We thank Raquel Fernandez, Jim Brander, Barbara Spencer, Bob Staiger, Jim Markusen, Jacques*

Thisse, Jonathan Eaton, Giorgia Giovanetti, Ramon Marimon, Steve Parente and seminar participants atBerkeley, Birbeck, Bocconi, Bonn, Georgetown, IMF, Louvain, LSE, Lund, MIT, Northwestern, NYU,Paris, Princeton, Stockholm, Vancouver and Wisconsin for comments and suggestions. The SwissNational Science Foundation Grant 12-50783.97 provided support for this research. Ottaviano also thanksthe TMR of the European Commission for financial support.

Global Income Divergence, Trade and Industrialization:The Geography of Growth Take-Offs

Richard E.Baldwin, Graduate Institute of International Studies, Geneva and CEPRPhilippe Martin, CERAS-ENPC, Paris and CEPR

Gianmarco I.P. Ottaviano, University of Bologna and CEPR*

This version: October 1999

ABSTRACTThis paper takes a step towards formalizing the theoretical interconnections among four post-Industrial Revolution phenomena – the industrialization and growth take-off of rich 'northern'nations, massive global income divergence, and rapid trade expansion. Specifically, we present astages-of-growth model in which the four phenomena are jointly endogenous and all are triggeredby a gradual fall in the cost of trading goods internationally. In the first stage, while trade costs arehigh, industry is dispersed and growth is low. In the second stage, the north industrializes rapidly,growth takes off and the south diverges. In the third stage, high growth and global divergencebecome self sustaining. We then show that, in a fourth stage, when transaction costs on “trading”ideas have sufficiently decreased, the south can quickly industrialize and converge.

JEL Classifications: F43, F01, O 19, N13

Key words: Growth Take-Off, Industrial Revolution, Economic Geography, Endogenous Growth,Trade and Development

1

1. Introduction

In 1990 the world's richest nation was 4500% richer than the poorest; in 1870 the figure was

900% and, before the first Industrial Revolution (mid-18th century), West European per capita incomes

were only 30% above those of China and India (Maddison 1983; Bairoch 1993). While some scholars

disagree with Bairoch's estimate of 18th century income gaps, none disputes the conclusion. Authors

as diverse as Braudel, Kuznets, Baumol and Maddison assert that the big north-south income

divergence appeared with the first Industrial Revolution. For example, Kuznets (1965 p.20) says:

“Before the 19th century and perhaps not much before it, some presently underdeveloped countries,

notably China and parts of India, were believed by Europeans to be more highly developed than

Europe, and at that earlier time their per capita incomes may have been higher than the then per capita

incomes of the presently developed countries”. Thus, by the time frame of human history, "the current

wide disparities – between rich and poor countries – are recent" (Kuznets 1966, p.393).

The Industrial Revolution caused this rapid income divergence by triggering industrialization and

a growth take-off in Europe while incomes stagnated in the now poor nations. At the same time, the

world experienced a rapid expansion of international trade.

We present a stages-of-growth model in which these four phenomena - northern

industrialization and growth take-off, income divergence, and trade expansion - are jointly endogenous.

They are triggered by a gradual, exogenous fall in the cost of international trade driven by lower

transportation cost as well as by market opening initiatives. In the first stage – while trade costs are still

high – falling trade costs have the usual static effects on prices, trade and welfare, but no growth effects.

Growth in this stage may be positive or zero, but in any case it proceeds at a fairly low rate since

growth-driving externalities, which are localized, are dampened by geographical dispersion of industry.

In the second stage – when trade costs have just crossed a certain threshold - industrial agglomeration

occurs very rapidly and, to be specific, say it occurs in the north. Agglomeration triggers a take off in

northern growth because geographic concentration of industry amplifies the exploitation of localized

technological externalities. In our model, agglomeration implies that the south has no incentive to invest

and innovate while the incentive to innovate in the north increases. Therefore, it not only generates

industrialization and a growth take-off, it also produces income divergence. In fact, in our model, global

divergence is both the cause and the consequence of the growth take-off in the north. Yet despite this,

* There is strong evidence (see section 2) that after WWII, the cost of trading ideas went down faster than thecost of trading goods.

* Other papers include Azariadis and Drazen (1990) and Zilibotti (1995)

2

the south may still benefit in welfare terms. In the third stage, high growth becomes stable and self-

sustaining.

The fact that externalities are localized, or to put it another way, that the cost of “transporting

ideas” is high, is one element that makes the agglomeration phenomenon sustainable. We show that,

when the cost of transporting goods asymptotes towards some natural limit but that the cost of “trading

ideas” decreases, then the model can generate a fourth stage . This entails rapid industrialization in the*

south, driven by heightened access to northern technologies. The emergence of southern industry slows

global growth somewhat and forces a relative de-industrialization in the north.

Relation to early literature Our model captures some elements of the informal analyses

of the classic growth scholars such as Kuznets and Rostow. Despite their disagreements on important

points, both think of the Industrial Revolution as a structural break. Kuznets (1966) divides growth into

two types: traditional growth (pre-1750) and modern economic growth (post-1750). The distinctive

feature of modern growth, according to Kuznets, is the rapidity of the shifts in industrial structure (he

talks of sweeping structural changes) and their magnitude when cumulated over decades. Rostow

(1960) goes further, identifying five stages in economic growth: the traditional society, the preconditions

for take-off, the take-off, the drive to maturity and the age of high mass-consumption. Finally, both

Kuznets and Rostow view modern economic growth as a sustained and non-reversible process.

Relation to recent literature The existing formal endogenous growth models deal with

modern economic growth, to use Kuznets' phrase. Models such as Romer (1990), Aghion and Howitt

(1992), and Grossman and Helpman (1991) do not model the emergence of growth preconditions nor

do these models consider the forces that initiate the transition to an endogenous, sustained growth

process (see Crafts,1995, however, for an analysis of the British Industrial Revolution in the light of

endogenous growth models). Lucas (1988 and 1993) insists on the role of localized social interactions

and positive spillovers, that are essential for growth, but does not model growth stages.

The 'big push' literature of Murphy, Shleifer and Vishny (1989a,b) points out that due to**

pecuniary externalities, an economy may be marked by two growth equilibria: one in which investment

3

and therefore growth is nil, since the economy is too small, and one in which agents invest in anticipation

of growth. The jump from one equilibrium to the other generates sudden industrialization. These

models do not, however, imply that one of the equilibria must precede the other and as such are not

models of growth stages. There is also no clear reason why the economy could not jump back

(because of a war for example) to the zero growth equilibrium. Furthermore, as made clear by

Matsuyama (1991) and Krugman (1991b), differences in initial conditions (history) or in self-fulfilling

expectations explain why an economy experiences a take-off or stays in a poverty trap. Hence, the

difference in the path taken by poor and rich economies, the divergence phenomenon, lies out of the

model.

Other models of take-off include Goodfriend and McDermott (1995), Acemoglu and Zillibotti

(1997) and Galor and Weil (1998 and 1999). The first paper shows that industrialization can by driven

by increased population and maket size. In the second one, accumulation of capital is facilitated by

financial development. Galor and Weil show that at a certain level of population, a virtous circle

develops for which higher human capital raises technological progress which in turn raises the value of

human capital. The focus of these papers is therefore quite different from ours and in particular they do

not relate the take-off to the divergence phenomenon.

The literatures on uneven development, formalized by Krugman (1981), Faini (1984), and on

economic geography (see Krugman and Venables, 1995), analyze trade and global income divergence.

None of these papers endogenises growth, so the long-run growth rate is never affected and growth

stages are not modeled. In Krugman (1981) the divergence is driven by technological externalities; in

Krugman and Venables (1995) it arises due to pecuniary externalities. The interesting recent

contribution of Kelly (1997) is the closest to ours as it shows that the expansion of market size through

a gradual improvement of transport linkages can lead to a sudden takeoff of the economy. This model

does not, however, explain why certain economies have experienced such a take-off and others have

not, i.e. the divergence phenomenon. Furthermore, Kelly's take-off is only temporary (the long-run

growth rate is exogenous) despite the fact that the sustainability of growth is the most important feature

of the transition from traditional to modern growth identified by early growth scholars. Goodfriend and

McDermott (1998) show that the extent to which technological spillovers are localized is key to the

divergence phenomenon, a theme that is also present in our paper. However, their model does not

feature an endogenous take-off, that is an abrupt jump in northern growth which is both the

4

Figure 1: Trend Growth in Industrial Production (Crafts1994).

consequence and the cause of global income divergence.

The remainder of this paper is in five parts. Section 2, reviews the historical evidence on the

four phenomena that we want to model and on the diminution of transaction costs which triggers them.

Section 3 introduces the basic model and equilibrium conditions. The fourth section studies the stability

properties of the model, establishing that the gradual reduction of trade costs eventually produces a

catastrophic agglomeration of industry. Section 5 studies the growth and divergence implications. The

sixth section discusses the conditions under which the South can rapidly converge and the last section

presents our concluding remarks.

