Endogenous Ranking and Equilibrium Lorenz Curve
Across (ex-ante) Identical Countries: A Generalization
By Kiminori Matsuyama1

September 2013

Abstract: This paper proposes a symmetry-breaking model of trade with a finite
number of identical countries and a continuum of tradeable consumption goods,
which differ in their dependence on nontradeable intermediate inputs, “producer
services”. Productivity of each country is endogenous due to country-specificexternal economies of scale in its service sector. It is shown that, in any stableequilibrium, the countries sort themselves into specializing in different sets oftradeable goods and that a strict ranking of countries in per capita income, TFP,
the service sector share, and the capital-labor ratio emerge endogenously.
Furthermore, the distribution of country shares, the Lorenz curve, is unique andanalytically solvable in the limit, as the number of countries grows unbounded.
Using this limit as an approximation allows us to study what determines the shapeof distribution, perform various comparative statics and to evaluate the welfare
effects of trade. In doing so, this paper extends the analysis of Matsuyama (2013)
for more general and flexible forms of scale economies. It turns out that thetechnique introduced in Matsuyama (2013) is useful for the equilibrium
characterization in this general case as well. Although some results ofcomparative statics and on welfare inevitably need to be modified, they change inways that illuminate the underlying mechanism of symmetry-breaking.
Keywords: Endogenous Comparative Advantage, Endogenous Dispersion,
External Economies of Scale, Globalization and Inequality, Symmetry-Breaking,Polarization, Power-law, Lorenz-dominant shifts, Log-supermodularity, Logsubmodularity
JEL Classification Numbers: F12, F43, O11, O19

1

Email: k-matsuyama@northwestern.edu; Homepage: http://faculty.wcas.northwestern.edu/~kmatsu/. This paper isprepared for International Workshop in Economic Theory: “Where Do We Stand?”, held on June 28-29, 2013, at
Ca’ Foscari, in Venice. I thank Guido Cazzavillan and Department of Economics at University of Venice for thehospitality during my stay and the Workshop participants for their comments. The final version also benefits fromthe feedback received during the seminars given at Chicago Fed, Institut für Höhere Studien in Vienna, andNorthwestern University.

©Kiminori Matsuyama, Equilibrium Lorenz Curve: A Generalization

1.

Introduction
Rich countries tend to have higher TFPs and higher capital-labor ratios than the poor.

Such empirical regularities are generally viewed as a causality running from TFPs and/or capitallabor ratios to per capita income, often under the maintained hypotheses that these countries
offer independent observations and that cross-country variation in per capita income would
disappear without any variations in some exogenous variables. However, there is acomplementary approach, popular in trade and economic geography, that suggests a two-way
causality. According to this approach, trade (and factor mobility) among countries/regions, even
if they were ex-ante identical, could lead to the instability of the symmetric equilibrium in which
they would remain identical. With such symmetry-breaking, cross-sectional dispersion andcorrelation in per capita income, TFPs, and capital-labor ratios, emerge endogenously as only
stable patterns.2 This approach does not try to argue that countries are ex-ante homogenous, northat exogenous heterogeneities are unimportant. On the contrary, it suggests that even small
heterogeneity or shocks could be amplified to create large productivity and income differences,which makes this approach appealing as a possible explanation for “Great Divergence” and“Growth Miracles.”The existing studies of symmetry-breaking, however, demonstrate this insight in a twocountry/region setup, which makes it unclear what the message of this approach is when applied
to a multi-country/region world. For example, does a symmetry-breaking mechanism split theworld into the rich and poor clusters, as the narrative of this literature, such as “core-periphery,”or “polarization” might suggest? Or does it keep splitting the world into finer clusters until the
distribution becomes more disperse, possibly generating a power-law like distribution, as
observed in the size distribution of metropolitan areas? More generally, which features of theeconomic environment determine the shape of distribution? Not only the existing studies on
symmetry-breaking are unable to answer these questions, but also generate little analyticalresults on comparative statics and welfare. Motivated by these concerns, Matsuyama (2013) has
2

See Fujita, Krugman, and Venables (1999) and Combes, Mayer, and Thisse (2008) in economic geography andEthier (1982), Helpman (1986, p.344-346), Krugman and Venables (1995) and Matsuyama (1996) in international
trade. The view that trade itself could magnify inequality among nations was discussed informally by Myrdal(1957) and Lewis (1977). See Matsuyama (2011) for more references. Symmetry-breaking is a circular mechanismthat generates stable asymmetric outcomes in the symmetric environment due to the instability of the symmetricoutcome. Although most prominent in economic geography, it has found applications in other areas of economics:
see a New Palgrave entry on “symmetry-breaking” by Matsuyama (2008) as well as a related entry on “emergence”
by Ioannides (2008).
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©Kiminori Matsuyama, Equilibrium Lorenz Curve: A Generalization

recently proposed an analytically tractable symmetry-breaking model of trade as a framework inwhich one could address these issues. The present paper offers a generalization of that analysis.
More specifically, consider a world with a (large but) finite number of (ex-ante) identicalcountries. In each country, the representative household supplies a single composite of primaryfactors, such as capital, labor, land, etc., and has Cobb-Douglas preferences over a continuum of
tradeable goods, as in Dornbusch, Fischer, and Samuelson (1977). These tradeables areproduced by two nontradeable inputs, the (composite) primary factor and the intermediate inputs,“producer services,” which are in turn produced by the (composite) primary factor. The two key
assumptions are that tradeable sectors differ in their dependence on its service sector and that
productivity of the service sector is endogenous due to country-specific external economies ofscale in its service sector. This creates a circular mechanism between the patterns of trade andcross-country productivity differences. Having a larger and hence more productive service
sector not only makes a country more productive. It also gives a country comparative advantage
in tradeable sectors that are more dependent on those services. This in turn means a larger
demand for services and, as a result, the country ends up having a larger and more productive
service sector.
With (a continuum of) tradable goods vastly outnumbering (a finite number of) countries,
this circular mechanism sorts different countries into specializing in different sets of tradeable
goods (endogenous comparative advantage) and leads to a strict ranking of countries in percapita income, TFP, the service sector share in GDP, and (in an extension that allows for variablefactor supply) capital-labor ratio in any stable equilibrium. Furthermore, the equilibrium
distribution of country shares, the Lorenz curve, is unique and analytically tractable in the limit,
as the number of countries grows unbounded. Using this limit as an approximation allows us to
study, among other things, what determines the shape of distribution and how various forms ofglobalization or technical change affect inequality across countries, and to evaluate the welfare
effects of trade (e.g., when trade is Pareto-improving, and when it is not, what fraction ofcountries might lose from trade).
The present paper generalizes the analysis of Matsuyama (2013), which models theservice sector as a Dixit-Stiglitz (1977) type monopolistic competitive sector, whose sectoral
TFP responds to a change in the variety of differentiated services that become available.
Although such a specification has advantage of offering a micro foundation for scale economies-2-©Kiminori Matsuyama, Equilibrium Lorenz Curve: A Generalizationand endogenous productivity, it comes at the cost of imposing the restriction that the degree ofscale economies, measured by the elasticity of the service sector TFP with respect to a change in
its total input, has to be constant; particularly, it has to be independent of the scale of the service
sector. The alternative specification in this paper- the service sector is competitive but subject tocountry-specific external economies of scale-, allows for the degree of scale economies being afunction of the scale of the service sector. Even under such more general and flexible forms ofscale economies, the technique introduced in Matsuyama (2013) turns out to be useful forcharacterizing the equilibrium distribution. Of course, some results on comparative statics andwelfare effects inevitably need to be modified, but they change in ways that illuminate theunderlying mechanisms of symmetry-breaking. This paper reproduces all of the results of
Matsuyama (2013) as a special case of the present model. It also resuscitates many discussions
that existed in earlier versions of Matsuyama (2013) but had to be removed from its publishedversion due to the space constraint.
Section 2.1 introduces the baseline model, which assumes that all consumption goods aretradeable and all primary factors are in fixed supply. Section 2.2 derives a single-country (orautarky) equilibrium. Section 2.3 derives a stable equilibrium with any finite number ofcountries, whose associated Lorenz curve is characterized by the second-order difference
equation with two terminal conditions. Section 2.4 explains why any equilibrium in which somecountries remain identical ex-post is unstable. Section 2.5 shows that, as the number of countries
grows unbounded, the Lorenz curve converges to the unique and analytically solvable solution of
the second-order differential equation with two terminal conditions. Armed with the explicit
formula for the limit Lorenz curve, this subsection examines when the distribution is bimodal orsatisfies a power-law. It also shows how various parameters causes a Lorenz-dominant shifts ofthe distribution, leading to greater or less inequality across countries, by making use of logsuper(sub)modularity. Section 2.6 studies the welfare effects of trade. Section 3 discusses two
extensions. In section 3.1, a fraction of the consumption goods are assumed to be nontradeable.
This extension allows us to study the effects of globalization through trade in goods. In section
3.2, one of the primary factors is allowed to vary in supply either through factor mobility andaccumulation. This extension not only generates the correlation between the capital-labor ratio
and per capita income and TFP, but also it allows us to study the effects of technical change that-3-©Kiminori Matsuyama, Equilibrium Lorenz Curve: A Generalization

increases the relative importance of human capital in production and of globalization throughfactor mobility.

