Answer Key

EconS425 - Homework #4 (Due on April 8th, 2015)
1. Exercise 2 from Chapter 9 in Shy (page 249)
Answer
a) The expected profit of a firm engaged in R&D, given that the other two firms are also
engaged in R&D, is the prize times the sum of the probability that the firm discovers
while the other two do not, plus twice the probability that it discovers while one
competing firm discovers and one does not, plus the probability that all the three
discover simultaneously, minus innovation cost. Formally, for each firm i, i  1, 2,3
111
1 1 1V
1 1 1V
7V
V
(2) 
I 
I
222
222 2
222 3
24
Thus, all the three firms find it profitable to engage in R&D if V  24I / 7  24 / 7 (since
I  1).
b) Now the single firm can operate zero, one, or two labs.
E i 
Operating a single lab: In this case, E 1 lab  V / 2  I
Operating two labs: In this case, the probability of discovery is one minus the
probability that none of the labs discovers. Thus,
11
4V
E 2 labs  (1 
)V  2 I 
 2I
22
3
E 2 labs  E 1 lab V  4I
Now
if
which constitutes a sufficient condition for having the
firm choosing to operate two labs.
2. A monopoly with constant marginal costs m = $10 faces the inverse demand curve p =
50 – Q. The interest rate is r = 10%. An inventor discovers a way to reduce marginal
costs to m = $6 (with no additional fixed costs) and receives a permanent patent for that
invention. Up to how much is the monopoly prepared to pay for a permanent license to
use the invention
(i) if it is certain that the inventor will not offer the invention to any other firm?
(ii) if it knows that the inventor may offer the invention to a potential entrant instead?
Answer
(i)
If the monopoly is certain that the inventor will not offer the invention to any other
firm, it will maximally pay the present value of the difference between its future
monopoly profits with and without the invention, or
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Instructor: Ana Espinola
(ii)
If the monopoly knows that the inventor may offer the invention to a potential
entrant instead, it will maximally pay the present value of the difference between its
future monopoly profits with the invention and its duopoly profits if a rival firm
obtains the license and enters its market, or
The potential entrant will maximally pay the present value of its future duopoly profits, or
3.
A government wants to stimulate research and development and must choose between two
policies to do so. The first one consists of increasing the length of the patent from 17 to 18
years, while the relevant interest rate is projected to stay at 7%. The second policy would leave
patent length at 17 years, but reduce the interest rate from 7% to 6%. Say the inverse annual
demand for the yet-to-be invented new good is p = 100 – 2Q, is constant over the years, and
marginal cost is constant at 20. Which policy is more effective? Do you need all the information
above to answer that question?
Answer
In a simple model, we can normalize the monopoly profits at $1 per year for the length of the
patent. (Other than the assumption that profits are constant over the years, the information
about marginal cost and demand is not required to answer this question.) The value to the
patent holder of a one-year extension is thus the present value of $1 during the eighteenth year.
The value of that dollar is PV(18th year for 1.00)=1.00×(1/1.07)18 ≅ 0.296
The value to the patent holder of a reduction in interest rate (making the present value of future
profits larger) is the difference between two streams of payments. We can calculate the present
value of annuities of $1 during 17 years by subtracting the value of a perpetual annuity starting
in 18 years from one starting today:
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Instructor: Ana Espinola
The difference between the two is 0.769, larger than 0.296, so a reduction in the interest rate is
more valuable to the patent holder (more effective as an incentive to innovation) than a oneyear extension of the patent.
4. An increase in R&D induced by, say, government subsidies could result in a higher rate of
product obsolescence, with new products replacing older ones at a faster rate. How does this
“side-effect” of the subsidies affect firms’ incentives to invest in research? (Hint: distinguish
between firms with and without market power.)
Answer
A higher level of R&D by potentially rival firms reduces the expected value of existing patents, by
increasing the likelihood that newly-developed products will become obsolete before their
patents run out. Competitive firms therefore have less incentive to invest in R&D with the
