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University of ZurichDepartment of Business AdministrationChair of EntrepreneurshipPlattenstrasse 14, CH-8032 Zurichwww.business.uzh.ch
Exercises in Innovation Economics
Daniel Halbheer and Michael Ribers
Part I: Daniel Halbheer∗
Problem Sets 1–6
Session Dates: 27.2., 6.3., 13.3., 20.3, 27.3, and 10.4.
Part II: Michael Ribers∗∗
Problem Sets 7–121
Session Dates: 17.4., 24.4., 8.5., 15.5., 22.5., and 29.5.
∗My office hours are on Tuesdays, 5-7 p.m., and by appointment. Email: [email protected].∗∗My office hours are on Tuesdays, 5-7 p.m., and by appointment. Email: [email protected] are based on problem sets prepared by Susan Mendez.
List of Exercises
Problem Set 1: A Primer in Oligopoly 41.1 Cournot Competition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41.2 Investment in the Cournot Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Bertrand Competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Problem Set 2: Innovation and Market Structure 72.1 Monopoly (Tirole, 1988) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Competition (Tirole, 1988) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Oligopoly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11
Problem Set 3: Auctions and Contests 123.1 The Vickrey Auction (Mas-Collel et al., 1995) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Vickrey Auctions to Choose among Ideas (Scotchmer, 2004) . . . . . . . . . . . . . . . . . . . . . . . 133.3 Prototype Contests to Choose among Ideas (Scotchmer, 2004) . . . . . . . . . . . . . . . . . . . . . 14
Problem Set 4: Patents I 164.1 Optimal Patent Length (Tirole, 1988) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2 Pooling of Complementary Patents (Motta, 2004) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Problem Set 5: Patent Races 205.1 Research Intensity and Market Structure (Tirole, 1988) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.2 Patent Value and the Number of R&D Attempts (Scotchmer, 2004). . . . . . . . . . . . . . . . 22
Problem Set 6: Research Joint Ventures 246.1 Cooperative R&D (d’Aspremont and Jacquemin, 1988) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Problem Set 7: Patents II 277.1 Patent Breadth and the Ratio Test (Scotchmer, 2004) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.2 Optimal Patent Length and Breadth (Gilbert and Shapiro, 1990) . . . . . . . . . . . . . . . . . . . 30
Problem Set 8: Licensing 338.1 Licensing Basic Research (Scotchmer, 2004) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.2 Anti–Competitive Cross–Licensing (Motta, 2004) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Problem Set 9: Licenses and Litigation 389.1 Profit Neutrality in Licensing (Maurer and Scotchmer, 2004) . . . . . . . . . . . . . . . . . . . . . . . 389.2 Litigation (Scotchmer, 2004) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Problem Set 10: Private-Public Partnership 4210.1 Private-Public Incentives (Scotchmer, 2004). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4210.2 The Government Grant Process (Scotchmer, 2004) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Problem Set 11: Strategic Patenting, Patent Pools and Mergers 4611.1 Strategic Patenting (Belleflamme and Peitz, 2010) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
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11.2 Patent Pools and Mergers (Belleflamme and Peitz, 2010) . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Problem Set 12: Diffusion 5112.1 Diffusion (Tirole, 1988) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5112.2 Open Source Software (Belleflamme and Peitz, 2010). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3
Problem Set 1: A Primer in Oligopoly
Exercise 1.1: Cournot CompetitionTwo firms produce homogenous goods and compete in quantities. The industry’s (inverse)demand function is P (Q) = a− bQ, where Q is total industry output, equal to q1 + q2. Thecost function for each firm is given below:
C1(q1) = F1 + c1q1
C2(q2) = F2 + c2q2.
(a) Assuming that F1 and F2 are small, compute the Nash equilibrium.
Firm i solvesmaxqi
πi = P (qi + qj) · qi − Ci(qi).
First-order condition:
∂πi∂qi
= 0 ⇒ a− 2bqi − bqj − ci = 0.
Firm i’s reaction function is therefore given by
Ri(qj) = qi =1
2b(a− bqj − ci).
We thus have two equations and two unknowns. Solving yields
q1 =1
3b(a− 2c1 + c2) and q2 =
1
3b(a− 2c2 + c1).
(b) If c1 > c2, which firm produces more? Does this depend on the fixed costs?
The quantity produced by a firm falls as its own cost increase and rises asits rival’s cost increase. If c1 > c2, firm 2 will thus produce more. Theproduction decision does not depend on F1 and F2 (these are paid regardlessof the output level and are not influenced by a rival’s production decision).If these fixed costs were sufficiently large, however, they might influence thefirms’ decisions to produce anything at all.
(c) Graph the reaction functions for c1 = c2. Suppose c1 rises. Graphically show the effectin the firms’ reaction functions and the equilibrium quantities.
4
Exercise 1.2: Investment in the Cournot ModelConsider two firms that compete in quantities. The (inverse) demand function is given byP (Q) = 3 − Q, where Q = q1 + q2. Assume that firm 1 makes an observable investmentdecision before the firms set quantities. If firm 1 decides not to invest, it pays nothing andincurs a marginal cost of 1. If firm 1 decides to invest, it pays F > 0 and incurs a marginalcost of 0. In any event, firm 2’s marginal cost is 1.
(a) Compute the equilibrium when (i) firm 1 does not invest and (ii) firm 1 invests. Whatis firm 1’s profit in each case?
If firm 1 does not invest, both firms solve
maxqi
(3− qi − qj − 1) qi.
In equilibrium,
q1 = q2 =2
3, p =
5
3, π1 = π2 =
4
9.
If firm 1 does invest, firm 1 solves
maxq1
(3− q1 − q2) q1 − F ,
while firm 2 solvesmaxq2
(3− q1 − q2 − 1) q2
as before. In equilibrium,
q1 =4
3, q2 =
1
3, p =
4
3, π1 =
16
9− F , π2 =
1
9.
(b) Given your answer in part (a), when will firm 1 invest?
Comparing firm 1’s profit in the two cases, firm 1 will invest if and only if
16
9− F >
4
9⇒ F <
12
9.
(c) How does the investment decision affect firm 2’s output and profit levels?
In Cournot games, quantities are “strategic substitutes”: if one firm increasesits output, the other firm wishes to decrease its output. By investing, firm 1reduces its production costs and wishes to produce more (due to the higher“markup”); firm 2 thus wishes to produce less, which reduces its profit.
5
Exercise 1.3: Bertrand CompetitionTwo firms produce heterogeneous goods and compete in prices. The demand and cost functionsfor each firm are as follows:
q1 = 2− 2p1 + p2 and C1(q1) = q1 + q21;
q2 = 2− 2p2 + p1 and C2(q2) = αq2 + q22 .
(a) Calculate each firm’s reaction function.
Firm 1 solves
maxp1
π1 = p1 · q1(p1, p2)− C1(q1(p1, p2)).
First-order condition:
∂πi∂pi
= 0 ⇒ R1(p2) = p1 = 1 +5
12p2.
By a similar calculation, one may obtain firm 2’s reaction function
R2(p1) = p2 =5 + α
6+
5
12p1.
(b) Given α = 1, calculate the equilibrium prices.
Given α = 1, R2(p1) = 1+ (5/12)p1. To determine the equilibrium, solve forthe intersection of the reaction functions:
p1 = 1 +5
12
(1 +
5
12p1
),
whence follows p1 = p2 = 12/7.
(c) Given α = 1, graph the reaction functions.
(d) Suppose α rises. Graphically show the effect on the firms’ reaction functions and theequilibrium prices.
If α rises, firm 2’s reaction function shifts rightwards. As a result of theincrease in firm 2’s cost, both prices rise (in the Bertrand model, prices are“strategic complements”).
(e) What are the Bertrand-Nash equilibrium prices in a market for a homogenous good?Discuss the intuition. What happens if c1 > c2?
If the firms are symmetric, prices pi are set equal to marginal cost c, and eachfirms makes zero profit. This result is known as the “Bertrand Paradox”. Inthis case pi = c1. Note that firm 2 makes a positive profit.
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Problem Set 2: Innovation and Market Structure
Exercise 2.1: Monopoly (Tirole, 1988)Consider a process innovation to understand the monopolist’s incentive to innovate offeredby the product market. In particular, assume that the innovation lowers the monopolist’s(constant) unit production cost from an initial high level cH to a level cL < cH . Let D(p) bethe (downward-sloping) demand for the good produced by the monopoly.The innovation is protected by a patent of unlimited duration. Letting the constants r andvm denote the interest rate and the monopolist’s value of the innovation per unit of time,respectively, the monopolist’s “pure” incentive to innovate is given by
V m =
∫ ∞
0e−rtvmdt =
vm
r.
(a) To construct a benchmark for evaluating the incentives offered by the product market,consider the firm’s incentive to innovate when it is run by a social planner.Show that the social planner’s incentive to innovate is given by
V s =1
r
∫ cH
cL
D(c)dc.
The planner sets a price equal to marginal cost, i.e., cH before innovation andcL afterwards. Thus, the innovation generates an additional net social surplusper unit of time equal to
vs =
∫ cH
cL
D(c)dc,
whence follows that
V s =vs
r.
(b) Now assume the monopoly is run by a profit-maximizing owner. Let Πm(c) be themonopolist’s profit when the output is sold at the monopoly price pm(c) given marginalcost level c.2
Show that the monopolist’s incentive to innovate is given by
V m =1
r
∫ cH
cL
D (pm(c)) dc.
Show further that V m < V s and discuss the intuition for this result.
2The monopoly price pm(c) is a solution to
maxp
πm(p, c) = (p− c)D(p).
Now observe that for each different value of c there will typically be a different optimal price. We thereforedefine Πm(c) = πm(pm(c), c), which tells us what the optimized value of πm is for different choices of c.
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Hint: Note that
vm = Πm(cL)−Πm(cH) =
∫ cH
cL
(−dΠm(c)
dc
)dc
and apply the envelope theorem to obtain an expression for the integrand.
