A Schumpeterian Model of Entrepreneurship, Innovation, andRegional Economic Growth1

byAmitrajeet A. Batabyal2
andPeter Nijkamp3

1

We thank participants in the Tinbergen Institute Workshop in Amsterdam, The Netherlands, on May 17-18, 2010, for their comments
on a previous version of this paper. 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, Free University, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands. Internet
pnijkamp@feweb.vu.nl

1

A Schumpeterian Model of Entrepreneurship, Innovation, andRegional Economic Growth
Abstract
We provide the first theoretical analysis of a one sector, discrete time, Schumpeterian modelof growth in a regional economy in which consumers are risk neutral, there is no population growth,monopolistic entrepreneurs produce intermediate goods, and a single consumption good is produced
competitively. Our analysis generates several new results. In the deterministic model, R&D in timesurely leads to an innovation in time

In this setting, we show that relative to the balanced

growth path (BGP) equilibrium, the social planner always allocates more labor to R&D and hence
achieves a larger size of innovation and a higher growth rate. Next, in the stochastic model, R&D
in timeprobabilistically leads to an innovation in time

In this setting, we first define theequilibrium and the steady state BGP allocations. Second, we generalize the notion of the steadystate and determine the number of unemployed workers. Third, we show that our regional economy
experiences bursts of unemployment followed by periods of full employment. Finally, we show thata decline in the time discount rate increases the average growth rate and the average unemployment.
Keywords: Creative Destruction, Dynamic, Entrepreneur, Innovation, Regional Economy,
Stochastic
JEL Codes: R10, E10, O31

2

1. Introduction
Few topics in the past three decades have evoked as much research interest among
economists as has the topic of economic growth. In addition, with the passage of time, researchers
have increasingly focused on the region as a salient unit of analysis and they have now convincingly
shown that regional growth and development are very closely related to the activities of competing
entrepreneurs. In the words of Fischer and Nijkamp (2009, p. 184), “[e]ntrepreneurship...is central
to regional economic development.” In addition, contend Fischer and Nijkamp (2009, p. 183), an“entrepreneurial culture is a prerequisite for the wealth of regions...”Present day research on entrepreneurship in the context of regional economic growth anddevelopment has emphasized three noteworthy points.4 First, entrepreneurship involves anintertemporal process in which new firms are being created, existing firms are growing, andunsuccessful firms are winding down their operations. Second, entrepreneurship involves controlof this intertemporal process by the entrepreneur/owner who also serves as a manager of risk.
Finally, entrepreneurship entails innovation in a probabilistic and competitive market environment.
In addition to emphasizing the above three points, the sizeable but mainly empirical and casestudy based literature on the trinity of entrepreneurship, innovation, and regional economic growth
has addressed a number of other pertinent issues. For instance, Lorentzen (2008) has used network
theory and empirical results to argue that firms find knowledge sources on different spatial scales
and that global networks and knowledge sources are very beneficial to them. After pointing out that
innovations and the capacity to innovate are crucial factors for regional development, Cornett (2009)4

See O’Farrell (1986), Malecki (1997), Verheul et al. (2002), Fischer et al. (2006), and Fischer and Nijkamp (2009) for a more detailed
corroboration of this claim.

