a multi-region model of economic growth with human capital ... batabyal.pdf · theoretical model...

Electronic copy available at: http://ssrn.com/abstract=2125366

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For their helpful comments on a previous version of this paper, we thank the Co-Editor Luigi Orsenigo, an anonymous referee, andsession participants in (i) the 2011 annual meeting of the North American Regional Science Council (NARSC) in Miami, Florida,(ii) the 2012 annual meeting of the Southern Regional Science Association (SRSA) in Charlotte, North Carolina, and (iii) the 2012World Congress of the Regional Science Association International (RSAI) in Timisoara, Romania. In addition, Batabyalacknowledges financial support from the Gosnell endowment at RIT. The usual disclaimer applies.

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Department of Economics, Rochester Institute of Technology, 92 Lomb Memorial Drive, Rochester, NY 14623-5604, USA. Phone585-475-2805, Fax 585-475-5777, Internet aabgsh@rit.edu

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Department of Spatial Economics, VU University, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands. Internetp.nijkamp@vu.nl

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A Multi-Region Model of Economic Growth With Human

Capital and Negative Externalities in Innovation1

by

Amitrajeet A. Batabyal2

and

Peter Nijkamp3

Electronic copy available at: http://ssrn.com/abstract=2125366

2

A Multi-Region Model of Economic Growth With Human

Capital and Negative Externalities in Innovation

Abstract

We use a multi-region model and provide the first theoretical analysis of the effects of human

capital use and a particular kind of innovative activity on economic growth. In each of the

heterogeneous regions in our model, consumers have constant relative risk aversion preferences,

there are negative externalities in innovation, and there are three kinds of manufacturing activities

involving the production of blueprints for inputs or machines, the inputs or machines themselves,

and a single final good for consumption. Our analysis generates four salient findings. First, for each

of the regions, we define a balanced growth path equilibrium, we characterize the market clearing

factor prices, and we determine the free entry condition in the R&D sector. Second, we show that

without growth in human capital, there is no sustained economic growth in any of the regions.

Third, we show that human capital growth generates sustained economic growth in each of the

regions. Finally, when discussing the above three findings, we shed light on the spatial dimensions

of economic growth in our multi-region aggregate economy.

Keywords: Economic Growth, Human Capital, Innovation, Multi-Region Economy, Negative

Externality

JEL Codes: R11, J24, O30

4

In the remainder of this paper, we shall use the words “input” and “machine” interchangeably.

5

A seminal paper on the incentives for and the effects of innovation is Arrow (1962).

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1. Introduction

The last three decades have given rise to a considerable amount of discussion about the role

that human capital can and does play in enhancing the economic growth of regions. According to

Faggian and McCann (2009), this state of affairs has arisen because there is increasing recognition

that knowledge is the key driver of growth in most contemporary regional economies and that highly

skilled workers are the key providers of this knowledge. An implication of this perspective is that

the available human capital in a region directly affects the efficient production of final consumption

goods. In addition, this human capital also indirectly influences the production of these final goods

by participating in R&D. In turn, this participation leads initially to the invention of blueprints for

new inputs and subsequently to the production of these new inputs.4

In recent times, researchers such as Fischer and Nijkamp (2009), Baumol (2010), and

Batabyal and Nijkamp (2012a) have noted that in addition to the important role played by human

capital in enhancing the economic growth of regions, innovative activities are also very significant

drivers of regional growth and development. In the words of Fischer and Nijkamp (2009, p. 186)

policymakers now understand that “the presence of successful entrepreneurship and of a favourable

business and innovation climate will bring high benefits to the host region.” A key conclusion

arising from this line of research is that regional growth and development are very closely connected

to the activities of innovative entrepreneurs.5

Several researchers have now examined the nexuses between human capital use and regional

economic growth. Florida et al. (2008) concentrate on educational and occupational measures of

4

human capital and show that the occupational measure outperforms the educational measure when

one attempts to account for regional labor productivity measured in wages. In contrast, the

educational measure is the better measure if one’s objective is to account for regional income.

Brunow and Hirte (2009) focus on Germany and show that there are age specific human capital

effects and that a momentary increase in regional productivity can occur during what these

researchers call the demographic transition. Fleisher et al. (2010) contend that investments in human

capital in the “less-developed areas” of China are warranted because they promote efficiency and

contribute to a decrease in regional inequalities. Hammond and Thompson (2010) examine trends

in human capital accumulation in the United States and find sparse evidence of convergence in

college attainment across metropolitan and non-metropolitan areas. Banerjee and Jarmuzek (2010)

suggest that policy ought to focus on the ways in which human capital accumulation might be

increased if one is to comprehend the problem of widening regional disparities in Slovakia.

