Trade, Human Capital and Inequality
Spiros Bougheas
University of Nottingham

Raymond Riezman
University of Iowa

February, 2005

Abstract
We develop a two-country, two-sector model of trade where the only
diﬀerence between the two countries is their distribution of humancapital endowments. We show that even if the two countries haveidentical aggregate human capital endowments the pattern of tradedepends on the properties of the two human capital distributions. We
also show that the two distributions of endowments also completely
determine the eﬀects of trade on income inequality. Then, we provethat there are long-term gains from trade if the marginal utility ofincome is constant or as long as losers from trade are compensated bywinners. Finally, we look at a simple majority voting model. It turns
out depending on the distribution of human capital, autarky and free
trade with and without compensation may be the outcome of majority
voting.

1

1

Introduction

In recent years the impact of trade on income inequality has been a topic
widely discussed in both academic and policy forums. What has triggeredinterest in this topic is a growing concern among industrialized nations about
their ability to sustain high standards of wellbeing in the face of competition
from low wage countries. Naturally, at the theoretical level these issues
have been addressed within models where trade is motivated because of
diﬀerences in technologies and aggregate endowments.1 However, it has also
been noticed that a large volume of international trade takes place between
countries with similar technologies and aggregate endowments and work inthis area has focused in explaining their patterns of trade.2
In this paper, we argue that the implications for inequality and welfare ofthis second type of trade have not been fully explored. It is well understoodthat trade produces both winners and losers. It is also well understoodthat if it is accompanied by an appropriate compensation scheme it will
Pareto-dominate autarky. However, it is not clear at all whether, in models
with agent heterogeneity, there exist socially acceptable mechanisms that
can implement such Pareto optimal outcomes. For example, when both thetrade regime and any redistribution schemes are decided by a majority rule.
In order to address these issues we develop a two-country, two-sector
model of trade where the only diﬀerence between the two countries is in theirdistribution of human capital endowments. Their technological capabilities
and the preferences of their consumers are identical. In each country there
is a primary sector where output is produced using labor and a high-tech
sector that uses human capital as its input. We will demonstrate that even if
the two countries have identical aggregate human capital endowments they
will trade with the patterns of trade depending on the properties of the two
human capital distributions. In fact, in our framework diﬀerences in size
do not aﬀect the patterns of trade. Knowledge of the two human capital
distributions is alone suﬃcient.3
1

Both the theoretical and empirical literatures are extensive and have recently beenreviewed by Feenstra and Hanson (2001).
2
See Brander (1991), Davis (1995), Grossman and Maggi (2000) and Krugman (1979)
for theoretical attemts to account for this observation.
3
Grossman and Maggi (2000) using production technologies where workers’ talents
can be complementary in some sectors and substitutable in others have found that the
distribution of human capital can potentially matter for a country’s patterns of trade. This
is in contrast with our paper where as long as the distributions diﬀer the two countries
can benefit from trade.
In addition, they exclusively focus on trade patterns while we are also interested on

2

We will also show that together the two distributions of endowments
also completely determine the eﬀects of trade on income inequality. Morespecifically, we will find that the latter always rises in the country thatexports the high-tech product and declines in the country that exports theprimary commodity. However, knowledge of endowment distributions alone
is not suﬃcient for drawing general conclusions about the impact of tradeon the world income distribution.
Next, we explore the welfare implications of our model. Not surprisingly, we prove that there are long-term gains from trade as long as losers
are compensated.4 However, designing and introducing appropriate compensation schemes can both be problematic. Design problems arise becauseof the diﬃculty not only of separating those who gain from those who lose
but also assessing the exact size of the corresponding gains and losses. But
even in the absence of design problems compensation schemes might not be
introduced because in models with agent heterogeneity compensation might
not be the outcome preferred by the majority.5 With these observations
in mind, we compare total welfare under autarky with the corresponding
welfare under trade and find that, unless the marginal utility of income isconstant, there exist free-trade equilibria that are welfare reducing.
Then it is natural to ask what is the relationship between the initial
distribution of endowments and aggregate welfare assuming that the issuesof openness and compensation are decided by a majority vote. We provide acomplete characterization of politico-economic outcomes and find that there
exist equilibria that do not enhance welfare. This happens either because
the majority decides that the country should not trade or because in someof these cases where the majority decides that it should trade it also decides
against any compensation schemes.
trade’s consequences for inequality and welfare.
Grossman (2004) also finds that distributions matter but in his model firms are not
perfectly informed about workers’ productivity and their output is not verifiable by theiremployees.
4Here, we completely ignore any short-term adjustment costs as the economy moves
from one regime to another. See Davidson and Matusz (2002, 2004) for interesting workin this area.
5
See Davidson, Matusz and Nelson (2004) for an in-depth analysis of these issues withina framework similar to ours.

