The Evolution of Markets and the Revolution of Industry:
A Uni…ed Theory of Growth
Klaus Desmet
Universidad Carlos III
and CEPR
Stephen L. Parente
University of Illinois
and CRENoS
February 2009
Preliminary and Incomplete
Abstract
This paper puts forth a uni…ed theory of growth. According to this theory, an
economy’s transition from a Malthusian era with stagnant living standards, to a modern era, with sustained and regular increases in income per capita, requires markets
to reach a critical size, and competition to reach a critical level of intensity. An
evolution in markets is thus a precondition for a revolution in industry. By allowing
an economy to produce a greater variety of goods, a larger market makes goods more
substitutable, raises price elasticity of demand, and lowers mark-ups. With lower
mark-ups …rms become larger in order to break even, which raises the return to innovation as a larger …rm is able to ammortize R&D costs over a larger of quantity of
goods. We demonstrate our theory in a dynamic general equilibrium model and use
it to demonstrate how various factors emphasized in the literature a¤ect the starting
date of this transition.
JEL Classi…cation: 014, N
Keywords: Industrial Revolution, Innovation, Competition, Market Revolution
We thank Oded Galor, Douglas Gollin, Berthold Herrendorf, Peter Howitt, Chad Jones, Richard
Rogerson, and David Weil, as well as seminar participants at the 2008 NBER Meeting on Economies across
space and time, 2008 Midwest Macro Meetings, and 2008 SED Meetings. The usual discalmer applies.
Desmet: Department of Economics, Universidad Carlos III. E-mail: klaus:desmet@uc3m:es; Parente:
Department of Economics, University of Illinois at Urbana-Champaign. E-mail: parente@illinois:edu:
1
Introduction
Being perhaps the single most important economic event in world history, the Industrial
Revolution has generated a vast body of research that studies its causes. Despite this
great e¤ort, the causes of the Industrial Revolution remain, in the words of Clark (2003),
“one of history’s great mysteries”, primarily because the theories o¤ered to date either
rely on mechanisms that are empirically implausible or fail to quantitatively account for
the transition from Malthusian to modern growth.1 This paper aims to change this state
of a¤airs. In particular, it puts forth a novel theory of the Industrial Revolution and
shows that it is empirically plausible by calibrating the model to the historical record of
England and by providing micro-evidence for the mechanism at the heart of our theory.
The paper’s central hypothesis is that markets need to become su¢ ciently large
and competition su¢ ciently strong before an economy can take o¤. An evolution in
markets is thus a precondition for a revolution in industry. In particular, a gradual
increase in market size during the Malthusian era, associated with population growth
and lower transport costs, spurs greater competition, and sets the industrial revolution
in motion, allowing the economy to move from a state of no growth to one of high and
constant growth.
Clearly, the novelty of our theory lies in the mechanism by which market size
matters, and not in the idea that the expansion of markets was a necessary condition
for take-o¤. Indeed, this idea is as old as economics itself, dating back to Adam Smith
(1776).2 Instead, what sets our theory apart is the mechanism linking market size, competition and innovation. This mechanism works as follows. A larger market allows an
economy to sustain a greater variety of goods, making them more substitutable. As a
result, price elasticities of demand increase, and mark-ups drop. These changes raise the
return to innovation. With a drop in mark-ups, …rms must sell more units of output in
order to break even. Hence, …rms become larger, allowing them to amortize the …xed in1
See Clark (2003) for a critical review of the literature and a description of its failures.
2
See, e.g., Szostak (1993) and Shiue and Keller (2007) for empirical evidence of the importance of
market size.
1
novation costs over a greater quantity of goods. Therefore, only after the market reaches
a critical size and competition is strong enough, does innovation endogenously take o¤.
To generate this elasticity channel, we use preferences based on Lancaster’s (1979)
model of trade in ideal varieties. The Lancaster construct, based on Hotelling’s (1929)
spatial model of horizontal di¤erentiation, assumes that each household has an ideal
variety of a manufactured good identi…ed by its location on the unit circle. As goods ‘…ll
up the circle’, neighboring varieties become closer substitutes, so that the price elasticity
of demand increases and mark-ups fall (Helpman and Krugman, 1985; Hummels and
Lugovskyy, 2005). Using these preferences, Desmet and Parente (2010) propose a oneperiod one-sector model that allows for both process and product innovation, and show
that a larger market is associated with larger rates of technological innovation.
In this paper, we transform the model of Desmet and Parente (2010) into a model
of the industrial revolution. We extend their model in three main ways. First, we make
the model dynamic. Second, we introduce a farm sector that produces a subsistence
consumption good. Third, we include children in the parents’ preferences, and assume
a lower cost of child rearing on the farm than in the city.3 With these features, the
model generates a transition from a period of almost stagnant living standards to one
with a constant and robust rate of economic growth. Additionally, it leads to a structural transformation with a declining agricultural share and a demographic transition
with population growth initially rising with the advent of the industrial revolution and
subsequently falling.
The model works as follows. On account of the subsistence constraint and the
low initial agricultural productivity, most of the population starts o¤ residing in the farm
sector. As the industrial population is small, the number of industrial varieties produced
in the economy is limited. Goods are not particularly substitutable, implying a high
mark-up and a small …rm size. As a result, …rms do not …nd it pro…table to incur a …xed
R&D cost to lower their marginal costs. Although living standards are nearly stagnant,
3
Consistent with this view, the laws restricting child labor in 19th century England applied only to
factory work, and explicitly excluded farm work (Doepke and Zilibotti, 2005).
2
exogenous increases in agricultural TFP allow the economy to sustain a slowly growing
population. Eventually, the population reaches a critical size, making industrial …rms
su¢ ciently large to warrant process innovation. At this point, …rms endogenously start
innovating, and an industrial revolution begins.
Process innovation in the industrial sector then sets o¤ a demographic transition
and a structural transformation. Process innovation increases wages in both the farm and
industrial sectors. Since children have a positive e¤ect on the utility of parents, both rural
and urban households have more children, and the population growth rate accelerates.
This corresponds to the …rst phase of the demographic transition. At the same time, the
increasing living standard relaxes the subsistence constraint, leading to a decrease in the
share of agriculture in economic activity. Assuming the cost of child rearing is higher in
the city than on the farm,4 the structural transformation implies a compositional e¤ect
that tends to lower population growth. This corresponds to the second phase of the
demographic transition.
In the limit, as living standards continue to rise, the subsistence constraint disappears, and the economy converges to constant agricultural and industrial labor shares.
Under certain parametric conditions, the population growth rate converges to zero. In
this case, price elasticity and …rm size asymptotically are constant. As a result, in the
limit …rms innovate at a constant rate every period. Thus, the economy converges to a
modern growth era with a constant positive growth rate of per capita GDP.
We calibrate the model to the historical experience of England over the 1300 to
2000 period. We …nd that the model does remarkably well in accounting for the main
features of England’s experience, especially with regard to its demographic transition
and the length of the Industrial Revolution. We then use the calibrated model to explore
how a number of factors identi…ed in the literature as having contributed to the Industrial
Revolution a¤ects its timing. In this sense, our paper relates to the work of Stokey (2001)
and Harley and Crafts (2000), who likewise calibrate models of the Industrial Revolution
4
Consistent with this view, the laws restricting child labor in 19th century England applied only to
factory work, and explicitly excluded farm work (Doepke and Zilibotti, 2005).
3
to answer questions about the e¤ect of certain factors on the timing. On some dimensions,
such as the number of sectors, these other models are richer than ours, and hence can
address the e¤ect of certain speci…c factors, such as improvements in energy productivity.
On other dimensions, our model is far richer. For example, we do not have to limit our
calibration to steady state comparisons as in Stokey (2001).
Given that the elasticity channel is essential to our theory, it is important to
justify the empirical plausibility of the mechanism, and thus our choice of HotellingLancaster preferences, compared to the more standard Spence-Dixit-Stiglitz construct.
