Download Economic Growth with Trade in Factors of Production

Economic Growth with Trade in Factors of Production
Karine Yenokyan*
Nazarbayev University
John J. Seater
North Carolina State University
Maryam Arabshahi
Chester…eld MO
December 2012
Abstract
We study the world trading equilibrium in a Ricardian model where factors of production
are themselves produced and tradable rather than endowed and non-tradable, corresponding to
the three-quarters of international trade that is in intermediate and capital goods. We show
that trade a¤ects economic growth purely through comparative advantage, even in the absence
of technology transfer, research and development, and international investment, and also in the
absence of aggregate scale e¤ects. Trade may raise a country’s growth rate or leave it unchanged.
When a world balanced growth rate exists, trade always raises the growth rate of both trading
partners. Otherwise, either one partner’s growth rate is increased and the other is una¤ected or
neither partner’s growth rate is a¤ected, depending on the patterns of comparative and absolute
advantage. Trade’s e¤ect on a country’s growth rate depends on the nature of the imported good
and not the exported good, a result contrary to the direction of many countries’export policies.
Trade in factors of production e¤ectively transfers technology by producing an equilibrium identical
to that which would obtain if technology had been transferred between trading partners even
though no such transfer actually occurs, which suggests that existing empirical evidence on the
relation between trade and technology transfer may not be evidence that trade facilitates actual
technology transfer. We show the conditions under which factor price equalization, the StolperSamuelson theorem, and the Rybczynski theorem hold. We perform a numerical analysis of the
transition dynamics, which appear to be saddle-point stable but may be monotonic or oscillatory
in converging to the balanced growth path.
Keywords: trade, growth, comparative advantage, world income distribution, e¤ective technology transfer
JEL classi…cation codes: O4, F15
*We thank three very professional and helpful referees for comments that greatly improved the
paper and also Areendam Chanda, Robert Kane, and Pietro Peretto for earlier comments that
helped form the paper..
0
1
Introduction
Does trade promote economic growth? Romer’s (1986) seminal article on endogenous growth
theory provided economists with the proper framework for addressing that important question.
Investigators soon started using that framework to study the growth e¤ects of trade. Among the
early contributions were Grossman and Helpman (1990, 1991), Rivera-Batiz and Romer (1991),
and Young (1991). Later insights were provided by Taylor (1993), Feenstra (1996), Barro and
Sala-i-Martin (1997), Ventura (1997), and Redding (1999). Research on the topic has continued
up to the present in work such as Connolly (2000), Howitt (2000), Acemoglu and Ventura (2002),
Galor and Mountford (2006), and Coe, Helpman, and Ho¤maister (2008). That literature’s answer
to the question of whether trade promotes growth generally has been "Yes," with trade’s e¤ect
working through two channels: an aggregate scale e¤ect and technology transfer. The scale e¤ect
arises from the increase in …rms’market size that opening to trade enables. A larger market size
makes …rms more pro…table and so leads them to do more of the activities that cause economic
growth. The technology transfer channel arises from trade’s facilitating knowledge spillovers as
countries set up lines of communication with their trading partners.
Surprisingly, the existing work actually leaves unanswered the question of whether trade itself
promotes growth. The presence of aggregate scale e¤ects has been decisively rejected by the data,
starting with the well-known article by Backus, Kehoe, and Kehoe (1992), so that .channel for
trade to in‡uence growth can be dismissed, taking with it all the explanations of trade’s growth
e¤ects that depend on the scale e¤ect. The second-generation fully endogenous growth literature
(Peretto, 1998; Howitt, 1999) provides the theoretical reason for scale e¤ects to be absent. The
data are much kinder to technology transfer facilitated by trade, which seems to be statistically
and economically signi…cant (Coe and Helpman, 1995; Coe, Helpman, and Ho¤maister, 2008).
However, in that mechanism it is not trade that a¤ects economic growth but the technology
transfer that trade facilitates. Without the technology transfer, trade would have no e¤ect on
growth. Conversely, without trade, technology transfer still would a¤ect growth as long as some
of it can occur independently of trade, which seems realistic. Interestingly, there also is evidence
that trade may a¤ect growth directly rather than through technology transfer. Alcala and Ciccone
(2004), for example, …nd strong and robust e¤ects of trade on growth. Their measure of openness
to trade does not depend in any way on technology transfer and so suggests that trade in and of
itself boosts growth. This conclusion is buttressed by Wacziarg and Welch’s (2003) …nding that
increased trade liberalization is positively associated with increased growth. Although these results
are not decisive, they do suggest that trade per se a¤ects economic growth. What is missing is a
theoretical argument showing why trade should have such an e¤ect.
The question we want to address, therefore, is precisely whether trade per se a¤ects growth
in a model without empirically-rejected aggregate scale e¤ects. Surprisingly little literature has
examined that question. Indeed, we are aware of a single article that uses the second-generation
framework (which has no aggregate scale e¤ect) to study the interaction of trade and growth:
Peretto (2003), In that model, the growth e¤ects of trade arise through knowledge spillovers,
which are a passive form of technology transfer. Trade itself does not promote growth.
We present a model of trade and growth that has neither aggregate scale e¤ects nor technology
transfer. We show that trade, in and of itself, can raise growth through the same comparative
advantage mechanism that raises the income level in static models without growth. The crucial
elements are that growth is endogenous and that the factors of production are tradable. The trade
part of the model is Ricardian, with trade driven by cross-country technology di¤erences. The
model di¤ers from standard Ricardian models in that factors of production are not endowed but
rather are produced. The growth part of the model is the general two-sector model described by
Barro and Sala-i-Martin (2004, chapter 5). There are two goods, both of which can be produced
by each country. The two goods are produced in separate sectors. In the main analysis, each
good can be used as a factor of production, and both goods are essential for production in both
1
sectors. Allowing the factors of production to be produced permits endogenous growth. Allowing
the factors of production to be traded generates growth e¤ects of trade. The model is su¢ ciently
tractable to allow analysis of the transition dynamics as well as the balanced growth path (hereafter,
BGP).1
We obtain several interesting results. First, trade working solely through comparative advantage can raise countries’ growth rates. That is not to say, of course, that other channels, such
as technology transfer, are unimportant. What it does say is that the most important channel
through which trade has economic e¤ects in traditional static models also operates on growth rates,
something for which there has been surprisingly little theoretical support heretofore. Second what
matters for the e¤ect of trade on a country’s growth rate is the type of good it imports, not the
type it exports. Speci…cally, importing a factor of production increases a country’s growth rate,
whereas importing a consumption good has no e¤ect on the growth rate. The type of good that
is exported is irrelevant to the exporting country’s growth rate, irrespective of whether or not it is
a factor of production. Third, trade can equalize countries’growth rates and therefore lead to a
stable distribution of GDPs across countries, but that is not a necessary outcome. Growth rates
may remain permanently unequal after previously autarkic economies open to trade, leading to a
permanently widening of the gap between their levels of GDP. The case of equal growth rates is
the same as Acemoglu and Ventura’s (2002) result, and it occurs when the world is in an interior
Ricardian trade pattern, with each country completely specializing in producing one of the two
factors of production and trading to obtain the other factor. The case of unequal growth rates
occurs when the world is in a corner Ricardian solution, in which one country does not specialize,
and is a case Acemoglu and Ventura did not consider. The model thus o¤ers a generalization of
theirs that seems to correspond to some observed cases in the historical data. Fourth, trade in
factors of production leads to a world equilibrium that is either identical or similar to the equilibrium that would prevail if countries transferred technology to their partners, even though in the
model no technology transfer actually occurs. The identical case arises in the interior solution,
and the similar case arises in the corner solution, mentioned in the previous result. Thus we have
a sort of technology equalization result, similar to the factor price equalization theorem, according
to which trade can e¤ectively equalize technology in whole or in part without any exchange of technology. Fifth, we show that factor price equalization holds under conditions exactly opposite those
required in a Hecksher-Ohlin model with endowed factors and that neither the Stolper-Samuelson
e¤ect nor the Rybczynski theorem hold in this kind of model. The analysis casts new light on factor
price equalization, showing that it arises when technology is the same across countries, either by
assumption (as in Hecksher-Ohlin) or because trade makes it so (as in the Ricardian model used
here). Sixth, our numerical analysis of the transition dynamics suggests that the balanced growth
path is saddle point stable and that the transition paths may be monotonic or oscillatory. The
endogenous nature of the world relative price is important in determining the transition dynamics,
which are completely di¤erent from the dynamics of the two sector closed economy endogenous
growth model, despite the fact that current model is based on the same structure as the closed
model. Trade introduces important new elements.
2
Model Speci…cation
The analysis is based on an extension of the standard two-sector growth model discussed by Barro
and Sala-i-Martin (2004, chapter 5) to allow trade between two countries. Indeed, the economic
structure of each of the two countries in our model is identical to that of the standard model. The
1 In all the literature on endogenous growth with international trade, only the pioneering but unfortunately
unknown study by Bond and Trask (1997) relies on pure comparative advantage to generate e¤ects of trade on
growth in a model without scale e¤ects or technology transfer. We compare their results with ours below.
2
only new element is that we allow trade. Because the speci…cation is mostly standard, we consign
all derivation and mathematical detail to the Appendix.2
2.1
Production and Preferences
Country i produces two goods, YiCK and YiH . Good YiCK can be used for consumption C or
can be used as gross investment I K in one kind of capital K. Good Y H can be used only as
gross investment I H in a second type of capital H produced in a di¤erent sector from Y CK by a
di¤erent technology. Both YiCK and YiH are tradeable. Each sector’s production technology is
Cobb-Douglas with country-speci…c parameter values:
YitCK = Ai (vit Kit ) i (uit Hit )1
YitH = Bi [(1
(1)
i
uit )Hit ]1
vit )Kit ] i [(1
(2)
i
CK
where v and u are the fractions of total K and H used in the Y
-producing sector and Ai , Bi ,
,
and
are
constants.
Both
K
and
H
are
freely
mobile
between
the two sectors, so v and u
i
i
can take any value in [0; 1] at any time t. Each of these production functions is homogeneous
of degree 1 in K and H, thus satisfying the critical requirement for endogenous growth that the
marginal products of the reproducible factors of production be bounded away from zero (Jones
and Manuelli, 1990).3 Both K and H depreciate at rate , which is the same for both countries.
