ENDOGENOUS GROWTH (PRODUCT VARIETY MODEL)

ENDOGENOUS GROWTH (PRODUCT VARIETY MODEL)

Endogenizing Technological Change

Product variety expands gradually because discovering how to produce a larger range of products take real resources, including time. For each new product there is a sunk cost of product innovation that must be incurred just once, when the product is first introduced, and never again. The sunk costs can be thought of as costs of research, an activity that results in innovations that add to the stock of technological knowledge. In this case, technological knowledge consists of a list of blueprints, each one describing how to produce a different product, and every innovation adds one more blueprint to the list.

What makes this approach different from an AK model is not just the sunk cost of product development, but also the fact that fixed costs make product markets monopolistically competitive rather than perfectly competitive .

A Simple Variant of the Product-Variety Model

There is a fixed number L of people, each lives forever and has constant flow of labour.

His utility each period only depends on consumption.

And discount rate on utility is ρ. This means that in steady state the growth rate g and the interest rate r must obey the Euler equation, which can be written as

g=(r-ρ)/ε ………(2)

final output is produced under perfect competition using labour and a range of intermediate inputs in interval [0,Mt]. Mt is measure of product variety.

So, final goods production function is given as

where Yt is output, and each xi is the amount of intermediate product i used as input. Labour input is always equal to the fixed supply L. each unit of intermediate product i produced requires the input of one unit of final good.

Let, Xt be the total amount of final good used in producing intermediate products. So Xt equals to

Assume that each intermediate product is produced in the same amount x.

Then x = Xt / Mt, substituting this into production function we get,

The final good is used for consumption and investment. Its only other use is in producing intermediate products. So the economy’s gross domestic product (GDP) is final output Yt minus the amount used in intermediate production:

Each intermediate product is monopolized by the person who created it. The monopolist seeks to maximize the flow of profit at each date, measured in units of final good.

(Where p denotes the price in unit of final good.)

The price of an input to a perfectly competitive industry is the value of its marginal product, so we have

Substituting Xt =MtX into the production function we get that final good output and the economy’s GDP will both be proportional to the degree of product variety.

Therefore the growth rate of GDP is proportional to growth rate of product variety.

Product variety grows at a rate that depends on the amount Ri of final output that is used in research. So we have

(where λ is a (positive) parameter indicating the productivity of the research sector.)

As the research sector of the economy is perfectly competitive so there is zero profit. Each blueprint is worth π/r to its inventor, which is the present value of the profit flow π discounted at the market interest rate r. Hence the flow of profit in research is,

which is just the flow of revenue (output λRt time price π/r) minus cost Ri. For this to be zero we need a rate of interest that satisfies the “research-arbitrage equation”:

r = λπ ………..(15)

That is, the rate of interest must equal the flow of profit that an entrepreneur can receive per unit invested in research.

By putting the value of r from 15 in equn 2 we get.

g = (λπ- ρ)/ ε …………(16)

Substituting the value of π from equation 10 we get,

We see that growth increases with the productivity of research as measured by the parameter λ and with the size of the economy as measured by labour supply L, and decreases with the rate of time preference ρ.

Limitations and Conclusion

 In this model, growth is driven by innovations that lead to the introduction of new (input) varieties. Productivity growth is driven both by the increased specialization of labor that works with an increasing number of intermediate inputs and by the research spillovers whereby each new innovator benefits from the whole existing stock of innovations. Ideas are nonrival, which means they can be freely used by new innovators in their own research activities. And they are excludable in the sense that each new innovation is rewarded by monopoly rents. It is the prospect of these rents that motivates research activities aimed at discovering new varieties.

A limitation of this model is that it does not capture the role of exit or turnover in the growth process, which Schumpeter refers to as creative destruction. 

Reference: Aghion & Howitt.