As we will discuss in detail in Section 2, the micro-evidence suggests that larger markets
are associated with higher elasticities of demand, lower mark-ups, and larger …rms. These
predictions are consistent with Hotelling-Lancaster, and inconsistent with Spence-DixitStiglitz.5
Hotelling-Lancaster preferences have the additional virtue of putting both process
and product innovation at the forefront of the Industrial Revolution. Previous models of
the Industrial Revolution almost exclusively emphasize technological progress, ignoring
product innovation. In the last decade, however, numerous economic historians have
argued rather convincingly that the Industrial Revolution was as much about product
innovation as process innovation. England during the 18th and 19th centuries did not
only experience an acceleration in technological progress, it also witnessed a signi…cant
increase in the varieties of consumer commodities, a so-called Consumer Revolution that
started sometime at the end of the 16th century (Styles, 2000; Berg, 2002).
In contrast to our model, standard endogenous growth models, based on the
Spence-Dixit-Stiglitz construct, are unable to generate both process and product innovation. Indeed, if the number of varieties is kept constant, as in the early endogenous
growth models, larger markets lead to more process innovation, but there is no product
5
Hotelling-Lancaster is not the only construct to generate these e¤ects. Salop (1979) does so in a
model similarly based on Hotelling (1929), Ottoviano, Tabuchi and Thisse (2005) do so using a quasilinear preference structure with a quadratic sub-utility, and Feenstra (2003) uses a translog expenditure
function. The same e¤ect also arises in oligopoly models, such as the one studied by Gali and Zilibotti
(1999). In addition, this elasticity e¤ect can also arise in Spence-Dixit-Stiglitz provided …rms take into
account how their price decisions a¤ect aggregate demand (Yang and Heijdra, 1993, and Jaimovich, 2007).
4
innovation. If, instead, the number of varieties is endogenized, an increase in the size of
the market size is exactly o¤set by a proportional increase in the number of varieties,
leaving process innovation una¤ected. This absence of a scale e¤ect is the key insight of
a second generation of endogenous growth models (e.g., Young, 1998). Taken together,
this earlier work implies that an increase in market size leads to either product innovation
or product innovation, but not to both. In light of the historical evidence, its failure to
account for both types of innovation makes it un…t to study the industrial revolution.
The number of theories of the Industrial Revolution that have been put forth is
vast. Clark (2003) identi…es three categories of such theories: exogenous based, multipleequilibria based, and endogenous based. Our theory belongs …rmly in the third category
of such theories as the driving force, market size, is endogenous in our model. Important
theories in this category are Kremer (1993), Goodfriend and McDermott (1998), Galor
and Weil (2000), Galor and Moav (2002), Voightlander and Voth (2006), and Clark
(2007). Clearly, the main di¤erence with our work lies in the novelty of the mechanism
we emphasize, and in its capacity to capture a number of important features of the
Industrial Revolution, such as the co-existence of process and product innovation and the
increase in …rm size, both largely ignored by previous work.
While our theory …rmly belongs in the third category, it also relates to several theories in the …rst category such as North and Thomas (1973), North and Weingast (1989),
Schultze (1968), Diamond (1997), and Findlay and O’Rourke (2007). These theories
argue that changes in exogenous factors such as institutions, agricultural productivity,
and trade costs a¤ected the timing of the Industrial Revolution.
A further virtue of
our work is that it does not invalidate many of the factors identi…ed in the literature as
having contributed to the Industrial Revolution. Population size, agricultural productivity, transportation costs, openness, and institutions all a¤ect the market size, and hence
the date at which …rm size reaches the critical level needed to make process innovation
pro…table.
The rest of the paper is organized as follows. Section 2 provides empirical support for the mechanism put forth in this paper, and hence serves as motivation. Section
5
3 describes the model and characterizes the optimal decisions of agents. Section 4 de…nes the equilibrium and shows algebraically that under certain conditions the economy
converges to a balanced growth path. Section 5 calibrates the model and describes a set
of numerical computations intended to fully illustrate the equilibrium properties of the
model and its mechanics. Section 5 concludes the paper.
2
Empirical Motivation
The contribution of our paper is the mechanism it puts forth. To be precise, its novelty
is not the argument that market size and …rm size matter for innovation,6 but rather
the general equilibrium mechanism that links market size to …rm size through the price
elasticity of demand and the importance this mechanism lends to both product and
process innovation. In what follows, we therefore exclusively focus on the evidence that
allows us to distinguish our mechanism from alternative mechanisms. Such evidence
relates to secular trends in price elasticity of demand, mark-ups, and …rm size. It also
requires analyzing the evidence on product innovation.
Product innovation Consistent with the historical record, our model leads to both
product and process innovation. A growing body of literature argues that product innovation was every bit as important to the Industrial Revolution as process innovation. The
creation of new consumer goods, and the increase in varieties, was an essential feature of
the Industrial Revolution and the period leading up to it. Berg (2002) analyzes the nature of British patents for the period 1627-1825 in a subset of industries, including metal
wares, glass, ceramics, furniture and watches. Of the 1,610 patents, over one-quarter
speci…ed new products or variations of existing ones. In a narrower study, Gri¢ ths et al.
(1992) document that roughly half of the 166 patented and non-patented improvements
in the textile industry between 1715 and 1800 concerned product innovation. Similarly,
De Vries (1993), using records from probate inventories, documents increasing variety in
6
The importance of market size goes back to the writings of Adam Smith (1776), and the relevance of
…rm size is prominent in the work of Joseph Schumpeter (1949) and Kenneth Arrow (1973).
6
household durables through the 18th century, despite relatively stagnant wages.
The creation of new consumer goods and new varieties is not a post 17th century
phenomenon. Referring to the 1500-1700 period, Styles (2000) lists a number of products
that were either entirely English inventions or dramatic remodeled goods from other
societies, such as pocket microscopes, drinking cups made from lead glass, and watches.
Other new products for English consumers, such as Delftware plates, Venetian glass, and
upholstered chairs, originated in Europe. Still others, such as porcelain, tea, tobacco,
sugar, lacquered cabinets, and painted calicos, came from Asia and the New World.
Price elasticity of demand.
Our mechanism is based on larger markets generating
higher price elasticities of demand. The most direct support for our theory would be
evidence of a secular rise in the price elasticity of demand. Unfortunately, time series
estimates for the price elasticity of individual products do not exist. Even in the crosssection, such estimates are uncommon. We know of only two such studies, both consistent
with our model. Barron et al. (2008) compute price elasticities in U.S. gasoline markets
and …nd that larger markets are associated with more elastic demand, whereas Hummels
and Lugovskyy (2006) document that import demand in larger markets is more responsive
to changes in trade costs.
Mark-ups.
Our mechanism implies a secular decline in mark-ups, a result of the in-
crease in the price elasticity of demand as markets expand. Estimates of mark-ups are
more readily available, although most studies focus on short time series. The one longrun study that we are aware of is Ellis (2006), who estimates mark-ups in the United
Kingdom for the period 1870-2003 and in the United States for the period 1980-2003.
Between 1870 and 2003 he …nds a 67 percent decline in the mark-up for the UK, whereas
for the US non-farm business sector he estimates a 10 percent drop between 1980 and
2003. Short-run studies also suggest that the mark-up is inversely related to market size.
Within the trade literature, Tybout (2003) documents that mark-ups generally fall following liberalization, and within the business cycle literature, Chevalier and Scharfstein
(1995) and Chevalier et al. (2003) …nd that mark-ups are countercyclical both at the
7
aggregate and the industry level.
Firm size.
Another implication of our mechanism is that …rm size increases with market
size. Here the studies are much more abundant. There is a large and extensive literature
that documents increases in establishment size since industrialization for both England
and the United States. For example, Lloyd-Jones and Le Roux (1980) document that the
median number of workers in cotton …rms in Manchester more than tripled between 1815
and 1841. In the case of pig iron, Feinstein and Pollard (1988) report that in England
production per furnace increased from 400 tons in 1750 to 550 in 1790. Using data from
the U.S. Census of Manufacturing, Sokolo¤ (1984) and Atack et al. (1999) …nd more of
the same in manufacturing industries over the 19th century, while Granovetter (1984)
documents this pattern continued into the 20th Century.