The authors of the individual chapters in Capello and Nijkamp (2009) have nicely discussed

the primarily empirical and case study based literature on the connections between innovative

activity and regional economic growth and development. However, in recent times, a small group

of theoretical papers have emerged that rigorously analyze the conditions under which alternate

innovative activities enhance regional economic growth and development. In this regard, the

contribution of Batabyal and Nijkamp (2012b) is of particular interest to us. These authors analyze

a dynamic model of an aggregate economy made up of regions. In the region,

consumers have constant relative risk aversion preferences, there is no human capital growth, and

there are three kinds of manufacturing activities involving the production of blueprints for inputs

or machines, the inputs or machines themselves, and a single final good for consumption. In this

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Dietzenbacher and Los (2002) contend that negative externalities of R&D can be usefully studied using the concept of “forwardmultipliers” in the literature on input-output analysis. Even so, the reader should note that here is no overlap between their analysisand the analysis we conduct in this paper.

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setting, these authors show that, inter alia, a policy of offering perpetual patent protection in the

region does not necessarily maximize social welfare in this region.

In the Batabyal and Nijkamp (2012b) model, even though there is no growth in human

capital, sustained economic growth arises in the region because the final consumption good in

this region is used systematically for R&D. Put differently, sustained growth is possible because the

region progressively invests more and more resources in the R&D sector. This is clearly one way

in which sustained economic growth can arise in a region and hence this sort of “lab equipment” or

“expanding varieties” model is now prominently discussed in standard texts on economic growth

such as Acemoglu (2009, pp. 433-457).

However, we have known at least since Romer (1990) that knowledge spillovers from past

R&D are a salient determinant of endogenous economic growth. This point is now well understood

and hence Acemoglu (2009, p. 444) has noted that “knowledge spillovers play an important role in

many models of economic growth...” This notwithstanding, two points are now worth emphasizing.

First, in the original Romer (1990) paper and in subsequent work—see Grossman and Helpman

(1991)—inspired by Romer’s paper, the knowledge spillovers were proportional or linear and this

linearity gave rise to endogenous economic growth. Second, as Roy (1997) and others have rightly

pointed out, the idea that knowledge spillovers are linear is far-fetched. Instead, Roy (1997) credibly

argues that it makes more sense to think of knowledge spillovers as being subject to negative

externalities.6 In this view, the greater the number of inputs or machines, the more costly it is to

innovate a new machine. The reader will note that this perspective on R&D also corresponds to a

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We do not focus on inter-regional trade related issues in this paper. Some of the issues we focus on in the context of trade arediscussed in Beladi and Oladi (2011) and in Oladi and Beladi (2008).

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view in which ideas for innovation in a particular region are drawn from a common pool and

innovation today creates a “fishing out effect” and makes future innovations more difficult.

To the best of our knowledge, the implications of negative externalities in innovation for

economic growth either in a single region or in a setting with multiple regions have not been studied

previously in the literature. Therefore, we use a multi-region model and provide the first theoretical

analysis of the effects of human capital use and negative externalities in innovation on economic

growth. In each of the heterogeneous regions in our model, consumers have constant relative risk

aversion preferences, there are negative externalities in innovation, and there are three kinds of

manufacturing activities involving the production of blueprints for inputs or machines, the inputs

or machines themselves, and a single final good for consumption. Our analysis leads to four

significant findings. First, for each of the regions under study, we define a balanced growth path

(BGP) equilibrium, we characterize the market clearing factor prices, and we ascertain the free entry

condition in the R&D sector. Second, we show that without growth in human capital, there is no

sustained economic growth in any of the regions. Third, we show that human capital growth

generates sustained economic growth in each of the regions under study. Finally, when discussing

the above three findings, we shed light on the spatial dimensions of economic growth in our multi-

region aggregate economy.7

The rest of this paper is organized as follows. Section 2 describes our multi-region

theoretical model that is related to previous work by Rivera-Batiz and Romer (1991) and Acemoglu

(2009, pp. 433-456). The model we analyze is related to the model employed by Batabyal and

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For concreteness, the reader may want to think of the aggregate economy as the European Union (EU), the regions as the variousnations in the EU, and the spatial units as the provinces within these individual EU member nations. In an alternate interpretation,the aggregate economy would be the United States, the regions would correspond to the various US states, and the spatial units woulddenote the counties in the individual US states.