2.

Baseline Model:

2.1

Key Elements of the Model
Following Matsuyama (2013), consider the world economy with J (ex-ante) identicalcountries, where J is a positive integer, and a continuum of tradeable consumption goods,
indexed by s  [0,1]. There may be multiple nontradeable primary factors of production, such as
capital (K), labor (L), etc., which can be aggregated to a single composite as V = F(K, L, …). For
now, it is assumed that these component factors are in fixed supply and that the representative
consumer of each country is endowed with the same quantity of V.
The representative consumer has Cobb-Douglas preferences over a continuum of
tradeable consumption goods. This can be expressed by an expenditure function,1s
E  exp   log( P ( s ))dB ( s )U , where U is utility; P(s) > 0 the price of good-s; B(s) =   (u )du
 0

0

the expenditure share of goods in [0,s], satisfying B' ( s)   (s) > 0, B(0)  0 , and B(1)  1 . By
denoting the aggregate income by Y, the budget constraint is then written as(1)

1
Y  exp   log( P ( s ))dB ( s )U .
 0

Each tradeable good is produced competitively by combining the two types of
nontradeble inputs; the composite of the primary factors and the intermediate inputs, “producer
services”, using a Cobb-Douglas technology with γ(s)  [0,1] being the share of producer
services in sector-s. The unit production cost of good-s can thus be expressed as:(2)

C (s )   (s )( )1  ( s ) ( PN ) ( s ) ,where ω is the price of the (composite) primary factor; PN the price of services; and  (s) is ascale parameter. Services are produced competitively, using the (composite) primary factor as
the only input, but its technology is subject to country-specific external economies of scale.
More specifically, its unit cost function is given by:
(3)

PN 


,
A(n)

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©Kiminori Matsuyama, Equilibrium Lorenz Curve: A Generalization

where A(n) is TFP in the service sector, which is treated as a constant by each firm in theservice sector but is increasing in its total input in the sector, n. For technical reasons, we
assume that A(n) is continuously differentiable, so that the elasticity of the sectoral TFP withrespect to n,(4) (n) A' (n)n 0,
A(n)is a well-defined and continuous function of n, which shall be referred to as the degree of scaleeconomies. Combining (2) and (3) yields,(5)

C (s)   (s), A(n)  ( s )which shows that the extent to which each tradeable goods sector benefits from a higher TFP in
the service sector depends on the share of services in its production, γ(s). With no loss ofgenerality, we may order the tradeable goods so that γ(s) is increasing in s  [0,1], so that highindexed sectors benefit more from scale economies in the service sector. Again, for technicalreasons, it is assumed to be continuously differentiable.
Before proceeding, note that we may set,
(6)

B(s) = s for all s  [0,1],

so that β(s) = 1 for all s  [0,1] by choosing the tradeable goods indices, without further loss ofgenerality. 3 In words, we measure the size of (a set of) sectors by the expenditure share of thegoods produced in these sectors. With this indexing, the size of sectors whose γ is less than orequal to γ(s) is equal to s. (In other words, γ(s) is the inverse of the cumulative distribution of γ
across sectors.) Furthermore, a country’s share in the world income is equal to the measure of
the tradeables for which the country ends up having comparative advantage in equilibrium.
2.2

Single-Country (or Autarky) Equilibrium (J = 1)
First, let us look at the equilibrium allocation of a one-country world (J = 1). This canalso be viewed as the equilibrium allocation of a country in autarky in a world with multiple
countries, which would serve as the benchmark for evaluating the welfare effects of trade.
3

To see this, starting from any indexing of the goods s'  [0,1] satisfying i) ~ ( s' )  [0,1] is increasing in s'  [0,1],

1 ~~~ii)  (s ' ) > 0 for s'  [0,1], and iii)   (s ' )ds '  1 , re-index the goods by s  B (s ' ) 0s' ~  (u)du . Then,0~~ ( s )  ~ ( B 1 ( s)) is increasing in s  [0,1] and ds   (s ' )ds ' , so that β(s) = 1 for s  [0,1].

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©Kiminori Matsuyama, Equilibrium Lorenz Curve: A Generalization

Because of Cobb-Douglas preferences, the economy must produce all the consumption
goods in the absence of trade, which means that their prices must be equal to their costs; that is,(7)

P( s)  C (s)  ( s)
 A(n)  ( s )for all s  [0,1]

Since the representative consumer spends β(s)Y = Y on good-s, and sector-s spends 100γ(s)% of
its revenue on services, the equilibrium size of the services sector, measured in its total input
payment, is
1

(8)

1

ωAnA =   ( s )  ( s )Yds =  A Y,

where

0

 A    ( s )ds .
0(Superscript A stands for autarky, here.) Since the aggregate income is accrued entirely to theprimary factors in equilibrium, 4(9)

YA = ωAV = ωAF(K,L,…),the equilibrium value of n in autarky is given as follows:(10)n A   AV

Eq. (9) shows that, with all the primary factors, capital (K), labor (L), etc. being aggregated into
the composite, V =F(K,L,…), the equilibrium price of the composite factor, ωA, is nothing but theaggregate TFP, as measured in GDP accounting. Eq. (10) shows that the equilibrium share of theservice sector, nA/V, is equal to the share of services in the total expenditure, which is equal to A in autarky.
2.3

Stable Multi-Country Equilibrium (J ≥ 2)
Let us now turn to the case with J ≥ 2. Since these countries are ex-ante identical, theyshare the same values for all the exogenous parameters. However, the stability of equilibrium
requires that no two countries share the same value of n, as explained later. This allows us torank the countries such that n j j 1 is (strictly) increasing in j. (The subscript here indicates theJrank, not the identity, of a country in a particular equilibrium.) Then, from (5), the relative cost
between the j-th and the (j+1)-th countries,

4

Note that the service sector firms in this model earn zero profit because they are competitive with the linear
technology, even though the sector as a whole is subject to economies of scale. The service sector firms in themodel of Matsuyama (2013) also earn zero profit because of the free entry condition, even though they aremonopolistically competitive.
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©Kiminori Matsuyama, Equilibrium Lorenz Curve: A Generalization(11)

 A(n j ) C j 1 ( s )  A(n j 1 ) C j ( s) ( s ) j   j 1 is increasing in s for any j = 1, 2, ..., J−1, for any combination of the factor prices  j j 1 , asJillustrated by upward-sloping curves in Figure 1. In words, a country with a larger service sector
has comparative advantage in higher-indexed sectors, which are more dependent on services.
Furthermore,  j j 1 must adjust in equilibrium so that each country becomes the lowest costJ

producer (and hence the exporter) of a positive measure of the tradeable goods.5 This implies
that a sequence, S j j 0 , defined by S0 = 0, SJ = 1, andJ A(n j ) C j 1 ( S j )  A(n j 1 ) C j (S j ) ( S j ) j 

 1
 
j

1



(j = 1, 2, ..., J−1),

is increasing in j. 6 As showed in Figure 1, the tradeable goods, [0,1], are partitioned into J
subintervals of positive measure, (Sj−1, Sj), such that the j-th country becomes the lowest costproducer (and hence its sole producer and exporter) of s  (Sj−1, Sj).7 Note also that thedefinition of S j j 1 can be rewritten to obtain:J 1

 (S j )(12)  j 1   A(n j 1 )     A(n ) 
j
j

 


1

(j = 1, 2, ..., J−1)

so that  j j 1 is also increasing in j; that is, a country with a large service sector is moreJ

productive and richer because it has a more productive service sector.
Since the j-th country specializes in (Sj−1, Sj), 100(Sj−Sj−1)% of the world income, YW, isspent on its tradeable sectors, so that its aggregate income is(13)

Y j  jV  (S j  S j 1 )Y W

(j = 1, 2, ...,J )Furthermore, since its sector-s in (Sj−1, Sj) spends 100γ(s)% of its revenue on its services, thetotal input payment of its service sector is
5