Sokolo¤’s (1984) study is particularly relevant because it shows that an increase
in …rm size was occurring prior to 1860, considered the starting year of the industrial revolution in the United States. In addition, he …nds that …rms also grew in size in industries
that did not mechanize, such as tanning, hats, boots and shoes. This is consistent with
our theory, which predicts that the increase in …rm size predates the economy’s take-o¤.
The Sokolo¤ study further supports our mechanism by uncovering a positive correlation
between market size, …rm size, and the level of industrialization. More densely populated
areas had larger …rms within a given industry. For example, in the wool textiles industry
in 1850 average …rm size was 38.7 in New England, compared to 14.5 in the Mid Atlantic, and 6.5 in the rest of the country. Additionally, it is those more dense areas that
industrialized …rst. Sokolo¤ documents the co-existence of artisan shops and factories
during the …rst half of the 19th century, with artisan shops located in rural areas with
low population density and high transportation costs.
There do not seem to be hard numbers of …rm size in England predating the
Industrial Revolution. Nevertheless, a burgeoning literature on proto-industrialization
suggests that …rm size increased substantially before the onset of the revolution. Protoindustrialization refers to the period between 1500 and 1700, when non-agricultural goods
8
were produced in the countryside for large regional, and even international, markets. It
is associated with the rise of the putting-out system, consisting of merchant capitalists
who would sell inputs to rural households and buy …nished good in return. Under this
system merchants controlled and centralized a number of activities, such as marketing
and …nance, whereas production was decentralized to rural households. If one interprets
the putting-out network as an organization, then the size of organizations was clearly
increasing before the Industrial Revolution. The only di¤erence, compared to the factory
system, is that not all tasks were performed under the same roof.
While the putting-out system was not truly a factory system, the evidence shows
that the transition from cottage production to the factory system was gradual and predated the Industrial Revolution. The centralization of production initially a¤ected speci…c
parts of the production process. For example, in the cotton industry spinning became
mechanized and centralized, when weaving was still being done by cottage industries.
Similarly, the printing of fabrics became centralized early on. Calico printing workshops
were proto-factories, and some of the leading calico printers were associated with the
introduction of mechanized spinning and weaving. In the woolen industry, the artisan
system was retained by clothiers using cooperative mills that centralized part of the production process. In the knitting industry, apart from sophisticated putting out networks,
the industry boasted centralized workshops from early in the eighteenth century. Large
workshops employing over forty parish apprentices existed in Nottingham as early as the
1720s. By the time that Hargreaves and Arkwright went to Nottingham, the concentration of labor in factories was a fairly familiar idea (Berg, 1994).
Many researchers, particularly Mendels (1972), argue that the rise of the cottage
industry was a critical step in the eventual industrialization of the British economy.
Indeed, Mendels claims that the proto-industrialization period was a critical transition
phase from the feudal world of the Middle Ages to the capitalist world of the modern
era. Our work compliments this area of research. In our theory increases in …rm size not
only predate the start of the Industrial Revolution, they are necessary for the revolution
to start.
9
Firm size and process innovation.
The idea that …rm size facilitates process inno-
vation has a long history in economics, going back as far as Schumpeter (1942). There
is much empirical evidence supporting this view. For example, Atack et al. (2008) …nd
that larger …rms were more likely to use steam power in the 19th century. Hannan and
McDowell (1984) reach a similar conclusion when analyzing the relationship between the
size of banks and the adoption of ATMs in the 1970s. In terms of R&D expenditures,
Cohen and Klepper (1996) …nd that they rise with …rm size, with a greater share being
allocated to process innovation.
3
The Model
In this section we describe the structure of the model economy. Time is discrete and
in…nite. There are three sectors: a farm sector, an industrial sector, and a household
sector. The farm sector is perfectly competitive and produces a single non-storable consumption good. The farm technology uses labor and land and is subject to exogenous
technological change. The industrial sector is monopolistically competitive and produces
a …nite set of di¤erentiated goods, each of which has a unique address on the unit circle.
There is both product and process innovation in the industrial sector. The household
sector consists of one-period lived agents, each of whom derives utility from consumption
of the agricultural good, consumption of the di¤erentiated industrial goods, and children.
For each household, there is a variety of the di¤erentiated good that it prefers above all
others. Households earn income by either working in the farm sector or the industrial
sector. In addition to working, households use their time to rear children, who constitute
the household sector in the next period. In what follows we describe the model structure
and the relevant optimization problems encountered by agents in each sector.
3.1
Household Sector
Endowments.
At the beginning of period t there is a measure Nt of households. Each
household is endowed with one unit of time, which it uses to rear children and to work
in either the farm or the industrial sector. There are no barriers to migration, so that
10
a household is free to work in either sector. Denote by Ntf and Ntx the measure of
households employed in agriculture and industry. Thus,
Nt = Ntf + Ntx :
(1)
Both types of households are uniformly distributed around the unit circle.
Preferences.
A household derives utility from the number of children it raises, nt ,
consumption of the agricultural good, cat , and consumption of the di¤erentiated industrial
goods, fcvt gv2Vt , where Vt denotes the set of di¤erentiated goods produced at time t. A
household located at point v~ on the unit circle has the following Cobb-Douglas preferences:
U = [(cat
ca )1
[g(cvt jv 2 Vt )] ] (nt )1
;
(2)
where
g(cvt jv 2 Vt ) = max[
v2Vt
cvt
1 + dv~v
]:
(3)
The subutility g(cvt jv 2 Vt ) follows Lancaster (1979) by assuming that each household
has a variety of the di¤erentiated good that it prefers above all others. This ideal variety
corresponds to the household’s location on the unit circle. The further away an industrial
variety v lies from a household’s ideal variety v~, the lower the utility it derives from consuming a unit of variety v. In particular, the quantity of variety v that gives the household
the same utility as one unit of its ideal variety v~ is given by 1 + dv~v , where dv~v denotes
the shortest arc distance between v and v~, and
> 0 is a parameter that determines how
fast a household’s utility diminishes with the distance from its ideal variety.7 Following
the literature on structural transformation and the demographic transition, each household has an agricultural subsistence constraint, represented by ca in the utility function.
Departing from the literature on the demographic transition, we assume that household
utility does not depend on the quality of children.
7
The expression 1 + dv~v is known as Lancaster’s compensation function.
11
Demographics.
Households live for one period. Let nit denote the number of children
of a household employed in sector i 2 ff; xg. The law of motion for the population is
then
Nt+1 = nft Ntf + nxt Ntx :
There is a time cost of rearing children, denoted by
(4)
i,
that depends on the sector
i 2 ff; xg in which the household works. We assume that this time cost is higher in the
city than in the countryside, i.e.,
x
f.
This is important for generating the second
phase of the demographic transition, characterized by a declining population growth rate.
Having a higher time cost in the city than in the country is consistent with the historical
record. For one, in the country it was possible to simultaneously watch children and
tend vegetables. In the city, factory work generally did not allow for this possibility.
For another, during the industrial revolution the cost of having a child surviving into
adulthood was higher in urban environments. For the United States, Becker et al. (2008)
document that prior to the introduction of urban sanitation in the 1920s the infant
mortality rate was higher in cities than in less densely populated areas.8 For England,
the evidence during the 19th century is similar: mortality was higher in cities and the
natural increase of population was lower (Williamson, 1985).9
Utility Maximization.
The di¤erential cost of rearing children in the city and on
the farm implies that household income will be di¤erent across sectors, even though
households are free to move. We therefore distinguish between an agricultural household’s
income per unit of time worked, ytf , and an industrial household’s income per unit of time
worked, ytx . The budget constraint of a household working in sector i 2 ff; xg is then
yti (1
i i
nt )
ciat +
X
pvt civt
(5)
v2V
8
See, also, Jones and Tertilt (2007), who report that the number of children per woman was higher in
non-urban areas and on farms throughout the 19th century.
9
Of course, the higher number of children in rural areas may be due to them being able to work more
easily on the farm than in the factory. See Rosenzweig and Evensen (1977) and Doepke (2004). In our
model, however, children do not participate in the labor force.
