Vertical Specialization and theInterdependence of Nations
Arnaud Costinot

Jonathan Vogely

Su Wang

MIT and NBER

Columbia University and NBERMITPRELIMINARY AND INCOMPLETE
December 7, 2010

Abstract
This paper develops an elementary theory of trade with sequential production.
In spite of its extreme simplicity, our theory is consistent with a number of stylized
facts and able to deliver a rich set of predictions regarding how vertical specialization
shapes the interdependence of nations. Among other things, we show that TFP growth
in any country participating in a global supply chain leads all its trading partners to“move up” the chain, and in turn, always increases inequality between countries atthe bottom of the chain. Our results point towards the importance of modelling thesequential nature of production for understanding the consequences of technologicalchanges in developing and developed countries on their trading partners worldwide.

We thank Bob Gibbons and Gordon Hanson. Costinot thanks the Alfred P. Sloan foundation for …nancial
support. Vogel thanks the National Science Foundation (under Grant SES SES-0962261) for research support.
Any opinions, …ndings, and conclusions or recommendations expressed in this paper are those of the authorsand do not necessarily re‡ect the views of the National Science Foundation or any other organization.
yCorresponding author: jvogel@columbia.edu

“One man draws out the wire, another straightens it, a third cuts it, a fourth
points it, a …fth grinds it at the top for receiving the head; to make the headrequires two or three distinct operations; to put it on is a peculiar business; towhiten the pins is another; it is even a trade by itself to put them into the paper;and the important business of making a pin is, in this manner, divided into
about 18 distinct operations, which, in some manufactories, are all performed by
distinct hands, though in some others the same man will sometimes perform twoor three of them,”Adam Smith (1776).

1

Introduction

Most production processes consist of a large number of sequential stages. In this regard the
production of pins in late eighteenth century England is no di¤erent from today’s production of tee-shirts, cars, computers, or semi-conductors. Today, however, production processes
increasingly involve vertical supply chains spanning multiple countries, with each country
specializing in particular stages of a good’s production sequence, a phenomenon which Hummels, Ishii, and Yi (2001) refer to as “vertical specialization.”
This global phenomenon has attracted a lot of attention among policy makers, businessleaders, and trade economists alike. On the academic side of this debate, a large literature has
emerged to investigate how the possibility to fragment production processes across borders
may a¤ect the volume, pattern, and consequences of international trade; see e.g. Feenstra
and Hanson (1996), Yi (2003), and Grossman and Rossi-Hansberg (2008). In this paper,
we propose to take a …rst look at a distinct, but equally important question: Conditionalon production processes being fragmented across borders, how do structural changes in one
country a¤ect other countries located at di¤erent stages of the same supply chain? In otherwords, how does vertical specialization shape the “interdependence of nations?”
Theoretically, deriving sharp and intuitive comparative static predictions in general equilibrium models with an arbitrary number of goods and countries is notoriously di¢ cult.1 In
order to make progress on this question, we therefore start by developing an elementary
theory of trade with sequential production. We consider a world economy with multiple
countries, one factor of production (labor), and one …nal good. Production is sequential andsubject to mistakes, as in Sobel (1992) and Kremer (1993). Production of the …nal goodrequires a continuum of intermediate stages. At each of these stages, production of one unit
1

Ethier (1984) o¤ers a review of theoretical results in high-dimensional trade models.

1

of an intermediate good requires one unit of labor and one unit of the intermediate good
produced in the previous stage. Mistakes occur along the supply chain at a constant Poisson
rate, which is an exogenous technological characteristic of a country. When a mistake occurs
at some stage, the intermediate good is entirely lost. By these stark assumptions, we aim
to capture the more general idea that because of less skilled workers, worse infrastructure,or inferior contractual enforcement, both costly defects and delays in production are more
likely in some countries than others.
In spite of its extreme simplicity, our theory of trade with sequential production is consistent with a number of stylized facts and able to deliver a rich set of predictions regarding how
vertical specialization shapes the interdependence of nations. Among other things, we showthat TFP growth in any country participating in a global supply chain leads all its trading
partners to “move up”the chain, and in turn, always increases inequality between countries
at the bottom of the chain. Our results point towards the importance of modelling thesequential nature of production for understanding the consequences of technological changes
in developing and developed countries on their trading partners worldwide.
Section 3 describes the properties of the free trade equilibrium in our basic environment.
Although our model allows for any …nite number of countries, the unique free trade equilibrium is fully characterized by a simple system of …rst-order non-linear di¤erence equations.
This system can be solved recursively by …rst determining the assignment of countries todi¤erent stages of production and then computing the wages and export prices sustainingthat allocation as an equilibrium outcome. In our model, the free trade equilibrium always
exhibits vertical specialization: countries with higher TFP, which have a lower probability of making mistakes, specialize in later stages of production, where mistakes are morecostly. Compared to standard Ricardian models, absolute rather than relative productivity
di¤erences are a source of comparative advantage among nations.
As mentioned before, our theory of trade with sequential production is consistent with anumber of stylized facts. First, since less productive countries produce and export at earlier
stages of production, our model predicts that poor countries have higher shares of primary
production in value added.2 Second, since rich countries tend to specialize in later stagesof production while poor countries tend to specialize in earlier stages, our model implies
that rich countries tend to trade relatively more with other rich countries (from whom they
import their intermediates and to whom they export their output) while poor countries tend
to trade relatively more with other poor countries, as documented by Hallak (2010). Third,
2

This observation was originally made by Kremer (1993) in the context of his closed-economy model.

2since intermediate goods produced in later stages have higher prices and countries producing
in these stages have higher wages, our model implies that rich countries both tend to import
goods with higher unit values, as documented by Hallak (2006), and to export goods withhigher unit values, as documented by Schott (2004), Hummels and Klenow (2005), andHallak and Schott (2010).3
Section 4 presents two comparative static exercises: population growth and TFP growth.
While interesting in its own right, our …rst comparative static exercise will also help us shedlight on how the sequential nature of production a¤ects the impact of technological changes
around the world. In a standard Ricardian model without sequential production, population
and TFP growth in one country should have the exact same e¤ects on its trading partners.
In our model, they will not. In fact, at the bottom of the chain, population and TFP growthin one country will have the exact opposite e¤ects on its trading partners.
After population growth in one country, market clearing requires this country to perform
more stages and all other countries to perform fewer stages. Hence, countries at the bottomof the chain move further down, whereas countries at the top move further up in response
to foreign population growth. At the bottom of the chain, these changes in the pattern ofvertical specialization are accompanied by declining inequality between nations. By moving
down and performing fewer stages, countries import and export intermediate goods withhigher labor intensities, i.e., goods with higher labor cost shares. Since poor countries arethose with a comparative advantage in these goods, their wages increase relative to the wagesof richer countries. By contrast, population growth has a non-monotonic e¤ect on inequality
between nations at the top of the chain, re‡ecting two con‡icting forces. On the one hand,countries are moving up, which tends to lower the labor cost shares of the intermediate goods
that they trade. On the other hand, countries are performing fewer stages, which tends toraise labor cost shares. Since poorer countries have a comparative advantage in stages withhigher labor intensities, the …rst force increases inequality between nations, whereas thesecond force, like at the bottom of the chain, decreases it.
The consequences of TFP growth are quite di¤erent. In this case, market clearing requires
all countries to perform more stages at the bottom of the chain, but fewer stages at the top.
Hence, all countries move up the supply chain in response to foreign productivity growth.
Changes in the pattern of vertical specialization at the bottom of the chain are, therefore,
3

Following Linder (1961), a large theoretical literature has o¤ered demand-based explanations of such
patterns emphasizing the role of non-homothetic preferences; see e.g. Markusen (1986), Flam and Helpman(1987), Bergstrand (1990), Stokey (1991), Murphy and Shleifer (1997) Matsuyama (2000), Fieler (2008), andmore recently, Fajgelbaum, Grossman, and Helpman (2009). We come back to this literature in Section 3.3.

3the exact opposite of what they were after population growth. Using the same logic as in
our …rst comparative static exercise, we can then show that changes in inequality between
countries at the bottom of the chain are the exact opposite as well. By moving up andperforming more stages, countries import and export intermediate goods with lower labor
intensities. Since these are the goods in which richer countries have a comparative advantage,
inequality between countries increases at the bottom of the chain. Finally, like in the caseof population growth, TFP growth has a non-monotonic e¤ect on inequality at the top of
the supply chain, re‡ecting the fact that these countries are both moving up and performing
fewer stages.
Section 5 discusses how more realistic features of global supply chains may be incorporated within our simple theoretical framework. Our …rst extension allows for the coexistenceof multiple supply chains. This generalization of our model nests the standard Ricardianmodel, when each chain is located in a single country. Our second extension allows forheterogeneity in failure rates across di¤erent stages, thereby recognizing that intermediate
goods may di¤er by more than the order in which they are performed. Put together, our …rst
and second extensions give us a simple way to introduce …nal assembly into our model andto explain why it tends to occur in countries with low TFPs in practice. Our …nal extension
introduces iceberg trade costs, which naturally rationalizes why vertical specialization tends
to be a regional phenomenon in practice (e.g. within NAFTA, the EU, or East Asia).
Our paper is related to several strands of the literature. First, we draw some ideas fromthe literature on hierarchies in closed-economy (and mostly partial-equilibrium) models.
Important early contributions include Lucas (1978), Rosen (1982), Sobel (1992), Kremer(1993), Garicano (2000) and Garicano and Rossi-Hansberg (2006). As in Sobel (1992) andKremer (1993), we focus on an environment in which production is sequential and subject
to mistakes, though we do so in a general equilibrium, open-economy setup. Models ofhierarchies have been applied to the study of international trade issues before, but withvery di¤erent goals in mind. For instance, Antràs, Garicano, and Rossi-Hansberg (2006)
use the knowledge economy model developed by Garicano (2000) to study the matching ofagents with heterogeneous abilities across borders and its consequences for within-country
inequality. Instead, countries are populated by homogeneous workers in our model.4
In terms of techniques, our paper is also related to a growing literature using assignment ormatching models in an international context; see, for example, Grossman and Maggi (2000),
4

Other examples of trade papers using hierearchy models to study within-country inequality include
Kremer and Maskin (2006), Sly (2010), Monte (2010), and Sampson (2010).

