Download Capabilities, Wealth and Trade

Capabilities, Wealth and Trade
John Sutton
London School of Economics
and
Daniel Trefler∗
Rotman School of Management, University of Toronto; CIFAR; NBER
August 11, 2012
ABSTRACT: This paper re-explores the relation between a country’s
level of wealth and the mix of products it exports. We argue that both
are simultaneously determined by countries’ capabilities i.e. by countries’ productivity and quality levels for each good. Our theoretical
setup has two features. (1) Some goods have fewer high-quality producers/countries than others i.e. there is Ricardian comparative advantage. (2) Imperfect competition allows high- and low-quality producers
to coexist, which we refer to as ‘product ranges.’ These two features generate a very particular non-monotonic, general equilibrium relationship
between a country’s export mix and its GDP per capita. We show that this
non-monotonicity permeates the international data on trade and GDP
per capita.
Key words: quality, economic development, international trade
JEL classification: F12, O10
∗. We thank Q. Zhang, Michelle Liu and especially Leilei Shen for excellent research assistance. We have benefited
from seminar presentations at the Canadian Institute for Advanced Research (cifar), the LSE, Princeton, Stanford,
Toronto, and the World Bank and are grateful of comments from Daron Acemoglu, Philippe Aghion, Bernardo Blum,
Avner Greif, Elhanan Helpman, David Hummels, Peter Morrow and Bob Staiger. Trefler thanks cifar and the Social
Sciences and Humanities Research Council of Canada (sshrc) for financial support. E-mails: [email protected] and
[email protected]
1. Introduction
A country’s capability — meaning the set of goods the country is able to produce and its quality and
productivity in producing them — drives its per capita income and the sectoral mix of its exports.
Aspects of the relationship between quality, income, and the sectoral mix of exports have been analyzed by a number of researchers. Hummels and Klenow (2005) estimate the impact of a country’s
per capita income and size on export quality. Hausmann, Hwang and Rodrik (2007) explore how
the process of ‘cost discovery’ affects the sectoral mix of exports, which in turn affects per capita
incomes. Flam and Helpman (1987) and Fajgelbaum, Grossman and Helpman (2011) examine the
co-determination of quality, income, and the sectoral mix of exports in models where demand-side
consumer heterogeneity plays a central role. In contrast, we use a supply-side Ricardian model to
show how the general equilibrium logic of comparative advantage provides important theoretical
and empirical insights into how quality capabilities simultaneously affect per capita incomes and
the sectoral mix of exports (as well as prices, markups and profits at the firm level).
To bring out these insights as clearly as possible we focus theoretically and empirically on
characterizing the range of countries exporting a specific good, as in Schott (2004), and in characterizing how these countries’ market shares vary with their incomes. A standard intuition for
the relationship between market shares and incomes runs through quality. Hummels and Klenow
(2005) show that rich countries must have high-quality exports because, at the aggregate level,
rich countries have high prices and high world market shares. A related inference appears in
Khandelwal (2010), Hallak and Schott (2011), and Baldwin and Harrigan (2011). We show both
theoretically and empirically that this aggregate insight does not carry over in general equilibrium
to the sectoral level because of Ricardian comparative advantage. For example, even though the
United States is a high-quality producer of stainless steel, it has a small world market share: It
cannot compete with markedly inferior Chinese stainless steel because U.S. wages have been bid
up by high demand for military aircraft, virtualization software, and other hard-to-make goods and
services that only a handful of rich countries are capable of producing.
Our model has two key elements. (a) We make the Ricardian assumption that products can be
ordered by the scarcity of quality capabilities. Specifically, if a country has high quality in good g
then it has high quality in all goods ranked below g. This means that low-g goods are ones for which
most countries have high quality and are in this sense ‘easy’ to make. In contrast, high-g goods are
ones for which few countries have high quality: they are ‘hard’ to make. This assumption captures
the notion of relative (in a Ricardian sense) scarcity of quality capabilities. (b) We assume that
goods are differentiated only by quality (pure vertical differentiation) and are supplied in markets
characterized by Nash equilibrium in quantities (Cournot competition). We use this assumption to
ensure that differing levels of quality will co-exist in equilibrium. Elements (a) and (b) generate a
correlation between a country’s income and its export mix. A country with high quality in just a few
goods will only be able to survive in a few markets and these will be the low-g or ‘easy’ markets.
As a result, derived demand for the country’s labour will be low and wages will be low. Thus,
low-wage countries will export low-g goods. A country with high quality in many goods will have
a high derived demand for its labour and have high wages. High wages will make the country a
high-cost producer of low-g goods. Hence, a high-wage country will only survive in high-g markets.
This Ricardian sorting generates an inverted-U relationship between income and market shares
at the sectoral level. To understand why, consider a country whose capabilities improve at the
sectoral level: that is, the country improves its quality in a g-ranked good until it reaches the
world quality frontier and then it improves its quality in the next, higher-ranked, good. During this
quality-improvement process, demand for the country’s labour rises, as does its wages. As quality
rises in good g, the country’s world market share of the good rises initially because quality must rise
faster than wage costs. This ‘direct’ or ‘quality’ effect underpins the Hummels and Klenow aggregate
correlation. However, as quality then rises in a higher-ranked good, wages continue to rise, thus
killing off the country’s competitiveness in good g: even though the country is a high-quality
producer of good g, its world market share must decline as capabilities rise in higher-ranked,
tougher-to-make sectors. This familiar inter-sectoral, general equilibrium feedback through the
labour market is what we call the Ricardian or ‘wage’ effect. It is the reason for the downward-sloping
section of the inverted-U relationship between income and market share. The model generates
a large number of other theoretical predictions which we describe below, but our empirics are
concentrated on this inverted-U relationship.
Turning to our empirical work, we investigate the sectoral-level, inverted-U relationship between
income and market shares using data for 94 countries in 2005. Data are from COMTRADE (4-digit
SITC and 6-digit HS) and, to a lesser extent, the U.S. imports file (10-digit HS). The theory states that
the inverted-U relationship is driven by labour-market spillovers across sectors. We thus confine
our attention to country-good pairs for which the good is important in the country’s export basket
2
and, by implication, in the country’s labour market. For each good separately, we build a ‘product
range,’ i.e., a range of incomes defined by the income levels of the poorest and richest exporters of
the good. Product ranges are related to Khandelwal’s (2010) quality ladders: the latter describes a
range of qualities while the former describes a range of incomes. We will not be estimating quality
and hence will have nothing to say about quality ladders.1 Schott’s (2004) work on ’overlap’ leads
us to expect that product ranges will be large, and this is indeed what we find. (The finding is
not driven by China.) We then non-parametrically estimate the relationship between income and
world market shares and show that it is exactly as predicted by the theory. (1) For those products
produced only by the richest countries, the relationship is positive: this is the direct or quality effect.
(2) For those products produced only by the poorest countries, the relationship is negative: this is the
Ricardian or wage effect. (3) For the remaining ‘middle’ products, the relationship is inverted-U, as
first the quality effect and then the wage effect kicks in. Re-stated, Ricardian comparative advantage
based on relative scarcity of quality capabilities leads to general equilibrium wage effects that are
central for understanding the cross-country, cross-sector relationship between quality, per capita
income, and the sectoral mix of exports.
Our paper has four key elements: (1) multiple sectors that are ranked based on Ricardian scarcity
of quality capabilities, (2) an imperfectly competitive market structure that supports the co-existence
of differing levels of quality, (3) endogenous income so that there can be general equilibrium
spillovers across sectors via the labour market (wages), and (4) empirical work relating income to
market shares at the sectoral level. With this in mind, we relate our paper to the existing literature.
Our results are driven entirely by supply-side considerations. Demand considerations play no
role in our work. Allowing for demand-side heterogeneity and demand for quality that rises with
income has yielded important insights for comparative advantage and per capita incomes e.g., Flam
and Helpman (1987), Hallak (2006, 2010), Choi, Hummels and Xiang (2009) and Fajgelbaum et al.
(2011).
The endogeneity of income allows us to bring issues of economic development to the forefront
of our research.The relationship between per capita income and the mix of exports has been the
subject of investigation at least since the discussion of ladders of development by Chenery (1960)
and more recently by Leamer (1984, 1987), Michaely (1984), and Schott (2003). In these papers, as
in ours, sectors are asymmetric and ordered. For example, Schott orders sectors by labour intensity.
1. The fact that we are not estimating quality means that our agenda is very different from that of Khandelwal (2010)
and Hallak and Schott (2011).
3
However, these papers do not consider quality, which is how we order sectors. Lall, Weiss and
Zhang (2006), Hausmann et al. (2007) and Schott (2008) provide policy-oriented discussions of the
thesis that ‘what your export matters.’ We do not discuss policy in this paper. Implicitly, however,
our work shifts the policy prescription away from getting the right mix of exports and towards
raising the quality of what is exported.
This paper builds on a series of papers by Sutton. In Sutton (1991, 1998, 2007a,b), firms optimally
invest in building quality capabilities and, once these capabilities are developed, firms engage in
Cournot competition. This leads to a world in which the relative scarcity of capabilities is an
endogenous equilibrium outcome. We can motivate this idea of relatively scarce capabilities by
reference to a key idea in the modern ‘market structure’ literature: if firms must incur fixed and
sunk outlays to develop their capabilities, then the number of firms that find it profitable to develop
these capabilities will be limited: the greater the elasticity of quality (or productivity) responses to
R&D or other fixed outlays, the greater the degree to which firms ‘escalate’ their R&D spending
in competing with rivals, and the fewer the number of producers that survive in the market. As a
result, capabilities are scarce and scarcer in some markets than in others: relative scarcity emerges
endogenously. This argument holds for a broad class of models, of which the Cournot model with
quality is the simplest and most tractable example; it does not hold for CES models with atomistic
firms. For a concise review, see Sutton (2007a). For a general equilibrium analysis of the mechanism
of entry and R&D competition leading to this see Sutton (2007b). In this paper, we simply take as a
given that some capabilities are relatively scarce.2
3
Sutton (2007b) provides an international trade model with two countries that produce final
goods, and a third country that produces raw materials. He shows that when there are raw materials
that are internationally traded, quality and productivity are not isomorphic and, in particular, that
there is a minimum level of quality (independent of productivity or wages) that must be attained if a
firm or country is to enter world export markets. Thus, there are limits on how much low wages can
offset low quality. Hallak and Sivadasan (2009) also break the quality-productivity isomorphism by
2. Cournot competition, unlike models of CES monopolistic competition with atomistic firms, explicitly allows for
‘large’ firms and this is increasingly recognized as a crucial feature of international trade. The idea that firms are not
a point in a continuum appears in di Giovanni and Levchenko (2011) and Eaton, Kortum and Sotelo (2012). As the former
note, a single New Zealand firm (Fonterra) is responsible for one-third of global dairy exports and Samsung is responsible
for 23% of Korean exports. See Melitz and Trefler (2012) for other examples. Cournot competition in international trade
models appears in Neary (2003) and Neary and Tharakan (2011).
3. Sutton’s (1991; 1998; 2007a) work is related to the literature on the endogenous choice of quality e.g., Verhoogen
(2008), Kugler and Verhoogen (2012), Khandelwal (2010), Baldwin and Harrigan (2011), and Johnson (2012).
4
postulating the existence of a minimum quality threshold needed for exporting. In our paper, by
contrast, quality and productivity are isomorphic.
The paper is structured as follows. Section 2 sets up the model. In section 3, we consider a limited
form of international differences in quality capabilities in order to develop a benchmark model with
assortive matching of countries to goods. Section 4 develops the more general model and the key
results. Section 5 draws out the empirical implications of the model that are then examined in
sections 6–9.
2. Set-Up
2.1. Consumer Choice
All countries are of the same size and composed of a unit mass of workers. All individuals have
identical Cobb-Douglas utility functions defined over G goods, indexed by g,
U=
∏(u g x g )δ
g
(1)
g
where
∑ δg
= 1, and u g and x g denote the quality and quantity of good g consumed. It follows
g
from the form of the utility function that each consumer spends fraction δg of income on good g. We
assume that all profits accrue to a separate group of individuals, who also have a utility function
of the form (1). From this it follows that we can treat all firms in the global market for g as facing
a unit-elastic market demand schedule, i.e. the total global expenditure on good g is a constant,
which we denote as Sg , independently of equilibrium prices. We note that the Sg are proportional
to the δg . We will assume throughout that all the δg are equal (δg = δ), and so all the Sg are equal
(Sg = S).
2.2. Equilibrium in the Product Markets
We characterize product market competition using the standard ‘Cournot model with quality’ introduced in Sutton (1991). In this model, firms are characterized by a level of capability, consisting of a
quality level and a productivity parameter denoting the number of worker hours per unit of output
5
produced,4 together with a (‘local’) wage rate specific to the country in which the firm is located.
