Trade, Product Cycles and Inequality Within andBetween Countries

Susan Chun Zhu∗

October 12, 2003

Abstract

This paper incorporates Northern product innovation and product-cycle-driventechnology transfer into the continuum-of-goods Heckscher-Ohlin model. The creationof very skill-intensive goods induces the North to transfer production of older, lessskill-intensive goods to the South. These relocated goods must be the most skill in-tensive by Southern standards. Thus, product cycles raise the relative demand forskilled workers and thus wage inequality within both regions. This runs contrary tothe Stolper-Samuelson theorem, but accords well with the fact that wage inequalityhas risen in both Northern and Southern countries. Moreover, product cycles increaseincome inequality between countries. Although technology transfer narrows the North-South income gap, this effect is more than offset by the effect of product innovation.

This paper also examines welfare implications of product cycles. Product cyclesbenefit both the North and the South and make skilled workers in both regions betteroff. Further, an increase in the supply of Southern skilled labor can raise the rate oftechnology transfer and narrow the North-South income gap. (JEL classification: F1,Keywords: international trade, product cycles, inequality)

∗Department of Economics, Michigan State University, Marshall Hall, East Lansing, MI, 48824, USA.email: zhuc@msu.edu. I am indebted to Dan Trefler for his encouragement and many helpful comments. Ialso thank Nancy Gallini, Wolfgang Keller, Angelo Melino, Diego Puga, Martin Richardson, Aloysius Siow,and Nadia Soboleva. Financial support from the Social Sciences and Humanities Research Council of Canadais gratefully acknowledged.

1. Introduction

The product cycle is not a new concept. It was minted by Vernon (1966) nearly forty years

ago. Since Krugman (1979) first formalized it by using a one-factor model featuring ex-

ogenous technical change, many other authors have extended various aspects of his model.

Jensen and Thursby (1987) endogenized the innovation process. Dollar (1986) incorporated

capital movement in a three-factor model. Flam and Helpman (1987) focused on quality

upgrading rather than expansion of goods varieties. Grossman and Helpman (1991) endo-

genized both innovation and technology transfer. Although these authors examined many

implications of the product cycle, they neglected its impacts on the domestic distribution of

income.1

There is increasing evidence that in recent decades, inequality has grown not only in

many Northern countries, but also in some Southern countries ( e.g., Feenstra and Hanson

1996, 1997, Robbins 1995, Cragg and Epelbaum 1996, Hanson and Harrison 1999, Berman

et al. 1998). The pervasiveness of rising inequality leads people to think international trade

might be an important factor (e.g., Wood 1994, Leamer 1996). However, the traditional

Stolper-Samuelson theorem predicts that when inequality increases in the North, it should

decline in the South. Thus, some authors claim that the traditional Heckscher-Ohlin model,

the backbone of the Stolper-Samuelson theorem, fails (e.g., Robbins 1995).

One purpose of this paper is to show that widening inequality in both regions can be

explained within a Heckscher-Ohlin framework. To this end, I incorporate technical change

1Dinopoulos and Segerstrom (1999) present a dynamic model of North-North trade that has implicationsfor wage inequality in Northern countries. However, it does not deal with Southern countries.

1

into the continuum-of-goods Heckscher-Ohlin model (Dornbusch et al. 1980). There are

two countries (the North and the South), two factors (skilled and unskilled labor), and a

continuum of goods which can be uniquely ranked in order of increasing skill intensity. I

focus on the complete specialization equilibrium in which the South specializes in less skill-

intensive goods and the North specializes in more skill-intensive goods. In contrast to the

main concern of Dornbusch et al. about endowment changes, I consider a world economy

driven by Northern product innovation. Since my goal is to examine the impact of technical

change rather than its determinants, I assume that product innovation is exogenous. In light

of recent work on technology-skill complementarities (e.g., Goldin and Katz 1998, Autor et

al. 1998), I further assume that new goods use relatively more skilled labor than old goods.

However, technology transfer is endogenized and driven by product innovation.

The core result is that product innovation and technology transfer increase inequality

within both regions. When new goods are introduced, the relative demand for Northern

skilled labor increases, thus raising Northern inequality. If the aggregate elasticity of sub-

stitution between Northern skilled labor and unskilled labor is sufficiently large, the intro-

duction of new goods makes the North less competitive in low-end goods. The result is

technology transfer − the North moves production of its older, less skill-intensive goods to

the South. Such technology transfer reduces the relative demand for Northern unskilled

labor and aggravates the wage gap in the North. At the same time, since the transferred

Northern goods are more skill intensive than the previously produced Southern goods, the

relative demand for Southern skilled labor increases. Thus inequality also rises in the South.

The basic insight of this core result was initially suggested by Feenstra and Hanson’s

2

(1996) observations on outsourcing and inequality in the United States and Mexico. How-

ever, there are several major differences between my work and theirs. First, the driving force

is different. In Feenstra and Hanson’s three-factor model (capital, skilled labor and unskilled

labor), a higher rate of return on capital investment in the South causes production reloca-

tion. In my two-factor model, Northern product innovation drives all the changes in labor

markets and trade patterns. I am describing product cycles. Second, Feenstra and Hanson

close their model with an exogenous, upward sloping supply of labor to their single industry.

I close my model with a general equilibrium trade balance equation. Third, in Feenstra and

Hanson there is no substitutability between skilled and unskilled labor. This is incompatible

with the focus on such elasticities that dominates the literature on wage inequality (e.g.,

Katz and Murphy 1992). In my model, the substitutability between skilled and unskilled

workers plays a central role in generating product cycles.

Product cycles increase inequality not only within countries, but also between coun-

tries. Product innovation and endogenous technology transfer have opposing effects on the

North-South income gap. Unlike product innovation, technology transfer narrows the gap.

However, the product innovation effect, being more direct, must dominate. This result dif-

fers sharply from Krugman (1979). The difference arises from the way technology transfer is

handled: in Krugman, technology transfer is exogenous. Krugman considers the case where

there is an increase in the rate of technology transfer while keeping the rate of product inno-

vation constant, and concludes that the income gap can be narrowed. In my model, however,

technology transfer is induced by product innovation and, in equilibrium, may not catch up

with it.

3

The simplicity of the model allows me to derive a rich set of welfare implications of

product cycles. The terms of trade move in favor of the North and against the South. The

North further benefits from a wider range of new products. Strikingly, the South also benefits

from product cycles. This is because the loss to the South from worsened terms of trade is

entirely offset by the gain from new goods. At the same time, since income is redistributed

from unskilled to skilled workers, skilled workers in both regions are better off.

The above model can be extended by including an increase in the supply of Southern

skilled labor. This extension helps to explain the income catch-up by some newly indus-

trialized countries. An increase in the supply of Southern skilled labor raises the rate of

technology transfer and narrows the North-South income gap.

The paper is organized as follows. Section 2 describes the basic model. Section 3 il-

lustrates the effects of product innovation and technology transfer on labor markets and

concludes that inequality can rise within and between countries. Section 4 examines the wel-

fare implications of product cycles. Section 5 extends the model by allowing for an increase

in the supply of Southern labor. Section 6 draws conclusions.

