THREE EQUATIONS GENERATING AN INDUSTRIAL
REVOLUTION?
MICHELE BOLDRIN, LARRY E. JONES AND AUBHIK KHAN
Abstract. In this paper we evaluate quantitatively the relationship between economic and demographic growth. We use simple
models of endogenous growth and fertility, featuring human capital
investment at the individual level in conjunction with either of two
models of fertility choice, the B&B dynastic motive, and the B&J
late-age-security motive. We find that exogenous improvements
in survival probabilities, lead to increases in the rate of return to
human capital investment. This leads, in turn, to an increase in
the growth rate of output per capita much like what is typically
seen when countries go through industrialization. In the models
implemented so far, we also find that this is accompanied by a permanent increase in the growth rate of population. Historical data
show that the increase in the growth rate of population during the
takeoff was temporary, and that the transition eventually lead to
a stationary population.
1. Introduction
The idea we pursue in this paper is simple, still ambitious. We lay
out a simple model (in fact: two family of models) of fertility choice,
production, and capital accumulation in which an increase in life expectancy may, in principle, cause a sustained increase in the growth
rates of productivity and income per-capita. We use different variants
of these models to assess the quantitative relevance of a contention that
has frequently been advanced in the literature on economic growth, and
in the context of studies of the British Industrial Revolution especially.
It can be summarized as follows: exogenous improvements in survival
probabilities provide the incentive for investing more in human capital
and productive skills, hence leading to an increase in the productivity
of labor; the latter, through some “endogenous growth” mechanism,
leads to a sustained increase in the growth rate of output. In other
words, the productivity take-off that characterizes most development
processes may be mostly due to exogenous improvements in the life
expectancy of the population. Furthermore, while the initial increase
in lifetime expectancy brings about also an increase in population and,
1
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
2
possibly, of its growth rate, both the increase in the productivity of
(parental) labor time and the enhanced incentives to invest in human
capital (of children) are likely to activate a quantity-quality tradeoff
that leads to a reduction in fertility and, eventually, to a newly stagnant population.
The intuition is straightforward, and it may actually work. Indeed,
it does seem to work in a variety of different theoretical formulations of
the basic intuition, as an extensive literature (discussed in Section 3)
has shown in the recent past. The intuition may also work in practice,
i.e., once solid historical evidence has been used to discipline calibration, a fully specified model of endogenous growth and fertility may
deliver (most of) an industrial revolution and a demographic transition using the historically observed changes in (young’s and adult’s)
life expectancies as the only exogenous “causes”. In fact, a number of
papers have compared numerical simulations of simplified models with
data and claimed that the ‘basic’ stylized facts of economic takeoffs
and demographic transitions are replicated. The most relevant among
these attempts we also discuss in Section 3; in our judgement, things
are less clear cut than the literature so far seems to believe and the
jury is still out on the overall capability of a well specified model of
the ‘survival hypothesis’ to generate an industrial revolution and a demographic transition. As we argue in Section 3 on the basis of the
historical evidence summarized in Section 2, it remains an open question if using historically observed time series of (age specific) survival
probabilities, and realistically calibrated technologies, one can actually
generate time series of output, capital stock, wages, rental rates, and
fertility that resemble those observed in UK during the 1700-2000 time
interval. This paper aims straight at this question, and the preliminary
answer is a qualified ‘maybe’, leaning more toward yes than no. Lots
of non-trivial work still needs to be carried out, by us and by other, to
reach a convincing answer.
In spite of its current appearances, our paper is not a theoretical
one; this research is meant, instead, to be an exercise in quantitative
economic history. For this reason, we do not claim that our model
is able to capture “all” industrial revolution-like sustained increases in
the growth rate of income per-capita. We concern ourselves with a specific historical episode: the English Industrial Revolution of the eighteenth and nineteenth centuries, and use more or less contemporaneous
episodes in other European countries purely to test the robustness of
certain stylized facts. This is not to rule out beforehand that the causal
explanation examined here may turn out to have “universal” value. In
fact, we suspect it may, in so far as the majority of eighteenth and
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
3
nineteenth century take-off episodes we are aware of, were proceeded
by substantial changes in the patterns of mortality, and by a more or
less visible increases in life expectancy. Before drawing such conclusion,
though, the ability of a differently calibrated version of this model to
replicate other episodes of economic and demographic growth should
be assessed. What we have learned so far is that, when calibrated to
the data for England 1700-2000, our two models exhibit an endogenous change in the rates of physical and human capital accumulation,
fertility, and labor productivity which is similar (but not of the same
magnitude of) those observed in the data.
Before moving on to discuss motivational evidence and previous literature, we should indulge a bit longer in a discussion of the basic
mechanism inspected in this paper, and of the modeling choices we
have made. Among the many differences between physical and human
capital, one is attributable to human mortality: when people die, their
human capital (h) goes away with them, while their physical capital (k)
is left behind to their descendants. Hence, everything else the same,
the k/h ratio an individual acquires is decreasing in the length of time
one expect to be able to use h. An increase, at the time of human capital investment decisions, in the expected future flow of workable hours
implies that the rate of return on human capital investment goes up.
If the underlying production function has the form F (k, h), an increase
in h/k increases the return on k in turn, thereby increasing the overall
productivity of the economy. This change causes the growth rates of
output per capita to increase in models where output growth is endogenous and determined by equilibrium conditions of the type: growth rate
of consumption equal to a monotone increasing transformation of the
discount factor times the marginal productivity of capital. Hence, our
choice of a linear homogeneous production function, in which output
per capita y = F (k, h), and the two inputs can be reproduced and accumulated without bound from one period to the next. This choice of
technology rules out changes in the relative price of h and k, something
that may well have taken place but which does not seem to follow, either directly or indirectly, from the ‘survival hypothesis’. We have also
chosen to abstract from ‘exogenous’ TFP in our model, be it neutral
or, as in other exercises, ‘biased’ toward skilled labor. This is only for
a matter of simplicity and not because we believe that changes in per
capita human capital alone can possibly explain the whole evolution
of U.K. aggregate productivity from 1700 to 2000. In fact, as Section
2 acknowledges in reviewing the cumulated historical evidence on the
sources of productivity growth, this cannot clearly be the case, and
a more complete account of the British Industrial Revolution should
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
4
also model and quantify the key technological changes that took place
during those two centuries.
In order to work, the causal mechanism described here needs to satisfy an additional couple of requisites. First, a generic increase in life
expectancy would not do, as the latter can be determined, for example, by a drop in infant or child mortality or by an increase in late age
survival: in both cases the amount of time during which productive
human capital can be used would not change. The survival rate that
matters is the one between the age at which productive skills are acquired and the age of retirement from the work force, that is between
10-15 and 50-60 years of age two centuries ago, and between 20-25 and
60-65 years of age nowadays. This explains why we have paid a substantial degree of attention to the life cycle and the timing of human
capital acquisition/utilization. Second, if the survival mechanism has
to work, the change in survival rates must eventually induce a reduction
in total fertility, that is they must be correlated to a movement along
the quantity/quality dimension among offsprings. Ideally, what one
would like to have is that births fall, survivors increase, investment per
survivor rises, time spent by survivors in productive activities increase,
hence incentive to invest in both h and k also increase. A sharp enough
reduction in birth rates is not an obvious consequence of an increase in
child and youth survival probabilities, as the ‘higher return on children’
the latter implies may, in reasonable circumstances, lead to an increase
in the number of children; something that is well known to happen, at
realistic parameter values, in models of endogenous fertility. This is a
crucial feature of the problem at hand, which has lead us to inspect
two different (almost opposite) rationalizations of why people have children and why they may or may not invest in their training. The first
is the well known ‘dynastic altruism’ motivation of Becker and Barro
(1988), while the second is commonly known as the ’late-age-insurance’
motive, and we use the formalization advanced in Boldrin and Jones
(2002). There is, therefore, a sense in which part of our paper consists of a “horse race” between different views of endogenous fertility.
As mentioned, we know from previous work (i.e. Boldrin and Jones
(2002)) that the B&B explanation tends to perform badly when used
to model endogenous fertility reactions to changes in mortality rates
in models of exogenous growth. We are interested to see how it does
when the same changes in mortality rates are used to generate changes
in human capital accumulation over the life cycle and increases in the
growth rate of output and labor productivity. Also, we know that the
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
5
B&J explanation, in its non-cooperative version,1 tends to predict levels of fertility that are low by historical standards, and much more
elastic (downward) to reductions in mortality rates than the cooperative version, which in turn predicts higher fertility rates. Again, such
features of the B&J model appear, at realistic parameter values, when
growth is driven by exogenous TFP, and how performances change in
an endogenous growth context is neither clear nor obvious.2
Additionally, as we like to impose upon our quantitative exercise the
discipline of historical measurement, we do abstain from introducing
into our models a number of ‘external effects’ that are frequently encountered in this literature, and that often become the main driving
force behind the final results. This is not meant to imply we are completely convinced that a number of external effects, especially when it
comes to the impact of public health and education, cannot have played
a role in the evolution of mortality rates and overall productivity during the British Industrial Revolution. In principle, they may have been
quite relevant, but no quantitative measurement of their nature, size
and impact is anywhere to be found. In this sense, ‘human capital
externalities’ are a bit like phlogiston, a fascinating and theoretically
useful, but hardly measurable entity. Our choice of sticking to models
where external effects are altogether absent means, then, that at least
when trying to perform an exercise in quantitative economic history, we
deem it appropriate to adhere to the rule ‘whereof one cannot measure,
thereof one should not assume.’
2. Motivational and Historical Evidence
The idea that there is a connection between economic development
and population dynamics is an old one. It appears in many forms
in the literature in both areas. Indeed much of the literature on the
Demographic Transition has focused on this link as a causal one —
the reduction in fertility is caused by the increase in per capita GDP
that goes along with the development process. The reasoning that
the relationship is causal is tenuous at best and really comes down to
the observation that the two series are co-integrated. Figure 1A below
shows the basics of the relationship in a historical setting in the UK for
1We
apologize to the reader for the, temporarily, cryptic language, that should
become less so once the formal models are introduced. The details of the cooperative
and non-cooperative case are discussed in Boldrin and Jones (2002).
2In this preliminary version of the paper, it remains unclear what the answer to
this question is. Much to our chagrin, we are still far from satisfactorily calibrating
and simulating the B&J model for versions other than the simplest one, and even
then for BGPs only.
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
6
Figure 1A: UK Population and GDP per capita, 1565 to 1990
12
ln(Pop), ln(GDP)
10
8
ln(GDP)
ln(Pop)
6
4
2
0
1550
1650
1750
1850
1950
Year
Figure 2.1
the period 1565-1990; data are reported in logs to facilitate the visual
capture of changes in the underlying trends.3
For output, it can be seen that there is a slow but steady climb over
the period from 1550 to about 1830 (or 1850). At that point output
growth accelerates as indicated by the change of slope in the series.
Population also shows a slow steady climb at first, lasting until about
1750. After that point, there are two distinct changes. First is the
acceleration that occurred between 1750 and about 2000, and then the
dramatic slowdown after that, with the series becoming almost flat
since 1960 or so.
Figure 1B tells a similar story, but in growth rates rather than loglevels. These series (naturally) show much more volatility than those
in Figure 1A, where the visual image is swamped by growth. Even still
one can easily see the changes discussed above as the average growth
rate of output shifts clearly upward (albeit volatilely so) and the growth
rate of population first goes up, then later, around 1900, falls again. An
3Data
sources are as follows. Population data, 1541-1866: Wrigley, Davies, Oeppen and Schofield, ”English population history from family reconstitution”; 18711970: Mitchell, ”European Historical Statistics”. GDP data are taken from Michael
Bar and Oksana Leukhina (2005). The basic sources are Clark from 1565 to 1865,
and Maddison from 1820 to 1990. Clark is normalized so that 1565 = 100, Maddison
is renormalized so that Maddison = Clark in 1865.
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
7
Figure 1B: GDP and Pop Growth Rates: UK 1565 to 1990
1.025
1.02
Growth Rate
1.015
1.01
GDP Growth
Pop Growth
1.005
1
0.995
0.99
1550
1600
1650
1700
1750
1800
1850
1900
1950
2000
Year
Figure 2.2
important additional feature of the UK data, which is only weakly replicated in data from other European countries (see Appendix II) is the
following: around 1770 the population growth rate increases substantially, and remains high for about a century and a half, before dropping
to levels near zero. This increase in population growth anticipates the
increase in output growth of between 50 and a 100 years4, but output
growth remains high after that or even increases till our days.5 These
two differences are crucial for our testing of the ‘survival hypothesis’:
(i) population growth increases before income growth does, (ii) the increase in population growth rate is temporary, and not accelerating,
while that in per capita GDP is permanent, and accelerating for about
two centuries.
Since this is of interest for our modeling choices, and relevant to assess the ability of different models to replicate the actual causal mechanism, if any, between sruvival probabilities and economic growth, we
also include here the historical record on survival probabilities at different ages.6 Shown in Figure 1C are time series of the probability of
4For
more details about output growth between 1700 and 1900, and on the
modifications of historical estimates brought about by recent work, see later in this
Section.
5Clearly, we are abstracting from the impact of the Great Depression, which is
visible in Figure 1B.
6Survival Data sources are, 1541-1866: Wrigley et al.
(); 1840-1906:
http://www.mortality.org; 1906-1990: Mitchell ().
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
8
Figure 1C: Survival Proabilites in the UK: 1565 to 1970
1
Probability
0.9
Surv5W&S
Surv15W&S
Surv5Mort
Surv15Mort
Surv20Mort
Surv5Mitch
Surv15Mitch
Surv20Mitch
0.8
0.7
0.6
0.5
1550
1600
1650
1700
1750
1800
1850
1900
1950
2000
Year
Figure 2.3
surviving to ages 5, 15 and 20 since 1565. As can be seen survival rates
to working age have shown a dramatic increase in 1880 or so. But also
note that the survival rate to age 15 may have picked up substantially
earlier, around 1750 or so.
[SURVIVAL PROBABILITIES TO 40 AND 60 YEARS TO BE
ADDED]
The same kind of relationship IS seen in cross sectional data in more
modern times. For example, cross sectional data on the average, over
the 1960 to 2000 period, rate of growth of per capita GDP and the
average, again over the 1960 to 2000 period, rate of growth of population for a sample of 98 countries taken from the Penn World Tables is
shown in Figure 2.
The relationship between γ y (per capita GDP growth) and γ n (population growth) is statistically and economically significant. Statistically, the estimated relationship is:
γ y = 0.0343 − 0.826γ n
The R2 of this regression is 25.4%, and the standard error of the
regression coefficient is 0.1444, giving a t-ratio of -5.72. Thus, with
every decrease of 1% in population growth rates, there is an associated 0.83% increase in per capita output growth rates, economically
a very large change. The strong correlation evidenced here is purely
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
9
Figure 2: Real GDP per capita Growth - Population Growth
0.07
Real GDP per capita Growth (1960-2000)
Korea
0.06
Hong Kong
0.05
Romania
Japan
China
Thailand
Malaysia
0.04
Republic of Congo
0.03
Syria
USA
UK
0.02
Jordan
Uruguay
0.01
Jamaica
0
0
0.005
0.01
-0.01
0.015
0.02
Mozambique
0.025
0.03
Madagascar
0.035
0.04
0.045
0.05
Nicaragua
Niger
-0.02
Sample: 98 countries
Source: Penn World Table ver 6.1
Average Population Growth Rate (1960-2000)
Figure 2.4
motivational, and we should focus next on more detailed UK data for
the period 1700-2000. Still, the reader should also consult () and ()
for further analysis of recent cross country data and the claim that the
causal mechanism investigated here is actually relevant to understand
current development experiences.