2. Motivation and historical evidence

2.1. Growth, technological innovation and divergence

Perhaps the most striking feature of the

Industrial Revolution concerns the increase in

growth rates. For example, Great Britain's per

capita income rose by 14% between 1700 and

1760, by 34% between 1760 and 1820, and by

100% between 1820 and 1870 (Maddison

1983). Population growth also increased sharply

during this period (due chiefly to better economic

conditions), so Britain's GDP rose even faster

than the per capita figures. In their recent re-

analysis of growth in this period, Crafts and Harley (1992) and Crafts (1995) have revised downward

Maddison's growth figures. Nevertheless, they confirm Maddison's basic message by finding a

structural break in the trend growth of industrial production in Great Britain around 1776 (see Figure

1). Additionally, economic historians (Macer 1993; Crafts 1995) argue that the growth take-off was

accompanied by a rapid increase in innovation seen as a profit seeking activity. Sullivan (1989) for

example finds that the growth rate of patenting was 0.5% before 1754 and 3.6% thereafter. Ashton

(1971), in his memorable phrase on the Industrial Revolution, sums it up in this way: "About 1760 a

wave of gadgets swept over England" and later "It was not only gadgets, however but innovations of

* Hawke (1980), Rosenberg (1994), Macer (1993) and Crafts (1995) explicitly stress the importance of localizedcumulative learning processes in their accounts of the Industrial Revolution.

5

various kinds - in agriculture, transport, manufacture, trade and finance- that surged up with a

suddenness for which it is difficult to find a parallel at any other time or place".

Economic historians differ on the role of technological progress in the Industrial Revolution in

Britain. On the one hand, Crafts and Harley (1992), view the Industrial Revolution as the result of

mostly exogenous technical change in a few industries (cotton and iron). On the other hand, the more

traditional and broad view of the Industrial Revolution, as illustrated by the above quote of Ashton and

the image of the unbound Prometheus of Landes (1969), is that of widespread change and technical

progress in many industries. Even though this view gives an important role to a few “macro” inventions,

it also insists on the role of the many small technological innovations and improvements that took place

in many industries during this period and on the importance of learning by doing . The pervasiveness*

of technological change across many sectors makes this “broad” view inconsistent with an explanation

of the Industrial Revolution solely based on a few big exogenous technological shocks limited to a few

industries. Perhaps paradoxically, the traditional “broad” view is therefore more consistent with a

theoretical framework where innovation is not solely the result of a few exogenous shocks but the

endogenous product of economic incentives, such as in some models of the “new” growth theory.

The “broad” view is corroborated by the work of Mokyr (1990) and Temin (1997). Using

British trade data, Temin discriminates between the two views. The “narrow” view implies that Britain

should have had a growing comparative advantage in a few industries (cotton and iron) which is

inconsistent with the trade data. On the contrary, Temin shows that British comparative advantage was

clear in a wide variety of manufacturing industries.

Historical data on growth in other countries are scarce, but the available estimates indicate that

19th century per capita GDP in India stagnated (Maddison 1971), or actually regressed (Bairoch 1993;

Braudel 1984). This, coupled with higher growth rates in the North, explains the divergence in

incomes. This is confirmed by Pritchett (1997) who by calculating a lower bound for income per capita

in poor countries also concludes that divergence was a very sudden and dramatic phenomenon during

the 19th century.

2.2 Industrialization and de-industrialization

* Until the mid 19th century, on Asian seas, only Indian built boats were used. They were however forbidden inEnglish harbours (Braudel, 1984).

6

During this same period, the sectoral composition of British GDP shifted radically, transforming

Britain from a predominantly agrarian country into the world's most industrialized nation (Crafts 1989).

In 1700, 18.5% of the labor force was in industry. In 1800, this percentage was 29.5% and in 1840

it was 47.3%. During the same period, Great Britain became a large food importer and a large

exporter of industrial goods. Structural changes in what came to be known as the Third World were

no less dramatic during this period. India, for instance, switched from a net exporter of manufactures

to a net exporter of raw materials (Chaudhuri 1966). To take a specific example, during the 18th

century the Indian textile industry was the global leader in terms of quality, diversity, production and

exports (Braudel 1984). Indian textiles were not only exported in huge quantities to Europe, but also

via Europe to the American continent. 18th century India and China also produced the world’s highest

quality silk and porcelain. These manufactured goods were exchanged for silver since many European

manufactures were uncompetitive in the East (Barraclough, 1978).

Yet at the end of the 19th century, more than 70% of Indian textile consumption was imported,

mainly from Great Britain (Bairoch 1993). A similar, but less dramatic, story can be told for the Indian

shipbuilding and steel industries. Similar cases can be found across Latin America and the Middle East*

(Braudel, 1984 and Batou, 1990).

These dramatic shifts in specialization suggest that the massive industrialization in the now rich

'north' may have been accompanied by a de-industrialization in the now poor 'south'.

2.3. Trade

Historians disagree on trade's exact role in the Industrial Revolution (on this debate see

Engermann, 1996 and O’Brien and Engermann 1991). None, however, disputes its rapid increase

during this period (Figure 2, from Deane 1979 illustrates this rapid increase in English foreign trade

during the period of the first Industrial Revolution). At least part of this rise was due to the decrease

in trade costs during the 18th century (see next sub-section).

7

Figure 2: English Foreign trade (18th century) (net imports plusdomestic exports; official values).

On one side of the debate, the

fact that exports accounted for a modest

fraction of GNP suggests a limited role for

trade. On the other side, it is important to

note that, during the nineteenth century,

Britain (which industrialized first) exported

an unusually high proportion of its total

output, around 25% compared with 10-

12% for France (Crafts 1984, 1989).

Also, even before the Industrial

Revolution, Britain was a more open economy than any of the continental countries, exporting close to

15% of its GNP. The role of trade was especially important for several leading industries that exported

a third of their output product in 1800 (Mokyr 1993). The cotton sector, depended for more than half

of its sales on foreign markets. Cotton was also cheap to transport at a time where most goods were

not (Crafts 1989). Braudel (1984) takes trade to be a key factor, noting, that between 1700 and 1800

British industries that focused on domestic sales increased output by 50% while those that produced

for export expanded by 500%. Deane (1979) also stresses the role of trade speaking of a Commercial

Revolution which transformed London during the eighteen century into “the center of the wide, intricate,

multilateral network of world trade”. According to her, trade helped to precipitate the industrial

revolution because it created a demand for the products of the British industry: without access to a

world market, Britain was trapped in the vicious circle of a closed economy in which it is not possible

to “obtain economies of scale and experience...Increasing exports of British manufactures encouraged

new industrial investment and innovation and generated increased domestic purchasing power”. The

analytical framework we present is actually very close in spirit to this view. It gives this causal role to

trade, but we actually show that trade does not need to be quantitatively large to be causal in the

making of the take-off. Hence, our model can also be seen as a contribution to this debate on the role

of trade in the Industrial Revolution.

Deane also noted that British trade during this period expanded much more rapidly with the

West and East Indies and Africa than with Europe, its traditional trade partner. In 1700, 85% of

English exports (excluding re-exports) were directed towards Europe, and 9% towards the West and

Figure 3: Transportation and Communication Costs, 1920-1950

0

10

20

30

40

50

60

70

80

90

100

1920 1930 1940 1950 1960 1970 1980 1990Source: World Bank (1995).

Ocean freightAirSatelliteTransatlantic phone

8

East Indies and Africa. Thus, at the end of the nineteenth century, Europe's share amounted to only

30% while that of the West and East Indies was 38%.

2.4 Transaction costs: goods and ideas

Our result of a take-off (the Industrial Revolution) coupled with global income divergence,

depends crucially on a gradual exogenous decrease in transaction costs on goods. The possibility of

later rapid convergence and industrialization of the South is conditioned by a decrease in the cost of

transacting “ideas”. Because there is a threshold effect in the level of transaction costs that triggers the

take-off and then the rapid convergence, the decrease in transaction costs (first on trading goods and

then on “trading” ideas) does not need to be large.

North (1968) offers precise evidence of the decrease of the cost of shipping during the period

between 1600 and 1850. He shows that this decrease was gradual and that prior to 1800, it was the

decline of piracy “permitting ships to reduce both manpower and armament, which contributed to most

of the fall.” Moreover, this decrease was more important on “routes still infested by pirates (East

Indies, Mediterranean, West Indies) than on routes free of pirates (Baltic, North Europe, North

Atlantic).” As a result of this increased security of shipping, insurance costs also declined and by 1770

they “had fallen to approximately two-thirds of the 1635 rate and continued to drop throughout the first

half of the nineteenth century.” Increased shipping speed was not a very important factor in the

decrease of the shipping cost during the whole period. Hence, a significant part of the decrease of

transaction costs prior and during the

first Industrial Revolution can be

thought as an exogenous phenomenon.