(Firms are wage takers.) At equilibrium, some subset of firms are active in the production of the
good. For each active firm, indexed by i, its output level is related to the inverse of its productivity
ci , its quality ui , and its (local) wage rate wi . Solving for a Nash equilibrium in quantities (Sutton,
1998, Appendix 14.1), we obtain the firm’s quality-adjusted equilibrium price,
wj cj
1
pi
=
∑
ui
Ng − 1 j u j
(2)
and its quality-adjusted output level,



Ng − 1
wi ci /ui 
xi ui = S
1 − ( Ng − 1)
wj cj
wj cj


∑j
∑j
uj
(3)
uj
where Ng (≥ 2) denotes the total number of firms that are active in the global market for good g,
S is total expenditure on good g and the sum ∑ j is taken over all active firms. One can see from
equation (2) that pi /ui is the same for all active firms. The condition for firm i to be active, i.e. to
have strictly positive output at equilibrium, is that
wj cj
wj cj
Ng
wi c i
1
<
=
(
)
∑
ui
Ng − 1 j u j
Ng − 1 u j
where (
wj cj
uj )
denotes the mean of
wj cj
uj
(4)
over all active producers. We will refer to wi ci /ui as firm i’s
‘effective cost level.’
It is useful to plug equation (2) into equation (3) to obtain an alternative expression for output:
xi =
1
pi
1−
wi c i
pi
S.
(30 )
Thus, a firm is active in equilibrium if its price pi exceeds its marginal cost wi ci .
Note that the right-hand side of equations (2) and (3) depend on ui and ci only through the ratio
ui /ci , which we refer to as the ‘capability’ of firm i. It follows that some key relationships between
capabilities and wages developed below will depend only on ui /ci and not on the absolute levels
4. Thus all costs are labour costs, and fixed costs are sunk, and so do not enter the present (short-run) analysis. Materials
cost, though of crucial importance in general, are here ignored in order to keep the analysis as clear as possible. This issue
is examined in depth in Sutton (2007b) who shows that the key point is this: in the absence of material cost, low-wage
countries can become viable in world markets even at low quality once their wage costs are sufficiently low: only the
ratio of unit costs (wages times labour input) to quality matters to viability, and shortcomings in quality can be offset by
a low value of the wage. But once material inputs as well as labour are required, a fall in the wage can only reduce unit
costs to the world-market value of the material input. This places a floor on price, and so establishes a corresponding
minimum quality level, independent of local wages, that is required for viability. Deficiencies in productivity can always
be compensated for by low wages, but deficiencies in quality cannot. This is an important reason for emphasizing the
role of quality in our present discussion.
6
Figure 1: Types of Countries and Groups of Goods
1, . . . ,N1
| {z }
1, . . . ,Nk
| {z }
1, . . . ,NK
| {z }
1
z }| {
1, . . . ,m
k
z }| {
1, . . . ,m
K
z }| {
1, . . . ,m
Types of Countries
Groups of Goods
of ui and ci . Since our empirical focus is on quality ui , without loss of generality we set ci = 1 for
the remainder of the paper and periodically remind the reader that our comments about quality are
also germane to productivity.
3. Theory I: The Perfect Sorting Baseline
Our aim is to establish a baseline equilibrium in which Ricardian comparative advantage leads to
assortative matching between countries and goods. We proceed in steps.
3.1. The Scarcity of Quality Capabilities
Our key assumption is that countries differ only in terms of their quality capabilities for different
goods. Assume for the moment that there are k = 1, ..., K goods. A type-k country is a country
whose firms can produce goods 1 to k at a ‘standard’ level of quality u, but cannot produce goods
k + 1 to K. The interpretation is that goods with higher indexes require capabilities that are scarcer.
They are ‘harder’ to make. Countries differ only in this capability dimension. There are thus K types
of countries.
We next generalize this slightly by assuming that there are K groups of goods. Within each group
there are m identical goods and the above applies to each good: Specifically, a type-k country has
firms that can produce each of the goods in groups 1 to k (at quality u ) but cannot produce any of
the goods in groups k + 1 to K. See figure 1. There are mK goods.
Following up on our introductory comments on the scarcity of capabilities, we assume that each
type-k country is endowed with a finite number of firms. For simplicity alone, we assume that each
country has exactly one firm per group-k good. It thus has m firms. (Whether a firm is active or not
is endogenous.)
Let Nk be the number of type-k countries. In an equilibrium with assortative matching, group-k
goods are produced by and only by firms from type-k countries. Given the symmetry in the model,
7
it is easy to show that all firms will be active so that in an equilibrium with assortative matching
Nk is also the number of firms worldwide producing each group-k good. Thus, for there to be
fewer equilibrium producers of hard-to-make goods, Nk must be decreasing in k. In fact, something
slightly stronger is needed. Recall that a firm from a type-k country can produce a good either
from group k or from any group ranked below k. To ensure that in equilibrium the group-k good is
chosen, we need Nk to be decreasing in k sufficiently rapidly:
Assumption 1 (Ricardian Relative Scarcity) Nk ≥ Nk+1 + 4.
Finally, in this section u does not vary across firms; such variation is introduced in section 4 and is
essential to our argument.5
3.2. Equilibrium in a Type-k Country
All type-k countries are identical. Let wk be the common wage. Let g index the mK goods. Our
assumptions above imply that in equation (2), w j = wk , u j = u, c j = 1 and Ng = Nk . Hence, for
any good g in group k that is produced in a type-k country, the equilibrium price in equation (2)
becomes
p gk =
Nk
w
Nk − 1 k
0
(2 )
and the equilibrium quantity in equation (30 ) becomes:
x gk
1
=
p gk
w
1− k
p gk
S=
Nk − 1 S
.
Nk2 wk
00
(3 )
Since one unit of output requires one unit of labour, the total demand for labour in a type-k
country is the output per good (x gk ) times the number of goods (m): LkD = mx gk . We assume that
there are N workers in each country, each of whom supplies one unit of labour so that labour supply
is LSk = N. Hence
LkD = LSk ⇐⇒ x gk =
N
.
m
(5)
00
From equations (3 ) and (5), the labour-market clearing wage is:
wk =
Nk − 1 mS
.
Nk2 N
(6)
5. Allowing u to vary with k does not alter anything because quality comparisons are always done within groups of
goods, not across groups of goods.
8
0
Equilibrium in product markets is already built into equation (2 ). Hence, plugging equation (6)
0
into (2 ) and noting that the result is independent of g, equilibrium product prices are given by
pk =
1 mS
.
Nk N
(7)
Income per worker yk is wages plus profits (i.e., revenues) summed over the m goods:
yk =
mpk x gk
.
N
(8)
For each good g that a type-k country produces, its net exports are:
Net Exports = (1 − δm) x gk > 0 .
(9)
This follows from the fact that the value of good-g net exports equals the value of production (pk x gk )
minus the value of consumption (δyk N = δmpk x gk ).6
An equilibrium is a set of wages {wk }k and product prices { pk }k such that consumers maximize
utility, producers maximize profits, labour markets clear nationally, product markets clear internationally, and trade is balanced. Under our assumption 1 restriction on the Nk s there will be perfect
sorting in equilibrium.
Lemma 1 Under assumption 1, all equilibria display perfect sorting i.e., group-k goods are produced by
type-k countries and only by type-k countries. Wages wk , GDP per worker yk , product prices pk , markups
pk /wk , and net exports are unique and given by equations (6)–(9). In addition, wk , yk , pk , and pk /wk are
all increasing in k i.e., countries with scarce capabilities have higher wages, higher GDP per worker, charge
higher prices, and have higher markups.
All proofs appear in Appendix A.
In this section we set up a perfect sorting equilibrium. Note that the Cournot model played no
essential role in this. Its role is central however in the next section, where we introduce quality
differences across different firms (countries) operating in an industry.
4. Theory II: Quality Differences
In this section, we relax the assumption characterizing the baseline model, that all producers of
a good have the same quality (and productivity.) The central aim is to characterize the band of
6. Note that δ is the share of expenditures spent on each good i.e., δ = 1/(mK ). This implies 1 − δm > 0.
9
countries, in terms either of their real wage rate or GDP per worker, that will be active in a given
industry at equilibrium. With this in mind, we examine one of the countries initially of type k − 1,
whose capability in the production of all goods of group k advances, in the sense that its quality,
denoted vk , rises from zero to the standard quality u. We will refer to this country as the ‘developing’
country. As vk rises, the developing country’s mix of output will gradually shift from the production
of goods of group k − 1 to goods of group k. This change will, in general, affect the equilibrium wage
rate of all countries of adjacent types, and the total income (and expenditure) in each country and
market.
We start with some notation. For a typical type-k country, let wk , be its wage, let pk be its price,
let x k be its quantity, and let u be its quality. Let wk−1 , pk−1 , x k−1 , and u be the corresponding
variables for a typical type-(k − 1) country. For the developing country, which may be producing
both group-(k − 1) and group-k goods, let w be its wage, let pk−1 and pk be its prices, let xk−1 and
xk be its quantities, let u be its quality in group-(k − 1) goods, and let vk be its quality in group-k
goods.
Income or GDP per worker for our three types of countries is:
yk−1 = mpk−1 x k−1 /N,
yk = mpk x k /N
and
y = m { pk−1 xk−1 + pk xk } /N
(10)
i.e., revenue per good times the number of goods (m) divided by the workforce (N).
Consider the situation in which vk has risen to the point where the home country is producing
both groups of goods. All producers of a good charge the same quality-adjusted price so that:
p
p k −1
= k −1
u
u
(11)
p
pk
= k .
vk
u
(12)
For each group-(k − 1) good there are Nk−1 − 1 typical producers with wage wk−1 and the one
developing-country producer with wage w. Hence multiplying equation (2) through by the common u,
p k −1 =
1
Nk−1 − 1
[( Nk−1 − 1)wk−1 + w] .
(13)
The developing country’s presence in both the k − 1 and k industries means that it now has
2m firms, the original m firms producing each of the group-(k − 1) goods and the m new firms
10
producing each of the group-k goods.7 The total number of firms producing each type-k good is
therefore Nk + 1: Nk firms have wage wk and quality u while the one developing-country firm has
wage w and quality vk . Hence,
pk
1
=
u
Nk
w
w
Nk k +
u
vk
.
(14)
To determine prices and wages, we need simply look at the labour-market clearing conditions.
Recalling that labour supply is given by N and that labour demand is given by total output,
labour-market clearing for a typical type-(k − 1) country is N = mx k−1 . Plugging the equation
(30 ) expression for x k−1 into N = mx k−1 yields:
N=m
1
p k −1
w
1 − k −1
p k −1
S,
(15)
where we have used the fact that every firm in a typical type-(k − 1) country charges price pk−1 and
has wage wk−1 .
For a typical type-k country, labour-market clearing is likewise given by:
1
N=m
pk
w
1− k
pk
S.
(16)
For the developing country, labour-market clearing is N = mx k−1 + mx k or
N=m
1
p k −1
1−
w
p k −1
S+m
1
pk
1−
w
pk
S.
(17)
Equations (11)–(17) are seven equations in the four prices and three wages. It is very easy to
solve explicitly for these seven variables in terms of a numeraire and this is done in Appendix
A.2, where the numeraire is expenditures per good S. This establishes that for vk in the range
where the developing country produces both groups of goods, there is a unique equilibrium and in
this equilibrium there are closed-form expressions for all the equilibrium prices. Since our focus is
comparative statics with vk , we write all equilibrium outcomes as functions of vk e.g., w(vk ), pk (vk ),
xk (vk ) and S(vk )
We now explore the effect of raising the exogenous variable vk from the critical value at which
xk (vk ) becomes positive, past the critical value at which xk−1 (vk ) becomes zero, and through to the
value vk = u at which the developing country becomes identical to a type-k country.
7. An alternative interpretation is as follows: the group-k good is produced by the same firm that produces the corresponding group-(k − 1) good. Each firm now has two independent businesses, making k and k − 1. Given Cobb-Douglas
preferences, constant marginal costs, and the fact that the firm takes S as given, it follows that the firm’s profit function
is additively separable in its activities in the k − 1 and k markets so that the firm sets the Cournot output in each of these
markets.
11
Lemma 2 Suppose assumption 1 holds. Then there are constants vkL and vkH with 0 < vkL < vkH < u such
that:
vkL < vk < vkH ⇐⇒ xk−1 (vk ) > 0 and xk (vk ) > 0;
vkH < vk ≤ u ⇐⇒ xk−1 (vk ) = 0 and xk (vk ) > 0.
We refer to the situation where the developing country is producing both xk−1 (vk ) and xk (vk ) as
Phase I. We refer to the situation where the developing country is producing only xk (vk ) as Phase
II. A third phase, where the developing country has gained the capability of producing goods in
group k + 1, will be described later. These three phases appear at the top of figure 2. Note that when
vk < vkL , the developing country’s quality is so low that it specializes in group-(k − 1) goods i.e., we
are in the perfect sorting case of the previous section.