The historical evidence we consider next is of three types: (i) estimates of the historical evolution of per capita GDP in the U.K. from
1700 to the early 1900s (DATA FOR XX CENTURY TO BE ADDED);
(ii) data on the demographic transition in the U.K. during the same
period, with a particular attention to the evolution of age-specific mortality rates and fertility; (iii) additional evidence, both from the U.K.
and other countries, of the timing of demographic and economic change.
2.1. Output Growth and its Sources. The following table is constructed using estimates reported in Crafts (1995), and Crafts and
Harley (1992). The factor shares used to compute the estimates are,
respectively, K = 0.40, L = 0.35 and H = 0.25, where K is the aggregate capital stock, H is a measure of human capital7 and L is total raw
labor. While this parameterization of the aggregate technology is not
exactly equivalent to the one we use in our modeling and calibration
exercise, it is the closest available in the literature, and should provide
7Add
details on how this measure is constructed.
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
10
a reasonable ‘target’ for the kind of detailed facts the formal model
should be able to match in order to be called ‘successful’.
Table 1: Output Growth, 1700-1913 (% per year)
Period
∆Y /Y ∆K/K ∆L/L ∆H/H ∆T F P
1700-1760
1760-1780
1780-1801
1801-1831
1831-1873
1873-1899
1899-1913
0.7
0.6
1.4
1.9
2.4
2.1
1.4
0.25
0.45
0.70
0.90
0.80
0.80
0.20
0.35
0.40
0.45
0.30
0.30
0.10
0.40
0.55
0.70
0.50
0.50
0.05
0.10
0.25
0.35
0.50
-0.20
The most striking feature of Table 1 is the very small growth rate
of TFP all along the two centuries under consideration, and especially
until 1850. This finding, that aggregate TFP growth was quite small
during the initial two centuries of the British Industrial Revolution, and
that accumulation of factors was the main driving force, characterized
Crafts and Harley fundamental contribution to the economic history
literature, and has helped to substantially alter the view of most economic historians about the nature of the Industrial Revolution. When
first published, the Crafts and Harley estimates of output and productivity growth were received with doubts, and gave rise to a fairly intense
debate. Their basic contention was that substantial technological improvements did take place during the period, some of which did imply
revolutionary changes, but such improvements were restricted to very
few sectors of the UK economy. More precisely, their historical work
shows that dramatic increases in productivity took place in the textile
(especially cotton), shipping, and metallurgical industries, the size of
which were, nevertheless, too small to generate the increase in aggregate TFP that previous researchers has estimated. Our readings of the
literature that ensued, and of the current consensus among economic
historians, is that the Crafts and Harley estimates, including minor
recent revisions, are the best available and, short of new additional
evidence, should be taken as summarizing the basic facts any model of
the British Industrial Revolution should be capable of matching. This
is, at least, the position adopted in this paper. In summary, from Table
1 we learn two things our modeling exercise must be able to account
for:
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
11
(1) Output growth did not increase suddenly from the previous
0.3 − 0.5% per year rate8 to 2% a year. The increase was slow
and progressive, and even more so (see below Table ??) if one
consider output per employee or output per capita (recall from
above that the population growth rate is close to 1.5% per year
between 1770 and 1890.)
(2) The contribution of physical and human capital accumulation
to output growth was quantitatively more important than that
of TFP during the whole period. It is only at the very end
of the nineteenth century that TFP growth starts to become
substantial.
Table 2, also based on Crafts and Harley estimates as reported in
Clark (2004, Chapt. 9), confirms and qualify these findings by using an
aggregate production function that excludes direct measures of human
capital but introduces land (Q). Again, TFP growth is slow at the
beginning and progressively accelerates, and, again, a sharp increase
in capital to labor intensity accounts for most of the growth in labor
productivity until the second half of the nineteenth century. In these
estimates, the factor shares adopted were of 35% for capital, 15% for
land and 50% for labor. For detailed estimates of factors share, see
Stokey (2001:69), showing that labor share goes from .45 to .47 between
1780 and 1850, capital share goes from .35 to .44 and land share goes
from .20 to .09 during the same period.
Table 2: Per Capita Productivity Growth (% per year)
Period
Y /L K/L Q/L T F P
1700-1760
0.30 0.31 -0.29 0.24
1801-1831
0.45 0.36 -1.35 0.53
1831-1861
1.09 0.59 -1.42 1.10
1760-1861
0.54 0.37 -1.12
0.58
Finally Table 3, adapted from Clark (2001, 2004) reports a number
of other measures of factor prices and productive efficiency for the
period of our interest. Wages are computed for a 10 hour day, land
rent is in $ per acre, and the return on capital is in percentage points.
The index of productive efficiency is normalized to a value of 100 for
the decade 1860-1869. It provides a further confirmation of what we
already said: TFP growth was slow at the beginning, accelerating only
well into the nineteenth century. Notice also that, while the return
8For
the most recent and reliable estimates of the growth rate of output pre-1700
in the UK see Clark (200?) and Feinstein (199?).
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
12
on capital is quite stable, the return on land increases dramatically, as
does the wage rate per day of work after 1800.
Table 3: Efficiency Growth in the Industrial Revolution
Decade
W R on Q R on K Input Costs Efficiency
1700-9
11.7
0.45
4.67
60.7
1710-9
12.1
0.48
4.96
60.9
1720-9
12.3
0.52
4.38
62.9
1730-9
12.7
0.51
4.14
69.7
68.9
1740-9
12.9
0.47
4.24
71.9
64.8
1750-9
12.9
0.59
4.26
78.5
65.6
1760-9
13.7
0.59
4.04
87.5
64.4
1770-9
14.6
0.69
4.15
102.2
64.4
1780-9
15.3
0.68
3.95
112.1
64.3
1790-9
18.2
0.83
4.10
142.9
66.3
1800-9
25.1
1.15
4.38
215.5
70.4
1810-9
34.1
1.47
4.63
296.5
81.6
1820-9
31.3
1.29
4.48
309.4
88.3
1930-9
32.8
1.22
4.85
341.5
95.0
1840-9
33.6
1.15
4.28
398.0
97.4
1850-9
35.4
1.24
4.10
462.8
98.9
1860-9
39.8
1.37
4.27
581.4
100.00
Clark (2002), also shows that agricultural wages decline from 1540 all
the way till 1640, when a slow recovery begins that is reversed again
about a century later. It is only by the middle of the 19th century
that real agricultural wages go back to their level of 300 years before,
and grow monotonically to new heights every period after that. The
indices he computes for the productivity of capital in the agricultural
sector show an almost parallel dynamics, even if the levels are different
and, in particular, the capital productivity level of 1550 is reached and
surpassed many times between 1620 and 1850; nevertheless, it is not
until 1800 that a monotone increasing trend sets in. Interestingly, the
estimated rates of return decrease from historical highs of 6.2 % in
1600-1620 to 4.2% in 1800 and then to 3.9% in 1900.
Various other data sources suggest that accumulation of productive
factors played a major role in ouptut growth, at least until 1850 and
probably after that. For example, the share of consumption over income decreased from 93% in 1700 to 80% in 1840, while the investment
to output ratio more than doubled, from 4.0% to almost 11% in 1840
(Feinstein 1988). Population and net fixed capital formation growth
rates are similar and raising, period after period, between 1760 and
about 1810. After that the population growth rate decreases, from
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
13
a pick of about 1.4%, while capital formation growth rate accelerates
and reaches almost 3% by 1840. (Nick von Tunzelmann, in Floud and
McCloskey, eds. (1994), p. 290). What this implies is that the K/L
ratio increases, slowly but steadily, from the middle of the eighteenth
century onward, and that after 1810 or so its growth rate accelerates
substantially.
MORE ON MEASURES OF LABOR PRODUCTIVITY AND FACTOR SHARES TO BE ADDED
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
14
2.2. Mortality and Fertility. What happened, during the same period, to mortality and fertility rates? Figure 3 summarizes the answer:
(a) mortality rates remained high well into the eighteenth century, improving only marginally toward its end, and then only for adult people;
(b) fertility tagged along with mortality rate as before, but then took
off substantially in the second half of the eighteenth century, leading
to a dramatic increase in the growth rate of the population; (c) infant
(and child) mortality did not improve seriously only until well into the
nineteenth century, and it is only in the very second part of this century
that dramatic, and dramatically fast, improvements take place.
45
250
40
200
35
30
150
25
20
100
15
10
50
5
Crude Birth rate
Crude death rate
1961
1946
1931
1916
1901
1886
1871
1856
1841
1826
1811
1796
1781
1766
1751
1736
1721
1706
1691
1676
1661
1646
1631
1616
1601
1586
1571
1556
0
1541
0
Infant Mortality
Figure 2.5. Figure 3: Britsh Crude Birth, Death
and Infant Mortality Rates 1541-1970
These mortality and fertility patterns generate the compounded population annual growth rates reported in Table 4 (from Wrigley and
Schofield (1981: 208-9)), and the Gross Reproduction Rates9 and expectation of life at birth reported in Table 5 (also, adapted from Wrigley
and Schofield (1981:231)). Both ventennial and decennial values have
been obtained by averaging over the quinquennial values that Wrigley
and Schofield report. There is one important message one wants to
learn here: life expectancy did not increase by any substantial amount
between 1700 and 1870, and this is because infant and child mortalities
9The
GRR is, basically, the same as the TFR restricted to female offsprings. It
does, therefore, take into account the fact that male infants are slightly more likely
than female infants.
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
15
did remain very high until late in the eighteenth century. On the other
hand, as Figure 1C above shows, survival probabilities from age five
to age twenty did increase of a visible amount during the same period.
It is with this quantitative restrictions on the actual improvements in
mortality rates that a model testing the ’survival hypothesis’ for the
British Industrial Revolution has to cope. Life expectancy is not only
the inappropriate theoretical measure to look at, but, historically, it
started to grow only when the IR was well under way; expectation of
life at birth, between 1681 and 1741 oscillates from a minimum of 27.9
years to a maximum of 37.1 years, from there it slowly raises to reach
40 years in 1826-1831 and 41.3 in 1871.
Table 4:English Population compound annual growth rates
1681-1701 1701-1721 1721-1741 1741-1761 1761-781
0.25
0.28
0.21
0.49
0.68
1781-1801 1801-1821 1821-1841 1841-1861 1861-1881
1.04
1.42
1.08
1.18
1.26
Table 5: GRR and Life Expectancy
1681-91 2.06 30.1
1691-01 2.17 34.5
1701-11 2.29 36.8
1711-21 2.15 36.5
1721-31 2.24 32.5
1731-41 2.28 32.3
1741-51 2.25 33.5
1751-61 2.33 37.0
1761-71 2.38 34.6
1771-81 2.51 36.9
1781-91 2.56 35.3
1791-01 2.76 37.0
1801-11 2.81 37.3
1811-21 2.97 37.8
1821-31 2.92 39.6
1831-41 2.56 40.5
1841-51 2.43 39.9
1851-61 2.42 40.0
1861-71 2.53 40.7
Our allegation, TO BE BETTER DOCUMENTED, is that such increase in survival rates, first slow and limited to the adult segments
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
16
of the population and then much more dramatic, generalized and particularly strong among children and infant, is to be considered as an
exogenous technological (or medical) shock and not as the effect of the
improvement in the living conditions of the British people that took
place before it. The fundamental motivation for our position is that,
on the one hand, an overwhelming number of demographic studies (see
Livi Bacci (2000) for an excellent summary) have documented that the
range of caloric-proteinic intake and living conditions compatible with
human survival is so large that the purely economic improvement in
the living conditions of the British people that took place between 1700
and 1800 cannot possibly be the cause for the increase in survival rates
that ensued. On the other hand, our reading of the specific causes of
the TO BE ADDED The late 1720s is the last time in which England
is affected by epidemics in any relevant form; in 1731 the total population is at 5.263 million. After that, the population growth rate starts
accelerating, peaking at 1.55% per year in the quinquennium 1826-1831
and then falling to around 1.2% a year until 1871, at which point the
population of England had quadrupled to 21.501 million.
Razzell (1993) contains a strong advocacy of the idea that “Population growth in 18th century England was due mainly to a fall in
mortality, which was particularly marked during the first half of the
century. The fall affected all socioeconomic groups and does not appear
to have occurred for primarily economic reasons.”
The idea that is is income growth that causes a decline in mortality,
and not viceversa, finds strong support, at least prima face, from the
observation that it is only in the second half of the 19th century that life
expectancy increases substantially, and this is at least a century after
income per capita had started to visibly increase in Europe, and in the
U.K. in particular. Further, this is not true only for the U.K. but seems
to be a general feature of most countries: for all European countries
life expectancy trend changes between one and half a century after per
capita GDP trend changes, and the growth rate in life expectancy after
the take off is three to six times higher than before; see, e.g. Easterlin
(1999, Tables 1 and 2). One should hasten to add, though, that looking
only at life expectancy at birth is rather distorting, and that the sharp
increase
Further, the same data also show that this fact is not true for late
growing countries such as Brazil or India: in their case, life expectancy
and economic growth move either simultaneously or the former earlier
than the latter, suggesting that life expectancy is mostly driven by the
spread of advanced medical practices, and that the latter, while vaguely
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
17
correlated with income levels, tend to follow a dynamics that is mostly
independent from that of per capita income growth.
TO BE ADDED
3. Review of Previous Literature
Libraries have been written, both on the Industrial Revolution and
on the Demographic Transition in the United Kingdom between 1600
and 2000, hence attempting a true review of the literature would result
in an impossible task. We focus here only on the recent literature, say
post 1980, and, within this, on three kinds of works: (i) those providing
data and empirical support to the particular causation mechanism we
are testing; (ii) those modeling such causation mechanism, in particular
those works that have attempted to bridge the gap between theory
and data; (iii) those works that explicitely question the direction of
causality we are concerned with and suggest that either the opposite is
true or that changes in age-specific survival probabilities had little or
nothing to do with the increase in the growth rate of output per capita
in the U.K after 1750.
One of the first researchers to begin addressing the potential causal
link between health and economic growth is Samuel Preston, see Preston (1975). His conclusion, based on data for the period 1900-1960,
was that health improvements were mostly due to autonomous technological change in the ‘health production function’, with little contribution from increases in income per capita per-se. On the opposite
side, and practically at the same time, Thomas McKeown, McKeown
(1976), uses data for England and Wales from the middle of the 19th
century to argue that it could not possibly be the case that improvements in medical technology were capable of explaining the increase in
life expectancy observed at the time in those areas.
McKeown’s argument, which was interpreted mostly as an argument
for the market as a provider of health services, has been widely criticized ever since, see Mercer (1990) and Easterly (1999) for recent examples and review of other works. This criticism TO BE ADDED
Among the most interesting recent economic papers that investigate
the same causation channel as we do, the following are closer to our
approach: Boucekkine, de la Croix and Licandro (2002. 2003), Cervellati and Sunde (2002), Kalemli-Ozcam (2002, 2003), Soares (2005),
and Zhang, Zhang and Lee (2001). DISCUSSION TO BE ADDED.
Hazan and Zoabi contains a sharp theoretical criticism to most of
the modeling strategies adopted in the previous literature. Correctly,
we believe, they point out that “[...] greater longevity raises children’s
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
18
future income proportionally at all levels of education, leaving the relative return between quality and quantity unaffected. [...] Our theory
also casts doubts on recent findings about a positive effect of health on
education. This is because health raises the marginal return on quality
and quantity, resulting in an ambiguous effect on the accumulation of
human capital.” This criticism does not apply to the models studied
here, as it is the incentive to invest in physical versus human capital
that is affected by the changes in survival probabilities. Further, we
investigate age specific survival probabilities directly, and allow for the
quantity/quality tradeoff to be resolved at the quantitative level.