4

Grossman (2004), Yeaple (2005), Ohnsorge and Tre‡er (2007), Blanchard and Willmann(2008), Nocke and Yeaple (2008), Costinot (2009), and Costinot and Vogel (2010). Here, likein some of our earlier work, we exploit the fact that the assignment of countries to stagesof production exhibits positive assortative matching— i.e., more productive countries are
assigned to later stages of production— in order to generate strong and intuitive comparative
static predictions in an environment with a large number of goods and countries.
In terms of focus, our paper is motivated by the recent literature documenting the importance of vertical specialization in world trade. On the empirical side, this literature builds
on the in‡uential work of Hummels, Rappoport, and Yi (1998), Hummels, Ishii, and Yi(2001), and Hanson, Mataloni, and Slaughter (2005).5 Our focus on how vertical specialization shapes the interdependence of nations is also related to the work of Kose and Yi (2001,
2006), Burstein, Kurz, and Tesar (2008), and Bergin, Feenstra, and Hanson (2009) who
study how production sharing a¤ects the transmission of shocks at business cycle frequency.
On the theoretical side, the literature on fragmentation is large and diverse. A nonexhaustive list of papers on this topic include Dixit and Grossman (1982), Sanyal (1983),
Jones and Kierzkowski (1990, 2001), Feenstra and Hanson (1996), Arndt (1997), Deardor¤(2001a,b), Egger and Falkinger (2003), Yi (2003, 2010), Kohler (2004), Baldwin and RobertNicoud (2007), Rodríguez-Clare (2010), and Grossman and Rossi-Hansberg (2008, 2010); see
Antràs and Rossi-Hansberg (2009) for a recent overview. Among the previous papers, ourtheoretical framework is most closely related to Dixit and Grossman (1982), Sanyal (1983),
and Yi (2003, 2010) who also consider environments with sequential production. None ofthese papers, however, investigate how structural changes in one country may di¤erentially
impact other countries located at di¤erent stages of the same supply chain. This is the main
focus of our analysis.

2

A Simple Trade Model with Sequential Production

We consider a world economy with multiple countries, indexed by c 2 C
f1; :::; Cg, one
factor of production, labor, and one …nal good. Labor is inelastically supplied and immobile
across countries. Lc and wc denote the endowment of labor and wage in country c, respectively. Production of the …nal good is sequential and subject to mistakes. To produce the…nal good, a continuum of stages s 2 S (0; 1] must be performed. At each stage, producing
one unit of intermediate good requires one unit of the intermediate good produced in the5

See also Johnson and Noregua (2010) for a generalization of Hummels, Ishii, and Yi’s (2001) methodology.

5

previous stage and one unit of labor. For expositional purposes, we assume that “intermediate good 0” is in in…nite supply and has zero price.6 “Intermediate good 1” corresponds tothe unique …nal good mentioned before. Mistakes occur along the supply chain at a constant
Poisson rate, c > 0, which is an exogenous technological characteristic of a country. It
measures total factor productivity (TFP) at each stage of the production process. Whena mistake occurs on a unit of intermediate good at some stage, that intermediate good isentirely lost. Formally, if a …rm from country c combines q(s) units of intermediate good swith q(s)ds units of labor, its output of intermediate good s + ds is given by
q (s + ds) = (1c ds) q (s) .

(1)

For technical reasons, we further assume that if a …rm produces intermediate good s+ds, thenit necessarily produces a positive measure of intermediate goods around that stage.7 Thisimplies that each unit of the …nal good is produced by a …nite, though possibly arbitrarily
large number of …rms. Countries are ordered such that c is strictly decreasing in c. Thus
countries with a higher index c have higher total factor productivity. All markets are perfectlycompetitive and all goods are freely traded. p(s) denotes the world price of intermediate
good s. We use the …nal good as our numeraire, p (1) = 1.

3

Free Trade Equilibrium

3.1

De…nition

In a free trade equilibrium, all …rms maximize their pro…ts taking world prices as given andall markets clear. Pro…t maximization requires that for all c 2 C,
p (s + ds) (1 +p (s + ds) = (1 +c ds) p (s) + wc ds,
00c ds) p (s) + wc ds, if Qc (s ) > 0 for all s 2 (s; s + ds],

6(2)

Alternatively, one could assume that “intermediate good 0” can be produced using labor only. In thissituation, the price of “intermediate good 0” would also be zero since only a measure zero of workers would
be required to perform this measure-zero set of stages. Assuming that “intermediate good 0” is in in…nite
supply allows us to avoid discussions of which country should produce this good. Such considerations are,of course, irrelevant for any of our results.
7
Formally, for any intermediate good s + ds, we assume the existence of s2
s + ds > s1 such that if
q (s + ds) > 0, then q (s0 ) > 0 for all s0 2 (s1 ; s2 ].

6where Qc (s0 ) denotes total output at stage s0 in country c. Condition (2) states that theprice of intermediate good s + ds must be weakly less than its unit cost of production, withequality if intermediate good s + ds is actually produced by a …rm from country c. To see
this, note that the production of one unit of intermediate good s + ds requires 1= (1c ds)units of intermediate good s as well as labor for all intermediate stages in (s; s + ds]. Thus
the unit cost of production of intermediate good s + ds is given by [p (s) + wc ds] = (1c ds)which is equal to (1 + c ds)p (s) + wc ds since ds is in…nitesimal. Good and labor market
clearing further require thatPC

c=1 Qc (s2 )PC

c=1 Qc (s1 )

Z S
0

=

Z s2
s1

PC

c=1

c Qc (s) ds, for all s1

Qc (s) ds = Lc , for all c 2 C,

s2 ,

(3)
(4)

Equation (3) states that the change in the world supply of intermediate goods between
stages s1 and s2 must be equal to the amount of intermediate goods lost due to mistakes inall countries between these two stages. Equation (4) states that the total amount of laborused across all stages must be equal to the total supply of labor in country c. In the rest of
this paper, we formally de…ne a free trade equilibrium as follows.
De…nition 1 A free trade equilibrium corresponds to output levels Qc ( ) : S ! R+ for all
c 2 C, wages wc 2 R+ for all c 2 C, and intermediate good prices p ( ) : S ! R+ such thatconditions (2)-(4) hold.

3.2

Existence and UniquenessWe …rst characterize the pattern of international specialization in any free trade equilibrium.
Lemma 1 In any free trade equilibrium, there exists a sequence of stages S0
0 < S1 <::: < SC = 1 such that for all s 2 S and c 2 C, Qc (s) > 0 if and only if s 2 (Sc 1 ; Sc ].
According to Lemma 1, there is vertical specialization in any free trade equilibriumwith more productive countries producing and exporting at later stages of production. The
intuition behind Lemma 1 can be understood in two ways. One possibility is to note thatsince new intermediate goods require both intermediate goods produced in previous stages
and labor, prices must be increasing along the supply chain. Thus intermediate goods
produced at later stages are less labor intensive, which makes them relatively cheaper to7

produce in countries with higher wages. In our model these are the countries with anabsolute advantage in all goods. Alternatively, one can look at Lemma 1 through the lensof the hierarchy literature; see e.g. Lucas (1978), Rosen (1982), and Garicano (2000). Since
countries that are producing at later stages can leverage their productivity on larger amounts
of inputs, e¢ ciency requires countries to be more productive at the top. A result similar
to Lemma 1 in an environment with a discrete number of stages can also be found in Sobel(1992) and Kremer (1993).
We refer to the vector (S1 ; :::; SC ) as the “pattern of vertical specialization” and denote
by Qc Qc (Sc ) the total amount of intermediate good Sc produced and exported by countryc. Using the previous notation, the pattern of vertical specialization and export levels canbe jointly characterized as follows.
Lemma 2 In any free trade equilibrium, the pattern of vertical specialization and export
levels satisfy the following system of …rst-order non-linear di¤erence equations:1

S c = Sc 1

ln 1

c

Qc = e

c (Sc

Sc

1)

c Lc

Qc 1

, for all c 2 C,

Qc 1 , for all c 2 C,

(5)
(6)

with boundary conditions S0 = 0 and SC = 1.
Lemma 2 derives from the goods and labor market clearing conditions (3) and (4). Equation (5) re‡ects the fact that the exogenous supply of labor in country c must be equal tothe amount of labor demanded to perform all stages from Sc 1 to Sc . This amount of labor
depends both on the rate of mistakes, c , as well as the total amount Qc 1 of intermediate
good Sc 1 imported from country c 1. Equation (6) re‡ects the fact that intermediate
goods get lost at a constant rate at each stage when produced in country c.
In the rest of this paper, we refer to the vector of wages (w1 ; :::; wC ) as the “world income
distribution” and to pc p (Sc ) as the price of country c’s exports (which is also the priceof country c + 1’s imports under free trade). Let Nc
Sc Sc 1 denote the measure ofstages performed by country c within the supply chain. In the next lemma, we show thatthe measures of stages being performed in all countries (N1 ; :::; NC ) entirely summarize howchanges in the pattern of vertical specialization a¤ect the world income distribution.
Lemma 3 In any free trade equilibrium, the world income distribution and export prices

