Exercises in Innovation Economics

Transcription

1 University of Zurich Department of Business Administration Chair of Entrepreneurship Plattenstrasse 14, CH-8032 Zürich 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 Part II: Michael Ribers Problem Sets Session Dates: 17.4., 24.4., 8.5., 15.5., 22.5., and My office hours are on Tuesdays, 5-7 p.m., and by appointment. daniel.halbheer@business.uzh.ch. My office hours are on Tuesdays, 5-7 p.m., and by appointment. michael.ribers@business.uzh.ch. 1 Exercises are based on problem sets prepared by Susan Mendez.

2 List of Exercises Problem Set 1: A Primer in Oligopoly Cournot Competition Investment in the Cournot Model Bertrand Competition...6 Problem Set 2: Innovation and Market Structure Monopoly (Tirole, 1988) Competition (Tirole, 1988) Oligopoly...11 Problem Set 3: Auctions and Contests The Vickrey Auction (Mas-Collel et al., 1995) Vickrey Auctions to Choose among Ideas (Scotchmer, 2004) Prototype Contests to Choose among Ideas (Scotchmer, 2004) Problem Set 4: Patents I Optimal Patent Length (Tirole, 1988) Pooling of Complementary Patents (Motta, 2004) Problem Set 5: Patent Races Research Intensity and Market Structure (Tirole, 1988) Patent Value and the Number of R&D Attempts (Scotchmer, 2004) Problem Set 6: Research Joint Ventures Cooperative R&D (d Aspremont and Jacquemin, 1988) Problem Set 7: Patents II Patent Breadth and the Ratio Test (Scotchmer, 2004) Optimal Patent Length and Breadth (Gilbert and Shapiro, 1990) Problem Set 8: Licensing Licensing Basic Research (Scotchmer, 2004) Anti Competitive Cross Licensing (Motta, 2004) Problem Set 9: Licenses and Litigation Profit Neutrality in Licensing (Maurer and Scotchmer, 2004) Litigation (Scotchmer, 2004) Problem Set 10: Private-Public Partnership Private-Public Incentives (Scotchmer, 2004) The Government Grant Process (Scotchmer, 2004) Problem Set 11: Strategic Patenting, Patent Pools and Mergers Strategic Patenting (Belleflamme and Peitz, 2010)

3 11.2 Patent Pools and Mergers (Belleflamme and Peitz, 2010) Problem Set 12: Diffusion Diffusion (Tirole, 1988) Open Source Software (Belleflamme and Peitz, 2010)

4 Problem Set 1: A Primer in Oligopoly Exercise 1.1: Cournot Competition Two firms produce homogenous goods and compete in quantities. The industry s (inverse) demand function is P (Q) =a bq, whereq is total industry output, equal to q 1 + q 2.The cost function for each firm is given below: C 1 (q 1 )=F 1 + c 1 q 1 C 2 (q 2 )=F 2 + c 2 q 2. (a) Assuming that F 1 and F 2 are small, compute the Nash equilibrium. Firm i solves max π i = P (q i + q j ) q i C i (q i ). q i First-order condition: π i q i =0 a 2bq i bq j c i =0. Firm i s reaction function is therefore given by R i (q j )=q i = 1 2b (a bq j c i ). We thus have two equations and two unknowns. Solving yields q 1 = 1 3b (a 2c 1 + c 2 ) and q 2 = 1 3b (a 2c 2 + c 1 ). (b) If c 1 >c 2, 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 as its rival s cost increase. If c 1 > c 2, firm 2 will thus produce more. The production decision does not depend on F 1 and F 2 (these are paid regardless of the output level and are not influenced by a rival s production decision). If these fixed costs were sufficiently large, however, they might influence the firms decisions to produce anything at all. (c) Graph the reaction functions for c 1 = c 2. Suppose c 1 rises. Graphically show the effect in the firms reaction functions and the equilibrium quantities. 4