3

2

The Model

There are two countries: A and B. Each country is populated by a continuum of agents of measure 1. Each agent (i for country A and j for countryB) is endowed with one unit of labor and some level of human capital,hi (hj ), randomly drawn from the interval [1, hMAX ]. Let fA and fB denote the density functions and FA and FB the corresponding human capital
distribution functions of countries A and B respectively.
There are two goods X and Y . Good X is a primary commodity andeach unit produced requires one unit of labor. In contrast, good Y is ahigh-tech product and each unit produced requires, in addition, one unit ofhuman capital. The amount of good Y produced by an agent corresponds totheir level of human capital. So, an agent with human capital hi produces
hi units of good Y .
All agents derive utility from the consumption of both goods and they
have identical homothetic preferences.

2.1

AutarkyIn this section, we derive the equilibrium under autarky. Without any lossof generality we concentrate on country A.
2.1.1

Production Possibilities Frontier

The maximum amount of good X that can be produced is equal to 1. Eachagent uses her single labor unit endowment to produce one unit of the primary good. The slope of the PPF at the point where it intersects the x axis
is equal to −(1/hMAX ). This is because eﬃciency requires specialization according to comparative advantage, and the agent with the most comparativeadvantage in producing Y is agent hMAX . However, as production of thehigh-tech product increases the PPF gets steeper because the new producers
have lower human capital endowments. The maximum amount of good Y
that the economy can produce, ĥA is attained when all the agents produce
good Y , hence, is equal to the average endowment of human capital. That
is,
ĥA =

Z hM AXhi fA (h)dh1

At the point where the PPF crosses the y axis its slope is equal to −1. In
general, the marginal rate of transformation (the absolute value of the slope)4

is equal to 1/h0 where h0 is equal to the human capital endowment of theagent with the highest endowment among those producing good X. Figure
1 shows the graph of the PPF.

X
MRT=1/h’1h^YFigure 1: Production Possibilities Frontier

2.1.2

Equilibrium

Define as pA the relative price (i.e. the price of good X measured in unitsof good Y ), qA (X) the quantity produced of good X and qA (Y ) the corresponding quantity of good Y . Then,
Proposition 1 Equilibrium under Autarky:
The equilibrium price satisfies 1 < pA < hMAX and there exists a criticallevel of human capital endowment, h∗A , such that pA = h∗A , all agents withhi < h∗A produce good X, all agents with hi > h∗A produce good Y , qA (X) =R hM AXR h∗A
hfA (h)dh.
1 fA (h)dh, and qA (Y ) = h∗A

Proof. The proposition follows from straightforward arbitrage arguments.
5Notice that agents with human capital endowments equal to h∗A are
indiﬀerent between producing X or Y .
2.1.3

Income Distribution

In order to measure incomes we need a numeraire. Any welfare comparisons
before and after a change in relative prices will be aﬀected by the choice ofnumeraire. However, as long as we are interested in changes in inequality
the choice of numeraire is inconsequential. With this in mind we use good
Y as the numeraire. Then, for each type of equilibrium we can derive thecorresponding income distribution of the economy. Let yi denote the incomeof agent i. Then,
Proposition 2 Income Distribution under Autarky:
Under Autarky, yi = h∗A for all i such that hi 6 h∗A , and yi = hi > h∗Afor all i such that hi > h∗A . The proportion of agents with income exactly
equal to h∗A is given by FA (h∗A ) and the proportion of agents with income
higher than h∗A (h∗A < hi < hMAX ) is given by 1 − FA (h∗A ).
We next illustrate what autarky equilibrium looks like for a particularutility function.
Example 1 Suppose that preferences are described by the utility function:
U (X, Y ) = U (X, Y ) = AX γ Y δ . For a given price pA , those agents withhi > pA (producers of Y ) maximize the above utility subject to the budget
constraint: hi − pA X − Y = 0 that yields the following demand functions:
X=

hi γ
δ
, Y = hipA γ + δ
γ+δwhile those agents with hi 6 pA (producers of X) maximize the same utility
subject to the budget constraint: pA − pA X − Y = 0 that yields the demand
functions:δγ, Y = pA
X=γ+δγ+δThe equilibrium price h∗A is such that the supply of X (demand for Y ) isequal to the demand for X (supply of Y ); in other words it satisfies thefollowing equality:
Z h∗

A

1

δ
fA (h)dh =γ+δ

Z hMAXh∗A

6hi γ
fA (h)dhh∗A γ + δγNotice that each producer of X produces 1 unit, consumes γ+δ
units andδ
supplies γ+δ units of X. Simplifying the above expression we get:R hM AXhi fA (h)dhγ qA (Y )
γ
h∗A
=
(1)
pA = h∗A =∗δFA (hA )δ qA (X)

2.2

Free Trade

We next turn to consideration of opening up to international trade. In our
two country model the only way that the two countries diﬀer is in theirdistributions of human capital endowments. In general, this implies thatpA 6= pB (h∗A 6= h∗B ) which means that autarky prices diﬀer in the two countries. Diﬀerent relative autarky prices imply that there are opportunities for
trade. Thus, countries with identical amounts of human capital but diﬀerent
distributions of human capital will still have an incentive to trade.
2.2.1