From the envelope theorem,
dΠm(c)
dc= −D (pm(c)) .
To see this, differentiate
Πm(c) = πm(pm(c), c)
with respect to c:
dΠm(c)
dc=
∂πm(pm(c), c)
∂p
dpm(c)
dc+
∂πm(pm(c), c)
∂c
=∂πm(pm(c), c)
∂c.
Here, the last equality uses the fact that pm(c) is the choice that maximizesπm, so
∂πm(pm(c), c)
∂p= 0.
A better way to write this is
dΠm(c)
dc=
∂πm(p, c)
∂c
∣∣∣∣p=pm(c)
.
With this in mind,
dΠm(c)
dc=
d
dc(p − c)D(p) = −D(pm(c)).
Putting the pieces together, we have
V m =1
r
∫ cH
cL
D (pm(c)) dc.
Note that pm(c) > c, and therefore D(pm(c)) < D(c). Thus, a comparisonof V m and V s reveals that V m < V s. This is easy to understand, becausemonopoly pricing at any cost level yields underproduction as compared withthe social optimum. Therefore, the monopolist’s cost reduction pertain to asmaller number of units.
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Exercise 2.2: Competition (Tirole, 1988)Consider now a situation in which a large number of firms produce a homogenous good witha technology exhibiting (constant) unit production costs cH . These firms are initially involvedin Bertrand price competition, so that the market price is cH and all the firms are earning zeroprofit. The firm that obtains the new technology cL is awarded a patent of unlimited duration.The (downward-sloping) industry demand function is denoted by D(p).Of course, the innovating firm’s product market price cannot be above cH , which calls for thedistinction of two possible cases: pm(cL) > cH and pm(cL) � cH . In the second case, in whichthe innovation is called drastic or major, the innovating firm enjoys monopoly power and theother, less efficient firms produce nothing. In the first case, the innovator is constrained tocharge cH because of potential competition from firms equipped with the old technology. Theinnovation is then called non-drastic or minor.
(a) Non-drastic process innovation. Show that the incentive to innovate is
V c =1
r
∫ cH
cL
D (cH) dc,
and establish that V m < V c < V s.
In the case of a non-drastic innovation, the innovator is constrained to set aprice equal to cH . His profit per unit of time is given by
Πc = (cH − cL)D(cH) =
∫ cH
cL
D (cH) dc.
Thus, the incentive to innovate in the competitive situation is
V c =1
r
∫ cH
cL
D (cH) dc.
We now compare V c to a monopolist’s innovation incentives.3 Notice thatcH < pm(cL) � pm(c) for all c � cL and thus that D(cH) > D(pm(cH)).From this, we derive
V m =1
r
∫ cH
cL
D (pm(c)) dc <1
r
∫ cH
cL
D (cH) dc = V c.
To compare V c to the social planner’s incentives, observe that D(cH) < D(c)for all c < cH , and therefore
V c =1
r
∫ cH
cL
D (cH) dc <1
r
∫ cH
cL
D(c)dc = V s.
(b) Drastic process innovation. Show that the incentive to innovate is
V c =1
r(pm(cL)− cL)D(pm(cL)),
3Observe that the innovator is a monopolist before innovation and afterwards.
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and establish further that V m < V c < V s.
In the case of a drastic innovation, the innovator sets the monopoly pricepm(cL). His profit per unit of time is given by
Πc = (pm(cL)− cL)D(pm(cL)).
Thus, the incentive to innovate in the competitive market environment is
V c =1
r(pm(cL)− cL)D(pm(cL)).
We now compare V c to a monopolist’s innovation incentives. Using the defi-nition of vm yields
V m =1
r{(pm(cL)− cL)D(pm(cL))− (pm(cH)− cH)D(pm(cH))} < V c.
Use a graphical argument to show that V s > V c.
(c) Why does a monopolist gain less from innovating than does a competitive firm? Providean intuitive explanation.
The result follows from different initial situations: The monopolist “replaceshimself” when he innovates whereas the competitive firm becomes a monopoly.In the literature, this property is called the replacement effect (due to Arrow,1962).
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Exercise 2.3: OligopolyConsider a symmetric n-firm Cournot oligopoly with a linear inverse demand function P (Q) =1−Q, with Q =
∑i qi, and constant average cost c.
(a) Show that each oligopolist produces an output equal to
qC(n, c) =1− c
n+ 1,
and hence earns a profit
πC(n, c) =
(1− c
n+ 1
)2
.
Each firm i solves
qCi ∈ argmaxqi
{πi(qi, qC−i)|qi � 0}.
The first-order conditions are
(1− qCi −∑i �=j
qCj − c)− qCi = 0.
Applying symmetry (i.e., qC = qCi = qCj ), we have
qC(n, c) =1− c
n+ 1.
By substitution,
p(n, c) = 1− nqC =1 + nc
n+ 1and πC(n, c) =
(1− c
n+ 1
)2
.
(b) Show that the private value of a drastic innovation that reduces cost from c to cL < c is
V C(n) =1
r
((1− cL
2
)2
−(1− c
n+ 1
)2).
Observe that πC(1, cL) is the innovator’s monopoly profit. Thus, the innova-tor’s profit per unit of time is
vC(n) = πC(1, cL)− πC(n, c).
By substitution,
V C(n) =1
r
((1− cL
2
)2
−(1− c
n+ 1
)2).
(c) How does V C(n) change with n? Discuss the intuition.
The value of an innovation is increasing in n, so innovation incentives arelarger in more competitive industries. Intuitively, the result reflects the factthat each oligopolist earns a positive profit before innovation takes place. Incase of a successful innovation, the innovating oligopolist “partially replaceshimself”, which reduces his incentives to undertake R&D (see above).
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Problem Set 3: Auctions and Contests
Exercise 3.1: The Vickrey Auction (Mas-Collel et al., 1995)
Consider the following auction (known as a second price, or Vickrey, auction). An object isauctioned off to I bidders. Bidder i’s valuation of the object (in monetary terms) is vi. Theauction rules are that each player submit a bid (a nonnegative number) in a sealed envelope.The envelopes are then opened, and the bidder which has submitted the highest bid gets theobject but pays the auctioneer the amount of the second-highest bid. If more than one biddersubmits the highest bid, each gets the object with equal probability. Show that submitting abid of vi with certainty is a weakly dominant strategy for bidder i.
Suppose not. Assume bidder i bids bi > vi. Then if some other bidder bidssomething larger than bi, bidder i is just as well off as if he would have bid vi. Ifall other players bid lower than vi, then bidder i obtains the object and pays theamount of the second highest bid. If the second highest bid is bj < vi, this resultsin the same payoff for player i as if he bid vi. However, suppose that the secondhighest bid of the other is bj > vi. Then, by bidding bi bidder i will win the objectand obtain a negative payoff. By bidding vi he will not win the object and obtaina payoff of zero. Therefore, bidding bi > vi is weakly dominated by bidding vi.
Suppose bidder i bids bi < vi. Then if all other bidders bid something smaller thanbi, bidder i is just as well off as if he would have bid vi. He will win the object andpay the second highest bid. If some other player bids higher than vi, then bidderi does not win the object regardless whether he bids bi or vi. However, supposethat nobody bids higher than vi and the highest bid of the other player is bj withbi < bj < vi. Then by bidding bi bidder i will not win the object, therefore gettinga payoff of zero. By bidding vi, he would win the object, pay bj < vi, and thusobtain a payoff of vi − bj > 0. Therefore, bidding bi < vi is weakly dominated bybidding vi. This argument implies that bidding vi is a weakly dominant strategy.
Thus, each bidder has an incentive to report faithfully his valuation of the object.
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Exercise 3.2: Vickrey Auctions to Choose among Ideas (Scotchmer, 2004)We now shed light on the conditions under which the Vickrey auction mechanism identifiesthe best idea for a targeted objective. From a social planner’s view, the most attractive ideawould be the one providing the greatest social surplus.Let the pair (vi, ci) describe and idea. Define prospective social surplus of idea i as thedifference between the observable discounted social value vi/r and the cost ci of developingthe idea into an innovation:
si =vir− ci.
Suppose that the sponsor asks each prospective innovator i to report social surplus si, andthat he chooses the firm that reports the highest surplus to invest. Label the highest and thesecond-highest report si and sj, respectively.
(a) Show that by promising a payment of vi/r− sj, a sponsor can safely pick the firm thatclaims the highest surplus.
If firm i is chosen to invest, it ends up with a profit equal to
vi/r − sj − ci = si − sj .
Using the above result, each firm i’s weakly dominant strategy is to reportfaithfully the surplus si it can deliver. Thus, the sponsor will ask the firmreporting the highest surplus to invest.
Observe that if si is close to sj, the payment to the winning firm will be closeto cost.
(b) In what sense does the Vickrey auction mechanism yield an efficient outcome? Whathappens if the value vi is not observable ex post?
The Vickrey auction mechanism elicits efficient investment in the sense thatthe high-surplus firm is chosen to invest, and that there is no duplication ofcosts (only the innovating firm pays its cost ci).
If vi is not observable ex post, payments cannot depend on delivered value.In that case, the mechanism does not work. However, note that the Vickreyauction mechanism involves a difficulty even in the case of observable values.If the value delivered is not verifiable by a court ex post, the innovator mayfear that the sponsor will renege, or give a smaller price than he deserves,which will negatively affect investment incentives.
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Exercise 3.3: Prototype Contests to Choose among Ideas (Scotchmer, 2004)Here we shall shed light on how the sponsor can circumvent the problem of ending up withpayments depending on the delivered value. The idea is to let the firms demonstrate theirideas by developing prototypes, and then choose ex post between them.Assume two firms i = 1, 2 have identical ideas (v, c). Each firm i announces ex ante the priceρi at which it is selling the innovation ex post. After naming prices, the firms and the sponsorwrite contingent contracts.4 Then, firms invest and deliver prototypes. If firm i’s prototype ischosen by the sponsor ex post, it receives the specified price ρi.
