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 and suggestions.

Original Article

# ECONOMIC GROWTH WITH TRADE IN FACTORS OF PRODUCTION

Article first published online: 22 JAN 2014

DOI: 10.1111/iere.12047

© (2014) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association

Additional Information

#### How to Cite

YENOKYAN, K., SEATER, J. J. and ARABSHAHI, M. (2014), ECONOMIC GROWTH WITH TRADE IN FACTORS OF PRODUCTION. International Economic Review, 55: 223–254. doi: 10.1111/iere.12047

#### Publication History

- Issue published online: 22 JAN 2014
- Article first published online: 22 JAN 2014
- Manuscript Revised: MAY 2012
- Manuscript Received: MAY 2011

- Abstract
- Article
- References
- Cited By

### Abstract

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. MODEL SPECIFICATION
- 3. TRADE BETWEEN TWO LARGE COUNTRIES
- 4. TRANSITION DYNAMICS
- 5. CONCLUSION
- APPENDIX
- References

We study the world trading equilibrium in a Ricardian model, where factors of production are produced and traded. Even in the absence of technology transfer, international investment, research and development, and aggregate scale effects, trade affects economic growth through comparative advantage. Trade may raise the growth rate or leave it unchanged, depending on the patterns of comparative and absolute advantage. Trade in factors of production can effectively equalize technology even when technology transfer does not occur. Factor price equalization may hold, but the Stolper–Samuelson and Rybczynski theorems do not. The transition dynamics can be monotonic or oscillatory.

### 1. INTRODUCTION

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. MODEL SPECIFICATION
- 3. TRADE BETWEEN TWO LARGE COUNTRIES
- 4. TRANSITION DYNAMICS
- 5. CONCLUSION
- APPENDIX
- References

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 effects 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 et al. (2009). That literature's answer to the question of whether trade promotes growth generally has been “Yes,” with trade's effect working through two channels: an aggregate scale effect and technology transfer. The scale effect arises from the increase in firms' market size that opening to trade enables. A larger market size makes firms more profitable 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 effects has been decisively rejected by the data, starting with the well-known article by Backus et al. (1992), so that channel for trade to influence growth can be dismissed, taking with it all the explanations of trade's growth effects that depend on the scale effect. The second-generation fully endogenous growth literature (Peretto, 1998; Howitt, 1999) provides the theoretical reason for scale effects to be absent. The data are much kinder to technology transfer facilitated by trade, which seems to be statistically and economically significant (Coe and Helpman, 1995; Coe et al., 2009). However, in that mechanism it is not trade that affects economic growth but the technology transfer that trade facilitates. Without the technology transfer, trade would have no effect on growth. Conversely, without trade, technology transfer still would affect growth as long as some of it can occur independently of trade, which seems realistic. Interestingly, there also is evidence that trade may affect growth directly rather than through technology transfer. Alcala and Ciccone (2004), for example, find strong and robust effects 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 (2008) finding that increased trade liberalization is positively associated with increased growth. Although these results are not decisive, they do suggest that trade *per se* affects economic growth. What is missing is a theoretical argument showing why trade should have such an effect.

The question we want to address, therefore, is precisely whether trade per se affects growth in a model without empirically rejected aggregate scale effects. 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 effect) to study the interaction of trade and growth: Peretto (1998). In that model, the growth effects 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 effects 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 differences. The model differs 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 sectors. Allowing the factors of production to be produced permits endogenous growth. Allowing the factors of production to be traded generates growth effects of trade. The model is sufficiently 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 effects 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 effect of trade on a country's growth rate is the type of good it imports, not the type it exports. Specifically, importing a factor of production increases a country's growth rate, whereas importing a consumption good has no effect 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 permanent 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 offers 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 effectively 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 effect 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 BGP 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 different 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 SPECIFICATION

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. MODEL SPECIFICATION
- 3. TRADE BETWEEN TWO LARGE COUNTRIES
- 4. TRANSITION DYNAMICS
- 5. CONCLUSION
- APPENDIX
- References

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 only new element is that we allow trade. Because the specification is mostly standard, we consign all derivation and mathematical detail to the Appendix.2

#### 2.1. Production and Preferences

Country *i* produces two goods, and . Good can be used for consumption *C* or can be used as gross investment in one kind of capital *K*. Good can be used only as gross investment in a second type of capital *H* produced in a different sector from by a different technology. Both and are tradable. Each sector's production technology is Cobb–Douglas with country-specific parameter values:

