©Kiminori Matsuyama, Engel’s Law in the Global Economy

Engel’s Law in the Global Economy: Demand-Induced
Patterns of Structural Change and Trade across Countries*

By Kiminori Matsuyama**

March 31, 2017

Abstract: The expenditure shares are more skewed towards higher income-elastic
sectors in richer countries. Through such an endogenous demand composition due tononhomothetic demand (Engel’s Law), economic growth and globalization affect i)
sectoral compositions in employment and in value-added, ii) variations in innovation
rates and in productivity change across sectors, iii) intersectoral patterns of tradeacross countries, and iv) migration of industries from rich to poor countries. These
effects, though interconnected, have been studied separately, which can bemisleading. This paper offers a unifying perspective on how economic growth andglobalization affects interdependent patterns of structural change and trade acrosscountries in the presence of Engel’s Law, capturing all these effects as well as theirinteractions. Among the main messages is that globalization magnifies, instead of
reducing, the power of endogenous domestic demand composition differences as adriver of structural change.
Keywords: Implicitly directly additive isoelastic nonhomothetic CES, Dixit-StiglitzKrugman monopolistic competition model of production and trade, The Schmookler
effect and directed productivity change, Home market effect in employment and inpatterns of trade, The Linder effect, Product cycles, Terms of trade change and factor
price convergence, Leapfrogging, Trade patterns reversal, Log-supermodularity,
Monotone likelihood ratio, Monotone comparative statics*This paper expands a section from Matsuyama (2015). As such, it benefitted from comments on that paper by P.
Antras, S. Athey, G. Barlevy, K. Behrens, R. Benabou, F. Buera, A. Costinot, J. Dingel, D. Donaldson, P.
Fajgelbaum, G. Grossman, E. Helpman, J. Kaboski, F. Lippi, R. Lucas, E. Luttmer, J. Markusen, M. Mestieri, N.
Stokey, B. Strulovici, J. Ventura, X. Vives, J. Vogel, I. Werning, F. Zilibotti, J. Zweimueller, the referees as well asworkshop participants at Chicago Fed, LSE, 2015 NBER Summer Institute (Income Distribution and Macro), Yale,RIETI, Hitotsubashi, Tokyo, Keio, CREI-Pompeu Fabra, Autonoma, MIT, New Economic School, HSESt.Petersburg, Columbia, and LUISS.
**Homepage: http://faculty.wcas.northwestern.edu/~kmatsu/; Email: k-matsuyama@norhtwestern.edu.

1©Kiminori Matsuyama, Engel’s Law in the Global Economy

1.

Introduction
Income elasticity differs across sectors. And the expenditure shares are more skewed

towards higher income elastic sectors in richer countries, both in times series and in cross-sectionof countries. Such an endogenous demand composition due to the nonhomotheticity of demandacross sectors, which we shall call Engel’s Law for the brevity, is an important channel throughwhich economic growth and globalization affect the patterns of structural change and tradeacross countries. For example, Engel’s Law plays a central role in accounting for changing
sectoral shares in employment and in value-added. It could also affect variations in productivity
growth rates across sectors, since the market size is a crucial factor in providing incentives forinnovations, as pointed out by Schmookler (1966) and many others. It also affects theintersectoral patterns of trade between rich and poor countries. As Linder (1961) argued, thestructure of an economy responds and adjusts relatively more to its domestic markets than to its
export markets, so that the difference in the demand composition across rich and poor countries
causes the rich (poor) to develop comparative advantage in the higher (lower) income-elastic
sectors, while relying on importing more from the poor (rich) in the lower (higher) income
elastic sectors. It could also play a crucial role in determining the migration patterns ofindustries from rich to poor countries. Vernon (1966), in particular, proposed the product cycle
hypothesis; new products are initially introduced in high-income countries, where they findmuch of their demand, and then migrate to low income countries, as the world economy grows.
As discussed in the literature review at the end of this section, some of these effects havebeen a subject of previous studies, but they have been mostly treated separately. This could be
misleading, as these effects are interconnected. For example, many studies in the structuraltransformation literature use models where productivity change in each sector is exogenously
given and ask to what extent the changing patterns of sectoral shares in employment and invalue-added can be accounted for by the demand nonhomotheticity or by productivity growthdifferences across sectors. Such an “income elasticity versus productivity growth” approach is afalse dichotomy in the presence of the Schmookler effect, because the relative productivity
growth rates across sectors respond endogenously to changes in the relative market sizes causedby economic growth due to the Engel’s Law. Furthermore, the existing studies typically use
closed economy models, where the domestic supply is necessarily equal to the domestic demand2©Kiminori Matsuyama, Engel’s Law in the Global Economy

sector-by-sector. Since the domestic supply composition no longer need to be equal to thedomestic demand composition in an open economy, one might think, intuitively, thatinternational trade would make the domestic demand composition less important as a driver ofstructural change. If so, one’s intuition is faulty. First, international trade would generate
productivity gains. The resulting income effect would cause a further sectoral shift in the
expenditure through Engel’s Law. Second, in the presence of the Linder effect, rich (poor)
countries would allocate even more resources in higher (lower) income-elastic sectors under
trade than under autarky, which means that the domestic demand composition would have morethan proportional effects on the domestic supply composition. Furthermore, product cycles, or
migration of industries from rich to poor countries could be a mechanism through which bothrich and poor countries achieve sectoral shifts towards more income-elastic, because thoseindustries that migrate from rich to poor may be less (more) income-elastic relative to the others
operating in rich (poor) countries. Viewed this way, product cycles should be regarded as anintegral part of the interdependent patterns of structural change across rich and poor countries.
For all these reasons, globalization could amplify, instead of reducing, the power of Engel’s Law
and the endogenous domestic demand composition differences across countries as a driver ofstructural change in the global economy.
The goal of this paper is to offer a unifying perspective on how economic growth andglobalization affect the interdependent patterns of structural change and trade across countries
due to Engel’s Law. With this goal in mind, it develops a model, which is rich enough to capture
all these effects of Engel’s Law as well as their interactions. At the same time, it deliberately
abstracts from all other factors in order to isolate the effects of Engel’s Law. Section 2
introduces this model and derives its equilibrium conditions. It considers the world economy, inwhich there is a single nontradeable factor of production, labor, and two countries, which differ
in the population size, N, and labor productivity, h, which also means that they differ in size,
measured in the total effective labor supply, L = hN. There is a continuum of nontradeable
consumption goods sectors, indexed by s  I , where I is a real interval, and preferences overthese goods are implicitly directly additive, isoelastic, and nonhomothetic with constant elasticity

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©Kiminori Matsuyama, Engel’s Law in the Global Economy

of substitution (CES).1 This class of preferences, due to Hanoch (1975) 2 , has several
advantages. First, it contains the standard homothetic CES as a limit case. Second, it allows forany number of sectors with sector-specific income elastic parameters,  (s) , while keeping theconstant elasticity of substitution across sectors,  , as a separate parameter. 3 Furthermore, the
CES parameter,  , can be greater than one (the case of gross substitutes) or less than one (thecase of gross complements).4 Third, income elasticity differences across sectors are independent f (c )ds , where M  is amonotone transformation. They are implicitly directly additive if it can be written as M  f (u; c )ds  1 .
1Preferences are explicitly directly additive if they can be written as u  M 

I

s

s

I

s

s

Clearly, explicit direct additivity implies implicit direct additivity. Implicit direct additivity imposes less restriction,
because a change in u can affect the relative weights attached on different consumption goods under implicit direct
additivity but not under explicit direct additivity. For example, explicitly directly additive CES, /( 1)u   s (cs )11/  ds
 I




with   1 , is homothetic. (The converse is also true. Under the explicit direct

additivity, CES implies homotheticity and vice versa; this is known as Bergson’s Law.) In contrast, implicitly /( 1)

11 / ds
directly additive CES,  s (u)(cs )




I



= 1 with   1 , is nonhomothetic if  log s (u) / u depends ons. We further assume that s (u) is isoelastic in u (i.e., a power function of u ), hence  log s (u) / u is
independent of u . This allows us to define the sector-specific income elasticity parameters.
2
Hanoch (1975), as well as its recent applications in Comin, Lashkari, and Mestieri (2015), considers the case where
the set of consumption goods is finite. Here, we consider the case where the set of consumption goods, I, is a realinterval (that is, it is a totally ordered set with a continuum of elements), as it facilitates the characterization of theequilibrium and comparative statics, as in Dornbusch, Fischer, and Samuelson (1980).
3
As known from the work of Houthakker (1960), Goldman and Uzawa (1964) and many others, Stone-Geary,
Constant Ratio of Income Elasticity (CRIE) or any explicitly (directly) additive preferences would impose therestriction, known as Pigou’s Law, that the ratio of the income elasticity and the price elasticity is constant across all
the sectors. (This restriction holds approximately for a finite set of sectors, but exactly for a continuum of sectors if
the expenditure share of any single sector is infinitesimal.) Pigou’s Law is not only rejected empirically, as shownby Deaton (1974) and others. It also makes explicit direct additive preferences conceptually unsuited for our purpose,
because the effects of the income elasticity differences across sectors are inseparable from those of the priceelasticity differences across sectors. In contrast, under implicit direct additivity, the income elasticity and the priceelasticity of different sectors can be “disentangled” from each other. (This might remind the reader of the problem inmacro-finance that intertemporally additive preferences impose the link between the intertemporal elasticity of
substitution and the risk tolerance, and that delinking these parameters requires the use of recursive preferences, as
pointed out by Epstein and Zin (1989). We thank J. Markusen and I. Werning for this analogy.)
4
The ability to deal with the case of gross complementarity across sectors,   1 , is important, as the empiricalestimates of Engel’s curves are in the range of   0.7  0.8 . In contrast, nonhomothetic preferences that rely onsome notion of vertical differentiation across goods, used by Flam and Helpman (1987), Stokey (1991), andFajgelbaum, Grossman, and Helpman (2011), necessarily imply that goods are gross substitutes. These preferences,which are designed to capture the nonhomotheticity of demand across goods with different quality levels within asector, are not suited for studying Engel’s Law, i.e., nonhomotheticity of demand across sectors that produce
complementary goods. At the same time, the ability to deal also with the case of gross substitutes,   1 , might