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Nijkamp (2012b). However, it is important to understand that there are two key differences between

our paper and the Batabyal and Nijkamp (2012b) paper. First, we allow the stock of human capital

in each of our regions to grow. Second, the specification of the innovation possibilities

frontier—section 2.2 below for full details—assumes that negative externalities in innovation are

a key driver of endogenous economic growth and not progressive investment of resources in R&D.

Section 3 first delineates the BGP equilibrium, then describes the market clearing input prices, and

lastly determines the free entry condition in R&D. Section 4 demonstrates that without growth in

human capital, there is no sustained economic growth in any of the regions of interest. Section

5 shows that human capital growth leads to sustained economic growth in the regions under

study. Finally, section 6 concludes and then discusses potential extensions of the research delineated

in this paper.

2. The Theoretical Framework

2.1. Preliminaries

Consider an aggregate economy consisting of heterogeneous and non-overlapping regions.

We index these regions with the subscript where Each of these regions is itself

composed of distinct spatial units which we index with the superscript where 8 In the

remainder of this paper, without loss of generality, we shall focus our analysis on the region. As

we shall see, this focus on the region will not preclude us from discussing the spatial dimensions

of our subsequent results.

The region has an infinite horizon economy in which there is human capital use and

9

See Blanchard and Fisher (1989, pp. 43-45) for more on the properties of CRRA utility functions.

8

innovative activity marked by the existence of negative externalities (on which more below). The

representative household in region displays constant relative risk aversion (CRRA) and its CRRA

utility function is denoted by where is

consumption in time is the time discount rate, and is the constant coefficient of relative

risk aversion and the reciprocal of the elasticity of intertemporal substitution.9 The region

possesses human capital in each of its distinct spatial units. The human capital in the spatial

unit in region at time is denoted by Clearly, the total stock of human capital in region

at time is given by We suppose that the stock of human capital in the region

or grows exponentially at rate The salience of this exponential growth in the stock of

human capital in region will become clear in the analysis we conduct below in sections 4 and 5

of the paper.

The single final good for consumption in region is produced competitively with the

production function

(1)

where is the human capital input in region denotes the different number of the varieties

of machines that are used to produce the final good at time is the total amount of the

machine of variety that is used at time and is a parameter of the production function.

We assume that the depreciate fully after use. This assumption is made for two reasons.

First, with this assumption, we can think of these as generic inputs. Second, because these

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Since we are working with a model of endogenous technology, firms and individuals in region must ultimately have a choicebetween different kinds of technologies and, in this regard, greater effort, investment, or R&D spending ought to lead to the inventionof better technologies. These features tell us that there must exist a meta production function or a “production function overproduction functions” which tells us how new technologies are generated in region as a function of the various inputs. FollowingAcemoglu (2009, p. 413), we refer to this meta production function as the “innovation possibilities frontier.”

9

depreciate immediately, in our subsequent mathematical analysis, we will not have to work with

additional state variables. Note that for a given number of varieties of machines or the region

production function in (1) exhibits constant returns to scale. In addition, we normalize the price of

the final consumption good at all time points to equal unity. Our next task is to discuss how new

machines in the region are first invented and then produced.

2.2. Machine invention and production

Once the blueprint for a particular variety of machine has been invented, one unit of this

machine can be produced at marginal cost equal to units of the final consumption good. The

so called “innovation possibilities frontier”10 in our region is given by

(2)

where is a flow parameter, is the negative externality parameter, and is the total

expenditure on R&D in region The magnitude of can be thought of as an indicator of the

strength of the negative externality. The total expenditure on R&D is the sum of the R&D

expenditures incurred by each of the distinct spatial units in region In symbols, this means that

Equation (2) tells us that, ceteris paribus, the greater the R&D expenditure incurred by any one of

the distinct spatial units into which region is divided, the greater is the rate of production of new

machine varieties in region Because the exponent of the number of machine varieties on the

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right-hand-side (RHS) of (2) is negative, the term captures the idea that there are negative

externalities in innovation in region In other words, the greater the existing number of machines,

the more costly it is to innovate a new machine.