Phase I:
We begin by characterizing the equilibrium properties of Phase I. These are illustrated in Figure 2
and stated in the following proposition.
Proposition 1 (Phase I) Let assumption 1 hold and consider vk ∈ (vkL , vkH ) so that xk−1 (vk ) > 0 and
xk (vk ) > 0. Then as vk rises from vkL to vkH :
1. xk−1 (vk ) falls to 0 and xk (vk ) rises from 0 i.e., the developing country reduces output of group-(k − 1)
goods and increases output of group-k goods;
2. pk−1 (vk )/w(vk ) falls and pk (vk )/w(vk ) rises i.e., markups charged by developing-country firms fall
for group-(k − 1) goods and rise for group-k goods;
3. w(vk )/S(vk ) rises i.e., the wage in the developing country rises relative to the numeraire;
4. y(vk )/S(vk ) rises, i.e. GDP per worker in the developing country rises relative to the numeraire.
The core Phase I insight is that as the developing country’s quality capabilities improve in
group-k goods, the developing country experiences rising demand for its output and labour and
hence rising wages. This creates two effects. (1) Improved quality improves the country’s competitiveness in group-k goods. This is the direct or ‘quality’ effect. (2) Higher wages reduce the country’s
competitiveness in group-(k − 1) goods. This is the general equilibrium Ricardian or ‘wage’ effect.
While not used for our empirics, we note that there are cross-country general equilibrium
impacts. For example, as China improves its quality capabilities in autos we would expect that
12
Quality of
Goods k, k + 1
Figure 2: Advancing Quality
Phase
Phase
Phase
I
II
III
•
•
•
vk
•
v k +1
•
•
•
Wages or
Markup
w or y/N
•
•
•
•
•
•
•
of Good k
GDP per worker
•
•
pk /w
Output of
Goods k − 1, k + 1
•
•
• x k −1
• x k +1
Good k
Output of
•
•
•
•
•
xk
•
vkL
•
vkH
u
vkL+1
vkH+1
u
Quality
vk
v k +1
Notes: The top panel shows the developing country’s quality in group-k goods advancing from vk = 0 to vk = u, and
then its quality in group-(k + 1) goods advancing from vk+1 = 0 to vk+1 = u. The critical value of vk at which production
of group-k goods becomes viable is labelled vkL , and is marked by the left-most dashed vertical line. The critical value of
vk at which production of group-(k − 1) goods becomes unviable is labelled vkH , and is marked by the dashed vertical
line separating Phases I and II. The second and third panels show how equilibrium income (wages and GDP per worker)
and the markup rise as qualities rise. The bottom two panels show how the output of goods in groups k − 1, k, and k + 1
change.
13
clothing manufacturers in Bangladesh would be better off and auto manufacturers in the United
States would be worse off. The next proposition makes this point.
Proposition 2 (Phase I) Let assumption 1 hold. Then as vk rises from vkL to vkH : (1) wk−1 (vk )/S(vk ) rises
and wk (vk )/S(vk ) falls i.e., relative to the numeraire, the wage rises in type-(k − 1) countries and falls in
type-k countries; and (2) w(vk )/wk−1 (vk ) rises i.e., the wage rises faster in the developing country than in
type-(k − 1) countries.
Phase II:
When quality capabilities rise above vkH , our developing country’s wage becomes so high that its
firms are no longer viable (profitable) in the markets for group-(k − 1) goods. Equations (12)–(17)
continue to hold with only two modifications. The labour-market equilibrium condition for the
developing country (equation 17) now becomes:
N = mxk = m
1
pk
1−
w
pk
S.
(17–II)
Also, the price of group-(k − 1) goods (equation 13) must be modified because there are now only
Nk−1 − 1 producers, all with wage wk−1 and quality u. Hence, from equation (2) with Ng = Nk−1 − 1,
p k −1 =
Nk−1 − 1
w
.
Nk−1 − 2 k−1
(13–II)
Equations (12), (13–II), (14), (15), (16) and (17–II) are six equations in the three prices and three
wages. It is very easy to solve explicitly for these six variables in terms of the numeraire S and
this is done in Appendix A.3. This establishes existence and uniqueness and provides a complete
characterization of equilibrium in Phase II.
The impact of vk rising to u appears in figure 2 as the early part of Phase II (where vk+1 is still 0),
and is stated in the next proposition.
Proposition 3 (Early Phase II) Let assumption 1 hold and consider vk ∈ (vkH , u) so that xk−1 (vk ) = 0 and
xk (vk ) > 0. Then as vk rises from vkH to u:
1. pk (vk )/w(vk ) rises i.e., the markup charged by developing-country firms rises;
2. w(vk )/S(vk ) rises i.e., the wage in the developing country rises relative to the numeraire;
3. y(vk )/S(vk ) rises i.e., GDP per worker in the developing country rises relative to the numeraire.
14
Phase III:
We now allow the developing country’s quality capabilities in group-(k + 1) goods, which we
denote by vk+1 , to rise from 0 to u (figure 2, top panel). All of our lemma 2 and proposition 1 results
relating changes in vk to changes in variables subscripted by k − 1 and k now apply when relating
changes in vk+1 to changes in variables subscripted by k and k + 1, respectively. In particular, there
is a critical value vkL+1 at which the developing country begins producing goods in group k + 1 and
a critical value vkH+1 at which the developing country stops producing goods in group k. During
this process markups rise for group-(k + 1) goods (the direct or ‘quality’ effect). Further, wages and
GDP per worker rise relative to the numeraire, and this reduces the markups for group-k goods (the
Ricardian or ‘wage’ effect). See figure 2.
5. Towards Empirics: Implications for Exports
To examine these predictions empirically, we will use international data and therefore need expressions for exports. We start by noting that, in equilibrium, all producers charge the same
quality-adjusted price so that consumers do not care which producer they buy from. We therefore
assume that in equilibrium there is no cross-hauling of goods across international borders i.e., a
good is either imported or exported, but not both.
Let X k−1 (vk ) and X k (vk ) be the value of exports for typical countries of type k − 1 and k, respectively. Note that a lower-case x is the quantity of output and an upper-case X is the value of exports.
Also note that these are values (price times quantity) since that is what we observe in the data. The
developing country’s value of exports for a good in group k − 1 and a good in group k is denoted
by Xk−1 (vk ) and Xk (vk ), respectively.8 The next lemma states that exports behave like production.
Lemma 3 Let assumption 1 hold.
1. As vk rises in Phase I, Xk−1 (vk )/S(vk ) and Xk−1 (vk )/X k−1 (vk ) fall to 0.
2. As vk rises in Phases I and II, Xk (vk )/S(vk ) and Xk (vk )/X k (vk ) rise from 0.
In our empirics we will examine the value of exports of group-(k − 1) and group-k goods as a
share of world exports. For the developing country these shares are given by:
θkX−1 (vk ) ≡
X k −1 ( v k )
Xk−1 (vk ) + ( Nk−1 − 1) X k−1 (vk )
and
θkX (vk ) ≡
Xk ( v k )
.
Xk (vk ) + Nk X k (vk )
(18)
8. Typical type-(k − 1) countries produce and export a single good so that exports are both gross and net. For some
values of vk , the developing country produces two goods and one of these is produced in such small amounts that the
good is imported. Thus, for the developing country, exports are gross rather than net.
15
θkX+1 is the same as θkX but with the k index incremented by 1.
The evolution of these world export shares follow immediately from lemma 3. In Phase I, the
developing country is shifting out of group-(k − 1) production and into group-k production. This
benefits type-(k − 1) countries and hurts type-k countries so that θkX−1 falls and θkX rises. In Phase
II, the developing country is specialized in group-k production and getting better at it, which hurts
type-k countries. Thus, θkX rises.
We now make the transition from theory to empirics. We do not observe capabilities, but changes
in capabilities induce observable changes in GDP per worker and the value of exports. These are
illustrated in figure 3, which plots export shares against GDP per worker for the developing country.
In Phase I, y(vk ) increases, θkX−1 (vk ) falls to 0, and θkX (vk ) rises from 0. Note that θkX (vk ) does not start
rising as soon as Phase I is entered because at this point production of group-k goods is so small that
these goods are still imported. The point at which exporting of group-k goods starts is indicated in
figure 3 by ymin ,k . This ymin ,k is very important for our empirics.
In the first part of Phase II where vk is rising to u, both y(vk ) and θkX (vk ) rise. In the second
part of Phase II, where capabilities in k + 1 are rising but have not yet reached a level at which the
developing country can enter group-(k + 1) markets, nothing happens. See figure 2. It follows that
the system is ‘stuck’ at the point where y = y(u).
In Phase III, the developing country enters group-(k + 1) markets and grabs world market share
i.e., θkX+1 rises. This drives up wages and makes the developing country less competitive in group-k
markets. As a result θkX (vk ) falls. As the process continues, the developing country reaches the point
where it consumes the small amount of xk that it is still producing i.e., θkX (vk ) = 0. By definition,
Phase III ends when xk = 0, so θkX (vk ) goes to 0 just before the end of the Phase. This is indicated
on figure 3 by the point ymax ,k , which is also important for our empirics. It is straightforward to
calculate closed-form solutions for ymin ,k and ymax ,k .9
Two key empirical points emerge from figure 3. First, poorer, low-quality countries can produce
the same good as richer, higher-quality countries. Low-quality countries compete because in equi9. There are two minor points in the figure that are not used in what follows. (1) At the start of Phase I, all countries
are identical so that each of the type-(k − 1) countries has a world market share of θkX−1 = 1/Nk−1 . At the end of Phase
II, all countries are again identical so that each type-k country has a world market share of θkX = 1/( Nk + 1). At the end
of Phase III it is θkX+1 = 1/( Nk+1 + 1). (2) Production always go to 0 at the end of a phase. Since gross exports go to 0
before production does (i.e., at the point where all domestic production is consumed domestically), gross exports always
go to 0 before the end of a phase. Likewise, at the start of a phase, production starts and is consumed domestically so that
exporting starts just after the start of a phase.
16
Figure 3: Empirics: Quality and the Income-Export Nexus
Phase I
Phase II
Phase III
Export Shares
(Direct or ‘Quality’ Effect)
0
(Ricardian or ‘Wage’ Effect)
θkX+1
θkX
θkX−1
ymin ,k
y(vkH )
y(u)
ymax ,k
GDP per
Worker
Product Range (k)
Notes: The figure illustrates two of the central empirical predictions of the model. First, a single good can be produced
both by low-wage, low-quality countries and by high-wage, high-quality countries. ymin ,k and ymax ,k are the GDP per
worker of the poorest and richest countries producing a group-k good. Second, world market shares will be an inverted-U
shaped function of GDP per worker.
librium they have low wages. This implies that there are ‘product ranges,’ that is, ranges of GDP per
worker compatible with viability in the market. In figure 3, the product range for group-k goods
is (ymin ,k , ymax ,k ). Second, whereas in a single-sector model, world market shares increase with
quality, in a multi-sector world there are Ricardian forces which lead to inverted-U shaped world
market shares. As quality rises, wages rise and this leads to a loss of competitiveness in low-k
goods. As a result, export shares eventually ‘turn down.’ These are the two main empirical facts that
we will examine. A third fact, that prices rise with quality and hence with income, is an important
implication of the model; however, we explore it empirically only in Appendix C because it has
17
been examined elsewhere e.g., in Schott (2004).10
11
We can restate this in a way that makes one of the key points of our thesis crystal clear. A poor
country can advance out of low-ranked goods and still remain poor: this happens when the country enters as
a low-quality producer into goods with wide product ranges. Since, as we shall show empirically, most
goods have wide product ranges, we might expect this type of no-growth shift in product mix to be
common. By the same token, a country may move from being poor to being rich without changing its product
mix: this happens when it improves the quality of the wide-product-range goods that it already exports. In
short, GDP per worker depends not just on what a country produces (as in Hausmann et al. 2007),
but on the quality of what is produced.
6. Data
Trade data are from COMTRADE for 2005. We use the 4-digit SITC Revision 2 classification (henceforth SITC4).12 To verify that all of our cross-sectional results hold for more detailed commodity
breakdowns we also use the 2005 COMTRADE data at the 6-digit HS level (1996 revision, henceforth
HS6) and the 2005 U.S. import data at the 10-digit HS level (henceforth HS10). We exclude countries
whose population was less than two million in 2005 and/or whose territorial integrity changed
substantially between 1980 and 2005 e.g. the USSR. (The exception is Germany, which we include.)
This leaves us with the 94 countries listed in Appendix B. GDP per capita and population data are
from the United Nations. We do not use a PPP adjustment because we are interested in nominal
price competition in world product markets.
10. Additionally, we note that our model, which has endogenous wages, delivers a clear statement about how prices rise
both because of quality improvements and because of rising marginal costs (wages). Many models of international trade
and quality treat wages as exogenous and so cannot give such a clear statement.