In most of the literature on demographic behavior and economic
growth, the problem is almost invariably cast in the following very
stylized form: before the second half of the 18th century European
countries, and the rest of the world indeed, were at a stationay Malthusian equilibrium with low (approximated to zero) growth in income
per capita, high mortality, high fertility, low (approximated to zero)
growth of population and small or nihil investment in human capital
and productive skills. A transition to a new stationary state of sustained growth in income per capita, low fertility and mortality (in fact,
constant population) and high investment in productive human capital
begins around 1750 and is completed roughly two century later. During the transition, mortality and fertility decline somewhat in parallel,
human capital accumulation progresses and the growth rate of income
per capita increases till it stabilizes at around two per cent per year
after WWII, finally, population multiplies by a factor of five before
becoming constant again. While there is lots of truth in this stylized
story, we actually believe it is a bit too much stylized. Because it is
too much stylized it is consistent with very many different and incompatible explanations, among which it is hard, not to say impossible, to
discriminate when the set of “facts” to be accounted by the theory is
restricted to those we just listed.
Additionaly, a variety of othe authors have investigated in isolation
one of more of the mechanisms we assemble here. The one most often encountered is the idea that an increase in life expectancy incrases
the incentive to acquire human capital (Blackburn and Cipriani [2004]
). Cavalcanti and Abreu de Pessoa (2003) look at the interaction between longevity and government fiscal and education policies. TO BE
ADDED.
It may not be unfair to say that the “dominant” view is the one
summarized in the ‘Malthus to Solow’ parable of Hansen and Prescott
(2002) and in the ‘development miracle’ Lucas (2002). While we concur
with the general description of the facts presented in these two papers,
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
19
and in the many other that follow similar lines, we find the underlying theory lacking in so far as some ’exogenous miracle’ becomes the
essential driving force, be it exogenous TFP growth or a particularly
engineered demand for children or a pervasive and aggregate human
capital externality. TO BE ADDED
4. Models of Endogenous Fertility and Growth
We will now present two classes of models of endogenous growth
and endogenous fertility: the ‘dynastic’ model of an infinitely lived
family (due originally to Becker and Barro (1988), and the ‘late age
insurance’ model, theorized among other by Caldwell (1978). We label
the first the B&B model, and the second the B&J model, as we use the
particular formalization of the late age insurance motive introduced by
Boldrin and Jones (2002).
4.1. The Cost of Children Under Certainty Equivalence. One
of the key tricks that we will be exploiting is a way of modeling childcare
costs in a certainty equivalent environment. The costs of raising a child
are many and complex. Moreover, in an uncertain world, like that in
the 18th and 19th centuries in the (currently) developed world, and
in much of the developing world still today, the cost of raising a child
depends on how long it survives. For example, if schooling is only
provided for those children reaching age 5, it is only a cost, ex ante, for
those expected to survive. In a certainty equivalent world, this can be
introduced into standard models of fertility choice by having different
costs for different ’ages’ of the kids. The formulation we adopt is based
on Schoonbroodt (2004).
The key step amounts to properly defining θt , the total cost of raising
one child. For simplicity, assume that there are 3 stages of life for a
child. Roughly, they correspond to ’newborn,’ ages 0 to 1, ’child,’
ages 1 to 5, and ’school aged,’ ages 5 to 15. This is supposed to
represent a ’rough’ division, finer ones would be possible, and will be
considered in future versions of this work. Let π 2t and π 3t denote the
probability of survival to the beginning of the second and third phases
of childhood conditional on being born. Thus, π 2t = (1 − IMRt ), and
π 3t = (1 − CMRt ). Then, we assume that the (expected) cost of a
newborn is given by:
m
θt = [b1 + π2t b2 + π 3t b3 ] ht wt + [α1 + π 2t α2 + π3t α3 ] cm
t = bt ht wt + αt ct
where, bi is the amount of time required from the parents for a child in
phase i, and αi is the amount of physical consumption good, relative
to parents consumption, required for a child of phase i. Thus, in this
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
20
formulation, b2 is only spent on those children that actually survive to
phase 2, and so forth. Then, denoting with µt = π 3t the probability
that a born child survives to the age in which he is educated and/or
trained in some productive activity, the total cost of rearing nt children
is given by:
nt [θt + µt ht+1 ]
.
Finally, let π t denote the probability to survival to adulthood. The
reason that this is a convenient formulation is that, from the point of
view of the differential effects of changing survival rates on the incentives for human capital formation, what we probably want to focus on
is the data analog of changes in π/π 3 , that is, the conditional probability to survival to working age given survival to age 5, since no human
capital investment is made for those children that die in infancy.
What we do next is to add this formulation into the B&B and B&J
models. Throughout this version, we will assume that α1 = α2 = α3 =
0 for simplicity.
4.2. Different Specifications. There is a large variety of different
specifications of the B&B and the B&J models that one would like to
consider, eight of which are reported in Appendix I (to becomes sixteen
as the T=4 cases are added!) Some of these versions are implementable
in both models, while other are not. Those that are common to both
B&B and the B&J fertility theories, are the following: (a) different
modeling of the length of time spent working versus the length of time
spent in training and the length of time spent in retirement, (b) different specifications for the aggregate production function, (c) different
specifications for the market arrangement in the provision of child care
services. The different versions that are special only to the B&J model
are due to (d) different solution concept for the donations game played
among the members of the middle age generation. Let us illustrate the
structure of the various versions, starting with point (d).
In the B&J model, when they become middle-age children internalize old parents utility from consumption, and therefore are motivated
to provide them with some transfer of consumption goods, which we
label “donations” and denote with dt . How much per capita donation is
chosen, in equilibrium, is nevertheless not uniquely determined by how
much children like their parents, which we denote with ζu(cot ), as the
within-siblings agreeement about how individual donations should be
arranged also matters. Two polar cases are relevant: in the cooperative
one siblings maximize joint utility by choosing a common donation d;
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
21
in the noncooperative case, each person take the other siblings’ donation as given and chooses its best response, with the equilibrium being
symmetric Nash. In the cooperative case, ceteris paribus, donations
are higher, and so is fertility, with respect to the cooperative equilibrium; further, because per capita donations decrease less rapidly in the
cooperative than in the noncooperative, the (negative) elasticity of endogenous fertility to an increase in the survival probability of children
is smaller (in absolute value) in the cooperative case; more detailes
about the two cases and their properties can be found in Boldrin and
Jones (2002).
With respect to (c), we look at two basic case: child care services
are produced directly at home (which we label the one sector case) or
are produced by the market (the two sector case). In both cases the
technology for rearing children is the same, and it uses both goods and
time as specified in the previous subsection. The difference between
the two models is subtle and has to do with the incentives of parents
to invest in the human capital of the children: in the one sector case, by
increasing the human capital of their own children, parents affect both
their future labor income and their opportunity cost of having children,
in so far as the latter requires parental time. In the two sector model,
parental investment in the human capital of their own children only
increases the labor income of the latter, as the cost of rearing children
is determined by market prices, and cannot be influenced by individual
variations in parental human capital.
With respect to (b), we consider both the general CES production
function y = F (k, e
h) = A [ηk ρ + hρ ]1/ρ , −∞ < ρ < 1, and the special
Cobb-Douglas case y = F (k, e
h) = Ak α h1−α 0 < α < 1. The reason for
studying also a general CES is that, for the latter, when ρ 6= 0, variations in factor intensities induce endogenous variations in factor shares.
The historical evidence we have exhamined suggests that sizeable variations in the share of income going to capital, labor, and human capital,
did occurr in the period under consideration.
Finally, (a) is an obvious requirement for two reasons. First, one
always looks for the technically simplest model that can do the job;
meaning, in this case, the model with the minimum number of periods that allow facing the data. Our current guess is that models in
which individuals live for four periods, of twenty years each, should be
enough for our purposes. Second, as our contention is that changes
in age specific survival probabilities have substantially different effects
on investment, human capital accumulation and fertility, one needs to
work with a model that is rich enough to treat variations in age specific
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
22
mortality rates independently. Furthermore, the historical evidence itself says that the growth in life expectancy was not the outcome of
a homogeneous drop in mortality rates at all ages, but that, instead,
different mortality rates changed at different points in time.
5. The B&B Model with Endogenous Growth
5.1. The Simplest B&B Model. The Household Problem is given
by:
U=
∞
X
β
t
g(Nts )u(cst )
t=0
subject to:
Rt Kt 10
=
∞
X
t=0
β
t
g(π t Ntb )u(cst )
=
∞
X
β
t
g(π t Ntb )u
t=0
b
b
s s
Ct + at Nt+1
+ ŵt ˆcc
t Nt+1 + Ht+1 + Kt+1 ≤ wt Nt ht
µ
Ct
π t Ntb
¶
m
s cc
t + ŵt Nt t +
ˆcc ≥ bt
t
m
t
+ cc
t ≤ 1
where Ct is aggregate spending it period t by the dynasty on conb
sumption, Nt+1
is the number of births in period t, Ht+1 , and Kt+1 are
aggregate spending on the dynasty on human and physical capital in
period t, Nts = π t Ntb is the number of surviving members in the dynasty
in period t, is the effective labor supply per surviving period t middle
aged person, hst is the human capital per surviving middle aged person
in period t, ˆcc
t is the number of hours of child care purchased per new
child, at is the goods requirement per child by the dynasty in period t,
m
t is the number of hours the typical dynasty member spends in the
market sector working, cc
t is the number of hours the typical dynasty
member spends in the childcare sector working, Rt is the gross return
on capital, wt is the wage rate for one unit of time with one unit of
human capital (so wt hst is the wage rate per unit of time for labor with
hst units of human capital), and ŵt is the wage rate in the childcare
sector.
10Here
we have used the simplifying assumption that H is produced using
the market technology, as discussed below. This amounts to an aggregation
assumption— if H is produced in a separate sector with the same, CRS, production
function, this will follow automatically. Many models of human capital investment
do not satisfy this assumption, however— often assuming that labor’s share is higher
in the H sector. See Lucas (1988) for example. It is of interest to explore other
alternatives as well.
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
23
For hst , we assume that investments are made for all children surviving to age 5, thus investments per person are hst = Ht /π 3t Ntb . Thus,
Nts hst = π t Ntb Ht /π3t Ntb = ππ3tt Ht .
Firms face static problems:
max
Yt − Rt Ktf − wt Lft
Yt ≤ F (Ktf , Lft )
s.t.
Goods and Labor market clearing (per dynasty) require that:
(F1)
Ktf = Kt
(F2)
Lft = Nts hst
(F3)
ˆcc N b
(F4)
b
Ct + at Nt+1
+ Ht+1 + Kt+1 = Yt
t
t+1
m
t
= bt Nts
cc
t
Note that interiority implies immediately that ŵt = wt hst that is,
wages are equalized across the two sectors. Because of this, letting
θt = at + bt ŵt , we can rewrite constrainst of the individual household
problem as:
Rt Kt
Rt Kt
b
b ˆcc
s s
+ ŵt Nt+1
Ct + at Nt+1
t + Ht+1 + Kt+1 ≤ wt Nt ht
m
s cc
t + ŵt Nt t +
b
b
Ct + at Nt+1
+ ŵt Nt+1
bt + Ht+1 + Kt+1 ≤ wt Nts hst
m
t
b
Ct + θt Nt+1
+ Ht+1 + Kt+1 ≤ wt Nts hst
b
Ct + θt Nt+1
+ Ht+1 + Kt+1 ≤ wt Nts hst
m
t
+ wt hst Nts
+ ŵt Nts
cc
t
cc
t
+
+ Rt Kt
m
s s
m
t + wt ht Nt (1 − t ) + Rt Kt
b
Ct + θt Nt+1
+ Ht+1 + Kt+1 ≤ wt Nts hst + Rt Kt
b
Ct + θt Nt+1
+ Ht+1 + Kt+1 ≤ wt ππ3tt Ht + Rt Kt .
This is the version of the BC we will use below. Thus, the Household
Problem is given by:
U=
∞
X
t=0
subject to:
β
t
g(π t Ntb )u
µ
Ct
π t Ntb
¶
b
+ Ht+1 + Kt+1 ≤ wt ππ3tt Ht + Rt Kt
Ct + θt Nt+1
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
24
Note that there is also a conceptual difficulty in assuming that at
is positive in an endogenous growth model. If we assume that at is
’constant,’ i.e., converges to a constant, de-trended at → 0. If we
assume that it is growing at some exogenous rate, at = (1 + g)t a0
then if the output of the economy (per capita) grows at a lower rate,
b
then at Nt+1
/Yt → ∞ which is not possible. If on the other hand
the output of the economy (per capita) grows at a higher rate, then
b
at Nt+1
/Yt → 0, again not very sensible. This is something that comes
out of using an endogenous growth model rather than an exogenous one
(as in the original B&B model), since there, we can assume that (1 +g)
is the exogenously given rate of growth of technology, and neither of
these issues arises. The only real solutions to this problem are to either
assume that at = 0 or to assume that the goods consumption of children
enter the utility of the planner. The second option is probably better,
b
since this would guarantee that at Nt+1
/Yt is a constant fraction of
output per person on a BGP (with the right choice of utility function).
Since this would raise a host of issues that we do not want to deal with
here, we will take the less appealing route and assume that at = 0.
Thus, θt = bt ŵt .
Because of this, it follows that feasibility becomes:
(F4)
Ct + Ht+1 + Kt+1 = Yt .
b
b
Note that θt Nt+1
does not appear in this equation! That is, θt Nt+1
=
s b
bt wt ht Nt+1 can be thought of as entering the individual dynasty budget
constraint since the household could either buy this service, or provide
it at home, but, it does not require market output.
Assuming that u(c) = c1−σ /(1 − σ), and g(N) = N η , we can write
this problem as:
U=
∞
X
t=0
·
¸1−σ X
∞
¤
¤
βt £
βt £
Ct
b η
b η+σ−1
=
N
[Ct ]1−σ
π t Nt
π
t
t
b
(1 − σ)
(1 − σ)
π t Nt
t=0
subject to:
(β t λt )
b
Ct + Nt+1
θt + Ht+1 + Kt+1 ≤ wt ππ3tt Ht + Rt Kt
First Order Conditions (FOCs, from now on) are:
Ct :
β t (1 − σ)Ut /Ct = β t λt
Ht :
β t λt wt ππ3tt = β t−1 λt−1
Kt :
β t λt Rt = β t−1 λt−1
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
β t (η + σ − 1)Ut /Ntb = θt−1 β t−1 λt−1
Ntb :
Where, Ut =
1
(1−σ)
£
¤η+σ−1
[Ct ]1−σ
πt Ntb
Ct :
(1 − σ)Ut /Ct = λt
Ht :
wt ππ3tt = Rt
Kt :
βRt UUt =
Ntb :
β t (η + σ − 1)Ut /Ntb = θt−1 β t−1 (1 − σ)Ut−1 /Ct−1
Ntb :
t
t
β(η + σ − 1) UUt−1
= θt−1 (1 − σ) CNt−1
Ntb :
t−1 Nt
t
β(η + σ − 1) UUt−1
= θt−1 (1 − σ) Ct−1
Nb
Ct
Ct−1
t−1
b
Nb
Ut−1
b
t−1
Now,
Ut
25
=
1
(1−σ)
1
(1−σ)
η+σ−1
[πt Ntb ]
[Ct ]1−σ
η+σ−1
b
[πt−1 Nt−1
]
[Ct−1 ]1−σ
=
h
πt
π t−1
iη+σ−1
γ η+σ−1
γ 1−σ
C ,
N
Thus, assuming that Balanced Growth Path (BGP, from now on)
holds, the FOCs become:
Ct :
(1 − σ)Ut /Ct = λt
Ht :
wt ππ3tt = Rt
Kt :
βRγ η+σ−1
γ 1−σ
= γC
N
C
Ntb :
So,
t−1
β(η + σ − 1)γ η+σ−1
γ 1−σ
= θt−1 (1 − σ) Ct−1
γN
C
N
Kt :
γ η+σ−1
γ 1−σ
=
N
C
Ntb :
t−1
β(η + σ − 1)γ η+σ−1
γ 1−σ
= θt−1 (1 − σ) Ct−1
γN
N
C
Ht :
wt ππ3tt = Rt
Nb
γC
βR
Nb
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
26
These are the 3 equations referred to in the title,11 they come out of
individual dynasty optimization plus intertemporal competitive equilibrium. As we will see, increases in ππ3tt , can, at least in some circumstances, generate and increase in the growth rate of per capita output.
