世界经济与政治：lecture notes on intrade

Lecture notes in International Trade
Istv′an K′onya
Dept,of Economics
Boston College
Fall 2001
Introduction
What is international trade theory?
International trade theory is basically an exercise in applied general equi-
librium modeling,This means that in many cases one can use the general
theorems that apply for any Arrow-Debreu economy,most notably about
the existence and welfare properties of equilibria,What makes international
trade a separate field is that it utilizes some assumptions that allow for much
sharper positive and normative results than what are possible in the theory
of general equilibrium (hence the phrase,applied”).
What are the special assumptions? Since there are many different mod-
els,let us try to summarize the common elements,The most basic postulate
is that the world consists of separate geographic entities (regions or coun-
tries),Goods and factors of production are more mobile within countries
than among them,because of physical,political and psychological barriers
of movement,In most models factors and goods are perfectly mobile within
countries,but factors are immobile across borders,The mobility of goods
across countries is an important variable,and ranges from none (autarchy)
to complete (free trade),Some of the basic results in trade theory come from
analyzing the effect of goods mobility.
Questions and answers
A good theory has both positive and normative applications,and interna-
tional trade is usually useful for both purposes,The most important ques-
tions in trade theory include the following:
Positive:
i
ii
Why do nations trade?
What do they trade?
Can we use a small set of parameters to predict trade flows?
What is the effect of trade barriers on trade?
Why do the man-made trade barriers exist?
Can trade in goods substitute for factor mobility?
What is the effect of trade on factor rewards?
Normative:
Is free trade better than autarchy?
Is free trade optimal for a country?
What are the effects of trade on income distribution?
If there are winners and losers,can the former compensate the latter?
If nations gain from trade,how are the gains distributed?
If we change parameters,how are these gains redistributed,both within
and across countries?
What are the welfare effects of various trade policies?
The most basic question is probably the first one,why do nations trade?
There are three possible answers yet,comparative advantage,division of labor
and oligopolistic conduct,The first concept,originated by David Ricardo,is
one of the nicest insights in economics,It roughly says that trade is due to
autarchy price differences,Because of differences in the underlying param-
eters,some countries can produce some goods relatively more efficiently,so
when trade is allowed they will specialize in those goods,Good anecdotes
abound,for example the professor and the secretary1,Much of traditional
1The story is as follows,Suppose the professor is more productive in both research and
envelope stuffing than the secretary,Still,if his advantage is relatively bigger in research,
he should specialize in it and leave envelopes to the secretary.
iii
trade theory is concerned with explaining the cause of autarchy price differ-
ences,Candidates include technology,factor endowments and tastes.
The second concept is even older,and (like almost everything in eco-
nomics) goes back to Adam Smith,His idea was that nations trade in order
to exploit economies of scale that arise from specialization,The finer the
division of labor is,the richer countries are,By increasing the size of the
market,trade allows for more specialization,and hence leads to gains for the
participants,This argument,in contrast to comparative advantage,works
even if the trading countries are identical in all respects,This explains why
it received much attention in the last two decades,when trade is increasingly
among seemingly similar countries,In fact specialization due to increasing
returns forms the basis for most of the,New Trade Theory”.
The third concept is fairly recent,and it was first advanced by James
Brander in the early ’80s,While specialization explains trade in similar but
differentiated products,oligopolies can lead to trade in identical goods,If,
for example,two firms in two countries compete in a Cournot-setting,they
will export to each other’s markets even if their products are identical,If
there are transportation costs,firms will charge a lower price abroad than at
home,which can lead to reciprocal dumping,While theoretically intriguing,
trade in identical products and reciprocal dumping are usually thought to
be less important in explaining trade than comparative advantage and spe-
cialization,Nevertheless,such models yield useful insights that should be
incorporated in a general model of trade.
Plan of the course
Most of the course will be about answering the positive questions,although
we will venture a little into the normative ones,The first part will focus on
comparative advantage,or the,classical” theory of trade,Twenty-five years
ago this was pretty much the trade theory,and it still forms the backbone
of the field,We will inquire into the questions about the pattern of trade,
income distribution,factor rewards etc,There are various interesting special
cases that received much attention,and we will look at them in some detail.
We will also summarize some of the evidence about trade models based on
comparative advantage,and the Heckscher-Ohlin model in particular,As
we will see,there is fairly solid support for the theory in general,but once
one looks at the data in more detail,many puzzling questions emerge,Thus
iv
there is a more or less general consensus among researchers that comparative
advantage is an important,but partial explanation for international trade.
The second part of the course will deal with,New Trade Theory”,Most
of it elaborates the division of labor explanation for trade,with a detour
towards oligopolies and reciprocal dumping,As we will see,comparative
advantage and increasing returns can coexist peacefully,each explaining a
different pattern of the data,There is some supportive evidence for this
richer model,presented at the end of the topic,An important branch of
“New Trade Theory” is the,New Economic Geography”,that is concerned
about the location of economic activity,It is a fascinating topic,although
it is a bit easy to get carried away,Nevertheless,it deserves some serious
attention.
Most of trade theory is essentially static,There are some questions,
however,that require us to explicitly consider dynamics,First,in many cases
comparative advantage seems to be acquired,and not given by unchanging
fundamentals,Thus we will incorporate learning-by-doing into trade models,
and see how specialization emerges in the long run,Second,trade can have a
large influence on economic growth,Open economy extensions of endogenous
growth theory shed some light on how international trade and growth are
related,and we will study such models,Third,much of trade (and even
more trade policy debate) is between rich and poor countries,thus we will
look at models of North-South trade,Finally,some evidence will be presented
about the connections among trade,technology and growth.
The last topic can be thought of as an exercise in applying the tools
learned in the first three parts,The world seems to be more and more pre-
occupied by,globalization”,and who else but trade theorists can provide
some insight into the debate,We will limit ourselves to questions that in-
terested researchers in the rich world,Much attention has been focused on
the trinity of immigration,trade and factor prices (wages),There is ample
evidence in the US (and also in some other industrial countries) of increasing
wage inequality,which coincides with globalization in the last two or three
decades,Thus this part will mainly look at theory and evidence on how
immigration,trade and wages are related,There is no definite answer yet,
but it is important to understand the framework of the debate,and - as I
said - it is also a good exercise.
Contents
I The classical theory of international trade 1
1 Basic issues 2
1.1 Comparative advantage with two goods,,,,,,,,,,,,,2
1.2 Explaining comparative advantage,,,,,,,,,,,,,,,,3
2 Analytical tools 6
2.1 The revenue function,,,,,,,,,,,,,,,,,,,,,,,6
2.2 The cost function,,,,,,,,,,,,,,,,,,,,,,,,,7
2.3 Consumer choice,,,,,,,,,,,,,,,,,,,,,,,,,9
2.4 The Meade utility functions,,,,,,,,,,,,,,,,,,,10
3 Equilibrium and the gains from trade 11
3.1 Defining the equilibrium,,,,,,,,,,,,,,,,,,,,,11
3.2 Gains from trade,,,,,,,,,,,,,,,,,,,,,,,,,12
4 Factor price equalization 15
4.1 General results,,,,,,,,,,,,,,,,,,,,,,,,,,15
4.1.1 Comparative advantage,,,,,,,,,,,,,,,,,,15
4.1.2 Factor proportions,,,,,,,,,,,,,,,,,,,,16
4.1.3 Factor prices,,,,,,,,,,,,,,,,,,,,,,,17
4.2 Factor price equalization,,,,,,,,,,,,,,,,,,,,,18
4.2.1 More factors than goods,,,,,,,,,,,,,,,,,20
4.2.2 At least as many goods as factors,,,,,,,,,,,,20
4.3 The pattern of trade under FPE,,,,,,,,,,,,,,,,,20
5 Comparative statics and and welfare 23
5.1 The transfer problem,,,,,,,,,,,,,,,,,,,,,,,23
5.2 The effect of a small tariff,,,,,,,,,,,,,,,,,,,,26
v
CONTENTS vi
5.3 Growth in factor endowments,,,,,,,,,,,,,,,,,,28
5.4 Technological change,,,,,,,,,,,,,,,,,,,,,,,28
6 Simple trade models 30
6.1 The Heckscher-Ohlin model – the role of factor endowments,30
6.2 The generalized Ricardian model – the role of technology,,,33
6.3 The specific factors model – income distribution,,,,,,,,36
7 Empirical strategies 39
7.1 The basic equation,,,,,,,,,,,,,,,,,,,,,,,,40
7.2 Extensions with FPE,,,,,,,,,,,,,,,,,,,,,,,40
7.3 Results without FPE,,,,,,,,,,,,,,,,,,,,,,,41
II Increasing returns and the,New Trade Theory” 45
8 External economies of scale 47
8.1 Gains from trade,,,,,,,,,,,,,,,,,,,,,,,,,47
8.2 An example,,,,,,,,,,,,,,,,,,,,,,,,,,,,48
8.3 Factor price equalization,,,,,,,,,,,,,,,,,,,,,51
9 Oligopoly,dumping and strategic trade 54
9.1 Reciprocal dumping,,,,,,,,,,,,,,,,,,,,,,,,54
9.2 Strategic trade policy,,,,,,,,,,,,,,,,,,,,,,,57
10 Monopolistic competition 59
10.1 Basics,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,59
10.1.1 Consumption,,,,,,,,,,,,,,,,,,,,,,,59
10.1.2 Production,,,,,,,,,,,,,,,,,,,,,,,,61
10.2 The trading equilibrium,,,,,,,,,,,,,,,,,,,,,61
10.2.1 The integrated equilibrium,,,,,,,,,,,,,,,,61
10.2.2 Factor price equalization,,,,,,,,,,,,,,,,,62
10.3 Transport costs and the home market effect,,,,,,,,,,63
10.3.1 Autarchy,,,,,,,,,,,,,,,,,,,,,,,,,64
10.3.2 Trade equilibrium,,,,,,,,,,,,,,,,,,,,,65
11 The New Economic Geography 67
11.1 A model of agglomeration,,,,,,,,,,,,,,,,,,,,67
11.2 Specialization in international trade,,,,,,,,,,,,,,,71
CONTENTS vii
12 Empirical strategies 75
12.1 Testable predictions,,,,,,,,,,,,,,,,,,,,,,,,75
12.2 The gravity equation,,,,,,,,,,,,,,,,,,,,,,,77
III Trade and growth 80
13 Trade,growth and factor proportions 81
13.1 The model,,,,,,,,,,,,,,,,,,,,,,,,,,,,,81
13.2 A small open economy,,,,,,,,,,,,,,,,,,,,,,83
13.3 A large country,,,,,,,,,,,,,,,,,,,,,,,,,,83
14 Learning-by-doing 85
14.1 A Ricardian model,,,,,,,,,,,,,,,,,,,,,,,,85
14.1.1 The model,,,,,,,,,,,,,,,,,,,,,,,,,85
14.1.2 Dynamics and the steady state,,,,,,,,,,,,,86
14.1.3 Industrial policy,,,,,,,,,,,,,,,,,,,,,88
14.2 Agriculture and the Dutch Disease,,,,,,,,,,,,,,,88
14.2.1 The closed economy,,,,,,,,,,,,,,,,,,,89
14.2.2 A small open economy,,,,,,,,,,,,,,,,,,90
14.3 North-South trade,,,,,,,,,,,,,,,,,,,,,,,,91
14.3.1 The model,,,,,,,,,,,,,,,,,,,,,,,,,91
14.3.2 Autarchy,,,,,,,,,,,,,,,,,,,,,,,,,93
14.3.3 Free trade,,,,,,,,,,,,,,,,,,,,,,,,,94
15 Endogenous growth and trade 98
15.1 Autarchy,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,98
15.2 International knowledge diffusion,,,,,,,,,,,,,,,,102
15.3 Trade with knowledge diffusion,,,,,,,,,,,,,,,,,103
15.4 Trade with no knowledge diffusion,,,,,,,,,,,,,,,,105
15.5 Imitation and North-South trade,,,,,,,,,,,,,,,,107
Part I
The classical theory of
international trade
1
Chapter 1
Basic issues
The basic questions in the classical theory of international trade can be ana-
lyzed in the two-by-two model,Thus we will assume here that there are two
factors of production (indexed by i) and two goods (indexed by j),Factors
are mobile across sectors but not across countries,both goods can be traded.
We do not always have to worry about the trade equilibrium explicitly,when
we do,we will assume that there are two countries (Home and Foreign).
1.1 Comparative advantage with two goods
The doctrine of comparative advantage links autarchy price ratios with trade
patterns,We can illustrate it by a simple exchange economy with one rep-
resentative agent.1 Let good 1 be the numeraire,and let pa and pt stand
for the autarchy and free trade relative price of good 2,We can summarize
an equilibrium by the net import vector m of the agent and the equilibrium
price vector p.
Let ma and mt be the autarchy and free trade net import vectors,A
competitive equilibrium is efficient,so that consumption maximizes utility
given the value of endowments,At any price vector,agents can consume
their endowment,so that ma must be on the budget line,In particular:
ma1 + ptma2 = mt1 + ptmt2.
Since ma is affordable at free-trade prices,mt must be revealed preferred to
1You can find a graphical treatment in DN Ch,1,p.7.
2
CHAPTER 1,BASIC ISSUES 3
it:
ma1 + pama2 < mt1 + pamt2.
Combining the equality and the inequality,using that ma = 0 and that
ptmt = 0,we have that:
(pt?pa)mt2 < 0.
Thus the home country will import good 2 if and only if its relative autarchy
price is higher than in the trade equilibrium,With two countries,the same is
true for the foreign country,Since trade must be balanced,as a corollary we
get that the free-trade price must be between the two autarchy price ratios.
Of course,to determine the equilibrium price,we also need to know demand
patterns.
1.2 Explaining comparative advantage
In the pure exchange model above,there might be two reasons why autarchy
prices differ across countries,One is demand and the other endowments.
Example,same endowment but different taste,or same taste but different
endowment,In a more general production model,we can look at tastes,
technology and factor abundance,Tastes work the same way,so let us deal
with the other two.
First,technology,This is the Ricardian explanation for trade and can
be illustrated with one factor,labor,Suppose that consumers want to con-
sume both goods in positive quantities,Then in autarchy,a country has to
produce both goods 1 and 2,Let aj indicate the unit labor requirement to
produce good j,and let w stand for the wage rate,Competition and the re-
quirement that both goods are produced ensures that price equals marginal
(and average) cost in both sectors:
a1wa = 1
and
a2wa = pa.
Dividing the second equation by the first,we get that:
pa = a2a
1
,
CHAPTER 1,BASIC ISSUES 4
thus comparative advantage is determined by the relative efficiency of a coun-
try to produce goods,The equilibrium free-trade price vector will be between
a2/a1 and A2/A1,This is Ricardo’s famous insight,trade patterns are de-
termined by relative,and not absolute,advantage.
Second,factor abundance,Assume identical technologies,two factors
(we need at least two) and fixed technologies,Let bij indicate the amount of
factor i to produce one unit of good j,and let vi be the amount of factor i
available and xa the autarchy production vector,Assuming that both factors
are fully employed,we have that:
xa1b11 + xa2b12 = v1
and
xa1b21 + xa2b22 = v2.
Divide the second equation by the first and solve for the ratio xa2/xa1 to get:
xa2
xa1 =
b11v2/v1?b21
b22?b12v2/v1.
It is easy to see that:
d(xa2/xa1)
d(v2/v1) =
b11b22?b12b21
(b22?b12v2/v1)2.
This expression is positive if and only if the numerator is positive,which can
be rewritten as b22/b12 > b21/b11,In words,the production of good 2 relative
to good 1 will be a positive function of the relative endowment of factor 2 if
and only if its production is relatively intensive in factor 2,Without loss of
generality we can assume this to be the case.
The final step in the chain of argument that relates factor endowments
and autarchy prices comes from demand,We rule out demand differences
in order to focus on factor abundance,This is,however,not enough,The
problem is that the consumption ratio (which in autarchy must equal the
production ratio) is a function of not just the relative price,but also income.
Thus to avoid complications with income effects,we have to assume identical
homothetic preferences,Then ca2/ca1 will be a decreasing function of pa alone,
and thus pa will depend on v2/v1 negatively,Thus we can conclude that
with identical homothetic preferences,a country will have a comparative
advantage in producing a good that uses its abundant factor intensively.
CHAPTER 1,BASIC ISSUES 5
Notice how much weaker this statement is than its Ricardian counterpart.
We need homothetic preferences,an unambiguous definition of factor intensi-
ties (not trivial when input coefficients are not fixed),and two factors,Even
with these,we will see that the factor abundance theory does not readily
generalize to higher dimensions,Some other features of the theory,such as
factor price equalization,however,do.
Chapter 2
Analytical tools
2.1 The revenue function
An extremely useful tool in trade theory is the revenue (or GDP) function.
It is an envelope function defined as follows:
r(p,v) = max
x
{px|(x,v) ∈ Y},
where Y is the production possibility set for the economy,x is the production
vector,v is the factor endowment vector and p indicates prices,In words,
the revenue function indicates the maximum amount of GDP a country can
achieve given its factor supply and prices.
The following properties are easy to prove,First,for r(p,v) as a function
of p:
r(p,v) is convex in p,Take any p1,p2 and let pα = αp1+(1?α)p2,Also,
let x1,x2 and xα be the corresponding optimal output vectors,Then
r(pα,v) = αp1xα + (1? α)p2xα ≤ αp1x1 + (1? α)p2x2 = αr(p1,v) +
(1?α)r(p2,v).
If r is differentiable in p,then x = rp(p,v) – the Envelope Theorem.
r(p,v) is homogenous of degree one in p,so that prp(p,v) = r(p,v).
Follows from the definition of r.
If r is twice differentiable in p,rpp is positive semi-definite (convexity)
and rppp = 0 (homogeneity).
6
CHAPTER 2,ANALYTICAL TOOLS 7
(p1?p2)(x1?x2) ≥ 0 – supply functions are positively sloped,Follows
from p1x1 ≥ p1x2 and p2x2 ≥ p2x1.
Now fix p and look at v:
If Y is convex,r(p,v) is concave in v,Proof similar to above,just note
that if (x1,v1) ∈ Y and (x2,v2) ∈ Y then (αx1 + [1?α]x2,αv1 + [1?
α]v2) ∈ Y.
If r(p,v) is differentiable in v,than rv(p,v) = w,Thus the shadow
prices of factors (which in competitive markets equal actual factor
prices) are given by the gradient rv,Envelope Theorem.
If Y has constant returns to scale (Y is a cone),r(p,v) is linearly
homogenous in v and vrv = r(p,v),For any λ > 0,suppose that
λr(p,v) > r(p,λv),or λpx(p,v) > r(p,λv),But c.r.s means that
(λx[p,v],λv) ∈ Y,so that r(p,λv) cannot be optimal – a contradiction.
Other direction follows similarly.
If r is twice differentiable in v,rvv is negative semidefinite,When Y is
c.r.s.,rvvv = 0.
(v1? v2)(w1? w2) ≤ 0,that is factor demand curves have negative
slope.
Finally,for cross effects:
If r(p,v) is twice differentiable,we have?wi?pj =?xjvi,Follows from rpv =
rvp.
w(p,v) is linearly homogenous in p,and thus pwp = prvp = w(p,v).
Proof,w(λp,v) = rv(λp,v) = λrv(p,v) = λw(p,v).
If Y is c.r.s,then x(p,v) is linearly homogenous in v,and thus vxv =
vrp,v = x(p,v).
2.2 The cost function
Notice that the revenue function is defined for a very general production
structure,We worked with the production possibility set,which allows for
CHAPTER 2,ANALYTICAL TOOLS 8
joint production and arbitrary returns to scale,If we rule out joint produc-
tion,it is often more convenient to work with the cost function,It is defined
as follows:
cj(w,xj) = min
vj
{wvj|fj(vj) = xj},
where vj is the vector of factors used to produced good j and f is the pro-
duction function,In addition,you should know and prove that with c.r.s.
cj(w,xj) = bj(w)xj,where bj(w) is the unit cost function for good j,We will
work with bj instead of cj,so it is useful to list its properties (prove them!):
bj(w) is concave in w.
The optimum choice of input coefficients aj is given by aj(w) = bjw(w).
bj(w) is homogenous of degree one in w and thus ajw = bj.
The optimal choice of inputs to produce xj is given by vj(w) = aj(w)xj.
There is a connection between the cost and the revenue functions,which
should not surprise you as it comes from duality:
r(p,v) = min
w {wv|?j,b
j(w) ≥ pj}.
Thus the revenue function can alternatively defined as the value function for
a problem where we minimize factor payments when unit costs are at least
as large as output prices,For proof,see DN Ch.2,p.45,It is enough to note
that for both representations of the revenue function we can write down the
Kuhn-Tucker sufficient conditions,We will always assume that all factors
are fully employed,so that:
summationdisplay
j
ajixj = vi,
and
j,bj(w) ≥ pj,xj ≥ 0,[bj(w)?pj]xj = 0.
The second condition allows for the possibility that not all goods are actually
produced,and we will see that happen in many important cases.
CHAPTER 2,ANALYTICAL TOOLS 9
2.3 Consumer choice
We will represent consumers’ choice mainly by the expenditure function:
e(p,u) = minc {pc|f(c) ≥ u},
where f(c) is now the utility function and c is the consumption vector,The
problem is mathematically the same as cost minimization,so we can just list
the properties of the expenditure function as follows:
e(p,u) is increasing and concave in p.
