An R&D Model of growth

Xavier Sala-i-Martin

Columbia University

Demand for new products

The Demand for a potential product to be invented (let’s call it product xi) is:

where Y represents the income of the customers (the size of the market), and pit is the price of good i at time t.

)1/(1 itit pYx

Demand

p=mc=1

p= 1/α>1

pi

xi

x*

R&D Firms

Two Step Decision: Should we invent in R&D?

Answer if R&D cost > PV(future profits), then no. Otherwise, yes.

Once I have the invention, what price will I be able to charge? Depends of the intellectual property right structure If perpetual patent, then you can charge “monopoly

prices” forever.

Solve backwards: first, step 2 Solve backwards: Start with Step 2: Assume you already have invented and you are

granted the monopoly, what price? Monopoly pricing: choose price so as to maximize profits.

Profits are equal to price minus marginal cost times quantity sold, and quantity sold is given by the demand function above

Using depand function above in profit function we get

itit xmcp )(

)1/(1)( itit pYmcp

Step 2

Take derivatives of profit and equalize to zero and get:

That is, price is a constant markup over the marginal cost. Notice that since α<1 the price is above marginal cost .

mc

pit

Step 2

Notice also that the quantity demanded in this case is

which is less than we would sell if price were to be equal to marginal cost,

Notice that the yearly profit is given by

The PDV of all future profits is:

)1/(1)1/(1 /ˆ mcYx

Yx *

)1/()1/(111

mcY

...

111 33

221

rrrV

Step 1: Should we invent?

Notice that we know that if we invent, the value of our firm (the value of all future profits is given by V.

The key question is: what are the COSTS of R&D?

Assume they are the constant amount of cookies given by η (which is constant).

Decision is, therefore: Do not invest in R&D if V< η Invest otherwise

Free Entry

Finally, assume there is free entry into the business of R&D. Free entry will make sure that V= η

Equilibrium in Financial Sector

Also, equilibrium in the asset market will make sure that the rate of return to bonds is equal to the rate of return to investment in R&D. The latter is given by profits (dividends) plus capital gains

Since V= η and η is constant, so r=π/ η.

V

Vr

0V

Growth

Thus, the Rate of Return in our economy Therefore, the growth rate of the economy is

given by the RATIO of profits to R&D costs).

)1/()1/(111

1Ymc

Growth is positive only if price is larger than marginal cost: profits need to be guaranteed

Marginal cost affects growth negatively (efficiency in production is good)

Growth is affected negatively by larger R&D costs:

R&D Costs should be understood broadly to include costs of setting up business, bureaucracy, corruption costs, entrepreneurial spirit, education system, etc

)1/()1/(111

1Ymc

Growth is less than optimal (optimal x is the one that we would have if price were equal to marginal cost and actual x is less because we have monopoly pricing). Thus, we have a DISTORTION from the granting of monopoly rights to inventors.

Growth is positively related to the SIZE of the market (scale effects).

)1/()1/(111

1Ymc

Policies

R&D policy would get the right growth rate, but notice that would not get the right quantity (if we subsidize R&D but keep p>MC, the quantity sold will still be too small –See Figure 1 above).

The correct policy is to SUBSIDIZE the purchases of x:

R&D firms receive p=1/α>1 Customers pay p=mc=1. The difference is financed by a public transfer.

Policies

Notice that R&D subsidies could actually be BAD if: R&D costs decrease with number of inventions There is obsolescence (quality ladders and

creative destruction)