12
Maximizing (2) subject to (5) yields the following …rst order necessary conditions:
ciat =
X
)(yti
(1
pvt civt =
(yti
ca ) + ca
(6)
ca )
(7)
v2Vt
i i
nt
= (1
)(1
ca
)
yti
We make assumptions on the technology parameters to ensure that yti (1
all t
(8)
i ni )
t
> ca for
0.
To further characterize the optimal consumption of the di¤erentiated goods, we
exploit the property of the linearity of the subutility function (3) with respect to the set of
di¤erentiated goods. This property implies that each agent consumes a single industrial
variety. The variety v 0 that an agent with ideal variety v~ buys is the one that minimizes
the cost of an equivalent unit of the agent’s ideal variety, pvt (1 + dv~v ), so that
v 0 = argmin[pvt (1 + dv;~v )jv 2 Vt ]:
(9)
Using (7), a household with ideal variety v~ and working in sector i therefore buys the
following quantity of variety v 0 :
civ0 t =
(yti ca )
pv 0 t
(10)
Its demand for all other varieties v 2 Vt is zero.
3.2
Industrial Sector
The industrial sector is monopolistically competitive, and produces a set of di¤erentiated
goods, each with a unique address on the unit circle. As in Lancaster (1979), …rms can
costlessly relocate on the unit circle. The technology for producing industrial goods uses
labor as its only input. The existence of a …xed cost, which takes the form of labor, gives
rise to increasing returns. Each …rm chooses its price and technology, taking aggregate
variables and the choices of other …rms as given. There is free entry and exit, so that the
number of …rms will adjust to ensure all …rms make zero pro…ts.
13
Production. Let Qvt be the quantity of variety v produced by a …rm; Lvt the units of
labor it employs; Avt its technology level, or production process; and
vt
its …xed cost in
terms of labor. Then the output in period t of the …rm producing variety v is
Qvt = Avt [Lvt
Both the …xed labor cost,
vt ,
vt ]
(11)
and the technology level, Avt , depend on the …rm’s rate of
process innovation, gvt . In particular, the …xed labor cost is given by
vt
= e
gvt
:
(12)
Thus, there are two components to the …xed cost: an innovation cost, represented by
e
gvt ,
that is increasing in the size of process innovation, gvt , and an operating cost, ,
that is incurred even if there is no process innovation. The …rm’s technology level, Avt ,
is given by
Avt = (1 + gvt )Axt :
(13)
where Axt is the benchmark technology in period t, taken to be the average technology
used by industrial …rms in period t
1:
Axt =
X
v2Vt
1
Av;t
mt 1
1;
(14)
Therefore, if gvt = 0, so there is no process innovation, a …rm uses the industrial benchmark technology, Axt , whereas if gvt > 0, the …rm uses a technology that is (1 + gvt ) times
greater than the benchmark technology.
Pro…t Maximization.
The …xed operating cost implies that each variety, regardless
of the technology used, will be produced by a single …rm. In maximizing its pro…ts,
each …rm behaves non-cooperatively, taking the choices of other …rms as given. Pro…t
maximization determines the price and quantity to be sold, the number of workers to
be hired, and the technology to be operated. As is standard in models of monopolistic
competition, …rms take all aggregate variables in the economy as given.10
10
In principle this requires …rms to be of measure zero, a condition that is not satis…ed. See Desmet
and Parente (2010) for a discussion of how …rms could be made of measure zero, without changing any
of the results.
14
Using (11), the pro…ts of the …rm producing variety v,
vt
= pvt Qvt
wxt [ e
gvt
+
vt ,
can be written as
Qvt
];
Axt (1 + gvt )
(15)
where wxt is the wage in the industrial sector, and pvt is the price of variety v.
The problem of the …rm producing variety v is to choose (pvt ; gvt ) to maximize
(15), subject to the aggregate demand for its product. As usual, the pro…t maximizing
price is a markup over the marginal unit cost wxt =[Axt (1 + gvt )], so that
pvt =
"vt
wxt
;
Axt (1 + gvt ) "vt 1
(16)
where "vt is the price elasticity of demand for variety v:
"vt =
@Cvt pvt
:
@pvt Cvt
The …rst order necessary condition associated with the choice of technology, gvt , is
e
gvt
+
Cvt
Axt (1 + gvt )2
0
(17)
where the strict inequality case in the above expression corresponds to a corner solution,
i.e., gvt = 0.
3.3
Farm Sector
The farm sector is perfectly competitive. Farms produce a single, non-storable consumption good, that serves as the economy’s numéraire. The farm technology is constant
returns to scale, and uses labor and land. There is a measure one of farms.
Production. Let Qat denote the quantity of agricultural output of the stand-in farm,
and Lat the corresponding agricultural labor input. Without loss of generality, we normalize the land owned by the stand-in farm to 1. The production function is Cobb-Douglas
in land and labor with a labor share of 1
> 0. Hence,
Qat = Aat Lat
Agricultural TFP, Aat , grows at a rate gat
(18)
0 during period t, so that
Aat+1 = Aat (1 + gat ):
15
(19)
During the Malthusian phase agricultural TFP grows at a constant exogenous rate,
a
>
0.11 However, once the industrial sector starts innovating, we allow for agricultural TFP
growth to accelerate. In particular, we assume that
gat = maxf
a;
Axt Axt
Axt 1
1
g
(20)
This expression captures the large secular rise in the growth rate of agricultural TFP
since the Industrial Revolution, as documented by Federico (2006). Implicitly, it re‡ects
the importance of innovations in industry in the form of farm equipment and synthetic
fertilizers, that are associated with high rate of growth in farm-labor productivity.12
Pro…t Maximization. The pro…t maximization problem of farms is straightforward,
as they are price takers. The pro…t of the stand-in farm is
at
= Aat Lat
wat Lat
(21)
where wat is the agricultural wage rate. Farms choose Lat to maximize equation (21).
This yields the standard …rst order condition
wat = Aat (Lat )
1
(22)
Total pro…ts (or land rents) are thus,
at
and pro…ts per unit of time worked,
at
= (1
at ,
= (1
)Aat (Lat )
(23)
are
)Aat (Lat )
1
(24)
11
To be consistent with the historical record of a slowly increasing population during the pre-industrial
era, agricultural TFP growth must be positive if there are decreasing returns to land.
12
Alternatively, though at the cost of substantial complexity, the same qualitative results could be obtained by having farms use industrial goods as intermediate inputs, rather than assuming that agricultural
TFP growth depends on technological progress in the industrial sector. As technological improvement
in industry lowers the relative price of industrial goods, farms would use more industrial intermediate
inputs, thereby increasing farm labor productivity. Results for this setup are available from the authors
upon request.
16
Pro…ts (or land rents) are rebated to the farm households in proportion to their time
worked. Hence, total income of a farm household per unit of time worked is the sum of
wages per unit of time worked and pro…ts per unit of time worked:
ytf = Aat (Lat )
1
(25)
Urban households, therefore, do not receive any farm pro…ts. Their income per unit of
time worked is
ytx = wxt
4
(26)
Equilibrium
As is standard in this literature, we focus exclusively on symmetric Nash equilibria. In
such an equilibrium, all …rms use the same technology, charge the same price, and are
equally spaced around the unit circle. This section de…nes a symmetric Nash equilibrium
for our economy, and explores the issue of uniqueness and limiting properties. It consists
of three parts. In the …rst part, we derive the aggregate demand for each good in the
symmetric case, and use this to simplify the …rst order pro…t maximization conditions.
In the second part, we de…ne a symmetric equilibrium. In the last part, we discuss the
limiting properties of the an equilibrium. In particular, we establish a set of parametric
restrictions that ensure that the economy converges to a balanced growth path.
4.1
Aggregate Demand
We …rst determine the aggregate demand for each industrial good. Demand comes from
both types of households. Since in a symmetric Nash equilibrium all varieties produced
are equally spaced around the unit circle, aggregate demand for a given variety depends
only on the locations and the prices of its closest neighbors to its right and its left on
the unit circle. Let dt denote the distance between two neighboring varieties in period t.