2. The Basic Model

The set-up is based on Dornbusch et al. (1980). There are two regions (the North and the

South), two factors (skilled and unskilled labor), and a continuum of goods indexed by z in

the interval [0, n]. I make the standard Heckscher-Ohlin assumptions. Production functions

are quasi-concave and exhibit constant return to scale. Goods markets and labor markets

are perfectly competitive. Furthermore, there are no trade barriers in the model. Trade is

4

always balanced.

I focus on the complete specialization equilibrium where the North produces more skill-

intensive goods and the South produces less skill-intensive goods. To this end, I make two

additional assumptions. First, the North is relatively abundant in skilled labor and the South

is relatively abundant in unskilled labor. Further, the difference in labor endowment is so

large that the relative wage for skilled labor is lower in the North than in the South. Second,

there are no factor intensity reversals. This implies that goods can be ranked uniquely by

skill intensity independently of factor prices. I use a higher z to index a more skill-intensive

good. Then, as shown in lemma 1 in appendix A.1, the North has a comparative advantage in

skill-intensive goods (z, n] while the South has a comparative advantage in unskilled-intensive

goods [0, z). Good z is the ‘competitive margin’. It is the only good that is produced in both

regions. Under the assumption that there are no trade barriers, the value of z is determined

by

pN(z) = pS(z) (1)

where pi(z) is the price of good z̄ in region i (= N, S).

On the demand side, I assume that all individuals have identical preferences which are

represented by the CES utility function

U =

[∫ n

0

x(z)σ−1

σ dz

] σσ−1

(σ > 1) (2)

where x(z) is the consumption of good z and σ is the elasticity of substitution between

goods.

5

Equilibrium is characterized by conditions of balanced trade and labor market clear-

ing. Let Yi be national income in region i. Let pi(z) be the price of good z. Let P ≡[∫ z̄

0pS(z)1−σdz +

∫ n

z̄pN(z)1−σdz

] 11−σ

be the aggregate price index. With identical CES pref-

erences, the balance-of-trade condition is

YN

∫ z̄

0

[pS(z)

P

]1−σ

dz = YS

∫ n

z̄

[pN(z)

P

]1−σ

dz. (3)

The left-hand side is the value of Northern imports and the right-hand side is the value of

Northern exports. Combining the balance-of-trade condition with equation (1) yields

B(z) ≡ ln

{YN

pN(z̄)

∫ z̄

0

[pS(z)

pS(z̄)

]1−σ

dz

}− ln

{YS

pS(z̄)

∫ n

z̄

[pN(z)

pN(z̄)

]1−σ

dz

}= 0. (4)

Using good z̄ as numeraire simplifies the following analysis.

Let Hi and Li be the supply of skilled and unskilled labor, respectively. Let wHi and

wLi be the wages of skilled and unskilled workers, respectively. Define wi ≡ wHi/wLi. Let

Hi(wi, z) and Li(wi, z) be the amount of skilled and unskilled labor, respectively, required to

produce one unit of good z. Finally, define hi ≡ Hi/Li and hi(wi, z) ≡ Hi(wi, z)/Li(wi, z).

To simplify notation, in the following I will drop wN and wS as arguments. However, I am

not assuming that substitution between the two types of labor is restricted. The aggregate

demand for Northern skilled labor is HdN =

∫ n

z̄x(z)HN(z)dz. With CES preferences, world

demand for Northern good z is x(z) = [pN(z)/P ]1−σ (YN + YS)/pN(z). With zero profits,

pN(z) = wHNHN(z) + wLNLN(z). Plugging x(z) into HdN , the condition of Northern skilled

6

labor market clearing can be expressed as

HN = HdN =

∫ n

z̄

[pN(z)

P

]1−σ(YN + YS)HN(z)

wHNHN(z) + wLNLN(z)dz. (5)

Similarly, the condition of Northern unskilled labor market clearing is

LN = LdN =

∫ n

z̄

[pN(z)

P

]1−σ(YN + YS)LN(z)

wHNHN(z) + wLNLN(z)dz. (6)

Since I am interested in the wage gap between skilled and unskilled labor, I combine equations

(5) and (6) and express the labor market clearing conditions in terms of wN . To this end,

define N (z̄) ≡ HdN/HN − Ld

N/LN . N (z̄) is the excess demand for skilled labor relative to

unskilled labor in the North. With some manipulation, N (z̄) = 0 can be written as2

N(z) ≡∫ n

z

[pN(z)

pN(z̄)

]1−σ[hN(z)− hN ]

1 + wNhN(z)dz = 0. (9)

Similarly, the corresponding Southern labor market clearing condition S (z̄) = HdS/HS −

2The skilled labor market equilibrium condition (5) can be rewritten as

HdN

HN=

YN + YS

wLNHNP 1−σ

∫ n

z̄

pN (z)1−σ hN (z)1 + wNhN (z)

dz = 1. (7)

Similarly, the unskilled labor market equilibrium condition (6) yields

LdN

LN=

YN + YS

wLNHNP 1−σ

∫ n

z̄

pN (z)1−σ hN

1 + wNhN (z)dz = 1. (8)

Equation (7) minus equation (8) results in the labor market equilibrium condition (9).

7

LdS/LS can be written as

S(z) ≡∫ z

0

[pS(z)

pS(z̄)

]1−σ[hS(z)− hS]

1 + wShS(z)dz = 0. (10)

The competitive margin z and relative wages (wN , wS) are determined simultaneously

by equations (4), (9) and (10).3 z̄ serves as a link between the two labor markets. In this

model prices of goods and national income are also endogenized. The proof of existence and

uniqueness of the equilibrium in Dornbusch et al. (1980) can be applied with only minor

modifications.

3. Product Cycles and Inequality

In contrast to the concern of Dornbusch et al. with endowment changes, I focus on the

impact of technical change on labor markets and trade patterns. In this paper technical

change takes the form of product innovation in the North. This is consistent with the fact

that new goods are mainly invented and first produced in a few industrial countries. In the

1980s the United States, Japan, Germany, Great Britain and France accounted for 91% of

total R&D expenditures in the OECD area, and the United States alone accounted for more

than half of total R&D expenditures (calculated using the OECD ANBERD Database 2000).

Krugman (1979) and Grossman and Helpman (1991) also make a similar assumption.

I make two additional assumptions about Northern product innovation. First, since

3Both Yi/pi(z̄) and pi(z)/pi(z̄) (i = S, N) can be expressed in terms of wi : Yi/pi(z̄) =[(wihi + 1) Li] / [(wihi(z̄) + 1) Li(z̄)] , and pi(z)/pi(z̄) = [(wihi(z) + 1) Li(z)] / [(wihi(z̄) + 1) Li(z̄)] .

8

my goal is to examine the impacts of innovation rather than its determinants, I assume

that product innovation is exogenous. Second, in light of recent work on technology-skill

complementarities,4 I assume that new goods use relatively more skilled labor than old goods.

Note that this assumption is made for analytical convenience. It is shown in appendix A.3

that all results hold under a weaker assumption that on average new goods are more skill

intensive than existing Northern goods.

In this section I will provide a weak condition guaranteeing that as the North produces

more new goods, it loses competitiveness in less skill-intensive goods. As a result, older less

skill-intensive Northern goods migrate South. Following Krugman (1979) I will use the term

‘technology transfer’ to refer to this process of production relocation. I will also examine the

implication of product cycles for inequality within and between countries.

