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Regional external economies and economic growth under asymmetry.

1. Introduction

Much research in macroeconomics has studied the various factors responsible for promoting economic development. As has been emphasized by a number of economists, one of the most important sources of economic growth is the continued accumulation of knowledge. In particular, Lucas (1988) demonstrates that external effects of human capital may be responsible for accelerating a country's rate of growth. In addition, recent evidence finds that spillovers of knowledge flow across countries. (1) Consequently, it is important to consider the impact of both local and regional external economies on economic growth. However, despite their potentially significant influence, most models have failed to address how regional considerations affect the process of development. (2)

Furthermore, because of the exogenous savings rate, the model has limited implications for policy coordination across locations. See also Tamura (1991).

The objective of this article is to incorporate the impact of neighboring economic activity on a country's rate of development in an endogenous growth model. (3) This is accomplished by introducing spillovers of knowledge across locations into the Lucas (1988) model of economic growth. In contrast to Lucas, we consider an economy with two different countries. We allow for the extent of spillovers both within and across countries to vary between the home and foreign economies.

There are a number of plausible reasons for such asymmetries. To begin, the transmission of information within a country is likely to be affected by the strength of intellectual property rights. For example, Matutes, Regibeau, and Rockett (1996) explain that patent policies promote the diffusion of knowledge but also protect innovative activity. Hopenhayn and Mitchell (2001) study optimal patent design through the breadth of patent protection, along with the length of time that innovators are protected. (4) In contrast, Lee and Mansfield (1996) discuss how differences in intellectual property rights across countries affect their ability to attract foreign direct investment.

Furthermore, the diffusion of knowledge within a country is related to the extent of labor mobility. In particular, Franco and Filson (2006) point out that many new firms are established by hiring employees from existing firms. These employees transfer their human capital to new employers. As a result, labor market policies that promote labor mobility within a country, such as unemployment insurance, enhance the transmission of ideas.

Finally, the transmission of information across borders is affected by a country's degree of openness to international markets. Notably, Coe, Helpman, and Hoffmaister (1997) find that developing countries experience more productivity growth by trading with countries that have a large amount of research-and-development activity. In contrast, Miller and Upadhyay (2000) emphasize that export activity is the key to raising productivity they conclude that countries that are more export oriented (a higher ratio of exports to gross domestic product) have higher levels of total factor productivity. Along these lines, Ederington and McCalman (2008) demonstrate that exporting countries are more productive because they have higher rates of technological adoption. In addition, Miyagiwa and Ohno (1995) contend that countries with more trade protection are slower to adopt new technologies.

Our benchmark model consists of a world in which there are two countries: home and foreign. The framework includes both local and regional external economies in the production of final output. We show that knowledge spillovers from the foreign country unambiguously raise productivity in the home country and therefore contribute to higher rates of progress. In contrast, spillovers within the home country may adversely affect growth in the presence of substantial knowledge flows from the foreign country. (5) Although the transmission of ideas across borders to the home country raises domestic productivity, it also discourages the accumulation of knowledge at home because individuals can free-ride on foreign ideas.

The framework also allows us to examine the impact of an increase in spillovers within the foreign economy on growth at home. As expected, an increase in the diffusion of knowledge within the borders of the foreign country weakens the incentives of foreign individuals to acquire human capital. Because agents at home rely on foreign knowledge, the domestic rate of growth suffers.

Moreover, there are interesting insights if there are cross-country spillovers in both directions. In particular, if spillovers from home to the foreign country increase, the domestic growth rate will be lower. That is, the home country is worse off if it promotes the diffusion of knowledge to foreign agents. The mechanism is similar to the impact of spillovers within the foreign country--as knowledge to the foreign country is transferred more easily, the incentives to accumulate human capital in the foreign country decline. Since individuals in the home country utilize foreign knowledge, domestic productivity falls.

The analysis from our benchmark structure yields novel insights for policies designed to affect the diffusion of knowledge. According to the standard, closed-economy Lucas (1988) model, an increase in domestic spillovers would unambiguously contribute to economic growth. From this perspective, strong protection of intellectual property rights would adversely affect development. In contrast, in our framework, protecting home property rights may stimulate productivity if the country is sufficiently open to inflows of knowledge from abroad. This is necessary in order to continue to provide individuals with incentives to develop new ideas. Furthermore, the home country benefits the most from trading with countries that also have strong intellectual property rights. Notably, this prediction mirrors the evidence from Coe, Helpman, and Hoffmaister (1997).

We further recognize that spillovers may also occur in the production of human capital. Consequently, we extend the benchmark model to include spillovers in the human capital accumulation process. However, the results reinforce our previous insights--external economies that weaken the incentives to develop knowledge adversely affect an economy's rate of progress. Interestingly, externalities in human capital accumulation may not affect the stability properties of the economy's balanced growth path (BGP).