The equations for the accumulation of K and H are
K_it = Ai (vt Kit )
H_ it = Bi [(1
i
1
i
(ut Ht )
vit )Kit ] i [(1
Ct
uit )Hit ]1
Kit
i
(3)
Hit
We can de…ne gross domestic product:
Yit = YitCK + pYitH
(4)
where p is the price of Y H in terms of Y CK . Utility is CRRA, so lifetime utility is:
U=
Z
0
2.2
1
1
Cit
1
1
e
t
dt
Relation to Other Models
Endogenous growth models fall into four broad classes. The …rst class comprises R&D models of
variety expansion or quality improvement. Those can be divided into three sub-classes: (1) the
…rst-generation fully endogenous models of Grossman and Helpman (1991) or Barro and Sala-iMartin (2004, chapters 6 and 7), and the like, (2) semi-endogenous growth models derived from
Jones (1995), and (3) second-generation fully endogenous growth models of Peretto (1998), Howitt
(1999), and their o¤spring. The second class comprises models based on learning-by-doing with
knowledge spillovers, such as Romer (1986). The third class contains the two-sector models, such
as Barro and Sala-i-Martin (2004, chapter 5). The fourth class comprises models based on CES
production with a high elasticity of substitution between a reproducible factor, such as capital,
2 There
are two mathematical appendices, one short and one long. The short appendix is included at the end of
this paper and addresses only a few major points of the analysis. The long appendix is available from the authors
upon request and is more complete.
3 This 2-sector model is not an AK model. AK models have no transition dynamics, whereas a two-sector model
such as this one has very rich dynamics. We discuss the transition dynamics below. See Chapter 5 in Barro and
Sala-i-Martin (2004) and Bond, Wang, and Yip (1996) discussions of the two-sector model’s dynamics for a closed
economy.
3
and non-reproducible factor, such as labor, as discussed in Barro and Sala-i-Martin (2004, chapter
1). The AK model is a special case of the CES model with an in…nite elasticity of substitution
and a coe¢ cient on labor of zero.4
Even though there is no R&D in our model, we interpret our framework as an approximation
to a second-generation fully endogenous R&D growth model. That interpretation is motivated by
the way we think of H. One common interpretation of H is as human capital, as in Uzawa (1965),
Lucas (1988), or Barro and Sala-i-Martin (2004, chapter 5). However, another interpretation of
H that is more suitable for our purposes is technical progress embodied in physical capital that
augments labor. Gort, Greeenwood, and Rupert (1999) have shown that technical progress of that
type is important, accounting for about 52 percent of economic growth. Often technical progress is
treated as augmenting the factor in which it is embodied, but embodiment and augmentation have
no necessary connection to one another. The factor that embodies a technology is not necessarily
the factor augmented by that technology. For example, consider a quilt maker using a traditional
sewing machine. Machine quilting requires a considerable amount of skill on the part of the quilter,
who must move the fabric under the needle to produce the patterns of stitching. Quilting sewing
machines have recently become available that operate by moving the machine over the fabric, which
turns out to require far less skill on the part of the user. Thus a given quilt can be produced by
a given person who either acquires a traditional machine and su¢ cient skill to use it or acquires
a quilting machine and does not bother with the skill. The technical progress embodied in the
quilting machine acts exactly like skill embodied in the worker, and so such progress should enter
the production function in the same way: as a labor-augmenting reproducible factor - that is, as
H - even though it is not embodied in the worker. Aghion and Howitt (2005) and Peretto (2007)
present models in which technical progress augments labor but is embodied in goods. That is just
the framework we need for our analysis, but we simplify it because it is very complicated for a
model that also has trade in factors of production. We therefore outline the approach and then
propose a simpli…ed version that is suitable for our purposes.
Peretto (2007) considers an economy in which …nal goods F are produced with a variety of
intermediate goods Ri and labor L. The production technology for a …nal good F is
F =
Z
N
Ri Zi Z 1
Li
1
di
0
where and are constants between zero and one, Ri is the quantity of intermediate good i, N
is the number of varieties of intermediate goods, Li is the quantity of labor using Ri , Zi is the
quality of Ri , and Z is the average value of the Zi , given by
Z=
1
N
Z
N
Zj dj
0
Notice that the quality Zi is embodied in Ri but augments labor in …nal goods production. The
intermediate goods Ri are produced by monopolistic competitors, who also do R&D to raise the
quality Zi of their product Ri . Increases in Zi raise the demand for Ri and thus also raise the price
of Ri , which in turn increases the monopoly pro…t of Ri ’s producer. That (incipient) increase in
pro…t induces new …rms to pay a sunk cost to enter the intermediate goods industry, raising N .
Peretto’s model provides many useful insights, but its complexity renders it di¢ cult to use
in analyzing trade. It e¤ectively has three sectors - …nal goods, intermediate goods, and R&D together with a mixture of perfect and imperfect competition, endogenous entry, and both variety
expansion and quality improvement. We therefore simplify the model in the following ways.
What we need for our subsequent work are two tradable goods that are factors of production,
4 Ventura (1997) uses a CES model to discussion of how trade, through comparative advantage, distributes growth
among countries. In Ventura’s model, trade has no e¤ect on the world’s balanced growth rate (see his equation 11).
4
one of which augments labor. We do not need an expanding variety of intermediate goods for
what we are doing, so we …x the number of varieties at 2. The two varieties are types of physical
capital. One corresponds to Peretto’s R and is standard physical capital that enters the production
function in the usual way. Converting a non-durable intermediate good into a durable capital good
always is feasible, as Peretto remarks. The other capital good takes the place of Peretto’s Z, which
already is a capital-like stock variable, and enters the production function as a labor-augmenting
factor. In the Aghion and Howitt and the Peretto frameworks, replacement of intermediate goods
by capital goods complicates the analysis, whereas in our case, it simpli…es it by allowing us to use
the symmetry inherent in the two-sector model. We also suppress labor so that we do not have
to deal with the issue of having more claimants on output than there is output to be distributed.
Solving that problem requires introducing imperfect competition so that factors are not necessarily
paid their marginal product, but in a two-sector model imperfect competition is di¢ cult to deal
with. We thus treat H as a second type of capital that enters the production function in a way
that is complementary to K, giving constant returns to scale.
Several articles in the literature use the two-sector model to study trade and growth. See, for
example, Bond and Trask (1997), Bond, Trask, and Wang (2003), Farmer and Lahiri (2005, 2006),
and Hu, Kemp, and Shimomura (2009). The distinguishing feature of our model is that we allow
for two tradable factors of production, not just one, something that has important implications for
the e¤ects of trade on growth.
3
Trade Between Two Large Countries
We now introduce foreign trade. We have two countries, 1 and 2, with di¤erent …xed production
technologies, discussed momentarily. In our framework of a two-factor Cobb-Douglas production
function, cross-country technology di¤erences are captured in di¤erent values of the total factor
productivities A and B and of the factor share parameters and . We assume that C and new
units of K and H are tradable but that the existing stocks of K and H are not tradable. In other
words, investment goods (new units of K and H) are free to move about the world, but once the
investment has been put in place, the resulting stocks of K and H are immovable. A factory
is an example. The materials to build the factory can be shipped abroad. Once the factory is
assembled, however, it is immovable. We restrict attention here to sale of investment goods of
one type in exchange for investment goods of the other type, that is, exchange of new units of
K for new units of H. Under this assumption, a country that exports new units of K gives up
ownership of those units and accepts in return ownership of the new units of H that it receives as
imports. We thus exclude net foreign investment, in which new units of capital are sent abroad
but ownership is retained. The reason is mathematical tractability. If we allowed net foreign
investment, we would have four stocks of capital in each country - K and H owned by domestic
residents and K and H owned by foreigners. That would lead to eight state variables in the model,
and the analysis would be totally intractable. By restricting attention to pure trade of K and H,
we keep the number of state variables down to four, which is tractable.
It is traditional to begin (and often end) discussion of trade with the case of a “small”country,
in which the country studied is an atomistic agent in the world economy. Here, however, it
is preferable to follow Grossman and Helpman (1991, chapter 9) and analyze two countries of
arbitrary size. The small country is a straightforward special case of this more general framework
that we leave to the reader to work out.5 To avoid complications of bilateral monopoly, we
suppose that each country’s economy consists of a large number of competitive …rms with identical
5 We do provide one major hint. With two large countries, the world price p responds to what both countries are
doing, and that response a¤ects the world’s dynamic path. With a small country, the world price does not respond
to what the small country is doing.
5
production functions, preventing either country from acting as a monopolist and thus guaranteeing
a competitive solution.6
Let X denote exports of Y CK , so that p
straints for country i are
1
X is imports of Y H . Then the accumulation con-
K_it = Ai (vit Kit ) i (uit Hit )1
H_ it = Bi [(1
i
+ Xit
uit )Hit ]1
vit )Kit ] i [(1
Cit
i
Kit
1
+ Xit
p
(5)
Hit
(6)
When economies are closed, X = 0. When a country i is open, it must choose X along with
everything else. Countries 1 and 2 are linked through trade, so the solutions for their growth rates
must be determined simultaneously. The key variable that guarantees world general equilibrium
is the price p, which now is determined to guarantee international trade balance. Because neither
country is small, equilibrium p depends on what both countries do, so that p, too, must be determined as part of the simultaneous solution for the two countries. We …nd the world general
equilibrium in two steps. First, we solve each country’s quasi-central planning problem, taking p
as given; then we impose the trade balance condition X1 = X2 to …nd the equilibrium value of
p7 .
It is important to note here that scale e¤ects, research and development, and technology transfer
all are absent from this model. Consequently, any e¤ects that trade has on growth will not be due
to those in‡uences but rather to comparative advantage alone.
3.1
Individual Country Solutions
With trade, country i’s Hamiltonian is
Vi =
Ci1
1
i
e
t
+
Bi ((1
i
Ai (vi Ki ) i (ui Hi )1
i
ui )Hi )1
i
vi )Ki ) i ((1
Ci
Hi +
Ki
Xi
p
Xi +
(7)
where and are the costate variables. The control variables are C, X, v, and u. The important
necessary condition for discussion here is the …rst-order condition for X:
@Vi
=
@Xi
i
+
i
p
=0
(8)
The other necessary conditions are given in the Appendix.
The Hamiltonian is linear in X, so the …rst-order condition (8) for X does not depend on
any control variable. We thus have bang-bang control for Xi . When
i + ( i =p) is positive,
equation (8) cannot be satis…ed; the marginal value of X equals
+
( i =p) and always is
i
positive, irrespective of the value X. Consequently, country i sets Xi as high as possible, which
it does by producing only Y CK and exporting some of it to obtain H. The opposite holds when
H
and
i + ( i =p) is negative. Country i then sets Xi as low as possible, producing only Y
obtaining Y CK solely through imports (so that Xi is negative). When
+
(
=p)
equals
zero,
i
i
country i does not engage in trade, and X is zero. To see this, note that
i + ( i =p) = 0 is
6 Analysis of the case where countries’national governments act as representatives for their countries’…rms and
bargain with other governments would be interesting but is beyond the scope of the present paper.
