Positive and Negative Externalities in Innovation, Trade, andRegional Economic Growth1

byAmitrajeet A. Batabyal2
andPeter Nijkamp3

1

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. 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

Positive and Negative Externalities in Innovation, Trade, andRegional Economic Growth
Abstract
We analyze the implications of the interactions between positive and negative externalities
in innovation and trade for economic growth in a region when this region is part of a two region
world economy. In both regions consumers have constant relative risk aversion preferences, there
is human capital use, and there are three kinds of manufacturing activities involving the productionof blueprints for inputs or machines, the inputs or machines themselves, and a single final good for
consumption. We study two cases. In the first case, there is no growth in the human capital stock but
innovative activities give rise to positive externalities or knowledge spillovers in two ways. In thissetting, we study whether and under what circumstances opening a region to trade results in an
increase in this region’s equilibrium growth rate. In the second case, there is growth in the humancapital stock but there are negative externalities in innovation. In this scenario, we show that
opening a region to trade leads to more innovation but to no change in its long run growth rate.

Keywords: Innovation, Knowledge Spillover, Negative Externality, Regional Economic Growth,
Trade

2

1. Introduction
What is the effect of trade on the growth rate of a regional economy? Although this question
has not been answered definitively, it is fair to say that when the region under consideration is acountry, researchers now generally “believe that trade promotes growth, and there is both micro andmacro evidence consistent with this belief” (Acemoglu, 2009, p. 678; also see Jones and Romer(2010)). Does this belief change when the region under consideration and the unit of analysis is notnecessarily a country but an economic entity that is either larger or smaller than a country?
Several researchers have analyzed this salient question. Rodriguez-Pose and Gill (2006) havefocused on eight major world economies. Their empirical analysis shows that as trade in primary
products loses its importance in the composition of total trade, disparities between the various
regions being studied are likely to increase. Gonzalez Rivas (2007) has conducted a statisticalanalysis for various regions in Mexico. She shows that the regions that benefit the most by opening
up to trade are those with lower levels of education. Thus, in this analysis, trade tends to reduce
economic inequality between the various regions in Mexico. In contrast, Boschma and Iammartino(2009) concentrate on provinces and sectors in Italy and point out that regional growth is not
influenced by simply being well connected to the outside world or by having a high variety ofknowledge flow into a region.
Naude et al. (2010) use export data for magisterial districts in South Africa and contend that,in general, regions that export do far better than regions that either do not export at all or not asmuch as the exporting regions under study. Using a two region model, Minniti and Parello (2011)
show that the impact of trade integration on economic growth and the temporal extent of this impact
depend greatly on the nature of R&D spillovers. Soukiazis and Antunes (2011) focus on regions in

3

Portugal and empirically demonstrate that along with human capital, exposure to trade is a salient
variable that explains regional growth and convergence. Finally, Saito et al. (2011) show that tradeliberalization leads to a reduction in the gap in economic welfare between the various domestic
regions under study. This finding leads them to conclude that trade liberalization promotes regional
economic development.
Three points are now worth emphasizing. First, the studies discussed in the preceding two
paragraphs have certainly enhanced our understanding of the many and complex connections
between trade openness and economic growth at the level of a region when the word region does notnecessarily refer to a nation. Second, the work of Faggian and McCann (2009) and Batabyal and
Nijkamp (2012a) tells us that the stock of human capital in a region is a key driver of economic
growth in this same region. Third, Fischer and Nijkamp (2009), Baumol (2010), and Batabyal and
Nijkamp (2012b) have all stressed the powerful role of innovative activities in enhancing regional
economic growth.
Even though the three points mentioned in the previous paragraph are now well understood,
we still know very little about the theoretical circumstances in which trade either does or does notpromote regional economic growth. This is because there are virtually no theoretical studies of trade
openness and economic growth in regions that are not necessarily nations and that are characterized
by the existence of innovative activities. Given this state of affairs, in our paper we study the effectsof the interactions between positive and negative externalities in innovation—on which more below
in section 2—and trade for economic growth in a region when this region is part of a two region
world economy. In both regions consumers have constant relative risk aversion preferences, there
is human capital use, and there are three kinds of manufacturing activities involving the production4

