A Multi-Region Model of Economic Growth With Human
Capital and Negative Externalities in Innovation1

byAmitrajeet A. Batabyal2
andPeter Nijkamp3

1

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 2012
World Congress of the Regional Science Association International (RSAI) in Timisoara, Romania. In addition, Batabyal
acknowledges financial support from the Gosnell endowment at RIT. The usual disclaimer applies.
2

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

3

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

1

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

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 humancapital use and a particular kind of innovative activity on economic growth. In each of theheterogeneous 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 eachof theregions, 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 thatwithout growth in human capital, there is no sustained economic growth in any of theregions.

Third, we show that human capital growth generates sustained economic growth in each of theregions. Finally, when discussing the above three findings, we shed light on the spatial dimensionsof economic growth in our multi-region aggregate economy.

Keywords: Economic Growth, Human Capital, Innovation, Multi-Region Economy, Negative
Externality
JEL Codes: R11, J24, O30

2Electronic copy available at: http://ssrn.com/abstract=2125366

1. Introduction
The last three decades have given rise to a considerable amount of discussion about the rolethat human capital can and does play in enhancing the economic growth of regions. According toFaggian 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 thatthe 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), andBatabyal and Nijkamp (2012a) have noted that in addition to the important role played by humancapital 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 connectedto 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

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).

3

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, theeducational 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 capitaleffects and that a momentary increase in regional productivity can occur during what theseresearchers call the demographic transition. Fleisher et al. (2010) contend that investments in humancapital in the “less-developed areas” of China are warranted because they promote efficiency andcontribute 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 incollege 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 beincreased 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 groupof theoretical papers have emerged that rigorously analyze the conditions under which alternate
innovative activities enhance regional economic growth and development. In this regard, thecontribution of Batabyal and Nijkamp (2012b) is of particular interest to us. These authors analyze
a dynamic model of an aggregate economy made up ofregions. In theregion,consumers have constant relative risk aversion preferences, there is no human capital growth, andthere 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 this4

setting, these authors show that, inter alia, a policy of offering perpetual patent protection in theregion does not necessarily maximize social welfare in this region.
In the Batabyal and Nijkamp (2012b) model, even though there is no growth in humancapital, sustained economic growth arises in theregion because the final consumption good inthis region is used systematically for R&D. Put differently, sustained growth is possible because theregion 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 growthsuch 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 understoodand hence Acemoglu (2009, p. 444) has noted that “knowledge spillovers play an important role inmany 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 thislinearity 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 toinnovate a new machine. The reader will note that this perspective on R&D also corresponds to a6

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

5view in which ideas for innovation in a particular region are drawn from a common pool andinnovation 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 foreconomic 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 theoreticalanalysis of the effects of human capital use and negative externalities in innovation on economic
growth. In each of theheterogeneous regions in our model, consumers have constant relative risk

aversion preferences, there are negative externalities in innovation, and there are three kinds ofmanufacturing 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 theregions 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 nosustained economic growth in any of theregions. Third, we show that human capital growth

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

the above three findings, we shed light on the spatial dimensions of economic growth in our multiregion 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 and7

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 are
discussed in Beladi and Oladi (2011) and in Oladi and Beladi (2008).

6

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 capitalin each of ourregions 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, andlastly determines the free entry condition in R&D. Section 4 demonstrates that without growth inhuman capital, there is no sustained economic growth in any of theregions of interest. Section

5 shows that human capital growth leads to sustained economic growth in theregions understudy. Finally, section 6 concludes and then discusses potential extensions of the research delineatedin this paper.
2. The Theoretical Framework
2.1. Preliminaries
Consider an aggregate economy consisting of
We index these regions with the subscript
composed ofwhereheterogeneous and non-overlapping regions.
Each of these

distinct spatial units which we index with the superscript

regions is itself

whereremainder of this paper, without loss of generality, we shall focus our analysis on the
we shall see, this focus on the

8

In theregion. As

region will not preclude us from discussing the spatial dimensionsof our subsequent results.
The

region has an infinite horizon economy in which there is human capital use and8

For concreteness, the reader may want to think of the aggregate economy as the European Union (EU), the regions as the various
nations 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 would
denote the counties in the individual US states.

