within a dynasty, namely age 20 to 30, to
determine the productivity shock P
this cohort’s fertility choice is a¤ected
P10 tby. Thus, the shock
10
t
where = E(ln( t=1 e )). To approximate
we are interested in is ln s^t = ln
t=1 e
its standard deviation, s , we assume that t follows an AR1 process and estimate
t
=
0
+
1 t 1
+
t;
(3)
where t
N (0; ), simulate a long series of f t g, compute a series fln st g and calculate
its standard deviation to get s = 0:07. Of course, the series, ln st , so constructed is not
exactly normally distributed. However, for all simulations, the Kolmogorov-Smirnov test
produces a p-value of about 0.4. This means that, at any reasonable signi…cance level, the
test would accept the null hypothesis that the distribution of ln st is normal and hence this
should provide a good approximation.
Costs of Children: Given preference and productivity parameters, we then calibrate child
costs, ( ); to match an annual population growth rate of 0.645 percent per year (see Haines,
Nb +N 1 +N 2
1994, Table 1). In the model, population growth corresponds to N
1 +N 2 +N 3 : In a steady
state with no uncertainty, this is given by the steady state level of fertility choice – the n
satisfying n0 (n; 1) = n: Thus, the target is n0 (n; 1) = 1:0064520 : This corresponds to a steady
state fertility level of 2.27 children per woman. In the goods cost case, we …nd = 0:1932,
while with time costs we …nd b = 0:1927. Since the wage for young workers is normalized
to 1, these costs imply that it takes about 20 percent of a young worker’s income or time to
produce a child. This means that that a two person household can at most have ten children.
Summary: These parameters are summarized in Table 2. We describe sensitivity to the
parameter values in Section 5.
4.2
Model Impulse Responses
Given the parameter values from the previous section we calculate the decision rules from the
model. These can then be used to estimate the model responses to di¤erent size productivity
shocks. The results are summarized in Table 3. The rows of the table correspond to the two
alternative types of cost structures for children, goods and time. The …rst two columns report
the elasticities (at the steady state) of (a) current and (b) lagged fertility to productivity
13
An alternative detrending method would be to use a Hodrick-Prescott …lter, instead. However, it is
not clear what value to use for the smoothing parameter since we are interested in long ‡uctuations, rather
than quarter-to-quarter or annual deviations from trend. In any case, for low enough smoothing parameter
resulting productivity shocks would typically be smaller. Assuming that the growth rate of productivity is
constant in the model but using HP productivity shocks would result in smaller fertility responses in the
experiments at the end of this and the next section.
18
Table 2: Parameter Values, Baseline
w1
Parameter
Value
3:00
0:9620
w2
1:00
1:25
(goods cost)
b
(time cost)
0:1932
0:1927
s
1:016120
0:07
shocks. The next two columns give the (c) contemporaneous and (d) lagged change in
fertility levels that the model predicts will result from a one percent increase in productivity.
Column (e) recalls the steady state level of completed fertility (CTFR). As can be seen
Table 3: Impulse Response (in percent and levels), Baseline
Cost
Type
Goods
Time
% Deviations
CTFR Levels
Initial
(a)
Lagged
(b)
Initial
(c)
Lagged
(d)
Steady St.
(e)
1.741
1.039
-1.561
-0.940
2.314
2.298
2.239
2.253
2.274
2.274
from this table, the fertility response to a 1-percent productivity shock generates a 1.74
percent contemporaneous increase in fertility (column (a)) in the goods cost case with a
lagged decrease of 1.56 percent one period later (column (b)). In the time cost case, the
corresponding percentage changes are a 1.04 percent contemporaneous increase and a 0.94
decrease one period later.
These magnitudes can be compared to the regression results shown in Table 1. In particular, column (a) is to be compared with 1 and column (b) with with 2 . Thus, if the model
…t the data perfectly we should see 1 0:83, and 2
0:84 (these values are taken from
the …rst row of Table 1). Although the model quantities are not exactly the same, they are
remarkably similar.
Comparing the two cases, the initial e¤ect is 70 percent larger in the goods cost case
than it is in the time cost case. The reason for this di¤erence is that when child costs are
primarily in terms of time there are two o¤setting e¤ects. First, when times are good there
is a natural tendency to save for the future to equalize marginal utilities. Here, the only way
to do this is by increasing family size. On the other hand, since the cost of raising children is
positively related to the wage, when times are good, children are relatively more expensive
– a force in the opposite direction. When the costs of children are in terms of goods, this
second e¤ect is not at work and hence, the overall e¤ect is larger in this case. As can be seen
19
in the table, as the theory predicts, the …rst e¤ect dominates if the intertemporal elasticity
of substitution is low enough ( > 1, see Proposition 1).
Further, since in both cases, the ratio of the initial response to the lagged response is
about -90 percent, the half-life (in absolute value) of the e¤ect is 7 periods (or generations)
in both cases. In particular, the e¤ect 7 periods later is a decrease in fertility of 0.8 percent
in the goods cost case and 0.5 percent in the time cost case (i.e., half of the initial responses).
To get a sense about the size of these e¤ects note that a one-percent decrease in productivity for a 10-year period roughly corresponds to a recession where GDP is 5 percent below
trend for two years. This would decrease fertility by about 0.02 to 0.04 children per woman
with a subsequent baby boom of similar size. Thus, the quantitative e¤ects on fertility of
a normal sized recession would be quite modest. As we shall see below, the model predicts
that a recession the size of the Great Depression gives a much larger response.
4.3
A Historical Episode: The U.S. Great Depression
In this section we use the historical record of the actual series of productivity shocks in the
U.S. to study the predicted response of fertility to the productivity busts and booms that
have occurred over the last 100 years.
We …rst focus on the dynasty most a¤ected by the great depression, i.e., the cohort
making fertility decisions in the 1930s (followed by the 1950s cohort, the 1970s cohort and
the 1990s cohort).
Second, we consider other dynasties also a¤ected by the Great Depression but less so.
That is, those making fertility choices either in the late 1920s and early 1930s (i.e., between
1926 and 1935, etc.) or in the late 1930s and early 1940s (between 1936 and 1944, etc.).
Finally, we consider the dynasty making fertility choices in the 1920s (between 1921 and
1930, etc). By construction, the Great Depression has no e¤ect on the fertility choices of
this last dynasty. Thus, we end up with four separate dynasties. Cohorts within these
dynasties make overlapping fertility decisions.
Finally, the measure of fertility constructed in this way –using di¤erent shocks for different cohorts –corresponds to CTFR since it represents completed fertility for each cohort.
We also construct an analog for TFR, i.e., period fertility, by averaging over the two cohorts
making their fertility decisions in any …ve-year period. In doing this, we solve the model
separately for the di¤erent dynasties and then aggregate.
4.3.1
CTFR: Most A¤ected Dynasty (M.A.D.)
First, to focus on the e¤ects of the Great Depression, we start with the dynasty making
fertility decisions in the 1910s, 1930s, 1950s, 1970s and 1990s. This dynasty is the most
a¤ected by the Great Depression. As a benchmark, we assume that the initial state of the
dynasty is at its steady state age-distribution. In other words, the dynasty has been facing
average shocks (s = 1) for a long time so that its current age-distribution is stationary.
De…ne nss n(s) = n(1) = n0 (n(1); 1) = n0 (n(s); s).
20
For this dynasty, the relevant series of productivity shocks, st , is shown in Table 4.
The results are shown in Figure 4. With a goods cost, the model predicts that fertility is
19.1 percent below trend during the 1930’s; with a time cost the prediction is 11.8 percent
below trend. In the data, CTFR actually fell by 18.5 percent. Thus, the model captures
a signi…cant fraction of this movement in either case. This is primarily due to the large,
negative shock of 12.4 percent during the Great Depression of the 1930s – the dynasty’s
young-to-old worker ratio is initially almost at its steady state value and hence e¤ects from
this are minimal.
Table 4: Productivity shocks (in percent), Baseline (M.A.D.)
Decade
Productivity Shock
1910s
-1.8
1930s
-12.43
1950s
6.89
1970s
3.90
1990s
-8.97
Figure 4: Percent Deviations in CTFR, Baseline (M.A.D.)
0.4
Percent Deviations CTFR
0.3
0.2
Model: Goods cost
Model: Time cost
CTFR: Data
Cohort TFP Shock
0.1
0
-0.1
-0.2
-0.3
-0.4
1900
1920
1940
1960
1980
Birth Year of Mother + 23 years
2000
As noted above, there is also a lagged e¤ect in the model from the Great Depression due
to the misalignment of the age structure that results from low fertility during the 1930s.