If e is differentiable in p,then c(p,u) = ep(c,u).
e(p,u) is linearly homogenous in p,and thus pep = e(p,u).
If preferences are homothetic,e(p,u) = ˉe(p)u,ˉe(p) is also called the
true price index,because it gives the required expenditure to buy one
unit of utility,We will use it a lot later.
epp(p,u) is negative semidefinite,and epp(p,u)p = cp(p,u)p = 0.
(p1?p2)(c1?c2) ≤ 0 – compensated demand functions are downward
sloping.
The expenditure function gives us the compensated demand function,
but in many cases (for example when doing comparative statics) we need the
uncompensated one,There is well-known connection between the two,and
it leads to the following properties:
c(p,u) = d[p,e(p,u)],compensated and regular demand equal each
other if the income level – y – is given by the expenditure function
evaluated at u
cp(p,u) = dp(p,y) + dy(p,y)d(p,y)T (the Slutsky-Hicks equation).
dy(p,y) = epu(p,u)/eu(p,u),where y = e(p,u).
CHAPTER 2,ANALYTICAL TOOLS 10
2.4 The Meade utility functions
A useful tool is the Meade (or direct trade) utility function that condenses
the information found in the various envelope functions,It is particularly
useful for analyzing the effects of tariffs and for normative purposes,We will
not use it much,but some of the literature does,so you should be familiar
with it,It is defined as follows:
φ(m,v) = maxx {f(x + m)|(x,v) ∈ Y},
where the notation is as before,In particular,Y is a convex production set
and f is a quasi-concave utility function,Thus φ(m,v) shows the maximum
utility when production is feasible,factor endowments are given by v and the
import vector is m,In essence we,optimize out” the production vector to
concentrate on net trade and endowments,We used this construct – without
mentioning its name – in the pure exchange model at the beginning.
As before,we can list the properties of φ as follows:
φ(m,v) is increasing in (m,v) (obvious).
φ(m,v) is quasi-concave in m,Let x1 and x2 be the optimal plans
corresponding to m1 and m2,Since Y is convex,1/2(x1 + x2) is fea-
sible,Then φ[1/2(m1 + m2),v] ≥ f[1/2(x1 + x2) + 1/2(m1 + m2)] =
f[1/2(x1 + m1) + 1/2(x2 + m2)] ≥ min{f(x1 + m1),f(x2 + m2)} =
min{φ(m1,v),φ(m2,v)}.
φm(m,v) ∝ p – the gradient of φ w.r.t,m is proportional to prices
(Envelope Theorem and FOC of competitive equilibrium).
φv(m,v) ∝ w.
Actually there is another trade utility function,which is called the indirect
trade utility function,It is defined as follows:
H(p,b,v) = max
c
{f(c)|pc ≤ r(p,v)?b,c ≥ 0}.
It gives the maximum utility that an economy can attain given prices,factor
endowments and trade balance (which does not have to be restricted to 0).
Since DN does not use it,we will not either,but you should know that it
exists.1
1You can learn more from the following paper,A.D,Woodland,Direct and indirect
trade utility functions,The Review of Economic Studies,Oct,1980.
Chapter 3
Equilibrium and the gains from
trade
3.1 Defining the equilibrium
Here we will establish some basic properties of the international equilibrium.
In most cases we will assume a representative consumer and fixed factor
supply,The latter can be relaxed fairly easily,but in most of the literature it
is not,DN deals with the flexible factor supply case,so you can take a look
there,About the former,heterogeneity is interesting when we look at gains
from trade (see below) and it can be managed fairly easily,In most other
cases,however,we have to revert to the representative consumer assumption.
The problem is,of course,that we can say very little about aggregate demand
functions for a general utility function,because of the aggregation problem.
Thus we need to make the heroic assumption of a representative consumer.
Sometime we even have to go further,and assume homothetic preferences,I
will remind you when this is the case.
Let us write down the conditions for autarchy,Using the revenue and
expenditure functions,it is an easy task:
e(p,u) = r(p,v)
ep(p,u) = rp(p,v),(3.1)
The first equation is the identity of GDP and national income,The second
is actually a vector equation,and it gives us market-clearing conditions for
all goods,We know by Walras’ law that one equation is redundant and that
11
CHAPTER 3,EQUILIBRIUM AND THE GAINS FROM TRADE 12
we can normalize the price of one good,We will specify which one when
necessary.
The free trade equilibrium is similarly easy to characterize,Let us keep
our convention of using upper-case letters for foreign variables,then we have
the following:
e(p,u) = r(p,v)
E(p,U) = R(p,V) (3.2)
ep(p,u) + Ep(p,U) = rp(p,v) + Rp(p,V).
We can easily relax the assumption of a representative consumer,Let h
index consumers and assume that each of them owns vh amount of factors.
Then,recalling that factor prices are given by rv,we have:
eh(p,uh) = rv(p,v)vh
EH(p,UH) = RV (p,V)V H (3.3)summationdisplay
h
ehp(p,uh) +
summationdisplay
H
EHp (p,UH) = rp(p,v) + Rp(p,V).
Notice that with identical homothetic utility functions there is a well-defined
aggregate demand function that takes the same form as the individual de-
mand functions.
3.2 Gains from trade
Let us start with the representative consumer case,Here a simple revealed
preference argument shows that there are gains from trade,We don’t in fact
have to use the equilibrium conditions,just compare utility at autarchy and
free trade prices (pa and pt),The argument is as follows:
e(pt,ua) ≤ ptca
= ptxa
≤ r(pt,v)
= e(pt,ut)
Since e(p,u) is an increasing function of u,utility at free trade must be at
least as high as in autarchy,Notice that there are actually two inequalities
CHAPTER 3,EQUILIBRIUM AND THE GAINS FROM TRADE 13
in the chain of argument,The first is the gain from having able to consume
at different prices and the second is the gain from having able to produce at
different prices,If one of the inequalities is strict,so will be the comparison
of utilities.
An extension of the argument above adds tariffs (or subsidies) with a net
revenue of T,In this case the home price vector (?p) will be different from the
rest of the world’s and there is a net revenue (or loss) generated by the tariffs.
Thus the national income identity has to be modified to e(?p,u) = r(?p,v)+T.
It is easy to see that as long as T ≥ 0,managed trade is preferable to
autarchy,since the inequalities above do not change,This is true regardless
of the fact that home faces different prices than the rest of the world,As
long as trade subsidies are not very large,home will benefit from trade.
Now we introduce heterogeneity,In this case it can obviously happen
that some people are better off with trade but others are hurt,Thus the
only thing we can hope for is the existence of a compensating mechanism
through which a Pareto-improvement can be achieved,The most powerful
such tools are lump-sum transfers,and we can show that if the government
can redistribute income,everybody can be made better off,One way to do
that is to show that a scheme that makes the autarchy consumption level
just affordable for all consumers generates positive revenue,Let τh stand for
the lump-sum transfers and let p be the resulting equilibrium price vector.
τh is defined as:
τh = (p?pa)cah + (wa?w)vh,
and it is easy to see that
wvh + τh = wavh?pacah + pcah = pcah,
where the second equality uses the autarchy budget constraint,Thus the
autarchy consumption vector satisfies the budget constraint at the free trade
prices p and transfers τh,We only need to see that the government generates
non-negative revenue:
summationdisplay
h
τh = p
summationdisplay
h
cah?w
summationdisplay
h
vh
= pxa?wv
≤ px?wv
= 0.
CHAPTER 3,EQUILIBRIUM AND THE GAINS FROM TRADE 14
Thus the transfers are feasible,and the consumers are at least as well off as
in autarchy (possibly better if they choose a different consumption vector).
Lump-sum transfers are usually not politically possible,so it is interesting
to ask whether some other type of taxes can achieve the desired result,As
DN show,commodity and income taxes can also be used,The proof is similar
to the one above,except that now we guarantee people their autarchy utility
levels and show that the government can achieve positive revenue,The idea
is that the government will set taxes in such a way that prices and factor
rewards equal the autarchy levels for consumers,pa and wa,Facing the
same prices,they will make the same choices as in autarchy,On the other
hand,producers’ decision will be based on the world equilibrium prices,so
the country is able to reap the gains from trade on the production side.
Formally,let T be the government’s tax revenue,(p,w) the equilibrium price
and factor price vectors and x the equilibrium output vector:
T = (pa?p)
summationdisplay
h
cah + (w?wa)
summationdisplay
h
vh.
But this is exactly the same revenue as above,which we know is non-negative.
The difference between this outcome and the one above is that now consumers
will not consume a different bundle,because we changed not only their in-
come but the prices they face,Thus the only gains come from the production
side,as government revenue.
There is one question that you should ask yourself,what happens with
the surplus in the two cases? DN is quite sloppy about this,and in the lump-
sum case I think they are not quite correct,This is why I used the proof
in Feenstra,which show that even if the government dumps the proceeds,
people are likely to be better off,In the commodity tax case,you cannot
argue the same way,but DN shows in a paper1 that under some conditions
you can redistribute the revenue and make everyone strictly better off.
1Dixit-Norman,Gains from trade without lump-sum compensation,Journal of Inter-
national Economics,August 1986.
Chapter 4
Factor price equalization
4.1 General results
It is time now to try to see what kind of general results emerge from our
model,We will look at comparative advantage,factor proportions and factor
rewards.
4.1.1 Comparative advantage
Generalizing comparative advantage is quite easy,given the properties of
trade equilibrium,In particular,we have that:
paxt ≤ r(pa,v),pact ≥ e(pa,ut) and e(pa,ut) ≥ e(pa,ua)
pa(ct?xt) = pamt ≥ 0.
Now we can use the facts that a similar inequality holds for the foreign
country,and that in equilibrium mt =?Mt,Combining these with the
above,we have that:
(pa?Pa)mt ≥ 0.
Thus,on average,a country will import a good for which it had a higher
autarchy price,Notice,however,that this does not have to be true for a
particular good and DN gives a counterexample.
15
CHAPTER 4,FACTOR PRICE EQUALIZATION 16
4.1.2 Factor proportions
What about explanations for comparative advantage? We will get back to
technology when we discuss the generalized Ricardian model,so let us for
now focus on the factor proportions explanation,As we discussed earlier,we
need to assume identical technologies and uniform homothetic preferences.
Then we can write the expenditure functions as e(p)u,Since we can choose an
arbitrary normalization of prices,it is convenient to have e(pa) = e(Pa) = 1.
Then from the autarchy equilibrium conditions we have that
ua = r(pa,v)
and
Ua = r(Pa,V).
But we saw that for an arbitrary price vector utility is higher than in autarchy,
so in particular we have
r(Pa,v) ≥ r(pa,v)
and
r(pa,V) ≥ r(Pa,V).
Combining these,we get
[r(pa,v)?r(Pa,v)]?[r(pa,V)?r(Pa,V)] ≤ 0.
This is a general result about the connection between autarchy prices and
factor endowments,If r(p,v) was linear in (p,v),we could get a correlation
similar to comparative advantage above,In the absence of linearity,we can
approximate the above inequality when (Pa,V) and (pa,v) are sufficiently
close together,Then the inequality can be rewritten as follows:
(pa?Pa)rpv(v?V) ≤ 0,
where rpv is the matrix of cross-derivatives of the revenue function,To prove
this,just note that because r(p,v) is homogenous of degree one in (p,v) if
technology is CRS,
r(p,v) = prpv(p,v)v.
You can relate rpv to the notion of factor intensities we discussed earlier,see
DN for more details,Thus for small changes we have a negative correlation
between autarchy prices and factor endowments,when we relate the two with
the concept of factor intensities.
CHAPTER 4,FACTOR PRICE EQUALIZATION 17
4.1.3 Factor prices
We can say something about factor prices if we assume identical technologies
and rule out joint production,An immediate result comes from the property
of the revenue function that factor demand curves must be downward sloping.
Applying this to the factor endowments and prices in Home and Foreign
(noting that goods prices are equalized through trade),we get that
(w?W)(v?V) ≤ 0.
Thus a country will have on average lower factor prices for factors it is rela-
tively well endowed with.
Another important question concerns factor rewards in free trade vs
autarchy,Given that goods prices are equalized in the free trade equilib-
rium,we would expect factor prices to move closer together,Unfortunately
this need not be the case,We would like to show that (v?V)(wa?Wa) ≤
(v?V)(w?W),that is factor prices at free trade are,closer” than they were
in autarchy,Since free trade is preferable to autarchy,using the homothetic
equilibrium conditions above we have that
wav ≤ wv
and
WaV ≤ WV.
Moreover,both w and W satisfy the constraint in the alternative definition
of the revenue function,since the output price vector is the same in the two
countries,This gives us
wv ≤ Wv
and
WV ≤ wV.
If we new that Wv ≤ Wav and that wV ≤ waV,we could write down the
two chains of inequalities that complete the argument:
wav ≤ wv ≤ Wv ≤ Wav
and
WaV ≤ WV ≤ wV ≤ waV,
CHAPTER 4,FACTOR PRICE EQUALIZATION 18
and we would have the desired result,But the last two inequalities need not
hold in general,so we cannot conclude that commodity trade leads to dimin-
ishing factor price differences,Although the notion is intuitively appealing,
trade in goods and factor mobility are not always substitutes,Thus policy
arguments based on that notion have no solid theoretical foundations.
4.2 Factor price equalization
The question of factor price equalization (FPE) is related to the previous
discussion,When factor prices are equalized through trade,they are obvi-
ously closer together than in autarchy,We know that FPE is not a general
property of a free trade equilibrium,but it is nevertheless important to see
under what circumstances it can result,There are two reasons for such in-
terest in FPE,First,if there is FPE,there are no incentives for factors to
move and trade in goods is a perfect substitute for trade in factors,Second,
when FPE prevails it is much easier to describe trade patterns,But when
are factor prices indeed equalized?
We assume no joint production,identical technologies and constant re-
turns to scale,so that we are able to use the unit cost functions derived
earlier,Let w and W be the equilibrium factor price vectors,then the free
trade equilibrium conditions are as follows:
b(w) ≥ p and x ≥ 0
b(W) ≥ p and X ≥ 0
a(w)x = v
a(W)X = V
x + X =
summationdisplay
h
dh(p,wvh) +
summationdisplay
H
DH(p,WV H).
FPE means that the two factor prices,w and W are identical,This means
that unit costs are the same for each good in the two countries,Assuming
that all goods are essential and thus have to be produced somewhere,for
each j the nonpositive profit condition has to hold with equality,Let us use
w and?p for the common factor price and price vectors,?x for total production
(i.e,?x = x+X),and let us add up the two factor market clearing conditions.
CHAPTER 4,FACTOR PRICE EQUALIZATION 19
Then we get that
b(?w) =?p
a(?w)?x = v + V
x =
summationdisplay
h
dh(?p,?wvh) +
summationdisplay
H
DH(?p,?wV H).
If you look at the second set of equalities,you can see that these would be
the equilibrium conditions for a world where both factors and goods are mo-
bile,in other words where there are no countries,We will call this construct
the integrated world equilibrium,Thus,in essence we have proved that when
factor prices are equalized,the world can achieve the integrated equilibrium
through trade in goods alone,Thus even if factor movements were possible,
they would not take place when FPE prevails,The construct of integrated
equilibrium also shows us when factor price equalization will occur,The first
set of equations (no pure profits) must hold in a free-trade equilibrium with
equal factor prices,The last set of equations (goods markets clear) is also
the same in the integrated equilibrium and in free trade,The only difference
is that with two countries factor markets have to clear separately,with x and
X between 0 and?x,Thus a trade equilibrium with equal factor prices in the
two countries exists when
a(?w)x = v,x ∈ [0,?x]
has a solution,In words,if using the techniques of production that prevail in
the integrated equilibrium (a[?w]) we can split production into two nonnega-
tive parts that exhaust factor supplies in both countries,we can have FPE.
Otherwise,we cannot.
Formally,the condition for FPE is a condition on the distribution of factor
endowments,Assuming the integrated equilibrium choices of?w,?p and?x are
unique,the set of endowments that are consistent with FPE is given by:
Ψ = {v|v = a(?w)x,x ∈ [0,?x]}.
Of course if v is in Ψ,the equivalent condition on the foreign country’s
endowment is also satisfied,Thus FPE depends on the likelihood of v falling
into Ψ,In the next chapters we will look at that likelihood in different cases.
CHAPTER 4,FACTOR PRICE EQUALIZATION 20
4.2.1 More factors than goods
In this case FPE is a measure zero event,To see this,note that the di-
mensionality of a(?w) is at most n,the number of goods,Then Ψ will be a
subspace of the n dimensional space,whereas the factor endowment space
has a dimension of m > n,Thus it is very unlikely that factor endowments
will fall into Ψ,and we can rule out FPE as accidental in this case.
See graph at lecture!
4.2.2 At least as many goods as factors
In this case the dimensionality of Ψ will be m,assuming that technologies for
producing different goods are different,that is al(?w) negationslash= aj(?w),We will assume
this to be the case,Then FPE will have positive measure,and its numerical
probability will depend on details of technology,The graphs in DN are very
instructive! An interesting problem emerges when n > m,In this case there
are m equations in a(?w)x = v,which means that the production plan is
not unique,Thus many production vectors are compatible with the same
distribution of endowments,On the other hand,world output is uniquely
determined by demand,so the integrated equilibrium is unique,There is a
discussion in DN about the effect of adding more goods,you can read it there.
In general,adding more goods might increase or decrease the likelihood of
FPE,HK has a chapter on adding non-traded goods,the main point is that
we need at least as many traded goods as factors for the FPE set to have
positive measure.
4.3 The pattern of trade under FPE
We saw earlier that in general we can only show a correlation result between
autarchy prices and trade pattern,and the link between factor endowments
and prices is even weaker,We will now show that with FPE we have much
stronger results,To focus on endowments,we will have no joint production,
identical technologies and identical homothetic preferences,Since FPE is
unlikely when there are more factors than goods,we will only look at the
other case,that is n ≥ m,Since when n > m production patterns are inde-
terminate,it is futile to have results on commodity trade,Even when m = n
there is no strong relationship between factor endowments and commodity
CHAPTER 4,FACTOR PRICE EQUALIZATION 21
trade patterns,unless n = m = 2,On the other hand,we have very nice
results on the factor content of trade,and this is what we will look at now.
With identical homothetic preferences,consumers will spend a share of
their income on each good,where the share only depends on relative prices.
Let tkv be the vector of factors embodied in country k’s imports,This is the
difference between the factor content of consumption and the factor content
of production in country k,The latter,of course,is just vk,the factor
endowment of country k,For the factor content of consumption,we know
that spending on each good is a function only of the equilibrium prices,p.
Since preferences are identical,each country will spend the same share of its
income on a particular good,Then,for a particular good j,market clearing
implies the following:
summationdisplay
k
pjckj =
summationdisplay
k
sjwvk? sj = pjcjwv,
where cj is world consumption of good j and v is world endowment of fac-
tors (and hence wv is world income),Then the factor content of country k
consumption is given as follows:
a(w)ck =
summationdisplay
j
aj(w)ckj
=
summationdisplay
j
aj(w)sjwvk/pj
=
summationdisplay
j
ajcjwv
k
wv
= wv
k
wv v.
Using the notation sk = wvk/(wv) for country k’s share of world income,we
have that
tkv = skv?vk.
The equation tells us that a country exports the services of factors with
which it is relatively well endowed compared to the world,If there is balanced
trade,then some elements of the net factor import vector will be positive
and others negative,If we rank factors by their relative endowment size
(i.e,vki /vi),there will be a cutoff such that all factors above the cutoff are
CHAPTER 4,FACTOR PRICE EQUALIZATION 22
exported and the others imported,This is the famous Vanek chain argument
for the factor content of trade,Notice that you can construct such a chain
even if trade is not balanced,but then we have to use the country’s share
in world spending,and it is possible that a country exports or imports all
factor services.
Chapter 5
Comparative statics and and
welfare
5.1 The transfer problem
The simplest comparative statics exercise turns out to be the transfer prob-
lem,The question is the following,when a country receives a transfer of
goods from another,will its terms-of-trade worsen or improve? In the for-
mer case,can the terms-of-trade worsen to such an extent that it is actually
worse off with the transfer? After WWI,that question had an important
application for the German reparation payments,and no less than Keynes
and Ohlin were involved in the debate,The comparative statics exercise will
show us not just the right answer,but also the power of mathematics to
clarify an issue that could not be decided with intuitive reasoning.
Before we start,we need to clarify the choice of the numeraire,For
simplicity,let there be n + 1 goods,and we will normalize the price of good
0 in both countries,We need to drop one equation as well,let this be the
market clearing condition for good 0,The price vector of the remaining n
goods will be p,and we assume that the home country is the recipient of a
transfer of [g0,g],Since ours is a competitive setting,Home can resell these
goods on the market and spend the proceeds in any way it likes,Thus the
23
CHAPTER 5,COMPARATIVE STATICS AND AND WELFARE 24
market clearing conditions become:
e(1,p,u) = r(1,p,v) + g0 + pg
E(1,p,U) = R(1,p,V)?g0?pg
ep(1,p,u) + Ep(1,p,U) = rp(1,p,v) + Rp(1,p,V).
Now we take total differentials with respect to p,u,U (the endogenous
variables) and [g0,g] (the exogenous variables),Let us introduce some no-
tation,m = ep? rp? g =?(Ep? Rp + g) for Home’s net import and
Foreign’s net export vector,ξ = g0 + pg for the value of the transfer and
S = epp + Epp? rpp? Rpp,Then we can write the differential system as
follows:
mdp + eu du = dξ
mdp + EU dU =?dξ
Sdp + epu du + EpU dU = 0.
