Marketing Exercises Part 7 Suppose That The Three Patent Holders Set Their License Fee Independently And

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that d

I= (1=9) (a2cI+c)2and d

E= (1=9) (a2c+cI)2. We can now

compute the incentive to acquire the patent for each firm. For the incumbent,

the value of the patent is given by the difference m

3. The previous result can be seen as the evidence of strategic patenting. It may

Exercise 8 The patent thicket3

Suppose that Nfirms, i= 1; : : : ; N, each own a patent that is essential to

the production of a given product. For simplicity, we assume that there is a

competitive industry that produces this product, buying and assembling the

necessary components from each of these Nfirms. Firm isets a license fee for

the use of its patent by the competitive assembly industry. The cost to firm i

per unit (for licensing its patent to assemblers) is denoted by ci. For simplicity,

assume that ci= 0 8i. The license fee charged by firm iis denoted by pi. The

price of the final product is denoted by p. We assume that the assembly firms

incur no other assembly cost in addition to paying royalties. Competition in

the assembly industry therefore ensures that p=PN

i=1 pi. Demand for the final

product is given by q=ap.

1. Suppose that the Npatent holders set their license fee independently and

non cooperatively. Derive the Nash equilibrium in prices, compute (i) the

total license fee for the Npatents (denote it by ps) and (ii) the profit of

each patent holder at equilibrium (denote it by s).

2. Suppose now that all firms form a patent pool and choose a common

license fee pto maximize their joint profit. Assembly firms that pay p

have access to the whole pool of patents. Derive the optimal p. Supposing

that the pool’s profit is equally distributed among the Nfirms, compute

each individual firm’s profit (denote it by p).

3. Comparing your answers to the previous two questions, show that the

formation of the pool (i) reduces the total license fee (p<ps) and (ii)

increases each patent holder’s profit (p> s). Discuss.

4. Consider that kfirms, with 1< k < N , form a patent pool and coordinate

their pricing decision, while the other Nkfirms continue to set their fee

independently.

(a) Derive the equilibrium profit of each firm, independent and pool

member, at equilibrium.

(b) Suppose that N3and show that if the pool has to be formed

through simultaneous individual decisions, no pool is formed, i.e.,

it is a Nash equilibrium for all firms to set their fee independently.

Discuss.

Solutions to Exercise 8

1. Firm i’s maximization program is maxpipi(apiPi), where Pidenotes

the sum of the license fees of all firms but i. The first-order condition yields the

reaction function pi(Pi) = 1

2(aPi). As firms are symmetric, they all set

2. The pool’s maximization program is maxpp(ap). From the first-order con–

dition, it is easily found that the optimal price is

3. Comparing total license fee, we observe that

psp=Na

N+ 1 a

2=(N1) a

2 (N+ 1) >0:

The total license fee is larger when firms set their fees independently. The

4. Pool formation

(a) There are now Nk+1 independent price setters (the Nkindependent

firms and the pool). Repeating the analysis of question 1, we easily find

(b) For a pool of kfirms to be stable, it must be that no pool member has an

incentive to unilaterally leave the pool, which is equivalent to

Exercise 9 Formation of patent pools [included in 2nd edition of the book]

Suppose that three firms (noted i= 1;2;3) each own a patent that is essential

to the production of a given final product. For simplicity, we assume that there is

a competitive industry that produces this final product, buying and assembling

the necessary components from each of these three firms. We assume that the

assembly firms incur no other assembly cost in addition to (i) paying royalties

for the use of the three essential patents, (ii) incurring transaction costs when

inquiring about the license fees. Regarding the latter transaction costs, it is

assumed that they are inversely related to the number of different license fees

that are set by the patent holders. Patent holders have indeed the possibility to

form so-called “patent pools” whereby they coordinate their decisions to set a

unique license fee that allows assembly firms to access two (or three, if all firms

join) patents at once. This is modeled as follows.

The price of the final product is denoted by p. Demand for the final

product is given by q=ap.

Competition in the assembly industry therefore ensures that pbe equal to

the the marginal cost of assembly, which depends on the patent pool that

patent holders may have formed. In particular, three options are possible:

p=8

r1+r2+r3+ 3; if no pool is formed,

rij +rk+ 2; if firms iand jform a pool,

rp+; if the three firms form a pool,

where 0 < a=3is the cost per transaction, riis the license fee set by

firm ifor accessing its patent, rij (resp., rp) is the common fee set by the

pool formed by firms iand j(resp., all firms) for accessing the bundle of

patents iand j(resp., all patents).