This implies, a baby boom for the next cohort in this dynasty which takes place in the
1950s. With a goods cost, the model prediction is an increase in fertility above trend of 34.9
percent, while with a time cost, the corresponding model prediction is 19.7 percent above
trend. These model predictions result from a combination of the fertility response to the low
young-to-old worker ratio due to the baby bust and the 8.4 percent productivity boom in
21
the 1950s. For the purposes of comparison, the peak of the baby boom in the CTFR data is
23 percent above trend.
In the model, the Great Depression has continuing impacts on the fertility choices in this
dynasty –there are continued ‡uctuations with a decrease of 11 to 18 percent below trend
in the 1970s (due to the lagged response to the baby boom but dampened by the 3.9 percent
upward ‡uctuation in TFP in the 1970s); this is followed by an increase of 2.7 to 4.2 percent
above trend in the 1990s –again a response to the low young-to-old worker ratio dampened
by the negative productivity shock of 8 percent during the 1990s.
This is an extreme version of the baby bust and boom in the model because it is constructed so that the cohort’s entire fertile period falls during the Great Depression. For other
cohorts or dynasties, the e¤ects are mitigated because at least part of their fertile period
is una¤ected by the Great Depression. This is also true for the baby boom: the dynasty
considered here is most a¤ected by the low young-to-old worker ratio and has a large positive
productivity shock in the post-war period.
4.3.2
CTFR: All Dynasties and Cohorts
Table 5 shows how we construct our four dynasties. It illustrates the state variables and
shocks for each cohort. For each dynasty, we assume that the initial value of the state
variable is n = nss ; the balanced growth level of fertility. The …rst dynasty (d1 ) starts with
women born between 1876-1880 (labeled 1880), cohort 1 (c1 ). They face a productivity
shock, s1 ; averaging over the period 1901-1910 when they are roughly age 20 to 30. Their
fertility choice is nc1 = n0 (nss ; s1 ): The second dynasty (d2 ) starts with women born …ve
years later, between 1881-1885 (labeled 1885), cohort 2 (c2 ). Their …rst productivity shock,
s2 ; is obtained by averaging over 1906-15. Their fertility choice is nc2 = n0 (nss ; s2 ): The
third dynasty (d3 ) starts with women born between 1886-1890 (labeled 1890), cohort 3 (c3 ).
This is the most a¤ected dynasty described above. The fourth dynasty (d4 ) starts with
women born between 1891-1895 (labeled 1895), cohort 4 (c4 ). Their …rst productivity shock
s4 ; comes from averaging over the period 1916-1925. Cohort 5, born between 1896-1900
(labeled 1900) are the next generation in d1 . Their initial productivity shock s5 ; comes from
averaging over the period 1921-30. Thus, their fertility choice is given by nc5 = n0 (nc1 ; s5 ):
Proceeding further in this fashion leads to Cohort 9, born between 1916-1920 (labeled 1920),
the third generation in dynasty 1 making fertility decisions between 1941 and 1950. The
shock that this cohort faces is not a¤ected by the Great Depression at all. We expand this
series to Cohort 20, born between 1971-1975 (labeled 1975), the third generation in dynasty
1 making fertility decisions between 1996 and 2005.
The fertility choice in the last column of Table 5 is what we plot in Figure 5. Again,
we show the results of the model for goods and time costs cases separately. Also plotted is
CTFR data from Jones and Tertilt (2008). As expected, we see that dynasties other that
the M.A.D. are less a¤ected by the the Great Depression. Still, there is some e¤ect –there
are smaller deviations from trend both down and up, 20 years later.
22
Figure 5: Percent Deviations in CTFR, Baseline (All Cohorts)
0.4
Percent Deviations CTFR
0.3
0.2
Model: Goods cost
Model: Time cost
CTFR: Data
Cohort TFP Shock
0.1
0
-0.1
-0.2
-0.3
-0.4
1900
4.3.3
1920
1940
1960
1980
Birth Year of Mother + 23 years
2000
TFR
Next, we compute an analog of TFR, i.e., period fertility, by averaging over the two cohorts
making their fertility decisions in any …ve-year period. That is, we compute TFR in any
…ve-year period by computing the average of fertility of the two cohorts who are making
half their fertility decision in that …ve-year period, weighted by their size (i.e., their parent’s
n n +n n
fertility choice). For example, T F R1931 35 = c6 ncc2 +ncc7 c3 : Note that, if all dynasties are
2
3
in a steady state CT F R = T F R = nss : One advantage of this fertility measure is that we
have data until 2000, while for completed fertility the last cohort in Jones and Tertilt (2008)
makes fertility decisions in the 1980s.
The results are shown in Figure 6. Since deviations in TFR in the data are larger than
deviations in CTFR and since we are averaging over less a¤ected cohorts, the model accounts
for a smaller fraction of these deviations.
With a goods cost, the model predicts that fertility would be 12.7 percent below trend
in the early and late 1930s; with a time cost, the analog is 7.8 percent. In the data, TFR is
20 percent below trend in the early 1930s and 22 percent below trend in the late 1930s. As
with CTFR, this result is mainly due to the negative shock of 11.2 percent during the Great
Depression but here the results are mitigated because some women don’t make their entire
fertility decision during this period. In the data, the peak of the baby boom occurs in the
late 1950s when fertility is 31 percent above trend. With a goods cost, the model predicts
TFR will be 28.5 percent above trend while with a time cost, the prediction is 16.4 percent.
Again, this result is a combination of the fertility response to the low young-to-old worker
23
Figure 6: Percent Deviations in TFR, Baseline
0.4
Percent Deviations TFR
0.3
0.2
Model: Goods cost
Model: Time cost
TFR: Data
TFP Shock
0.1
0
-0.1
-0.2
-0.3
-0.4
1900
1920
1940
1960
1980
2000
Year
ratio due to the baby bust and the productivity boom in the 1950s and early 1960s. Hence,
the model accounts for 38.5 to 62.6 percent of the baby bust and for 53 to 92 percent of the
baby boom in TFR.
In subsequent periods, TFR in the model keeps ‡uctuating –11.9 to 18.9 percent below
trend in the late 1970s (due to the prior baby boom but dampened by the 3.9 percent increase
in TFP). The corresponding movements in the data on TFR is 20.7 percent below trend.
The increase in fertility in the 1990s of 7.1 percent above trend in the data is also partly
captured by the model which predicts a 2.8 to 4.7 percent increase.
5
Sensitivity
In this section, we discuss the sensitivity of our quantitative results above to (1) changes in
the age range used to assign TFP shocks to di¤erent cohorts, (2) using labor productivity
(LP) instead of TFP and (3) changes in base parameter values.
The …rst two parts take the parameters in Table 2 as given but change the way productivity shocks are computed. Hence, elasticities and impulse responses remain the same but the
size of shocks di¤ers. Results are reported in Figures ?? to 12 and Tables 9 to 11. Increasing
the age range tends to decrease the e¤ect of the great depression on fertility because it is
diluted across more cohorts. Therefore, the predicted size of the baby boom is also smaller.
It also a¤ects the timing of the baby bust and boom. The results using labor productivity
are very similar to the baseline, except for the cohort making fertility decisions in the 1910s
24
who experience a productivity boom and those making fertility decisions in the 1920s who
experience a signi…cant negative shock. Due to endogenous ‡uctuations, these changes work
themselves through the fertility ‡uctuations later on in the century.
The third part reports how elasticities and fertility levels change when we change parameter values in Table 2. Results are reported in Table 6. For each parameter change, we
report two cases. First, we give results in the case where costs of children remain at their
baseline value, which leads to varying fertility levels in steady state. Second, we recalibrate
to the target population growth by adjusting costs of children accordingly. Results show that
within a reasonable range, parameter choices don’t change elasticities and impulse responses
very much whether we recalibrate or not. In line with Proposition 1, decreasing the relative
wage of old workers, decreases the elasticity to past productivity shocks signi…cantly.
5.1
Age range of productivity shocks
One problem with the assumption that the only relevant shock for the dynasty is when their
youngest cohort is age 20 to 30 is that it implicitly assumes that the shock lasts for the entire
period of 20 years. In this section, we describe model results when the shock is set to the
average productivity shock over the period where the cohort is age 20 to 40.
Figure 7 shows cohort productivity shocks for this experiment compared to the baseline.
Overall shocks are smoother and smaller because some busts and booms are smoothed out
due to the longer fertile period. Also, Cohort c5 making part of their fertility decisions in
the 1920s faces a relatively large negative shock due to the Great Depression in the 1930s
when c5 is age 30 to 40. Therefore, the baby bust starts too early and is signi…cantly smaller
than in the baseline model as can be seen in ?? and 9 and in Tables 10 and 11. Because of
the smaller baby bust in the 1930s and smaller shocks, the baby boom is also smaller and so
are subsequent ‡uctuations.