Otherwise, the factor price would be zero for a positive fraction (at least 1/J) of the world population.
To see why, Sj ≥ Sj+1 would imply Cj (s) > min{Cj–1(s), Cj+1(s)} for all s  [0,1], hence that the j-th country is not
the lowest cost producers of any tradeable good, a contradiction.
7
In addition, S0 is produced and exported by the 1st country and SJ by the J-th country. The borderline sector, Sj (j =1, 2,…, J−1), could be produced and exported by either j-th or (j+1)-th country or both. This type of indeterminacy
is inconsequential and ignored in the following discussion.
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©Kiminori Matsuyama, Equilibrium Lorenz Curve: A Generalization

 Sj

njωj =    (s )ds  YW = (Sj−Sj−1)ГjYW
 S j 1(14)(j = 1, 2, ...,J )

where
Sj

(15)

1
 j  ( S j 1 , S j )  ( s )ds .
S j  S j 1 Sj 1(j = 1, 2, ...,J )is the average of  over (Sj−1, Sj). Note that  j j 1 is increasing in j, since  is increasing in s.
J

From (13) and (14),(16)

n j  jV ;(j = 1, 2, ...,J )Because  j j 1 is increasing in j, eq.(16) shows that n j j 1 is also increasing in j, as has beenJJ

assumed. Eq.(13) shows that  j represents TFP of the j-th poorest country, and s j  S j  S j 1 ,the size of the tradeable sectors in which this country has comparative advantage, is also equal toj

its share in the world income. Thus, S j  k 1 sk is the cumulative share of the j poorest
countries in world income. By combining (12), (13), (15), and (16), we obtain:
Proposition 1: Let S j be the cumulative share of the j poorest countries in world income.
Then, S j j 0 solves the second-order difference equation with two terminal conditions:J (S j ) A(V( S j , S j 1 )) 

(17)

S j  S j 1  A(V (S j 1 , S j )) S j 1  S j

where ( S j 1, S j ) 1with S 0  0 & S J  1 ,Sj1
 ( s )ds .
S j  S j 1 S j 1

Figure 2 illustrates a solution to eq. (17) by means of the Lorenz curve,  J : [0,1]  [0,1] ,defined by the piece-wise linear function, with  J ( j / J )  S j . From the Lorenz curve, we canrecover s j j 0 , the distribution of the country shares and vice versa. 8 A few points deserveJemphasis. First, because  j   (S j 1 , S j ) is increasing in j, s j 1 / s j  ( S j 1  S j ) / ( S j  S j 1 ) >1. Hence, the Lorenz curve is kinked at j / J for each j = 1, 2, ..., J−1. In other words, the8This merely states that there is a one-to-one correspondence between the distribution of income shares and theLorenz curve. With J ex-ante identical countries, there are J! (factorial) equilibria for each Lorenz curve.
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ranking of the countries is strict.9 Second, since both income and TFP are proportional tos j  S j  S j 1 , the Lorenz curve here also represents the Lorenz curve for income and TFP.

Third, we could also obtain the ranking of countries in other variables of interest that arefunctions of s j j 0 . For example, the j-th country’s share in world trade, is equal toJs  s  /  s  s  , which is increasing in j . The j-th country’s trade dependence,
2

j

j

J

k 1

2

k

k

defined by the volume of trade divided by its GDP, is equal to 1  s j , which is decreasing in j.
Note also that Proposition 1 of Matsuyama (2013) can be obtained as a special case of theabove proposition, with A(n)  (n) (i.e.,  (n)   > 0), which simplifies the second-orderdifference equation, eq. (17), to:(18) ( S j )  ( S j , S j 1 ) S j  S j 1  ( S j 1, S j ) S j 1  S j

 1.

This shows that the constant degree of scale economies was crucial for the equilibrium
distribution in Matsuyama (2013) to be scale-free, i.e., independent of V.
2.4

Symmetry-Breaking: Instability of Equilibria without Strict Ranking of Countries
In characterizing the above equilibrium, we have started by imposing the condition thatno two countries share the same value of n, and hence the countries could be ranked strictly sothat n j j 1 is increasing in j, and then verified later that this condition holds in equilibrium.
J

Indeed, there are also equilibria, in which some countries share the same value of n, and without
strict ranking, n j j 1 is merely nondecreasing in j. For example, consider the case of J = 2 andJsuppose n1 = n2 in equilibrium. Then, from (5), C1 (s ) / C 2 ( s )  1 /  2 , which is independent ofs. Thus, the condition under which each country produces a positive measure of goods issatisfied only if C1 (s ) / C 2 ( s )  1 /  2 = 1. This means that, in this equilibrium, the consumers
everywhere is indifferent as to which country they purchase tradeable goods from. In otherwords, the patterns of trade are indeterminate. If exactly 50% of the world income is spent oneach country’s tradeable goods sectors, and if this spending is distributed across the two
9This is in sharp contrast to Matsuyama (1996), which shows a non-degenerate distribution of income acrosscountries, but with a clustering of countries that share the same level of income. The crucial difference is that thecountries outnumber the tradeable goods in the model of Matsuyama (1996), while it is assumed more realistically,
here and in Matsuyama (2013), that the tradabable goods outnumber the countries.
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countries in such a way that the services sector of each country ends up receiving exactly  A / 2
fraction of the world spending, then they would operate exactly at the same scale, n1 = n2 = nA.
Thus, the two countries remain identical ex-post. However, it is easy to see that this symmetricequilibrium, which replicates the autarky equilibrium in each country, is “sitting on the knifeedge”, in that the required spending patterns described above must be met exactly in spite of theconsumers’ indifference. Furthermore, this symmetric equilibrium is unstable with respect to theusual Marshallian quantity adjustment dynamics. 10 To see this, imagine that small random
perturbations make n1 slightly lower than n2. This would leads to a change in the spending
patterns that reduces demand for the service sector in country 1 relative to that in country 2,
creating a force towards a further decline in n1 and a further increase in n2 , pushing the world
further away from the symmetric allocation. This logic carries over to the case of J > 2 with nj =nj+1 for some j, because that would imply C j (s ) / C j 1 ( s )   j /  j 1  1 so that, for a positive
measure of goods, the consumers would be indifferent between buying from the j-th or (j+1)-thcountry, which would generate the same knife-edge property. For this reason, we restrict
ourselves only to the equilibrium with a strict ranking of countries.11
10

Alternatively, one could introduce dynamics through learning-by-doing, as done in Matsuyama (2002), byassuming that TFP of the service sector depends on its (discounted) cumulative total input. In the model of
Matsuyama (2013), where the service sector is monopolistically competitive, one could also introduce dynamics
through firms entering (exiting) in response to positive (negative) profit.
11
The logic behind the instability of equilibrium without strict ranking of countries is similar to that of the mixed
strategy equilibrium in games of strategic complementarity, particularly the game of the battle of the sexes. The
assumption of a finite number of countries is crucial, but the assumption of zero trade cost in tradeable goods is not.
To understand the latter, consider the case of the constant degree of scale economies θ(n) = θ and J = 2. Imagine
that these two countries are ex-ante identical, but repeatedly hit by small random shocks such that the realized
parameter values cause the ratio of nA of the two countries to fluctuate over a small support around one, [e–ε, eε] with
ε > 0. (For example, the relative size of the two countries, V1/V2, might fluctuate around one, due to small shocks toexogenous components of TFP.) Now, assume an iceberg trade cost, such that one unit of the good shipped shrinks
to e–δ < 1 when it arrives, where δ > 0. Then, one could show that, even with a small ε > 0, symmetry breaking
occurs eventually if δ < εθ(γ(1)–γ(0))/2. (The logic here should be familiar to those who are exposed to the notion ofstochastic stability of dynamical systems with random perturbations, where the long run stability of equilibrium
depends on the size of its basin of attraction.) This extension also suggests that the world undergo a symmetrybreaking bifurcation, when the trade cost declines from δ > εθ(γ(1)–γ(0))/2 to δ < εθ(γ(1)–γ(0))/2. Of course, thismeans that, with a small positive cost δ > 0, infinitesimal perturbations ε  0 cannot break symmetry. However, thisis a mere technicality with no substantive issue at stake. Symmetry-breaking captures the idea that the symmetricoutcome is more vulnerable to small shocks than the asymmetric outcomes, so that the asymmetric outcomes arelikely to be observed. What matters is that, the smaller the trade cost, smaller shocks are enough to break symmetry.
Incidentally, there are a couple of ways to extend the model that could ensure that equilibrium without strict
ranking is unstable even to infinitesimal shocks in spite of a small trade cost. For example, the present model
assumes for simplicity, like any standard Ricardian model, that the goods produced by different countries are perfect
substitutes within each sector. This means that introducing a small iceberg trade cost causes a discontinuous shift in
the demand across countries, which is why small (but not infinitesimally) shocks are needed for breaking symmetry.
Instead, consider the Armington specification that the goods produced by different countries within each sector are- 10 -