- (1)

- (2)

where *v* and *u* are the fractions of total *K* and *H* used in the -producing sector and , , , and are constants. Both *K* and *H* are freely mobile between the two sectors, so *v* and *u* 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 δ, which is the same for both countries. The equations for the accumulation of *K* and *H* are

- (3)

We can define gross domestic product:

- (4)

where *p* is the price of in terms of . Utility is Constant Relative Risk Aversive, so lifetime utility is

#### 2.2. Relation to Other Models

Endogenous growth models fall into four broad classes. The first class comprises research and development (R&D) models of variety expansion or quality improvement. Those can be divided into three subclasses: (i) the first-generation fully endogenous models of Grossman and Helpman (1991) or Barro and Sala-i-Martin (2004, chapters 6 and 7) and the like, (ii) semiendogenous growth models derived from Jones (1995), and (iii) second-generation fully endogenous growth models of Peretto (1998), Howitt (1999), and their offspring. 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 a Constant Elasticity of Substitution (CES) production with a high elasticity of substitution between a reproducible factor, such as capital, and a nonreproducible 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 infinite elasticity of substitution and a coefficient 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 et al. (1999) have shown that technical progress of that type is important, accounting for about 52% 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 sufficient 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 simplified version that is suitable for our purposes.

Peretto (2007) considers an economy in which final goods *F* are produced with a variety of intermediate goods and labor *L*. The production technology for a final good *F* is

where α and θ are constants between 0 and 1, is the quantity of intermediate good *i*, *N* is the number of varieties of intermediate goods, is the quantity of labor using , is the quality of , and *Z* is the average value of the , given by

Note that the quality is embodied in but augments labor in final goods production. The intermediate goods are produced by monopolistic competitors, who also do R&D to raise the quality of their product . Increases in raise the demand for and thus also raise the sales of , which in turn increases the monopoly profit of 's producer. That (incipient) increase in profit induces new firms 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 difficult to use in analyzing trade. It effectively has three sectors—final 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, one of which augments labor. We do not need an expanding variety of intermediate goods for what we are doing, so we fix the number of varieties at two. 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 nondurable 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 simplifies 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 difficult 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 et al. (2003), Farmer and Lahiri (2005, 2006), and Hu et al. (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 effects of trade on growth.

### 3. TRADE BETWEEN TWO LARGE COUNTRIES

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. MODEL SPECIFICATION
- 3. TRADE BETWEEN TWO LARGE COUNTRIES
- 4. TRANSITION DYNAMICS
- 5. CONCLUSION
- APPENDIX
- References

We now introduce foreign trade. We have two countries, 1 and 2, with different fixed production technologies, discussed momentarily. In our framework of a two-factor Cobb–Douglas production function, cross-country technology differences are captured in different values of the total factor productivities *A* and *B* and of the factor share parameters α and η. 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 firms with identical production functions, preventing either country from acting as a monopolist and thus guaranteeing a competitive solution.6

Let *X* denote exports of , so that is imports of . Then the accumulation constraints for country *i* are

- (5)

- (6)

When economies are closed, . 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 find 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 to find the equilibrium value of *p*.7

It is important to note here that scale effects, R&D, and technology transfer all are absent from this model. Consequently, any effects that trade has on growth will not be due to those influences but rather to comparative advantage alone.

#### 3.1. Individual Country Solutions

With trade, country *i*'s Hamiltonian is

- (7)

where ϕ and ψ are the costate variables. The control variables are *C*, *X*, *v*, and *u*. The important necessary condition for discussion here is the first-order condition for *X*:

- (8)

The other necessary conditions are given in the Appendix.

