Baby Busts and Baby Booms:
A Cross-Country Study of Fertility Responses
to Depressions and War Capital Build-Ups∗
Larry Jones
University of Minnesota
Alice Schoonbroodt
University of Southampton
February 2007
Preliminary draft
∗
The authors thank the National Science Foundation and the Spanish Ministry of Ed-
ucation, under grants BEC2002-04294-C02-01 and SEC2005-08793-C04-01 for financial
support. We also thank Michele Boldrin, V.V. Chari, Deborah Levison, Ellen R. McGrattan, Henry Siu and Michèle Tertilt for helpful comments.
1
Abstract
In the demography literature, the large 20th century fluctuations in fertility have
often been linked 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
these ideas in a stochastic growth model with endogenous fertility. In this model we
study the effects of productivity shocks as well as one-time shocks to capital-people
(labor) ratios (a combination of mortality, depreciation and war capital built-ups)
on fertility levels. We first show that a ‘catching-up’ phenomenon occurs. The idea,
though, is different from the one advanced by most demographers. There will be
catching up, not because of previously low fertility rates per se, but because of off
balanced growth dynamics in capital-people ratios. Next, we show that this model
exhibits pro-cyclical fertility in most cases. Using data on U.S. TFP and capitalpeople ratios before and after WWII, the model accounts for 70 to 85 percent of
the pre-WWII baby bust (mostly due to the negative productivity shock of the
Great Depression) and for 60 to 80 percent of the U.S. baby boom (mostly driven
by off balanced growth dynamics in capital-people ratios). In turn we analyze cross
country differences in sizes of depressions, baby busts as well as the effect of WWII
on respective capital-people ratios generating baby booms. Preliminary results
show that large and long-lasting depressions tend to be accompanied with below
average fertility before WWII and post-WWII capital-people ratios are positively
correlated with sizes of ensuing baby booms.
2
1
Introduction
The demographic transition is almost invariably rendered as a monotone, if
nonlinear, process along which countries move from high to low mortality
and fertility levels. During an intermediate phase, because fertility is still
high but mortality has already dropped, a possibly substantial increase in
population takes place. When looking at data through these spectacles,
almost the entire 20th century appears to constitute, for the United States
and for most other developed countries, a period of unusual deviations around
an otherwise declining trend in fertility: first, during 1920-1940, the baby
bust, an acceleration of the decrease in fertility, followed by an upswing –
the baby boom during 1940-1965 – and subsequently a second slump to even
lower fertility levels, 1965-1985.
Demographers often link these large fluctuations in fertility to the series of
economic ‘shocks’ that occurred with similar timing – the Great Depression,
WWII and the economic expansion that followed, and then the productivity
slow down of the 1970’s. 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), Freedman et al. (1959), Goldberg et al. (1959), and
Whelpton et al. (1966)).
These two hypotheses can be combined to yield the following: current
fertility is a stable function of productivity levels (or trends) and the current
stock of children/people in the economy. That is, unusual (relative to trend)
drops in income cause fertility to fall, other things equal; vice versa, unusual
(relative to trend) increases in income 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
3
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.
This paper aims at investigating the theoretical and quantitative implications of such a function in a 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.1
We first show that the model exhibits a ‘catching-up’ phenomenon in
a non-stochastic environment. The idea, though, is different from the one
advanced by most demographers. There is catching up, not because of previously low fertility rates per se. Instead, the intuition for the ‘catching-up’
result is as follows. Suppose we start on a balanced growth path (BGP).
That is, the capital to population ratio is at its optimal value. Then a (one
time unexpected) shock to both capital depreciation and mortality occurs.
If these shocks destroy different proportions of K and N, the (detrended)
K/N ratio is off its balanced growth path level. This implies a relatively
high return on either investments in capital or people (fertility). It follows
1
Although it seems unlikely that fertility decisions are affected by quarter to quarter
fluctuations in productivity (as addressed in the Business Cycle literature), longer fluctuations, 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 in the seventies, are likely to affect parents outlook on their children’s future well being, and therefore their current fertility decisions. In
the present context, temporary shocks should therefore be understood as extended swings
around a very long term trend.
4
that either investment or fertility will be relatively high for a while to bring
the K/N ratio back to its BGP value.
Applied to the baby boom following World War II throughout the developed world, we experiment with an initial quantitative version of the model
in which we trace out the Transition Dynamics in response to a ‘surprise’
shock to see the effects on fertility when some portion of N and/or K are
destroyed (deaths and depreciation). The idea is that this will be useful in
thinking about the fertility effects of a ‘shock’ such as WWII to fertility. In
particular, this has the potential of being of interest to study the differential
effects on the time path of fertility and hence the size of any baby boom
when the war happens on ones own soil and both N and K are destroyed
versus when the country sends people to a war but the destruction in K is
very limited. For example, in the U.S., there was virtually no destruction of
K and the baby boom was large, while in much of Europe, the destruction
of BOTH N and K was large and the baby boom was comparatively small.
The results are significant, but additional considerations, namely war capital
built ups, which we interpret as exogenous increases of the capital stock are
important.
Next we turn to a version of the model with stochastic productivity and
government investments in war capital. We show that if capital income and
labor income are taxed at the same rate and governments lead a balanced
budget, shocks to government spending financed through such taxation is
equivalent to productivity shocks. We then show that fertility and investment
are pro-cyclical.
As a benchmark, we then analyze the U.S. experience in the 20th century
through the spectacles of our model. First, we present data on war capital built up in the U.S. in the early 1940s. Next, we present fertility and
productivity fluctuations for the period 1910 to 1970. Here we derive the
necessary properties for numerical implementation. We then calibrate the
5
model as before and perform the following experiment. We use decade by
decade productivity shocks as calculated from the data. To this we add a
one time shock to government expenditure for war-capital build-up in the
1940s. A fraction of this capital is dropped back into the economy in the
1950s generating the exogenous increase in K/N. The great depression in
the 1930s and the sudden tax increase for government war capital built-up in
the early 1940s produce the downswing in fertility in the model. It accounts
for 2/3 of the one observed in the data. The increase in the K/N ratio in
the late 1940s and 1950s generates an upward swing in fertility of the order
of 65% of the observed Baby Boom. The fact that the baby Boom drags out
into the 1960s is accounted for by above trend productivity levels.
In Section 5 we use this theoretical mechanism to study cross-country
differences in sizes of pre-WWII depressions and the possibility of a linkage
between the destruction of people and property in WWII on the subsequent
baby booms around the world. In contrast to the U.S., Canada and Australia, in much of Europe, depressions were milder and both property and
people were destroyed in the war. The model predicts that the down-swing
in fertility as well as the baby boom resulting from this should be small,
or non-existent. Indeed, we find that the correlations between sizes of depressions, changes in capital-labor ratios and changes in Total Fertility Rate
(TFR, pre-WWII trough to post-WWII peak) are substantial.
The outline of the paper is as follows. In Section 2, we briefly review the
literature. In Section 3 we lay out the basic model, derive off-balanced growth
dynamics in the non-stochastic version and pro-cyclicality of fertility in the
stochastic version. Section 4 analyzes the U.S. experience as a benchmark
with data on war capital built-up, TFP and TFR fluctuations and numerical implementation. Section 5 shows preliminary results for cross-country
comparisons.
6
2
Literature Review
Let us briefly review the literature. The original demographic literature on
the Baby Bust-Boom-Bust was mostly qualitative in nature. Similar versions
of the ‘Catching Up’ can be found in examples such as Whelpton (1954),
Freedman et al. (1959), Goldberg et al. (1959), and Whelpton et al. (1966).
Among recent ones, perhaps the best known conjecture relating cyclical
movements in income and fertility is the one 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 (low) relative to individual aspirations, fertility is high (low).
Since the baby boom mothers grew up in bad times and therefore had low
material aspirations while they were making fertility decisions during good
times, i.e. expected lifetime income was high, they had many children, 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.
Operationally, low income for today’s generation implies high fertility for
the next generation, especially if its expected lifetime income is particularly
high.2
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 with high (low) fertility are, in fact, ‘dynamic variations’ on 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 fixed natural ratio
between available economic resources and population size. When income per
capita - i.e. labor productivity - increases above this natural level, fertility
2
One version of this would be to assume that there is habit formation in consumption,
as in much recent literature on asset pricing.
7
also increases until the long run ratio is re-established, and vice versa for
periods of economic crisis. Hence 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. In this view, periods of very harsh economic conditions, in which
per capita income decreases below the natural level, are also periods in which
fertility decreases.3
From the perspective of a theorist of economic growth, the Malthusian
view is cast in terms of a ‘stationary’ model, one in which there is no persistent growth in income per capita but there is an essential fixed factor,
e.g, land. In such a setting, summarized as Yt = F (L, Nt ) where L is the
time invariant stock of land and Nt is the time-varying population, standard
neoclassical properties imply that when Nt is below (above) its stable long
run level, N ∗ , the wage rate wt = FN (L, Nt ) increases above (below) its ‘natural’ level w ∗ = FN (L, N ∗ ).
Also, from a growth-theoretical perspective,
the view exemplified by the work of Easterlin describes a ‘growth’ model
Yt = F (Xt , Nt ), in which labor productivity wt = FN (Xt , Nt ) grows at a rate
γ t = ργ t−1 +εt , with εt a possibly random disturbance, and the ‘other inputs’
Xt are either not essential in production or fully reproducible. In general,
this can either be an exogenous growth model, in which the growth rate γ t is
determined outside the model, or an endogenous one, in which γ t is in fact a
function of the rate at which the reproducible ones among the inputs Xt are
accumulated. Either way, oscillations in population are mean reverting, to
the long run natural level in the first case and to the long run natural growth
rate in the second, at least as long as the growth rate of income tends toward
some long run average value.4
3
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 below.
4
Again, in both cases the crucial steps when taking the theory to the data consist in
8
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 filling this gap.
More recently, there have been two interesting studies on the baby boom
using Dynamic General Equilibrium (DGE) techniques. These are the papers
by Greenwood et al. (2005) and Doepke and Maoz (2005). Greenwood et al.
(2005) analyze a DGE model of fertility choice where the precipitating event
is a reduction in the cost of raising children in the post WWII period due
to an acceleration in the rate of productivity growth in the home. They
find that much of the upswing in the baby boom can be accounted for by
this mechanism in their model. Doepke and Maoz (2005) also use DGE
techniques but focus on the entry of women in the labor force during WWII
and the long lasting effect this had for that cohort due to the extra-normal
human capital that they accumulated from this experience. They also have
significant successes in matching the quantitative magnitudes of the boom.
In contrast, we focus on these same events as responses to medium run shocks
to the income process of a dynasty.
making additional operational assumptions about (a) what the long run growth rate (level)
of income is, and (b) how economic agents make predictions about it, and the degree of
time persistence in the deviations from the supposedly stable long run value. Our study
makes no exception to this rule, hence the substantial attention we dedicate to both (a)
and (b) in the discussion below, and the sensitivity that the quantitative predictions of
the formal model display to variations in either (a) or (b).