What are R, w and θ? As discussed above,
θt = bt wt hst =
bt wt Ht
.
π 3t Ntb
is constant, or that θt is not
Note that this implies that either H
N
(assuming that wt is, along a BGP) even when the parameters bt and
π 3t are constant. Substituting this into the equations above gives:
γC
γ η+σ−1
γ 1−σ
= βR
Kt :
C
N
Ntb :
or Ntb :
or Ntb :
b
Nt−1
bt−1 wt−1 Ht−1
(1
−
σ)
γ
b
π 3t−1 Nt−1
Ct−1 N
η+σ−1 1−σ
bt−1 wt−1 Ht−1
1)γ N
γ C = π3t−1 Ct−1 (1 − σ)γ N
H
η+σ−1 1−σ
1)γ n
γ c = bw
(1 − σ)γ n
π3 C
β(η + σ − 1)γ η+σ−1
γ 1−σ
=
N
C
β(η + σ −
β(η + σ −
Ht :
wt ππ3tt = Rt
Total capital (per dynasty) in period t is just Kt . Total effective
labor supply (per dynasty) is Nts hst while the amount used for child
b
rearing is bt Nt+1
hours in the dynasty. Thus, labor market clearing
requires that effective labor supply in the market activityhis:
i
b
£ s s
¤
£
¤
Nt+1
s
b
s
b
b
s b
Lm
π
=
h
h
−
b
N
N
−
b
N
N
−
b
N
=
h
π
=
h
t t+1
t t
t t+1
t
t Nb
t
t
t t
t
t t
t
h
i
h
i
b
b
b
N
N
Ht Nt
πt
bt
t+1
t+1
= π3t N b π t − bt N b = Ht π3t − π3t N b ,
t
since, hst =
t
Ht
.
π 3t Ntb
t
Note that the term π t /π 3t is the conditional
probability of surviving to working age conditional on surviving to
age 5; according to our interpretation this is the relevant measure of
longevity we take as the main “cause” of the industrial revolution, at
least in this simple version of the model. In a more complete version,
we also look at the expected length of the period during which dynasty
members are of working age.
Thus,
³
h
i´
³
h
i´
Ht
πt
H
π
−
b
γ
−
bγ
1, K
=
F
1,
t n
n
³ t hπ3t
i´
³ K hπ3
i´
Ht
πt
H
π
R = Fk 1, Kt π3t − bt γ n = Fk 1, K π3 − bγ n
w=F
11Indeed,
every other model we consider in this paper takes more than three
equations to ‘generate’ an industrial revolution.
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
27
Finally, we must also have:
h
³
Ct + Ht+1 + Kt+1 = F Kt , Ht ππ3tt −
b
bt Nt+1
π 3t Ntb
Dividing by Ht gives:
Ct
Ht
+
Ht+1
Ht
+
Kt+1
Ht
=F
or, assuming BGP,
C
H
+ γH +
K
γ
H H
=F
³
³
h
πt
π 3t
π
π3
−
Kt
,
Ht
K
,
H
h
−
b
bt Nt+1
π 3t Ntb
b
γ
π3 N
i´
,
i´
i´
Thus, the system of equations that we want to solve is:
γ η+σ−1
γ 1−σ
=
N
C
Kt :
γC
βR
γ σC γ 1−η−σ
= βR
N
h iσ
γC
γ 1−η−σ
= βR
N
γ
or,
or,
N
Ntb :
β(η + σ −
h
i
γC
below).
γN
H
1)γ η+σ−1
γ 1−σ
= bw
(1
N
C
π3 C
(see the relevance of
− σ)γ N
w ππ3 = R
³
h
i´
H
π
w = F 1, K π3 − bγ n
³
h
i´
H
π
−
bγ
R = Fk 1, K
n
π3
Ht :
w:
R:
i´
³ h
C
K
K
π
b
F eas :
+
γ
+
γ
=
F
,
−
γ
H
H
H H
H
π3
π3 N
Variables are: R, w, C/H, K/H, γ H , γ N , γ C . Note that γ C = γ H (or
C/H and/or K/H must go to zero) since γ C is the growth rate of C
not c. These are the growth rates in aggregates. Thus, the growth
rates in per capita terms are given by:
s
cs
C
/Nt+1
Nts
γ c = t+1
= t+1
= CCt+1
= γ C /γ N
s
cs
Ct /N s
t N
t
t
t+1
Again, what we want to do with this is to do comparative statics with
respect to changing π, and remember that b depends on the whole array
of π 0 s.
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
28
5.2. B&B as a Planning Problem. Rewriting the B&B Competitive
Equilibrium Problem as a Planner’s Problem, we obtain (assuming that
u(c) = c1−σ /(1 − σ), and g(N) = N η ):
U=
∞
X
t=0
·
¸1−σ X
∞
¤
¤η+σ−1
βt £
βt £
Ct
b η
π t Nt
π t Ntb
=
[Ct ]1−σ
b
(1 − σ)
(1
−
σ)
π t Nt
t=0
subject to:
(β t λt )
³
´i
h
Nb
Ct + Ht+1 + Kt+1 ≤ F Kt , (π t Ntb )ht 1 − b πtt+1
Nb
t
³
´
Nb
Here, the second term, (π t Ntb )ht 1 − b πtt+1
is derived from the fact
Nb
t
that there are (π t Ntb ) number of people surviving to working age, with
Nb
) is the
ht human capital per surviving person, and that (1 − b πtt+1
Ntb
amount of time remaining, per survivor, after the time spent doing
child care for the next generation is accounted for.
Human capital per worker is ht =
Ht
,
µt Ntb
this gives us a labor input
b
Nt+1
)
π t Ntb
b
Ht π t Ntb − bNt+1
(
)
= (π t Ntb )
µt Ntb
π t Ntb
µ
¶
b
π t Ntb − bNt+1
πt
=
Ht
µt
π t Ntb
Lt = (π t Ntb )ht (1 − b
Thus, the constraint for the problem is:
³ b
´i
h
π N −bN b
(β t λt )
Ct + Ht+1 + Kt+1 ≤ F Kt , πµt Ht t πt t N b t+1
t
or
(β t λt )
Ct + Ht+1 + Kt+1 ≤ F [Kt , L1t ht ]
and
b
≤ L2t
bNt+1
L1t + L2t ≤ π t Ntb
Which can be written as:
t
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
(β t λt )
and
29
i
h
Ct + Ht+1 + Kt+1 ≤ F Kt , L1t µHNt b
t
t
b
≤ L2t
bNt+1
L1t + L2t ≤ π t Ntb
TO BE ADDED: SHOW THAT PLANNER PROBLEM AND EQUILIBRIUM PROBLEM HAVE EQUIVALENT SOLUTIONS.
6. The B&J Model with Endogenous Growth
6.1. Assumptions and Notation. The model studied in this part
has T-period lives (T will be either 3 or 4, for the time being), altruism from children toward parents but not viceversa, and certaintyequivalence within each family (i.e., we look at a ”representative family” and make the bold assumption that both an economy-wide insurance for individual death risk, and bequest mechanisms of the specific
form specified below, are automatically implemented.) Utility is arbitrary, in principle, but we specialize it to CES to get analytical solutions
and then calibrate the model. Population in each period is denoted,
respectively, N y for the young, N m for middle age (N m1 and N m2 in
the model with four periods, distinguishing between ”early” and ”late”
middle age), and N o for the old. Age specific survival probabilities to
the next period of life are denoted as π y , π m , and π o for young, early
middle-age and late middle-age people, respectively; they range in the
interval [0, 1], with π o = 0 in the three period model.
Production function is arbitrary neoclassical, but we assume either
Cobb-Douglas or general CES to make progresses with algebra. As
argued in the motivational section, the empirically most relevant case
appears to be that of a CES with an elasticity of substitution between
human and physical capital less than one, i.e., with less substitutability
than the Cobb-Douglas. The aggregate production function is denoted
as Yt = F (Kt , Ht ) = Lt F (kt , ht ), where Lt is the total amount of raw
labor supplied in period t, and F has the usual neoclassical properties.
People can only accumulate human capital during the first period, when
young. Two different technologies to accumulate human capital will be
considered. In the first, labelled the one sector case, for each child
parents invest goods, in a quantity a ≥ 0, and a fraction b ∈ [0, 1]
of their own time to rear her to working age. In the second, the two
sector case, both goods and time are purchased on the market, from a
competitive child-care sector that uses this same technology.
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
30
People can procreate only during their middle age (early middle age,
when T=4), and the number of newborns in period t, per middle age
person, is nbt . The total cost of having a child is denoted θt + µt ht+1 ,
where θt = a + bt wt ht , µt ∈ [π yt , 1] is the exogenous probability that
a child born in period t survives till when she can receive professional
training/education, and ht+1 is her endogenous level of human capital,
chosen by the parents.
Notice that, with this timing of events and, in particular, under
the assumption that old people do not work, the role played by the
survival probability π m in determining the investment in human capital
is ambiguous. For sure, when π m increases, total investment by middle
age agents should increase. Total investment, though, can take three
forms: more children, more capital per worker, and a higher human
capital for each child. Of these three kinds of investment, the first
decreases the (growth rate of) per capita income, while the other two
should both increase it. The conjecture is, therefore, that an increase
in π m will lead to higher aggregate investment in all three directions
(k, n and h) with the increase in k and in the h/k ratio depending
on relative values of the technological parameters. An increase in π y ,
which, as we know from previous work, leads to less fertility, should
increase h unambiguously, while it is unclear if it leads also to higher
levels of k; intuition and basic algebra suggests that, in any case, the
net effect on the h/k ratio should be positive.
6.2. The Simplest B&J Model. Here we illustrate the analytically
simplest version of the B&J model, which is the one adopted so far in
the calibration exercises summarized in the next section. This version
has three period lives, positive mortality only for the young, middle age
agents behaving non-cooperatively in determining donations to parents,
a Cobb-Douglas production function, and two distinct sectors, one for
market output and for childcare services. In this case, parents do not
use their own time to rear children but purchase the services from the
market, at the going equilibirum price. The optimization problem is
o
o
Max{nbt ,dit ,ht+1 kt+1 } u(cm
t ) + ζu(ct ) + βu(ct+1 )
subject to:
b
i
ˆcc
cm
t + kt+1 + nt (at + ŵt t + µt ht+1 ) + dt ≤ wt ht
o
i
ct ≤ Rt kt + dt + Dt
cot+1 ≤ Rt+1 kt+1 + π t nbt dt+1
m
cc
t + t ≤ 1
ˆcc ≥ bt .
t
m
t
+ ŵt
cc
t
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
31
cc
Market clearing in the childcare sector implies that ˆcc
t = t , for the
representative agent in the cohort. Similarly, interiority will imply that
ŵt = ht wt , and hence, it follows that θt = at + bt ht wt , and thus, since
at = 0, θt = bt ht wt . Thus, output, in period t + 1, per old person in
the same period, is given by:
yt+1 = F (kt+1 , e
ht+1 ) = F [kt+1 , π t nbt ht+1 (1 − bt nbt+1 )],
and rental rates and wage rates are given by:
Rt+1 = Fk (kt+1 , π t nbt ht+1 (1 − bt nbt+1 ))
and,
respectively.
wt+1 = Fhh (kt+1 , π t nbt ht+1 (1 − bt nbt+1 ))
6.2.1. Donations and Old Age Consumption. Assuming that u(c) =
c1−σ /(1 − σ), first order conditions for donations are:
cot = ζ 1/σ cm
t ,
from which we find that:
dt =
cot =
cm
t =
ζ 1/σ (wt ht − It ) − Rt kt
ζ 1/σ + πt−1 nbt−1
£
¤
ζ 1/σ
b
n
(w
h
−
I
)
+
R
k
π
.
t−1
t
t
t
t
t
t−1
ζ 1/σ + π t−1 nbt−1
ζ
1/σ
£
¤
1
πt−1 nbt−1 (wt ht − It ) + Rt kt
b
+ π t−1 nt−1
∂cot
ζ 1/σ
= 1/σ
Rt
∂st−1
ζ + π t−1 nbt−1
h
i
∂cot
π t−1 ζ 1/σ
1/σ
=h
i2 ζ (wt ht − It ) − Rt kt
∂nbt−1
ζ 1/σ + π t−1 nbt−1
∂cot
ζ 1/σ
= 1/σ
π t−1 nbt−1 wt
b
∂ht
ζ + π t−1 nt−1
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
32
6.2.2. Intertemporal First Order Conditions. (FOCn)
0
u
(cm
t )(θ t
0
+ µt ht+1 ) = βu
∂cot+1
o
(ct+1 )
∂nbt
(FOCh)
b
0 o
u0 (cm
t )µt nt = βu (ct+1 )
∂cot+1
∂ht+1
(FOCk)
0
u
(cm
t )
0
= βu
∂cot+1
o
(ct+1 )
∂st
Reducing the system to two equations and two unknowns. From (FOCh)
and (FOCk) we observe that,
³
´
³
´
µt−1 Fk kt , e
ht = π t−1 Fhh kt , e
ht .
Using y = Ak αe
h1−α , the latter equality yields
nbt
1
=
bt
µ
1−
¶
1−α b
kt ,
αµt−1 nbt−1
So, because nbt−1 is predetermined, the intertemporal equilibrium relation between capital intensity and fertility is one to one. We also
have
¸1−α
·
π t−1 (1 − α)
Rt = αA
αµt−1
¸α
·
αµt−1
wt = (1 − α)A
π t−1 (1 − α)
¶1−α
µ
π t−1 (1 − α)
b
bt b
ybt = A
kt = A
kt
αµt−1
e
ht
(1 − α)π t−1 b
b
e
=
kt .
ht =
ht
αµt−1
µ
¶
1−α b
1
b
b
1−
kt (wb + µt γ h,t )
It = kt+1 +
b
αµt−1 nbt−1
The remaining variables can be computed accordingly.
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
33
6.2.3. Computing the Intertemporal Competitive Equilibrium. We exploit the consumption ratio
b
cot
= ζ 1/σ .
m
b
ct
When the latter is plugged in FOCk, this can be solved to express the
growth rate of per capita output as a function of parameters and state
variables only:
·
¸1−α
π t (1 − α)
βζ 1/σ−1
σ
³
´ αA
γ h,t =
αµ
ζ 1/σ + πbt 1 − αµ 1−αnb b
kt
t−1 t−1
From the latter, the whole competitive equilibrium sequence can be
computed recursively starting from given initial conditions for {k0 , h0 , n−1 }
and parameters.
6.2.4. BGP Solutions. Because,
h ³
´
i
ζ 1/σ
b
b
b
c = 1/σ
πn w − I + Rk .
ζ + πn
h ³
´
i
1
m
b
b
πn w − I + Rk
b
c = 1/σ
ζ + πn
o
we have that.
b
co
= ζ 1/σ
b
cm
bb
The eight variables to be determined are {d,
k, n, γ h , b
cm , b
co , R, w}.Starting
with the donation function, the eight BGP equations we use are
³
´
1/σ
b
ζ
w − I − Rb
k
d=
ζ 1/σ + πn
FOCn
´
i
πζ 1/σ−1 h 1/σ ³
σ−1
b
b
[γ h ]
ζ
w − I − Rk
(bw + µγ h ) = β 1/σ
[ζ + πn]2
FOCh
πζ 1/σ−1
σ
[γ h ] µ = β 1/σ
w
ζ + πn
FOCk
ζ 1/σ−1
σ
[γ h ] = β 1/σ
R
ζ + πn
BCMa
b
cm + γ h b
k + n(bw + µγ h ) + db = w
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
BCOld
MCK
MCH
34
b
co = Rb
k + πndb
R = Fk (b
k, πn(1 − bn))
k, πn(1 − bn))
w = Fhh (b
These can be manipulated, as in the intertemporal competitive equilibrium case, to find that
αµn(1 − bn)
b
k=
1−α
·
¸1−α
π(1 − α)
R = αA
αµ
¸α
·
αµ
w = (1 − α)A
π(1 − α)
µ
¶1−α
π(1 − α)
b
bb
yb = A
k=A
kt
αµ
Last step, using FOCk solve for γ h as a function of n and then use
FOCn to solve the nonlinear equation for the BGP value of n.