8satisfy the following system of …rst-order linear di¤erence equations:
wc+1 = wc + ( cc+1 ) pc , for all c < C,

p c = e c Nc p c 1 + e c Nc

1 (wc = c ) , for all c 2 C,

(7)
(8)

with boundary conditions p0 = 0 and pC = 1.
Lemma 3 derives from the zero-pro…t condition (2). Equation (7) re‡ects the fact thatfor the “cut-o¤” good, Sc , the unit cost of production in country c, (1 + c ds) pc + wc ds,
must be equal to the unit cost of production in country c + 1, (1 + c+1 ds) pc + wc+1 ds.
Equation (8) directly derives from the zero-pro…t condition (2) and the de…nition of Nc andpc . It illustrates the fact that the price of the last intermediate good produced by countryc depends on the price of the intermediate good imported from country c 1 as well as thetotal labor cost in country c.
Combining Lemmas 1-3, we can establish the existence of a unique free trade equilibrium
and characterize its main properties.
Proposition 1 There exists a unique free trade equilibrium. In this equilibrium, the patternof vertical specialization and export levels are given by equations (5) and (6), and the world
income distribution and export prices are given by equations (7) and (8).
The proof of Proposition 1 formally proceeds in two steps. First, we use Lemma 2
to construct the unique pattern of vertical specialization and vector of export levels. In
equations (5) and (6), we have one degree of freedom, Q0 , which corresponds to total input
used at the initial stage of production and can be set to satisfy the …nal boundary condition
SC = 1. Once (S1 ; :::; SC ) and (Q0 ; :::; QC 1 ) have been determined, all other output levels
can be computed using equation (1) and Lemma 1. Second, we use Lemma 3 together withthe equilibrium measure of stages computed before, (N1 ; :::; NC ), to characterize the uniqueworld income distribution and vector of export prices. In equations (7) and (8), we still have
one degree of freedom, w1 , which can be used to satisfy the other …nal boundary condition,
pC = 1. Finally, once (w1 ; :::; wC ) and (p1 ; :::; pC ) have been determined, all other prices canbe computed using the zero-pro…t condition (2) and Lemma 1.

3.3

Discussion

In spite of its extreme simplicity, our theory of trade with sequential production delivers arich set of predictions. First, since less productive countries produce and export at earlier
9stages of production, our model predicts that poor countries have higher shares of primary
production in value added. Second, since rich countries tend to specialize in later stagesof production while poor countries tend to specialize in earlier stages, our model implies
that rich countries tend to trade relatively more with other rich countries (from whom they
import their intermediates and to whom they export their output) while poor countries tend
to trade relatively more with other poor countries, as documented by Hallak (2010). Third,
since intermediate goods produced in later stages have higher prices and countries producing
in these stages have higher wages, our model implies that rich countries both tend to import
goods with higher unit values, as documented by Hallak (2006), and to export goods withhigher unit values, as documented by Schott (2004), Hummels and Klenow (2005), andHallak and Schott (2010).
Following Linder (1961), the last two previous stylized facts have traditionally been rationalized using non-homothetic preferences; see e.g. Markusen (1986), Flam and Helpman(1987), Bergstrand (1990), Stokey (1991), Murphy and Shleifer (1997) Matsuyama (2000),
Fieler (2008), and Fajgelbaum, Grossman, and Helpman (2009). The common starting pointof the previous papers is that rich countries’ preferences are skewed towards high quality
goods, so they tend to import goods with higher unit values. Under the assumption thatrich countries are also relatively better at producing high quality goods, these models canfurther explain why rich countries tend to export goods with higher unit values and why
countries with similar levels of GDP per capita tend to trade more with each other.8
The complementary explanation o¤ered by our elementary theory of trade with sequential
production is very di¤erent. According to our model, countries with similar per-capita
incomes are more likely to trade with one another because they specialize in nearby regionsof the same supply chain. Similarly, countries with higher levels of GDP per capita tend tohave higher unit values of imports and exports because they specialize in higher stages inthe supply chain, for which inputs and outputs are more costly.
8

In Fajgelbaum, Grossman, and Helpman (2009), such predictions are obtained in the absence of any
exogenous relative productivity di¤erences. In their model, a higher relative demand for high-quality goods
translates into a higher relative supply of these goods through a “home-market” e¤ect.

10

4Comparative Statics

4.1

De…nitions

Before turning to our two main comparative static exercises, it is useful to introduce someformal de…nitions describing the changes in the pattern of vertical specialization and theworld income distribution in which we will be interested.
De…nition 2 Let (S10 ; :::; SC0 ) denote the pattern of vertical specialization in a counterfactual
free trade equilibrium. A country c 2 C is moving up (resp. down) the supply chain relative tothe initial free trade equilibrium if Sc0 Sc and Sc0 1 Sc 1 (resp. Sc0 Sc and Sc0 1 Sc 1 ).
Formally, a country is moving up or down the supply chain if we can rank the set of stages
that it performs in the initial and counterfactual free trade equilibria in terms of the strong
set order.9 This simple mathematical notion will allow us to formalize a major concern ofpolicy makers and business leaders in developed countries, namely the fact that China andother developing countries are “moving up the value chain”; see e.g. OECD (2007).
De…nition 3 Let (w10 ; :::; wC0 ) denote the world income distribution in a counterfactual free
trade equilibrium. Inequality is increasing (resp. decreasing) among a given group fc1 ; :::cn g00of adjacent countries if wc+1
=wc0 wc+1 =wc (resp. wc+1
=wc0 wc+1 =wc ) for all c1 c cn .
Formally, inequality is rising or decreasing within a given group of adjacent countries
if we can rank the vector of wages in the initial and counterfactual free trade equilibria interms of the monotone likelihood ratio property. Since wages correspond to GDP per capita
in our model, this property o¤ers a simple way to conceptualize changes in the world income
distribution in an economy with multiple countries.

4.2

Population Growth

In this section we study the impact of an increase in the total endowment of labor, Lc0 , of agiven country c0 , which we refer to as “population growth.”We proceed in two steps. First,
we use Lemma 2 to characterize the changes in the pattern of vertical specialization. Second,
we turn to Lemma 3 to characterize the associated changes in the world income distribution.
Our …rst comparative static result can be stated as follows.
9

Of course, a given country may be neither moving up nor down after a change.

11

Proposition 2 Population growth in country c0 increases the measure of stages performedin this country and decreases the measure of stages performed in any other country. In turn,all countries c < c0 move down the supply chain and all countries c > c0 move up.
The broad intuition behind Proposition 2 is simple. An increase in the supply of laborin one country tends to raise total output at all stages of production. Since labor supply
must remain equal to labor demand, this increase in output levels must be accompanied bya decrease in the measure of stages Nc performed in country c 6= c0 . Proceeding by iterationfrom the bottom and the top of the supply chain, we can then show that this change in
Nc can only occur if all countries below c0 move down and all countries above c0 move up.
Finally, since the total measure of stages must remain constant, the measure of stages Nc0
performed in country c0 must increase.
Let us now use the previous result to analyze the consequences of population growth on
the world income distribution.
Proposition 3 Population growth in country c0 decreases inequality among countries c 2
f1; :::; c0 1g, increases inequality among countries c 2 fc0 ; :::; c1 1g, and decreases inequality among countries c 2 fc1 ; :::; C 1g, with c1 2 fc0 + 1; :::; Cg.
The logic of Proposition 3 can be sketched as follows. From Lemma 3, we know that
relative wages satisfy
wc+1
c
c+1
=1+
, for all c < C.
(9)wc
(wc =pc )Thus, wc+1 =wc is decreasing in the labor intensity, wc =pc , of country c’s export. From
Proposition 2, we also know that countries at the bottom of the chain: (i) are moving down
into lower stages, which tend to have lower export prices; and (ii) are performing fewerstages, which tends to reduce export prices (for a given price of imported inputs). Bothe¤ects tend to reduce the price of intermediate goods traded in that region of the supply
chain, and in turn, to increase their labor intensity. This explains why inequality between
nations declines at the bottom of the chain. The non-monotonic e¤ects on inequality at
the top of the chain re‡ect two con‡icting forces. On the one hand, countries are moving
up, which tends to increase the price of intermediate goods traded in that region of the
supply chain, and in turn, to decrease their labor intensity. On the other hand, countries are
performing fewer stages, which tends to reduce the price of their exports and increase theirlabor intensity.1010Note that since c1 2 fc0 + 1; :::; Cg, the third group of countries, fc1 ; :::; C 1g, is non-empty if c1 < C,
but empty if c1 = C. In our simulations, we have encountered both situations.

12

The core mechanism behind Proposition 2 is reminiscent of the mechanism underlying
terms-of-trade e¤ects in a standard Ricardian model; see e.g. Dornbusch, Fischer, andSamuelson (1977) and Krugman (1986). From an economic standpoint, equation (9) captures
the basic idea that the wage of country c+1 should decrease relative to the wage of country c
if and only if it moves into sectors in which it has a comparative disadvantage. In our model,
since country c + 1 has a higher wage, these are the sectors with higher labor intensities. In
a standard Ricardian model, this would be the sectors in which country c + 1 is relatively
less productive instead.
There is, however, one important di¤erence between our elementary theory of tradewith sequential production and a standard Ricardian model. In our model, the pattern ofcomparative advantage depends on endogenous di¤erences in labor intensity across stages.
In a standard Ricardian model, the same pattern only depends on exogenous productivity
di¤erences. This subtle distinction explains why our simple model is able to predict nonmonotonic changes in inequality among the richest countries, whereas a standard Ricardianmodel would not.11

4.3

TFP Growth

We now turn our attention to the consequences of a decrease in the failure rate c0 of agiven country c0 , which we refer to as “TFP growth.”For simplicity we restrict ourselves toa small change in c0 , in the sense that it does not a¤ect the ranking of countries in termsof failure rates. As in the previous section, we start by describing changes in the pattern ofvertical specialization.
Proposition 4 TFP growth in country c0 increases the measure of stages performed in all
countries c < c0 and decreases the measure of stages performed in all countries c > c0 . In
turn, all countries c 6= c0 move up the supply chain.
According to Proposition 4, all countries move up the supply chain in response to productivity growth in one country. Note that the consequences of TFP growth in country c0
are the same as the consequences of population growth at the top of the chain, but the exact
11

To see this, consider the closed economy model of Costinot and Vogel (2010). By relabelling “workers”
as countries, this model can be reinterpreted as a standard Ricardian model with a large number of countries
and goods, but no sequential production. The pattern of international specialization in this model also
consists of a ladder of countries, with “high skill” countries specializing in the “high skill” goods. In thisenvironment, unlike in the present model, population growth in one country can decrease inequality only
among low skill countries and increase inequality only among high skill countries; see pages 758-759.