5 Exercise 1.2: Investment in the Cournot Model Consider two firms that compete in quantities. The (inverse) demand function is given by P (Q) =3 Q, whereq = q 1 + q 2. Assume that firm 1 makes an observable investment decision before the firms set quantities. If firm 1 decides not to invest, it pays nothing and incurs a marginal cost of 1. If firm 1 decides to invest, it pays F>0 and incurs a marginal cost 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. What is firm 1 s profit in each case? If firm 1 does not invest, both firms solve In equilibrium, If firm 1 does invest, firm 1 solves max q i (3 q i q j 1) q i. q 1 = q 2 = 2 3, p = 5 3, π 1 = π 2 = 4 9. max q 1 (3 q 1 q 2 ) q 1 F, while firm 2 solves as before. In equilibrium, max q 2 (3 q 1 q 2 1) q 2 q 1 = 4 3, q 2 = 1 3, p = 4 3, π 1 = 16 9 F, π 2 = 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 increases its output, the other firm wishes to decrease its output. By investing, firm 1 reduces 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

6 Exercise 1.3: Bertrand Competition Two firms produce heterogeneous goods and compete in prices. The demand and cost functions for each firm are as follows: q 1 =2 2p 1 + p 2 and C 1 (q 1 )=q 1 + q1 2 ; q 2 =2 2p 2 + p 1 and C 2 (q 2 )=αq 2 + q2 2. (a) Calculate each firm s reaction function. Firm 1 solves First-order condition: max p 1 π 1 = p 1 q 1 (p 1,p 2 ) C 1 (q 1 (p 1,p 2 )). π i p i =0 R 1 (p 2 )=p 1 = p 2. By a similar calculation, one may obtain firm 2 s reaction function R 2 (p 1 )=p 2 = 5+α 6 (b) Given α =1, calculate the equilibrium prices p 1. Given α =1, R 2 (p 1 )=1+(5/12)p 1. To determine the equilibrium, solve for the intersection of the reaction functions: p 1 =1+ 5 ( 1+ 5 ) p 1, whence follows p 1 = p 2 =12/7. (c) Given α =1, graph the reaction functions. (d) Suppose α rises. Graphically show the effect on the firms reaction functions and the equilibrium prices. If α rises, firm 2 s reaction function shifts rightwards. As a result of the increase 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 c 1 >c 2? If the firms are symmetric, prices p i are set equal to marginal cost c, and each firms makes zero profit. This result is known as the Bertrand Paradox. In this case p i = c 1. Note that firm 2 makes a positive profit. 6

7 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 offered by the product market. In particular, assume that the innovation lowers the monopolist s (constant) unit production cost from an initial high level c H to a level c L <c H.LetD(p) be the (downward-sloping) demand for the good produced by the monopoly. The innovation is protected by a patent of unlimited duration. Letting the constants r and v m 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 = 0 e rt v m dt = 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 c L D(c)dc. The planner sets a price equal to marginal cost, i.e., c H before innovation and c L afterwards. Thus, the innovation generates an additional net social surplus per unit of time equal to v s = ch c L D(c)dc, whence follows that V s = vs r. (b) Now assume the monopoly is run by a profit-maximizing owner. Let Π m (c) be the monopolist s profit when the output is sold at the monopoly price p m (c) given marginal cost level c. 2 Show that the monopolist s incentive to innovate is given by V m = 1 r ch c L D (p m (c)) dc. Show further that V m <V s and discuss the intuition for this result. 2 The monopoly price p m (c) is a solution to max π m (p, c) =(p c)d(p). p Now observe that for each different value of c there will typically be a different optimal price. We therefore define Π m (c) =π m (p m (c),c), which tells us what the optimized value of π m is for different choices of c. 7