Notice that S is a negative semidefinite matrix,because e,E are concave
and r,R are convex in p,We will in fact assume that S is negative definite,
which will be the case when there is some substitutability in demand or
production between the numeraire and non-numeraire goods,This means
that S is invertible,and we can express dp from the last equality as follows:
dp =?S?1epu du?S?1EpU dU.
Substituting this into the first two equations,and using that cy(p,y) = epu/eu
(and the same for Foreign)1,we can write down the two matrix equations
bracketleftbigg 1?mS?1c
y?mS?1CY
mS?1cy 1 + mS?1CY
bracketrightbiggbracketleftbigg e
u du
EU dU
bracketrightbigg
=
bracketleftbigg dξ
dξ
bracketrightbigg
We can solve the matrix equation in the usual way,by multiplying both
sides by the inverse of the left-hand matrix (assuming it exists),Let the
determinant of that matrix be
D = 1 + mS?1(CY?cy),
1c(p,y) is now the uncompensated demand function,It is a bit confusing,but blame
DN.
CHAPTER 5,COMPARATIVE STATICS AND AND WELFARE 25
then we end up with two equations for the changes in utility:
bracketleftbigg e
u du
EU dU
bracketrightbigg
= 1D
bracketleftbigg dξ
dξ
bracketrightbigg
.
We can sign D by assuming that the equilibrium is stable in the Walrasian
sense,If you remember the mechanics of a comparative statics exercise,the
left-hand side matrix is just the Jacobian with respect to p of the system.
In economic terms,it is the price derivative of the world excess demand
function,For Walrasian stability,this has to be negative definite (so that
the system returns to equilibrium after a small perturbation in p),which
implies that D > 0,Thus assuming that the value of the transfer is positive
(dξ > 0),it will be beneficial for Home if and only if the equilibrium is stable.
Since a non-stable equilibrium is measure zero anyway,we can conclude that
a positive transfer cannot harm the home country and must hurt the foreign
country,While in principal a terms-of-trade loss could offset the direct gain
from the transfer,a careful mathematical analysis shows that in practice this
possibility can be ruled out.
A related question,which was the one that occupied Keynes and Ohlin,
concerns the change in the terms-of-trade (t.o.t),which for the foreign coun-
try would be mdp,Keynes claimed that in addition to the transfer,Foreign
would experience a deterioration in its terms of trade,Ohlin argued for the
opposite,We can easily show,that in principle both possibilities can arise:
mdp = mS
1(CY?cy)
1 + mS?1(CY?cy) dξ.
Thus the foreign country’s t.o.t will deteriorate if and only if:
mS?1(CY?cy) < 0
To see what this means,let there be only two goods,of which the non-
numeraire good is exported by Foreign,Then S?1 is a negative scalar,and
thus Foreign’s t.o.t will deteriorate if its marginal propensity to consume
its export good is higher than Home’s marginal propensity to consume its
import,Although this does not have to be the case,there is a presumption
that CY > cy,Thus,here at least,Keynes was probably right.
CHAPTER 5,COMPARATIVE STATICS AND AND WELFARE 26
5.2 The effect of a small tariff
The next exercise we look at is the effect of a small tariff on the welfare of
Home,We will assume that Foreign does not retaliate,it simply accepts
whatever price emerges in equilibrium,Thus if the tariff home sets is t,we
have P = p?t,Our equilibrium conditions are
e(1,p,u) = r(1,p,v) + t[ep(1,p,u)?rp(1,p,v)]
E(1,p?t,U) = R(1,p?t,V)
ep(1,p,u) + EP(1,p?t,U) = rp(1,p,v) + RP(1,p?t,V).
Taking total differentials and using the assumption of a small tariff (that is,
a deviation from free trade,so that t = 0),we have
mdp + eu du = mdt
mdp + EU dU =?mdt
Sdp + epu du + EpU dU = (S?s)dt,
where s = epp?rpp,The change in prices again can be calculated from the
third set of equations:
dp = (I?S?1s)dt?S?1cyeudu?S?1CY EUdU,
so that the matrix equation for utility changes is given by
bracketleftbigg 1?mS?1c
y?mS?1CY
mS?1cy 1 + mS?1CY
bracketrightbiggbracketleftbigg e
u du
EU dU
bracketrightbigg
=
bracketleftbigg mS?1sdt
mS?1sdt
bracketrightbigg
We can easily solve the system for the effect on utilities to get
bracketleftbigg e
u du
EU dU
bracketrightbigg
= 1D
bracketleftbigg mS?1sdt
mS?1sdt
bracketrightbigg
.
This in general will be non-zero,Moreover,Home can select tariffs and
subsidies in such a way that dt is positive when mS?1s > 0,and gain from
such a policy,In two dimensions,this means (since S,s < 0) a tariff on
Home’s import,Thus we can see that as long as Foreign does not retaliate,
free trade is not optimal for home.
A related question concerns the possibility that the t.o.t of Home improves
to such an extent that the price of its import actually falls,This has been
CHAPTER 5,COMPARATIVE STATICS AND AND WELFARE 27
know as the Metzler-paradox,To see the change in prices,substitute the
utility changes back to the price equation to get
dp = [I?S?1s + S?1(CY?cy)mS?1s/D]dt
With two goods we can show that the change in the relative price of the
non-numeraire good is given by
dp/dt = 1? sS + (D?1)sSD
= 1? sSD.
Since both S and s are smaller than zero,this is negative iff SD > s,which
we can simplify (using the definitions of S,D and s) to
SD?s = Epp?Rpp?(cy?CY )m
= (Epp?Rpp?CY M) + cyM
= Mp + cyM
> 0.
The last equality comes from the fact that M(p,Y) = C[p,R(p,V)]?X(p,V),
and therefore
Mp = Cp + CY Rp?Xp = Epp?CY C + CY X?Rpp = Epp?Rpp?CY M,
where we used the Slutsky-Hicks equation to get the second equality,In
order to get the Metzler paradox,we therefore need
pMp
M + pcy < 0.
Normally,we would expect the first term,the supply elasticity of exports in
Foreign,to be positive,The second term is also positive if both goods are
normal in Home,Thus for the paradox to arise,either the foreign export
supply elasticity has to be sufficiently negative or the good has to be inferior
in Home,These are both theoretically possible,but seem relatively unlikely
in practice.
CHAPTER 5,COMPARATIVE STATICS AND AND WELFARE 28
5.3 Growth in factor endowments
The possibility that a country can be worse off when its endowment of factors
increases has been raised for developing countries,and it has been dubbed
as the case of immiserizing growth,Let us now look at the conditions for
such a phenomenon to arise,Suppose Home experiences a change of dv in
its endowments,which leads to the following system of equations:
mdp + eu du = rv dv
mdp + EU dU = 0
Sdp + epu du + EpU dU = rpv dv.
The solution can be calculated easily,and it is
bracketleftbigg e
u du
EU dU
bracketrightbigg
= 1D
bracketleftbigg (1 + mS?1C
Y )rv dv?mS?1rpv dv
mS?1cyrv dv + mS?1rpv dv
bracketrightbigg
.
To see what this involves,let us again have only two goods (of which the
non-numeraire one is imported by Home) and a change in the endowment of
only one factor,After some manipulation (see DN,or derive yourself) we get
that
eu du < 0 iff?Mpp/M + mpp/m?(1?pcy)?[(?w/?p)(p/w)?1] < 0,
where w is the price of the factor whose endowment has changed,If both
goods are normal,the first three terms will be negative,Thus immiserizing
growth can arise only if the last term is sufficiently positive,We will return
to that possibility when we talk about specific models.
5.4 Technological change
Technological change can be modeled by introducing a shift parameter θ in
Home’s revenue function,with the property that rθj(1,p,v,θ) > 0,Then we
end up with the following system of differentials:
mdp + eu du = rθ dθ
mdp + EU dU = 0
Sdp + epu du + EpU dU = rpθ dθ.
CHAPTER 5,COMPARATIVE STATICS AND AND WELFARE 29
This can be solved easily to yield
bracketleftbigg e
u du
EU dU
bracketrightbigg
= 1D
bracketleftbigg (1 + mS?1C
Y )rθ dθ?mS?1rpθ dθ
mS?1cyrθ dθ + mS?1rpθ dθ
bracketrightbigg
.
A special case is when θ is the TFP parameter,so that xj = θjfj(vj).
We can show from the alternative definition of the revenue function that in
this case r(p,v,θ) = r(θp,v),and therefore rθ = px,rpθ = x,Assuming two
goods and technological progress only in the non-numeraire good,we have
that bracketleftbigg
eu du
EU dU
bracketrightbigg
= 1D
bracketleftbigg [px?mx(1?pC
Y ))/S]dθ
mx(1?pcy)/Sdθ
bracketrightbigg
.
If both goods are normal,0 < pcy < 1,Thus Foreign will benefit from a
technological change in Home if and only if it imports the good in which the
change occurred,In other words,Foreign will benefit from growth in Home’s
export sector and loose from growth in Home’s import competing sector.
Since the world has to benefit from growth (eu du+EU dU = rθ dθ = pxdθ),
Home will benefit in the latter case,When the change occurs in Home’s
export sector,it will loose if the t.o.t loss (the second term) is greater than
the direct gain (px),You can work out the condition for that,but it is not
very illuminating.
Chapter 6
Simple trade models
Here we will look at special cases of the general trade model we developed
earlier,The main simplification will be to limit the number of goods and fac-
tors,which lead to very sharp – if highly model specific – conclusions,Much
of international trade was taught in terms of these simple models,mostly the
Hecksher-Ohlin variety,They do indeed yield many useful insights,but you
have to bear in mind their limitations,Now that we have seen the general
model,we are better equipped to decide just how special the assumptions
are and what features of the results survive.
6.1 The Heckscher-Ohlin model – the role of
factor endowments
There are two goods produced by two factors,We will assume that in equi-
librium both goods are produced,which gives us two conditions for factor
market clearing and two for zero profits,Let a(w) be the matrix of unit input
coefficients,than we have
a(w)x = v
and
wprimea(w) = p.
The first result we note is the Factor Price Equalization Theorem,which
requires that both countries produce the two goods in equilibrium,We have
already seen that this is a restriction on factor endowments,so we do not
have to assume no specialization separately,This shows the advantage of
30
CHAPTER 6,SIMPLE TRADE MODELS 31
the integrated equilibrium approach,since we traced back FPE to model
fundamentals,We also saw that in the 2x2 case FPE has positive measure.
The next result is the Rybczynski Theorem,which gives the connection
between factor endowments and output levels,given unchanging commodity
(and hence factor) prices,With constant output (and factor) prices,x is a
linear function of endowments,and we can calculate the change in production
levels easily,Let us introduce the,hat” notation for percentage changes,so
that for any variable y we have?y = dy/y,Taking logs of the factor market
clearing conditions and using λij = aijxj/vi for the share of factor i used in
sector j,we have
v1 = λ11?x1 + λ12?x2
v2 = λ21?x1 + λ22?x2.
We can solve for the percentage changes in production to get
x1 = λ22?v1?λ12?v2λ
11λ22?λ12λ21
x2 = λ11?v2?λ21?v1λ
11λ22?λ12λ21
.
It is easy to show the the denominator is proportional to the difference in
relative factor intensities (λ11λ22?λ12λ21 = x1x2v1v2 [a11a22?a12a21]),so without
loss of generality we can assume that it is positive,In words,we assume that
good 1 is more intensive in the use of factor 1 for the given factor prices.
Then it is easy to show that1
v1 >?v2x1 >?v1 >?v2 >?x2,
which is Jones’s famous magnification result,It says that changes in factor
endowments show up magnified in production,A special case is the Ry-
bczynski Theorem,when only one factor endowment changes,If,say,?v2 = 0,
we have
x1 >?v1 > 0 >?x2.
Thus the percentage change in the production of good 1 (that uses factor 1
intensively) is bigger than the percentage change in the endowment of factor
1,and the production of good 2 falls.
1Do it,using the fact that λi1 + λi2 = 1.
CHAPTER 6,SIMPLE TRADE MODELS 32
The mirror image of this result is the Stolper-Samuelson Theorem that
gives the connection between factor prices and commodity prices,We dif-
ferentiate the logarithm of the zero-profit conditions,use the result that
aww = 0 and the notation θij = aijwi/pj (the share of factor i in sector j
revenue) to get
p1 = θ11?w1 + θ21?w2
p2 = θ12?w1 + θ22?w2.
The solution for the factor price changes is given by
w1 = θ22p1?θ21p2θ
11θ22?θ12θ21
w2 = θ11p2?θ12p1θ
11θ22?θ12θ21
,
and the denominator again is proportional to the difference in factor inten-
sities (assumed to be positive),Thus we have the analogous magnification
result that
p1 >?p2w1 >?p1 >?p2 >?w2.
In words,we have that the return of one factor (which is used intensively
in the production of the good whose price saw a larger increase) increases
in terms of both goods,whereas the real return of the other factor falls.
The Stolper-Samuelson Theorem emerges as a special case when only one
commodity price changes.
The final canonical result is the Heckscher-Ohlin Theorem on trade pat-
terns,It is actually a corollary of the magnification result in production,To
see it,note that as long as FPE holds,factor prices are fixed at the integrated
equilibrium levels,and thus production is a linear function of endowments.
In fact,we know that
x1x2 = 1|λ|(?v1v2),
where |λ| is the denominator in the equations for production changes (and
proportional to the factor intensity difference) and λi1 + λi2 = 1,Thus
assuming that good 1 is relatively intensive in the use of factor 1,the country
with a higher ratio of v1/v2 will have a higher ratio of x1/x2,With homothetic
preferences and common prices,however,both countries will consume the two
goods in the same ratio,Thus if Home has relatively more of factor 1,it has
CHAPTER 6,SIMPLE TRADE MODELS 33
to export good 1 and vice versa,Notice that it is the same argument that
we made with fixed coefficients at the first lecture,and it holds for variable
coefficients as long as FPE prevails.
6.2 The generalized Ricardian model – the
role of technology
We saw the simple Ricardian model at the beginning of the class,One
problem with the two-good/one factor setting is that the revenue function
is not differentiable in a non-trivial subset of the parameter values,To see
this,note that r can be written as (normalizing the price of good 1)
r(p,v) = max
braceleftbigg v
a1,
pv
a2
bracerightbigg
.
This function has a kink at p = a2/a1,which means that r is not differen-
tiable there,Although such a relative price might seem exceptional,it can
in fact result in a trade equilibrium with positive measure,This means that
comparative statics is difficult,since one cannot use the standard tools.
To remedy this,we will use the continuum good extension of the Ricardian
model,developed by Dornbusch,Fischer and Samuelson,It turns out to be
surprisingly simple,and very suitable to analyze the effects of technology.
Suppose there is one factor,labor,and a continuum of goods indexed by
z ∈ [0,1],The unit labor coefficient is a(z) for good z in the home country
and a?(z) for the foreign country and we define
A(z) = a
(z)
a(z),
Without loss of generality,we arrange goods in such a way that Aprime(z) < 0,
i.e,the home country is relatively more efficient in the production of goods
with low index,The home country will produce good z if it will have a cost
advantage in it,that is,if
ω < A(z),
where ω = w/w? (Home’s wage rate relative to Foreign),Given our assump-
tion on A(z),Home will produce goods 0 ≤ z ≤ ζ(ω),where ζ(ω) = A?1(ω)
and hence ζprime(ω) < 0,By the same argument Foreign will specialize in the
production of all the other goods,ζ ≤ z ≤ 1.
CHAPTER 6,SIMPLE TRADE MODELS 34
On the demand side we will use the generalized Cobb-Douglas prefer-
ences2,which imply that expenditure share of each good is constant,In
particular,we will have
p(z)c(z)
Y = b(z)
for both countries,Let us define the fraction of world expenditure spent on
Home goods as
ν(ζ) =
integraldisplay ζ
0
b(z)dz,
with the property that
νprime(ζ) = b(ζ) > 0.
In equilibrium,given full specialization,national income in Home must
equal spending on Home goods,Let us normalize the world population to
1 and let L be the population of Home,Then the following equilibrium
condition must hold:
wL = ν(ζ)[wL + w?(1?L)],
from which we get that
ω = ν(ζ)1?ν(ζ)1?LL,
It is easy to check that the right-hand side is increasing in ζ.
We now close the model with the reduced-form equilibrium condition that
determines ζ:
A(ζ) = ν(ζ)1?ν(ζ)1?LL,
Since the left-hand side decreases and the right-hand side increases withζ,the
equilibrium is unique and stable,From the knowledge of the cutoff good we
can solve for all the other variables,notably for ω and the commodity prices
p(z) = min{a(z)w,a?(z)w?},Before we move to comparative statics,let us
note that the relative wage ω is a measure of well-being in both countries.
Indirect utility in Home is given by
v =
integraldisplay ζ
0
b(z)log[b(z)/a(z)]dz +
integraldisplay 1
ζ
b(z)log[b(z)ω/a?(z)]dz,
2The utility function is given by u = integraltext1
0 b(z)logc(z)dz,with
integraltext1
0 b(z)dz = 1.
CHAPTER 6,SIMPLE TRADE MODELS 35
which is increasing in ω3,You can derive the similar expression for Foreign
to show that it decreases with ω.
An obvious comparative statics result concerns L,the relative size of
the home country,An increase in L will lead to an increase in ζ,which
means that the relative wage ω goes down,Thus Home utility will decrease
and Foreign utility will increase,The reason is an unfavorable shift in the
terms-of-trade for Home,which results from the fact that at an unchanged
relative wage Home supply increases,but world demand does not change.
Thus to eliminate the inequilibrium ω must decrease,which will lead to an
increased demand for Home goods and to an increased range of goods produce
there,Notice that although Foreign,lost” some marginal goods to Home,
it is nevertheless better off with the change,In this model,small is indeed
beautiful!
The next change we consider is in technology,In particular,let us assume
that Foreign unit labor requirement is λa?(z) for any z,and there is a decrease
in the parameter λ,Our task is a bit complicated now,since indirect utility
now depends directly on λ,To see more clearly,let us still use the notation
A(ζ) = a?(ζ)/a(ζ),and let B(ζ) = ν(ζ)/[1? ν(ζ)],Using the equilibrium
condition and the,hat” notation,we can easily show that
ζ =?λ
epsilon1B?epsilon1A,
where epsilon1i is the elasticity of the particular function,and we know that epsilon1A < 0
and epsilon1B > 0,We also have that
ω =?λ + epsilon1A?ζ = epsilon1Bepsilon1
B?epsilon1A
λ,
which means that?λ <?ω < 0,Thus an improvement in Foreign’s technology
will lead to a lower relative wage in Home,but the percentage decrease will
be smaller than the percentage drop in λ,This means that Foreign will gain
both because of increased efficiency and a higher relative wage,Home will
also gain,because its relative wage decreases by less than the increase in
foreign efficiency,and thus its purchasing power in terms of foreign goods
increases (whereas it does not change in its own goods).
3Note that v depends on ω indirectly through ζ,but that indirect derivative will be
zero,since vζ = 0.
CHAPTER 6,SIMPLE TRADE MODELS 36
A final change we study is a convergence in technology between Home and
Foreign,Suppose initially Home had the higher wage rate,that is ω > 1,We
compare this with complete convergence,when ω = A(z) ≡ 1,We can show
that such a change results in a loss for Home and in a gain for Foreign,Notice
that when technologies are the same,there is no reason to trade,This means
that the real wage in Home in terms of any good is given by 1/a(z),On the
other hand,when Foreign had an inferior tecnology,real wage in terms of an
imported good was ω/a?(z) > 1/a(z) (since it was more efficient to produce
the good in Foreign) and real wage in terms of an exported good was 1/a(z).
Thus Home’s real wage was higher in the original situation,Foreign,on the
other hand,enjoys the productivity gain for goods produced there and the
terms-of-trade gain which is just the opposite of Home’s loss,The reason
for such stark result is that such a convergence must be biased towards the
import competing sector in Foreign,since it had a larger technological gap
in those goods (almost by definition),And as we saw earlier,a growth in the
import competing sector must hurt the other country.
6.3 The specific factors model – income dis-
tribution
The specific factor model can be thought of as another attempt to make the
Ricardian model suitable for comparative statics,since labor has a decreasing
marginal product in each use,The most plausible reason for this is the
existence for another factor that is,specific” to a sector and is immobile
between different uses,One can think of buildings and machinery that cannot
be converted for use in a different industry,Thus we will assume the existence
of two sectors and three factors,One (labor) is mobile between the sectors,
but the other two are not,The endowments of the factors in the Home
country are L,K1 and K2,respectively.
On the production side we have various conditions,Let Fi(Li,Ki) be the
production function in sector i,which we can also write as KiFi(Li/Ki,1) =
Kifi(li),We assume the standard properties and the Inada conditions for Fi,
which means that fi is increasing and concave in li and limli=0 fprimei(li) = ∞.
Let w be the wage rate and pii be the return to specific factor i,From profit
maximization we have that
p1fprime1(l1) = w = p2fprime2(l2),
CHAPTER 6,SIMPLE TRADE MODELS 37
and pi
i
pi = [fi(li)?lif
prime
i(li)].
An important result for future reference is that
pii
pi
prime (l
i) =?lifprimeprimei (li) > 0
The factor market clearing condition is simply K1l1 +K2l2 = L,We will use
the model to look at changes in the factor returns in response to a change in
L,Ki and the relative price,For the latter we normalize the price of good 1
and write p = p2/p1.
