The Schumpeterian Multi Sector Model

The Schumpeterian Multi Sector Model

In this model, there are multiple innovation sector in the economy.

The final good production function is given as

 

where each xit is the flow of intermediate product i used at t, and the productivity parameter Ait  reflects the quality of that product.

According to equation (1) the final output produced by each intermediate product is determined by the production function

Each intermediate product has its own monopoly, and its price equals its marginal product in the final sector,

The monopolist in sector i chooses the quantity xit that maximises her profit:

The aggregate behaviour of the economy depends on the aggregate productivity parameter which is the unweighted numerical average of all the individual productivity parameters.

So again the economy’s GDP is proportional to its effective labour supply AtL.

Innovation and Research Arbitrage

Innovation in each sector takes place exactly as in the one-sector model. Specifically, there is a single entrepreneur in each sector who spends final output in research and innovates with probability,

Where η is the research expenditure in sector I relative to the target productivity in sector i:

 

  Where  is the productivity parameter if he succeeds.

The entrepreneur chooses the research expenditure Rit that maximizes her net benefit:

This is the same as the research arbitrage equation in the one-sector model, so it solves for the same constant productivity-adjusted research and frequency of innovation.

One important feature of this model is that the probability of innovation m is the same in all sectors, no matter what the starting level of productivity Ai,t−1.

Growth

As per capita GDP is again proportional to the aggregate productivity parameter At, therefore the economy’s growth rate is again the proportional growth rate of At:

 

The aggregate growth rate is no longer random, because bad luck in some sectors will be offset by good luck in others.

In each sector i we have

 By the law of large numbers we know that the fraction of sectors that innovate each period will be m. Therefore, the economy-wide average At can be expressed as

The average A2t among sectors that did not innovate at t is just last period’s economy-wide average At−1,

 

Thus, the growth rate in each period will be equal to

g = μ(γ-1) ………(17)

which is the same as the long-run average growth rate of the one-sector model.

Conclusion and limitations

Compared to the AK model, both the Schumpeterian model and the product-variety model have the advantage of presenting an explicit analysis of the innovation process underlying long run growth. Compared to the product-variety model, the Schumpeterian model assigns an important role to exit and turnover of firms and workers, which, as we argued at the end of the previous chapter, is consistent with an increasing number of recent studies demonstrating that labor and product market mobility are key elements of a growth-enhancing policy near the techno- logical frontier.

This is not to say that the Schumpeterian model is free of problems. We have already discussed the problem of the scale effect of increased population on growth, but we have also argued that this difficulty can be resolved within the Schumpeterian paradigm. Another difficulty with the model as presented so far is the absence of capital.

Another problem with the theory is the assumption of perfect financial markets; in reality, R&D firms rely very much on capital markets, which seem to work much better in some countries than in others. 

Reference: Aghion & Howitt