(a) Consider the minimum prices(ρ1, ρ2) = (2c, 2c)
to cover costs in expectations (assuming that the tie-breaking rule is to randomize).Show that these prices cannot be sustained in equilibrium.
The equilibrium bids ρi are such that neither firm has an incentive to reviseits bid, assuming that the other firm’s bid is fixed. Since each firm would winwith probability 0.5, expected revenues are
1
2ρ1 =
1
2ρ2 = c.
Given firm 2 demands 2c in the event it is chosen, firm 1 can improve profitby reducing its price to ρ1 = 2c− ε. Then, firm 1 makes profit
ρ1 − c = c− ε > 0
instead of 0. Thus, the prices (ρ1, ρ2) cannot be sustained in equilibrium.
(b) Let
F (ρ) =
{ρ
(v/r) if 0 � ρ � c
1− cρ + c
(v/r) if c � ρ � vr
,
be the cumulative distribution of the firms’ prices, and consider the strategies “do notdeliver a prototype if the draw ρ turns out to be smaller than the threshold level c” and“innovate and demand price ρ if the draw exceeds c”. Show that the mixed strategiesF1 = F2 = F are a Nash equilibrium.
The probability that a firm does not innovate is
F (c) = c/(v/r).
If firm 1 develops the innovation and demands any price in [c, v/r], firm 1’sexpected profit is
ρ[F2(c) + 1− F2(ρ)]− c = 0.
This is an equilibrium because each price in the support of the distributionyields the same expected profit as any other price, namely zero.
Remark: The term [F2(c) + 1 − F2(ρ)] represents the probability that firm1 wins the prototype contest. With probability F2(c) firm 2 does not invest,and with probability 1−F2(ρ) firm 2 invests but demands a higher price thandoes firm 1.
4The contracts ensure that inventors are not subject to hold-up.
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(c) In what sense do prototype contests yield an inefficient outcome?
With positive probability, the sponsor does not get the innovation, and evenif he gets it, there is a large probability that the costs will be duplicated. Thisis the social price for the inability to observe or verify value.
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Problem Set 4: Patents I
Exercise 4.1: Optimal Patent Length (Tirole, 1988)An industry is initially competitive. The price is equal to the firms’ marginal cost, c. Theindustry’s demand function is D(p) = 1 − bp, with b > 0. Assume that one firm has accessto a cost-reducing technology which allows production at marginal cost c < c by spendingφ(c) = K(c−c)2/2, withK sufficiently large so that the process innovation remains nondrastic.The new technology is implemented at time 0, and the patent lasts T units of time (after whichthe technology c can be used freely be all firms). The firms compete a la Bertrand, and therate of interest is r.
(a) Denote by Δ ≡ c − c the magnitude of the cost reduction chosen by the firm thatimplements the new technology. Show that
Δ(τ) =τD
rK,
where τ ≡ 1− e−rT and D ≡ 1− bc.
Hint: The firm will choose Δ so as to
maxΔ
(∫ T
0Δ(1− bc)e−rtdt− KΔ2
2
).
Given patent life T , the inventor chooses the Δ that maximizes the differencebetween the benefits of implementing the new technology and the cost ofR&D:
maxΔ
(∫ T
0Δ(1− bc)e−rtdt− KΔ2
2
)
= maxΔ
(ΔD
∫ T
0e−rtdt− KΔ2
2
)(D ≡ 1− bc)
= maxΔ
(τ
rΔD − KΔ2
2
). (τ ≡ 1− e−rT )
The first-order condition reads
τD
r−KΔ = 0,
which implies
Δ(τ) =τD
rK.
Observe that Δ(τ) is increasing in τ (and hence T ), so that a longer patentlife will lead to a higher cost reduction.
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(b) Show that the new technology increases welfare by
W (τ) =
(∫ T
0Δ(1− bc)e−rtdt− KΔ2
2
)+
(1− τ
r
)(DΔ(τ) +
b
2Δ(τ)2
).
Hint: The generated consumer surplus is equal to
∫ ∞
T
(DΔ(τ) +
b
2Δ(τ)2
)e−rtdt =
(1− τ
r
)(DΔ(τ) +
b
2Δ(τ)2
).
Use a graphical argument to derive the formula for the generated consumersurplus. Note that the firm’s profit is part of the consumer surplus after thepatent expires (redistribution).
(c) Show that optimal patent life, τ , lies in (0, 1), so the optimal patent length T is finite.
Hint: From the envelope theorem,
d
dτ
(maxΔ
{∫ T
0Δ(1− bc)e−rtdt− KΔ2
2
})=
Δ(τ)D
r.
Optimal patent life, τ , is a solution to
dW (τ)
dτ=Δ(τ)D
r+
(1− τ
r
)(DΔ′(τ) + bΔ(τ)Δ′(τ)
)
− 1
r
(DΔ(τ) +
b
2Δ(τ)2
)= 0.
Substituting Δ(τ) and Δ′(τ), the first-order condition simplifies to
f(τ) :=3
2bτ2 + (rK − b)τ − rK = 0.
Note that f(0) < 0 and f(1) > 0. As f(τ) is a convex function, the (positive)solution lies in the interval (0, 1). Thus, the optimal patent length is finite.
Remark: Although the size of the innovation is positively related to patentlength, patent life shouldn’t be set at an infinitely large number. The reason isthe need to balance off the benefits of a larger invention against the inefficiencyof monopoly pricing.
17
Exercise 4.2: Pooling of Complementary Patents (Motta, 2004)
To produce a certain (homogenous) final good, n manufacturers need two complementarytechnologies, whose patents are owned by two firms A and B, who separately license thetechnologies at a unit royalty fee wi (i = A,B). The game is as follows. In the first stage, thepatentholders independently and simultaneously decide the royalty level. In the second stage,the manufacturers compete a la Bertrand, and incur unit costs c + wA + wB , where c < 1.They face market demand q = 1−p (as usual, if several firms all charge the same lowest price,demand is equally shared among them; zero demand goes to firms having higher prices).
(a) What are the equilibrium values of royalties and prices? Calculate each patentholder’sequilibrium profit.
In the last stage, given that manufacturers compete in prices, the Bertrandequilibrium applies: the market price will be
p = c+ wA + wB ,
and final demandq = 1− (c+ wA + wB).
In the first stage, each patentholder decides the royalty fee so as to
maxwi
πi = wi(1− c− wA − wB).
From ∂πi/∂wi = 0, it follows that the symmetric equilibrium royalty rate is
w∗ =1− c
3,
and the final price (by substitution) is
p∗ =2 + c
3.
Patentholders’ profits are
π∗ =(1− c)2
9.
(b) Consider an alternative situation where the two patentholders assign the right of ex-ploitation of their patents to a patent pool. It is now the pool which sets the value ofboth royalties. Find equilibrium values of royalties and final prices under the patent pooland compare them with the previous case. What is the equilibrium profit of the patentpool?
Under the patent pool, there is joint-profit maximization of the patentholders.The pool’s problem is therefore to
maxwA,wB
πP = wA(1− c− wA − wB) + wB(1− c− wB − wA).
Solving the first-order conditions gives the symmetric solution
wP =1− c
4.
18
By substitution, the market price is
pP =1 + c
2,
and the pool’s total profit is
πP =(1− c)2
4.
(c) Show that forming the patent pool is both profitable for the patentholders and good forconsumers.
It is straightforward to see that the patent pool Pareto dominates the situationwhere the two patents are licensed independently. Final prices (as well asroyalties) are lower (therefore, consumers are better off) and patentholder’sprofits are higher. Observe that manufacturers in this example always get zeroprofits.
19
Problem Set 5: Patent Races
Exercise 5.1: Research Intensity and Market Structure (Tirole, 1988)Consider a symmetric patent race involving n firms, which initially make no profit. Each of thefirms has a private value V of the patent. Patent life is infinitely long, time is continuous, andthe rate of interest is r. The probability of success by firm i at given time, t, is an exponentialfunction: If ti represents firm i’s (random) success date, then Prob {ti � t} = 1− e−hit.5 Thechoice variable for each firm i is a flow of expenditure xi that yields a probability of discoveryhi = h(xi) per unit of time. Assume that h′ > 0, h′′ < 0, h(0) = 0, h′(0) = +∞, andh′(+∞) = 0.6
(a) Let firm 1 choose expenditure y ≡ x1, and let firms 2, 3, . . . , n choose expenditurex ≡ xi. Show that the probability that none of the firms has discovered by time t is
e−[h(y)+(n−1)h(x)]t.
The probability that firm i has not discovered until time t is
Pr{ti � t} = 1− Pr{ti � t} = e−hit.
Thus, assuming independence of R&D activities, the probability that none ofthe firms has discovered until time t is
e−h(x1)t · e−h(x2)t · ... · e−h(xn)t = e−[h(y)+(n−1)h(x)]t.
(b) Show that firm 1’s expected intertemporal profit is
h(y)V − y
h(y) + (n− 1)h(x) + r.
Hint: Note that∫ ∞
0[h(y)V − y]e−[h(y)+(n−1)h(x)]te−rtdt =
h(y)V − y
h(y) + (n− 1)h(x) + r.
Suppose that none of the firms has discovered before time t. By spendingamount y on R&D, firm 1 is the first to innovate with probability h(y), andearns, starting from that moment, V . Thus, firm 1’s present discounted valueof the expected profit over time is
∫ ∞
0[h(y)V − y]e−[h(y)+(n−1)h(x)]te−rtdt =
h(y)V − y
h(y) + (n− 1)h(x) + r.
5Note that hi is the conditional probability of success, given no success to date. Further, the expected timetill success for firm i is the reciprocal of the hazard rate; that is E(ti) = 1/h(xi).