Let us start with a change in p,We can see that there must be a re-
allocation of labor from sector 1 to sector 2,which leads to an increase in
w and to a decrease in w/p (since marginal value products equalize in the
two sectors),We can also see that pi1 must fall,because it is an increasing
function of l1,On the other hand,pi2/p must rise,since it is an increasing
function of l2,Thus real wage rises in terms of one good and falls in terms of
the other,whereas the return of specific factor 1 (2) decreases (increases) in
terms of both goods,It is also apparent that the output of good 1 increases
and the output of good 2 decreases,If the price increase comes from a tariff,
we have the intuitive result that capitalists in the protected sector will be
better off (except,of course,when Metzler’s paradox arises).
Now let us take a look at a change in the endowment of the mobile factor,
L,To accommodate the extra labor,the marginal value product of labor
must fall in both sectors,so that w falls,On the other hand,both l1 and
l2 will rise,so the returns to both specific factors must increase,Thus labor
is unambiguously hurt and the specific factors are unambiguously better off
with the change,When the amount of a specific factor changes (say in sector
1),the marginal product of labor there rises,To restore equilibrium,labor
has to flow into sector 1,Suppose that this process continues until the original
labor/capital ratio l1 is restored,But since l2 have fallen,the value marginal
products are not equalized across sectors,Thus l1 will also decrease,and
the wage rate will rise,This implies that pii must fall for i = 1,2,Thus
an increase in the endowment of a specific factor benefits the owners of the
mobile factor,and hurts the owners of both specific factors.
Before we finish,let us look at the output changes in the various cases.
When p increases,production in sector 1 increases and production in sector
2 falls,but by a smaller proportion than the change in p,If L rises,both
CHAPTER 6,SIMPLE TRADE MODELS 38
outputs increase,but by less than L,Finally,an increase in a specific factor
endowment leads to a smaller proportional increase in the production of that
sector and a fall in the production of the other sector,Thus the magnification
result we saw in the Heckscher-Ohlin model does not arise here.
Chapter 7
Empirical strategies
Im the notes I do not want to discuss specific results from various papers,you
are referred for those to Feenstra,Helpman (1998) and Harrigan (2001),The
latter two are survey articles that cite many original papers and summarize
their results,Instead,I will focus on the empirical strategies researchers have
used to test the theory of comparative advantage,As a first note,let me quote
Harrigan by saying that almost no empirical work has tested the doctrine of
comparative advantage directly,This is,to some extent,inevitable,since we
rarely (if ever) observe autarchy prices together with net trade vectors,There
are natural experiments that can be used,one notable example is Japan 150
years ago (Bernhofen and Brown,see Feenstra).
This means that empirical work usually tests not comparative advantage,
but explanations for it,Various restrictions are made in order to get rela-
tionships between measurable variables,By far the most common model to
be tested is the Heckscher-Ohlin model,and its various generalizations,At
its most restrictive form,it assumes two goods and factors,identical tech-
nologies and identical homothetic preferences to predict trade patterns,A
more general result links factor endowments and the factor content of trade,
as long as we assume FPE,Since recent work focuses on the Vanek equation,
let us thus first look at the testable implications with FPE and then see what
possibilities and problems arise without it.
39
CHAPTER 7,EMPIRICAL STRATEGIES 40
7.1 The basic equation
We saw earlier that in the case of identical technologies,identical homothetic
preferences and FPE the factor content of trade is uniquely determined by
endowments,In particular,we have
tkv = skv?vk,
with the previously introduced notation,This is an equation in which we can
measure all variables (there are no parameters to estimate),so we cannot use
regression analysis to test it,Instead,one can compute a rank-correlation
measure for the left- and right-hand sides,Alternatively,we can check
whether the sign of the two sides coincides,Although these are fairly weak
tests,the evidence is not very supportive (see Bowen-Leamer-Sveikauskas).
In particular,there are three ways how the data fails to support the basic
factor abundance hypotheses (see Trefler 1993,1995),These are:
The measures of factor content are compressed towards zero.
Poor countries have systematically larger values of tkv,For rich countries
the opposite is true.
Poor countries tend to be abundant in more factors than rich countries.
Let us see what people have tried to do to recouncile the model with these
empirical failures.
7.2 Extensions with FPE
One way to extend the previous equation is by allowing differences in tech-
nology,If these are factor augmenting,countries with different technologies
that have the same amount physical units of a factor can have different en-
dowments in efficiency units,Then FPE refers not to the price of physical,
but of efficiency units and we can use the Vanek argument,as long as en-
dowments are not too far away,The problem in this case,however,is that
we only observe physical units,To see why this is a problem,for each factor
write endowments in efficiency units as piki vki,where piki is the technology co-
efficient for factor i in country k relative to some benchmark country,Then
CHAPTER 7,EMPIRICAL STRATEGIES 41
the factor content of trade for factor i can be written as
tkvi = sk
summationdisplay
l
pilivli?piki vki,
The problem is that we do not know the coefficients piki,and we do not
have degrees of freedom to estimate them,There is a different technology
parameter for each country-factor pair,and this is the number of equations we
have,This means that piki can be calculated as a residual,and we can always
explain the pattern of trade with the calculated technological differences.
Of course the calculated technology parameters have to be,plausible”.
If,for example,Albanian technology parameters would come out consistently
greater than German ones,we would be suspect our results,Still,we need
some benchmark to which we measure the plausibility of the parameters.
One possibility is to assume that piki = pik,that is technology is country,but
not factor specific,Then if country 1 is the benchmark country,we can easily
show that pik = wki /w1i for any i,and we can use the observable factor price
ratios as measures of technology parameters,Alternatively,we can assume
that there exist groups of countries,such that within a group all countries
share the same technology,but technology is different across groups,An
obvious example to such a group is the OECD,another could be NICs,while
a third group could be LDCs,With this restriction we gain degrees of freedom
to estimate the pi’s and see if they lead to plausible numbers.
To conclude,in the presence of FPE we can test the factor abundance
using the Vanek result for the factor content of trade,preferably augmented
by some restricted form of technological differences,One can also relax the
assumption of identical homothetic preferences and allow for home bias in
consumption,although in the classical theory of trade there is no really good
reason why such bias would occur,The results in Helpman or Harrigan show
that such an augmented model performs reasonably well,but some important
problems remain.
7.3 Results without FPE
There is some independent evidence that FPE does not hold in many cases.
The efficiency unit argument is a nice way to introduce technological differ-
ences,but it also makes FPE less likely,Remember that for FPE we need
CHAPTER 7,EMPIRICAL STRATEGIES 42
that factor endowments are not very different across countries,This is es-
pecially hard for labor endowments,which might be similar in the physical
sense but very different if interpreted as human capital,Thus we need to
investigate the implications of a failure of FPE.
The main problem with estimating the factor content of trade when FPE
does not hold is that techniques of production differ across countries in a
way that cannot be fixed by introducing efficiency units,Even with identical
technologies,different factor prices lead to different input vectors,as we
indicated it when we wrote the unit input coefficient matrix as a(w),In his
famous work leading to the Leontief-paradox1,Leontief used US input-output
coefficients to measure the factor content of US trade,This is only correct
when there is FPE,otherwise foreign technology will not be the same and
the calculated factor content will be biased.
An illustration of the problem can be found in Helpman,Suppose that
there are two goods,two countries and two factors but no FPE,Let country
A have a higher capital-labor ratio,and hence a higher wage rate and a
lower rental rate on capital,Then we know that aAKi > aBKi and aALi < aBLi
for i = 1,2,Let good 2 be the more capital intensive,and suppose that
both countries specialize,country A in good 2 and country B good 1,Given
identical homothetic preferences,the net import vector of country A is given
by
mA =
bracketleftbigg sAxB
1
sBxA2
bracketrightbigg
.
If we use A’s technology matrix to calculate the factor content of its net
import vector,we get
tAv =
bracketleftbigg sAaA
K1x
B
1?s
BaA
K2x
A
2
sAaAL1xB1?sBaAL2xA2
bracketrightbigg
=
aAK1
aBK1s
AKB?sBKA
aAL1
aBL1s
ALB?sBLA
.
On the other hand,
sAv?vA =
bracketleftbigg sA(KA + KB)?KA
sA(LA + LB)?LA
bracketrightbigg
=
bracketleftbigg sAKB?sBKA
sALB?sBLA
bracketrightbigg
1Leontief tried to measure the factor content of US trade after WWII,His results
indicated that the US is a net exporter of labor and an importer of capital,Since at
that time the US had the most advanced economy,Leontief considered his results as
paradoxical.
CHAPTER 7,EMPIRICAL STRATEGIES 43
gives the,real” factor content of trade measured by the differences in factor
endowments.
Thus the factor content of A’s net import vector will be compressed to-
wards zero when A’s technology is used,This can explain the first empirical
regularity called missing trade,because measured factor content will be bi-
ased downward,The reason for this is that when we use A’s techniques to
calculate imports from B,we understate the amount of labor and overstate
the amount of capital used in producing good 1,To get the correct measure,
we need to use the techniques for the exporter for any good.
It is possible to get a weak,but testable restriction on the pattern of
bilateral trade without FPE,Assume still that technology is the same in the
two countries,that is,the revenue function has the same form,Let xk be
the production vector of country k and tkl be the factor content of imports
from country l to k using the exporter’s techniques a(wl),The main insight
is that given the factor endowment vk + tkl,the production vector xk + mkl
would be feasible,It is not necessarily optimal,however,since exploiting its
differing factor prices country k can use different techniques to achieve higher
revenue:
p(xk + mkl) ≤ r(p,vk + tkl)
≤ r(p,vk) + rv(p,vk)tkl
= pxk + wktkl,
where the second inequality follows from the concavity of r in v,Comparing
the first and the last lines yields pmkl ≤ wktkl,In the exporting country,on
the other hand,the value of exports equals the payment of the factors used
in producing them,therefore pmkl = wltkl,Combining these two yields
(wk?wl)tkl ≥ 0.
Repeating the exercise for country l using the techniques of country k,we
also get that
(wl?wk)tlk ≥ 0,
and that
(wk?wl)(tkl?tlk) ≥ 0.
These inequalities imply that a country will on average export a factor whose
price is relatively lower there,They are weak in a sense that they only predict
CHAPTER 7,EMPIRICAL STRATEGIES 44
correlations,but this is all we can hope as a general result,You can check that
in the case of two factors the correlation actually becomes a clear prediction,
but just as with many results this does not generalize.
Part II
Increasing returns and the
“New Trade Theory”
45
46
The,New Trade Theory” is not so new now,since it originated around
1980 and was developed pretty much by the middle of that decade,The
main reasons for its emergence were both theoretical and empirical,On the
theoretical side,there was a long tradition in trade theory that emphasized
increasing returns and specialization as causes of trade,but economists could
not handle imperfect competition – that is a natural consequence of increas-
ing returns – very well until the late 1970’s,With advances in game theory
and industrial organization,the necessary modeling tools to analyze imper-
fect competition were available,On the empirical front,it was observed
that much of trade is in varieties of similar goods,which is known as,intra-
industry” trade,Also,most of world trade are between seemingly similar
nations,that is the within OECD,The theory of comparative advantage,on
the other hand,predicts that trade should be greatest between countries who
have very different endowments.
Thus theorists started to think about models of increasing returns and
imperfect competition,As folk wisdom says,however,there are many ways
to be imperfectly competitive,and as of today there is no generally accepted
way to model market power,Thus attention turned to specific models that
highlight one particular aspect of reality,in the hope that a few of these
simple models would give us enough intuition about the general issues in-
volved,We will study three types of such models,increasing returns due to
externalities,oligopolic markets and monopolistic competition,Let us start
with the first.
Chapter 8
External economies of scale
This approach has the longest tradition to model increasing returns,Strictly
speaking it does not quite fit into the,New Trade Theory“,because it avoids
the question of market structure,The attraction of externalities is that firms
still perceive themselves as price takers,and hence perfect competition is
preserved,This modeling technique is somewhat out of fashion,but it still
deserves some mention partly because of its earlier popularity and partly
because it is still the best way to study some important issues.
8.1 Gains from trade
We will start with describing the autarchy and trade equilibria and we will
take a look at the possible gains from trade,To simplify a bit,we do not
allow for joint production,thus production of each good can be described by
a production function:
xj = fj(vj,ξ).
The notation is as before,and ξ is the vector of external effects,Examples
include overall production in the sector,or the average level of human cap-
ital in the country,or world population,There are two issues that will be
important,first,externalities can be local or global in scope,and second,the
variables that affect the production function might be different in free trade
than in autarchy,We will return to these issues later.
Since perfect competition still applies,we can represent the total value
of production by the revenue function,just as before,Now it can be defined
47
CHAPTER 8,EXTERNAL ECONOMIES OF SCALE 48
as follows:
r(p,v,ξ) = maxx
braceleftBiggsummationdisplay
j
pjfj(vj,ξ)|
summationdisplay
j
vj ≤ v
bracerightBigg
.
Given the consumption side,the equilibrium in autarchy and free trade can
be defined exactly as before,The only difference is that ξ must be consistent
with equilibrium,If,say,ξ is the vector (0,0,...,xj,...,0) for industry j,then
the xj in ξ must equal actual production.
An important question concerns the gains of trade,It is no longer true
that a country necessarily gains from trade,and we will see an example for
the opposite,For now we give a sufficient condition for positive gains,which
also reveals the extra channel that operates now,The sufficient condition is
summationdisplay
j
pjfj(vja,ξ) ≥
summationdisplay
j
pjfj(vja,ξa),
where vja is the factor input vector in industry j in autarchy,To see that the
condition is sufficient,we show that with it the autarchy production (and
consumption) vector is affordable at free trade:
pxa ≤
summationdisplay
j
pjfj(vja,ξ) (from our condition)
≤
summationdisplay
j
pjfj(vj,ξ) (vja is feasible)
= r(p,v,ξ).
The condition states that productivity is greater at the free trade value
of the external effects than at the autarchy value,It reveals an additional
source of gain in trade,which can either come from larger scale or exposure to
foreign external economies,Note that even if the condition does not hold,the
country can gains from trade through the traditional channels (a possibility
to trade at a price different from the autarchy one),Thus trade is harmful
only if the loss due to less favorable externalities outweigh the static efficiency
gains,Let us see an example that illustrates these possibilities.
8.2 An example
Let us take a simple Ricardian model with two goods and two countries,We
choose units in such a way that the unit labor requirement is one in both
CHAPTER 8,EXTERNAL ECONOMIES OF SCALE 49
sectors,Productivity in sector 1 also depends on aggregate production there,
so that the production functions can be written as
x1 = ˉx1/21 L1
x2 = L2,
where ˉx1 is aggregate production in sector 1,Preferences take the Cobb-
Douglas form,with an α < 1/2 fraction of consumer spending falling on
good 1,We assume that technology and tastes are the same for the two
countries.
There is a unique autarchy equilibrium,which can be derived as follows.
The wage rate must equal unity,given the zero profit condition in sector
2,Then the relative price of good 1,pa,is given by pa = 1/L1 (since in
equilibrium,ˉx1 = x1),Using the two demand conditions,we have that
(1?α)L = L2
αL = L1.
Thus the autarchy equilibrium is given by the following:
wa = 1
pa = 1/(αL)
xa1 = (αL)2
xa2 = (1?α)L
For future reference,note that in the homothetic case utility is proportional
to the real wage,where the price deflator is the,true price index”,and
depends on the precise form of the utility function,In this case it is just
pα,so that indirect utility is proportional to 1/p,It is noteworthy that here
utility increases with the size of the country,due to the external economies
of scale.
Now we turn to free trade,It is easy to see that no trade is an equilibrium,
since at the autarchy production and consumption levels the relative price is
the same in the two countries,Since relative endowments and technologies
are the same,this is not surprising,What is interesting,however,is that
there are two other equilibria where the production of good 1 agglomerates
in one country,Since they are completely symmetric,let us look at the case
CHAPTER 8,EXTERNAL ECONOMIES OF SCALE 50
when this is the home country,Our first observation is that good 2 must
be produced in both countries,Suppose this is not the case,and Foreign
specializes in good 2,It then has a wage rate of one,and to rule out arbitrage
the wage must be at least one at Home,But then world demand for good 2
is (1?α)(L+wL) > 2(1?α)L > L,since α < 1/2,Thus country 2 cannot
provide good 2 alone.
This means that we have factor price equalization (FPE),since both
countries produce good 2 and the zero profit conditions for sector 2 tie down
wages at unity,Using the demand conditions,the production function and
the consistency condition for the externality we have:
w = 1 and W = 1
p = 1/(2αL) and (= P)
x1 = (2αL)2 and X1 = 0
x2 = (1?2α)L and X2 = L.
The important thing is that both countries gain from trade (compare
the two prices),but our sufficient condition above does not hold for the
foreign country,The productivity in sector 1 in Foreign declines (to zero
in this case),but this loss is compensated by the price drop that results
from the larger scale of production in Home,Moreover,the utility levels are
the same in the two countries,despite the fact that Foreign lost the,high
productivity” industry,This points to the fact that it is welfare that matters,
and not production patterns,We will return to this issue when we look at
endogenous growth and trade.
You can easily verify that as long as there is FPE the same conclusion
applies,regardless of the size of the countries,It is possible,however,to
construct an equilibrium where Foreign is worse off with trade,For that we
need that Home specializes in good 1 and Home is smaller than Foreign,To
be more precise,we have to step outside of the FPE region,which requires
that
L < α1?αL?,
where L? is now the foreign labor force and population,You can verify that
CHAPTER 8,EXTERNAL ECONOMIES OF SCALE 51
the following constitutes the equilibrium allocation:
w = α1?αL
L > 1 and W = 1
p = wL and (= P)
x1 = L2 and X1 = 0
x2 = 0 and X2 = L?.
Home is better off than in autarchy,since its real wage is higher in terms
of both goods,For Foreign,real wage in terms of good 2 does not change,
and in terms of good 1 it is given by 1/p,If the free trade price is higher
than the autarchy price in Foreign,it will be worse off,This is a condition
on the relative sizes of the two countries:
L
L? <
α√
1?α.
The reason for this result is that the IRS industry ends up in the,wrong”
country,which is too small to sustain it at an efficient scale,Since Home
cannot provide world demand in good 1 at p = 1,the price (and hence the
wage rate in Home) has to go up,leading to a deterioration in Foreign’s
terms-of-trade,Notice,however,that there is an FPE equilibrium with the
same parameter values when good 1 agglomerates in Foreign,The problem
is that there is nothing that guarantees the efficient equilibrium to emerge.
8.3 Factor price equalization
Following HK we will concentrate on the case of industry-,country- and
output-specific externalities,That is,we have 1,..,I industries with external
economies of scale,and I +1,..,n c.r.s,industries,The production function
for the former is given by
xj = fj(vj,ˉxj),j = 1..I,
where ˉxj is industry output in sector j,We will not write down the equi-
librium equations (see HK),but it is easy to see that the i.r.s,industries
have to agglomerate in one country,The reason for this is that to reproduce
the integrated equilibrium production scale in those industries has to equal
CHAPTER 8,EXTERNAL ECONOMIES OF SCALE 52
world demand,which is only possible when only one country produces an
i.r.s,good,The c.r.s industries can be distributed freely across countries,
given that production must be non-negative,Thus the FPE region1 is given
by the following set:
Ψ =
braceleftBig
v|v = a(?w)x,j = 0..I,xj ∈ {0,?xj}andj = I + 1,..,n,xj ∈ [0,?xj]
bracerightBig
.
HK has nice pictures with three goods and two factor,One interesting
consequence of i.r.s,is that the diagonal need not belong to the FPE set,if
demand for the i.r.s,goods is large,In this case similarities in factor endow-
ments do not make FPE more likely,in fact you need some dissimilarities
to get it,To give a simple example,recall the Ricardian model above with
equal country sizes,but assume now that α > 1/2,FPE (or more precisely,
IWE) requires that the production of good 1 is concentrated in one country.
But this requires that one country satisfies world demand,yielding
2αL < L? α ≤ 1/2,
which does not hold now,Thus IWE cannot be reproduced with identical
country sizes,and you need enough dissimilarity in endowments to be in the
FPE set.
Second,in order to get FPE with positive measure,we need at least as
many c.r.s goods as factors,You can see this from the definition of the set,
or intuitively from the fact that trade only in i.r.s,goods will not equalize
factor prices even if there are enough equations,because input productivity
also depends on scale,Third,even though the i.r.s sector is the most capital
intensive in the HK picture,the country which is endowed with more capital
need not have it,The reason again is scale,in order to be able to produce
the integrated equilibrium quantity,the country also has to be large,On the
other hand,as long as tastes are homothetic,the Vanek chain is still valid,
so that the factor content of trade is determined.
Forth,there is beneficial trade in goods (if not in factor content) even
if endowments are identical,due to specialization (remember our earlier ex-
ample),Fifth,the FPE equilibrium is not necessarily unique - there might
1To be more precise,this is the region where the IWE can be reproduced,The Ricardian
example showed that there can be FPE without reproducing the IWE,We are interested
in the IWE set,which for convenience we still call the FPE set,because there all the gains
from integration can be exhausted through trade in goods.