3focuses on Denmark and studies the factors that facilitate regional growth and the mechanisms that
stimulate innovative behavior in large, small, and medium sized enterprises. Michael and Pearce(2009) note that through innovation, entrepreneurship creates wealth for both individuals andnations. Therefore, these authors contend that it makes sense for governments to support
entrepreneurship because by doing so, these same governments are actually encouraging innovation.
Nijkamp (2009) pays particular attention to what he calls the “regional action space ofentrepreneurs” and then he surveys the nexuses between entrepreneurship and regional economic
growth.
Schwartz and Gothner (2009) empirically analyze the extent to which business incubators
have been successful in promoting entrepreneurship, innovation, and regional economic
development. Valliere and Peterson (2009) use data from the Global Entrepreneurship Monitor and
the Global Competitiveness Report and show that in developed nations, a significant fraction of theeconomic growth rates can be attributed to high expectation entrepreneurs who exploit national
investments in knowledge creation and regulatory freedom. Focusing on the links between
knowledge and innovation, Vaz and Nijkamp (2009) develop a “knowledge circuit model” thattakes into account all the pertinent stakeholders and then offers a framework for research in applied
policy problems. Finally, Henderson and Weiler (2010) first sketch the salient relationships between
entrepreneurship, innovation, and economic growth and then they empirically assess the relationshipbetween entrepreneurship and job growth across United States labor market areas and counties.
The papers discussed in the preceding two paragraphs have certainly enhanced ourunderstanding of the many nexuses between the trinity of entrepreneurship, innovation, and regional
economic growth. This notwithstanding, to the best of our knowledge, there are virtually no4theoretical studies that are both dynamic and stochastic in nature and that study the ways in which
entrepreneurship and innovation endogenously influence economic growth in a regional economy.
Given this state of affairs, in our paper, we analyze deterministic and stochastic versions of astylized one sector Schumpeterian model of growth in a regional economy in discrete time in which
consumers are risk neutral, there is no population growth, monopolistic entrepreneurs produce
intermediate goods of distinct qualities, and a single consumption good is produced competitively.
More specifically, in the deterministic version of the model, R&D in time necessarily leads
to a vertical, process innovation in time

In this setting, we first characterize the balanced

growth path (BGP) for our regional economy and then we compare the BGP growth rate with thePareto optimal growth rate. Next, in the stochastic version of the model, R&D in time
probabilistically leads to a vertical, process innovation in time

In this setting, we undertake

four analytical tasks. First, we define the equilibrium and the steady state BGP allocations. Second,
we generalize the notion of the steady state and determine the number of unemployed workers.
Third, we show that our regional economy experiences bursts of unemployment followed by periodsof full employment. Finally, we point out that a decline in the time discount rate increases theaverage growth rate and the average unemployment in our regional economy.
The rest of this paper is organized as follows. Section 2.1 first describes the deterministic
version of the one sector Schumpeterian model that is adapted from the prior work of Grossman andHelpman (1991a, 1991b) and Aghion and Howitt (1992). Section 2.2 describes the BGP and thenspecifies restrictions on the parameters so that the pertinent transversality condition is satisfied.
Section 2.3 compares the BGP growth rate with the Pareto optimal growth rate and shows that thequality or size of innovations is always too small relative to the size of innovations in the Pareto5

optimal allocation. Section 3.1 delineates the stochastic version of the section 2.1 model. Section
3.2 through 3.5 provide a detailed discussion of the four analytical tasks described in the previous
paragraph. Section 4 concludes and then discusses potential extensions of the research delineatedin this paper.
2. The Deterministic Schumpeterian Model
2.1. Preliminaries
We begin by focusing on a stylized regional economy that is subject to the forces ofinnovation. To this end, consider an infinite horizon regional economy in which only a single sector,
that is the subject of study in this paper, experiences quality improvements over time. Time isdiscrete. Since we are interested in working with a model of endogenous technology, firms andindividuals in our regional economy must ultimately have a choice between different kinds oftechnologies and, in this regard, greater effort, investment, or research spending ought to lead to theinvention of better technologies. These features tell us that there must exist a meta production
function or a “production function over production functions” which tells us how new technologies
are generated as function of various inputs. Following Acemoglu (2009, p. 413), we shall refer tothis meta production function as the “innovation possibilities frontier.” It is important to note that
we are using the term “inputs” in a very general sense. In other words, these inputs can beintermediate goods, machines, or even capital. With this caveat in mind, in the remainder of thispaper, we shall refers to these inputs as “intermediate goods.”
Consumers in our regional economy are risk neutral and their constant relative risk aversion(CRRA) utility function in time

is

where

6

is consumption in timeandis the constant coefficient of relative risk aversion.5 In what follows, we shall abstract awayfrom issues related to population growth in our regional economy. The consumption good sector in
our regional economy has the production function

(1)

where

is the output of the consumption good,used in production,andis the quality of the unique intermediate goodis the quantity of this intermediate good used in time

is a parameter,is the amount of labor used in the production of the consumption good in time
The total endowment of labor (workers) in our regional economy is

