The Schumpeterian Model
In this model, growth is generated by a random sequence of
quality-improving innovations. It is called Schumpeterian because it embodies the
force that Schumpeter (1942) called “creative destruction”; that is, the
innovations that drive growth by creating new technologies also destroy the
results of previous innovations by making them obsolete.
A One-Sector Model
The Setup
There is a sequence of discrete time period t = 1, 2, 3…. In each
period fixed number L of Individuals, which supplies one unit of labour in that
period inelastically. People consume only one good, called the final good.
The production function in the Cobb-Douglas form is given as
Where y denotes the output of the final good.
A denotes the productivity of the intermediate input in that
period.
X denotes the amount of intermediate product used.
L is the labour supply used in production.
The intermediate product is produced by a monopolist each period,
using the final good as all input, one for one. Final output that is not used
for intermediate production is available for consumption and research, and it
constitutes the economy’s gross domestic product.
Production and Profit
The monopolist at t maximizes expected
consumption by maximizing her profit ∏t,
measured in units of the final good:
Where p is the price of the
intermediate product relative to the final good.
Xt is the output. So this equation
gives (profit = Total revenue – input cost)
Thus the monopolist’s price will be the marginal product of her
intermediate product in the final sector, which is
that are both proportional to the effective labour supply.
Substituting from equation (6) into the production function (1) we
get,
Innovation
In each period there is one person who has an opportunity to
attempt an innovation. If she succeeds, the innovation will create a new
version of the intermediate product, which is more productive than previous
versions. he productivity of the intermediate good in use will go from last
period’s value At−1 up to At = γAt−1, where γ>1. If she fails, then
there will be no innovation at t and
the intermediate product will be the same one that was used in t −1, so
At = At−1. But the
more the entrepreneur spends on research, the more likely she is to innovate. The
probability μt
that an innovation occurs in any period t depends positively
on the amount Rt
of final good spent on research, according to the innovation
function
is the productivity of the new intermediate
product that will result if the research succeeds. The
reason why the probability of innovation depends inversely on A∗t is
that as technology advances it becomes more complex and thus harder to improve
upon.
Where λ denotes the productivity
of the research sector and σ its elasticity which lies between 0 and 1.
Thus the marginal
product of research in generating innovations is positive but decreasing.
Research
Arbitrage
If the entrepreneur at t successfully innovates, she will
become the intermediate monopolist in that period, because she will be able to
produce a better product than anyone else. Otherwise, the monopoly will pass to
someone else chosen at random who is able to produce last period’s product. The
reward to a successful innovator is the profit π∗t. As
the probability of success is Φ(nt), So expected reward is
As the research costs Rt irrespective of success or failure. So
the benefit from research is given as
The research arbitrage equation implies that the
productivity-adjusted level of research nt will be a constant n, and
hence the probability of innovation μt will also be a constant Φ(n).
So, we have
Growth
The rate of economic growth is the proportional growth rate of per
capita GDP (GDPt/L),
which according to equation (9) is also the proportional growth rate of the
productivity parameter At:
It follows that growth will be random. In each period, with
probability μ the entrepreneur will innovate, resulting in
and with the probability (1−μ) he will fail, resulting in
The growth rate will be governed
by this probability distribution every period, so by the law of large numbers
the mean of the distribution
will be the economy long
run growth rate.
Thus, In the long run, the economy’s average growth rate equals
the frequency of innovations times the size of innovations.
Using equation (14) we get average
growth rate as
This equation gives the following results
Growth increases with the productivity of innovations l. This
result points to the importance of education, and particularly higher education,
as a growth enhancing device. Countries that invest more in higher education
will achieve a higher productivity of research, and will also reduce the
opportunity cost of research by increasing the aggregate supply of skilled labour.
2 Growth
increases with the size of innovations, as measured by the productivity
improvement factor g.
An
increase in the size of population should also bring about an increase in
growth by raising the supply of labour L.
References: Aghion & Howitt
Hi. I’m Vivek Sharma. I do write Blog on different issues and topics related to Economy, Exams and Political issues. Inspired to make things looks better.
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