The problem with this alternative is that it implicitly assumes that fertility is uniform
between age 20 and 40. Hence, the fertility decision of women born at the beginning of the
century is heavily a¤ected by the Great Depression in this experiment when in reality most
of them had completed their fertility in the 1920s.
Since this alternative is at the opposite extreme of the baseline, the two experiments
together should give us reasonable bounds on how much of the ‡uctuations in fertility over
the 20th century in the US the model can account for.14
5.2
Labor Productivity versus Total Factor Productivity
Since there is no capital in our model, it is not clear wether we should use labor productivity
(LP) or total factor productivity (TFP) to infer productivity shocks. The argument for using
TFP is that if capital were included in the model, fertility would respond to TFP alongside
investment. The argument for using LP is that most people don’t own much capital (besides
14
In the Additional Appendix available online we perform an experiment where TFP shocks are weighted
by average age-speci…c fertility rates.
25
Figure 7: Cohort TFP Shocks, Age 20-40 versus Age 20-30
0.2
Percent Deviations TFP
0.15
TFP Shock, 20-40
TFP Shock, 20-30
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
1900
1920
1940
1960
1980
Birth Year of Mother + 23 years
2000
Figure 8: Percent Deviations in TFR, Age 20-40, Time Cost
0.4
Percent Deviations TFR
0.3
Experiment
Baseline
TFR: Data
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
1900
1920
1940
1960
Year
26
1980
2000
Figure 9: Percent Deviations in TFR, Age 20-40, Goods Cost
0.4
Percent Deviations TFR
0.3
Experiment
Baseline
TFR: Data
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
1900
1920
1940
1960
1980
2000
Year
durable goods and houses) so that their fertility response is to changes in LP.
Here we perform the same experiment as in the main text but use deviations from trend
in LP instead of TFP. Again, we …t a linear trend through the LP data. The resulting
productivity shocks for each cohort are compared to the baseline in Figure 10 and in Table 9.
As can be seem in the …gure, there are two main di¤erences between the LP compared to
the TFP series of shocks. The very …rst cohort making fertility decisions in the early 1900s
experience a productivity boom and those making fertility decisions in the 1910s and 1920s
experience a signi…cant negative shock. Therefore, the baby bust is already well underway
in the at this time. Despite the similar size of the shock in the 1930s, fertility decreases less
in the LP case than in the TFP case. This is because of the mitigating e¤ect of the low
young-to-old worker ration due to low fertility in the 1910s. Subsequently, the baby boom
starts earlier but is of similar size as the baseline due to the larger LP shock in the 1950s
and 1960s compared to TFP. Overall, however, the results are quite similar.
Note that the growth rate of LP in the data is 2.23 percent per year on average, while
it is only 1.61 percent per year for TFP. Also, the estimated standard deviation of shocks
is 0.08 in most simulations while it is only 0.07 for TFP. In the experiment above, we held
these two parameters constant so that we perform only one change at a time. In the next
subsection, we perform sensitivity with respect to all parameters, including the productivity
growth rate, and s , and analyze LP with respect to these changes there.
27
Figure 10: Cohort TFP Shock, LP versus TFP
0.2
Percent Deviations LP and TFP
0.15
LP Shock
TFP Shock
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
1900
1920
1940
1960
1980
Birth Year of Mother + 23 years
2000
Figure 11: Percent Deviations in TFR, Labor Productivity, Time Cost
0.4
Percent Deviations TFR
0.3
Experiment
Baseline
TFR: Data
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
1900
1920
1940
1960
Year
28
1980
2000
Figure 12: Percent Deviations in TFR, Labor Productivity, Goods Cost
0.4
Experiment
Baseline
TFR: Data
Percent Deviations TFR
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
1900
1920
1940
1960
1980
2000
Year
5.3
The e¤ects of including physical capital
In this section we show that including physical capital as an alternative investment into the
model does not change the contemporaneous elasticity to productivity shocks for reasonable
values of the capital depreciation rate across generations.
To do this while keeping the size of the parameter space small, we consider the special
case where w2 = 0 and (st ) = (i.e. the goods cost case) and compare two speci…cations
of the production function for the dynasty:
1. F (K; w1 N 1 +w2 N 2 ) = w1 N 1 , the simpli…ed version of the baseline case without capital;
2. F (K; w1 N 1 + w2 N 2 ) = K (w1 N 1 )1
In case 1, assuming shocks are i:i:d:,
Bellman equation 1 is equivalent to:15
V (s)
maxn0
[sw
b1 bn0 ]1
1
+ (n0 )1
, the case including capital.
= 1 and using the homogeneity properties, the
E V (s0 )
Taking the …rst-order condition, fertility, n0 , as a function of the productivity shock, s,
is given by:
0
1
w=T
As
n0 (s) = @
s
(4)
=T
=T + (1 )E
[V (s0 )]
15
See Appendix A.2 for details.
29
where, since E V (s0 ) is independent on s, is just a constant. Since E(s) = 1, we have
E(n0 (s)) = . Percent deviations, are therefore given by:
n0 (s) E(n0 (s))
=s
E(n0 (s))
1
Using the dynastic head’s utility derived before but introducing savings in the budget
constraint, the problem the dynasty solves in case 2 is given by:
i i
hP
h
1
t PT
a t
1 a
a t a
(s
))
js0
(s
))u(c
(
)
g(N
max
U0T = E
P ( ; ; fN0a g ; K0 ; s0 )
t
t
t=0
a=1
subject to:
P3
a t a a t
)ct (s ) + Ntb (st ) + Kt+1 (st ) st (Kt (st 1 ))
a=1 Nt (s
Nt1 (st 1 ) = Ntb 1 (st 1 ) is the number of births in period t
Nta (st a ) = Nta 11 (st 1 (a 1) ) for a = 2; :::; T ;
Nta (st a ) = 0 for a > T ;
N0a given, a = 1; :::; T .
1
( t w1 Nt1 (st 1 ))
1;
;
where st = (s0 ; s1 ; ::::; st ) is the history of shocks up to and including period t:
Assuming that = 1, + = 1, costs are in terms of goods, t = t , and shocks are
i:i:d, and dividing Ct and Kt by t , P can be written recursively as:
n 1
o
C
1
1
10
0 0
T 1
(BE)
V (N ; K; s) = max
+
E (V (N ; K ; s ))
10
0
(C;N ;K )
subject to:
T C + N 10 + K 0 = sF (K; N 1 ) + (1
)K
Now, it is straightforward to show that, as long as investment and fertility choice are
positive, if there is a unique solution, k 0 , to the equation EVK ((1; k 0 ; s0 ) = EVN ((1; k 0 ; s0 )
then the capital-labor ratio is independent on the current shock, s. Also, fertility is only
dependent on the stock of population, if the latter is not in the right relation to the current
stock of capital. For example, after epidemics or wars with a high death but low destruction
rates. As long as w2 = 0, productivity shocks, unlike the baseline model, do not set o¤
dampened oscillations.
Further simplifying, we get:
n 1
o
c
1
0 1
0 0
+
(n
)
E
(v(k
;
s
))
(be)
v(k; s) = max
1
0 0
c;n ;k
subject to:
0
c
sf (k)
k 0 n0 + (1
)k
30
n0
k 0 n0
(1
)k
0
N0 ; K0 ; s0 given
Taking …rst-order conditions, fertility, n0 , as a function of the productivity shock, s, is
given by:
0
n (s) =
(
1
1
(1
( k0 + ) + ( k0 + ) (
)Ev(k 0 ; s0 ))
1
(1
1
)Ev(k 0 ; s0 ))
1
[sf (k) + (1
)k] :
(5)
In the case where capital fully depreciates after one 20-year period, = 1, we see that,
since k 0 is independent on s, n0 (s) = s. Therefore, just as in the model without capital,
percent deviations, are also given by:
n0 (s) E(n0 (s))
=s
E(n0 (s))
1:
Thus, the fertility response to productivity shocks is the same in the two models, namely 1.
If < 1, this is not the case. We use the same parameters as in the baseline, except
w2 = 0. Further, we follow Jones and Schoonbroodt (2010b) and set = 0:33 and =
1 (1 0:088)20 . Finally, we calibrate = 0:0214 to get an annual population growth rate
of 0.65. As a fraction of one person’s income, this corresponds to =w = 0:1609, which is the
relevant magnitude to be compared to in the baseline model.
We …nd that with these parameters, the fertility response to a 1 percent deviation in
productivity decreases to 0.9938. Hence, introducing capital into the model with one period
working life tends to dampen the e¤ects of productivity shocks ever so slightly. In the
additional appendix to this paper, we show that in a model with stochastic ageing (useful to
obtain reasonable working life lengths while keeping the state space small), we show that the
elasticity is larger (smaller) than 1 if the adult survival rate is larger (smaller) than (1
).