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©Kiminori Matsuyama, Engel’s Law in the Global Economyof per capita income level. 5 Furthermore, with their income elasticity parameters being the only
fundamental heterogeneity across sectors, the sectors can be indexed such that the income
elasticity is increasing in the index. Then, holding prices given, a higher per capita income shifts
the expenditure shares towards higher-indexed goods.6 On the production and trade side, wedeliberately use the standard monopolistic competition model of trade due to Dixit-Stiglitz(1977) and Krugman (1980), in order to isolate the role of Engel’s Law. Each nontradeable
consumption good is produced by a competitive sector, which assembles tradable differentiatedintermediate inputs, using the Dixit-Stiglitz (CES) aggregator. Each differentiated intermediate
input is supplied by a monopolistically competitive firm, using labor for both production andentry (or innovation). These differentiated inputs are tradable, subject to the iceberg trade cost,
as in Krugman (1980). One key feature of this setup is that productivity of each consumption
goods sector in each country is endogenously determined, as it depends on the availability ofdifferentiated inputs, which change through entry/exit of monopolistically competitive firms aswell as through trade.7
Section 3 looks at the closed economy equilibrium. An increase in h (or N) improves
welfare. In other words, it leads to a higher standard-of-living. This shifts the relative market
sizes towards the higher-indexed sectors due to Engel’s Law, causing a proportional change inmake our preferences potentially useful as a reduced form way of capturing the nonhomotheticity of demand acrossgoods with different quality levels within a sector.
5
Comin, Lashkari, and Mestieri (2015) review the empirical evidence that income elasticity differences acrosssectors are stable over a wide range of per capita income levels. This is in strong contract to the Stone-Geary
preferences, which implies that income elasticity differences across sectors decline with per capita income. This
makes Stone-Geary unsuited for modelling North-South trade, as well as long run developing processes, as pointed
out by Buera and Kaboski (2009). See Matsuyama (2016) for a more extensive discussion on the restrictive nature ofStone-Geary and other explicitly directly additive nonhomothetic preferences.
6
Formally, by indexing sectors such that  log s (u) / u is monotonically increasing in s, s (u) becomes logsupermodular in u and s. Then, holding prices given, a higher expenditure implies a higher u, which in turn shifts
the density function of the expenditure shares across sectors to the right in the sense of the monotone likelihood ratio(MLR) and shifts its cumulative distribution function to the right in the sense of the first-order stochastic dominance(FSD). See Athey (2002) and Vives (1999; Ch.2.7) for log-supermodularity and monotone comparative statics andCostinot (2009) and Costinot and Vogel (2012, 2015) for applications to international trade.
7
There is an alternative interpretation of our model, as was used in Matsuyama (2015, Section 2). Consumers have
implicitly directly additive isoelastic nonhomothetic CES preferences over a continuum of consumption categories.
And the utility of consuming each category is given by a Dixit-Stiglitz aggregator of a variety of tradeable
differentiated consumer products. And each tradeable differentiated consumer product is supplied by a monopolisticcompetitive firm and subject to the iceberg cost. Here, we instead follow Ethier (1982) and Romer (1990) andinterpret the tradeable differentiated products as intermediate inputs to the production of nontradeable consumption
goods for two reasons. First, it allows us to talk about endogenous sectoral productivity, which makes it easier todiscuss its implications on the Schmookler effect and structural change. Second, much of global trade in
manufacturing consists of intermediate inputs, not consumption goods; see, for example, Antras (2015).

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©Kiminori Matsuyama, Engel’s Law in the Global Economy

the sectoral shares in employment. This change in the relative market sizes also leads to someentries of input producers to the higher-indexed sectors as well as to some exits from the lowerindexed sectors. The resulting change in the relative productivity across sectors (the Schmookler
effect) makes the higher-indexed goods relatively cheaper, which moderates (amplifies) the
sectoral shift when the consumptions are gross complements (gross substitutes).
Section 4 characterizes the cross-country variations in a trade equilibrium. The wage rate
(per efficiency unit) is lower in the country smaller in size at any positive trade cost, because thesmaller economy has disadvantage of having the smaller domestic market. In contrast, thecountry ranking (i.e., which country is richer measured in welfare or standard-of-living) is
endogenously determined. When the trade cost is sufficiently high, the country with higher laborproductivity could be poorer if it is sufficiently smaller in size, because of its disadvantage ofbeing smaller. But it is richer when the trade cost is sufficiently low, which makes thisdisadvantage sufficiently small. At any given equilibrium, the domestic demand composition ismore (less) skewed towards the higher-indexed sectors in the country that is richer (poorer) atthat equilibrium, due to Engel’s Law. With a positive trade cost, this cross-country difference inthe domestic market compositions leads to relatively more input producers to enter in thehigher(lower)-indexed sectors in the richer(poorer) country. As a result, the richer country has
relatively higher productivity in higher income-elastic sectors (the Schmookler effect) andallocate disproportionately more labor in those sectors (the Home Market Effect in employment).
This disproportionate effect on the cross-country difference in the domestic demand composition
also shows up in the inter-sectoral patterns of intra-sectoral trade. Although there are two-way
flows of differentiated inputs within each sector, there is a unique cutoff sector, sc  I , such thatthe richer country runs a trade surplus in the sectors above the cutoff and the poorer runs a trade
surplus in the sectors below the cut-off. Thus, the richer (poorer) becomes a net exporter in thehigher (lower) income-elastic sectors, because its domestic demand composition is more skewed
towards higher (lower) income-elastics (the Linder effect).88Note that it is not the relative country size but the relative standard-of-living that determines the direction of thepatterns of intersectoral trade in this model. The relative country size does matter but only indirectly through its
effect on the relative standard-of-living. For example, imagine that one country, say Switzerland, is much smaller
but its consumers enjoy higher standard-of-living than those living in the country that is much larger, say China.
Our model predicts that Switzerland is a net-exporter in high income elastic sectors, even though the Chinese
domestic markets might be larger than the Swiss domestic markets in all sectors, including high income elastic

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©Kiminori Matsuyama, Engel’s Law in the Global Economy

Section 5 conducts some comparative statics of the trade equilibrium. Section 5.1 looks
at labor productivity growth uniform across countries, which does not affect the terms-of-trade(the relative wage) and the ranking of the two countries. It shifts both the expenditure andemployment shares towards higher-indexed sectors in both countries. It also shifts the cutoffsector, sc  I . Thus, the richer country switches from a net exporter to a net importer in somemiddle sectors, generating something akin to Vernon’s product cycles. The intuition behind thisresult is easy to grasp. As both countries become richer and shift their expenditure towards
higher-indexed sectors, the weights of the higher indexed sectors, in which the richer country
runs a surplus, become higher. In order to keep the overall trade account between the twocountries in balance, the sectoral trade account of the richer country must deteriorate in eachsector. This is why its sectoral trade balances switch from being positive to negative in somemiddle sectors. Furthermore, migrating sectors in the middle range from the richer to the poor
countries causes the sectoral shares in employment to shift up in both countries. How welfare
gains from such a change are distributed across the two countries depends on the elasticity of
substitution across sectors. By increasing the relative market sizes of high-indexed sectors andhence by reducing the relative prices of those sectors in which the richer country hascomparative advantage through the Schmookler effect, a uniform labor productivity growthwidens (narrows) the welfare gap between the two countries, when the consumption goods
produced in different sectors are substitutes (complements).
The effects of globalization, captured by a reduction in the iceberg trade cost uniform
across sectors, are similar to uniform labor productivity growth, except there are additional termsof trade effects when the two countries differ in size, measured in the total effective labor supply(or equivalently in GDP). When the two countries are equal in size (Section 5.2), the wage rates
are always equalized across the countries and hence the terms of trade are not affected by areduction in the trade cost. This means that the country with higher labor productivity has higher
per capita income and higher standard-of-living. And without causing any terms of trade change,
the effects of globalization are isomorphic to those of uniform labor productivity growth. The
intuition is, again, easy to grasp. A lower trade cost allows both countries to have better accesssectors. This is because the Linder effect is due to the difference in the domestic demand composition, not in theabsolute size of the domestic demand in each sector.

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©Kiminori Matsuyama, Engel’s Law in the Global Economy

to the differentiated inputs produced abroad, which generates productivity gains isomorphic tolabor productivity growth. This income effect of productivity gains from trade causes bothcountries to shift their expenditure towards higher-indexed sectors, and the richer (the poorer) toswitch from a net exporter (importer) to a net importer (exporter) in some middle sectors,
generating product cycles, despite that the decline in the trade cost is uniform across sectors.
And again, a globalization widens (narrows) the welfare gap between the two countries when the
consumption goods produced in different sectors are substitutes (complements).
When the two countries are unequal in size (Section 5.3), the factor price is lower in thesmaller country, reflecting its disadvantage of being smaller in this world of aggregate increasing
returns due to the variety effect. Globalization reduces (but never eliminates) this disadvantage,
and causes the factor prices to converge (but never completely equalize) and hence the terms oftrade to change in favor of the smaller country. 9 This generates some additional effects. If laborproductivity is lower in the smaller country--which includes the case where the two countries
have the equal population size--, this country has lower standard-of-living regardless of the tradecost. However, if labor productivity is higher in the smaller country, globalization causes aleapfrogging due to a terms-of-trade change and a factor price convergence. At a high trade cost,the smaller country might have a lower standard of living in spite of their higher laborproductivity, because of their disadvantage of having the smaller domestic markets.
Globalization reduces this disadvantage enough so that its standard-of-living becomes higher. In
our setup, this leads to a reversal of patterns of trade. The smaller country with higher laborproductivity is a net exporter in the lower income-elastic sectors at a higher trade cost, and a net
exporter in the higher income-elastic sectors at a lower trade cost.

Relations to the Existing Studies
As this paper aims to offer a unifying perspective on the role of Engel’s Law in theinterdependent patterns of structural change and trade across countries, there are many related
papers in both structural change and trade literatures.

9This effect of globalization on the terms of trade is not due to the nonhomotheticity, as shown by Matsuyama(2015, section 3) in an alternative model, which differs from the present model only in that the domestic demand
composition difference across the two countries is due to the difference in taste.

8©Kiminori Matsuyama, Engel’s Law in the Global Economy

For models of structural change, see Matsuyama (2008) and Herrendorf, Rogerson, andValentinyi (2014). The latter also offers an extensive review of the empirical regularities on thechanging patterns of sectoral shares in employment and in value-added. Engel’s Law plays the
key role in most studies in this literature, see, e.g, Kongsamut, Rebelo, and Xie (2000) and Bueraand Kaboski (2012), just to name a few. A relatively few studies, Baumol (1967) and Ngai and
Pissaridis (2007), focus on an exogenous difference in productivity growth rates across sectors as
an alternative driver of structural change. Both the income-elasticity and the exogenous
productivity growth differences across sectors are incorporated in some recent studies, such as
Matsuyama (2009) and Comin, Lashkari, and Mestieri (2015). In particular, the latter isnoteworthy as it derives a clear decomposition of the income effect and the price effect as thetwo competing drivers of structural change with an arbitrary number of sectors, thanks to its useof the implicitly directly additive isoelastic nonhomothetic CES. Most studies in this literature
consider only closed economy models. Although Matsuyama (2009) and Uy, Yi, and Zhang(2013) studied open economy implications of structural change, they are primarily interested in
the questions of how structural change in one country spillovers to another country. Furthermore,
the use of Stone-Geary preferences makes these studies unable to isolate the role of Engel’s Law.
In all these models, the Schmookler effect is absent; the sectoral difference in productivity
growth rates is assumed to be exogenous and unresponsive to changes in the relative market
sizes across the sectors.
The Schmookler effect is central to the directed technical change literature; see, e.g.,
Acemoglu (1998, 2002), Acemoglu and Zilibotti (2001), Gancia and Zilibotti (2005, 2009), andAcemoglu (2009, Ch.15). In these models, the relative market sizes are given exogenously. The
idea of linking Engel’s Law to the Schmookler effect was pursued by Murphy, Shleifer and
Vishny (1989) and Matsuyama (1992, 2002), among others, but these studies have a quite
different goal. They are primarily interested in the role of nonhomotheticity on the country’s
aggregate growth performance. By considering models where the sectors differ not only in theincome elasticity but also in the productivity growth potential, these studies showed how an
endogenous shift in the demand composition towards sectors with more (less) productivity
growth potential would accelerate (slow down) the aggregate growth of the economy.
Furthermore, these studies use forms of nonhomothetic preferences, where the effects of income
9©Kiminori Matsuyama, Engel’s Law in the Global Economy