Our analysis concludes by extending the benchmark model to look at migration decisions across countries. In the presence of asymmetries, growth in one country (the leader) will be higher than in the other. Although immigration restrictions could preclude individuals from taking advantage of differences in productivity across countries, limited amounts of natural resources, such as land, are also likely to affect location decisions. As a result, many individuals may decide to migrate to the country with a lower growth rate to take advantage of lower land prices and other forms of congestion costs.

The article is organized as follows. Section 2 describes the benchmark model in which there are regional external economies in the production of final output. Section 3 extends the analysis to study the implications of external economies in the production of human capital. Section 4 investigates the stability properties of the BGP. Section 5 looks at the possibility of migration between the two countries. Section 6 concludes and provides additional extensions for future research. Proofs of major results are included in the Appendix.

2. The Benchmark Model

We begin our investigation by incorporating regional external economies within the Lucas (1988) model of human capital and economic growth. That is, we initially consider that human capital spillovers affect productivity. In order to study the spatial aspects of knowledge transfers, there are two different locations in the economy. We refer to these locations as "country 1" and "country 2." Alternatively, country 1 may be viewed as the "home" country, while country 2 is the "foreign" country. In each location, individuals are endowed with some positive amount of country-specific knowledge. Agents in country 1 have an initial endowment of type 1 human capital but are born without any type 2 knowledge. As a result, we write [h.sub.1](O) > O, [h.sub.2](O) = 0. Similarly, for agents in the second country, we have [h.sub.2](O) > O, [h.sub.1](O) = 0. [H.sub.1](t)([H.sub.2](t)) represents the average stock of knowledge among agents in country 1(2).

Production depends on labor input and physical capital. In particular, final output by type 1 agents is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1)

in which [u.sub.1](t) represents the fraction of time that individuals in country 1 devote to production. With the remaining amount of time, agents acquire additional human capital. In considering the effects of uncompensated knowledge spillovers, [[gamma].sub.1] represents the external effects of knowledge that take place within country 1, while [[gamma].sup.*.sub.1] describes the extent to which spillovers flow across borders from country 2. The same details apply to production in location 2:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Interestingly, we allow for the degree of spillovers to vary across countries--external economies occur both within and across countries. That is, we study economies in which [[gamma].sub.1] [not equal to] [[gamma].sub.2] and [[gamma].sup.*.sub.1] and [[gamma].sup.*.sub.2] . There are a number of plausible reasons for such asymmetries. Notably, differences between [[gamma].sub.1] and [[gamma].sub.2] can represent institutional differences that affect the transmission of knowledge within each country. In particular, they may reflect different levels of intellectual property rights protection. Moreover, it is also likely that the strength of regional external effects varies between the home and foreign countries. For example, differences between [[gamma].sup.*.sub.1] and [[gamma].sup.*.sub.2] could reflect the extent of trade activity by each country. If country 1 is more export oriented than country 2, agents in the home country could acquire more foreign knowledge through the process of trade.

We turn to the optimization problem for individuals in country 1 (type 1 agents). The representative individual has the following lifetime utility function:

U = [[integral].sup.[infinity].sub.0] [e-.sup.-[rho]t] [[([c.sub.1](t)).sup.1-[alpha]/1-[alpha]]dt, (2)

where [rho] > 0 and 1/[alpha] > 0 are the discount rate and the intertemporal elasticity of substitution, respectively. Production in the economy is homogeneous with respect to consumption and capital, so that the evolution of physical capital in the economy is governed by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

For simplicity, there is no population growth or depreciation of physical capital.

The model is based on the idea that knowledge is country specific. That is, individuals in each country can acquire additional location-specific knowledge only through studying. In this manner, the accumulation of knowledge reflects a strong degree of comparative advantage. Thus, the evolution of human capital is as follows:

[h.sub.1](t) = [PHI](1 - [[mu].sub.1](t))[h.sub.1](t). (4)

[PHI] > 0 represents the growth rate of human capital if individuals devote all their time to human capital accumulation.

To solve the individual's optimization problem, we apply Pontryagin's maximum principle by writing down the expression for the current-valued Hamiltonian:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where [[lambda].sub.1](t) and [[mu].sub.1](t) represent the costate variables for physical capital and human capital for agents in country 1, respectively. We discuss the necessary conditions for the problem later. (7)

First, individuals must choose how much to consume in each period:

[(c.sub.1](t)).sup.-[alpha]] = [[lambda].sub.1](t). (6)

As [[lambda].sub.1](t) represents the value of an additional unit of physical capital in utils, Equation 6 is a standard intertemporal consumption condition--in maximizing lifetime utility, individuals trade off additional utility between the present and the future.

In addition to the savings decision, individuals must also decide how to allocate their time between the production and education sectors:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

For a given level of consumption, the choice of working time represents an investment problem. That is, individuals in each period determine how much physical and human capital to accumulate. The left-hand side reflects that allocating more time to the production sector yields additional units of physical capital. Each unit of physical capital generates additional [[lambda].sub.1](t) utils. In contrast, the right-hand side shows the value of time devoted to human capital investment. In order to maximize lifetime utility, Equation 7 demonstrates that individuals acquire additional capital so that the return across sectors is the same. Therefore, we will also refer to the choice of working time as a "no-arbitrage" condition.