Since in the present model, survival in the workforce is zero, this is an extreme special case.
Thus, introducing capital in general does not change the fertility response to productivity
shocks very much.16
5.4
Three Period Working Life (3PWL)
While the fertility response to productivity shocks is very similar whether w2 > 0 or w2 = 0,
the dampened oscillations e¤ect only occurs if w2 > 0. In this subsection, we add one more
period of working life to show that similar dampened oscillations occur in this case. That
is, we assume that w3 > 0. Since this increases the number of state variables by one, we
abstract from productivity shocks thereby reducing the size of the state-space back to two.
In the experiments, we take the initial fertility deviation in and around the 1930s as given
and evaluate the oscillations that follow.
16
See the Additional Appendix, Jones and Schoonbroodt (2010a), for details.
31
Under the assumptions that
and the BE in (1) becomes:
V (N 1 ; N 2 ; N 3 )
s.t.
T C + N b0
max T
C;N b
P3
a=1
= 1, as discussed in Section 3.2, C a = C a0 ,
, and
=1
C1
1
+ V (N 10 ; N 20 ; N 30 )
(6)
wa N a ;
N a0 = N a 1 , for a = 2; 3;
N 10 = N b .
Using the homogeneity property of V , the above BE is equivalent to the following one in
0
terms of fertility per household, n = N b =N 1 ; with Vb (n; 1; 1=n ) = V (N 1 =N 2 ; 1; N 3 =N 2 )=T ,
2
N1
where n = N
=N
:
2 and n
N3
Vb n; 1; n1
2
6
maxn0 4
w1 n+w2 +w3 1
n
T
n0 n
!1
+ n1
1
3
7
Vb n0 ; 1; n1 5
Taking the …rst order condition with respect to n0 and rearranging gives
2
(F OC)
LHS(n0 ) =
1
T Vb1 (n0 ;1; n
)
=
3
w1 + wn + w
nn
T
n0
= RHS(n0 )
Then, LHS is increasing in n0 , with LHS(0) = 0 if E V^1 (0; s) = 1. Also, RHS(n0 ) is
decreasing in n0 with a positive intercept at n0 = 0. Thus, there is a unique solution.
Next, we want to derive comparative statics with respect to the state variables, last
period’s fertility, n, as well as fertility two periods ago, n .
As before, RHS shifts down when last period’s fertility per capita, n, or fertility two
periods ago, n , increase. While LHS is independent of n and hence n0 decreases when n
increases, LHS now depends on n. Which way it shifts following an increase in n, depends
on the cross-partial derivative with respect to N 1 and N 3 , V13 . In particular, if V13 > 0
then LHS shifts up when n increases and, hence, n0 unambiguously decreases. However,
if V13 < 0, then LHS shifts down when n increases and, hence, the overall e¤ect on n0 is
ambiguous.
In what follows, we compare simulations of this model to simulations in the baseline
as well as a non-stochastic version of the baseline model (i.e., a special case of the model
presented in this section where w3 = 0). This intermediate simulation allows us to disentangle
how much of the di¤erence comes from having a third period of working life and how much
comes from ignoring productivity shocks.
In the deterministic three period working life model, we assume the model period is 15
years. That is, people work from age 15 to age 60. Hence, we will have three, instead of
four dynasties, …ve years apart from each other. We keep all the parameters the same as
32
in the baseline, except for w3 and the cost of children, .17 We set w3 = 1:125. That is,
45-60 year old workers earn 25 percent less than workers age 30-45. Calibrated costs don’t
change much from the baseline ( = 0:1932 or b = 0:1927). Here, we …nd = 0:1946 to
match the steady state population growth rate of 0.65 percent per year as before. In the
intermediate deterministic version with w3 = 0, = 0:1901. Incidentally, note that in all
these simulations V13 < 0 so that the e¤ects of last period’s fertility on current fertility are
ambiguous.
The simulation consists of o¤ BGP dynamics, taking the fertility deviation in and around
the 1930s from the baseline. That is, everything plotted until 1940 is exogenous here and
we are interested in the baby boom and ‡uctuations thereafter.
Figures 13 and 14 show CT F R for all cohorts for time and goods costs, respectively,
while Figures 15 and 16 show T F R. As can be seen, magnitudes don’t di¤er much between
the deterministic model with two versus three periods of working life. Hence, most of the
di¤erence to the baseline results are due to the lack of additional productivity shocks, reinforcing the baby boom (and hence subsequent ‡uctuations). However, the baby boom
happens slightly earlier. This is less of an issue of length of the period as it is one of length
of generations. In the three period model, parents and children are 15, instead of 20 years
apart. However, interpreting the model as having 20-year periods, would imply that people
are productive until age 80.
In sum, then the dampened oscillations are not an artefact of having two periods working
life but remain even in the three period case.
17
Note that in the deterministic version of the model, whether costs are in terms of time or goods is
irrelevant.
33
Figure 13: Percent Deviations in CTFR, 3PWL, Time Cost
0.4
Percent Deviations CTFR
0.3
0.2
Deterministic 3
Deterministic 2
Baseline
CTFR: Data
0.1
0
-0.1
-0.2
-0.3
-0.4
1900
1920
1940
1960
1980
Birth Year of Mother + 23 years
2000
Figure 14: Percent Deviations in CTFR, 3PWL, Goods Cost
0.4
Percent Deviations CTFR
0.3
0.2
Deterministic 3
Deterministic 2
Baseline
CTFR: Data
0.1
0
-0.1
-0.2
-0.3
-0.4
1900
1920
1940
1960
1980
Birth Year of Mother + 23 years
34
2000
Figure 15: Percent Deviations in TFR, 3PWL, Time Cost
0.4
Percent Deviations TFR
0.3
0.2
Deterministic 3
Deterministic 2
Baseline
TFR: Data
0.1
0
-0.1
-0.2
-0.3
-0.4
1900
1920
1940
1960
1980
2000
Year
Figure 16: Percent Deviations in CTFR, 3PWL, Goods Cost
0.4
Percent Deviations TFR
0.3
0.2
Deterministic 3
Deterministic 2
Baseline
TFR: Data
0.1
0
-0.1
-0.2
-0.3
-0.4
1900
1920
1940
1960
Year
35
1980
2000
5.5
Parameter sensitivity
Next, we compute how elasticities and fertility levels change when we change parameter
values in Table 2. Results are reported in Table 6. For each parameter change, we report
two cases. First, we give results in the case where costs of children remain at their baseline
value, which leads to varying fertility levels in steady state. Second, we recalibrate to the
target population growth by adjusting costs of children accordingly. Results show that
within a reasonable range of parameter values, the elasticity of contemporaneous fertility to
productivity shocks ranges from 1.6 to 1.9 in the goods cost case and 0.95 to 1.2 for the time
cost case, while the elasticity of lagged fertility ranges from -1.25 to -1.9 in the goods cost
case and -0.7 to -1.2 for the time cost case, whether we recalibrate or not. Thus, the results
don’t seem to be very sensitive to parameter choices.
A few interesting cases are worth mentioning, however. First, we know from the theory
that the choice of is important to determine the contemporaneous fertility response to
productivity shocks. For a = 2:5, the result is still signi…cant but we know that it tends
to zero as ! 1.
Second, since in the detrended version of the model, the e¤ective discount factor is 1 ,
an increase in has the same e¤ect as a decrease in as long as > 1. As expected, elasticities change accordingly. The more patient people are, the less they respond to productivity
shocks. For very low values of , of 0.6 percent per year, the recalibration exercise reveals
a non-linearity. In this case, the response becomes very large with an elasticity of 2.2 in
current fertility and -2.4 one period later. From Jones and Schoonbroodt (2010b) we know
that as long as > 1, a decrease in productivity growth generates an increase in fertility.
Hence, in the recalibration exercise costs are required to increase signi…cantly (to about 30
percent of a person’s time) in order to hit the target fertility level. This may be the reason
for the high elasticity.
Third, as mentioned in the previous subsection, LP grows at 2.3 percent per year on
average over the 20th century and the estimated standard deviation is 0.8. Table 6 shows
elasticities for = 1:02620 and s = 8. For = 1:02620 , the elasticity of current fertility is
1.2 to 1.9 depending on the cost structure and lagged fertility is -1.1 to -1.8. For s = 0:08,
the elasticities are only very slightly higher than those in the baseline. Hence, the e¤ects of
LP shocks shown above would be higher if we changed and s accordingly.