elasticity difference cannot be disentangled from those of price elasticity differences. Both
Murphy, Shleifer, and Vishny (1989) and Matsuyama (2002) considered only the closed
economy case. An open economy version of Matsuyama (1992) assumes no trade cost, so that
producers everywhere face the same tradeable goods prices, which means the cross-country
difference in the domestic demand composition cannot play any role in the allocation of
resources. Taken together, these studies might have left some readers with the impression thatthe role of Engel’s Law would have to be less important in open economies.
In the international trade literature, many models of trade with nonhomothetic
preferences have been proposed to study the role of Engel’s Law in explaining the patterns oftrade between rich and poor countries; Markusen (2013) for a survey. Because merely replacing
homothetic preferences by nonhomothetic preferences in the standard neoclassical trade models
would, ceteris paribus, make rich countries consume more and import more in high income
elastic sectors, virtually all existing models of trade with nonhomothetic preferences postulatethat the rich (poor) countries have comparative advantages in higher (lower) income elastic
sectors. For example, in their Ricardian models of trade, Matsuyama (2000) and Fieler (2011),
the technological superiority of rich countries are assumed to be greater in high income elastic
sectors. In their factor proportion models of trade, Markusen (1986) and Caron, Fally, andMarkusen (2014), rich countries are assumed to be relatively more abundant in the factors used
relatively more intensively in high income elastic sectors. Such correlations between the
differences on the supply side and the demand side are not causally linked in these models.
Instead, they hold by assumption. In other words, all these models suggest that rich countries areexporters in high income elastic sectors, despite their domestic demand composition is moreskewed towards such sectors. This is contrary to the Linder argument that rich countries areexporters in high income elastic sectors because their domestic demand composition is moreskewed towards such sectors, which is central to our analysis. All these studies use
nonhomothetic forms, in which the effects of income elasticity differences cannot be
disentangled from those of price elasticity differences. Furthermore, the ranking of countries, i.e.,
which country is richer, is exogenously determined. In our analysis, leapfrogging can occur,
because globalization allows the smaller country with higher labor productivity to catch up andtake over the other country.
10©Kiminori Matsuyama, Engel’s Law in the Global Economy

The idea that, in the presence of small but positive trade costs, the structure of an
economy responds and adjusts more to the domestic markets than to the export markets, andhence the cross-country difference in the demand composition could become a source ofcomparative advantage was first formalized by Krugman (1980), who called it “the Home
Market Effect.” In Krugman (1980), the cross-country difference in the demand composition isdue to the exogenous variations in taste across countries. 10 In our analysis, on the other hand, it
arises endogenously due to Engel’s Law.
There have been some attempts to model product cycles, or migration of industries fromrich to poor countries. In Krugman (1979), they occur exogenously, as new products areinnovated in the rich country as an exogenous rate, and products produced in the rich country are
migrated to the poor country at an exogenous rate. Grossman and Helpman (1991) endogenized
this process by assuming that the rich has comparative advantage in innovation, while the poor
has comparative advantage in imitation. In both models, product cycles are driven by technology
diffusions. Furthermore, all products enter symmetrically and product cycles affect the relative
number of products produced in the two countries, which remains constant along the balanced
growth path. Hence, there is no structural change, and the income elasticity difference acrosssectors, or Engel’s Law, is not a factor in these models. In the Ricardian model of Matsuyama(2000), different sectors produce complementary consumption goods, which are ranked
according to the priority, and hence the richer country has higher expenditure shares in sectorsthat produce low-priority goods. It is also assumed that the richer countries has comparativeadvantages in those sectors. In this setup, it is shown that uniform labor productivity growthcauses structural change, i.e., labor allocations shifts towards sectors producing lower-priority
goods in both countries, and this is achieved partly by product cycles, i.e., migration of sectors
producing middle-priority goods from the richer to the poor countries. However, the effects of a10

Krugman (1980) demonstrated the Home Market Effect in a two-country, two-sector model, in which the world
demand for the two sectors are equal in size, but distributed unevenly across the two countries of equal size, what hecalled “the mirror-image” assumption. This unfortunately left the factor behind the Home Market Effect ambiguous.
This is because, under this assumption, the sector in which one country develops comparative advantage is not only
the sector in which this country expenditure share is larger than the other country’s, but it is also the sector in which
this country’s expenditure is larger than the other country’s, and it is also the sector in which this country’s
expenditure is larger than in the other sector. To resolve this ambiguity, Matsuyama (2015, section 3) extends
Krugman’s model to the case where there are many sectors of unequal size and two countries of unequal size, andshowed that the country develops comparative advantage in sectors, neither because its expenditures in these sectors
are larger than the other country’s nor because they are larger than those in the other sectors, but because its
expenditure shares in these sectors are larger than the other country’s expenditure shares in the same sectors.

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trade cost reduction, or globalization, are not explored in Matsuyama (2000), because there is notrade cost in that model. To the best of our knowledge, the possibility that gains from a trade
cost reduction and the resulting income effect alone can cause structural change as well as
product cycles in the presence of Engel’s Law has never been demonstrated before. And the
Linder effect is absent in any of these existing product cycles models.
Before proceeding, some remarks should be made of two lines of research, both of whichare, in spite of the prominent role played by the nonhomotheticity of demand, quite orthogonal to
our analysis.
The first explores various alternatives to the Dixit-Stiglitz model of monopolisticcompetition, by using non-CES aggregators of differentiated products within a sector; e.g.,Behrens and Murata (2007), Bertoletti and Etro (2015), Foellmi and Zweimueller (2006),
Matsuyama (in preparation), and Zhelobodko, Kokovin, Parenti, and Thisse (2012). See Parenti,
Thisse, and Ushchev (2017) and Thisse and Ushchev (2017) for surveys. Some studies in thisliterature explore the implications on intra-industry trade; see Behrens and Murata (2012a,
2012b), Bertoletti, Etro, and Simonovska (2016), Foellmi, Hanslin, and Kohler (2014), Foellmi,
Hepenstrick, and Zweimueller (forthcoming), and Simonovska (2015). By departing from the
Dixit-Stiglitz CES aggregator, these models generate some income effects on the nature ofmonopolistic competition and intra-industry trade that are absent in the Dixit-Stiglitz-Krugman
model of trade. The nonhomotheticity in these models is all about the consumers’ “love forvariety” being dependent on their income. With its focus on the intra-sectoral allocations and on
the issues like variable mark up and “pricing to the market,” this literature abstracts from intersectoral issues by using models with a single sector. In contrast, we abstract from their issues bykeeping the Dixit-Stiglitz-Krugman structure of monopolistic competition and trade within eachsector, and instead focus on the implications of Engel’s Law, by using the implicitly additive
nonhomothetic CES preferences across sectors.
The second literature studies the patterns of intra-industry trade between the rich andpoor countries with quality differentiated products. See, e.g., Flam and Helpman (1987), Stokey(1991), and Fajgelbaum, Grossman, and Helpman (2011). Motivated by the observations thatthe rich (poor) countries tend to export higher (lower) quality products within a sector, thesestudies developed nonhomothetic demand systems that rely on the idea that, as their incomes go
12©Kiminori Matsuyama, Engel’s Law in the Global Economy

up, more consumers switch from lower-quality goods to higher-quality goods. Hence, byconstruction, products of different quality levels are gross substitutes, which makes their demandsystems unsuitable for studying Engel’s Law. 11 Indeed, with their focus on the intra-industry
trade, they abstract from the patterns of intersectoral trade. Nevertheless, Fajgelbaum, Grossman
and Helpman (2011) deserves special mention. Unlike the other models of intra-industry tradewith quality differentiation, which all assume that the rich country has comparative advantage inhigher quality goods, they used a monopolistic competition model with costly trade to generatedemand-induced patterns of intra-industry trade, which they also attribute to Linder (1961). Dueto the presence of an outside competitive sector that produces numeraire good, their model
predicts that the country becomes a net-exporter of the quality levels for which it has larger
domestic market than the other country. One important implication of this prediction is that,when the two countries are sufficiently similar in size, the rich (poor) country becomes a netexporter of high (low) quality products, due to the nonhomotheticity of the aggregate demand. 12
Indeed, their analysis and ours are nearly perfect complements to each other. Their analysis isabout intra-sectoral trade, designed to address IO-trade issues. They focus on within-sector
quality specialization and its implications on within-country inequality. To this end, they11

For example, Fajgelbaum, Grossman and Helpman (2011) considers a single monopolistic competitive industry,
which produces indivisible products, say the automobile industry. These indivisible products come in two quality
levels, H & L, and different products are horizontally differentiated within each quality segment. In addition, there isan outside competitive sector that produces the divisible numeraire good tradeable at zero cost, which is big enoughto kill any general equilibrium effect or the terms-of-trade effect. Each agent consumes one unit of a particular
product from either H or L. Building on the discrete choice model of consumer behaviors, they derive a nested logit
demand system, with the property that the rich consumers are more likely than the poor to choose an H-productunder the assumption that marginal utility of the numeraire good is higher when combined with an H- product,
which generates nonhomotheticity of demand. As is well-known, any demand system generated by a discrete choicemodel of consumer behaviors necessarily imply that different products have to be substitutes. In contract, Engel’s
Law is about the nonhomotheticity of demand across complementary goods. Food and footwear are low income
elastic, and pharmaceutical products and automobiles are high income elastic, not because food is as not good as
drugs or shoes are not as good as cars. As consumers become richer, they may spend more on drugs and cars. They
may also switch from low-quality food and shoes to high-quality food and shoes. However, they do not stop eating
food or wearing shoes.
12
Notice that the Home Market Effect works very differently in their model. In our model, similar to Krugman(1980), the Home Market Effect is due to the cross-country difference in the domestic market composition, while, in
FGH, similar to Helpman and Krugman (1985, Ch. 10.4), it is due to the cross-country difference in the absolute
domestic market size at each quality level. This is because, in FGH, different quality segments of the monopolisticcompetitive sector are not competing against each other in the factor market due to the presence of a large outside
good sector, which is big enough to kill the general equilibrium effect. Thus, between a very small but rich country(say Switzerland) and a very large but poor country (say, China), their model predicts that China could become anet-exporter at every quality level, including in high quality products, while Switzerland an exporter in the outside
good.

13

©Kiminori Matsuyama, Engel’s Law in the Global Economy

abstract from the patterns of trade across sectors and from any effects on cross-country inequality,
by fixing the terms of trade and the country ranking. In contrast, our analysis is about intersectoral trade, designed to address macroeconomic growth and structural change issues. We
focus on the patterns of trade across sectors producing complementary goods and its implications
on cross-country inequality with endogenous terms-of-trade and endogenous country ranking.
To this end, we abstract from within-sector quality specialization and within-country inequality.

2.

The Model
Imagine the world economy that consists of two countries, indexed by j or k = 1 or 2. .