13

opposite at the bottom. To understand this result, consider …rst countries located at the
top of the chain. Since total output of the …nal good must rise in response to an increase inproductivity in country c0 , countries at the top of the chain must perform fewer stages forlabor markets to clear. By a simple iterative argument, these countries must therefore movefurther up the supply chain, just like in Proposition 2.
The logic behind our results at the bottom of the chain is more subtle. The broad intuition
behind the opposite e¤ects of population and TFP growth can be understood as follows.
Holding the pattern of vertical specialization …xed, population growth in country c0 increases
the total labor supply of countries c c0 , but leaves their labor demand unchanged. Thus
labor market clearing requires countries at the bottom of the chain to reduce the number ofstages they perform, to move down the chain, and to increase their output, thereby o¤setting
the excess labor supply at the top. By contrast, TFP growth in country c0 increases thetotal labor demand of countries c c0 (since country c0 now produces more output at eachstage), but leaves their labor supply unchanged. As a result, countries at the bottom of thechain now need to increase the number of stages they perform, to move up the chain, andto reduce their output in order to o¤set the excess labor demand at the top.12
In sharp contrast to a standard Ricardian model without sequential production, ourmodel therefore predicts that population and TFP growth may have opposite e¤ects on thepattern of international specialization. Building solely on the idea that labor markets mustclear both before and after a given TFP shock, Proposition 4 illustrates how, through thefragmentation of the production process across borders, productivity growth in one country
may lead all its trading partners to move up the supply chain, even in the absence of TFP
growth in any of these countries.
The previous results have important implications for the world income distribution, which
we now describe.
Proposition 5 TFP growth in country c0 increases inequality among countries c 2 f1; :::; c0 1g,
decreases inequality among countries c 2 fc0 ; :::; c1 1g, increases inequality among countries c 2 fc1 ; :::; c2 1g, and decreases inequality among countries c 2 fc2 ; :::; C 1g, withc1 c2 and c1 ; c2 2 fc0 ; :::; Cg.
12

It is worth pointing out that while technological change in c0 has unambiguous e¤ects on Nc for allc 6= c0 , the e¤ect on Nc0 is ambiguous. According to Proposition 4, technological change in the country at
the bottom (top) of the chain necessarily increases (decreases) the measure of stages it performs. Thus Nc0
may either increase or decrease.

14

Like Section 4.2, changes in the pattern of vertical specialization naturally translate
into changes in inequality between countries. At the top of the chain, the consequences of
TFP growth for inequality are the same as the consequences of population growth, echoing
the results of Propositions 2 and 4. The non-monotonicity— with inequality rising among
countries c 2 fc1 ; :::; c2 1g and decreasing among countries c 2 fc2 ; :::; C 1g— arises fromthe same two con‡icting forces: countries move up the chain but produce fewer stages.
At the bottom and in the middle of the chain, by contrast, the consequences of TFP
growth for inequality are di¤erent from those of population growth. In the middle of thechain, inequality may decrease among countries c 2 fc0 ; :::; c1 1g because of the directe¤ect of a reduction in c0 , which tends to decrease inequality between c0 and c0 + 1, as
seen in equation (9). This force was absent from our previous comparative static exercise
since labor endowments did not directly a¤ect zero pro…t conditions. At the bottom of thechain, c < c0 , our simple model again leads to sharp predictions regarding the implications ofstructural changes on inequality between countries. The logic is the same as in the previous
section. From Proposition 4, we know that low TFP countries are both moving up the supply
chain and performing more stages. Both e¤ects tend to increase the price of the intermediate
goods that they trade, and in turn, to decrease their labor intensity. Since richer countries
are those with a comparative advantage in these goods, TFP growth in one country raises
inequality between nations at the bottom of the chain. Though poor countries are moving
up the supply chain— an observation which is often taken as “good news” among policy
makers— they end up relatively worse o¤.

5Extensions

Our elementary theory of trade with sequential production is special along several dimensions. A natural concern is the robustness of our main results to various modi…cations ofsome of our key assumptions. To address this issue we next present a number of extensionsof our basic framework that incorporate more realistic features of global supply chains.

5.1

Sequential and Simultaneous Production

Most production processes are neither purely sequential, as assumed in this paper, norpurely simultaneous, as assumed in most of the existing literature. Producing an aircraft,for example, requires multiple parts. Many of these parts, e.g., engine, seats, windows, are15

produced simultaneously before being assembled, but each of these parts also require a large
number of sequential stages, e.g. extraction of raw materials, re…ning, and manufacturing.13
With that in mind, we turn to a generalization of our original model that allows for thecoexistence of multiple supply chains, indexed by n 2 N
f1; :::; N g. Beside its added
realism, one bene…t of this extension is that it also provides a strict generalization of astandard Ricardian model model with simultaneous production.
Each supply chain is as described in Section 2, except for the fact that the amount oflabor Lnc , allocated to chain n in country c is now endogenous. We denote by nc > 0 theexogenous rate at which country c makes mistakes if participating in supply chain n.
[TO BE COMPLETED]

5.2

Heterogeneous Stages of Production

In order to focus attention in the simplest possible way on the novel aspects of an environment
with sequential production, we have assumed that stages of production only di¤er in one
dimension: the order in which they are performed. In practice, stages of production often alsodi¤er greatly in terms of factor intensity, with some stages being much more skill-intensive
than others. To capture such considerations within our simple theoretical framework, we
now allow failure rates to be an exogenous characteristic of both a stage and a country.
Among other things, this extension will give us a simple way to rationalize why assembly, a
low-skill part of the production process, tends to occur in poor countries in practice.
Formally, we assume that mistakes occur at a Poisson rate, c (s)(s) c , with (s)the “skill-intensity”of stage s. We come back to this terminology in a moment.
[TO BE COMPLETED]

5.3

Trade Costs

An important insight of the recent trade literature is that changes in trade costs a¤ect thepattern and consequences of international trade not only by a¤ecting …nal goods trade, butalso by a¤ecting the extent of production fragmentation across borders; see e.g. Feenstra
and Hanson (1996), Yi (2003), and Grossman and Rossi-Hansberg (2008). We now discuss
how the introduction of iceberg trade costs in our simple environment would a¤ect thegeographic structure of global supply chains, and in turn, the interdependence of nations.
[TO BE COMPLETED]13

Of course, “manufacturing” itself requires a large number of sequential stages.

16

6Concluding Remarks

In this paper, we have developed an elementary theory of trade with sequential production.
In spite of its extreme simplicity, the predictions of our model are consistent with a numberof stylized facts: (i) poor countries have higher shares of primary production in value added;(ii) rich countries tend to trade relatively more with other rich countries, while poor countries
tend to trade relatively more with other poor countries; and (iii) rich countries both importand export goods with higher unit values. In our theory, none of these predictions rely on
non-homothetic preferences, as commonly assumed in the existing literature. Instead theycapture the basic idea that countries with di¤erent levels of GDP per capita tend to operate
in di¤erent regions of the same supply chain.
Using this elementary theory, we have taken a …rst step towards analyzing how verticalspecialization shapes the interdependence of nations. Among other things, we have shownthat TFP growth in any country participating in a global supply chain leads all its trading
partners to “move up”the chain, and in turn, always increases inequality between countries
at the bottom of the chain. These results point towards the importance of modelling thesequential nature of production for understanding the consequences of technological changes
in developing and developed countries on their trading partners worldwide.
Finally, while we have emphasized the consequences of vertical specialization for the interdependence of nations, we believe that our general results also have useful applications
outside of international trade. Sequential production process are pervasive in practice. They
may involve workers of di¤erent skills, as emphasized in the labor and organizations literature. They may also involve …rms of di¤erent productivity, as in the industrial organization
literature. Whatever the particular context may be, our theoretical analysis may help shed a
new light on how vertical specialization shapes the interdependence between di¤erent actorsof a given supply chain.

17

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22

A

Proofs (I): Free Trade Equilibrium

Proof of Lemma 1. We proceed in four steps.
Step 1: p ( ) is continuous.
Consider a stage s0 2 (0; 1]. By equations (3) and (4), we know that there must be at least
one country, call it c0 , producing intermediate good s0 . By assumption, this country must also be
producing all intermediate goods s 2 (s0 ds; s0 ]. Thus condition (2) implies
p (s0 ) = (1 +c0 ds) p (s0ds) + wc0 ds,

Taking the limit of the previous expression as ds goes to zero, we get
p (s0 ) = lim p (s) .

(10)

s!s0

This already establishes the continuity of p ( ) at 1. Now consider s0 2 (0; 1). By the same logic,
there must be at least one country, call it c0 again, producing all intermediate goods s 2 (s0 ; s0 +ds].
Condition (2) therefore also implies
p (s0 + ds) = (1 +c0 ds) p (s0 ) + wc0 ds,

Taking the limit of the previous expression as ds goes to zero, we then get
lim p (s) = p (s0 ) .