Free Trade Equilibrium

As is usually the case, the autarky prices will determine the pattern ofinternational trade.
Proposition 3 The country with the higher relative autarky price ratio will
export good Y and import good X.
Proof. Without loss of generality, consider the case where pA > pB (h∗A >Consider an agent in country A with endowment h∗A − ε (ε > 0 andsmall). Under autarky that agent produces one unit of X and consumes
any linear combination of one unit of X and h∗A units of Y . When tradeis allowed that same agent will produce h∗A − ε units of Y , and trade themfor (h∗A − ε) p1B > 1 units of X and consumes any linear combination of(h∗A − ε) p1B > 1 units of X and (h∗A − ε)pA /pB = (h∗A − ε) p1B h∗A > h∗A units of
Y . A similar argument shows that an agent in country B with endowmenth∗B + ε who produced good Y under autarky produces good X under free
trade. At the global equilibrium there is a cut-oﬀ level of human capital endowment, h∗G (h∗B < h∗G < h∗A ), such that all agents in both countries withendowments higher than h∗G produce good Y and all agents with endowments less than h∗G produce good X. This means that more Y is producedin country A and less Y is produced in country B implying that country A
will export good Y .
In order to calculate the free-trade price we need to construct the global
PPF.
h∗B ).

7

2.2.2Global PPF

Using each country’s human capital endowments and their identical technologies we can construct a global PPF. The construction of the global
frontier follows the same rules as those followed for the construction of eachindividual frontier using the global distribution of human capital endowments. The latter distribution denoted by G is given by:
FA (h) + FB (h)2
The maximum quantity of X that the two countries can produce is equal to
2 and the corresponding quantity of good Y is equal to ĥA + ĥB .
We use the same utility function of Example 1 and compute the free
trade price.
G(h) =

Example 2 Suppose that preferences are described by the utility function:
U (X, Y ) = AX γ Y δ . The free trade price, pT , is equal to h∗G and is givenby:
R hM AXhi g(h)dhγh∗GpT = h∗G =δG(h∗G )
where g is the density of G.
2.2.3

The Eﬀect of Country Size

To this point, we have assumed that the two countries are of equal size.
However, in our model it is simple to derive the global trade equilibrium
when the sizes of the two countries are not equal. In this section, we change
the population of country B so that it is n times larger than country A.
The human capital distributions are assumed to be the same as above. Thisimplies that the measure of country B’s distribution is equal to n. Notice,as there are not any economies of scale in our model, that this change has
no eﬀect on country B’s autarky price. It is as if n countries with exactly
the same human capital distribution and of equal size attempt to trade.
However, there are not any gains from trade because all of them have thesame autarky price.
Our analysis in the previous section suggests that in order to find the new
global trade equilibrium price we need to derive the global human capital
distribution. We denote this new distribution by Gn (h) and is given by:
Gn (h) =FA (h) + nFB (h)n+18Lemma 1 As n −→ ∞, Gn (h) −→ FB (h).
The above lemma states that as the diﬀerence in size gets larger the
global trade equilibrium price approaches the autarky price of the larger
country. In the limit, the small country cannot influence the global tradeequilibrium price.
2.2.4

Patterns of Trade

Suppose that the two countries are of equal size and their human capital
distributions have the same mean which implies that their aggregate endowments are equal. If the two human capital distributions are diﬀerent thenthe autarky prices will be diﬀerent. This implies that aggregate endowments
may not be accurate predictors of the patterns of trade. This leads to twoquestions.
The first is under what conditions will a country that has a higher aggregate endowment in human capital export the human capital intensive good?
The second question is what properties of human capital distributions provide reliable guides to predict trade patterns? We answer the first question
with the following proposition.
Proposition 4 Suppose that preferences are Cobb-Douglas the sizes of thetwo countries are equal and let FB (h) dominate FA (h) in the sense of firstorder stochastic dominance. Then country B, that is the human capitalabundant country, will export the human capital intensive good.
Proof. We need to show that h∗A < h∗B . First-order stochastic domiRhRhnance implies that FA (h) > FB (h) and h0M AX hi fA (h)dh < h0MAX hi fB (h)dh
for every h0 . Then the inequality follows directly from the autarky priceequilibrium condition (1).
This proposition identifies the patterns of trade for the case in which one
human capital distribution dominates the other in the sense of first-orderstochastic dominance. In this case the variances of the two distributions
do not matter. Hence, the pattern of trade depends only on aggregate
endowments as in the Heckscher-Olin-Samuelson model. Figure 2 illustrates
the above result.
Given that the populations of the two countries are equal both production possibilities frontiers intersect the x axis at the same point. However,if the two countries produce only the high-tech good then country B, thatis the country with the higher endowment, will produce more. What the9X

