Answers to Chapter 15 Exercises
Review and practice exercises
15.1. Perfect competition and innovation. “Perfect competition is not only impossible
but inferior, and has no title to being set up as a model of ideal efficiency.” Do you agree?
Why or why not?
Answer: In a static sense, perfect competition is the most efficient industry structure,
since it maximizes social welfare. In a dynamic sense, however, perfect competition is not
necessarily the ideal model, since it may be less conducive to technological progress than
situations that allow for some (temporary) degree of market power. The latter would not
necessarily be monopoly markets (in a static sense) but rather forms of oligopoly, in which
firms compete not only in quantity (or price) but also in R&D, so that they can outpower
rivals in the future and gain some market power.
15.2. Innovation and market structure. “R&D competition implies a dynamic system
whereby industries tend to become more and more concentrated.” Do you agree? Why or
why not?
Answer: If preemption motives are very strong, then the tendency is for technology leaders
to cement their leadership. However, if the replacement e↵ect is strong; or if organizational
inertial is very important; then laggards are more likely to innovate than leaders.
Challenging exercises
15.3. Cost reduction with Bertrand competition. Two firms are engaged in Bertrand
competition. There are 10,000 people in the population, each of whom is willing to pay at
most 10 for at most one unit of the good. Currently, both firms have a constant marginal
cost of 5.
(a) What is the equilibrium in this market? What are the firms’ profits?
Answer: Both firms charge p = 5 and earn ⇡ = 0.
(b) Suppose that one firm can adopt a new technology that lowers its
marginal cost to 3. What is the equilibrium now? How much would
this firm be willing to pay for this new technology?
Answer: The firm with the lower cost technology charges a fraction of a cent less than p = 5
and sells to all 10,000 customers. Its profits are ⇡ = [(5 3) · 10, 000] = 20, 000. It would
be willing to pay up to 20,000 for this technology.
(c) Suppose the new technology mentioned in (b) is available to both
firms. The cost to a firm of purchasing this technology is 10,000. The
game is now played in two stages. First, the firms simultaneously
decide whether to adopt the new technology or not. Then, in the
second stage, firms set prices simultaneously. Assume that each firm
knows whether or not its rival acquired the new technology when
choosing its prices. What is (are) the Nash equilibrium (equilibria) of
this game?
Answer: There are two pure-strategy equilibria: (1) firm 1 invests in the low-cost technology
and firm 2 does not, and (2) firm 2 invests in the low-cost technology and firm 1 does not.
It is not an equilibrium for both firms to invest or for neither firm to invest. (There is also
a mixed strategy equilibrium in which each firm invests with probability 0.5.)
15.4. Word-of-mouth di↵usion of innovations. Show that the model of di↵usion by wordof-mouth communication implies the adoption path given by (15.2). (Note: this problem is
quite challenging mathematically speaking.)
Answer: A new user adopts if and only if a user is matched with a non-user. This happens
with probability 2 xt (1 xt ). This implies that the growth in xt is proportional to the
product xt (1 xt ).
@ xt
/ xt (1 xt )
@t
where / means “proportional to.” Suppose that xt has the functional form:
xt =
where
1
1 + exp
(t
(15.3)
)
is a constant. Then
@ xt
@t
1
=
1 + exp
Since
1
I conclude that
xt =
@ xt
@t
(t
)
!2
exp(t
exp
(t
)
1 + exp
(t
)
= xt (1
2
xt )
)
which confirms (15.3) satisfies the desired property. Knowing that xt = x0 when t = t0 , I
know that
1
x0 =
1 + exp
(t0
)
which implies
1 + exp
(t0
exp
(t0
(t0
1
x0
1 x0
) =
x0
) = ln(1 x0 )
) =
ln(x0 )
and finally
= t0 + ln(1
x0 )
ln(x0 )
15.5. Strength and length of patent rights.27 Many standardization agreements require
that patent holders cross-license their patents on reasonable and non-discriminatory terms
(sometimes denoted by the acronyms RAND or FRAND). Suppose consumer demand for a
patented innovation is given by D(p). Let c be production cost and pM monopoly price.
Suppose that, if the patent is licensed to a competitor, then firms compete in prices (a la
Bertrand), knowing that the second firm’s cost includes the per-unit license f to pay the
patent holder. Finally, let the patent last for T periods.
(a) Show that, if f = pM c, then firm profits and consumer welfare are
the same with and without patent licensing.
Answer: Under this weak patent system, the monopolist is actually indi↵erent between
selling directly to the consumer or receiving license fees from rival competitors. Specifically,
with respect to the strong patent case, the patent holder loses profits corresponding to area
A and gains profits corresponding to area C. If the weak patent is just a little weaker than
the strong patent — that is, if the license fee is just a little lower than pM minus c —, then
areas A and C are approximately equal. In other words, the patent holder su↵ers very little
from a slight weakening of the patent.