8 Hint: Note that v m =Π m (c L ) Π m (c H )= ch c L ( ) dπm (c) dc dc and apply the envelope theorem to obtain an expression for the integrand. From the envelope theorem, dπ m (c) dc = D (p m (c)). To see this, differentiate Π m (c) =π m (p m (c),c) with respect to c: dπ m (c) dc = πm (p m (c),c) p dp m (c) dc + πm (p m (c),c) c = πm (p m (c),c). c Here, the last equality uses the fact that p m (c) is the choice that maximizes π m,so π m (p m (c),c) =0. p A better way to write this is With this in mind, dπ m (c) dc = πm (p, c) c p=p m (c). dπ m (c) dc = d dc (p c)d(p) = D(pm (c)). Putting the pieces together, we have V m = 1 r ch c L D (p m (c)) dc. Note that p m (c) >c, and therefore D(p m (c)) <D(c). Thus, a comparison of V m and V s reveals that V m <V s. This is easy to understand, because monopoly pricing at any cost level yields underproduction as compared with the social optimum. Therefore, the monopolist s cost reduction pertain to a smaller number of units. 8

9 Exercise 2.2: Competition (Tirole, 1988) Consider now a situation in which a large number of firms produce a homogenous good with a technology exhibiting (constant) unit production costs c H. These firms are initially involved in Bertrand price competition, so that the market price is c H and all the firms are earning zero profit. The firm that obtains the new technology c L 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 c H, which calls for the distinction of two possible cases: p m (c L ) >c H and p m (c L ) c H. In the second case, in which the innovation is called drastic or major, the innovating firm enjoys monopoly power and the other, less efficient firms produce nothing. In the first case, the innovator is constrained to charge c H because of potential competition from firms equipped with the old technology. The innovation is then called non-drastic or minor. (a) Non-drastic process innovation. Show that the incentive to innovate is and establish that V m <V c <V s. V c = 1 r ch c L D (c H ) dc, In the case of a non-drastic innovation, the innovator is constrained to set a price equal to c H. His profit per unit of time is given by Π c =(c H c L )D(c H )= ch c L D (c H ) dc. Thus, the incentive to innovate in the competitive situation is V c = 1 r ch c L D (c H ) dc. We now compare V c to a monopolist s innovation incentives. 3 Notice that c H <p m (c L ) p m (c) for all c c L and thus that D(c H ) >D(p m (c H )). From this, we derive V m = 1 r ch c L D (p m (c)) dc < 1 r ch c L D (c H ) dc = V c. To compare V c to the social planner s incentives, observe that D(c H ) <D(c) for all c<c H, and therefore V c = 1 r ch c L D (c H ) dc < 1 r ch c L D(c)dc = V s. (b) Drastic process innovation. Show that the incentive to innovate is V c = 1 r (pm (c L ) c L )D(p m (c L )), 3 Observe that the innovator is a monopolist before innovation and afterwards. 9

10 and establish further that V m <V c <V s. In the case of a drastic innovation, the innovator sets the monopoly price p m (c L ). His profit per unit of time is given by Π c =(p m (c L ) c L )D(p m (c L )). Thus, the incentive to innovate in the competitive market environment is V c = 1 r (pm (c L ) c L )D(p m (c L )). We now compare V c to a monopolist s innovation incentives. Using the definition of v m yields V m = 1 r {(pm (c L ) c L )D(p m (c L )) (p m (c H ) c H )D(p m (c H ))} <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? Provide an intuitive explanation. The result follows from different initial situations: The monopolist replaces himself when he innovates whereas the competitive firm becomes a monopoly. In the literature, this property is called the replacement effect (due to Arrow, 1962). 10