(11)

s!s+
0

The continuity of p ( ) at s0 2 (0; 1) directly derives from equations (10) and (11).
Step 2: p ( ) is strictly increasing.
We proceed by contradiction. Suppose that there exists a pair of stages, s1 and s2 , such thats1 < s2 and p (s1 ) p (s2 ). Since p ( ) is continuous, there must also exist a stage s0 2 (s1 ; s2 ] andan " > 0 such that p (s) p (s0 ) for all s 2 (s0 "; s0 ]. As in Step 1, we know that there must be
at least one country, call it c0 , producing intermediate good s0 . This, in turn, requiresp (s0 ) = (1 +c0 ds) p (s0ds) + wc0 ds > p (s0For ds small enough, the previous inequality contradicts p (s)ds) .

p (s0 ) for all s 2 (s0"; s0 ].

Step 3: If c2 > c1 , then wc2 > wc1 .
We proceed by contradiction. Suppose that there exist two countries, c2 > c1 , such thatwc2 wc1 . In a free trade equilibrium, equations (3) and (4) require country c1 to produce at least
one intermediate good in (0; 1), call it s1 . By equation (1), this country must also be producing all23

intermediate goods s 2 (s1

Since

c2 <

ds; s1 ]. Thus condition (2) implies
p (s1 ) = (1 +c1 ds) p (s1ds) + wc1 ds,(12)

p (s1 )

c2 ds) p (s1ds) + wc2 ds,

(13)

(1 +

c1 , equation (12) and inequality (13) imply wc2 > wc1 , which contradicts wc2

wc1 .

Step 4: If c2 > c1 and Qc1 (s1 ) > 0, then Qc2 (s) = 0 for all s < s1 .
We proceed by contradiction. Suppose that there exist two countries, c2 > c1 , and two intermediate goods, s1 > s2 > 0, such that c1 produces s1 and c2 produces s2 . By equation (1), c1 produces
all intermediate goods s 2 (s1 ds; s1 ], whereas c2 produces all intermediate goods s 2 (s2 ds; s2 ].
Thus condition (2) implies
p (s1 ) = (1 +c1 ds) p (s1ds) + wc1 ds,

p (s2 ) = (1 +c2 ds) p (s2ds) + wc2 ds,

p (s1 )

(1 +c2 ds) p (s1ds) + wc2 ds,

p (s2 )

(1 +c1 ds) p (s2ds) + wc1 ds.

Combining the four previous expressions, we get[(1 +c2 ds) p (s1ds) + wc2 ds] [(1 +c1 ds) p (s2ds) + wc1 ds][(1 +c1 ds) p (s1ds) + wc1 ds] [(1 +c2 ds) p (s2ds) + wc2 ds] ,which can be rearranged as

By Step 2, we know that p (s1(1 +c2 ds) [p (s1

ds)

p (s2

ds)] wc1(1 +c1 ds) [p (s1

ds)

p (s2

ds)] wc2

ds)

p (s2

(1 +

c2 ds) wc1ds) > 0. Thus the previous inequality implies(1 +c1 ds) wc2 .

(14)

Since c2 < c1 , inequality (14) implies wc1 > wc2 , which contradicts Step 3.
To conclude the proof of Lemma 1, let us de…ne Sc sup fs 2 SjQc (s) > 0g for all c 2 C. By
equation Step 4, we must have S0 0 < S1 < ::: < SC = 1, and for all s 2 S and c 2 C, Qc (s) > 0
if Sc 1 < s < Sc and Qc (s) = 0 if s < Sc 1 or s > Sc . Since Qc (s) > 0 requires Qc (s0 ) > 0 for all
s0 2 (s ds; s], we must also have Qc (Sc ) > 0 and Qc (Sc 1 ) = 0 for all c 2 C. Thus Qc (s) > 0 if
and only if s 2 (Sc 1 ; Sc ]. Finally, by equations (3) and (4), country C must produce stage 1, sothat SC = 1. QED.

24

Proof of Lemma 2. We …rst consider equation (6). Lemma 1 and equation (3) imply
Qc (s2 )Qc (s1 ) =

c

Z s2

Qc (s) ds, for all s1 ; s2 2 (Sc 1 ; Sc ].

s1

(15)

Taking the derivative of the previous expression with respect to s2 , we get
dQc (s) =ds =c Qc (s) , for all s 2 (Sc 1 ; Sc ].

The solution of the previous di¤erential equation must satisfy
Qc (Sc ) = e

Sc

c (Sc

1)

lim Qc (s) .

(16)

s!Sc+ 1

Lemma 1 and equation (3) also imply
Qc (Sc 1 + ds)

Qc 1 (Sc 1

"

ds) =

c

lim Qc (s) +s!Sc+ 1c 1 Qc 1 (Sc 1#ds) ds.

Taking the limit of the previous expression as ds goes to zero, we get
lim Qc (s) = lim Qc 1 (s) = Qc 1 (Sc 1 ) .

s!Sc+ 1

s!Sc

(17)

1

Equation (6) derives from equations (16) and (17) and the de…nition of Qc Qc (Sc ).
Let us now turn to equation (5). By Lemma 1 and equation (4), we know that
Z Sc
Sc

1

Qc (s) ds = Lc , for all c 2 C.

(18)

By equations (15) and (17), we also know that
Z Sc
Sc

Qc (s) ds =1

[Qc 1 (Sc 1 )Qc (Sc )] .

(19)

Qc (Sc )] , for all c 2 C.

(20)

c

1

Equations (18) and (19) imply
Lc =

1
c

[Qc 1 (Sc 1 )Equation (5) derives from equations (6) and (20) and the de…nition of Qc Qc (Sc ). The boundary
conditions S0 = 0 and SC = 1 have already been established in the proof of Lemma 1. QED.

25

Proof of Lemma 3. We …rst consider equation (7). Lemma 1 and condition (2) imply
p (Sc + ds)(1 +c+1 ds) p (Sc )p (Sc )

(1 +c ds) p (Scwc+1 ds

ds)

wc ds

p (Sc + ds)(1 +p (Sc )

c+1 ds) p (Sc(1 +c ds) p (Sc )wc ds,

ds)

wc+1 ds.

After simpli…cations, the two previous inequalities can be rearranged as( cc+1 ) p (Sc )

wc+1

wc

( c

c+1 ) p (Scds) .

Since p is continuous, taking the limit of the above chain of inequalities as ds goes to zero we get
wc+1

wc = ( cc+1 ) p (Sc ) , for all s 2 (Sc 1 ; Sc ].

which is equivalent to equation (7) by the de…nition of pc p (Sc ).
Let us now turn to equation (8). Lemma 1 and condition (2) implyp (s + ds) = (1 +c ds) p (s) + wc ds

Taking the limit of the previous expression as ds goes to zero, we get
dp (s) =ds =c p (s) + wc , for all s 2 (Sc 1 ; Sc ].

The solution of the previous di¤erential equation must satisfyp (Sc ) = ec (Sc Sc1)

lim p (Sc 1 ) +s!Sc+ 1"e c (Sc Sc 1 )c1#(wc = c ) ,which is equivalent to equation (8) by the continuity of p ( ) and the de…nitions of Nc Sc Sc 1
and pcp (Sc ). The boundary conditions derive from the fact that p0 = p (S0 ) = p(0) = 0 and
pC = p (SC ) = p (1) = 1. QED.
Proof of Proposition 1. We proceed in four steps.
Step 1: (S0 ; :::; SC ) and (Q0 ; :::; QC ) satisfy equations (5) and (6) if and only if
S c = S0 +Qc = Q0

Pc

1

c0 =1

Pc

c0 =1

ln

c0

"

Q0
Q0

Pc0 1

c00 =1c0
c00 =1

P

c00

Lc00

c00 Lc00

c0 Lc0 , for all c 2 C.

#

, for all c 2 C,

(21)
(22)

Let us …rst show that if (S0 ; :::; SC ) and (Q0 ; :::; QC ) satisfy equations (5) and (6), then they satisfy
equations (21) and (22). Consider equation (22). Equations (5) and (6) imply
Qc = Qc 1

c Lc , for all c 2 C,26

By iteration we therefore have
Qc = Q0Pc

c0 =1

c0 Lc0 , for all c 2 C.

Now consider equation (21). Starting from equation (5) and iterating we get
Pc

Sc = S 0

c0 =1

1

ln 1

c0

c0 Lc0
Qc0 1, for all c 2 C.

Equation (21) directly derives from the previous expression and equation (22). It is a matter ofsimple algebra to check that if (S0 ; :::; SC ) and (Q0 ; :::; QC ) satisfy equations (21) and (22), thenthey satisfy equations (5) and (6).
Step 2: There exists a unique pair of vectors (S0 ; :::; SC ) and (Q0 ; :::; QC ) satisfying equations(5) and (6) and the boundary conditions: S0 = 0 and SC = 1.
PC
Q0 , then there does not exist a pair of vecLet Q0c=1 c Lc . By Step 1, if Q0
tors (S0 ; :::; SC ) and (Q0 ; :::; QC ) that satisfy equations (5) and (6). Otherwise (Q0 ; :::; QC ) and(S0 ; :::; SC ) would also satisfy equations (21) and (22), which cannot be the case if Q0 Q0 . Now
consider Q0 > Q0 . From equation (21), it is easy to check that @SC =@Q0 < 0 for all Q0 > Q0 ;limQ0 !Q+ SC = +1; and limQ0 !+1 SC = S0 . Thus conditional on having set S0 = 0, there ex0
ists a unique Q0 > Q0 such that (S0 ; :::; SC ) and (Q0 ; :::; QC ) satisfy equations (21) and (22) and
SC = S. Step 2 derives from Step 1 and the previous observation.
Step 3: For any (N1 ; :::; NC ), there exists a unique pair of vectors (w1 ; :::; wC ) and (p0 ; :::; pC )satisfying equations (7) and (8) and the boundary conditions: p0 = 0 and pC = 1.
For any (N1 ; :::; NC ), w1 , and p0 , there trivially exists a unique pair of vectors (w2 ; :::; wC ) and(p1 ; :::; pC ) that satisfy equations (7) and (8). Thus taking (N1 ; :::; NC ) as given and having set
p0 = 0, we only need to check that there exists a unique w1 such that pC = 1. To do so, we …rst
establish that pC is strictly increasing in w1 . We proceed by iteration. By equation (8), we knowthat p1 is strictly increasing in w1 . Thus by equation (7), w2 must be strictly increasing in w1 as
well. Now suppose that pc 1 and wc are strictly increasing in w1 for c < C. Then pc must be strictlyincreasing in w1 , by equation (8), wc+1 must be strictly increasing in w1 , by equation (7). At thispoint we have established, by iteration, that pC 1 and wC are strictly increasing in w1 . Combining
this observation with equation (8), we obtain that pC is strictly increasing in w1 . To conclude, let
us note that, by equations (7) and (8), we also have limw1 !0 pC = 0 and limw1 !+1 pC = +1.
Since pC is strictly increasing in w1 , there therefore exists a unique w1 such that pC = 1.
Steps 1-3 imply the existence and uniqueness of (S0 ; :::; SC ), (Q0 ; :::; QC ), (w1 ; :::; wC ), and(p0 ; :::; pC ) that satisfy equations (5)-(8) with boundary conditions S0 = 0, SC = 1, p0 = 0, and
pC = 1. Now consider the following output levels and intermediate good prices
Qc (s) = e

c (s

Sc

1)