6This assumption of a “memoryless” R&D technology implies that a firm’s probability of making a discoveryand obtaining a patent at a point in time depends only on this firm’s current R&D experience and not on itspast R&D experience.
20
(c) Show that the first-order condition for the symmetric Nash equilibrium expenditure isgiven by
[(n− 1)h(x) + r][h′(x)V − 1]− [h(x) − h′(x)x] = 0.
Assume that the left-hand side of the first-order condition strictly decreases with x.Show that there exists a unique equilibrium research intensity x∗(n).
Hint: Focus on a symmetric equilibrium with y = x.
Differentiating with respect to y, we obtain the first-order condition
[(n − 1)h(x) + r][h′(y)V − 1]− h(y) + h′(y)y = 0.
Imposing symmetry, we obtain
[(n− 1)h(x) + r][h′(x)V − 1]− [h(x) − h′(x)x] = 0.
Observe that the left-hand side of the preceding equation is positive at x = 0(because h′(0) = +∞) and negative at x = +∞ (because h′(∞) = 0 andh′′ < 0). Hence, there exists a unique equilibrium research intensity x∗(n).
(d) Suppose that the objective function is strictly concave. Show that
dx∗(n)dn
> 0.
Provide an intuitive explanation for the result.
From the implicit function theorem, we know that
dx∗(n)dn
= −h(x∗)[h′(x∗)V − 1]
[–].
Here, [–] denotes a negative expression (the negativity follows from the second-order condition). From the concavity of h and the fact that h(0) = 0,
h(x) > xh′(x).
Therefore, the first-order condition implies
h′(x∗)V − 1 > 0.
We thus getdx∗(n)dn
> 0.
21
Exercise 5.2: Patent Value and the Number of R&D Attempts (Scotchmer, 2004)
Let the triple (v, c, p) denote an idea, where v is the value of achieving the specified objective,c is the cost of the research approach, and p is the probability that the approach fails. Althoughthe ideas are symmetric, we assume that the innovators employ different approaches, so thatthe successes and failures of the different attempts are independent. Therefore, if n approachesare taken, the probability that all of them fail is pn, so that the probability of at least onesuccess is 1−pn, which is denoted by P (n).7 Let r denote the rate of interest, and let S = v/rdenote the social value of the innovation. Further, let Π denote the patent value and assumethat Π � S.
(a) Show that the equilibrium number of firms that enter the patent race, ne, satisfies
1
neΠP (ne) � c.
For each n, the expected per-firm profit is
1
nΠP (n).
Firms will enter the patent race up to the point where an additional firm wouldnot make profit. Thus, ne satisfies
1
neΠP (ne) � c.
(b) Show that the social optimal number of entrants, n∗, satisfies
S[P (n∗)− P (n∗ − 1)] � c � S[P (n∗ + 1)− P (n∗)].
The social value provided by the nth entrant is
S[P (n)− P (n− 1)]− c.
The social optimal number of participants, n∗, can be described as the numberwhere the marginal entrant would add as much social value as social cost c,but his or her successor would not:
S[P (n∗)− P (n∗ − 1)] � c � S[P (n∗ + 1)− P (n∗)].
(c) Why do the private and the social incentives to enter the patent race differ? Provide anintuitive explanation.
The reason is that the marginal entrant receives the average profit rather thanthe marginal profit, but it is the marginal profit that determines the optimalnumber of participants (this problem is usually discussed as the “problem ofthe commons”).
In our setting, the average profit is greater than the marginal profit due tothe concavity of P (·). Thus, the private value of entry can be positive evenif the social value of entry is negative.
7Note that, in contrast to the previous model, a failure does not lead to a renewal of efforts.
22
(d) Show that the patent value Π∗ to induce the optimal number of attempts should bechosen such that
P (n∗)Π∗ = cn∗.
Hint: Graph the functions P (n)S and P (n)Π as a function of n.
See Figure 4.2 in Scotchmer (2004).
23
Problem Set 6: Research Joint Ventures
Exercise 6.1: Cooperative R&D (d’Aspremont and Jacquemin, 1988)Consider a duopoly selling a homogenous product. Let the inverse market demand function bep = a−Q, where Q = q1+ q2 is the total production and a > 0. Each firm has marginal costsci = c − xi − λxj, where xi is the amount of research that firm i undertakes, and λ ∈ [0, 1]is a parameter indicating the spillover that results from the R&D investment xj made by theother firm.8 The cost of R&D is given by
κ(xi) =g
2x2i , with g >
4
3.
We consider two different two-stage games, in which the investments xi are the strategies ofthe first stage, followed by the quantities qi in the second stage.
(a) Competition in both stages (let the superscript C denote ‘competition’).
(i) Show that the Nash-Cournot equilibrium output is given by
qCi =a− c+ (2− λ)xi + (2λ− 1)xj
3.
At the last stage of the game, each firm i chooses qi so as to
maxqi
(a− qi − qj − ci(xi, xj)) qi.
Note that the profit-maximization problem is the same as in a typical(one-shot) Cournot game with heterogenous firms. Therefore,
qCi (ci(xi, xj), cj(xi, xj)) =a− 2ci(xi, xj) + cj(xi, xj)
3
=a− c+ xi(2− λ) + xj(2λ− 1)
3.
(ii) Show that the unique (and symmetric) R&D level for each firm is given by
xC =2(a− c)(2 − λ)
9g − 4− 2λ+ 2λ2.
Recall that each firm’s product-market profit is equal to(qCi
)2. Thus,
each firm chooses xi to
maxxi
πi(xi, xj) =
(a− c+ xi(2− λ) + xj(2λ− 1)
3
)2
− g
2x2i .
8When the rival invests xj , it is as if firm i had done that investment itself and reduced its cost by λxj .
24
By taking the first derivative and applying symmetry (xi = xj = xC) oneobtains xC as given above.
Remark: Note that the second-order condition requires g > 2(2 − λ)2/9.The stability conditions for the R&D stage require (see, e.g., Shapiro,1989, p. 335)
∂2πi/∂xi∂xj∂2πi/∂x2i
=
∣∣∣∣−2(2− λ)(2λ− 1)
2(2 − λ)2 − 9g
∣∣∣∣ < 1
and are satisfied for
g >2(2− λ)(λ− 1)
3
A sufficient condition for stability to be met is that g > 4/3, which alsosatisfies the second-order condition.
(iii) Show that
qC =3(a− c)g
9g − 4− 2λ+ 2λ2and πC =
(a− c)2g(9g − 8 + 8λ− 2λ2)
(9g − 4− 2λ+ 2λ2)2.
The results follow by substitution.
(iv) Show that consumer surplus and welfare are given, respectively, by
CSC =18(a − c)2g2
(9g − 4− 2λ+ 2λ2)2and WC =
4(a− c)2g(9g − 4 + 4λ− λ2)
(9g − 4− 2λ+ 2λ2)2.
Recall that CSC =(a− pC
)QC/2 and that WC = CSC + 2πC . The
results then follow by substitution.
(b) The Research Joint-Venture (let the superscript J denote ‘Joint-Venture’).
(i) Show that maximizing joint profits at the first stage of the game yields
xJ =2(a− c)(1 + λ)
9g − 2(1 + λ)2.
Hint: Maximize∑2
i=1 πi(xi, xj) and focus on the symmetric equilibrium.
The firms solve
maxx1,x2
=
2∑i=1
[(a− c+ xi(2− λ) + xj(2λ− 1)
3
)2
− g
2x2i
].
Taking first derivatives and focusing on the symmetric equilibrium, theresult follows.
25
(ii) Show that
qJ =3(a− c)g
9g − 2(1 + λ)2and πJ =
(a− c)2g
9g − 2(1 + λ)2.
The results follow by substitution.
(iii) Show that consumer surplus and welfare are given, respectively, by
CSJ =18(a − c)2g2
(9g − 2− 4λ− 2λ2)2and W J =
4(a− c)2(9g − 1− 2λ− λ2)
(9g − 2− 4λ− 2λ2)2.
Recall that CSJ =(a− pJ
)QJ/2 and that W J = CSJ + 2πJ . The
results then follow by substitution.
(c) When spillovers are large enough, that is, λ � 1/2, d’Aspremont and Jacquemin (1988)show that (i) xJ > xC , (ii) qJ > qC , and (iii) W J > WC . Provide an intuitiveexplanation for these results.
See d’Aspremont and Jacquemin (1988).
26
Problem Set 7: Patents II
Exercise 7.1: Patent Breadth and the Ratio Test (Scotchmer, 2004)
In a protected market, that is, a market shielded from competition (either naturally, by firmscolluding, government intervention, or labour unions), there is an innovator and a number ofimitators that potentially enter the market. Entry is assumed to be costly. If no entry occurs,the innovator sets the monopoly price. In contrast, if entry occurs, the imitators can sell theirproduct as perfect substitutes for the patented technology and the price would be lower thanthe monopoly price but higher than the competitive price (pm > p > pc) since entry is costly.Furthermore, assume that the market demand is given by x(p) and that marginal productioncosts are normalized to zero.
(a) Let K be the cost of entering the protected market and assume that there are n activefirms in the market (including the innovator). Given discounted patent length T , findthe condition at which entry stops expressed by K,T, x(p(n)), p(n).
Hint: Entry occurs as long as an additional entrant makes a positive profit.
Because we assumed symmetric marginal cost for all firms, each active firmproduces the same quantity. Therefore, the profit of each entrant is theaggregate profit, x(p(n))p(n), divided by the number of firms, n, minus entrycost, K. Entry stops when firms are making positive profits and one additionalentrant makes profit turn negative:
1
n+ 1[Tx(p(n+ 1))p(n + 1)]−K < 0 � 1
n[Tx(p(n))p(n)]−K
(b) Interpret patent breadth as the cost K of entering the protected market. K can beduplication cost for inventing around the protected technology or, alternatively, as alicensing fee charged by the patentholder to allow entry. Does it make a difference whichinterpretation is used? Discuss this issue from the point of view of the patentholder.