3.1. Outline

The creation of new goods has a direct impact on the trade balance. The utility function in

equation (2) implies that for given income and prices, consumers would be better off if they

have a wider range of goods. Thus, as new goods become available, consumers allocate some

of their budget to new goods and spend less on all old goods. This leads to a shift in demand

from Southern goods to Northern goods. Ceteris paribus, the North develops a trade surplus.

To restore the trade balance, Northern national income must rise (see equation 3).

4Goldin and Katz (1998) document that technology-skill complementarities existed in manufacturing earlyin this century. Krueger (1993) and Berman et al. (1994) report that use of computers raises the demandfor skills. Bresnahan, Brynjolfsson and Hitt (2002) find that information technology and the correspondingchange in firm organization also increase the demand for skilled labor. The theoretical work on technology-skill complementarities includes Galor and Tsiddon (1997).

9

The creation of new goods also directly affects the Northern labor market. Since new

goods are more skill intensive than all existing Northern goods, the creation of new goods

raises the relative demand for skilled labor. Thus, the relative wage of skilled labor rises.

This, together with the increase in national income just discussed, implies that the wage

of Northern skilled labor must rise. At the same time, the creation of new goods leads to

an excess supply of Northern unskilled workers. Whether the wage of unskilled labor rises

or falls in equilibrium hinges on the degree of substitutability between Northern skilled and

unskilled labor. Unskilled labor can be substituted for skilled labor in two ways. First,

cheaper unskilled labor replaces skilled labor within production of each good, i.e., within-

good substitution. Second, by the Rybczynski effect, production of less skill-intensive goods

expands, i.e., between-good reallocation. When Northern skilled and unskilled workers are

sufficiently substitutable, the oversupplied unskilled labor can be absorbed without lowering

their wage. In this case, the North loses its competitiveness in less skill-intensive Northern

goods so that these goods move South. In the above I have sketched out a simple story

about product cycles. In the following section I will formalize this idea.

3.2. Product Cycles

I introduce new goods as follows. Let t index the state of technology and let n be the highest

goods index given t. As technology evolves from t to t + dt, new goods are introduced over

the range (n, n + dn). I consider the impact of changes in t and hence n on wS, wN , and z.

As discussed above, the conclusion that Northern innovation leads to product cycles

(i.e., raises z) hinges on the degree of substitutability between skilled and unskilled labor

10

in the Northern production. Let εaN ≡ −d ln(Hd

N/LdN)/d ln wN be the aggregate elasticity

of substitution between Northern skilled and unskilled labor. Theorem 1 formalizes this

observation.

Theorem 1. There exists a constant ε ∈ (0, σ) such that εaN > ε ⇔ dz/dt > 0. That is,

if and only if skilled and unskilled labor are sufficiently substitutable in the production of

Northern goods, Northern innovation sets up product cycles in which new goods are initially

produced in the North and then relocated to the South.

Proof. See appendix A.4.

The critical degree of substitutability between Northern skilled and unskilled labor (ε)

depends on factor intensity differences across goods. If the differences are large, the creation

of very skill-intensive goods imposes more pressure on the unskilled labor market. In order

to absorb the oversupplied unskilled labor, a larger value of ε is required.

Theorem 1 also implies that εaN > σ is a sufficient condition for product cycles to occur. In

particular, when σ approaches 1 (i.e., preferences are represented by the Cobb-Douglas utility

function), the sufficient condition becomes εaN > 1. Empirically, this sufficient condition is

likely to be satisfied. Most estimates of the aggregate elasticity of substitution are between

1 and 2 (Johnson 1970, Freeman 1986, Katz and Murphy 1992, Heckman et al. 1998, and

Krusell et al. 2000).

3.3. The Role of the Elasticity of Substitution

I have discussed how product cycles rest on the substitutability between skilled and un-

skilled workers. This section further develops this point and shows that care is needed in

11

thinking about the aggregate elasticity of substitution. The aggregate elasticity of substi-

tution incorporates direct factor substitution within goods as well as indirect factor sub-

stitution via changes of output mix (i.e., the Rybczynski effect). The elasticity of sub-

stitution between skilled and unskilled labor in the production of good z is defined as

εN(z) ≡ −d ln hN(z)/d ln wN . (Recall that wN ≡ wHN/wLN is the relative wage of Northern

skilled labor and hN(z) ≡ hHN(z)/hLN(z) is the relative employment of skilled labor in good

z.) Using the labor demand equations in (5) and (6), εaN can be expressed as a weighted

average of within-good elasticities of substitution between factors (εN(·)) and the elasticity

of substitution between goods (σ):

εaN =

∫ n

z[pN(z)]1−σ θLN(z)θHN(z)εN(z)dz

YLNYHN

∫ n

z[pN(z)]1−σ dz

+ σ

[1−

∫ n

z[pN(z)]1−σ θLN(z)θHN(z)dz

YLNYHN

∫ n

z[pN(z)]1−σ dz

](11)

where θLN(z) ≡ wLNLN(z)/pN(z) is the cost share of Northern unskilled labor in good z,

θHN(z) ≡ wHNHN(z)/pN(z) is the cost share of Northern skilled labor, YLN ≡ wLNLN/YN is

the national income share of unskilled labor, and YHN ≡ wHNHN/YN is the national income

share of skilled labor. Note that the second term in equation (11) is always positive.5

Equation (11) is a continuum-of-goods version of the aggregate elasticity of substitution

discussed in Jones (1965). As εN(·) increase, εaN becomes larger. When εN(·) = 0 (i.e., no

within-good substitution), labor market adjustment depends on a Rybczynski-style realloca-

5Using hN (z)−hN

wN hN (z)+1 = θHN (z)−YHN

wN YLN= YLN−θLN (z)

wN YLN, equation (9) can be rewritten as∫ n

z̄[pN (z)]1−σ [θLN (z)− YLN ] dz = 0. Combining this with θLN (z)2 = [θLN (z)− YLN ]2 +

2 [θLN (z)− YLN ] YLN + Y 2LN yields

∫ n

z[pN (z)]1−σ

θLN (z)2dz > Y 2LN

∫ n

z[pN (z)]1−σ

dz. It fol-lows that

∫ n

z[pN (z)]1−σ

θLN (z)θHN (z)dz =∫ n

z[pN (z)]1−σ

θLN (z)dz − ∫ n

z[pN (z)]1−σ

θLN (z)2dz <

YHNYLN

∫ n

z[pN (z)]1−σ

dz.

12

tion of output between skill-intensive and unskilled-intensive goods. This is captured by the

second term in equation (11). A bigger value of σ implies a stronger substitution effect via

output reallocation. This is because with a bigger σ, consumers are more willing to substi-

tute cheaper goods for more expensive ones. Therefore, as the creation of new goods raises

the relative wage of Northern skilled workers and with it the relative prices of skill-intensive

goods,6 a larger value of σ facilitates output reallocation from skill-intensive goods to less

skill-intensive ones. This increases the capacity to absorb oversupplied unskilled Northern

workers. Equation (11) thus helps one to understand what has been estimated by researchers

such as Katz and Murphy (1992). It also helps one understand how the Feenstra-Hanson

model equilibrates: even though they assume that there is no substitution between skilled

and unskilled labor within goods production (εN(·) = εS(·) = 0), equilibration occurs via a

Rybczynski-style reallocation.