The Hamiltonian is linear in *X*, so the first-order condition (8) for *X* does not depend on any control variable. We thus have bang-bang control for . When is positive, Equation (8) cannot be satisfied; the marginal value of *X* equals and always is positive, irrespective of the value *X*. Consequently, country *i* sets as high as possible, which it does by producing only and exporting some of it to obtain *H*. The opposite holds when is negative. Country *i* then sets as low as possible, producing only and obtaining solely through imports (so that is negative). When equals zero, country *i* does not engage in trade, and *X* is zero. To see this, note that is equivalent to . The price *p* is the international price for in terms of , that is, the ratio of marginal utilities of and . The costate variables and are, respectively, country *i*'s marginal utilities of goods and goods from internal production. Their ratio is the marginal value of goods in terms of goods if country *i* produces both goods; that is, the ratio is country *i*'s internal price for in terms of . If this internal price equals the external (world) price *p*, country *i* can obtain the same number of units of in exchange for from its own internal operations as it can by trading on the world market. Country *i* is indifferent on the margin between trade and autarky. The borderline case of indifference to trade will prevail no more than momentarily because, as and vary over time, they generally will not satisfy the equality . Consequently, we ignore the knife-edge case henceforth.

Comparisons of the world price *p* with the internal prices are central to all that follows. The results of the previous paragraph suggest that country *i* specializes in producing when and in producing when .

#### 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. BGPs under autarky

The solution for the balanced growth rate under autarky is standard. The growth rates of *C*, *K*, *H*, , and *Y* are equal. Their common value γ is

- (9)

The value of *p* is

- (10)

See the Appendix and the references cited there for details.

##### 3.2.2. BGPs with trade: Derivation

The equilibrium world price *p* must fall between and ; 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 is arbitrary, so we assume without loss of generality that . Since the ratio of costate variables represents the internal price for in terms of , the ratio is equivalent to the autarkic price level in each country given by (10). We then have

- (11)

For *p* to be strictly in the interior of the closed interval , country 1 sets , specializes in production of , and trades to obtain . Country 2 does the opposite, producing only and trading for . When *p* is on the boundary of the interval, equaling either *p*_{1} or *p*_{2}, 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.

The first-order conditions for *C* and *X* are unchanged from the unconditional problem, but the first-order condition for *X* now holds with equality. Having accepted the world price *p* and agreed to specialize in producing , country 1 now chooses ϕ and ψ to satisfy (8) exactly. The same kind of manipulations as for the autarkic model show that the growth rates of *C*, , *K*, and *Y* all equal the growth rate of consumption. Because Equation (8) now always holds, we have . Trade balance constrains *p* to lie in the closed interval . The growth rates for country 1 and 2 now are

- (12)

- (13)

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 therefore also is constant, implying that the growth rates of ϕ_{1} and ψ_{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:

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

- (14)

Balanced growth thus requires that

- (15)

It is straightforward to show that

- (16)

Substituting into (12) or (13) gives the common growth balanced growth rate

- (17)

where

We discuss stability of the BGP below.

##### 3.2.3. BGPs with trade: Implications

The foregoing results lead to two interesting and important implications.

###### 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:

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 inefficient domestic production. From the world's point of view, productive resources in each country have been shifted from inefficient to efficient uses. Trade acts like technical progress, raising world total factor productivity (TFP) 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 defined in (11), trade does not increase both countries' growth rates. However, it never lowers either partner's growth rate.

###### Effective 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 effectively share each other's relative efficiency. For example, before trade, country 1 produces both and . Domestic production of uses the comparatively inefficient technology of the domestic sector. When trade starts, country 1 shuts down its own sector and relies instead on country 2's relatively efficient sector. Country 2 is in a symmetric position, shutting down its sector and relying instead on its partner's relatively efficient 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 effect, each country has transferred to itself the more efficient technology of its trading partner, even though no technology transfer actually occurs.

Another way to think of effective 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 benefit of the technology embodied in the good it buys without having to learn the process for creating that technology.

Effective technology transfer through trade raises an interesting question for the interpretation of existing empirical work on technology spillovers. Coe and Helpman (1995), Coe et al. (2009), 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 efficient production on their partner's soil for their own less efficient 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.

#### 3.3. Unbalanced Growth with Trade

As we have seen above, trade balance requires that the world price *p* fall in the closed interval because otherwise the two countries would try to specialize in and export the same good. Balanced growth requires that *p* equal the quantity Ω on the right side of (14). However, nothing guarantees that Ω falls between *p*_{1} and *p*_{2}. The only restriction we have imposed so far is that (to guarantee that trade occurs and to specify the direction of the trade flows). That restriction puts no limits on the value of Ω. We now analyze the effect of trade when Ω falls outside the critical interval.