The patent holding firm would prefer licensing, because by charging a licensefee of K, it can at least earn (n− 1)K and compensate part of the loss fromgiving up its monopoly position.
(c) Put yourself in the social planner’s shoes and assume that K is large enough to coverthe patentees’ cost. Let
T =
∫ τ
0e−rtdt
denote discounted patent length from time 0 to τ , and suppose that there are two policiesto choose from: (Tm,K) is a short enough patent, so that no single competitor entersthe market, but long enough to cover the patentholder’s cost. (T c,K) is a long livedpatent that allows entry in the protected market.Let the discounted profit of the patentholder be πm = pmx(pm)Tm under the policy(Tm,K) and π = px(p)T c under the policy (T c,K). Graphically illustrate the per-periodprofit of the patentholder and the deadweight loss caused by each policy.
27
The profit of the patentholder and the associated deadweight loss can be seenbelow (filled for the short lived patent):
x
p
pm
p
x(pm) x( p)
(d) Which policy will consumers prefer?
Hint: Use the ”Ratio Test” and proceed along Technical Note 4.7.1 (Scotchmer 2004).Define consumer surplus as a function s(p) and deadweight loss as l(p) for all p.It is an assumption in the ”Ratio test” that both policies are equally profitable forthe firm.
Consumers are better off with the policy that generates a larger consumersurplus. They will prefer policy (T c,K) to (Tm,K) if
T cs(p) +
(1
r− T c
)s(0) > Tms(pm) +
(1
r− Tm
)s(0). (S1)
Rearranging, we have
T c (s(p)− s(0)) > Tm (s(pm)− s(0)) ,
or, equivalently,
T c (s(0)− s(p)) < Tm (s(0)− s(pm)) .
In exercise (c), we saw that the sum of the profit and the deadweight lossis equal to the loss in consumer surplus. Thus, we can write s(0) − s(p) =px(p) + l(p) for p ∈ [0, pm]. Inserting this above gives us:
T c [px(p) + l(p)] < Tm [pmx(pm) + l(pm)] .
Making use of the fact that both policies are equally profitable if pmx(pm)Tm =px(p)T c, we can rewrite the above condition as
T c [px(p) + l(p)] <px(p)T c
pmx(pm)[pmx(pm) + l(pm)] .
28
Rearranging terms, the condition boils down to
pmx(pm)
l(pm)<
px(p)
l(p)(S2)
The policy that generates lower prices p is (T c,K) and it is preferred by theconsumers if inequality (S1) holds. This is the case if and only if (S2) holds,where in each period the ratio of profit to deadweight loss is greater.
29
Exercise 7.2: Optimal Patent Length and Breadth (Gilbert and Shapiro, 1990)A social planner aims at finding the optimal combination of patent length and patent breadth.Let T denote patent length and suppose that patent breadth, π, is the flow rate of profit thatthe patentee obtains during patent protection. Assume that a broader patent increases thepatentee’s market power and therefore the associated deadweight loss. In other words, if welet W (π) denote the per period social welfare, we have W ′(π) < 0. When a patent expires,the flow of profits decline to π and social welfare rises to W = W (π). Time is continuous andthe interest rate is r.The social planner’s goal is to maximize total social welfare Ω(T, π) by optimally choosingT and π subject to achieving a cost-effective reward for the innovation with value V , that isV (T, π) � V where V (T, π) is the patentholder’s total discounted profit.Assume for simplicity that the social planner observes a stationary and predictable environment,that is there is only one innovation and no uncertainty about future conditions.
(a) Write down the discounted social welfare and the present value of the patentee’s profitsas functions of T and π.
Hint: Ω(T, π) is the discounted social welfare during patent protection plus the dis-counted social welfare after the patent expires. The present value of the paten-tee’s profits V (T, π) is the sum of the discounted profits obtained during patentprotection and the discounted profits after patent expiration.
Discounted social welfare and the present value of the patentee’s profit aregiven by
Ω(T, π) =
T∫0
W (π)e−rtdt+
∞∫T
W e−rtdt
and
V (T, π) =
T∫0
πe−rtdt+
∞∫T
πe−rtdt,
respectively.
(b) Define π = φ(T ) as the flow of profits needed to obtain V . For a given T > 0 we canthen write V as:
V ≡T∫0
φ(T )e−rtdt+
∞∫T
πe−rtdt.
Solve the integral and differentiate with respect to T to find an expression for φ′(T ).
Solving the integrals in V ≡T∫0
φ(T )e−rtdt+∞∫T
πe−rtdt, we find
V = φ(T )1− e−rT
r+ π
e−rT
r.
Differentiating with respect to T yields
0 = e−rT (φ(T )− π) + φ′(T )1− e−rT
r,
30
which can be rewritten as
φ′(T ) = −e−rT (φ(T )− π)r
1− e−rT. (S3)
(c) Show that social welfare is given by
Ω(T, π) = −W (π)e−rT
r+W (π)
1
r+ W
e−rT
r.
Derive the first-order conditions.
Total welfare is given by
Ω(T, π) =
T∫0
W (π)e−rtdt+
∞∫T
We−rtdt
= −W (π)e−rT
r+W (π)
1
r+ W
e−rT
r.
The first-order conditions read:
∂Ω
∂T=W (π)e−rT − W e−rT =
(W (π)− W
)e−rT (S4)
∂Ω
∂π=−W ′(π)
e−rT
r+W ′(π)
1
r= W ′(π)
1− e−rT
r. (S5)
(d) Show that dΩ/dT > 0 when the flow of profits is set optimally π = φ(T ). If welfareincreases with patent length, how long should patent length be?
Hint: Note that the welfare function can be rewritten as Ω(T, φ(T )) by construction.Use the chain rule to find
dΩ
dT=
∂Ω
∂T+
∂Ω
∂πφ′(T ).
Then make use of the results obtained in (b) to substitute for the expressionφ′(T ). Finally, assume that patent breadth is increasingly costly in terms of socialwelfare, i.e. W ′(π) < 0 and W ′′(π) < 0 on [π, φ(T )]. Graph the welfare functionto understand the argument.
Using (S3), (S4), and (S5) we find
dΩ
dT=
[W (φ(T ))− W −W ′(φ(T ))(φ(T ) − π)
]e−rT .
The concavity of the welfare function implies (graph the welfare function tosee the argument)
W (φ(T ))− W
(φ(T )− π)> W ′(φ(T ))
or equivalently:
W −W (φ(T )) < −W ′(φ(T ))(φ(T ) − π).
31
Thus, dΩdT > 0 and increasing T always has a positive effect on welfare, so the
optimal patent length is infinite. The idea is that increasing patent breadthraises market power and hence deadweight loss. Therefore, increasing thereward on a flow basis is costly in terms of social welfare.
(e) Why might this result not apply in practice? Discuss.
An infinitely-lived patent is optimal under a predictable and stationary en-vironment. If the environment is not predictable and there is uncertaintyabout future demand and costs, risk averse firms might prefer shorter andbroader patents in order to share risk. In an non-stationary environment, Cu-mulative innovation plays an important role. Since inventions build on eachother, patents with infinite length is likely to have deterrent effects on firms’incentives to invest in related research.
32
Problem Set 8: Licensing
Exercise 8.1: Licensing Basic Research (Scotchmer, 2004)Consider two independent firms. Firm 1 produces a basic innovation and firm 2 has an idea fora second-generation product that emerges with no delay after the first innovation is made.9
The per-period value of the first innovation to end users is given by x and the per-periodincrement to market value generated by the application is y. Let c1 and c2 represent the costof undertaking basic research and to develop the second-generation product, respectively. Theinterest rate is r and T denotes the discounted patent length. Further let π and l denotefractions of the total per-period value to end users. That is, π(x+ y) is the per-period profitat the proprietary price (due to patent protection) and l(x+ y) is the deadweight loss.10
(a) Suppose that a governmental institution has both ideas. Further suppose that it financesthe innovations through a lump-sum tax and offers the product for free. Write down thesocial value obtained from both innovations, W ∗.
If both innovations were public and sold at price p = 0, the social value ismaximal (as there is no deadweight loss). In that case:
W ∗ =x+ y
r− c1 − c2.
(b) Write down the social value, W p, and consumer surplus, Sp, obtained when the innova-tions are performed by the two firms under the patent regime.
The social value when both innovations are performed by the two firms is thediscounted total value to end users less the deadweight loss due to patentprotection, and less invention cost. Therefore,
W p =(x+ y)
r− (x+ y)lT − c1 − c2.
The consumer surplus is given by
Sp =(x+ y)
r− (x+ y)(l + π)T − c1 − c2.
(c) Suppose the firms coordinate their R&D activities. What is their joint profit, Π?
The firms obtain a fraction π of the value generated by both innovationsduring patent protection less the cost of turning both ideas into innovations.Hence,
Π = (x+ y)πT − c1 − c2.
9Assume that most of the profit is due to the second-generation product and both firms have blockingpatents on the application.
10Thus, 1− π − l is the fraction of (x+ y) defining consumer surplus.
33
(d) Now suppose the firms do not coordinate their research activity but instead the firmscan either sign a license agreement ex-ante, before the second innovator invests c2 (butafter the first innovation has been made), or ex-post, after the second innovator investsc2. Illustrate the bargaining situation.
An illustration of the bargaining situation is seen below (including payoffs thatwill be explained in the following exercises):
Ex-antelicense?
Payoffs with licence:πTx− c1+ 1
2πTy, 12πTy− c2or
πTx− c1+ 12(πTy− c2), 12(πTy− c2)
Firm 2:invest?
Ex-postlicense?
πTx− c1, 0
πTx− c1+ 12πTy, 12πTy− c2 πTx− c1,−c2
Yes No
Yes No
Yes No
(e) In the game, each firm’s threat point is defined as the expected profit it can guaranteeitself if it does not license. Find the threat points and bargaining outcome for each firmat the ex-post licensing node under the assumption that the bargaining surplus is splitequally. Indicate the pay-off’s in the diagram.