9
3
A Model of the Depression and WWII:
Productivity Shocks, Capital-People Ratios
and Fertility
In this section, we present versions of the standard Barro-Becker which includes mortality and depreciation. We first set up the general model.
In the second subsection we derive on and off balanced growth path (BGP)
dynamics for the non-stochastic case. We show that if the capital people ratio
is above its BGP level, fertility will be high for one period to bring the K/N
ratio back to its BGP value. We also present an initial quantitative version
of the model in which we trace out the Transition Dynamics in response to a
‘surprise’ shock to see the effects on fertility when some portion of N and/or
K are destroyed (deaths and depreciation).
In subsection 3.3 we deal with stochastic generalizations of this model.
We first derive homogeneity properties of the model. Second, we show that if
capital income and labor income are taxed at the same rate and governments
lead a balanced budget, shocks to government spending financed through
such taxation is equivalent to productivity shocks. Third, we show that under
certain assumptions, the capital-labor ratio is independent of the current
productivity shock but that both, investment and fertility are pro-cyclical.
3.1
Basic Setup of The Model
In this Section, we revisit versions of the standard Barro-Becker model in
which each person goes through two stages of life, childhood and adulthood.
In childhood, no decisions are made. In adulthood each person has children
at the beginning of the period. Parents care about the average utility of
their children as well as the number of children. This leads to the following
preferences for the dynasty head:
10
P∞
t=0
Ct
),
β t g(Nt )u( N
t
where Ct is the aggregate consumption in period t by the dynasty and Nt is
the number of people in adulthood (and hence, alive, working and reproducing in period t). To simplify notation in what comes below, let c =
C
,
N
i.e., c
is per capita consumption in the dynasty, and there is an implicit assumption
that all dynasty members will share the aggregate consumption, C, equally
in each period. Furthermore, let g(N) = N η and u(c) =
c1−σ
1−σ
and define
U(N, C) = N η+σ−1 u(C).
Next, we describe the feasibility constraint. Let Nbt denote the total number of births for the dynasty this period and Xt dynasty investment in physical capital, K. Following Alvarez (1999), we write this in terms of aggregate
variables to avoid non-convexities due to per child bequest choices. That is,
for given initial conditions and the assumption of CRS production function,
the following set 
is convex for any sequence of productivities, {At }∞
t=0 .
∞

 (Nt , Kt , Ct )t=0 |∀t ≥ 0, Ct + θt Nbt + Xt ≤ At F (Nt , Kt )

Λ(N0 , K0 ) =
Nt+1 = πNt + Nbt



K = (1 − δ)K + X
t+1
t
t







Thus, in this version, it is well understood what conditions on U(N, C) are
sufficient to guarantee uniqueness of a solution. These sufficient conditions
are that U(N, C), is strictly increasing in C, weakly increasing in N and
quasi-concave in (N, C).
There are two sets of parameter restrictions that satisfy the natural monotonicity and concavity restrictions for this functional form, both in terms of
the aggregate, or dynasty variables, (C, N), and in terms of per capita valC
ues, (N, c) = (N, N
). The first one is the standard assumption in the fertility
literature with 0 < η < 1, 0 < σ < 1 and 0 ≤ η + σ − 1 < 1. Note that in
2
this case U > 0 for all (N, C) ∈ R+
. The other one allows for intertempo-
ral elasticities of substitution in line with the standard growth and business
cycle literature with σ > 1, η + σ − 1 ≤ 0 which implies η < 0. Utility is
11
negative in this case. Note that in the case where, η + σ = 1 (allowed under
both configurations), utility becomes a function of aggregate consumption
only. Hence, conditions for monotonicity and concavity of U involve UC and
UCC only. In this case, N is exactly like a capital good since the two utility
effects of increasing N exactly cancel out. These two effects are: 1. the
direct benefit of having extra children – g(N) = N η – in the utility function;
2. the direct cost of having children by diluting per capita consumption –
C 1−σ
/(1 − σ) – in the utility function. In calibrated examples we use this
N
simplification extensively and find that the second parameter configuration
which is more common in the growth and business cycle literatures fits the
facts better.
3.2
Catching up to Balanced Growth:
Non-stochastic Version of the Model
In this subsection we show use a non-stochastic version of the model to show
how the detrended capital-people ratio adjusts from an off balanced growth
level back to its balanced growth value. That is, suppose the capital-people
ratio is above its BGP value (e.g. due to a recent shock to mortality), then
fertility ’catches up’ to bring the ratio back to its BGP level. Thus, we will
study solutions to maximization problems of the form:
P (γ, β; N0 , K0 )
Max{Ct ,Nt ,Kt }
U0 ({Ct , Nt , Kt }) =
subject to:
Ct + θt Nbt + Xt ≤ F (Kt , γ t Nt ),
Kt+1 ≤ (1 − δ)Kt + Xt ,
12
P∞
t=0
β t Ntη+σ−1
Ct1−σ
1−σ
Nt+1 ≤ πNt + Nbt
(N0 , K0 ) given.
In what follows we assume that the costs of children are in goods only
and make the simplifying assumption that η + σ − 1 = 0. Also, we allow for
an exogenous trend in labor productivity.
Assumptions: The growth rate, γ ≥ 1, is an exogenous constant, θt = γ t a,
F (K, N) = AK α N 1−α and η + σ − 1 = 0.
The next lemma states that the detrended capital stock next period is
constant – independently of this period’s stocks of capital and people.
Lemma 1. If there is a solution to the maximization problem, then for any
date t such that investment and fertility are strictly positive, Xt > 0 and
Kt+1
Nbt > 0, γ t+1
= Z ∗ . That is, the detrended capital-people ratio is conNt+1
stant. Z ∗ is the unique solution to the following equation:
αAZ α−1 + 1 − δ =
(1 − α)Aγ α
Z + πγ
θ
(1)
Proof. See Appendix.
Lemma 1 shows that along the balanced growth path, the capital-people
ratio is constant. Moreover, starting with a capital-people ratio that is off
the BGP, the lemma says that – under mild assumptions – the capital-people
ratio will be back to the BGP in one period. From equation 1, one can also
see that if mortality, 1 −π, or the base cost of children, θ, increase forever the
BGP capital-people ratio will increase. On the other hand, if depreciation, δ,
or the growth rate of technology, γ, increase forever, the BGP capital-people
ratio will decrease. The effect of A, the scale parameter, depends on the
relative sizes of πγ and (1 − δ).5
5
The simplest case to consider is when 1 − δ = πγ. This corresponds to the equal
13
3.2.1
BGP dynamics of interior solutions
Let K̂t =
Kt
.
γt
Whenever
K̂t
Nt
= Z ∗ , i.e. on the BGP, output that period is
given by:
Yt = A (Z ∗ )α γ t Nt
That is, output is exactly the same as it would be in a non-stochastic
e = A (Z ∗ )α .
ANt model, but with A
Next, we show that the growth rate in aggregate consumption, γ Ct is
independent on t.
. If there is a solution to the maximization probLemma 2. Let γ Ct = CCt+1
t
lem, then for any date t such that investment and fertility are strictly positive, Xt > 0 and Nbt > 0, and the capital-people ratio is at its BGP value,
Kt
= Z ∗ , then γ Ct = γ ∗C = β αA [Z ∗ ]α−1 + 1 − δ . That is, the growth rate
γ t Nt
in consumption is constant.
Proof. See Appendix.
Note that Lemma 2 gives the growth rate as a function of the deep parameters of the model since Z ∗ can be found from equation 2, and it depends
only on the deep parameters. From the lemmas, we can completely characterize the path of the solution to the model for interior parts of the path just
by using the restrictions given from technology. This is that:
1. γ t Nt , Kt and Yt all grow at the same rate as Ct , the rate γ ∗C =
1/σ
β αA [Z ∗ ]α−1 + 1 − δ
,
depreciation case in the (K, H) model (see Jones et al. (2005)). In this case, equation 2
becomes:
(1 − α)Aγ ∗(α)
LHS(Z ∗ ) = αAZ ∗(α−1) =
Z
= RHS(Z ∗ ).
θ
αθ
And so we have a closed form solution for Z ∗ given by Z ∗ = (1−α)γ
. Note that this is
independent of ‘scale’ in this case (i.e., it doesn’t depend on A.)
14
2. that, Ct /Yt , Xt /Yt and θt Nbt /Yt are constant, and,
3. finally, Kt /(γ t Nt /) = Z ∗ for all t.
Let φC = Ct /Yt , φX = Xt /Yt and φN = θt Nbt /Yt .
K0
∗
0
Proposition 1. Assume that (N0 , K0 ) satisfies γK
where Z ∗ is
0N = N = Z
0
0
the unique solution to:
Z α + πγ
αAZ α−1 + 1 − δ = (1−α)Aγ
θ
Then, the solution to the maximization problem satisfies:
1.
2.
Ct+1
Ct
=
Yt+1
= KKt+1
Yt
t
∗ α−1
β αA [Z ]
Xt
Yt
= φX =
3.
θ t Nbt
Yt
4.
Ct
Yt
= XXt+1
=
1/σt
+1−δ
,
γ ∗ −(1−δ)
A(Z ∗ )α−1
θ t+1 Nbt+1
θ t Nbt
= γ ∗C for all t where γ ∗C =
for all t,
= φN = θ (Z ∗ )−α
γ ∗ −πγ
γA
for all t,
= φC = 1 − φN − φX for all t,
5. The growth rate of population, γ N is γ C /γ = γ ∗ /γ,
6.
Yt
Ct
t
, N
, X
,
Nt
Nt
t
t
and K
all grow at the common rate of exogenous labor
Nt
augmenting technological change, γ,
7. C0 = φC AK0 [Z ∗ ]α−1 ,
8. V (K0 , N0 ) =
C01−σ
1
,
(1−σ) 1−βγ ∗(1−σ)
0
9. If (N0 , K0 ) satisfies γK
= K0
= Z ∗ , then (λN0 , λK0 ) also satisfies
0N
N0
0
λK0
= λK0
= Z ∗ . It follows that the solution to the problem given
γ 0 λN0
λN0
initial condition (λN0 , λK0 ) is simply λ times the solution given initial
condition (N0 , K0 ), that the Value of the problem is λ1−σ = λη times
the value from initial condition (N0 , K0 ), and that the growth rates,
output shares, etc., given in parts 1-8 also hold if the initial condition
is (λN0 , λK0 ).
15
Proof. See Appendix A.
This proposition describes the entire path of all variables. There is an
implicit requirement that the non-negativity restrictions on investment be
non-binding at the solution given in Proposition 1. In numerical implementations, we check that these hold.
3.2.2
Off-BGP dynamics and transition paths
The next step is to determine the off-BGP dynamics when
K0
N0
6= Z ∗ . For this,
we still know that Lemma 1 must hold for any path in which the solution is
interior. Thus, if (N0 , K0 ) is such that the resulting optimal path has Xt > 0
and Nbt > 0 for all t, it will be true that
all subsequent t too) even if
N0
K0
K1
γN1
= Z ∗ (and
Kt+1
γ t+1 Nt+1
= Z ∗ for
6= Z ∗ . Given this, Proposition 1 tells us
what the dynamics of the path will be for periods t = 1 on for any initial
conditions such that the solution is interior for all periods.