In other words, in any time
sectorandworkers in this economy can work either in the consumption goodor in the R&D sector

where entrepreneurs with research firms engage ininnovative activities and thereby attempt to invent new intermediate goods. An entrepreneur—witha research firm—who successfully generates an innovation can use this innovation to effectively
monopolize the intermediate good sector. Note that this “monopolistic entrepreneur” remains amonopolist only until the appearance of the next innovation. Put differently, we can think of thismonopolist as having acquired a patent on the new intermediate good that lasts for one time period.
There is a linear technology for the production of the intermediate goods so that an intermediate
good, once invented, can be produced at the constant marginal costin terms of the consumption

good. The use of this intermediate good as an input permits more efficient methods to be used in the
production of the consumption good (see equation (1)).
5

See Acemoglu (2009, p. 308) for more on the properties of CRRA utility functions.

7

The “engine” of regional economic growth in our paper is process innovations that lead toquality improvements. As such, the R&D technology in our regional economy is such that the useofworkers in timenecessarily leads to an innovation in time

and the number of workers

used in research determines the quality or the size of the innovation via the function
differently, if datehas quality

quality of an intermediate good isthen the new datePutintermediate good

Now, to stress the salience of labor use in the R&D sector for the occurrenceof innovations, we suppose that innovations in our regional economy occur only when
addition, for mathematical tractability, we also suppose that the functionIn

is strictly increasing,differentiable, strictly concave, and that it satisfies the so called Inada conditions.6 With thisbackground in place, the task for us now is to characterize the balanced growth path (BGP) for our
regional economy. In the course of doing so, we shall specify restrictions on the parameters of ourmodel so that the pertinent transversality condition is satisfied.
2.2. The equilibrium and the BGP
As a precursor to characterizing the BGP, let us first describe an equilibrium in our regional
economy. To this end, given the current quality
denote at timeof the intermediate good, letandthe price of the intermediate good, the interest rate, and the wage, respectively.

Now, given the current quality and a deterministic path for this quality

an equilibrium is acollection of time paths or trajectories of allocations and prices(2)with the property that the representative consumer maximizes utility, the consumption good sector
maximizes profits given prices, the monopolistic intermediate good producer chooses quantities and6

See Acemoglu (2009, pp. 33-34) for more on the Inada conditions.

8

prices to maximize profits, the R&D sector hires researchers to maximize profits, and all markets
clear.
The representative consumer’s optimization problem gives us the Euler equation
(3)
where

is the time discount rate, and the transversality condition
(4)

where

is a value function that represents the net present discounted value of owning theblueprint of an intermediate good of quality

in timeSince the representative consumer is risk

neutral, we can tell that the coefficient of relative risk aversion

Hence the Euler equation(equation (3)) is satisfied if and only if the interest rate is the inverse of the time discount rate. In
symbols, the pertinent condition is(5)
An obvious implication of equation (5) is that the interest rate is constant in equilibrium.
The consumption good producers’ maximization problem gives us the following demand
function for intermediate goods(6)
Now, to reduce the number of cases that we need to study and also to clearly bring out the key
Schumpeterian aspect of our model, we suppose that once a new intermediate good is invented, theold vintage is destroyed through obsolescence—this is the Schumpeterian creative destruction in ourmodel—and hence the new “monopolistic entrepreneur” faces no competition from previous
incumbents and he can price his intermediate good at the unconstrained monopoly price.7 Since the7

In the language of Aghion and Howitt (1992, p. 328), the innovations we are studying in this paper are “drastic” innovations.

9monopolistic entrepreneur or the intermediate good producer faces a demand function that isisoelastic, his pricing decision satisfies(7)

which also tells us thatand that per period profits are

Knowing theselast two expressions, we can tell that wages are now given by(8)and the output of the consumption good sector is given by(9)

In this deterministic Schumpeterian model, the current monopolistic entrepreneur gets
replaced by a different monopolistic entrepreneur in the next time period with certainty. Therefore,
the present monopolistic entrepreneur’s value function is only the period profits and this is given by(10)
Given current quality

the R&D sector solves

(11)

10

The first order necessary condition for an optimum to the above maximization problem is(12)

The reader should note that in our deterministic Schumpeterian model, the R&D sector makes profits
in equilibrium.8 As such, we suppose that the shares of these R&D firms are held equally by the
consumers in our regional economy so that profits accrue to the representative consumer.
Having described the equilibrium for our regional economy, we are now in a position todelineate a BGP equilibrium in which the allocation of labor is constant over time, i.e.,
for all