This distance is inversely proportional to the number of varieties, namely,
dt =
1
:
mt
17
(27)
Since the nearest competitors to the right and to the left of the …rm producing variety v
are each located at the same distance dt from it, we do not need to di¤erentiate between
them, and thus denote each competitor by vc and their prices by pct .
To begin, we derive the demand by agricultural households for variety v. The
…rst step is to identify the location of the household on the unit circle that is indi¤erent
between buying varieth v and variety vc . The agricultural household that is indi¤erent
between buying varieties v and vc is the one whose cost of a quantity equivalent to one
unit of its ideal variety in terms of v equals the cost of a quantity equivalent to one
unit of its ideal variety in terms of vc . Consequently, the agricultural household that is
indi¤erent between v and vc is the one located at distance dvt from v, where
pct [1 + (d
dvt ) ] = pvt [1 + dvt ]:
(28)
Given this indi¤erence condition applies to agricultural households both to the right
and to the left of v, the uniform distribution of agricultural households around the unit
circle implies that a share 2dvt of them consumes variety v. Since each household spends
(ytf
ca ) on the industrial good, the total demand for v by agricultural households is
f
Cvt
= 2dvt Ntf cfvt =
2dvt Ntf
(ytf
ca )
:
(29)
:
(30)
pvt
By analogy, total demand by industrial households is
x
Cvt
= 2dvt Ntx cxvt =
2dvt Ntx
(ytx
pvt
ca )
Given that all …rms are spaced evenly in the symmetric equilibrium, it follows
that
2dvt = dt :
(31)
Aggregate demand for variety v, Cvt , is the sum of (29), (30). Hence,
Cvt = dt (Ntf cfvt + Ntx cxt ) =
dt
(ytf Ntf + ytx Ntx
pvt
Nt ca )
(32)
With this demand in hand, we can solve for the price elasticity in a symmetric
Nash equilibrium. This involves three steps. First, from (32) it is easy to show that
@Cv p v
=1
@pv Cv
18
@dv pv
@pv dv
(33)
Next, by taking the total derivative of the indi¤erence equation (28) with respect to pvt ,
we solve for @dvt =@pvt , and substituting this partial derivative in (33) yields
"vt = 1 +
[1 + dv ]pv
1
[pv dv
+ pc (d
dv )
1 ]d
(34)
v
Finally, we invoke symmetry, i.e., pvt = pct and 2dvt = dt , so that (34) reduces to
"vt = 1 +
1
1 2
( ) +
2 dt
2
(35)
Aggregate demand for the agricultural good is easy to determine. Individual
household’s demand, (6), implies that aggregate demand is
)(ytf Ntf + ytx Ntx
Cat = (1
4.2
Nt ca ) + Nt ca
(36)
Symmetric Equilibrium
We next de…ne a symmetric Nash Equilibrium for our economy. Because the decisions
of households, industrial …rms and farms are all static, the dynamic equilibrium for the
model economy is essentially a sequence of static equilibria that are linked through the
laws of motion for the population, the benchmark technology in the industrial sector, and
TFP in the farm sector.
As is standard, the equilibrium must satisfy pro…t maximization, utility maximization, and market clearing conditions. It must also be the case that each household is
indi¤erent between working in the farm sector and the industrial sector. More speci…cally,
for a household with ideal variety v the utility associated with consumption and children
should be the same across sectors:
U (cfvt ; cfat ; nft ) = U (cxvt ; cxat ; nxt ):
(37)
It must also be the case that …rms in the industrial sector earn zero pro…ts in equilibrium.
This is a consequence of there being free entry. Thus,
pvt Qvt
wxt [ e
gvt
+
Qvt
]=0
Axt (1 + gvt )
(38)
This condition e¤ectively determines the number of varieties and the distance between
varieties.
19
The zero pro…t condition (38), together with mark-up equation (16) and the
elasticity equation in the symmetric equilibrium (35), provides the key to understanding
the positive relation between market size and …rm size. From the elasticity expression (35)
it is apparent that as the number of varieties increases, and the distance between …rms
decreases, the price elasticity of demand increases. This result is easy to understand: by
increasing the number of varieties, the unit circle gets more crowded, making neighboring
varieties more substitutable. From the price expression (16), it follows that the greater
elasticity leads to tougher competition, reducing the markup. The zero pro…t condition
(38) then implies that the size of …rms, in terms of production, must increase: given the
same …xed cost, a …rm must sell a greater quantity of units in order to break even. As
we will see later, larger …rms …nd it easier to bear the …xed cost of innovation, leading to
a positive relation between market size and technological progress.
We now de…ne the dynamic Symmetric Equilibrium.
De…nition of Symmetric Equilibrium. A Symmetric Equilibrium is a sequence of
household variables fcfvt ; cfat ; nft ; ytf ; Ntf ; cxvt ; cxat ; nxt ; ytx ; Ntx g, a sequence of farm variables
fQat ; Lat ;
at g,
a sequence of industrial …rm variables fQvt ; Lvt ; pvt ; gvt ; "vt ; Avt g, and a
sequence of aggregate variables fVt ; wxt ; mt ; wat ; dt ; Nt ; Axt ; Aat g that satisfy
(i) utility maximization conditions given by (6), (7) and (8).
(ii) farm pro…t maximization conditions given by equations (18), (22), (24) and (25).
(iii) industrial pro…t maximization conditions given by (11), (13), (12), (26), (16), (17),
and (35)
(iv) market clearing conditions
(a) industrial goods market: equation (11) = equation (32)
(b) industrial labor market:
mt Lvt = Ntx (1
x x
nt )
(c) farm goods market: equation (18) = equation (36)
20
(39)
(d) farm labor market:
Lat = Ntf (1
f f
nt )
(40)
(v) aggregate laws of motion for Axt given by (14); for Nt given by (4); and for Aat
given by (19) and (20)
(vi) zero pro…t condition of industrial …rms given by (38)
(vii) indi¤ erence condition of households given by (37)
(viii) population feasibility given by (1)
4.3
Properties
We conclude this section by addressing the limiting properties of the model, namely,
whether the economy converges to a balanced growth path.
We do this because the
general long run pattern of development is characterized by an initial period of roughly
constant living standards, followed by an industrial revolution with low and increasing
rates of growth, followed by a modern economic growth with a higher and constant growth
rate.
Proposition 1. The economy converges to a balanced growth path with constant technological progress in industry provided that (i) ( 1
gat > 0 for all t, and (iii)
f
1)( 1
)(
x
f
1
)
+ (1
f
1) = 0, (ii)
is su¢ ciently close to 1.
Proof. We start by recalling three equilibrium conditions: at all times utility should be
equal across farming and industrial households,
(ytf
ca ) (
1
ca =ytf
f
)1
= (ytx
ca ) (
ca =ytx
1
x
)1
(41)
and income should equal expenditure in both sectors,
ytf Ntf (1
f f
nt )
= (1
ytx Ntx (1
x x
nt )
)(ytf Ntf + ytx Ntx ) + (1
=
(ytf Ntf + ytx Ntx )
21
(1
))ca (Ntf + Ntx )
ca (Ntx + Ntx )
(42)
(43)
We now show that ytf in the limit goes to in…nity. To do so, it su¢ ces to show that
the growth in ytf is strictly positive in each period. Since ytf = Aat Lat 1 , this amounts to
showing that
gat
(1
)
L_at
> 0:
Lat
(44)
Expression (8) implies that population growth is …nite, and therefore growth in the hours
worked in agriculture is also …nite, so that if
is close enough to 1, expression (44) is
satis…ed.