7 Equivalently, we could solve for world general equilibrium as a world central planning problem, obtaining u ,
1t
u2t , v1t , v2t , C1t , C2t , Xt , and pt in a single step. The two-step approach gives a more intuitive view of what each
country is doing.
6
equivalent to p = i = i . The price p is the international price for Y H in terms of Y CK , that is,
the ratio of marginal utilities of Y H and Y CK . The costate variables i and i are, respectively,
country i’s marginal utilities of Y CK -goods and Y H -goods from internal production. Their ratio
is the marginal value of Y H -goods in terms of Y CK -goods if country i produces both goods; that
is, the ratio is country i’s internal price for Y H in terms of Y CK . If this internal price equals
the external (world) price p, country i can obtain the same number of units of Y CK in exchange
for Y H from its own internal operations as it can by trading on the world market. Country i is
indi¤erent on the margin between trade and autarky. The borderline case of indi¤erence to trade
will prevail no more than momentarily because, as i and i vary over time, they generally will
not satisfy the equality p = i = i . Consequently, we ignore the knife-edge case henceforth.
Comparisons of the world price p with the internal prices i = i are central to all that follows.
The results of the previous paragraph suggest that country i specializes in producing Y H when
p > i = i and in producing Y CK when p < i = i .
3.2
Balanced Growth Rates in World Equilibrium
We begin with the BGPs for the two countries and for the world as a whole. We address the
transition dynamics later.
3.2.1
Balanced Growth Paths under Autarky
The solution for the balanced growth rate under autarky is standard. The growth rates of C, K,
H, Y CK , and Y are equal. Their common value is
=
1
i
i+ i
1
Ai
1
1
Bi
i
i+ i
1
i i
i+ i
(1
i
i)
(1
1
i) i
i+ i
(1
1
i) i
i+ i
i
(1
(1
i)
1
i )(1
i)
i+ i
(9)
The value of p is
pi =
Ai
Bi
1
i)
1
i)
(1
i
i (1
i
i
i
1
1
i+ i
(10)
i
See the Appendix and the references cited there for details.
3.2.2
Balanced Growth Paths with Trade: Derivation
The equilibrium world price p must fall between 1 = 1 and 2 = 2 ; otherwise, both countries
would try to specialize in and export the same good, violating international balance of trade.
Which country has the higher value of i = i is arbitrary, so we assume without loss of generality
that 1 = 1 > 2 = 2 . Since the ratio of costate variables represents the internal price for Y H in
terms of Y CK , the ratio is equivalent to the autarkic price level in each country given by (10). We
then have
p2 =
A2
B2
(1
2
2 (1
2
2
1
2)
1
2)
2
1
1
2+ 2
p
2
A1
B1
(1
1
1 (1
1
1
1
1)
1
1)
1
1
1
1
1+ 1
= p1
(11)
For p to be strictly in the interior of the closed interval [p2 ; p1 ], country 1 sets v = u = 1, specializes
in production of Y CK , and trades to obtain Y H . Country 2 does the opposite, producing only
Y H and trading for Y CK . When p is on the boundary of the interval, equaling either p1 or p2 , we
have extra complications. For now, we restrict attention to the interior (i.e., the case where the
inequalities in (11) are strict), discussing the corner cases afterward.
7
The …rst-order conditions for C and X are unchanged from the unconditional problem, but the
…rst-order condition for X now holds with equality. Having accepted the world price p and agreed
to specialize in producing Y CK , country 1 now chooses and to satisfy (8) exactly. The same
kind of manipulations as for the autarkic model show that the growth rates of C, Y CK , K, and
Y all equal the growth rate of consumption. Because equation (8) now always holds, we have
p = 1 = 1 . Trade balance constrains p to lie in the closed interval [p2 ; p1 ]. The growth rates for
country 1 and 2 now are
"
#
1
1
1
1
1
1
1
A1 1 (1
(12)
1)
1;T =
p
2;T
1
=
B2
2
2
1
2)
(1
2
p
(13)
2
where the subscript T indicates that this growth rate pertains when the countries trade.
Balanced growth requires that everything that grows must do so at a constant rate. The only
growth rate for p consistent with both these requirements is zero, so p must be constant along the
BGP. The ratio 1 = 1 therefore also is constant, implying that the growth rates of 1 and 1 are
equal. It is straightforward to show that, for each country to have balanced growth individually,
the two countries must have the same growth rate:
1;T
=
2;T
With the two countries growing at the same rate, we have balanced growth for the world. Equating
(12) and (13) and solving for p gives
A1
B2
p=
(1
2
2 (1
1
1
1
1)
1
2)
1
1
1+ 2
1
(14)
2
Balanced growth thus requires that
A2
B2
(1
2
2 (1
2
2
1
2)
1
2)
1
2+ 2
1
2
A1
B1
2
1
1)
1
1)
(1
1
1 (1
1
1
1
1
1
1+ 1
(15)
1
It is straightforward to show that
1;T
Substituting p =
as p Q
2;T
(16)
into (12) or (13) gives the common growth balanced growth rate
=
1;T
where
=
R
1
1
1 2
1+ 2
(1
1)
(1
1
1) 2
1+ 2
=
(1
1
2
=
2;T
1) 2
1+ 2
1
[
]
(1
(1
2)
1
1 )(1
2)
1+ 2
(17)
A11
We discuss stability of the BGP below.
3.2.3
Balanced Growth Paths with Trade: Implications
The foregoing results lead to two interesting and important implications.
8
2
1+ 2
1
B21
1
1+ 2
Response of growth rates to trade In the interior solution that we are discussing here, where
the world price p is inside the bounds given in (11), we obtain the important result that trade
increases the growth rate of both trading partners. That is easily seen by comparing the growth
rates under trade with those under autarky:
"
#
"
#
1
1
1
1
1
1
1
1
1
1
1
1
1
1
=
A1 1 (1
> 1;Au =
A1 1 (1
1)
1)
1;T
p
p1
2;T
=
1
B2
2
2
(1
1
2)
2
p
2
>
2;Au
=
1
B2
2
2
(1
1
2)
2
p2 2
The economic intuition is straightforward. In the interior trading equilibrium, each country
specializes in the good of its comparative advantage and abandons production of the other good,
obtaining it through trade instead of relatively ine¢ cient domestic production. From the world’s
point of view, productive resources in each country have been shifted from ine¢ cient to e¢ cient
uses. Trade acts like technical progress, raising world Total Factor Productivity by leading to
abandonment of the lower values of A and B. With higher TFP, growth rates rise. As we will see
momentarily, however, that result holds only in the interior. When the world price is on either
boundary of the interval de…ned in (11), trade does not increase both countries’ growth rates.
However, it never lowers either partner’s growth rate.
E¤ective technology transfer Equation (17) shows that, in the interior, trade not only raises
both countries’growth rates but also equalizes them. The economic intuition is that in the trading
equilibrium the two countries e¤ectively share each other’s relative e¢ ciency. For example, before
trade, country 1 produces both Y CK and Y H . Domestic production of Y H uses the comparatively
ine¢ cient technology of the domestic Y H sector. When trade starts, country 1 shuts down its
own Y H sector and relies instead on country 2’s relatively e¢ cient Y H sector. Country 2 is in a
symmetric position, shutting down its Y CK sector and relying instead on its partner’s relatively
e¢ cient Y CK sector. The result is that each country ends up relying on exactly the same mix of the
two countries’technologies, leading to equal growth rates. In e¤ect, each country has transferred
to itself the more e¢ cient technology of its trading partner, even though no technology transfer
actually occurs.
Another way to think of e¤ective technology transfer arises from our previous discussion of
the interpretation of H. We argued that in our framework H is a proxy for technical progress
embodied in tradable factors of production. In keeping with that view, we can think of trade in
factors of production as a mechanism for transferring technology without the need for the recipient
country to learn the production techniques of its trading partner. The recipient gets the bene…t
of the technology embodied in the good it buys without having to learn the process for creating
that technology.
E¤ective technology transfer through trade raises an interesting question for the interpretation
of existing empirical work on technology spillovers. Coe and Helpman (1995), Coe, Helpman,
and Ho¤maister (2008), and Keller and Yeaple (2009), among others, present evidence that trade
in goods facilitates technology transfer. The argument is that trade makes it easier for trading
partners to adopt each other’s technology. The foregoing results suggest another possibility, that
trade allows the partners not to import their partner’s technology but rather to substitute more
e¢ cient production on their partner’s soil for their own less e¢ cient production on their own soil.
The Coe-Helpman and Keller-Yeaple approach does not allow one to distinguish between the two
mechanisms. Of course, both mechanisms could be at work. It is not immediately clear how to
sort out the relative importance of the two channels, an issue left to future research.
9
3.3
Unbalanced Growth with Trade
As we have seen above, trade balance requires that the world price p falls in the closed interval
[p2 ; p1 ] because otherwise the two countries would try to specialize in and export the same good.
Balanced growth requires that p equals the quantity on the right side of (14). However, nothing
guarantees that
falls between p1 and p2 . The only restriction we have imposed so far is that
p1 > p2 (in order to guarantee that trade occurs and to specify the direction of the trade ‡ows).
That restriction puts no limits on the value of . We now analyze the e¤ect of trade when falls
outside the critical interval.
3.3.1
When
Relation between growth rates
falls outside the closed interval
"
1
A2 2 2 (1
2)
[p2 ; p1 ] =
2
1
B2 2 (1
2)
2
1
1
2+ 2
;
2
A1
B1
1
1)
1
1)
(1
1
1 (1
1
1
1
1
1
1
1+ 1
#
the world price p cannot equal it because trade balance restricts p to be in the interval. The
world price p then equals whichever interval boundary is closer to , and the world economy is
at a corner. Recall that the endpoints of the interval are the internal, autarkic prices for the two
counties. When the world is at a corner solution, the growth rate of the country whose price
de…nes that corner is the same under trade as under autarky, the growth rate of the other country
is higher under trade than under autarky but lower than the growth rate of its trading partner,
and balanced growth for the world and for the two countries individually is impossible.