¸1−α
·
βζ 1/σ−1
π(1 − α)
σ
[γ h ] = 1/σ
αA
αµ
ζ + πn
7. Quantitative Implementations of the Models
In this section, we describe the results of calibrating and simulating
some (two, at the time being) among the simple models described in
the preceding sections. To this point in time, we have only conducted
full blown simulations of the Barro and Becker version of the model,
and “comparative BGP simulations” of one version of the B&J model,
the one with two sector, Cobb-Douglas production function and noncooperative behavior of siblings in the donations game.
What we find in the case of the B&B framework, is that versions
of the model that are calibrated to realistic values of the parameters
give rise to quantitatively interesting and realistic changes in the rate
of growth of output per capita when survival probabilities are changed
as they have changed historically in the UK between 1700 and 2000.
This is, we think, evidence that the mechanism we are highlighting
here — improved survival rates increasing the rate of return to human
capital investments and increasing the technological frontier in such a
way as to increase the growth rate of output — is worthy of more careful
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
35
development. The model does not do so well when it comes to fertility
decisions per se, however. Here, in contrast to what is seen in both
the time series data for the UK and in the country cross section data
described in Section 2, increased survival rates permanently increase
fertility rates and thereby asymptotic population growth rates. Although not huge, these increases are of significant size economically. In
particular, what the various versions of the models we have simulated
so far miss is the following historical demographic pattern: fertility
does neither increase nor decrease to a large extent when survival rates
(among adults, especially) begins to improve; which leads to a first long
period of increasing growth rates of population. Eventually, and especially when children and infant mortality rates drop, fertility also drops
rapidly and dramatically, slowing down population growth. Should we
extend the analysis to the whole of of the twentieth century, then one
should require the model to yield a zero rate of population growth. The
demographic dynamics generated by the B&B model does not satisfy
all such requirements, as population growth rates are always increasing together with the growth rate of per capita GDP as mortality rates
decrease. As we show below, when calibrated to (second half of the)
XX-century-like parameter values, the B&B model yields a BGP with
an endogenous growth rate of output per capita equal to about 2%,
which is pretty much what we seem to observe, and a zero growth rate
of the population, which is also what we seem to observe.
The behavior of the B&J model is, in some sense, the mirror image
of the B&B: it does better on the side of fertility, but more poorly
on the side of output, as it tends to exagerate the increase in the the
growth rate of output that follows a drop in mortality. We have so
far calibrated and simulated only the non-cooperative version, hence
things may appear different once we succeed simulating the transition
path for the cooperative version. In the same circumstances as the B&B
model, i.e. when calibrated to XX-century-like values, it generates a
2% growth rate of output per capita but fails to achieve a constant
population, predicting instead a population that declines at the rather
rapid pace of 8% per year.
7.1. Calibrating the B&B Model. Recall that the B&B version of
the model can be summarized through the choice of utility function,
the production function for output and the cost of newborn children.
In keeping with the above, we choose:
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
U=
∞
X
t=0
and
36
·
¸1−σ X
∞
¤
¤η+σ−1
Ct
βt £
βt £
b η
=
[Ct ]1−σ
π t Nt
π t Ntb
b
(1 − σ)
(1
−
σ)
π t Nt
t=0
F (K, L) = AK α (Lh)1−α
Thus, the parameters we need to pick are: β, σ, η, b, A, α.For this, we
choose α = 0.36 following the RBC literature and select combinations
of the remaining parameters, β, σ, η, b, and A so as to match interest
rates, population growth rates and output growth rates in the UK in
1750 given the survival rates that were in effect at that time. Since it
is not obvious whether calibrating to early in the period (e.g., 1750) or
later (e.g., current) is more reasonable, we constructed two versions of
the calibration, one matching the 1750 facts, the other matching the
facts as they are currently.
To do this, we must first choose the length of a period. For this,
we choose T = 20 years as a baseline. Given this, we need data on
the survival rates to age 5 and 20. Unfortunately, as can be seen from
the data presented in Figure 1C, the Wrigley, et. al., data can only be
used to calculate directly survival probabilities to age 15, not 20, for the
period 1750 to 2000. The data from www.mortality.org (from 1840 to
1905) and Mitchell (from 1905 to 1970) can be used to calculate survival
rates to 20 directly. Survival rates to age 5 are directly available from
these sources and can be used to construct a continuous series for the
entire 1750 to 2000 period. The data from www.mortality.org can be
used to compute estimated survival probabilities to both age 15 and
20, and these are shown in Figure 1C. To adjust for the fact that data
on survival to age 20 is not available for the 1750 to 1840 period, we
use the fact that the ratio of the probability of survival to age 20 to the
probability of survival to age 15 was .96 in 1840, and rose steadily to
1.0 over the period from 1840 to 1970. Thus, using the linear function
π t = 0.96 ∗ Surv15W &St
seems a conservative approximation. This gives an initial value of
π 1750 ≈ 0.6 which is what we use for both out calibration and out quantitative exercises. That µ1750 ≈ 0.75 comes directly from the Wrigley,
et. al. tables. Summarizing then, the actual values of µt and πt that
we use are given in Table 6 below.
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
37
Table 6: Survival Probabilities
Year µt
πt
1735 0.696 0.609
1765 0.744 0.666
1795 0.746 0.678
1825 0.771 0.705
1850 0.728 0.660
1870 0.735 0.684
1890 0.770 0.733
1910 0.829 0.824
1930 0.919 0.919
1950 0.962 0.961
1970 0.977 0.976
Over the period from 1650 to 1750, population in the UK grew
at about 0.1% per year (Wrigley, et. al. (****)) while per capita
GDP grew at about 0.3% per year (Clark (****)). Thus, for our
1700 calibration, we will choose parameters so that on an initial BGP,
γ y = γ n ≈ 1.0 given that µ1750 = 0.75, π 1750 = 0.60 (Wrigley and
Schofield (****)). We choose β = 1.03−1 on an annual basis so that
real rates of interest with constant consumption would be about 3%.
This gives us 2 extra degrees of freedom that we will remove in future vintages of the paper. We adopt a similar strategy for the 2000
calibration. Again we use β = 1.03−1 and use the more recent values
of µ2000 = π2000 = 1.0, and γ n ≈ 1.0, γ y = 1.02.The values of the
parameters we have chosen for our base case are shown in Table 7.
Table 7: Calibration of the B&B Model
1750 Calibration 2000 Calibration
α
0.36
0.36
β
(1.03)−1
(1.03)−1
η
0.20
0.20
σ
0.85
0.85
b
0.06
0.10
A
4.0
4.65
µ1750 0.75
0.75
π 1750 0.60
0.60
µ2000 1.00
1.00
π 2000 1.00
1.00
Table 8 shows the targeted quantities and how close the calibrated
version of the model comes to matching them.
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
Table 8: Model versus Data,
Target 1750 Calibration Data 1750
R
2.99%
4.5%
γn
1.001
1.001
γy
1.001
1.003
Calibrated Moments
Source 2000 Calibration
C&H 4.52%
W&S 1.001
Clark 1.018
38
Data 2000
4%
1.003
1.02
7.2. Results From the Calibration.
Figures 3C and 3D show the results of the predicted time series of
population and per capital income, both from the model and the
actual time series from the UK, and both for the 1750 and the 2000
calibrations. For the model quantities, we show BGP levels only, not
a full transition in the probabilities. Not surprisingly, they show
several common features:
Figure 3C: Model vs. Data; 1750 Calibration, Levels
12
ln(Pop), ln(GDP)
10
8
ln(Pop)
ln(GDP)
ln(PopMod)
ln(GDPMod)
6
4
2
0
1750
1800
1850
1900
1950
2000
Year
Figure 7.1
(1) When calibrated to either the 1750 or the 2000 BGP, the model
is able to match the targeted values with reasonable parameter
values.
(2) When simulated ‘forward’ (or ‘backward’, from 2000) the model
also tracks reasonably well the observed changes in the growth
rate of output at low frequencies. If anything, it predicts more
growth in income than observed.
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
39
(3) The model does not do well, indeed it performs rather poorly,
in tracking the overall pattern of population growth rates. For
both calibrations, the model predicts an increase in γ n of about
1% per year between 1750 and 2000, whereas in the data, γ n,1750 ≈
γ n,2000 ≈ 1.00. This gets at the mechanism that is driving the
model results and points to both its strengths and its weaknesses. That is, in the model, increases in survival probabilities
increase the rate of return to human capital formation, thus
increasing output growth rates. For the same reason however,
children become better investments. This increases fertility and
thereby increases population growth rates permanently.
(4) The model is weak at tracking the higher frequency movements
in output growth and population growth. This is perhaps not
surprising since the driving force is only survival probabilities
and it is unlikely that reductions in these are the cause of the
Great Depression or the two World War.
7.3. Calibrating the B&J Model. As with the B&B model, we need
to calibrate an aggregate production function and a utility function
that has three arguments, own middle and old age consumption, and
consumption of parents. Thus, the parameters we need to pick are:
β, σ, ς, b, A, α.For this, we choose α = 0.36 following the RBC literature
and select combinations of the remaining parameters, β, σ, η, b, and A
so as to match observables for either 1750 or 2000. As the mortality
data used are the same as for the B&B, the description is not repeated
here. The values of the parameters we have chosen for our base case
are shown in Table 9, while Table 10 shows the targeted quantities and
how close the model comes to matching them .
Table 9: Calibration of the B&J Model
1750 Calibration 2000 Calibration
α
0.36
0.36
β
(1.03)−1
(1.03)−1
ς
1.57
1.80
σ
0.50
0.50
b
0.06
0.06
A
7.9
10.00
µ1750 0.75
0.75
π 1750 0.60
0.60
µ2000 1.00
1.00
π 2000 1.00
1.00
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
Table 10: Model versus Data, Calibrated Moments
Target 1750 Calibration Data 1750 Source 2000 Calibration
R
6.56%
4.5%
AK
8.6%
γn
1.001
1.001
W&S 1.0009
γy
1.0006
1.003
Clark 1.019
40
Data 2000
4%
1.003
1.02
7.4. Results From the Calibration. The results for the B&J model
are, in some sense, similar but complementary to those of the B&B
model. While both can be calibrated to match targeted quantities at
either 1750 or 2000, both appear to have troubles at matching observed
dynamic patterns. The B&J model, at least in the cooperative version
adopted here, seems to have an additional troubles, i.e. it requires
an unreasonably high value for the parameter ς to match observed
data. Without a weigth that large on the utility of parents, the rate of
return on having children is too low, compared to that of just investing
in physical capital, and parents end up having an unrealistically low
number of children.12
Moving to the dynamics, for the B&B it is population that behaves
poorly, while for the B&J it seems to be output. This can be seen, more
clearly, in Figure 4D, where the B&J model substantially overpredicts
the growth rate of output. Notice that this may not be a ‘structural’
weakness of the B&J model, but may steam from the calibration of A,
which, as noted in the previous footnote, is very high and yields a rate
of return on capital, in both calibrations, that is substantially higher
than the value in the data. In other words, for reasons that still escape
us, investing in capital is too much profitable respect to investing in
children in the B&J model, hence the mismatch with data.
8. Directions for Future Work
The experiments run so far point to several potentially fruitful further directions for research.
12We
admit being puzzled by the behavior of the B&J model along this dimension, which makes us suspicious of our own algebra. This is because we must, at the
same time, use a very large value for A in the production function, which generates
an abnormally high rate of return on capital, and hence the problem with children’s
donations noted in the text.
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
41
Figure 4C: B-J Model vs. Data; 1750 Calibration, Levels
12
ln(Pop), ln(GDP)
10
8
ln(Pop)
ln(GDP)
ln(PopMod)
ln(GDPMod)
6
4
2
0
1750
1800
1850
1900
1950
2000
Year
Figure 7.2
Figure 4D: B-J Model vs. Data; 2000 Calibration, Levels
12
ln(Pop), ln(GDP)
10
8
ln(Pop)
ln(GDP)
ln(PopMod)
ln(GDPMod)
6
4
2
0
1750
1800
1850
1900
1950
2000
Year
Figure 7.3
(1) Get a version of the B&J model working without assuming an
unreasonably large weight for the utility of the old parents.
Based on past experience with these models, it is quite possible that this will offer improvements in the tracking of fertility.
Aside for possible mathematical mistakes, we expect the cooperative version to make investing in children much more attractive, allowing a more realistic calibration of the production
function and, hence, of the rate of return on capital.
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
42
(2) Do the full model with transition dynamics (for both the B&B
and B&J models).
(3) Explore other, more complex models of the formation of human
capital and the changes that a CES production function may
induce in factor shares and the rate of return on capital.
(4) Add one more period (two, for the B&B) in the life times of
the representative agents, and introduce the relevant mortality
rates in the transition from one period to the other. This latter
is a particularly important improvement, as it may generate a
temporary increase in fertility in the first part of the transition
(when the survival rate of adults increases) and a subsequent decrease in fertility later on (when the survival rates of infant and
children increase). While this may not eliminate the structural
tendency of the B&B model to make output and population
growth rates increase together, it may allow the B&J model to
better match the historical patterns.
9. Bibliography
Becker, Gary S., and Robert J. Barro (1988), “A Reformulation of
the Economic Theory of Fertility,” Quarterly Journal of Economics
103, 1—26.
Becker, Gary S. and H.G. Lewis (1973), “On the Interaction Between
the Quantity and Quality of Children,” Journal of Political Economy
81, S279—S288.
Becker, Gary S., Kevin M. Murphy and Robert Tamura (1990), “Human Capital, Fertility, and Economic Growth”, Journal of Political
Economy 98, S12—S37.
Blackburn, Keith and Giam Pietro Cipriani (2004), “A Model of
Longevity, Fertility and Growth,” Journal of Economic Dynamics and
Control 26, 187-204.
Boucekkine, Raouf, David de la Croix and Omar Licandro (2002),
“Vintage Human Capital, Demographic Trends and Endogenous Growth,”
Journal of Economic Theory 104, 340-375.
Boucekkine, Raouf, David de la Croix and Omar Licandro (2003),
“Early Mortality Declines at the Dawn of Modern Growth,” Scandinavian Journal of Economics 105, 401-418.
Cavalcanti Ferreira, Pedro and Samuel de Abreu Pessoa (2003), “The
Cost of Education, Longeivty and the Poverty of Nations,” mimeo,
Fundation Getulio Vargas, Rio de Janeiro.
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
43
Cervellati, Matteo and Uwe Sunde (2002), “Human Capital Formation, Life Expectancy and the Process of Economic Development,”
Discussion Paper No. 585, IZA, Bonn, Germany.
Clark, Gregory (2001), “The Secret History of the Industrial Revolution,” mimeo, University of California at Davis.
Clark, Gregory. (2004), The Conquest of Nature: An Economic History of the World, forthcoming, Princeton University Press.
Crats, Nick F.R. (1985), Bristish Economic Growth During the Industrial Revolution, Oxford, UK: Oxford University Press.
Crats, Nick F.R. (1995), “Exogenous or Endogenous Growth? The
Industrial Revolution Reconsidered,” Journal of Economic History 55,
745-772.