©Kiminori Matsuyama, Equilibrium Lorenz Curve: A Generalization

The symmetry-breaking mechanism that renders equilibrium without strict ranking
unstable and leads to the emergence of strict ranking across ex-ante identical countries is a twoway causality between the patterns of trade and cross-country productivity differences. A
country with a larger services sector not only has higher TFP, but also comparative advantage intradeable sectors that depend more on services. Having comparative advantage in such sectors
means a larger demand for such services, which leads to an even larger services sector and hence
higher TFP. Since tradeable goods differ in their dependence of services, some countries end up
becoming less productive and poorer than others. Although many similar symmetry-breaking
mechanisms exist in the trade and geography literature, they are usually demonstrated in models
with two countries/regions. One advantage of the present model is that, with a continuum ofgoods, the logic extends to any finite number of countries/regions.
2.5

Equilibrium Lorenz Curve: Limit Case (J  ∞)Even though eq. (17) fully characterizes the equilibrium distribution of country shares, it

is not analytically solvable. Of course, one could try to solve it numerically. However,numerical methods are not useful for answering the question of the uniqueness of the solution orfor determining how the solution depends on the parameters of the model. Instead, in spirit
similar to the central limit theorem, let us approximate the equilibrium Lorenz curve with a large
but a finite number of countries by lim J   J   .12 It turns out that, as J  ∞, eq.(17)converges to the second-order differential equation with two terminal conditions, whose solution
is unique and can be solved analytically. 13 This allows us to study not only what determines theshape of the Lorenz curve, but also conduct various comparative statics, and to evaluate thewelfare effects of trade.
Here’s how to obtain the limit Lorenz curve, lim J   J   . The basic strategy is totake Taylor expansions on both sides of eq. (17).14 First, by setting x  j / J and x  1 / J ,highly but not perfectly substitutable. This would make the property of the model continuous at zero trade cost, sothat even infinitesimal perturbations would be enough for symmetry-breaking even in spite of a small trade cost.
12

Note that this is different from assuming a continuum of countries, as eq.(17) is derived under the assumption that
there are a finite number of countries.
13
From this, the Lorenz curve is unique for a sufficiently large J. I conjecture that the Lorenz curve is unique for any
finite J, but the proof has been elusive, and it remains an open question.
14
Initially, I used a different approximation strategy to obtain the limit Lorenz curve in Proposition 2 of Matsuyama(2013). I am grateful to Hiroshi Matano for showing me this (more efficient) method, without which I might nothave been able to achieve the two extensions in Matsuyama (2013) and generalizations studied in this paper.
- 11 -

©Kiminori Matsuyama, Equilibrium Lorenz Curve: A Generalization

x

2

 
 o  x  ,2

S j 1  S j   ( x  x)  ( x)   ' ( x)x  " ( x ) 2  o x ,x

S j  S j 1   ( x )   ( x  x)   ' ( x )x  " ( x) 2

2

2

from which the LHS of eq. (17) can be written as:
LHS =S j 1  S j

 1

S j  S j 1" ( x )x  o x  .
 ' ( x)Likewise, ( S j , S j 1 ) ( S j , S j 1 )  ( x  x )

 ( s )ds1
  ( ( x ))   ' ( ( x )) ' ( x)x  o( x )
 ( x  x )   ( x)
2
(x)



 ( x)

 ( s )ds1
  (( x))   ' ( ( x))' ( x)x  o( x )
 ( x)  ( x  x )2
 ( x  x )so thatAV( S j , S j 1 )  AV ( ( x)) V
A' V (( x)) ' ( ( x)) ' ( x )x  o( x )2

V
A' V ( ( x )) ' ( ( x)) ' ( x )x  o( x ) ,2from which the RHS of eq.(17) can be written as:AV( S j 1 , S j )   AV ( ( x )) 

 (S j )  ( x )  A(V( S j , S j 1 )) 


VA
'
(
V

(

(
x
)))

RHS =  1  ' ( ( x))' ( x) x  o x  A(V (S , S )) 
A
(
V

(

(
x
)))


j

1
j


 1   (V (( x ))) ' ( ( x )) ' ( x )x  o x 

where we recall  ( n) A' (n )n 0 is the degree of scale economies. By equating the LHS andA( n)RHS of eq.(17),1" ( x)x  o x   1   (V (( x))) ' ( ( x)) ' ( x)x  o x  .
 ' ( x)By letting x  1 / J  0 , we obtain the second-order differential equation:(19)" ( x )  (V ( ( x))) ' ( ( x))' ( x)' ( x)By integrating the above once,- 12 -

©Kiminori Matsuyama, Equilibrium Lorenz Curve: A Generalization

(V (( x)))  (V ( ( x ))) c c 0  exp ' ( x)  e 0VVwhere c0 is an integral constant andlog  ' ( x)  n

( n)    (u ) du0

is an increasing and continuously differentiable function. By integrating the above once again, ( x)  (V ( s )) c
 ds  c1  e 0 x ,V exp
0where c1 is another integral constant.
From the two terminal conditions,  (0)  0 and  (1)  1 , the two integral constants, c0
and c1, are pinned down as:1

  (V (s )) e   exp ds ;V0c0c1  0 .

From this, we obtain to ( x)

1  (V (s )) 
  (V ( s ))  
exp
ds

exp



0  V  0  V ds x .

This can be further rewritten as follows:
Proposition 2: The limit equilibrium Lorenz curve, lim J   J =  , is given by: ( x)(20)hˆ( s )  (V ( s )) with hˆ( s )  expx  H  ( x )   h( s )ds , where h( s )  1

V


ˆ
0
h
(
u
)
du

0

Figure 3 illustrates eq. (20). As shown in the left panel, h(s) is positive and decreasing in s  [0,1]. Thus, its integral, x  H (s) , is increasing and concave. Furthermore, h(s) is normalized insuch a way that H (0)  0 and H (1)  1 , as shown in the right panel. Hence, its inverse function,
the Lorenz curve, s  ( x)  H 1  x  is increasing, convex, with  (0)  0 and  (1)  1 . It isalso worth noting that the Lorenz curve may be viewed as the one-to-one mapping between a setof countries (on the x-axis) and a set of the goods they produce (on the s-axis).
Proposition 2 of Matsuyama (2013) can be obtained as a special case of the above
proposition, where A(n)  (n) , which implies  (n)   > 0 or ( n)  n , and hence- 13 -

©Kiminori Matsuyama, Equilibrium Lorenz Curve: A Generalizationhˆ( s )  exp  ( s )  .

Again, this shows that the constant degree of scale economies was crucial for the equilibrium
distribution in Matsuyama (2013) to be scale-free, i.e., independent of V.
From s  ( x)  H 1  x  , one could obtain GDP of the country at 100x% (with WorldGDP normalized to one), y   ' ( x) , its cumulative distribution function (cdf), x   ( y ) ( ' ) 1 ( y ) , and its probability density function (pdf),  ( y )   ' ( y ) . Table shows one suchcalculation for  (n)   > 0 (i.e., A(n)  (n) ), and an algebraically tractable one-parameter (λ)
family of  functions, which turns out to generate power-law (e.g., truncated Pareto).15 Notethat the power in the pdf is λ/θ  2.
Table: Power-Law Examples
Example 1:

Example 2:

 ( s)  s
Inverse Lorenz
Curve:
Lorenz Curve: ( s )  log 1  (e  1) s

1  e s1  e x  H (s )

Example 3:

1





log 1  (e  1) s 



x

e 1
e  1s   (x )
Cdf:

1
1


1 e
y

 e  1 
1
log
y 

 


1
y 2

1
y

1  e ye  1




e
y 
e  1e 1Pdf:

 ( y )  ' ( y)Support:[ y Min , y Max ]



1





 ( s )  log 1  (e   1) s  (  0;   )1  (e  1)s 

1



e
1



log 1  (1  e  ) x

x   ( y)
 (  ' ) 1 ( y )11  (e


 

 




1

1





 1) x    1

e 1


1

1

 y  y 
  1
1  yy Min 
 1   Maxe    1
1  e   2( /  )  1 ( y)(  /  ) 1
 ( y Min ) ( /  )1 
 ( y Max )    e  1 
  y  e  1         e  1  
e
 

 e  1     

15

Example 1 and Example 2 may be viewed as the limit cases of Example 3, as λ → 0 and λ → θ, respectively. In
addition to being algebraically tractable, the power-law examples have some empirical appeal when “countries” areinterpreted as “cities” or “metropolitan areas”: see, e.g., Gabaix and Ioannides (2004). I am grateful to Fabrizio
Perri for suggesting to me to construct power-law examples.
- 14 -