In turn, the Euler equations reflect capital gains or losses from physical and human capital over time:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

Notably, since physical capital is only used in the production of final output, the Euler equation for physical capital is determined entirely by the value of physical capital at each point in time, [[lambda].sub.1](t). In contrast, human capital is an input in both the production and the education sectors. Consequently, capital gains from human capital accumulation also depend on its contribution in the production sector. As a result, the Euler equation for human capital depends on both the value of human capital, [[mu].sub.1](t), and the value of physical capital, [[lambda].sub.1](t). Finally, we have imposed the equilibrium condition that a representative individual's human capital stock and the average stock of knowledge within each country are the same ([h.sub.i](t) = [H.sub.i](t) and [h.sub.2](t) = [H.sub.2](t)).

Although the Euler equation for human capital reflects the contribution of human capital across both sectors, the no-arbitrage condition implies that individuals require the same rate of return to both factors of production. Thus, Equation 7 can be expressed to determine the allocation of human capital:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)

Additional human capital can be applied to the production sector to yield additional physical capital. Alternatively, it can be applied to the education sector so that individuals acquire more knowledge. Since the value of human capital in the production sector can be expressed in terms of the value of human capital in the education sector, capital gains from the accumulation of knowledge can be written entirely in terms of the value of human capital, [[mu].sub.1](t):

[[mu].sub.1](t) = [[mu].sub.1](t)[rho] - [PHI]). (11)

In this way, the capital gains for each type of capital are expressed in an analogous manner. The Euler equation for physical capital, Equation 8, depends only on the value of additional physical capital, [[lambda].sub.1](t). A similar result occurs in Equation 11.

We seek to understand how the economy will grow over time. In what follows, let [[theta].sub.x](t) denote the growth rate for the variable x in period t. As in any standard growth model, the growth rate for consumption among agents in country 1 is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

From the evolution equations, the growth rates for human and physical capital are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (14)

In particular, we are interested in determining the growth rates for all variables along the economy's BGP. This equilibrium concept imposes that all variables grow at constant rates over time. The growth rate of variable x along the BGP is [[theta].sup.B.sub.x].

We begin with the no-arbitrage condition. Not only does it express agents' required rates of return to both types of capital, but it also can be used to show how the accumulation of factors occurs:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

Along the BGP. the amount of working time must remain constant. This yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

In its present form, the no-arbitrage condition relates how the capital gains from both types of capital must evolve in relationship to both capital stocks along the BGP. However, Equations 8 and 11 demonstrate how the values of each type of capital grow in relationship to each factor:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

Notably, Equation 17 shows that the productivity of physical capital on the BGP depends on the accumulation of physical capital and human capital in both countries.

Furthermore, as observed by the growth rate for consumption in Equation 12, the marginal product of physical capital pins down the growth rate for consumption. In order for the growth rate of consumption to be constant along the BGP, the marginal product of physical capital must be constant--this also leads to restrictions on the accumulation of capital:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (18)

At this point, we have imposed the conditions implied by the BGP on all equations in the system with one exception--the evolution equation for physical capital. From Equation 14, it is observed that physical capital and consumption must grow at the same rate, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. As a result, the BGP for country 1 is governed by the following system of equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (19)

Analogously, we have a similar set of conditions for growth rates in country 2. The analysis yields the following proposition:

PROPOSITION 1. The competitive BGP is governed by the following system of equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (20)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (21)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (22)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (23)

From Proposition 1, it is easy to see that regional external economies can play an important role in the development process. As observed in Equation 20, human capital accumulation in the foreign country may accelerate growth at home. Moreover, Equation 22 demonstrates that the accumulation of foreign knowledge affects human capital accumulation domestically. The following lemmas highlight these patterns of interaction.

We begin by considering a simple case in which human capital spillovers occur only in country 1. That is, spillovers of knowledge take place within the home country and from the foreign country to the home country. One can interpret that in country 2, the foreign country, there is full domestic protection of intellectual property rights. In addition, the foreign country is not export oriented. In contrast, the home country has a lower degree of intellectual protection and is export oriented. Lemma 1 describes the development process in this situation:

LEMMA 1. Assume that [[gamma].sub.2] = [[gamma].sup.*.sub.2] = 0. Furthermore, let [alpha] > 1 and [PHI] > [rho]. In this case, a unique BGP for the world exists, which is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (24)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (25)

The growth rate of country 1 is increasing in [[gamma].sup.*.sub.1]. However, local spillovers will lead to lower growth if [[gamma].sup.*.sub.1] > [beta] ([alpha]/[alpha] - 1).

PROOF. See Appendix.

Interestingly, different sources of external economies can produce conflicting effects on growth. On the one hand, the diffusion of knowledge from the foreign country increases domestic productivity and contributes to higher growth. However, if domestic external effects are stronger, information at home will be transferred more easily, contributing to a free-rider problem. If [[gamma].sup.*.sub.1] is high enough, domestic individuals will rely a great deal on knowledge from the foreign country. This causes human capital accumulation in the home country to decline significantly, leading to a lower rate of growth.