Both εN(·) and σ affect not only the sign of dz/dt, but also its magnitude. The more

substitutable unskilled labor is for skilled labor, the bigger is the rate of technology transfer.

With small values of εN(·) and σ, oversupplied unskilled Northern labor can only partially

be absorbed by factor substitution within goods and output reallocation between existing

goods. Most of the absorption must come from expanding the production of the least skill-

intensive goods, i.e., by reducing z. The higher εN(·) and σ are, the weaker the pressure to

lower z and, hence, the more effective the product-cycle pressure to raise z. The elasticities

of substitution in the South (εS(·) ≡ −d ln hS(·)/d ln wS) affect dz/dt in a similar way. To

formalize these observations, write εi as a function of a shift variable βi (i = S,N). βi shifts

6Consider two Northern goods z1 and z2. The change in the relative good price is d ln[pN (z2)/pN (z1)]/dt =[θHN (z2)− θHN (z1)]d ln wN/dt. This is positive when z2 > z1, i.e., z2 is more skill intensive than z1.

13

the entire εi(·, βi) schedule up. That is, a large βi makes the εi more elastic. The results

are summarized in theorem 2.

Theorem 2. dz/dt is increasing in βS, βN and σ. That is, an increase in either region’s

within-good elasticities of substitution between factors or the elasticity of substitution be-

tween goods raises the rate of technology transfer.

Proof. See appendix A.4.

3.4. Rising Wage Inequality Within Countries

In recent decades many developed countries and some developing countries have experienced

rising wage inequality. This poses a challenge to traditional trade theory. The Stolper-

Samuelson theorem appeals to shifting terms of trade to predict that rising inequality in

developed countries will go hand in hand with falling inequality in developing countries. This

contradiction has led some authors to doubt the relevance of the Heckscher-Ohlin framework

(e.g., Robbins 1995). Interestingly, the next theorem shows that this puzzle can be resolved

by embedding technical change into the continuum-of-goods Heckscher-Ohlin model. Recall

that wi = wHi/wLi (i = N, S) is the wage of skilled relative to unskilled workers. It is the

measure of within-country inequality in my model.

Theorem 3. (i) dwN/dt > 0. (ii) Assume εaN > ε̄. Then dwS/dt > 0. That is, product

cycles raise inequality in the North. If Northern skilled and unskilled labor are sufficiently

substitutable, then product cycles also raise inequality in the South.

Proof. See appendix A.4.

14

The creation of new, skill-intensive goods raises the relative demand for Northern skilled

labor. Thus, the relative wage in the North rises. It is worth noting that this result holds

whether z rises or falls. Even if z falls, thereby increasing the relative demand for Northern

unskilled labor, this effect is entirely offset by the effect of newly created goods. This is

consistent with the empirical finding that domestic skill-biased technical change is likely the

main factor behind rising inequality in developed countries (e.g., Berman et al. 1994, Autor

et al. 1998, Berman et al. 1998).

In contrast, technology transfer (rising z) is the only factor affecting the relative labor

demand in the South. The goods that migrate South may be the least skill intensive from

a Northern perspective, but they are the most skill intensive from a Southern perspective.

Technology transfer therefore raises the relative demand for skilled labor in the South, leading

to wage inequality there.7 The condition εaN > ε̄ in theorem 3 ensures z to rise.

3.5. Rising Inequality Between Countries

Product cycles increase inequality not only within countries, but also between countries.

Northern product innovation and endogenous technology transfer have opposing effects on

the North-South income gap. Unlike product innovation, technology transfer narrows the

North-South gap. This raises the concern about whether technology transfer will help the

South to catch up with the North. The next theorem shows that this concern is unwarranted.

Since the effect of product innovation is more direct, it offsets the negative effect of technology

7Zhu (2003) examines empirically the extent to which increasing demands for skilled workers can beexplained by product cycles, i.e., by U.S. innovation and the subsequent relocation of production to U.S.trading partners. She finds strong evidence that product cycles are significantly and positively correlated torising demand for skilled workers in a large panel of industries and countries.

15

transfer on Northern income. Thus, product cycles widen the North-South income gap.8

Theorem 4. d(YN/YS)/dt > 0. That is, product cycles widen the North-South income gap.

Proof. See appendix A.4.

Theorem 4 further implies that when the elasticity of substitution between goods (σ) is close

to unity, the rate of technology transfer (d ln z/dt) can never exceed the rate of product

innovation (d ln n/dt).9

Theorem 4 stands in sharp contrast to Krugman (1979). The difference mainly arises

from the way technology transfer is handled. In Krugman, technology transfer is exogenous.

Krugman considers the case where there is an increase in the rate of technology transfer

while keeping the rate of Northern innovation constant, and concludes that the North-South

gap can be narrowed. In my model, however, technology transfer is endogenous. Since

the rate of technology transfer lags behind the rate of innovation, the effect of technology

transfer on the income gap is of a second order compared to the effect of product innovation.

This completes the discussion of the implications of product cycles for inequality within and

between countries. I now turn to welfare analysis.

8An extension of the model by including the increasing supply of Southern skilled labor can help oneexplain the income catch-up by some newly industrialized countries in recent decades. See section 5 for moredetail.

9When σ approaches 1, the balance-of-trade condition in equation (3) implies

d ln(YN/YS)dt

=d ln(n− z)

dt− d ln z

dt.

By theorem 4 , d ln(YN/YS)/dt > 0. Thus, d ln n/dt > d ln z/dt.

16

4. Welfare Analysis

The simplicity of this model allows me to derive a rich set of welfare implications of product

cycles. In this section I will first examine change in the terms of trade. This is the basis for

further analysis of national welfare and labor welfare.

4.1. Terms of Trade

In a model with a continuum of goods, it is natural to define terms of trade as a price index

using initial net exports as the weights. By this definition, an increase in the terms of trade

implies that the initial consumption bundle is cheaper at the new prices than at the old ones.

This further implies that improved terms of trade always benefit consumers. (See Dixit and

Norman 1980, p.132.)

Using the above definition, the change in Northern terms of trade may be written as

dPN/dt = YS

∫ n

z̄[pN(z)/P ]1−σ [d ln pN(z)/dt] dz − YN

∫ z̄

0[pS(z)/P ]1−σ [d ln pS(z)/dt] dz. (Re-

call that P ≡[∫ z̄

0pS(z)1−σdz +

∫ n

z̄pN(z)1−σdz

] 11−σ

is the aggregate price index.) The change

in Southern terms of trade is dPS/dt = −dPN/dt. Theorem 5 states that product cycles

improve the terms of trade in the North while worsening the terms of trade in the South.

Theorem 5. Assume εaN > ε̄. Then dPN/dt > 0 and dPS/dt < 0. i.e., the terms of trade

rise in the North and fall in the South.

Proof. See appendix A.4.