Once again, results are symmetric for high and low values of , so without loss of generality
consider the case where is larger than the upper boundary of the critical interval:
p1 =
A1
B1
(1
1
1 (1
1
1
1
1)
1
1)
1
1
1
1+ 1
<
1
The world price p then equals the upper boundary p1 . Using that value for p in the growth rate
formulae (12) and (13) gives the solutions
8
9
"
# 1 (1 +1 )
>
>
1
<
=
1
1
1
1
1
A1 1 (1
1)
1
1
1
=
=
A
(1
)
1 1
1
1;T
1;A
1
1
1
>
>
B1 1 (1
:
;
1)
8
9
"
# 1 2+
1
=
1
1
1
1
1<
A
(1
)
1
1
1
1
2
2
=
B
(1
)
2 2
2;T
2
1
1
:
;
B1 1 1 (1
1)
The growth rate for country 1 is the same under trade as under autarky, the growth rate for
country 2 is di¤erent than the autarkic rate, and the two growth rates clearly are di¤erent from
each other.
The mathematical reason that country 1 grows at its autarkic rate even under trade is that
the boundaries of the critical interval are the internal relative prices that would prevail under
autarky, so when the world price hits the upper boundary, it equals the autarky price for country
1. Substituting that price into the general growth rate formula (12) then returns the autarky
growth rate. The economic intuition behind the mathematics is that at the corner country 1 does
not specialize in producing just one good but instead produces both (as it must because its solution
is the same as under autarky). On the margin, it uses the same technologies under trade that it
uses under autarky, so the growth rate under trade equals the growth rate under autarky.
10
The growth rate for country 2 is not the same as under autarky. Country 2 continues to
specialize in one good and trade for the other good. By abandoning production of one of the goods
and trading for it instead, country 2 continues to reap the e¢ ciency gains from trade which raise
its growth rate just as in an interior solution. As a result, country 2’s trade growth rate exceeds
its autarky rate. To see that result formally, note that
(
)
2
1
2
1
2
2+ 2
A
(1
)
1
2
2
1
2
2
2
=
B2 2 (1
2;A
2)
1
2
B2 2 2 (1
2)
8
9
"
# 1 2+
1
<
=
1
1
1
1
1
A1 1 (1
1)
1
2
<
B2 2 2 (1
2)
1
1
1
:
;
B1 1 (1
1)
=
2;T
because by assumption
A2
B2
(1
2
2 (1
2
2
1
2)
1
2)
1
2
1
2+ 2
A1
B1
<
2
1
1)
1
1)
(1
1
1 (1
1
1
1
1
1
1+ 1
1
Even though country 2’s growth rate increases with trade, it remains below country 1’s growth
rate. We can see that result from (16) together with p < .
In the corner case, then, trade leaves unchanged the growth rate of the country that does not
specialize and raises the growth rate of the country that specializes. The specialized country’s
growth rate remains below the unspecialized country’s growth rate forever, so the corner solution
is stable. The specialized country’s production becomes a smaller and smaller fraction of world
output, which comes to be dominated by the unspecialized country. The unspecialized country
continues to import capital (either H or K, depending on whether p <
or p > ) from the
specialized country but it also operates both sectors forever.8
To some extent, the reason that the world is in the corner solution with unbalanced growth is
that the specialized country’s total factor productivity is too low. The condition for the world to
be in the corner with country 2 specialized in the production of Y H is
A1
B1
(1
1
1 (1
1
1
1
1)
1
1)
1
1
1
1+ 1
<
=
1
A1
B2
(1
2
2 (1
1
1
1
1)
1
2)
1
1
1
1+ 2
2
Some trivial algebra reduces this condition to the simpler expression
B2
2
2
(1
1
2)
2
< B1
1
1
(1
1
1)
1
If country 2 could increase its TFP parameter B2 in the Y H industry, it could reverse this inequality
and move the world out of the corner to the interior where world balanced growth with equal growth
rates for all countries is possible. Of course, an increase in B2 alone would alter comparative
advantages, but an equal increase in A2 and B2 would leave comparative advantage unchanged
(see (11), the trade balance condition) and still move the world toward the interior. To that extent,
we can say that unbalanced growth results from "excessively" low total factor productivities across
all industries in one of the countries. The output/factor elasticities i and i also play a role in
8 Bond, Trask, and Wang (2003) analyze a three-good, dynamic Hecksher-Ohlin model with two accumulating
factors of production, one of which is not tradeable, and with all countries having identical production technolgies for
each good. Identical technologies make it possible for both countries to be incompletely specialized because trade
guarantees factor price equalization, as in the standard static Hecksher-Ohlin model. In contrast, in our model
incomplete specialization by both countries is not possible because di¤ering technolgies imply that factor prices are
not equalized whenever either country operates both sectors. See the discussion of factor price equalization below.
11
comparative advantage and the value of . The e¤ects of changes in and are of ambiguous
sign, depending on whether and are greater or smaller than 1 in magnitude, so we cannot say
in general whether an increase in or would raise or lower the expressions for trade balance and
for . In contrast, the e¤ects of the TFP parameters Ai and Bi are unambiguous. We thus can
say that the country that falls behind the rest of the world (i.e., the specialized country) does so
at least in part because its productivity is too low, an intuitively reasonable conclusion.
3.3.2
World income distribution
The possibility that trade does not equalize growth rates contrasts with Acemoglu and Ventura’s
(2002) conclusion that trade leads to a stable world income distribution. Acemoglu and Ventura
present a model in which all countries converge to the same growth rate. Once growth rates are
equal, relative incomes do not change, leading to their conclusion that the world income distribution
stabilizes. The reason for the di¤erence between their results and ours is that they restrict their
model in such a way that it necessarily yields the equivalent of our interior solution, in which all
growth rates are equal. In particular, they specify that each country is endowed with a monopoly
in the production of a subset of intermediate goods, which no other country ever is permitted to
produce. Each country therefore is specialized from the outset by assumption. The specialization
is imposed exogenously. Comparative advantage plays no role in determining it. Given that
exogenous …xed pattern of specialization, trade improves and equalizes all countries’growth rates
by allowing each country to use all intermediate goods that it cannot produce itself. The resulting
equilibrium is mathematically equivalent to the interior solution of the model developed here, and
corner solutions are excluded a priori. In the less constrained analysis of the present model, in
contrast, the pattern of production is determined endogenously by comparative advantage, and
corner solutions are possible outcomes. In the corners, world balanced growth does not occur, and
the world income distribution is not stable.
The possibility that growth rates fail to converge has practical value. Table 1, taken from
Maddison (2001), presents data on growth rates in various regions of the world going back a
thousand years. Over that entire time, Africa’s growth rate has lagged behind the rest of the
world. That is true even if one restricts attention to the more reliable data for the last 200 years.
Our theoretical result of non-convergent growth rates o¤ers a possible explanation for the historical
behavior of African growth rates.
3.3.3
E¤ective technology transfer again
When the world is in a corner, it is incompletely specialized. In that case, there still is e¤ective
technology transfer through trade but only in one direction: from the unspecialized country (the
one in the corner) to the specialized country (the one not in the corner).
3.4
Trading factors and non-factors of production
The results so far have been derived for the case where both traded goods are factors of production.
We now examine the case where one of the goods is not a factor of production, which leads to an
important conclusion concerning what it is about trade that can increase a country’s growth rate.
12
3.4.1
Interior and Corner Solutions
We suppose that Y CK -type goods (in the form of K) are not useful in production, only in consumption. That assumption requires that = = 0. The two production functions then are of
the AK form:
YiCK
YiH
= Ai ui Hi
= Ci
= Bi (1 ui ) Hi
Going through the usual steps yields the autarkic growth rate for country i:
i;A
=
1
(Bi
)
Because Y CK is not a factor of production, its TFP parameter A has no e¤ect on the growth rate.
The condition (11) for trade balance simpli…es to
p2 =
A2
B2
p
A1
= p1
B1
Recall that we are assuming that country 1 has a comparative advantage in producing Y CK and
country 2 has an advantage in good Y H . The balanced growth condition (15) becomes
A2
B2
A1
=
B2
A1
B1
(18)
With trade, country 1 specializes in Y CK , and country 2 specializes in Y H . The growth rate
for the interior solution is
1
= 1;T = 2;T = [B2
]
The growth rate of both countries is determined only by TFP in the Y H -sector of country 2.
Country 1’s TFP parameter A1 plays no role in the growth rate. When (18) is not satis…ed, the
world price p will be at one of the boundary values A1 =B1 or A2 =B2 , and the world is in a corner
solution with one country specializing in production of one good and the other country producing
both goods and not specializing. When p equals A2 =B2 > , country 1 produces only Y CK and
imports Y H , and country 2 produces both Y CK and Y H and imports Y CK . The growth rates for
the two countries are
1
B2
A1
1;T =
A2
2;T
=
1
[B2
]
A little algebra shows that country 1’s growth rate 1;T is larger than under autarky, but it is
smaller than country 2’s growth rate 2;T , consistent with relation (16) and p > . In this corner,
then, trade has no e¤ect on the growth rate of country 2 (the country that does not specialize),
raises the growth rate of country 1 (the country that does specialize), and leaves the specialized
country’s growth rate below that of the unspecialized country.
Results are slightly di¤erent at the other corner. When p equals A1 =B1 < , country 1 produces
both Y CK and Y H and imports Y H , and country 2 produces only Y H and imports Y CK . The
growth rates are the same as under autarky:
1;T
=
1
[B1
13
]
2;T
=
1
[B2
]
In this case, trade does not change either country’s growth rate. The mathematical constraints
placed on the problem (i.e., A1 =B2 > A1 =B1 = p > A2 =B2 ) imply that B1 > B2 , so that
1;T > 2;T , again consistent with relation (16) because now p < .
3.4.2
Imports: The Driver of Trade-Enhanced Growth
These results make clear exactly how trade a¤ects growth of output and lead to an important
conclusion: What matters for output growth is not the good that is exported but rather the good
that is imported. Growth rates depend on TFP in sectors that produce factors of production.
Trade can raise a country’s output growth rate by allowing that country to substitute another
country’s higher TFP for its own. It is the importation of a factor of production that can raise a
country’s growth rate; it does not matter what good is exported in payment9 .
The economic intuition for this result goes to the heart of what makes perpetual growth possible.
Ultimately, economic growth is driven by augmenting the non-reproducible factors of production
in such a way as to make the production function linearly homogeneous in the reproducible factors
of production. For example, in the Solow-Swan model, perpetual growth in income per person
is possible if and only if technical progress is labor augmenting, as Phelps (1966) showed long
ago. In the standard model, physical capital and labor-augmenting technical progress are the two
reproducible factors of production, and the labor-augmenting nature of technical progress makes the
production function linearly homogeneous in capital and technical progress. The crucial element
for our purposes here is to recognize that it is the nature of the production function with respect to
the reproducible factors of production that makes perpetual growth possible. Similarly, trade can
raise growth permanently if it makes the production of the reproducible factors of production more
e¢ cient. The increase in e¢ ciency arises from comparative advantage, with each trading partner
specializing in the factor in which it has a comparative advantage (i.e., in which it is the relatively
e¢ cient producer). So, when each country stops producing one factor itself and instead obtains it
by trading with its partner, it is making itself more e¢ cient in creating factors of production, and
that raises the growth rate. In contrast, importing a good that is not a factor of production does
nothing for increasing the e¢ ciency of making the goods that drive economic growth.