Notably, our insights differ from the standard, closed-economy Lucas (1988) framework. In his model, uncompensated spillovers unambiguously contribute to higher growth, implying that policies promoting protection of intellectual property rights (lower 71) would adversely affect economic development. By comparison, in our model, such policies may be growth enhancing if the home country has a strong outward orientation.

The framework also allows us to examine the impact of an increase in spillovers within the foreign country on growth in each economy. For simplicity, we continue to assume that knowledge does not spillover from the home country to the foreign economy:

LEMMA 2. Assume that [[gamma].sub.2] > 0 and [[gamma].sup.*.sub.2] = 0. Furthermore, let [alpha] > 1 and [PHI] > [rho]. In this case, a unique BGP for the world exists, which is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (26)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (27)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (28)

If [beta] > [[gamma].sup.*.sub.1], an increase in local spillovers of knowledge is growth enhancing in both countries. However, local spillovers of knowledge within country 2 adversely affect growth in country 1.

Lemma 2 demonstrates that an increase in external economies within the foreign country leads to adverse consequences for growth in the home country. Since there are external effects of knowledge within country 2, agents will free-ride on their country-specific human capital stock. Consequently, human capital accumulation among foreign individuals will decline. As the accumulation of foreign knowledge is lower, the productivity of physical capital in country 1 falls. This leads to lower growth at home. One possible interpretation of our result is that export-oriented countries will benefit from trading with countries that have strong intellectual property rights (low values of [[gamma].sub.2]). This prediction mirrors the evidence from Coe, Helpman, and Hoffmaister (1997) productivity will increase more if countries trade with partners that invest more in human capital.

We conclude by considering the case of cross-country spillovers in both directions. In order to highlight the effects of knowledge transfers across borders, we assume that domestic spillovers do not occur:

LEMMA 3. Assume [[gamma].sub.1] = [[gamma].sub.2 = 0. Also, assume that [alpha] > 1, [beta] > [[gamma].sup.*.sub.2] and [PHI] > [rho]. Under these conditions, a unique BGP exists, which is governed by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (29)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (30)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (31)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (32)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is increasing in [[gamma].sup.*.sub.1] and is decreasing in [[gamma].sup.*.sub.2]. Similar results hold for country 2.

Lemma 3 shows that if spillovers from home to the foreign country increase, the domestic growth rate will be lower. That is, the home country is worse off if it promotes the diffusion of knowledge to foreign agents. The mechanism is similar to the impact of spillovers within the foreign country as knowledge in the foreign country is transferred more easily, the incentives to accumulate human capital in the foreign country decline. Since individuals in the home country utilize foreign knowledge, domestic productivity falls.

3. External Economies in Human Capital Accumulation

In our benchmark framework, we consider that human capital spillovers directly affect the production of final output. However, new ideas also contribute to existing knowledge in an economy. Empirical research indicates that both types of mechanisms are at work. That is, numerous studies have identified that human capital spillovers may raise productivity and affect society's stock of knowledge. For example, Rauch (1993) observes that individuals in markets with higher average levels of human capital tend to earn higher wages. This result is consistent with the productivity-enhancing effects of knowledge spillovers in the Lucas (1988) model. There is also evidence that human capital spillovers contribute to knowledge in different areas. Using patent citations, Jaffe, Trajtenberg, and Henderson (1993) conclude that spillovers are statistically more likely to come from the same geographic region. Over time, knowledge spillovers to other locations occur. In addition, Carlino, Chatterjee, and Hunt (2007) observe that human capital accumulation is important for innovative activity. Notably, they conclude that urban areas with higher levels of educational attainment have higher average per capita patent rates.

This evidence points to two different types of effects of human capital spillovers. Rauch (1993) demonstrates that spillovers may lead to higher levels of labor productivity. In contrast, Jaffe, Trajtenberg, and Henderson (1993) show that spillovers can augment society's stock of knowledge. To capture both effects, in this section, we expand our model to consider the impact of spillovers in the production of human capital. For simplicity, we consider that the spillovers are symmetric across countries. That is, [[gamma].sub.1] = [[gamma].sub.2] = [gamma] and [[gamma].sup.*.sub.1] = [[gamma].sup.*.sub.2] = [[gamma].sup.*]. In addition, growth of knowledge in country 1 is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

In contrast to the benchmark model, the accumulation of knowledge depends on the average stock of human capital in each country. While [psi] reflects the degree of spillovers from the home country, [[psi].sup.*] represents externalities from the neighboring country. The functional form follows Tamura (1991), who studies income convergence within an endogenous growth model. It has a convenient property that the production of human capital exhibits constant returns to scale at the worldwide level. While human capital is the only factor of production in Tamura's work, both physical and human capital contribute to final output in our framework. In this manner, we are able to compare the effects of external economies in the production sector to those in the education sector. Moreover, we study the implications of spillovers within and across countries. (8) All other aspects of the model are the same as the previous section.