The gradual decrease in the cost of

trading goods continued during the

nineteenth century. For example

Bairoch (1989) estimates that the

transport cost as a percentage of

production costs for a 800km trade

shipment of manufactured iron goods was 27% in 1830, 21% in 1850, 10% in 1880 and 6% in 1910.

As illustrated on figured 3, in the twentieth century, transport costs on goods continued to decrease until

9

1960 (for oceanic shipping costs) and until 1980 for airfreight. Communication costs, however,

continued to plunge. Caincross (1997) further documents the decrease in the cost of “trading” ideas

internationally. The cost of a 3-minute call from New-York to London fell from about $250 in 1930

to a few pennies today.

3. The Basic Model

The basic logic of our growth take-off – namely, that catastrophic agglomeration speeds growth

in the presence of localized learning externalities – would, we conjecture, make sense in a very broad

class of models. However, few such models could be solved analytically; Fujita, Krugman and

Venables (1999) show that most models with catastrophic agglomeration must be solved numerically.

To illustrate the interplay of economic forces as sharply as possible, we want analytic results and this

leads us to adopt explicit functional forms and some severe simplifying assumptions. In particular, our

model combines Martin and Ottaviano (1999) and Baldwin (1999) and as such it adopts functional

forms and simplifying assumption from the standard product-innovation growth model and from the

economic geography literature.

3.1 Basic Assumptions

Consider a world economy with two regions (north and south) each with two factors (labor L

and capital K) and three sectors: manufactures M, traditional goods T, and a capital-producing sector

I. Regions are symmetric in terms of preferences, technology, trade costs and labor endowments. The

Dixit-Stiglitz M-sector (manufactures) consists of differentiated goods where production of each variety

entails a fixed cost (one unit of K) and a variable cost (a units of labor per unit of output). Its costM

function, therefore, is B+wa m, where B is K's rental rate, w is the wage rate, and m is total outputM i i

of a typical firm. (Numbered notes refer to the attached 'Supplemental Guide to Calculations').1

Traditional goods, which are assumed to be homogenous, are produced by the T-sector under

conditions of perfect competition and constant returns. By choice of units, one unit of T is made with

one unit of L.

Regional labor stocks are fixed, but each region's K is produced by its I-sector (I is a

mnemonic for innovation when interpreting K as knowledge capital, for instruction when interpreting

0K ' QK'L I

aI

, F ' waI ; aI/1

K wA, A/2K%8(1&2K) , 2K/

KK w

g /0K

K'

LI A

2K

10

(3-1)

(3-2)

K as human capital, and for investment-goods when interpreting K as physical capital). The I-sector

produces one unit of K with a units of L. To individual I-firms, a is a parameter, however followingI I

Romer (1990) and Grossman and Helpman (1991), we assume a sector-wide learning curve. That is,

the marginal cost of producing new capital declines (i.e., a falls) as the sector's cumulative output rises.I

Many justifications of this learning are possible. Romer (1990), for instance, rationalizes it by referring

to the non-rival nature of knowledge.

The specific production and marginal cost functions assumed are:

where Q and L are I-sector output and employment, F is I-sector marginal cost (in equilibrium F isK I

the M-sector's fixed cost), K /K+K* where K and K* are the northern and southern cumulative I-w

sector production levels, and 8 (a mnemonic for learning spillovers) is a parameter governing the

internationalization of learning effects. Southern technology is isomorphic with a*=1/A*K andIw

A*/82 +1-2 . Finally, following Romer (1990) and Grossman and Helpman (1991), depreciationK K

of knowledge capital is ignored, so K0 =Q . The regional K's therefore represents three quantities:K

region-specific capital stocks, region-specific cumulative I-sector production (i.e. learning), and region-

specific numbers of varieties (recall that there is one unit of K per variety).

The early trade-and-endogenous-growth literature (e.g., Grossman and Helpman 1991 and

Rivera-Batiz and Romer 1991) considered only the extreme cases of 8=1 and 8=0. However recent

empirical studies – such as Eaton and Kortum (1996), Caballero and Jaffee (1995), Ciccone (1997)

– indicate that international learning spillovers are neither perfect nor nonexistent. We therefore assume

partially localized learning externalities, i.e. 0<8<1. When 8<1, regional I-sector labor productivities,

1/a and 1/a*, depend on a common global element K and a regional element, namely A/2 +8(1-2 )I I K Kw

for the north and A*/82 +1-2 for the south.K K

Given (2-1), the growth rate of north's K is related to L, 8 and 2 according to:I K

U'm4

t'0

e&DtlnQ dt , Q / C 1&"T C "

M , CM/ m K%K (

i'0c 1&1/F

i di1

1&1/F F>1

cj 'sj"E

p j

; sj /p 1&F

j

mK%K(

i'0p 1&F

i di

We view K as knowledge capital and note that while some aspects of knowledge are easily transferred*

internationally others are not. Some easily-transferred aspects of knowledge are captured by 8>0 in our model,but we also assume that important aspects of variety-specific knowledge are 'tacit' in the sense that they areembodied in the skill and know-how of workers. This component makes K difficult to trade. To keep our resultsas sharp as possible, we make the simplifying assumption that K is nontraded.

11

(3-3)

(3-4)

The corresponding expression for K*'s growth is g*=L *A*/(1-2 ). I K

To keep the analysis tightly focused on key issues, we assume an infinitely-lived representative

consumer (in each country) with preferences:

where D is the time preference, Q is a consumption composite of C and C (respectively, consumptionT M

of T and a CES composite of M-varieties), and c is consumption of variety i. Each regionali

representative consumer acts atomistically even though she owns all her region's L and K. Northern

income, Y, is wL+BK. Southern income is w*L+B*K* (recall that L=L*).

While goods (M and T) are traded, factors (L and K) are not. For goods, we adopt the*

standard simplifying assumptions (Krugman 1991a; Krugman and Venables 1995) that T-trade is

costless but trade in M is impeded by frictional (i.e., 'iceberg') import barriers (see Fujita, Krugman and

Venables 1999 for detailed discussion of this assumption). Specifically, J$1 units of M must be

exported to sell one unit abroad. J is viewed as reflecting all costs of doing business abroad. These

include everything from the mundane – shipping costs and trade policy barriers (tariffs, etc.) – to more

exotic factors such as the cost of providing after-sales services.

3.2 Intermediate Results

Utility optimization implies that a constant fraction " of northern consumption expenditure E falls

on M-varieties with the rest spent on T. Northern optimization also yields unitary elastic demand for

T and the CES demand functions for M varieties:

B ' B "E w

FK w; B /

2E

2K%N(1&2K)%

N(1&2E)

N2K%1&2K

, 2E / EE w

02K ' 2K(1&2K)(g&g()

12

(3-5)

(3-6)

where s is variety j's share of expenditure on all M-varieties in the north, E is northern expenditure andj

the p's are consumer prices. The optimal northern consumption path satisfies the Euler equation E0/E=r-

D (r is the north's rate of return on investment) and a transversality condition. Southern optimization

conditions are isomorphic.

On the supply side, free trade in T equalizes nominal wage rates as long as both regions

produce some T (i.e. if " is not too large). Taking home labor as numeraire and defining p as T'sT

price, p =p *=w=w*=1. As for the M-sector, we choose units such that a =1-1/F. As usual M-T T M2

sector optimal pricing is given then by p=1 and p*=J where p and p* are typical local and export

market prices, respectively. Southern M-firms have analogous pricing rules.3

With monopolistic competition, equilibrium operating profit is the value of sales divided by F. 4

Rearranging (2-4), using the optimal pricing rules :5

where E is world expenditure, 2 is north's share of E , and N/J . Here N is a mnemonic for thew w 1-FE

'free-ness' (phi-ness) of trade since trade gets freer as N rises from N=0 (prohibitive trade costs) to

N=1 (costless trade). Also, B is a mnemonic for the 'bias' in northern M-sector sales since B measures

the extent to which the value of sales of a northern variety (namely, pc+p*c*) exceeds average sales

per variety worldwide (namely, "E /K ). The expression for B* is analogous. Note that the definitionw w 6

of B permits a decomposition of B changes into global developments (measured by K0 ) and localw

developments (measured by 20 and 20 ). K E

Finally, differentiating its definition, the law of motion for 2 is:K

Note that the search for steady states is simplified by inspection of (2-6). By definition 20 =0 in steadyK

state, so the model has only two types of long-run equilibria: those in which g=g* (nations accumulate

capital at equal rates), and those in which 2 equals either unity or zero. We refer to these two typesK

as, respectively, the interior and core-periphery outcomes. (To avoid repetition, we consider only the

V 'B

D%g, V (

'B(

D%g

q 'B / (D%g)

F, q ( '

B( / (D%g)

F (

13

(3-7)

(3-8)

core-in-north case, i.e. 2 =1.)K

3.3 Long-Run Equilibria

The simplest way of analyzing this model is to take L as numeraire (as assumed above) and L,I

L *, and 2 as state variables. The L's indicate labor devoted to creating new K, so they are theI K I7

regional levels of real investment. While there may be many ways of determining investment in a general

equilibrium model, Tobin's q-approach (Tobin, 1969) is a powerful, intuitive, and well-known method

for doing just that. The essence of Tobin's approach is to assert that the equilibrium level of investment

is characterized by the equality of the stock market value of a unit of capital – which we denote with

the symbol V – and the replacement cost of capital, P . Tobin takes the ratio of these, so what tradeK

economists would naturally call the M-sector free-entry condition (namely V=P ) becomes Tobin'sK

famous condition q/V/P =1. K

The denominator of Tobin's q is the price of new capital. Due to I-sector competition, northern

and southern prices of K are F and F* (respectively). Calculating the numerator of Tobin's q (the

present value of introducing a new variety) requires a discount rate. In steady state, E0=0 in both

nations, so the Euler equations imply that r)=r)*=D, ('bars' indicate steady-state values). Moreover from

(2-5), the present value of a new variety also depends upon the rate at which new varieties are created.