©Kiminori Matsuyama, Equilibrium Lorenz Curve: A Generalization

Intuitively, with a smaller  , the use of local services is more concentrated at higher-indexed
sectors, so that only a smaller fraction of countries specializes in such “desirable” tradeables.
Hence, a smaller  makes the pdf decline more sharply at the upper end.
Eq.(20) can also be used to investigate when a symmetry-breaking mechanism of thiskind leads to, say, a bimodal distribution, as the narrative in much of this literature, (“coreperiphery” or “polarization”) seems to suggest. When the increasing and continuously
differentiable function (V ()) can be approximated by a two-step function, the corresponding
pdf becomes bimodal, as illustrated in Figure 4. 16 This could occur in two different ways. First,

 () may be approximated by a two-step function. 17 Thus, the world becomes polarized into
the rich core and the poor periphery, when the tradeable goods can be classified into two
categories in such a way that they are roughly homogeneous within each category. Second,() may be approximated by a two-step function, or equivalently,  () has a sharp single peak,that is to say, when the scale economies in the service sector are subject to the single thresholdexternalities. 18 Generally, a symmetry-breaking mechanism of this kind leads to N “clusters” ofcountries if (V ()) can be approximated by an N-step function.
Another advantage of Eq.(20) is that one could easily see the effect of changing V. To
see this, note that the numerator of h(s) , hˆ( s )  exp (V ( s )  / V ) , satisfies  2 log(hˆ( s )) sgn    sgn ' (V ( s )) ( s ) ' (s )   sgn ' (V ( s )).
 Vs 
In words, hˆ( s ) is log-submodular in V and s, for  ' ()  0 and log-supermodular in V and s, if

 ' ()  0 .19 For example, the arrows in Figure 3 illustrates the effect of a higher V for  ' ()  0 .
16

Formally, consider a sequence of (increasing and continuously differentiable) (V ()) functions that converges

point-wise to a two-step function, (V (s ))   L for s  [0, s ) and (V (s ))   H   L for s  (s ,1] . Then, thesequence of the corresponding cdf’s converges to the cdf,  ( y ) = 0 for y  1  (1  s )(1  e  ) ; ( y ) = [1  (1 / s  1)e  ]1 for 1  (1  s )(1  e  )  y  1  s (e  1) , and  ( y ) = 1 for y  1  s (e  1) , where  H  L  0 .
17
Note that this is different from assuming that  is a two-step function, which is equivalent to assume that there
are only two tradeable goods. Then, the equilibrium distribution would not be unique; see Matsuyama (1996). To
obtain the uniqueness, it is essential that  is increasing, which means that the set of the tradeable goods is acontinuum, and hence outnumbers the set of the countries for a large but finite number, J.
18
Obviously, this second possibility is absent in Matsuyama (2013), with the constant degree of scale economies.
19
See Topkis (1998) for mathematics of super-(and sub-)modularity and Costinot (2009) for a recent application tointernational trade.

- 15 -

©Kiminori Matsuyama, Equilibrium Lorenz Curve: A Generalization

In this case, hˆ( s ) is log-submodular, which means that a higher V shifts the graph of hˆ( s ) down
everywhere but proportionately more at a higher s. Since h(s) is a rescaled version of hˆ( s ) to
keep the area under the graph unchanged, the graph of h(s) is rotated “clockwise” by a higher V,
as shown in the left panel. This “single-crossing” in h(s) implies that a higher V makes theLorenz curve more “curved” and move further away from the diagonal line everywhere (i.e.,
without any “Lorenz-crossing”), as shown in the right panel. In short, a higher V causes a
Lorenz-dominant shift of the Lorenz curve away from the diagonal. Thus, any Lorenz-consistentinequality measure, such as the generalized Kuznets Ratio, the Gini index, the coefficients ofvariations, etc. all agree that a higher V leads to greater inequality, when the degree of scaleeconomies is an increasing function. Likewise, the arrows in Figure 3 also illustrates the effectof a lower V for the log-supermodular case,  ' (n)  0 . In other words, a higher V in this casecauses a Lorenz-dominant shift of the Lorenz curve in the direction opposite of the arrows, thatis, closer to the diagonal, hence less inequality. 20
2.6

Welfare Effects of Trade
The mere fact that trade creates ranking of countries, making some countries poorer thanothers, does not necessarily imply that trade make them poorer. It could simply mean that theygain less from trade than others. To see whether trade would make some countries poorer, weneed to compare the utility levels under trade and under autarky. From eq.(1), the welfare underautarky is

 



1







log U A  log  AV   log P A ( s ) ds .
0

Likewise, the welfare of the country that ends up being the j-th poorest can be written as
1log U j   log  jV    logP (s ) ds ,0

where the tradeable goods prices satisfy

P( s )   k  A(nk ) P A ( s )   A  A(n A )  ( s )   A(Vk )   kA A    A(V )  ( s )for s  (Sk−1, Sk) for k = 1, 2, …, J.

20

Likewise, any upward shift in Θ(●) would lead to greater inequality. The effect of θ, discussed in Matsuyama(2013) may be viewed as a special case of this observation with Θ(n) ≡ θn.

- 16 -

©Kiminori Matsuyama, Equilibrium Lorenz Curve: A Generalization

Combining these equations yields
Uj   j  1  P( s) log A   log A    log A dsU    0  P ( s) Sk  j  J  Sk
 A(Vk )   k 
 ds  , log A      log A ds    ( s ) log
A 
 A(V )     k 1  Sk 1   
S k 1

which can be further rewritten as follows:
Proposition 3 (J-country case): The country that ends up being the j-th poorest under tradegains from trade if and only if:
(21)

J
J
Uj 
j 
 A(Vk ) ( S k  S k 1 ) > 0log A  =  log S k  S k 1  +  k log
A 
k 1
k 1
 A(V ) U  k 

Eq. (21) in Proposition 3 offers a decomposition of the welfare effects of trade. The first term isthe country’s TFP relative to the world average, and it is increasing in j , negative at j = 1 andpositive at j = J. The second term captures the usual gains from trade (i.e., after controlling forthe income and TFP differences across countries) and it is always positive. However, aside fromrather obvious statements like “a country gains from trade if the second term dominates the firstterm,” or “a country gains from trade if its income (and TFP) ends up being higher than theworld average,” Proposition 3 offers little insight without an explicit solution for eq. (17).
As J   , the task of evaluating the overall welfare effect becomes greatly simplified.
By setting x* = j/J and x = k/J in eq. (21) and noting that  j / k  ' ( x*) / ' ( x) andS k  S k 1   ' ( x)dx as J   , eq.(21) converges to:11 A(V ( ( x)))  ' ( x*)  U ( x*)  ' ( x)dx
 ' ( x)dx    ( ( x)) log
log
  log
A 
A
(
V

)
 U  0  ' ( x ) A0(V (( x)))
 c 0 , this can be rewritten as:V11 A(V (( x)))  U ( x*)  (V ( ( x*)) (V (( x)) d ,log d    ( ( x)) log
A 
V
V
A
(
V

)
 U  0A
0Since eq. (19) implies log  ' ( x)  11 A(V ( s )) (V (s*))
 (V ( s ))  ds .
 
ds    ( s ) log
VV A(VA ) 00

To summarize:- 17 -

©Kiminori Matsuyama, Equilibrium Lorenz Curve: A Generalization

Proposition 4 (Limit case; J  ∞): The country that ends up being at 100x* percentile under
trade gains from trade if and only if:11
 A(V ( s )) 
 U ( x*)  (V (s*))
 (V ( s ))  ds > 0,
log

 
ds    ( s ) log
A 
V
V
 U 

 A(VA ) 
0
0

(22)

where s*   ( x*) or x*   1 ( s*) .
Proposition 4 offers a decomposition of the welfare effects of trade, similar to Proposition 3. In
fact, Proposition 4 allows us to say a lot more about the overall welfare effects of trade, as shownin the following two Corollaries.
Corollary 1: All countries gain from trade if and only if
11
 A(V (s )) (V (0))
 (V ( s )) ds .
ds
 ( s ) log0  V V A(VA ) 
0

To see what is involved in this condition, consider the case A(n)  (n) , or  (n)   , as in
Matsuyama (2013). Then, this can be rewritten as:1
 ( 0)  ( s)    ( s) (23)
1  A    A  log A  ds = Diversity (Theil index/entropy) of γ.
    0
Note that the sufficient and necessary condition under which all countries gain from trade, (23),depends solely on γ(•). In particular, it is independent of θ, which only plays a role ofmagnifying the gains and losses from trade. LHS of eq.(23) shows how much the share of theservice sector, and hence productivity, declines in the country that ends up being the poorest.
RHS of eq. (23) is the Theil index (or entropy) of  , which measures its dispersion. This
corollary thus states that trade is Pareto-improving (i.e., even the country that ends up being thepoorest would benefit from trade) when the tradeable goods are sufficiently diverse, as measured
by the Theil index of  , and hence the gains from specialization (by making countries ex-post
heterogeneous) is sufficiently large.
Corollary 2: Suppose that the condition of Corollary 1 fails. Then, for sc > 0, defined by11 A(V ( s )) (V ( sc )) (V ( s )) ds .
 
ds    ( s ) log
VV A(VA ) 00a): All countries producing and exporting goods s  [0, sc) lose from trade, while all countries
producing and exporting goods s  (sc, 1] gain from trade.