B’s PPF

A’s PPFY

Figure 2: Comparative Advantage under First-Order Stochastic Dominanceproposition demonstrates is that when the two production possibilities fron(Y )(Y )< qqBB(X). Thus, given homothetictiers have the same slope then qqAA(X)preferences, country B exports the high-tech good.
The above result contrasts with the no-trade result that Grossman andMaggi (2000) derive under first-order stochastic dominance. In their model,
both sectors exhibit constant returns to scale in talent (our human capital)
and thus the slopes of the two PPFs at points where any ray through theorigin crosses them are equal. Then homotheticity implies that under autarky the two countries produce exactly the same ratio of quantities of thetwo goods and thus there is no comparative advantage and hence, no trade.
In contrast, in our model the returns to human capital vary across sectors
and thus, there are gains from trading.
To provide an answer to the second question we consider the case of
two distributions of human capital that have the same mean but diﬀerent
variance (mean-preserving spreads) with the additional restriction that theircumulative distribution functions cross only once. Let the variance of coun-10try A’s distribution be higher than that of country B’s. In the terminology
of Grossman and Maggi (2000) the country A’s distribution is more diverse
than country B’s. Figure 3 shows the two production possibilities frontiers.

X
A’s PPF

B’s PPFY

Figure 3: Comparative advantage under Mean-Preserving Spreads
Notice that the two PPFs share the same intercepts. This is because (a)the populations of the two countries are equal that implies that the maximum amount of good X that they can produce is the same, and (b) aggregate
endowments are the same which implies that the maximum amount of goodY that they can produce is also the same. Notice that country B’s PPF
lies inside country A’s PPF. This follows from the fact that if FA (h) is moreRhRh
diverse than FB (h) then h0MAX hfA (h)dh > h0M AX hfB (h)dh. That is, if
the two countries produce both goods and also produce the same quantityof X then country A will produce a higher quantity of Y .
The implications for the trade pattern are clear. First note that in thevicinity of the x intercept country A’s PPF is flatter than B’s while it is
steeper in the vicinity of the y intercept. Then,∗ continuity
implies that(Y )qB (Y ∗ )=)there exists a unique ray through the origin ( qqAA(X∗)
qB (X ∗ ) such that at11

the points where it crosses the two PPFs their slopes are equal (M RTA =M RTB = M RT ∗ ). If we are to the left of that ray, i.e.M RS < M RT ∗ , thenqA (Y )qB (Y )qA (X) > qB (X) meaning that Country A exports y, the high-tech good. If
(Y )
(Y )
we are to the right of that ray, i.e.M RS > M RT ∗ , then qqAA(X)< qqBB (X)andcountry B exports the high-tech good.
One implication of the above discussion is that when countries only diﬀer
in the distributions of their endowments, these distributions alone are not
suﬃcient to determine the pattern of trade. We also need to know the exact
specification of preferences even when they are the same for all agents. Once
more, this result contrasts with the corresponding result in Grossman andMaggi (2000). In their model the more diverse country always exports thegood that is produced using a process characterized by input substitutability while the less diverse country exports the good that is produced using aprocess characterized by input complementarity. Therefore, the two distributions completely determine the pattern of trade. The following propositionsummarizes our results for the case of uniformly distributed endowments.

Proposition 5 Suppose that the preferences of agents are described by theutility function U (X, Y ) = AX γ Y δ (let k ≡ γδ ), the sizes of the two countries
are equal, the endowments of country A are uniformly distributed on [1, 2]
and those of country B are uniformly distributed on [1 + x, 2 − x]; where(0 < x < 0.5). Then for each x there exists a k∗ such that for every k < k∗country B will export the human capital intensive good while for every k > k∗country A will export the human capital intensive good.
1
and FB = h−1−x
Proof. Notice that fA = 1, FA = h − 1, fB = 1−2x
1−2x .
Then substituting these expressions in (1) we can solve for the two autarky
prices:p
1 + 1 + 4(1 + 0.5k)2k∗
pA = hA =2+kandp
1 + x + (1 + x)2 + 4(1 + 0.5k)(2(1 − x) + 0.5x2 )k∗pB = hB =2+k

We observe that, since x < 0.5, both prices are monotonically increasing inh∗h∗k and that lim h∗A < 1 and lim h∗A > 1. These conditions are suﬃcient for
k→0

B

k→∞

B

the existence of k ∗ which completes the proof of the proposition.
Above we consider two countries with identical aggregate endowments
and preferences. However, country A’s distribution of human capital en12