This leaves two things to be determined then:
1. For initial conditions such that the solution is interior, determine what
(N1 , K1 ) are, and,
2. Under what conditions, on (N0 , K0 ) is it true that the solution is interior?
This is what we do in this section. To do this, we will need to know the value
of the program from starting on the BGP from period 1 onwards. If the
optimal plan from such a starting point is to have (N1 , K1 ) satisfy
K1
γN1
= Z ∗,
it follows from the usual Dynamic Programming results that the continuation
plan is characterized by Proposition 1 above.
Thus, our strategy is guess and verify, that for some initial values of
(N0 , K0 ) with
K0
N0
6= Z ∗ , the optimal plan will satisfy
K1
γN1
= Z ∗ for some
choice of (N1 , K1 ) and then see for what choices of (N0 , K0 ) this guess is
correct. The following proposition answers the above questions.
16
Proposition 2. There is a unique, positive solution, K1∗ , to the equation:
AK0α N01−α − [K1 − (1 − δ)K0 ] − θ0
K1
− πN0
γZ ∗
=
"
1+
θ0
γZ ∗
(1 − σ)D
#1/σ
K1 .
If this K1∗ satisfies K1∗ − (1 − δ)K0 ≥ 0 and γZ1 ∗ K1∗ − πN0 ≥ 0, then, the
solution to the Problem is interior with the first period given by (K1∗ , N1∗ )
K∗
where γN1∗ = Z ∗ , AND all variables grow at rate γ ∗C from period 1 on.
1
Proof. See Appendix A.
Proposition 2 implies that if at some point in time, t∗ the capital-people
ratio is above its BGP value (Z ∗ ), fertility relative to investment in physical
capital must be high to bring the capital-people ratio back to BGP (and
vice-versa if it is below its BGP value). Thus, due to the misalignment in
K/N a ‘catching-up’ phenomenon occurs. In a decentralized environment,
this phenomenon occurs because, off-BGP, the marginal product of labor and
hence the wage rate is high. Having children is then an attractive investment
since parents care about children’s well-being.
3.2.3
Numerical implementation: US versus Europe?
In this section, we present findings of computing an example of the model
in the previous sections. The idea here is to simulate what the effects are
on the optimal path of population and capital formation when an event like
WWII occurs, destroying a significant fraction of N and/or K in a country.
For this we use the version of the model outlined in section 3 above.
Calibration of parameters We set the length of a period as 10 years. We
need to choose β, δ, π, γ, A, a, σ, α. To choose these we use some standard
facts from the RBC literature in conjunction with the BGP equations given
in Section 2 along with some additional facts. Thus, we have:
17
1. α = .33,
2. δ ann = 0.06 per year, or δ = 1 − (1 − .06)10 = 0.46,
3. β ann = 0.97 per year, or β = 0.74,
4. γ ann = 1.017 per year, or γ = 1.18,
5. π = 0.80,
6. σ = 2.0.
This leaves the two parameters, A and a not yet determined. To set these,
we calibrate so that γ N = 1.0872. There are (probably) many combinations
of A and a that will do this. One possibility is to choose the (A, a) combination that simultaneously matches γ N and φX . The difficulty that will appear
if we try to do this is how to match φX . That is, should we choose φX to
match observed investment rates in the data with
X
Y
in the model, or
X
?
C+X
The values that we use are: A = 3 and a = 0.5. These generate reasonable
values for investment rates.
What we simulate is a situation in which the economy is on the BGP and
it is hit with a surprise reduction in either N or in both N and K at the
beginning of period t∗ . We need to choose how big a shock to use, i.e., how
’misaligned’ is (Kt∗ , Nt∗ ) from the BGP solution? For this, we run two sets
of experiments. For both of these experiments, we assume that
K0
N0
= Z ∗.
For the first experiment, we assume that:
(Kt∗ , Nt∗ ) = (γ ∗ Kt∗ −1 , (1 − χ)γ N Nt∗ −1 ).
In this case, it follows that
K t∗
γ t ∗ Nt ∗
∗
= (1 − χ)Z ∗ . This choice triggers the
transition dynamics from date t forward as described in Section 3.2.2. This
corresponds to the situation in which there is no K destroyed, and χ represents the death rate in the population as a result of the shock.
18
In the second experiment, we assume that both N and K are hit with
the same size shock. Thus, we assume that:
(Kt∗ , Nt∗ ) = ((1 − χ)γ ∗ Kt∗ −1 , (1 − χ)γ N Nt∗ −1 ).
In this case, it follows that
K t∗
γ t ∗ Nt ∗
= Z ∗ . Hence, the system is always on the
BGP, even immediately following the shock.
Experiment #1, shock to N only: US experience?
We consider one time surprise shocks to the size of the population between
0 percent and 20 percent at date t∗ = 3. The results are shown in Figures 1
to 3. Figure 1 shows the time path of output, which dips below its BGP
level at the time of the shock and follows a BGP at the same growth rate as
before beginning in period 4. Figures 2 and 3 show the time paths of
X
,
Y
Nb
N
and
respectively.
Figure 1: Time Path of Aggregate Output (log) with Period 3 Deaths Shock
to N
10
Aggregate Output (log)
9.5
9
Deaths = 10 %
Deaths = 5 %
BGP
8.5
8
7.5
7
0
1
2
3
4
5
6
7
8
9
10
Period
As can be seen in Figure 2, the misalignment between N and K generates
a baby boom in the period after the shock. The size of this boom naturally
depends on the size of the shock. If the shock to N is 10 percent, the General
19
Figure 2: Time Path of Nb /N with Period 3 Deaths Shock to N
0.32
0.315
0.31
Deaths = 10 %
Deaths = 5 %
BGP
b
Fertility, N /N
0.305
0.3
0.295
0.29
0.285
0.28
0
1
2
3
4
5
6
7
8
9
10
Period
Reproduction Rate (GRR) increases from 0.20 to 0.236, an increase of about
20 percent. If the size of the shock to N is 20 percent, the GRR increases
from 0.20 to 0.273, an increase of about 33 percent.
It is not clear what the best comparison is for the data counterpart for
this experiment. Clearly the size of the decrease in N overall in the US was
considerably less than 10-20 percent. However, the destruction in N was also
not uniform across age and sex. Rather it was highly concentrated in males
in the 18 to 35 year age group. In 1940, there were about 11 million males in
the US population between the ages of 20 and 30. Of these, about 500,000
died in WWII. Thus, 5 percent seems like an upper bound to the effective
death rate in the data. As a rough comparison, during the U.S. baby boom
completed fertility (i.e. children ever born, CEB) increased from about 2.3
to about 3.2 children per woman, an increase of about 40 percent. Thus,
although quantitatively significant, the predicted baby boom in the model
seems quite a bit smaller than that in the US data, 10 percent in the model,
vs. 40 percent in the data.
20
Figure 3: Time Path of X/Y with Period 3 Deaths Shock to N
0.148
0.146
Investment, X/Y
0.144
0.142
0.14
0.138
Deaths = 10 %
Deaths = 5 %
BGP
0.136
0
1
2
3
4
5
6
7
8
9
10
Period
Experiment #2, proportional shock to both N and K: Europe’s experience?
Here we alter the experiment done above by simultaneously decreasing
the stocks of N and K proportionally. Clearly, this leaves the K/N ratio on
the BGP. Figure 4 shows the time path of output from this shock. As can
be seen, the time paths resulting from a shock of this type is quite different
from that seen above. In particular, there is no baby boom at all in this case.
Rather, levels of all variables drop, but growth rates are never affected since
we remain on the BGP.
3.3
Pro-cyclical Fertility:
Stochastic Version of the Model
In this section we add stochastic productivity and government investments to
the basic Planner’s Problem studied in Section 3.2. We show that if capital
income and labor income are taxed at the same rate and governments lead
a balanced budget, shocks to government spending financed through such
21
Figure 4: Time Path of Aggregate Output (log) with Period 3 Proportional
Deaths Shock to K and N
10
Deaths = 10 %
Deaths = 5 %
BGP
Aggregate Output (log)
9.5
9
8.5
8
7.5
7
0
1
2
3
4
5
6
7
8
9
10
Period
taxation is equivalent to productivity shocks. Furthermore, we show that
fertility (and investment) is pro-cyclical.
3.3.1
The Planner’s problem with government investments
In this section, we consider the stochastic version of the model presented
in Section 3 and add government spending, Gt . The original maximization
problem is of the form
P1 (N0 , K0 , s0 )
Max
t
{Ct (st ),Nt+1 (s ),Kt+1 (st )}
U0 ({Ct , Nt , Kt }) = E
subject to:
hP
∞
t η+σ−1
t=0 β Nt
Ct1−σ
|s0
1−σ
Ct (st ) + θt Nbt (st ) + Xt (st ) + Gt (st ) ≤ st F (Kt (st−1 ), γ t Nt (st−1 )),
Kt+1 (st ) ≤ (1 − δ)Kt (st−1 ) + Xt (st ),
Nt+1 (st ) ≤ πNt (st−1 ) + Nbt (st )
22
i
Gt (st , τ t ) = (1−δ)Gt−1 (st−1 , τ t−1 )+τ t st F (Kt (st−1 , τ t−1 ), γ t Nt (st−1 , τ t−1 ))
(N0 , K0 , G0 , s0 ) given.
where Ct denotes dynasty consumption in period t, Nt dynasty size, Kt the
private capital stock, Nbt the number of newborns in the dynasty, Xt private
investments and st is the sequence of productivity shocks up to period t.
Let VP 1 (N0 , K0 , s0 ) denote the maximized value of the objective at a solu∗
∗
tion (assuming one exists) and let {(Ct∗ (N0 , K0 , s0 ), Nt+1
(N0 , K0 , s0 ), Kt+1
(N0 , K0 , s0 ))}∞
t=0 ,
denote the solution itself.
Assumption 1: F is concave and homogeneous of degree 1 in (K, N).
Proposition 3. Under Assumption 1, with equal tax rates on capital and
labor income, τ kt = τ nt = τ t and a balanced budget every period, P1 is
equivalent to P2 , where P2 (N0 , K0 , s0 , τ 0 ) ≡
hP
i
1−σ
∞
t η+σ−1 Ct
Max
U
({C
,
N
,
K
})
=
E
β
N
|s
0
t
t
t
0
t
t=0
1−σ
t t
t t
t t
{Ct (s ,τ ),Nt+1 (s ,τ ),Kt+1 (s ,τ )}
subject to:
Ct (st , τ t )+θt Nbt (st , τ t )+Xt (st , τ t ) ≤ (1−τ t )st F (Kt (st−1 , τ t−1 ), γ t Nt (st−1 , τ t−1 )),
Kt+1 (st , τ t ) ≤ (1 − δ)Kt (st−1 , τ t−1 ) + Xt (st , τ t ),
Nt+1 (st , τ t ) ≤ πNt (st−1 , τ t−1 ) + Nbt (st , τ t )
Gt (st , τ t ) = (1−δ)Gt−1 (st−1 , τ t−1 )+τ t st F (Kt (st−1 , τ t−1 ), γ t Nt (st−1 , τ t−1 ))
(N0 , K0 , s0 , τ 0 ) given,
in the following sense. If {Ct , Nt+1 , Kt+1 } solves P2 (N0 , K0 , s0 , τ 0 ), then it
also solves P1 (N0 , K0 , (1 − τ 0 )s0 ).