Again, we solve for the BGP of the world by applying Pontryagin's maximum principle of optimal control. The current-valued Hamiltonian for a representative individual in country 1 is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The necessary conditions are

[([c.sub.1](t)).sup.-[alpha] = [[lambda].sub.1](t), (34)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (35)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (36)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (37)

For brevity, we focus on providing interpretation for the different equilibrium conditions compared to the benchmark model. While Equations 34, 35, and 36 follow analogously from the previous analysis, the Euler equation for human capital reflects that the value of additional knowledge in the home country depends on the average stock of human capital in both countries. This takes place because spillovers from both countries affect the productivity of human capital investment.

As in the benchmark economy, we seek to express capital gains from each type of capital in terms of the value of capital in each individual sector. Since physical capital is used only in the production sector, the Euler equation for physical capital depends only on [[lambda].sub.1](t). However, the stock of human capital in both countries affects returns in each sector of the economy--again, capital gains from human capital depend on its contribution in both the production and the education sectors. Nevertheless, the no-arbitrage condition, Equation 35, relates the value of additional human capital in the education sector to its value in the production sector:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (38)

As a result, the Euler equation can be expressed in terms of the value of human capital, [[mu].sub.1](t):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In addition, from the evolution equation of human capital, Equation 33, capital gains from human capital depend on the extent of human capital accumulation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (39)

From this point, the analysis follows the approach in the benchmark model. In order for the no-arbitrage condition to hold along the BGP,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (40)

In contrast to the benchmark model, in order for the growth rates of human capital to be constant along the BGP, they must also be equal. (9) Therefore, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Consequently, the no-arbitrage condition simplifies to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (41)

Many of the remaining BGP conditions are the same as the previous section. In order for the growth rate of physical capital to be constant, the growth rate of consumption must be the same as the growth rate of physical capital, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In addition, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], so that the marginal product of physical capital is constant. On substitution into Equation 41,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

However, since the growth rates of human capital are the same in both countries,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (42)

This yields the following proposition:

PROPOSITION 2. The growth rate of consumption in each country along the BGP is

[[theta].sup.B.sub.c] = (([beta] + [gamma] + [[gamma].sup.*])/[beta]([alpha] + [psi] + [[psi].sup.*]) + ([alpha] - 1])( [gamma] + [[gamma].sup.*])) ([PHI] - [rho]). (43)

If [alpha] > 1 and [PHI] > [rho], a unique BGP for the world exists.

At this juncture, it is interesting to compare the effects of spillovers from different sources. As in the standard Lucas (1988) model, production externalities can promote economic growth. However, spillovers in the accumulation of knowledge (from either country) adversely affect growth. The ability to free-ride on the available stock of knowledge lowers agents' incentives to invest in education. Interestingly, these results echo themes introduced in the benchmark model. From Lemmas 2 and 3, if knowledge spillovers affect productivity directly (i.e., through the production technology), they may be growth enhancing. However, if they work indirectly, by reducing agents' incentives to acquire human capital, then knowledge spillovers are likely to lower growth.

4. Dynamics

As we introduce external economies into the accumulation of human capital, it is interesting to consider whether these sources of spillovers affect the stability properties of the model. Mulligan and Sala-i-Martin (1993) emphasize that the stability properties of the BGP are difficult to determine analytically if variables grow at different rates. Therefore, we focus our attention on the economy with spillovers in the accumulation of knowledge but not in the production of final output. That is, [gamma] = [[gamma].sup.*]. However, [psi] and [[psi].sup.*] may be positive. In this setting, the economy's BGP is also a common growth path.

To begin, the dynamical system for the economy depends on six endogenous variables: [c.sub.1](t), [h.sub.1](t), [k.sub.1](t), [[lambda].sub.1](t), [u.sub.1](t), and [u.sub.1](t). In addition, along the BGP, all the variables except [c.sup.B.sub.1] are not constant. Consequently, we must redefine the variables in the system so that they are all constant. We choose the following transformation: [z.sub.1](t) = [h.sub.1](t)/[k.sub.1](t), [z.sub.2](t) = [c.sub.1](t)/[k.sub.1](t), and [z.sub.3](t) = [[lambda].sub.1](t)/[[mu].sub.1](t) Since the amount of time devoted to human capital accumulation is not likely to be constant outside the BGP, we must solve for [u.sub.1](t) as a function of[z.sub.1](t), [z.sub.2](t), and [z.sub.3](t). This can be obtained from the no-arbitrage condition:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Assuming that both countries have the same initial endowments and evolve in a symmetric manner, [h.sub.1](t) = [h.sub.2](t) = h(t),

u([z.sub.1](t),[z.sub.3](t)) = [[A[beta]/[PHI][z.sub.3](t)].sup.1/1 - [beta]] (1/[z.sub.1(t)). (44)