CHAPTER 8,EXTERNAL ECONOMIES OF SCALE 53
be an equilibrium with no FPE for the same parameter values,Again,see
earlier example,HK has pictures,too,Sixth,there might be FPE in an
equilibrium that does not reproduce the IWE,in our example no trade was
an equilibrium with equal wages,Thus belonging to Ψ is necessary but not
sufficient for FPE,For more on these issues,see W,Ethier:,Decreasing
costs in international trade and Frank Graham’s argument for protection”,
Econometrica,Sept,1982.
Chapter 9
Oligopoly,dumping and
strategic trade
As you know very well,the theory of oligopoly is far from being complete.
There are a variety of models even in a static framework,and once you allow
for dynamics,things are hopelessly diffuse,For this reason I do not find the
“general” treatment in HK particularly useful or interesting,Instead,we
will look at two issues that arise only in an oligopolistic setting,One is the
“reciprocal dumping”,which provides an additional motive for trade besides
comparative advantage and specialization,It also highlights an additional
channel through which trade can be beneficial,which no other model delivers.
The second issue is,strategic trade policy”,that a government can actively
influence an industry in order to help the domestic firm capture international
rents,The two papers that cover these are,Brander and Krugman,“A
Reciprocal Dumping Model of International Trade”,JIE 1983 and Brander
and Spencer,“Export Subsidies and Market Share Rivalry”,JIE 1985.
9.1 Reciprocal dumping
Suppose there are two identical countries,Home and Foreign,Both has one
firm producing an identical good with labor,The firms in the two countries
view themselves as Cournot competitors,and there is an,iceberg” trans-
portation cost of τ > 1,Marginal cost is constant and equals c for do-
mestic production,while the iceberg form of transportation costs mean that
marginal cost of export is cτ,There is also a fixed cost F that is independent
54
CHAPTER 9,OLIGOPOLY,DUMPING AND STRATEGIC TRADE 55
of production,Let x indicate the domestic and x? indicate export production
of the home firm,and let Foreign’s export and domestic production be y and
y?,Inverse demand is given by p(z),where z is total consumption of the
good in a country.
Since the problem is symmetric,it is enough to concentrate on the Home
market,The total profits of the two firms are given as
pi = xp(x + y) + x?p(x? + y?)?cx?cτxF
pi? = yp(x + y) + y?p(x? + y?)?cycτy?F,
and the first-order conditions for Home production are
p + xpprime = c
p + ypprime = τc.
The second-order conditions are also need to be satisfied,In fact,we will
need something that is a bit stronger:
xpprimeprime + pprime < 0
ypprimeprime + pprime < 0.
The meaning of these conditions is that marginal revenue is decreasing in the
other firm’s output,and they guarantee that the equilibrium is stable.
Let σ = y/(x + y) and let epsilon1 be the absolute value of the elasticity of
demand,Then the FOC’s can be rewritten as
p = epsilon1cepsilon1 + σ?1
p = epsilon1τcepsilon1?σ,
which yield
p = epsilon1c(1 + τ)2epsilon1?1
σ = τ?epsilon1(τ?1)1 + τ
Notice that the equilibrium price has to be non-negative,which from
above implies that epsilon1 > max{σ,1? σ} > 1/2,Then it is easy to check
that σ < 1/2,i.e,the Foreign company has a lower market share,This is
CHAPTER 9,OLIGOPOLY,DUMPING AND STRATEGIC TRADE 56
the standard Cournot result that the company with a higher marginal cost
will have lower sales,More interestingly,as long as epsilon1 < τ/(τ? 1),σ will
be positive,Thus if transport costs are not very large,the foreign firm will
export to the home market,Since the analysis of Foreign is symmetric,in this
case there will be two-way trade in an identical product,Thus here trade
is due to neither comparative advantage nor specialization,but to market
segmentation in an oligopolistic industry,The reason why this trade is called
“dumping” is that each firm has a lower markup on its export than on its
home sales,due to the presence of transportation costs,This is true as long
as τ > 1 and transport costs are not prohibitive.
Clearly such trade is not Pareto optimal,since transporting goods wastes
resource,It is possible,however,that trade is better than autarchy,because
the latter has monopoly distortions as well,In particular,trade has a pro-
competitive effect,which might be larger than the loss in transport,In fact,
it is easy to see that when transport costs are close to zero,trade must be
beneficial,The reason is that in such case the loss from transport costs will
be second-order,but there will be a discrete drop in prices due to increased
competition.
One can also show that when utility is quasilinear in z,such that u =
v(z) + k,a slight drop in τ from the prohibitive level will decrease welfare.
With quasilinear utility aggregate welfare can be written as
W = 2[u(z)?cz?c(τ?1)y?F],
and thus dW
dτ = 2[(p?c)
dz
dτ?cy?c(τ?1)
dy
dτ].
But at the prohibitive level y = 0 and p = cτ,so that
dW
dτ
vextendsinglevextendsingle
vextendsinglevextendsingleτ= epsilon1
epsilon1?1 = 2(p?c)
dx
dτ > 0,
Thus a small decrease in τ that enables two-way trade is welfare decreasing.
An interesting extension is when there is free entry and profits are driven
to zero,In this case welfare only depends on consumer surplus,which in turn
rises when the price goes down,Thus trade will be beneficial if and only if
trade leads to a drop in prices,Before trade,the first-order and zero-profit
conditions for each firm are
xpprime + p = c
(p?c)x = F
CHAPTER 9,OLIGOPOLY,DUMPING AND STRATEGIC TRADE 57
Assume that after trade,the price rises,From the FOC above,it is easy to
calculate that dx
dp =
xpprimeprime?pprime
(pprime)2,
given that x =?(p?c)/pprime and dz/dp = 1/pprime,Profits now are given by
pi = (p?c)x?F + (p?cτ)x?,
which will be positive since the first term is positive (both x and p are higher
than in autarchy) and the second is also non-negative if trade takes place.
But this is a contradiction,since profits are zero by the free-entry condition.
Thus trade must lower prices and thus raise welfare,The reason is the
presence of fixed costs,although trade will lower domestic sales x,x + x?
rises and thus average cost falls,Without fixed costs trade would be neutral,
since free entry would result in marginal cost pricing even without it.
9.2 Strategic trade policy
We will look at a very simple model that captures the essence of the argument.
Suppose there are two countries,each of which has two sectors,One sector
produces the numeraire with perfect competition,The other sector has one
firm in each country,who compete in a Cournot fashion on a third market.
This assumption is not essential,but it simplifies welfare analysis,There
are two stages,in the first the government chooses whether to subsidize the
oligopolistic firm and in the second the two companies compete,There is
one factor,labor,and we choose units such that the unit labor requirement
is one in both sectors,so that the wage rate equals unity.
We solve the model by backward induction,Assume that the government
chooses a per unit subsidy s,The first-order condition for the Home company
is then
xpprime + p = c?s.
The government maximizes profit minus the amount of the subsidy,or W =
xp(x + x?)? (c? s)x? sx = xp? cx,Instead of calculating the optimal
subsidy,let us show that a small subsidy increases welfare,This is easily
done as follows:
dW
ds
vextendsinglevextendsingle
vextendsinglevextendsingles=0 = (p + xpprime?c)dx
ds + xp
primedx
ds = xp
primedx
ds > 0,
CHAPTER 9,OLIGOPOLY,DUMPING AND STRATEGIC TRADE 58
since dx?/ds < 0 (in the Cournot-model,companies with smaller marginal
cost produce more).
The reason for this result is that production in Home and Foreign are
strategic substitutes,When the home company increases production due to
the subsidy,the foreign firm will contract,Thus the home firm gains market
share to the expense of the other,and hence its profits increase by more
than the amount of the subsidy,Hence,strategic trade policy” increases the
welfare of Home from its free trade level.
Note that the results are quite specific to the market structure,If the
companies are strategic complements (such as in the differentiated Bertrand
case),export taxes are optimal.1 Second,we assumed that the foreign coun-
try does not retaliate,If it does,market shares will not change and the
subsidy will just be a transfer from taxpayers to shareholders,Third,if
home consumers demand the oligopolistic good,consumer surplus must also
be taken into account,Finally,the fact that strategic trade is a theoretical
possibility does not mean that it is empirically feasible,In fact,even in the
case of Airbus and Boing (which confirm pretty well to the assumptions) the
calibrations are mostly pessimistic.
1This is demonstrated by Grossman and Eaton 1986 (QJE),More generally,they show
that the Brander-Spencer result is not robust to changes in market conduct and the optimal
policy depends on the precise nature of oligopolistic behaviour.
Chapter 10
Monopolistic competition
The monopolistic competition is the workhorse of,New Trade Theory”,It
was originally developed by Chamberlain in the 1930’s,but it was made
tractable for mathematical analysis by Dixit and Stiglitz,It is attractive
because increasing returns are internal to the firms,so the problem of multiple
equilibria does not arise (as it did in the externality approach),On the
other hand,by assuming firms are very small,we don’t have to worry about
strategic interactions between companies that make any general treatment of
oligopolies impossible,On the other hand,a tractable model of monopolistic
competition is also quite special,and relies heavily on specific functional
forms,Thus we should view it as a complement rather than a substitute for
the other models of economies of scale.
10.1 Basics
10.1.1 Consumption
We assume that there are 2 sectors in the economy,Food and Manufacturing.
The food sector produces a homogenous product (which we choose as the nu-
meraire),but manufacturing goods are differentiated,There are N different
varieties available,and consumers view them as imperfect – but symmetric
– substitutes,To be more precise,agents have the following utility function:
u = u(y,C),
59
CHAPTER 10,MONOPOLISTIC COMPETITION 60
where u is a homothetic function,y is consumption of food and C is a man-
ufacturing aggregate,given by
C =
bracketleftbiggintegraldisplay N
0
c(i)1?1/σ di
bracketrightbigg σσ?1
.
Consumption of each variety is given by c(i),and we assume that there are
a continuum of such goods,It is easy to check that this specification means
consumers like variety,just set c(i) = c and note that the resulting expression
is increasing in N.
Let the income of the representative agent be E,and her spending on
manufacturing goods Em,Since u is homothetic,we can do the optimization
problem in two steps,First,given spending on manufacturing,we can solve
for the demand functions of an individual variety,Second,we can determine
spending on food and manufacturing goods,For the first step,you can check
that the demand function for variety i can be written as
c(i) = p(i)
σ
P1?σ Em,
where P is the true price index for manufacturing,and it is given by
P1?σ =
integraldisplay N
0
p(i)1?σ di.
The important fact is that a price change in each variety has an infinitesimal
effect on the price index,the elasticity of demand is simply σ,We will use
this many times later.
Now we turn to the problem of finding the optimal manufacturing spend-
ing Em,Since u is homothetic,the share of spending that falls on manufac-
turing depends only on the price index P,We assume that all varieties fetch
the same price p (which will be the case in equilibrium),then we can write
P1?σ = Np1?σ.
Thus manufacturing spending can be written as
Em = α(pN 11?σ)E,
where the functional form of α(·) depends on the specification of u.
CHAPTER 10,MONOPOLISTIC COMPETITION 61
10.1.2 Production
The food sector is c.r.s and perfectly competitive,with a unit cost function
of c(w,r) (w is wage,r is rental rate for capital),The zero profit condition
requires that
c(w,r) ≥ 1,
with equality if food is produced,For the manufacturing factor we assume
that each variety has the same cost function,C(w,r,x),where x is the pro-
duction level of the variety,We also assume that the potential number of
varieties is large,so that a firm can always enter the market with a new prod-
uct,Then it is easy to see that each differentiated good will be produced by
only one firm,who can price the good as a monopolist (taking prices of other
goods as given),Since the elasticity of demand for a good is σ,the pricing
equation becomes
p(i)
bracketleftbigg
1? 1σ
bracketrightbigg
= Cx(i)[w,r,x(i)].
You can see that price will be marked up above marginal cost by a constant,
which depends only on the elasticity of demand,σ.
We also assume free entry in manufacturing,which means that if there
are positive profits a firm will enter with a new variety,This will lead to the
elimination of pure profits,so that p(i)x(i) = C[w,r,x(i)],Using the pricing
equation from above,we get that
Cx(w,r,x)x
C(w,r,x) = 1?
1
σ.
Notice that I dropped the index i for varieties,because the zero profit con-
dition pins down production scale at the same level for each good (assuming
there is a unique solution to the equation),This also means that the price
charged will be the same for any variety,so our assumption at the end of the
consumption chapter was justified.
10.2 The trading equilibrium
10.2.1 The integrated equilibrium
We describe the integrated world equilibrium,which is basically the equilib-
rium in a closed economy,Many of the equations we already have,except for
CHAPTER 10,MONOPOLISTIC COMPETITION 62
the factor market clearing conditions,For the food sector demand for labor
and capital are given by
Lf = ycw(w,r)
Kf = ycr(w,r),
and for the manufacturing sector we have
Lm = NCw(w,r,x)
Km = NCr(w,r,x).
With these the equilibrium can be summarized by the following system
of equations:
1 = c(w,r)
p = C(w,r,x)x
1? 1σ = Cx(w,r,x)xC(w,r,x)
L = ycw(w,r) + NCw(w,r,x)
K = ycr(w,r) + NCr(w,r,x)
α(pN 11?σ) = pxNy + pxN.
We have six equations in the six unknowns,x,N,y,p,w and r,so all the
endogenous variables are determined.
10.2.2 Factor price equalization
Now we distribute the amounts of capital and labor between two countries,
and examine if trade can reproduce the integrated equilibrium,Assuming
both goods are produced,trade will lead to FPE,since the first three equa-
tions above must hold in both countries,They also pin down the scale of
production at the IWE level,Thus the question for FPE boils down to
the question of whether the 4th and 5th equations can be solved for a non-
negative y and N in both countries,Observe that given x,the two equations
are identical to the ones in the c.r.s,case,except that N plays the role of
production in the manufacturing sector,But we assumed that N is a con-
tinuous variable,so you can see that the conditions for FPE in this model
CHAPTER 10,MONOPOLISTIC COMPETITION 63
are identical to that in the c.r.s,one,Thus if endowments are sufficiently
similar enough,we have FPE,because both countries can produce food and
manufactures using the techniques of the integrated world equilibrium.
This means that the sectoral pattern of trade is determined the same way
as before,if food production is relatively more labor intensive,the country
with relatively more labor will export food,The Vanek chain is also intact,
even when there are more goods than factors,so that the factor content of
trade can be predicted from relative factor endowments (we of course need
homothetic preferences),The difference from the c.r.s model is that now we
have intraindustry trade,manufactures will be produced and exported by
both countries,This is the case even if factor endowments are the same,so
that there is no comparative advantage,In that case countries gain from
trade because they specialize in the production of varieties,and free trade
makes it possible to consume more of them.
Without FPE we cannot say much about trade patterns,because with
different factor prices the scale of production in general will be different,The
exception is when the cost function can be written as c(w,r)f(x),which is not
a very realistic assumption (it rules out fixed costs),We can give a sufficient
condition,however,for gains from trade,The condition is as follows:
c(w,r,x)
c(w,r,xa)
parenleftbiggNa
N
parenrightbigg 1
σ?1
≤ 1,
which states that,average” productivity does not decline in the manufac-
turing sector,There are two components in the left-hand side,which reveal
two source of potential gains from trade,First,if the scale of production
increases with trade,countries gain through economies of scale,Second,if
the number of available varieties increases,consumers are better off,The
two can be traded off against each other,and since this is only a sufficient
condition,against the traditional gains from trade,HK shows the proof that
the condition is indeed sufficient and has more on how to relate it to model
parameters.
10.3 Transport costs and the home market
effect
One of the nice things about the Dixit-Stiglitz structure is that it is very
easy to introduce,iceberg” transportation costs into it,We will write down
CHAPTER 10,MONOPOLISTIC COMPETITION 64
a simple model with transportation costs,and see what they imply for spe-
cialization and welfare,We will retain the demand specification from the
previous chapters,and we will specify the utility function as Cobb-Douglas,
so that the share of spending on manufactures equals α,There is only one
factor of production now,labor,The unit labor requirement in food pro-
duction is one,and we again normalize the price of food to one,In the
manufacturing sector the cost function is specified as
c(w,x) = (a + b)wx,
so that there is a fixed and variable component of the labor requirement.
We assume that food can be transported costlessly and that both coun-
tries produce it after trade,This guarantees FPE so that w = W = 1,and
can be ensured by choosing α sufficiently small (why?),Manufactured goods
bear an iceberg cost τ > 1,Then a good whose domestic price is p will
be sold abroad for τp,This means that the elasticity of demand will be σ
regardless of where the good was produced,and the pricing equation can be
written for any good as
p = σσ?1b.
We can choose units of goods arbitrarily,and we do it in such a way that
σ/(σ?1)b = 1,Thus we have that
p = w.
The zero profit condition pins down the unique size of production as we saw
above,In this case the result is
x = aσ,
once we use the condition that relates σ and b.
10.3.1 Autarchy
In this case the country must produce both food and manufactured goods,If
country size is L,demand and hence labor requirement for food is (1?α)L.
Labor demand in manufacturing is given by N(a + bx) = aσN,This must
equal the the part of labor force not devoted to food production,αL,which
gives us
N = αLaσ,
CHAPTER 10,MONOPOLISTIC COMPETITION 65
This closes the model,since we have all endogenous variables,The utility of
a representative agent will be proportional to the real wage,where the price
index is Pα,Since w = 1,utility will simply be a monotonic transformation
of aσP1?σ/α,which is given by (see earlier discussion)
aσP1?σ/α = L.
Thus a larger country will be better off,because it can produce more varieties.
10.3.2 Trade equilibrium
We derive the equilibrium when both countries produce manufactures and
see what the conditions are for it,The difference from autarchy is that goods
markets clear at the world level,so that world demand for food determines
labor devoted to it,In particular,we have that Lf + L?f = (1?α)(L + L?).
Total demand for manufacturing labor is Lm +L?m = aσ(N +N?),which has
to equal α(L + L?),Thus we have that
N + N? = α(L + L
)
aσ,
The final step to close the model is to write down the market clearing
conditions for a manufacturing goods,Notice that although there are N+N?
such goods,there are only two different prices,because all varieties produced
in a country fetch the same price,Thus there will be only two different
market clearing conditions,one for goods produced in the home country and
another for those produce in Foreign,Moreover,by Walras’ Law we can
ignore one of them,so let us write down the condition for Home goods,Two
things will simplify things,first,let us define ρ = τ1?σ and second,let us
introduce n = N/(N +N?),With these and using the condition for the total
number of varieties from above,and using the demand function and the price
index define earlier,we have1
L
n + ρ(1?n) +
ρL?
ρn + 1?n = L + L
.
We can solve this equation for n,and check that the solution lies between
zero and one (since n is a share),We end up with the following:
n = L?ρL
(1?ρ)(L + L?) if ρ ≤
L
L? ≤
1
ρ.
1You must also realize that if demand for an export good is c,τc units must be shipped
in order for c units to arrive,so we must multiply Foreign demand for home goods by τ.
CHAPTER 10,MONOPOLISTIC COMPETITION 66
Thus if the size of the two countries does not differ too much,both will have
a positive share in manufacturing,If not,the smaller one will specialize in
food production,so that either n = 1 or n = 0,Thus the full equilibrium is
given by
n =
0 if LL? < ρ
L?ρL?
(1?ρ)(L+L?) if ρ ≤
L
L? ≤
1
ρ
1 if LL? > 1ρ
.
From the knowledge of n we can calculate everything else,so the equilib-
rium is indeed determined,The interesting thing is that the larger country
will have proportionately more of the manufacturing sector,since it is easily
checked that n > L/L? iff L > L?,This is one version of the so-called home
market effect,which says that a country will export goods for which it has
larger demand,In this case this is true at the sectoral level,the larger coun-
try will be a net exporter of manufactured goods,Transportation costs are
essential for this result,since we need some segmentation of the markets in
order to talk about home and foreign demand for goods (and not just world
demand).
We can calculate indirect utilities to compare trade with autarchy,Since
we still have w = 1,utility will be a monotonic function of the price index,
and we can calculate that
P1?σ ∝ n + ρ(1?n) ∝
ρ(L + L?) if n = 0
(1 + ρ)L if 0 < n < 1
L + L? if n = 1
.
Thus in any equilibrium the larger country will be better off,but both coun-
tries gain from trade.2 This is similar to the conclusion in the externality
case,even if a country completely de-industrializes it will be better off,be-
cause it can enjoy the gains from specialization in the other country,In
this case this means that the number of varieties will increase as an effect
of trade,and the gain will compensate for the loss caused by transportation
costs,Thus again,we have to look at welfare and not specialization patterns
when we evaluate the effect of trade.
2Notice that when n = 0,ρ(L + L?) ≥ L(L + L?)/L? > L.
Chapter 11
The New Economic Geography
Economic geography is the study of the location of economic activity,It has
a long tradition,but there was a dramatic resurgence in the 1990’s,Based
on the monopolistic competition model of trade theory,Paul Krugman and
others developed a set of models that can explain the emergence of agglom-
eration,both in population and in sectoral specialization,These models are
pretty much variations on the same theme,but given their relative simplic-
ity,the results are remarkably complex,The basic question is as follows:
given symmetric locations,is there a feedback mechanism that can lead to
a spontaneous concentration of economic activity? And if yes,what are the
key parameters that predict the emergence of such concentration? Thus the
question is phrased in such a way,that geography actually does not matter
(in a sense that there are no natural differences in,say,access to markets),
but the arising location pattern is explained entirely by endogenous forces.
We will look at two papers,one that predicts population agglomeration be-
cause people are mobile,and another the predicts industrial agglomeration
because people are immobile,The first can be viewed as a model of regional
activity within countries,and the other as a model of international trade.