##### 3.3.1. Relation between growth rates

When Ω falls outside the closed interval,

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 Ω, and the world economy is at a corner. Recall that the endpoints of the interval are the internal, autarkic prices for the two countries. When the world is at a corner solution, the growth rate of the country whose price defines 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 Ω, so without loss of generality consider the case where Ω is larger than the upper boundary of the critical interval:

The world price *p* then equals the upper boundary *p*_{1}. Using that value for *p* in the growth rate formulas (12) and (13) gives the solutions

The growth rate for country 1 is the same under trade as under autarky, the growth rate for country 2 is different than the autarkic rate, and the two growth rates clearly are different 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.

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 efficiency 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

because by assumption

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 .

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 or ) 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 TFP is too low. The condition for the world to be in the corner with country 2 specialized in the production of is

Some trivial algebra reduces this condition to the simpler expression

If country 2 could increase its TFP parameter *B*_{2} in the 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 *B*_{2} alone would alter comparative advantages, but an equal increase in *A*_{2} and *B*_{2} 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 and also play a role in comparative advantage and the value of Ω. The effects of changes in α and η are of ambiguous sign, depending on whether α and η are greater or smaller than 1/2 in magnitude, so we cannot say in general whether an increase in α or η would raise or lower the expressions for trade balance and for Ω. In contrast, the effects of the TFP parameters and 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 difference between their results and ours is that they restrict their model in such a way that it is equivalent to 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 fixed 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 nonconvergent growth rates offers a possible explanation for the historical behavior of African growth rates.

Years | |||||||
---|---|---|---|---|---|---|---|

Region | 1000–1500 | 1500–1820 | 1820–70 | 1870–1913 | 1913–50 | 1950–73 | 1973–98 |

Source: Maddison (2001), Table B-22. | |||||||

W. Europe | 0.13 | 0.15 | 0.95 | 1.32 | 0.76 | 4.08 | 1.78 |

U.S. | 0.36 | 1.34 | 1.82 | 1.61 | 2.45 | 1.99 | |

Japan | 0.03 | 0.09 | 0.19 | 1.48 | 0.89 | 8.05 | 2.34 |

Asia/Japan | 0.05 | 0.00 | −0.11 | 0.38 | −0.02 | 2.92 | 3.54 |

Africa | −0.01 | 0.01 | 0.12 | 0.64 | 1.02 | 2.07 | 0.01 |

##### 3.3.3. Effective technology transfer again

When the world is in a corner, it is incompletely specialized. In that case, there still is effective 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 Nonfactors 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.

##### 3.4.1. Interior and corner solutions

We suppose that -type goods (in the form of *K*) are not useful in production, only in consumption. That assumption requires that . The two production functions then are of the AK form:

Going through the usual steps yields the autarkic growth rate for country *i* :

Because is not a factor of production, its TFP parameter *A* has no effect on the growth rate.

The condition (11) for trade balance simplifies to

Recall that we are assuming that country 1 has a comparative advantage in producing and country 2 has an advantage in good . The balanced growth condition (15) becomes

- (18)

With trade, country 1 specializes in , and country 2 specializes in . The growth rate for the interior solution is

The growth rate of both countries is determined only by TFP in the -sector of country 2. Country 1's TFP parameter *A*_{1} plays no role in the growth rate. When (18) is not satisfied, the world price *p* will be at one of the boundary values or , 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 , country 1 produces only and imports , and country 2 produces both and and imports . The growth rates for the two countries are

A little algebra shows that country 1's growth rate is larger than under autarky, but it is smaller than country 2's growth rate , consistent with relation (16) and . In this corner, then, trade has no effect 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 different at the other corner. When *p* equals , country 1 produces both and and imports , and country 2 produces only and imports . The growth rates are the same as under autarky:

In this case, trade does not change either country's growth rate. The mathematical constraints placed on the problem (i.e., imply that , so that , again consistent with relation (16) because now .