Ex-post, the producer of the basic innovation has a threat point of πTx− c1and the second generation producer has a threat point of −c2. The bargainingsurplus is πTy and it is shared equally between the firms. Payoffs can be seenat the bottom of the diagram in exercise (d).
(f) Find the threat point and bargaining outcome for each firm at the ex-ante licensing nodeunder the assumption that the bargaining surplus is split equally. Indicate the payoffs inthe diagram.
Following failed ex-ante licensing negotiations, firm 2 is better off by invest-ing than not investing if 1
2πTy − c2 > 0. The thread points at the ex-antebargaining node are therefore πTx − c1 +
12πyT and 1
2πyT − c2 for firms 1and 2 respectively. Since the firms are bargaining for πTx − c1 + πyT − c2there is nothing left to divide and the bargaining surplus is zero. The ex-antelicensing agreement payoffs are simply the thread points as indicated in thefirst line of the upper-left branch on the diagram in exercise (d).
34
On the other hand, the costs c2 could be so high that 12πTy − c2 < 0. Firm
2 then anticipates that it will be held up ex-post and that it will not be ableto cover costs. Firm 2 will therefore not invest following failed negotiationsfor ex-ante licensing. The thread point are then given by πTx− c1 and 0 forfirms 1 and 2 respectively. The firms are bargaining for πTx− c1 + πyT − c2which gives a bargaining surplus of πyT − c2. Splitting the bargaining surplusequally gives the ex-ante licensing agreement payoff indicated in the secondline of the upper-left branch on the diagram in exercise (d).
35
Exercise 8.2: Anti–Competitive Cross–Licensing (Motta, 2004)Consider two firms that play the following game. In the first stage, they jointly decide whetherthey want to cross-license their technologies. The technologies are assumed to be perfectsubstitutes and are used to produce the same homogenous good. At this stage, if cross-licensing is agreed upon, they also jointly decide the same per-unit of output royalty cL for thecross-license. In the following stage, they compete in quantities. Assume for simplicity thatthe only unit cost, if any, is given by cL. Assume linear demand p = 1−Q, where Q is totaloutput.
(a) Find each firm’s equilibrium output and the associated per-firm profit if no cross-licensingis agreed upon.
When no cross-license is agreed upon, each firm solves
maxqi
πi = (1− qi − qj)qi.
The reaction functions are given by
qi(qj) =1− qj
2.
In a symmetric equilibrium, each firm’s output is qNL = 1/3 and the associ-ated per-firm profit is πNL = 1/9.
(b) Assuming that cross-licensing is agreed upon, write down the maximization problem ofeach firm and find equilibrium outputs and per-firm profits.
Firm i solvesmaxqi
πi = (1− qi − qj − cL)qi + cLqj.
The term cL appears both as cost and as revenue, because each firm has topay the other a unit royalty. The reaction functions are
qi(qj) =1− qj − cL
2and qj(qi) =
1− qi − cL2
.
Equilibrium outputs and per-firm profits are,
qL(cL) =1− cL
3
and
πL(cL) =(1 + 2cL)(1− cL)
9,
respectively. Note that here is an example where profit is not equal to quantitysquared for all cL.
(c) In the first stage, firms not only decide whether to cross-license or not but also theydetermine the level of cL. Find the equilibrium level of the unit royalty.
To find the equilibrium level of cL, we need to find the value at which thefunction πL(cL) reaches its maximum. Solving
dπL
dcL=
2(1 − cL)− (1 + 2cL)
9
!= 0,
36
we obtain c∗L = 1/4.
(d) Compute each firm’s equilibrium output and profit.
The output of each firm is
qL(1/4) =1− 1/4
3= 1/4
and the per–firm profit is
πL(1/4) =(1 + 2(1/4))(1 − 1/4)
9= 1/8.
(e) Are the firms going to cross-license their technologies?
The joint output and profit under cross-licensing correspond to the joint profitmaximization outcome, i.e. the monopoly solution. To see this, consider themonopolists problem:
Π = (1−Q)Q ⇒ ∂Π
∂Q= 1− 2Q = 0 ⇒ QM =
1
2
Since total output is given by Q = qi + qj and qi and qj are equal, we seethat under cross licensing the firms do indeed end up producing the monopolyquantity. Note that the monopoly profit is higher than the sum of Cournotprofits when no cross-license is agreed upon (see (a)).
37
Problem Set 9: Licenses and Litigation
Exercise 9.1: Profit Neutrality in Licensing (Maurer and Scotchmer, 2004)Consider a situation where a new product is invented. The product must be produced efficientlyto reach its full value but in many cases the patentholder may not be necessarily the mostefficient firm to produce it. Assume that the firm faces a convex and positive increasingmarginal cost’s function denoted by γ(·). If marginal costs are increasing, the average cost ofsupply can be lowered by producing in several plants. Furthermore, let C be the set up costfor each production plant. Thus the cost of producing q units of the product in a single plantis given by
Γ(q) = C +
∫ q
0γ[q]dq.
Assume linear demand for the proprietary good p(q) = 1− q, where q denotes total output.
(a) Since set up is costly, efficient production requires a finite number of plants. Assume forsimplicity that the firm is able to produce in two plants. Find the total available profitas a function of total output if each plant covers half of the production. What is theresulting benchmark output, price and profit in equilibrium?
The total available profit is given by
ΠT = p(q)q − 2Γ[q/2],
which can be rewritten as
ΠT = (1− q)q − 2C − 2
∫ q/2
0γ[q]dq.
Solving the maximization problem (use Leibnitz integral rule) leads to
∂ΠT
∂q= 1− 2q − γ[q/2]
!= 0.
The profit-maximizing total supply and the resulting price is
q∗ =1
2(1− γ[q∗/2]) and p(q∗) =
1
2(1 + γ[q∗/2]).
The associated benchmark profit is ΠT = p(q∗)q∗ − 2Γ[q∗/2].
(b) Assume that the firm licenses all production to two firms. Each licensee is chargedthe same per-unit royalty fee, ρ, so that they face the same marginal cost. Show thatthe benchmark output can also be achieved by licensing all production if both licenseescompete in quantities and each supplies q∗/2. Set the royalty fee to be ρ = q∗/2.
If both firms face the same marginal cost, they will supply the same outputin equilibrium. Each firm solves
maxqi
πi = (1− qi − qj)qi − C −∫ qi
0γ[q]dq − ρqi
38
and obtains∂πi∂qi
= 1− 2qi − qj − ρ− γ[qi]!= 0.
Assuming that qi = qj = q∗/2 and ρ = q∗/2, we obtain
q∗ =1
2(1− γ[q∗/2]),
which is the same benchmark output.
(c) Consider now a situation where the patentholder produces in one plant and there is onlyone licensee. Is it possible to obtain the benchmark profit-maximizing quantities with aroyalty fee?
Hint: Start by solving the maximization problem of each firm
The licensee maximizes the following profit function
π1 = (1− q1 − q2)q1 − Γ[q1]− ρq1,
obtaining
q1(q2) =1
2(1− q2 − γ[q1]− ρ).
The profit function of the patentholder is given by
π2 = (1− q1 − q2)q2 − Γ[q2] + ρq1.
After solving the maximization problem the licensor obtains
q2(q1) =1
2(1− q1 − γ[q2]).
If ρ > 0, the equilibrium does not satisfy q1 = q2, since
q2 + γ[q2] = q1 + ρ+ γ[q1]
If the patentholder licenses only to one firm at a royalty fee ρ, the effectivemarginal cost of the licensee is higher than that of the licensor, hence inequilibrium the licensee produces less and production is inefficient. In thissituation the benchmark profit can not be reached.
39
Exercise 9.2: Litigation (Scotchmer, 2004)When a product patent is infringed the court can mete out damages. The infringer musteither pay the patentholder his lost profit or return his unjust enrichment. These remediesserve two purposes: compensate the infringed patentholder and deter infringement. Assumelinear demand for the proprietary good p(Q) = 1 − Q and constant marginal cost c. If aninfringer enters the market, both firms compete in quantities.
(a) Find first the patentholder’s profit under monopoly πM . Then, assuming that both firmsare competing in the market, find the profit of the patentholder πp
I , and the profit of theinfringer (C=competitor) πc
I . Show the results in a diagram.
No infringement (Monopoly)The firm maximizes
maxQ
πM = (1−Q)Q− cQ.
In equilibrium the output, price and profit are
QM =1
2(1− c)
pM =1
2(1 + c)
πM =1
4(1− c)2
Infringement (Competition)Each firm maximizes
maxqi
πi = (1− qi − qj)qi − cqi
and obtains
qi = qj =1
3(1− c).
The equilibrium output, price and total profit are
QC =2
3(1− c)
pC =1
3(1 + 2c)
πC =2
9(1− c)2
The profit of the patentholder and the profit of the infringer are
πpI = πc
I =1
9(1− c)2
If the infringer enters the market, the patentholder loses market share and themarket price decreases. See Figure 7.1 in Scotchmer (2004).
40
(b) Find the patentholder’s lost profit and the infringer’s unjust enrichment
The patentholder’s lost profit would be
πM − πpI =
5
36(1− c)2.
The infringer’s unjust enrichment would be
πpI =
1
9(1− c)2.
(c) In the case of punishment, does the patentholder prefer to get back his lost profit or theunjust enrichment of the infringer?
Because the monopoly profit is larger than the oligopoly profit, it follows that
πM > πpI + πc
I .
Thus,πM − πp
I > πcI .
The patentholder prefers the lost-profit rule to the unjust-enrichment rule.
(d) Does the lost-profit penalty deter infringement?
Yes. With the lost-profit rule the infringer would have to pay (πM − πpI ).
This is more than the profit he would achieve when infringing (πcI). The
unjust-enrichment rule might also deter infringement, since the infringer is setindifferent between infringing or not, when he has to disgorge his profit.