There is a single nontradeable factor of production, which shall be called labor. Country j ispopulated by N j homogenous agents, each of whom supplies h j units of effective labor
inelastically at the wage rate, w j . Thus, the income (and the expenditure) of each agent in j is

E j  w j h j and the total effective labor supply is Lj  h j N j . The population size, N j , and its
effective labor supply per agent or labor productivity, h j , which may also be interpreted as
human capital, are the only possible sources of heterogeneity across the two countries.
2.1

Nonhomothetic Preferences and Expenditure Shares:

There is a continuum of sectors, indexed by s  I , each producing a nontradeable consumption~kgood, also indexed by s  I , where I  R is an interval. The preferences of each agent, U =

U Csk , s  I , are given implicitly by

(1)

1



~
I ( s ) U ( s ) 
k


 

 1
k 
s



C  ds


 1



 1 ;  s  0 ,   0 and   1 ,

~k



k



with  ( s )    / 1    0 , which ensures that U = U Cs , s  I is globally monotone increasingkand globally quasi-concave in Cs , s  I . Without further loss of generality, we normalize  (s )such that  (s)ds  1 . The utility function (1) is implicitly directly additive, isoelastic, constantI

elasticity of substitution (CES), a class of utility functions introduced by Hanoch (1975). The
standard homothetic CES preferences,

14

©Kiminori Matsuyama, Engel’s Law in the Global Economy


1
1
 1
~k 
k 1

U   ( s ) Cs
ds ,
 I


 

is a special case of (1), where  ( s)  1 for all s  I . By letting  (s) dependent on s, this class ofutility functions allows for the income elasticity to differ across sectors, while keeping the priceelasticity, η, constant and the same across sectors. In what follows, we assume that the sectors1

 ( s ) 
~k
 ~k

can be ordered such that  (s) is strictly increasing in s  I . Then,  s,U  ( s ) U
,



 1
 in (1), is log-supermodular in s and

 
k
s

the coefficient on the term C

1



 

~
U k . By applying Lemma~k~ k  ( s) ~k1 (See Appendix), for gˆ (s;U )  (  s ) U
, this implies that, as U goes up, the agent

 cares more about the higher-indexed goods in the sense that the density function of the weights
attached to different sectors satisfies the monotone likelihood ratio (MLR) property and hence
that its cumulative distribution function satisfies the first-order stochastic dominance (FSD)
property.

~kk

Each agent in k chooses Cs , s  I , to maximize U , subject to (1) and the budget
constraint:
(2)

k
s

k
s

k

k

k

 P C ds  E  w h ,Iktaking Ps , s  I , the nontradeable consumption goods prices in k, given. The solution can bekexpressed in terms of the expenditure share on good s, ms :    k  ( s ) (3)s U k 

k 1

Ps
Psk Csk  s Um  k 
1


E
Ek
k
s

 ( s )   U k  ( t ) 

I

t

P 
,
P  dt
k 1
s
k 1

with

k
s

 m ds  1
I

t

~k

k

where U is the maximized value of U and given by the indirect utility function of (1):(4)

  U k  ( s ) 

I

s

1
1

P / E  ds
k
s

k 1

 1,

15

©Kiminori Matsuyama, Engel’s Law in the Global Economy

which is implicitly additively separable.13 Recall the parameter restriction,  ( s )    / 1    0 ,which ensures the global monotonicity of the utility function, (1). This restriction implies thatk

LHS of (4) is strictly increasing in U k , and hence U is strictly increasing in E k .

 P  , is log-supermodular in s and U .

 P  , eq.(3) shows that, holding the

Notice that the numerator of eq.(3),  s Ukk  ( s ) 

Hence, by applying Lemma 1 for gˆ (s,U )   s U

k  ( s ) 

k 1
s

k

k 1
s

prices constant, the agent with a higher E k (and hence a higher U k ) allocates larger shares of
their expenditure towards higher-indexed sectors in the sense that the density function of theexpenditure share across sectors satisfies the MLR property and that its cumulative distributionfunction satisfies the FSD property. Note also that, from eq.(3), one could express the relative
expenditure shares of any two sectors as:log(msk / msk' ) = log(s /  s ' )   ( s)   (s' )log(U k ) + (1   ) log(Psk / Psk' ) .
and the relative demands of any two sectors as:log(Csk / Csk' ) = log(s /  s ' )   ( s)   (s' )log(U k )   log(Psk / Psk' ) .
This shows not only that the relative demand for a higher-indexed sector has higher incomekkkelasticity. It also shows that the slope of the Engel curve,  log(Cs / Cs ' ) /  log(U ) = (s)   (s' ) , is independent of the real aggregate consumption per capita (or the standard-ofliving), U k .14 Furthermore, these income elasticity parameters are not linked to the price

13

This means that, with  ( s)  1 , (1) does not satisfy what is commonly called “indirect additivity,” or more

precisely “explicit indirect additivity,” because its indirect utility function is not explicitly additive.
14

 (s)ds  1 , P , defined by P     U  P  ds , could be viewedkNote that, with the normalization,

I

k

k 1

k  ( s ) 1

I

k

s

k 1

s

k

as the aggregate price index, and U  E / P could be viewed as the real aggregate consumption per capita.
Furthermore, eq.(3) could be written as:    
 Psk  ( s)Pskk  ( s)C  s k
U
 s
Ek
,
P 
k  ( s )
P 
so that  (s) is interpreted as an approximately income elasticity. It is approximate because the aggregate price
k
s

k

 

k

k

index, P , depends not only on the prices of all consumption goods, but also on U , and hence E .

16©Kiminori Matsuyama, Engel’s Law in the Global Economy

elasticity, unlike in the explicitly directly additive nonhomothetic preferences that are subject toPigou’s Law. 15
2.2

Production and Trade:
We keep the rest of the model deliberately standard, using the Dixit-Stiglitz-Krugman

monopolistic competitive model of production and trade, in order to isolate the role of Engel’s
Law.
2.2.1 Competitive Nontradeable Consumption Goods Sectors:
Each nontradeable consumption good, s  I , is produced in a competitive sector, also
indexed by s  I , by assembling a continuum of tradable differentiated inputs, indexed by  s , with the Dixit-Stiglitz aggregators,Ysk   qsk ( )
 s



(5)

k

1
1


 d  1; s  I ; σ > Max{1, η},kwhere Ys is the output; qs ( ) the quantity of input variety   s used in sector s  I ; s the12j

set of tradeable differentiated inputs available for s  I , and s  s  s , where s ( j = 1 or
2) are the (disjoint) sets of differentiated inputs produced in country j in equilibrium. The
restriction on σ, the elasticity of substitution between inputs within each sector, implies not onlyσ > 1 but also σ > η, so that differentiated inputs are closer substitutes within each sector thankacross sectors.16 Given ps ( ) , the unit price of input variety v  s in k, each competitive
15

For example, by using the Constant Ratio of Income Elasticity (CRIE) preferences, which are explicitly directly
additive, both Fieler (2011, eq.(9)) and Caron, Fally, and Markusen (2014, eq.(20)) derive the relative expenditure
curve, which can be rewritten as :log(msk / msk' ) = const.   (s)   (s' ) log(k ) + (1   (s)) log(Psk )  (1   (s' )) log(Psk' ) ,and hence its relative demand curve as:log(Csk / Csk' ) = const.   (s)   (s' ) log(k )   (s) log(Psk )   (s' ) log(Psk' )kkwhere  , the Lagrange multiplier associated with the budget constraint, is inversely related to E at any given
prices. Thus, under CRIE, it is infeasible to disentangle the effects of income elasticity differences and those of
price elasticity differences. Furthermore, Deaton (1974) and many others who have estimated such log-linear
consumption demand systems have rejected the Pigou’s Law (i.e., the proportionality of the income elasticity andprice elasticity across sectors), but have not been able to reject a common price elasticity of substitution, implied byour preferences. Comin, Lashkari, and Mestieri (2015) reviews the empirical evidence in support of the log-linear
Engel curves implied by implicitly directly additive CES and against those implied by explicit direct additivity.
16
For the empirical more relevant case of gross complements, η < 1, this imposes no additional restriction. For thecase of gross substitutes, it is necessary to assume σ > η > 1. If η > σ > 1, two differentiated inputs used in the same

17

©Kiminori Matsuyama, Engel’s Law in the Global Economy

consumption good sector chooses the input combination to minimize its cost, which yields theunit cost (and hence the unit price) of the consumption good s  I in k:
1

(6)





1
1
P   psk ( ) d  ,  skswhich is the Dixit-Stiglitz price aggregator, the dual of (5), and the quantity of input varietyv s used in k:

(7) pk ( ) q ( )   s k  Ysk .
 Ps 
ks

2.2.2 Iceberg Costs and Demand for Differentiated Inputs:
The unit price of each input variety, psk ( ) ,   s , depends on k, because of the(iceberg) trade costs; To deliver one unit of   sj in country k,  jk units need to be shippedfrom j. Thus, with the unit factory price, psj ( ) ,   sj , psk ( ) =  jk psj ( )  psj ( ) . Then,kkkfrom (3) and (7), and using Ys  N Cs , the demand for   sj by country k is: jk qsk ( )   jk s N k E k  U k k ( s ) 

1 

(8)

k  s  U =  jk N  s E
where  jk   jk 

P   p ( )    N E  U k k  ( s ) 



k
s

k

jk

k 

s

k  ( s ) P   p ( )
k  
s

jk

j
s



P   p ( ) ,k  sjs

 1 . Thus, the aggregate demand for   sj can be expressed as:jj

Ds ( ) = As ( ps ( ))



,

where
(9)

Asj   k  jk bsk ;(10)

bsk   s E

  U k k  ( s )  

N k Psk   P  N P  s U k

 (s)

k 

k

k  
swhere Asj may be interpreted as the aggregate demand shift parameter for a variety produced in
sector-s in country j; bsk as the aggregate demand shift parameter for sector-s in country k; andsector would become Hicks-Allen complements and the entry of two monopolistic competitive firms into the same
sector would become strategic complements, leading to multiple equilibria for the same reason discussed at length inMatsuyama (1995).

18

©Kiminori Matsuyama, Engel’s Law in the Global Economy

 jk is the weight attached to the aggregate spending by country k of varieties produced incountry j. Eqs. (8)-(10) show that the demand curve for each variety has a constant priceelasticity with its demand shift parameter, Asj , which depends on the trade costs in a mannerfamiliar in the Dixit-Stiglitz-Krugman monopolistic competition models of trade. What is new iskthat the utility level, U , has differential impacts on the demand shift parameters across sectors
due to the nonhomotheticity of preferences.
For the remainder of this paper, we follow Krugman (1980) and many others byassuming that  11   22  1 and  12   21    1 , so that(11)

11   22  1 and 12   21     1 < 1.

Thus,   [0,1) measures how much each country spends on an imported variety relative to what
it would spend in the absence of the trade cost; it is inversely related to  , with   0 for   and   1 for   1 .
2.2.3 Production and Pricing by Monopolistically Competitive Firms:
Each differentiated input variety is produced by a monopolistically competitive firm.
Producing one unit of each differentiated input in sector-s requires  s units of labor, so that themarginal cost is equal to w j s for    sj . Since Eq. (8) shows that the price elasticity ofdemand for each input is constant,  , all the input varieties are priced with the same mark-up
rate:
(12)

j
s

p ( ) 

w j s1  1/  psj for all    sj ,and hence they are produced by:(13)

ysj  Asj ( psj )  .

By inserting (12) into (6),
(14)

P 

k 1
s



s

 p ( ) d =    p ( ) d =  V  ( p )
k
s

1

j

 sj

jk

k
s

1

j

j

s

jk

j 1
s

where Vs j is the Lebesgue measure of  sj , the equilibrium measure of varieties produced (and ofactive firms) in sector-s of country j.
2.2.4. Free Entry Conditions and Sectoral Shares in Employment:
19

©Kiminori Matsuyama, Engel’s Law in the Global Economy

This equilibrium measure, Vs j , is determined by the free entry condition. To enter sectors, all monopolistically competitive firms need to pay the setup cost per variety, s , in labor, andthey have incentive to do so, as long as the profit is non-negative. Thus, in equilibrium, either apositive measure of firms (and varieties) enter, in which case they all break even ( Vs j  0 

psj ysj  w j  s ysj  s ), or no firms (and varieties) enter, because they would earn negative profit

if they were to enter ( psj ysj  w j  s ysj  s  Vs j  0 ). Using eqs. (12) and (13), this free entry
condition can be written as the complementarity slackness condition:

Vs j  0 ;ysj  Asj ( psj )   (  1)s / s .