One important parameter is the relative wage of old to young workers because it determines the e¤ective depreciation rate into retirement. Since w1 was normalized to 1, we
only move w2 : In the case where w2 = 0; the model exhibits no dependence on last period’s
fertility because the depreciation rate is the same every period, 100 percent. Table 6 shows
how fast the lagged e¤ect disappears as we set w2 to 1.5, 1 and 0.5.
36
6
International Evidence
In this section, we show that countries with a smaller Depression in the 1930s also tend to
have smaller baby busts and that those with smaller baby busts tend to have smaller baby
booms in the 1950s, conditional on the economic circumstances in the 1950s. Before we start,
several data related issues should be addressed. These relate to data availability, detrending
methods and dealing with war periods. For details on data sources and detrending methods,
see Appendix A.5.
Ideally, we would like to use the same measures of fertility and productivity as we do for
the U.S.. However, for the 20th century complete TFR series and consistent TFP measures
are only available for few countries (see e.g., Chesnais (1992) for TFR and Cette, Kocoglu,
and Mairesse (2009) for TFP). We therefore use Crude Birth Rates (CBR) mostly from
Mitchell (1998) and GDP per capita from Barro and Ursua (2008), instead . This gives us
17 countries for which we have data from 1900 until 1993.18
As to detrending, for CBR, we use the same detrending method as we did for TFR in
the U.S., namely a Hodrick-Prescott (HP) …lter. While for the U.S. a linear detrending
method for log-TFP poses few problems, this is not true for all countries we consider. In
particular, countries such as Portugal or Japan show a permanent negative deviation early
in the century and a permanent positive one later on. Besides the linear detrending method,
we therefore also use an HP …lter in comparison. CBR and GDP deviations in the 1930s
and 1950s are shown in Tables 12 and 14.
Furthermore, the two world wars a¤ected fertility levels in several ways that are unrelated
to our mechanism. For example, men being away to …ght prevents some women from having
children. Typically, there is rapid catching up as soon as the men return, if they do (e.g.,
see the period 1943-1946 in U.S. data shown in Figure 3). The longer the war involvement
and the larger the death rate of soldiers, the more unpredictable this e¤ect on fertility rates
becomes. Similar issues emerge for GDP data during war periods. For example, a large
part of the increase in output in Germany during the war was not for private consumption
but rather war related. We therefore also use dummies for war and immediately after war
periods to adjust our measures of deviations in CBR and GDP in comparison to deviations
resulting from a simple linear trend and the HP …lter. Resulting CBR and GDP deviations
in the 1930s and 1950s are shown in Tables 13 and 15.
Given these provisos, we are now ready to analyze sizes of depressions, baby busts and
baby booms. First, we aim to determine the correlation between the size of the Depression
in the 1930s and the size of the baby bust at the time across countries. Since the timing of
the Depression was di¤erent across countries, we divide the 1930s into two periods: 1930 to
1934 and 1935 to 1939. We then run eight cross-country OLS regressions, one for each time
bti denote the percent deviation
period combined with one of four detrending methods. Let X
18
The main issues are. CBR as opposed to TFR is sensitive to mortality in certain groups of the population
such as wars which a¤ect total population but not the number of fertile women. GDP per capita as opposed
to TFP or LP is only an indirect measure of the productivity of the labor force and therefore corresponds
less well to our model.
37
from trend in variable X in country i in …ve-year period t:
i
[t =
CBR
1;t
+
i
\
2;t GDP t
+ "i
where t = f30 34; 35 39g: The results for each detrending method are given in Table 7.
As can be seen, the coe¢ cient, 2;t , is positive for all detrending methods and both time
periods. It is larger (with a lower p-value) whenever an HP …lter is used rather than linear
detrending and whenever the e¤ects of wars are taken into account in detrending. Also, for
most detrending methods, 2;t is larger in the early 1930s than in the late 1930s.
Second, we aim to determine whether a larger baby bust in the 1930s implies a larger
baby boom in the 1950s, controlling for the GDP deviation in the 1950s. To do this, we
run eight cross-country OLS regressions, one for each time period combined with one of four
detrending methods.
i
[t =
CBR
3;t
+
i
[
4;t CBRt 20
+
i
\
5;t GDP t
+ "i
where t = f50 54; 55 59g: The results for each detrending method are given in Table 8.
As can be seen, the coe¢ cient, 4;t , is negative for all detrending methods and both
time periods. It is larger in absolute value (with a lower p-value) whenever an HP …lter
is used rather than linear detrending and, for the late 1950s, whenever the e¤ects of wars
are taken into account in detrending. Also, 4;t is larger in the early 1930s than in the late
1930s when the e¤ects of wars are not taken into account in detrending, smaller when they
are. Similar to 2;t , the coe¢ cient 5;t is positive for all detrending methods and both time
periods. However, p-values are larger, especially in the early 1950s.
In sum then, international evidence on the sizes of depressions and baby busts in the
1930s strongly supports our theory while the evidence for the size of the subsequent baby
boom, though rightly signed, is somewhat weaker.
7
Concluding remarks
TBW
38
A
Appendix
A.1
Transformation of the value function in Section 3.3
V (N 1 ; N 2 ; A)
s.t.
T
maxN 10 ;C
C + (s)N 1
C1
T
0
s
0
N 2 = N 1.
0
) + E V (N 1 ; N 20 ; s0 )js
=(1
P2
wa N a ;
a=1
0
The homogeneity property and the fact that N 2 is a function of s only implies:
1
(N 2 )
s.t.
V (N 1 =N 2 ; 1; s)
T
C + (s)N 1
0
N 2 = N 1.
Letting c =
1
(N 2 )
C
,
N2
T C1
maxN 10 ;C
0
s
P2
1
0
0
E V (N 1 =N 2= ; 1; s0 )js
we can write this as
V (N 1 =N 2 ; 1; s)
maxN 10 ;c
T
1
T
N2
)+
wa N a ;
a=1
+ (N 2 )
s.t.
=(1
N2
c2 + (s)N 1
0
P2
s
0
(c2 )1
1
20
N
N2
=(1
)
0
0
E V (N 1 =N 2 ; 1; s0 )js
wa N a ;
a=1
N 2 = N 1.
1
(N 2 )
Or,
1
(N 2 )
V (N 1 =N 2 ; 1; s)
maxN 10 ;c
T
1
+ (N 2 )
s.t.
T
N2
10
N1
N2
N2
c2 + (s) NN 1
1
(N 2 )
N1
N2
1
(c2 )1
=(1
)
0
E V (N 1 =N 1 ; 1; s0 )js
h
i
1
2
2 2
s w1 N
N
+
w
N
:
N2
Or,
1
(N 2 )
V (N 1 =N 2 ; 1; s)
maxN 10 ;c
s.t.
T
N2
1
(N 2 )
T
10
N1
N2
N2
c2 + (s) NN 1
(c2 )1
=(1
)+
h
i
1
2
2 2
s w1 N
N
+
w
N
:
N2
39
N1
N2
1
0
E V (N 1 =N 1 ; 1; s0 )js
1
Clearly, the (N 1 ; c) that solve the RHS of this identity are unchanged if the (N 2 )
is
removed from the OBJ and the N 2 is removed from the constraint. Thus, the following
functional equation must be satis…ed by V we have:
V (N 1 =N 2 ; 1; s)
s.t.
maxN 10 ;c
10
c2 + (s) NN 1
T
T
N1
N2
1
Also, if V solves this then (N 2 )
10
b
(c2 )1
=(1
)+
N1
N2
1
0
E V (N 1 =N 1 ; 1; s0 )js
h
i
1
2
s w1 N
+
w
:
N2
V solves the original BE.
1
N
N
Let n0 = N
be fertility per household (multiply by 2 to get fertility
and n = N
1 =
N2
N 20
per woman as in CTFR) as in the original utility function***. Accordingly, de…ne Ve (n; s)
V (n; 1; s). Then, it is enough to …nd the function Ve (n; s) that solves:
h
h
ii
Ve (n; s) max 0 ;c
T (c2 )1 =(1
) + n1 E Ve (n0 ; s0 )js
s.t.
c2 + (s)n0 n
T
s [w1 n + w2 ] :
h
i
h
i
0 0
0 0
e
e
Assuming that the s are i.i.d. simpli…es this in that E V (n ; s )js = E V (n ; s ) , so
that we have:
h
h
ii
Ve (n; s) maxn0 ;c
T (c2 )1 =(1
) + (n)1 E Ve (n0 ; s0 )
s.t.
c2 + (s)n0 n
T
s [w1 n + w2 ] :
Let Vb (n; s) = Ve (n; s)=T; b(s) = (s)=T; w
b1 = w1 =T; w
b2 = w2 =T to get:
h
h
ii
Vb (n; s) maxn0 ;c
(c2 )1 =(1
) + n1 E Vb (n0 ; s0 )
s.t.
c2 + b(s)n0 n
s [w
b1 n + w
b2 ] :
Taking FOC with respect to n0 in (be) gives
(F OCn0 )
(BC)
(s)
n
T
= n1
s[w1 n+w2 ]
T
(s)n0 n
(c2 )
c2 =
E Vb1 (n0 ; s0 )
or
(F OCn0 )
(s)
n
T
=
s[w1 n+w2 ]
T
(s)n0 n
n1
E V^1 (n0 ; s0 )
or
(F OCn0 )
(s)
n
T
=
s[w1 n+w2 ]
T
(s)n0 n
E V^1 (n0 ; s0 )
40
A.2
Transformation of the value function and …rst-order conditions in Section 5.3: F (K; w1 N 1 + w2 N 2 ) = w1 N 1
In the model presented in Section 3, suppose w2 = 0.