Qc 1 , for all s 2 (Sc 1 ; Sc ],hip (s) = e c (s Sc 1 ) pc 1 + e c (s Sc 1 ) 1 (wc = c ) for all s 2 (Sc 1 ; Sc ].

27

By construction, [Q1 ( ) ; :::; QC ( )], (w1 ; :::; wC ), and p ( ) satisfy conditions (2)-(4). Thus a free
trade equilibrium exists. Since (S0 ; :::; SC ), (Q0 ; :::; QC ), (w1 ; :::; wC ), and (p0 ; :::; pC ) are unique,the free trade equilibrium is unique as well by Lemmas 1-3. QED.

B

Proofs (II): Population Growth

Proof of Proposition 2. Throughout this proof, we use the two following relationships
c Lc

Qc 1 =

1

e
L
c ce
1 e

Qc =

c Nc

, for all c 2 C

(23)

c Nc
c Nc

, for all c 2 C,

(24)

Equation (23) derives from equation (5) and the de…nition of Nc
Sc Sc 1 . Equation further
uses equation (6). For future reference, note that @Qc 1 =@Nc < 0 and @Qc =@Nc < 0, whereas@Qc 1 =@Lc > 0 and @Qc =@Lc > 0. The rest of our proof proceeds in six steps.
Step 1: If L0c0 > Lc0 and c0 6= C, then NC0 < NC .

P
0
We …rst establish that Q0C > QC . By Lemma 1, we know that Q0C Q0C (1) = C
c=1 Qc (1). By
the First Welfare Theorem, we also know that the allocation in a free trade equilibrium is Pareto
optimal. Thus Q0C must be the maximum output level of the …nal good attainable given the new
resource and technological constraints, i.e.,
Q0C =subject toPCe

c=1 Qc (s2 )PC

arg maxe 1 ( );:::;QeC ( )Qs1

e c (s) ds
Qe

c=1 Qc (1) ,R s2 PCec=1 Qc (s1 )
Z S
0

PC

c=1

e

c Qc (s) ds, for all s1s2 ,L0c , for all c 2 C,

(25)
(26)

e 1 ( ) ; :::; Qe C ( ) such that
where L0c0 > Lc0 and L0c = Lc for all c 6= c0 . Now consider Qande c (s)Q0

Qc0 (s) +e c (s)
Q

1

c0 e

c0 s

e

c0 S

L0c0

Lc0 , for all s 2 S,

Qc (s) , for all s 2 S and c 6= c0 .

Since Q1 ( ) ; :::; QC ( ) satis…es the initial resource and technological constraints, as described bye 1 ( ) ; :::; Qe C ( ) must satisfy, by construction, the new resource and techconditions (3) and (4), Q
nological constraints, as described by conditions (25) and (26). Since L0c0 > Lc0 , we must also

28

have
e C (1) =e c (1) + Q
Q
0

1

c0 e

c0 S

e

c0 S

L0c0

Lc0 + QC > QC .

e C (1), the previous inequality implies Q0 > QC . Combining this observatione c (1)+ Q
Since Q0C Q
0Cwith equation (24) and the fact that c0 6= C, which implies L0C = LC , we get NC0 < NC .

Step 2: If L0c0 > Lc0 , then Nc0 < Nc for all c > c0 .

We proceed by iteration. By Step 1, we know that if L0c0 > Lc0 and c0 6= C, then NC0 < NC .
Let us now show that if Nc0 < Nc and c > c0 + 1, then Nc0 1 < Nc 1 . Since c > c0 + 1, we knowthat L0c = Lc . Thus Nc0 < Nc and equation (23) imply Q0c 1 > Qc 1 . Since c > c0 + 1, we also knowthat L0c 1 = Lc 1 . Thus Q0c 1 > Qc 1 and equation (24) imply Nc0 1 < Nc 1 . This completes theproof of Step 2.
Step 3: If L0c0 > Lc0 and c0 6= 1, then N10 < N1 .
We proceed by contradiction. Suppose that N10
N1 . Since c0 6= 1, we know that L01 = L1 .
Thus N10 N1 and equation (24) imply Q01 Q1 . Let us now show that if Nc0 Nc and Q0c Qc for0
c < c0 1, then Nc+1
Nc+1 and Q0c+1 Qc+1 . Since c < c0 1, we know that L0c+1 = Lc+1 . Thus00
Qc and equation (23) imply Nc+1
Nc+1 , whereas Nc+1
Nc+1 and equation (24) imply
Q0c0
Qc+1 Qc+1 . At this point, we have established, by iteration, that Nc0 Nc and Q0c Qc for all
countries c < c0 . Let us now turn to countries c c0 . First consider country c0 . By assumption, weknow that L0c0 > Lc0 . Thus Q0c0 1 Qc0 1 and equation (23) imply Nc00 > Nc0 , whereas Nc00 Nc0and equation (24) imply Q0c0 < Qc0 . Using the same logic as before, let us now show that if0> Nc+1 and Q0c+1 < Qc+1 . Since c c0 , we know that
Nc0 > Nc and Q0c < Qc for c c0 , then Nc+10000
> Nc+1 , whereas Nc+1
> Nc+1 andLc+1 = Lc+1 . Thus Qc < Qc and equation (23) imply Nc+10
equation (24) imply Qc+1 < Qc+1 . At this point, we have therefore established, by iteration, that
Nc0 Nc and Q0c Qc for all countries c 2 C, with strict inequality for all countries c c0 . This
P
P
0
0
0
S00 = SC S0 = 1 by
S00 > SC S0 = C
implies C
c=1 Nc , which contradicts SC
c=1 Nc = SC
Lemma 2.
Step 4: If L0c0 > Lc0 , then Nc0 < Nc for all c < c0 .
We proceed by iteration. By Step 3, we know that if L0c0 > Lc0 and c0 6= 1, then N10 < N1 .
Since c0 6= 1, we know that L01 = L1 . Thus N10 < N1 and equation (24) imply Q01 > Q1 . Let us0
now show that if Nc0 < Nc and Q0c > Qc for c < c0 1, then Nc+1
< Nc+1 and Q0c+1 > Qc+1 . Since0c < c0 1, we know that L0c+1 = Lc+1 . Thus Q0c > Qc and equation (23) imply Nc+1
< Nc+1 ,00whereas Nc+1 < Nc+1 and equation (24) imply Qc+1 > Qc+1 . This completes the proof of Step 4.
Step 5: If L0c0 > Lc0 , then Sc0

Sc for all cc0 .

We proceed by iteration. By Lemma 2, we know that SC0 = SC = S. Thus Sc0 Sc is satis…edfor c = C. Let us now show that if Sc0
Sc and c > c0 , then Sc0 1
Sc 1 . Since c > c0 ,000
Lc0 > Lc0 and Step 2 imply Nc < Nc . Combining this observation with Sc Sc and the de…nitionof Nc Sc Sc 1 , we obtain Sc0 1 Sc 1 . This completes the proof of Step 5.
Step 6: If L0c0 > Lc0 , then Sc0 1

Sc 1 for all c

29

c0 .

We proceed by iteration. By Lemma 2, we know that S00 = S0 = 0. Thus Sc0 1
Sc 1 is00satis…ed for c = 1. Let us now show that if Sc 1 Sc 1 and c < c0 , then Sc Sc . Since c < c0 ,Sc 1 and theL0c0 > Lc0 and Step 4 imply Nc0 < Nc . Combining this observation with Sc0 1
0de…nition of Nc Sc Sc 1 , we obtain Sc Sc . This completes the proof of Step 6.
PCP0
To conclude, note that Step 2, Step 4, and the fact that C
c=1 Nc = 1, by Lemma
c=1 Nc =002, imply Nc0 > Nc0 and Nc < Nc for all c 6= c0 . Finally, De…nition 2, Step 5, and Step 6 imply that
all countries c < c0 move down the supply chain and all countries c > c0 move up. QED.
Proof of Proposition 3. We start by demonstrating the following Lemma.
Lemma 4 For any country 1 < c < C, if Nc0 < Nc and (wc =wc 1 )0 < (wc =wc 1 ), then (wc+1 =wc )0 <(wc+1 =wc ).
Proof of Lemma 4. Consider a country 1 < c < C. Equations (7) and (8) imply
wc+1=1+( cwc

e c Ncc+1 )1+ e c Nc

c

wc 1
wc

pc 1
wc 1

(27)

By equation (7), we also know thatwc
=1+( c 1
wc 1

c)

pc 1
wc 1

,

(28)

which further implies
wc 1
wc

pc 1
wc 1

Since (wc =wc 1 )0 < (wc =wc 1 ) and

=

c 1 >

(pc 1 =wc 1 ).
1+( c 1c ) (pc 1 =wc 1 )(29)

c , equation (28) immediately implies

pc 1
wc 1

0

<

pc 1
wc 1

.