of blueprints for inputs or machines, the inputs or machines themselves, and a single final good for
consumption.4 We study two cases. In the first case, there is no growth in the stock of human capitalbut innovative activities give rise to positive externalities or knowledge spillovers in two ways.5 In
this setting, we study whether and under what circumstances opening a region to trade results in an
increase in this region’s equilibrium growth rate. In the second case, there is growth in the stock ofhuman capital but there are negative externalities in innovation. In this setting, we show that openinga region to trade leads to more innovation but to no change in this region’s long run growth rate.6
The rest of this paper is organized as follows. Section 2 describes our theoretical model ofa two region world economy in which each region is open. This model builds on the prior work ofRivera-Batiz and Romer (1991), Grossman and Helpman (1991), Acemoglu (2009, pp. 678-680),
and Batabyal and Nijkamp (2012c). Section 3 focuses on the first case of the preceding paragraphand provides a detailed analysis of the circumstances in which trade openness leads to an increasein a region’s equilibrium growth rate. Section 4 concentrates on the previous paragraph’s secondcase and points out that trade openness results in more innovation but in no change in the long run
growth rate of the region under study. Section 5 concludes and then discusses potential extensionsof the research delineated in this paper.

4

In the remainder of this paper, we shall use the words “input,” “machine,” and “technology” interchangeably.
5

Although our paper is not concerned with migration, it should be noted that social ties, through migration, can also give rise toknowledge spillovers. See Gosens and de Vaal (2010) for additional details on the implications of this way of generating knowledge
spillovers.
6

This research agenda is entirely consistent with the suggestions for future research contained in a recent paper by Batabyal and
Nijkamp (2012c). See Doring and Schnellenbach (2006) for a survey of knowledge spillovers and the connections between such
spillovers and the economic growth of regions.

5

2. The Theoretical Framework
2.1. Preliminaries
Consider a stylized, infinite horizon world economy that is made up of two distinct regions.
We index these two regions with the subscript

whereEach of these two regions iscomposed of distinct spatial units which we index with the superscript

where

In eachof the two regions, the preferences of the representative household and the production functions are
identical. In addition, there is human capital use and innovative activity in each of the two regions.
The representative household in each region displays constant relative risk aversion (CRRA) and
its CRRA utility function—see Mas-Colell et al. (1995, p. 194) for more details—is given by
where
discount rate, andAt any time

is consumption in timeis the timeis the constant coefficient of relative risk aversion.
the human capital in the

consumption good sector

region oris employed either in the final

or in the R&D sector

In addition, in region

This means thatthese two broad categories of human capital are givenby the sum of the final good and research sector human capital in each of thein the region. Mathematically, this means thatand that

distinct spatial units
In the subsequent

analysis in section 3 (4), we assume that the stock of human capital in each of the two regions doesnot (does) grow. In addition, we suppose that the total available human capital is suppliedinelastically in both the regions under study.
The single final good for consumption is produced competitively in each region with the
production function

6

(1)

where

is the human capital input employed in the final consumption good sector in region

denotes the different number of the varieties of machines (inputs) that are used to produce the final
good in region

at timeandis a parameter of the production function. Because

it is clear that each spatial unit in region

supplies its own human capital and thereby

contributes positively to the production of the final consumption good

The price of the final

consumption good in each region at all time points is normalized to unity. Our task now is to discuss
how new machines in each of the two regions are invented and then produced.
2.2. Machine invention and production
The so called “innovation possibilities frontier”7 in theregion that describes the

production of new machines or inputs before this region is opened up to trade is given by

(2)

where

is a flow parameter,

capital at timeandis the portion of region

humanthat is employed in the research (R&D) sector.

Four implications of equation (2) are worth emphasizing. First, the termon the right-

hand-side (RHS) of this equation captures the idea that there are positive externalities or knowledge

7

Since we are working with a model of endogenous technology, firms and individuals in each of the two regions under study 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 invention of better technologies. These features tell us that in each region there must exist a meta production
function or a “production function over production functions” which tells us how new technologies are generated as a function ofvarious inputs. Following Acemoglu (2009, p. 413), we refer to this meta production function as the “innovation possibilities
frontier.”