It is worth noting that product innovation improves the terms of trade in the North. This

is in contrast to the conventional result that technical progress in the export sector generally

17

worsens the terms of trade. When product cycles raise the relative wages of skilled labor in

both regions, the goods that use relatively more skilled labor become more expensive within

each region (see footnote 6.) Note that good z̄ is the least skill intensive from a Northern

perspective while it is the most skill intensive from a Southern perspective. Thus, in terms

of good z̄, the relative price of any Northern good increases and the relative price of any

Southern good decreases, implying that the terms of trade must rise in the North and fall

in the South.

4.2. National Welfare

Since each region has two types of labor, care is needed in measuring national welfare. If the

government can do a lump-sum transfer within a region, or can redistribute domestic income

optimally through a complete set of indirect taxes, then a higher indirect utility, which is

derived using national income, will imply that all types of labor in the region can be made

better off (see Dixit and Norman 1980, p.20).

When all types of labor have the same CES preferences, it is straightforward to derive

the Northern indirect utility as UN = YN/P . Then the change in Northern welfare can be

further derived as

d ln UN

dt=

1

σ − 1

[pN(n)

P

]1−σdn

dt+

1

YN

dPN

dt. (12)

The first term reflects the gain from consuming new goods. The second term captures the

effect of terms of trade. Since the terms of trade improve in the North (see theorem 5),

equation (12) suggests that the North will always benefit from product cycles.

This differs markedly from Krugman (1979) who finds that technology transfer must make

18

the North worse off. The difference arises from the different impact of technical change on

the terms of trade. In Krugman, when the rate of technology transfer catches up with the

rate of product innovation, the terms of trade in the North will fall. In contrast, in my model,

although technology transfer shifts income from the North to the South, it is a second-order

effect compared to the effect of product innovation. As shown in theorem 5, product cycles

improve the terms of trade in the North. Thus, even when technology transfer reduces

Northern income, the North is still better off.

Likewise, the change in Southern welfare (US) can be written as

d ln US

dt=

1

σ − 1

[pN(n)

P

]1−σdn

dt+

1

YS

dPS

dt. (13)

Equation (13) suggests that new goods benefit the South to the same extent as the North.

As shown in appendix A.4, the gain from new goods always outweighs the loss to the South

from worsened terms of trade. Therefore, the South is also better off from product cycles.

Theorem 6 summarizes these results.

Theorem 6. Assume εaN > ε̄. Then dUN/dt > 0 and dUS/dt > 0. That is, product cycles

improve welfare in both the North and the South.

Proof. See appendix A.4.

4.3. Labor Welfare

Labor welfare is measured by the indirect utility of each type of labor. When national welfare

improves and wage inequality rises, skilled labor in both the North and the South must be

19

better off.10 For unskilled workers, it is less certain whether they would be better or worse

off.

The creation of new goods benefits all types of workers. However, technology transfer

has very different effects on skilled and unskilled workers. As discussed, technology transfer

shifts income from the North to the South and raises wage inequality in both regions. The

former effect benefits all types of Southern labor and hurts all types of Northern labor. The

latter effect benefits skilled labor in both regions and hurts unskilled labor in both regions.

Therefore, technology transfer makes Southern skilled labor better off and makes Northern

unskilled labor worse off.

5. Increasing Labor Supply

The above model can be extended by including an increase in the supply of Southern skilled

labor. This extension helps to explain the income catch-up by some newly industrialized

countries in recent decades. This is also similar in spirit to the new growth literature in

which human capital accumulation plays a central role in promoting economic growth (e.g.,

Romer 1990). In the following I assume that the supply of Southern skilled labor is increased

exogenously. Specifically, let γHS ≡ d ln HS/dt be the rate of increase in the supply of

Southern skilled labor. I further assume that the increasing supply of Southern skilled

workers does not violate the assumption that the relative supply of skilled labor is much

10Note thatdUi

dt= Hi

d(wHi/P )dt

+ Lid(wLi/P )

dt= Hi

dUHi

dt+ Li

dLi

dt(i = S, N)

where UHi and ULi are the indirect utility of skilled and unskilled labor, respectively. When wage inequalityrises, dUHi/dt must be larger than dULi/dt. Therefore, if dUi/dt > 0, then dUHi/dt > 0.

20

lower in the South than in the North, so that wS > wN still holds.

An increase in the supply of Southern skilled workers raises the rate of technology transfer.

The reasons are simple. First, an increasing supply of skills causes the wage of skilled workers

to fall, thus making the South more competitive in skill-intensive goods. Second, by the

Rybczynski effect, the South expands production of more skill-intensive goods.

An increase in the supply of Southern skilled labor further affects wage inequality in

both regions. When the rate of technology transfer is larger (due to γHS), the demand for

Northern unskilled labor is less. Thus, the negative effect of product cycles on Northern

inequality is augmented. In the South, however, although the demand for Southern skills

increases, its effect on the relative wage is offset by the more direct effect of an increasing

supply of skilled labor. Thus, the negative effect of product cycles on Southern inequality is

mitigated.

Theorem 7. (i) dz/dt is increasing in γHS. (ii) dwN/dt is increasing in γHS and dwS/dt is

decreasing in γHS. That is, an increase in the supply of Southern skilled labor raises the rate

of technology transfer. Further, it augments the negative effect of product cycles on Northern

inequality and mitigates the negative effect of product cycles on Southern inequality.

Proof. See appendix A.4.

More importantly, the increased supply of Southern skills enables the South to catch up

with the North. The next theorem formalizes this point.

Theorem 8. There exists a positive constant γ such that γHS > γ ⇔ d(YN/YS)/dt < 0.

That is, if the increase in the supply of Southern skilled labor is large enough, the North-

South income gap is narrowed.

21

Proof. See appendix A.4.

Theorem 8 establishes that when the increase in the supply of Southern skilled labor is

big enough, the North-South gap can be narrowed.11 This result follows the observation that

an increasing supply of Southern skilled workers can raise the rate of technology transfer.

6. Conclusions

Previous work on the product cycle neglects its implications for domestic income distri-

bution. However, in recent decades, inequality has risen in both developed countries and

some developing countries. Since this contradicts the Stolper-Samuelson theorem, it poses a

challenge to traditional trade theory.

This paper incorporates product innovation and technology transfer into the continuum-

of-goods Heckscher-Ohlin model. It shows that inequality can rise in both regions. The

intuition is simple. The creation of new, highly skill-intensive goods in the North raises the

relative demand for Northern skilled labor and hence raises inequality. When the aggre-

gate elasticity of substitution between skilled and unskilled labor in the North is sufficiently

enough, the creation of new goods causes the North to lose competitiveness in older goods.

These older Northern goods thus migrate South. Since these older goods are the least skill in-

tensive of the goods produced in the North and the most skill intensive of the goods produced

in the South, technology transfer reduces the relative demand for Northern unskilled labor

and increases the relative demand for Southern skilled labor. This aggravates inequality in

11Note that the effect of increasing labor supply on the North-South gap can arise in the absence ofNorthern product innovation (i.e., dn/dt = 0). In this case, the cutoff γ̄ becomes zero.

22

the North and also creates inequality in the South.

Product cycles increase inequality not only within countries, but also between countries.

Product innovation widens the North-South income gap while technology transfer narrows

the gap. Since the effect of technology transfer is entirely offset by the effect of product

innovation, product cycles increase the North-South gap.