11 Exercise 2.3: Oligopoly Consider a symmetric n-firm Cournot oligopoly with a linear inverse demand function P (Q) = 1 Q, withq = i q i, and constant average cost c. (a) Show that each oligopolist produces an output equal to q C (n, c) = 1 c n +1, and hence earns a profit ( ) 1 c 2 π C (n, c) =. n +1 Each firm i solves qi C arg max{π i (q i,q i) q C i 0}. qi The first-order conditions are (1 qi C qj C c) qc i =0. i j Applying symmetry (i.e., q C = qi C = qj C ), we have q C (n, c) = 1 c n +1. By substitution, p(n, c) =1 nq C = 1+nc n +1 and π C (n, c) = ( ) 1 c 2. n +1 (b) Show that the private value of a drastic innovation that reduces cost from c to c L <cis ( (1 V C (n) = 1 ) 2 ( ) ) cl 1 c 2. r 2 n +1 Observe that π C (1,c L ) is the innovator s monopoly profit. Thus, the innovator s profit per unit of time is v C (n) =π C (1,c L ) π C (n, c). By substitution, ( (1 V C (n) = 1 ) 2 cl r 2 ( ) ) 1 c 2. n +1 (c) How does V C (n) change with n? Discuss the intuition. The value of an innovation is increasing in n, so innovation incentives are larger in more competitive industries. Intuitively, the result reflects the fact that each oligopolist earns a positive profit before innovation takes place. In case of a successful innovation, the innovating oligopolist partially replaces himself, which reduces his incentives to undertake R&D (see above). 11

12 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 is auctioned off to I bidders. Bidder i s valuation of the object (in monetary terms) is v i. The auction 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 the object but pays the auctioneer the amount of the second-highest bid. If more than one bidder submits the highest bid, each gets the object with equal probability. Show that submitting a bid of v i with certainty is a weakly dominant strategy for bidder i. Suppose not. Assume bidder i bids b i >v i. Then if some other bidder bids something larger than b i, bidder i is just as well off as if he would have bid v i.if all other players bid lower than v i, then bidder i obtains the object and pays the amount of the second highest bid. If the second highest bid is b j <v i, this results in the same payoff for player i as if he bid v i. However, suppose that the second highest bid of the other is b j >v i. Then, by bidding b i bidder i will win the object and obtain a negative payoff. By bidding v i he will not win the object and obtain a payoff of zero. Therefore, bidding b i >v i is weakly dominated by bidding v i. Suppose bidder i bids b i <v i. Then if all other bidders bid something smaller than b i, bidder i is just as well off as if he would have bid v i. He will win the object and pay the second highest bid. If some other player bids higher than v i, then bidder i does not win the object regardless whether he bids b i or v i. However, suppose that nobody bids higher than v i and the highest bid of the other player is b j with b i <b j <v i. Then by bidding b i bidder i will not win the object, therefore getting a payoff of zero. By bidding v i, he would win the object, pay b j <v i, and thus obtain a payoff of v i b j > 0. Therefore, bidding b i <v i is weakly dominated by bidding v i. This argument implies that bidding v i is a weakly dominant strategy. Thus, each bidder has an incentive to report faithfully his valuation of the object. 12

13 Exercise 3.2: Vickrey Auctions to Choose among Ideas (Scotchmer, 2004) We now shed light on the conditions under which the Vickrey auction mechanism identifies the best idea for a targeted objective. From a social planner s view, the most attractive idea would be the one providing the greatest social surplus. Let the pair (v i,c i ) describe and idea. Define prospective social surplus of idea i as the difference between the observable discounted social value v i /r and the cost c i of developing the idea into an innovation: s i = v i r c i. Suppose that the sponsor asks each prospective innovator i to report social surplus s i,and that he chooses the firm that reports the highest surplus to invest. Label the highest and the second-highest report s i and s j, respectively. (a) Show that by promising a payment of v i /r s j, a sponsor can safely pick the firm that claims the highest surplus. If firm i is chosen to invest, it ends up with a profit equal to v i /r s j c i = s i s j. Using the above result, each firm i s weakly dominant strategy is to report faithfully the surplus s i it can deliver. Thus, the sponsor will ask the firm reporting the highest surplus to invest. Observe that if s i is close to s j, the payment to the winning firm will be close to cost. (b) In what sense does the Vickrey auction mechanism yield an efficient outcome? What happens if the value v i is not observable ex post? The Vickrey auction mechanism elicits efficient investment in the sense that the high-surplus firm is chosen to invest, and that there is no duplication of costs (only the innovating firm pays its cost c i ). If v i is not observable ex post, payments cannot depend on delivered value. In that case, the mechanism does not work. However, note that the Vickrey auction 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 may fear that the sponsor will renege, or give a smaller price than he deserves, which will negatively affect investment incentives. 13

