# ECON 6555 Homework 4

ECON 6555 Homework 4

1.  Suppose the demand curve for widgets is given by p=100 − 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 − 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 − 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 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 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 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 + 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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