purpose of obtaining market power. Monopolistic firms whose market power is based solely on
an existing patent, on the other hand, may have more incentive to invest in R&D, so as to
maintain their innovative lead over potential rivals.
5. Considers a monopolist with the following demand curve : P=390-2Q. The monopolist faces
MCM=ACM=30.
a. Solve for the profit-maximizing level of monopoly output, price, and profits.
Q=90; P=210 and profits=16,200
b. Suppose a potential entrant is considering entering, but the monopolist has a cost
advantage. The potential entrant faces costs MCPE=ACPE=40. Assuming the monopolist
continues to profit-maximize, solve the residual demand curve for the entrant.
P=210-qpe
c. Assume the potential entrant follows the Cournot assumption about the monopolist’s
output. Solve for the potential entrant’s output, price, and profits in this scenario. What
are the new monopoly profits?
qpe=42.5; P=125; profits=3,612.5; profits to monopolist = 8,550
d. Is there a price the monopolist could charge to deter entry? Solve for the limit price and
output that will completely deter entry. What is monopoly profit at this point?
P=40; q=175 and profits = 1,750
6. Consider the problem described in 5.
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a. Make an extensive form of the game based on your solutions in problem 5
b. Solve for the Nash equilibrium of the game. If the monopolist threatens ahead of time
to limit price, is the threat credible? Explain
Answer
a. The numbers for the figure are from the solution to Problem 5
b. The Nash equilibrium is that in each case, the monopolist would produce q = 90.
Given that the entrant knows this, the entrant will enter and earn 3,612.5 in profits,
rather than stay out and earn 0 in profits. The threat to limit price is not credible.
(Note that the possibility of forgoing current profits for future gain is not built into
this game since it is played only once.)
c. If the potential entrant trembled and stayed out, the monopolist would choose to
produce 90 (not limit price) and earn $16,200 in profits. A limit-pricing policy would
not be reasonable, because it reduces the monopolist’s profits.
7. Let us analyze an entry-deterrence game in the automobile market where an incumbent (firm 1)
operates as a monopolist during the first period and a potential entrant (firm 2) must decide
whether or not to enter to this market in the second period. The incumbent’s costs are privately
know by the incumbent, they can be low (c1=0) or high (c1=2). The potential entrant does not
exactly know the cost structure of the incumbent, but it knows the probability distribution of
cost functions. Firm 2 know that the unit cost of firm 1 satisfies:
h
h
Finally, the market demand function in each period is given by p=10-Q, where Q is the aggregate
amount sold to consumers, firm 2’s unit-production cost is c2=1 and firm 2 has to pay an entry
cost of F2=9 if it enters at t=2.
a. Draw the two-period, signaling, entry-deterrence game
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Instructor: Ana Espinola
b. Solve the game assuming a high-cost incumbent: obtain firm 2’s expected profit. If entry
occurs in t=2, what is firm 1’s best strategy? (calculate firm 1’s profits and output level
in period 1)
c. Solve the game assuming a low-cost incumbent: obtain firm 2’s profit. Discuss the signal
firm 1 uses to deter entry.
Answer
a. See figure 8.10 (page 204) in Shy’s textbook.
b.
Assuming High-Cost incumbent:

First, identify firm 2’s profits when it enters and the incumbent is High-Cost:
We know that
=
and
=
≅
Hence:

Second, identify firm 2’s profits when it enters and the incumbent is Low-Cost:
We know that
Hence:
=
and
=
≅
Firm 2 maximizes expected profit,
Hence, firm 2 enters into the automobile market.
Therefore, given that entry occurs and the firms play Cournot in t=2, the best firm 1 could do is to
maximize the first-period profit by playing the monopoly’s output in t=1. We know that
=
Hence its profit is:
and
. Therefore, firm 1’s profits in both periods is
c. If firm 1 is low-cost, then firm 2’s profits is:
. But since firm 2 does not know for
sure that firm 1 is a low-cost one, the incumbent has the incentive to reveal it to firm 2. Hence firm 1
has to select an output level that can only be produced by a low-cost firm. The high-cost incumbent
obtains negative profits when it produces such a quantity. Therefore, the high-cost firm would never
produce this quantity. For instance, the low-cost incumbent has to produce something above its
monopoly output level (q>qM, that is q>5).
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Instructor: Ana Espinola