- 18 -

©Kiminori Matsuyama, Equilibrium Lorenz Curve: A Generalizationb): Consider a shift parameter, σ > 0, such that A(n; )  [ A(n;1)] .21 Then, s c is independent ofσ, and the fraction of the countries that lose, xc  H ( s c ; ) , is increasing in σ with lim x c  s c 0

and lim xc  1 .
 Corollary 2a) follows immediately from eq.(22) and the definition of sc. The independence of scfrom σ follows from that A(n; )  [ A(n;1)] implies (n; )  (n;1) , and hence the definitionof sc can be rewritten as
11 A(V (s ); ) (V ( sc ); ) (V (s ); ) ds  
ds    ( s ) log
VV A(VA ; ) 0
01 (V ( sc );1) 1  (V ( s );1)  A(V ( s );1)  ds   0 .
  ds    ( s ) log
V
V
A
(
V

;
1
)



A
 
0
0


Corollary 2 is illustrated in the right panel of Figure 3. Corollary 2a) states that all
countries that end up specializing in [0, sc) lose from trade and they account for xc fraction of
the world. 22 Corollary 2b) states that, a higher σ shifts the Lorenz curve but sc is unaffected sothat xc goes up vertically. As varying σ from 0 to ∞, xc increases from sc to 1. So, perhaps
surprisingly, when  () and A(;1) satisfy the condition that some countries lose from trade,virtually all countries could lose from trade, as σ goes to ∞.

3.

Two Extensions
The above model can be generalized in many directions. This section offers two

extensions. The first allows a fraction of the consumption goods within each sector to be
nontradeable. By reducing the fraction, this extension enables us to examine how inequality
Obviously, for the case discussed in Matsuyama (2013), A(n)  (n) , hence θ works as such a shift parameter.
At one of the seminars where Matsuyama (2013) was presented, someone commented that Corollary 2a) implies acomplete unraveling of trade if the countries choose whether to trade. He argued that, after xc > 0 fraction of thecountries decide to go back to autarky, the xc fraction of the remaining countries would also go back to autarky, andthis process would continue unless no countries would be left to trade. This comment is, however, false, because
Corollary 2a) is a limit result, which offers a good approximation for a sufficiently large J. As more countries leave
and J becomes smaller, this approximation becomes less and less accurate. Indeed, with a small J, even the country
that ends up being the poorest might produce a sufficiently wide set of tradeables, which could ensure that even thepoorest gains from trade, in which case no country has an incentive to stop trading. Indeed, in her insightfulcomments, Emily Blanchard suggested to me that, under the condition that ensures xc > 0, the model of Matsuyama(2013)—hence this model as well—, may offer a model of the equilibrium size of the trading bloc. It seems,
however, that characterizing the equilibrium size would be feasible only numerically.
21
22- 19 -

©Kiminori Matsuyama, Equilibrium Lorenz Curve: A Generalization

across countries is affected by globalization through trade in goods.23 The second allows
variable supply in one of the components in the composite of primary factors, either throughfactor accumulation or factor mobility. This extension not only generates the correlation
between the capital-labor ratio and per capita income and TFP. By changing the share of thevariable primary factor in the composite, this extension also allows us to examine how inequality
across countries is affected by technological change that increases importance of human capitalor by globalization through trade in factors.
3.1

Nontradeable Consumption Goods: Globalization through Trade in Goods

In the model of section 2, all consumption goods are assumed to be tradeable. Assume now thateach sector-s produces many varieties, a fraction τ of which is tradeable and a fraction 1−τ isnontradeable, and that they are aggregated by Cobb-Douglas preferences. 24 The expenditure
function is now obtained by replacing log( P( s)) with  log( PT ( s ))  (1   ) log( PN (s )) for each s [0,1], where PT ( s )  Min{C j ( s ))} is the price of each tradeable good in sector-s, common
across all countries, PN ( s )  C j (s ) is the price of each nontradeable good in sector-s, which isequal to the unit of cost of production in each country.
Instead of going through the entire derivation of the equilibrium, only the key steps will
be highlighted below. Again, let n j j 1 be a monotone increasing sequence. As before,J (S j )(12) j 1  A(n j 1 )  j  A(n j ) (13)

Y j  jV  (S j  S j 1 )Y W .

 1.

(j = 1, 2, ..., J−1)
(j = 1, 2, ...,J )However, the equilibrium size of the service sector is now given by, instead of (16):
(24)





n j   j  (1   ) A V .

Combining these equations yields

23

Emi Nakamura suggested another interpretation of this thought experiment. By increasing τ one could also see
how inequality across regions change, as one looks at more and more disaggregated levels of geographical units.
24
This specification assumes that the share of nontradeable producer services in sector-s is γ(s) for both nontradeable
and tradeable consumption goods. This assumption is made because, when examining the effect of globalization bychanging τ, we do not want the distribution of γ across all tradeable consumption goods to change. However, forsome other purposes, it would be useful to consider the case where the distribution of γ among nontradeable
consumption goods differ systematically from those among tradeable consumption goods. For example, Matsuyama(1996) allows for such possibility to generate a positive correlation between per capita income and the nontradeable
consumption goods prices across countries, similar to the Balassa-Samuelson effect.
- 20 -

©Kiminori Matsuyama, Equilibrium Lorenz Curve: A Generalization

Proposition 5 (J-country case): Let S j be the cumulative share of the J poorest countries.
Then, S j j 0 solves the 2rd order difference equation with two terminal conditions:
J







 (S j ) A V [ (S j , S j 1 )  (1   ) A ] 
S j  S j 1  A V [ (S j 1 , S j )  (1   ) A ] S j 1  S j

 1 with S 0  0 & S J  1 ,Sj

1where ( S j 1 , S j )  (s )ds .
S j  S j 1 Sj1
This equilibrium converges to a collection of J identical single-country (autarky) equilibria as τ
 0 and to the J-country trade equilibrium shown in Proposition 1, as τ  1.
Now, let us calculate the limit Lorenz curve by following the steps similar to those used in
section 2.5. As before, by setting x  j / J and x  1 / J ,LHS =S j 1  S j
S j  S j 1

( S j , S j 1 ) ( S j 1 , S j ) " ( x)x  o x  .
 ' ( x)

1
 ( x  x )

 ( s )ds1
  ( ( x))   ' ( ( x))' ( x )x  o( x )
 ( x  x)  ( x)2
 ( x)( x) ( s )ds1
  ( ( x))   ' ( ( x )) ' ( x)x  o( x )
 ( x)   ( x  x)
2
 ( x  x )so thatA V [ j  (1   ) A ]







= A V [ ( ( x))  (1   ) A ] + V A' V [ ( ( x))  (1   ) A ] d ( ( ( x))) x  o( x )

A V [j 1  (1   ) A ]

2

dx







= A V [ ( ( x))  (1   ) A ]  V A' V [ ( ( x))  (1   ) A ] d ( (( x))) x  o( x )2

dxfrom which
RHS = 1  V (( x)) A' V [ (( x))  (1 A  ) ] d ( ( ( x))) x  o( x )A

A V [ ( ( x))  (1   ) ]

dx

= 1   V [ (( x ))A  (1   ) ] d ( ( ( x))) x  o( x )A

1  (1   ) /  ( ( x))dx- 21 -

©Kiminori Matsuyama, Equilibrium Lorenz Curve: A Generalization

Combining these yields

1A" ( x)x  o x  = 1   V [ (( x ))A  (1   ) ] d ( ( ( x))) x  o( x )
 ' ( x)1  (1   ) /  ( ( x))dxHence, as J   , x  1 / J  0 ,A
" ( x )=  V [ ( ( x))A  (1   ) ] d ( ( ( x))) ' ( x)1  (1   ) /  ( ( x))dx

Integrating once, (  ( x ))log  ' ( x)  0  ( ( x ))  V [v  (1   ) A ] 
 V [v  (1   ) A ]
dv

c

exp
dv ' ( x) e c0 .
 