1. Suppose that the three patent holders set their license fee independently

and non cooperatively. Derive the Nash equilibrium in fees, compute (i)

the price of the final product (denote it ps) and (ii) the profit of each

patent holder at equilibrium (denote it s).

2. Repeat the previous analysis by assuming that firm iand jcoordinate

their decisions to set a common fee rij that has to be paid for acquiring

the right to use patents iand j. Firm k, on the other hand, still acts

separately and sets its license fee rk. Compute again the price of the

final product (denote it p2), as well as the equilibrium profits of the firms.

Denote the profit of firms iand j in (supposing that they divide their

joint profit equally) and the profit of firm k,out.

3. Suppose now that the three firms form a patent pool and choose a common

license fee rpto maximize their joint profit. Assembly firms that pay rp

have access to the whole set of patents. Derive the optimal rp. Supposing

that the pool’s profit is equally distributed among the three firms, compute

each individual firm’s profit (denote it by p). Compute also the price of

the final product (denote it pp).

4. Comparing your answers to the previous three questions, show that, even

in the absence of transaction costs (= 0), the more there are patents

in the pool, the smaller the price of the final product, and the larger the

sum of the patent holders’ profits. Discuss the economic intuition behind

these results.

5. Consider now the following pool formation game. Before they set the

license fees, patent holders simultaneously decide whether to join a patent

pool or not (only one pool can form). Set a= 120 and characterize the

Nash equilibrium of this game for all admissible values of the transaction

cost (i.e., 0 < 40). Discuss the economic intuition behind your

results.

Solutions to Exercise 9

1. Firm i’s maximization program is maxriri(ariri3), where ride-

notes the sum of the license fees of all firms but i. The first-order condition yields

the reaction function ri(ri) = 1

2. The pool formed by firms iand jand firm khave, respectively, the following

maximization programs are maxrij rij (a2rij rk)and maxrkrk(a2rij rk).

The first-order conditions yield the following reaction functions: rij (rk) =

3. The pool’s maximization program is maxrprp(arp). From the first-order

condition, it is easily found that the optimal price is rp= (1=2) (a). It

4. Comparing prices of the final product and total profits, we easily conclude:

pp=1

2(a+)< p2=2

3(a+)< ps=3

4(a+);

5. All three firms join the pool iff pout, which is equivalent to

12 (120 )21

9(120 2)2,1321200+ 14400 0

Exercise 10 Patent pools and mergers

Consider a vertical market structure with 2 upstream firms (A and B) and

2 downstream firms (a and b). The downstream firms require the input of each

of the upstream firms, who demand linear royalties (rA

a,rA

b,rB

a,rB

b) charged

for each unit the respective downstream firm sells. Downstream firms face the

inverse demand P(q) = abq; where q=qa+qb. Assume that the royalties

accruing to the upstream firms are the only costs that the downstream firms face

(i.e., downstream marginal costs are ca=cb= 0) and all costs of the upstream

firms are sunk.

1. Brie‡y explain in one or two sentences why perfectly complementary

upstream products are a good way to describe a set of complementary

patents.

2. Solve for the symmetric subgame-perfect Nash equilibrium in which the

upstream firms set non-discriminatory royalties (rI

i=rI

jrI) in the first

stage and downstream firms engage in Cournot-competition in the second

stage.

3. Now assume that firms Aand amerge (vertical merger) and maximize

joint profits. Solve for a subgame-perfect Nash equilibrium in which the

upstream firms set (non-discriminatory) royalties in the first stage and

downstream firms engage in Cournot-competition in the second stage.

How does the merger affect total royalties charged and quantities sold?

4. Starting from the original (separate) setup, now assume that firms A

and Bmerge (horizontal merger) and maximize joint profits. Solve for

a subgame-perfect Nash equilibrium in which the upstream firm(s) set

nondiscriminatory royalties in the first stage and downstream firms en-

gage in Cournot-competition in the second stage. How does the merger

affect total royalties charged and quantities sold?

Solutions to Exercise 10

1. If producing a certain product requires the use of multiple patented ideas, then

the producer needs to obtain licenses from each of the patentees. In most cases,

2. The downstream interactions are of standard Cournot-type. Downstream firms

individually maximize. In particular, a=qa(ab(qa+qb)rArB). One

obtains the reaction functions

3. Denoting the profits of the merged firm as A, we get the following profit func–

tions (focusing on the internal solution; in the alternative possible solution A

sets rAso high to price bout of the market, and we would have a downstream

and an upstream monopoly): B=rBq,A=rAqb+qa(abq rB)and

b=qb(abq rArB). While rAdoes not affect the output decision of

3b. Total production levels are q=qa+qb=2arA2rB

3b.