dowments has a higher variance than country B’s corresponding distribution. The production possibilities frontiers of the two countries are depicted
graphically in figure 3. At the point where the PPFs cross the horizontal
axis the PPF of country A has a slope equal to 2 while the one of country
B’s has a slope equal to 2 − x. In contrast, at the point where the PPFs
cross the vertical axis the PPF of country A has a slope equal to 1 while theone of country B’s has a slope equal to 1 + x.
What happens to the patterns of trade depends on the relative preferences for each good and on the tails of the two distributions. As an extreme
example, consider the case where there is a very strong preference for thehuman capital intensive good. Then the primary good will only be producedin country A by those agents with very low endowments of human capital.
Similarly, if there is a very strong demand for the primary commodity thenthe human capital intensive good will once more only be produced in country
A but now by those agents on the upper tail of the distribution.
Example 3 Consider an example based on the above proposition. Let x =0.2. Then k∗ = 1.55 and for that value the autarky price in both countries
is the same and equal to 1.612. For k < 1.55 country B will export thehigh-tech product while for k > 1.55 country A will export it.
The analysis of the patterns of trade so far was restricted to the case ofcountries of equal size. It turns out that this restriction is inconsequential
because size variations do not aﬀect the patterns of trade. The results ofsection 2.2.3 suggests that the patterns of trade are completely determinedby diﬀerences in autarky prices which in turn are completely determined by
the distributions of human capital endowments. Put diﬀerently, two countries that are not of equal size but have the same human capital distribution
will also have the same autarky prices.
2.2.5

Trade and Inequality

We compare the income distributions of each country under autarky and after trade. In this section, we are only interested in changes in inequality andthus the choice of numeraire does not matter. We begin with the following
lemma.
Lemma 2 Trade increases inequality in the country that exports the hightech product and reduces inequality in the country that exports the primarycommodity.

13

Proof. Without any loss of generality, assume that h∗B < h∗G < h∗A .
therefore at the global equilibrium country A exports the high-tech productwhile country B exports the primary commodity. Using again good Y as
the numeraire, we observe that, after trade, country A’s income distribution
is as follows: yi = h∗G for all i such that hi 6 h∗G , and yi = hi > h∗G for alli such that hi > h∗G . Comparing this distribution to the corresponding one
obtained under autarky we find that all agents with hi 6 h∗G (proportion
equal to FA (h∗G )) have experienced a decrease in income equal to h∗A − h∗G ,those agents with h∗G < hi < h∗A (proportion equal to FA (h∗A )−FA (h∗G )) haveexperienced a decrease in income equal to h∗A − hi , while the income of therest of the agents (proportion equal to 1 − FA (h∗A )) has remained the same.
Therefore, the poor have experienced the greatest relative loss in income, theloss of the middle-income group has been more moderate, while the incomesof those agents in the high-income group has remained unchanged. Similarly,
comparing country B’s after trade income distribution to the corresponding
one obtained under autarky we find that the income of the poor (proportion
equal to FB (h∗B )) has increased by h∗G − h∗B , the income of those in the
middle income group (proportion equal to FB (h∗G ) − FB (h∗B )) has increasedby h∗G − hi , while the incomes of those agents in the high-income group(proportion equal to 1 − FB (h∗G )) has remained unchanged.6
We can now prove the main result of this section.
Proposition 6 Part1: Suppose that country B’s distribution dominatescountry A’s distribution in the sense of first-order stochastic dominance.
Then trade will increase inequality in country B and decrease inequality incountry A.
Part 2: Suppose that countries A and B have the same aggregate endowments but country A’s distribution is more diverse than country B’s.
Then trade will increase inequality in country A if there is relatively strong
demand for the primary good while inequality in country B will increase if
there is relatively strong demand for the high-tech good.
Proof. The proof of part 1 follows from Proposition 4 and Lemma 2 andthat of part 2 (for the case of uniform distributions) follows from Proposition
6

In general, we need to be cautious with inequality comparisons because one needs totake into account not only relative income changes but also absolute ones. For example,
an increase in the gap between rich and poor does not necessarily imply an increase ininequality if it is also accompanied by an increase in per capita income that is uniformly
distributed. Nevertheless, such concerns are clearly irrelevant for our model. When inequality increases, depending on the numeraire used either the rich get richer and the poor
stay the same or the poor get poorer and the rich stay the same.

14

5 and Lemma 2.
Trade and human capital unemployment Here, as in the HeckscherOlin-Samuelson model, we also find that in the country that imports thehuman capital intensive good the return to human capital falls and thereverse happens in the country that exports it. However, in our model, in
addition to this price eﬀect, we have a quantity eﬀect that works in the same
direction. In the Heckscher-Olin-Samuelson model there is full employmentof all inputs in both countries both before and after trade. In contrast, in ourmodel, human capital is only partially employed. In fact, its employment
increases in the country that exports the human capital intensive good while
it declines in the other country.

2.3

The World Income Distribution

Before we examine the welfare properties of our economy we compare thepre-trade world income distribution with the corresponding post-trade one.
Under the assumption that the populations of the two countries are equal
the world income distribution under autarky is given by:
Proportion of Agents
FB (PB )2FB (PA )−FB (PB )2
FA (PA )2
A (PA )
1 − FB (PA )+F
2

IncomePB
PB < hi < PA
PA
PA < hi < hMAX

The post trade world income distribution is given by:
Proportion of Agents
FB (P ∗ )+FA (P ∗ )2 ∗∗A (P )
1 − FB (P )+F
2