Combining this observation with equation (29)— the right-hand side of which is increasing in(Pc 1 =wc 1 )— we obtain
wc 1
wc

0

0

pc 1
wc 1

<

wc 1
wc

pc 1
wc 1

.

(30)

0

To conclude, note that Nc0 < Nc implies e c Nc < e c Nc . Thus equation (27) and inequality (30)
imply (wc+1 =wc )0 < (wc+1 =wc ). QED.
Proof of Proposition 3 (Continued). The rest of the proof of Proposition 3 proceeds in threesteps.
Step 1: If L0c0 > Lc0 and c0 6= 1, then (w2 =w1 )0 < (w2 =w1 ).

30

Equation (7), equation (8), and p0 = 0 imply
w2
1
=1+
( 1
w1
1

2)

e 1 N11 .

(31)

By Proposition 2, we know that L0c0 > Lc0 implies N10 < N1 if c0 6= 1. Combining this observation
with equation (31), we obtain (w2 =w1 )0 < (w2 =w1 ) if c0 6= 1.
Step 2: (wc+1 =wc )0 < (wc+1 =wc ) for all c < c0 .

By Proposition 2, we know that L0c0 > Lc0 implies Nc0 < Nc for all c < c0 . Therefore, we can
invoke Step 1 and Lemma 4 to establish, by induction, that (wc+1 =wc )0 < (wc+1 =wc ) for all c < c0 .
Step 3: If there exists ecfor all ec c < C.

c0 such that (wec+1 =wec )0 < (wec+1 =wec ), then (wc+1 =wc )0 < (wc+1 =wc )Suppose that there exists e
c c0 such that (wec+1 =wec )0 < (wec+1 =wec ). By Proposition 2, we knowthat L0c0 > Lc0 implies Nc0 < Nc for all c > ecc0 . Therefore, we can again invoke Lemma 4 to0establish, by induction, that (wc+1 =wc ) < (wc+1 =wc ) for all ec c < C. This concludes the proofof Step 3. At this point, we can de…ne c1 inf c c0 j (wc+1 =wc )0 < (wc+1 =wc ) . Step 2 implies
that if L0c0 > Lc0 , then wc+1 =wc falls for all c < c0 , while Step 3 and the de…nition of c1 implies
that wc+1 =wc rises for all c0 c < c1 and falls for all c1 c < C. In order to establish Proposition
3, the only thing left to show is that c1 > c0 , which is what we establish in our …nal step.
Step 4: If L0c0 > Lc0 and c0 6= C, then (wc0 +1 =wc0 )0 > (wc0 +1 =wc0 ).

We proceed by contradiction. Suppose that (wc0 +1 =wc0 )0 (wc0 +1 =wc0 ). Then for any " > 0,~c < L~0~0~ c j < " andthere must also exist Lc0 L
L0c0 such that jL
L
c0
c0
0
0
(w
~c0 +1 =w~c0 )0(w~c0 +1 =w~ c0 ) ,(32)

0 ) denote the world income distribution if labor endowments inwhere (w
~1 ; :::; w~C ) and (w~10 ; :::; w~Ce c and L~ 0c , respectively. For jL~ 0c~ c j small enough, Lemma 2 implies
country c0 are equal to L
L
0
0
0
0

S~10 < S~1 < S~20 < ::: < S~c0 0 1 < S~c0 1 < S~c0 < S~c0 0 < S~c0 +1 < ::::,(33)

where S~1 ; :::; S~C and S~10 ; :::; S~C0 denote the pattern of vertical specialization if labor endowmentse c and L~ 0 , respectively. First, note that for any c < c0 1, sincein country c0 are equal to L
c00
S~0 < S~c < S~0 < S~c+1 , condition (2) implies that
c

c+1

0
p~0 S~c+1
0
w
~c+1
0
p~ S~c+1

w
~c+1

=

~0
~
e c+1 (Sc+1 Sc ) p~0 S~c

e
=

0
w
~c+1
~0
S~c )
c+1 (S
c+1

~0

+

~

e c+1 (Sc+1 Sc )

1

,

c+1

p~ S~c

w
~c+1

~0

+

~

e c+1 (Sc+1 Sc )

1

,

c+1

where p~ ( ) and p~0 ( ) denote the schedule of prices if labor endowments in country c0 are equal to31

~ 0 , respectively. Since S~0 > S~c , the two previous equations further imply that for any
e c and L
L
c0
0
c+1
c < c0 1,
0
0
0
0
=w
~c+1 .
(34)=w~c+1
p~ S~c+1~c+1 ) p~0 S~c+1~c+1
p~ S~c =w
p~0 S~c =w
Second, note that for any c < c0 , since S~c 1 < S~c0 < S~c , condition (2) also implies that
p~0 S~c

=

0
w
~c+1

~ ~0
e c+1 (Sc Sc ) p~0 S~c0
0
w
~c+1
S~0 )

~

e c (Sc

p~ S~c
=

w
~c+1

c

+

~0

1

,

(35)

c+1

p~ S~c0

~

+

w
~c+1

Let us now show that if p~0 S~c0 =w~c0~

e c+1 (Sc Sc )~0e c (Sc Sc )

1

c

w
~c
w
~c+1

.

(36)

p~ S~c0 =w~c , then~ ~0
e c+1 (Sc Sc ) p~0 S~c0e c (Sc Sc ) p~ S~c0

0
w
~c+1
~ ~0
e c+1 (Sc Sc )

w
~c+1
~
~0
e c (Sc Sc ) 1

~

1

c+1

~0

,

(37)

w
~c
w
~c+1

c

.

(38)

We start with inequality (37), which can be rearranged as

e

c+1

0

~c
(S~c S~c0 ) w
0
w
~c+1

p~0 S~c0
w
~c0

w
~c
~ ~0
e (Sc Sc )w
~c+1

p~ S~c0p~(S~c0 )w~ccp~ S~c0w~c.

By equation (7), we know that
w
~c
~ ~0
e (Sc Sc )
w
~c+1
c

e

c+1

0

~c
(S~c S~c0 ) w
0
w
~c+1

Under the assumption that p~0 S~c0 =w
~c0

e

~ ~0
c+1 (Sc Sc )0 ~0w
~c0 p~ Sc
0
w
~c+1
w
~c0

w
~c
p~0 S~c0
w
~c0

=
1+(

p~(S~c )cc+1 ) w~c
p~0 (S~0 )

,

(39)

.

(40)

c

w
~c0

=
1+( c

c+1 )

p~0 (S~c0 )w~c0p~ S~c0 =w~c , equations (39) and (40) imply
p~(S~c0 )
w
~c

1+( c

w
~c
~ ~0
> e (Sc Sc )p~(S~c0 )w~c+1

c+1 ) w~cc

p~ S~c0w~c,where the second inequality also uses the fact that S~c0 < S~c . Thus inequality (37) holds. Let us32now consider inequality (38), which can be rearranged as
~

c

~0

e c+1 (Sc Sc )1

~ ~0
e c (Sc Sc )

1

c+1

w
~c
.
w
~c+1

(41)

h
~ ~
~c
e c (Sc Sc 1 )Since S~c 1 < S~c0 for any c < c0 , condition (2) implies p~c S~c =whi~ ~0e c (Sc Sc ) 1 = c . Combining the previous inequality with equation (7), we obtain
w
~c
=
w
~c+1

1
p~(S~c )c+1 ) w
~c

1+( c

c

h
~ ~0
c (Sc Sc )
)
c+1 e

<
c+( c

1

i

i
1 = c >

(42)

By inequalities (41) and (42), a su¢ cient condition for inequality (38) to hold is
h

~

~0

c+1 (Sc Sc )
c e
h
~ ~0
c (Sc Sc )
c+1 e

i
1
i
1

c
c+( c

c+1 )

~

~0

e c (Sc Sc )

i,
1

h
i
~ ~0
c+1 (Sc Sc )=1e
. The previous inc+1x
is increasing in x for t > 0. At this point, we
1 e tx~c . Combining thisp~ S~c0 =w~c0have established that inequalities (37) and (38) hold if p~0 S~c0 =whwhich can be rearranged as c = 1 eequality necessarily holds since f (x)c(S~c S~c0 )ihobservation with equations (35) and (36), we further have that for any c < c0 ,~c0p~0 S~c0 =w

0~c+1
p~ S~c0 =w~c ) p~0 S~c =w

Since p~0 (0) = p~ (0) = 0, we know that p~0 S~0 =w~10(34) and (43) to establish, by iteration, thatw
~c0 0
p~ S~c0
w
~ c0

=

=

(43)

p~ S~0 =w~1 . Thus we can use implications

p~0 S~c0 1

p~ S~c0 1

w
~c0 0

w
~ c0

Since Sec0 < Sec0 0 , we know from condition (2) that
p~0 S~c0p~ S~c =w~c+1 .

~~
e c0 (Sc0 Sc0 1 ) p~0 S~c0 1

w
~c0 0

.