7

spillovers from the total stock of research ideas resulting in the production of machines (inputs). In
this regard, the greater isregionor

the more productive is the human capital employed in research inSecond, because

capital in any one of the

we see that the greater the research humanspatial units that comprise region

the greater is this spatial unit’s

contribution to the region-wide creation of new machines. Third, equation (2) says that the positive
externalities in innovation in region

are general in the sense that they are created by the entire setof available machines (inputs) in the two region world economy oravailable in regionor8

and not just by the machines

Finally, because of the way in which we have modeled the existenceof positive externalities in equation (2), opening up region

to trade will not increase the existing

positive externalities.
In each of the two regions under study, there is free entry into R&D activities. The numberof initial machine varieties in regionoris supplied by monopolists. A firm that invents a

new machine of variety is the profit maximizing monopolistic supplier of this variety. The demand
for a machine of variety

in region

is given by maximizing the net total profit of the final

consumption good sector.
Let us denote the net present discounted value of owning the blueprint of a machine ofvarietyin regionby the differentiable in time function

The assumed timedifferentiability of our value function means that we can write this function in the form of a8

Whether positive externalities in innovation are general or specific—in a sense to be made precise in section 3.2 below—will havea great bearing on whether opening either one of our two regions to trade influences the equilibrium growth rate in this region. See
section 3 below for additional details.

8Hamilton-Jacobi-Bellman (HJB) equation given by9
(3)
where

is the profit of the monopolist producing the machine of variety

andis the interest rate at timeat timein region

With this background in place, the task for us now is tocharacterize the balanced growth path (BGP) equilibrium for our two regions and to then show thatwhen the positive externalities in innovation in theregion are general, opening this region to

trade has no effect on its equilibrium growth rate. In undertaking this and other related exercises in
section 3 below, we shall adapt the solution techniques employed by Peters and Simsek (2009, pp.
289-292).
3. Positive Externalities in Innovation and the Effects of Trade
3.1. General positive externalities
3.1.1. Autarky
We characterize the BGP equilibrium growth rate of each of our two regions in autarky first
and then in the presence of trade. To this end, letand letdenote the stock of human capital in region

denote the “world” or two region stock of human capital. We wish to

study a BGP equilibrium in which each region’s stock of knowledge or, equivalently, each region’s
stock of machines

grows at the same rate, say,

We now determine the common “world”

growth rate.
Adapting equation 13.27 in Acemoglu (2009, p. 445) to our problem, we deduce that the perperiod profit of a monopolist machine seller in region

is given by

9

See theorem 7.10 and the related discussion in Acemoglu (2009, p. 244, pp. 435-436) for more on the technical details of thisprocedure.

9(4)Now on a BGP, the interest rate must be constant and identical in both regions of our world
economy. Let us denote this constant interest rate byThen, standard computations tell us that thefunction denoting the net present discounted value of owning the blueprint of a machine of variety
in region

can be written as(5)

Since there is free entry in the R&D sector in each region, we can adapt equation 13.28 in
Acemoglu (2009, p. 445) and write the pertinent free entry condition for our problem as
From equation 13.13 in Acemoglu (2009, p.437) we deduce that the wage paid
to human capital in our regions is

Now, substituting equation (5) in the above

free entry condition and then using the equation forthat we have just stated, we get(6)Adding up equation (6) for each of the two regions under study, we get an equation linking the world
human capitaland the world human capital in research

to the interest rate

That equation

is

(7)

10

Maximizing the utility of the representative household in either region

gives us thestandard consumption Euler equation. Summing equation (2) over both regions

we getNow, substituting this last equation in the consumption Euler equation—see
equation 13.16 in Acemoglu (2009, p. 437) for additional details—gives us an expression relating
andThat expression is(8)

Combining equations (7) and (8), we can solve forThis gives us(9)Substituting this value offrom equation (9) into equation (8), we get(10)

as the growth rate of our two region world economy.
To ascertain the allocation of human capital within each region

note thatis constant. This constancy and equation (2) together tell us that

In similar

fashion, dividing equation (6) for region 1 by equation (6) for region 2 gives us
Combining these results for each region

11

yields(11)Our analysis thus far leads to two conclusions. First, we see that the technology or machine levels
in each region are proportional to their stock of human capital or