11. Some readers will have noticed that the output-quality or output-income relationship in figure 3 looks like the
Heckscher-Ohlin output-capital or cones-of-diversification relationship (see Leamer, 1984; Schott, 2003). One might
wonder, then, why was our theory needed? For one, improvements in quality are very different empirically from capital
deepening. More importantly, this prediction is just one of several predictions that arise in our model, and all these
predictions flow from the fact that there is imperfect competition. Imperfect competition is needed to ensure that different
qualities co-exist, that prices are a non-constant markup over marginal cost, that prices are correlated with quality, and
that export shares are correlated with income in ways that reflect quality. In short, our output-quality relationship is just
one of several predictions. The output prediction in isolation can be modelled more simply; however, we are interested in
a bundle of predictions that require us to append an imperfectly competitive market structure onto a trade model. Thus,
a cones-of-diversification Heckscher-Ohlin model delivers at best only a small part of what is needed and a more natural
trade model in our setting is the Ricardian model, which emphasizes the role of technological capability for delivering
quality.
12. This allows us to go back to 1980 for a large number of countries and check that our results hold for these earlier
years.
18
7. Product Ranges
A key prediction of our theory is that in general equilibrium countries with different quality capabilities may nevertheless export the same good. See the product range in figure 3. While we do not
observe quality, an observable implication is that at least some goods will be produced both by rich
and poor countries. To investigate, for each product g we identify the poorest and richest exporters
of the product. Denote these by ymin,g and ymax,g , respectively. To avoid ‘noise’ associated with small
reported export values, a problem to which trade data are notoriously prone, for each good we look
only at the set of countries for whom the good is a ‘significant’ export, in the sense that the value of
its exports in that good constitute at least 1% of the value of exports of the country’s principal export
good.13 An important theoretical reason for using this 1% cut-off is that it ensures that the good is
sufficiently important to the exporter to generate the general equilibrium wage impacts upon which
our theory rests.
Product ranges are displayed in figure 4. Each point corresponds to a unique SITC4 good (g) and
the figure plots (ymin,g ,ymax,g ). A point therefore shows the range of income levels of countries for
which g is a significant export. All the points necessarily lie above the 45◦ line. For reference, along
the axes we show the log GDP per capita of Nepal, China, Poland and the United States.
The striking feature of figure 4 is the preponderance of points in the top left corner, i.e. the
preponderance of products for which the income range is very wide. To get a clearer sense of
magnitudes, consider the points lying in the top left corner where ln ymin,g < 8.25 and ln ymax,g > 10.
If a product is in this region then its richest significant exporter is at least 5.8 times richer then its
poorest significant exporter (e10−8.25 = 5.8). For the median product in the region the corresponding
difference is 79-fold (e4.4 = 79). These are huge differences. And there are a lot of goods in this
region: the region contains 81% of all products displayed in the figure and accounts for 70% of
world trade.
We will shortly show the reader that this observation about wide product ranges is robust,
and holds even in the most detailed trade data (HS10). However, we first draw three economic
insights from the wideness of product ranges. The first deals with Hausmann et al. (2007). Their
13. More formally, let l index countries, let g index goods, let Xgl be the value of country l’s exports of good g, and let
∑ g Xgl be country l’s total exports. Identify the good that accounts for the largest share of country l’s exports i.e., the good
with the largest Xgl /Xl . Call this good g(l ). Then good g is a significant export of country l if Xgl /Xl > 0.01Xg(l ),l /Xl .
Next, let K ( g) be the set of countries for which g is a significant export. Then ymin,g is the poorest country in K ( g) and
ymax,g is the richest country in K ( g).
19
Figure 4: Product Ranges
Richest (GDP per Capita) Producer of Good g: ln(ymax,g)
12
USA
10
POL
8
CHN
6
NPL
NPL
POL
CHN
USA
4
4
6
8
10
12
Poorest (GDP per Capita) Producer of Good g: ln(ymin,g)
Notes: Each point represents an SITC4 product. The horizontal axis is ln ymin,g , the poorest country for
which the product is a significant export. The vertical axis is ln ymax,g , the richest country for which
the product is a significant export.
20
exercise uses all goods in a country’s export basket even though products with wide ranges are
‘uninformative’ about a country’s income in the sense that knowing that a wide-range product is a
significant contributor to a country’s export basket tells us little about the country’s income. Figure
4 shows that such ‘uninformativeness’ is the norm rather than the exception.
Second, our theory emphasizes that for each product, multiple quality levels can coexist in
equilibrium. One can therefore interpret the wide ranges as support for the theory provided that
one is willing to accept that product ranges are the result of quality differences. As is well known,
quality is difficult to identify without detailed data about product characteristics. Since we do not
have this information we refer to the ranges as product ranges rather than as quality ranges and
take the weaker position that wide product ranges are implied by the theory but do not imply the
theory.
Third, there are two distinct groups of points that lie far from the top-left corner in figure 4.
These are ‘informative’ products. The first group lies to the bottom left (ln ymax,g < 10) and consists
of those goods exported only by relatively low- and middle-income countries (the ‘L group’). The
second group lies to the top right (ln ymin,g > 8.25) and consists of those goods exported only by
relatively high-income countries (the ‘H group’). On our present interpretation, L-group goods are
not produced by high-income countries because these countries’ wage costs are too high, whereas
H-group goods are not produced by low-income countries because their quality capabilities are too
low.
The reader will and should be skeptical about the wide product ranges in figure 4. For the
remainder of this section we anticipate five possible objections to the figure.
1. It is all aggregation bias: One would expect that the large product ranges displayed in figure
4 would become much narrower with finer product-level data. This is not the case. In figure 5 we
repeat the exercise using HS6 data (world trade data from COMTRADE) and using HS10 data (U.S.
import data). The distribution of product ranges in figures 4 and 5 are very similar. In particular,
product ranges remain large and in both panels just over 70% of total exports are accounted for by
the uninformative products in the top left (ln ymin,g < 8.25 and ln ymax,g > 10).14
2. Finer disaggregation is always better: The fact that nothing changes when moving to finer levels
14. In figure 5 there are thousands of points, many of which lie on top of each other. To make the figure clearer, instead
of plotting ln ymax,g on the vertical axis we have plotted ln ymax,g + e where e is a uniformly distributed random variable
on (−0.05,0.05). This adds a tiny random vertical shift to the data, which helps the reader see where the bulk of points
are located. Likewise, we have added a tiny random horizontal shift to ln ymin,g .
21
Figure 5: Product Ranges: Insensitivity to Aggregation
Notes: Each panel in this figure is constructed in the same way as figure 4, but with different data. Figure 4 used the
SITC4 classification and COMTRADE (world) data. The left-hand panel of the current figure uses the HS6 classification
and COMTRADE data. The right-hand panel uses the HS10 classification and U.S. import data.
of product disaggregation may seem puzzling, since if the move to a finer level of aggregation
involved the breaking up of technologically disparate sub-industries into individual industries, we
might expect the range to narrow as we move to this new level of aggregation. An examination
of the way in which industries are broken up in the HS6 and HS10 data throws light on why
disaggregation beyond SITC4 does not alter the distribution of ranges. In some cases the SITC4
industry is as disaggregated as the HS6 and even the HS10 industries e.g. ‘New tires for motor cars’
is a single category in both SITC4 and HS6. In other cases, the disaggregation is based only on size
or value, without any reference to capabilities e.g. ‘New tires for motor cars’ feeds into seven HS10
codes that distinguish between technology-ambiguous differences in the diameter of the tire. In yet
other cases the SITC4 code is disaggregated only by introducing a capability-irrelevant ‘parts of’
HS6 or HS10 code. This is pervasive e.g. see the HS6 categories associated with SITC4 7817 ‘Nuclear
reactors.’ Finally, in those cases where a technology-based disaggregation of products is introduced
it is often unclear whether this disaggregation conveys any information about differences in required capabilities: for example, SITC4 7252 ‘Machinery for making paper pulp, paper, paperboard;
Cutting machines’ is disaggregated in HS10 into a number of industries, including ‘Machines for
22
making paper bags etc.’ and ‘Machines for making paper cartons etc.’ Thus, finer disaggregation is
typically not more informative about quality capabilities. Were an ideal disaggregation of industries
to be constructed on the basis of the quality capabilities required, this would doubtless lead to some
narrowing in the relevant ranges. However, the limitations of the published data are quite serious
even at the most disaggregated level.15
3. Estimation error: Another possible objection to our wide product ranges is that we have not
sig
reported standard errors. Let Ng be the number of countries for which g is a significant export.16
sig
It is possible that products with wide ranges are products for which Ng is small i.e. for which
there are very few observations and hence large standard errors. This is not the case; indeed, the
sig
opposite is true. The correlation between Ng and the product range ln ymax,g − ln ymin,g is positive
sig
(0.57) and, for example, products with Ng ≥ 20 (one quarter of all products) all have large product
ranges. However, to deal with this objection in the simplest way possible, in figures 4–5 we have
sig
only displayed those products for which there are at least three significant exporters (Ng
≥ 3).
That is, we displayed only 547 of the possible 746 SITC4 goods. These 547 products account for
98.3% of world trade so that we are only excluding very minor products. We conclude from this
that wide product ranges are not an artifact of statistical uncertainty. To be safe though, we will
sig
continue throughout this paper to restrict attention only to products for which Ng ≥ 3.
4. Wide product ranges are an artifact of using a 1% cut-off for ‘significant exporters’: Again, this is
not the case. On-line appendix figure A1 shows that the inference we have drawn from figures
4–5 is not sensitive to the choice of cut-offs. It repeats figure 4 for a low percentage cut-off (0.1%),
a high percentage cut-off (10%), and cut-offs based on mixtures of percentages and dollar values
(x gk > $5 million and x gk > $50 million). In every single case the pattern displayed in figures 4–5 is
repeated.17
5. Wide product ranges are driven by China: Omitting China does not alter the impression that
15. For what we are doing, the relevant market is never equatable with an item in a government commodity classification,
be it SITC4, HS6 or HS10. Sometimes the relevant market is more detailed than HS10 (as in many electronic parts) and
sometimes the relevant market is less detailed than SITC4 (as in many apparel products). Thus, all of our conclusions
must be thought of relative to a definition of the market that is determined by the commodity classification, not the actual
product producers.
sig
16. Ng is the dimension of K ( g) in footnote 13.
17. There is a minor technical point about figure 5 that should be reviewed. Since the United States is far from most
countries and since trade costs increase in distance we expect that countries’ exports to the United States will be more
concentrated on a few goods than their exports to the world. This is indeed the case. Therefore, for the HS10 panel of
figure 5, which is based on U.S. data, we use a 0.1% cut-off instead of a 1% cut-off. This results in far more points in the
figure, but does not alter the distribution of points in the figure. See on-line appendix figure A1 for the HS10 figure using
a 1% cut-off.
23
product ranges are wide. Indeed, the reader can omit China from these figures simply by deleting
all points for which either ln ymin,g = 7.5 or ln ymax,g = 7.5. (China’s log GDP per capita is 7.5.)
Having established the robustness of figure 4, we can now restate our conclusion. Our theory
implies that there will be product ranges: the empirical surprise is that product ranges are often so
large.
8. Market Share Predictions
Figure 3 presented our predictions about a country’s share of world exports. Underlying that figure
is a comparative static in which a country that previously specialized in producing good k − 1 first
sees its quality in good k rise up from a very low level to that of the world standard and then sees its
quality in good k + 1 rise up from a very low level to that of the world standard. This comparative
static highlighted two mechanisms affecting world export shares. First, as capabilities rise in good
k, the country produces more of k and gains an increasing share of world exports. This is the direct
or quality effect. Second, as quality rises for good k + 1 wages are pushed up, which erodes the
country’s competitiveness in good k. This is the general equilibrium Ricardian or wage effect. These
two mechanisms lead to the world export share predictions in figure 2. For middle-capability goods
(k), world export shares display an inverted-U shaped relationship with income as first the quality
effect and then the wage effect come into play. For low-capability goods (k − 1), only the wage effect
is relevant and world export shares decline in income. For high-capability goods (k + 1) only the
quality effect is relevant and world export shares are increasing in income. See figure 3.
We operationalize these distinct export-share predictions of goods k − 1, k, and k + 1 as follows.
Recall that informative goods were those with ln ymin,g > 8.25 (the H group) and ln ymax,g < 10 (the
L group). We associate good k + 1 with goods in the H group, good k with goods in the top half
of the L group (6 < ln ymax,g < 10), and good k − 1 with goods in the bottom half of the L group
(ln ymax,g < 6). This gives us three groups: the ‘High’ group (k + 1), the ‘Middle’ group (k), and the
‘Low’ group (k − 1).