Since the innovation function

andsatisfies the Inada conditions, equation (12)will always have an interior solution and, in addition, a particular condition is satisfied. Using
equations (5), (8), and the labor market clearing condition

we can write this particularcondition as(13)
Equation (13) tells us that in the regional economy under study, the BGP allocation of labor to the
R&D sector or

depends on the monopoly markup, the time discount rate, and the R&D

technology. In particular, this BGP allocation does not depend on the quality of the existing
intermediate good, because, on the one hand, higher quality intermediate goods generate more profits
but, on the other hand, higher quality intermediate goods raise wages and hence make additionalinnovations costlier.

8

For an alternate model in which the R&D sector is characterized by free entry and hence makes zero profits in equilibrium, see
Acemoglu (2009, pp. 468-472).

11

The BGP equilibrium is completely characterized by equation (13). In fact, once wedetermine the values of

andoutput (of the consumption good) is determined by equation (9)and consumption is given by the net output. In other words,(14)

In each time period, the quality, and therefore output, consumption, and wages, all grow by thefactor of

The transversality condition—see equation (4)—in our model will be satisfied aslong as(15)
From the above discussion it follows that the constant growth path we have described is anequilibrium with positive growth whenever the parametric conditionis satisfied.

This completes the discussion of the equilibrium and the BGP equilibrium for our regional economy.
We now proceed to compare the BGP growth rate with the Pareto optimal growth rate for our
regional economy.
2.3. The BGP and the Pareto optimal growth rates
Let us first compute the optimal choice of intermediate good production by the social planner
given that the quality of the intermediate good isconsumption good isand that employment in the production of the

For this static problem, the social planner solves

(16)

12

which tells us that(17)and that(18)
Now, let us analyze the dynamic tradeoff for the social planner and then determine theallocation of labor between the R&D and the other sectors. The social planner’s dynamic
optimization problem is(19)
subject to
(20)
(21)
and
(22)
The first order necessary condition for an optimum for the choice variable

is
(23)

Let us conjecture a solution to the above first order necessary condition in whichandare constant for all time

Using this conjecture, the above first order necessary

condition simplifies to(24)
which has a unique solution and hence our conjecture is verified. Note that our social planner’s
optimization problem is weakly concave in the relevant choice variables. Therefore, it follows thatthe conjectured path that satisfies the first order conditions is indeed optimal if the transversality

13

condition

holds. Then, the social planner’s allocation of workers in the R&D sector isalso constant and given by the solution to equation (24). Further, quality, output, and consumption,
all grow by the factor of
Recall that the function

is a strictly decreasing function. Using this piece of information

along with equations (13) and (24), we conclude that(25)
The above two results tell us that the social planner always allocates more labor to R&D and hence
achieves a larger size of innovation and a higher growth rate. The reason for this outcome is thefollowing. The social planner’s static allocation is unaffected by monopoly distortions that arecaptured by theterm in equation (13) and that is absent in equation (24). Therefore, the social

planner produces more intermediate goods for a given quality level. This means that every unit ofquality that is innovated is more valuable to the social planner than to an equilibrium firm which,in turn, implies that the social planner innovates more and achieves a higher growth rate. Thisconcludes our comparative discussion of the BGP and the Pareto optimal growth rates in our
regional economy. We now move on and conduct a detailed analysis of the stochastic version of thebasic, deterministic model of this section.
3. The Stochastic Schumpeterian Model
3.1. Preliminaries
The one sector Schumpeterian model of this section is essentially the same as the one weanalyzed in section 2. However, there is one key difference. Now, the function
probability of innovation. Each innovation improves an intermediate good’s quality
Further, when a new innovation arrives a fraction

14

denotes the
to

where

of workers employed in the consumption goodsector are unable to adapt to this new technology and hence become unemployed for one time period.
During this period, these unemployed workers “retrain” or “retool” themselves. Our task now is todefine the equilibrium and the steady state BGP allocations for this stochastic Schumpeterian model.
Recall from the analysis in section 2 that risk neutrality on the part of consumers in our
regional economy implies that the interest rate is given by(26)
and that the equilibrium wages and profits are given by