IncomeP∗
P ∗ < hi < hMAX

Comparing the two tables we observe that trade reduces the gap between
the rich and the poor and that there is income convergence within sectors.
However, without any information about the two human capital distributions we cannot make any general statements about world inequality. In
fact, propositions 5 and 6 together suggest that whether the gap between
two countries’ inequality measures increases or decreases after they trade depends on the patterns of trade which in turn, depend on the two endowment
distributions and preferences.
15

3

WelfareIn this section we are going to demonstrate that uncompensated trade does
not necessarily enhance social welfare. We know that not all agents gainfrom trade. But here we are going to prove a stronger result; namely that if
the losers are not compensated then trade might reduce social welfare. We
measure welfare by using a standard additive social welfare function:
Z hM AXU (X(h, p), Y (h, p))f (h)dh(2)
W (h, p) =1

Let pa denote the equilibrium relative price under autarky. If trade is welfare

improving then the following must be true:½Z hMAX¾
a
U (X(h, p), Y (h, p))f (h)dhp = arg min1

That is, if trade is welfare improving then the social welfare must be minimized when agents trade at autarky prices. We can prove the following
result:Proposition 7 Suppose that the preferences of agents are described by theutility function U (X, Y ) = AX γ Y δ . Then, unless γ + δ = 1, there exists a
set of prices such that if the country trades at those prices its welfare will
decrease.
Proof. See the Appendix.
To understand the intuition for this result consider the postulated ‘weighted
utilitarian’ social welfare function. One can think of this welfare function as
representing the expected utility of an agent whose endowment is randomly
drawn from a distribution that is the same as the distribution of aggregate
endowments. Suppose we change γ and δ but we keep the ratio γδ constant.
We know that such a change will only aﬀect the marginal utility of income
leaving equilibrium prices and quantities unaltered. However, expected utility valuations are aﬀected by changes in the marginal utility of income.
We next identify the relationship between the marginal utility of income
and the social welfare minimizing prices to better understand the circumstances under which uncompensated trade can reduce social welfare. The
next proposition completely characterizes the prices for which social welfare
falls.7
7

It will become clear that the result must hold for any atomless distribution with aconvex domain. However, our method of proof cannot be applied for general specificationsof distribution functions.

16

Proposition 8 Suppose that the preferences of agents are described by theutility function U (X, Y ) = AX γ Y δ and that endowments are uniformly distributed on the interval [1, 2]. If γ +δ < 1 then there exists an interval (p, pa )such that if the country trades at a price in that interval its social welfare
will be lower relative to autarky. Similarly, if γ + δ > 1 then there exists aninterval (pa , p) such that if the country trades at a price in that interval its
social welfare will be lower relative to autarky.
Proof. See the Appendix.
Putting Propositions 7 and 8 together the intuition is straightforward.
When θ < 1 the marginal utility of income is decreasing in income. We
know that when the equilibrium free trade price is below the autarky priceinequality increases. What happens in this case is that trade transfers income from the relatively poor to the relatively rich. But, given that agents
marginal utility of income is decreasing in income, the absolute value of thewelfare losses of the poor are higher than the welfare gains of the rich. In
contrast, when θ > 1 the marginal utility of income is increasing in income.
When the equilibrium free trade price is above the autarky price inequality
decreases. In this case, trade transfers income from the relatively rich to
the relatively poor. But given that agent’s marginal utility of income isincreasing the absolute value of the welfare losses of the rich are higher than
the welfare gains of the poor.
Example 4 Suppose that the preferences of agents and their endowment
distribution have the same specifications as those used in the above proposition. Using the equilibrium condition under autarky given by equation(1), we find that pa = 1.53518. We will consider two cases. Case (1):γ = δ = 0.1 (decreasing marginal utility of income). The social welfare
under autarky is equal to 1.05744. The price that minimizes social welfare
is equal to 1.50743, the corresponding social welfare is equal to 1.05738 andp = 1.47985. Case (2): γ = δ = 0.9 (increasing marginal utility of income).
The social welfare under autarky is equal to 1.67189. The price that minimizes social welfare is equal to 1.55999, the corresponding social welfare isequal to 1.67118 and p = 1.58507.

3.1

Gains from Trade

We have demonstrated that moving from autarky to free trade without compensating those whose welfare is reduced by such a move can be welfare
reducing. Of course, we expect that if trade is accompanied by the appropriate income redistribution then it will be not only welfare enhancing
17

but also Pareto-improving. We show that this is true for the case of CobbDouglas preferences.
Proposition 9 Suppose that preferences are described by the utility function
U (X, Y ) = AX γ Y δ . Then trade always Pareto dominates autarky if it isaccompanied by the appropriate income redistribution.
Proof. See the Appendix.