Second, eachregion allocates the same share of its human capital between R&D and production or
These two conclusions tell us that the allocation of human capital within the two regions under study
is given by the expression for the total human capital in research orin equation (9). Finally, thereader should note that the BGP resource allocation that we have just described will, in fact, be an
equilibrium if the so called transversality condition holds. In our model, this condition is satisfied
10

if
3.1.2. Trade

Let us begin our analysis of the effects of trade by assuming that each of the two regions
under study produces different inputs or machines and that there is trade in inputs. The key point tocomprehend now is that our aggregate, two region “world economy” is fundamentally like either oneof the two regions in autarky that we studied in section 3.1.1. As such, suppose that there exists a
BGP equilibrium in which the economy of each region grows at the time invariant rate
Then, analogous to equation (4), the per period profit of a monopolist machine seller inregionis given by(12)

Recall that in each region

we have

This relationship and equation (12) tell us that

10

Note that in our model, we have
Therefore, if the levels of technology are not proportional to the level of humancapital at timethen the technology levels will need to adjust to the level we have just specified along a transition path and theregional economies in our model will display transition dynamics.

12

there is a clear connection between the human capital in R&D in each of thecomprising each region
smaller (larger) is

spatial units

and the profits of the monopolist machine sellers. Ceteris paribus, thethe more positive (negative) is the contribution that the

spatial unit’s

research human capital makes to the profits of the monopoly machine sellers.
We know that the interest rateis constant on the BGP. Hence we deduce that the valuefunction is now given by(13)

Substituting equation (13) into the free entry condition

we get(14)

Observe that equation (14) in this section is the analog of equation (6) in section 3.1.1. Using
equation (14) we can solve for the constant interest rate. We get
result in the consumption Euler equation we get

Equation (2) and theanalysis in section 3.1.1 together tell us thatfor

Using this last

Substituting

in the equationin the previous line and then simplifying the resulting equation gives us an expression for thegrowth rate

and that expression is

(15)

13

Note that the BGP resource allocation with the growth rate described above by equation (15) will,
as in section 3.1.1, be an equilibrium as long as the transversality condition given by the inequality
is satisfied.11
3.1.3. Results
Inspecting equations (10) and (15) we see that the growth rates of the two regions in autarky
and with trade are identical. This means that there are no dynamic gains from trade. The reason forthis negative result can be understood by comparing equations (6) and (14) and comprehending threeeffects. First, note that each input or machine resulting from innovation with trade is more profitable
than the machine resulting from innovation in autarky because trade leads to a larger market size.
Mathematically, this can be seen from the fact that the left-hand-side (LHS) of (14) is greater thanthe LHS of (6) because

Call this the “larger market” effect.

Second, note that the RHS of equation (14) is also greater than the RHS of equation (6)because a greater number of inputs is produced with trade and hence

In turn, this meansthat wages paid and the cost of undertaking R&D with trade resulting in superior inputs in thecompeting production sector is also greater than the corresponding cost in autarky. Call this the“higher cost” effect. Finally, as noted in the paragraph following equation (2), opening up region
to trade does not increase the existing positive externalities. Put differently, the two regions are
already integrated as far as knowledge flows are concerned. Therefore, the “larger market” and the“higher cost” effects offset each other in the BGP equilibrium and the growth rates remain identical.
Even though there are no dynamic gains from trade, the reader should note that trade does11

In contrast with the scenario analyzed in section 3.1.1, there are no transition dynamics in the two region world equilibrium with tradethat we have been studying in this section. In particular, beginning with any values forandthe two regional economies
begin growing immediately at the rate specified in equation (15).

14lead to static gains. This means that each region’s output of the final consumption good is larger at
all points in time after this region has been opened up for trade. This claim is easily verified. In thisregard, note that each region

production of the final good in autarky is given by equation (1).

