The Schumpeterian One-Sector Model

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