An R&D Model of growth

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.

pi p= 1/α>1 p=mc=1 xi x* Demand

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

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 .

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:

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.

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).

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

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).

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)

An R&D Model of growth