##### 3.4.2. Imports: The driver of trade-enhanced growth

These results make clear exactly how trade affects 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 payment.9

The economic intuition for this result goes to the heart of what makes perpetual growth possible. Ultimately, economic growth is driven by augmenting the nonreproducible 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 efficient. The increase in efficiency 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 efficient producer). So, when each country stops producing one factor itself and instead obtains it by trading with its partner, it is making itself more efficient 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 efficiency 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 efficiency of trade with respect to growth rates

Looking back over the various results we have obtained for the effect 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 effect on growth is guaranteed to be weakly Pareto efficient (not hurting any growth rate) and likely to be strongly Pareto efficient (raising at least one growth rate without lowering any growth rate). That conclusion obviously has the strong implication for the policy debate that opening to trade is beneficial 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 different factor endowments. In contrast, the present model assumes a dynamic Ricardian framework in which two large open countries have different 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 modification because the framework of analysis is so different from that underlying the conditions under which the theorems originally were derived. Indeed, Jones (1965, p. 563) shows that his “magnification effect” need not hold in a general equilibrium framework where just final 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 sector and country 2 only the sector. Thus

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, and , where . Comparing marginal products across countries, we see that we have factor price equalization:

The situation is different in the corner solution. There, country 1 operates both sectors, and country 2 operates only the sector. The production functions are

Examining marginal products shows that

so that factor price equalization does not hold in the corner. These results are interesting for at least two reasons.

First, it might seem that trading the factors is sufficient 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 different 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 Hecksher–Ohlin framework, factor price equalization requires that both countries operate in the interior of their cones of diversification, 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 Hecksher–Ohlin framework, that requires that both countries produce both goods so that the relevant cost functions can take on the same value (see Feenstra, 2003). In this dynamic Ricardian framework, technology is effectively equalized by trade, but only for an interior solution. It is only in the interior that trade generates a full effective 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 effective 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 Hecksher–Ohlin or Ricardian model, rates of return cannot be equalized.

##### 3.5.2. Stolper–Samuelson

The Stolper–Samuelson theorem shows that in the standard Hecksher–Ohlin 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 effect works through the change in price? The easiest parameters to study are the total factor productivities and , so we start with them. Their effects are identical except for sign, so we restrict attention to an increase in *A*_{1}.

In the interior, both countries are specialized, so we know immediately that the Stolper–Samuelson theorem does not apply. We cannot ask the question it addresses, namely, the effects of a change in the relative price of two goods produced within a country, but we can ask about the effect of a change in the world relative price of the two traded goods on the factor returns in each country. The gross returns to *K* and *H* in each country are

- (19)

- (20)

- (21)

- (22)

From (14) we see that *p* is a positive function of *A*_{1} , so an increase in *A*_{1} has a direct effect on and and an indirect effect (through *p*) on all four rates of return. The indirect effect is what interests us here, and it is negative for and and positive for and , irrespective of which good is intensive in which factor (that is, the relation between α_{1} and η_{2}). The economic interpretation is straightforward. An increase in TFP in the 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 effect at play.

The economy's full response to a change in *A*_{1} (the direct and indirect effects) 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:

and international capital market equilibrium requires that each type of capital has the same return across countries:

The full solution for is

which is a positive function of both TFP parameters *A*_{1} and *B*_{2}. 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 first 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 effect 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 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 in the relative price of the two goods changes 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 *p*_{1}, given by (11). Here, we assume that the increase in *A*_{1} is not large enough to move the world out of the corner. An increase in *A*_{1} affects the rates of return both directly and indirectly, just as in the interior. The Stolper–Samuelson effect is the indirect effect coming through the change in *p*_{1}. An increase in *A*_{1} raises *p*_{1} (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 effect is absent here. As in the interior, the full effect of an increase in *A*_{1} is to raise both rates of return and .

##### 3.5.3. Rybczynski

The Rybczynski theorem addresses the effect 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 effect 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 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 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 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, and countries want to hold them 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 is to increase production of the factor in relatively short supply and shut down investment in the other factor altogether.

### 4. TRANSITION DYNAMICS

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. MODEL SPECIFICATION
- 3. TRADE BETWEEN TWO LARGE COUNTRIES
- 4. TRANSITION DYNAMICS
- 5. CONCLUSION
- APPENDIX
- References

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), Faig (1995), Bond et al. (1996), and Mino (1996). Mulligan and Sala-i-Martin consider a general model that exhibits constant returns to scale at the private level but also incorporates increasing or decreasing returns at the social 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 effects of fiscal policies and stochastic shocks. Bond et al. derive 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 differential 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 parameters. The asymptotic adjustment toward steady state can be driven by either adjustments in physical capital or by the stabilizing forces of the price level. 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 et al. that dynamics of the economy and effects of capital income taxation depend on assumptions regarding the relative factor intensities in both sectors of production. These studies all contribute significantly 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 field” (p. 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 differs from that of Bond and Trask in three ways that affect 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 differences 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 , and examines the transitional dynamics in the neighborhood of the BGP with complete specialization, where country 1 specializes in the production of good and country 2 specializes in production of good . 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 and are