This implies that each active firm in sector-s based in country j hires  s ysj  s = s units oflabor, so that labor demand by sector-s in country j is Lsj  sVs j and its share in employment is

f s j  Lsj / Lj  sVs j / Lj . Thus, the above complementarity condition can be further rewritten as:(15)

f s j  Lsj / Lj  sVs j / Lj  0 ; ysj  Asj ( psj )   (  1)s / s ,

with
(16)

j

 f ds  1,Iswhich is nothing but the labor market clearing condition,  Lsj ds  Lj  h j N j .
I

In what follows, we use the following normalizations to simply the notation. First,
choose the unit of each differentiated input in sector-s such that  s  1 1/  . This simplifies(12) to(12’)

psj ( )  psj  w j

for all    sj and all s  I .

Second, choose the unit of variety in sector-s such that s  1/  . These two normalizations
together simplifies (15) to:(15’)

f s j  Vs j / Lj  0 ;ysj  Asj ( w j )   1

for all s  I and j = 1 and 2.

In other words, without loss of generality, we choose the units of measurement such that eachactive firm produces by ysj  1 , hires labor by  s ysj  s  1 and sells its output at psj  w j , tobreak even in equilibrium, and the labor demand by sector-s of country j is Lsj  Vs j .
20©Kiminori Matsuyama, Engel’s Law in the Global Economy

2.3

Equilibrium Conditions:
We are now ready to consolidate all the equilibrium conditions. First, using (11), (12’)

and (15’), eq. (14) becomes(14’)( Ps1 )1  f s1L1 (w1 )1  f s2 L2 (w2 )1 ; ( Ps2 )1  f s1L1 (w1 )1  f s2 L2 ( w2 )1 , s  I ,By introducing   w1 / w2 , the relative wage or the factoral terms of trade (and also the relative
prices of input varieties produced in the two countries in the same sector), (14’) can be further
simplified to:(17)(w1 / Ps1 ) 1  f s1L1  f s2 L2 () 1 ;j

(w2 / Ps2 ) 1  f s1L1()1  f s2 L2 , s  I ,jwhere w / Ps is the TFP of sector-s in country j.
Second, from (9), (11), (15’), the complementary slackness condition for free entry insector-s in each country can be written as(18)

f s1  0 ; (bs1  bs2 )( w1 )   1&

f s2  0 ; ( bs1  bs2 )( w2 )   1 .

&f s2  0 ;  ( ) d s1  d s2  1 ,This can be further rewritten as(19)

f s1  0 ; d 1s   ( )  d s2  1 ;sI .

by introducing d sj  (bsj )( w j )  , which is the domestic market’s share in the revenue of adifferentiated input producer based in j.17 This variable, d sj  (bsj )( w j )  , can be expressed in
two different ways from (3) and (10). First, by eliminating Psk from (3) and (10), 1 k ( ( s )  )  1 (20) d  b w   h  N  U 
k
s

k
s

k 

k 

k

s



   k  1 

,
s

m 

k = 1 and 2.

Alternatively, by eliminating U k from (3) and (10) and using E k N k  w k h k N k  wk Lk ,

  = m L (w / P )d sk  bsk wk



k
s

k

k

k 1 
s.

Using (17), this can be further rewritten to:17

On the other hand, the export market’s share in the revenue of an input producer in j is  ( w j / w k )  d ks , ( k  j ) ,which is equal to the domestic market’s share in the revenue of an input producer in k , d sk , multiplied by( w j / w k )  , due to the relative price between these two producers, and multiplied by  due to the trade cost.

21

©Kiminori Matsuyama, Engel’s Law in the Global Economy

(21)

1 1
s
2 2
s

  = f L  mf LL ( ) ;d 1s  bs1 w1



1 1
s

1

2
s
1

  = f L (m) L  f L

d s2  bs2 w2



1 1
s

2 2
s

Finally, the expenditure share or the market size distribution, as well as the employment
share across sectors must add up to one in each country.
k
s

(22)

 m ds  1 for k = 1 and 2.

(23)

 f ds  1 for j = 1 and 2.

I

j

I

s

Note that eqs. (19), (20), and (21) impose the conditions on six functions of s  I , andthat eqs. (22) and (23) impose four additional conditions, but one of them is redundant due to theWalras’ Law. These conditions, eqs. (19)-(23), altogether determine six endogenous functions ofs  I ; d sk (the domestic market share in the revenue of an input producer in each sector in eachcountry), msk (the market size distribution in each country), and f sk (the employment share ineach country), as well as three endogenous variables, U k (the standard-of-living in each country)
and  (the terms-of-trade).
Before proceeding to solve for the equilibrium, it is worth pointing out one notable (orperhaps unusual) feature of this set of the equilibrium conditions; it contains U k , k = 1 and 2.
Normally, when we analyze a general equilibrium model, we first solve for the equilibrium
allocations (and prices) by conducting a positive analysis. Then, we plug those equilibrium
allocations into the utility functions to obtain the welfare levels by conducting a normative
analysis. Here, due to the implicit nature of the utility function, the consumer demand depends
on the welfare level, which in turn affect the equilibrium allocations, which in turn affect thewelfare level. Therefore, it is more efficient to solve for the equilibrium allocations and prices
and for the welfare levels together, without the separation of the positive and normative analyses.
Indeed, when solving for the equilibrium below, U k , k = 1 and 2 are among the first endogenous
variables that will be pinned down.

3

Patterns of Structural Change in a Closed Economy Equilibrium

22

©Kiminori Matsuyama, Engel’s Law in the Global Economy

First, let us consider the case of ρ = 0, where each country must produce all differentiatedinputs used in every sector. Thus, for all s  I and for k = 1 and 2, f sk > 0, and hence, from (18),
d sk = 1. Inserting this to (19) and (20) yields

(24)



f  m  h  N
k
s

k
s

k 

  1 k      1 k ( ( s )  )    .
0  U skSubscript “0” is added here to indicate that U 0 is the equilibrium utility level, or the standard-ofliving, achieved when ρ = 0. Note that eq. (24) shows that the employment (and value-added) is
distributed proportionately with the market size across sectors in a closed economy.
k

By integrating (24) across all the sectors and using (22) or (23), we can pin down U 0 as
  1 k     h  N k I  1 k ( ( s )  )    0 U s

ds  1 ,which can be written more compactly as
(25)

 

U0k  u x0kk  N  h  L ,with x0  hk k  1 kkwhere u is defined implicitly by(26)

 1    x 

  1 ( ( s )  )    







  s u( x)
I

ds .

Lemma 2-i) in the appendix shows that u , defined in eq.(26), is a strictly increasing function.
kThus, the utility level, or the standard-of-living, in the closed economy increases with x0 

h  N  h  L . (Again, subscript “0” is used to indicate ρ = 0.) Eq.(25) shows thatU  u x  increases not only in labor productivity, h , but also in N . This is due to the
k 

k
0

k  1 k

k

k
0

k

k

aggregate increasing returns to scale in the presence of “love for variety” and the fixed cost, afamiliar feature of the Dixit-Stiglitz monopolistic competition model. This can be also seen,kk  1from eq.(17) and ρ = 0, (w / Ps )

 f sk Lk , so that the sectoral TFP is increasing in the total
1

1

2

2

employment in that sector. Notice also that the condition for U0  u ( x0 ) < U0  u( x0 ) can be  L  h  L , which occur even if h  h when L / L  h / h 

expressed as h1

 1 12  1 2

1

2

1

2

1

2 1

 1 . In

other words, the country with higher labor productivity may have a lower standard-of-living
23

©Kiminori Matsuyama, Engel’s Law in the Global Economy

when it is sufficiently smaller. This is because those living in a small country has disadvantage
in the presence of aggregate increasing returns.18
Next, plugging (25) and (26) into (24) yields the equilibrium density functions of
employment and market sizes across sectors as follows:
k
s

k

s

f m

(27)

  1 ( ( s ) )     u( x )
s

k
0



 1 k    


0

x 

  1 ( ( s ) )     u(x )   u(x ) 
I

s

k
0

t

k
0

  1 ( (t ) )    



.
dt

k

The numerator of (27) is log-supermodular in s and x0 . Thus, by applying Lemma 1 for
k
0



k
0

gˆ (s, x )   s u( x )  1 ( ( s )  )    







1

1

2

2

, eq.(27) shows that, for U0  u( x0 )  U0  u( x0 ) , country 2,

whose standard-of-living is higher than country 1, spend relatively more on higher-indexed
1

2

goods in the sense that ms / ms is strictly decreasing in s (that is, the density functions ofequilibrium market size distribution across sectors satisfies the MLR property) as well as in thesense that the cumulative distribution function for country 2 first-order stochastically dominates(FSD) the cumulative distribution function for country 1.
kNotice the difference between the two expressions of ms , eq.(3) and eq.(27), in particular
how it depends on the utility. Eq.(3) implies that, holding the prices given, the relative market
size of two sectors, s  s ' , responds to an increase in U k as

 log(msk / msk' )=  ( s)   (s ' ) .
 log U k 

In contrast, eq.(27) shows that, in equilibrium, the relative market size of two sectors responds as

d log(msk / msk' )  1 =  ( s )   (s ' ) .
kd log u( x0 )
   This is due to the Schmookler effect. A change in the relative market size causes some entries
into higher-indexed sectors and some exits from lower-indexed sectors, which leads to a higher

18

This result does not contradict what we noted earlier, i.e., eq.(4) shows that the agent’s utility is increasing in

E k and hence in h k , holding the prices given. When comparing the two countries in equilibrium, the prices differ
across the two countries because the measure of varieties used in each sector in each country is endogenously
determined by the free entry condition.

24

©Kiminori Matsuyama, Engel’s Law in the Global Economy(lower) productivity in higher-(lower)-indexed sectors, which reduces the relative prices ofkk  1higher-indexed goods. Formally, by setting ρ = 0 in eq.(17), (w / Ps )

Psk' / Psk  f sk' / f sk

1 /(1 ) msk' / msk

1 /(1 ) f sk Lk , so that.

This change in the relative price amplifies (moderates) the shift in expenditure shares if different
sectors produce substitutes (complements). Indeed, from eq.(3) and using the above expression,
the total effect can be calculated as

d log(msk / msk' )  log(msk / msk' )d log(Psk / Psk' ) d log(msk / msk' )
(
1


)
=
+
d log u( x0k ) log u( x0k )d log msk / msk' d log u( x0k )

















 log(msk / msk' ) 1   d log(msk / msk' )=+, log u( x0k )1   d log u( x0k )









from which

d log(msk / msk' )  log(msk / msk' )    1 = ,d log u( x0k ) log u( x0k )     









which is exactly what we obtained from eq.(27). Here, (  1) /(   ) > 1 for η > 1 and(  1) /(   ) < 1 for η < 1 capture amplification and moderation due to the Schmookler effect.
In the literature of structural transformation, it is common to treat the income elasticity
difference across sectors and the productivity growth difference across sectors as two separate
exogenous causes of structural change. The above result suggests that, in the presence of the
Schmookler effect, such a dichotomy can be misleading, as some productivity growth differences
may be induced by the income elasticity differences.
The above amplification or moderation effect also affects the welfare impact of a change
in x0 . From Lemma 2-ii) shown in the appendix, d logux  / d log  = xu' x / u(x)k  (x) is increasing (decreasing) in x if η >(<) 1. In words, welfare gains from a percentagekincrease in x0 is higher (lower) at a higher x if η >(<) 1. This implies, among other things, that a
uniform labor productivity growth, h1 / h1  h2 / h2  0 , magnifies (reduces) the relative gap in
2

1

2

1

the standard-of-living between the two countries, U0 / U0  u( x0 ) / u( x0 )  1, if different sectors
produce substitutes (complements).
25

©Kiminori Matsuyama, Engel’s Law in the Global Economy

Before proceeding, it is worth pointing out that, when ρ = 0, the equilibrium conditions,eqs. (19), (20), (21), (22) and (23), do not depend on ω. In other words, ω is indeterminate; there
is nothing to pin down the relative wage of the two countries that are isolated from each other.
This is no longer the case, when ρ > 0. Indeed, as the first step to solve for a trade equilibrium,
we need to determine the relative wage or the terms-of-trade between the two countries.