Since
f
x,
>
condition (41) implies that if ytf goes to in…nity in the limit, then
ytx also goes to in…nity in the limit. This, in turn, implies that in the limit both types of
households have a constant (though di¤erent) number of kids:
1
ni =
(45)
i
where i 2 ff; xg. As the number of children is constant in the limit, the utility indi¤erence
condition (41) implies that
yf
=(
yx
f
1
x
(46)
)
Moreover, if we divide equation (42) by (43), it follows that in the limit
yf N f
1
=
x
x
y N
(47)
Substituting (46) into (47) then gives the following limit expression:
Nf
1
=
x
N
x
(
f
1
)
(48)
Thus, the shares of agricultural and industrial households converge to …xed numbers. This
together with (45), implies that in the limit population growth is constant. Substituting
(48) and (45) into the expression for population growth, (nf
1)N f =N + (nx
1)N x =N
gives the following expression:
(
Therefore, if ( 1
1
1
f
f
1)( 1
1)
1+
)(
x
f
(
1
1
)
x
f
(
1
)
x
f
+(
1
)
+( 1
f
1
x
1
1)
1+
1
(
x
f
1
(49)
)
1) = 0, as stated in condition (ii), population
growth converges to zero. With a constant population, a constant number of children in
22
each sector, and a constant share of households employed in the industrial sector in the
limit, it follows that the total number of hours worked in industry also converges to a
constant.
It is now easy to show that in the limit gv is a constant. The zero pro…t condition,
Qv =
e
gv A
x (1
+ gv )("
1), together with the …rst order condition for technological
progress, (17), implies that gv is a positive function of the price elasticity of demand:
gv =
"v
Since the total production of each …rm is e
gv "),
1
gv A
x (1
(50)
+ gv )("
1) and the total number
N x in the limit, it follows that the number of …rms is
of hours worked in industry is
m = N x =( e
1
where m = 1=d. Substituting into (35) gives
"=1+
1
1 2 Nx
(
)
+
g
2
2
e v"
(51)
Now re-write (51) as
2 "
+1
(2 N x = e
(2 + 1)"
gv
) =0
(52)
and take the total derivative of this expression with respect to gv . This yields
@"
=
@gv
(2 N x )
2 ( + 1)"t
e
gv
(2 + 1) "
1
(53)
From (35) we know that " > 1, so that this derivative (53) is strictly negative. Given that
(51) implies that " is decreasing in gv and (50) implies that " is increasing in gv , and given
that N x is constant in the limit, there is a unique, and constant, gv , and thus a unique,
and constant, ". Therefore, if N x is constant, gv is also constant. The economy therefore
converges to a balanced growth path with constant growth in the industrial sector.
In Proposition 1 we have shown that if population growth converges to zero, the
economy converges to a balanced growth path, with constant technological progress in
the industrial sector. In Proposition 2, we show that in the limit technological progress
is a positive function of population, N , and a negative function of the cost of innovation,
.
23
Proposition 2. Technological progress in the balanced growth path is a positive function
of population, N , and a negative function of the cost parameter of innovation, .
Proof. As argued in the proof of Proposition 1, on the balanced growth path, expressions
(50) and (51) determine the rate of technological progress and the price elasticity of
demand. Expression (50) does not depend on N x , whereas expression (51) does. To see
this, re-write (51) as (52) and totally di¤erentiate, keeping gv …xed. This gives
@"
(2 = e
=
x
@N
2 ( + 1)"
gv )
(N x ) 1
(2 + 1) "
1
(54)
Since " > 1, the above partial derivative is strictly positive, so that an increase in N x
leads to a greater elasticity of demand for any given gv . Recall that expression (50)
implies that the elasticity is upward sloping in gv , whereas expression (51) implies that
the elasticity is downward sloping in gv . This, together with the fact that a greater value
of N x causes an upward shift in expression (51), allows us to conclude that gv is increasing
in N x . Since N x is a …xed share of N , gv is therefore also increasing in the size of the
population.
To show that gv is decreasing in , we use a similar argument. Re-write (51) as
(52) and totally di¤erentiate with respect to , keeping N x and gv constant. This gives
@"
=
@gv
(2 N x )
2 ( + 1)"
gv e gv
(2 + 1) "
1
Since " > 1, the above partial derivative is strictly negative, so that an increase in
(55)
leads
to a smaller elasticity of demand for any given gv . By analogy with the above argument,
this implies that gv is decreasing in .
The intuition for the positive relation between the size of the limit population and
the steady-state rate of technological progress is straightforward. A greater population
leads to a larger number of households employed in the industrial sector. The greater
size of the industrial sector, and the larger number of varieties produced, implies lower
mark-ups, and tougher competition. To break even, industrial …rms must become larger.
These larger …rms then endogenously choose to innovate more. This is obvious from the
…rst order condition on technology choice, (17), which exhibits two e¤ects: an increase
24
in innovation raises a …rm’s …xed cost and lowers its marginal cost. The …rst (negative)
e¤ect is independent of …rm size, whereas the second (positive) e¤ect is increasing in …rm
size. As a result, larger …rms innovate more.
Proposition 2 leads to a number of conclusions. Starting o¤ in a situation with no
technological progress in industry, two situations can arise. If population reaches the critical size for take-o¤ before population growth converges to zero, we will get an industrial
revolution, and the economy will converge to a balanced growth path with strictly positive technological progress in industry. However, if population growth converges to zero
before that critical size is reached, we have an industrialization trap, and the industrial
sector never innnovates. As Proposition 2 suggests, this industrialization trap becomes
increasingly likely, the higher is . This is easy to see when considering the extreme case
of
being in…nite. Then obviously there will never be any take-o¤.13
5
Numerical Experiments
In this section, we …rst calibrate the model to the historical record of England over the
period 1300-2000. We then examine within this calibrated structure how the timing of the
industrial revolution is a¤ected by a number of factors emphasized by other researchers
as being important for understanding why England was the …rst country to industrialize.
5.1
Calibration
The calibration strategy is to assign parameter values so that the model equilibrium is
characterized initially by a Malthusian-like era and in the limit by a modern growth
era. Empirically, the key observations of the Malthusian era targeted in the calibration
are: (i) a constant living standard, (ii) a constant rate of population growth, and (iii) a
dominant share of agricultural activity in the economy. Empirically, the key observations
of the modern growth era targeted in the calibration are: (i) a constant, positive rate of
growth of per capita GDP, and (ii) a dominant share of industrial activity in the economy.
13
We do not call this a development trap because unless we make assumptions that a in equation
(21) goes to zero in the limit, then there will be increases in agricultural output per person in the
industrialilzation trap.
25
Theoretically, for the model to generate this modern growth era with constant per capita
output growth, the population growth rate must converge to zero in the limit. This is
another key restriction in the calibration exercise.
Before assigning parameters, it is necessary to identify the empirical counterpart
of a model period. Given that households live for one period, during which they raise
their o¤spring, it is reasonable to interpret a period as the time that elapses between
generations. In models that employ a two-period generational construct, a period is
typically thought of as being 35 years in length. However, since our model is a model of
the last millennium and as life expectancies were far shorter before the 20th century, we
choose 25 years for the period length.
Table 1 lists the parameter values and provides brief comments on how each was
assigned. A few additional words are warranted in the case of some of the parameters.
As stated in Table 1, the initial population, N0 , is set so that an industrial revolution
starts in 1750. Here, we point out that the start of the industrial revolution should not be
interpreted as the …rst period in which industrial …rms innovate, but rather as the period
that is followed by a rapid acceleration in the growth rate of per capita GDP and a rapid
movement of employment into industry. As stated in Table 1, the rural time cost of
rearing children,
f,
is set to satisfy the zero population growth condition as described by
Proposition 2. This restriction implies that the time rearing in the countryside is roughly
20 percent of the time rearing cost in the city as estimated by de la Croix and Doepke
(2004). This may seem a relatively small cost, but it is in line with time estimates by
Ho (1979) for rural Philippine households. Finally, as stated in Table 1, the parameter
in Lancaster’s compensation function, which determines how fast utility diminishes as
a household moves away from its ideal variety is set so that the mark-up of price over
marginal cost in the limit is consistent with estimates by Jaimovich and Floetotto (2008).