- (23)

- (24)

where as usual *p* is the world relative price determined by Equation (14). The transition dynamics about the steady state are difficult even with linearization because we have a five-dimensional system consisting of the dynamic equations for *k*_{1}, *k*_{2}, , , 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 differ from their BGP values:

- Case 1: ,
- Case 2: ,
- Case 3: ,
- Case 4: ,

The first 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 affect the world relative price. Changes in the world price transmit the effects 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: ,

In this case, country 1 specializes in production of good on the BGP but deviates from the BGP value of its 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 from country 2 along the transition to the BGP. This conclusion follows from the accumulation condition for *H* in country 1:

which implies that *H* depreciates at rate δ. 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 and so does not have an incentive to trade for . 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

- (25)

The dynamics of that model are discussed in Barro and Sala-i-Martin (2004, chapter 5). The dynamic system for country 1 is written in terms of the variables and and expressed in terms of the following two-dimensional system:

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 (2004, chapter 5) 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 , at which point country 1 opens to trade again.

Now, consider country 2. On the BGP country 2 specializes in producing and imports from country 1. The imported 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 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 and trading to obtain from country 1.

The relative price for in country 1 is determined by the ratio of marginal products of the two types of capital:

- (26)

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 . 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 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: ,

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 different 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

- (27)

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 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 can also 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 first. 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: ,

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*:

- (28)

The domestic stock of *K* depreciates at rate δ:

- (29)

Accumulation of *H* in the presence of complete specialization is given by

- (30)

The usual derivation shows that the growth rate of consumption in country 1 is

- (31)

Equations (29)–(31) determine the paths of *C*_{1}, *H*_{1}, and *K*_{1}.

A similar analysis for country 2 gives the following three equations for determining the paths of *C*_{2}, *H*_{2}, and *K*_{2}:

- (32)

- (33)

- (34)

To complete the solution, we need to determine the growth rate of the world relative price, *p*. The balanced trade condition is

Total differentiation of that condition and some algebra provide the growth rate of the world relative price:

where is constant along the transition to the BGP because investment in both *K*_{1} and *H*_{2} is zero during the transition and *K* and *H* both depreciate at rate δ.11

The transition behavior of this world economy can be expressed in terms of the following five variables: *p*, , , , . The corresponding five-dimensional system is

where the are various combinations of the parameters α_{1}, η_{2}, *A*_{1}, *B*_{2}, ρ, θ, δ, the BGP values of *k*_{1}and *k*_{2}, and the initial value *k*_{12} (which is constant on the transition path).12 The system can be written as = , where is the five-dimensional vector of the growth rates of the variables, *p*, *c*_{1}, *k*_{1}, *c*_{2}, and *k*_{2}, and is a vector of deviations of the variables from their steady-state values. The system's solution can be approximated as

where *M* is the matrix of the eigenvectors of matrix *N*, κ is the diagonal matrix of eigenvalues of matrix *N*, and *z*_{0} 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 , , , , , and . The value of α_{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 η_{2} is based on the assumption that α_{1} > η_{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 implies that the annual real interest rate is 1% (consistent with the average real rate of return on U.S. Treasury 1-year bills), and the value of implies an annual depreciation rate of 10%. The values of θ and *A*_{1} are commonly used in calibration exercises (e.g., see Mulligan and Sala-i-Martin, 1993).