There is free entry into R&D activities in region The number of initial machine varieties

is supplied by monopolists. Note that because many different firms in the region are engaging

in expenditures on R&D activities, there is no aggregate uncertainty in the innovation process and

hence (2) holds deterministically in our model. A firm in region that invents a new machine of

variety is the monopolistic supplier of this variety and, therefore, at any time it sets a price

to maximize profit. Because of our assumption of full depreciation of machines, this price

can also be interpreted as the user cost of this machine. The demand for a machine of variety is

given by maximizing the net total profit of the final consumption good sector. Finally, the resource

constraint affecting the region at any time is

(3)

where have been explained previously and is total spending or investment

on machines. With this theoretical framework in place, our next task is to characterize the BGP

equilibrium for our innovative region. While undertaking this exercise, we shall adapt some

results in Peters and Simsek (2009, pp. 159-162) to our multi-region case and we shall also specify

the market clearing input prices and determine the free entry condition in R&D.

3. The BGP Equilibrium

3.1. Preliminaries

Let and denote the interest rate and the wage paid to the human capital input in

the region at time Then, an equilibrium in region is a collection of time paths of

11

consumption levels, machine expenditures, R&D expenditures wages, prices

for machines, value functions and interest rates such

that all markets clear, the representative household maximizes utility, firms maximize profit, the

evolution of the number of machine varieties is consistent with the innovation possibilities

frontier given by (2), and the value function—on which more below in section 3.3—is consistent

with free entry into R&D.

3.2. Market clearing input prices

From the discussion in section 3.2 of Batabyal and Nijkamp (2012b), we infer that the

demand for machines from the producers of the final consumption good in region is isoelastic and

given by

(4)

Equation (4) tells us that the demand for machines depends only on the price (user cost) of the

machine and on the equilibrium supply of human capital in region This demand does not depend

directly on the interest rate the wage paid to human capital and the total number of

machine varieties Given the isoelastic demand in (4), straightforward algebra tells us that the

price of machines is

(5)

The last equality in (5) follows from our normalization

We now adapt equation 13.10 in Acemoglu (2009, p. 436) to our problem. From profit

maximization it follows that the demand for machines from the final good sector in region can be

11

See theorem 7.10 in Acemoglu (2009, p. 244) for more on the technical details of this procedure.

12

expressed as

(6)

Because the human capital input within region we see that there is a distinct spatial

dimension to the generation of the region-wide demand for the final consumption good. Specifically,

(6) clearly tells us that each of the spatial units contributes human capital additively to generate

the region-wide demand for the final consumption good. Further, the greater the human capital

contribution of the spatial unit in region the greater is the region-wide demand for the final

consumption good.

The human capital market in region is competitive. Therefore, the wage paid to human

capital is given by the marginal product of human capital. Differentiating the RHS of (1) with

respect to the human capital input and then using (6), the marginal product of human capital in

region is

(7)

3.3. Free entry condition in R&D

To derive the pertinent free entry condition, we will need to work with a particular value

function. To this end, let us denote the net present discounted value of owning the blueprint of a

machine in region of variety by the time differentiable function The assumed time

differentiability of our value function means that we can write this function in the form of a

Hamilton-Jacobi-Bellman (HJB) equation given by11

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(8)

where is the per period profit function of the various research firms in region

Because the human capital input in region we see that each of the distinct

spatial units comprising region provides its human capital and thereby contributes positively to

the profit of the various research firms in this region.

Now, on the BGP we know that the interest rate in the region must be constant. Let

denote this constant interest rate. Then, it follows that on the BGP, the value function satisfies

(9)

where is the growth rate of the profit of the research firms in region and the last equality

follows from the facts that and that the profits grow at the same rate as the

region-wide stock of human capital

Finally, using (2) we infer that the free entry condition in the region under study is given

by

(10)

To understand this free entry condition in R&D in region note the following two points. First, one

unit of the final consumption good when invested in research yields a flow rate of innovation equal

to Second, the value of each innovation is given by This completes our

discussion of particular features of the BGP equilibrium in region Our next task is to demonstrate

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that without growth in human capital, there is no sustained economic growth in this region.

4. Implications of No Growth in Human Capital

We begin by considering the case where the stock of human capital in region is constant.

Mathematically, this means that and that In this case, the free entry condition

given in (10) above requires that on the BGP equilibrium path, we have

(11)

If this condition is satisfied as a strict inequality then it is clear that total expenditure on R&D in

region or and this last equality means that the number of machine varieties or is

constant. Note that since the total expenditure on R&D in region is the sum of the research

expenditures undertaken by the individual spatial units making up region means that not

a single one of the spatial units comprising region is incurring any R&D expenditure. This is

the sense in which there is a distinct spatial dimension to the free entry condition in region On the

other hand, if the condition in (11) holds with equality, then, once again, is constant at level

The last two findings in the preceding paragraph tell us that on the BGP equilibrium path,

we have

(12)

Inspecting (12) it is clear that there is no sustained growth in region and total output is constant.