(27)

and
(28)
The basic new feature in the model of this section is the labor market. The specification oftechnology in the first paragraph of this section tells us that capturing this technological progress isthe only way to generate growth in our model. However, this feature also imposes challenges on our
regional economy in the short run. We can think of these challenges as either changes in the sectoralcomposition of labor or as the required skill level of the workforce. In what follows, we shall capture
these features in a straightforward manner by simply supposing that a fractionof the workers

employed in the consumption good sector will be unemployed in order to get retrained. Let us denote
unemployed workers byso that total labor

To capture the retraining aspect of our model, we will need to define and work with a newstate variable. To this end, letdenote the state of the economy. That is, we shall denote15

a state where there has been no innovation in the last period by
where an innovation occurred in the last time period by(full employment) and a state(unemployment). Mathematically, we

thus have(29)

Using the notation in equation (29), we can express the number of unemployed people as(30)

The timing of the various activities in our stochastic model is as follows. At time
allocated according to
time

and

With probability

so that tomorrow’s quality is given bylabor isthere is an innovation in

In this case, some people in the labor

force allocated to work in the consumption good sector will be unemployed as they have to learn towork with the new intermediate good of higher quality. If there is no innovation then we haveand all workers allocated to the consumption good sector can be used for production.
In other words,
Letdenote the total spending on intermediate goods in our regional economy. Then, an

equilibrium in our regional economy now consists of time paths or trajectories of allocations
a value function

and prices

such that the representative consumer maximizes utility taking prices as given, the final(consumption) good producers maximize profits at given prices, the monopolistic entrepreneur

16

producing the intermediate good maximizes profits, the R&D sector hires the optimal amount oflabor (researchers) given the value function, and all markets clear. A BGP allocation is an allocation
where the flow rate or probability of innovation is constant. Note that in our regional economy, thisdoes not mean that all variables grow at a constant rate. Along the BGP, the probability of an
innovation is constant but its actual occurrence is still a random variable. Finally, note that both thevalue function and intermediate good prices are functions of the state variable

but wages are not.

This completes our description of the equilibrium and the steady state BGP allocations for thestochastic Schumpeterian model under study in this section.
3.2. The steady state and unemployed workers
We now wish to generalize the notion of the steady state and then determine the number of
unemployed workers in our regional economy. To this end, let us first solve the model described in
section 3.1. First, we need to find a value function to characterize the equilibrium behavior of theresearch firms. This naturally leads to the following question. What is the value of being themonopolistic entrepreneur with an intermediate good of quality

This entrepreneur’s current

profits are given by equation (28). In addition, Ken Arrow’s so called “replacement effect” implies
that the current incumbent (monopolistic entrepreneur) will not be active in research.9 Therefore,
from this incumbent’s point of view, the probability that there is an innovation in timeis the sameas the probability that this incumbent will be replaced in the next time period. Now, denoting the
replacement probability for an incumbent with quality

in statein timebythe valueof being a monopolistic entrepreneur is given by the functional equations

9

The “replacement effect” refers to the idea that a monopolist has lower incentives to undertake innovation than does a competitive
firm because the innovation will replace its own extant profits. See Arrow (1962) and Acemoglu (2009, pp. 420-421) for more on
this effect.

17

(31)

and
(32)

In writing equations (31) and (32), we have used the facts that the interest rate is constant and thatthe per period profits depend on the state variable

via the available labor supply (also see equation(28)).
To understand why we need two functional equations to pin down the value of innovation,note that in the first period of being a monopolistic entrepreneur, the state of our regional economy
issince this monopolistic entrepreneur himself had the innovation in the last period. Hence,in his first period of using his innovation, the profits are lower since the economy is characterized
by unemployment. In case this monopolistic entrepreneur does not get replaced (which happens withprobability
value), he remains the sole provider of the intermediate good, and he obtains aThis captures the fact that the quality stays the same because the incumbent

monopolistic entrepreneur does not engage in research and, conditional on survival, the regional
economy’s state isas there was no innovation in time

have been replaced). The value of being the monopolist in state
per period profits

and the continuation value(otherwise the monopolist wouldwith quality

then consists of thewhich arises with probability

Let us now describe the BGP allocation in our regional economy. In section 3.1, we havedefined the BGP as an equilibrium in which the probability of an innovation is constant. The