From the evolution equations for physical and human capital,

[[??].sub.1](t)/[z.sub.1] = [[theta].sub.h](t) - [[theta].sub.k](t) = [PHI](1 - [u.sub.1]( [z.sub.1](t),[z.sub.3](t))) + [z.sub.2](t) - A[z.sub.1][(t).sup.[beta]][u.sub.1][([z.sub.1](t),[z.sub.3](t)).sup.[beta]]. (45)

The lifetime-utility-maximizing amount of consumption, along with the evolution of physical capital, produces the growth rate of [z.sub.2](t):

[[??].sub.1](t)/[z.sub.2] = [[theta].sub.c](t) - [[theta].sub.k](t) = A[z.sup.[beta].sub.1]u[(z.sub.1](t),[z.sub.3](t)).sup.[beta]] [1 - ([alpha] + [beta])/[alpha]] + [z.sub.2](t) - [rho]/[alpha] (46)

Finally, from the Euler equations for physical and human capital,

[[??].sub.3](t)/[z.sub.3] = [[theta].sub.[lambda]](t) - [[theta].sub.[mu]](t) = A(1 - [beta] [z.sub.1][(t).sup.[beta]]u[(z.sub.1](t),[z.sub.3](t)).sup.[beta]] + [PHI](1 - [psi] - [[psi].sup.*] + [PHI](1 - [psi]u([z.sub.1](t),[z.sub.3](t))([PHI] + [[psi].sup.*]). (47)

The Jacobian for the system is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

on evaluating at the BGP. The expressions for the unsigned entries are provided in the Appendix. At this juncture, we offer the initial observation:

LEMMA 4. The determinant of [OMEGA] is negative.

By Lemma 4, the dynamical system is stable. However, it remains to be determined whether the system is saddle-path stable or globally indeterminate. The following proposition provides a sufficient condition in which the system is saddle-path stable, as in the Lucas (1988) model, without any externalities:

PROPOSITION 3. Define [??] = [psi] + [[psi].sup.*]. In addition, consider the value of [??] that satisfies

[([[??].sub.0]).sup.2] + (1/[PHI])[([alpha] - [beta])[phi] - (1 - [beta])([PHI] - [rho])][[??].sub.0] - [alpha][beta] = 0.

If [alpha] > 1, then [[??].sub.0] exists. As long as[??] [greater than or equal to] [[??].sub.0], the dynamical system is saddle-path stable.

In the standard Lucas (1988) model without any external effects, the BGP is saddle-path stable. (10) However, as discussed in the work by Jaffe, Trajtenberg, and Henderson (1993) and Carlino, Chatterjee, and Hunt (2007), the production of human capital is likely to involve spillovers of knowledge. In a number of two-sector growth models, externalities may lead to multiple equilibria with different stability properties. (11) Interestingly, Proposition 3 shows that the stability properties of the Lucas (1988) growth model appear to be robust--if there are sufficiently strong external economies in the production of knowledge, the BGP will be saddle-path stable.

5. Migration

The preceding analysis produces a number of important insights regarding the sources of economic growth across countries. In particular, the strength and source of knowledge spillovers affect productivity, along with the incentives to acquire additional human capital. Notably, in the presence of asymmetries, growth in one country (the leader) will be higher than in the other. Why would perpetual differences in growth occur? A simple answer presumes that barriers to mobility, such as immigration laws, prevent individuals from implementing location decisions to take advantage of differences in productivity across countries. However, limited amounts of natural resources, such as land, also affect migration across countries. For example, Rauch (1993) and Palivos and Wang (1996) discuss that locational choices are determined by two important features. First, human capital externalities act as an agglomerative force in which productivity leads to higher wages. Second, the availability of land implies that larger populations lead to higher land prices and housing costs.

In this section, we outline an extension of our benchmark model, in which a country's growth rate is determined along with its population size. While the world's total population is fixed at measure 1, the distribution of individuals who choose to reside in each country is determined endogenously. For simplicity, there are no costs associated with migration. Instead, following Berliant, Reed, and Wang (2006), the choice of location depends on a country's fixed residential costs. Assuming that the landmass of each country is the same, we posit that the cost function associated with the population in each country is also the same: [v.sub.1](N) = [v.sub.2](N) = v(N). In order to obtain a closed-form solution for the equilibrium population mass of each country, v(N) = vN. Consequently, higher values of v represent higher residential costs for a given population--for example, if each country has a smaller endowment of land.

Assuming that each country is on its BGP in the initial period, locational choice (i.e., the migration decision) can also be determined in the initial time period. Recognizing that the growth rate of consumption in country is 1 (2) is given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], discounted lifetime utility (net of residential costs) in the home country is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (48)

Furthermore, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] < [rho] (the condition for bounded lifetime utility), the expression for net lifetime utility at home simplifies to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (49)

Since the total population size of the world is equal to one, net lifetime utility in the foreign country is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (50)

Individuals will choose to migrate to each country in order to reach the highest lifetime utility net of residential costs. Therefore, the equilibrium population size of the home country is described by the following:

LEMMA 5. Let [alpha] > 1 and [PHI] > [rho]. In this case, a unique BGP for the world exists. In addition, the population mass of the home country is equal to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If each country has the same growth rate, the initial amounts of consumption in each country would also be equal. As a result, each country would be inhabited by one-half of the world's population. Alternatively, if the initial amounts of consumption in each country are the same and the foreign country grows at a faster rate than country 1, the home country would have a smaller population size than the foreign country.