Crats, Nick F.R and Knick C. Harley (1992), “Output Growth and
the British Industrial Revolution: A Restatement of the Crafts-Harley
View,” Economic History Review, 45, 703-730.
Easterly, Richard A. (1999), “How Beneficient is the Market? A Look
at the Modern History of Mortality,” European Review of Economic
History 3, 257-294.
Ehrlich, Isaac and Francis T. Lui (1991), “Intergenerational Trade,
Longevity, and Economic Growth,” Journal of Political Economy 99,
1029—1060.
Feinstein, Charles H. and Sidney Pollard, eds. (1988), Studies in
Capital Formation in the United Kingdom, 1750-1920, Oxford, UK:
Oxford University Press.
Floud, Robert and Donald McCloskey, eds. (1994), The Economic
History of Britain since 1700, Cambridge, UK: Cambridge University
Press.
Galor, Oded and David N. Weil. (2000), “Population, Technology
and Growth: From Malthusian Stagnation to the Demographic Transition and Beyond,” American Economic Review 90, 806-828.
Harley, Knick C. (1982), “British Industrialization Before 1841: Evidence of Slower Growth During the Industrial Revolution,” Journal of
Economic History 42, 267-289.
Hazan, Moshe and Hosny Zoabi (2005), “Does Longevity Cause
Growth?” mimeo, Hebrew University and CEPR, March.
Kalemli-Ozcan, Sebnem (2002), “Does the Mortality Decline Promote Economic Growth?” Journal of Economic Growth 7.
Kalemli-Ozcan, Sebnem (2003), “A Stochastic Model of Mortality,
Fertility, and Human Capital Investment,” Journal of Economic Development 62.
Landes, David S. (1998), The Wealth and Poverty of Nations, W.
W. Norton and Company.
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44
Livi Bacci, Massimo (2000), The Population of Europe. A History,
London, UK: Blackwell Publishers.
Lucas, Robert E.(2002), Lectures on Economic Growth, Cambridge,
MA: Harvard University Press.
McCloskey, Donald (1994), “1780-1860: A Survey,” in Floud, R. and
D. McCloskey (eds.) The Economic History of Britain since 1700, Vol.
1, Chapter 10; Cambridge, UK: Cambridge University Press.
McKeown, Thomas (1976), The Modern Rise in Population, London,
Rutledge.
Mercer, Alex (1990, Disease, Mortality and Population in Transition,
Leicester, UK: Leicester University Press.
Mitchell, (), European Historical Statistics
Preston, Samuel H. (1975), “The Changing Relation Between Mortality and Level of Economic Development,” Population Studies 29,
213-248.
Razzell, Peter (1993), “The Growth of Population in EighteenthCentury England: A Critical Reappraisal,” The Journal of Economic
History 53, 743-771.
Schofield, Robert (1973) “Dimensions of Illliteracy, 1750-1850,” Explorations in Economic History 10, 437-454
Soares, Rodrigo R. (2005), “Mortality Reductions, Educational Attainment, and Fertility Choice,” mimeo, Catholic University of Rio de
Janeiro and University of Maryland, forthcoming in American Economic Review.
Voth, H.-J. (????), “Living Standards During the Industrial Revolution: An Economist’s Guide,” mimeo, Universitat Pompeu Fabra,
Barcelona.
West, Edwin G. (1978), “Literacy and the Industrial Revolution,”
Economic History Review, XXXI,
Wrigley, E.A., R.S. Davies, J.E. Oeppen, and Robert S. Schofield
(1997), English Population History from Family Reconstitution 15801837, Cambridge, UK: Cambridge University Press.
Wrigley, E.A. and Robert Schofield (1981), The Population History
of England 1541-1871, Cambridge, MA: Harvard University Press
Zhang, Junsen, Zhang Jie, and Ronald Lee (2001), “Mortality Decline and Long Run Economic Growth,” Journal of Public Economics
80, 485-507.
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
45
10. Appendix I: Algebra and Variations on Basic Models
10.1. The Aggregate Production Function.
Let production be defined at the individual, disaggregated level,
with yti = F (kts (i), m
t (i)), for each member i of the working age cohort,
which is composed of Nts people in the B&B notation, and of Ntm in the
B&J notation. Here we stick to the B&B notation. Aggregate output
is then
Z
Nts
Yt =
F (kts (i),
m
t (i))di,
0
and thus,
R Ns
Yt = 0 t F (kts (i),
m
t (i))di
= Nts F (kts ,
m
t )
= π t Ntb F (kts , hst (1−bt nt+1 ))
y
Ht
s
b
t
= π t Ntb F ( πK
b , π N b (1 − bt nt+1 )) = F (Kt , (Nt − bt Nt+1 )ht+1 ) =
t Nt
3t t
b
)hyt ) =
F (Kt , (π t Ntb − bt Nt+1
b
= F (Kt , (π t Ntb − bt Nt+1
) π3tHNt b ) = F (Kt , Ht ( ππ3tt −
t
b
bt Nt+1
)).
π 3t Ntb
Thus, this can be thought of as production happening at the individual level.
10.2. B&J, Three Periods, One Sector, Youth Mortality Only.
Because in this subsection π y ∈ (0, 1) and π m = 1, let π just denote
π y ; in θt = a + bt wt ht set a = 0, and when appropriate, write: It =
nbt (θt + µt ht+1 ) + st . Each middle aged agent, i, in period t solves:
Max{dit ,nbt ,ht+1 ,st }
o
o
u(cm
t ) + ζu(ct ) + βu(ct+1 )
subject to:
b
i
cm
t ≤ wt ht − st − nt (θ t + µt ht+1 ) − dt
o
i
ct ≤ Rt kt + dt + Dt
cot+1 ≤ Rt+1 kt+1 + π t nbt dt+1 .
In our notation, kt+1 = st , Dt are the total donations of all surviving siblings but agent i, Rt = Fk (Kt , Ht ), wt = Fh (Kt , Ht ), Kt =
m
st−1 Nt−1
= st−1 Nto , and Ht = ht Ntm . It is convenient to formulate
this in terms of per old person in period t. Effective labor supply e
ht+1
in period t + 1, per middle aged person in period t (that is, per old
person in period t + 1, in force of the survival assumptions made here)
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
46
is πt nbt ht+1 (1 − bt nbt+1 ). Thus, output, in period t + 1, per old person
in the same period, is given by:
yt+1 = F (kt+1 , e
ht+1 ) = F [kt+1 , π t nb ht+1 (1 − bt nb )],
t
t+1
and rental rates and wage rates are given by:
and,
Rt+1 = Fk (kt+1 , π t nbt ht+1 (1 − bt nbt+1 ))
wt+1 = Fhh (kt+1 , π t nbt ht+1 (1 − bnbt+1 ))
respectively.
Substitute the constraints in the utility functions, to get the maximization problem:
£
¤
Max{nbt ,dit ,ht+1 ,st } u( wt ht − st − nbt (θt + µt ht+1 ) − dit )+
+ζu(Rt kt + dit + Dt ) + βu(Rt+1 kt+1 + π t nbt dt+1 ).
10.2.1. Donations and Old Age Consumption. We solve for donations
first, and for both the cooperative and the non-cooperative equilibrium.
First order conditions for donations are:
b
0 o
u0 (cm
t ) = ζπ t−1 nt−1 u (ct )
for the cooperative solution, and
0 o
u0 (cm
t ) = ζu (ct )
for the noncooperative one. Assuming that u(c) = c1−σ /(1 − σ) we get:
¡
¢1/σ m
cot = ζπ t−1 nbt−1
ct
for the cooperative, and
cot = ζ 1/σ cm
t
for the noncooperative case, respectively.
Cooperative Donations and Consumptions. Substituting in the budget
constraints and imposing symmetry (i.e., that dt = dit ) gives:
¡
¢1/σ
π t−1 nbt−1 dt + Rt kt = ζπ t−1 nbt−1
(wt ht − dt − It ) .
Solving this for dt gives:
¢1/σ
¡
(wt ht − It ) − Rt kt
ζπ t−1 nbt−1
.
dt =
¡
¢
1/σ
+ π t−1 nbt−1
ζπ t−1 nbt−1
Plugging this in the budget constraint for the old
cot = π t−1 nbt−1 dt + Rt kt
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
47
¢1/σ
¡
b
£
¤
n
ζπ
t−1
t−1
cot = ¡
πt−1 nbt−1 (wt ht − It ) + Rt kt .
¢
1/σ
+ π t−1 nbt−1
ζπ t−1 nbt−1
h
i−1
1/σ
1/σ
Υt + πt nbt .
To shorten notation, set Υt = ζπt nbt and Πt = Υt
Compute next the partial derivatives of old age consumption with respect to saving, fertility, and education.13
∂cot
= Πt−1 Rt
∂st−1
¤
£
∂cot
= Πt−1 π t−1 (wt ht − It ) + πt−1 nbt−1 (wt ht − It ) + Rt kt ×
b
∂nt−1
h
i
h
i
1/σ
1/σ
1/σ−1
1/σ−1
1
1
b
Υ
−
Υ
ζπ
Υ
+
π
n
ζπ
Υ
+
π
t−1
t−1
t−1
t−1
t−1
t−1
t−1
t−1
t−1 σ
σ
×
=
h
i2
1/σ
b
Υt−1 + πt−1 nt−1
¤
π t−1 Πt−1 ( σ1 − 1) £
i π t−1 nbt−1 (wt ht − It ) + Rt kt .
= Πt−1 π t−1 (wt ht − It )+ h
1/σ
Υt−1 + π t−1 nbt−1
Hence:
·µ
¸
¶
π t−1 nbt−1
∂cot
Πt−1 π t−1
1−σ
1/σ
Υt−1 +
(wt ht − It ) +
Rt kt ,
= 1/σ
σ
σ
∂nbt−1
Υt−1 + π t−1 nbt−1
and
∂cot
= Πt−1 π t−1 nbt−1 wt (1 − bnbt ).
∂ht
Notice that the latter expression is derived using the fact that an individual parent affects the stock of per-capita human capital accumulated
by its own children, but not the average. For later usage, report here
also the consumption of the middle age
£
¤
1
cm
πt−1 nbt−1 (wt ht − It ) + Rt kt
t =
1/σ
Υt−1 + π t−1 nbt−1
13In
what follows, we have assumed, as per usual that
∂nt
∂nt−1
=
∂st
∂nt−1
=
∂ht
∂nt−1
=
∂nt
∂kt−1
=
∂kt
∂kt−1
=
∂ht
∂kt−1
=
∂nt
∂ht−1
=
∂kt
∂ht−1
=
∂ht
∂ht−1
= 0.
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
48
NonCooperative Donations and Consumptions. In this case, following
the same steps as before, we get
dt =
cot =
Thus,
∂cot
∂nbt−1
and,
ζ 1/σ (wt ht − It ) − Rt kt
ζ 1/σ + πt−1 nbt−1
£
¤
ζ 1/σ
b
n
(w
h
−
I
)
+
R
k
π
.
t−1
t
t
t
t
t
t−1
ζ 1/σ + π t−1 nbt−1
∂cot
ζ 1/σ
= 1/σ
Rt
∂st−1
ζ + π t−1 nbt−1
h
i
π t−1 ζ 1/σ
1/σ
ζ
=h
(w
h
−
I
)
−
R
k
i2
t t
t
t t
ζ 1/σ + π t−1 nbt−1
∂cot
ζ 1/σ
= 1/σ
πt−1 nbt−1 wt (1 − bnbt )
b
∂ht
ζ + π t−1 nt−1
Also in this case, for later use, report the consumption of middle age
people
£
¤
1
b
cm
n
(w
h
−
I
)
+
R
k
π
t−1
t
t
t
t
t
t = 1/σ
t−1
ζ + π t−1 nbt−1
Notice that, everything else equal, donations and old age consumption are higher in the cooperative case when πnb > 1, while middle age
consumption is lower, and viceversa.
10.2.2. Intertemporal First Order Conditions. The first order conditions determining the level of the three components of total investments
have the same formal representation in both the cooperative and the
noncooperative case. They are
(FOCn)
∂cot+1
0 o
u0 (cm
)(θ
+
µ
h
)
=
βu
(c
)
t
t t+1
t
t+1
∂nbt
(FOCh)
∂cot+1
0 m
b
0 o
u (ct )µt nt = βu (ct+1 )
∂ht+1
(FOCk)
∂cot+1
0 o
u0 (cm
)
=
βu
(c
)
t
t+1
∂st
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
49
Reducing the system to two equations and two unknowns. The target here is to rewrite everything as a function of γ h,t = ht+1 /ht and
b
kt = kt /ht . In what follows, for any variable z, let zbt = zt /ht and
k0 is given, we will solve for the
γ z,t = zt+1 /zt . Because the initial b
competitive equilibrium by finding values of all ”hatted” variables plus
(one plus) the growth rate of ht in period t. This, I believe, simplifies
both notation and algebra.
From (FOCh) and (FOCk) we get that, in both the cooperative and
non cooperative world
³
´ (1 − bnb )π
³
´
t−1
t
e
e
Fhh kt , ht .
Fk kt , ht =
µt−1
In the Cobb-Douglas case, Fk (t) = αFkt(t) and Fhh (t) =
hence, because e
ht = π t−1 nbt−1 ht (1 − bnbt ), the fertility rate is
nbt =
(1−α)F (t)
,
h
ht
(1 − α)b
kt+1
,
αµt
From the last one we can derive two useful conclusions. First, that
either b
k increases a lot when µt drops, or the effect of a decrease in
children’s mortality is a decrease of fertility, not one to one but on the
e
k
scale of (1−α)
times the elasticity of b
k to µ minus one. Second, we can
αµ2t
write rate of return, wage, and per capita output as a function of b
kt+1
b
and parameters. Recall that kt+1 is being chosen in period t.
Ã
!#1−α
"
b(1 − α)b
kt+1
(1 − α)π t−1
1−
Rt = αA
αµt−1
αµt
wt = (1 − α)A
ybt = Ab
kt
"
"
(1 − α)πt−1
αµt−1
π t−1 (1 − α)
αµt−1
Ã
Ã
b(1 − α)b
kt+1
1−
αµt
b(1 − α)b
kt+1
1−
αµt
!#1−α
!#−α
bt b
=A
kt
bt is actually a
Notice an interesting thing: the productivity factor A
function of current and past parameters, and of the capital labor ratio
chosen today for tomorrow! We can now write the growth rate of output
yt+1
= γ A,t
e × γe
k,t .
yt
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
50
Plugging the fertility rate back into the definition of effective human
capital.
Ã
!
b
e
b
k
h
k
(1
−
α)π
b
t
t−1
t
t+1
e
1 − b(1 − α)
.
=
ht =
ht
αµt−1
αµt
Next, we use our expression for nbt to rewrite all remaining variables as
a function of predetermined variables, b
kt+1 and γ h,t .
·
¸
¢
(1 − α) ¡
b
b
It = kt+1 γ h,t +
bwt + µt γ h,t
αµt
Cooperative donations
³
´1/σ ³
´
b
wt − It − Rt b
kt
C
b
dt = ³
´
et 1/σ
ζπ t−1 (1−α)k
(1−α)e
kt−1
+ πt−1αµ
αµ
ζπt−1 (1−α)e
kt
αµt−1
t−1
t−1
Cooperative middle age consumption
b
cm
t = ³
ζπ t−1 (1−α)e
kt
αµt−1
1
´1/σ
+
ζπ t−1 (1−α)kt
αµt−1
ζπ t−1 (1−α)e
kt
αµt−1
´1/σ
+

π t−1 (1−α)e
kt
αµt−1
Cooperative old age consumption
´
³
e 1/σ
b
cot = ³

π t−1 (1−α)e
kt
αµt−1
NonCooperative donations
C
dbN
=
t


³
´
b
πt−1 (1 − α) wt − It
αµt−1
´
³
π t−1 (1 − α) wt − Ibt
αµt−1

+ Rt  b
kt
+ Rt  b
kt
³
´
ζ 1/σ wt − Ibt − Rt b
kt
ζ 1/σ +
πt−1 (1−α)e
kt
αµt−1
NonCooperative middle age consumption
³
´


b
(1
−
α)
w
−
I
π
t−1
t
t
1

kt
b
cm
+ Rt  b
t =
e
πt−1 (1−α)kt
1/σ
αµ
t−1
ζ +
αµt−1
NonCooperative old age consumption
³
´


bt
1/σ
(1
−
α)
w
−
I
π
t−1
t
ζ

+ Rt  b
cot =
kt .