11.1 A model of agglomeration
There are two types of goods,just as before,food and differentiated manu-
facturing products,The utility function is Cobb-Douglas,with food’s share
in consumption equal to 1? μ,The manufacturing aggregate is the previ-
ous CES,with an elasticity of substitution of σ,There are two regions in a
67
CHAPTER 11,THE NEW ECONOMIC GEOGRAPHY 68
country,with a total population of one,There are two factors of production,
each specific to an industry,peasants and workers,We assume that the
former are immobile,but the latter can move freely between the two regions.
The number of peasants is (1?μ)/2 in each region,and the total number of
workers is μ,This is a normalization,which will lead to simple equilibrium
numbers (see below).
Food will be the numeraire,produced with a c.r.s technology,and we
choose the unit peasant requirement to be one,Thus peasant wage is also
unity,The manufacturing sector is the same as previously,Again we choose
units such that pi = wi,for each region (w is workers’ wage),Then the
zero profit condition pins down the scale of production at x = aσ,Since
workers are the only input in manufacturing,the number of varieties ni can
be calculated from the factor market clearing condition for workers:
ni(a + bx) = Li? ni = Li/(aσ).
Let us write down the market clearing conditions for a given distribu-
tion of workers,Before that,however,let us define the manufacturing price
indexes in the two regions as follows:
P1?σ1 = n1w1?σ1 + n2ρw1?σ2
P1?σ2 = n1ρw1?σ1 + n2w1?σ2,
where ρ = τ1?σ is our measure of transportation costs (we again assume that
food is costlessly tradable),Suppose that regional incomes are given by Yi,
which are defined as
Yi = 1?μ2 + wiLi.
Then the market clearing conditions for goods produced in regions 1 and 2
are given by:
aσ = μw
σ
1 Y1
P1?σ1 +
μρw?σ1 Y2
P1?σ2
aσ = μρw
σ
2 Y1
P1?σ1 +
μw?σ2 Y2
P1?σ2,
We can combine the sets of equations,and simplify them,Let us introduce
a new variable,λ = L1/μ (region 1’s share of workers),Then we have the
CHAPTER 11,THE NEW ECONOMIC GEOGRAPHY 69
following:
wσ1 = Y1λw1?σ
1 + (1?λ)ρw
1?σ
2
+ ρY2λρw1?σ
1 + (1?λ)w
1?σ
2
wσ2 = ρY1λw1?σ
1 + (1?λ)ρw
1?σ
2
+ Y2λρw1?σ
1 + (1?λ)w
1?σ
2
.
If we substitute the equations for regional incomes Yi,we have two equations
for two unknowns (w1,w2),given the distribution of workers,λ,Thus we are
left to see how that distribution is determined.
Instead of fully characterizing the possible equilibria,we ask two more
limited questions,We are interested in two situations,one of full symmetry
(λ = 1/2) and the other is full agglomeration (λ = 1 or λ = 0),The
question we ask is whether these equilibria are stable,and if the answer is
ambiguous,how does it change with parameter values,In particular,we
want to know what happens when the transportation cost parameter τ (and
hence ρ) changes,Since the model is statics,stability is a somewhat heuristic
concept,To check for the stability of the core-periphery equilibrium (when
manufacturing is agglomerated in,say,region 1),we do the following,Note
that in such a case,the wage equation for region 2 defines potential wage,
since there is no manufacturing there,If however,this potential wage gives
a higher real wage than the one in region 1,a firm can,defect” from region
1 to region 2 and attract workers by offering higher real wages,Thus the
core-periphery pattern is stable if
w1
Pμ1 >
w2
Pμ2,
This condition leads to a fairly simple expression,When λ = 1,we can
solve the model analytically for the wage rates,which will then give us the
price indexes,You can check that w1 = 1,wσ2 = ρY1 + 1/ρY2,P1?σ1 = 1 and
P1?σ2 = ρ,Real wage is given by nominal wage divided by the full price index,
which also takes into account food prices,and is given by Pμi,Substituting
for these in the stability condition,we have that the asymmetric equilibrium
is stable if and only if
1 + μ
2 τ
1?σ?μσ + 1?μ
2 τ
σ?1?μσ ≤ 1.
The left-hand side is one at τ = 1 (since with no transportation costs,location
does not matter) and it is decreasing at τ = 1,Thus the core-periphery
CHAPTER 11,THE NEW ECONOMIC GEOGRAPHY 70
pattern is stable for small transportation costs,What happens when we
increase τ? It depends on the other parameter values,When τ → ∞,the
first term goes to zero,The second,however,goes to infinity provided that
μ < 1? 1/σ,the no black hole condition.1 Then there will be a value of τ
when the left-hand side becomes greater than one,so that the concentration
equilibrium becomes unstable,We call this value τs,the sustain point.
The next step is to check the stability properties of the symmetric equi-
librium,It is easy to see that when λ = 1/2,we have w1 = w2 = 1 and
P1 = P2,which means that complete symmetry is an equilibrium,Stabil-
ity requires that the relative real wage,(w1/Pμ1 )/(w2/Pμ2 ) is decreasing in λ
around the symmetric equilibrium,so that a small increase in the population
of region 1 leads to a decrease in its real wage relative to region 2,We totally
differentiate the two wage equations with respect to λ,w1 and w2,substitute
for the symmetric equilibrium values,and use ω1 for the real wage of region
1,Notice that since we evaluate things around the symmetric equilibrium,
each change in region one will be accompanied by an equal negative change
in region 2,so that dω1 =?dω2,Thus it is enough to evaluate dω1 in order
to see how relative real wages change,Thus we get that
dω1
dλ = 2zP
μ 1
σ?1
bracketleftbiggμ(2σ?1)?(σ + μ2σ?1)z
σ?μz?(σ?1)z2
bracketrightbigg
,
where z = (1?ρ)/(1 + ρ).
The symmetric equilibrium is stable if and only if the above expression
is negative,You can check that the sign depends on the numerator,which
leads to the following condition:
ρ < (1 + μ)(σ?1 + μσ)(1?μ)(σ?1?μσ).
Since ρ ∈ (0,1),the condition can be satisfied only if μ < 1?1/σ – the no
black hole condition,Assuming that it holds,the right-hand side gives us τb
– the break point –,at which symmetry must be broken.
To sum up,we have two critical values,τs and τb,and it is possible
to show that τs > τb,Thus when the transportation cost is very high,
only the symmetric equilibrium is stable,When we lower τ,there comes
1It is called like that because if it does not hold,increasing returns are so strong
that manufacturing always agglomerates,We are not interested in such economies,so we
assume that the condition holds.
CHAPTER 11,THE NEW ECONOMIC GEOGRAPHY 71
a point when the core-periphery pattern and symmetry are also stable,In
this region we have three stable equilibria,the symmetric one and two with
complete agglomeration,Finally,as the transportation cost falls further,the
symmetric equilibrium becomes unstable,and symmetry must be broken.
See pictures in class or in FKV,They look really nice.
11.2 Specialization in international trade
Now we look at a model where agglomeration comes not in the form of
population,but industrial concentration,This is more realistic when regions
are countries,between which labor movement is very limited,It also applies
more to Europe,where even within countries people are far less mobile,
than in the US,The main idea is fairly simple,manufacturing firms use the
products of other firms as intermediate inputs,This leads to the presence of
forward and backward linkages,It is good to be close to other firms because,
first,there are more intermediates available without transportation costs
(forward linkage),and second,there are is more demand as an intermediate
input for the firm’s product (backward linkage),As we will see,these linkages
give rise to very similar forces than the core-periphery model of the previous
chapter.
There are two sectors as before,agriculture and industry,Food is the
numeraire,and it is produced with a c.r.s technology,The unit labor require-
ment is one,so that the price of food equals the wage rate,Now there is only
one primary factor of production,but manufacturing also uses intermediate
inputs,For simplicity we assume that industrial demand for intermediates
takes the same form as consumer demand,so that producing a variety re-
quires the CES aggregate of all varieties,To be more precise,we assume that
the cost function in manufacturing is
c(x) = (a + bx)w1?αPα,
where P is the price index in the region and it is defined as before,Price is
set as a constant (σ/[σ?1]) markup over marginal cost,and again we choose
units so that
p = w1?αPα.
Next,we normalize the size of the labor force in each country to one,and let
λi stand for manufacturing’s share of labor in country i,Then the following
CHAPTER 11,THE NEW ECONOMIC GEOGRAPHY 72
must hold:
wiλi = (1?α)nipix,
since 1? α share of total revenue is paid out in wages,This gives us the
number of varieties produced in a region,ni:
ni = wiaσ(1?α)p
i
λi.
Let us write down the price indexes using this and the pricing equation:
aσ(1?α)P1?σ1 = λ1w1?σ(1?α)1 P?ασ1 + λ2ρw1?σ(1?α)2 P?ασ2
aσ(1?α)P1?σ2 = λ1ρw1?σ(1?α)1 P?ασ1 + λ2w1?σ(1?α)2 P?ασ2,
where again ρ = τ1?σ,Notice that the price indexes are now also on the right-
hand side,because manufactures’ prices include the cost of intermediates,
which are in turn given by the price indexes.
Let us define region i’s expenditure on manufactures as Ei,It comes from
two sources,final demand by consumers and intermediate demand by local
firms,Thus we can write it as follows:
Ei = μwi + αnipix =
parenleftbigg
μ + αλi1?α
parenrightbigg
wi,
where we used the fact that an α share of total cost (and,by the zero profit
condition,total revenue) is spent on intermediates by firms,We also know
from previous chapters that x = aσ and pi is defined above,The final piece,
as before,is market clearing on the goods market:
aσ(w1?α1 Pα1 )σ = E1P1?σ
1
+ ρE2P1?σ
2
aσ(w1?α2 Pα2 )σ = ρE1P1?σ
1
+ E2P1?σ
2
,
with Ei and Pi defined above.
Our goal is to find out when manufacturing agglomeration and complete
symmetry are stable equilibria,We assume for now that μ < 1/2,which
means that both countries have to produce some food in equilibrium to satisfy
world demand,Then we will have factor price equalization,and in both
countries (since food is the numeraire) wi = 1,This implies that λ1 = 2μ,
CHAPTER 11,THE NEW ECONOMIC GEOGRAPHY 73
since this is world demand for manufactures,Then the price indexes reduce
to
P1?σ+ασ1 = 2μ(1?α)aσ and P2 = τP1.
With these it is easy to check that the market clearing condition for country 1
goods holds,The condition for country 2 goods,however,is only an implicit
one (since no production actually takes place),But it can still be solved
for the hypothetical manufacturing wage that the country could offer,The
condition that defines the sustain point τs is that this hypothetical wage is
lower than unity,and this yields
1 + α
2 τ
1?σ?ασ
s +
1?μ
2 τ
σ?1?ασ
s = 1.
The interesting thing about this condition is that it looks exactly the same
as in the previous chapter,except that μ is replaced with α,Thus we do
not need to derive its properties again,if the no black hole condition holds,
agglomeration will be sustainable when τ < τs.
To find the break point,we need to know how the manufacturing wage re-
sponds when we transfer workers from agriculture to industry,starting from
the symmetric equilibrium,Thus we totally differentiate the wage equations
(the goods market clearing conditions) around λ1 = λ2 = μ,and then evalu-
ate the sign of that expression,Without going into details (see FKV),let us
state the result for the break point τb:
τ1?σb = (1 + α)(σ?1 + ασ)(1?α)(σ?1?ασ).
This is also the same as in the core-periphery model,with α instead of μ.
Thus regardless of the different forces that lead to concentration,the two
model’s conclusions are the same,For high transportation costs,symmetry
is the unique stable equilibrium,For medium level transportation costs con-
centration becomes feasible,And for small levels of τ,it will be the only
stable equilibrium,Thus as transport costs fall,the core-periphery pattern
(either in the form of population agglomeration or in the form of industrial
concentration) will spontaneously arise.
Let us briefly deal with the case of μ > 1/2,Then manufacturing can-
not be concentrated entirely in one country,but agriculture can,Thus we
no longer have factor price equalization in the asymmetric equilibrium,and
CHAPTER 11,THE NEW ECONOMIC GEOGRAPHY 74
region 2 may or may not produce manufacturing,If μ is large enough,there
will be industrial production in both countries,but the one with only manu-
facturing will enjoy a higher wage rate,Krugman and Venables (1995) solve
numerically for the break and sustain points,which no longer have simple
analytic forms,What is more interesting,they also plot real wages in the two
regions,assuming stable equilibria,They show that when the core-periphery
pattern forms,the periphery must suffer an absolute decline in living stan-
dards,As transport costs fall further,however,at some point the living
standard of region 2 will start rising,both in absolute and relative terms,At
the point of no shipping costs,τ = 1,the two countries will have the same
wealth,It is possible,however,that in the final stage of convergence country
1 experiences a decline in its real wage,Thus if development is captured by
falling transport costs,the world economy will first experience a divergence,
then a convergence in national inequalities,Hence the model can rationalize
both the fear of the South form deindustrialization and the fear of the North
from low-wage competition in the South,See FKV for more details.
Chapter 12
Empirical strategies
12.1 Testable predictions
We will derive testable hypotheses from the monopolistic competition model
without transportation costs,The first one concerns the volume of trade,
VT,Suppose there are two countries,each producing the same type of
goods,but they specialize to different varieties,This is the simplest version
of the Dixit-Stiglitz model,Since both countries specialize,and preferences
are homothetic,they will consume each other’s goods according to their
share in world GDP,Let these be s and 1?s for Home and Foreign,x is the
production of a single variety,p is its price (common to all goods) and n and
n? the number of varieties produced in Home and Foreign,Using y and y?
for the two countries’ GDP,we have:
y = pnx
y? = pn?x.
The volume of trade is given by
VT = sy? + (1?s)y = 2yy
yw,
where yw stands for world GDP,and we substitute the definition of s,This
is the simplest form of the Gravity Equation and it says that trade increases
with the similarity of the two countries.
A more general result emerges when we consider more than two countries.
The prediction we derive concerns the volume of trade within a group of
75
CHAPTER 12,EMPIRICAL STRATEGIES 76
countries,A,We introduce the following notation:
ya =
summationdisplay
i∈A
yi,ea = ya/yw,?i ∈ A,eia = yi/ya.
Let VTa be the volume of trade within group A,Then we can derive the
following:
VTa =
summationdisplay
i∈A
summationdisplay
j∈A,jnegationslash=i
siyj
=
summationdisplay
i
si(ya?yi)
= ya
summationdisplay
i
ea[eia?(eia)2]
= yaea[1?
summationdisplay
i
(eia)2].
Thus the share of trade in group GDP can be written as
VTa
ya = e
a[1?summationdisplay
i
(eia)2],
which increases with the relative importance of group A in the world economy
and with its homogeneity in terms of the size of its member countries.
Finally,we look at the more general model with two sectors,one pro-
ducing homogenous goods (z) and the other producing differentiated ones
(x),We prove that the share of intra-industry trade declines as the two
countries become more dissimilar in their factor endowments,To measure
intra-industry trade,we use the Grubel-Lloyd index,which for countries k
and l is defined as
Gkl = 2
summationtext
j min{IM
kl
j,IM
lk
j }summationtext
j (IM
kl
j + IM
lk
j )
,
where IMklj is the value of imports of sector j output from country k to
country l,The index is between zero and one by construction,and it increases
with the sectoral overlap in bilateral imports.
We can simplify the index using the special structure of our model,First,
we assume that FPE holds,so that prices and demand patterns do not
change,neither does world GDP,Second,we are interested only in changes in
CHAPTER 12,EMPIRICAL STRATEGIES 77
factor composition,so that we keep relative country sizes constant,This can
be achieved by a movement along the constant factor price line,Third,we
assume that manufacturing is capital intensive and the home country has a
higher capital intensity,In such a scenario the homogenous good is exported
by Foreign,and Home only exports manufactured goods,The Grubel-Lloyd
index can be simplified to the following:
G = 2spn
x
2(1?s)pnx =
sn?
(1?s)n,
since by construction Home produces and exports more differentiated goods
than Foreign (and assuming balanced trade).
We look at a change where Home acquires more capital relative to For-
eign,so the two become even more dissimilar,The effect of the change will
be an increase in manufacturing production in Home,an equal decrease in
manufacturing production and an increase in the production of the homoge-
nous good in Foreign,By assumption s,p and the aggregate production
variables do not change,the only effect is a reallocation of production – a
pure Rybczynski effect,Now it is easy to check that the numerator of G
decreases,and the denominator increases with the change,Thus the share
of intra-industry trade declines as the two countries become more dissimilar.
This is a natural results,since as factor proportion differences increase,the
role of inter-industry trade becomes larger,even though total trade increases.
12.2 The gravity equation
We already saw a very simple form of the gravity equation in the previous
chapter,Now we derive a more general formula with transportation costs.
There are two ways to do this,and they are almost equivalent,The main
assumption is that countries are completely specialized in production,either
because of product differentiation a la Dixit-Stiglitz,or because consumers
perceive that otherwise homogenous goods are different because of their ori-
gin (the Armington assumption),The first derivation is due to Anderson
(1979),who used the latter formulation,We will follow a recent derivation
by Anderson and Van Winkoop,except that we work with the monopolistic
competition model.
Thus assume there is only one sector of production,which produces dif-
ferentiated goods,The goods are tradable,but bear,iceberg” transportation
CHAPTER 12,EMPIRICAL STRATEGIES 78
cost,Using the demand function we derived earlier,we can write country i
exports to country j as:
pixij = ni(piτij)
1?σ
P1?σj yj,
with
P1?σj =
summationdisplay
k
nk(pkτjk)1?σ.
Since each country is specialized in different varieties,the sum of the
value of exports of i into all countries (including itself) must equal i’s GDP.
This condition can be written as
yi = nip1?σi
summationdisplay
j
(τij/Pj)1?σyj.
Substituting for nip1?σi into the export equation above,and using the earlier
notation si = yi/yw (where yw is world income),we get that
pixij = yiyjy
w
parenleftbigg τ
ij
ΠiPj
parenrightbigg1?σ
,
where Πi is given by
Π1?σi =
summationdisplay
k
(τik/Pk)1?σsk.
If we substitute for nip1?σi into the definition of the price index,we get
that
P1?σi =
summationdisplay
k
(τik/Πk)1?σsk.
We can see that if trade costs are symmetric (so that τij = τji),which we
assume,the last two equations have a unique solution,Pi = Πi (up to scale,
which allows for the normalization of one price),Substituting for Πj in the
export equation,we get the general Gravity Equation as
pixij = yiyjy
w
parenleftbigg τ
ij
PiPj
parenrightbigg1?σ
.
The,canonical” version of the GE does not include the price index terms,
only distance (as a proxy for trade costs) and the two countries’ GDP,We
CHAPTER 12,EMPIRICAL STRATEGIES 79
can see the advantage of deriving the equation from a full general equilibrium
model,It shows that not only bilateral distance,but distance from the rest
of the world (captured by the price index terms) has to be included,The
“remoteness” variables are important because more remote countries have to
offer a lower supply price to the rest of the world in order to compensate for
trade costs,so gross demand for their goods will be higher,Thus if we take
two countries that have the same attributed except their distance to the rest
of the world,the more remote one will trade more in general.
Part III
Trade and growth
80
Chapter 13
Trade,growth and factor
proportions
Strictly speaking,neoclassical growth models (such as the Solow or Ramsey
models) do not answer questions about trade and growth,The reason is
that steady state growth in these models is exogenous,so that trade does
not have a long-term growth effect,It does have an effect on the level of
aggregate variables,and it might influence transitional dynamics,For this
reason,endogenous growth models are better suited to study the long-term
effect of openness on growth,a question of great importance to policymakers.
Nevertheless,there are still useful insights emerging from neoclassical models.
An important shortcoming of static trade models is that factor supplies are
constant,which might be true for primary factors such as labor or land.
But once we think about capital that can be accumulated,it is important
to endogenize its stock and relate it to model fundamentals,Thus in this
chapter we describe a generalization of the Solow model to explore the effect
of capital accumulation on comparative advantage.
13.1 The model
We need at least two sectors to get trade,so we have one producing consump-
tion goods (C) and another producing investment goods (Z),There are two
factors,capital and labor,The latter grows exogenously at the rate n,while
the former can be accumulated,with investment requiring investment goods.
First we can describe the structure of production given the stock of labor
81
CHAPTER 13,TRADE,GROWTH AND FACTOR PROPORTIONS 82
and capital,and then look at the laws of motion,Notice that at any point
of time the model is a conventional Heckscher-Ohlin one,The two goods are
produced by c.r.s production functions:
C = Fc(Kc,Lc)
Z = Fz(Kz,Lz).
We can rewrite the production functions in intensive form,using the
notation
c = C/L,z = Z/L,k = K/L,ki = Ki/Li,λ = Lc/L,
with K,L being the aggregate stock of labor and capital,The usual con-
ditions in a competitive economy hold,linking factor rewards and marginal
productivities,As long as both goods are produced,we have the following
equalities:
c = fc(kc)
z = fz(kz)
r = pfprimec(kc) = fprimez(kz)
w = p[fc(kc)?kcfprimec(kc)] = fz(kz)?kzfprimez(kz)
k = λkc + (1?λ)kz,
where the last is the resource constraint in the economy and p is the relative
price of the consumption good.
For each sector,the relative factor reward ω = w/r can be written as
ω = fi(ki)fprime
i(ki)
ki.