Interestingly, although there may be persistent differences in growth rates between the two countries, some individuals would choose to live in the country with a lower growth rate. If residential costs are significantly affected by population size (higher values of v), they will play an important role in migration decisions. Consequently, many individuals could decide to migrate to the slower country to take advantage of lower land prices and other forms of congestion costs.

6. Conclusions

This article studies the effects of knowledge spillovers on growth in an open-economy setting. Interestingly, the framework can be used to study the impact of domestic and foreign sources of information on a country's rate of progress. Notably, the strength of local and regional external economies can vary across countries. As a result, the model provides novel insights regarding the diffusion of knowledge compared to standard, closed-economy versions of the Lucas (1988) growth model.

There are a number of promising issues that warrant further attention. In particular, the population size in the economy is fixed. Introducing population growth would lead to migration dynamics along the BGP. In addition, each country produces a homogeneous consumption good. The model could be extended so that countries manufacture specialized consumption goods. Rather than studying the movement of factors of production (such as labor) across countries, one could investigate the flow of goods between them. Similar to the endogenous coordination costs in Tamura (1996), transportation costs should depend on the stock of physical capital in each economy. As a result, an expanding volume of trade will be an important feature of the world's BGP. Moreover, trade policies will influence development patterns.

Appendix

The proofs for some of the major results are provided here.

PROOF OF LEMMA 1. The effect of [[gamma].sup.*.sub.1] on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is given by

[partial derivative][[theta].sup.[beta].sub.c1]/[partial derivative][[gamma].sup.*.sub.1] = ([PHI] - [rho])/[[alpha][beta] + ([alpha] - 1)[[gamma].sub.1]][alpha].

If [PHI] > [rho] and [alpha] > 1, the derivative is positive. The impact of [[gamma].sub.1] on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is

[partial derivative][[theta].sup.[beta].sub.c1]/[partial derivative][[gamma].sub.1] = ([phi] - [rho])([alpha][beta] - [[gamma].sup.*.sub.1]([alpha] - 1)/[alpha][[[alpha][beta] + ([alpha] - 1)[[gamma].sub.1]].sup.2].

If [[gamma].sup.*.sub.1] > ([alpha]/alpha] - l) [beta], the derivative is negative.

PROOF OF LEMMA 4. The Jacobian matrix is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

in which each element of the matrix is evaluated along the economy's BGP. The expressions for each element are listed as follows: [a.sub.11] = [PHI] u([z.sub.1](t),[z.sub.3](t))/[z.sub.1](t), [a.sub.12] = 1, [a.sub.13] = -[PHI] 1/1 - [beta] 1/[z.sub.3](t) u [([z.sub.1](t),[z.sub.3](t)) -[beta] A[z.sub.1][(t).sup.[beta]]u[[z.sub.1](t)).sup.[beta]] 1/1 - [z.sub.3](t), [a.sub.21] = 0], [a.sub.22] = 1], [a.sub.23] = [1 - (alpha] + [beta]/[alpha]] A[z.sub.1][(t).sup.[beta]][beta]u[[z.sub.1](t)),[z.sub.3](t)).sup.[beta]] 1/1 - [beta][z.sub.3](t), [a.sub.31] = -[PHI] ([psi] + [[psi].sup.*])1/[z.sub.1](t)u([z.sub.1](t),[z.sub.3](t)), [a.sub.32] = 0, and [a.sub.33] = -[beta] A[z.sub.1][(t).sup.[beta]]u[[[z.sub.1](t)),[z.sup.3](t)).sup.[beta]] 1/[z.sub.3](t) + [PHI]([psi] + [[psi].sup.*])1/1 - [beta] 1/[z.sub.3](t)u([z.sub.1](t), [z.sub.3](t)). Furthermore, the partial derivatives of u([z.sub.1](t), [z.sub.3](t)) are [partial derivative]u([z.sub.1](t), [z.sub.3](t))/[partial derivative]u([z.sub.1](t) = -1/([z.sub.1](t)u([z.sub.1](t),[z.sub.3](t)) < 0, and [partial derivative]u([z.sub.1](t), [z.sub.3](t))/[partial derivative]u([z.sub.3](t) = -1/1 - [beta] 1/[z.sub.3](t) u([z.sub.1](t),[z.sub.3](t)) > 0.

The determinant of the Jacobian is

Det([OMEGA]) = [a.sub.11][a.sub.33] + [a.sub.31]([a.sub.23] - [a.sub.13]).

Define[??]=[psi]+[[psi].sup.*]. The determinant is negative if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This condition reduces to

[alpha] > - [??].