π t−1 (1−α)e
kt
1/σ
αµ
t−1
ζ +
αµt−1

THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
51
i1/ρ
h
ρ
ρ
e
e
In the CES case, writing y = F (k, h) = A ηk + h
= Az 1/ρ ,
ρ−1
for −∞ < ρ < 114, we compute Fk (t) = ηk ztF (t) and Fhh (t) =
Using again the ratio between (EGCh) and (EGCk), we have
¸1/(1−ρ)
·
[π t (1 − bnbt+1 )]ρ
b
b
kt+1 .
nt =
ηµt
h
hρ−1 F (t)
.
zt
Write rate of return, wage, and per capita output as a function of nbt+1
and parameters (quite interestingly, b
kt+1 drops out everywhere).
"
¸ρ/(1−ρ) #1/ρ−1
·
π t−1 (1 − bnbt )
Rt = Aη η +
µt−1
" µ
#1/ρ−1
¶ρ/(1−ρ)
ηµt−1
wt = A η
+1
π t−1 (1 − bnbt )
"
µ
¶ρ/(1−ρ) #1/ρ
b
(1
−
bn
)
π
t−1
t
ybt = Ab
kt η +
= At (nt+1 )b
kt
ηµt
So, in the CES we have an interesting ”Ak”, where A is a, time
varying, function of tomorrow’s fertility and physical to human capital
ratio. The other quantities can all be computed as before, by substituting into the definitions.
10.2.3. Computing the Intertemporal Competitive Equilibrium. At this
point we are left with two equations, (FOCn) and (FOCk), to solve
for the two remaining choice variables, b
kt+1 and γ h,t ; all the remaining
variables are then uniquely determined. To do this, and hic sunt leones,
³
´1/σ
ζπ t−1 (1−α)e
kt
we take the ratio between b
cot and b
cm
,
that
is
equal
to
t
αµ
t−1
and to ζ 1/σ , in the cooperative and noncooperative case respectively.
Error Equation Approach. The problem we face, in both cases, is that
the equilibrium choice of ht+1 depends on equilibrium values a further
period ahead, i.e. in period t + 2, because the rate of returns on human
capital investment and fertility depends on future values of capital and
human capital (i.e. on the disposable income of next period middle
age, after allowing for their own choices of investment). When the
production function is CES, this is compounded by the fact that current
fertility also depends on future fertility. This allows, in principle, for
multiplicity of equilibria.
14We
are ruling out the two extreme cases, i.e. fixed coefficients and linear,
corresponding respectively to ρ = −∞, and ρ = 1.
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
52
Take ratios between (FOCn) and (FOCk) to obtain, in the cooperative case
π t γ h,t
Rt+1 (bwt + µt γ h,t ) = ¡
×
¢1/σ
+ π t nbt
ζπ t nbt
·µ
¸
¶
´ 1−σ
¡
¢
π t nbt ³
b 1/σ
×
ζπt nt
+
wt+1 − Ibt+1 +
Rt+1 b
kt+1
σ
σ
and in the noncooperative case
³
´
i
h
π t γ h,t
1/σ
b
b
³
´
Rt+1 (bwt + µt γ h,t ) =
wt+1 − It+1 − Rt+1 kt+1 .
ζ
ζ 1/σ + π t nbt
In either case, we choose candidate equilibrium sequences for {kt , ht }t=T
t=0
by minimizing the appropriate error, for all t = 0, 1, ..., T , and then proceed to substitute back into all the previous equations. Notice that,
depending on which case is being solved (CD or CES) different expressions for nbt must be plugged in the equations above.
Recursive Approach — Cobb-Douglas. While the procedure above applies works for general production function (as well as for general utility function) in the case of the special functional forms adopted here,
a more straightforward approach seems possible.
Notice that the cooperative consumption ratio is
!1/σ
Ã
b
cot
kt
ζπ t−1 (1 − α)b
=
b
cm
αµt−1
t
and the noncooperative consumption ratio is
b
cot
= ζ 1/σ .
b
cm
t
Using (FOCk), in the cooperative case we have
Ã
!1/σ
!1/σ
Ã
kt
kt+1
ζπ t−1 (1 − α)b
ζπt (1 − α)b
= βαA
×
γ h,t
αµt−1
αµt
−1
Ã
!1/σ
b
b
π t (1 − α)kt+1 
ζπ t (1 − α)kt+1
+
×
×
αµt
αµt
"
(1 − α)π t−1
×
αµt−1
Ã
b(1 − α)b
kt+1
1−
αµt
!#1−α
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
53
Using (FOCk), in the noncooperative case we have
"
Ã
!#1−α
1/σ
b
βζ
αA
(1
−
α)π
b(1
−
α)
k
αµ
t−1
t+1
t
γ h,t ζ 1/σ =
1−
1/σ
αµt−1
αµt
αµt ζ + π t (1 − α)b
kt+1
These are both fairly ugly expressions, but they are both solvable for
γ h,t as an explicit function of b
kt+1 . Plug this function back into (FOCn)
and solve FOCn numerically for b
kt+1 . If the latter has a unique admissible solution, the equilibrium is unique and, given the initial b
k0 it can
be computed by forward recursion.
Recursive Approach — CES. The cooperative consumption ratio is
!1/σ
Ã
b
cot
kt
ζπ t−1 (1 − α)b
=
b
cm
αµt−1
t
and the noncooperative consumption ratio is
b
cot
= ζ 1/σ .
b
cm
t
Using (FOCk), in the cooperative case we have
µ
¶1/σ
h
i
b
ρ 1/(1−ρ)
[π
(1−bn
)]
t
t+1
ζπ t b
kt+1
ηµt
×
γ h,t = µ
¶
1/σ
h
h
i
i
b
b
ρ 1/(1−ρ)
ρ 1/(1−ρ)
[π
(1−bn
)]
[π
(1−bn
)]
t
t
t+1
t+1
ζπ t b
+
πtb
kt+1
kt+1
ηµ
ηµ
t
"
·
π t (1 − bnbt+1 )
×Aηβ η +
µt
t
¸ρ/(1−ρ) #1/ρ−1 Ã
kt
ζπ t−1 (1 − α)b
αµt−1
!−1/σ
Using (FOCk), in the noncooperative case we have
"
¸ρ/(1−ρ) #1/ρ−1
·
π t (1 − bnbt+1 )
βζ 1/σ−1
γ h,t =
Aη η +
i
h
µt
[π t (1−bnbt+1 )]ρ 1/(1−ρ)
1/σ
b
ζ +
π t kt+1
ηµ
t
In this case, because of the nature of the solution for current fertility,
we have an explicit solution for the current growth rate, but it depends
on the choice of future fertility. Hence, in the CES case, we must ’guess’
a candidate sequence for fertility and then proceed.
10.2.4. BGP Solutions. A BGP is a solution to the previous equations
in which all the ‘per capita’ variables are growing at some common rate
t m
γ h , (i.e., cm
t = γ h c0 , etc.), the size of the middle age generation is also
growing at a constant rate γ n = πn, and the two price variables, R,
and w are constant. In what follows, for any variable zt let zb = zt /ht .
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
54
Cooperative BGP. A couple of definitions apply to both to CobbDouglas and CES production functions. From
(ζπn)1/σ
b
co =
h ³
´
i
πn w − Ib + Rb
k
(ζπn)1/σ + πn
h ³
´
i
1
b + Rb
πn
w
−
I
k
b
cm =
(ζπn)1/σ + πn
we get
b
co
= (ζπn)1/σ = Υ1/σ .
m
b
c
£
¤−1
Recall that Υ = ζπn, and Π = Υ1/σ Υ1/σ + πn
bb
The eight variables to be determined are {d,
k, n, γ h , b
cm , b
co , R, w}.
Donation function
1/σ
db =
FOCn
σ−1
ζn [γ h ]
FOCh
FOCk
(ζπn)
³
´
b
wh − I − Rb
k
(ζπn)1/σ + πn
.
·³
´ 1−σ ¸
Π
πn ´ ³
1/σ
b
Υ +
w−I +
Rb
k
(bw+µγ h ) = β 1/σ
Υ + πn
σ
σ
ζ [γ h ]σ µn = βΠw(1 − bn)
ζπn [γ h ]σ = βΠR
BCMa
BCOld
MCK
MCH
k + n(bw + µγ h ) + db = w
b
cm + γ h b
b
co = Rb
k + πndb
R = Fk (b
k, πn(1 − bn))
w = Fhh (b
k, πn(1 − bn))
Solution Procedure
From FOCk and FOCh, replacing the definitions for R and w
µFk (b
k, πn(1 − bn)) = π(1 − bn)Fh (b
k, πn(1 − bn))
h
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
55
Cobb-Douglas
(1 − α)b
k
µα
Hence, either we get a very small capital to human capital ratio, but
otherwise n cannot be too small. In particular, for current day parameter values, with (1−α)
= 2, b
k = 1, and µ = 1, we actually get a very
α
high level of fertility, i.e. n = 2. In general, because we are measuring
human and physical capital in units such that their relative price is
constant at one, whatever that means, I suspect that b
k << 1, so that
n << 2, but not zero. Let’s proceed. We can now write R, w, and y
as functions of b
k and parameters. From
n=
R = αAb
k α−1 [πn(1 − bn)]1−α , w = (1 − α)Ab
kα [πn(1 − bn)]−α
we get:
Ã
!#1−α
b(1 − α)b
k
π(1 − α)
1−
R = αA
µα
µα
"
Ã
!#−α
π(1 − α)
b(1 − α)b
k
w = (1 − α)A
1−
µα
µα
"
Ã
!#1−a
b
(1
−
α)π
b(1
−
α)
k
yb = Ab
k
.
1−
αµ
αµ
"
Plugging the fertility rate back into the definition of effective human
capital.
´
k³
b (1 − α)πb
b
e
αµ
−
b(1
−
α)
k
.
h=
(αµ)2
Next, we use our expression for n to rewrite aggregate investment as a
function of predetermined variables, b
k and γ h .
·
¸
(1
−
α)
Ib = b
k γh +
(bw + µγ h )
αµ
Again, we have two ways of approaching the final step (i.e. using FOCn
and FOCk to solve for the unknowns b
k and γ h .)
First approach: take ratios between FOCn and FOCk (I am not
replacing the expressions for R, w, Ib here)
πγ h
R(bw + µγ h ) = ³
×
´
1/σ
ζπ(1−α)e
k
π(1−α)e
k
+ µα
µα
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
Ã
ζπ(1 − α)b
k
× 
µα
!1/σ
56


³
´
b
π(1 − α)k 
1 − σ b
+
Rk
w − Ib +
µασ
σ
k (or viceversa) and plug the
Solve the latter for γ h as a function of b
solution back into FOCk, to get the last unknown.
As in the intertemporal case, FOCk is, after replacing for R and n
from above,
Ã
!#1−α
"
b
b(1
−
α)
k
ζπ(1 − α)b
k
π(1
−
α)
[γ h ]σ = βΠαA
1−
,
µα
µα
µα
which leads to the second procedure. That is, solve the FOCk for
γ h as a function of b
k, this gives

1/σ(1−α)/σ
h
i−1
ζπ(1−α)e
k 1/σ
ζπ(1−α)e
k 1/σ
π(1−α)e
k
2
( µα ) + µα
µα A 
 β( µα )
×
γh = 

ζπ(1 − α)b
k
"
π(1 − α)
×
µα
Ã
b(1 − α)b
k
1−
µα
!#
CES
i1/ρ
h
hρ
=
Recall our notation, which is: y = F (k, e
h) = A ηk ρ + e
Az 1/ρ , Fk (t) =
h ρ−1
ηkρ−1 F (t)
zt
and Fhh (t) = h ztF (t) . Then
¸1/(1−ρ)
·
ηµ
b
.
k=n
[π(1 − bn)]ρ
"
·
¸ρ/(1−ρ) #1/ρ−1
π(1 − bn)
R = Aη η +
ηµ
" µ
w=A η
" ·
yb = An η
ηµ
π(1 − bn)
ηµ
[π(1 − bn)]ρ
¶ρ/(1−ρ)
¸ρ/(1−ρ)
#1/ρ−1
+1
+ [π(1 − bn)]ρ
#1/ρ
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
57
As we are expressing everything in term of fertility, effective human
capital per unit of human capital is just.
b
e
h = πn(1 − bn).
Next, we rewrite aggregate investment as a function of predetermined
variables, n and γ h .
¸1/(1−ρ)
·
ηµ
b
I = n(bw + µγ h ) + γ h n
[π(1 − bn)]ρ
We are again back to FOCn and FOCk and the two unknowns, which
in this case are n and γ h . I just report the final equation for the second
solution procedure, the one that uses FOCk to solve for γ h as a function
of n, the rest is as before.
"
¸ρ/(1−ρ) #1/ρ−1
·
¤
£
π(1
−
bn)
−1
Aη η +
[γ h ] = β(ζπn)1/σ−1 (ζπn)1/σ + πn
ηµ
σ
NonCooperative BGP. Also in this case, from
h ³
´
i
ζ 1/σ
b
b
b
c = 1/σ
πn w − I + Rk .
ζ + πn
h ³
´
i
1
m
b
b
b
c = 1/σ
πn w − I + Rk
ζ + πn
o
we get
b
co
= ζ 1/σ
m
b
c
bb
The eight variables to be determined are {d,
k, n, γ h , b
cm , b
co , R, w}.
Donation function is
³
´
ζ 1/σ w − Ib − Rb
k
d=
ζ 1/σ + πn
FOCn
h
³
´
i
πζ 1/σ
ζ [γ h ]σ−1 (bw + µγ h ) = β 1/σ
ζ 1/σ w − Ib − Rb
k
[ζ + πn]2
FOCh
πζ 1/σ
σ
ζ [γ h ] µ = β 1/σ
w(1 − bn)
ζ + πn
FOCk
ζ 1/σ
ζ [γ h ]σ = β 1/σ
R
ζ + πn
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
BCMa
BCOld
MCK
MCH
58
k + n(bw + µγ h ) + db = w
b
cm + γ h b
k + πndb
b
co = Rb
k, πn(1 − bn))
R = Fk (b
k, πn(1 − bn))
w = Fhh (b
Solution Procedure
From FOCk and FOCh, replacing the definitions for R and w
k, πn(1 − bn)) = π(1 − bn)Fh (b
k, πn(1 − bn))
µFk (b
h
Cobb-Douglas
(1 − α)b
k
µα
Same solution for fertility as in the cooperative case. Hence, factor
prices and output have the same expressions as a function of b
k (obviously, the latter will have a different value in noncooperative equilibrium.)
"
Ã
!#1−α
π(1 − α)
b(1 − α)b
k
R = αA
1−
µα
µα
Ã
!#−α
"
b(1 − α)b
k
π(1 − α)
1−
w = (1 − α)A
µα
µα
"
Ã
!#1−a
b
(1
−
α)π
b(1
−
α)
k
yb = Ab
k
.
1−
αµ
αµ
n=
Plugging the fertility rate back into the definition of effective human
capital.
´
k³
b (1 − α)πb
b
e
αµ
−
b(1
−
α)
k
.
h=
(αµ)2
Next, we use our expression for n to rewrite aggregate investment as a
function of predetermined variables, b
k and γ h .