Proof. Omitted. See Lucas (1990).
23
Given this proposition, we will treat war investments as ‘productivity
shocks’ to the private economy. If the government decides to use its capital
(G) for private sector production. This is reflected as an increase in the
capital stock. The latter is then off its balanced growth path level and
triggers a baby boom. This is what we will argue in Section 4.3 based on the
intuition in Section 3.2.
In what follows, we show that investments in capital and people (i.e. fertility) are procyclical. Hence higher expenditures that imply higher tax rates
will negatively affect fertility. However, the capital-people ratio is independent on productivity shocks and hence this kind of policy. As long as agents
don’t forecast that this investment will become useful in private production,
it will not affect decisions about capital labor ratios.
3.3.2
Pro-cyclical fertility and investment
First, we derive homogeneity properties for P1 .
Proposition 4. The value function is homogeneous of degree η in (N0 , K0 ),
V (λN0 , λK0 , s0 ) = λη V (N0 , K0 , s0 ), and policy functions are
homogeneous of degree one in (N0 , K0 ),
∗
∗
{(Ct∗ (λN0 , λK0 , s0 ), Nt+1
(λN0 , λK0 , s0 ), Kt+1
(λN0 , λK0 , s0 ))}∞
t=0
∗
∗
∗
∞
= λ{(Ct (N0 , K0 , s0 ), Nt+1 (N0 , K0 , s0 ), Kt+1 (N0 , K0 , s0 ))}t=0 .
Proof. See Appendix.
This follows from a standard homogeneous/homothetic type argument.
Below, we use this theorem to reduce the problem to one in which there is
only one endogenous state variable, namely, the capital-people ratio. This
simplifies numerical implementation significantly.
Assumption 2: Shocks are a stationary first-order Markov process.
Under this assumption, the problem can be written recursively as:
24
(BE)
V (N, K, s) =
max
′
(C,N ′ ,K )∈Γ(N,K,s)
n
o
1−σ
N η+σ−1 C1−σ + βE (V (N ′ , K ′ , s′ ))
where Γ(N, K, s) =
(
(C, N ′ , K ′ ) : K ′ − (1 − δ)K ≥ 0, N ′ − πN ≥ 0, (N0 , K0 , s0 ) given
0 ≤ C ≤ sF (K, N) − K ′ + (1 − δ)K − θ (s) N ′ + θ (s) πN
Assumption 3: Assume that θ t = γ t θ for all t.6
The next proposition states that capital-people ratios are independent on
the current shock. This is similar to the result in Section 3.2 where K/N
ratios were independent on time. It also implies that tax shocks as described
in P2 will not affect the choice of next period’s K/N ratio. The proof of the
next proposition requires the following lemma.
Lemma 3. If V is homogeneous of degree η in (N, K), then the partial
derivatives VN and VK are homogeneous of degree η − 1.
Proof. See Appendix.
Proposition 5. If in addition to Assumptions 1 to 3, shocks are i.i.d. and
there is a unique solution, k ∗′ , to
EVK ((1, k ′ , s′ )θ = EVN ((1, k ′ , s′ )
then
K ′ (s)
N ′ (s)
is independent on s and X/N and N’/N are procyclical.
Proof. See Appendix.
6
Note that Assumption 3 excludes the time cost case since θ does not depend on shocks.
25
)
4
The U.S. Experience, 1900-2000
In this section, we present data on war capital built up in the U.S. in the
early 1940s. We then present fertility and productivity fluctuations for the
period 1910 to 1970.
In Section 4.1 we present some evidence on the evolution of capital stocks
in the U.S.. We find that total (i.e. private and government) fixed assets
double between 1929 and 1950. When decomposing this change, it becomes
apparent that most of this increase was due to public investments in fixed
assets. Moreover, the largest increase occurred in industrial structures. We
view this increase as exogenous to this model and assume that war investments have been used in combination with labor for (non-war) production
after 1945. Together with data on war casualties, this allows us to derive
a range of the implied change in capital-people (or capital-labor) ratios between 1929 and 1950. The magnitudes for K/N in 1950 lie between 1.5 and
2 times the K/N in 1929.
We then lay out the basic facts about the time paths of TFP and fertility
in the U.S. over the 20th century. We find that the correlation and timing of
events are, qualitatively, very much in line with the predictions of the model.
We then calibrate the model as before and perform the following experiment. We use decade by decade productivity shocks as calculated from the
data. To this we add a one time shock to government expenditure for warcapital build-up in the 1940s. A fraction of this capital is dropped back into
the economy in the 1950s generating the exogenous increase in K/N. The
great depression in the 1930s and the sudden increase in government war
capital built-ups in the early 1940s produce the downswing in fertility in the
model. It accounts for 2/3 of the one observed in the data. The increase
in the K/N ratio in the late 1940s and 1950s generates an upward swing in
fertility of the order of 65% of the observed Baby Boom. The fact that the
26
baby Boom drags out into the 1960s is accounted for by high productivity
levels relative to trend during these years.
4.1
Data: War capital build-up in the U.S.
In the preceding sections, we have seen that the misalignment in capitalpeople ratios resulting from the death toll of young males accounts for less
than 25 percent of the U.S. baby boom. In this section we show that not only
was capital spared from destruction in the U.S., but instead the total capital
stock almost doubled between 1929 and 1950. Decomposing this change into
private and public investments, we see that while private investment rates
stagnated during the war years, national defense investments skyrocketed.
Among those, the largest increases occurred in industrial structures. In the
next section, we argue that these investments – exogenous to our model –
increased labor productivity and therefore triggered the baby boom.
We use data from the U.S. Department of Commerce, Bureau of Economic
Analysis (2003).7 In Figures 5 and 6 , we plot net stock of fixed asset indices
series (2000=100) decomposed into private and government fixed assets from
1925 to 2000. Zooming in to 1929 to 1950, several observations can be made:
1. Total fixed assets grew at a faster rate from 1940 to 1945.
2. Private fixed assets did not grow much between 1929 (16.5) and 1945
(17.1).
3. Government fixed assets skyrocketed during the war, then decreased
from 1945 to 1950 to come back to their earlier trend growth thereafter.
These observations call for a closer look at the types of government investments during this period. Figure 7 plots national defense investment
7
The data is also available online at http://www.bea.gov/bea/dn/FA2004/SelectTable.asp
27
Figure 5: Chain-Type Quantity Indexes for Net Stock of Fixed Assets (from
BEA, Table 1.2.), 1929-2000
120
Net Stock
k of Fixed Assets
(Chain-type quantity index, 2000=100)
100
80
60
40
20
0
Year
Fixed assets
Private
Government
indices between 1933 and 1950. The increase in investments in structures is
larger than the increase in equipment and software. Within structures, the
increase in buildings was larger than military facilities. The largest increase
among buildings was clearly in industrial buildings, not residential. In fact,
the index (2000=100) rose from 6.7 in 1933 to 5,300 in 1944. In Figure 8 we
show the corresponding capital stocks over the same period. Here, the index
for industrial capital increased from 10 in 1940 to 188 in 1945. After the
war, equipment and software was either sold or depreciated quickly, while
government structures remained at very high levels.
To assess the increase in the economy-wide capital-people ratio between
1929 and 1950, we proceed as follows. In 1929, total fixed assets in the
economy were at a chain-type index of 14.77. In 1950, they were at 22.171.
Detrending the latter at 1.01721 and dividing by the 1929 value gives
28
Figure 6: Chain-Type Quantity Indexes for Net Stock of Fixed Assets (from
BEA, Table 1.2.), 1929-1950
40
Net Stock of Fixed Assets
(Chain-type qua
antity index, 2000=100)
35
30
25
20
15
10
5
0
Year
Fixed assets
∆K(1929 to 1950) =
K1950
γ (1950−1929)
K1929
Private
Government
= 1.05336
On the other hand, as we have assessed in the previous section, the death
toll of WWII was of about 5 percent of the workforce. Assuming that population stayed otherwise constant, we get:
1929
∆(K/N)(1929 to 1950) = ∆K(1929 to 1950) N
=
N1950
1.05336
0.95
= 1.11
We will use this assessment in the quantitative experiment in the next
Section. This calculation could of course be done in many different ways.
We will address some alternatives in Appendix A.
29
Figure 7: Chain-Type Quantity Indexes for National Defense Investment in
Fixed Assets (from BEA, Table 7.6A.), 1933-1950
5,000
(Chain-type quantitty index, 2000=100)
Investment in Goverrnment Fixed Assets
6,000
4,000
1. National defense
1.1 Equipment and software
1.2 Structures
3 000
3,000
121B
1.2.1
Buildings
ildi
1.2.1.1 Residential
1.2.1.2 Industrial
2,000
1.2.2 Military facilities \4\
1,000
0
Year
Figure 8: Chain-Type Quantity Indexes for Net Stock of National Defense
Fixed Assets (from BEA, Table 7.2A.), 1933-2000
200
160
(Chain-type qua
antity index, 2000=100)
Net Stock of Gov
vernment Fixed Assets
180
140
1. National defense
120
1.1 Equipt and softw.
100
1.2 Structures
1.2.1 Buildings
80
1.2.1.1 Residential
60
1.2.1.2 Industrial
1.2.2 Military facil.
40
20
0
Year
30
4.2
Data: Productivity and fertility fluctuations
In this section, we lay out the basic facts about the time paths of TFP and
fertility in the U.S. over the 20th century. We begin with the facts pertaining
to the growth in productivity as laid out in Kendrick (1961) and Kendrick
(1973). As most economists know, this period is one of more or less continued
growth in productivity with a few significant breaks. The most significant
of these is the Great Depression. Figure 9 shows Kendrick’s compilation of
Total Factor Productivity (TFP) for the U.S. over the period from 1889 to
1969.
Figure 9: Total Factor Productivity, 1889-1969
250
TFP index
x, 1929=100
200
150
100
50
0
1889 1894 1899 1904 1909 1914 1919 1924 1929 1934 1939 1944 1949 1954 1959 1964 1969
Year
Source: 1889-1928: Kendrick (1961), 1929-1969: Kendrick (1973)
The obvious facts about TFP over this period are:
1. The continual upward trend,
2. The marked decline below trend that took place in the 1920s and 1930s,
31
3. The return to trend following WWII.
The exact sizes of these features of the data depend on how one treats the
trend growth in productivity over the period. For example: Was there a
common, exogenous growth rate in TFP over the entire period with higher
frequency fluctuations (albeit highly autocorrelated fluctuations) around this
trend? Or, were there two regimes of growth, one prior to WWII and one
after, with higher growth in the second regime? These questions will have
an impact on the analysis we present below and because of this, we will try
several alternatives. (See more discussion on this in the Appendix.)
After fitting an exponential trend to this series, we obtain the following
shocks to productivity over this period. (See Figure 10.)