In this case, since w2 = 0 by assumption, we have that V (N 1 ; N 2 ; s) = V (N 1 ; N 2 ; s) for
any N 2 ; N 2 . Because of this, we will write V as depending only on (N 1 ; s).
The Bellman equation becomes:
Ve (N 1 ; s)
s.t.
maxC;N b T
TC + Nb
C1
1
+
sw1 N 1 ;
h
E Ve (N 10 ; s0 )js
i
N a0 = N a 1 , for a = 2; 3;
N 10 = N b .
Assuming further that shocks are i.i.d. and using the homogeneity property of V , the
BE is equivalent to:
h
i
1
+ (N 10 )1 E Ve (1; s0 )js
(N 1 )1 Ve (1; s) maxC;N b T C1
s.t.
TC + Nb
sw1 N 1 ;
N a0 = N a 1 , for a = 2; 3;
N 10 = N b .
Dividing the BE by T (N 1 )1
V (s)
maxc1 ;n0
s.t.
c1 + bn0
(c1 )1
1
maxn0
(n0 )1
+
E V (s0 )js
sw
b1 ;
where V (s) = Ve (1; s)=T .
Substituting out c1 , we get
V (s)
and the budget constraint by T N 1 , we get:
[sw
b1 bn0 ]1
1
+
(n0 )1
E V (s0 )js
Taking the …rst-order condition with respect to n0 , we get:
[sw
b1
bn0 ]
b = (1
)E[V (s0 )](n0 )
Solving for n0 as a function of s, we get equation 4 in the main text.
41
A.3
Transformation of the value function and …rst-order conditions in Section 5.3: F (K; w1 N 1 + w2 N 2 ) = K (w1 N 1 )1
A.4
Sensitivity Tables: Cohort Shocks, CTFR and TFR
42
A.5
A.5.1
International Evidence: Details
Data Sources
CBR data form 1900 to 1993 are obtained from Mitchell (1998) for all countries, except
Germany. German data was combined using Chesnais (1992, Appendix A1) for the 1900 to
1943 period and Statistisches Bundesamt (2007) for 1946 to 1993, linearly interpolating 1944
and 1945. All GDP per capita data comes from Barro and Ursua (2008) and is in millions
1990 International Geary-Khamis dollars.
A.5.2
Detrending: Linear OLS and HP
CBR deviations reported in Table 12, columns 1 and 2, and Table 14, columns 1 to 4, are
obtained by running an HP …lter (smoothing parameter, w = 20; 000) on annual CBR data
for each country with implied annual deviations averaged over …ve years.
GDP deviations reported in Table 12, columns 3 and 4, and Table 14, columns 5 and
6, are obtained by running an OLS regression of the form ln GDPti = 0i + 1i t + "it using
annual GDP per capita
data for eachi country, i, with …ve-year deviations computed as
h
i
P
Ps+4 "i
i
i
\s (s+4) = exp ln
t
t
where i = E(ln( s+4
GDP
t=s e )).
t=s e
GDP deviations reported in Table 12, columns 5 and 6, and Table 14, columns 7 and 8,
are obtained by running an HP …lter (smoothing parameter, w = 20; 000) on ln GDP per
capita data (smoothing parameter, w = 20; 000) for each country, i, with …ve-year deviations
computed as above using HP residuals.
A.5.3
Detrending: War Dummies
To account for the e¤ect of wars in Table 13 and Table 15, we …rst create four dummy
variables, dik , k = 1; :::4, for the time periods t = f1 : 1914 1918; 2 : 1919 1923; 3 :
1940 1945; 4 : 1946 1950g for each country, i. We then run the following time series
regression on annual data for X i = fGDP i ; CBRi g for each country, i.
ln Xti
=
i
0
+
i
1t
+
4
X
i
i
k+1 dkt
+ "it
k=1
i
e i = exp ln X i P4
We then compute X
t
t
k=1 k+1 dkt . The deviations reported in Table 13
and Table 15 are then calculated exactly as those in Table 12 and Table 14, described in
eti instead of Xti .
Section A.5.2 except that we use X
A.5.4
Observations about Cross-Country Busts and Booms
From Tables 12 and 13 we make the following observations. Countries that, similar to
the U.S., have large depressions and large baby busts throughout the 1930s are Canada,
Australia and New Zealand. Countries with slightly smaller depressions and baby busts
43
are Germany, the U.K. and France. Note that France has a late depression (late 1930s)
and correspondingly, a late baby bust while Germany and the U.K. have large depressions
in the early 1930s and correspondingly large baby busts early on. For the Netherlands,
Belgium and Switserland, the depression is noticeable only if the e¤ects of wars are taken
into account when detrending (Table 13). For Southern European countries, Italy, Portugal
and Spain, both the depression as well as the baby bust are small or non-existent, depending
on detrending method. For Northern European countries, Denmark, Norway and Sweden,
large baby busts can be observed, despite the fact that the depression is small or non-existent
depending on detrending method. These observations rationalize results in Table 7.
Because of the last observation, when analyzing baby booms in the main text, we use
[
\1930s as the independent variable. This allows us to determine
CBR1930s rather than GDP
whether baby busts are correlated with subsequent baby booms, independently of their cause.
Indeed, from Tables 14 and 15, all three Northern countries display substantial baby
booms in the 1950s, though potentially mitigated by a recession in the 1950s. Similarly,
Central European and Southern European countries, all have small baby booms in the late
1950s with GDP signi…cantly below trend for most detrending methods. Southern European
countries have somewhat of a baby boom. The largest baby booms are to be observed in
the U.S., Canada, Australia and New Zealand where the baby bust was particularly large.
These observations rationalize the weaker results in Table 8.
44
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Table 5: Dynasties and Cohorts, Baseline
Cohort
Label
Birthyear
Label
Fertile
Period
Productivity
Shock
State var., n
Dynasty
Fertility
Choice
Label
Value
d1
d2
d3
d4
...
...
...
...
nss
n c1
n c2
n c3
n c4
...
...
...
n c4
n c5
n c6
n c7
n c8
...
...
...
n c8
n c9
nc10
nc11
nc12
...
...
...
nc12
nc13
nc14
nc15
nc16
13.61
7.93
1.68
0.75
-17.76 -11.34
-19.82 -12.48
...
...
...
nc16
nc17
nc18
nc19
nc20
-18.31 -11.31
-11.58 -6.83
4.25
2.67
5.10
2.90
c1
c2
c3
c4
1880
1885
1890
1895
1901-10
1906-15
1911-20
1916-25
s1
s2
s3
s4
0.50
-2.49
-0.45
4.48
nss
...
...
...
...
nss
c5
c6
c7
c8
1900
1905
1910
1915
1921-30
1926-35
1931-40
1936-45
s5
s6
s7
s8
3.02
-5.74
-11.19
-0.29
n c1
...
...
...
...
n c2
c9
c10
c11
c12
1920
1925
1930
1935
1941-50
1946-55
1951-60
1956-65
s9
s10
s11
s12
7.21
7.14
8.40
10.46
n c5
...
...
...
...
n c6
c13
c14
c15
c16
1940
1945
1950
1955
1961-70
1966-75
1971-80
1976-85
s13
s14
s15
s16
12.71
10.45
5.37
-0.17
n c9
...
...
...
...
nc10
c17
c18
c19
c20
1960
1965
1970
1975
1981-90
1986-95
1991-00
1996-00
s17
s18
s19
s20
-3.14
-4.49
-6.67
-3.15
nc13
...
...
...
...
nc14
48
...
...
...
...
...
nss
...
...
n c3
...
...
n c7
...
...
nc11
...
...
nc15
...