Since the steady state is marked by time-invariant L's, (2-2) implies that the growth rate of K is time-Iw

invariant in steady state. In particular, the growth rate will either be the common g)=g)* (in the interior

case), or north's g (in the core-periphery case). In either case, the steady-state values of investing in

new units of K are :8

Given this the regional q's depend only on parameters and state variables, i.e.:

Optimizing consumers set expenditure at the permanent income hypothesis level in steady state. That9

2E 'L % D2K / A

2L % D 2K /A % (1&2K)/A (

2K '12

, 2K '12

1± 1%81&8

1%871&87

; 7 / 1 &2DN(1&8N)

[(8(1%N2)&2N]L

&1

LI '2K

A

"F

2L%D2K

A%

1&2K

A (AB & D

L I ' L (

I '"(1%8)L&D(F&")

F(1%8)

14

(3-9)

(3-10)

(3-11)

(3-12)

is, they consume labor income plus D times their steady-state wealth, FK= 2 /A, and, F*K*= (1-) ) ) )

K

2 )/A* in the north and in the south respectively . Thus:) )

K10

This relation between 2 and 2 can be thought as the optimal savings/expenditure function since it is) )

E K

derived from intertemporal utility maximization.

3.3.1 Interior Steady States

Consider first interior steady states where both nations are investing (innovating), so q)=1 and

q)*=1. Using (2-5) and (2-1) in (2-8), q)=q)*=1 gives a second relation between 2 and 2 which we) )

K E

can think of as the optimal investment relation. Together with the optimal saving relation of (2-9), it

produces three solutions:

The first is the symmetric case. The second and third roots – which correspond to interior, non-

symmetric steady states – are economically relevant only for a narrow range of N. In particular, the

second and third solutions converge to 1/2 as N approaches a particular value which we call N (forcat

reasons that become clear below). For levels of N below N , the second and third solutions arecat

imaginary and so are irrelevant. For levels of N above another critical value (defined explicitly below),

the second solution is negative and the third solution exceeds unity, so both are economically irrelevant.

Given (2-10), the remaining aspects of the interior steady state are easily calculated. In

particular, solving q)=1 for g) and then using (2-2):

L * is found by a similar procedure. Note that for the symmetric case (2 =1/2):) )

I K

q ( ' 8 (1%N2)L%N2D(2L%D)N

NCP ' 2L%D & (2L%D)2&482L(L%D)28(L%D)

LI '"2L&D(F&")

F, L

(

I ' 0

1/2

0 (= no trade) (= costless trade)

Interiornon-Symmetric (NB: g=g*)

Core-Periphery (core in north)

Interior Symmetric (NB: g=g*)

MCP

2K

N

15

(3-13)

(3-14)

(3-15)Figure 4

Using the second and third roots from (2-10) in (2-11) yields analytic solutions for L in the interior non-)

I

symmetric cases, but the expressions are too unwieldy to be revealing.

3.3.2 Core-Periphery Steady States

Expression (2-10) yields values of 2 that are economically irrelevant when N exceeds a critical)

K

level. For such N's,2 has two types of solutions: the core-periphery outcome (2 =0 or 1), or the) )

K K

symmetric outcome (note that 2 =1/2 solves q)=q)*=1 for all N). The critical value, call it N (a)

KCP

mnemonic for core-periphery), is established by noting that at the core-periphery outcome 2 =1, q)=1)

K

and q) *<1. That is, continuous innovation is profitable in the north since V=F, but V*<F* so no) ) ) )

southern M-firm would choose to setup. Using (2-1), (2-2), (2-5), (2-8) and 2 , q)* with 2 =1) )

K K

simplifies to :11

The N that solves q)*=1 defines the endpoint of the core-periphery set, namely:

Note that although there are two roots, only one is economically relevant. 12

Using 2 =1, the remaining aspects of the core-periphery steady state are simple to calculate.)

K

In particular, since 2 =1, q)=1 and q)*<1, we have:)

K

3.3.3 A Map of Steady States

Figure 4 summarizes the various steady states and their

dependence on trade costs. North's share of world K,

2 , is on the vertical axis and all conceivable levels ofK

trade free-ness are shown with the [0,1] interval on the

horizontal axis. As noted above, the symmetric case

exists for all N, but the core-periphery outcome (either

Production does not literally 'shift' since factors are internationally immobile. Nevertheless 2 rises since capital*K

accumulation is encouraged in the north and discouraged in the south.

16

2=1 or 0) is an equilibrium only for N>N . The final type of steady state, the interior, non-symmetric)K

CP

case, is shown as the bowed line.

4. Stability and Catastrophic Agglomeration

Although an equal division of M-varieties is always an equilibrium, it need not be stable, as the

economic geography literature has emphasized. Indeed, in our model two cycles of 'circular causality'

tend to de-stabilize the symmetric equilibrium.

The first is the well-known demand-linked cycle in which production shifting leads to

expenditure shifting and vice versa. The particular variant present in our model is based on the

mechanism introduced by Baldwin (1999). To see the logic of this linkage, consider a perturbation that

exogenously shifts one M-sector firm from the south to the north. Firms are associated with a unit of

capital and capital-earnings are spent with a local bias because of transaction costs, so 'production

shifting' leads to 'expenditure shifting' . Other things equal, this expenditure shifting raises northern*

operating profits and lowers southern operating profits due to a market-size effect. This tends to raise

q and lower q* thereby speeding north's accumulation and retarding south's. The initial exogenous13

shift thus leads to another round of production shifting and the cycle repeats. As we shall see, if trade

costs are sufficiently low, demand-linked circular causality alone can de-stabilize the symmetric

equilibrium.

The second link is the growth-linked circular causality introduced by Martin and Ottaviano

(1999). When I-sector technological externalities are transmitted imperfectly across borders,

production shifting leads to 'cost shifting' in the I-sector. For instance, suppose an exogenous

perturbation increased 2 slightly. Given localized knowledge spillovers, this shock lowers the northernK

I-sector's marginal cost and raises that of the south. Other things equal, this raises q and lowers q*,

so the initial production shifting raises north's rate of investment and lower south's. Of course, this

'growth shifting' further increases 2 and the cycle repeats. Again, if trade costs are low enough,K

growth-linked circular causality alone can yield to total agglomeration.

Mq /qM2K *

*2K'1/2

' 2 1&N1%N

d2E

d2K **2K'1/2

% 41%8

1%N2

(1%N)2

& (1&N)2

1%N2% 1&8

17

(4-1)

The sole force opposing agglomeration here is the local competition effect. Namely, raising 2K

(north's share of varieties) tends to raise local competition in the north and lower it in the south. Since

competition is bad for profits, raising 2 tends to lower q and raise q*.K14

4.1 Stability of the Symmetric Interior Steady State

The appendix shows that in the neighborhood of the symmetric equilibrium, the linearized

system has two positive and one negative real roots when N is less than a critical value. For this range

of N's the system is saddle path stable, since only 2 is a nonjumper. For N beyond the critical value,K

the linearized system has three positive eigenvalues, so the symmetric equilibrium is unstable. As it turns

out, however, an informal approach to stability provides the same answer with greater intuition.