Figure 10: Total Factor Productivity: Deviations from Exponential Trend
1.2
TFP index, 1929=100
1.15
1.1
1.05
1
0.95
0.9
0.85
1967
1964
1961
1958
1955
1952
1949
1946
1943
1940
1937
1934
1931
1928
1925
1922
1919
1916
1913
1910
1907
1904
1901
1898
1895
1892
1889
0.8
Year
As can be seen in the figure, there was a fairly long and deep fall in productivity from trend that took place from 1910 to 1940. In the deepest part
32
of the depression, productivity was about 17 percent below trend. Since that
time there was a steady increase up until the late 1960s when productivity
was about 10 percent above trend.
This timing of the movements of productivity around trend fits well with
the movements in fertility seen in the data. Figure 11 shows the time path of
the Total Fertility Rate (TFR) over the period from 1860 to 1968. The figure
plots two time series for TFR. The first 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. 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 1920 to 1930, as can be seen in the Haines data. Putting together the
NSA and Haines data, it appears that a good description would be:
1. High, and fairly constantly decreasing fertility until 1920-21, when it
reaches a TFR of about 3.2 children per woman,
2. Acceleration of the rate at which fertility is falling between 1922 and
1932 (from TFR=3.2 to TFR=2.2),
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 1941 to 1956, with TFR going from 2.2 up
to 3.6,
5. High, stable fertility from 1957 to 1962 at about TFR=3.6,
6. Falling fertility over the remainder of the period from a TFR of 3.6 in
1963 to about 1.8 around 1980.
33
Figure 11: Total Fertility Rate, 1850-1990
6
5.5
5
TF
FR
4.5
4
3.5
3
2.5
2
1.5
1850
1860
1870
1880
1890
1900
1910
1920
1930
1940
1950
1960
1970
1980
1990
Year
TFR (NSA (1970))
TFR (Haines (1994))
7. Slight recovery in TFR from 1.8 in 1980 to 2.07 in 2000.
Figure 12 shows a similar picture of fertility over this period. It shows the
deviations from a fitted, linear trend from 1900 to 1990 from Haines. The
deviations at annual frequencies are calculated by subtracting the NSA data
for the yearly observations from the fitted trend from Haines.8 It shows a
similar pattern to that described above.
Figure 13 shows the two series of deviations plotted on the same graph.
Although it is not perfect, there is an impressive coincidence in timing. The
coefficient of correlation between the two annual series for the years 1917 to
Fitting a trend to the NSA data and calculating deviations from this
trend gives virtually identical results. Using the average fertility rate from
the NSA data, we find that deviations from the mean are positive until 1926,
the baby bust starts later and is slightly smaller while the baby boom is
smaller in the 1960s and ends in 1965 - as opposed to 1969 as in Figure 10.
8
34
Figure 12: Total Fertility Rate: Deviations from Linear Trend
0.5
0.4
TFP index,, 1929=100
0.3
0.2
0.1
0
-0.1
-0.2
1968
1965
1962
1959
1956
1953
1950
1947
1944
1941
1938
1935
1932
1929
1926
1923
1920
-0.4
1917
-0.3
Year
1968 is 0.67, which suggests that the U.S. TFR is strongly procyclical during
this time period. In sum, TFP was below trend from about 1910 to about
1940, while fertility was below trend from about 1925 to about 1940. They
were both above trend over the 1945 to 1965 period with the TFP growth
extending beyond this time.
Finally, to compare model responses versus actual time series of fertility
deviations, we need the realizations of shocks. From Kendrick’s TFP data,
the estimated series for productivity shocks (1 is the mean) decade by decade
(beginning with 1910 to 1919) is (0.935, 0.966, 0.905, 0.987, 1.026, 1.079). From
TFR data, the same series of deviations in fertility (0 is the mean) are
(−0.02, −0.06, −0.26, −0.06, 0.29, 0.23).
This is not the whole story about the timing of the baby bust and boom,
obviously. One possibility is that the baby boom was simply a by-product
of delayed fertility of those women whose fertility was low during the period
35
Figure 13: Total Factor Productivity and Total Fertility Rate: Deviations
from Trend
0.5
0.4
TFP index
x, 1929=100
0.3
0.2
0.1
0
-0.1
-0.2
1968
1965
1962
1959
1956
1953
1950
1947
1944
1941
1938
1935
1932
1929
1926
1923
1920
-0.4
1917
-0.3
Year
of the baby bust. This is not true, however. In fact, women born roughly
between 1905 and 1925 had lower completed fertility than did their mothers
or daughters. This can be seen in Figure 14, which shows completed fertility
by birth cohort over the period.9 This reaches a minimum with the 1908
birth cohort then increases back up to a peak with the 1938 birth cohort,
falling thereafter. This pattern implies that if the baby boom is a catching
up of fertility from the baby bust, it is at the aggregate level, not at the
level of the individual woman. See also Greenwood et al. (2005), Doepke and
Maoz (2005) and Jones and Tertilt (2006) for more on the make-up, across
birth cohorts, of the fertility pattern during the baby boom.
9
The data is from Jones and Tertilt (2006), Table A6, column 1.
36
Figure 14: Children Ever Born, 1815-1960 Birth Cohorts
6
Children
n Ever Born
5
4
3
2
1
0
Birth Year of Mother
4.3
Numerical implementation for the U.S experience
In this section we show the results of the numerical analysis of our model
applied to the U.S.. First, we derive a numerically tractable version of the
recursive Planner’s Problem. The Planner’s problem we solve is the one
in Section 4.3.3. This requires only current productit=vity shocks and the
detrended capital-people ratio as state variables. Second, we must pick parameters. To do this, we borrow insights from previous sections. After
calibrating to several facts from the data, we calculate decision rules as a
function of (detrended) capital-people ratios and productivity shocks (we
will omit ‘detrended’ henceforth: all variables are in per capita terms and
detrended). In particular, we compute fertility decisions. From those, we can
then read off deviations in fertility given observed productivity shocks and
37
(unanticipated) capital-labor ratio shocks. Details on the solution algorithm
and computation tools are available upon request from the authors.
The model predicts a fertility drop of 22 percent during the Great Depression compared to 26 percent in the data. For the post-WWII baby boom,
the model generates a 19 percent increase compared to 29 percent in the data
for the 1950s. This is mostly due to the high capital-people ratio. For the
1960s, the model also captures a 19 percent increase in fertility compared to
23 percent in the data. The latter is mostly due to the productivity being 8
percent above trend during this decade.
4.3.1
Detrending
Assumption 4: Assume that η + σ − 1 = 0.
In accordance with our data treatment, we want to work in detrended
bt = Xtt , K
b t = Ktt , θt = γ t θ and F (K, N) =
bt = Ctt , X
variables. To do so, let C
γ
γ
γ
AK α N 1−α . Then we can rewrite the problem in detrended variables as:
hP
∞
1−σ t
bt , Ntb , X
bt }) = E
P3
≡
Max
U0 ({C
)
t=0 (βγ
subject to:
bt (st ),Ntb (st ),X
b t (st )}
{C
bt (st ) + θNbt (st ) + X
bt (st ) ≤ st F (K
b t (st−1 ), Nt (st−1 )),
C
b t+1 (st ) ≤ (1 − δ)K
b t (st−1 ) + X
bt (st ),
γK
Nt+1 (st ) ≤ πNt (st−1 ) + Nbt (st )
b 0 , s0 ) given.
(N0 , K
38
b 1−σ
C
t
|s0
1−σ
i
4.3.2
Recursive formulation with exogenous growth
Assumption 5: Assume that the st are i.i.d.
Think of K, C, and X as detrended.
Under Assumptions 1 to 5, the maximization problem P3 can be written
recursively as in
(BE)
V (N, K, s) =
where Γ(N, K, s) =
(
max
′
(C,N ′ ,K )∈Γ(N,K,s)
n
C 1−σ
1−σ
+ βγ 1−σ E (V (N ′ , K ′ , s′ ))
o
(C, N ′ , K ′ ) : 0 ≤ C ≤ sF (K, N) − γK ′ + (1 − δ)K − θN ′ + θπN
γK ′ − (1 − δ)K ≥ 0, N ′ − πN ≥ 0, (N0 , K0 , s0 ) given
′
′
1−σ
+θπN )
. Let
Proposition 6. Let U(N, K, N ′ , K ′ ) = (sF (K,N )−γK +(1−δ)K−θN
1−σ
C(X) be the set of
homogeneous of degree (1 − σ) functions. Then the operator
T : C(X) → C(X)
T V (N, K, s) = ′ ′max
{U(N, K, N ′ , K ′ ) + βE (V (N ′ , K ′ , s′ ))}
(N ,K )∈Γ(N,K,s)
is a contraction mapping.
Proof. 1.) C(X) is complete.
2.) If V (λN, λK, s) = λ1−σ V (N, K, s) then T V (λN, λK, s) = λ1−σ T V (N, K, s)
4.3.3
Formulation of the stochastic ANK model with exogenous
growth and one endogenous state variable
Again, think of K, C, and X as detrended.
Let c =
C
,k
N
=
K
, k′
N
=
K′
,n
N′
=
N′
, f (k)
N
= F (k, 1) and v(k, s) =
V (1, k, s). Using homogeneity and dividing both sides by N, the problem
in (BE) can be rewritten as
39
)
(be)
v(k, s) =
where Γ(k, s) =
max
′
(c,n,k )∈Γ(k,s)
(
n
c1−σ
1−σ
+ βγ 1−σ n1−σ E (v(k ′ , s′ ))
o
(c, n, k ′ ) : 0 ≤ c ≤ sf (k) − γk ′ n + (1 − δ)k − θn + θπ
γk ′ n − (1 − δ)k ≥ 0, n − π ≥ 0, (N0 , K0 , s0 ) given
Solving this problem implies decision rules for per capita consumption,c =
c(k, s), next period’s capital-people ratio, k ′ = k ′ (k, s), population growth,
n = n(k, s) and investment per capita, x = x(k, s) = γk ′ − (1 − δ)k. In the
next Subsection we implement this solution numerically.
4.3.4
Calibration of parameters
As in the deterministic case, we set the length of a period as 10 years. We
need to choose β, δ, π, γ, A, a, σ, α and σ ε (standard deviation of the shock).
To choose these we use values for β, δ, π, γ, σ, α the values as in the nonstochastic case. Thus, we have:
1. α = .33,
2. δ ann = 0.06 per year, or δ = 1 − (1 − .06)10 = 0.46,
3. β ann = 0.97 per year, or β = 0.74,
4. γ ann = 1.017 per year, or γ = 1.18,
5. π = 0.80,
6. σ = 2.0.
7. σ ε = 0.0687
Again, this leaves the two parameters, A and a not yet determined. To set
these, we calibrate so that population growth, n(k, s) =
steady state capital-people ratio k ∗ =
40
Kt
Nt γ t
N′
N
at the (detrended)
with an average shock, s = 1,
)
satisfies n(k ∗ , s = 1) = 1.0872. The values that we use are: A = 1 and
a = 0.278 in this case.
4.3.5
Decision rules
In this section we describe calibration results and decision rules for investment and population growth. First, as stated in Proposition 5, the capitalpeople ratio next period is independent of this period’s productivity shock.