Label Goods
Time
-1.57
-6.71
-3.20
5.25
-0.94
-4.06
-1.93
3.12
4.23
2.52
-6.20 -3.79
-19.05 -11.79
-7.41 -4.50
6.04
16.17
34.88
23.43
3.55
9.50
19.68
13.56
Table 6: Impulse Response (in percent and levels), Parameter Sensitivity
Cost Type
Perc. Dev.
CTFR Lev.
Perc. Dev. CTFR Lev.
Cost
Init.
Baseline
Goods
Time
( = 3;
1.74
1.04
Sensitivity
= 3:5
Goods
Time
= 2:5
Goods
Time
= 0:9720
Goods
Time
= 0:9520
Goods
Time
= 1:02620
Goods
Time
= 1:00620
Goods
Time
s = 0:08
Goods
Time
s = 0:06
Goods
Time
w2 = 1:5
Goods
Time
w2 = 1
Goods
Time
w2 = 0:5
Goods
Time
Lag.
Init.
= 0:9620 ;
-1.56
-0.94
2.31
2.30
Lag.
SS
= 1:01620 ;
2.24
2.25
Init.
s
2.27
2.27
Lag.
Init.
Lag.
(Rec.)
= 0:07; w1 = 1; w2 = 1:25)
1.74
1.04
No Recalibration
-1.56
-0.94
2.31
2.30
2.24
2.25
0.193
0.193
Recalibration
1.81
1.16
-1.73
-1.12
2.17
2.16
2.10
2.11
2.13
2.14
1.83
1.20
-1.72
-1.13
2.32
2.30
2.24
2.25
0.150
0.150
1.66
0.88
-1.36
-0.73
2.52
2.50
2.44
2.46
2.48
2.48
1.66
0.86
-1.41
-0.74
2.31
2.29
2.24
2.26
0.246
0.247
1.68
0.99
-1.40
-0.83
2.47
2.46
2.40
2.42
2.43
2.44
1.65
0.96
-1.40
-0.82
2.31
2.30
2.24
2.26
0.242
0.243
1.81
1.09
-1.75
-1.06
2.15
2.14
2.08
2.10
2.12
2.12
1.82
1.11
-1.70
-1.04
2.32
2.30
2.24
2.25
0.151
0.152
1.88
1.14
-1.93
-1.18
2.03
2.02
1.95
1.97
1.99
2.00
1.88
1.17
-1.82
-1.13
2.32
2.30
2.23
2.25
0.122
0.123
1.61
0.94
-1.25
-0.73
2.63
2.62
2.55
2.57
2.59
2.59
2.22
1.29
-2.38
-1.48
2.32
2.30
2.22
2.24
0.294
0.295
1.75
1.04
-1.57
-0.94
2.31
2.30
2.24
2.26
2.27
2.28
1.75
1.04
-1.57
-0.94
2.31
2.30
2.24
2.25
0.193
0.194
1.74
1.04
-1.57
-0.94
2.31
2.30
2.23
2.25
2.27
2.27
1.74
1.04
-1.56
-0.94
2.31
2.30
2.24
2.25
0.191
0.192
1.84
1.11
-1.88
-1.15
2.33
2.32
2.25
2.27
2.29
2.30
1.83
1.10
-1.87
-1.14
2.32
2.30
2.23
2.25
0.198
0.199
1.61
0.95
-1.21
-0.71
2.28
2.27
2.22
2.24
2.25
2.25
1.61
0.95
-1.20
-0.71
2.31
2.30
2.25
2.26
0.186
0.187
1.29
0.73
-0.51
-0.29
2.23
2.23
2.19
2.20
2.21 1.30
2.21 0.73
49
-0.51
-0.29
2.30
2.29
2.26
2.27
0.176
0.177
Table 7: Cross-Country: Depressions and Busts, 1930s
Dependent
Independent
Variable
Variable
i
[ 30
CBR
34
i
[ 35
CBR
39
Detrending Method
Linear
HP
Linear
No war
HP
No war
Constant ( 1;30 34 )
i
\30 34 ( 2;30 34 )
GDP
(p-value)
-0.0522
-0.0839
-0.0365
-0.0446
0.3512
(0:059)
0.4253
(0:008)
0.3933
(0:084)
0.6480
(0:008)
Constant ( 1;35 39 )
i
\35 39 ( 2;35 39 )
GDP
(p-value)
-0.0856
-0.1211
-0.0716
-0.0806
0.2118
(0:441)
0.5743
(0:005)
0.2482
(0:335)
0.5554
(0:004)
Table 8: Cross-Country: Busts and Booms, 1950s vs. 1930s
Dependent
Independent
Variable
Variable
i
[ 50
CBR
54
i
[ 55
CBR
59
Constant ( 3;50 54 )
i
[ 30 34 ( 4;50 54 )
CBR
(p-value)
i
\50 54 ( 5;50 54 )
GDP
(p-value)
Constant ( 3;55 59 )
i
[ 35 39 ( 4;55 59 )
CBR
(p-value)
i
\55 59 ( 5;55 59 )
GDP
(p-value)
Detrending Method
Linear
HP
Linear
No war
HP
No war
0.0477
0.0203
0.0853
0.0506
-0.2795
(0:229)
-0.3680
(0:058)
-0.1527
(0:634)
-0.2792
(0:372)
0.2181
(0:171)
0.3316
(0:092)
0.2571
(0:208)
0.2034
(0:428)
0.0960
0.0296
0.0680
0.0296
-0.0973
(0:714)
-0.2854
(0:181)
-0.4048
(0:143)
-0.5748
(0:020)
0.6052
(0:012)
1.0424
(0:004)
0.4029
(0:045)
0.4887
(0:046)
50
Table 9: Productivity Shocks (in percent) for Experiments
TFP
Cohort
LP
Baseline
Age 20-40
c1
c2
c3
c4
-0.90
-3.85
-1.84
3.02
-1.10
-0.14
0.14
-1.75
5.30
-0.63
-6.04
-4.55
c5
c6
c7
c8
1.58
-7.06
-12.43
-1.68
-5.17
-4.11
-3.09
2.26
-2.00
-6.41
-10.84
-4.79
c9
c10
c11
c12
5.72
5.65
6.89
8.92
6.59
7.57
9.31
9.21
1.65
4.80
8.86
11.60
c13
c14
c15
c16
11.14
8.91
3.90
-1.56
7.81
3.95
-0.03
-3.43
13.17
11.59
7.73
2.39
c17
c18
c19
c20
-4.49
-5.82
-7.97
-8.86
-5.97
-6.58
-7.71
-8.62
-2.25
-6.44
-10.69
-12.48
51
Age 20-30
Table 10: CTFR response (in percent) from Experiments
Baseline
Cohort
TFP, age 20-40
LP, age 20-30
Goods
Time
Goods
Time
Goods
c1
c2
c3
c4
-1.57
-6.71
-3.20
5.25
-0.94
-4.06
-1.93
3.12
-1.92
-0.25
0.25
-3.05
-1.15
-0.15
0.15
-1.83
9.22
-1.09
-10.54
-7.94
5.44
-0.65
-6.42
-4.81
c5
c6
c7
c8
4.23
-6.20
-19.05
-7.41
2.52
-3.79
-11.79
-4.50
-7.31
-6.94
-5.61
6.87
-4.43
-4.20
-3.39
4.07
-11.05
-10.24
-9.30
-0.85
-6.77
-6.23
-5.78
-0.56
c9
c10
c11
c12
6.04
16.17
34.88
23.43
3.55
9.50
19.68
13.56
19.12
20.47
22.10
9.63
11.12
11.89
12.80
5.60
14.36
19.23
25.56
21.02
8.40
11.16
14.86
12.23
c13
c14
c15
c16
13.61
1.68
-17.76
-19.82
7.93
0.75
-11.34
-12.48
-2.17
-9.23
-16.58
-13.73
-1.57
-5.85
-10.41
-8.38
9.96
3.78
-6.55
-12.08
5.66
1.96
-4.44
-7.63
c17
c18
c19
c20
-18.31
-11.58
4.25
5.10
-11.31
-6.83
2.67
2.90
-8.51
-2.79
3.30
-1.74
-4.93
-1.52
1.94
-1.26
-12.02
-14.36
-12.93
-10.79
-7.21
-8.55
-7.52
-6.50
52
Time
Table 11: TFR response (in percent) from Experiments
Baseline
Time Period
TFP, age 20-40
LP, age 20-30
Goods
Time
Goods
Time
Goods
1901-05
1906-10
1911-15
1916-20
-0.78
-4.14
-4.96
1.03
-0.47
-2.50
-2.99
0.60
-0.48
-0.54
-0.48
-1.24
-0.29
-0.32
-0.29
-0.75
4.61
4.06
-5.82
-9.24
2.72
2.39
-3.54
-5.61
1921-25
1926-30
1931-35
1936-40
4.71
-0.84
-12.74
-12.99
2.80
-0.59
-7.84
-8.06
-2.72
-4.43
-5.80
-3.31
-1.65
-2.69
-3.51
-2.01
-9.66
-10.66
-9.79
-5.01
-5.90
-6.51
-6.02
-3.13
1941-45
1946-50
1951-55
1956-60
-0.34
10.70
25.01
28.54
-0.25
6.38
14.42
16.42
3.46
10.44
17.37
17.61
2.06
6.15
10.12
10.28
6.63
16.81
22.41
23.19
3.84
9.78
13.02
13.49
1961-65
1966-70
1971-75
1976-80
17.94
7.85
-8.23
-18.86
10.47
4.41
-5.37
-11.93
11.40
3.67
-5.41
-10.56
6.67
2.07
-3.46
-6.60
15.10
6.81
-1.52
-9.26
8.71
3.77
-1.32
-6.00
1981-85
1986-90
1991-95
1996-00
-19.00
-15.24
-4.35
4.70
-11.85
-9.19
-2.33
2.79
-11.99
-10.37
-5.55
-2.58
-7.35
-6.22
-3.20
-1.49
-12.05
-13.15
-13.68
-11.89
-7.43
-7.86
-8.06
-7.03
53
Time
Table 12: Depressions and Baby Busts across Countries
Country
[ 1930s
CBR
\1930s , Linear
GDP
\1930s , HP
GDP
1930-34 1935-39
1930-34 1935-39
1930-34 1935-39
U.S.
