**ECON**** 6555 Homework
4**

1.
Suppose the demand curve for
widgets is given by *p*=100
− 2*q*. One firm owns the
patent on the widget, but licenses
its patent
to two
manufacturers (and does not produce
any widgets itself). Assume each manufacturer
has a
total cost curve given by
TC(*q*)
= *q*2, and there are
no fixed
costs*.*If
the two
licensees compete by choosing quantities
(a la
Cournot), what royalty rate should
the patent
holder set? What fixed fee
should it charge each licensee?

2.
Assume the demand curve for
widgets is given by *p*=100
− 2*q*. Assume that there are
two firms that each owns a
patent on perfectly substitutable widgets. Each has a
constant marginal cost per
widget of $20, and no
fixed costs.

(a) If the two firms compete by choosing quantities, what will their profits be?

(b) If the two firms enter an illegal cross-licensing agreement to share their patents, what common royalty rate should they charge each other to maximize their profits?

(c) Illustrate the two outcomes from (a) and (b) on the same market demand curve.

3.
Suppose the demand curve for
widgets is given by *p*=100
− *q *where
*p
*is the price, and *q *is
the quantity.

(a) If the
market is served by a
single monopolist with constant marginal
cost of
*mc*1=$80,
what is
its incentive
(or additional
profit) from developing a cost-saving
process innovation that reduces
marginal cost to *mc*2=$20? Be sure to
include a diagram to illustrate your answer.

(b) If
the market
is competitive, and firms sell
widgets at a price equal
to constant
marginal cost *mc*1=$80, what is an individual firm’s
incentive to develop the same
cost- saving process innovation (for which
it obtains
a patent
to exclude
other firms) that reduces marginal cost
to *mc*2=$20? Be sure
to include
a separate
diagram to illustrate your answer.

4.
Suppose the number of potential
adopters of a new technology
is *N*=21,
and *β*=0.07. (a) Assuming a Central
Source Model, calculate the number
of adopters
of the
new technology for *t*=0, 1, 2,…,
30. Assume
that the
“central source” is one of
the 21
adopters such that *D*(0)=1.

(b) Now
assume an Epidemic Model with
*N*=21, and
*β*=0.07. Calculate
the number
of adopters of the new technology
for *t*=0, 1, 2,…, 30.
Assume that *D*(0)=1.

(c) Graph the two adoption series on the same chart. Which model predicts faster adoption

of the technology? Why?

5.
Suppose the demand curve for
a new
technology is given by *p*=100
– *q*.
The patent
holder’s total cost function
is TC(*q*) = 500 + 40*q*.

(a) What are profits if the firm chooses the profit-maximizing price?

(b) What
are profits
if the
firm chooses
a penetration
price equal to marginal cost?
(c) What are profits
if the
firm chooses
an extreme
penetration price equal to zero? *Be** sure to include a diagram
to aid
your answers!*