Since we must pool across exporters and products, we will need to consider normalizations of
export shares that are designed to control for country and product size. Consider country `’s share
of world exports of good g. We normalize the income level of ` by reference to the income levels y
g
and y g of the poorest and richest exporters, respectively, of g; we represent the position of log GDP
24
per capita within this range as (ln y` − ln y )/(ln y g − ln y ).18
g
g
We also need to adopt some normalization for the level of exports. This will be affected, as the
theory indicates, by product-market size and country size. The global market size for product g is
given by Sg (or equivalently δg ) in the theory. See section 2.1. To control for Sg , we scale country `’s
exports of good g, Xg` by world exports of g, Xg ≡ Σ` Xg` . Country size is constant in the theory.
In pratice, larger countries will produce a larger subset of the m products in any given capability
group. To control for this we scale Xg` /Xg by its average Σ g ( Xg` /Xg )/n` where n` is the number of
goods exported by country ` i.e. the number of goods for which Xg` > 0. Summarizing, we plot
(Normalized GDP per Capita) g` ≡
ln y` − ln y
ln y g − ln y
g
(19)
g
against
(Normalized World Export Share) g` ≡
( Xg` /Xg )
1
n` Σ g ( X g` /X g )
(20)
where the numerator is country `’s share of world exports of g and the denominator is country `’s
average share of world exports.
Figure 6 plots normalized world export shares against normalized GDP per capita for our three
groups of goods. The first thing to note about the plots is the preponderance of points on or very
near the horizontal axis. This reflects the fact that poorer countries have zero exports of many goods.
This fact could have been built into the model by having our small country produce only one of m
possible products in group k. It is also a feature of the Eaton and Kortum (2002) model where a
country can potentially produce all goods but draws high productivity for only a subset of goods.
Hence in interpreting these scatters, our focus of interest lies not on means — which tend to be
dominated by the many ( g,`) pairs with near-zero exports — but on the upper bound of the scatter.
With this in mind, we estimate a quantile regression (the 90th quantile). This appears as the curve
shown in each of the panels of figure 6.19
The upper, middle, and lower panels correspond to Low-group goods (k − 1), Middle-group
goods (k), and High-group goods (k + 1), respectively. The panels bear out the world-export-share
predictions of the model. World export shares are decreasing in income for the Low group, increas18. Note that y differs from ymin,g . The former is the minimum across all countries that export any positive amount of
g
good g while the latter is the minimum across any country for which g is a significant export. Likewise for the difference
between y g and ymax,g .
19. We use the SAS QUANTREG procedure with a 6th order polynomial.
25
Figure 6: Normalized World Market Shares
Notes: Each point in the plot corresponds to a product-country ( g,`) pair. The vertical axis is country `’s share of
world exports of good g, normalized as in equation (20). The horizontal axis is country `’s income, normalized as
in equation (19). The curves are the 90th quantile regressions.
26
ing in income for the High group, and display an inverted-U relationship for the Middle group.
This is exactly as predicted in figure 3.
We round out this section with a discussion of specification issues. In constructing figure 6 we
made five choices. First, we defined the low, middle and high groups by reference to the cut-offs
ln ymin,g = 6, ln ymin,g = 8.25 and ln ymax,g = 10. The choice of cut-offs does not matter. As on-line
appendix figure A2 shows, figure 6 is not substantially altered by lowering the ln ymin,g = 6 cut-off
to 5.25 or raising it to 6.75; by lowering the ln ymin,g = 8.25 cut-off to 7.25 or raising it to 9.25; or
by lowering the ln ymax,g = 10 cut-off to 9. Changing the cut-offs beyond these ranges results in
groups that include either too few observations or too many wide product-range, uninformative
goods. The only sensitivity occurs when ln ymin,g = 6 is raised to 6.75, in which case the interior
peak for the Low group shifts from 0.05 to 0.20. This is because the Low group now contains so
many Middle group goods. Thus, figure 6 is robust to changes in cut-offs.
Second, we used SITC4 data but the figures look virtual identical using HS6 data. See on-line
appendix figure A3. The exception is that the Low group again has an interior peak near 0.20. Third,
we chose the normalization (ln yk − ln y )/(ln y g − ln y ). This choice of normalization plays almost
g
g
no role empirically because y g − y does not vary much across goods i.e. it is not much different
g
than normalizing by a constant. Specifically, across all goods its maximum is 6.4, its median is 6.0
and its 5th percentile is 5.2. Fourth, in equation (20) we normalized world export shares Xg` /Xg by
1
average world export shares n−
k Σ g ( X g` /X g ). We have experimented extensively with alternative
normalizations of Xg` /Xg , including the 50th, 75th, 90th, and 99th percentiles of each country’s
world export shares.20 All of these normalizations produce curves with the same shapes as those
in figure 6. Fifth, we reported quantile regressions based on the 90th quantile. The curves do not
change when using the 85th, 95th, or 99th quantiles.21
9. Development Ladders in the Cross-Section: The Role of Product Ranges
Wide product ranges and their correlation with unit values suggest that a country’s wealth depends
not just on what goods it produces, but also on the quality of the goods produced. To investigate
20. For each country ` these are percentiles of the set { X1` /X1 , . . . , Xg` /Xg , . . . , xG` /xG }.
21. There are a few extreme ‘vertical’ outliers that would ‘squash’ figure 6 down to the horizontal axis if displayed.
Rather than leave them off the figure entirely, we shrink them towards the horizontal axis as follows. In the top panel, if
a vertical point y exceeds 9, then it is replaced by 9 + f (∆) where ∆ = y − 9 and f (∆) = ln(1 + ∆)/5 so that f (0) = 0 and
f 0 > 0. Likewise for the middle and bottom panels, but with 4 instead of 9. This does not have and cannot have an effect
on the position of the quantile regressions.
27
further, return to figure 4 and, as before, divide goods into three groups: (i) the L group to the lower
left (ln ymax,g < 10); (ii) the H group to the upper right (ln ymin,g > 8.25); and, (iii) the ‘uninformative’
group, comprising the remaining goods. In what follows we confine attention to the L and H goods
because these are the only ones that are informative.22
We show in figure 7 the relationship of a country’s GDP per capita with (a) the share of L goods
in its export basket (left panel) and (b) the share of H goods in its export basket (right panel). We see
a clear fall in the share of L goods and a rise in the share of H goods as GDP per capita increases.
But an important feature of figure 7 lies in the fact that the relation between the product mix and
income is not bi-directional: while significant exporters of H goods are necessarily rich, it is not the
case that rich countries are necessarily significant exporters of H goods. A very low contribution of
H goods is consistent with a relatively high level of GDP per capita. (See Malaysia in the right-hand
panel). Similarly, while a high share of L goods necessarily implies that a country is poor, many
poor countries have a low share of L goods. (See Zimbabwe or Bangladesh in the left-hand panel.)
We can restate this in a way that makes one of the key points of our thesis crystal clear. A
poor country can advance out of L goods and still remain poor: this happens when the country enters as a
low-quality producer into uninformative goods i.e. goods with wide quality ranges. Since most goods are
uninformative, we might expect this type of no-growth shift in product mix to be common. By the
same token, a country may move from being poor to being rich without changing its product mix: this happens
when it improves the quality of the uninformative, wide-quality-range goods that it already exports.
10. Conclusions
The aim of this paper has been to explore the way in which advances in wealth are associated both
with changes in the product mix and with changes in quality (and productivity) within a given
set of industries. The central point relates to the fact that the range of income levels of significant
exporters of most products is very wide. At a theoretical level, one reason for the wide product
ranges lies in aggregation of disparate sub-industries (though the result is not sensitive to the level
of aggregation); another reason lies in the fact that within any industry, there will, in a general
22. The conclusions of this section are in no way sensitive to the choice of 8.25 and 10 as cut-offs: we have chosen these
because they represent break points in figure 4. Also, in figure 4 there is one good that is in both the L and H groups. This
is cameras (SITC4 8732). It lies at the point (ln ymin,g , ln ymax,g ) = (8.9,9.8). We place this in the H group, though where it
goes makes no difference because it is only one of hundreds of goods.
28
Figure 7: The Share of L and H Goods in Each Country’s Export Basket
0.12
Low (L) Group
0.50
Share of H Goods in a Country's Export Basket
Share of L Goods in a Country's Export Basket
0.60
0.40
0.30
0.20
0.10
0.00
ZWE
4
High (H) Group
0.10
0.08
0.06
0.04
0.02
MYS
BGD
0.00
6
8
10
12
4
6
log GDP per Capita, ln(yk)
8
10
12
log GDP per Capita, ln(yk)
Notes: Each point represents a country (there are 94 points in each panel). The horizontal axis is log GDP per capita in
2005. The vertical axis is a country’s exports of L goods (right panel) or H goods (left panel) as a share of the country’s
total exports. Using the product ranges in figure 4, a good is an L good if ln ymax,g < 10 and an H good if ln ymin,g > 8.25.
equilibrium multi-country setting, be a viable range of producer income levels. In this viable range,
poor low-quality exporters compete with rich high-quality exporters.
The central property of product ranges is that, in a multi-market general equilibrium setting,
the relation between quality and price on the one hand, and output and global market share on
the other, is non-monotonic. There is at equilibrium a range of producer qualities (and so wealth
levels) that are viable in a given industry. As quality rises, the country moves into the production
of higher-ranked goods, and its equilibrium wage (and GDP per capita) rises. But this means that
its output and global market share all exhibit an inverted-U relationship with quality, and so with
GDP per capita. As quality rises, market share rises, and wages rise also. As the country advances
into the production of higher-ranked products, the rise in wage causes its effective cost level to rise,
and its global market share in this industry to fall. It is this inverted-U relation that is the basis of
the selection effect that links a country’s wealth to its product mix.
29
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31
Appendix A. Proofs
Appendix A.1. Proof of Lemma 1 (Perfect Sorting)
In the main text we established that the wk and pk of equations (6) and (7) uniquely clear labour
and product markets. It is straightforward to show that trade is balanced (the value of total
exports equals the value of total imports). This is built into the above equations via the consumer
optimization problem i.e., via Walras’ Law. Thus, all of the requirements of equilibrium are satisfied
except one. Specifically, in deriving equations (6) and (7) we have assumed that firms from country
group k maximize profits by producing within product group k. We must now verify that these
firms could not earn higher profits or even positive profits by producing in some other product
group.
By assumption, a firm from country k0 has quality 0 when producing a good in group k > k0 .
Hence, it will never produce a group-k good. Consider a firm from country k0 = k + i (i > 0)
that is considering whether to produce a good from group k. The firm earns lower profits doing
so if and only if the firm faces wages wk+i (> wk ) that are so high as to render the firm unviable
in the production of good type-k goods. Using equation (4) with the inequality reversed and with
u j = u for all j (recall that u denotes the common standard of quality shared by all active firms in
the market), the firm is not viable in type-k goods if wk+i > ( Nk wk + wk+i )/( Nk − 1 + 1) or
Nk
w k +i
>
.
wk
Nk − 1
(21)
Substituting in wages from equation (6) and simplifying, inequality (21) is equivalent to Nk >
Nk2+i /( Nk+i − 1) or, upon subtracting Nk+i from both sides,
Nk − Nk+i >
Nk+i
.
Nk+i − 1
(22)
By assumption 1, the left-hand side is strictly greater than 3. By the assumption that there are at
least two producers globally, Nk+i ≥ 2 and the right-hand side is less than or equal to 2. Hence
inequality (22) holds. Thus, a firm from country k + i (i > 0) earns negative profits producing a
group-k product.23
Finally, equations (5)-(8) establish that wk , yk , pk and pk /wk are decreasing in Nk and hence, by
assumption 1, increasing in k.
23. Notice that we only used Nk ≥ Nk+1 + 3, which is weaker than assumption 1. In order to ensure that inequality (22)
is still satisfied when we abandon the perfect sorting equilibrium and allow one country to move from type k − 1 to type
k, we will need the stronger assumption 1 that Nk ≥ Nk+1 + 4.
32
Appendix A.2. Equilibrium Conditions and Outcomes in Phase I
A typical type-(k − 1) country producing a group-(k − 1) good has wage wk−1 and price pk−1 . Rearranging equation (13) as 1 − wk−1 /pk−1 = w/[ pk−1 ( Nk−1 − 1)] and plugging this into equation
(15) yields:
N=m
w
p2k−1 ( Nk−1
− 1)
S
or
p k −1 =
w
mS
N Nk−1 − 1
1/2
(150 )
.
Likewise re-arranging equation (14) as 1 − wk /pk = [wu/vk ]/[ pk Nk ] and plugging this into
equation (16) yields:
N=m
w
S
(vk /u) p2k Nk
or
pk =
w
mS
N (vk /u) Nk
1/2
(160 )
.
For the developing country, equations (11)–(12) imply pk−1 = pk−1 and pk = (vk /u) pk . Plugging
these into equation (17) yields:
N
1
=
mS
p k −1
1−
w
p k −1
1
+
(vk /u) pk
w
1−
(vk /u) pk
.