The reason mark-up estimates restrict the value of
is that the mark-up is a function
of the price elasticity of demand, and the price elasticity is a function of , as shown in
equation (35).
alpha= .98000 mu= .91250 abar=1.55000 beta= .50000 kappa=1.55000 gam-
26
maA= .00380000 phi= 6.0000 tauR= .0215 tauU= .0943 theta= .870 Ax(0)= 1.000
Aa(0)= 4.751 N(0)= 384.000
Table 1: Parameter values
1. Population
N0 = 384 start of industrial revolution in 1750
2. Industrial technology parameters
Ax0 = 1
normalization
= 1:55
median percentage of ratio of non-production to production workers in US manufacturing outside central o¢ ces (Berman et al.,
1994)
= 6:00
limiting growth of per capita GDP between 1-5-2.0% (Maddison,
2008)
3. Agricultural technology parameters
Aa0 = 4:75 constant agricultural living standard in pre-1500 era
pre-1500 average annual population growth rate of .025% (Mada = 0:004
dison, 2008)
= 0:87
free
in 1750.
4. Preference parameters
ca = 1:55 agricultural share of employment in 1600 (Allen, 2004)
= 0:98
2% limiting share of agriculture’s share of employment (Mitchell,
1992)
= 0:913 total fertility rate in 2007 for London of 1.80.
= 0:50
mark-up estimates between 5-15% in the limit (Jaimovich and
Floetotto, 2008)
5. Child rearing parameters
f = 0:022
zero population growth growth in the limit
x = 0:094
estimates between 7.5-15% per household (de la Croix and Doepke,
2004)
Figures 1-4 present the equilibrium path for the model economy and the English
economy over the 1300-2000 period along four dimensions: technological progress in the
industrial sector, the growth rate of GDP per capita, the growth rate of population,
and agriculture’s share of employment. Growth rates for both the model economy and
England are expressed in annual terms. Thus, for example, a growth rate of 1.5 percent
in Figure 2 implies that in the model period of 25 years output per capita increased by
a factor 1.45. Data on the growth rates of GDP per capita and population come from
27
Annual Rate Technological Progress Industry
2.0%
Technological Progress (Model)
Annual Growth Rate
1.5%
1.0%
0.5%
0.0%
1300
1400
1500
1600
1700
1800
1900
2000
2100
Year
Figure 1: Technological Progress (Benchmark)
Maddison (2008). Data on the agricultural share of employment come from multiple
sources: for the 1300-1800 period, we use estimates provided by Allen (2000), whereas
from 1800 onwards we use data from Mitchell (1992).14
As can be seen, the calibrated model matches the historical experience of the
English economy extremely well. We did not calibrate the model economy to England’s
Industrial Revolution. We calibrated to the pre-1500 Malthusian era, the modern growth
era, and a starting date of 1750 for the industrial revolution’s start. Thus, the model’s
ability to match the path of England’s growth rate of per capita GDP, its population
growth rate, and its agriculture’s share of employment for the period 1750-1900 so closely
represent three successful tests of our theory.
Additionally, the model’s predictions for the time paths of both the price of industrial goods relative to agricultural goods and mark-ups are consistent with available
14
Clark (2002) also provide estimates for the 1500-1700 period for England. They are lower than those
of Allen (2002), and can be interpreted as a lower bound. Calibrating to Clark’s …gures is not a problem.
Another alternative is to use the estimates of Allen (2002) for the rural population, rather than the
agricultural population. Those …gures provide an upper bound.
28
Annual Growth GDP per Capita
2.5%
GDP per Capita Growth(Model)
GDP per Capita Growth (Data England)
Annual Growth Rate
2.0%
1.5%
1.0%
0.5%
0.0%
1300
1400
1500
1600
1700
1800
1900
2000
2100
Year
Figure 2: Growth GDP per Capita (Benchmark)
Annual Growth Population
1.5%
Population Growth (Model)
Annual Growth Rate
Population Growth (Data England)
1.0%
0.5%
0.0%
1300
1400
1500
1600
1700
1800
1900
Year
Figure 3: Growth Population (Benchmark)
29
2000
2100
Agricultural Employment Share
100.0%
Agricultural Share (Model)
Agricultural Share (Data England)
Annual Growth Rate
80.0%
60.0%
40.0%
20.0%
0.0%
1300
1400
1500
1600
1700
1800
1900
2000
2100
Year
Figure 4: Agricultural Employment Share (Benchmark)
data. With respect to the relative price of industrial goods, the model predicts a secular
decline that converges to a constant in the limit. The limiting prediction is due to the
result that the rate of agricultural TFP growth eventually equals the constant rate of
process innovation in industry. This is the pattern documented by Yang and Zhu (2008)
for England. They report a decline in the relative price of 50 percent between 1700 and
1820 before leveling o¤. Consistent with those observations, in our model the relative
price is essentially the same in 1825 and 2000. However, compared to the data, it exhibits a slightly smaller decline of 30 percent over the 1700-1825 period. With respect to
the mark-up, the model generates a secular decline that converges to a constant in the
limit. Ellis (2006) documents that mark-ups have declined in England over the 1870-1985
period, but does not provide evidence that it is converging to a constant. The model predicts a smaller percentage decline in the mark-up over the comparable period: whereas
Ellis (2006) reports a 67 percent decline between 1870 and 1985, the model predicts a 15
percent decline.
In terms of other relevant statistics, the calibrated model predicts a 3-fold increase
30
in the number of varieties and a 40-fold increase in the size of manufacturing …rms since
1750. Although time series data on establishment size exist, they do not go back far
enough in time to say whether the 40-fold prediction of the model over the 1750-2000
period is plausible. A 3-fold increase in the number of varieties might be on the low side.
In summary, given the rather stark structure of the model, and given the very long
and diverse period of analysis, the model’s ability to match the data is fairly remarkable.
In light of this success, it is informative to investigate how factors that are likely to di¤er
across societies a¤ect the timing of the industrial revolution. This is the subject we
analyze next.
5.2
The Timing of the Industrial Revolution
We now explore how certain parameters a¤ect the timing of the industrial revolution.
We do not consider variations to all parameters in this subsection. Rather, we consider
variations to the values of those parameters that correspond most closely to factors identi…ed by various researchers as being important for understanding why the Industrial
Revolution happened …rst in 18th century England.
Among the list of factors heavily emphasized in the literature are: institutions,
agricultural productivity, fertility, population size, trade and transportation costs. We
only explicitly take up the …rst three in the analysis. We do not directly consider the
importance of trade and transportation costs because our model is closed. Clearly, trade
and transportation costs a¤ect the size of the market, and hence the timing of the industrial revolution.15 For this reason, one should be careful about using a country’s domestic
population as a measure of the size of its market.16 For judging the market size of actual
economies, one needs to not only consider the size of the domestic population, but its
openness to trade with other countries and the cost of transporting goods within the
15
In Desmet and Parente (2010) we explore the e¤ects of the reduction in trade barriers and the
improvement in transportation infrastructure on innovation in a static model with Hotelling-Lancaster
preferences. Not surprisingly, we …nd that freer trade and lower transport costs have the same positive
e¤ects on innovation as an increase in population size.
16
Voth (2003) is critical of uni…ed growth models that rely exclusively on population size to generate
the industrial revolution, and argues that historical evidence points to market size being a more relevant
factor.
31
country’s borders.
We do not consider how the starting date of the industrial revolution responds
to a change in the initial population size either because such a change only causes a
temporary departure from the Malthusian steady state. More speci…cally, changing the
initial population parameter, N0 , while maintaining all other parameter values constant,
means the economy does not start in a Malthusian steady state. As in the standard
Malthusian model, a departure from the steady state is quickly reversed via higher fertility
in the case where the living standard is above the steady state level or via lower fertility
in the opposite case. Lowering the population by, say, 10 percent has no e¤ect on the
start of the industrial revolution because the system returns to its Malthusian steady
state in a few periods.
5.2.1
Agricultural Productivity
How much later would the Industrial Revolution in England have occurred if agricultural
productivity had been lower? Several researchers, such as T.S. Schultz (1968) and Jared
Diamond (1997), have argued that high agricultural productivity has been critical for
long-run development. Towards the goal of answering this question, we consider two
experiments. The …rst lowers the value of initial agricultural TFP, Aa0 , by 4.2 percent,
whereas the second reduces the growth rate of agricultural TFP during the pre-industrial
period from 0.015 percent to 0.012 percent per annum (a reduction of 20 percent). The
results for the process innovation are displayed in Figure 5.17
As shown in Figure 5, a lower starting level of agricultural TFP delays the onset
of the industrial revolution: a 4.2 percent reduction in the initial level of agricultural
productivity postpones the start of the industrial revolution by about 275 years. The
reason why the higher initial agricultural TFP in the benchmark case hastens the start
of the industrial revolution is easy to understand. Higher agricultural TFP implies a
higher population during the Malthusian era, which implies that the economy produces a
greater variety of di¤erentiated goods. Good are, thus, more substitutable, and mark-ups
17
In these experiments, we adjust the initial population so that the economy continues to display a
Malthusian era steady state.