Specifically, to study the symmetric equilibrium's stability, we exogenously increase 2 by aK

small amount and check the impact of this perturbation on the regional q)'s, allowing expenditure shares

to adjust according to (2-8). In particular, using (2-9), (2-1), (2-2), (2-5) and (2-8), the steady-state

q can be expressed as a function 2 and L . Holding L constant for the moment, the partial derivativeK I I

of interest is Mq)/M2 from q)=[2 ,L ,2 [2 ];N]. The symmetric equilibrium is stable, if and only ifK K K E K) ) ) )

Mq) /M2 is negative for a simple reason. If a unit of capital 'accidentally' disturbed symmetry, theK

'accident' lowers Tobin's q in the north and raises it in the south (by symmetry Mq)*/M2 and Mq)/M2K K

have opposite signs). Moreover, these incipient changes in Tobin's q are sufficient statistics for changes

in regional investment levels (see Baldwin and Forslid 1997a Proposition 1 for details). Thus when

Mq)/M2 <0, the perturbation generates self-correcting forces in the sense that L falls and L* rises. IfK I I

the derivative is positive, the 'accident' boosts L and lowers L*, thus amplifying the initial shock to 2 .I I K

Plainly the symmetric equilibrium is unstable in this case. We turn now to signing Mq)/M2 . K

Differentiating the definition of q with respect to 2 , we have:K

Using (2-8) to find d2 /d2 =2D8/[L(1+8)+D](1+8), we see that the system is unstable for sufficiently)

E K

low trade costs (i.e. N .1). And under weak regularity conditions, the system is stable, i.e. (3-1) is

negative, for sufficiently high trade costs (N .0). 15

Ncat '[L(1%8)%D] & (1&82)[L(1%8)%D]2%82D2

8[L(1%8)%2D]

* Hence, divergence would not occur in a set of countries for which technological spillovers are sufficiently strong. This isreminiscent of Goodfriend and McDermott (1998).

18

(4-2)

(3-1) illustrates the three forces affecting stability. The first and third term are positive, so they

represent the destabilizing forces, namely the demand-linked and growth-linked circular causalities

(respectively). The negative second term reflects the stabilizing local-competition effect. Clearly,

reducing trade costs (dN>0) erodes the stabilizing force more quickly than it erodes the destabilizing

demand-linkage.

To isolate the two distinct cycles of circular causality, suppose, for the sake of argument, that

the demand-linkage is cut, so d2 /d2 =0. In this case, Mq)/M2 is positive and the system is unstable)

E K K

when 8<2N/(1+N ). This shows that growth-linked circular causality can by itself produce total2

agglomeration when trade costs are low enough. (Recall that 0#N#1 is a measure of the free-ness of

trade, so N=1 indicates costless trade). To see the dependence of growth-linked circular causality on

localized knowledge spillovers, note that with 8=1 and d2 /d2 =0, the symmetric equilibrium is always)

E K

stable . At the other extreme, when spillovers are purely local (8=0), the symmetric equilibrium is never*

stable even without the demand linkage.

Finally the critical level of N at which the symmetric equilibrium becomes unstable is defined

by the point where (3-1) switches sign, namely Mq)/M2 =0. This expression is quadratic in N, so it hasK

two roots. The economically relevant one is :16

Observe that the range of unstable N's, (N ,1], gets smaller as the internationalization ofcat

learning effects (as measured by 8) increases. Since the growth-linkage becomes weaker as 8 rises,

it is easy to understand why the range of trade costs that leads to instability shrinks as 8 rises. We

come back in section 6 on the role of rising 8 for possible convergence in the South. Additionally, the

instability set expands as the discount rate D rises since this amplifies the demand linkage. That is, the

equilibrium return to capital rises with D, so a higher D amplifies the expenditure shifting that

accompanies production shifting.

4.2 Stability of Other Steady States

1/2

0 (= no trade)

1 (= costless trade)

Interiornon-Symmetric

(stable)

Core-Periphery(stable)

Symmetric(unstable)

(trade

free-ness)

Symmetric(stable)

2K

Ncat NCP

N

CC

19

Figure 5

The stability test for the core-periphery equilibrium case is slightly different since it entails 2 =1,)

K

q)=1 and q)*<1. The procedure, is to find the range of N where q)*<1, when 2 =1. Since this is exactly)

K

the procedure used in Section 2 to determine the range of the core-periphery, we see that the core-

periphery steady state is stable wherever it exists.

To examine the stability of the interior-non-symmetric steady state, we adopt the same

procedure. Namely, we study Mq)/M2 evaluated at 2 given by (2-10). Given the complex nature ofK K)

Mq)/M2 and (2-10), we cannot sign the derivative analytically. However, for reasonable values of theK

parameters the derivative is negative. This finding is robust to sensitivity analysis on parameter values.17

Finally, we turn to showing that there

is no overlap in the zones of stability. With

some difficulty, it is possible to show that

N >N and that the endpoints of theCP cat

interior-non-symmetric steady state are

exactly equal to N and N . Using thesecat CP 18

results, Figure 5 summarizes the model's

stability properties in a diagram with N and

2 on the axes. More precisely, as alreadyK

argued, ‘stable’ corresponds to a saddle

while .‘unstable’ corresponds to an unstable

node/focus. The fact that interior-non-symmetric steady states must correspond to saddles follows from

continuity arguments as the dynamical system undergoes a supercritical pitchfork bifurcation when N

crosses N : the symmetric steady state loses its stability to the two new neighboring steady states.cat

5. Three Stages of Growth

As history would have it, the cost of doing business internationally has declined sharply since

the 18th century. While this trend seems obvious and irreversible with hindsight, it was not obviously

predictable in advance, nor was it monotonic. For example, we have seen in section 2 that just before

the Industrial Revolution, this decrease was mostly exogenous and not primarily related to technical

progress. During many periods, and sometimes for decades at a time, the trend was reversed. From

20

1929 to 1945, for example, international trade became increasingly difficult. Restoration of peace and

the founding of GATT allowed the trend to resume, but this was not a foregone conclusion in, say,

1938. Hence, even though we think of transaction costs as decreasing with time, we assume that agents

take the level of the transaction cost to be constant in their private decisions. If we were to take into

account forward expectations on future lower transaction costs, we would need to specify an

exogenous rate of decrease of J with time. This would greatly complicate this analysis and would

introduce “expectations” driven equilibria. Following Krugman and Venables (1995), this section

considers the implications of lowering the cost of trade (as captured by the parameter J).

5.1 Location and Trade Costs: A Punctuated Equilibrium

When trade costs are high the symmetric equilibrium is stable and gradually reducing trade costs

(dN>0) produces standard, static effects – more trade, lower prices, and higher welfare (more on this

below). There is, however, no impact on industrial location, so during an initial phase, the global

distribution of industry appears unaffected by N.

As trade free-ness moves beyond N , however, the world enters a qualitatively distinct phase.cat

The symmetric distribution of industry becomes unstable, and northern and southern industrial structures

begin to diverge; to be concrete, assume industry agglomerates in the north. If 2 could jump, it wouldK

be on the interior-non-symmetric equilibrium (shown as the CC locus in Figure 5). Since CC is vertical

at N , the impact on location would be catastrophic. That is to say, an infinitesimal change in tradecat

costs would produce a discrete change in the steady-state global distribution of industry.

Since 2 cannot jump, crossing N triggers transitional dynamics in which northern industrialKcat

output and investment rise and southern industrial output and investment fall. Moreover, in a very well

defined sense, the south would appear to be in the midst of a 'vicious' cycle driven by backward

(demand) linkages and forward (cost) linkages. The demand linkages would have southern firms

lowering employment and abstaining from investment, because southern wealth is falling, and southern

wealth is falling since southern firms are failing to invest. The cost linkages would lead to an increase

in the cost of southern investment/innovation relative to the north as 2 rises (due to localized learningK

externalities), and to an increase in 2 since the cost of southern investment/innovation rises relative toK

the north. By the same logic, the north would appear to be in the midst of a 'virtuous' cycle. Rising 2K

would expand the north's relative market size and reduce its relative cost of investment/innovation.

21

Although we cannot analytically characterize the transitional dynamics of a system with three

non-linear differential equations, we can say that a continuing rise in trade free-ness would raise 2 untilK

the core-periphery outcome is the only stable long-run equilibrium. Of course, southern knowledge

never disappears entirely, so the core-periphery outcome is only reached asymptotically (the number

of southern varieties remains fixed, but the value of these drops forever towards zero due to the

ceaseless introduction of new northern varieties).

Once the core-periphery outcome is reached – or more precisely, once we can approximate

2 as unity – the world economy enters a third distinct phase. For trade costs lower than this point,K

the world economy behaves as it did in the first phase. That is to say, making trade less costly has the

usual static effects, but no location effects.

Plainly, the location equilibrium in this world would appear as a punctuated equilibrium. In the

first and third phases, lower trade costs have no impact on the distribution of world industry, but in the

second phase, the north's share of world industry increases rapidly.

We can also note that catastrophic agglomeration will not occur for countries for which

technological spillovers are important (high 8) . This may be the case because of physical, political,

linguistic and cultural proximity. Our model is therefore consistent with a process of divergence

between North and South while no divergence occurs among countries in the “North” (Europe, North

America) for which technological spillovers were certainly quite high even in the 18th and 19th century.

5.2 Growth Stages

Long-run growth in this model is driven by the ceaseless accumulation of knowledge capital

resulting in an ever greater range of M-varieties. Given preferences, this ceaseless expansion of variety

raises real consumption continually. While technology and output in the traditional sector is stagnant,

the expansion of M-varieties forces up the price of T relative to that of the composite good C . TheM

value of the two sectoral outputs thus grows in tandem.