From this equation it is easy to check that the capital intensity in each sector,
ki,is increasing in ω,Moreover,if there is no specialization,by Stolper-
Samuelson?r ≥?p ≥?w,so that ω is decreasing in p,Using this,and the first
two and the last equations above,we can see that
cp > 0,zp < 0.
Assuming that the consumption sector is more capital intensive at any ω,at
unchanged prices ck > 0,zk < 0 by the Rybczynski Theorem.
CHAPTER 13,TRADE,GROWTH AND FACTOR PROPORTIONS 83
13.2 A small open economy
In this case p is exogenous for the country,since it is given by the world
market,This means that when there is no specialization the factor intensities
are also fixed,since they only depend on p through ω,But then λ is a linear
function of k (see the resource constraint),and the per capita GDP function
y(k) = λpfc(kc) + (1?λ)fz(kz)
is also linear in k,The last bit is to determine specialization patterns,which
depend only on k,Since k is a convex combination of the exogenous ki,
both goods can be produced only if kz ≤ k ≤ kc,If k < kz,only the labor
intensive good,z is produced,If k > kc,the economy specializes in the
capital intensive consumption good.
The savings decision is simply given by the Solow-assumption of a con-
stant saving rate,s:
˙k = sy(k)?(δ + n)k,
where δ is the exogenous depreciation rate,We just derived that y is a
concave function of k,with a linear portion between kc and kz,Thus the
model works pretty much as the conventional Solow-model,and there is a
unique and globally stable steady state,The economy ends up where the
linear line (δ +n)k and y(k) intersect,just as in the one-sector model,This
can be at any portion of y,depending on the parameter values n,s,δ relative
to the world,If the country is not very different form the rest of the world,
it will not specialize and factor prices will be the same as elsewhere,The
effect of parameters on specialization is easy to derive,For example,if the
country is more patient than the rest of the world,it will export the capital
intensive good c and import investment goods.
13.3 A large country
Now we assume that there are two countries,Home and Foreign,We first
describe the autarchy steady state,and then see what happens when the two
countries start to trade,Assuming that the initial capital stock is below its
steady state value,both goods have to be produced,Now the GDP function
will also be a function of the relative price p,which is endogenous,The
CHAPTER 13,TRADE,GROWTH AND FACTOR PROPORTIONS 84
equilibrium conditions can be written as
sy(k,p) = z(k,p)
˙k = sy(k,p)?(δ + n)k.
To draw the phase diagram,note that the first equation defines a downward
sloping schedule in the (k,p) space,The system must always be on this
schedule,Setting the second equation,the law of motion for k,equal zero
defines the other schedule,We need the condition
syk < δ + n
to be satisfied,then the schedule is upward sloping,This implies a unique
steady state (the intersection),and a unique path converging towards the
steady state along the first schedule,You can see that as k increases,the
price of the capital intensive consumption good declines.
Now we introduce trade,and assume that both countries are in steady
state initially,with sh > sf,This means that the steady state capital stock
is higher,and the relative price of the consumption good is lower in the home
country,Thus the momentary impact of trade will be to raise p in Home and
lower it in Foreign,Home starts exporting c and importing z,This is not
the end of the story,however,The change in p will induce changes in the
capital stocks,In Home,the price is now higher than it is consistent with
the steady state capital stock,which means further investment and a growth
in k,The opposite will happen in Foreign,Thus trade amplifies differences
in factor proportions! As long as the two economies are not very different in
terms of their parameters n,s,δ there will still be FPE,but it is less likely
now than in the static framework.
Chapter 14
Learning-by-doing
14.1 A Ricardian model
In this chapter we take a look at the dynamics of comparative advantage
from a different perspective,The paper I describe is Paul Krugman’s,The
narrow moving band,the Dutch Disease and the competitive consequences
of Mrs,Thatcher”,The idea is very simple,productivity depends on ex-
perience,which in a Ricardian setting means that comparative advantage is
self-reinforcing,If a nation specializes in a set of goods,because of learning
it will become more productive in those goods,On the other hand,complete
specialization in the (continuum good) Ricardian model means that other
countries will not produce Home’s goods,and they do not accumulate expe-
rience in them,Thus productivity differences grow over time,unless there is
a shock or government intervention that shakes up the established pattern.
14.1.1 The model
The model is as follows,In each period it is isomorphic to Dornbusch-Fischer-
Samuelson (DFS),Thus the production function of a good z in Home and
Foreign is given by1
x(z,t) = a(z,t)l(z,t)
X(z,t) = A(z,t)L(z,t).
1We revert back to our earlier tradition of using uppercase letters for foreign variables.
85
CHAPTER 14,LEARNING-BY-DOING 86
Thus at each point in time Home has a comparative advantage in good z if:
A(z,t)
a(z,t) > w,
where w is the relative wage in Home (we normalize the Foreign wage to
unity).
We assume that productivity depends on experience k(z,t) and K(z,t):
a(z,t) = k(z,t)epsilon1
A(z,t) = K(z,t)epsilon1,
with 0 < epsilon1 < 1,Experience depends on cumulative production,We allow for
spillovers over borders,but their effect is smaller than the effect of production
in the country,To be precise,we have
k(z,t) =
integraldisplay t
∞
[x(z,s) + δX(z,s)]ds
K(z,t) =
integraldisplay t
∞
[X(z,s) + δx(z,s)]ds,
with 0 ≤ δ < 1.
The final step to close the model is to specify demand,First,we assume
that the labor force in Home and Foreign (l and L) grow at the exogenous rate
n,The demand functions take the usual logarithmic form,and for simplicity
we assume that the coefficients are unity,Thus each good receives one unit
of expenditure,with total expenditure equal to the wage rate,Then the
demand equation is given by
w = ˉz1? ˉz Ll,
where ˉz is the border commodity that determines specialization patterns.
14.1.2 Dynamics and the steady state
The relative productivity in a sector is given by [K(z,t)/k(z,t)]epsilon1,To get the
growth rate of this expression,take logs and differentiate,Then using the
definitions for k and K,we get
dlogK(z,t)/k(z,t)
dt =
X(z,t) + δx(z,t)
K(z,t)?
x(z,t) + δX(z,t)
k(z,t),
CHAPTER 14,LEARNING-BY-DOING 87
Looking at the production functions,we can see that if the relative labor
allocation is fixed,the expression will converge to a steady state,To see
this,define χ(z,t) = K(z,t)/k(z,t) and λ(z) = l(z)/L(z),Then the law of
motion for χ can be written as:
˙χ
χ = lk(z,t)
epsilon1?1
bracketleftbiggχ(z,t)epsilon1?1
λ(z) +
δ
χ(z,t)?1?
δχ(z,t)epsilon1
λ(z)
bracketrightbigg
,
and the steady state value of χ is implicitly given by
[χ(z)]epsilon1?1 = λ(z)
bracketleftbigg1?δ/χ(z)
1?δχ(z)
bracketrightbigg
You can check that when χ(z,t) is below (above) this value,˙χ is positive
(negative),Thus the steady state is globally stable,One can also see that
the steady state value of χ(z) and thus relative productivity is a monotonic
function of λ(z),say
χ(z) = α[λ(z)].
In particular,χ(z) is decreasing in λ(z),α(0) = 1/δ and α(∞) = δ.
Now we are ready to describe the long-term equilibrium of the model.
Suppose that at some starting point the relative technology coefficients are
given,Then the pattern of specialization simply follows the rule in DFS,
so if we assume that relative productivity is declining in z,there will be a
marginal good ˉz that separates goods produced in Home and Foreign,This
means that at this point in time,for any z < ˉz we have λ(z) = ∞ and for
all z > ˉz we have λ(z) = 0,Thus for goods produced in Home,relative
productivity of Foreign will converge monotonically to δ,and for the other
goods it will tend to 1/δ.
The long-term pattern of specialization is now straightforward,Whatever
the initial pattern was,it will be reinforced over time,The downward sloping
relative productivity curve will become a step function,where the step is at
the initial border commodity ˉz,The wage rate will be between the two
extremes (δepsilon1,δ?epsilon1),depending on the relative labor force of Home (which is
constant over time) and the initial pattern of comparative advantage,Thus
initial conditions matter in this model,and there is a whole range of possible
steady state values for ˉz,In particular,ˉz can take on any value between
(δepsilon1/[1+δepsilon1],1/[1+δepsilon1]) (see the demand condition),Thus the more industries
Home can initially grab,the better its terms-of-trade and welfare will be in
the long run.
CHAPTER 14,LEARNING-BY-DOING 88
14.1.3 Industrial policy
We now use the model to look at industrial policy,In this framework,there
is a possibility for a government to target industries for export success,This
is the old infant industry argument for protection,and can be rationalized
by the model as follows,Suppose the two countries are in steady state,and
a pattern of specialization is locked in,Now the Home government imposes
prohibitive tariffs on a set of industries,which we can assume w.l.o.g to be
in the (right) neighborhood of ˉz,These goods will become non-tradable,and
each country will satisfy its demand:
l(z) = l
L(z) = L.
If the home country is larger than Foreign,its productivity growth in these
sectors will be faster,and after some time period it will catch up,Protection
needs to be continued until the productivity improvement is sufficient enough
to give Home a comparative advantage in the protected goods,At this time
trade might be resumed,with ˉz moved to the right and Home experiencing an
improvement in its terms-of-trade and welfare,Some attributed the success
of Japan to such industrial policies,where the government targeted successive
industries for competition on the world market,Notice that for such a policy
to work the Home market has to be large,otherwise Foreign will continue
to have a productivity advantage even without trade,Second,as Home
targets new industries,it will become harder to achieve the desired result.
This is because with each intervention the Home wage rate rises,and its
comparative disadvantage becomes larger in the remaining industries,Thus
protection needs to last longer and longer,and in the limit the maximum
wage rate that can be achieved is given by α(l/L),since this is the maximum
productivity advantage Home can achieve in a non-traded sector.
14.2 Agriculture and the Dutch Disease
Now we discuss a related model,which can be found in Kiminori Mat-
suyama’s,Agricultural productivity,comparative advantage,and economic
growth” (JET 1992),There are two main differences between the previous
model and this one,First,learning-by-doing takes place only in one sector.
This means that the growth of the economy will be driven by this sector,
CHAPTER 14,LEARNING-BY-DOING 89
which we assume to be manufacturing (“the engine of growth”),Second,
preferences are non-homothetic,in particular the income elasticity of the
stagnant sector (agriculture) is less than one,As we will see,these features
of the model can explain two phenomena of interest,One is the possibility
that increasing agricultural productivity releases resources into manufactur-
ing,thereby accelerating growth,The other is just the opposite,a more
productive agriculture draws resources away from manufacturing,thereby
slowing growth,Which one of the two possibilities will occur,depends cru-
cially on the openness of the economy.
14.2.1 The closed economy
As we said,there are two sectors,food and industry,Both are produced
by two factors,one specific to the sector,Thus we can view the production
functions as having decreasing returns to the mobile factor:
xm = mf(l) fprime > 0,fprimeprime < 0
xa = ag(1?l) gprime > 0,gprimeprime < 0
where l is labor allocated to manufacturing and the total labor force is one.
Agricultural productivity a is constant,but manufacturing productivity m
depends on aggregate experience in the sector:
˙m = δxm.
Let p be the relative price of manufactures,Then competition for labor
ensures that its marginal product is equalized between sectors:
agprime(1?l) = pmfprime(l).
There is no borrowing or lending2,thus consumers maximize the instan-
taneous utility function
u = β log(ca?γ) + logcm.
We assume that the economy is productive enough to supply people with
at least the subsistence level,so that ag(1) > γ > 0,Then the optimality
conditions imply that
ca = γ + βpcm.
2Or,because agents are identical,the interest rate adjusts such that their holdings of
bonds equal zero in each period.
CHAPTER 14,LEARNING-BY-DOING 90
For a closed economy,production must equal consumption for each good.
Solving the condition for equal marginal products and substituting into the
demand equation we get
γ = a[g(1?l)?βgprime(1?l)f(l)/fprime(l)],
where the right-hand side is decreasing in l,For a given value of a,the
equation defines the equilibrium share of employment in manufacturing,l(a).
The important observation is that lprime(a) > 0,so that an improvement in
agricultural productivity leads to an increase in manufacturing employment,
higher manufacturing output,and a higher rate of growth in that sector.
Since the food sector is stagnant,the growth rate of the economy will rise
and it is easy to see that utility rises as well.
This argument depends entirely on the fact that γ > 0,so that preferences
are non-homothetic,This assumption is plausible,and it is consistent with
the observation that agriculture’s share in employment and GDP declined
drastically,Since food is a necessity,a more productive agriculture can supply
people’s needs with fewer resources,This is a popular argument for the
British industrial revolution,where displaced agricultural workers flocked to
the city,thereby fueling the Industrial Revolution,The problem with this
argument is that it completely unravels in an open economy.
14.2.2 A small open economy
Let us now consider the opposite case of a small open economy,In this
case the relative price of manufactures is given by the world markets,and
it is exogenous for the home country,This means that the product market
equilibrium condition can be written as
agprime(1?l)
mfprime(l) = p.
Since the left-hand side is increasing in l,now agricultural productivity has a
negative effect on manufacturing employment,Thus the growth of the econ-
omy slows down with an increase in a,since resources are reallocated towards
agriculture,The intertemporal welfare of consumers may still increase,but
if agents are patient enough,it will decrease,Thus an abundance of natural
resources is a mixed blessing,cautionary tales include the antebellum South
or Interwar Argentina.
CHAPTER 14,LEARNING-BY-DOING 91
The assumptions made are quite special,In particular,learning-by-doing
takes place only in industry,and its effects are localized,Thus the growth
effect of an increase in agricultural productivity is not robust,There are ex-
amples of successful industrialization through food production,one can think
of Denmark,What is robust,however,is the unravelling of the argument for
industrialization we made in the previous chapter,In an open economy,it is
impossible that industry is the engine of growth,and an increase in a releases
resources into the dynamic sector,The reason is,of course,comparative ad-
vantage,a productivity improvement will cause a sector to expand,because
the consumption and production decisions are separable.
The model also illustrates formally the phenomenon of the Dutch Disease.
Reinterpret the agricultural sector as natural resources,and suppose a coun-
try discovers new fields (which we capture by an increase in a),This effect is
temporary,since the new fields will be eventually exhausted,Nevertheless,
in the present framework the temporary change will have a permanent effect.
The reason is learning-by-doing,since while a is higher,labor is drawn away
from manufacturing and the sector contracts,Since industrial productivity
depends on cumulative production,the country will experience a permanent
setback in industry.
14.3 North-South trade
We will describe a simplify version of Alwyn Young’s,Learning-by-doing and
the dynamic effects of international trade”,The model is fairly complicated,
but the intuition is not,We will use the specific functional forms Young uses
when he discusses the effect of trade,you can read his more general setup in
the paper.
14.3.1 The model
The production side of the model is somewhat non-standard,but Ricardian
in essence,At each period of time,there are an infinite amount of potential
goods available,in a sense that the blueprints for these goods are given.
Their unit labor requirements,however,are different,We order the goods in
a way that their potential productivity decreases with their index,but their
actual productivity is infinite as the index goes to infinity,The reason for
this difference is learning-by-doing,that is bounded for each good but can go
CHAPTER 14,LEARNING-BY-DOING 92
on indefinitely,due to the availability of new goods,But before the economy
can produce the new goods,it has to go through a period of learning in lower
indexed goods,Since we assume that learning-by-doing spills over to other
goods,experience in lower indexed goods paves the way for producing higher
indexed ones.
To be precise,we assume that at some period zero,productivity is given
by
a(s,0) =
braceleftbigg ˉae?s if s ≤ T(0)
ˉae?T(0)es?T(0) if s > T(0),
Thus at time zero there are a range of goods in which learning-by-doing has
already been exhausted,those with index below T(0),All other goods are
still subject to improvement,but actual productivity decreases with distance
from T(0),You can check that productivity is symmetric around T(0),with
a[T(0)?α] = a[T(0) + α],Thus the goods that can be produced cheapest
are in a neighborhood of T(0).
Now we introduce learning-by-doing,We assume in general that there are
spillover across goods,but only goods that has not exhausted their potential
can generate spillovers for all other similar goods,You can see Young’s
paper for a general formulation,but for our purposes it is enough to write
his specific form:
˙a(s,t)
a(s,t) =
integraldisplay
s∈S
2L(v,t)dv,
where S is the set of goods with learning-by-doing not yet exhausted,You
can check that with this formulation productivity remains symmetric around
an ever increasing T(t),with the same functional forms as we defined for
T(0):
a(s,t) =
braceleftbigg ˉae?s if s ≤ T(t)
ˉae?T(t)es?T(t) if s > T(t),
Moreover,from this we can see that
dT(t)
t =
1
2
˙a(s,t)
a(s,t) =
integraldisplay
s∈S
L(v,t)dv,
thus the dynamics of production is described with the evolution of T(t).
There is no borrowing or lending,so the consumers spend their income
in each period,They maximize the instantaneous following utility function:
u =
integraldisplay
s
log[c(s) + 1]ds,
CHAPTER 14,LEARNING-BY-DOING 93
which means that consumers value variety,but not infinitely (marginal utility
at zero is bounded),The first-order conditions are written as
c(s) + 1
c(z) + 1 =
p(z)
p(s).
Obviously the consumer wants to consume the cheapest good,say s?,Then,
since marginal utility is bounded,there is a price above which consumption
is zero,This defines a borderline commodity,M,whose consumption is zero,
but for any good with a price smaller than p(M)),consumption is positive.
Then for any good s,we have
p(s)c(s) = p(M)?p(s).
14.3.2 Autarchy
For any good produced,its price is given by the supply side,as in any Ricar-
dian model,Let us normalize the wage rate to one,then we have p(s) = a(s).
This means that for goods consumed in positive quantities,
a(s)c(s) = a(M)?a(s).
We know how the productivity variables look like,at each period of time,
they are symmetric around T(t),Thus the cheapest goods will be in a neigh-
borhood of T(t),and M will be the the smallest indexed product consumed
(and there will be a good N equidistant from T[t] that indicates the highest
indexed good consumed).
Let τ = T?M = N?T,then the budget constraint is written as
1 =
integraldisplay N
M
a(s)c(s) = 2ˉa(τ?1)e?M + 2ˉae?T,
which leads to
eT = 2ˉa(τ?1)eτ + 2ˉa.
You can easily see that 0 < dτ/dT < 1,which implies that dM/dT > 0.
Thus the range of goods consumed moves to the right as T increases due
to learning-by-doing,and the range also becomes wider,It is also easy to
see that dc(T)/dT > 0 and consumption of goods in the same distance from
T rises,Thus both the variety and the quantity of goods increases with T,
which means an unambiguous rise in utility as knowledge accumulates.
CHAPTER 14,LEARNING-BY-DOING 94
Let us now calculate the growth rate of the economy,We look at GDP
per capita growth at unchanged prices,and define it is
g(t) =
integraltext
s a(s,t)?x(s,t)/?tdsintegraltext
s a(s,t)x(s,t)ds
dL(t)/dtL(t),
Using the economy wide resource constraint and the autarchy equilibrium
conditions,we can rewrite this as follows:
g(t) =
integraltextN(t)
M(t)?a(s,t)/?tx(s,t)ds
L(t)
=
2dT(t)/dt integraltextN(t)T(t) a(s,t)x(s,t)ds
L(t)
= L(t)2,
where we used the fact that consumption (and production) is symmetric
around T(t),so that half of the labor force is engaged in the production of
goods with learning-by-doing potential,With constant population L,the
economy grows at a constant rate of L/2.
14.3.3 Free trade
We look at trade between two economies,one (the DC) more advanced than
the other (the LDC),We capture this by assuming that Tdc?Tldc = X > 0
when they start trading,The production side of the model can be described
analogously to DFS,For goods below Tldc,both countries exhausted learning-
by-doing,For goods between Tldc and Tdc,the LDC still learns,but the DC
does not,Finally,for goods above Tdc both countries still learn,Thus relative
productivity can be written as:
adc(s,t)
aldc(s,t) =
1 if s ≤ Tldc(t)
e?2(s?Tldc) if Tldc(t) < s < Tdc(t)
e?2X if s > Tdc(t)
.
This curve is similar to the step function in Krugman’s paper,except that the
middle range is downward sloping (and not vertical),Obviously the relative
wage of the DC must be between the two extremes,w ∈ [1,e2X].
CHAPTER 14,LEARNING-BY-DOING 95
The equilibrium wage and production patterns are determined by the
demand side,There are three equations that need to be solved for the final
equilibrium,the budget constraints for the representative consumer in each
country and the balanced trade condition:
integraldisplay
s∈ldc
aldc(s)cldc(s)ds +
integraldisplay
s∈dc
adc(s)cldc(s)ds = 1
integraldisplay
s∈ldc
aldc(s)ccd(s)ds +
integraldisplay
s∈dc
adc(s)cdc(s)ds = w
integraldisplay
s∈dc
Lldcadc(s)cldc(s)ds?
integraldisplay
s∈ldc
Ldcaldc(s)cdc(s)ds = 0
The solution to these conditions together with the supply side determines
the relative wage rate at the DC,the range of goods consumed in the two
countries,and the pattern of specialization.