The steady state value we find is k ∗ = 0.088. Figures 15 to 17 show plots
of the decision rules. The capital-people ratio is on the x-axis. Each curve
represents a cut of the policy function for a particular contemporary productivity shock this period (between s=0.85 and s=1.18).
Figure 15: Decision rule: Capital-people ratio next period as a function of
this period’s capital-people ratio, for various sizes of current productivity
shocks
0.18
Capital−People Ratio Next Period, K’/N’
0.16
s=0.85
s=0.92
s=1.01
s=1.09
s=1.18
0.14
45 degree
0.12
0.1
0.08
0.06
0.04
0.8
1
1.2
1.4
1.6
Relative Capital−People Ratio, K/N: 1939−1950
41
1.8
2
Figure 15 shows capital-people ratios chosen next period given the capitalpeople ratio this period and the current productivity shock. We also plot the
45 degree line to show that, no matter what current k = K/N or s, the
economy always converges back to the steady state value next period.
Figure 16: Decision rule: Population growth as a function of capital-people
ratios, for various sizes of productivity shocks
1.5
1.4
s=0.85
s=0.92
s=1.01
s=1.09
s=1.18
Population Growth, N’/N
1.3
1.2
1.1
1
0.9
0.8
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Capital−People Ratio, K/N
From Figure 16, the population growth rate is increasing in k = K/N.
That is if K is too large relative to N, investment in children increases. This
is the same intuition as in the non-stochastic case. Furthermore, population
growth (and hence fertility) is procyclical. That is, fix k on the x-axis. The
higher the productivity shock, the larger is population growth.
From Figure 17, investment per capita – as opposed to population growth
– is decreasing in k = K/N. Again, the intuition is the same as in the nonstochastic case. If the capital-people ratio is high, investment in capital is low
enough so that next period’s capital stock is back to its steady state value.
42
Figure 17: Decision rule: Investment per capita as a function of capitalpeople ratios, for various sizes of productivity shocks
0.09
Investment Per Person, X/N
0.08
0.07
0.06
0.05
0.04
0.03
0.04
s=0.85
s=0.92
s=1.01
s=1.09
s=1.18
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Capital−People Ratio, K/N
Similarly to population growth, however, investment is also procyclical.10
4.3.6
Results
Figure 18 shows the policy function for fertility in percent deviations. On
the x-axis, we express the (detrended) capital-people ratio relative to the
steady state capital-people ratio. Hence a value of 1 means that the current
capital-people ratio is also at its steady state level.
Assuming that the (detrended) capital-people ratio was at its steady state
value in 1929, we can then combine the 10 percent negative shock in productivity during the Great Depression in the 1930s (s1930s = 0.9) with a
10
We briefly discuss corner solutions in the solution algorithm in Appendix A. In the
present example corner solutions only occurred for extreme values of k and s, none of
which are relevant for the observed magnitudes in the data.
43
∆(K/N)1930s = 1 on the x-axis to find a 22 percent downward deviation
in fertility on the y-axis. This compares well to the 26 percent baby bust
observed in the data.
Figure 18: Percent deviations in fertility decisions as a function of relative
capital-people ratios, for various sizes of productivity shocks
0.6
0.4
Percent Deviation in Fertility
Post WWII: 1960s
Post WWII: 1950s
0.2
0
Great Depression
−0.2
−0.4
s=0.85
s=0.92
s=1.01
s=1.09
s=1.18
−0.6
−0.8
0.85
0.9
0.95
1
1.05
1.11.11
Relative Capital−People Ratio, K/N: 1929−1950
Similarly, for the immediate post-war period (see label “Post WWII:
1950s”) we combine the productivity shock of (s1950s = 1.03) with a ∆(K/N)1929−1950 =
1.11 (see Section 4.1) on the x-axis to find a 19 percent upward deviation in
fertility on the y-axis. The observed increase in the data is 29 percent.
Finally, we assume that in the 1960s the (detrended) capital-people ratio
is back to its steady state value, i.e. ∆(K/N)1929−1960s = 1, and we know
that the productivity shock is of the order of 8 percent above trend. Again,
44
we find an upward deviation in fertility of about 19 percent. In the data, the
upward swing was 23 percent in the 1960s.
4.4
Summary
This is a significant success for several reasons:
1. This model captures more than 65 percent of the baby bust and entire
baby boom.
2. The model captures the long lasting baby boom starting at the end of
the 1940s through the 1960s.
3. For the immediate post-war period (late 1940s and 1950s) most of the
increase in fertility comes from off-BGP dynamics in N and K, while for
the 1960s the 8 percent productivity shock is the main driving force.
Next, we compare sizes of depressions and baby busts, and cpital-people
reatios and baby booms across countries.
5
Depressions, Baby Busts, WWII and Baby
Booms Around the Globe
In this section we take a look at several other countries through the spectacles
of our theory. The countries we consider are Australia, Canada, France, the
Netherlands, Italy, the United Kingdom, Germany and Japan.
First, we report Total Fertility Rate (TFR) series between 1850 and 2000
from Chesnais (1992) and calculate the size of baby busts and booms. For
each of these countries, we also present data on real output per capita between 1915 and 1945 to assess country-specific sizes of the Great Depression
45
(relative to the U.S.)11 Finally, we derive a ranking of pre- to post-WWII
changes in capital-people ratios using data from Maddison (1982). Overall,
we find that the size of the depression together with changes in capital-people
ratios correlate strongly with changes in TFR.
5.1
TFR and the Great Depression
Figures 20 to 27 show TFR series for all countries listed above. In Table 1 we
summarize this data identifying the years and levels of the lowest pre-WWII
and the highest post-WWII TFR observation. This gives us an idea of the
length and size of baby booms in the selected countries. Figures 28 to 35
plot Real GDP per capita for the same countries. As a reference, we plot
U.S. data on each graph as well.
5.1.1
Australia and Canada
The fertility experience in Australia and Canada (Figures 20 and 21, respectively) is very similar to that of the U.S., with small exceptions. Australia’s
pre-war trough happened earlier and the difference in TFR from trough to
peak was slightly smaller. For Canada, the level of fertility in 1900 was
higher, with a faster decrease until the 1930s but a smaller difference in
trough to peak TFR through the 1960s. This fits quite well with our theory
regarding productivity shocks: both countries experiences severe and long
lasting depressions; the depression started earlier in Australia. Moreover,
WWII was not fought on either of these countries own soil – presumably
destroying little capital.
11
We thank Cole et al. (2005) for providing the data.
46
Table 1: Change in Total Fertility Rate, Selected Countries
Country
Year of Trough
pre-WWII a
TFR
at Trough
Year
of Peak
TFR
at Peak
(TFR Peak)
– (TFR Trough)
US
Australia
Canada
1938
1933
1938
2.18
2.19
2.69
1958
1958
1958
3.70
3.41
3.90
1.52
1.22
1.21
Netherlands
France
Italy
UK
Germany
1938
1938
1953
1938
1933
2.61
2.07
2.30
1.79
1.84
1948
1948
1963
1963
1963
3.48
2.98
2.50
2.83
2.50
0.87
0.91
0.20
1.04
0.66
Japan
none
none
none
none
0.00
Source: Chesnais (1992), Table A2.1.
5.1.2
Europe
The pattern of fertility in Europe during the 20th century is, with the appropriate adjustments for the specific aspects of European economic history,
altogether similar to that reported earlier for the U.S., with one important
quantitative exception: the size of the post WWII baby boom. Consider the
data about TFR in Figures 5.3 to 5.8. In France, Germany and the U.K. the
First World War brings about a substantial drop in fertility, as its impact
on the economic conditions and capital stocks of those countries is rather
substantial; this seems to fit rather well with the general causal link we are
exploring in this paper. Similarly, the ‘depression era’ was neither as dra47
matic nor as long lasting for Europe as it was in the U.S., which makes the
much lower drop in fertility during that period also easy to understand from
our point of view. So far so good, but things become less straightforward in
the second part of the century.
The post WWII recovery was much stronger (in relative terms) in continental Europe than in the U.S., and especially so in France, Germany, and
Italy; the 1950s and 1960s witnessed the European convergence phenomenon,
with strong and above average growth in both income per capita and TFP.
One would therefore expect an equally larger European baby boom, at least
compared to the one in the U.S.. But this never materialized. We address
this phenomenon in the next section.
5.1.3
Japan
Japan, did not see a Great Depression comparable to the U.S. (see Figure 35)
and TFR (see Figure 27) did not fall particularly rapidly during that time.
It actually stagnated then increased in the late 1930s and early 1940s when
Japan’s real GDP per capita experienced a boost. Later on, fertility dropped
rapidly, a phenomenon we reconcile with per- to post-war capital-people ratios in the next Section.
5.2
TFR and capital-people ratios
Since data on capital stocks across countries is hard to come by, we use
data on output and labor inputs to back out changes in capital-people ratios
between 1929 and 1950. To do this, we proceed as follows.
5.2.1
Basic Structure
From Maddison (1982), there is data, both time series and cross section, on:
1
Total GDP;
48
2
Labor Force Size, Total;
3
Labor Force Participation Rates for– by sex: males, females; by
age: under 15, between 15 and 64, 65 and over;
4
Labor Force Employment;
5
Annual Hours worked per person;
6
Labor Productivity – GDP per Man-hour in 1970 US Relative Prices.
From these we want to get some idea what happened to K/N ratios.
Below, we go through several possible sets of assumptions to get at this.
There is NO data on K directly, and it is not clear what the data counterpart of what we call N in the model is. This is because the model does not
distinguish between males and females, different ages, or any other reason
for not participating in the labor market.
5.2.2
Basic Assumptions
Assume that total output in country i in year t is given by:
Yit = Kitα (Ait Hit )1−α
where Kit is the capital stock, Ait is the level of labor productivity and
Hit is total hours.
49
5.2.3
Algebra
Assume that Hit = φit Nit .
In this case,
Yit = Kitα (Ait Hit )1−α = Kitα (Ait φit Nit )1−α .
Thus, it follows that productivity is given by:
h iα
Yit
1−α Kit
=
(A
φ
)
;
it
it
Nit
Nit
or,
or,
h
Kit
Nit
Kit
Nit
iα
=
=
h
Yit
(Ait φit )α−1 ;
Nit
Yit
Nit
i1/α
(Ait φit )(α−1)/α .
Hence, for any two years t and τ , we have:
Kit
Nit
Kiτ
Nitτ
=
h
i
Yit 1/α
(Ait φit )(α−1)/α
Nit
h
i1/α
Yiτ
(Aiτ φiτ )(α−1)/α
Niτ
=
h
Ait φit
Aiτ φiτ
i(α−1)/α Yit
Nit
Yiτ
Niτ
1/α
.
If φiτ = φit for all τ and t, we have that:
Kit
Nit
Kiτ
Nitτ
=
h
Ait
Aiτ
i(α−1)/α Yit
Nit
Yiτ
Niτ
1/α
.