12

See Acemoglu (2009, pp. 202-209) or Aghion and Howitt (2009, pp. 36-37) for additional details on this consumption Euler equation.

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From the representative household utility maximization—see section 2.1—problem we obtain the

so called consumption Euler equation12 and this equation tells us that because consumption

too is constant in the region under study. The reader should note that the constancy of

consumption also follows from the fact that (2) implies that expenditure on R&D or once

the number of machine varieties reaches its long run or steady state level given by (12). Now,

from the resource constraint given by (3), we can deduce that with no human capital growth,

consumption in region is given by

(13)

To understand the results in the preceding paragraph intuitively, note that when the stock of

human capital in region is constant, the profits of machine producers are also constant over time.

This notwithstanding, R&D becomes more and more expensive as the flow rate of innovation is

decreasing in the current level of machine varieties. Therefore, there is no endogenous growth in

region as long as the stock of human capital is constant.

Equation (13) shows that the number of machine varieties or positively influences

consumption in region In addition, (13) nicely demonstrates the two spatial dimensions that affect

the magnitude of consumption. First, from an intra-region perspective, because human capital in

region or consumption in this region clearly depends positively on the provision of

human capital by each of the spatial units that comprise region In particular, because of the

additive relationship between and the greater the human capital contribution of each

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of the spatial units, the greater is total consumption in region Second, from an inter-region or

“aggregate economy” perspective, we know that consumption and that This

means that the magnitude of consumption and the human capital input in the aggregate economy are

positively affected by consumption and the stock of the human capital input in each of the regions

under consideration. Specifically, the human capital input is salient because of two reasons. First,

it directly and positively influences consumption in region Second, it indirectly and

positively affects consumption in the aggregate economy under consideration.

Before concluding this section, we need to say something about the existence of transitional

dynamics in our model for any arbitrary region In this regard, note that whenever

the number of machine varieties will converge to the long run value This

means that there are transitional dynamics in our model. In particular, along the transition path,

will gradually increase to and the interest rate will steadily decline to its long run value

Finally, when the free entry condition given in (11) will be slack. However, because

there is no depreciation of machines in our model, the number of machine varieties will remain

at the higher level and, as a result, region will have a continuum of steady states. We now proceed

to show that human capital growth engenders sustained economic growth in region

5. Sustained Equilibrium Growth

Consistent with an observation of ours in section 2.1, it is now time to analyze the

implications of the stock of human capital in region growing exponentially at rate We can

use the free entry condition given in (10) to ascertain the joint evolution of the number of machine

varieties and the human capital stock On an equilibrium path with positive R&D in

region we have and hence the free entry condition in (10) will hold as an equality. This

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tells us that

(14)

Differentiating (14) with respect to time gives us

(15)

Using the HJB equation in (8), we infer that

(16)

Now, along the BGP we know that the interest rate in region is constant and that the number of

machine varieties grows at the constant rate Therefore, we get

and this last equality tells us that Given these results, we now use

(15) to conclude that

(17)

Equation (17) clearly shows that the region now experiences sustained economic growth. This

is because growth in the stock of human capital in region increases per period profits and this, in

turn, makes R&D more valuable over time. This positive effect counteracts the fact that over time,

R&D actually becomes more costly because of the negative externalities in innovation denoted by

Total output of the final consumption good in region is given by

13

Because the two region growth rates in equations (17) and (19) do not depend on the stock of human capital in this region, themodel we are analyzing in this section does not have a scale effect of the sort discussed by Jones (1995).

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(18)

Inspecting (18) and recalling that the human capital input we see that within region

there is a clear and positive geographic connection between the total output of the final

consumption good and the human capital provided by each of the distinct spatial units

that comprise region In particular, an increase in the provision of human capital by any one of the

spatial units into which region is divided raises the total output of the final consumption good.

Similarly, from an inter-region or “aggregate economy” standpoint, we have aggregate output of the

consumption good or This last expression clearly tells us

that the greater the provision of human capital from each of the regions under consideration, the

greater is the aggregate output of the final consumption good.

