Download 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:
1/(1 )
it
it
where Y represents the income of the
customers (the size of the market), and pit is
the price of good i at time t.
x Y p

Demand
pi
p= 1/α>1
p=mc=1
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
  ( pit  mc) xit

Using depand function above in profit function we get
1/(1 )
it
  ( pit  mc)Y p
Step 2

Take derivatives of profit and equalize to zero
and get:
pit 

mc

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
xˆ  Y 1/(1 ) / mc1/(1 )
which is less than we would sell if price were
to be equal to marginal cost, x*  Y
Notice that the yearly profit is given by
 1  1/(1 )
    1
Y mc /(1 )
 

The PDV of all future profits is:
V
1
1 r

2
1  r 
2

3
1  r 
3
 ...
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
r

  V
V
Since V= η and η is constant, V  0 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).
  1  1/(1 )



/(
1


)
   1

Ymc
1   

 









  1  1/(1 )

 /(1 )
   1

Ymc
1   

 









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/(1 )



/(
1


)
   1

Ymc
1   

 











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)