(b) Show that lowering the license fee infinitesimally from f = pM
increases consumer welfare without decreasing patent value.
c
Answer: This is a consequence of the fact that profits are a concave function of price and
pM , the monopoly price, is the value that maximizes profits. In the neighborhood of the
optimal solution, the derivative of profit with respect to price is approximately equal to
zero: a slightly lower (or higher) price leads to an approximately equal profit.
(c) Show that, by decreasing the license fee and increasing patent length,
a new patent system can be obtained that provides the same reward
to patent holders and makes consumers strictly better o↵.
Answer: To summarize: in terms of post-innovation value, society has much more to gain
from a weakening of the patent system than the monopolist has to lose. This contrasts with
changing the duration of the patent: in terms of post-innovation value, what a monopolist
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has to gain from a longer patent is proportional to what society has to lose: double the
patent life and you double both the patent value and total allocative inefficiency. Putting
these facts together suggests that an optimal patent system should make patents very long
but very weak. This would seem to be the best compromise between making patents valuable
and minimizing the allocative inefficiency that they induce.
(d) What aspects of the innovation reward system may the above analysis
miss out (open question)?
Answer: One of the purposes of patents, in addition to giving incentives for innovation, is
to publicize the relevant information regarding the product or process in question. In fact,
it is a requirement for a patent that it should contain “a written description . . . in such
full, clear, concise and exact terms as to enable any person skilled in the art . . . to make
and use the same” (U.S. Patent Code, Section 112). Among other things, this allows the
information possessed by one party to be available to everyone else, thus accelerating the
aggregate rate of innovation.
The disclosure requirement may, however, create the wrong incentives for i̊nvestors.
Suppose that a laboratory has made a basic innovation, one that can be used as a basis
for several other derived innovations. For example, a technique for locating and purifying
human genes (basic innovation) may be used in the development of a series of medical drugs
(derived innovations). If patent protection is very weak — in the sense described above —,
then the inventor will have little incentive to patent. In fact, by keeping its invention secret,
it will have a better chance at winning the race to develop the derived innovations that flow
from the basic one. But, from a social point of view, such delay in making information
public may be very costly. In fact, it implies that only one firm — as opposed to several
firms — will be engaged in research directed at derived innovations.28 An optimal patent
system should trade o↵ this e↵ect against the allocative efficiency e↵ect considered above.
15.6. Patent thickets. Firm X produces a certain smartphone for which market demand
is given by Q = a p. Production cost consists of licensing n patents required to produce the
gadget. Each patent is owned by a di↵erent firm and all license fees fi are set simultaneously.
Given the values of fi (which we assume are per-unit fees), firm X sets the smartphone price
and consumers decide how much to pay.
(a) Show that, in equilibrium, each patent’s license fee is given by f =
a
n+1 .
P
Answer: Let c =
fi be the firm’s cost, that is, the sum of all license fees it must pay per
unit that it sells. From Section 3.2, we know that optimal price is given by (a + c)/2. This
implies that q = (a c)/2 smartphones will be sold. Therefore, patent i license revenues
are given by
⇣
X ⌘
a c
⇡i = f i
= 12 fi a
fk
2
k
Maximizing ⇡i with respect to fi implies
⇣
X ⌘
1
a
fk
2
k
4
1
2
fi = 0
By symmetry, fi = f and ⇡i = ⇡. This implies
1
2
(a
1
2
n f)
f =0
or thus
a
n+1
✓
◆2
1
a
⇡=
2 n+1
f=
Now suppose that the n patent holders form a pool and jointly set license fees.
(b) Determine the optimal license fees set by the pool.
Answer: Assuming that there is a license pool is equivalent to assuming that there is only
one license fee to pay, the license for the bundle of n patents. This corresponds to setting
n = 1 in the above equations, which yields
a
2
1 ⇣ a ⌘2
⇡p =
2 2
fp =
where the subscript p stands for “patent pool.”
(c) Show that patent holders, firm X and smartphone buyers are all better
o↵ if a patent pool is formed.
Answer: Each individual patent holder receives 1/n of the total license fee, that it,
⇡p
1
⇡=
=
n
2
✓
2
a
p
n
◆2
This is a greater number than under no patent pool if and only if
p
2 n<n+1
which is true for all n > 1. Firm X’s cost under no patent pool is given by
c = nf =
na
n+1
This is greater than fp , firm X’s cost when there is a patent pool, if and only if
n
1
>
n+1
2
which is true for all n > 1. Finally, since cost is lower under a patent, so is the final product
price, and so consumers are also better o↵.
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