7

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 byconsumption in time

whereis the time discount rate, andisis the constant coefficient of relativerisk aversion and the reciprocal of the elasticity of intertemporal substitution.9 The

regionpossesses human capital in each of its

spatialunit in region

at timedistinct spatial units. The human capital in theis denoted byat timeis given byorgrows exponentially at rate

Clearly, the total stock of human capital in region

We suppose that the stock of human capital in thehuman capital in region

regionThe salience of this exponential growth in the stock ofwill become clear in the analysis we conduct below in sections 4 and 5of 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 varietiesof machines that are used to produce the final good at timemachine of variety
We assume that thethat is used at timeandis the total amount of theis a parameter of the production function.

depreciate fully after use. This assumption is made for two reasons.

First, with this assumption, we can think of theseas generic inputs. Second, because these9

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

8depreciate immediately, in our subsequent mathematical analysis, we will not have to work withadditional 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 ofthe final consumption good at all time points to equal unity. Our next task is to discuss how new
machines in theregion 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 thismachine can be produced at marginal cost equal tounits of the final consumption good. The

so called “innovation possibilities frontier”10 in our

region is given by

(2)

where

is a flow parameter,

expenditure on R&D in regionis the negative externality parameter, and
The magnitude ofis the total

can be thought of as an indicator of thestrength 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 ofthedistinct spatial units into which region

machine varieties in region

is divided, the greater is the rate of production of new

Because the exponent of the number of machine varietieson the10Since we are working with a model of endogenous technology, firms and individuals in region must ultimately have a choice
between 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. Following
Acemoglu (2009, p. 413), we refer to this meta production function as the “innovation possibilities frontier.”

9

right-hand-side (RHS) of (2) is negative, the term
externalities in innovation in region

captures the idea that there are negative

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 theregion are engaging

in expenditures on R&D activities, there is no aggregate uncertainty in the innovation process andhence (2) holds deterministically in our model. A firm in region
varietythat invents a new machine ofis the monopolistic supplier of this variety and, therefore, at any timeit sets a priceto maximize profit. Because of our assumption of full depreciation of machines, this pricecan also be interpreted as the user cost of this machine. The demand for a machine of variety

isgiven by maximizing the net total profit of the final consumption good sector. Finally, the resource
constraint affecting theregion at any time

is
(3)

where

have been explained previously andis 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 someresults 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
the

and

region at timedenote the interest rate and the wage paid to the human capital input in
Then, an equilibrium in region10

is a collection of time paths ofconsumption levels, machine expenditures, R&D expenditures
for machines, value functions

wages, prices
and interest rates

suchthat all markets clear, the representative household maximizes utility, firms maximize profit, theevolution 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 thedemand for machines from the producers of the final consumption good in regionis isoelastic andgiven by(4)
Equation (4) tells us that the demand for machines depends only on the price (user cost) of themachine and on the equilibrium supply of human capital in region
directly on the interest rate
machine varieties

the wage paid to human capitalThis demand does not depend
and the total number ofGiven the isoelastic demand in (4), straightforward algebra tells us that theprice 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

11

can be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 thespatial units contributes human capital additively to generate

the region-wide demand for the final consumption good. Further, the greater the human capital
contribution of thespatial unit in region

the greater is the region-wide demand for the final

The human capital market in region

is competitive. Therefore, the wage paid to humanconsumption good.

capital is given by the marginal product of human capital. Differentiating the RHS of (1) withrespect to the human capital input
regionand then using (6), the marginal product of human capital in

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 valuefunction. To this end, let us denote the net present discounted value of owning the blueprint of amachine in regionof variety

by the time differentiable function

The assumed timedifferentiability of our value function means that we can write this function in the form of a
Hamilton-Jacobi-Bellman (HJB) equation given by11
11

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

12

(8)
where

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

Because the human capital input in region
spatial units comprising region

we see that each of the

distinctprovides its human capital and thereby contributes positively tothe profit of the various research firms in thisregion.

Now, on the BGP we know that the interest rate in theregion 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

follows from the facts thatand the last equality

and that the profits grow at the same rate

as theregion-wide stock of human capital
Finally, using (2) we infer that the free entry condition in theregion 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
toSecond, the value of each innovation is given by

discussion of particular features of the BGP equilibrium in region

13

This completes our

Our next task is to demonstratethat without growth in human capital, there is no sustained economic growth in thisregion.

4. Implications of No Growth in Human Capital
We begin by considering the case where the stock of human capital in region
Mathematically, this means thatand thatis constant.