5.2.1 Stage-One's Growth and Investment Rates

By definition, the initial interior solution entails symmetry, i.e.,2 =2 =1/2 and, as long as N) )

E K

<N , this outcome is stable. The steady-state rate of K accumulation during this phase is found usingcat

the expression for L from (2-12) in (2-2), to get:)

I

Stage I : g ' g ( '"(1%8)L&D(F&")

F

P ' K w"

1&F2K % N(1&2K)

"1&F , P ( ' K w

"1&F

N2K % 1&2K

"1&F

Stage I : g income ' g (income '

"2(1%8)L & D"(F&")F(F&1)

Stage I :LI

Y'

"(1%8)L & D(F&")(F%")(1%8)L % "D

22

(5-1)

(5-2)

(5-3)

(5-4)

This common rate of K-accumulation is unaffected by the level of trade costs, N.

Steady-state growth in real income is nominal Y divided by the perfect consumption price

index, P. Given preferences, and p =p *=1, the perfect price index is P , where P is the CES pricez z M M"

index . Thus P and P* are:19

In steady state, nominal Y and 2 are time-invariant, yet the P's falls on the steady-state growth pathK

since K rises at the common rate of g). Thus P falls at g)/(F-1) and real income grows at "g)/(F-1).wM

Using (4-1): :20

where g is the real income growth rate. By inspection, the growth rate rises with 8 and ", but fallsincome

with D and F (all of which are standard results in the trade and endogenous growth literature). Again,

trade costs do not play a role as long as the economy remains at the symmetric equilibrium.

Consider next, the rate of investment in innovation. With labor as numeraire, the rate of

investment in steady state is L/Y. Using (2-12) and the definition of Y, we have :) ) )

I21

Again, the ratio is rising in 8 and ", falling in D and F, and unaffected by N.

These results are simple to establish since stage-one is a steady state. The stage-three growth

rate is similarly simple to establish, so we turn to it next.

5.2.2 Stage-Three's Growth and Investment Rates

Once the stage-three steady state is reached (or at least when 2 is close enough to unity toK

Stage III : g '"2L&D(F&")

F

Stage III :L I

Y' "2L & D(F&")

(F%2")L % "D

23

(5-5)

(5-6)

approximate the steady-state 2 as equal to unity), (2-2) and (2-15) imply:K

Importantly, this g) exceeds the stage-one g) only to the extent that spillovers are localized, i.e. 8<1. The

common real income growth rate, viz. g)"/(F-1) where g) is given by (4-5), is also higher than the stage-

one growth rates (as long as 8<1). Observe that the south – which is completely specialized in the

traditional sector – engages in no innovation, and indeed makes no investment of any kind.

Nevertheless, the south experiences the same rate of growth as the north due to continual terms of trade

gains: the price of T (which south exports) is time-invariant but the price of C (which south imports)M

falls.

The stage-three northern investment ratio is:

As long as growth is positive in the third stage, this is greater than that of the first stage even with 8=1

since all investment/innovation occurs in the north (or which ever nation acquires the core).

Finally, notice that while further reduction of N raises both nations' real income trajectories (via

one-off drops in the perfect price index), liberalization has no affect on the common slope of their

growth paths.

5.2.3 Stage-Two: Growth and Investment During the Take-Off

The growth rate during the take-off period is given by (2-2) where L is given by (2-11) andI

2 by the second root in (2-10). This growth rate is difficult to characterize analytically but can beK

easily simulated numerically. Because the difference between N and N is very small, the rate ofcat CP

investment and economic growth rise abruptly to that of the third stage. In a numerical example for

which J .2.18 and J = 2.2, growth increases between these two trading costs from 0 to +6,2%. Thecat cp

investment rate increases in the north from 0 to 5.2%. Furthermore, since 2 cannot jump the take-offK

also features transitional dynamics: the rate of productive investment (in human, knowledge, and/or

physical capital) rises, and one or more manufacturing sectors develop with a high rate of growth. After

the take-off, the third phase of our model is marked by a high and stable rate of economic growth as

0 (= no trade) (= costless trade)

Global Income Divergence

'Take-Off''modern growth'

stage

'pre-moderngrowth'

stage

1

Ncat NCP

N

24

Figure 6

well as by a more stable sectoral composition of output.

5.3 Income Divergence

Figure 6 shows how the ratio of steady-

state real income levels (north's divided by

south's) varies with trade costs. The figure

approximates the actual predicted path by

assuming that the system is always at the stable

steady state that corresponds to each level of

trade free-ness. There are clearly three phases

in the figure. In the first phase, trade cost

reductions have no impact on this ratio. Per capita income levels are identical since 2 =2 =1/2 and,) )

E K

by symmetry, northern and southern price indices are identical. In the second phase, where N>N ,cat

industry begins to agglomerate in the north. This has two effects, both promoting income divergence.

First, as 2 rises, northern steady-state wealth rises while southern steady-state wealth falls. Second,)

K

due the 'home market' effect, the shift in industry location has a favourable impact on the northern price

index and a dilatory impact on the south's price index. That is, as long as trade is not costless, southern

consumers face higher consumer prices since all trade costs are passed on to consumers. In the final

phase, some of the divergence is reversed. The reason is that although the north's wealth is higher than

the south's, lowering trade costs reduces the difference in north and south price indices up to the point

N=1, where the two price indices are identical.

5.4 Welfare

As pointed out above, our model is an example of uneven development in the sense that the

agglomeration of industry in the north produces immediate divergence with the south. However,

because agglomeration generates a take-off which materializes itself into an acceleration of the world

rate of innovation, the take-off also produces benefits for the south. The tension between the negative

effect of agglomeration and the positive effect of the increase in the rate of innovation is what makes

the welfare effect of the take-off ambiguous for the south. Northern and southern steady-state welfare

U'c0"g

D2(F&1)ln L%

D2K

A2K%N(1&2K)

"F&1 , U('

c0"g

D2(F&1)ln L%

D1&2K

A ((N2K%1&2K)

"F&1

0 (= no trade)

Steady-State Welfare

'Take-Off' 'modern growth' stage

'pre-moderngrowth' stage

North

North and South

South high "

South medium "

South low "

25

(5-7)

Figure 7

(i.e., the present value of the utility flows) as functions of N are, respectively:

where g) and 2 depend upon N as described above, and c captures terms that do not depend upon)

K o

N. Notice that while a rise in g) is welfare enhancing in both regions, raising 2 raises northern welfare,)

K

but lowers that of the south. As discussed above, the impact of raising the steady-state 2 is twofold:K

it shifts wealth from south to north and it lowers the northern price index relative to the southern price

index, as long as N<1.

Figure 7 plots the levels of welfare

corresponding to the higher arm of the pitchfork

represented in Figure 5. We have simulated

these levels of welfare as a function of trade

costs and found three generic cases (again

considering only steady states). Two elements

are constant in all cases. First, both regional

welfare levels rise as N rises (imports get

cheaper) during stage-one (the pre-modern

growth phase), and second the north's welfare is insensitive to N in the final stage. Symmetry explains

the first element and the fact that 2 =1 in the final phase accounts for the second element. The cases)

K

differ only in the south's welfare level in the third phase.

Note first that when transaction costs are sufficiently high (N is below the threshold level), a

decrease in transaction costs has the usual static effects in both the south and the north. It raises welfare

because it lowers the real price of traded manufactured goods. At the point of the take-off, north and

south welfare diverge. The north benefits from agglomeration and a higher growth rate. The south

benefits only from higher growth; agglomeration actually harms the south. This explains why post-take-

off welfare is always lower in the south.

The positive growth effect of the take-off explains why the comparison of welfare before and

after the take-off is ambiguous. If the share of manufacturing goods is low enough, the increase in the

growth rate of the manufacturing sector does not have a large welfare impact. In this case, the south

VT'1J"2K

E wN(1&2E)

N2K%1&2K

26

(5-8)

loses due to agglomeration and its welfare never reaches the level it had before the take-off. In the

intermediate " case, the south first loses but eventually attains a welfare level that exceeds its pre-take-

off level. Finally, when manufacturing is sufficiently high, the positive growth effect dominates and the

take-off benefits both the south and the north. Similar results are obtained when we vary other

parameters: the welfare impact is more favourable to the south the higher the growth effect of

agglomeration, that is the larger the market size (the higher L), the more local the spillovers (the lower

8), the more the economies of scale (the lower F) and the lower the subjective discount rate D.

Importantly, after the take-off, lowering transaction costs always improves welfare in the south

because it decreases the price of goods imported from the north. Thus, even though the south may

have been made worse off by agglomeration in the north, resisting further reductions in transaction costs

is not welfare improving.

5.5 Trade

At the symmetric steady state, north and south engage in pure intra-industry trade in

differentiated products only. When 1/2<2 <1, the regions have different relative factor stocks andK

some Heckscher-Ohlin-trade occurs (north is the net exporter of capital-intensive M goods). The last

case is when 2 equals unity and only inter-industry trade occurs; the north exports M-varieties inK

exchange for T.