An implication of (18) is that the growth of output in region or equals

(19)

which is constant.13 We now want to show that both R&D expenditures and total consumption

expenditures in region grow at the same rate. First, note that from (2) and (17), we get

(20)

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Equation (20) tells us that the ratio has to be constant. Knowing this, we deduce that

(21)

Equation (21) implies that total R&D expenditure in region is proportional to the output of

the final consumption good Therefore, we can write and

this allows us to express the resource constraint (3) confronting region as

(22)

Equation (22) brings out the positive connection between total consumption in region and

the human capital contribution of each of this region’s spatial units. Specifically, we see that as

the human capital contribution of spatial unit where rises (falls), total consumption in

region also rises (falls). Equation (22) also implies that total consumption in region is

proportional to the multiplicative term In turn, this means that total consumption in

region grows at rate which can be expressed as

(23)

In addition, per capita consumption in region grows at the same rate as the number of machine

varieties In symbols, we have

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(24)

where the last equality follows from the consumption Euler equation. The reader should understand

that the path we have just described will correspond to a BGP equilibrium with positive growth in

region if the condition

(25)

holds. Also, it should be noted that the second inequality in (25) guarantees that the transversality

condition in our multi-region model is satisfied.

As in section 4, once again, there are transitional dynamics in region with growth in the

stock of human capital. In this regard, (14) and (24) tell us that on a BGP, we have

(26)

Now consider the more interesting case in which at time the ratio is below the

level given in (26). This means that the region begins with a low level of technology relative to

its stock of human capital. In this case will, at the outset, grow faster than the ratio and

hence the ratio will gradually rise towards the BGP value given in (26). Intuitively,

what this means is that the region under study initially has a higher incentive to

innovate—because the negative externalities in innovation given by have not kicked in

yet—and hence grows faster on the transition path.

Is the equilibrium with growth in the stock of human capital that we have been discussing

in this section thus far Pareto optimal? Adapting the analysis in Acemoglu (2009, pp. 440-442) to

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our problem, it can be shown that the answer is no but, having said this, it is important to understand

that the socially planned region will not always feature higher growth compared to the BGP

equilibrium. To see this, note that in region there are monopoly distortions and

negative technological externalities in innovation. If the monopoly distortions are the only

distortions present in region then this region will grow at a slower rate because new entrants in

R&D do not fully capture the surplus from an innovation. In contrast, the negative technological

externalities create an opposing force because, with these externalities, each innovating firm does

not take into account the fact that it is making innovation more difficult for future firms. Given this

state of affairs, a social planner will want to internalize this second externality effect and hence may

want to slow down both innovation and economic growth in region

6. Conclusions

In this paper, we used a dynamic, multi-region model to conduct, to the best of our

knowledge for the first time, a theoretical analysis of the effects of human capital use and innovation

with negative externalities on economic growth in the regions under consideration. For an

arbitrary region where we first defined a balanced growth path (BGP) equilibrium,

we delineated the market clearing factor prices, and we ascertained the free entry condition in

research. Second, we showed that without growth in human capital, there is no sustained growth in

region Third, we established that human capital growth leads to sustained economic growth in

region Finally, making intra-region and inter-region comparisons, we showed how the spatial

dimension of our model affected the equilibrium values of the key model variables such as the rate

of production of new machine varieties and the consumption of the final good.

The analysis in this paper can be extended in a number of directions. Some of the most

22

interesting extensions that, inter alia, would also yield insights into phenomena occurring over space

involve analyzing a multi-region model of the sort studied in this paper in which trade between the

different regions is explicitly modeled. To see the kinds of questions that might be analyzed in such

a scenario, consider first a simplified version of the multi-region model of this paper in which

In other words, there are only two distinct regions under consideration but these two regions trade

with each other. Now, modifying (2) slightly, if we specify the pertinent innovation possibilities

frontier as where then two questions that would be interesting

to study concern the impact of opening each region to trade. In particular, does trade openness lead

to more or less innovation in each of the two regions? In addition, what impact does trade have on

the long run growth rate of these same two regions?

As a second possible extension, let us dispense with negative externalities in innovation and

replace (2) with an alternate innovation possibilities frontier in which where

as in the preceding paragraph and represents human capital working in the

R&D sector in region One possible interpretation of this innovation possibilities frontier

is that knowledge spillovers are created by the entire range of available inputs in the two region

economy. In this setting, one can ask and answer several interesting questions. Here are two

examples. Does trade raise, lower, or have no effect on the equilibrium growth rate in each of the

two regions? On a related note, if trade has no effect on the equilibrium growth rate in each of these

two regions, then under what circumstances does trade have a positive impact on the two relevant

equilibrium growth rates? Studies that incorporate the features delineated in this and in the previous

paragraph into the analysis will increase our understanding of the ways in which the interactions

between innovative activities and trade influence the growth and development of multi-region

23

aggregate economies.

24

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