(170 )
Plugging equations (150 )–(160 ) into equation (170 ) establishes that the equilibrium wage in the developing country is:
w(vk )
m
=
S(vk )
N
"
#2
p
( Nk−1 − 1)1/2 + Nk1/2 / vk /u)
.
Nk−1 + Nk /(vk /u)
(23)
Plugging this back into equations (150 )–(160 ) yields closed-form solutions for the equilibrium prices
pk−1 (vk )/S(vk ) and pk (vk )/S(vk ). (Recall that all solutions are normalized by the numeraire S(vk ).)
Plugging w(vk ), pk−1 (vk ) and pk (vk ) into equations (11)–(14) yields closed-form solutions for the
remaining equilibrium wages and prices, namely, pk−1 (vk ), pk (vk ), wk−1 (vk ), and wk (vk ), all relative
to the numeraire S(vk ).
Appendix A.3. Equilibrium Conditions and Outcomes in Phase II
Plugging pk−1 from equation (13–II) into equation (15) and rearranging yields:
w k −1 ( v k )
m Nk−1 − 2
.
=
S(vk )
N ( Nk−1 − 1)2
33
(15–II)
For closed-form solutions to all of the equilibrium prices and wages in Phase II proceed as
follows. Plugging equation (15–II) into (13–II) yields a closed-form solution for pk−1 (vk )/S(vk ).
The equation (160 ) expression for pk holds in Phase II, as does the equation (11) expression pk =
1/2
w
(vk /u) pk . Hence, pk = (vk /u) pk = (vk /u) mS
. Plugging this into equation (17–II) and
N (v /u) N
k
k
rearranging yields:
w(vk )
m
α
v /u
=
where α ≡ k
<1.
2
S(vk )
N (1 + α )
Nk
(24)
Plugging equation (24) into equation (160 ) yields a closed-form equilibrium solution for
pk (vk )/S(vk ). Plugging this and w/S from equation (24) into (14) yields a closed-form equilibrium
solution for wk (vk )/S(vK ). Thus, we have provided closed-form equilibrium solutions for all of the
endogenous prices in Phase II.
Appendix A.4. Proof of Lemma 2
Let vkL be the value of vk at which the developing country is about to produce a type-k good, but has
not yet started. At this point its wage w(vkL ) is that of a typical type-(k − 1) country i.e., w(vkL ) =
wk−1 where wk−1 is given in equation (6). Equating the equation (6) expression for wk−1 with the
equation (23) expression for w(vkL ) and solving for vkL yields:
vkL =
( Nk−1 − 1) Nk
u.
Nk2−1
(25)
Let vkH be the value of vk at which the developing country stops producing goods of type-(k − 1)
n
o
w(v H )
i.e., xk−1 (vkH ) = 0. From equation (30 ), xk−1 (vkH ) = p 1(v H ) 1 − p (kv H ) so that xk−1 (vkH ) = 0
k −1
k
k −1
k
implies pk−1 (vkH ) = w(vkH ). Plugging this into equation (150 ) yields:
w(vkH )
1
m
.
=
H
N ( Nk−1 − 1)
S(vk )
(26)
Equating this to w(vkH )/S(vkH ) from equation (23) yields:
r
p
vk p
vk
− Nk−1 − 1 Nk
+ Nk = 0
(27)
u
u
√
which is quadratic in vk /u. The larger of the two roots exceeds 1 and hence is not a candidate for
√
√
√
calculating vkH because we require vk < u.24 The other root is ( Nk /2)
Nk−1 − 1 − Nk−1 − 5
so that
vkH = u
2
p
Nk p
Nk−1 − 1 − Nk−1 − 5 .
4
(28)
p
p
√
24. This root is ( Nk /2)
Nk−1 − 1 + Nk−1 − 5 . Since Nk ≥ 2, assumption 1 implies Nk−1 ≥ 6. Together these
imply that this root exceeds 1.
34
It is straightforward but tedious to verify that vkL < vkH < u. See the on-line Appendix E.1 for a
step-by-step derivation.
Appendix A.5. Monotonicity of Wages in Phase I and II
Lemma 4 (1) w(vk )/S(vk ) is increasing in vk for vk in the range vkL < vk < u. (2)
w(vk )/S(vk )
vk /u
is decreasing
in vk for vk in the range vkL < vk < vkH .
Proof of part: Consider Phase I where vkL < vk < vkH . Differentiating equation (23) with respect to
vk yields
√ √
∂w(vk )/S(vk )
= αk ξ + − vk
vk − ξ −
∂vk
(29)
where αk > 0 is a constant (i.e., independent of vk ) and
ξ+ =
u
i
Nk1/2 h
( Nk−1 − 1)1/2 − (2Nk−1 − 1)1/2 < 0
Nk−1
(30)
u
i
Nk1/2 h
( Nk−1 − 1)1/2 + (2Nk−1 − 1)1/2 > 0.
Nk−1
(31)
√
ξ− =
√
It is tedious but straightforward to show that Nk ≥ 2 and assumption 1 imply
q
vkH < ξ + . See
q
< vkL . Hence
on-line Appendix E.2 for a step-by-step derivation. Since ξ − < 0, it must be that ξ −
q
q
√
vkH ). Hence, ξ − < vk < ξ + in Phase I. Hence
the interval (ξ − , ξ + ) contains the interval ( vkL ,
the right-hand side of equation (29) is positive and w(vk )/S(vk ) is increasing in vk .
Next consider Phase II where vkH < vk < u. From equation (24), w(vk )/S(vk ) is increasing in α
and hence in vk .
Proof of part 2: (vkL < vk < vkH ): Dividing equation (23) by (vk /u)S(vk ) yields:
w(vk )/S(vk )
m
=
vk /u
N
"
#2
"
#2
√
√
( Nk−1 − 1)1/2 + Nk1/2 / vk /u
m ( Nk−1 − 1)1/2 vk /u + Nk1/2
√
√
=
.
N
Nk−1 (vk /u) + Nk
Nk−1 vk /u + Nk / vk /u
Differentiating,
∂
w(vk )/S(vk )
vk /u
∂vk
s
)
r
vk
Nk
vk
Nk
= −αk (vk )
+2
−
u
Nk−1 − 1 u
Nk−1 − 1
s
s
"r
# "r
#
√
√
vk
Nk
vk
Nk
= −αk (vk )
+ ( 2 + 1)
− ( 2 − 1)
u
Nk−1 − 1
u
Nk−1 − 1
where αk (vk ) ≡ 2(m/N)1/2 (
(25) and Nk−1 > 2 imply that
(
w(vk )/S(vk ) 1/2
) Nk−1 ( Nk−1
vk /u
q
− 1)1/2 /[ Nk−1 /(vk /u) + Nk ]2 > 0. Equation
p
√
vkL /u > ( 2 − 1) Nk /( Nk−1 − 1). Hence in Phase I both bracketed
terms are positive and the derivative is negative.
35
Appendix A.6. Proof of Proposition 2
Part 1 (wk−1 /S increasing in vk ): Equating the right-hand sides of equations (13) and (150 ) yields:
1/2 1/2
m
w k −1 ( v k )
1
w(vk )
w(vk )
1
=
.
(32)
−
S(vk )
N Nk−1 − 1
S(vk )
Nk−1 − 1 S(vk )
√
√
√
The right-hand side of equation (32) is quadratic in w/S and thus increasing in w/S for w/S ≤
√
√
m/N Nk−1 − 1/2. It thus suffices to show that this last inequality is satisfied in Phase I. Since
w/S is increasing in vk (lemma 4), it suffices to show that the inequality holds at vk = vkH , which
m
1
1/2 ≤ [ m ( N
1/2 /2 or 4 ≤ ( N
2
from equation (26) implies [ N
k −1 − 1)]
k −1 − 1) or Nk −1 ≥ 3. But
Nk−1 −1 ]
N
Nk ≥ 2 and assumption 1 together imply Nk−1 > 3. Thus, the inequality holds at vkH .
Part 1 (wk /S decreasing in vk ): We start with a preliminary result. At vk = vkL the developing
country does not produce xk . Hence we are in the perfect sorting equilibrium of section 3: w(vkL ) =
wk−1 (vkL ) which, from equation (6), gives us w(vkL )/S(vkL ) =
Nk−1 −1 m
.
Nk2−1 N
Further, from equation (25),
w(vkL )/S(vkL )
N
− 1 m ( Nk−1 − 1) Nk
1 m
= k−21
/
=
.
Nk N
Nk−1 N
Nk2−1
vkL /u
(33)
We now turn to the core of the proof. Equating the right-hand sides of equations (14) and (160 )
yields:
wk ( v k )
=
S(vk )
m 1 w(vk )/S(vk )
N Nk
vk /u
1/2
−
1 w(vk )/S(vk )
.
Nk
vk /u
This is just equation (32) with Nk replacing Nk−1 − 1 and (w/S)/(vk /u) replacing w/S. Thus,
p
p
wk /S is increasing in (w/S)/(vk /u) if (w/S)/(vk /u) ≤ (m/N) Nk−1 /2. Since (w/S)/(vk /u)
is decreasing in vk (part 2 of lemma 4), the inequality holds throughout Phase I if it holds at vk = vkL .
But from equation (33), this means
1 m
Nk N
≤
Nk m
4 N
or 4 ≤ Nk2 , which always holds. Hence wk /S
is increasing in (w/S)/(vk /u). But (w/S)/(vk /u) is decreasing in vk (part 2 of lemma 4). Hence
wk (vk )/S(vk ) is decreasing in vk .
Part 2: Dividing both sides of equation (32) by w(vk )/S(vk ) establishes that wk−1 /w is decreasing in
w/S and hence, by lemma 4, decreasing in vk .
Appendix A.7. Proof of Proposition 1
Part 1: From equation (30 ), , xk−1 = (S/pk−1 )(1 − w/pk−1 ). From equation (150 ), pk−1 /S is increasing
in w/S. Dividing equation (150 ) by w(vk ),
p k −1 ( v k )
=
w(vk )
m
1
S(vk )
N ( Nk−1 − 1) w(vk )
36
1/2
(34)
which is decreasing in w/S. Hence xk−1 (vk ) is decreasing in w/S and, by lemma 4, decreasing in
vk . By the definition of vkH , xk−1 (vkH ) = 0 so that xk−1 (vk ) falls to 0. By labour-market clearing,
mxk−1 + mxk = N so that xk is increasing in vk . By the definition of vkL , xk (vkL ) = 0 so that xk (vk )
rises from 0.
Part 2: From equations (11) and (34), pk−1 (vk )/w(vk ) is decreasing in w/S and hence, by lemma 4,
pk−1 (vk )/w(vk ) is decreasing in vk .
From equation (160 ),
pk (vk )
=
w(vk )
m 1 1
1
N Nk vk /u w(vk )/S(vk )
1/2
.
(35)
From equation (11), pk = (vk /u) pk . Hence:
(vk /u) pk (vk )
pk (vk )
=
=
w(vk )
w(vk )
m 1
N Nk
1/2 w(vk )/S(vk )
vk /u
−1/2
(36)
which, by lemma 4, is increasing in vk .
Part 3: See lemma 4.
Part 4: Revenue for any firm i is Ri = pi xi . Multiplying the expression for xi in equation (30 ) by pi
yields Ri /S = pi xi /S = 1 − wi /pi . Hence, for a developing country producing a type-(k − 1) good,
R k −1 ( v k )
w(vk )
= 1−
= 1−
S(vk )
p k −1 ( v k )
N
w(vk )
( Nk−1 − 1)
m
S(vk )
1/2
,
(37)
where the second equality follows from equation (11) (pk−1 = pk−1 ) and the equation (150 ) expression for pk−1 (vk ). For a developing-country firm producing a type-k good,
w(vk )
Rk (vk )
= 1−
= 1−
S(vk )
pk (vk )
N w(vk )/S(vk )
N
m k
vk /u
1/2
(38)
where the second equality follows from equation (12) (pk = (vk /u) pk ) and the equation (160 )
expression for pk−1 (vk ).
GDP per capita of the developing country is given by y = (mRk−1 + mRk )/N (see equation 10)
so
R k −1 ( v k ) R k ( v k )
+
S(vk )
S(vk )
√
1/2
m
m 1/2 p
N
w(vk )
=2 −
Nk−1 − 1 + √ k
.
N
N
S(vk )
vk /u
y(vk )
m
=
S(vk )
N
37
Substituting into this the equation (23) expression for w(vk )/S(vk ) and simplifying yields:
√ √
√
m
m 1 − 2 Nk−1 − 1 Nk u/vk
y(vk )
=
+
·
.
S(vk )
N N
Nk−1 + Nk (u/vk )
bx
The second expression on the right-hand side is a rational function of the form ca−−dx
2 with a, b, c,
√
d > 0. It is positive and decreasing in x (= u/vk ) on the domain 0 ≤ x ≤ a/b, or 0 ≤ vuk ≤
4( Nk−1 − 1) Nk and so y(vk )/S(vk ) is increasing in vk on this domain. Finally, we note that the
relevant domain (vkL ,vkH ) is a subset of this domain; for this it suffices to note from the expression
(25) for vkL that
vkL
( Nk−1 − 1) Nk
=
≤ 4( Nk−1 − 1) Nk
u
Nk2−1
which follows from the fact that Nk−1 ≥ 2. This completes the proof of Proposition 1.