This simple model also has rich welfare implications of product cycles. Product cycles

benefit both the North and the South. At the same time, since income is redistributed from

unskilled to skilled workers, skilled workers in both regions are better off from product cycles.

Finally, an increase in the supply of Southern skilled labor raises the rate of technology

transfer. This narrows the North-South income gap and can explain some of the trends in

cross-country convergence.

23

A. Appendix

Let Hi and Li be the supply of skilled and unskilled labor, respectively, in region i (=S, N). Let wHi and wLi be the wages of skilled and unskilled workers, respectively. Definewi ≡ wHi/wLi. Let Hi(wi, z) and Li(wi, z) be the amount of skilled and unskilled labor,respectively, required to produce one unit of good z. Define hi ≡ Hi/Li and hi(wi, z) ≡Hi(wi, z)/Li(wi, z). Let pi(z) be the price of good z. Let θHi(wi, z) ≡ wHiHi(wi, z)/pi(z)be the cost share of skilled labor. Let θLi(wi, z) ≡ wLiLi(wi, z)/pi(z) be the cost shareof unskilled labor. Let YHi ≡ wHiHi/Yi be the national income share of skilled labor.Let YLi ≡ wLiLi/Y be the national income share of unskilled labor. Finally, let εi(z) ≡−∂ ln hi(z)/∂ ln wi be the elasticity of substitution between skilled and unskilled labor ingood z. To simplify notation, I drop wN , wS and t as arguments.

A.1. Comparative Advantage

Lemma 1. Let C(wHi, wLi, z) be the unit cost of producing good z in region i. Assume thatC(·) is twice differentiable. If wS > wN , then ∂ [C(wHN , wLN , z)/C(wHS, wLS, z)] /∂z < 0.

Proof. Since C(wHi, wLi, z) is homogenous degree of one in wHi and wLi, C(wHi, wLi, z) =wLi · C(wi, 1, z). Differentiating this with respect to wLi yields ∂C(wHi, wLi, z)/∂wLi =C(wi, 1, z)−wi∂C(wi, 1, z)/∂wi. By Shepard’s Lemma, I have Li(z) = ∂C(wHi, wLi, z)/∂wLi.Combining the above two equations yields ∂C(wi, 1, z)/∂wi = Hi(z). It follows that

∂

∂wi

[∂ ln C(wHi, wLi, z)

∂z

]=

∂

∂wi

[∂ ln[wLi · C(wi, 1, z)]

∂z

]=

∂

∂wi

[∂ ln C(wi, 1, z)

∂z

]

=∂

∂z

[∂ ln C(wi, 1, z)

∂wi

]=

∂

∂z

[Hi(z)

Li(z) + wiHi(z)

]

=∂

∂z

[hi(z)

1 + wihi(z)

]=

∂hi(z)/∂z

[1 + wihi(z)]2> 0.

Hence, if wN < wS, then ∂ ln C(wHN , wLN , z)/∂z < ∂ ln C(wHS, wLS, z)/∂z, which implies∂ [C(wHN , wLN , z)/C(wHS, wLS, z)] /∂z < 0.

Therefore, C(wHN , wLN , z) and C(wHS, wLS, z) will intersect only once. I claim thatthe intersection z must be in the interval [0, n]. Suppose that z were not in the interval[0, n]. Then schedule C(wHN , wLN , z) would be either below or above C(wHS, wLS, z) inthe interval [0, n], which implies that either the North or the South would produce all thegoods. This further implies that wages would be zero for the region which did not produceany goods. In this case it would be optimal to allocate some goods production to thisregion. This contradicts the assumption that z was not in the interval [0, n]. Therefore,there is a unique z ∈ [0, n] satisfying C(wHN , wLN , z) = C(wHS, wLS, z). Further, for z < z,C(wHN , wLN , z) > C(wHS, wLS, z); for z > z, C(wHN , wLN , z) < C(wHS, wLS, z).

24

A.2. Downward Sloping Aggregate Labor Demand Curve

Lemma 2. Given z, ∂N(z)/∂wN < 0 and ∂S(z)/∂wS < 0.

Proof. I only consider the Northern labor market. Let me start with the case where thetechnology is Leontief, i.e., εN(·) = 0. Differentiating N(z) in equation (9) with respect towN yields

dN(z)

dwN

=σ

w2NYLN

∫ n

z

[pN(z)

pN(z̄)

]1−σ

[θHN(z)− YHN ]θLN(z)dz. (14)

Since∫ n

z

[pN (z)pN (z̄)

]1−σ

[θHN(z)− YHN ]dz = 0, 12 and θHN(z)− YHN increases in z, there exists

z0 ∈ (z, n) such that (i) when z = z0, θHN(z) − YHN = 0;(ii) when z < z0, θHN(z) − YHN

< 0; and (iii) when z > z0, θHN(z)− YHN > 0. Further, since θLN(z) decreases in z, I have(i) when z ≤ z0, θLN(z) ≥ θLN(z0); and (ii) when z > z0, θLN(z) < θLN(z0). Therefore,equation (14) implies

dN(z)

dwN

<σ

w2NYLN

∫ z0

z

[pN(z)

pN(z̄)

]1−σ

[θHN(z)− YHN ]θLN(z0)dz

+σ

w2NYLN

∫ n

z0

[pN(z)

pN(z̄)

]1−σ

[θHN(z)− YHN ]θLN(z0)dz

=σθLN(z0)

w2NYLN

∫ n

z

[pN(z)

pN(z̄)

]1−σ

[θHN(z)− YHN ]dz = 0.

It is easy to see that for other types of technology allowing for substitution between skilledand unskilled labor i.e., εN(·) > 0, dN(z)/dwN < 0 still holds.

A.3. Total Differential Equation System

Totally differentiating equilibrium conditions (4), (9) and (10) yields the differential equationsystem

[cjk]

dzdwS

dwN

= [bj]dt, where [cjk] =

Bz BwSBwN

Sz SwS0

Nz 0 NwN

and [bj] =

−Bt

−St

−Nt

. (15)

The elements and signs of [cjk] and [bj] are as follows. Since the North has a comparativeadvantage in more skill-intensive goods, c11 is positive (see lemma 1.) Because the excess

12Using hN (z)−hN

wN hN (z)+1 = θHN (z)−YHN

wN YLN, equation (9) can be rewritten as

∫ n

z̄

[pN (z)pN (z̄)

]1−σ

[θHN (z)− YHN ] dz = 0.