4

Trade Equilibrium: Cross-Country Variations
This section focuses on how the two countries differ in the trade equilibrium for a given

set of the parameter values. The next section will deal with comparative statics.
4.1(Factoral) Terms of Trade
In what follows, let us focus on the case where f s1  0 and f s2  0 for all s  I . This

simplifies eq. (19) to d 1s   ( )  d s2  1 and  ( ) d s1  d s2  1 , from which(28)

d 1s 

1   ( ) 1 2

and d s2 

1   ( )
for all s  I .
1 2

Inserting (28) into (21) yields(29)

f s1L1  f s2 L2 () 1 

(1   2 ) L1m1s(1   2 ) L2 ms2
1 1
1
2 2

f
L
(

)

f
L
and

for s  I .
ss
1   ( ) 
1   ( )

Integrating these expressions across all sectors and using (22) and (23),L1  L2 () 1 

(1   2 ) L1
(1   2 ) L2
1
1
2

L
(

)

L
and

.
1   ( ) 
1   ( )

These two expressions are equivalent. Indeed, either of them can be rewritten as:
(30)


L1
2 1 1   ( )
=
,



;



(

)
L2
1   ( )

where ;   is strictly increasing in   (  1 /  ,  1 /  ) and satisfies lim1 /    ;    0 , 1;    1 , andlim   ;     . Figure 1 illustrates eq.(30), which determines the (factor)   1 / 

terms of trade   w1 / w2 as a function of the relative labor supply, L1 / L2 , for a given   (0,1) .
It shows that   w1 / w 2 is strictly increasing in L1 / L2 and   w1 / w 2  1 if and only if
L1 / L2  1 . Thus, the factor price is higher in the larger economy, which reflects the aggregate
26©Kiminori Matsuyama, Engel’s Law in the Global Economy

increasing returns to scale pointed out earlier.19 It also shows the lower and upper bounds for theterms of trade,   (  1 /  ,  1 /  ) . The arrows indicate the effects of an increase in ρ. As shown,
it flattens the graph, thereby causing a factor price convergence. This is because globalization,
captured by a reduction in τ and hence an increase in ρ, reduces the smaller country’s
disadvantage.
It is also worth pointing that, because  is strictly increasing in L1 / L2 with the range  (  1 /  ,  1 /  ) and  = 1 for L1 / L2 = 1 for any ρ  (0,1) , eq.(28) implies i) d s1 ( d s2 ) isstrictly increasing (decreasing) in L1 / L2 ; ii) d s1  0 and d s2  1 as L1 / L2  0 ; iii) d 1s  1 andd s2  0 as L1 / L2  1 ; and iv) d s1 = d s2 = 1/(1+ρ) > ½. Thus, the domestic market accounts

more for the revenue of the input producers based in the larger country; the domestic (export)
market account for most of the revenue of those operating in a very large (small) country, andwhen the two countries are equal in size, the domestic market accounts more than a half of theirrevenue in the presence of the trade cost.
4.2

Standard-of-Living and Market Size Distributions
Next, combining (28) and (20) yields  1 (31)  1  (1   2 )(h1 ) N 1    ( ( s )  )    
m1s   s U 1
,

1


(

)



  

  1   1  (1   2 )(h2 ) N 2    
2 ( ( s )  )    
ms2  

U
.
s



 1   ()


  



k

Here, the subscript “ρ” is added to indicate that U  , the equilibrium standard-of-living achievedin each country under trade, depends on ρ. By integrating (31) across all the sectors and using(22), we obtain

 



(32)

19

1


 
1


U u x ,(1   2 ) h1 N 1(1   2 ) x10
with x ;
1   ( )
1   ( )
1








Note that eq.(30) implies w1L1 / w 2 L2    ;    ( )   / ( )    , which is strictly increasing in 

(hence also in L1 / L2 ) and w1 L1 / w 2 L2 < 1 if and only if  < 1 (hence also if and only if L1 / L2 < 1). Thus, thelarger economy is larger regardless of whether it is measured in the total labor supply or in the aggregate GDP.

27

©Kiminori Matsuyama, Engel’s Law in the Global Economy 

(1   2 ) h2 N 2(1   2 ) x02
with x ,
1   ()1   ()

 

2


2


2


U u x ,where u is the same increasing function defined implicitly by (26). Note that the welfarekeffects of globalization on each country are summarized by a single index, x . Note also that thelower and upper bound on the terms of trade established earlier,   (  1 /  ,  1 /  ) , which can be11seen in Figure 1, ensures gains from trade for both countries;    1 /  implies x  x0 , hence

 

 

 

 

U 1  u x1 > U01  u x10 and    1 /  implies x2  x02 , hence U 2  u x2 > U02  u x02 .
Plugging (32) back into (31) and using the definition of u , given by (26), yields theequilibrium density function of the market size distribution across sectors in each country as
follows.

(33)

msk 

 u(x )

 

s





 1 k    




x 
k


Note that  s u( x )k  1 ( ( s )  )    



k


 

I

  1 ( ( s )  )    







k


for gˆ (s, x )   s u( x )

s

t

k





  1 ( ( t )  )    

for k = 1 and 2.

dtkis log-supermodular in s and x . Hence, by applying Lemma 1

  1 ( ( s )  )      1 ( ( s )  )     u(x )
  u(x )
k




1

 
1

2

 
2

, it follows from eq. (33) that, for U   u x < U   u x ,country 2, whose standard-of-living is higher than those in country 1, spend relatively more on
12

higher-indexed in the sense that ms / ms is strictly decreasing in s (that is, the density functionsof the equilibrium market size distribution across sectors satisfies the MLR property) as well as
in the sense that the cumulative distribution function for country 2 first-order stochasticallydominates (FSD) the cumulative distribution function for country 1. In shorts, the domestic
demand composition is more skewed towards the higher-indexed in the country with higher
standard-of-living. The MLR property can also be seen by taking the ratio from (33) to obtain(34)  1 




1 
1
m  x      u( x ) 
 u( x2 ) m  x2 
  
1
s
2
s

  1 ( ( s )  )    .

28

©Kiminori Matsuyama, Engel’s Law in the Global Economy

 

1

1

 

2

2

Clearly, this is strictly decreasing in s if U   u x < U   u x and strictly increasing in s if

 

 

U 1  u x1 > U 2  u x2 .
4.3

Home Market Effect in Employment and in Patterns of Trade
Unlike in the closed economy case, the employment distribution in each country is no1

longer proportional to the market size distribution in that country. By solving (29) for f s andf s2 and using (30), we obtain the equilibrium density function of employment across sectors ineach country as follows:(35)

f s1 

m1s   () ms2> 0;
1   ()

which requires  ( )

f s2 

ms2   () m1s> 0,
1   ()

 m1s / ms2   1 ( ) . Furthermore, the ratio of the two,f s1  1   ( )  m1s / ms2   () 
f s2  1   ()  1   () m1s / ms2 

(36)

1

2

1

2

1

2

1

2

1

2

is increasing in ms / ms and satisfies f s / f s  ms / ms  1 , f s / f s  ms / ms  1, orf s1 / f s2  m1s / ms2  1 .

 

1

1

2

 
2

Figure 2 illustrates eq.(34) and eq.(36) for the case of U   u x < U   u x . In this
1

2

1

2

case, ms / ms is strictly decreasing in s. This causes f s / f s to be also strictly decreasing in s.
1

2

1

2

1

2

Furthermore, there is a unique cutoff sector, sc  I , such that f s / f s  ms / ms  1 holds below
1

2

1

2

the cutoff and f s / f s  ms / ms  1 above the cutoff. And the graph of f s / f s is steeper than
12

the graph of ms / ms . Thus, disproportionately larger fractions of labor are employed in lower(higher) income elastic sectors in the country with lower (higher) standard-of-living, precisely
because its domestic demand composition is more skewed towards markets in lower (higher)
income elastic sectors. This is in strong contrast to the closed economy case, where labor isallocated across sectors proportionately to the market size distribution across sectors, so thatf s1 / f s2  m1s / ms2 . In other words, international trade magnifies the power of the domestic
demand composition in dictating the allocation of resources across sectors.
29

©Kiminori Matsuyama, Engel’s Law in the Global Economy

This result might come as a surprise to those who address the questions of structuralchange within a closed economy setting. In a closed economy, the domestic supply isnecessarily equal to the domestic demand in each sector, and hence a change in the domestic
demand composition across sectors would cause a proportional change in the composition of
production and hence the sectoral allocation of resources. Many people seem to believe that, inan open economy, the domestic demand composition becomes less important, because thedomestic supply need not be equal to the domestic demand in each sector. This logic is false.
That the domestic supply is no longer equal to the domestic supply in each sector means that the
impact of the domestic demand composition is no longer proportional; instead, it could be morethan proportional. Indeed, as long as the trade cost is not zero, the difference in the domestic
demand composition across countries give different incentives for entry of firms (or moregenerally innovation) across sectors in different countries in the presence of the Schmookler
effect. Through such differential Schmookler effects across countries, the richer (poorer)country develops comparative advantage in higher (lower) income-elastic sectors, which is the
Linder effect. And a lower trade cost causes the richer (poorer) country to allocate even moreresources towards higher (lower) income-elastic sectors by importing even more from the poorer(richer) country in lower (higher) income-elastic sectors. Hence, globalization magnifies, insteadof weakening, the power of the domestic demand composition differences in dictating thepatterns of structural change.
4.4

The Linder Effect, or the Home Market Effect in Inter-Sectoral Patterns of Intra-

Sectoral Trade
As the above paragraph suggests, the disproportional effect of the market size distribution
on the employment distribution under trade manifests itself in the inter-sectoral patterns of intrasectoral trade. Indeed, they are the two sides of the same coin. As indicated in Figure 2, thecountry with higher (lower) standard-of-living becomes a net exporter (importer) above thecutoff and a net importer (exporter) above the cutoff. To see this, recall that country k spends

bsk ( psk )1  bsk ( psj )1  bsk (w j )1 per variety produced in sector-s of country j ≠ k. With thejmeasure of varieties produced in this sector, Vs , the total gross export value from j to k in
j

k

j 1

sector-s is Vs bs (w ) f s j bsk (w j )1 Lj . Thus, the net export value from 1 to 2 in sector-s is30

©Kiminori Matsuyama, Engel’s Law in the Global Economy

1



2

1 2

1 1

given by NX s   NX s =  f s bs (w )

L1  f s2bs1 (w2 )1 L2 . Using (28), (30) and (35), this canbe further rewritten as:(37)

NX s1   NX s2 1

w2 L2
w1 L1
1
2

m

m

m1s  ms2  .
=
s
s


()  ( )  

2

1

2

1

 
1

Thus, NX s   NX s > 0 for s < sc and NX s   NX s < 0 for s > sc when U   u x < U 2  u x2 . This is nothing but the Linder effect. It may also be viewed as a variant of theHome Market Effect of Krugman (1980). The key difference is that the cross-country differencein the market size distribution across sectors is due to nonhomothetic preferences in this model,
not due to the exogenous cross-country variations in taste as assumed in Krugman (1980).
1