The choice of values *B*_{2} and *k*_{12} 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 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 *B*_{2} by moment matching. We therefore try different values of *B*_{2} to generate the annual growth rate in the presence of trade in the range of 2%–3%. For the given values of other parameters, the values of *B*_{2} consistent with this rate of growth are 0.17–0.2. Note that this does not necessarily imply that *B*_{2} cannot take larger values. We can find another combination of the parameter values that will generate annual growth rate in the acceptable range for higher values of *B*_{2}. Similarly, we have no direct measure of *k*_{12}, making it impossible to know reasonable values to choose, so again we explore the model's behavior for different values of *k*_{12}. Note that *B*_{2} and *k*_{12} are very different in meaning. *B*_{2} is a parameter of the model, capturing elements of the production technology. In contrast, *k*_{12} is the ratio of two endogenous variables. Its initial value reflects how close the initial value of one type of capital (*K*) in the first economy is compared to the other type of capital (*H*) in the second economy. It is difficult to know what is a reasonable range of values for such a ratio. Consider the related ratio of *K*_{1} to *H*_{1}. 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 reflection 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 United States, the value of producer durables and structures is about the same, suggesting that the ratio of *K*_{1} to *H*_{1} is about 1. Because we are assuming that country 1 and country 2 are large relative to each other, we would expect *k*_{12}, which is the ratio of *K*_{1} to *H*_{2}, to be the same order of magnitude.

For the values of *B*_{2} in the range of 0.17–0.2, we tried different values of *k*_{12} 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, there is also some asymmetry in the pattern of convergence between countries specializing in *K*-type 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 *k*_{12} equal to 5 and higher yield real eigenvalues and overall monotonic approach of the variables to their BGP values, whereas the values of *k*_{12} in the range closer to 1 yield complex eigenvalues with oscillatory convergence to the BGP. We just argued that values of *k*_{12} 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 *k*_{12} yield qualitative results similar to those for *k*_{12} = 5 and above, so it seems that reasonable values of *k*_{12} 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: ,

In this final 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 *C*_{2}, *K*_{2}, and *H*_{2} are determined by the following set of the equations:

- (35)

- (36)

- (37)

Now the growth rates of country 1 and country 2 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

where . Note that this ratio is not the same as the ratio in case 3 above. However, like *k*_{12}, 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 five-dimensional system:

where the elements again are combinations of the parameters α_{1}, η_{2}, *A*_{1}, *B*_{2}, ρ, θ, δ, the BGP values of *k*_{1}and *k*_{2}, and the initial value *k*_{21} (which is constant on the transition path).13 This system can be written as = , where is the five-dimensional vector of the growth rates of the variables *p*, *c*_{1}, *k*_{1}, *c*_{2}, and *k*_{2} and where is a vector of deviations of the variables from their steady-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 α_{1}, η_{2}, θ, ρ, δ, and *A*_{1} as in case 3: , , , , , and . Here, as well, we keep the values of *B*_{2} in the range 0.17–0.2 and try different values of *k*_{21}. Values of *k*_{21} at all near 1 produce real eigenvalues and relatively monotonic convergence. However, the values of *k*_{21} 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

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. MODEL SPECIFICATION
- 3. TRADE BETWEEN TWO LARGE COUNTRIES
- 4. TRANSITION DYNAMICS
- 5. CONCLUSION
- APPENDIX
- References

We have seen that trade in goods can raise the growth rates of both trading partners through comparative advantage without there being any scale effects, technology transfer, R&D, 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 effect as technology transfer and in one important special case has exactly the same effect as technology transfer. We thus have the major implications that (i) trade generally increases growth rates, (ii) trade need not increase a given country's growth rate and need not lead to growth convergence, and (iii) trade can lead to growth outcomes that are equivalent to what would emerge from technology transfer. These effects of trade on growth mean that the use of closed-economy models to analyze cross-country data are likely to be misleading. We also show that trade in goods that are not factors of production does not affect 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 unaffected. 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 Hecksher–Ohlin framework both countries must not specialize. The unifying principle is that trade will equalize factor prices if it leads to effective 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 do not 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.

### APPENDIX

- Top of page
- Abstract
- 1. INTRODUCTION
- 2. MODEL SPECIFICATION
- 3. TRADE BETWEEN TWO LARGE COUNTRIES
- 4. TRANSITION DYNAMICS
- 5. CONCLUSION
- APPENDIX
- References

#### A.1. Solution with Trade

##### A.1.1. Individual country solutions

With trade, country *i*'s Hamiltonian is

- (A.1)

where ϕ and ψ are the costate variables. The necessary conditions are

- (A.2)

- (A.3)

- (A.4)

- (A.5)

- (A.6)

- (A.7)

plus initial and transversality conditions, which are unneeded in what follows.