25

relative demand for skilled labor decreases in the relative wage of skilled labor (see lemma 2),c22 and c33 are negative. The signs of other elements in [cjk] are implied by the conventionof goods ranking that a higher indexed good uses relatively more skilled labor.

c11 =

∫ z̄

0pS(z)1−σdz +

∫ n

z̄pN(z)1−σdz∫ z̄

0pS(z)1−σdz

∫ n

z̄pN(z)1−σdz

− σ∂

∂zln

CN(wHN , wLN , z)

CS(wHS, wLS, z)> 0

c12 =σ

wS

[θHS(z)− YHS] > 0 c13 =σ

wN

[YHN − θHN(z)] > 0

c21 =θHS(z)− YHS

wSYLS

> 0

c22 =1

w2SYLS

∫ z

0

[pS(z)

pS(z̄)

]1−σ

{θLS(z)θHS(z) [σ − εS(z)]− σθLS(z)YHS} dz < 0 c23 = 0

c31 =YHN − θHN(z)

wNYLN

> 0 c32 = 0

c33 =1

w2NYLN

∫ n

z

[pN(z)

pN(z̄)

]1−σ

{θLN(z)θHN(z) [σ − εN(z)]− σθLN(z)YHN} dz < 0

b1 =pN(n)1−σ

∫ n

zpN(z)1−σdz

dn

dt> 0 b2 = 0

b3 = −[pN(n)

pN(z̄)

]1−σ[θHN(n)− YHN ]

wNYLN

dn

dt< 0

Since |cjk| > 0, [cjk]−1 exists. Thus, differential equations system (15) implies

dz

dt= |cjk|−1(c22c33b1 − c12c33b2 − c22c13b3) (16)

dwS

dt= |cjk|−1[−c21c33b1 + (c11c33 − c13c31)b2 + c21c13b3] (17)

dwN

dt= |cjk|−1[−c31c22b1 + c12c31b2 + (c11c22 − c21c12)b3] (18)

Note that all my results hold under the weaker assumption that on average new goodsare more skill-intensive than all old Northern goods, i.e., θHN(n) > YHN . This implies thatb3 is always negative. Since the following results depend on the sign of b3 rather than themagnitude of b3, the weaker assumption will not change my results qualitatively.

A.4. Proofs of Theorems

Proof of theorem 1:

26

Proof. The aggregate elasticity of substitution between skilled and unskilled labor in theNorth can be derived as follows. Combining equations (5) and (6) yields

HdN

LdN

=1

wN

∫ n

z

[pN (z)pN (z̄)

]1−σwNhN (z)

1+wNhN (z)dz

∫ n

z

[pN (z)pN (z̄)

]1−σ1

1+wNhN (z)dz

.

Thus,

εaN ≡ −∂ ln(Hd

N/LdN)

∂ ln wN

= 1 +∂ ln

∂ ln wN

∫ n

z

[pN(z)

pN(z̄)

]1−σ1

1 + wNhN(z)dz

− ∂ ln

∂ ln wN

∫ n

z

[pN(z)

pN(z̄)

]1−σwNhN(z)

1 + wNhN(z)dz.

Using ∂∂ ln wN

∫ n

z

[pN (z)pN (z̄)

]1−σ1

1+wNhN (z)dz =

∫ n

z̄

[pN (z)pN (z̄)

]1−σ

[εN(z)− σ] θLN(z)θHN(z)dz−(1− σ)θHN(z̄)YLN

∫ n

z̄

[pN (z)pN (z̄)

]1−σ

dz and ∂∂ ln wN

∫ n

z

[pN (z)pN (z̄)

]1−σwNhN (z)

1+wNhN (z)dz =

∫ n

z̄

[pN (z)pN (z̄)

]1−σ

[σ − εN(z)] θLN(z)θHN(z)dz +(1−σ)θLN(z̄)YHN

∫ n

z̄

[pN (z)pN (z̄)

]1−σ

dz, εaN can be further derived

as

εaN =

∫ n

z[pN(z)]1−σ θLN(z)θHN(z)εN(z)dz

YLNYHN

∫ n

z[pN(z)]1−σ dz

+ σ

[1−

∫ n

z[pN(z)]1−σ θLN(z)θHN(z)dz

YLNYHN

∫ n

z[pN(z)]1−σ dz

]. (19)

The first term in equation (19) represents within-good substitution, and the second termmeasures between-good reallocation.

Equation (16) implies that with b2 = 0, dz/dt > 0 if and only if c13b3 > c33b1. Pluggingc13, c33, b1 and b3 into c13b3 − c33b1, I obtain that c13b3 > c33b1 if and only if

εaN > ε ≡ σ

[YHN − θHN(z)] [YLN − θLN(n)]

YHNYLN

.

Since 0 < YHN − θHN(z) < YHN , and 0 < YLN − θLN(n) < YLN , I have ε ∈ (0, σ).In order to highlight the impact of industry skill intensities on ε, I consider two extreme

cases. In the first case, production of all the goods uses very similar technology such thatYHN − θHN(z) and YLN − θLN(n) approach 0, implying that ε also approaches 0. Thus,εa

N > ε can be easily satisfied. In the second case, production of each good uses very differenttechnology such that both θHN(z) and θLN(n) approach 0, implying that ε approaches σ. Ifthere is no within-industry substitution, then εa

N > ε does not hold.Proof of theorem 2:

27

Proof. dz/dt can be rewritten as

dz

dt=

b1 − c13b3/c33

c11 − c12c21/c22 − c13c31/c33

. (20)

Since c22 decreases in βS, from equation (20) it is easy to see that dz/dt increases in βS.Equation (20) also implies

d2z

dtdβN

=c13c22(c11c22 − c12c21)b3 − c13c31c

222b1

|cjk|2dc33

dβN

> 0

Similarly, since both c22 and c33 decrease in σ, d2z̄/dtdσ > 0.Proof of theorem 3:

Proof. Equation (18) implies that with b2 = 0, dwN/dt > 0 always holds. Combining b2 = 0with equations (16) and (17) yields

dwS

dt=

c21

|cij| [−c33b1 + c13b3] = −c21

c22

dz

dt.

Thus, dwS/dt and dz/dt have the same sign. By theorem 1, if εaN > ε̄, then dz/dt > 0. It

follows that dwS/dt > 0.Proof of theorem 4:

Proof. The balanced trade condition (3) implies

d ln(YN/YS)

dt=

pN(z)1−σ

∫ n

z̄pN(z)1−σdz

dn

dt−

∫ z̄

0pS(z)1−σdz +

∫ n

z̄pN(z)1−σdz∫ z̄

0pS(z)1−σdz

∫ n

z̄pN(z)1−σdz

dz̄

dt

+(1− σ)

∫ n

z̄pN(z)1−σ

[ddt

ln pN (z)pN (z̄)

]dz

∫ n

z̄pN(z)1−σdz

−∫ z̄

0pS(z)1−σ d

dt

[ln pS(z)

pS(z̄)

]dz

∫ z̄

0pS(z)1−σdz

.

Using∫ n

z̄pN(z)1−σ d

dt

[ln pN (z)

pN (z̄)

]dz =

∫ n

z̄pN(z)1−σdz

{[θLN(z̄)− YLN ] d ln wN

dt− ∂ ln pN (z̄)

∂z̄dz̄dt

}and

∫ z̄

0pS(z)1−σ d

dt

[ln pS(z)

pS(z̄)

]dz =

∫ z̄

0pS(z)1−σdz

{[θLS(z̄)− YLS] d ln wS

dt− ∂ ln pS(z̄)

∂z̄dz̄dt

}, the change

28

in the North-South gap can be rewritten as

d ln(YN/YS)

dt=

pN(z)1−σ

∫ n

z̄pN(z)1−σdz

dn

dt−

∫ z̄

0pS(z)1−σdz +

∫ n

z̄pN(z)1−σdz∫ z̄

0pS(z)1−σdz

∫ n

z̄pN(z)1−σdz

dz̄

dt

+ (1− σ)

{[θLN(z̄)− YLN ]

d ln wN

dt+ [YLS − θLS(z̄)]

d ln wS

dt+

∂ ln pS(z̄)pN (z̄)

∂z̄

dz̄

dt

}

(21)

Plugging equations (16), (17) and (18) into equation (21) yields

d ln(YN/YS)

dt=

1

σ

[b1 − c11c22(c33b1 − c13b3)

|cjk|]

+∂ ln [pS(z̄)/pN(z̄)]

∂z̄

dz̄

dt> 0.