14 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 with payments depending on the delivered value. The idea is to let the firms demonstrate their ideas by developing prototypes, and then choose ex post between them. Assume two firms i =1, 2 have identical ideas (v, c). Eachfirmi announces ex ante the price ρ i at which it is selling the innovation ex post. After naming prices, the firms and the sponsor write contingent contracts. 4 Then, firms invest and deliver prototypes. If firm i s prototype is chosen by the sponsor ex post, it receives the specified price ρ i. (a) Consider the minimum prices (ρ 1,ρ 2 )=(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. (b) Let The equilibrium bids ρ i are such that neither firm has an incentive to revise its bid, assuming that the other firm s bid is fixed. Since each firm would win with probability 0.5, expected revenues are 1 2 ρ 1 = 1 2 ρ 2 = c. Given firm 2 demands 2c in the event it is chosen, firm 1 can improve profit by reducing its price to ρ 1 =2c ɛ. Then, firm 1 makes profit ρ 1 c = c ɛ>0 instead of 0. Thus, the prices (ρ 1,ρ 2 ) cannot be sustained in equilibrium. F (ρ) = { ρ (v/r) 1 c ρ + c (v/r) if 0 ρ c if c ρ v r be the cumulative distribution of the firms prices, and consider the strategies do not deliver 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 strategies F 1 = F 2 = 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 s expected profit is ρ[f 2 (c)+1 F 2 (ρ)] c =0. This is an equilibrium because each price in the support of the distribution yields the same expected profit as any other price, namely zero. Remark: The term [F 2 (c) +1 F 2 (ρ)] represents the probability that firm 1 wins the prototype contest. With probability F 2 (c) firm 2 does not invest, and with probability 1 F 2 (ρ) firm 2 invests but demands a higher price than does firm 1. 4 The contracts ensure that inventors are not subject to hold-up., 14

15 (c) In what sense do prototype contests yield an inefficient outcome? With positive probability, the sponsor does not get the innovation, and even if he gets it, there is a large probability that the costs will be duplicated. This is the social price for the inability to observe or verify value. 15

16 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. The industry s demand function is D(p) =1 bp, with b>0. Assume that one firm has access to a cost-reducing technology which allows production at marginal cost c< c by spending φ(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 which the technology c can be used freely be all firms). The firms compete à la Bertrand, and the rate of interest is r. (a) Denote by Δ c c the magnitude of the cost reduction chosen by the firm that implements the new technology. Show that where τ 1 e rt and D 1 b c. Hint: The firm will choose Δ so as to ( T max Δ 0 Δ(τ) = τd rk, Δ(1 b c)e rt dt KΔ2 2 Given patent life T, the inventor chooses the Δ that maximizes the difference between the benefits of implementing the new technology and the cost of R&D: ( T max Δ =max Δ =max Δ ) Δ(1 b c)e rt dt KΔ2 0 2 ( T ) ΔD e rt dt KΔ2 0 2 ( τ KΔ2 ΔD r 2 ). (D 1 b c) ). (τ 1 e rt ) The first-order condition reads τd KΔ =0, r which implies Δ(τ) = τd rk. Observe that Δ(τ) is increasing in τ (and hence T ), so that a longer patent life will lead to a higher cost reduction. 16