4

Trade and Political Economy Equilibrium

In the previous section we showed that welfare results depend critically on
whether or not there is redistribution of income to compensate those agents
who suﬀer losses under free trade. In this section, we demonstrate that suchpolicies might be ruled out in a political economy equilibrium. In addition,
we are going to show that it is possible that the majority might vote for trade
without redistribution even when trade reduces aggregate welfare. We adopt
a very simple political economy model and assume that majority voting
decides (a) the choice between autarky and trade, and (b) any redistribution
policies.
We completely characterize the political economy equilibria for the caseof diminishing marginal utility of income (θ < 1)8 and prices in the interval(1, 2). Our first proposition characterizes equilibria when redistribution is
not on the political agenda while the second proposition characterizes theequilibria when both options are available. Let hm denote the human capitalendowment of the median voter; i.e. FA (hm ) = 0.5.
Remember that when the marginal utility of income is diminishing if
p < pT < pA uncompensated trade reduces social welfare. We need toconsider three cases. The first case is when 1 < pT < p < pA < 2. We knowthat the welfare of all those agents with human capital endowments suchthat h > pA is higher under trade and the welfare of all agents with humancapital endowments such that h < pT is lower under trade. Since utility
is weakly monotonic in endowments it implies that for those agents, withhuman capital endowments such that pT < h < pA there exists a thresholdlevel of endowment h1 such that the welfare of all agents with human capitalendowments such that pT < h < h1 is lower under trade and the welfare of
all agents with human capital endowments such that h1 < h < pA is higher
under trade.
8

Similar results can be obtained when θ > 1.

18

The second case is when 1 < p < pT < pA < 2. As in the previous case,there exists a threshold level of endowment h2 such that the welfare of allagents with human capital endowments such that pT < h < h2 is lower under
trade and the welfare of all agents with human capital endowments such thath2 < h < pA is higher under trade. The last case is when 1 < p < pA < pT <2. Now, the welfare of all those agents with human capital endowments suchthat h < pA is higher under trade and the welfare of all agents with humancapital endowments such that h > pT is lower under trade. Using a similarargument as above we can show that there exists a threshold level of income
h3 such that the welfare of all agents with human capital endowments suchthat pA < h < h3 is higher under trade and the welfare of all agents withhuman capital endowments such that h3 < h < pT is lower under trade.
When the political agenda does not include the option of redistribution we
have the following proposition:
Proposition 10 Characterization of Politico-Economic Equilibria without
redistribution for θ < 1.
Let 1 < pT < p
If h1 > hm then Autarky
If h1 < hm then Trade (Social welfare increases)
Let p < pT < pA
If h2 > hm then Autarky
If h2 < hm then Trade (Social welfare decreases)
Let pA < pT < 2
If h3 > hm then Trade (Social welfare increases)
If h3 < hm then Autarky
In contrast, when the option of redistribution is included in the political
agenda we get:
Proposition 11 Characterization of Politico-Economic Equilibria with redistribution for θ < 1.
Let 1 < pT < p
If h1 > hm then Trade with Redistribution
If h1 < hm then Trade without Redistribution (Social welfare increases)
Let p < pT < pA
If h2 > hm then Trade with Redistribution
If h2 < hm then Trade without Redistribution (Social welfare decreases)
Let pA < pT < 2
If h3 > hm then Trade without Redistribution (Social welfare increases)
If h3 < hm then Trade with Redistribution
19

Proof. The results follow directly from the fact that when the majorityof voters are better oﬀ under trade then the outcome of the first vote will be‘trade’ and when the second vote is available the outcome will be ‘without
redistribution’. In contrast, when the majority of voters are better oﬀ underautarky then the outcome of the first vote will be ‘autarky’ and when thesecond vote is available the outcome will be ‘with redistribution’.

5

Conclusion

In this paper, we have assumed that the distribution of human capital isexogenous. One obvious extension would be to allow for endogenous accumulation of skills. This can be accomplished by considering an economy inwhich 2-period lived agents spend their first period of their lives investing inskill accumulation while during the second period produce, trade and consume. In such a model, the agents’ investment in skills will depend on their
expectations about both government policies and the trade regime.
There are two types of government policies that would be worthwhile toconsider; namely redistribution policies and educational subsidies. There isa growing literature that examines issues related to the relationship between
skill accumulation and income inequality but the majority of the work inthis area has ignored government policies. Two exceptions are Deardoﬀ(1997) and Janeba (2000). However both papers focus on the optimalityof government policies ignoring their potential implementation in systems
where decisions are not taken by a social planner but rely on a majority
rule.
Another possible extension is to consider the problem that governments
face when they decide how to allocate a fixed budget for investments inhuman capital accumulation. In this case government policies completely
determine the distribution of human capital (there is no initial distribution
to begin with) which in turn will determine the patterns of trade and posttrade income distribution. A third extension would be to apply our model
to immigration issues. It is clear from our analysis that immigration oremigration of agents will aﬀect both welfare and income distribution.