Proof of theorem 5:Proof. Using the balance-of-trade condition (3), the change in the terms of trade in theNorth can be derived as

dPN

dt= YS

∫ n

~z

[pN(z)

P

]1−σ [d

dtln

pN(z)

pN(z̄)

]dz − YN

∫ z̄

0

[pS(z)

P

]1−σ [d

dtln

pS(z)

pS(z̄)

]dz

+YS

∫ n

z̄

[pN(z)

P

]1−σ

dzd

dtln

pN(z̄)

pS(z̄). (22)

Since pN(z̄) = pS(z̄) always holds, d [ln pN(z̄)/pS(z̄)] /dt = 0. Thus, the last term in equation

(22) vanishes. Note that ddt

ln pi(z)pi(z̄)

= [θLi(z̄)− θLi(z)] d ln wi

dt− ∂ ln pi(z̄)

∂z̄dz̄dt

. Plugging these results

together with the balance-of-trade condition into equation (22) yields

dPN

dt= YN

∫ z̄

0

[pS(z)

P

]1−σ

dz

{[θLN(z̄)− YLN ]

d ln wN

dt+ [YLS − θLS(z̄)]

d ln wS

dt+

∂ ln pS(z̄)pN (z̄)

∂z̄

dz̄

dt

}.

(23)

By theorem 3, when εaN > ε̄, both dwS/dt and dz̄/dt are positive. By lemma 1, ∂ ln [pS(z̄)/pN(z̄)] /∂z̄

> 0. At the same time, dwN/dt > 0, θLN(z̄) > YLN and YLS > θLS(z̄). Therefore, dPN/dt > 0.It follows that dPS/dt = −dPN/dt < 0.

Proof of theorem 6:Proof. With CES preferences, the Northern indirect utility is UN = YN/P , where P ≡[∫ z̄

0pS(z)1−σdz +

∫ n

z̄pN(z)1−σdz

] 11−σ

is the aggregate price index. The change in Northern

29

welfare can be derived as

d ln UN

dt=

d

dtln

YN

pN(z̄)+

1

σ − 1

ddt

∫ z̄

0

[pS(z)pS(z̄)

]1−σ

dz + ddt

∫ n

z̄

[pN (z)pN (z̄)

]1−σ

dz

∫ z̄

0

[pS(z)pS(z̄)

]1−σ

dz +∫ n

z̄

[pN (z)pN (z̄)

]1−σ

dz

=1

σ − 1

[pN(n)

P

]1−σdn

dt+

∫ z̄

0

[pS(z)

P

]1−σ

dz

{[θLN(z̄)− YLN ]

d ln wN

dt+ [YLS − θLS(z̄)]

d ln wS

dt− ∂

∂z̄

[ln

pN(z̄)

pS(z̄)

]dz̄

dt

}.(24)

Combining equations (23) and (24) yields

d ln UN

dt=

1

σ − 1

[pN(n)

P

]1−σdn

dt+

1

YN

dPN

dt.

When σ > 1, the first term is positive, suggesting that the North benefits from new goods.Theorem 5 implies that the second term is also positive. Thus, the North further benefitsfrom improved terms of trade.

Similarly, the change in Southern welfare can be derived as

d ln US

dt=

1

σ − 1

[pN(n)

P

]1−σdn

dt+

1

YS

dPS

dt.

In the following I will show that for the South, the gain from new goods can entirely offsetthe loss from worsened terms of trade.

d ln US

dt=

1

σ − 1

[pN(n)

P

]1−σdn

dt−

∫ n

z̄

[pN(z)

P

]1−σ

dz

{[θLN(z̄)− YLN ]

d ln wN

dt+ [YLS − θLS(z̄)]

d ln wS

dt−

∂ ln pN (z̄)pS(z̄)

∂z̄

dz̄

dt

}. (25)

Plugging equations (16), (17) and (18) into equation (25) yields

d ln US

dt=

∫ n

z̄

[pN(z)

P

]1−σ

dz

[b1

σ(σ − 1)+

∫ z̄

0pS(z)dz +

∫ n

z̄pN(z)dz

σ∫ z̄

0pS(z)dz

∫ n

z̄pN(z)dz

c22(c33b1 − c13b3)

|cjk|

]> 0.

Proof of theorem 7:Proof. (1) When the increase in the supply of Southern skilled labor is taken into account,

30

b1 and b2 in differential equation system (15) will be replaced with b′1 = b1 + γHSYHS and

b′2 =

∫ z

0

[pS(z)pS(z̄)

]1−σ

dzYLShSγHS. Equation (16) implies that

d2z

dtdγHS

= |cjk|−1

{c22c33YLShS

∫ z̄

0

[pS(z)

pS(z̄)

]1−σ

dz − c12c33YHS

}> 0.

(2) Equation (18) implies

d2wN

dtdγHS

= |cjk|−1

{−c22c31YHS + c12c31YLShS

∫ z

0

[pS(z)

pS(z̄)

]1−σ

dz

}> 0.

Equation (17) implies

d2wS

dtdγHS

= |cjk|−1

{−c21c33YHS + (c11c33 − c13c31)YLShS

∫ z

0

[pS(z)

pS(z̄)

]1−σ

dz

}

<c33YHS

wSYLS

[θLS(z) +

∫ z̄

0pS(z)dz∫ n

zpN(z)dz

YLS

]< 0.

Proof of theorem 8:Proof. Plugging equations (16), (17) and (18) into (21) yields

d ln YN/YS

dt=

b1

σ−

∫ z̄

0pS(z)dz +

∫ n

z̄pN(z)dz∫ z̄

0pS(z)dz

∫ n

z̄pN(z)dz

c22 (c33b1 − c13b3)

σ|cjk| − σ − 1

σYHSγHS

−∫ z̄

0pS(z)dz +

∫ n

z̄pN(z)dz∫ z̄

0pS(z)dz

∫ n

z̄pN(z)dz

γHS

σ|cjk|

{c22c33YHS − c12c33

∫ z̄

0

[pS(z)

pS(z̄)

]1−σ

dzYLShS

}.

Thus, d ln(YN/YS)/dt < 0 if and only if

γHS > γ ≡b1 −

R z̄0 pS(z)dz+

R nz̄ pN (z)dzR z̄

0 pS(z)dzR n

z̄ pN (z)dz

c22(c33b1−c13b3)|cjk|

(σ − 1)YHS +R z̄0 pS(z)dz+

R nz̄ pN (z)dzR z̄

0 pS(z)dzR n

z̄ pN (z)dzc33|cjk|

[c22YHS − c12

∫ z̄

0

[pS(z)pS(z̄)

]1−σ

dzYLShS

] > 0.

Without Northern product innovation (i.e., dn/dt = 0), both b1 and b3 become zero. In thiscase, γ̄ = 0.

31

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