Broadly,there are three different equilibria,depending on which part
of the relative productivity curve the economies land at (see figures at the
paper),One is when w = 1,in which case the cutoff good is to the left
of Tldc,Thus the LDC only produces goods which have exhausted their
potential,Since the wage rates are equal,demand and income in the two
economies are the same,This means that the ranges of goods consumed are
identical and that demand is symmetric around Tdc,Since it produces only
goods to the left of Tldc,the LDC will stop growing altogether,Because of
the demand pattern,half of world demand falls on DC goods with learning
potential,so that
dTdc(t)
dt =
Lldc + Ldc
2,
In the second type,the wage rate equals e2X and the cutoff in production
is to the right of Tdc,In this case world demand is symmetric around Tldc,and
the DC only produces goods with learning-by-doing,The rates of technical
progress in the two economies can be written as
dTdc(t)
dt = Ldc
and
dTldc(t)
dt =
Lldc?e2XLdc
2,
since half of world demand is for goods where the LDC has still learning
potential,but Ldc of the workforce in these sectors comes from the DC,Since
CHAPTER 14,LEARNING-BY-DOING 96
the wage rate is higher in the DC,its consumers will consume a wider range
of goods in both directions,Thus DC consumers will enjoy both cutting
edge products and very old-fashioned ones that people in the LDC no longer
consume (antiques?).
The last possibility is when the wage rate is between the two extremes,
and the cutoff good is between Tldc and Tdc,As you can see from the pictures,
it is possible that the two production ranges ar disjoint,so that the set of
produced goods is not connected,In this case the both countries will allocate
half of their workforce to goods with learning,and this will also be the rate of
their technological progress,When the product range is connected,it must
be the case that less than half of the LDC workforce,and more than half of
the DC’s workforce is engaged in producing goods with learning,Thus the
rate of technical progress in both cases is at most Lldc/2 and at least Ldc/2.
Summarizing our results,we get that in any possible equilibria the rate
of progress increases in the DC,but decreases in the LDC,Thus,although
statically efficient,free trade will cause dynamic losses for the LDC,There
is one possibility when this is not the case in the long run,when the rate of
progress is so much faster in the LDC in autarchy,that it grows faster even
with trade and overtakes the DC at some point in time,This can happen
when the difference in knowledge (X) is not very large,and the LDC is
much bigger than the DC,Thus the model can generate the phenomenon
of leapfrogging,when an economy starting from behind overtakes the leader.
This,however,is not a very likely outcome,and emerges more naturally from
other models.3
Focusing on the case when the LDC does not overtake,we can see that
the technological gap necessarily widens if Ldc ≥ Lldc,since dX/dt = dTdc/t?
dTldc/dt,and the former is above Ldc/2,while the latter is below Lldc/2,Even
if Ldc < Lldc,the technical gap eventually widens,unless the LDC overtakes.
Thus the model has a,knife-edge” property,technological differences will
widen over the long run,in either direction,The reason for this is that when a
country falls behind,it has a comparative advantage in goods whose potential
has already been exhausted,which leads to a further relative decline,Thus
the model predicts long-run divergence,since even when the LDC catches
up,it will become the DC and diverges from the other country afterwards.
3The model also does not explain why the LDC is behind in the first place,Moreover,
overtaking would be quicker without trade,For a model tailored more towards leapfrog-
ging,see Brezis et,al.
CHAPTER 14,LEARNING-BY-DOING 97
A final word of caution is in order,First,the model is very special in
that there are no international spillovers,It is possible,for example,that
by trading with the DC the LDC can use the knowledge which is already
accumulated there,The model does not allows for that,since the cost of
learning in the LDC does not decrease with trade,Second,we did not talk
about welfare,which is the primary measure when we compare free trade
and autarchy,Since the LDC still enjoys the usual static gains from trade,
its dynamic losses must be set against these gains,Thus welfare might very
well improve even if in the long run the LDC falls behind.
Chapter 15
Endogenous growth and trade
In this chapter we look at growth models where growth is truly endogenous,
i.e,it arises from rational decisions of economic actors,The main reference
we use is the book by Grossman and Helpman,“Innovation and Growth in the
Global Economy”,There are two different but related models in that book.
One uses the monopolistic competition framework of Dixit-Stiglitz,and it is
pretty much the same as the original contribution of Paul Romer,In the
second,growth arises not from an expansion of varieties,but from steady
improvements in existing goods,The two models share the same reduced
form equilibrium in autarchy,but there are differences when we introduce
trade,We use the variety approach,since we already know much of this
model and we do not really have time to explore the other,You are urged,
however,to study the quality upgrading approach as well.
15.1 Autarchy
The static model is basically the monopolistic competition framework,so
we can rush through quickly,Consumers maximize the intertemporal utility
function
U(t) =
integraldisplay ∞
t
e?ρ(τ?t) logD(t)dt,
where
D(t) =
bracketleftBiggintegraldisplay
n(t)
0
c(i,t)α di
bracketrightBigg1/α
,
98
CHAPTER 15,ENDOGENOUS GROWTH AND TRADE 99
so the elasticity of substitution is σ = 1/(1?α),The allocation of income to
different goods in any period is separable from the intertemporal allocation
of spending,so if period income is E(t),we have for any good that
p(i)c(i) = p(i)
1?σ
P1?σ E,
where P is the usual CES price index.
The intertemporal problem is to choose a spending pattern that maxi-
mizes utility,given that total income E(t) equals spending:
E(t) = P(t)D(t).
Substituting into the utility function,we can write the current value Hamil-
tonian as
H = logE?logP + q(rb + w?E),
where q is the dynamic multiplier,r is the interest rate,b is the representative
consumer’s holding of bonds and w labor income,The first order conditions
can be written as
E = 1q
˙q = (ρ?r)q
˙b = rb + w?E.
Taking the (log) time derivative of the first equation and using the second,
we can write ˙
E
E = r(t)?ρ.
A convenient normalization is to equate nominal spending E to 1 in each
period,which leads to
r(t) = ρ.
On the production side,we assume that each good has a patent that
never expires,thus they are supplied by monopolists,Production requires
only labor,and it has c.r.s,now,Normalizing the unit labor requirement to
unity,the pricing equation is
p = wα.
CHAPTER 15,ENDOGENOUS GROWTH AND TRADE 100
Thus each firm chooses the same price,which together with E = 1 implies
that profits are given by
pi = 1?αn,
Thus (not surprisingly) profits per firm decline with the number of companies,
but they are positive for any n < ∞.
The final step is to see how the number of varieties,n evolves,We assume
that companies can increase the number of varieties incrementally by using
a finite amount of resources,In addition to using labor,innovation efficiency
also depends on the amount of knowledge in the economy,We assume that
knowledge is a public good,and it is accumulated as a side effect of innova-
tion,Thus when firms introduce a new variety,they have monopoly rights to
produce the good,but they cannot appropriate the knowledge they generated
during innovation,In general,knowledge will be a function of the number
of varieties,and for simplicity we assume the two things are the same,Thus
when entrepreneur commits l units of labor to research,the number of new
varieties she generates is given by
˙n = lna,
where 1/a is the general productivity in research.
We assume that firms issue equity to finance innovation,The value of
a company holding a patent depends on the stream of dividends and the
capital gains it offers,Let v stand for the stock market value of a firm,then
the asset equation for v can be written as
rv = pi + ˙v.
The cost of innovation is wa/n,and its return is v,If the latter is larger
than the former,there will be an infinite demand for labor in research,since
we assume free entry into innovation,If the cost is greater than the return,
nobody will innovate,Thus we have
wa
n ≥ v and ˙n > 0
with complementary slackness,We also have a condition for labor market
clearing,together with a condition for non-negative employment in both
sectors,a˙n
n +
1
p = L and
1
p ≤ L,
CHAPTER 15,ENDOGENOUS GROWTH AND TRADE 101
where the second part is total labor demand in the manufacturing sector
(since E = 1).
We can now derive the equilibrium growth path from the conditions
above,Combining the labor market clearing condition,the pricing equa-
tion and the no positive profit condition in innovation,we can rewrite the
law of motion for the number of varieties:
˙n
n =
L
a?
α
vn if v >
αa
nL
0 if v ≤ αanL
.
Together with the asset equation above (substituting for the level of profits
and for r = ρ),these equations determine the evolution of n and v,It turns
out that we can introduce new variables that simplify the laws of motion.
Thus let
V = 1vn
be the inverse of the aggregate value of the stock market,and
g = ˙nn.
Then the dynamic equilibrium conditions can be written as
g =
L
a?αV if V <
L
αa
0 if V ≥ Lαa
and ˙
V
V = (1?α)V?g?ρ.
At any time the first condition must be satisfied on the equilibrium path.
Suppose that along this path the growth rate of knowledge g is positive.
Then it is easy to show that if ˙V is positive,g will eventually fall to zero and
V goes to infinity,If the amount of varieties is constant,V = ∞ means that
v = 0,But in such a situation profits are strictly positive,so that if stocks
are valued according to fundamentals (which must be if the transversality
condition holds),v must also be positive – a contradiction,Now let ˙V < 0.
Then g grows without bounds,and V falls towards zero,But if there are no
bubbles in stock valuation,we can integrate the asset equation to get
v(t) =
integraldisplay ∞
t
e?ρ(τ?t)1?αn(τ) dτ < 1?αρn(t),
CHAPTER 15,ENDOGENOUS GROWTH AND TRADE 102
which implies that V > 0 – again a contradiction.
This means that the system must jump to the steady state immediately,
and stay there forever,If there is positive growth in the steady state,from
the asset equation and from ˙V = 0 we can calculate
g = (1?α)La?αρ.
Thus an economy grows faster1 if at has a larger labor force,a lower discount
rate,a more productive research sector and a smaller elasticity of substitution
among goods (which also means more monopoly power),If the right-hand
side is negative,then g is zero and there is no innovation.
15.2 International knowledge diffusion
We start investigating the effect of openness on growth with alternative as-
sumptions,First we look at two economies that do not trade with each other,
but can draw from the pool of knowledge of the other country,In the absence
of international patent protection2,there is nothing to prevent innovators to
invent the same product in both countries,Thus in such a setting it is likely
that there will be a duplication of efforts to some extent,We allow for this
by assuming that there is larger country A whose discoveries are always new,
and a smaller country B for whom a Ψ share of available varieties duplicate
ones in A.
The only difference between this setting and the previous one is that the
stock of knowledge relevant for innovation is now nA + ΨnB,Thus the rate
of growth in varieties is given by
˙ni = l(nA + ΨnB)/a,
and the zero profit condition (assuming positive growth) in the research sector
is
vi = w
ia
na + ΨnB.
1We only calculated the growth rate of varieties,but it is easy to show that the growth
in the aggregate,good” D is proportional to g.
2For which there are no incentives,since companies do not sell their products in the
other country.
CHAPTER 15,ENDOGENOUS GROWTH AND TRADE 103
Substituting these into the labor market clearing condition and into the asset
equation,and using the conditions for r,pi and p,we can solve for the steady
state growth rate exactly as above:
gi = (1?α)L
A + ΨLB
a?αρ.
Thus the rate of innovation increases in both countries,and it increases by
more for the smaller one that can now draw form the large stock of knowledge
in A,Thus openness in the sense of no barriers to knowledge flows is beneficial
for both countries.
15.3 Trade with knowledge diffusion
Now we allow for trade in goods in addition to the perfect dissemination of
ideas,Let us write
E = EA + AB,
and normalize E = 1,We will look at a steady state with positive innovation
where the two countries’ share in world income is constant,Then the interest
rate in both countries must be ρ (see consumer problem above),The share
of i in income equals the share of spending on goods produced in i,and can
be written as
si = n
i(pi)1?σ
nA(pA)1?σ + nB(pB)1?σ.
Then the spending share of an individual good in country i is given by si/ni.
The pricing equations are the same in both countries,pi = wi/α,Using
the spending share from above,profits are given by pii = (1?α)si/ni,and
the asset equation is now written as
rivi = ˙vi + (1?α)s
i
ni,
The free entry condition in research is the same as for the closed economy,
except that now the stock of knowledge is the number of varieties worldwide
(with global competition there are no redundancies):
vi = w
ia
n,
CHAPTER 15,ENDOGENOUS GROWTH AND TRADE 104
where n = nA + nB,Finally,labor markets clear in country i if
a˙ni
n +
si
pi = L
i.
We again introduce the variable V i = 1/(vini) and look for a steady
state where V i is constant,Since ˙V i/V i =?˙vi/vi? ˙ni/ni,the inverse of the
aggregate value of the stock market is constant if
˙vi
vi =?
˙ni
ni ≡?g
i.
Substituting this into the asset equation and using ri = ρ we get
siV i = ρ + g
i
1?α.
Now we use the labor market clearing condition together with the zero profit
condition in research to write
ni
n (ag
i + αasiV i) = Li.
The last step is to conjecture that the growth rates in the two economies
are the same and to verify that conjecture,Thus we set gA = gB = g,and
substitute it into the last two equations,Since the same growth rate means
that sAV A = sBV B (see above),we can add up the last equation above for
A and B and use nA + nB = n to solve for g:
g = (1?α)L
A + LB
a?αρ.
Plugging this back into the other equilibrium conditions,we verify that si,
ni/n and siV i are constant along the balanced growth path,so we indeed
found the steady state,Thus the two economies grow at the same rate,which
depends on the common model parameters plus the total labor force in the
world.
Two things are important to note,First,the results are identical to the
ones we would get if we calculated them for the integrated world,Thus in
this setting trade in goods and the free flow of ideas reproduces the integrated
world equilibrium,Both countries grow faster than in autarchy,although the
CHAPTER 15,ENDOGENOUS GROWTH AND TRADE 105
gain is larger for the smaller country,Second,the growth rate is the same
as in the previous chapter when there is no duplication of research,so that
Ψ = 1,The two outcome,however,differ in an important way,When there
is no trade,consumers have access only to local varieties,whereas now they
can consume a wider range of goods,Thus they are strictly better off with
free trade than without,The increase in varieties is a level effect,since it
does not influence the growth rate of the economies,Access to the world
stock of knowledge,however,has a growth effect,since long run growth rates
depend on the world population instead of the local ones,Trade in goods has
a growth effect only to the extent it eliminates the duplication of research
effort.
15.4 Trade with no knowledge diffusion
We now explore the opposite possibility to the first one,when there is free
trade in goods but ideas do not travel across borders,We can still use some of
the equations from above with small modifications,which take into account
that productivity in research now depends on ni instead of n,In particular,
the labor market clearing equation becomes
agi + s
i
pi = L
i,
and the zero profit condition in research is
vi = aw
i
ni,
Using the asset equation,the pricing formula (these do not change) in
that equation gives
˙wi
wi = ρ + g
i? (1?α)s
i
awi =
gi? ˉgi
α,
where we get the second equality by using labor market clearing to substitute
for si/wi and
ˉgi = (1?α)L
i
a?αρ,
CHAPTER 15,ENDOGENOUS GROWTH AND TRADE 106
the autarchy growth rate,Next we differentiate the definition of si to get the
rate of change in that variable:
˙si
si = (1?s
i)
bracketleftbigg
(gi?gj) + (1?σ)
parenleftbigg ˙pi
pi?
˙pj
pj
parenrightbiggbracketrightbigg
.
Since markups are constant,we have ˙pi/pi = ˙wi/wi,which leads to
˙si
si = (1?s
i)[(1?σ)(gi?gj) + σ(ˉgi? ˉgj)].
Finally,differentiating the resource constraint and using the equation for
˙wi/wi we get
˙gi =
parenleftbiggLi
a?g
i
parenrightbiggbracketleftbigg1
α(g
i? ˉgi)? ˙s
i
si
bracketrightbigg
.
To find the steady state,we conjecture that the share of the larger country
(assumed to be A) approaches 1 in the long run,From the equation for ˙sA/sA
we can see that in steady state ˙sA = 0,which together with Li/a > gi implies
gAss = ˉgA.
It is easy to see that gA must approach its steady state value from above,
otherwise the system would not converge (see equation for ˙gA above),For
the small country,we can show that the rate of change in sB does not go to
zero,even if sB tends to a constant (zero),From above,we have
˙sB
sB → (1?σ)(g
B
ss?g
A
ss) + σ(ˉg
A? ˉgB)].
Then it is easy to calculate the steady state growth in the small country’s
knowledge stock:
gBss = ˉgB? α(1?α)1?α(1?α)(ˉgA? ˉgB).
Thus we conclude that the large country’s rate of innovation increases for
a while,but eventually it returns to the autarchy value,The small country’s
technical progress decreases,until it reaches the steady state value above.
The small country suffers dynamic losses from trade,as its rate of growth
decelerates,As usual,the welfare calculation is more ambiguous,since there
are still the static gains and the number of available varieties increase faster
for country B than they did in autarchy,But it is possible for B’s welfare to
decrease,In fact,it is possible to show that the autarchy growth rate is too
low in both countries,so that trade exacerbates the already existing market
failure in B.
CHAPTER 15,ENDOGENOUS GROWTH AND TRADE 107
15.5 Imitation and North-South trade
We study a model of imitation that is based on the variety approach above.
We have two economies,North and South,We assume that only the North
can discover original products,but the South is able to copy existing ones.
This might arise when international pattern protection is weak,and an ex-
ample is the widespread phenomenon of,reverse engineering”.
We use many of the building blocks from above,The pricing equation for
Northern firms is still the same pN = wN/α,and Northern profits are
piN = (1?α)pNxN,
where xN stands for the sales of a typical Northern firm,The zero-profit
condition applies for entering into research in the North,which gives
vN = aw
A
n,
with n being the stock of knowledge capital.
In the South,no new innovation is possible,but companies might learn
how to manufacture existing Northern products,As long as they face a lower
wage rate than the North,they can capture the market by underpricing the
innovator of the good,In fact we assume that wages are sufficiently low such
that the Southern firm can charge its monopoly price3,which is pS = wS/α
and thus has a profit
piS = (1?α)pSxS.
Imitation is a costly activity,and its unit labor cost is am/nS,Thus research
productivity in the South depends positively on the number of varieties that
have been successfully copied,Then free entry into research implies
vS = amw
S
nS,
The asset equations that relate the values of companies to their profits
and capital gains can be written as follows,In the South,once an imitator
succeeds it keeps the market forever,thus
ρvS = piS + ˙vS.
3It is possible to derive conditions on the primitive parameters of the model that
guarantee this,see GH for details.
CHAPTER 15,ENDOGENOUS GROWTH AND TRADE 108
A Northern innovator also faces the possibility that its product is imitated.
We assume that the South targets all products with equal intensity,which
means that the probability to imitate any particular product is ˙nS/nA,Then
the asset equation for the North is
ρvN = piN + ˙vN? ˙n
S
nA.
Notice that we assumed that the two countries have a constant share in
spending in the long run,which after normalizing world spending to 1 leads
to rN = rS = ρ.
Since the growth in Southern varieties is exactly offset by a decrease in
Northern ones,the growth in the knowledge stock equals the growth in nA.
Thus the labor market clearing conditions are
a˙n
n + n
NxN = LN
for the North and
am ˙nS
nS + n
SxS = LS.
Finally,demand for any good is given by
xi = p
i)?σ
nN(pN)1?σ + nS(pS)1?σ.
Now we are ready to characterize the steady state,We assume that the
share of each country in the total number of varieties ni/n is constant,This
implies that the number of varieties must grow at the same rate in the two
countries,since g = nA/ngA + nS/ngS,Thus we have
gi = g.
Let us introduce the variable m = ˙nS/nA,the steady state imitation rate.
Then it is easy to show that
nS
n =
m
g + m.
It can be verified retrospectively that the aggregate value of the stockmarket
in the North is constant,thus ˙vA/vA =?˙nA/nA,Substituting these into the
asset equation of the North we have
piN
vN = ρ + g + m.
CHAPTER 15,ENDOGENOUS GROWTH AND TRADE 109
Using the pricing equation and the labor market clearing for the North in
the profit equation,we get
piN = (1?α)w
N
α(1?nS/n)n(L
N?ag).
Combining these and the free entry condition for the North gives us an
equation linking g and m:
1?α
α
parenleftbiggLN
a?g
parenrightbigg g + m
g = ρ + g + m.
This equation gives us a positive relationship between g and m,Using the
same step for the South,we can calculate
1?α
α
parenleftbiggLS
am?g
parenrightbigg
= ρ + g.
The equilibrium values of g and m are given by these two equations,From the
second,we can determine the rate of growth in varieties in the two economies:
g = (1?α)L
S
am?αρ.
The rate of imitation m can be calculated from the first equation linking
g and m,It is not very illuminating,but it gives us some constraints that
need to be satisfied for g,m > 0,Thus we need
LN
a <
LS
am <
LN
a +
αρ
1?α.
To understand these conditions,suppose the two countries are of the same
size,Then we need that the cost of imitation is lower than the cost of
original innovation,but not very much lower,Given this condition (which is
not unreasonable),there innovation in the North and imitation in the South.
What happens to the growth rates of the two countries compared to
autarchy? Perhaps surprisingly,the rate of growth is greater in free trade for
both the North and the South,The former follows from the condition that
LS/am > LN/a and the latter from the natural assumption that imitation is
less costly in the South than original innovation,If that assumption holds,
it is natural that the South gains,since it can get new products cheaper
CHAPTER 15,ENDOGENOUS GROWTH AND TRADE 110
than in autarchy,In the North we have two opposing forces,First,Southern
imitation shortens the duration of a monopoly,which is a negative effect on
innovation,Second,imitation lessens the competition for Northern labor for
the remaining firms,which increases production,sales and profits,In our
setting the second force dominates,thus we have an increase in Northern
growth,But this result very much depends on the specific functional forms,
and does not generalize.