18

probability of an innovation is essentially the probability of replacement and this is given by
Because this replacement probability is only a function of the number of
employed researchers, forto be constant, we needfor allandPut

differently, we need the number of researchers to be constant along a BGP. Using the market
clearing condition for the labor market and equation (30), we can tell that the number of workers
producing the consumption good and the number of unemployed workers are given by
(33)
(34)
(35)
and
(36)
Therefore, along a BGP, employment in production of the consumption good andunemployment are only a function of the state of the regional economy
and the current quality

but independent of timeof the intermediate good. Using this fact and equation (28), we can solve

for the BGP per period profits. We get
(37)
and
(38)
Denoting the constant BGP innovation probability by

we see that the valuefunctions in equations (31) and (32) are independent of time and functions only of the current quality
and the state of the economy

Therefore, the value of being a monopolistic entrepreneur can be

written as

19

(39)

and

(40)

An implication of equation (40) is that
Using this last expression along with equations (33) and (34), equation (39) can be rewritten as(41)

Given the value function in equation (41), research firms select the number of researchers
to solve(42)

To understand the above maximization problem, note that if a research firm employs
it achieves an innovation with probability

This innovation has a value of

researchers,
but thisvalue materializes only in the next time period and hence it is discounted. However, the wage billhas to be paid for in the present time period. Now, the number of researchers hired satisfies the20following first order necessary condition10(43)which, upon further simplification, reduces to(44)

Along the BGP we haveandUsing these two conditions, we canrewrite equation (44) as(45)Note that equation (45) determines the BGP number of researchers

as a function only of theparameters of the underlying problem. In particular and as required for the BGP,
function of time and nor is it a function of the current quality

is neither a

Recall that by assumption

is a strictly concave function. Therefore, the left-hand-side (LHS) of equation (45) is strictly
decreasing in
namely,In addition, we suppose that an Inada type condition for the innovation function,holds. Then it follows that there exists a unique valuethat solves

equation (45).

10

Becauseof interest.

is a strictly concave function, the following first order necessary condition is also sufficient to characterize the optimum

21

If we compare equation (45) with the corresponding condition in section 2, we see that thestructure of the deterministic and the stochastic regional economies is very similar. In section 2, thekey equation which pinned down allocation in the labor market was equation (13). Now, if we let
the constant fraction of workers who are unemployed

and then we compare equations (13) and(45) then it is clear the only difference in the two equations stems from the product of the parameterand the discount rate

This last discount rate incorporates in it the fact that existing

patents now expire with probability

instead of with probability one which was implicitin the analysis in section 2. Our penultimate task is to show that in the probabilistic setting of thissection, our regional economy experiences bursts of unemployment followed by periods of full
employment.
3.3 Unemployment and employment patterns
We begin by noting that we have already characterized the dynamic behavior ofunemployment in our regional economy in equation (30). In the discussion leading up to equation(30) in section 3.1, we noted that unemployment is positive whenever the labor force has to be
retrained to adapt itself to the new technology and it is zero otherwise. However, in the one final
good sector Schumpeterian model that we are studying in this section, innovations occur
stochastically. Therefore, it is in this sense that our regional economy will feature bursts of
unemployment followed by periods of full employment.
In particular, whenever a new innovation occurs—and this happens with probability—the regional economy under study experiences unemployment in the following period. In
contrast, when no innovation takes place, all workers find jobs in the consumption good sector andthe economy experiences full employment. Our last task now is to demonstrate that a decline in the22

time discount rateincreases both the average growth rate and the average unemployment in our

regional economy.
3.4. Average growth rate and unemployment
Total output of the consumption good in our regional economy along a BGP is proportional
to quality. Mathematically, this means that(46)

Along the BGP, the number of workers

is given by equations (33) and (34). In other words,

the number of production workers changes probabilistically. Now, to describe the average growthrate of our regional economy, note that conditional on being in state

the expected growth of theeconomy along the BGP is given, after some steps of algebra, by(47)