In other words,

In contrast, each region

with trade isandproduction of the final good

becausePut differently, becausehouseholds in each region display a love for variety in the final consumption good sector, trade in
inputs increases output levels but not the output growth rates. Our next task is to study the effectsof trade in our two region world economy when there are positive externalities in innovation after
the two regions have been opened up to trade. This is the case in which the positive externalities ininnovation are specific in nature.
3.2. Specific positive externalities
3.2.1. Autarky
The key difference now lies in the form of the innovation possibilities frontier. Specifically,
this frontier is now no longer given by equation (2) but, instead, by(16)

Note from equation (16) that the positive externalities are now not general but specific because theevolution of the stock of knowledge in either region depends, inter alia, on the existing number ofmachines or inputs

in theregion.

Consider first the situation in autarky. Using the methodology employed in section 3.1, we
obtain an equation that is the analog of equation (6). Specifically, we get

15

(17)

Similarly, we can derive an equation for the growth rate of each region

that is the analog of

equation (10). The equation we seek is(18)

Because the positive externalities we are now studying are specific, the term denoting machines on
the LHS of equation (17) is given by

and the human capital term on the RHS of equation (18)

is given by
3.2.2. Trade
Some thought tells us that when the two regions are open to trade, the previous analysis in
section 3.1.2 continues to apply and the growth rate of the two region world economy under study
is given by equation (15).
3.2.3. Results
Comparing equations (15) and (18) it is clear that the BGP growth rate in both regions
increases with trade. This is because

Pursuing an alternate line of reasoning, let uscompare equations (6) and (17). Once again, in the case of specific positive externalities, the “larger
market” and the “higher cost” effects identified in section 3.1.3 apply and they offset each other.
However, now, trade creates additional knowledge spillovers which increase the incentives for R&D
in both regions. In this regard, observe that the termin equation (14) is larger than the termin equation (17) and hence the growth rate of the two region world economy is higher with16

trade.
The basic conclusion emanating from our analysis in section 3 can be stated in two ways.
First, using the language of positive externalities, trade increases the economic growth rate of aregion when positive externalities are specific before trade but general after trade. Second, using thelanguage of spillovers, trade enhances the growth rate of a regional economy when it also gives rise
to knowledge spillovers. This completes our discussion of the circumstances in which trade
openness leads to an increase in a region’s equilibrium growth rate when there are positive
externalities in innovation. We now proceed to analyze the effects of trade when there is growth in
the stock of human capital but there are negative externalities in innovation.
4. Negative Externalities in Innovation and the Effects of Trade
4.1. Autarky
There are two key modeling differences in the analysis that we undertake in this section.
First, the stock of human capital grows at rate

in both the regions under study. Second, following

Batabyal and Nijkamp (2012c), the innovation possibilities frontier in each region is now given by

(19)

where

and

are as defined in section 2.2,is the total expenditure on R&D in regionis the negative externality parameter, andThis total expenditure on R&D is the sum of the R&D

expenditures incurred by each one of the distinct spatial units in region In symbols, we haveThis stipulation means that the

spatial units that comprise theregion matter because the total

R&D expenditure in this region is the sum of the amounts expended on R&D in each of these17

individual spatial units.
We first analyze the autarky equilibrium in which, for analytical tractability, the stock ofhuman capital in both the regions under study is the same. This means thatfor eachNow, following the methodology described in Batabyal and Nijkamp (2012c),12 we deduce that the
profit of the input or machine producers in the
now grows at rate

region is given by

and that this profit

Focusing on a BGP equilibrium, we conjecture that on such a path, the total

number of machines in region

at timeorgrows at the common ratein each of the tworegions under study. This means that the total number of machines in the two region world economyoralso grows at rate

and that the interest rate in each regionoris constant at acommon level given by, say,
In a BGP equilibrium, the value function for a machine producing firm can be solved to give(20)

where the denominator on the RHS of equation (20) is
producers grows at rate

because the profit of the machine

Substituting equation (20) into the pertinent free entry condition—see

equation (10) in Batabyal and Nijkamp (2012c) for additional details—this free entry condition canbe written as

(21)

12

See Acemoglu (2009, pp. 436-446) for a textbook discussion of this methodology.