20

6

APPENDIX

6.1

Proof of Proposition 7

Using the demand functions that we derived in example 1, we find that we
can write the indirect utility function V of an agent with income h who
trades at price p as
V =c

hγ+δ
pγ

´γ ³
´δ
³
γ
δ
where c = A γ+δ
. Notice that the income of an agent who proγ+δ
duces the primary commodity is equal to p.
Then, using (2), social welfare is given by:
p

δ

Z p

f (h)dh + p−γZ hMAXhγ+δ f (h)dhp

1

The f.o.c. condition for a minimum is given by:µ¶
Z hM AX−1δ−γγ+δ
δp F (p) − γph f (h)dh = 0ppNotice that the s.o.c. is also satisfied. Rearranging the above expression wefind that if the social welfare minimizing price is given by the solution of thefollowing equation::
p

γ+δ

γ
=
δ

R hMAXpahγ+δ f (h)dh
F (p)

The proof is completed by adding the observation that unless γ + δ = 1 thesolution of the above equation will not be equal to pa .

6.2

Proof of Proposition 8

Let θ = γ + δ and k = γδ . From proposition (7) we know that the price thatminimizes social welfare is given by the solution to the following equation
R2 θ
p h dhθp =kp−1
In order to prove the proposition we need to show that this price increases
with θ, i.e. the marginal utility of income. The reason that this step is21

suﬃcient follows from (a) proposition 7, where we have shown that when
θ = 1 the minimum is attained at the autarky price, and (b) the continuityof the social welfare function with respect to p. After solving the integral
and rearranging the above expression we get
¶
µ
k
1 θ+1
1+θ
− pθ = k
2
1+
(A1)
p
1+θ
1+θ
The left-hand side of (A1), denoted by L, is strictly increasing in p while
the right-hand side, denoted by R, is independent of p. Then to completedR
the proof we need to show that dLdθ < dθ . Now,
¶
µ
k
dL
k
θ+1
=p
log p − pθ+1
1+
− pθ log pdθ1+θ
(1 + θ)2
and

dR
k
k θ+1
=−
2
2θ+1 +
log 22dθ(1 + θ)
1+θ

thus
dR dL
−
dθ
dθ

=

´
k ³ θ+1
k
log 2 − pθ+1 log p −(2θ+1 − pθ+1 )(A2)−21+θ
(1 + θ)2pθ (p − 1) log p
From (A1) we find thatpθ (p − 1)
k
= θ+1
1+θ
2
− pθ+1
We can substitute this expression in (A2) to get
dR dL
−
dθ
dθ

´
pθ (p − 1) ³ θ+1
1 θ
θ+1
p (p − 1) −log 2 − plog p −
2
2θ+1 − pθ+1
1+θpθ (p − 1) log p2θ+1 log 2 − pθ+1 log p
1=− log p
−
θ+1
θ+1
2
−p
1+θ
= (1 + θ)2θ+1 (log 2 − log p) − (2θ+1 − pθ+1 )=But this last expression is monotonically decreasing in p for 1 < p < 2 and
it is equal to 0 for p = 2 which completes the proof.

22

6.3

Proof of Proposition 9

Let pA and hA denote the equilibrium price and income under autarky andpT and hT the equilibrium price and income under trade. Then we can write
the indirect utility function under autarky as
VA =

hγ+δ
A
pγA

and the indirect utility function under trade as:hγ+δVT = Tγ
pT
Since preferences are homothetic a change in income at any given price level
will not aﬀect the shares of income spent on each good. Suppose that after
trade we adjust each agent’s income so that their post-adjustment indirect
utility is equal to their indirect utility under autarky. Let τ (hT ) denotethe tax (subsidy if negative) imposed on an agent whose post-trade income
is equal to h. By definition the tax (subsidy) must satisfy the following
equality:γ+δh(hT − τ (hT ))γ+δ
= Aγ , ∀hγ
pTpA
Rearranging the above expression we find that the tax (subsidy) must satisfy
τ (hT ) = hT − hA

µ

pT
pA

¶ γ

γ+δ

In order to prove the proposition we need to show that
Z hMAXτ (hT )f (h)dh > 0

(3)

(4)

1

That is, aggregate tax revenues must be higher than aggregate subsidy expenditures which implies that the tax revenues raised from those agents
whose welfare improves under trade is higher than the total amount of subsidies oﬀered to those agents whose welfare deteriorates. Substituting (6) in23

the left-hand side of (7) we get:µ ¶ γ !
µ ¶ γ !
Z pT Ã
Z pA Ã
pT γ+δ
pT γ+δ
pT − pAf (h)dh +h − pAf (h)dhpA
pA
1pTµ ¶ γ !
Z hMAX Ã
pT γ+δ
h−hf (h)dh
+
pA
pA
µ ¶ γ
Z hMAX
pT γ+δ
hf (h)dh − pA
F (pA )
= pT F (pT ) +pA
pTµ ¶ γ Z hMAX
pT γ+δ
−hf (h)dh
pA
pA
Notice that this expression is the same for pT > pA and pT < pA . The proof
will be complete if we can show that the above function is minimized whenpT = pA . After diﬀerentiating the above function with respect to pT , setting
pT equal to pA in the derivative and solving the final expression for pA wefind that the solution for pA is given by equation (1) which gives the solution
for the autarky price. Then the above function achieves a global optimum
when pT is equal to the autarky price. It is easy to verify that the S.O.C.
also holds.

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