Because factors are not traded, goods trade balances each period. The global volume of

exports is thus twice the north's exports of manufactures. From (2-4), the steady-state volume of

manufactures exports to the South (exclusive of transport costs) is:

Finally, we study the expansion of trade during the three phases. The global trade volume is graphed

in Figure 8 (using the same parameters as in the previous figure with "=0.3). Again there are three

distinct phases. In the initial phase, the level of trade is fairly low and all trade is intra-industry trade:

0 (= no trade)

1 (= costless trade)

Steady-State World Trade

'Take-Off''modern growth' stage'pre-modern

growth' stage

Ncat NCP

N

27

Figure 8(1/J)"N/(1+N)[L+D/(1+8)] . Furthermore,

lowering trade costs promotes trade in a

smooth, gradual fashion. Once N>N ,cat

agglomeration occurs rapidly in the north, so

the nature of trade shifts. The north becomes

a net exporter of industrial goods and a net

importer of traditional goods. Once all

industry is in the north, the south must satisfy all

its demand for industrial goods via imports and

exports of manufactures from the North (exclusive of trade costs) are: (1/J)"L. It can be shown that

indeed, trade is larger after the take-off than just before (when trade costs are J ). cat

Furthermore trade needs not be “large” before the takeoff to be at the origin of the take-off.

As a ratio of income in the North, exports just before the take-off are: (1 /J )"N /(1+N ). Becausecat cat cat

"<1, N <1 and J >1, this ratio can actually be very small. Just after the take-off, this ratio becomescat ca t

(" /J ) (1+D). For the same numerical example as in section 5.2.3 in which growth increases from 0cp

to 6.2% during the take-off, the ratio of exports to income goes from just 1.2% to 15.1%. The reason

that trade before the take-off does need to be quantitatively large to cause the take-off is that at the

threshold level of transaction costs, “catastrophic” agglomeration just requires that the demand and

growth linkages outweigh the local-competition effect so as to put into motion the circular mechanism

that produces the take-off.

6. Globalization and Industrialization of the South

While the radical income disparity between poor and rich countries is still a dominant feature,

the decades since WWII have also seen some spectacular examples of rapid convergence – what

Lucas (1993) calls 'miracles'. Here we show that our model can produce a 'miracle' in the south (with

a two region model a miracle in the south produces full income convergence). The key is to take a

broader view of international integration.

Up to this point, we have viewed integration as nothing more than the lowering of trade costs.

As discussed in section 2, the post-war integration, especially that of the past two or three decades,

8mir 'N(2L%D)

L(1%N2)%DN2

1/2

0 (= no spillovers)

Interiornon-Symmetric

(stable)

Core-Periphery(stable)

Symmetric(stable)

Technologicalspillovers

Symmetric(unstable)

2K

8mir

8

18mir’

Figure 9Miracle in the South

28

(6-1)

has lowered the cost of 'transporting' ideas more than it has lowered the cost of transporting goods.

In the context of our model, a decrease in the cost of communications and more generally an increase

in the speed of international diffusion of ideas is translated into an increase in 8, which measures the

internationalization of knowledge spillovers in the I-sector. To focus sharply on this trend in the relative

cost of trading goods and of trading

ideas, we make the simplifying

assumption that all recent integration

consist of rising 8. That is, we start from

the stage-three situation of full

agglomeration in the north and suppose

that trade free-ness N has risen to some

natural upper bound, but in a fourth

stage 8 rises towards unity, i.e. perfect

international transmission of learning

externalities.

Starting from a situation with full industrial agglomeration in the north (2 =1), the increase in)

K

8 initially has no impact on southern industry or on the global growth rate given by (4-5). However,

southern I-sector labour productivity rises with 8, so at some threshold level of 8 (call this 8 formir

'miracle'), the steady-state q* exceeds unity (see equation 3.13 where it can be checked that q*rises

with 8) . Beyond this, southern M firms find it profitable to invest in new ideas/varieties. The 8 that

sets q)*=1 for a given N defines the critical level of 8 beyond which the core-periphery outcome is no

longer stable. Using (3-13) this level is:

Clearly 8 rises with the free-ness of trade. mir 22

As in the case of falling trade barriers, there is a second critical value of 8 where the symmetric

equilibrium becomes stable. This value, denoted as 8 is the level of 8 where Mq)/M2 evaluated atmirK

2 =1/2, i.e. (3-21, becomes negative. (3-1) is quadratic in 8 and the economically relevant solutionK

defines 8 ´ . The different phases of the “miracle” are described in figure 9.mir

Stage IV : g ' g ( '"(1%8mir))L&D(F&")

F

29

(6-2)

As with the north's take-off, the miracle in the south would appear to be driven by two virtuous

circles. When the south invests, its capital stock and therefore permanent income begins to rise,

triggering demand-linked circular causality. Rising local expenditures boosts southern profits and this

in turn gives a new incentive to innovate/invest. Moreover, as K* rises, the southern I-sector begins

to benefit from localized learning externalities, and this triggers cost-linked circular causality. The net

effect is a drastic structural change as the south industrializes. During the transitional phase north and

south real incomes converge.

The miracle in the south, however, differs from the initial northern take-off in three ways. First,

the southern industrialization does not shut down northern innovation. It merely forces a shift of some

northern resources from the M-sector to the T-sector (here we think of the T-sector as including

services as well as agriculture). Second, the source of the south's take-off is quite different. The

miracle occurs due to the south's ability to learn from the north's experience in innovation rather than

trade openness per se. For this convergence process to occur, openness to foreign ideas and

technologies is key. This is consistent with the Japanese experience at the end of the 19th century and

the account of Rodrick (1995) on the success of the Asians dragons.

Finally, the convergence of the south pushes the global steady-state growth rate to a level that

is in-between the stage-one (pre-Industrial Revolution) rate and the stage-three (rich-north-poor-south)

rate. In particular, defining stage-four as the phase where 2 has returned to 1/2, the stage-four rate)

K

of innovation is:

By inspection this is higher than stage-I rate since 8 ´ exceeds the 8 in (4-1). But it is lower than themir

stage-three rate in (4-5) since 8 ´ <1 (assuming that some barriers prevent 8 from reaching unity). mir

7. Concluding Remarks

Our model has some interesting and intriguing political economy implications. The most obvious

one would, at first glance, appear to support notions of 'inequalizing trade'. In our model, the big

30

divergence between rich and poor countries is a necessary implication of Europe's Industrial Revolution

and the expansion of international trade triggered both. Indeed, the creation of a global core-periphery

situation is a necessary condition for the growth take-off. Moreover, our paper describes a purely

economic mechanism (i.e. other than cultural or political) that explains why the Industrial Revolution did

not spread to what we now call the Third World. The traumatic experience of massive de-

industrialization in India during the 19th century is consistent with our model and it also suggests one

reason why this country, along with most other Third World countries, kept an attitude of distrust

towards international trade.

Our model, however, departs sharply from the 'inequalizing trade' paradigm on several key

points. First, we showed that the present value of the south's welfare could have been increased the

Industrial-Revolution-cum-income-divergence. That is, the south could be even poorer than it is today,

had the Industrial Revolution not occurred. Second, our model posits that income divergence was

caused by lower trade costs, not only by plunder or imperialism. Third, in our model, trade

liberalization and more generally the reduction in transaction costs first generates massive divergence

of real incomes but then is conducive to a process of relative convergence. Finally, we show that to

the extent the recent decades of international integration have lowered the cost of trading ideas more

than it has lowered the cost of trading goods, integration can be the key to southern industrialization.

Finally, our model may also be taken as providing a long-term perspective on the convergence

literature (see Barro and Sala-i-Martin (1992) inter alia). That literature essentially takes the 19th

century global income disparity as given and seeks to measure whether this gap has narrowed in the

postwar period. Our model attempts to analyse the long term origin of the divergence between North

and South by linking it explicitly to the growth take-off of the Industrial Revolution.

e1 ' L(1%8)%D; , e2,3 'b ± b 2&4cF(1%8)

F(1%8);

b / F[e1(1&8)%28D] % 4"N(1&8N)e1

(1%N)2, c / &2"e1 e1[8&2N(1%8)

(1%N)2]&8D(1&N)

1%N

31

(A-1)

Appendix: Formal Stability Analysis of Section 2 (Symmetric Steady State)

This appendix uses standard stability tests involving eigenvalues. As in Baldwin (1999) and

Baldwin and Forslid (1997b), we find that the results generated from the more intuitive approach in the

text are identical to those of more formal techniques.