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 inregionor

and this last equality means that the number of machine varieties orconstant. Note that since the total expenditure on R&D in regionis the sum of the researchexpenditures undertaken by the individual spatial units making up region
a single one of thespatial units comprising regionismeans that not

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

On theis 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

14

and total output is constant.

From the representative household utility maximization—see section 2.1—problem we obtain theso called consumption Euler equation12 and this equation tells us that
too is constant in the

because consumption

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
the number of machine varieties

once

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 ofhuman 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 isdecreasing in the current level of machine varieties. Therefore, there is no endogenous growth inregionas long as the stock of human capital is constant.
Equation (13) shows that the number of machine varieties or

consumption in region

In addition, (13) nicely demonstrates the two spatial dimensions that affectthe magnitude of consumption. First, from an intra-region
regionorpositively influences

perspective, because human capital in

consumption in this region clearly depends positively on the provision ofhuman capital by each of thespatial units that comprise region

additive relationship between

andIn particular, because of thethe greater the human capital contributionof each12

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

15

of thespatial units, the greater is total consumption in region“aggregate economy” perspective, we know that consumption

Second, from an inter-region or
and thatThismeans 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 theregions

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 andpositively 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 valueThismeans that there are transitional dynamics in our model. In particular, along the transition path,will gradually increase to
Finally, when

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

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
at the higher level and, as a result, region

will remain

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 theimplications of the stock of human capital in region

growing exponentially at rate

We canuse the free entry condition given in (10) to ascertain the joint evolution of the number of machinevarieties
regionand the human capital stock
we have

On an equilibrium path with positive R&D in

and hence the free entry condition in (10) will hold as an equality. This16

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
machine varieties

grows at the constant rateis constant and that the number of

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 theregion now experiences sustained economic growth. Thisis because growth in the stock of human capital in region

increases per period profits and this, inturn, 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

17

is given by(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 thethat comprise region

In particular, an increase in the provision of human capital by any one of thespatial units into which region

distinct spatial units

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 theconsumption good orThis last expression clearly tells usthat the greater the provision of human capital from each of theregions under consideration, thegreater is the aggregate output of the final consumption good.
An implication of (18) is that the growth of output in regionor

equals(19)

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

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

(20)

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).

18

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
the final consumption goodin regionis proportional to the output of

Therefore, we can write

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

as

(22)

Equation (22) brings out the positive connection between total consumption in region
the human capital contribution of each of this region’s
the human capital contribution of spatial unit
regionwhere

spatial units. Specifically, we see that as
rises (falls), total consumption inalso rises (falls). Equation (22) also implies that total consumption in region

proportional to the multiplicative term
regionandgrows at rateis

In turn, this means that total consumption inwhich can be expressed as(23)

In addition, per capita consumption in region
varieties

grows at the same rate as the number of machine

In symbols, we have

19

(24)

where the last equality follows from the consumption Euler equation. The reader should understandthat the path we have just described will correspond to a BGP equilibrium with positive growth inregionif 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 thestock 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 timetheratio is below thelevel given in (26). This means that theregion begins with a low level of technology relative toits stock of human capital. In this casewill, at the outset, grow faster than thehence theratio andratio will gradually rise towards the BGP value given in (26). Intuitively,

what this means is that theregion under study initially has a higher incentive toinnovate—because the negative externalities in innovation given byhave not kicked inyet—and hence grows faster on the transition path.
Is the equilibrium with growth in the stock of human capital that we have been discussingin this section thus far Pareto optimal? Adapting the analysis in Acemoglu (2009, pp. 440-442) to20

our problem, it can be shown that the answer is no but, having said this, it is important to understandthat 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 andnegative 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 technologicalexternalities 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 thisstate 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 ourknowledge for the first time, a theoretical analysis of the effects of human capital use and innovation
with negative externalities on economic growth in thearbitrary region

where

regions under consideration. For an

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 inregionThird, we established that human capital growth leads to sustained economic growth inregionFinally, 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 rateof 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 most21

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 sucha 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 tradewith each other. Now, modifying (2) slightly, if we specify the pertinent innovation possibilities
frontier as

wherethen two questions that would be interesting

to study concern the impact of opening each region to trade. In particular, does trade openness leadto 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 andreplace (2) with an alternate innovation possibilities frontier in which
as in the preceding paragraph andR&D sector in region

whererepresents human capital working in the

One possible interpretation of this innovation possibilities frontieris 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 thetwo 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

22

aggregate economies.

23

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