41
Problem Set 10: Private-Public Partnership
Exercise 10.1: Private-Public Incentives (Scotchmer, 2004)
In the “ideas” model, an idea is a pair (v, c), where v represents per-period social value andc the cost. There are two ways to fund the idea and turn it into an innovation: funding it bythe public sector, or by the private sector through proprietary prices. If the idea is funded bythe public sector, v also represents the per-period consumer surplus with competitive supplyand its discounted social value is (1/r)v. If the idea is funded by the private sector under anintellectual property regime, the profit available under a patent that lasts for discounted lengthT is πvT , where π is the fraction of per-period value that goes to profit. Assume that v andc are known to the private sector but not to the public sector.
(a) Which ideas would the public sector like to support? Which ideas would the privatesector like to support?
The public sector’s objective is to invest in those ideas for which vr − c > 0,
but it cannot identify which ideas they are. The private sector will invest inthose ideas for which πvT − c > 0.
(b) There are some ideas with a commercial value that is not sufficient to cover costs, butat the same time still would still be worth doing from a social point of view. Theseideas are characterized by the following inequality 1
rv > c > πvT . In an attempt tosee these ideas implemented the government decides to offer a subsidy s to the privatesector. Show which ideas will be realized? Show the result graphically. Is a subsidy agood solution?
With a subsidy from the government the private sector will invest in thoseideas for which πvT + s > c. The amount of funded ideas will increase withthe subsidy, but there are low-v ideas for which the subsidy should not begiven. The subsidy should not be given to ideas that fall in the filled triangleon the figure below:
v
c vr
πvT
πvT + s
s
(c) Suppose the government insists that in order to claim the subsidy the private sponsormust make a matching commitment of funds in some amount m. In other words, the
42
private and the public sponsors jointly contribute m + s and invest in ideas suggestedby the industrial sponsor. If the chosen idea is such that c > s + m, then the privatesponsor must provide the required supplement. If the idea is such that c < s+m, thenthe surplus goes to supporting graduate students. Finally, assume for simplicity that theprivate sponsor receives πvT on the subsidized innovation.Which ideas are going to be funded by the private sponsor alone (i), by the governmentand the industrial sponsor together (ii), and which ideas are not going to be funded atall (iii)? Show the results graphically.
i) It is always cheaper for the private sector to claim the subsidy for ideaswhere c > m and it is always cheaper for the private sector to not claimthe subsidy if c < m. We therefore restrict attention to ideas wherec < m. Further, if the private sector does not claim the subsidy, thecommercial value has to be above the cost, i.e. πvT > c. The set ofideas are marked in light grey in the figure below.
ii) We now restrict attention to ideas where the private sector will claim thesubsidy, i.e. where c > m. We split the analysis in two. First, for ideaswhere c > s +m the private sector will only implement ideas for whichthe commercial value is larger than the cost minus subsidy, i.e. whereπvT > c− s. Second, for ideas where m < c < s+m the private sectorwill only implement ideas for which the commercial value is larger thanthe required supplement m, i.e. where πvT > m. This sets a lower limitfor the value of an idea given by v = m
πT . The set of ideas are marked indark grey in the figure below.
iii) The remaining ideas are not going to be funded.
A graphical illustration can be found in Scotchmer (2004) figure 8.4:
v
c πvT + s
s
πvT
m
s+m
v
A partnership with matching funds sets incentives to the industrial sponsor tofund ideas that otherwise would not have been funded and solves the moralhazard problem.
43
Exercise 10.2: The Government Grant Process (Scotchmer, 2004)Grants, as opposed to prizes or intellectual property rights, cover researcher expenditure be-fore they are incurred. In Switzerland, the largest governmental sponsor is the Swiss NationalScience Foundation (SNSF); it calls for proposals and awards it funds to researchers whoseideas are promising and worth implementing.
Consider the following discrete-time model. Researchers get ideas for projects with a frequencyλn = 1/n, which means a researcher gets ideas every n’th period (n ∈ 1, 2, ...). A researchercan implement an idea immediately in each time period, and all ideas have the same value andcost (v, c). Assume that grants are awarded with certainty. Finally, if a proposal is acceptedfor funding the researcher receives c < ρ < v. The timing in the model is illustrated below:
t0 1 2 n−1 n n+1 n+2
Idea and implementation
No idea No idea No idea
Idea and implementation
No idea No idea
(a) Assume that a researcher can always complete all ideas when she chooses to incur theeffort cost c. Find the discounted present value of completing the ideas conceived, for aresearcher with creativity λ. Suppose that the discount factor is δ ∈ (0, 1).
Hint: If δ < 1, the formula for the geometric series is∑∞
t=0 δt = 1
1−δ .
To grasp the intuition, we first calculate the discounted present value for aresearcher with λ = 1
3 . The payoff from investing in all research ideas is thengiven by:
U(λ)|λ= 13
= (ρ− c) + 0δ + 0δ2 + (ρ− c)δ3 + 0δ4 + 0δ5 + (ρ− c)δ6 + · · ·= (ρ− c)(1 + δ3 + δ6 + δ9 + · · · ).
Setting δ = δ3 we find:
U(λ)|λ= 13
= (ρ− c)(1 + δ + δ2 + δ3 + · · · )
=ρ− c
1− δ=
ρ− c
1− δ3.
For the general case, define
1Iλ =
{1, if t is a time period where the researcher has an idea0, otherwise.
The discounted present value for a researcher with productivity λ is then givenby:
U(λ) =
∞∑t=0
(ρ− c)δtIλ = (ρ− c)
∞∑t=0
δnt = (ρ− c)
∞∑t=0
(δn)t =ρ− c
1− δn.
44
(b) If the researcher is awarded the grant and decides not to incur the cost c, what is thepresent value of lost future grants?
The net loss from missing future grants is simply the infinite stream of profitdiscounted from the time the researcher would have her next idea:
δnUλ = δnρ− c
1− δn.
(c) The researcher’s net gain if she does not implement the idea is the saved cost c. Underwhat condition is the researcher going to perform instead of pocketing the money?
The researcher will perform if
c � δnρ− c
1− δn⇔ c � δnρ
. Taking log’s and noting that log(δ) < 0, we derive that:
n � log(c)− log(ρ)
log(δ).
Hence, only researchers with low n and thus high productivity λ can be ex-pected to perform. Further, raising the grant size, ρ, will make more “lazy”researchers stay in research. Impatient researchers and researchers for whichcosts of implementing projects are high are also seen to be more likely to shirk.
45
Problem Set 11: Strategic Patenting, Patent Poolsand Mergers
Exercise 11.1: Strategic Patenting (Belleflamme and Peitz, 2010)
Consider a market where demand is given by P (q) = a−q. An incumbent firm has a proprietarytechnology with a constant marginal cost of cI (with cI < a). One other firm could enter themarket as a Cournot duopolist, but the technology available to this firm does not allow it tomake any positive profit if it enters. More precisely, the marginal cost corresponding to theentrant’s technology, cE , is such that cE � (a+ cI)/2 ≡ c.
(a) Show that the entrant’s technology implies the non-positivity of the entrant’s quantityat the Cournot-Nash equilibrium.
The incumbent and entrant maximizes
πI = (a− qI − qE)qI − cIqI and πE = (a− qI − qE)qE − cEqE
respectfully. The lowest possible value of the entrant’s marginal cost is cE = cand, hence, it suffices to show that qE � 0 for this case. Plugging in cE =(a+ cI)/2 in the entrants profit yields:
πE = (a
2− qI − qE)qE − cI
2qE.
We obtain the reaction functions:
qI =a− cI − qE
2and qE =
a− cI − 2qI4
.
Solving for the entrant’s quantity yields qE = 0.
(b) Suppose now that an alternative technology becomes available with a constant marginalcost c comprised between cI and c. Specifically, assume that a = 10, cI = 6, cE = 8,and c = 7. Show that, although the incumbent has no incentive to switch to and usethis technology, it has a higher incentive to acquire a patent on it than the entrant has.
To show this, we need to compare the incumbent’s and the entrant’s profits inthe situations where either one patents the technology. Denote the situationin which the incumbent patents by superscript IP . If the entrant patents weuse the superscript EP . We need to show that πIP
I − πEPI > πEP
E − πIPE .
If the incumbent patents: This is equivalent to the initial situation, inwhich no alternative technology is available. Hence, the entrant stays out ofthe market and makes πIP
E = 0. The incumbent maximizes monopoly profitsπIPI = (a− qI)qI − cIqI . We obtain πIP
I = 4.
If the entrant patents: The entrant’s profit is πEPE = (10−qI−qE)qE−7qE
and the resulting reaction functions are qE = 3−qI2 and qI =
4−qE2 . We obtain
46
the equilibrium quantities qI =53 , qE = 2
3 , πEPI = 25
9 , and πEPE = 4
9 .
It is now straight-forward to obtain that the incumbent’s and the entrant’sgains from patenting are 11
9 and 49 , respectively. Hence, the incumbent has
higher incentives to patent the technology even if it will not be used. Thiscan be seen as strategic patenting and may help to explain why some firmsfile for many more patents than they actually intend to use.
47
Exercise 11.2: Patent Pools and Mergers (Belleflamme and Peitz, 2010)
Consider a vertical market structure with 2 upstream firms (firms A and B) and 2 downstreamfirms (firms a and b). The downstream firms require the input of each of the upstream firms,who demand linear royalties (rAa , r
Ab , r
Ba , r
Bb ) charged for each unit the respective downstream
firm sells. Downstream firms face the inverse demand P (Q) = 1 − Q; where Q = qa + qb.Assume that the royalties accruing to the upstream firms are the only costs that the downstreamfirms face and all costs of the upstream firms are sunk.
(a) Draw the market structure and indicate each firm’s profits.
