# Proprietary technology and Demand for the technology

Problem Set 1
1. A firm has a proprietary technology and demand for the technology is given by ๐๐ท = 5 โ 4๐,
where ๐๐ท is the quantity demanded for the licenses of the technology per year and P is the price
of the licenses. Please answer the following questions.
a. To maximize profits, how much should the firm charge for each license of the
technology?
b. Calculate (per-year) consumer surplus, producer surplus and deadweight loss (if there is
any)
c. Suppose according to the intellectual property law, patents expire after 20 years. The
interest rate is 4%. Calculate the private value of the patent.
d. What is the total deadweight loss of the patent over its life?
e. What is the total consumer surplus generated by the patent over its life (including the
time after the patent expires).
f. What is the social value of the patent?
g. Someone suggests the government purchase the patent from the firm. How much does
the government need to pay the firm for the patent? What are the benefits of doing so?
h. What does Michael Kremer (1998) suggest the government do in this case?
2. Suppose the government announces a prize for a solution to its need. Both firm 1 and firm 2
have an idea for a technology to satisfy the governmentโs need. Firm 1โs idea is (๐ฃ1, ๐1
) =
(8,10) and firm 2โs idea is (๐ฃ2, ๐2
) = (7, 2), where ๐ฃ is the per-period consumer surplus of the
technology if competitively supplied, and ๐ is the cost of developing the idea into a technology.
Interest rate is 10%. The government wants to hold a Vickery auction to pick the best idea.
a. Calculate the social values of firm 1โs idea, ๐ 1, and firm 2โs idea, ๐ 2. Which firmโs idea is
better?
b. For a Vickery auction to work, what values should the government observe?
c. If you are the CEO of firm 1 and someone you trust tells you that firm 2 will report a
social value of \$55. What is your profit if you win the bid? If you lose the bid?
d. If you are the CEO of firm 1 and someone you trust tells you that firm 2 will report a
social value of \$85. What is your profit if you win the bid? If you lose the bid?
e. If you are the CEO of firm 1, what range of social values should you report regardless of
what firm 2 does, and why?
f. If you are the CEO of firm 2, what range of social values should you report regardless of
what firm 1 does, and why?

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