2

It is worth emphasizing that country 1 becomes a net exporter in sectors where ms  ms
1

1 1

2

2 2

holds, which are not necessarily sectors where ms w L  ms w L holds. What determines the
direction of net sectoral trade flows in a general equilibrium model of the home market effect is
not the cross-country difference in the market size in each sector. What matters is the crosscountry difference in the demand compositions, i.e., in the cross-country difference in the market
size distributions across sectors. 20
4.5

Ranking the Countries: Trade-off between Human Capital and Country Size
Our remaining task is to rank the two countries in terms of the standard-of-living. This issimple when the two countries are in equal size, L1  L2  L . In this case, ω = 1 so that   L , and hence , x / x = h / h xk  (1   ) x0k  (1   ) hk N k  (1   ) hk

 1

1


2


1

2  1

20

The Home Market Effect is often described simply as “relatively large domestic demand gives competitive
advantages to exporting firms.” To this, we have heard some IO people say something to the effect that the share of
the domestic sale must be trivial for many firms based in small economies like Denmark or Switzerland. The result
here should explain why such a criticism is unwarranted. Even if the Swiss domestic market might be smaller thanthe Chinese domestic market in every sector in absolute terms, some sectors should account for larger shares in the
Swiss expenditure than in the Chinese expenditure, as long as the two countries differ in the demand composition.
And that is what determines the patterns of comparative advantage in a general equilibrium model of the Home
Market Effect. Matsuyama (2015, section 3) demonstrated this by extending the Krugman model with a continuumof sectors and with two countries of unequal size. In the context of the present model, even if Switzerland may bemuch smaller than China, and consequently the domestic market accounts for a tiny share of the revenue for the
Swiss firms operating in any sectors (recall the result shown earlier that d s1  0 as L1 / L2  0 ), Switzerland should
become a net-exporter in high-income elastic sectors and China a net-importer in low-income elastic sectors, as longas Switzerland has higher standard-of-living than China.

31

©Kiminori Matsuyama, Engel’s Law in the Global Economy= w1h1 / w2h2



 1

. Thus, the country with higher labor productivity has higher standard-of-living. This country also has higher per capita income.
Generally, the condition under which Country 1 has lower standard-of-living than Country
1

 
1

2

 
2

1

2

2, U   u x < U   u x or x  x can be written as: 1

1   () x10  h1  N 1  h1 
 2   2  2   2 
1   ( )
x0  h  N  h 

 1

1
L1
2 1  h  ()  2 
L2
h 

 1,

which can be further rewritten as: 1 211 
 h 
 ~  h1 L1(38) 2   ( ;  )     2 ;     2 ;   .
L h hTo understand this condition, it would be useful to compare it with the conditions under
1

 
1

2

 
2

which Country 1 is poorer under autarky, U0  u x0 < U0  u x0 and Country 1 has lower per
capita income, w1h1  w2h2 , which can be written as:1

L1  h1  
L2  h2 ,1  h1  1 
L1
   ;      h ;   ,


(

;

)


 h2   h 2 

L2





respectively. Figure 3 illustrates these conditions. The black curve depicts the graph of
~
1
1
2
2
L1 / L2   h1 / h 2 ;   on which U   u x = U   u x holds. It is downward-sloping, and

 

 

 

 

 

 

U 1  u x1 < U 2  u x2 holds below and to the left of this curve, and U 1  u x1 > U 2  u x2holds above and to the right of this curve. The red curve depicts the graph of L1 / L2  h1 / h 2 

1 

1

 
1

2

 
2

1

,

 
1

on which U0  u x0 = U0  u x0 holds. It is also downward-sloping and U0  u x0 <

 

 

 

U02  u x02 holds below and to the left of this curve, and U01  u x10 > U02  u x02 holds above
and to the right of this curve. The blue curve depicts the graph of L1 / L2   h1 / h 2 ;   , onwhich w1h1  w2 h 2 holds. It is also downward-sloping and w1h1  w 2 h 2 holds below and to the32©Kiminori Matsuyama, Engel’s Law in the Global Economy

left of this curve, and w1h1  w2 h 2 holds above and to the right of this curve. It is also easy to~verify that  1;   =  1;   = 1 and1  h1 ~  h1   h1 <
  2 ;    2 ;   <  2 hh h 

<1

for

h1
 1;
h2

>1

for

h1
< 1,
h2

1 

 h1
 ~  h1   h1   2 ;   >  2 ;   >  2 hh h 

as shown in Figure 3.
For L1 / L2 = 1, all three curves intersect at h1 / h 2 = 1. Hence, h1 / h 2 < 1 implies
U 01  U 02 , U 1  U 2 and w1h1  w 2 h 2 , while h1 / h 2 > 1 implies U 01  U 02 , U 1  U 2 andw1h1  w2 h 2 . Thus, when the two countries are equal in size, comparing labor productivity alone

can determine which country has higher standard-of-living, as already pointed out. When thetwo countries are unequal in size, these three conditions diverge. To see this, consider the caseof h1 / h 2 > 1. For L1 / L2  h1 / h 2 1 , U 01  U 02 , U 1  U 2 and w1h1  w2 h 2 . Thus, when the

country with higher labor productivity is not too smaller in size, it has higher standard-of-living
both under autarky and under trade. It also has higher per capita income. ForL1 / L2  h1 / h 2



1 

 1 , however, the country with higher labor productivity has lower standard-of-living in autarky. When the condition (38) holds, this country has lower standard-of-living
and is the net-exporter in the lower income elastic sectors. Notice that (38) is more stringent
than L1 / L2  h1 / h 2 

1 

1 
~
< 1. In other words, for  h1 / h 2 ;   < L1 / L2  h1 / h 2  < 1, thestandard-of-living in this country is lower in autarky but higher under trade, because trade
reduces this country’s disadvantage of being smaller. Notice also that the condition (38) is lessstringent than L1 / L2 <  h1 / h 2 ;   < 1, the condition under which its per capita income is~smaller. In other words, for  h1 / h 2 ;   < L1 / L2 <  h1 / h 2 ;   < 1, the standard-of-living inthis country is lower even when its per capita income is still higher in this country. This canoccur because this country benefits less from the variety effect due to its smaller size.

5

Trade Equilibrium: Comparative Statics
33

©Kiminori Matsuyama, Engel’s Law in the Global Economy

Having characterized the cross-country variations in a given trade equilibrium, we now
turn to comparative static exercises.
5.1

Uniform Labor Productivity Growth: Interdependent Patterns of Structural

Change Across Countries and Product Cycles
First, consider the effects of a uniform labor productivity growth. That is, laborproductivity goes up at the same rate in all the activities in both countries. This can be capturedby  log( h1 ) =  log( h 2 )   log(h) > 0. This keeps h1 / h 2 and L1 / L2 unchanged, with  log( L1 ) = log( L2 ) =  log(h) > 0. Therefore,   w1 / w2 is also unchanged, and so are x10 / x02 andx1 / x2 , with  log( x10 ) =  log( x02 ) =  log( x1 ) =  log( x2 ) =  log(h) > 0.

With  log( x1 ) =  log( x2 ) > 0, both U 1  u ( x1 ) and U 2  u ( x2 ) go up. With theirstandard-of-living improving, both countries shift their expenditure shares towards higherindexed sectors in the sense of both MLR and FSD. This can be seen from eq.(33) and applying
k


 

k


Lemma 1 for gˆ (s, x )   s u( x )  1 ( ( s )  )    







.

Even though x1 and x2 goes up at the same rate to keep x1 / x2 unchanged, the standardof-living in the two countries do not go up at the same rate. To see this, log(U 1 / U 2 )  log(u( x1 ))   log(u( x2 ))1
2==   x    x  .
 log(h) log(h)Hence, from Lemma 2-ii),
(39)

sgn

 log(U 1 / U 2 )= sgn(  1) sgn( x1  x2 ) .
 log(h)Thus, the standard-of-living goes up at a faster rate in the Richer country if   1 and in the
Poorer country if   1 . In words, welfare gaps widen (narrow) if the goods produced indifferent sectors are substitutes (complements) .
To see how the patterns of trade change, log-differentiate (34) to yield,
12

 log(m1s / ms2 )   1   log(U  / U  ) ( (s)   )
, log(h)      log(h)and then use (39) to obtain34

©Kiminori Matsuyama, Engel’s Law in the Global Economy

(40)

sgn

 log(m1s / ms2 ) sgn (s)   sgn(  1) sgn( x1  x2 ) = sgn( x2  x1 ) .
 log(h)from Lemma 2-ii) and by recalling the parameter restriction,  (s)    /(1   )  0 , that ensures
the global monotonicity of the utiliy function, (1).
Figure 4 illustrates this for U 1  u ( x1 ) < U 2  u ( x2 ) . In this case, the downward-sloping
1

2

curve, ms / ms , shifts up, which causes the cutoff sector, sc , to move up. As a result, the Rich’s
trade balances switch from net surpluses to net deficits in some middle sectors.21 The intuition
behind this result is easy to grasp. As the standard-of-living improves in both countries, both
shift their expenditure shares towards the higher-indexed sectors. In response, both countries
reallocate their resources towards higher-indexed sectors. In other words, the relative weights ofhigher-indexed sectors, in which the richer runs surpluses, go up and the relative weights oflower-indexed sectors, in which the poorer runs surpluses, go down. This means that, in order tokeep the overall trade account between the two countries in balance, the richer’s sectoral tradeaccount must deteriorate in each sector. This is why the richer switches from being a net
exporter to being a net importer in some middle sectors.
5.2

Globalization Without Terms of Trade Change: Interdependent Patterns ofStructural Change Across Countries and Product Cycles
Next, consider the effects of globalization, captured by a trade cost reduction, or a1higher   ( ). First, let us look at the case where the two countries are in equal size:

L1  L2  L . In this case, the factor price is always equalized, w1  w 2  w , or ω = 1,kk

  N = (1  )h  L , and hence ,independent of ρ, so that x  (1   ) x0 = (1   ) hk

x1 / x2 = x10 / x02 = h1 / h2kk  1

 , as noted earlier. That is, the country with higher standard-of 1living is the one with higher labor productivity and with higher per capita income. 22 Hence, thecountry with higher labor productivity is always a net exporter in higher-indexed sectors and a

21

For U 1  u ( x1 ) > U 2  u ( x 2 ) , the upward-sloping curve, m1s / ms2 , shifts down, which also leads to the cutoff

sector, sc , to move up. Either way, the Rich’s trade balances must switch from net surpluses to net deficits in somemiddle sectors.
22
In this case, the two countries have the same aggregate GDP, but differ in GDP per capita.

35

©Kiminori Matsuyama, Engel’s Law in the Global Economy

net importer in lower-indexed sectors, precisely because they have relatively larger expenditure
shares in higher-indexed sectors, which causes disproportionately larger shares of workers are
employed in higher-indexed sectors due to the home market effect.
Furthermore, in this case, the effects of globalization, a higher  , can be seen only bykk  L . Indeed, without causing any terms-of-trade change,

looking at x = (1   ) x0 = (1   ) hk

 1

the effects of a higher  is isomorphic to a uniform labor productivity growth, with
 log(1   ) > 0 equivalent to (  1) log( h1 ) = (  1) log( h 2 )  (  1)  log( h) > 0.