0A
1  (1   ) A / v
1

(
1


)

/

v
0



Integrating once more,  ( s )  V [  (1   ) A ] cexp0  0 1  (1   ) A / v dvds e 0 x  c1



( x)

From  (0)  0 &  (1)  1 , the two integral constants are pinned down to obtain:
Proposition 6 (Limit Case; J   ): The limit equilibrium Lorenz curve,  , is given by: ( x)x  H  ( x);    h( s; )ds ,0  ( s )  V [v  (1   ) A ] ˆwhere h( s; )  1
with h( s; )  exp  
dv  .
Aˆh(u; )du 0 1  (1   ) / vhˆ( s; )0

Proposition 5 of Matsuyama (2013) can be obtained as a special case of the above.
Again, Figure 3 illustrates the above proposition. For each  , h( s; ) is positive, anddecreasing in s , and it is normalized so that its integral from 0 to 1 is equal to 1. Thus, H ( s; )is increasing and concave in s , with H (0; )  0 and H (1; )  1 . Hence,  ( x; )  H 1  x;  isincreasing and convex in x, with  (0; )  0 and  (1; )  1 . Note also lim hˆ( s; )  1 . Thus, as 0

τ  0, each country converges to the same single-country (autarky) equilibrium and hence theLorenz curve converges to the diagonal line, and inequality disappears. Likewise, lim hˆ( s;  )  1exp (V ( s )) / V  . Thus, as τ  1, the Lorenz curve converges to the one shown in
Proposition 2.

- 22 -

©Kiminori Matsuyama, Equilibrium Lorenz Curve: A Generalization

As before, let us look at the log-super(sub)modularity of hˆ( s ) to see the effects ofparameter changes. From log(hˆ( s; )) V [ ( s )  (1   ) A ] ' ( s) ,s1  (1   ) A /  (s )we have  2 log(hˆ( s; )) Asgn    sgn  ' V [ ( s )  (1   ) ] .
Vs



 



Again, for  ' ()  0 , hˆ( s ) is log-submodular in V and s, so that a higher V causes a Lorenzdominant shift away from the diagonal line, implying a greater inequality across countries, whilefor  ' ()  0 , hˆ( s ) is log-supermodular in V and s, so that a higher V causes a Lorenz-dominant
shift toward the diagonal line, implying a less inequality acourss countries. For  (n)   , theLorenz curve is independent of V but it also implies  2 log(hˆ(s; )) / s  0 , so that a higher causes a Lorenz-dominant shift away from the diagonal, i.e., a greater inequality. Thus, theeffects of V obtained in section 2 for τ = 1 extends for τ < 1.
For the effect of τ, the above Proposition suggests that changing from τ = 0 to τ > 0
clearly causes a Lorenz-dominant shift away from the diagonal, hence a greater inequality.
However, this does not necessarily mean that any small increase in τ would cause a Lorenzdominant shift. Matsuyama (2013) already shows that  2 log(hˆ(s; )) / s  0 for  (n)   .
Another sufficient condition for  2 log(hˆ(s; )) / s  0 is  ' (n)  0 for n  V A and  ' (n)  0
for n  V A . When these conditions are met, a higher τ causes a Lorenz-dominant shift awayfrom the diagonal, suggesting that globalization through trade in goods leads to greater inequality
across countries.
3.2

Variable Factor Supply: Effects of Factor Mobility and/or Factor Accumulation
Returning to the case where τ = 1, this subsection instead allows the available amount of

the composite primary factors, V, to vary across countries by endogenizing the supply of one of
the component factors, K, as follows:(25)

Vj = F(Kj,L)with

ωjFK(Kj, L) = ρ.

- 23 -

©Kiminori Matsuyama, Equilibrium Lorenz Curve: A Generalization

where FK(Kj, L) is the first derivative of F with respect to K, satisfying FKK < 0. In words, thesupply of K in the j-th country responds to its TFP, ωj, such that its factor price is equalized
across countries at a common value, ρ. This can be justified in two different ways.
A. Factor Mobility: Imagine that L represents (a composite of) factors that are immobile acrossborders and K represents (a composite of) factors that are freely mobile across borders, whichseek higher return until its return is equalized in equilibrium. 25 According to this interpretation,
ρ is an equilibrium rate of return determined endogenously, although it is not necessary to solvefor it when deriving the Lorenz curve.26
B. Factor Accumulation: Reinterpret the structure of the economy as follows. Time iscontinuous. All the tradeable goods, s  [0,1], are intermediate inputs that goes into the1
production of a single final good, Yt, with the Cobb-Douglas function, Yt  exp   log( X t ( s ))ds 
 0

1
so that its unit cost is exp   log( Pt ( s ))ds  . The representative agent in each country consumes
 0





and invests the final good to accumulate Kt, so as to maximize  u (Ct )e  t dt s.t. Yt  Ct  K t ,0where ρ is the subjective discount rate common across countries. Then, the steady state rate of
return on K is equalized at ρ. 27 According to this interpretation, K may include not only physicalcapital but also human capital, and the Lorenz curve derived below represents steady state
inequality across countries.
Again, only the key steps will be shown. Let n j j 1 be monotone increasing. From (12)J

and (25),

25

Which factors should be considered as mobile or immobile depends on the context. If “countries” are interpreted
as smaller geographical units such as “metropolitan areas,” K may include not only capital but also labor, with L
representing the immobile “land.” Although labor is commonly treated as an immobile factor in the trade literature,
we will later consider the possibility of trade in factors, in which case certain types of labor should be included
among mobile factors.
26
Also, Yj = Vj = ωjF(Kj, L) should be now interpreted as GDP of the economy, not GNP, and Kj is the amount of K
used in the j-th country, not the amount of K owned by the representative agent in the j-th country. This also meansthat the LHS of the budget constraint in the j-th country, eq.(1), should be its GNP, not its GDP (Yj). However,calculating the distributions of GDP (Yj), TFP (ωj), and Kj/L does not require to use the budget constraint for eachcountry, given that all consumption goods are tradeable (τ = 1). The analysis would be more involved if τ < 1.
27
The intertemporal resource constraint assumes not only that K is immobile but also that international lending andborrowing is not possible. Of course, these restrictions are not binding in steady state, because the rate of return isequalized across countries at ρ
- 24 -

©Kiminori Matsuyama, Equilibrium Lorenz Curve: A Generalization(26)

 A(n j 1 )  j 1  
FK ( K j 1, L)  j  A(n j ) FK ( K j , L)

 (S j )1which implies that  j j 1 , K j j 1 , V j j 1 , and Y j j 1 are all monotone increasing in j, hence
J

J

J

J

they are (perfectly) positively correlated.

Since V j is now endogenous, eq.(13) and eq.(15) needbe modified as:(27)

n j   jV j   j F ( K j , L)(28)

Y j  jV j  j F ( K j , L)  ( S j  S j 1 )Y W .

These equations can so be summarized as: V
V  A j 1V j 1   j 1 j 1 = j 1 
Yj
S j  S j 1
 jV jV j  A jV j  

Y j 1

S j 1  S j

 (S j ) 1.

To see what is involved, suppose V = F(K, L) = ZKα with 0 <  < 1 /(1   ()) . Then,
Proposition 7 (J-country case): Let S j be the cumulative share of the J poorest countries inincome. Then, S j j 0 solves:JK j 1 Y j 1 S j 1  S j   j 1 




Kj
Yj
S j  S j 1   j 

1
1 







 (S j ) A Z K j 1  ( S j , S j 1 )  1
 1 with S 0  0 & S J  1 , A Z K  ( S , S ) 
j
j 1
j




Sj

1
where ( S j 1 , S j )  (s )ds .
S j  S j 1 Sj1
Two remarks are in order. First, S j j 0 here represents the cumulative shares in Y/L and in K/L,
J

not in TFP. However, the distribution of TFP can be obtained from the distribution of Y/L (orK/L) and vice versa, using a monotone transformation,  j 1 /  j  Y j 1 / Y j 1. Second, theabove condition does not fully characterize the equilibrium distribution. Generally, we need
another condition to pin down the level of Y (or K/L), because the symmetry-breaking
mechanism is not scale-free under a variable degree of scale economies, and the supply of
primary factors is now variable. For A(n)  (n) , this condition can be rewritten as the second
order difference equation in S j j 0 as follows, which fully characterizes the equilibrium LorenzJcurve:- 25 -

©Kiminori Matsuyama, Equilibrium Lorenz Curve: A Generalization

Corollary 3 (the J-country case): Let A(n)  (n) . Then, S j j 0 solves:J ( S j )S j 1  S j   (S j , S j 1 ) 1  ( S j ) 1 with S 0  0 & S J  1 ,S j  S j 1  ( S j 1 , S j ) Sj