##### A.1.2. Balanced growth path with trade

Knowing that country 1 specializes in allows us to write a simplified maximization problem for it, conditional on the fact that it produces no . Equations (A.1)–(A.7) still characterize the problem, but now we set . The Hamiltonian reduces to

- (A.8)

and the necessary conditions become

- (A.9)

- (A.10)

- (A.11)

- (A.12)

There no longer are first-order conditions for *v* and *u* because they already have been set to 1. The first-order conditions for *C* and *X* are unchanged from the unconditional problem, but the first-order condition for *X* now holds with equality. Having accepted the world price *p* and agreed to specialize in producing , country 1 now chooses ϕ and ψ to satisfy (A.12) exactly.

Differentiating (A.11) with respect to time, dividing both sides by ϕ, and manipulating yields the growth equation for consumption

- (A.13)

The same kind of manipulations as for the autarkic model show that the growth rates of , *K*, and *Y* all equal the growth rate of consumption; as before, we denote this common growth rate γ. We obtain the growth rate of ϕ_{1} from the remaining necessary conditions and the requirements of balanced growth. Because Equation (A.12) now always holds, we have . Trade balance constrains *p* to lie in the closed interval , 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 BGP. The ratio , therefore, also is constant, implying that the growth rates of ϕ_{1} and ψ_{1} are equal. We can use this fact together with (A.9) and (A.10) to solve for , obtaining

- (A.14)

We then substitute this expression into (A.9) and divide by ϕ_{1} to obtain the growth rate for ϕ_{1}:

- (A.15)

Finally, substituting this solution into (A.13) gives us what we are after, the growth rate of country 1 in the presence of trade:

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 , only , so its Hamiltonian is

- (A.16)

with corresponding necessary conditions. Going through the same steps as for country 1 yields the growth rate

- (A.17)

#### A.2. Factor Price Equalization

##### A.2.1. Interior

In country 1, only good is produced with the production function

The marginal product of *K*-type capital is

On the BGP, in the presence of trade, the ratio of *K* to *H* in country 1 is

where *p* is the world price, given by Equation (14) in the main text. Substituting the expression for *p* into the solution for the ratio and then into the above expression for marginal product, we get

In country 2 only good is produced with the production function

The marginal product of *K* in country 2 is

The ratio in country 2 is

Substituting into the expression for the marginal product and using the expression for *p* from Equation (14) gives

It then is straightforward to show that

Similar steps show that

so that

##### A.2.2. 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

With these expressions, it is straightforward to show that

where *p*_{1} is the world price in the corner, given by Equation (11) in the main text.

- 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 effects of trade on growth in a model without scale effects or technology transfer. We compare their results with ours below.

- 2
There are two mathematical appendices, one short and one long. The short appendix is included at the end of this article 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 two-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 et al. (1996) discussions of the two-sector model's dynamics for a closed economy.

- 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 effect on the world's balanced growth rate (see his equation 11).

- 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 affects the world's dynamic path. With a small country, the world price does not respond to what the small country is doing. - 6
Analysis of the case where countries' national governments act as representatives for their countries' firms and bargain with other governments would be interesting but is beyond the scope of this article.

- 7
Equivalently, we could solve for world general equilibrium as a world central planning problem, obtaining , , , , , , , and in a single step. The two-step approach gives a more intuitive view of what each country is doing.

- 8
Bond et al. (2003) analyze a three-good, dynamic Hecksher–Ohlin model with two accumulating factors of production, one of which is not tradable and with all countries having identical production technologies 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 differing technologies imply that factor prices are not equalized whenever either country operates both sectors. See the discussion of factor price equalization below.

- 9
Bond and Trask (1997) have a three-sector model with separate sectors for

*C*,*K*, and*H*. However, their analysis is substantially different from ours because they restrict attention to a single small open economy, assume that*H*is human capital and nontradable, and do not analyze world equilibrium. The existence of only one tradable 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 nontradable 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. - 10
See Yenokyan (2010) for the details.

- 11
See Yenokyan (2010) for the details of derivations.

- 12
See Yenokyan (2010) for the expressions for the .

- 13
See Yenokyan (2010) for the elements of matrix D.

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- Top of page
- Abstract
- 1. INTRODUCTION
- 2. MODEL SPECIFICATION
- 3. TRADE BETWEEN TWO LARGE COUNTRIES
- 4. TRANSITION DYNAMICS
- 5. CONCLUSION
- APPENDIX
- References

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