17 (b) Show that the new technology increases welfare by ( T ) ( W (τ) = Δ(1 b c)e rt dt KΔ2 1 τ + 2 r 0 Hint: The generated consumer surplus is equal to (DΔ(τ)+ b2 ) ( 1 τ Δ(τ)2 e rt dt = r T )(DΔ(τ)+ b2 Δ(τ)2 ). )(DΔ(τ)+ b2 Δ(τ)2 ). Use a graphical argument to derive the formula for the generated consumer surplus. Note that the firm s profit is part of the consumer surplus after the patent 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 T }) max Δ(1 b c)e rt dt KΔ2 dτ Δ 0 2 Optimal patent life, τ, is a solution to = Δ(τ)D. r dw (τ) dτ = Δ(τ)D ( ) 1 τ (DΔ + (τ)+bδ(τ)δ (τ) ) r r 1 (DΔ(τ)+ b2 ) r Δ(τ)2 =0. Substituting Δ(τ) and Δ (τ), the first-order condition simplifies to f(τ) := 3 2 bτ 2 +(rk b)τ rk =0. Note that f(0) < 0 and f(1) > 0. Asf(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 patent length, patent life shouldn t be set at an infinitely large number. The reason is the need to balance off the benefits of a larger invention against the inefficiency of monopoly pricing. 17

18 Exercise 4.2: Pooling of Complementary Patents (Motta, 2004) To produce a certain (homogenous) final good, n manufacturers need two complementary technologies, whose patents are owned by two firms A and B, who separately license the technologies at a unit royalty fee w i (i = A, B). The game is as follows. In the first stage, the patentholders independently and simultaneously decide the royalty level. In the second stage, the manufacturers compete à la Bertrand, and incur unit costs c + w A + w B,wherec<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 s equilibrium profit. In the last stage, given that manufacturers compete in prices, the Bertrand equilibrium applies: the market price will be and final demand p = c + w A + w B, q =1 (c + w A + w B ). In the first stage, each patentholder decides the royalty fee so as to max w i π i = w i (1 c w A w B ). From π i / w i =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 exploitation of their patents to a patent pool. It is now the pool which sets the value of both royalties. Find equilibrium values of royalties and final prices under the patent pool and compare them with the previous case. What is the equilibrium profit of the patent pool? Under the patent pool, there is joint-profit maximization of the patentholders. The pool s problem is therefore to max π P = w A (1 c w A w B )+w B (1 c w B w A ). w A,w B Solving the first-order conditions gives the symmetric solution w P = 1 c 4. 18

19 By substitution, the market price is and the pool s total profit is p P = 1+c 2, π P = (1 c)2. 4 (c) Show that forming the patent pool is both profitable for the patentholders and good for consumers. It is straightforward to see that the patent pool Pareto dominates the situation where the two patents are licensed independently. Final prices (as well as royalties) are lower (therefore, consumers are better off) and patentholder s profits are higher. Observe that manufacturers in this example always get zero profits. 19

20 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 the firms has a private value V of the patent. Patent life is infinitely long, time is continuous, and the rate of interest is r. The probability of success by firm i at given time, t, is an exponential function: If t i represents firm i s (random) success date, then Prob {t i t} =1 e h it. 5 The choice variable for each firm i is a flow of expenditure x i that yields a probability of discovery h i = h(x i ) per unit of time. Assume that h > 0, h < 0, h(0) = 0, h (0) = +, and h (+ ) =0. 6 (a) Let firm 1 choose expenditure y x 1, and let firms 2, 3,...,n choose expenditure x x i. 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{t i t} =1 Pr{t i t} = e h it. Thus, assuming independence of R&D activities, the probability that none of the firms has discovered until time t is e h(x 1)t e h(x 2)t... e h(xn)t = e [h(y)+(n 1)h(x)]t. (b) Show that firm 1 s expected intertemporal profit is Hint: Note that 0 h(y)v y h(y)+(n 1)h(x)+r. [h(y)v y]e [h(y)+(n 1)h(x)]t e rt dt = h(y)v y h(y)+(n 1)h(x)+r. Suppose that none of the firms has discovered before time t. By spending amount y on R&D, firm 1 is the first to innovate with probability h(y), and earns, starting from that moment, V. Thus, firm 1 s present discounted value of the expected profit over time is 0 [h(y)v y]e [h(y)+(n 1)h(x)]t e rt dt = h(y)v y h(y)+(n 1)h(x)+r. 5 Note that h i is the conditional probability of success, given no success to date. Further, the expected time till success for firm i is the reciprocal of the hazard rate; that is E(t i)=1/h(x i). 6 This assumption of a memoryless R&D technology implies that a firm s probability of making a discovery and obtaining a patent at a point in time depends only on this firm s current R&D experience and not on its past R&D experience. 20