Hence,

by going through the analysis as done in the previous subsection, one can show that the standardof-living improves (a higher U k ) in both countries and they shift their expenditure shares
towards higher-indexed sectors both in the sense of MLR and FSD. Furthermore, one can show:sgn

 log(U 1 / U 2 )= sgn(  1) sgn( x1  x2 ) .
 log(1   )so that globalization causes the welfare gap between the Rich and the Poor to widen (narrow) if
the goods produced in different sectors are substitutes (complements). One can also show:sgn

 log(m1s / ms2 )= sgn( x2  x1 ) , log(1   )so that the cutoff sector moves up (see Figure 4). Thus, the richer country, the country withhigher labor productivity, switches from a net exporter to a net importer in some middle sectors,
generating something akin to product cycles without any technology diffusion across countries.
In summary, when the two countries are equal in size, globalization causes no terms-oftrade change. And without any terms-of-trade change, globalization is isomorphic to the effects
of uniform labor productivity growth, because it allows the consumption goods sectors in eachcountry to better access to the varieties of inputs produced abroad. Such productivity gains fromtrade cause structural change through an endogenous change in the demand composition.
5.3

Globalization with Terms-of-Trade: Leapfrogging and Patterns of Trade Reversal
When the two countries are unequal in size, the factor price is lower in the smaller

country, due to the disadvantage of being smaller in the presence of aggregate increasing returns.
The larger the trade cost, the greater this disadvantage. Globalization reduces this disadvantage

36©Kiminori Matsuyama, Engel’s Law in the Global Economy

for the smaller country, thereby causing the terms of trade change in favor of the smaller country,
and a factor price convergence, as shown in Figure 1.
When the smaller country has lower labor productivity, this country always has lower
standard-of-living, regardless of the trade cost. However, when the smaller country has higher
labor productivity, it is possible that this country has lower standard-of-living at a high trade cost
but higher standard-of-living at a low trade cost. This possibility is illustrated in Figure 5, which
reproduces some parts of Figure 3. Below and to the left of the red curve, Country 1 has lower
standard-of-living than Country 2 in autarky. Below and to the left of the black curve, Country 1
has lower standard-of-living than Country 2 under trade. Globalization, a higher ρ, rotates theblack curve clockwise, as indicated by the arrows. As ρ approaches zero, the black curve
converges to the red curve, which is invariant to the trade cost. As ρ approaches one, the black
curve converges to the vertical line, h1 / h 2 = 1. Now, consider the case where country 1 has
higher labor productivity, i.e., h1 / h 2 > 1 but it is sufficiently smaller so that L1 / L2 < ( h1 / h 2 )1
< 1. Thus, we consider the point, ( h1 / h 2 , L1 / L2 ), located to the right of the vertical line, h1 / h 2= 1 and below the red curve. Then, with a sufficiently small ρ, the black curve passes above andto the right of this point, which means that Country 1 has lower standard-of-living. With asufficiently large ρ, the black curve passes below and to the left of this point, which means theCountry 1 has higher standard-of-living. Thus, closer to autarky, Country 1 is poorer due to its
disadvantage of being smaller in the presence of aggregate increasing returns, hence runningsurpluses in lower-indexed sectors. Globalization reduces the disadvantage of being smaller,causing a factor price convergence, which makes it richer, hence running surpluses in higherindexed sectors. This result thus suggests the possibility that some relatively small countries
with relatively highly educated labor forces, which might initially have lower standard-of-living
due to their remote locations and might be net-exporters in the low income elastic sectors, benefit
more from globalization and overtake other larger countries and emerge as net-exporters in thehigh income elastic sectors.

6.

Concluding Remarks
The endogeneity of the demand composition due to the nonhomotheticity of demand, orEngel’s Law, is an important channel through which economic growth and globalization affect37

©Kiminori Matsuyama, Engel’s Law in the Global Economy

sectoral patterns in employment, value-added, and productivity change, as well as intersectoral
patterns of trade and migration of industries across countries. Some of these effects have beenstudied in the past, but only separately, which could be misleading, as these effects areinterconnected.
This paper offered a unifying perspective on the role of Engel’s Law in the global
economy, by developing a model, which is rich enough to capture all these effects and theirinteractions and, at the same time, abstracts from all other factors in order to isolate the role ofEngel’s Law. The key ingredients of the model are i) two countries that differ in population sizeand labor productivity (and hence its size, measured in the total effective labor supply); ii)
implicitly additive, isoelastic, nonhomothetic CES preferences over a continuum of nontradeable
consumption goods, iii) endogenous productivity differences across sectors and across countries,
due to endogenous variety of differentiated inputs supplied monopolistically competitively with
the iceberg trade cost, as in Dixit-Stiglitz (1977) and Krugman (1980). In terms of cross-country
variations, it is shown among others that the country with higher standard-of-living, whose
domestic demand composition is more skewed towards higher income elastic sectors, allocatesdisproportionately larger shares of labor in higher income elastic sectors (the Home Market
Effect in employment) and becomes a net-exporter in those sectors (the Linder effect). In termsof comparative statics, it is shown among others that labor productivity growth (andglobalization in the case of the equal country size) cause structural change towards higher
income elastics in both countries; product cycles, in which the richer country switches from a
net-exporter to a net-importer of the sectors in the middle range of income elasticities; andwelfare gaps to narrow (widen) when sectors are complements (substitutes) through the market
size (Schmookler) effect on the relative productivity and price changes. In addition, when the
countries differ in size, globalization could help the smaller country with higher laborproductivity overtake the other (Leapfrogging), which leads to a reversal of the patterns of trade.
For all these reasons, globalization amplifies, instead of reducing, the power of Engel’s Law andthe endogenous domestic demand composition differences as a driver of structural change. 23
23

Both the endogeneity of the demand composition and that of the terms of trade are crucial for most of these results.
To clarify the former, Matsuyama (2015, section 3) considers an alternative model, where the domestic demand
composition differences are due to the exogenous differences in taste. (This model may be viewed as an extensionof the Krugman (1980)’s Home Market Effect model to the case of an arbitrary number of sectors with an arbitrary
exogenous variations in taste across the two countries which are not necessarily equal in size.) To clarify the latter,

38

©Kiminori Matsuyama, Engel’s Law in the Global Economy

It would have been impossible to capture all these effects of Engel’s Law and theirinteractions, if we had used Stone-Geary, CRIE or any other explicit direct additive form ofnonhomothetic preferences, which impose the strong restriction between the income and priceelasticities of the goods, known as Pigou’s Law. This restriction not only has been rejected
empirically, but also makes it impossible to disentangle the effects of income elasticity
difference from those of price elasticity differences across sectors, hence to isolate the role ofEngel’s Law. This becomes feasible only with the use of an implicitly directly additive form ofnonhomothetic preferences, as it allows for an arbitrary number of consumption goods with
good-specific income elasticity parameters, which can be controlled for independently of the(constant) price elasticity parameters. It seems that this class of preferences should find a wide
range of applications.

Appendix: Two Lemmas
This appendix offers two lemmas, which are used repeatedly in the analysis.
Lemma 1: For a positive value function, gˆ (; x) : I  R , with a parameter x, define a densitygˆ ( s; x )function on I by g (s; x) , and denote its distribution function by G s; x  . If gˆ ( s; x)
ˆ
g
(
t
;
x
)
dt

I

is log-supermodular in s and x, i.e.

 2 log gˆ ( s; x )
 0,
sx
g (s; x1 )is decreasing in s for x1  x2 ;g ( s; x2 )i)Monotone Likelihood Ratio (MLR):

ii)

First-order Stochastic Dominance (FSD): G ( s; x ) is decreasing in x.

For the proof, see Matsuyama (2015, Appendix). 24

Matsuyama (2015, section 4) adds a competitive outside sector, which produces a homogenous good that can be
traded at zero cost, and is large enough to kill any general equilibrium terms-of-trade effect.
24
The results in this lemma are not new. For example, they were used in Matsuyama (2013, 2014) without proof.
Furthermore, ii) follows from i). Indeed, they are special cases of more general properties of log-supermodularity
known in the literature: see, e.g., Athey (2002) and Vives (1999; Ch.2.7).

39

©Kiminori Matsuyama, Engel’s Law in the Global Economy

Lemma 2: For   1 , define u (x ) implicitly by x

1
 



 1
(  ( s )  )     s u ( x ) Ids . If( ( s )   ) /(1   )  0 ,i)u (x ) is strictly increasing in x;ii) ( x) 

xu ' ( x)is decreasing in x if η < 1 and increasing in x if η > 1.
u ( x)

Proof: Differentiating the definition yields
1

 1
 1      1    1 ( ( s )  )   1(  ( s )  1)

 x
 
   s u ( x )  s ( ( s )   )u ( x ) u ' ( x) dsI     

x

1
1
 





 1
   1  u ' ( x ) ( ( s )  )     s u ( x ) 
 

( ( s )   ) ds
 1    u ( x )  I







 1
  ( s)   1   1    ( s)   ( ( s )  )  
  1   
ds = (  1)    s u ( x )  f ( s; x) ds .
II ( x )      1   
 1  
x






Hence,(A1)1
  ( s)   = (  1)   dF ( s; x) > 0,I
 ( x)
 1   1( ( s )  )   u ( x)
where f ( s; x )   u ( x) stI is a density function, and F (s; x) is its cumulative
 dt

 1
(  ( t )  )  

distribution function.
First, (A1) shows  ( x ) 

xu ' ( x)> 0, hence u (x ) is strictly increasing. Second, because u (x ) isu ( x) 1(  ( s )  )  

strictly increasing,  s u ( x) is log-supermodular in s and x. Hence, from ii) of

Lemma 1, F ( s; x) satisfies FSD. For η < 1,

 (s )  
is increasing in s, so that RHS of (A1) is1 (s )  
is decreasing in s, so that1RHS of (A1) is decreasing in x, hence  (x ) is increasing in x. Q.E.D.
increasing in x, hence  (x ) is decreasing in x.

For η > 1,

40

©Kiminori Matsuyama, Engel’s Law in the Global Economy

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43©Kiminori Matsuyama, Engel’s Law in the Global Economy

Figure 1: (Factoral) Terms of Trade Determination: L1 / L2 =   ;  ω ≡ w1/w2(ρ)‒1/σ1(ρ)1/σL1/L2
O1Figure 2: Endogenous Market Size Distribution and the Home Market Effect in Employment and11
22Inter-sectoral Patterns of Intra-sectoral Trade: for U   u x < U   u x 

fs1/ fs2

 

ms1/ ms2

1

O

s

sc
1’s Net Exports

2’s Net Exports
44

©Kiminori Matsuyama, Engel’s Law in the Global Economy

Figure 3; Ranking the Countries: Trade-off between Labor Productivity and Country Size
L1 / L2~
L1 / L2   h1 / h 2 ;   U 1  U 2
L / L   h / h ;    w1h1  w 2 h 2
1

2

1

2

1



L1 / L2  h1 / h 2

O

1

(ρ)

‒1/σ

(ρ)

(1‒2σ)/σ(σ‒1)



1 

 U 01  U 02

h1 / h 2

Figure 4: Interdependent Patterns of Structural Change and Product Cycles: The Effects of An
Uniform Labor Productivity Growth and Globalization (when the two countries are equal in size)

fs1/ fs2

ms1/ ms2
1

O

s
sc

sc

The richer country’s sectoral trade balances
switch from surpluses to deficits

45

©Kiminori Matsuyama, Engel’s Law in the Global Economy

Figure 5: Leapfrogging and Reversal of Patterns of Trade: The Effects of Globalization (whenthe two countries are unequal in size)

L1 / L2

1

1


U U

2


U 01  U 02

U 1  U 2
O

1

U 01  U 02

(ρ)

(1‒2σ)/σ(σ‒1)

h1 / h 246