Since [alpha] > 1, the previous condition is satisfied regardless of parameter values. Therefore, the system is always stable because the Jacobian has a negative determinant.

PROOF OF PROPOSITION 3. As the determinant of the Jacobian is negative, the system will be saddle-path stable if the trace is positive. The trace is equal to

Tr([OMEGA]) = [a.sub.11] + [a.sub.22] + [a.sub.33]

We have that [a.sub.11] > 0 and [a.sub.22] = 1. Therefore, a sufficient condition occurs if [a.sub.33] > 0. It can be shown that the following is a sufficient condition for [a.sub.33] > 0:

([psi] + [[psi].sup.*][[PHI]([alpha] + [psi] + [[psi].sup.*]) - ([PHI] - [rho])](1 - [beta]) > [beta](1 - [psi] - [[psi].sup.*])[PHI]([alpha] + [psi] + [[psi].sup.*].

Algebraic manipulation yields the expression in the text.

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(1) Using patent citations, Jaffe, Trajtenberg, and Henderson (1996) find that spillovers are more likely to take place across nearby national borders. Coe and Helpman (1995) observe that foreign research and development raises total factor productivity among Organization for Economic Cooperation and Development countries. Using an exogenous growth model building on the work of Mankiw, Romer, and Well (1992), Chua (1993) concludes that regional external economies from physical and human capital accumulation are significant in explaining differences in growth rates across countries.

(2) Benabou (1996) demonstrates that fiscal spillovers across locations can affect human capital accumulation. In his work, knowledge spillovers can take place at the economy-wide level, but all locations have equal access to the economy's stock of human capital. Thus, he does not consider the effects of location on the extent of knowledge spillovers. Kubo (1995) constructs a model with interregional externalities from physical capital. In his work, each country has two different sectors: an agricultural good and a manufacturing good. He illustrates how two different countries within the same region can have different patterns of economic development. However, Kubo's model is not an optimizing growth model in an endogenous growth context. In particular, he imposes an exogenous savings rate and utilizes the steady state as the equilibrium concept. Thus, economic growth occurs only outside the steady state.

(3) Moreno and Trehan (1997) observe that growth tends to take place in clusters. That is, a country's rate of progress is strongly related to growth in neighboring countries, even after controlling for common shocks.

(4) Eswaran and Gallini (1996) demonstrate that patent protection affects the direction of technological change through incentives to develop new production methods rather than new products.

(5) Notably, this observation shares a similar theme with related research in international economics. In the presence of domestic market distortions, opening markets can exacerbate these distortions and lead to lower welfare. For example, inefficiencies may be the result of imperfectly competitive behavior in product markets. Markusen (1982) constructs a model of multinationals and shows that greater market access potentially lowers welfare. If opening markets contributes to increased market power, production may decrease. In our framework, domestic inefficiencies are more severe if there are more spillovers in the home country.

(6) External economies stem from the average stock of knowledge in each country. Consequently. none of our results depend on differences in population size. Thus, we circumvent problems resulting from scale effects, which have been criticized in the literature on endogenous growth. For more details, see Eicher and Turnovksy (1999), Jones (1999), and Papageorgiou (2003).

(7) For brevity, the transversality conditions for the stocks of human and physical capital are omitted.

(8) Chamley (1993) studies a version of the Lucas (1988) model in which externalities arise from investment in human capital. In his analysis, the production of human capital depends on the average amount of time that individuals in the economy spend studying. In contrast, we examine the implications of external effects from the stocks of knowledge among individuals. Moreover, we do so in the presence of both local and regional externalities.

(9) In equilibrium, the growth rate of human capital depends on the extent of human capital accumulation in both countries: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In order for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], to be constant, the stock of knowledge must be the same across countries at each point in time. This requires that the growth rates and initial stocks of human capital are also equal.

(10) Bond, Wang, and Yip (1996) provide a general framework to determine transitional dynamics for endogenous growth models without externalities. In their framework, the Jacobian matrix has a block-recursive structure that allows them to apply the Routh theorem to demonstrate that a BGP is generally saddle-path stable.

(11) See, for example, Boldrin and Rustichini (1994).

Dmytro Holod, Assistant Professor of Finance, Department of Finance, College of Business, State University of New York at Stony Brook, Stony Brook, NY 11794, USA; E-mail Dmytro.Holod@sunysb.edu.

Robert R. Reed, Assistant Professor of Economics, Department of Economics, Finance, and Legal Studies, Culverhouse College of Business, University of Alabama, Tuscaloosa, AL 35487, USA; E-mail rreed@cba.ua.edu; corresponding author.

We thank two anonymous referees for their insightful comments. We are grateful to the coeditor, Kent Kimbrough, for valuable feedback.

Received September 2006: accepted April 2008.
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Comment:Regional external economies and economic growth under asymmetry.
Author:Holod, Dmytro; Reed, Robert R.
Publication:Southern Economic Journal
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Date:Apr 1, 2009
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