To solve for the average growth rate we need to derive the unconditional probabilities
andthat our regional economy is in states

have to satisfy the equations

andrespectively. These probabilities

and

and

The first of these two

equations is the straightforward identity that there are only two states and the second equation
follows from the fact that the probability of being in stateis independent of the current state. Thisobservation tells us that
(48)

23

Now, using equation (47) the average growth rate in the economy is given by(49)
After some additional steps of algebra, equation (49) can be simplified to(50)

Now, to continue the analysis, let us assume that

This means that quality

improvements are sufficiently large so that the increase in labor productivity

dominates the effectof having a smaller labor force due to the necessity of retraining. Using this assumption and the fact
thatis an increasing function, it is clear that

Therefore, to analyze the impact of achange in the discount rate on the growth rate of the regional economy, we have to ascertain how
the equilibrium number of researchers changes when the discount rate declines.
The allocation of researchers in our regional economy is determined by equation (45) which
implies thatis decreasing in the time discount rate

To see this clearly, note that the interest

rate is increasing in the time discount rate (see equation (26)). For a given level of researchers
the LHS of equation (45) is decreasing inand the RHS is increasing inHence, if equation (45)is to be satisfied at higher interest rates, the number of researchers has to decline as the LHS isdecreasing inThis tells us thatand therefore we deduce that a decline in the time

discount rate increases the number of researchers employed and thus the growth rate of our regional
economy. To understand this result intuitively, note that as decreases, the interest rate declines sothat profits which materialize in the future are worth more today which is when researchexpenditures are incurred. This increases the incentive to invest in research which, in turn, increases

24our regional economy’s growth rate.
Finally, let us focus on the average unemployment rate in the regional economy under study.
Since the unemployment rate depends only on the state of the economy

and is given by(51)

using equation (48), the average unemployment rate along the BGP is(52)

Partially differentiating the LHS and the RHS of equation (52) with respect togives us(53)

We now claim that the last term on the RHS of equation (53) is positive. To see why thisclaim is true, observe thatmaximand is given bysolves the maximization problem in (42) and that the research firms’
where

Because(54)

it follows that

25

(55)

Inspection of equation (55) reveals that
The above analysis tells us that the average unemployment rate is increasing in the numberof researchers employed (see equation (53)). In the above analysis, we have also demonstrated thata decline in the time discount rate will increase the equilibrium number of researchers. Therefore,
the average unemployment rate is higher when the time discount rate and hence the equilibrium
interest rate declines. From an intuitive standpoint, the reader will note that in our model, there isunemployment only because the use of new technologies require retraining. Because a decline in thetime discount rate

leads to a higher probability of innovation, worker retraining occurs more often

and this results in a higher equilibrium unemployment rate.
4. Conclusions
In this paper, we used a Schumpeterian model to conduct, to the best of our knowledge forthe first time, an explicitly dynamic and stochastic analysis of the ways in which vertical innovations
and entrepreneurial activities interact to endogenously generate growth in a regional economy witha single final (consumption) good sector. We first characterized the equilibrium and the BGP for thedeterministic version of our model and then we conducted a comparative analysis of the BGP growthrate with the Pareto optimal growth rate for our regional economy. This analysis showed that relative
to the BGP allocation, the social planner always employed more labor in R&D, achieved a larger
size or quality of innovation, and hence a higher rate of growth.
Next, we studied the stochastic version of our model and this study led to four outcomes.
First, we defined the equilibrium and the steady state BGP allocations for our regional economy.

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Second, we generalized the notion of the steady state and we ascertained the number of unemployedworkers. Third, we showed that our economy experienced bursts of unemployment followed byperiods of full employment. Finally, we pointed out that a decline in the interest rate increased theaverage growth rate and the average unemployment rate in our regional economy.
The analysis in this paper can be extended in a number of directions. Here are twosuggestions for extending the research delineated in this paper. First, it would be useful to analyze
the innovation/entrepreneurship/growth nexus in a model in which there are multiple open regionsthat potentially trade with each other. Second, one could study a scenario of the sort studied in thispaper but with the key difference that the R&D sector is characterized by the presence of
agglomeration effects. Studies of the connections between innovations, entrepreneurial activities,
and economic growth that incorporate these features of the problem into the analysis will increase
our understanding of the many and varied forces that collectively lead to the development ofdynamic and stochastic regional economies.

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