Appendix A.8. Proof of Proposition 3
Part 2 follows immediately from part 1 of lemma 4. Since equations (11) and (160 ) hold in Phase
II, so do equations (36) and (38). Part 1 follows from equation (36) and part 2 of lemma 4. For
part 3, start by observing that Rk (vk )/S(vk ) rises in vk , which follows from equation (38) and part
2 of lemma 4. In Phase II all income comes from production of type-k goods so that y(vk )/S(vk ) =
(m/N) Rk (vk )/S(vk ), which is also rising in vk .
Appendix A.9. Proof of Lemma 3
Consumers spend a fraction δ of income on each good. It follows that the value of exports of a typical
type-k country is X k = pk x k − δmpk x k where mpk x k is income or GDP. Labour market equilibrium
is N = mx k . It follows that
X k (vk )
N pk (vk )
X k −1 ( v k )
N p k −1 ( v k )
= (1 − δm)
and
= (1 − δm)
S(vk )
m S(vk )
S(vk )
m S(vk )
(39)
where the right-hand equation follows from the left-hand equation by symmetry.
Turning to the developing country, consider Phase II where only group-k goods are produced
and exported. By the logic of equation (39), Xk /S = (1 − δm)(N/m)( pk /S), which is increasing
in pk /S and hence, by parts 1 and 2 of proposition 3, in vk . Further, Xk /X k = ( Xk /S)/( X k /S) =
pk /pk = vk /u where the last equality follows from equation (12). Hence Xk /X k is increasing in
vk . Next consider Phase I, where both goods are produced. Let Rk−1 ≡ pk−1 xk−1 and Rk ≡ pk xk
be revenues so that income or GDP is m ( Rk−1 + Rk ). It follows that the value of the developing
38
country’s exports of group-(k − 1) and group-k goods are given by:
Xk−1 (vk ) = max {0, Rk−1 (vk ) − δm [ Rk−1 (vk ) + Rk (vk )]} = max {0, (1 − δm) Rk−1 (vk ) − δmRk (vk )}
Xk (vk ) = max {0, (1 − δm) Rk (vk ) − δmRk−1 (vk )} .
(40)
(The max operator is needed because it is possible that one of the two goods is imported.) Revenue
for any firm i is Ri = pi xi . Multiplying the expression for xi in equation (30 ) by pi yields Ri =
pi xi = {1 − wi /pi }S. Hence for the developing country producing group-(k − 1) and group-k
goods, Rk−1 /S = (1 − w/pk−1 ) and Rk /S = (1 − w/pk ). From part 2 of proposition 1, a Phase I rise
in vk raises w/pk−1 and lowers w/pk so that Rk−1 /S falls and Rk /S rises. Hence from equation (40),
Xk−1 (vk )/S(vk ) falls and Xk (vk )/S(vk ) rises in Phase I.
Consider X k−1 /S. Equation (13) and proposition 2 imply that pk−1 /S is increasing in vk in Phase
I. Hence from equation (39), X k−1 /S is increasing in vk in Phase I, as required. But we have already
seen that Xk−1 /S falls. Hence Xk−1 /X k−1 falls. Finally, in Phase I, Rk−1 /S is falling so that by equation (40), Xk /S is rising faster than (1 − δm) pk xk /S. Compare this to X k /S = (1 − δm)( pk /S)(N/m)
from equation (39). By equation (12), pk /S is rising faster than pk /S. Also, xk is rising faster than
the constant N/m. Hence Xk /X k is rising.
Appendix B. Trade Data
COMTRADE reports each bilateral transaction twice, once by the importer and once by the exporter.
We always use the importer’s data as this is known to be more reliable for most countries.
The countries in our sample are (using ISO codes for brevity25 ): AFG, AGO, ALB, ARG, AUS,
AUT, BDI, BEN, BFA, BGD, BGR, BOL, BRA, CAN, CHE, CHL, CHN, CMR, COL, CRI, CUB, DEU,
DNK, DOM, DZA, ECU, EGY, ESP, FIN, FRA, GBR, GHA, GIN, GRC, GTM, HND, HTI, HUN, IDN,
IND, IRL, IRQ, ISR, ITA, JAM, JOR, JPN, KEN, KHM, LBN, LKA, MAR, MDG, MEX, MLI, MMR,
MOZ, MWI, MYS, NER, NGA, NIC, NLD, NOR, NPL, NZL, PAK, PER, PHL, PNG, POL, PRT, PRY,
ROU, RWA, SAU, SDN, SEN, SGP, SLE, SLV, SOM, SWE, TCD, TGO, THA, TUN, TUR, UGA, URY,
USA, VEN, ZMB, and ZWE. The only major countries not included in our list are Taiwan and Honk
Kong. Taiwan is excluded because there are no 1980 data. Hong Kong is excluded because, for
our purposes, it should be merged with China in 2005 and be by itself in 1980. None of our 2005
25. See on-line appendix table A1 for a full list of country names and GDPs per capita.
39
cross-section results are affected by the inclusion of Taiwan and Hong Kong (the latter either by
itself or merged with China).
We exclude live animals, meat, fish and dairy. These goods account for only 2.1% of trade and
including them does not affect our results at all; however, it is hard to relate trade in these goods to
the issues raised in this paper.
Price data p gk are from the U.S. historical imports CD, 2001–2005. This CD only reports what
is called the ‘first quantity’ and ‘first value’ so that all observations within an HS10 product have
the same quantity units. We sum U.S. imports and quantities by HS10 product and trading partner
(exporter to the United States). We calculate unit values with the summed data. In addition, we
winsorize the unit values below the 10th within-HS10 percentile and above the 90th within-HS10
percentile. Winsorizing makes virtually no difference to our results.
Appendix C. Price Ranges
The theory predicts that, for a single good, all producers of the good will share the same pricequality ratio. Since richer countries have higher quality, they should have higher prices. That is,
prices should be increasing in the income of the exporter. Since price data are not available, we
follow Schott (2004) in proxying for prices using HS10 unit values from the 2005 U.S. import file.
We emphasize that unit values are extremely noisy so that caution must be exercised in interpreting
them as prices. See Appendix B for a discussion of the data.
Let p gk be the unit value of good g exported by country k to the United States. We are interested
in how the p gk vary as we move through product ranges. The most familiar way of doing this is
Schott’s (2004, table V) famous regression ln p gk = α g + β ln yk where α g is an HS10 product fixed
effect. Re-estimating Schott’s regression using 2005 U.S. imports from our 94 exporters (187,363
observations), the OLS estimate of β is 0.29 (clustered t = 8.05) so that, as in Schott, there is indeed
a statistically significant positive correlation between unit values and exporter incomes.
A sharper prediction of our theory is as follows. Consider a single HS10 good g. Recall that
in the HS10 panel of figure 5 we plotted the income of the poorest and richest countries that had
significant exports of g i.e. we plotted (ln ymin,g , ln ymax,g ). Since for each g we know the identity of
the poorest and richest countries, we know these countries’ unit values. We denote them in obvious
40
fashion by pmin,g and pmax,g . We expect that
∆ g ≡ ln pmax ,g − ln pmin ,g > 0.
This inequality is sharp in that it is directly related to our product ranges, that is, to the poorest (min)
and richest (max) exporters that define the boundaries of our product ranges. It is also an inequality
that is unlikely to hold because we are examining two specific unit values (pmax ,g and pmin ,g ) even
though we know that such unit values are extremely noisy.
∆ g > 0 defines one inequality for each product range in the HS10 panel of figure 5. A nonparametric test of ∆ g > 0 is the sign test, which easily rejects the null hypothesis that the signs of the
∆ g are random (p-value of less than 0.0001). The mean value of ∆ g is 0.63 (t = 27.23) and, more
robustly with noisy data, the median value of ∆ g is 0.45. Since e0.45 − 1 = 0.57, this implies that
the richest significant exporter of the median product has a unit value that is 57% higher than the
corresponding unit value of the poorest significant exporter. Cautiously interpreting unit values as
prices, this means that prices are increasing as one moves through a product range in figure 5.
It is tempting to examine an even stronger prediction, namely, that unit-value ranges ∆ g are large
when product ranges ln ymax ,g − ln ymin ,g are large. While this is not a prediction of the model, it
can be generated by adding more restrictions on how scarcity varies across countries and products.
To examine this prediction we estimate the following regression:
(ln pmax ,g − ln pmin ,g ) = 0.15 + 0.18(ln ymax ,g − ln ymin ,g )
(clustered t = 11.23).
Thus, large product ranges in figure 5 are associated with large unit-value ranges.
41
On-Line Appendix – Not for Publication
Online Appendix to
“Deductions from the Export Basket:
Capabilities, Wealth and Trade”
On-Line Appendix – Not for Publication
Figure A1: Product Ranges: Sensitivity to Cut-offs
12
12
11
Richest GDP per Capita (ymax)
10
9
8
7
6
5
9
8
7
6
5
SITC4, 1% Cut‐Off or $5M in Exports
SITC4, 0.1% Cut‐Off
4
4
4
5
6
12
7
8
9
10
11
Poorest GDP per Capita (ymin)
12
Richest GDP per Capita (yymax)
11
Richest GDP per Capita (y
(ymax)
10
10
9
8
7
6
5
4
6
7
8
9
7
8
9
10
11
12
12
11
11
10
9
8
7
6
10
Poorest GDP per Capita (ymin)
11
12
10
9
8
7
6
5
HS10, 1% Cut‐Off
SITC4, 1% Cut‐Off or $50M in Exports
4
5
6
5
SITC4, 10% Cut‐Off
4
4
5
12
Richest GDP per Capita (yymax)
Richest GDP per Capita (ymax)
11
4
5
6
7
8
9
10
11
Poorest GDP per Capita (ymin)
4
12
4
5
6
7
8
9
10
11
12
Poorest GDP per Capita (ymin)
On-Line Appendix A. Robustness of Product Range Diagrams
There were several choices made in constructing the product ranges of figures 4–5. When defining
the product ranges we only looked at country-product pairs for which the product accounted
for a significant share of the country’s exports. As in the main text, let x g` /x` be the share
of country `’s exports accounted for by product g. We considered country-product pairs with
x g` /x` ≥ α maxg0 x g0 ` /x` where α = 0.01. In the top left and bottom left panels of figure A1 we
choose α = 0.001 and α = 0.10, respectively. As is apparent, the impression of the pervasiveness of
wide product ranges does not change.
An alternative to choosing cut-offs based on x g` /xk ≥ α maxg0 x g0 ` /x` is to use a dollar value
cut-off: x g` > $5,000,000 or x g` > $50,000,000. This has the advantage that it keeps more of the
trade of rich countries, and the disadvantage that it eliminates more of the trade of poor countries.
We therefore combine this dollar criterion with our previous α (or percentage) criterion. That is, an
observation ( g,`) is included if it meets either of the two criteria. The results appear in the right-hand
panels of figure A1. The top and bottom panels use $5,000,000 and $50,000,000, respectively.
On-Line Appendix B. Robustness of Market-Share Diagrams
In section 8 we claimed that the figure 6 market-share diagram was not very sensitive to the choice
of cut-offs ln ymin,g = 6, ln ymin,g = 8.25 and ln ymax,g = 10. Figure A2 shows the basis for this claim.
In the top pair of panels, ln ymin,g = 6 is lowered to 5.25. In the second pair of panels, ln ymin,g = 6 is
raised to 6.75. Here the monotonicity of the Low group is lost, but only because the Low group now
absorbs so many Middle-group goods. It remains true that the Low-group peak is far to the left of
the Middle-group peak. In the third pair of panels, ln ymin,g = 8.25 is lowered to 7.25. Monotonicity
i
On-Line Appendix – Not for Publication
is lost for the High group, but only because it now includes so many Middle-group goods. (Also,
the downturn at the far right of the High panel is entirely associated with Norway.) In the fourth
pair of panels, ln ymin,g = 8.25 is raised up to 9.25. In the bottom pair of panels, ln ymax,g = 10 is
lowered to 9. From figure A2 it is clear that the market-share predictions of figure 6 are not sensitive
to the choice of cut-offs, or where they are, this ‘sensitivity’ is entirely explainable.