BGP’s
Assume that (Kit , Niτ ) is on the BGP for some country i and some base
year τ and that it is also on the BGP in all years between τ and t. Let
γ A denote the (exogenous) rate of growth of labor augmenting technological
change – γ A =
As+1
As
population – γ N =
– and let γ N denote the (endogenous) rate of growth of
Ns+1
.
Ns
50
Note that output per capita, consumption per capita and capital stock
per capita all grow at the rate of technical change on this BGP – γ Y /N =
γ C/N = γ K/N = γ A – and that totals grow at the rate γ Y = γ C = γ K = γ N γ A
on the BGP.
Thus, if a country is on the BGP both in years τ and t we have:
Kit
Nit
Kiτ
Nitτ
=
h
Ait
Aiτ
i(α−1)/α (α−1)/α
γ t−τ
A
h
Yit Niτ
Yiτ Nit
Yit
Nit
Yiτ
Niτ
i1/α
1/α
(α−1)/α
= γ t−τ
A
Yit
Nit
Yiτ
Niτ
1/α
=
(α−1)/α t−τ τ −t 1/α
= γ t−τ
γY γN
A
(α−1)/α 1/α t−τ (α−1)/α t−τ 1/α
= γ t−τ
(γ A γ N )t−τ γ τN−t
= γA
γA
= γ t−τ
A
A .
In detrended variables and on the BGP, we have
∆(K/N)τ
to t
=
Kit
γ t Nit
A
Kiτ
γ τ Niτ
A
=1
Thus, in general,
∆(K/N)τ


=
to t
Yit
Nit
Yiτ
Niτ
γ t−τ
A
=
1/α
Kit
Nit
Kiτ
Nitτ
/γ t−τ
A
(α−1)/α
= γ t−τ
A
Yit
Nit
Yiτ
Niτ
1/α
/γ t−τ
A


gives a measure of how far off country i is from the BGP in period t assuming
that it was on the BGP in period τ . If it is 1 the country is on the BGP in
period t, if it is smaller than 1, the capital-people ratio is too low – and we
expect low fertility and high investment, vice versa if it is greater than 1.
51
Table 2: Change in capital-people ratios, Selected Countries, γ A = 1.017
Country
Y /H1929
Y /H1938
Y /H1950
K/N1938
/γ 38−29
A
K/N1929
K/N1950
/γ 50−29
A
K/N1929
USA
Australia
Canada
2.45
2.16
1.76
2.62
2.34
1.75
4.25
3.05
3.33
0.776
0.807
0.624
1.805
0.973
2.342
Netherlands
France
Italy
UK
Germany
1.82
1.31
0.98
1.7
1.19
1.80
1.69
1.28
1.84
1.47
2.27
1.85
1.37
2.40
1.4
0.614
1.362
1.413
0.804
1.196
0.671
0.974
0.945
0.973
0.563
Japan
0.64
0.87
0.59
1.594
0.271
Source: Maddison (1982).
5.2.4
Sample calculations
Figure 19 plots the ratios in Table 2 against changes in TFR given in Table 1.
The regression line relating the change in TFR and (log) change in detrended
K/N by country shows an R2 = 0.73 with a fairly high correlation. In the
calculations here, we use 1929 as a base year (i.e. on BGP) and a growth rate
of 1.7 percent per year (from Kendrick’s data). Assuming the base year is
1938 and/or the growth rate is 2 percent changes the magnitudes of capitalpeople ratios but not the ranking across countries. We view this as additional
evidence that one-time changes in capital-people ratios have a potential at
explaining fluctuations in fertility.
52
Figure 19: Change in TFR versus relative capital-people ratio, selected countries
1.6
1.4
TFR: Peak-Trough
1.2
(TFR:P-T)= 0.469Ln(relative K/N) + 0.8807
R 2 = 0.7362
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Relative K/N 1929-1950, detrended
5.3
Summary
In this section, we have seen that the analysis in previous sections may help
understanding cross-country differences in sizes of fluctuations in fertility.
One should keep in mind that this mechanism works through the ensuing
increase (or decrease) in labor productivity (and hence wages) and prospects
of future well-being of descendants which enters the utility function.
6
Conclusion
TBW
53
A
Appendix
A.1
Proof of Lemma 1
Proof.
Step 1:
Ct = F (Kt , γ t Nt ) − Xt − θt Nbt
= F (Kt , γ t Nt ) + (1 − δ)Kt − Kt+1 + θt πNt − θt Nt+1
Step 2:
FOC’s are:
(Kt+1 )
−σ
β t Ct−σ = β t+1 Ct+1
[FK (t + 1) + 1 − δ] ,
(Nt+1 )
−σ
β t Ct−σ θt = β t+1 Ct+1
[FL (t + 1) + θt+1 π] .
Step 3:
(Kt+1 )
h
h
Ct+1
Ct
Ct+1
Ct
iσ
iσ
= β [FK (t + 1) + 1 − δ] ,
=
β
θt
[FL (t + 1) + θt+1 π] .
Step 4:
[FK (t + 1) + 1 − δ] =
1
θt
[FL (t + 1) + θt+1 π] .
(Nt+1 )
Assume that the production function is of the form F (K, N) = AK α N 1−α .
Step 5:
αA
α
Kt+1
(γ t+1 Nt+1)
Kt+1
1−α
1−α
+1−δ =
α
t+1 N
t+1 )
(1−α)A Kt+1 (γ
θt
Nt+1
+ π θt+1
.
θt
Assume that costs of children grow at the exogenous rate of technological
t
change, θt = γ t θ, and define the detrended capital stock as K̂t = K
.
γt
54
Step 6:
αA
h
Nt+1
K̂t+1
i1−α
+1−δ =
α
t+1
) (γ t+1 )
(1−α)A Kt+1 (γ
γtθ
(γ t+1 )α Nt+1
αA
h
Nt+1
K̂t+1
i1−α
+1−δ =
α
t+1
(1−α)A K̂t+1 γ
α
t
γ θ
Nt+1
αA
h
Nt+1
K̂t+1
i1−α
+1−δ =
(1−α)Aγ
θ
Step 7:
Step 8:
Let Zt =
α
K̂t+1
.
Nt+1
h
K̂t+1
Nt+1
1−α
1−α
Nt+1
+ πγ.
+ πγ.
iα
+ πγ.
Step 9:
α−1
LHS(Zt+1 ) = αAZt+1
+1−δ =
(1 − α)Aγ α
Zt+1 + πγ = RHS(Zt+1 ). (2)
θ
LHS(Z) is a monotone decreasing function of Z which asymptotes at +∞ at
Z = 0 and asymptotes at 1−δ as Z → ∞. RHS(Z) is a monotone increasing
function of Z with intercept at πγ, which converges to +∞ as Z → +∞.
Thus, it follows that there is a unique Z ∗ such that LHS(Z ∗ ) = RHS(Z ∗ )
and that this Z ∗ does not depend on t.
A.2
Proof of Lemma 2
Proof. From the proof of Lemma 1, Step 3 we have
σ
Ct+1
σ
γC =
Ct
= β [FK (t + 1) + 1 − δ]
Ktα (γ t Nt )1−α
= β αA
+1−δ
Kt
"
#
t 1−α
γ Nt
= β αA
+1−δ
Kt
= β αA [Z ∗ ]α−1 + 1 − δ
55
A.3
Proof of Proposition 1
1. From the restrictions given from technology, we know that along the
BGP, γ t Nt , Kt and Yt must grow at the same rate as Ct .
2. From the law of motion for capital, we have:
Kt+1 = γ ∗ Kt = (1 − δ)Kt + Xt ;
Hence,
Xt Y t
Y t Kt
1−α
AKtα (γ t Nt )
= (1 − δ) + φX
Kt
1−α
t
A (γ Nt )
= (1 − δ) + φX
Kt1−α
γ ∗ = (1 − δ) +
= (1 − δ) + φX A (Z ∗ )α−1
Thus,
φX =
γ ∗ −(1−δ)
.
A(Z ∗ )α−1
This gives φX as a function of the deep parameters of the model since Z ∗
is determined from equation 2 and only depends on deep parameters.
3. From the law of motion for population, we have:
Nt+1 = πNt + Nbt ;
γ t+1 Nt+1 = πγγ t Nt + γγ t Nbt ;
56
Using the growth rate of γ t Nt , we get:
γ t+1 Nt+1 = γ ∗ γ t Nt
Combining the two:
γ ∗ γ t Nt = πγγ t Nt + γγ t φN Yt /θt ;
t
;
γ ∗ = πγ + γφN γ tYθN
t
∗
γ = πγ +
AK α (γ t Nt )
γφN t γ t θNt
γ ∗ = πγ + γφN
AKtα (γ t Nt )
θ
1−α
;
−α
;
AK α
γ ∗ = πγ + γφN (γ t Ntt)α θ ;
∗
γ = πγ +
γφN Aθ
h
Kt
γ t Nt
iα
;
γ ∗ = πγ + γφN Aθ (Z ∗ )α ;
∗
γ −πγ
φN = θ γA(Z
∗ )α ;
Again, this gives φN as a function only of the deep parameters.
4. Finally,
φC = 1 − φN − φX
5. Since γ ∗ =
γ t+1 Nt+1
γ t Nt
must be constant.
(from 1.), we get γ N =
Nt+1
Nt
=
γ∗
γ
6. The growth rate of per capita variables can be determined from 1. and
5. as follows:
Yt+1 /Nt+1
Yt /Nt
=
Yt+1 /Yt
Nt+1 /Nt
=
γ∗
γN
=γ
and analogously for consumption, capital and investment.
57
7. From above, it follows that:
C0 = φC Y0 = φC AK0α (γ 0 N0 )1−α
= φC A
= φC A
K0α K0 (γ 0 N0 )1−α
K0α K01−α
K0 (γ 0 N0 )1−α
K01−α
= φC AK0 [Z ∗ ]α−1
8. Also, Ct = [γ ∗ ]t C0 . So,
X [γ ∗t C0 ]1−σ
Ct1−σ
=
βt
1
−
σ
1−σ
t
t
X γ ∗t(1−σ)
X
t
= C01−σ
βγ ∗(1−σ) /(1 − σ)
= C01−σ
βt
1−σ
t
t
V (N0 , K0 ) =
=
X
βt
C01−σ
1
(1 − σ) 1 − βγ ∗(1−σ)
Here, both C0 and γ ∗ are functions of the parameters only along with
K0 (or N0 ).
NOTE: If βγ ∗(1−σ) > 1, this sum is infinite, and the expression given
in the last step is no longer correct.
9. The last part (9.) is standard homogeneous/homothetic reasoning and
is omitted.
A.4
Proof of Proposition 2
IF the guess that
K1
γN1
= Z ∗ is correct, then, the solution to the problem is as
stated in Proposition 1. That is, Ct = φC Yt for all t ≥ 1,
58
Kt
γ t Nt
= Z ∗ , for all
t ≥ 1, etc. Thus, it follows that the continuation value of the problem can
be calculated as:
V1 (N1 , K1 ) =
= C11−σ β
P∞
t
t=1 β
P∞
Ct1−σ
1−σ
∗t(1−σ)
tγ
t=0 β
1−σ
=
=
1−σ
P∞
t
t=1 β
[γ ∗(t−1) C1 ]
1−σ
βC11−σ
(1−σ)[1−βγ ∗(1−σ) ]
= C11−σ β
P∞
t=1
β t−1 γ
∗(t−1)(1−σ)
1−σ
.