1
where ( S j 1 , S j )  (s )ds .
S j  S j 1 Sj 1
Again, let us calculate the limit Lorenz curve. As before, by setting x  j / J andx  1 / J ,LHS =K j 1KjS j 1  S j
S j  S j 1

= 1" ( x)x  o x 
 ' ( x)1
( S j , S j 1 )   ( ( x ))   ' (( x))' ( x )x  o( x )2
1
( S j 1 , S j )   ( ( x ))   ' ( ( x))' ( x )x  o( x ) .
2

Furthermore, we have

K ( x  x)   K ( x) 1   " ( x) x   o x  ,' ( x )

from whichA Z K j 1   (S j , S j 1 )= AZ K ( x)   (( x)) 1   Z K ( x)   (( x)) "( x)   ' (( x))' ( x) x   o( x )

A Z K j  ( S j 1 , S j ) ' ( x)2 ( ( x))







= AZ K ( x)   (( x)) 1   Z K ( x)   (( x))   ' (( x))' ( x) x   o( x )

2 ( ( x))from which







 ( S j ) /(1  ) A Z K j 1  ( S j , S j 1 ) 
RHS = 
 A Z K  ( S , S ) 
j
j 1
j



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©Kiminori Matsuyama, Equilibrium Lorenz Curve: A Generalization= 1

 " ( x)  ' ( ( x )) ' ( x)  (( x))x  o( x )
 Z  K ( x)   ( ( x))  1
 ( ( x))   ' ( x)Combining these and let J   , x  1 / J  0 yields" ( x ) Z K  ' ( x )   (( x))d ( ( x )) ' ( x ) 1     ( ( x)) Z K  ' ( x )  (( x))
dx






where use has been made ofK ' ( x ) " ( x)or K ( x )  K  ' ( x) ,K ( x)  ' ( x)where K is the average of K. By setting V  Z (K ) ,Proposition 8 (Limit Case, J  ∞): The limit equilibrium Lorenz curve in income,  , solves:







" ( x ) V  ' ( x )   ( ( x )) ' ( ( x)) ' ( x ) ' ( x) 1     (( x)) V  ' ( x )  ( ( x ))with  (0)  0 &  (1)  1 .
Again, some remarks are in order. First,  here represents the Lorenz curve in Y/L and in K/L,
not in TFP. The Lorenz curve in TFP can be obtained from  , using a monotonex

1 transformation,  ( x)   ' (u ) 01

du /    ' (u ) 
 0

1 

du  . Second, the above differentialequation generally depends on V  Z (K ) . Under Interpretation A (Factor Mobility), K may beinterpreted as the initial endowment of the mobile factor in each country and hence be treated as
exogenous, so that the above equation fully characterizes the equilibrium Lorenz curve.
However, under Interpretation B (Factor Accumulation), K is endogenous so that we need
another condition to pin it down. Again, this is because the symmetry-breaking mechanism is
not scale-free under a variable degree of scale economies. Third, for α > 0, the above differential
equation does not have a closed form solution for a general A(n) or  (n) , because

 ' ( x ) appears inside  () .28 For A(n)  (n) ,  (n)  n , which enables us to solve it explicitly
as follows:28

For α = 0, it is equivalent to eq. (19).
- 27 -

©Kiminori Matsuyama, Equilibrium Lorenz Curve: A Generalization

Corollary 4 (Limit Case, J  ∞): Let A(n)  (n) . Then, the limit equilibrium Lorenz curve,
 , solves:" ( x )d ( ( x )) ' ( x ) 1     ( ( x ))dx
with  (0)  0 &  (1)  1 , whose unique solution is:1/ hˆ( s )
with hˆ( s )  1    ( s ) x  H  ( x )   h( s )ds , where h( s )  1
 1

0
 hˆ(u )du ( x)

0This reproduces Proposition 6 of Matsuyama (2013). Again, Figure 3 illustrates this. For each < 1 1 /  = 1 /(1   ) , h( s; ) is positive, and decreasing in s , and it is normalized so that its
integral from 0 to 1 is equal to 1. Thus, H ( s;  ) is increasing and concave in s , with

H (0; )  0 and H (1; )  1 . Hence,  ( x; )  H 1  x;  is increasing and convex in x, with
 (0;  )  0 and  (1; )  1 . It is also easy to check lim hˆ( s; )  exp   ( s ) .
 0Again, the effect of α can be seen by noting that
1/ 




hˆ( s; )  1 
 ( s)  1 
,

is log-submodular in α and s (and in θ and s). Thus, a higher α (and a higher θ) causes a Lorenzdominant shift away from the diagonal, as shown in the arrows in the right panel of Figure 3.
This result thus suggests that skill-biased technological change that increases the share of humancapital and reduces the share of raw labor in production, or globalization through trade in somefactors, both of which can be interpreted as an increase in α, could lead to greater inequality
across countries.

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©Kiminori Matsuyama, Equilibrium Lorenz Curve: A Generalization

References:
Combes, P.-P., T. Mayer and J. Thisse, Economic Geography, Princeton University Press, 2008.
Costinot, A., “An Elementary Theory of Comparative Advantage,”Econometrica, 2009, 1165-92.
Dixit, A. K., and J. E. Stiglitz, “Monopolistic Competition and Optimum Product Diversity,”
American Economic Review, 1977, 297-308.
Dornbusch, R., S. Fischer, and P. A. Samuelson, “Comparative Advantage, Trade, and Payments
in a Ricardian Model with a Continuum of Goods,” American Economic Review, 67(1977), 823-839.
Ethier, W., “Decreasing Costs in International Trade and Frank Graham’s Argument forProtection,” Econometrica, 50, 1982, 1243-1268.
Fujita, M., P., Krugman, P. and A. Venables, The Spatial Economy, MIT Press, 1999.
Gabaix, X., and Y.M. Ioannides, “The Evolution of City Size Distributions,” in J.V. Henderson
and J.F. Thisse (eds.) Handbook of Regional and Urban Economics, Elsevier, 2004.
Helpman, E., “Increasing Returns, Imperfect Markets and Trade Theory,” Chapter 7 in
Handbook of International Economics, edited by R. Jones and P. Kenen, 1986.
Ioannides, Y., “Emergence,” in L. Blume and S. Durlauf, eds., New Palgrave Dictionary of
Economics, 2nd Edition, Palgrave Macmillan, 2008.
Krugman, P. and A. Venables, “Globalization and Inequality of Nations,” Quarterly Journal of
Economics, 110 (1995), 857–80.
Lewis, W.A. The Evolution of the International Economic Order,Princeton University Press, 1977.
Matsuyama, K., “Why Are There Rich and Poor Countries?: Symmetry-Breaking in the World
Economy,” Journal of the Japanese and International Economies, 10 (1996), 419-439.
Matsuyama, K., “The Rise of Mass Consumption Societies,” Journal of Political Economy, 110,October 2002, 1035-1070.
Matsuyama, K., “Symmetry-Breaking” in L. Blume and S. Durlauf, eds., New Palgrave
Dictionary of Economics, 2nd Edition, Palgrave Macmillan, 2008.
Matsuyama, K., “Endogenous Ranking and Equilibrium Lorenz Curve Among (ex-ante)
Identical Countries,” May 2011 version, http://faculty.wcas.northwestern.edu/~kmatsu/.
Matsuyama, K., “Endogenous Ranking and Equilibrium Lorenz Curve Among (ex-ante)
Identical Countries,” Econometrica, Vol. 81:5 (September 2013), 2009-2031.
Myrdal, G. Economic Theory and Underdeveloped Regions, Duckworth, 1957.
Topkis, D., Supermodularity and Complementarity, Princeton University Press, 1998.

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©Kiminori Matsuyama, Equilibrium Lorenz Curve: A Generalization

Figure 1: Comparative Advantage and Patterns of Trade in the J-country World
Cj−1(s)/Cj(s)Cj(s)/Cj+1(s)

S j−1

Sj

1

O

1

s

sj

Figure 2: Equilibrium Lorenz curve, ΦJ: A Graphic Illustration for J = 4
1
s4

S3
s3
S2
s2
S1
O

1/4

2/4

3/4

1

- 30 -

©Kiminori Matsuyama, Equilibrium Lorenz Curve: A Generalization

Figure 3: Limit Equilibrium Lorenz Curve, Φ(x), and its Lorenz-dominant Shiftx=H(s) = Φ−1(s)
h(s)
1

xc
xc

s
O

O

1

sc

s=Ф(x)=H−1(s)1

Figure 4: Polarization or Bimodal Distributionh(s)x=H(s)

1O

Ф(x)=ss1

O1- 31 -