18

Equation (21) will be satisfied for all timenumber of machines

is given by

as long as the constant growth rate of the total

Then, adapting the analysis in Batabyal and Nijkamp(2012c) to our problem, we infer that a BGP equilibrium with the growth rate

exists as

long as our model parameters satisfy(22)
The satisfaction of the inequality in (22) above also ensures that the transversality condition holds
in each of the two regions under study.
The consumption Euler equation resulting from the representative household’s utility
maximization problem in each region tells us that
previous finding that

Combining this result with ourallows us to obtain an explicit equation for the common interest ratein either region. The equation we seek is(23)

Finally, substituting equation (23) into equation (21) gives us a ratio delineating the two region orworld level of machines divided by the stock of human capital. That ratio is(24)

What happens to the autarky BGP equilibrium growth rate in either one of our two regions when this19

region is opened to trade?13 We now proceed to answer this question.
4.2. Trade
In contrast to the autarky case, the profits of the machine or input producers are now given
byThis is because these producers now serve a larger market compared to the market

in autarky. As in section 4.1, once again we focus on a BGP equilibrium in which the total numberof machines in each region grows at rate
superscript

the interest rate is constant at level

and thedenotes trade.

In this BGP equilibrium, a machine producing firm’s value function can be solved to yield(25)

and the corresponding free entry condition tells us that

(26)

Equation (26) is satisfied for all timemachinesor

as long as the constant growth rate of the total number ofNow, employing reasoning analogous to that employed in section 4.1,

we deduce that a BGP equilibrium exists as long as the inequality in (22) is satisfied.
4.3 Results
The analysis thus far in this section and some thought together tell us that on the BGP
13

Note that transitional dynamics exist in the model of this section in the following sense. If the initial value of the ratio describing
the world level of machines divided by the stock of human capital oris different from the value given in equation (24)then thisratio will converge to its BGP equilibrium level.

20equilibrium, we must haveandThese last two results together tell us that

opening a region to trade does not change either the growth rate or the interest rate on the BGP. This
notwithstanding, rewriting equation (26), we see thatrewriting equation (21) gives us
and recalling that

Similarly,
Combining these last two expressions

we see that(27)

Inspecting equation (27), we see that even though the growth rate of machines or thetechnology in a region is identical along the BGP, the two region world level of technology to humancapital ratio is higher after a region is opened to trade. To see why this result arises, note thefollowing three step chain of events. First, starting from an initial state corresponding to the autarky
BGP equilibrium, i.e., starting fromthe level of the world technology-

human capital ratio steadily rises to its new BGP value given by

Second, right aftera region is opened up to trade, there is quicker innovation of machines in the world economy andhence the total number of machines

grows faster than the rate

growth rate steadily declines to the autarky growth rate

Finally, this more rapid

This completes our discussion ofnegative externalities in innovation and the impact of trade on a region’s equilibrium growth rate.
5. Conclusions
In this paper, we analyzed the implications of the interactions between positive and negative
externalities in innovation and trade for economic growth in a region when this region is part of a
two region world economy. We studied two cases in detail. In the first case, there was no growth in21

the stock of human capital but innovative activities gave rise to positive externalities or knowledge
spillovers that were either general or specific in nature. In this setting, we examined whether andunder what circumstances opening a region to trade resulted in an increase in this region’s
equilibrium growth rate. In the second case, there was growth in the human capital stock but there
were negative externalities in innovation. In this scenario, we showed that opening a region to trade
led to more innovation but the long run growth rate of each region remained unchanged.
The analysis in this paper can be extended in a number of directions. In this regard, consider
a world economy that consists of two groups of regions, namely, a “Northern” group and a“Southern” group. Suppose that the “Southern” group is larger than the “Northern” group. In
addition, there are two sectors in the economy of each group and these two sectors are an agricultural
sector and a manufacturing sector. In this setting, several questions would be worth researching.
Here are three examples of such questions. First, does trade between the “Northern” and the“Southern” groups of regions cause one group to specialize in a sector with a relatively low growthpotential? Second, is there any learning by doing in either one of the two groups of regions? Finally,
how do the answers to the preceding two questions influence the long run equilibrium of the world
economy with trade? Studies that incorporate these features of the problem into the analysis will
advance our understanding of the ways in which the trinity of human capital use, innovative activity,
and trade influence the growth and development of dynamic regional economies. In addition, thisunderstanding will also be enhanced by focusing on the differences in regional production
technologies, spatial absorption capacities, and consumer preferences.

22

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