This result has an interesting policy implication. It is quite correct for countries to formulate
trade policies that promote production and export of the goods in which the country has a comparative advantage. However, that is only half the necessary policy if the country seeks to increase
its growth rate. It also must ensure that at least some of the foreign exchange earned from the
exports is used to import factors of production in which the country does not have a comparative
advantage. Importing only consumption goods will do nothing for growth.
3.4.3
Pareto E¢ ciency of Trade with Respect to Growth Rates
Looking back over the various results we have obtained for the e¤ect of trade on growth, we see that
trade never reduces growth and in all but one case raises the growth rate of at least one trading
partner. Thus - in this model, at least - trade’s e¤ect on growth is guaranteed to be weakly Pareto
e¢ cient (not hurting any growth rate) and likely to be strongly Pareto e¢ cient (raising at least
9 Bond
and Trask (1997) have a three-sector model with separate sectors for C, K, and H. However, their analyis
is substantially di¤erent from ours because they restrict attention to a single small open economy, assume that H
is human capital and non-tradeable, and do not analyze world equilibrium. The existence of only one tradeable
factor of production makes Bond and Trask’s growth implications a special case of the present model, essentially
a combination of special cases considered here. The presence of a non-tradeable factor of production guarantees
that growth rates will not be equalized across countries unless the technology for producing that factor is the same
across countries.
14
one growth rate without lowering any growth rate). That conclusion obviously has the strong
implication for the policy debate that opening to trade is bene…cial because it will not hurt growth
and may help it.
3.5
Factor Price Equalization, Stolper-Samuelson, and Rybczynski
The famous theorems from standard trade theory assume the standard static Hecksher-Ohlin framework of two small open economies producing two goods with identical technologies but di¤erent
factor endowments. In contrast, the present model assumes a dynamic Ricardian framework in
which two large open countries have di¤erent technologies and in which the factors of production
are endogenous rather than endowed. We now see to what extent the "big three" theorems of
classical trade theory carry over to the present framework. We might expect at least some modi…cation because the framework of analysis is so di¤erent from that underlying the conditions under
which the theorems originally were derived. Indeed, Jones (1965, p.563) shows that his "magni…cation e¤ect" need not hold in a general equilibrium framework where just …nal goods prices are
endogenous, even taking the quantities of the factors of production as given. We now show that
the endogeneity of the factors of production and the requirements of capital market equilibrium
introduce considerations absent from the static Hecksher-Ohlin framework and alter the theorems.
Our results also provide a new perspective on the nature of the Factor Price Equalization theorem.
3.5.1
Factor Price Equalization
The dynamic Ricardian model has interesting implications for factor price equalization and reveals
the underlying reason for it.
In the present model, when the world is at an interior solution, country 1 operates only the
Y CK sector and country 2 only the Y H sector. Thus
Y1CK
= A1 K1 1 H11
Y2H
= B 2 K 2 2 H2
1
1
2
From these equations it is straightforward to derive the marginal products of K and H in each
country, which are the gross rates of return to each type of capital, rKi and rHi , where i = 1; 2.
Comparing marginal products across countries, we see that we have factor price equalization:
rK1
rH1
= prK2
= prH2
The situation is di¤erent in the corner solution. There, country 1 operates both sectors, and
country 2 operates only the Y H sector. The production functions are
Y1CK
= A1 K1Y
CK
Y1H
=
B1 K1H
1
Y2H
= B 2 K 2 2 H2
1
1
H1Y
1
H1H
CK
1
1
1
1
2
Examining marginal products shows that
rK1
rH1
6= prK2
6
=
prH2
so that factor price equalization does not hold in the corner. These results are interesting for at
least two reasons.
15
First, it might seem that trading the factors is su¢ cient to guarantee equal factor prices (as
measured by rates of return). The corner solution shows that not to be the case. There, the
two factors are traded, but they have di¤erent rates of return. The reason is that country 1
obtains its marginal unit of H from itself, not from country 2, and country 1 is disadvantaged in
the production of H. The constraint is binding in the corner, and equality of marginal rates of
return does not hold.
Second, the conditions that yield factor price equalization in this dynamic Ricardian model are
exactly the opposite of those required for the standard static Hecksher-Ohlin model. In the static
H-O framework, factor price equalization requires that both countries operate in the interior of
their cones of diversi…cation, which means that both countries produce both goods, that is, neither
country is specialized. In contrast, in this dynamic Ricardian model factor price equalization
requires that the world be in the interior region where each country produces only one good, that
is, both countries are specialized. The contrasting conditions for factor price equalization in fact
rely on the same underlying phenomenon. In both types of models, factor price equalization
requires that the two countries use the same technology. In the H-O framework, that requires that
both countries produce both goods so that the relevant cost functions can take on the same value
(see Feenstra, 2004). In this dynamic Ricardian framework, technology is e¤ectively equalized
by trade, but only for an interior solution. It is only in the interior that trade generates a
full e¤ective technology transfer that leaves the two countries producing as if they actually had
exchanged technology. Thus we see that the deep underlying condition for trade to bring about
factor price equalization is that it must lead to an e¤ective equalization of technology. Once that
happens, international trade provides the linkages necessary for the market to reallocate factors
of production until they earn equal rates of return across countries. Without equalization of
technology, as in the corner solution in either the H-O or Ricardian model, rates of return cannot
be equalized.
3.5.2
Stolper-Samuelson
The Stolper-Samuelson theorem shows that in the standard H-O framework an exogenous increase
in the relative price of a good increases the return to the factor of production used intensively in
that good and decreases the return to the other factor. There are no exogenous changes in prices
in the present model because neither country is "small" in the sense of taking prices as given,
so we cannot ask the question that the Stolper-Samuelson theorem addresses. We can ask two
related questions, though: How do the returns to the factors of production respond to a change
in an underlying parameter that causes a change in a relative price, and how much of the total
e¤ect works through the change in price? The easiest parameters to study are the total factor
productivities Ai and Bi , so we start with them. Their e¤ects are identical except for sign, so we
restrict attention to an increase in A1 .
In the interior, both countries are specialized, so we know immediately that the StolperSamuelson theorem does not apply. We cannot ask the question it addresses, namely, the e¤ects
of a change in the relative price of two goods produced within a country, but we can ask about the
e¤ect of a change in the world relative price of the two traded goods on the factor returns in each
16
country. The gross returns to K and H in each country are
rK1
rH1
rK2
rH2
=
= prK1 =
=
1 A1 p
2 B2 (K2 =H2 )
= prK2 =
2 B2 p
1
1
1 A1 (K1 =H1 )
=
1 A1 p
1
1
1
1
(19)
1
1
1
(20)
1
2
1
1
1
1
=
2 B2 p
2
2
1
1
(21)
2
2
2
1
(22)
2
From (14) we see that p is a positive function of A1 , so an increase in A1 has a direct e¤ect on rK1
and rK2 and an indirect e¤ect (through p) on all four rates of return. The indirect e¤ect is what
interests us here, and it is negative for rK1 and rK2 and positive for rH1 and rH2 , irrespective of
which good is intensive in which factor (that is, the relation between 1 and 2 ). The economic
interpretation is straightforward. An increase in TFP in the Y CK industry induces an increase
in the ratio of K to H, which necessarily reduces the marginal product of K and increases the
marginal product of H. In the interior, there is nothing like the Stolper-Samuelson e¤ect at play.
The economy’s full response to a change in A1 (the direct and indirect e¤ects) is to increase all
four rates of return. Domestic capital market equilibrium requires that the two types of capital
have the same rate of return, expressed in common units:
rK1
=
rK2
=
rH1
p
rH2
p
and international capital market equilibrium requires that each type of capital has the same return
across countries:
rK1
rH1
= prK2
= prH2
The full solution for rK1 is
rK1 =
"
A1 2
B2 1
1 2
1
2(
1
2
1
(1
1)
1)
(1
2 (1
(1
2)
1)
2 )(
1
1)
#1
1
1+ 2
which is a positive function of both TFP parameters A1 and B2 . The intuition is straightforward.
An increase in either TFP parameter raises the marginal products of both types of capital in that
production function. Capital market equilibrium then requires that the marginal products in the
other country equal those in the …rst country. In contrast to Stolper-Samuelson, rates of return
move in the same, not opposite, directions.
In the interior, specialization renders the simple, original version of the Stolper-Samuelson
e¤ect inapplicable because each country produces only one good. Also, the endogeneity of K and
H means that the factor "endowments" themselves respond to any change in the world relative
price p of the two goods rather than remaining constant as in the standard Hecksher-Ohlin model,
so that the K=H ratios in the two countries adjust to satisfy the requirements of capital market
equilibrium.
In the corner, one country does not specialize. As usual, consider the corner where it is country
1 that is unspecialized. For country 1, we can ask the Stolper-Samuelson question of how a change
17
in the relative price of the two goods change relative rates of return. The rates of return for K
and H are the same as in (19) and (20) except that p is replaced by p1 , given by (11). Here
we assume that the increase in A1 is not large enough to move the world out of the corner. An
increase in A1 a¤ects the rates of return both directly and indirectly, just as in the interior. The
Stolper-Samuelson e¤ect is the indirect e¤ect coming through the change in p1 . An increase in
A1 raises p1 (i.e., raises the relative price of H in terms of K). That reduces the rate of return to
K and raises the rate of return to H, irrespective of the factor intensities in production. So once
again the Stolper-Samuelson e¤ect is absent here. As in the interior, the full e¤ect of an increase
in A1 is to raise both rates of return rK1 and rH1 .
3.5.3
Rybczynski
The Rybczynski theorem addresses the e¤ect of an exogenous change in relative factor endowments
and concludes that an increase in one of the factors of production will lead to an increase in output
in the industry that uses that factor intensively and a decrease in the output of the other industry.
In the present model, no such e¤ect emerges. Instead, an increase in one factor leads to a decrease
in the output of the industry that produces that factor and an increase in the output of the industry
that produces the other factor. The initial shock may shut down trade, depending on the initial
pattern of trade.
Suppose the world is in the interior and on the BGP when a natural disaster reduces country
1’s stock of K. The reduction in K reduces the current K=H ratio but does not change the
ratio’s equilibrium value. Because of the bang-bang nature of investment, country 1 responds by
shutting down investment in H and devoting all investment to K until the K=H ratio is restored
to its equilibrium value. Under the assumptions we have been using, country 1 was exporting K
and importing H before the shock, something it no longer wants to do. Country 1 therefore also
shuts down trade until it restores the K=H ratio to the equilibrium value. None of this response
depends on which industry uses K intensively, so the Rybczynski theorem does not hold. The
same conclusion holds by similar logic if the world starts in the corner solution when the shock
occurs.
The reason the Rybczynski theorem does not hold is that the factors of production are not
endowments but rather are produced that countries want to hold in a ratio that depends on the
underlying parameters of the economy. A shock to the existing stock of a reproducible factor does
not change the desired factor ratio, so the response to a shock it to increase production of the
factor in relatively short supply and shut down investment in the other factor altogether.
4
Transition Dynamics
We now turn to a study of the model’s transition dynamics.
Most of the existing literature on the transition dynamics of the two-sector model mostly studies
only closed economies. Important contributions include Mulligan and Sala-i-Martin (1993), Bond,
Wang and Yip (1996), Faig (1995), and Mino (1996). Mulligan and Sala-i-Martin considered a
general model that exhibits constant returns to scale at the private level but also incorporates
increasing or decreasing returns at the social level. The analysis of the transitional dynamics of
the model is based on numerical simulations. Bond, Wang and Yip derived transitional dynamics
of a closed general two sector endogenous growth model. The dynamic properties of the model are
analyzed using the system consisting of three di¤erential equations in price (p), consumption per
unit of human capital (c) and the ratio of physical capital to human capital (k). As they show the
transitional dynamics depend on the assumptions about the relationship between factor intensity
18
parameters. The asymptotic adjustment towards steady state can be driven by either adjustments
in physical capital or by the stabilizing forces of the price level. Faig develops graphical tools to
analyze dynamic properties of the model with physical and human capital and then uses a derived
framework to analyze e¤ects of …scal policies and stochastic shocks. Mino focuses on the analytical
framework to analyze the dynamics of the two-sector model with physical and human capital in
the presence of capital income taxation. His arguments are in the same line with Bond, Wang and
Yip that dynamics of the economy and e¤ects of capital income taxation depend on assumptions
regarding the relative factor intensities in both sectors of production. These studies all contribute
signi…cantly to the understanding of the transitional dynamics of two-sector endogenous growth
models of closed economies.
Introducing trade into the two-sector model adds new dimensions to the transition dynamics.
As Mino (1996) remarks, “Since the literature on sectoral shifts has usually ignored the possibility
of endogenous growth, the open-economy version may provide interesting contributions to the
…eld” (page 247). Indeed, Ventura (1997) argues that the East Asian growth miracle can be
explained by “structural transformation”when faster accumulation of capital leads to the expansion
of the capital-intensive sector and contraction of the labor-intensive sector and not just continuing
production of both goods with more capital-intensive techniques. To the best of our knowledge, the
only contribution in the literature that follows Mino’s suggestion is that of Bond and Trask (1997).
They analyze a small open economy with three sectors: capital, consumption, and education.
Capital goods and consumption goods are tradable, and education is not tradable. They show
that the BGP for such an economy is saddle-point stable.
Our model di¤ers from that of Bond and Trask in three ways that a¤ect the economy’s transition
dynamics. First, in general both factors of production are tradable in our model. Second, we do
not restrict attention to a small open economy but rather consider countries that may be large.
Third, we analyze world general equilibrium. These key di¤erences lead to a substantially richer
pattern of transition dynamics than emerge in the closed-economy case and that are consistent
with Ventura’s argument of “structural transformation.”
Our analysis is based on the general case where 6= and examines the transitional dynamics
in the neighborhood of the BGP with complete specialization, where country 1 specializes in the
production of good Y CK and country 2 specializes in production of good Y H . An interesting
aspect of the transition dynamics is that their properties depend heavily on the initial deviations
of factor ratios in both countries from their BGP values. The BGP solutions for the two countries’
factor ratios k1 K1 =H1 and k2 K2 =H2 are
k1 =
k2 =
1
1
2
1
p
(23)
p
(24)
1
2
where as usual p is the world relative price determined by equation (14). The transition dynamics
about the steady state are di¢ cult even with linearization because we have a …ve-dimensional system consisting of the dynamic equations for k1 , k2 , c1 = C1 =K1 , c2 = C2 =K2 , and p. Nonetheless,
we can deduce some analytical results and obtain others numerically. There are four cases to
consider, depending on how the initial values of the factor ratios di¤er from their BGP values:
1. k1 < k1 , k2 > k2
2. k1 < k1 , k2 < k2
3. k1 > k1 , k2 < k2
4. k1 > k1 , k2 > k2
19
The …rst two cases are fairly easy because no trade occurs along the transition. That means the
countries’dynamics are independent of each other, allowing us to treat the two countries separately
and obtain an analytical solution for the transition dynamics. The remaining two cases are more
complicated because countries trade along the transition path. The countries are large relative
to each other, so they a¤ect the world relative price. Changes in the world price transmit the
e¤ects of one country’s actions to the other country, requiring a simultaneous solution for the two
countries. The high dimensionality of the problem requires a numerical solution.
The main conclusions from the following analysis are that the BGP is saddle-path stable and
that transition dynamics may exhibit oscillatory behavior.
4.1
Case 1: k1 < k1 , k2 > k2
In this case country 1 specializes in production of good Y CK on the BGP but deviates from the
BGP value of its K=H ratio by having too much H capital (the good that it does not produce on
the BGP) relative to K. Country 2 is in the opposite situation with too much K and too little H
compared to the BGP ratio. Because country 1 has relatively more of the H than its BGP value,
it sets investment in H to 0. With investment in H equal to 0, country 1 does not import Y H
from country 2 along the transition to the BGP. This conclusion follows from the accumulation
condition for H in country 1:
X1
=0
H_ 1 + H1 =
p
which implies that H depreciates at rate . Intuitively, country 1 has too much H, so the relative
price of H in country 1 is very low. Country 1 no longer has a comparative advantage in the
production of Y CK and so does not have an incentive to trade Y CK for Y H . With trade shut
down, the optimization problem faced by country 1 reduces to that of the closed economy one-sector
endogenous growth model. The present value Hamiltonian for country 1 becomes
V =
C11
1
1
e
t
+
1
A1 K1 1 H11
1
C1
K1
(25)
The dynamics of that model are discussed in Barro and Sala-i-Martin (chapter 5, 2004). The
dynamic system for country 1 is written in terms of the variables c1 = C1 =K1 and k1 = K1 =H1
and expressed in terms of the following two dimensional system:
"
#
1 2
( 1 1)(
1 )A1 k1
c_1 =c1
c1 c1
1
=
1 2
k
k1
k_1 =k1
1
1
A1 ( 1 1)k1
The phase diagram for country 1 in this case is similar to that for the closed economy one sector
model in Barro and Sala-i-Martin (chapter 5, 2004) and is shown in Figure 1. Along the transition
path country 1 depreciates H capital, accumulates K capital, and decreases consumption until it
reaches the BGP value of k1 , at which point country 1 opens to trade again.
Now, consider country 2. On the BGP country 2 specializes in producing Y H and imports
Y
from country 1. The imported Y CK is used for consumption and investment in K. As we
have just discussed, country 1 no longer has an incentive to trade, so country 2 must open a sector
producing good Y CK in order to have consumption. Country 2 also acts as a closed economy two
sector model until the price level in country 1 reaches the level consistent with the BGP in the
presence of trade. At that price level both countries will have incentives to trade, so country 2 will
resume specializing in Y H and trading to obtain Y CK from country 1.
CK
The relative price for Y H in country 1 is determined by the ratio of marginal products of the
two types of capital:
1
1
p1 =
k1
(26)
1
20
It follows that the dynamics of the price along the transition path are determined by the accumulation conditions for K and H. The price dynamics in the two countries are shown in Figure 2. As
we have already discussed the balanced trade condition requires world relative price to fall inside
the closed interval [p2 ; p1 ]. It follows from (26) that under the conditions of case 1 the relative
price of H capital in country 1 will be lower than the world relative price level in the presence of
trade. However, as country 1 accumulates K capital and depreciates H capital along the transition
path, the relative price level of good Y H increases until it reaches the world relative price level
consistent with the presence of trade along the BGP. When the price in country 1 reaches the level
of p determined by equation (14) both countries will open to trade.
4.2
Case 2: k1 < k1 , k2 < k2
Once again, country 1 has too much H relative to K, so the solution for country 1 is exactly the
same as in case 1. However, for country 2 the situation is di¤erent from the previous case. Now,
country 2 also has too much H relative to K, so it, too, has no incentive to accumulate H. Country
2 sets investment in H equal to 0 and accumulates only K. The present value Hamiltonian for
country 2 is
C1
1
e t + 2 A2 K2 2 H21 2 C2
K2
(27)
V = 2
1
This Hamiltonian looks exactly like its counterpart for country 1 given by (25), so the dynamic
behavior of country 2 will be similar to that of country 1. The two countries have qualitatively
similar phase diagrams that are like Figure 1.
In case 2 both countries start with a high level of H relative to K. Therefore, in both countries
the relative price of good Y H is lower than the expression for the world price level p given by
equation (14) that would prevail on the BGP in the presence of trade.. It also can be shown that
the starting relative price level in country 2 is lower than its autarkic price level associated with
the operation of both sectors.10 As the two countries accumulate K and depreciate H along the
transitional path, the relative price level in both countries increases. As explained earlier, the
BGP world relative price level in the presence of trade falls inside the closed interval given by the
autarkic BGP price levels in the two countries, with the upper and lower bounds of the interval
given by the autarkic price levels in country 1 and country 2, respectively. That means that, as
the relative price increases in both countries along the transition path, country 2 will achieve its
autarkic BGP price level …rst. At that point, country 2 will begin operating both sectors and will
stay in autarky until the price level in country 1 reaches p. When that happens, trade will begin,
and each country will specialize in the good for which it has a comparative advantage. Figure 3
shows the price adjustment paths.
4.3
Case 3: k1 > k1 , k2 < k2
We now suppose that country 1 has too much K relative to H and country 2 has too much H
relative to K. The important property of this case is that at the starting point each country has
more of the good for which it has a comparative advantage on the BGP. Because country 1 has
relatively more K than H, it sets investment in K to 0 and exports its entire production of K:
X1 = A1 K1 1 H11
1
C1
(28)
The domestic stock of K depreciates at rate :
K1t = K10 e
1 0 See
Yenokyan (2010) for the details
21
t
(29)
Accumulation of H in the presence of complete specialization is given by
H_ 1
A1 k1 1 c1 k1
=
H1
p
(30)
The usual derivation shows that the growth rate of consumption in country 1 is
C_1
1
=
C1
(1
1 )A1 k1
1
p
+
p_
p
(31)
Equations (29), (30), and (31) determine the paths of C1 , H1 and K1 .
A similar analysis for country 2 gives the following three equations for determining the paths
of C2 , H2 and K2 :
K_ 2
1
= pB2 k2 2
c2
(32)
K2
t
H2t = H20 e
1
C_2
=
p 2 B2 k2 2
C2
(33)
1
(34)
To complete the solution, we need to determine the growth rate of the world relative price, p.
The balanced trade condition is
A1 H1 k1 1
C1 = pB2 H2 k2 2
Total di¤erentiation of that condition and some algebra provide the growth rate of the world
relative price:
p_
p
=
(
( pB2 k2 + c1 k12 )
2
A1 k1
( pB2 k2 2 + c1 k12 )
1)
1
(1
k12
pB2 k2 2
h
1 )(A1 k1
1
p
2 pB2 k2
c1 k12 (1
2
1
1 )A1 k1
2 c2
1
c1 k1 )
i
p
( pB2 k2 2 + c1 k12 )
where k12 = K1 =H2 is constant along the transition to the BGP because investment in both K1
and H2 is zero during the transition and K and H both depreciate at rate .11
The transition behavior of this world economy can be expressed in terms of the following
…ve variables p, c1 = C1 =K1 , k1 = K1 =H1 , c2 = C2 =K2 , k2 = K2 =H2 . The corresponding …ve
dimensional system is:
2
3 2
32
3
p=p
_
n11 n12 n13 n14 n15
p p
6 c_1 =c1 7 6 n21 n22 n23 n24 n25 7 6 c1 c1 7
6
7 6
76
7
6 k_1 =k1 7 = 6 n31 n32 n33
6
7
0
0 7
6
7 6
7 6 k1 k1 7
4 c_2 =c2 5 4 n41
0
0 n44 n45 5 4 c2 c2 5
n51
0
0 n54 n55
k2 k2
k_2 =k2
where the nij are various combinations of the parameters 1 , 2 , A1 , B2 , , , , the BGP values of
k1 and k2 , and the initial value k12 (which is constant on the transition path).12 . The system can
be written as z_ = N zt , where z_ is the …ve-dimensional vector of the growth rates of the variables,
1 1 see
1 2 See
Yenokyan (2010) for the details of derivations
Yenokyan (2010) for the expressions for the nij .
22
p, c1 , k1 , c2 and k2 and zt is a vector of deviations of the variables from their steady state values.
The system’s solution can be approximated as:
zt = M e
t
M
1
z0
where M is the matrix of the eigenvectors of matrix N, is the diagonal matrix of eigenvalues of
matrix N and z0 is the vector of initial deviations of the variables from their steady state.
The high dimensionality of the above system does not allow an analytical solution, so we
calibrated the model and performed some simulations to study the system’s behavior. We present
only a brief summary of the simulation results here. We set the time unit to a quarter of a year.
To start the simulation exercise we imposed the values: 1 = 0:3, 2 = 0:25, = 0:0025, = 0:025,
= 2, and A1 = 1. The value of 1 is the usual capital share in a Cobb-Douglas production
function, taken as an average from the National Income and Product Accounts. The choice of the
initial value for 2 is based on the assumption that 1 > 2 , meaning that the share of physical
capital in the production of a labor-augmenting type of capital is smaller than in the production
of physical capital itself. This is the usual assumption that a factor’s share in its own production
is higher than in the production of other factors, or in other words that we do not have factor
intensity reversal. With quarterly time units, the value of = 0:0025 implies that the annual real
interest rate is 1 percent (consistent with the average real rate of return on US Treasury 1-year
bills), and the value of = 0:025 implies an annual depreciation rate of 10 percent. The values of
and A1 are commonly used in calibration exercises (e.g., see Mulligan and Sala-i-Martin, 1993).
The choice of values B2 and k12 is rather arbitrary. There is no division in the data between
industries that produce capital whose quality augments labor (H-type capital in the model) and
capital whose quality augments capital itself (K-type capital). We thus have no direct measure
of total factor productivity (TFP) in the H industry. The model is a very simple construct, convenient for exploring new dimensions of growth theory but not realistic enough to make believable
estimation of B2 by moment-matching. We therefore try di¤erent values of B2 to generate the
annual growth rate in the presence of trade in the range of 2 3 percent. For the given values of
other parameters the values of B2 consistent with this rate of growth are 0:17 0:2. Note that this
does not necessarily imply that B2 cannot take larger values. We can …nd another combination
of the parameter values that will generate annual growth rate in the acceptable range for higher
values of B2 . Similarly, we have no direct measure of k12 , making it impossible to know reasonable
values to choose, so again we explore the model’s behavior for di¤erent values of k12 . Note that B2
and k12 are very di¤erent in meaning. B2 is a parameter of the model, capturing elements of the
production technology. In contrast, k12 is the ratio of two endogenous variables. Its initial value
re‡ects how close the initial value of one type of capital (K) in the …rst economy is compared to
the other type of capital (H) in the second economy. It is di¢ cult to know what is a reasonable
range of values for such a ratio. Consider the related ratio of K1 to H1 . Would we expect that
to be large or small? Recall that we are thinking of H as types of capital that have embedded
in them technical advances that augment labor, whereas K is capital whose embodied technical
advances augment K itself. A bit of re‡ection suggests that a great deal of physical capital, such
as machinery and other producer durables, seems to be like H. Structures, such as factories and
storage sheds, seem more like K-type capital. In the US, the value of producer durables and
structures is about the same, suggesting that the ratio of K1 to H1 is about 1. Because we are
assuming that country 1 and country 2 are large relative to each other, we would expect k12 , which
is the ratio of K1 to H2 , to be the same order of magnitude.
For the values of B2 in the range of 0:17 0:2 we tried di¤erent values of k12 to study the
dynamic behavior of the system. There are three distinctive patterns in the transition behavior of
the system. First, the dynamic system can yield complex eigenvalues with oscillatory convergence
to the BGP. Second, the system can also produce monotonic convergence to the BGP. Finally,
23
there is also some asymmetry in the pattern of convergence between countries specializing in Ktype capital and H-type capital which arises from the fact that the good used as K-type capital
also is the good used for consumption.
It turns out that the values of k12 equal to 5 and higher yield real eigenvalues and overall
monotonic approach of the variables to their BGP values, whereas the values of k12 in the range
closer to 1 yields complex eigenvalues with oscillatory convergence to the BGP. We just argued
that values of k12 as high as 5 or 10 are unlikely to be consistent with our assumption of two
large economies. On the other side, very low values of k12 yield qualitative results similar to those
for k12 = 5 and above, so it seems that reasonable values of k12 are consistent with oscillatory
behavior. The model thus suggests that cyclicality is the expected pattern of dynamic adjustment
of the countries to the BGP. We will see more cyclicality in the pattern of the adjustment of the
two countries in the discussion of case 4, which we take up next.
4.4
Case 4: k1 > k1 , k2 > k2
In this …nal case, both countries have relatively too much K. The set of dynamic equations for
country 1 are again given by (31), (30) and (29). Country 2 sets investment in K to 0. Country
2 imports Y from country 1 and uses all of it for consumption. The paths of C2 , K2 and H2 are
determined by the following set of the equations:
C_2
1
=
(1
C2
2 )B2 k2
2
p_
p
(35)
t
(36)
c2 k2
p
(37)
K2t = K20 e
H_ 2
= B2 k2 2
H2
+
Now the growth rates of country 1 and country both depend on the transitional behavior of the
world relative price. As in the previous case, we use the balanced trade condition to solve for the
growth rate of the world relative price, obtaining
p_
p
=
A1 k1 1 1 (1
c2 k21 + c1
c2 k21
(1
c2 k21 + c1
1 )(A1 k1
1
c1 k1 )
p
2 )B2 k2
c1
c2 k21 + c1
(1
1 )A1 k1
1
p
2
where k21 K2 =K1 . Note that this ratio is not the same as the ratio in case 3 above. However,
like k12 , it remains constant along the transition path for the same reasons. As in the previous
case, the transitional behavior of this world economy is described by a …ve dimensional system:
2
3 2
32
3
p=p
_
d11 d12 d13 d14 d15
p p
6 c_1 =c1 7 6 d21 d22 d23 d24 d25 7 6 c1 c1 7
7
6
7 6
76
6 k_1 =k1 7 = 6 d31 d32 d33 0
6
7
0 7
6
7 6
7 6 k1 k1 7
4 c_2 =c2 5 4 d41 d42 d43 d44 d45 5 4 c2 c2 5
d51 0
0 d54 d55
k2 k2
k_2 =k2
where the elements dij again are combinations of the parameters 1 , 2 , A1 , B2 , , , , the BGP
values of k1 and k2 , and the initial value k21 (which is constant on the transition path).13 This
system can be written as z_ = Dzt , where z_ is the …ve-dimensional vector of the growth rates of the
variables p, c1 , k1 , c2 and k2 and where zt is a vector of deviations of the variables from their steady
1 3 See
Yenokyan (2010) for the elements of matrix D.
24
state values. Again the high dimensionality of the system does not allow an analytical solution, so
we resort to a numerical exploration.
We use the same initial values of 1 , 2 , , , , and A1 as in case 3: 1 = 0:3, 2 = 0:25, = 2,
= 0:0025, = 0:025, and A1 = 1. Here as well we keep the values of B2 in the range 0:17 0:2
and try di¤erent values of k21 . Values of k21 at all near 1 produce real eigenvalues and relatively
monotonic convergence. However, the values of k21 higher than 2 generates oscillatory behavior.
These results suggest that oscillatory behavior again is to be expected along the transition path,
as in case 3 above.