Firm A
π = ra qa+ rb qb
Firm B
π = ra qa+ rb qb
Firm a
πa = (1− (qa+qb)− ra − rb )qa
Firm b
πb = (1− (qa+qb)− rb − rb )qb
Consumers
(q) = 1− (qa+qb)
ra qa rb qbra qarb qb
(q)qa (q)qb
(b) Solve for the symmetric subgame-perfect Nash equilibrium in which the upstream firmsset non-discriminatory royalties (rI = rIi = rIj ) in the first stage and downstream firmsengage in Cournot-competition in the second stage.
The downstream interactions are of standard Cournot-type. Downstream firmsindividually maximize. In particular, πi = qi(1 − (qi + qj) − rA − rB). Oneobtains the reaction functions
qi(qj) =1− qj − rA − rB
2.
Therefore, in equilibrium, the downstream firms each choose quantity
q∗i =1− rA − rB
3.
48
Anticipating this, the upstream firms maximize
πI = rI(Q(rI , rJ )) = rI(21 − rI − rJ
3),
which yields the reaction functions
rI(rJ) =1− rJ
2.
We therefore obtain rA = rB = 13 and total output Q = 2
9 .
(c) Now assume that firms A and a merge (vertical merger) and maximize joint profits.Solve for a subgame-perfect Nash equilibrium in which the upstream firms set royaltiesin the first stage and downstream firms engage in Cournot-competition in the secondstage. How does the merger affect total royalties charged and quantities sold?
Denoting the profits of the merged firm as πA, we get the following profitfunctions:11
πB = rB(qa + qb)
πb = qb(1− (qa + qb)− rA − rB)
πA = rAqb + qa(1− (qa + qb)− rB).
While rA does not affect the output decision of downstream firm a anymore,it does still affect the choice of firm b (raising rivals cost effect). On theother hand, as firm a increases qa, this does not only lower the price in thedownstream market, it also lowers the amount of the good firm b produces.In a Cournot equilibrium, firm a will not take this into account, as it takes theamount produced by b as given (as opposed to, say, the Stackelberg case).The two firms’ reaction functions are:
qb(qa) =1− qa − rA − rB
2and qa(qb) =
1− qb − rB2
.
Solving these yields the equilibrium production levels
q∗a =1 + rA − rB
3and q∗b =
1− 2rA − rB3
.
Anticipating this, upstream firm B maximizes
πB = rB(2− rA − 2rB
3)
which yields the reaction function
rB(rA) =2− rA
4.
11We focus on the internal solution; Alternatively firm A sets rA so high to force firm b out of the marketand therefore anticipates the standard monopoly outcome in the second stage.
49
Firm A maximizes
πA = rA
(1− 2rA − rB
3
)+
(1 + rA − rB
3
)(1−
(2− 2rB − rA
3
)− rB
),
which yields the reaction function
rA(rB) =1− rB
2.
Therefore, in equilibrium, the royalty rates are rA = 2/7 and rB = 3/7. Thetotal quantity produced downstream is
Q∗ = q∗a(r∗A, r
∗B) + q∗b (r
∗A, r
∗B) =
2− rA − 2rB3
=2
7.
Note that while the total royalty has increased, downstream production stillincreases.
(d) Starting from the original (separate) setup, now assume that firms A and B merge (hori-zontal merger) and maximize joint profits. Solve for a subgame-perfect Nash equilibriumin which the upstream firm(s) set one nondiscriminatory royalties (rM = rMa = rMb )in the first stage and downstream firms engage in Cournot-competition in the secondstage. How does the merger affect total royalties charged and quantities sold? What isthe overall welfare implication of the merger compared to question (b)?
Now there is one upstream monopolist charging rM . The downstream Cournotequilibrium quantities (as in (b)) are q∗i = q∗j = 1−rM
3 where rM = rA + rB .
Therefore, the upstream firm maximizes πM = rM
(2(1−rM )
3
), which yields
the royalty rate r∗M = 12 and total quantity Q∗
M = 13 .
Since the merged upstream firm can always set rM = rA + rB it can as aminimum make the same profit as in (b). Therefore, the upstream firm isat least as good off as before. The intermediate firms are also better off asthey face lower marginal costs. Lastly, since quantities produced are higherconsumers are also better off.
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Problem Set 12: Diffusion
Exercise 12.1: Diffusion (Tirole, 1988)Consider an industry with an incumbent and a potential entrant. The incumbent makes flowprofit Πm
0 before adopting a process innovation (monopoly setting). If only the entrant hasadopted, the respective profits are Πd
1 and Πd2 (asymmetric duopoly). If only the incumbent
has adopted, it makes flow profit Πm1 > Πm
0 (monopoly setting). If both firms have adopted,each makes a flow profit of Πd (symmetric duopoly). Suppose that Πd
1 + Πd2 � Πm
1 andΠd
1 < Πd < Πd2. Time is continuous, and the rate of interest ir r. The cost of adoption, C(t),
is decreasing and convex, is “high” at date 0 (no one wants to adopt initially), and eventuallybecomes “low” (both firms end up adopting).
(a) Interpret the assumption about the flow profits.
The assumption Πd1 +Πd
2 � Πm1 requires that aggregate profits in a duopoly
are lower than in a monopoly setting (when taking the inefficient firm out).is the efficiency effect. Once the new technology is in the market, the profitobtained by an adopter incumbent is lower that the aggregate profit obtainedjointly by a passive incumbent and an adopter entrant.
The second assumption, Πd1 < Πd < Πd
2, suggests that the new technology issuperior to the incumbent’s initial technology.
(b) Show that the entrant is a “faster second” in the sense that it reacts earlier to preemptionthan the incumbent.
Given the competitor has innovated, the optimal date for adopting can bederived as follows:
• The incumbent solves:
maxt
[Πd −Πd
1
r− C(t)
]e−rt.
From the first order condition, we obtain
rC(TF1 ) + |C ′(TF
1 )| = Πd −Πd1.
• The entrant solves:
maxt
[Πd
r−C(t)
]e−rt.
From the first order condition, we obtain
rC(TF2 ) + |C ′(TF
2 )| = Πd.
To show that the entrant is a “faster second,” we use the results of themaximization problems. Since Πd > Πd −Πd
1, we have that
rC(TF2 ) + |C ′(TF
2 )| > rC(TF1 ) + |C ′(TF
1 )|.
51
Because cost of adopting are decreasing and convex, we finally know thatTF2 < TF
1 . The entrant will follow first, owing to the fact that the incumbentalready makes a profit before adopting the new technology.
52
Exercise 12.2: Open Source Software (Belleflamme and Peitz, 2010)Consider a market with n software firms competing in quantities. Each firm incurs a constantmarginal cost equal to ci and produces a differentiated product in quantities qi, sold at pricepi. Each firm faces linear inverse demand given by pi(qi, q−i) = a − qi − γ
∑j �=i qj where
γ ∈ [0, 1] is the inverse measure of product differentiation.
(a) Write down the firm’s problem and find the best response function and equilibriumproduction level.
The problem of a typical firm can be written
maxqi
πi = maxqi
(a− qi − γ∑j �=i
qj − ci)qi.
The FOC gives us firm i’s reaction function
qi(q−i) =1
2(a− γ
∑j �=i
qj − ci).
Summing over all i gives the total quantity in the Nash equilibrium
Q∗ =∑i
q∗i =1
2(na− γ(n− 1)Q∗ − C) ⇔ Q∗ =
na− C
2 + γ(n− 1),
where C =∑
i ci. By definition it is seen that
∑j �=i
q∗j = Q∗ − qi and C =∑j �=i
cj + ci.
Inserting the three results above into firm i’s reaction function gives us theequilibrium quantity
q∗i =a(2− γ)− ci(2 + γ(n − 2)) + γ
∑j �=i cj
(2 + γ(n− 1))(2 − γ).
It can be seen from firm i’s profit and best response function above that
π∗i = (q∗i )
2.
(b) In the pre-innovation stage, all firms use the technology and produce at marginal costci = c > 0. Suppose now that firm 1 has developed a software innovation that has theeffect of decreasing its marginal cost to c1 = c − x with 0 < x < c. Find the profit offirm 1 when the innovation is patented or kept secret.
Plugging the marginal costs into the result above gives the results:
πsecret1 =
(a(2− γ)− (c− x)(2 + γ(n− 2)) + γ(n− 1)c
(2 + γ(n− 1))(2 − γ)
)2
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(c) Alternatively, firm 1 can choose to disclose the source code of its innovation. Disclosurewill entail two contrasting effects. On the one hand, the quality of the software will beenhanced thanks to the efforts of open source developers. We model this positive effectby assuming that firm 1’s cost will be reduced further after disclosure: c1 = c − αx,with 1 < α < c/x as a measure of the contribution from the open source community.On the other hand, disclosure also means that firm 1’s competitors will have access tothe innovation as well. We assume that after disclosure, the other firms’ marginal costbecomes cj = c − αβx, where 0 < β < 1 measures the generality of firm 1’s softwareinnovation (if β = 0, the software is completely specific to firm 1’s production process;if β = 1, the software is completely general and yields identical benefits to all firms).Find the profit of firm 1 when the firm discloses the source code under open source.
Inserting the marginal costs under disclosure gives the results:
πdisclose1 =
(a(2 − γ)− (c− αx)(2 + γ(n− 2)) + γ(n− 1)(c − αβx)
(2 + γ(n− 1))(2 − γ)
)2
(d) Find the condition under which the firm disclose under open source and discuss theeffects of (i) competition, (ii) the generality of the software innovation, and (iii) thecontribution from the open source community.
The innovative firm will release the source code if profit is larger by doing so
πdisclose1 � πsecret
1 ⇔ α � 2 + γ(n − 2)
2 + γ(n− 2)− γβ(n− 1).
It is now seen that higher likelihood of source code release under open sourcegoes along with
i) lower competition on the product market (lower γ or lower n),
ii) higher specificity of the software for the innovating firm (lower β), or
iii) larger contributions from the open source community (larger α).
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