32
Annual Rate Technological Progress Industry
2.0%
Technological Progress (Benchmark)
Technological Progress (A0 4% lower)
Technological Progress (ga 20% lower)
Annual Growth Rate
1.5%
1.0%
0.5%
0.0%
1300
1400
1500
1600
1700
1800
1900
2000
2100
2200
2300
Year
Figure 5: E¤ect of Lower Agricultural TFP on Timing of Industrial Revolution
are smaller. This leads to larger …rms, and thus more pro…table process innovation.
The size of the delay, namely 275 years, may be somewhat surprising, but it really
is not. With the calibrated growth rate of agricultural TFP of 0.015 percent per annum in
the benchmark case, it takes about 275 years for agricultural TFP to rise by 4.2 percent.
In other words, the 275 year delay found in this experiment simply represents the time
it takes agricultural TFP to reach the initial benchmark level. The size of the delay
does suggest that a small increase in agricultural TFP can be extremely important for
an economy’s takeo¤, provided of course that the industrial sector is not characterized
by excessive barriers.
For similar reasons, a lower rate of agricultural TFP growth also delays the onset
of the industrial revolution. For an annual growth rate of 0.012 percent (instead of 0.015
percent), the start is only delayed by about 50 years. This shorter delay makes sense:
to achieve the same accumulated growth as with the benchmark TFP growth of 0.015
percent over 200 years takes takes about 50 years more with a TFP growth of 0.012
percent.
33
Annual Rate Technological Progress Industry
2.0%
Technological Progress (Benchmark)
Technological Progress (kappa 10% higher)
Technological Progress (phi 10% higher)
Annual Growth Rate
1.5%
1.0%
0.5%
0.0%
1300
1400
1500
1600
1700
1800
1900
2000
2100
2200
2300
Year
Figure 6: E¤ect of Worse Institutions on Timing of Industrial Revolution
5.2.2
Policy and Institutions
We next explore how the timing of England’s Industrial Revolution was a¤ected by institutional factors, a main theme in the research of North and Weingast (1989) and Ekelund
and Tollison (1981). In the real world, the …xed costs …rms incur to operate and to innovate depend to a large extent on institutions and policy. We therefore interpret larger
values for the …xed cost parameters,
that
and , as worse institutions and policies. Recall
is the …xed cost of operating the benchmark technology, whereas
determines
how much the …xed cost increases when better technologies are adopted.
Figure 6 shows what happens when we increase each …xed cost parameter separately by 10 percent. In the case of a higher …xed operating cost
, the industrial
revolution is delayed by 75 years; in the case of a 10 percent higher …xed adoption cost ,
the industrial revolution is delayed by 250 years. While raising either parameter delays
the start of the industrial revolution, the intuition for the delays is di¤erent. In the case of
, worse policy or institutions imply less varieties produced in the economy, meaning the
elasticity of demand is lower and innovation is unpro…table. In the case of a 10 percent
34
increase in , the number of varieties and …rm size are una¤ected. However, because the
cost of process innovation is higher, …rms have to be larger to …nd innovation pro…table.
5.2.3
Child Rearing Costs
We end this section by exploring the impact of the cost of rearing children on an economy’s
industrialization. Our motivation for doing this is the evolutionary theories of long-run
development recently put forth by Galor and Moav (2002) and Clark (2007) that emphasize the importance of reproduction rates of certain segments of the English population.
Galor and Moav (2002), for example, argue that evolution favored those individuals with
a stronger preference for quality of o¤spring versus quantity of o¤spring, whereas Clark
(2007) points to the higher reproduction rates of the thriftier types among English society
as being important for its escape from the Malthusian trap. While our model abstracts
from savings and quality of children, it is nevertheless informative to examine the e¤ect
of changing the relative reproduction rate of rural and urban households. We do this in
two ways. First, we increase the urban child rearing cost,
we lower the rural child rearing costs,
f
x,
by 10 percent, and second,
by 10 percent.18
Figure 7 shows these results. As expected, a lower rural child rearing cost accelerates the timing of the industrial revolution, whereas a higher urban rearing cost delays
the start of the industrial revolution. Note, however, that the 10 percent reduction in
the rural child rearing cost has a much larger e¤ect on the timing of the industrial revolution: the 10 percent reduction hastens the start of the industrial revolution by 125
years, whereas the 10 percent increase in the urban time cost of rearing children delays
the start of the industrial revolution by only 25 years.
These results are intuitive. As there are no barriers to migration, what matters
for the starting date is the growth rate of the domestic population. With most of the
population living in the country initially, the e¤ect of a change in the urban cost of
child rearing has little e¤ect on the size of the domestic population. This explains why
18
The time cost of child rearing in one of the sectors is the only parameter whose value is changed in
the experiment. Hence, the model need no longer predict zero population growth in the limit. Nor does
it necessarily imply that the economy begins in a Malthusian steady state.
35
Annual Rate Technological Progress Industry
2.5%
Technological Progress (Benchmark)
Technological Progress (taux 10% higher)
Technological Progress (tauf 10% lower)
Annual Growth Rate
2.0%
1.5%
1.0%
0.5%
0.0%
1300
1400
1500
1600
1700
1800
1900
2000
2100
Year
Figure 7: E¤ect of Child Rearing Costs on Timing of Industrial Revolution
changing the cost of urban child rearing has nearly no e¤ect on the starting date of the
industrial revolution. The opposite is true for the countryside. Clearly, in our model
the cost of raising children in the countryside is far more important in determining a
country’s industrialization date than the cost of raising children in the city.
Of course, this prediction only holds under the assumption that labor is free to
move between sectors. With perfect labor mobility, the relevant variable is the aggregate
fertility rate. In the context of the Industrial Revolution in England, our description is
reasonable: as argued by Williamson (1985) the growing relative importance of cities was
due to rural-urban migration. Of course, if migration were impeded, having a lower child
rearing cost in the countryside would not a¤ect the start of the industrial revolution. In
this case, growth in the urban population would have to come entirely from the fertility of
urban households. In the case of Latin America, for example, barriers to urban migration
were historically substantial. Clearly, these were not the only barriers to development in
Latin America, but our theory certainly gives them an important role in explaining Latin
America’s later development.
36
6
Conclusion
This paper proposes a novel mechanism whereby an economy transitions from a Malthusian era, with stagnant living standards, to a modern growth era, with sustained and
regular increases in income per capita. The paper’s main hypothesis is that markets
need to reach a critical size before the returns to innovating are high enough to cover the
…xed costs. Market size matters because a larger market can support a larger variety of
goods. More varieties mean that goods become more substitutable. As a result, demand
elasticity increases and mark-ups fall. To break even, …rm size increases. This is critical for innovation as larger …rms are more easily able to cover the …xed costs associated
with research and development. We have shown within the context of our theory how
numerous factors, such as institutions and agricultural TFP, impact the timing of the
industrialization. We have also provided evidence that this mechanism is consistent with
the historical record of development.
Clearly, the novelty of the paper lies in the mechanism by which larger markets
bring about the industrial revolution, rather than in the idea that markets were critical.
Other economists, such as Adam Smith, have argued that market size was important for
understanding why England was the …rst country to experience an industrial revolution.
But Smith and other economists who have argued this point have di¤erent mechanisms
in mind that typically involve increasing returns and specialization. Our mechanism does
not rely on either.
We see a number of areas where future research will be valuable. One such area is
to make the structure of the model economy richer, say, by allowing for savings. Another
area is to explore whether this model is capable of generating divergence, followed by
convergence, between countries, as industrialization spreads across the globe.
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