¸
·
(1 − α)
b
b
(bw + µγ h )
I = k γh +
αµ
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
59
Again, two procedures are possible for the last step. The first gives:
h
³
´
i
γ h πζ 1/σ
1/σ
b − Rb
ζ
w
−
I
k
R(bw + µγ h ) =
e
k
ζ 1/σ + π(1−α)
µα
k (or viceversa) and plug the soSolve the latter for γ h as a function of b
lution back into FOCk, to get the last unknown. The second procedure
gives,
"
Ã
!#1−α
1/σ−1
b
βζ
b(1
−
α)
k
π(1
−
α)
[γ h ]σ =
αA
1−
e
µα
µα
ζ 1/σ + π(1−α)k
µα
to get an explicit function for γ h as a function of b
k.
CES
The part concerned with solving for the physical to human capital
ratio as a function of fertility and then computing rate of return, real
wage, output, effective human capital and aggregate investment, is
identical to the cooperative case, hence:
·
¸1/(1−ρ)
ηµ
b
k=n
.
[π(1 − bn)]ρ
"
¸ρ/(1−ρ) #1/ρ−1
·
π(1 − bn)
R = Aη η +
ηµ
" µ
#1/ρ−1
¶ρ/(1−ρ)
ηµ
wt = A η
+1
π(1 − bn)
#1/ρ
" ·
¸ρ/(1−ρ)
ηµ
yb = An η
+ [π(1 − bn)]ρ
[π(1 − bn)]ρ
b
e
h = πn(1 − bn)
¸1/(1−ρ)
·
ηµ
b
I = n(bw + µγ h ) + γ h n
[π(1 − bn)]ρ
We just report the last step for the CES case using the procedure
that manipulates FOCk directly. In the noncooperative case with CES
production this approach gives immediately an explicit solution for γ h
as a function of n.
"
¸ρ/(1−ρ) #1/ρ−1
·
1/σ−1
βζ
π(1
−
bn)
[γ h ]σ = 1/σ
Aη η +
ηµ
ζ + πn
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
60
10.3. Three Periods, Two Sectors, Youth Mortality Only. In
this model there are two distinct sectors, one for market output and for
childcare services. In this case, parents do not use their own time to
rear children but purchase the services from the market, at the going
equilibirum price. The optimization problem is
o
o
Max{nbt ,dit ,ht+1 kt+1 } u(cm
t ) + ζu(ct ) + βu(ct+1 )
subject to:
b
i
m
cc
ˆcc
cm
t + kt+1 + nt (at + ŵt t + µt ht+1 ) + dt ≤ wt ht t + ŵt t
o
i
ct ≤ Rt kt + dt + Dt
cot+1 ≤ Rt+1 kt+1 + π t nbt dt+1
m
cc
t + t ≤ 1
ˆcc ≥ b.
t
cc
Market clearing in the childcare sector implies that ˆcc
t = t , for the
representative agent in the cohort. Similarly, interiority will imply that
ŵt = ht wt , and hence, it follows that θt = at + bht wt , and thus, since
at = 0, θt = bht wt .Thus, output, in period t + 1, per old person in the
same period, is still given by:
ht+1 ) = F [kt+1 , π t nb ht+1 (1 − bnb )],
yt+1 = F (kt+1 , e
t
t+1
and rental rates and wage rates are given by:
and,
respectively.
Rt+1 = Fk (kt+1 , π t nbt ht+1 (1 − bnbt+1 ))
wt+1 = Fhh (kt+1 , π t nbt ht+1 (1 − bnbt+1 ))
10.3.1. Donations and Old Age Consumption. Assuming that u(c) =
c1−σ /(1 − σ), first order conditions for donations are:
¡
¢1/σ m
ct
cot = ζπ t−1 nbt−1
for the cooperative, and
cot = ζ 1/σ cm
t
for the noncooperative case, respectively.
Cooperative Donations and Consumptions. We have:
¢1/σ
¡
(wt ht − It ) − Rt kt
ζπ t−1 nbt−1
dt =
.
¡
¢
1/σ
+ π t−1 nbt−1
ζπ t−1 nbt−1
¢1/σ
¡
b
£
¤
n
ζπ
t−1
t−1
πt−1 nbt−1 (wt ht − It ) + Rt kt .
cot = ¡
¢
1/σ
+ π t−1 nbt−1
ζπ t−1 nbt−1
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
61
£
¤
1
b
cm
n
(w
h
−
I
)
+
R
k
π
t−1
t
t
t
t
t
¢
t = ¡
t−1
1/σ
+ π t−1 nbt−1
ζπ t−1 nbt−1
h
i−1
1/σ
1/σ
b
b
Υt + π t nt , and It = st +
Recall that Υt = ζπ t nt , Πt = Υt
nbt (bwt ht + µt ht+1 ). Partial derivatives of old age consumption with
∂cot
).
respect to saving, fertility, and education are (note the change in ∂h
t
∂cot
= Πt−1 Rt
∂st−1
∂cot
Πt−1 π t−1
= 1/σ
b
∂nt−1
Υt−1 + π t−1 nbt−1
·µ
¸
¶
π t−1 nbt−1
1−σ
1/σ
Υt−1 +
(wt ht − It ) +
Rt kt ,
σ
σ
∂cot
= Πt−1 πt−1 nbt−1 wt .
∂ht
NonCooperative Donations and Consumptions.
dt =
ζ 1/σ (wt ht − It ) − Rt kt
ζ 1/σ + πt−1 nbt−1
£
¤
ζ 1/σ
b
n
(w
h
−
I
)
+
R
k
π
.
t−1
t
t
t
t
t
t−1
ζ 1/σ + π t−1 nbt−1
£
¤
1
b
n
(w
h
−
I
)
+
R
k
cm
π
t−1
t
t
t
t
t
t = 1/σ
t−1
ζ + π t−1 nbt−1
cot =
∂cot
∂nbt−1
∂cot
ζ 1/σ
= 1/σ
Rt
∂st−1
ζ + π t−1 nbt−1
h
i
π t−1 ζ 1/σ
1/σ
=h
i2 ζ (wt ht − It ) − Rt kt
ζ 1/σ + π t−1 nbt−1
∂cot
ζ 1/σ
= 1/σ
π t−1 nbt−1 wt
b
∂ht
ζ + π t−1 nt−1
10.3.2. Intertemporal First Order Conditions. (FOCn)
0 o
u0 (cm
t )(θ t + µt ht+1 ) = βu (ct+1 )
∂cot+1
∂nbt
(FOCh)
b
0 o
u0 (cm
t )µt nt = βu (ct+1 )
∂cot+1
∂ht+1
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
62
(FOCk)
∂cot+1
∂st
Reducing the system to two equations and two unknowns. From (FOCh)
and (FOCk) we get that, in both the cooperative and non cooperative
world
³
´
³
´
ht = π t−1 Fhh kt , e
ht .
µt−1 Fk kt , e
0 o
u0 (cm
t ) = βu (ct+1 )
In the Cobb-Douglas case, the fertility rate is
¶
µ
1
1−α b
b
1−
nt =
kt ,
b
αµt−1 nbt−1
So, because nbt−1 is predetermined, the intertemporal equilibrium relation between capital intensity and fertility is one to one. Hence, one
of the BGP roots must be not admissible (or totally unstable). Notice
that fertility decreases when capital intensity increases. We also have
(notice that rate of return and wage rate depend only uppon mortality
rates and parameters.
¸1−α
·
π t−1 (1 − α)
Rt = αA
αµt−1
¸α
·
αµt−1
wt = (1 − α)A
π t−1 (1 − α)
¶1−α
µ
π t−1 (1 − α)
b
bt b
ybt = A
kt = A
kt
αµt−1
e
ht
(1 − α)π t−1 b
b
e
=
kt .
ht =
ht
αµt−1
¶
µ
1
1−α b
b
b
It = kt+1 +
1−
kt (wb + µt γ h,t )
b
αµt−1 nbt−1
The remaining variables can be computed accordingly.
h
i1/ρ
In the CES case, recall again that y = F (k, e
h) = A ηk ρ + e
hρ
=
ρ−1
Az 1/ρ , for −∞ < ρ < 1, Fk (t) = ηk ztF (t) , and Fhh (t) =
"
µ ρ ¶1/(1−ρ) b #
π t−1
1
kt
nbt =
1−
b
µt−1 η
nbt−1
"
¶ρ/(1−ρ) #1/ρ−1
µ
π t−1
Rt = Aη η +
µt−1 η
h
hρ−1 F (t)
.
zt
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
63
" µ
#1/ρ−1
¶
ηµt−1 ρ/(1−ρ)
wt = A η
+1
π t−1
"
µ
¶ρ/(1−ρ) #1/ρ
πt−1
kt
ybt = Ab
kt η +
= At b
ηµt−1
10.3.3. Computing the Intertemporal Competitive Equilibrium.
Error Equation Approach. The ratio between (FOCn) and (FOCk), in
the cooperative case is
π t γ h,t
Rt+1 (bwt + µt γ h,t ) = ¡
×
¢1/σ
+ π t nbt
ζπ t nbt
¸
¶
·µ
´ 1−σ
¢
¡
π t nbt ³
b 1/σ
+
Rt+1 b
wt+1 − Ibt+1 +
kt+1
×
ζπt nt
σ
σ
and in the noncooperative case is
h
³
´
i
π t γ h,t
´ ζ 1/σ wt+1 − Ibt+1 − Rt+1b
Rt+1 (bwt + µt γ h,t ) = ³
kt+1 .
ζ 1/σ + π t nbt
Recursive Approach. In the recursive approach we exploit the consumption ratios, so let’s repeat them here. The cooperative consumption
ratio is
¡
¢1/σ
b
cot
b
=
ζπ
n
t−1
t−1
b
cm
t
and the noncooperative consumption ratio is
b
cot
= ζ 1/σ .
b
cm
t
Cobb-Douglas. In the cooperative case
γ σh,t (ζπ t−1 nbt−1 )
=
β(ζπ t nbt )1/σ
£
(ζπt nbt )1/σ
+
¤−1
π t nbt
·
π t (1 − α)
αA
αµt
¸1−α
Please, note that in the previous equation we have not replaced the
expression derived above for fertility, i.e.
¶
µ
1
1−α b
b
nt =
1−
kt ,
b
αµt−1 nbt−1
in place of nbt , purely to keep the expression from exploding. Once
you replace it, then the growth rate is immediately computed as a
function of parameter values, and of the state variables nbt−1 and b
kt .
This particular case is even simpler than the other ones.
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
64
In the noncooperative case
γ σh,t
=
ζ 1/σ +
πt
b
¸1−α
·
βζ 1/σ−1
π t (1 − α)
³
´ αA
αµ
1 − αµ 1−αnb b
kt
t−1 t−1
CES. Using (FOCk), in the cooperative case (again, no replacement)
£
¤−1
γ σh,t (ζπ t−1 nbt−1 ) = β(ζπ t nbt )1/σ (ζπt nbt )1/σ + π t nbt
×
"
µ
¶ρ/(1−ρ) #1/ρ−1
πt
×Aη η +
µt η
Using (FOCk), in the noncooperative case
"
¶ρ/(1−ρ) #1/ρ−1
µ
1/σ−1
βζ
π
t
Aη η +
γ σh,t = 1/σ
b
µ
ζ + π t nt
tη
Also with CES, as with CD, we have an explicit solution for the
current growth rate, in terms of state variables
10.3.4. BGP Solutions.
Cooperative BGP. Recall useful definitions.
h ³
´
i
(ζπn)1/σ
o
b + Rb
b
c =
πn
w
−
I
k
(ζπn)1/σ + πn
h ³
´
i
1
b + Rb
b
cm =
πn
w
−
I
k
(ζπn)1/σ + πn
b
co
= (ζπn)1/σ = Υ1/σ
b
cm
bb
The eight variables to be determined are {d,
k, n, γ h , b
cm , b
co , R, w}
Donation function
db =
FOCn
σ−1
ζn [γ h ]
FOCh
³
´
(ζπn)1/σ wh − Ib − Rb
k
(ζπn)1/σ + πn
.
·³
´ 1−σ ¸
Π
πn ´ ³
1/σ
Υ +
w − Ib +
Rb
k
(bw+µγ h ) = β 1/σ
Υ + πn
σ
σ
ζ [γ h ]σ µn = βΠw
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
65
FOCk
ζπn [γ h ]σ = βΠR
BCMa
BCOld
b
cm + γ h b
k + n(bw + µγ h ) + db = w
MCK
MCH
b
co = Rb
k + πndb
R = Fk (b
k, πn(1 − bn))
w = Fhh (b
k, πn(1 − bn))
Solution Procedure
µFk (b
k, πn(1 − bn)) = πFhh (b
k, πn(1 − bn))
Cobb-Douglas
µαAb
kα−1 [πn(1 − bn)]1−α = π(1 − α)Ab
kα [πn(1 − bn)]−α
αµn(1 − bn)
b
k=
1−α
¸1−α
·
π(1 − α)
R = αA
αµ
¸α
·
αµ
w = (1 − α)A
π(1 − α)
¶1−α
µ
π(1 − α)
b
bb
k=A
kt
yb = A
αµ
Last step, again using FOCk, solve for γ h as a function of n and then
use FOCn to solve the nonlinear equation for the BGP value of n.
¸1−α
·
β (ζπn)1/σ−1
π(1 − α)
σ
[γ h ] =
αA
αµ
(ζπn)1/σ + πn
CES
Recall our notation, which is: y = F (k, e
h) == Az 1/ρ , Fk (t) =
ρ−1
h
and Fhh (t) = h ztF (t) .
zt
h
i1/ρ−1
h
i1/ρ−1
hρ
= πAhρ−1 ηk ρ + e
hρ
µAk ρ−1 ηk ρ + e
ηkρ−1 F (t)
TO BE ADDED
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
NonCooperative BGP.
h ³
´
i
ζ 1/σ
b + Rb
πn
w
−
I
k
.
ζ 1/σ + πn
h ³
´
i
1
πn w − Ib + Rb
k
b
cm = 1/σ
ζ + πn
b
co
= ζ 1/σ
m
b
c
bb
The eight variables to be determined are {d,
k, n, γ h , b
cm , b
co , R, w}
Donation function is
³
´
ζ 1/σ w − Ib − Rb
k
d=
ζ 1/σ + πn
FOCn
´
i
πζ 1/σ−1 h 1/σ ³
ζ
w − Ib − Rb
k
[γ h ]σ−1 (bw + µγ h ) = β 1/σ
[ζ + πn]2
b
co =
FOCh
[γ h ]σ µ = β
FOCk
πζ 1/σ−1
w
ζ 1/σ + πn
ζ 1/σ−1
R
[γ h ] = β 1/σ
ζ + πn
σ
BCMa
k + n(bw + µγ h ) + db = w
b
cm + γ h b
BCOld
k + πndb
b
co = Rb
MCK
k, πn(1 − bn))
R = Fk (b
MCH
w = Fhh (b
k, πn(1 − bn))
Solution Procedure
Cobb-Douglas
µαAb
kα−1 [πn(1 − bn)]1−α = π(1 − α)Ab
kα [πn(1 − bn)]−α
αµn(1 − bn)
b
k=
1−α
66
THREE EQUATIONS GENERATING AN INDUSTRIAL REVOLUTION?
67
¸1−α
π(1 − α)
R = αA
αµ
¸α
·
αµ
w = (1 − α)A
π(1 − α)
µ
¶1−α
π(1 − α)
b
bb
yb = A
k=A
kt
αµ
Last step, again using FOCk, solve for γ h as a function of n and then
use FOCn to solve the nonlinear equation for the BGP value of n.
¸1−α
·
βζ 1/σ−1
π(1 − α)
σ
[γ h ] = 1/σ
αA
αµ
ζ + πn
CES
·
TO BE ADDED
Department of Economics, University of Minnesota and Research
Department, Federal Reserve Bank of Philadelphia