The market-share diagram (figure 6) was drawn using SITC4 data. In figure A3 the figure is
redrawn using HS6 data. Because there is much more data, we report the 95th quantile rather than
the 90th quantile that was used for the SITC4 data. As is apparent, figures 6 and A3 are almost
identical. The only difference lies in the Low group where the peak of the quantile is no longer at
the far left; however, its peak remains close to the extreme left and certainly much further left than
for the middle group.26
26. On a minor technical note, at the HS6 level there are a few goods that meet the criteria for being in both the Medium
and High groups. Where this is the case the goods are put in the High group; however, since there are so few of these
goods it makes no difference where they are put.
ii
On-Line Appendix – Not for Publication
Figure A2: Normalized World Market Shares, Sensitivity to Cut-Offs
er ln(ymin,g) = 6 to ln(ym
Lowe
min,g) = 5.25
Low
Middle
Raise ln(ymin,g) = 6 to ln(yymin,g) = 6.75
Low
Middle
Lower ln
n(ymin,g) = 8.25 to ln(yymin,g) = 7.25
High
Middle
Raise ln(ymin,g) = 8.25 to ln
n(ymax,g) = 9.25
Middle
High
Lower ln(ymax,g) = 10 to ln(yymax,g) = 9
Middle
Low
Notes: This figure reports the sensitivity of figure 6 to the choice of cut-offs defining the Low, Middle and High
groups of goods. SITC4 data are used. The only sensitivity occurs when ln ymin,g = 6 is raised to 6.75, in which case
the Low group has an interior peak. However, this peak is far to the left, at 0.20, and only appears because the Low
group now contains so many Middle group goods.
iii
On-Line Appendix – Not for Publication
Figure A3: Normalized World Market Shares, HS6 Data
Normalized World Export Shares
k ‐ 1 or Low
w‐Group Goods
Normalized World Export Shares
k or Mediu
um‐Group Goods
Normalized World Export Shares
k + 1 or High
h‐Group Goods
Normalized G
GDP per Capita
Notes: This figure repeats figure 6, but using HS6 COMTRADE data and the 95 quantile (since the HS6 data are
much finer than the SITC4 data).
iv
On-Line Appendix – Not for Publication
Table A1: List of Countries
GDP per Capita (2005)
Code Country
$US
BDI
MWI
ZWE
RWA
NER
MMR
SLE
AFG
NPL
SOM
MDG
UGA
MOZ
GIN
TGO
BFA
BGD
HTI
KHM
MLI
GHA
BEN
KEN
TCD
ZMB
SDN
IND
SEN
NGA
PAK
NIC
PNG
CMR
BOL
PHL
IRQ
HND
IDN
LKA
PRY
EGY
CHN
MAR
AGO
GTM
JOR
SLV
101
157
170
226
245
248
273
273
276
283
283
317
323
325
337
387
422
429
444
473
475
513
526
580
637
675
713
730
803
820
899
928
955
1,028
1,163
1,213
1,225
1,244
1,253
1,266
1,392
1,766
1,906
2,039
2,147
2,293
2,545
Burundi
Malawi
Zimbabwe
Rwanda
Niger
Myanmar
Sierra Leone
Afghanistan
Nepal
Somalia
Madagascar
Uganda
Mozambique
Guinea
Togo
Burkina Faso
Bangladesh
Haiti
Cambodia
Mali
Ghana
Benin
Kenya
Chad
Zambia
Sudan
India
Senegal
Nigeria
Pakistan
Nicaragua
Papua New Guinea
Cameroon
Bolivia
Philippines
Iraq
Honduras
Indonesia
Sri Lanka
Papua New Guin.
Egypt
China
Morocco
Angola
Guatemala
Jordan
El Salvador
GDP per Capita (2005)
ln y k
4.62
5.06
5.13
5.42
5.50
5.51
5.61
5.61
5.62
5.64
5.65
5.76
5.78
5.78
5.82
5.96
6.05
6.06
6.10
6.16
6.16
6.24
6.27
6.36
6.46
6.52
6.57
6.59
6.69
6.71
6.80
6.83
6.86
6.94
7.06
7.10
7.11
7.13
7.13
7.14
7.24
7.48
7.55
7.62
7.67
7.74
7.84
Code
Country
ALB
COL
ECU
THA
TUN
PER
DOM
DZA
BGR
JAM
CUB
BRA
ROU
CRI
ARG
TUR
URY
MYS
VEN
LBN
CHL
MEX
POL
HUN
SAU
PRT
ISR
GRC
ESP
NZL
SGP
ITA
DEU
FRA
CAN
JPN
AUS
AUT
GBR
FIN
NLD
SWE
USA
DNK
IRL
CHE
NOR
Albania
Colombia
Ecuador
Thailand
Tunisia
Peru
Dominican Rep.
Algeria
Bulgaria
Jamaica
Cuba
Brazil
Romania
Costa Rica
Argentina
Turkey
Uruguay
Malaysia
Venezuela
Lebanon
Chile
Mexico
Poland
Hungary
Saudi Arabia
Portugal
Israel
Greece
Spain
New Zealand
Singapore
Italy
Germany
France
Canada
Japan
Australia
Austria
United Kingdom
Finland
Netherlands
Sweden
USA
Denmark
Ireland
Switzerland
Norway
$US
2,691
2,739
2,794
2,797
2,846
2,911
3,073
3,115
3,441
3,622
4,093
4,260
4,557
4,616
4,728
4,969
4,996
5,098
5,374
5,436
7,297
7,365
7,923
10,942
13,119
17,457
19,389
25,562
25,947
26,789
26,968
30,053
33,718
33,862
35,071
35,646
36,321
36,760
36,954
37,307
38,512
39,539
41,348
47,839
48,373
49,282
63,704
ln y k
7.90
7.92
7.94
7.94
7.95
7.98
8.03
8.04
8.14
8.19
8.32
8.36
8.42
8.44
8.46
8.51
8.52
8.54
8.59
8.60
8.90
8.90
8.98
9.30
9.48
9.77
9.87
10.15
10.16
10.20
10.20
10.31
10.43
10.43
10.47
10.48
10.50
10.51
10.52
10.53
10.56
10.59
10.63
10.78
10.79
10.81
11.06
On-Line Appendix C. Full Country Names and GDP per Capita
Table A1 provide full country names for the ISO codes listed in Appendix B.
On-Line Appendix D. Additional Justification of Our Modelling Approach
D.1. Alternative Models Used in the International Trade Literature
There are a number of models of international trade that could have been used to deliver our
results e.g. Eaton and Kortum (2002), Bernard, Eaton, Jensen and Kortum (2003) and Melitz (2003).
Why have we not used these? The core of our model has two components. First, we assumed
that quality capabilities are scarce and asymmetrically distributed across countries. Second, in
equilibrium wages adjust to changes in quality capabilities so that rich countries are priced out
of low-k goods. One could obtain our main results in these other models, but it would be less
straightforward.
v
On-Line Appendix – Not for Publication
For one, in these other models, scarcity is described by the distribution of productivities (G ( a) in
Melitz, 2003; the Type II extreme value distribution in Eaton and Kortum, 2002 and Bernard et al.,
2003). To use these other models one would have to provide a detailed specification of how these
distributions vary across both countries and industries. This can be done, but it would require so
much asymmetry that the elegance of these models would be lost. Re-stated, these other models are
designed to handle within-industry heterogeneity and are less concerned with standard betweenindustry comparative advantage. In contrast, we have made the extreme assumption that there is
at most one firm per country and focussed instead on the cross-country, cross-industry distribution
of capabilities that are central to our Ricardian logic.
For another, in these other models wage adjustment plays a role in determining entry thresholds.
Beyond this, there is little discussion of the comparative advantage implications of wages or of why
some countries are rich and others poor. It is these latter issues that are our main concern.
In short, we have chosen the simplest model possible that focuses on (1) Ricardian asymmetries
in the distribution of quality capabilities and (2) the role of wages as an adjustment mechanism in a
multi-industry world populated by rich and poor countries.
D.2. The Endogeneity of Quality Capabilities
The driving assumptions on this paper are that (a) some capabilities are relatively scarce, and (b)
the relatively scarce capabilities are distributed asymmetrically across countries. (We have chosen
to take (a) and (b) as given, and explore their consequences.) It might seem natural to endogenize
the entry process, and so derive (a) and (b) from more primitive assumptions. We have chosen
not to do this for the following reasons. Endogenizing (a) is straightforward, and has been done
elsewhere (Sutton, 1991, 1998, 2007b). To model (b), however, requires that some assumption be
made regarding asymmetries between countries. This could be done by assuming that the (unobservable) relationship between the fixed and sunk costs of product development, and product
quality, differ across countries; but to do this would simply beg the question, why? This leads, then,
into the broad economic history of industrial development, and so to the issues that lie far beyond
our present scope; no single way of modelling the origin of these cross-country differences could
hope to command general acceptance. And so we have chosen to present the theory in its simplest
form, staying close to the empirical observables, by taking (a) and (b) as our primitives.
On-Line Appendix E. Detailed Proofs
E.1 Details of Proof That vkL < vkH < u
2
√
√
Simplifying
N√
k −1 − 1 + √Nk −1 − 5 √
√equation (28) using
2 Nk−1 − 1 Nk−1 − 5 = 2 Nk−1 − 3 + Nk−1 − 1 Nk−1 − 5 ,
=
( Nk−1 − 1) + ( Nk−1 − 5) +
p
p
vkH
N = k ( Nk−1 − 3) + Nk−1 − 1 Nk−1 − 5 .
u
2
√ √
√
Define z ≡ vk /u. Re-write equation (27) as ( Nk + z) = Nk−1 − 1 Nk z or
( Nk + z)2 = Nk ( Nk−1 − 1)z.
Cross-multiplying this equation, it can be further re-written as
LHS(z) ≡
1
Nk z
=
≡ RHS.
( Nk + z)2
Nk−1 − 1
vi
On-Line Appendix – Not for Publication
LHS(z) is increasing in z for z ∈ [0,1] and LHS(0) = 0 < RHS. I will show that LHS(1) > RHS,
which will establish that there is only one solution on the unit interval to this equation.
LHS(1) − RHS =
1
( Nk + 1)2 ( Nk−1
Nk ( Nk−1 − 1) − ( Nk + 1)2
Nk ( Nk + 3) − ( Nk + 1)2
Nk2 + 3Nk − Nk2 − 1 − 2Nk
− 1)
1
2
( Nk + 1) ( Nk−1 − 1)
1
=
2
( Nk + 1) ( Nk−1 − 1)
1
=
2
( Nk + 1) ( Nk−1 − 1)
>0
≥
[ Nk − 1]
where the inequality follows from our scarcity assumption: Nk−1 − 1 ≥ ( Nk + 4) − 1. Hence
vkH /u =
2
p
Nk p
Nk−1 − 1 − Nk−1 − 5 < 1.
4
(41)
We complete the proof by showing that vkL < vkH . From equations (25) and (41), vkL < vkH if and
only if
2
p
Nk p
( Nk−1 − 1) Nk
<
N
−
1
−
N
−
5
k
−
1
k
−
1
4
N2
√ k −1
p
1 p
Nk−1 − 1
<
Nk−1 − 1 − Nk−1 − 5
Nk−1
2
p
p
p
2 Nk−1 − 1 < Nk−1
Nk−1 − 1 − Nk−1 − 5
p
p
( Nk−1 − 2) Nk−1 − 1 > Nk−1 − 5
which
Nk−1 ≥
√ must hold because
√ our scarcity√assumption implies
√
√ Nk + 4 ≥ 6 so that ( Nk−1 −
2) Nk−1 − 1 ≥ (6 − 2) Nk−1 − 1 > 4 Nk−1 − 1 > 4 Nk−1 − 5 > Nk−1 − 5.
E.2 Details of Proof That
q
vkH /u < ξ +
From equations (28) and (31), vkH /u < ξ + if
√ h
√ i
p
p
Nk p
Nk p
Nk−1 − 1 − Nk−1 − 5 <
Nk−1 − 1 + 2Nk−1 − 1 .
2
Nk−1
√
√
Multiplying both sides by
Nk−1 − 1 + Nk−1 − 5 and simplifying:
h
i
p
p
Nk−1 [( Nk−1 − 1) − ( Nk−1 − 5)] < 2 ( Nk−1 − 1) + Nk−1 − 1 2Nk−1 − 1
hp
ip
p
+2
Nk−1 − 1 + 2Nk−1 − 1
Nk−1 − 5
p
p
4Nk−1 < 2( Nk−1 − 1) + 2 Nk−1 − 1 2Nk−1 − 1
hp
ip
p
+2
Nk−1 − 1 + 2Nk−1 − 1
Nk−1 − 5
p
p
( Nk−1 + 1) < Nk−1 − 1 2Nk−1 − 1
hp
ip
p
+
Nk−1 − 1 + 2Nk−1 − 1
Nk−1 − 5
vii
On-Line Appendix – Not for Publication
But 2Nk−1 − 1 > Nk−1 − 1 so that the first term on the RHS exceeds Nk−1 − 1. Further, since
√our
scarcity assumption implies that Nk−1 ≥ 6, the second term on the right hand side exceeds 5 +
√
1/2
< ξ +.
11 > 5. Hence the RHS exceeds Nk−1 − 1 + 5 = Nk−1 + 4. This establishes that vkH /u
viii