-0.167
-0.184
-0.264
-0.209
-0.206
-0.149
Canada
Australia
New Zealand
-0.127
-0.184
-0.181
-0.183
-0.185
-0.169
-0.256
-0.211
-0.243
-0.254
-0.124
-0.090
-0.179
-0.150
-0.177
-0.166
-0.043
-0.010
U.K.
Germany
France
-0.123
-0.137
-0.005
-0.111
0.087
-0.134
-0.115
-0.296
-0.014
-0.059
-0.076
-0.176
-0.058
-0.168
0.137
0.012
0.126
-0.001
Netherlands
Belgium
Switzerland
-0.084
0.009
-0.088
-0.121
-0.081
-0.135
0.016
0.039
0.017
-0.052
-0.046
-0.061
0.111
0.168
0.091
0.079
0.112
0.024
Italy
Spain
Portugal
-0.004
0.059
0.008
-0.003
-0.116
-0.031
-0.080
-0.041
-0.074
-0.060
-0.081
-0.178
0.019
0.039
0.086
0.094
0.030
0.007
Denmark
Norway
Sweden
-0.135
-0.184
-0.172
-0.094
-0.189
-0.128
0.027
-0.026
-0.083
-0.008
0.001
-0.014
0.093
0.040
-0.022
0.078
0.089
0.053
0.027
-0.010
-0.111
-0.083
0.029
0.138
Japan
54
Table 13: Depressions and Baby Busts across Countries, No War
Country
[ 1930s
CBR
\1930s , Linear
GDP
\1930s , HP
GDP
1930-34 1935-39
1930-34 1935-39
1930-34 1935-39
U.S.
-0.144
-0.163
-0.258
-0.202
-0.192
-0.129
Canada
Australia
New Zealand
-0.104
-0.170
-0.166
-0.158
-0.173
-0.147
-0.252
-0.211
-0.250
-0.251
-0.124
-0.098
-0.177
-0.152
-0.191
-0.165
-0.047
-0.032
U.K.
Germany
France
-0.137
-0.189
-0.047
-0.126
0.027
-0.160
-0.111
-0.305
-0.107
-0.055
-0.086
-0.251
-0.055
-0.178
-0.041
0.016
0.129
-0.181
Netherlands
Belgium
Switzerland
-0.065
-0.062
-0.101
-0.094
-0.141
-0.139
-0.064
-0.088
-0.036
-0.124
-0.156
-0.107
-0.035
-0.043
0.001
-0.085
-0.105
-0.067
Italy
Spain
Portugal
-0.021
0.046
0.013
-0.013
-0.126
-0.024
-0.093
-0.059
-0.119
-0.076
-0.098
-0.217
-0.028
0.003
-0.005
0.017
-0.011
-0.090
Denmark
Norway
Sweden
-0.126
-0.163
-0.147
-0.073
-0.169
-0.097
-0.016
-0.055
-0.106
-0.047
-0.028
-0.038
0.015
-0.017
-0.065
-0.011
0.018
0.004
0.080
0.052
-0.112
-0.087
0.010
0.100
Japan
55
Table 14: Baby Busts and Baby Booms across Countries
Country
[ 1950s
CBR
[ 1930s
CBR
\1950s , Linear
GDP
\1950s , HP
GDP
1950-55 1955-60
1930-35 1935-40
1950-55 1955-60
1950-55 1955-60
U.S.
0.144
0.167
-0.167
-0.184
0.049
0.009
0.072
0.016
Canada
Australia
New Zealand
0.155
0.102
0.093
0.198
0.097
0.121
-0.127
-0.184
-0.181
-0.183
-0.185
-0.169
-0.040
-0.085
0.006
-0.037
-0.077
0.057
0.033
-0.013
0.030
0.013
-0.022
0.056
-0.024
0.006
0.103
0.016
0.065
0.052
-0.123
-0.137
-0.005
-0.111
0.087
-0.134
-0.091
-0.314
-0.177
-0.068
-0.119
-0.112
-0.026
-0.172
-0.003
-0.014
0.017
0.025
0.039
0.060
0.013
0.045
0.100
0.042
-0.084
0.009
-0.088
-0.121
-0.081
-0.135
-0.152
-0.111
-0.104
-0.081
-0.102
-0.035
-0.021
0.040
-0.039
0.026
0.015
0.010
Italy
Spain
Portugal
-0.071
-0.066
-0.026
-0.042
0.018
0.025
-0.004
0.059
0.008
-0.003
-0.116
-0.031
-0.264
-0.133
-0.268
-0.177
-0.157
-0.237
-0.106
0.024
-0.087
-0.034
-0.013
-0.085
Denmark
Norway
Sweden
0.131
0.068
0.014
0.039
0.056
-0.025
-0.135
-0.184
-0.172
-0.094
-0.189
-0.128
-0.106
-0.061
-0.029
-0.103
-0.066
-0.016
-0.029
0.024
0.001
-0.049
0.002
-0.007
-0.020
-0.179
0.027
-0.010
-0.443
-0.353
-0.251
-0.168
U.K.
Germany
France
Netherlands
Belgium
Switzerland
Japan
56
Table 15: Baby Busts and Baby Booms across Countries, No War
Country
[ 1950s
CBR
[ 1930s
CBR
\1950s , Linear
GDP
\1950s , HP
GDP
1950-55 1955-60
1930-35 1935-40
1950-55 1955-60
1950-55 1955-60
U.S.
0.168
0.187
-0.144
-0.163
0.059
0.019
0.100
0.037
Canada
Bustralia
New Zealand
0.189
0.114
0.128
0.228
0.107
0.150
-0.104
-0.170
-0.166
-0.158
-0.173
-0.147
-0.038
-0.086
-0.003
-0.035
-0.078
0.048
0.034
-0.017
0.009
0.013
-0.025
0.038
-0.035
-0.026
0.105
0.009
0.040
0.058
-0.137
-0.189
-0.047
-0.126
0.027
-0.160
-0.089
-0.316
-0.243
-0.066
-0.120
-0.180
-0.021
-0.155
-0.170
-0.009
0.037
-0.112
0.083
0.021
0.025
0.083
0.073
0.054
-0.065
-0.062
-0.101
-0.094
-0.141
-0.139
-0.209
-0.195
-0.140
-0.140
-0.179
-0.071
-0.160
-0.134
-0.110
-0.091
-0.120
-0.049
Italy
Spain
Portugal
-0.066
-0.073
-0.016
-0.035
0.012
0.034
-0.021
0.046
0.013
-0.013
-0.126
-0.024
-0.281
-0.147
-0.298
-0.199
-0.170
-0.267
-0.182
-0.013
-0.168
-0.101
-0.042
-0.150
Denmark
Norway
Sweden
0.180
0.087
0.047
0.079
0.070
0.002
-0.126
-0.163
-0.147
-0.073
-0.169
-0.097
-0.138
-0.086
-0.048
-0.133
-0.091
-0.035
-0.103
-0.039
-0.037
-0.106
-0.048
-0.036
Japan
0.053
-0.126
0.080
0.052
-0.449
-0.361
-0.286
-0.200
U.K.
Germany
France
Netherlands
Belgium
Switzerland
57