It proves convenient to take as state variables, E, E* and 2 , with the two Euler equations andK

(2-9) as the system equations. Using r=B/F+F/F, where F/F=-L -8A*L*/A, and g=LA/2 , we can) )

I I I K

express the differential equations in terms of L, L * and 2 . Using the definition of E and E*, namelyI I K

E=L-L +"BE 2 /F and an analogous expression for E*, as well as E =(2L-L-L *)/(1-"/F), we canI K I Iw w

express the Ls in terms of E's and the two 2s. Finally, using 2 =E/(E+E*), we can express the systemI E

equations in terms of the state variables: E, E* and 2 . K

Linearizing the system around the symmetric steady state, the eigenvalues of the resulting

Jacobian matrix are:

The first eigenvalue is plainly positive. By inspection, given that c<0 at sufficiently low levels of N, we

know that at least in this range of trade costs the eigenvalues are all real. In this case, given the positive

radicals in e and e are positive, it follows that the eigenvalue that adds the radical – call this e – is2 3 2

always positive. The third eigenvalue changes sign at the point where c=0. Solving this for N, we get23

exactly (3-2).

Thus in the neighbourhood of the symmetric system, the linearized system has two positive and

one negative real roots for N<N . This makes it saddle path stable, because only one of the statecat

variables is a nonjumper. For N>N we have three positive eigenvalues and therefore an unstablecat

steady state.

32

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p 1&Fj "E

)& waM

p &Fj "E

)%

p ( 1&Fj "E

)(& (waMJ)

p ( &Fi "E

)(

) ' mK%K(

i'0p 1&F

i di , )( ' mK%K(

i'0p (1&F

i di

(1&F)cj & waM(&F)cj

p j

' 0 , (1&F)c(j & (wJaM)(&F)

c (j

p (j

' 0

p(1& 1F

) ' waM , p ((1& 1F

) ' wJaM

p & waM 'pF

, p ( & wJaM 'p (

F

(p & waM)c % (p ( & wJaM)c ( 'pcF

%p (c(

F

1. Note that K's rental rate B is the operating surplus (i.e. the Ricardian surplus). This is due to thefact that each unit of K is variety specific, so K's reward is like that of a specific factor.

2. This follows from perfect competition, constant returns and the choice of units in the T sector.

3. Using the demand functions, the flow of operating profit is:

The Dixit-Stiglitz monopolistic competitors take other firms' prices as given, so the first-orderconditions for a typical variety are:

where c and c * are sales to consumers in the local and export markets (for a typical variety). j j

Using a =1-1/F, simplification of these yields the pricing rules in the text.M

4. To see this, consider the first order conditions for a typical variety (dropping the varietysubscript for convenience):

Rearranging gives us operating profit per final sale:

so total operating profit is

5. Using a cumbersome, but precise notation, northern per-variety sale at consumer prices iss E +s E , where subscripts indicate region of origin and superscripts indicate markets. PluggingN N

N N S S

the optimal pricing rules p=1, p*=J into the formula for a variety's market share:

Supplemental Guide to Calculations:

"The geography of Growth Take-offs"

s NN '

1K%K (N

, s SN'

NNK%K (

BN ' ("E w

FK w)

2E

2K%N(1&2K)%

N(1&2E)

N2K%1&2K

B ( /N2E

2K%N(1&2K)%

(1&2E)

N2K%1&2K

V ' Bo m4

0

e&(D%g)tdt

q ( '"E wB (A (

F(D%g)

q ( ' 8 (1%N2)L%N2D(2L%D)N

where N=J . Utilizing the share formulas in the expression for operating profit (see previous note),1-F

we have (with some rearrangement):

where B /B is northern operating profit (for a typical variety), 2 =E/E and the superscript 'w'N Ew

indicates world totals. A similar procedure yields an expression for B /B*.S

6. Specifically, B* equals B*("E /K ), where w w

7. Since the model has only one primary factor L, expenditure allocation is tantamount to resourceallocation, so labour is the natural numeraire. Also the I-sector technology indicates that L, L * andI I

2 are the determinants of the regional rates of K accumulation, so these are the natural stateK

variables.

8. From (2-7) and the facts that E0 =20 =20 =0 in steady state, B is time invariant in steady state, soK E

B falls at the rate K grows. Using the steady-state discount rate D, we have:w

Solution of the integral yields the result in the text.

9. By definition expenditure is income less investment, i.e. L-L+BK and we wish to show that thisI

equals L+D2 /A in steady state. This boils down to showing that BK=L +D2 /A. From q)=1,K I K

B=F(D+g). Multiplying through by K and using growth rate form of the I-sector productionfunction yields the result after some manipulation.

10. To see this, note that at the symmetric steady state, K times V must equal K times F since q=1implies V=F. Using the definition of F, KF=K/(K+8K*)=2 /[2 +8(1-2 )]. The definition of AK K K

yields the result. The south steady-state wealth is derived similarly.

11. Given (2-1), (2-7), and (2-10):

With 2 =1, we have that E =2L+D, 2 =(L+D)/E and A*=8. Also B*=N2 +(1-2 )/N, so usingK E E Ew w

D+g)=B) /F with q)=1 (in the core-periphery steady state), we have:)

2L%D ± (2L%D)2&482L(L%D)28(L%D)

0 < 2L%D & (2L%D)2&482L(L%D)28(L%D)

< 1

2L%D & 28(L%D) < (2L%D)2&482L(L%D) <'>

(2L%D)2 % 4(8(L%D))2 & 48(L%D)(2L%D) < (2L%D)2&482L(L%D) <'>

8(2L%D) < (2L%D)

B ' B ("E w

FK w) , B( ' B ( ("E w

FK w) ;

B /2E

2K%N(1&2K)%

N(1&2E)

N2K%1&2K

, B(/N2E

2K%N(1&2K)%

(1&2E)

N2K%1&2K

;

2E / EE w

12. Solving for the N where q*=1 yields two solutions:

We wish to show: (1) the roots are both real, (2) the root listed in the text is bounded between zeroand unity, (3) the other root always exceeds unity.

To show (1), note that the term under the radical is minimized when 8=1, yet in this case it isD >0. To show (2), we need to demonstrate that:2

the left-hand inequality holds since 48 L(L+D)>0. To see that the right-hand inequality holds, we2

rearrange it to see that it holds if and only if:

which holds since 8<1.The third fact is established by similar algebraic manipulations.

13. Specifically,

The home market effect is, by definition, the impact of d2 on B and B*. By inspection, B isE

increasing and B* is decreasing in 2 , as long as trade is less than perfectly free, i.e. N<1. E

14. Specifically,

B ' B ("E w

FK w) , B( ' B ( ("E w

FK w) ;

B /2E

2K%N(1&2K)%

N(1&2E)

N2K%1&2K

, B(/N2E

2K%N(1&2K)%

(1&2E)

N2K%1&2K

;2E / EE w

Ncat '[D%L(1%8)]± D2%L(1%8)[2D%L(1%8)](1&82)

8[2D%L(1%8)]

D2%L(1%8)[2D%L(1%8)](1&82) > 8[2D%L(1%8)] & [D%L(1%8)]

28(1&8)[L(1%8)]2 % 6D8(1&8)[L(1%8)] % 4D28(1&8) > 0

P / p 1&"T P "

M , where PM / mK w

0

p 1&Fi

11&Fdi

The local competition effect is, by definition, the impact of d2 on B and B*. By inspection, B isK

decreasing and B* is increasing in 2 , as long as trade is less than perfectly free, i.e. N<1. K

15. In particular, we need that 8D<(1+8)L.

16. The two roots are:

The root that adds the radical is always greater than unity and so lies outside the economicallyrelevant range. To show this note that the assertion requires

Two cases must be considered. If the right side is negative, the assertion holds automatically sincethe left side is a positive real number. If the right side is positive, we can square both sides withoutswitching the inequality. Performing this and gathering terms we have:

which confirms the result. The same type of reasoning shows that the root that subtracts the radicalis always less than unity and never negative.

17. Specifically, we take L=1,D=.1,8=.5. These may be considered reasonable since from see (2-16) and (2-2), they imply – together with "=.3 and F=3 – that the symmetric rate of K growth is6%.

18. These results involve tedious calculations performed in Maple. The worksheet is availableupon request.

19. In using the perfect consumption price index, we are guided by microeconomic theory. That is,taking P as the perfect price index, E/P is the indirect utility function for a moment in time. In this way,real income is related to a welfare measure.

In particular, the perfect price and CES price indices are, respectively:

20. The time-invariant, steady-state Y equals wL+BK, which reduces to:

Y ' L %"F

[L% D1%8

] 'F%"F

L %"D

F(1%8)

PM ' K1

1&F(1%J1&F)1

1&F

Y 'F%"F

L %"D

F(1%8), and L I '

"F

L &D(F&")F(1%8)

LI

Y'

"(1%8)L & D(F&")(F%")(1%8)L % "D

e3 ' 0 <'> b ' b 2&4cF(1%8)

<'> 0'&4cF(1%8) <'> c'0

P reduces to:M

where symmetry of regions and varieties, and the optimal pricing rules are used to derive the secondexpression. Time differentiation of P (taking logs first) and the fact that " is the weight on theM

perfect price index yield the result in the text.

21. The components are:

so rearranging

22. In particular, d8 /dN equals (2L+D) [L(1-N )-DN ]/[L(1+N )+DN ] which is positive as longmir 2 2 2 2 2

as 8 is less than 1.mir

23. The condition is: