Trade and the Speed of Convergence

E. Young Song*
Sogang University

Abstract

This paper derives a convergence equation for a world integrated by trade. Factor priceequalization induced by trade reduces saving in capital-scarce economies, while it exercises the
opposite effect in capital-abundant economies. Thus income convergence among parametrically
identical economies is slower in a world integrated by trade than in a world of autarkic economies.
Under standard parameter values, the integrated world model generates low rates of convergencefrequently observed in empirical studies, without resorting to a counterfactual share of capital orhigh adjustment costs in investment.

Keywords: convergence, factor price equalization, growth, integration, trade
JEL Classification: F43, O41

I. Introduction
Research on income convergence exploded following the works of Mankiw, Romer, andWeil (1992) and Barro and Sala-i-Martin (1992a). These studies opened a stage for empiricalresearch on convergence rigorously based on neoclassical growth theory. A major attraction of
the approach is that the coefficients of a regression equation can be directly linked to thestructural parameters of the underlying theory, enabling researchers to assess the quantitative*

Department of Economics, Sogang University, C. P. O. Box 1142, Republic of Korea.
Tel: +82-2-705-8696, Fax: +82-2-705-8180, E-mail: eysong@sogang.ac.kr
1plausibility of the theory from estimation results.
A surprising feature of this literature is that regression equations are derived from a worldthat is composed of autarkic economies. Many trade economists find this lack ofinterdependence among economies quite disturbing (Findlay, 1996). As emphasized by classicaltrade theory, trade in goods is effectively trade in production factors embedded in goods. A world
linked by trade must show the pattern of convergence different from a world of autarkic
economies.
This paper derives a convergence equation for a world integrated by trade. We build aneoclassical model of growth in a world of economies with identical preferences andtechnologies, and examine how the growth rates of economies with different initial conditions
evolve over time under factor price equalization brought about by trade. The convergenceequation that we derive for the integrated world corresponds closely to the autarkic versions
obtained by Mankiw et al. (1992) and Barro and Sala-i-Martin (1992a). As theirs, our equation
can be used as a regression equation for estimating the rate of convergence, and the rate ofconvergence can be directly linked to the structural parameters of the model. However, there areimportant differences: In an integrated world, (a) the growth rates of individual economies are
positively affected by the growth rate of the world; (b) the rate of convergence increases as the2

world grows faster, and (c) the rate of convergence, under conventional parameter values, ismuch lower in an integrated world than in a world of autarkic economies.
Given the vast literature on dynamic Heckscher–Ohlin models, this research naturally
overlaps with a number of existing studies. The studies by Oniki and Uzawa (1965), Stiglitz(1970), and the works summarized in Findlay (1995) demonstrated the possibility that the gap
between a rich and a poor economy widens with trade. However, these results are dependent on
the assumption that poor countries are poor because they have inherently high subjective
discount rates. More closely related to our paper are the studies by Chen (1992), Ventura (1997)
and Atkeson and Kehoe (2000). As our paper, these studies investigate the world of economies
with identical time preference, and show that the difference in capital-labor ratio among
economies will persist under factor price equalization. However, none of these studies derive aconvergence equation and express the rate of convergence as a function of model parameters.
Researchers of convergence were bewildered by the fact that neoclassical growth theory,
under plausible parameter values, overpredicts the rate of convergence by a big margin. To
reduce the rate of convergence predicted by the model, various strategies had been adopted.
Mankiw et al. (1992) and Barro et al. (1995) assumed a counterfactually large share of capital inproduction. They justified this assumption by arguing that capital should be interpreted as3

including both physical and human capital. However, if we model human capital accumulation ina more rigorous way, the rate of convergence considerably increases, worsening theoverprediction problem (Ortigueira and Santos, 1997). Duczynski (2002) introduces adjustment
costs in investment to slow down convergence. However, adjustment costs are difficult toobserve and hard to calibrate. Our integrated world model generates low rates of convergencefrequently observed in data without imposing an extraneous assumption or restriction.
This paper is organized as follows. In Section II, we describe the model of a world
integrated by trade and make some general observations. In Section III, we calculate the rate ofconvergence predicted by the model under Cobb-Douglas technology. In Section IV, weconclude.

II. General Dynamics

A closed economy produces a final good and two intermediate goods, using capital and labor.
The representative household has the following objective function:∞1
C 1−θ exp[ −( ρ − n )t ]dt
1−θ
0

Max ∫

s .t.

K& = RK + WH − C − ( n + δ ) K ,H& = gH .

4(1)

1/θ is the intertemporal rate of substitution and ρ is the subjective discount rate. The number ofworkers, which we will denote by L, grows at the rate of n. We use C to denote consumption per
worker, K to denote physical capital per worker, and H to denote human capital per worker. Let R
and W be the rental price of physical capital and the wage rate per human capital. Physicalcapital depreciates at the rate of δ and human capital per worker grows at an exogenous rate of
g.1 To ensure that utility is bounded, we assume thatρ − n + (θ − 1) g > 0.

(2)

Let us use small letters to denote quantities per effective worker: k is equal to K/H and c is equal
to C/H. Using this notation, we can express the solution for (1) in the following way:1
c& = [ ( R − δ − ρ ) − g ]c ,(3)

k& = ( R − δ − n − g )k + W − c ,(4)lim k c −θ e −[ ρ − n +(θ −1) g ]t = 0.

(5)

θ

t →∞

We adopt the production structure introduced by Corden (1971) and recently used by
Ventura (1997). Competitive firms produce a final good Y by bundling two intermediate goods, 1
and 2. No capital or labor is used in bundling. The bundling technology can be described as a
CRS production function with diminishing marginal products. Let the unit cost of the final good1

In a companion paper, we investigate the case where g is endogenously determined by education and obt

ain results very similar to the present paper’s.
5

be e(p1, p2), where the arguments are the prices of intermediate good 1 and 2. We take Y as thenumeraire:e(p1, p2) = 1.

(6)

The final good is used both for consumption and for augmenting physical capital. Thus, the priceof consumption and the price of investment are both equal to unity, as we implicitly assumed in
the household problem above. The choice of numeraire also implies that R and W are the realrental price and the real wage. The real interest rate r is given by R − δ.
Competitive firms produce intermediate goods 1 and 2 using the following CRS technologies
with diminishing marginal products.
Q1 = G1(K1, H1),(7)

Q2 = G2(K2, H2),

(8)

where Qj denotes the output of intermediate good j and K1 + K2 = K and H1 + H2 = H. We assume
that good 1 is more physical-capital-intensive than good 2. Given the prices of intermediate
goods, the competitive economy allocates K and H such that GDP per worker Y, which is equalto p1 Q1 + p2 Q2, is maximized. Thus Y, Q1, and Q2 can be expressed as functions of p1, p2, K and
H. Because p2 is a decreasing function of p1 by (6), we can drop p2 and write these functions as
Y(p1, K, H), Q1(p1, K, H) and Q2(p1, K, H). Because these functions are linearly homogenous in K

6and H, we can express them in per-effective-worker form as follows:y(p1, k) ≡ Y(p1, K, H)/H,(9)

q1(p1, k) ≡ Q1(p1, K, H)/H,(10)

q2(p1, k) ≡ Q2(p1, K, H)/H.

(11)

We can easily obtain the following proposition.

[Proposition 1]
GDP per effective worker can be expressed as a function of capital per effective worker
alone:y(p1, k) = f(k)(12)

f’(k) = R,

(13)

f(k) − f’(k)k = W.

(14)f’’(k) < 0.

(15)

Furthermore,

The proof can be found in the appendix. Using proposition 1, (3) and (4) can be written as:

1
c& = [ ( f ' ( k ) − δ − ρ ) − g ]c ,(16)

k& = f ( k ) − (δ + n + g )k − c .

(17)

θ

7

In the following, we will focus on the case in which f(k) satisfies the Inada condition.2 The
economy converges to the unique steady state defined by:

R* = f ' ( k *) = δ + ρ + θg ,(18)

c* = f (k *) − (δ + n + g )k *.

(19)

We use asterisks to denote steady state values. The steady state interest rate is given by r* = R* −

δ. In short, using the function f(k), our two-sector economy can be reduced to the standard onesector model known as the Ramsey–Cass–Koopmans model.
We now consider a world composed of I countries. 3 All countries are identical to theeconomy described above, except in that they have different starting levels of K and H. Supposethat all these countries are closed. Then, denoting country by subscript i (i = 1, …, I), ci and ki
would be governed by the same dynamics as described by (16) and (17). ki would converge to k*
regardless of its initial level. In the long run, all countries would reach the same level of income
per effective worker and would also attain the same level of income per worker if the initiallevels of human capital were identical across countries.
Now suppose that these countries engage in costless trade of intermediate goods with eachother. The final good is not traded. Capital and labor do not move across countries, and no

If f(k)/k has a lower bound greater than δ+n+g, the economy asymtotically behaves like an AK-type endo
genous growth model. The Inada condition precludes this possibility.
3
Although we call them countries, they can be states, prefectures, or regions in a single country. We may prefer the latter interpretation because factor price equalization is more likely to hold in this environment.
822

international lending and borrowing occurs.4 Let σi be the share of country i in world effective
labor:σi = H i Li / ∑ H j L j

(20)

j

Li is the number of workers in country i. This share is constant, given our assumption that thegrowth rates of population and human capital are identical everywhere. We denote world
variables by dropping country subscripts. For example,c = ∑ σ i ci ,

(21)

i

k = ∑ σ i ki .

(22)

i

Note that using proposition 1, R /W can be expressed as a function of k. Given R/W, we candetermine the values of K1/H1 and K2/H2 that minimize the costs of producing intermediate good
1 and 2. Let us call these capital intensities k1(k) and k2(k). We assume that

k2(k) ≤ ki ≤ k1(k) for any i.

(23)

In other words, the factor endowments of all countries are inside the cone defined by k1(k) andk2(k). It is well-known in international trade (Dixit and Norman, 1980) that if this condition issatisfied, free trade among I countries replicates the equilibrium of the integrated economy, a4

These assumptions are not restrictive in the context of our model. Producing the final good does not cost any resources and can occur anywhere without affecting equilibrium. Factor prices are equalized across countries and households are indifferent between investing at home and lending abroad. Thus, the amount ofinternational lending is indeterminate. Factors can move across countries changing their levels of output, b
ut their levels of income are not affected. Infinitesimal costs in cross-border movements of factors would be enough to confine them within borders.
9hypothetical closed economy endowed with physical capital of ∑ K i and effective labor ofi∑ H L . Goods and factors prices are equalized across countries. There is no specialization iniii

production and all countries produce two intermediate goods. If country i is abundant in physicalcapital (ki > k), it exports the capital–intensive intermediate good 1. If country i is abundant inlabor (ki < k), it exports the labor-intensive intermediate good 2. c and k follow the same path asgiven by (16) and (17), converging to the steady state of c* and k*. The world as a whole
behaves just like a single closed economy.
Because factor prices are identical across countries, consumption and capital of country i
follow the following paths:

1
c&i = [ ( r − ρ ) − g ]ci ,(24)

k&i = ( r − n − g )ki + W − ci .

(25)

θ

Using (16), (17), (24) and (25), we can prove the following proposition.

[Proposition 2]
Suppose that the world is engaged in free trade and (23) is satisfied at every moment.
Then, the capital gap between country i and the world can be expressed astki (t ) − k (t ) = exp[ ∫ ( r − n − g − φ )ds ] ( ki (0) − k (0)) .
0

10

(26)

r − n − g − φ is positive during transition if k(0) < k*.

φ is the propensity to consume out of wealth. The proof can be found in the appendix.
Proposition 2 shows that the difference between country i and the world will be increasing in an
integrated world where output per effective worker is growing. It contains a strong implication
for the relationship between trade openness and growth. Suppose that world output per effective
worker is growing and a small country i has k i (0) < k (0) . If it does not trade with the world, its
output will reach y* = R* k* + W* in steady state. However, if it trades with the world, its long-run
output will be determined by yi* = R * k i* + w* , where∞ki* = k * + exp[ ∫ ( r − n − g − φ )ds ]( ki (0) − k (0)) < k*.

(27)

0

yi* < y* and country i attains a lower long-run level of output per worker under free trade. Thisimplies that a small country whose capital per effective worker is below (above) world average
will experience a lower (higher) average growth rate under free trade than in an autarky.5(26) also implies thattk i − k j = exp[ ∫ (r − n − g − φ )ds] (k i (0) − k j (0)) .

(28)

0

5

Output jumps up on the moment of opening trade due to static gains from trade. All countries gain fro
m trade as they face the path of prices different from the autarky one. Trade stimulates consumption of acountry whose capital is below average and this additional consumption reduces its long-run level of income.
11

In other words, the capital gap between any two countries is ever widening during transition in
an integrated world that is growing. Curiously, the widening gap in capital-labor ratio has notbeen noted in the literature.6 It is a direct implication of factor price equalization and thepermanent income hypothesis. We obtain it under general CRS technologies and generalisoelastic preferences.7 A potential problem that arises from this generalization is that (23) canbe violated during transition even if it is satisfied at time zero. If this happens, our dynamics is
no longer applicable. This problem can largely be avoided under Cobb-Douglas technology, as
we will see in the next section.

III. The Rate of Convergence under Cobb–Douglas Technology
We saw in the previous section that the absolute gap in capital between countries must be
widening in a growing world that is integrated by trade. However, income convergence is usually
defined in terms of the gap in logarithmic income log yi − log y j , which is approximately equal

to the percentage income gap yi / y j − 1 . The widening absolute gap can be consistent withconvergence in logarithmic income. In this section, we show that under Cobb–Douglas
technology with parameter values commonly used, the percentage gap in income indeed declines
6

Atkeson and Kehoe (2000) anticipated our result by showing that if a small country opens trade to a bi
g country that already is in steady state, its growth stops on the moment of opening. Note that this stagnatio
n occurs in our model when the world is in steady state and r − n − g − φ = 0.
7
Ventura (1997) used a logarithmic utility function and Leontief production functions.
12during transition despite the widening absolute gap in capital. Trade only slows down
convergence in logarithmic income.
The pattern of convergence in a world of autarkic economies is well understood for the
Cobb–Douglas case. By log-linearizing the system composed of equations (16) and (17) using
the production function f(k) = Ak α , Barro and Sala-i-Martin (1992a) derive the following
convergence equation for autarkic economies:1y (T ) 1y *= (1 − exp[ −λcT ]) log i ,log iTy i ( 0) T
y i ( 0)

(29)

where
1
2

λc = [ ( R * −δ − n − g ) 2 + 4(1 − α )R* R*(− δ − n − g ) − ( R * −δ − n − g )] .

θ

α

(30)

λc is the rate of convergence and is always positive. It measures the rate at which the gap
between current income (log yi) and steady state income ( log yi* ) shrinks in a single closed
economy. If countries in the world are parametrically identical and if they all are in autarky,logyi (T )y (0).
= exp[ −λcT ] log iy j (T )y j ( 0)(31)

Thus λc also determines the rate at which the percentage income gap between two countries
shrinks. The standard method for estimating λc is to run a cross-country regression of equation(29). Many empirical studies estimate λc to be around the value of .02.8
In a standard cross-country regression, y i* is allowed to vary among countries, implying conditional con
vergence. In a cross-regional regression, it is frequently assumed that regions approach the same steady state. So many growth regressions report λc close to 0.02 that some regard it as an empirical law. See Barroand Sala-I-Martin (2004, chaps. 11–12). However, estimation methods are controversial. Caselli, Esquivel, a138

It is well-known that neoclassical growth theory overpredicts the rate of convergence understandard parameter values. We take the following values as the benchmark case:θ = 1, ρ = 0.02, δ = 0.05, n = 0.01, g = 0.02.

(32)

λc is sensitive to α, capital share of income. The upper line in Figure 1 traces the value of λc as
α changes from zero to 1. At a conventional value of α, which is around 0.30, λc is well above
0.10. To reduce it down to 0.02, α should be as high as 0.80. This is the reason that Mankiw et at.
(1992) suggested that α should be interpreted as the share of broadly defined capital, including
both human and physical capital.
All this controversy proceeded with the assumption that countries in the world arecompletely separated from each other. We need to derive the convergence equation for a world
connected by trade. We manipulate (26) to obtain the following:ki (T )k ( 0)− 1 = exp[ −λoT ]( i− 1) ,k (T )k ( 0)

(33)

where

λo =

T

1
[ g k − ( r − n − g − φ )]dt .
T ∫0

(34)

nd Lefort (1996), Islam (1995), and many others use variants of fixed-effects panel estimation and find that λc is far greater than 0.02. The panel estimations eliminate omitted variable bias, but also introduce strong upward bias due to measurement errors. See Durlauf, Johnson, and Temple (2005) and Hauk and Wacziarg (2004) for an evaluation of this controversy.
14

Figure 1. Rate of Convergence in the Neoclassical Model

We use gk to denote the growth rate of k. Using the fact that yi = Rki + w and (33), we can convert(33) into an equation for output:Ty ( 0)yi (T )
− 1 = exp[ ∫ ( r − n − g − φ + g R − g y )ds ] ( i− 1) .
y ( 0)y (T )
0(35)

The following proposition is useful for dealing with our two-sector economy.

[Proposition 3]
Suppose that the production functions for the final good and the two intermediate goods are
given by the following Cobb-Douglas functions.
15

1− m

Y = A f Q1 Q2
m

α

,

(36)
1−α j

Q j = Aj K j j ( H j )(j = 1, 2).

(37)

Then
f ( k ) = Ak α ,(38)where A is a positive constant and α = m α1 + (1−m) α2.

The proof is in the Appendix. The result is convenient for us because we can interpret theaggregate Cobb–Douglas production function of Barro and Sala-i-Martin (1992a) as the one
derived from our two-sector economy. Then we can make a direct comparison between theirresults and ours. With the Cobb–Douglas assumption, Rk/y is constant at α . Thus, g R + g k = g y .
From (34) and (35),yi (T )y (0)− 1 = exp[ −λoT ]( i− 1) ,y (T )y (0)(39)

Using the Taylor approximation that log yi y ≈ yi y − 1 and rearranging terms, (39) can be
written as:y (T ) 1
1y (T ) 1y ( 0)log i≈ log+ (1 − exp[ −λoT ]) log.
Ty i ( 0) Ty ( 0) Ty i ( 0)(40)Compared with the convergence equation for autarkic countries, we can note two differences.
First, the growth rate of country i now depends on the world growth rate, 1/T log y(T)/y(0).

16Second, we have y(0) in place of yi*.9 In a cross-country regression where the world growth rate
and the initial world output can be treated as constant, we can estimate λo using the same
regression equation that is used in estimating λc. Thus in standard convergence regressions using
data from economies connected by trade, researchers are likely to obtain estimates for λo, not λc.
This misappropriation would be more likely when a regression is over regions in a single country,
which is far more likely to be integrated by trade.
To determine the sign of λo, we prove the following proposition in the Appendix.

[Proposition 4]

In the two-sector economy where all the production functions are of Cobb-Douglas form andcapital per effective worker is growing, λo > 0 if and only ifR*θ> α (δ + n + g ).

(41)

With this proposition at hand, we go back to the question whether condition (23) will be violated.
We can easily show that under Cobb–Douglas technology, k2/k and k1/k are constant. Thus, if λo> 0, by (33) k i / k − 1 decreases during transition, and (23) will never be violated if it issatisfied at time zero. Using (2) and (18), we can easily see that (41) always holds with θ = 1. If
9We show below an additional difference: the rate of convergence is an increasing function of the world
growth rate.
17θ = 2, with the other parameters at the benchmark levels, the condition holds for α < .69. With θ= 3, the condition holds for α < .54. Thus under parameter values commonly used, thepercentage income gaps will be shrinking during transition. In addition, factor price equalization
will hold throughout transition once it holds at the initial moment.
To obtain a numerical value of λo, we first examine the case of θ = 1. In this case, φ = ρ − n
and gk − (r− n− g− φ ) = gk − gc. Because y = Ak a , g k =gc =1− α R * 1

λc

θ α

1

α

g y . We can show thatg y on the convergent path of the log-linearized system. Plugging theseequations into (34), we find

λo =

1

α

(1 −

1−α

λc

R*) x ,

(42)

where
x=

1
y (T )
.
log
T
y (0)

The rate of convergence is increasing in x, the growth rate of world output per effective worker.

λo under various parameter values can be calculated from (42), once we know the value of x,
which can be obtained from data. According to the Penn World Table and Barro and Lee (2001),
world output grew at the annual rate of 0.0385 and world effective workers grew at the annual
rate of 0.023 between 1960 and 1990. Thus x for the world economy is given by .0155 and weuse 0.015 as the value of x.10 The lower line of Figure 1 traces the value of λo as α changes. The
10

The corresponding value for the US economy is 0.012. We can use this value of x if we study the18rate of convergence in the integrated world is much lower than that in the world of autarkic
economies. To have the rate of convergence equal to 0.02, we need α = 0.32, which is close to
the shares of physical capital observed in most countries.
For θ ≠ 1, we cannot obtain a simple formula like (42) as φ depends on the future path of theinterest rate. In this case, we numerically simulate the model to obtain λo. We obtain theconvergent paths of k and r and directly calculate the value of the integral in equation (34) on thepaths. We select the initial value of k such that x is equal to 0.015. The results for the cases whereθ =2 and θ =3 are reported in Table 1. We find again that the rate of convergence is much lower
in an integrated world than in a world of autarkic economies. Increasing the value of x reduces
the difference, but unless we adopt an absurdly large value of x, the order does not change.
We can see that with a lower intertemporal rate of substitution, the rate of convergencebecame somewhat lower compared to the case where θ = 1. With θ = 2, λo = 0.02 for α = 0.025.
With θ = 3, λo = 0.02 for α = 0.022. As we expected, λo becomes negative for α > 0.69 with θ =2, and for α > 0.54 with θ = 3. However, these negative values are virtually zero. Thus we do nothave to worry about the possibility that condition (23) will be violated during transition evenwhen λo is negative. We experimented with different choices for the values of ρ, δ, n, or g, but
we found that the results in table 1 are not sensitive to changes in their values.
convergence among the US states.
19

Table 1--Simulation Results for the Neoclassical Model

θ=2

θ=3

λc

λo

λc

λo

0.1

0.265

0.098

0.249

0.085

0.2

0.160

0.032

0.147

0.024

0.3

0.109

0.014

0.099

0.009

0.4
0.5

0.078
0.056

0.006
0.003

0.070
0.049

0.003
0.001

0.6

0.040

0.001

0.034

–0.0000.7

0.027–0.0000.023–0.001

0.8
0.9

0.016
0.007

–0.000–0.000

0.013
0.006

–0.000–0.000IV. Conclusion

Many believe that income convergence among countries is an overresearched topic. We caneasily understand why, seeing the vast literature developed around this single topic. Nevertheless,it still is surprising that the key theory that generated all this controversy has been derived in theabsence of trade among countries. This paper attempts to fill this remaining hole. We derive theconvergence equation for a world integrated by trade by combining the text book models of tradeand growth. We find that there is a general tendency that the absolute gaps in capital-labor ratio

20

between countries widen during transition. We also find that these widening gaps are compatible
with convergence in logarithmic income, and only reduce the speed of convergence. Our
approach is superior to the conventional approach based on autarkic economies in that our model
predicts the rate of convergence close to observed values, under standard parameter values andwithout imposing arbitrary assumptions.
This paper may provide some theoretical support for the studies such as Rodriguez andRodrik (2000) and Slaughter (2001), who argued that no robust evidence exists that trade speeds
up convergence. However, this paper has no intention to argue that trade will actually slow down
convergence in the real world. The model that we use is highly stylized and fails to reflect many
important aspects of growth, most seriously specialization and technology spillovers. However,anyone who would like to build a compelling theoretical argument that trade speeds up thegrowth of poor countries will have to fight against the forces identified by this paper.

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23

Mathematical Appendix[Proof for Proposition 1]

The competitive two-sector economy allocates resources as if it solves the following problem.

y = v ( p1 , p2 , k ) ≡Max

K1 , K 2 , H 1 , H 2

p1G1 ( K1 , H 1 ) + p2G2 ( K 2 , H 2 )(A1)s.t. K1 + K 2 ≤ k ,H 1 + H 2 ≤ 1.

Using the envelope theorem, we can show that∂v∂v∂v
= q1 ,= q2 ,
= R.
∂p1
∂p2
∂k

(A2)

Let ej be the partial derivative of e with respect to pj. For each intermediate good, output must be
equal to domestic demand. Using Shepherd’s lemma, this condition can be expressed as

q1 ( p1 , k ) e1 ( p1 , p2 ).
=q2 ( p1 , k ) e2 ( p1 , p2 )(A3)

The left-hand side of (A3) is the relative supply of intermediate good 1 and the right-hand side is
the relative demand. Recall that by (6), p2 is a decreasing function of p1. Then we can easily
show that the relative supply is increasing in p1 and the relative demand is decreasing in p1. Thus
(A3) implicitly defines p1 as a decreasing function of k. Then we can express v as a function of kalone:
y = v ( p1 ( k ), p2 ( p1 ( k )), k ) ≡ f ( k ) .
24

(A4)

From (6), e1dp1 + e2 dp 2 = 0 . Using this equation and (A2),
dy =

∂v
∂v
∂v
q e
dp1 +
dp2 + dk = ( q1dp1 + q2 dp2 ) + Rdk = ( 1 − 1 ) q2 dp1 + Rdk .
∂p1
∂p 2
∂k
q2 e2

(A5)

In autarky equilibrium, the term in the parenthesis is zero. Thus dy/dk = f’(k) = R. Because
producing the final and two intermediate goods does not generate any profits, y = Rk + W. Thus
W = y – Rk = f(k) − f’(k) k. Finally, using the Stolper–Samuelson theorem, we can express R as
an increasing function of p1. Since p1, in turn, is a decreasing function of k, R = f’(k) is
decreasing in k.

[Proof for Proposition 2]

From (24) and (5), we can obtain the following consumption function:ci = φ ( k i + η ) ,

(A6)

s
∞
1
φ = 1 / ∫ exp[− ∫ ((r (u ) − n − (r (u ) − ρ ))du ]ds ,
θ
t
t

(A7)

where

∞

s

η = ∫ W ( s ) exp[− ∫ ( r (u ) − n − g )du ]ds .
t

(A8)

t

φ is the propensity to consume out of wealth and η is the present value of wages. Under factor
price equalization φ and η are identical in all countries. Thus25

c = φ (k + η ) .

(A9)Using (17), (25), (A6) and (A9), we obtain the following equation:tk i − k = exp[∫ (r − n − g − φ )ds] (ki (0) − k (0)) .

(A10)

0

(A10) also implies thattk i − k j = exp[ ∫ (r − n − g − φ )ds] (k i (0) − k j (0)) .

(A11)

0

Suppose that k(0) < k* and the world economy is approaching the steady state from below.
Then c and k are increasing, and r is decreasing during transition. By (16), 1/θ (r − ρ) is greaterthan g during transition. Then
∞

s

∞

∞

s

t

t

1
.
r (t ) − n − gs11= ∫ exp[− ∫ (r (u ) − n − (r (u ) − ρ ))du ]ds ≥ ∫ exp[− ∫ (r (u ) − n − g )du ]ds
φ (t ) tθttt≥ ∫ exp[− ∫ (r (t ) − n − g )du ]ds =r − n − g − φ is strictly positive during transition and reaches zero in steady state.

[Proof for Proposition 3]

The unit cost functions for the final good and two intermediate goods are given by:1− me( p1 , p2 ) = ε f p1 p2mαd j ( R ,W ) = ε j R j W
26

1−α j

,

(A12)

(j = 1, 2),(A13)whereε f = A f −1m − m (1 − m) − (1−m ) ,ε j = A j −1α j −α j (1 − α j )− (1−α j ).

Zero profits in the production of two intermediate goods require that p j = d j ( R,W ) . Plugging
these equations into the numeraire constraint e( p1 , p2 ) = 1 ,ε f ε 1mε 21− m RαW 1−α = 1 ,

(A14)

where
α = mα1 + (1 − m)α 2 .

Equation (A3) can be expressed as:
p1q1
pe
m
.
= 11 =
p 2 q 2 p 2 e2 1 − m(A15)

From this, we can derive the following equations.
R = αyk −1 ,W = (1 − α ) y.

(A16)

Thus α is equal to capital share in income. Plugging (A16) into (A14), we obtain thaty = Ak α ,
where
−1

−m

A = ε f ε1 ε 2− (1− m )[Proof for Proposition 4]27

α −α (1 − α ) − (1−α ) .

(A17)

If we plug equation (A9) into equation (17) and divide both sides by k ,1
ηg k − ( r − n − g − φ ) = W (1 − φ ) ,
k
W

(A18)

where
∞

∫ exp[− ∫ (r − n − g − g )dv] ds .
φ =W∫ exp[− ∫ ( r − n − g − g )dv ] ds
η

t

s
t

W

s
t

c

∞

t

(A19)

gW is the growth rate of wage. Since W = (1−α) A kα, gW = gy. We can show that on theconvergent path with k(0) < k*, gc > gy iff

R*

θ

11

> α (δ + n + g ). Using (39), λo > 0 iff

R*

θ

R*

θ

> α (δ + n + g ). 11 By (A19), φ> α (δ + n + g ).

See Appendix 2C in Barro and Sala-i-Martin (2004).
28

ηW< 1 iff