Again, we need that βγ ∗(1−σ) < 1 for the last equality.
Here, C1 = φC Y1 = φC AK1α (γN1 )1−α = φC A
φC AK1 [Z ∗ ]α−1 .
Thus, given that
K1
γN1
α K 1−α
K1
1
1−α
1)
= φC A K1 (γN
1−α
K1
= Z ∗ , we have:
1−σ
V1 (N1 , K1 ) =
K1α K1 (γN1 )1−α
∗ α−1
]
β [φC AK1 [Z ]
(1−σ)
1−βγ ∗(1−σ)
= D (K1 )1−σ ,
1−σ
where D =
Now, if
K1
γN1
∗ α−1
]
β [φC A[Z ]
(1−σ)
1−βγ ∗(1−σ)
.
= Z ∗ , we have, from the law of motion of population that:
N1 = πN0 + Nb0 ;
K1
γZ ∗
= πN0 + Nb0 ;
Nb0 =
K1
γZ ∗
− πN0 .
And, from the law of motion for capital:
K1 = (1 − δ)K0 + X0 ;
X0 = K1 − (1 − δ)K0 .
Thus,
59
=
Nb0 =
1
γZ ∗
[(1 − δ)K0 + X0 ] − πN0 .
Hence, C0 is given by:
C0 = Y0 − X0 − θ0 Nb0 ;
C0 = Y0 − [K1 − (1 − δ)K0 ] − θ 0
C0 =
AK0α N01−α
h
K1
γZ ∗
i
− πN0 ;
− [K1 − (1 − δ)K0 ] − θ0
h
K1
γZ ∗
i
− πN0 .
Thus, IF the guess is correct, the choice of K1 must solve:
ii1−σ
h
h
K1
α 1−α
/(1−σ)+D (K1 )1−σ .
MaxK1 AK0 N0 − [K1 − (1 − δ)K0 ] − θ0 γZ ∗ − πN0
The FOC for this problem is:
ii−σ h
h
h
K1
−
πN
× 1+
AK0α N01−α − [K1 − (1 − δ)K0 ] − θ 0 γZ
∗
0
h
h
ii h
K1
AK0α N01−α − [K1 − (1 − δ)K0 ] − θ 0 γZ
−
πN
0 × 1+
∗
θ0
γZ ∗
θ0
γZ ∗
i
= (1−σ)DK1−σ ,
i−1/σ
= [(1 − σ)D]−1/σ K1 ;
h
h
ii 1+ θ0 1/σ
1−α
γZ ∗
K
AK0α N0 − [K1 − (1 − δ)K0 ] − θ 0 γZ1∗ − πN0 = (1−σ)D
K1 .
This is of the form, LHS(K1 ) = RHS(K1 ), where LHS is a linear function with negative slope and positive intercepts at K1 = 0, and where RHS
is a linear function with positive slope and zero intercept. It follows that
there is a unique solution and that it is positive.
Given this value for K1 , call it K1∗ , the rest of the variables are determined
solely from feasibility and the restriction that the capital-population ratio be
correct. That is:
X0 = K1∗ − (1 − δ)K0 ;
60
Nb0 =
1
γZ ∗
[(1 − δ)K0 + X0 ] − πN0 ;
AK0α N01−α − [K1∗ − (1 − δ)K0 ] − θ0
N1 = πN0 + Nb0 .
h
K1∗
γZ ∗
i
− πN0 ;
After period 0 the solution to the problem is as given in the Proposition 1
and notes following.
There are 2 more restrictions that must be satisfied which correspond to
non-negativity restrictions on investment in physical capital and population.
These are:
K1∗ − (1 − δ)K0 ≥ 0;
1
γZ ∗
A.5
[(1 − δ)K0 + K1∗ − (1 − δ)K0 ] − πN0 =
1
K∗
γZ ∗ 1
− πN0 ≥ 0.
Proof of Proposition 3
Proof. We know that V (λN, λK) ≡ λη V (N, K)
Therefore taking partial derivative with respect to N on both sides gives
λVN (λN, λK) ≡ λη VN (N, K)
VN (λN, λK) ≡ λη−1 VN (N, K)
Analogously,
VK (λN, λK) ≡ λη−1 VK (N, K)
61
A.6
Proof of Proposition 4
Proof.
Step 1:
{(Ct , Nt+1 , Kt+1 }∞
t=0 is feasible for P (N0 , K0 , s0 )
if and only if
{(λCt , λNt+1 , λKt+1 }∞
t=0 is feasible for P (λN0 , λK0 , s0 )
Step 2: U0 (λ{Ct , Nt+1 , Kt+1 }) = λη U0 ({Ct , Nt+1 , Kt+1 }).
Step 3: If the claim is false, then there is something better for P (λN0 , λK0 , s0 )
∗
∗
than λ{(Ct∗ (N0 , s0 ), Nt+1
(N0 , K0 , s0 ), Kt+1
(N0 , K0 , s0 ))}∞
t=0 .
′
′
′
∞
Call this feasible plan {(Ct , Nt+1 , Kt+1 )}t=0 . From Step 1, the plan
′
′
′
1
{(Ct , Nt+1 , Kt+1 )}∞
t=0 is feasible for P (N0 , K0 , s0 ). From step 2,
λ
′
′
′
1
{(Ct , Nt+1 , Kt+1 )}∞
t=0 , gives higher utility than
λ
∗
∗
∗
{(Ct (N0 , K0 , s0 ), Nt+1 (N0 , K0 , s0 ), Kt+1
(N0 , K0 , s0 ))}∞
t=0 ,
a contradiction.
A.7
Proof of Proposition 5
Proof. Taking first order conditions with respect to N ′ and K ′ in (BE),
we get
(F OC N′)
N η+σ−1 C −σ θ = βEVN ((N ′ , K ′ , s′)
(F OC K′)
N η+σ−1 C −σ = βEVK ((N ′ , K ′ , s′ )
Combining the two, we get
EVK ((N ′ , K ′ , s′ )θ = EVN ((N ′ , K ′ , s′ )
From Lemma 3, since V is homogeneous of degree η in (N, K), VK and
VN are homogeneous of degree η − 1. Thus we can rewrite the above as
(N ′ )η−1 EVK ((1, k ′ , s′)θ = (N ′ )η−1 EVN ((1, k ′ , s′ ),
62
or,
EVK ((1, k ′ , s′ )θ = EVN ((1, k ′, s′ )
IF the equation above has a unique solution for k ′ , then k ′ =
independent of today’s K and N stocks, ratio or shock, s.
A.8
K′
N′
Details on the numerical algorithm
Available from the authors upon request
A.9
Cross-country Figures
Figure 20: Total Fertility Rate, Chesnais (1992), Table A2.1, Australia
Australia and U.S.
4.5
4
3.5
TFR
3
2.5
Australia
U. S.
2
1.5
1
0.5
0
1840
1860
1880
1900
1920
Year
63
1940
1960
1980
2000
is
Figure 21: Total Fertility Rate, Chesnais (1992), Table A2.1, Canada
Canada and U.S.
6
5
TFR
4
Canada
3
U. S.
2
1
0
1840
1860
1880
1900
1920
1940
1960
1980
2000
Year
Figure 22: Total Fertility Rate, Chesnais (1992), Table A2.1, Netherlands
Netherlands and U.S.
6
5
TFR
4
Netherlands
3
U. S.
2
1
0
1840
1860
1880
1900
1920
1940
Year
64
1960
1980
2000
Figure 23: Total Fertility Rate, Chesnais (1992), Table A2.1, France
France and U.S.
4.5
4
3.5
TFR
3
2.5
France
U. S.
2
1.5
1
0.5
0
1840
1860
1880
1900
1920
1940
1960
1980
2000
Year
Figure 24: Total Fertility Rate, Chesnais (1992), Table A2.1, Italy
Italy and U.S.
4.5
4
3.5
TFR
3
2.5
Italy
U. S.
2
1.5
1
0.5
0
1840
1860
1880
1900
1920
Year
65
1940
1960
1980
2000
Figure 25: Total Fertility Rate, Chesnais (1992), Table A2.1, U.K.
U.K. and U.S.
6
5
TFR
4
England and Wales
3
U. S.
2
1
0
1840
1860
1880
1900
1920
1940
1960
1980
2000
Year
Figure 26: Total Fertility Rate, Chesnais (1992), Table A2.1, Germany
Germany and U.S.
6
5
TFR
4
Germany
3
U. S.
2
1
0
1840
1860
1880
1900
1920
1940
Year
66
1960
1980
2000
Figure 27: Total Fertility Rate, Chesnais (1992), Table A2.1, Japan
Japan and U.S.
6
5
TFR
4
Japan
3
U. S.
2
1
0
1840
1860
1880
1900
1920
1940
1960
1980
2000
Year
Figure 28: Real GDP per capita, 1925=100, Australia
U.S. and Australia
1.4
RGDP per capita; 1925=100
1.2
1
0.8
US
Australia
0.6
0.4
0.2
0
1915
1920
1925
1930
Year
67
1935
1940
1945
Figure 29: Real GDP per capita, 1925=100, Canada
U.S. and Canada
1.4
RGDP per capita; 1925=100
1.2
1
0.8
US
Canada
0.6
0.4
0.2
0
1915
1920
1925
1930
1935
1940
1945
Year
Figure 30: Real GDP per capita, 1925=100, Netherlands
U.S. and Netherlands
1.4
RGDP per capita; 1925=100
1.2
1
0.8
US
Netherlands
0.6
0.4
0.2
0
1915
1920
1925
1930
1935
Year
68
1940
1945
Figure 31: Real GDP per capita, 1925=100, France
U.S. and France
1.4
RGDP per capita; 1925=100
1.2
1
0.8
US
France
0.6
0.4
0.2
0
1915
1920
1925
1930
1935
1940
1945
Year
Figure 32: Real GDP per capita, 1925=100, Italy
U.S. and Italy
1.4
RGDP per capita; 1925=100
1.2
1
0.8
US
Italy
0.6
0.4
0.2
0
1915
1920
1925
1930
Year
69
1935
1940
1945
Figure 33: Real GDP per capita, 1925=100, U.K.
U.S. and U.K.
1.6
RGDP per capita; 1925=100
1.4
1.2
1
US
0.8
UK
0.6
0.4
0.2
0
1915
1920
1925
1930
1935
1940
1945
Year
Figure 34: Real GDP per capita, 1925=100, Germany
U.S. and Germany
1.6
RGDP per capita; 1925=100
1.4
1.2
1
US
0.8
Germany
0.6
0.4
0.2
0
1915
1920
1925
1930
Year
70
1935
1940
1945
Figure 35: Real GDP per capita, 1925=100, Japan
U.S. and Japan
1.6
RGDP per capita; 1925=100
1.4
1.2
1
US
0.8
Japan
0.6
0.4
0.2
0
1915
1920
1925
1930
Year
71
1935
1940
1945
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