(44)

~

+

~

e c0 (Sc0 Sc0 1 )

1

,

(45)

:

(46)

c0

~
~
e c0 (Sc0 Sc0 1 ) p~ S~c0 1

w
~ c0

~

+

~

e c0 (Sc0 Sc0 1 )
c0

33

1

Inequality (44) and equations (45) and (46) imply p~0 S~c0 =w~c0 0

p~ S~c0 =w~c0 . Finally, sinceSec0 < Sec0 0 , we also know that p~0 S~c0 0 =w
~c0 0 > p~0 S~c0 =w~c0 0 . Combining these two observations,
we get p~0 S~c0 0 =w
~c0 0 > p~ S~c0 =w~c0 . Together with equation (7), the previous inequality implies~c0 +1 =w~c0 ), which contradicts inequality (32). Thus (wc0 +1 =wc0 )0 > (wc0 +1 =wc0 ),(w~c0 +1 =w~c0 )0 > (wwhich implies c1 inf c c0 j (wc+1 =wc )0 < (wc+1 =wc ) > c0 . As mentioned above, Proposition 3
directly follows from Steps 2-4. QED.

C

Proofs (III): Technological Change

Proof of Proposition 4. Like in the proof of Proposition 2, we use equations (23) and (24),which both derive from Lemma 2, and are reported below for expositional purposes:
c Lc

Qc 1 =

c Nc

1

e
L
c ce
1 e

Qc =

, for all c 2 C

(23)

c Nc
c Nc

, for all c 2 C,

(24)

For future reference, note that @Qc 1 =@Nc < 0 and @Qc =@Nc < 0, whereas @Qc 1 =@ c > 0 and@Qc =@ c < 0. The rest of our proof proceeds in six steps.
0c0 <Step 1: If0c0 and c0 6= C, then NC < NC .

We …rst establish that Q0C > QC . By the same argument as in Step 1 of Proposition 2, Q0C
must be such thatPC e
Q0C = arg max
c=1 Qc (S) ,e 1 ( );:::;QeC ( )
Q

subject toPCe

c=1 Qc (s2 )

PC
0

where

0
c0 <

e c (s)Q00c =c0 ande ( c00
c0

R s2 PCec=1 Qc (s1 )Z S

s1e c (s) ds
Q0 ec Qc (s) ds, for all s1

c=1

s2 ,

Lc , for all c 2 C,
e

(47)
(48)

e

c for all c 6= c0 . Now consider Q1 ( ) ; :::; QC ( ) such that)(Sc0 s) Q (s)
c0
0
c0 e

+
1

e

0
c0 s
0
c0 S

!Z

S c0

S c0

1

h

1

e ( c00c0i)(Sc0 t) Q (t) dt, for all s 2 S,c0ande c (s)Q

Qc (s) , for all s 2 S and c 6= c0 .

Since Q1 ( ) ; :::; QC ( ) satisfy the initial resource and technological constraints, as described by34

e 1 ( ) ; :::; Qe C ( ) must satisfy, by construction, the new resource and techequations (3) and (4), Q
nological constraints, as described by equations (47) and (48). Since 0c0 < c0 , we must alsohavee C (1) =e c (1) + Q
Q
0

c0 e

c0 S

e

c0 S

1

Z Sc

0

Sc0

1

h

1

e ( c00c0i)(Sc0 t) Q (t) dt + Q > Q .
c0CCe C (1), the previous inequality implies Q0 > QC . Combining this observatione c (1)+ Q
Since Q0C Q
0Cwith equation (24) and the fact that c0 6= C, which implies 0C = C , we get NC0 < NC .

Step 2: If

0
c0 <

0
c0 , then Nc < Nc for all c > c0 .

We proceed by iteration. By Step 1, we know that if 0c0 < c0 and c0 6= C, then NC0 < NC .
Let us now show that if Nc0 < Nc and c > c0 + 1, then Nc0 1 < Nc 1 . Since c > c0 + 1, we knowthat 0c = c . Thus Nc0 < Nc and equation (23) imply Q0c 1 > Qc 1 . Since c > c0 + 1, we also knowthat 0c 1 = c 1 . Thus Q0c 1 > Qc 1 and equation (24) imply Nc0 1 < Nc 1 . This completes theproof of Step 2.
Step 3. If

0
c0 <

0
c0 and c0 6= 1, then N1 > N1 .

We proceed by contradiction. Suppose that N10
N1 . Since c0 6= 1, we know that 01 = 1 .
00Thus N1 N1 and equation (24) imply Q1 Q1 . Let us now show that if Nc0 Nc and Q0c Qc
0Nc+1 and Q0c+1 Qc+1 . Since c < c0 1, we know that 0c+1 = c+1 .
for c < c0 1, then Nc+1
000
Nc+1 , whereas Nc+1
Nc+1 and equation (24)
Qc and equation (23) imply Nc+1
Thus Qc
0
imply Qc+1
Qc+1 . At this point, we have established, by iteration, that if N10
N1 , then00
Nc Nc and Qc Qc for all countries c < c0 . Let us now turn to countries c c0 . First consider
country c0 . By assumption, we know that 0c0 < c0 . Thus Q0c0 1 Qc0 1 and equation (23) imply
Nc00 < Nc0 , whereas Nc00 < Nc0 and equation (24) imply Q0c0 > Qc0 . Using the same logic as before,
0c0 , then Nc+1
< Nc+1 and Q0c+1 > Qc+1 .
let us now show that if Nc0 < Nc and Q0c > Qc for c
0
Since c c0 , we know that 0c+1 = c+1 . Thus Q0c > Qc and equation (23) imply Nc+1
< Nc+1 ,00whereas Nc+1 < Nc+1 and equation (24) imply Qc+1 > Qc+1 . At this point, we have therefore
established, by iteration, that Nc0 Nc and Q0c Qc for all countries c 2 C, with strict inequalityPP
00
for all countries c c0 . This implies C
S00 < SC S0 = C
c=1 Nc = SC
c=1 Nc , which contradicts00
SC S0 = SC S0 = 1 by Lemma 2.
Step 4 If

0
c0 <

0
c0 , then Nc > Nc for all c < c0 .

We proceed by iteration. By Step 3, we know that if 0c0 < c0 and c0 6= 1, then N10 > N1 .
Since c0 6= 1, we know that 01 = 1 . Thus N10 > N1 and equation (24) imply Q01 < Q1 . Let us0
now show that if Nc0 > Nc and Q0c < Qc for c < c0 1, then Nc+1
> Nc+1 and Q0c+1 < Qc+1 . Since000c < c0 1, we know that c+1 = c+1 . Thus Qc < Qc and equation (23) imply Nc+1
> Nc+1 ,00whereas Nc+1 > Nc+1 and equation (24) imply Qc+1 < Qc+1 . This completes the proof of Step 4.
Step 5: If

0
c0 <

0
c0 , then Sc

Sc for all cc0 .

The proof is identical to the proof of Step 5 in Proposition 2 and omitted.
Step 6: If

0
c0 <

0
c0 , then Sc 1

Sc 1 for all c

35

c0 .

We proceed by iteration. By Lemma 2, we know that S00 = S0 = 0. Thus Sc0 1
Sc 1 is00satis…ed for c = 1. Let us now show that if Sc 1 Sc 1 and c < c0 , then Sc Sc . Since c < c0 ,Sc 1 and theL0c0 > Lc0 and Step 4 imply Nc0 > Nc . Combining this observation with Sc0 1
0de…nition of Nc Sc Sc 1 , we obtain Sc Sc . This completes the proof of Step 6.
To conclude, note that Steps 2 and 4 imply that Nc0 > Nc for all c < c0 and Nc0 < Nc for allc > c0 . Finally, De…nition 2, Step 5, and Step 6 imply that all countries c 6= c0 move up the supply
chain. QED.
Proof of Proposition 5. Following the same logic as in the proof Proposition 3, one can show
the following lemma.
Lemma 5 For any country 1 < c < C, if Nc0 > Nc and (wc =wc 1 )0 > (wc =wc 1 ), then (wc+1 =wc )0 >(wc+1 =wc ).
Proof of Lemma 5. The proof is similar to the proof of Lemma 4 and omitted.
Proof of Proposition 5 (Continued). The rest of the proof of Proposition 5 proceeds in threesteps.
Step 1: If

0
c0 <

0
c0 and c0 6= 1, then (w2 =w1 ) > (w2 =w1 ).

Like in the proof of Proposition 3, Equation (7), equation (8), and p0 = 0 imply
1
w2
=1+
( 1
w1
1

2)

e 1 N1

1 .

(49)

By Proposition 4, we know that 0c0 < c0 implies N10 > N1 . Since c0 6= 1, we also know that 01 =Combining the two previous observations with equation (31), we obtain (w2 =w1 )0 > (w2 =w1 ).

1.

Step 2: (wc+1 =wc )0 > (wc+1 =wc ) for all c < c0 .
By Proposition 4, we know that 0c0 < c0 implies Nc0 > Nc for all c < c0 . Therefore, we can
invoke Step 1 and Lemma 5 to establish, by induction, that (wc+1 =wc )0 > (wc+1 =wc ) for all c < c0 .
Step 3: If there exists e
c > c0 such that (wec+1 =wec )0 < (wec+1 =wec ), then (wc+1 =wc )0 < (wc+1 =wc )for all ec c < C.

Suppose that there exists e
c > c0 such that (wec+1 =wec )0 < (wec+1 =wec ). By Proposition 4, we know
00that c0 < c0 implies Nc < Nc for all c > e
c > c0 . Therefore, we can use the same strategy as inLemma 4 in the proof of Proposition 3 to establish that (wc+1 =wc )0 < (wc+1 =wc ) for all ec c < C.
This concludes the proof of Step 3.
To conclude, let us de…ne c1 inf c c0 j (wc+1 =wc )0 > (wc+1 =wc ) , and similarly, let us de…nec2 inf c > c1 j (wc+1 =wc )0 < (wc+1 =wc ) . Step 2 implies that if 0c0 < c0 , then wc+1 =wc rises for
all c < c0 . The de…nitions of c1 and c2 imply that wc+1 =wc falls for all c0 c < c1 and rises for all
c1 c < c2 . Step 3 and the de…nition of c2 imply that wc+1 =wc falls for all c2 c < C. Proposition
5 directly derives from the previous observations and De…nition 3. QED.

36