Anticipating this, upstream firm Bmaximizes B=rB(2arA2rB

3b)which

yields the reaction function r

B(rA) = (2arA)=4. Firm Aanticipates down–

4. Now there is one upstream monopolist charging r. The downstream Cournot

equilibrium gives us the quantities qi;j =ar

Exercise 11 Optimal copyright length

“The Adelphi Charter on Creativity, Innovation and Intellectual Property

responds to one of the most profound challenges of the 21st century: How to

ensure that everyone has access to ideas and knowledge, and that intellectual

property laws do not become too restrictive. The Charter sets out new prin-

ciples for copyrights and patents, and calls on governments to apply a new

public interest test. It promotes a new, fair, user-friendly and efficient way of

handing out intellectual property rights in the 21st century. The Charter has

been written by an international group of artists, scientists, lawyers, politicians,

economists, academics and business experts.”

Here is what is written in the Adelphi Charter about the length of IP pro-

tection: “The ‘term’ – the period over which a copyright or patent is effective

– has always been limited – and for very good reasons. Now, however, there is

a disturbing trend towards lengthening terms, particularly in the area of copy-

right. [. . . ] The earliest ‘term’ of copyright, given as a right to authors, was

14 years, with the option of extending for an extra 14 years. The length of

some jurisdictions, there is now pressure to lengthen terms to 90 or 120 years,

and a continual pressure for all governments to follow suit and ‘level-up’. There

are a number of problems associated with the lengthening copyright terms.”

(www.adelphicharter.org)

1. Set out the economic problem that a regulator must solve when choos-

2. Do you concur with the authors of the Adelphi Charter when they claim

that “there are a number of problems associated with the lengthening

your opinion.

Solutions to Exercise 11

Exercise 12 “Pay-for-delay” and generic drugs

Under the “pay-for-delay” deal the patent holder of a drug pays a maker of

generic drugs to delay its launch of a cheap copy.

2. Model the market for drugs as a homogeneous Cournot market with lin-

ear demand (it would be straightforward to include asymmetries between

3. Some branded drug makers have decided to offer lower-priced “authorized”

generic version shortly before the patent expires. Note that the first (non-

4. Some generic drug makers now agree to delay their launch of their generic

Exercise 13 Licensing of process innovation under horizontal product differ-

entiation

Consider the Hotelling model with linear transport costs where two firms, 1

and 2, are located at the extreme points of the unit interval. There is a unit

mass of consumers who are uniformly distributed over this interval. A consumer

located at x2[0;1] has utility r x p1if she buys one unit of product 1,

or r(1 x)p2if she buys one unit of product 2. Consumers have unit

demands; we assume that ris large enough so that each consumer buys one or

the other product.

1. Initially, both firms produce at a constant marginal cost c0. Compute the

profit of the two firms at the Nash equilibrium of the pricing game.

2. Suppose that firm 1 has found a process innovation that decreases its cost

from c0to c0k, with k < c0and k < 3. Compute the the profit of the

two firms at the Nash equilibrium of the pricing game if firm 1 is the sole

user of the innovation.

3. Firm 1 could sell a license on its innovation to firm 2. Show that fixed-fee

licensing is not profitable but that royalty licensing is. Give the economic

intuition behind your result.

Solutions to Exercise 13

1. The indifferent consumer is identified by ^xsuch that r^xp1=r(1 ^x)

p2, or ^x= 1=2 + (p2p1)=(2). The mass of consumers with x^x(resp.

2. Now, firm 1 chooses p1to maximize 1= (p1c0+k) ^x, while firm 2 chooses

p2to maximize 2= (p2c0) (1 ^x). From the F.O.C. for profit maximiza–

tion, we find the two firms’ reaction function as p1= (c0k++p2)=2

2+k

32

2k

32

3. Under fixed-fee licensing, both firms operate at the same cost c0k. The

equilibrium of the pricing game is the same as in Question 1 (which means that

the cost reduction is entirely passed on to the consumers): both firms achieve

Consider now royalty licensing. Firm 1 sets the maximum royalty that is accept-

able for firm 2: r=k. Hence, firm 2’s cost is equal to c0k+r=c0. The