5
Conclusion
We have seen that trade in goods can raise the growth rates of both trading partners through
comparative advantage without there being any scale e¤ects, technology transfer, research and
development, or international investment. Comparative advantage determines the pattern of
trade, that is, which good will be produced in which country. When a certain condition on the
model parameters is met, the world achieves an interior solution in which a world BGP exists, is
unique, and is globally asymptotically stable. When the condition is not met, the world achieves
a corner solution in which growth rates are not equalized. Consequently, in contrast to Acemoglu
and Ventura (2002), trade need not generate a stable world income distribution. In the interior
solution, trade raises the growth rates of both countries. In the corner solution, trade raises the
growth rate of the technologically smaller country but still leaves it below the growth rate of the
technologically larger country. Trade in factors has some of the same e¤ect as technology transfer
and in one important special case has exactly the same e¤ect as technology transfer. We thus have
the major implications that (1) trade generally increases growth rates, (2) trade need not increase
a given country’s growth rate and need not lead to growth convergence, and (3) trade can lead
to growth outcomes that are equivalent to what would emerge from technology transfer. These
e¤ects of trade on growth mean that the use of closed-economy models to analyze cross-country
data is likely to be misleading. We also show that trade in goods that are not factors of production
does not a¤ect a country’s growth rate. In particular, what determines whether trade increases a
country’s growth rate or leaves it unchanged is the type of good that the country imports. If the
imported good is a factor of production, trade will raise the country’s growth rate. Otherwise,
trade leaves the growth rate una¤ected. Factor price equalization holds in the interior but not
in the corners. The conditions for factor price equalization in this dynamic Ricardian model
are exactly the opposite from those required in the standard Hecksher-Ohlin framework. Here
we require that both countries specialize in a single good, whereas in the H-O framework both
countries must not specialize. The unifying principle is that trade will equalize factor prices if
it leads to e¤ective equalization of technology. Neither the Stolper-Samuelson theorem nor the
Rybczynski theorem holds in this kind of model.
The study of the transitional dynamics reveals that there can be four scenarios describing the
dynamic behavior of two trading economies, which are large relative to each other. In two of those
cases countries don’t trade along the transition. Deviations of the factor ratios from their BGP
values lead to the situation where countries accumulate the factor of production they are lacking
before opening to trade on the BGP. In the remaining two cases countries trade not only on the
BGP but also along the transition. The evolution of the world relative price in the presence of
trade depends on control and state variables of both countries, leading to complicated dynamic
systems describing the dynamic behavior of trading countries. The analysis suggests that the
BGP is saddle-path stable and that transition dynamics may be oscillatory.
25
References
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28
Figure 1: Phase diagram - Case 1
Figure 2: Price dynamics - Case 1
29
Figure 3: Price dynamics, case 2
Ta b le 1
R a te s o f G row th o f G D P p e r C a p ita
(a n nu a l ave ra g e c o m p o u n d g row th ra te s, p e rc e nta g e p o ints)
Ye a rs
R e g io n
10 00 -15 00
1500-1820
1 82 0-7 0
1 8 7 0 –1 9 1 3
1 91 3-5 0
1 95 0-7 3
1 97 3-9 8
W . E u rop e
0 .1 3
0 .1 5
0 .9 5
1 .3 2
0 .7 6
4 .0 8
1 .7 8
0 .3 6
1 .3 4
1 .8 2
1 .6 1
2 .4 5
1 .9 9
US
Japan
0 .0 3
0 .0 9
0 .1 9
1 .4 8
0 .8 9
8 .0 5
2 .3 4
A sia / J a p a n
0 .0 5
0 .0 0
-0 .1 1
0 .3 8
-0 .0 2
2 .9 2
3 .5 4
A fric a
-0 .0 1
0 .0 1
0 .1 2
0 .6 4
1 .0 2
2 .0 7
0 .0 1
S o u rc e : M a d d iso n (2 0 0 1 ), Ta b le B -2 2 .
30
Appendix
A
A.1
Solution with Trade
Individual Country Solutions
With trade, country i’s Hamiltonian is
Vi
where
and
=
Ci1
1
e
i
Bi ((1
t
+
Ai (vi Ki ) i (ui Hi )1
i
ui )Hi )1
vi )Ki ) i ((1
i
i
Ci
Ki
Hi +
Xi
p
are the costate variables. The necessary conditions are
"
#
i 1
v
K
(1
i
i
_ =
vi )
i vi Ai
i Bi i (1
i
i
u i Hi
(1
_ =
i
i (1
i
vi Ki
u i Hi
i )ui Ai
Bi (1
i
@Vi
= Ci e
@Ci
@Vi
=
@vi
@Vi
=
@ui
i Ai
i (1
i )Ai Hi
t
i
i Bi (1
@Vi
=
@Xi
i
+
i )Hi
i
p
(1
(1
1
i
(39)
vi )Ki
ui )Hi
i
(40)
(41)
(1
(1
i B i i Ki
vi Ki
u i Hi
ui )
vi )Ki
ui )Hi
(38)
=0
1
i
vi Ki
u i Hi
i Ki
i )(1
Xi +
vi )Ki
ui )Hi
(1
(1
i
1
=0
vi )Ki
ui )Hi
i
=0
(42)
1
=0
(43)
(44)
plus initial and transversality conditions, which are unneeded in what follows.
A.2
Balanced growth path with trade
Knowing that country 1 specializes in Y CK allows us to write a simpli…ed maximization problem
for it, conditional on the fact that it produces no Y H . Equations (38) - (44) still characterize the
problem, but now we set u = v = 1. The Hamiltonian reduces to
V1
=
C11
1
+
e
1
t
+
X1
p
1
A1 K1 1 H11
C1
1
K1
X1
(45)
H1
and the necessary conditions become
_ =
1
_ =
1
1
"
1 (1
1 A1
K1
H1
1 )A1
31
K1
H1
1
#
1
(46)
1
+
1
(47)
@V1
= C1 e
@C1
@V1
=
@X1
t
1
1
+
1
p
=0
(48)
=0
(49)
There no longer are …rst-order conditions for v and u because they already have been set to one.
The …rst-order conditions for C and X are unchanged from the unconditional problem, but the
…rst-order condition for X now holds with equality. Having accepted the world price p and agreed
to specialize in producing Y CK , country 1 now chooses and to satisfy (49) exactly.
Di¤erentiating (48) with respect to time, dividing both sides by , and manipulating yields the
growth equation for consumption
!
_
1
1
+
(50)
C1 =
1
The same kind of manipulations as for the autarkic model show that the growth rates of Y CK , K,
and Y all equal the growth rate of consumption; as before, we denote this common growth rate
. We obtain the growth rate of 1 from the remaining necessary conditions and the requirements
of balanced growth. Because equation (49) now always holds, we have p = 1 = 1 . Trade balance
constrains p to lie in the closed interval [p2 ; p1 ], and balanced growth requires that everything
that grows must do so at a constant rate. The only growth rate for p consistent with both these
requirements is zero, so p must be constant along the balanced growth path. The ratio 1 = 1
therefore also is constant, implying that the growth rates of 1 and 1 are equal. We can use this
fact together with (46) and (47) to solve for K1 =H1 , obtaining
K1
=p
H1
1
1
(51)
1
We then substitute this expression into (46) and divide by
_
1
=
p
1
1
1
1
(1
1
1)
1
to obtain the growth rate for
1
A1
1:
(52)
1
Finally, substituting this solution into (50) gives us what we are after, the growth rate of country
1 in the presence of trade:
"
#
1
1
1
1
1
1
A1 1 1 (1
1)
1;T =
p
where the subscript T indicates that this growth rate pertains when country 1 trades.
Country 2’s growth rate is found the same way. Country 2 produces no Y CK , only Y H , so its
Hamiltonian is
V2
=
C21
1
+
e
2
t
+
2
[ C2
B2 (K2 ) 2 (H2 )1
K 2 + X2 ]
2
H2 +
(53)
X2
p
with corresponding necessary conditions. Going through the same steps as for country 1 yields the
growth rate
1
1
2p 2
B2 2 2 (1
(54)
2;T =
2)
32
B
Factor Price Equalization
B.1
Interior
In country 1 only good Y CK is produced with the production function
Y1 = A1 K1 1 H11
1
The marginal product of K-type capital is:
rK1 =
@Y1
=
@K1
1
K1
H1
1 A1
1
On the BGP in the presence of trade the ratio of K to H in country 1 is
K1
=p
H1
1
1
1
where p is the world price, given by equation (14) in the main text. Substituting the expression
for p into the solution for the K=H ratio and then into the above expression for marginal product,
we get
! 1 1+
1
2
1)
2 (1
A1 2 1 1 2 (1
1)
rK1 =
(
1)
(1
2 )( 1 1)
B2 1 1 2 2 1
(1
2)
In country 2 only good Y H is produced with the production function
1
H_ + H = B2 K2 2 H2
2
The marginal product of K in country 2 is
@ H_ + H
rK2 =
=
@K
2 B2
K2
H2
2
1
The K=H ratio in country 2 is
K2
2
=p
H2
1
2
Substituting into the expression for the marginal product and using the expression for p from
equation (14) gives
! 1 1+
1
( 2 1)(1
1
2
1 ( 2 1)
1)
A1 2
(1
1)
1
rK2 =
2 2
(1
2 )( 1 2)
B2 1 2 2 2 1
(1
2)
It then is straightforward to show that
rK1 = prK2
Similar steps show that
1+
rH1 =
rH2 =
A1
1 (1+ 2 )
2
1
1
B2 1
A1 2
B2 1
2
2
1
2
1
(1
(1
1 2
(1
1
(
1)
1
2
(1+
1)
(1
2)
1)
(1
2 (1
(1
2)
so that
rH1 = prH2
33
2 )(1
2)
1)
1
1)
2 )(
1
1)
!1
!1
1
1+ 2
1
1+ 2
B.2
Corner
In the corner case, country 1 produces both goods. The rates of return are derived by the same
steps as in the interior, giving
Y
rK1
A1 1
=
B1 1
1+
Y
rH1
A1
=
2
1
A1 1
1
B1 1
2
B1 1
1
(1
rH2 = B2
2)
2
2
1 (1+ 1 )
1
(1
1
1
1
(1
1 1
1
1
A1 2
2
1)
1
1)
1 (1
1)
(1
1)
(1
1
1( 2
1)
(1
1( 2
1)
(1
1
2
!1
!1
!1
( 1 1)(1
1)
1)
(1
1 )( 1 2)
1)
(1
1)
1 )(
1)
1)
B1 2
1
2)
(1+ 1 )(1
1)
(1
1) 1
1(
1
2
(1
(1
1(
1
(1
1)
1)
1
1)
(1
1( 1
1
1 (1
1)
(1
1
A1 1
H
rH1
=
2
1
B1 1
H
rK1
=
rK2 = B2
1
1 1
(1
1
1 ( 1 1)
1
A1 2
1
B1 2
1
1 2
(1
1 2
(1
1)
1 )(
1)
1
!1
1
1+ 1
1
1+ 1
1
1+ 1
1
1+ 1
( 2 1)(1
1)
1)
(1
1 )( 2 2)
1)
2 (1
1)
(1
1)
1)
1) 2
!1
!1
1
1+ 2
1
1+ 2
With these expressions, it is straightforward to show that
rK2
rH2
Y
6= p1 rK1
Y
6
=
p1 rH1
where p1 is the world price in the corner, given by equation (11) in the main text.
34