# ECON 6555 Homework 4 Due Date: Tuesday, February 24 (at the beginning of class) 1. Suppose the demand curve for widgets is…

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ECON 6555 Homework 4

Due Date: Tuesday, February 24 (at the beginning of class)

1. Suppose the demand curve for widgets is given by p=100 âˆ’ 2q. 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) = q2, 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 âˆ’ 2q. 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 âˆ’ 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 mc1=$80,

what is its incentive (or additional profit) from developing a cost-saving process

innovation that reduces marginal cost to mc2=$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 mc1=$80, what is an individual firmâ€™s incentive to develop the same costsaving

process innovation (for which it obtains a patent to exclude other firms) that

reduces marginal cost to mc2=$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 Î²=0.07.

(a) Assuming a Central Source Model, calculate the number of adopters of the new

technology for t=0, 1, 2,â€¦, 30. Assume that the â€œcentral sourceâ€ is one of the 21 adopters

such that D(0)=1.

(b) Now assume an Epidemic Model with N=21, and Î²=0.07. Calculate the number of

adopters of the new technology for t=0, 1, 2,â€¦, 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 â€“ q. The patent

holderâ€™s total cost function is TC(q) = 500 + 40q.

(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!

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