21 (c) Show that the first-order condition for the symmetric Nash equilibrium expenditure is given 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 ( ) =0and h < 0). Hence, there exists a unique equilibrium research intensity x (n). (d) Suppose that the objective function is strictly concave. Show that dx (n) > 0. dn 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 secondorder 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 get dx (n) dn > 0. 21

22 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. Although the ideas are symmetric, we assume that the innovators employ different approaches, so that the successes and failures of the different attempts are independent. Therefore, if n approaches are taken, the probability that all of them fail is p n, so that the probability of at least one success is 1 p n, which is denoted by P (n). 7 Let r denote the rate of interest, and let S = v/r denote the social value of the innovation. Further, let Π denote the patent value and assume that Π S. (a) Show that the equilibrium number of firms that enter the patent race, n e, satisfies 1 n e ΠP (ne ) c. For each n, the expected per-firm profit is 1 ΠP (n). n Firms will enter the patent race up to the point where an additional firm would not make profit. Thus, n e satisfies 1 n e ΠP (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 n th entrant is S[P (n) P (n 1)] c. The social optimal number of participants, n, can be described as the number where 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 an intuitive explanation. The reason is that the marginal entrant receives the average profit rather than the marginal profit, but it is the marginal profit that determines the optimal number of participants (this problem is usually discussed as the problem of the commons ). In our setting, the average profit is greater than the marginal profit due to the concavity of P ( ). Thus, the private value of entry can be positive even if the social value of entry is negative. 7 Note that, in contrast to the previous model, a failure does not lead to a renewal of efforts. 22

23 (d) Show that the patent value Π to induce the optimal number of attempts should be chosen 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

24 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 be p = a Q, whereq = q 1 + q 2 is the total production and a>0. Each firm has marginal costs c i = c x i λx j,wherex i 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 x j made by the other firm. 8 The cost of R&D is given by κ(x i )= g 2 x2 i, with g> 4 3. We consider two different two-stage games, in which the investments x i are the strategies of the first stage, followed by the quantities q i 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 q C i = a c +(2 λ)x i +(2λ 1)x j. 3 At the last stage of the game, each firm i chooses q i so as to max q i (a q i q j c i (x i,x j )) q i. Note that the profit-maximization problem is the same as in a typical (one-shot) Cournot game with heterogenous firms. Therefore, qi C (c i(x i,x j ),c j (x i,x j )) = a 2c i(x i,x j )+c j (x i,x j ) 3 = a c + x i(2 λ)+x j (2λ 1). 3 (ii) Show that the unique (and symmetric) R&D level for each firm is given by x C = 2(a c)(2 λ) 9g 4 2λ +2λ 2. Recall that each firm s product-market profit is equal to ( qi C each firm chooses x i to ) 2. Thus, max x i π i (x i,x j )= ( a c + xi (2 λ)+x j (2λ 1) 3 ) 2 g 2 x2 i. 8 When the rival invests x j,itisasiffirmi had done that investment itself and reduced its cost by λx j. 24

25 (iii) Show that By taking the first derivative and applying symmetry (x i = x j = x C )one obtains x C 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) 2 π i / x i x j 2 π i / x 2 i = λ)(2λ 1) 2(2 2(2 λ) 2 9g < 1 and are satisfied for 2(2 λ)(λ 1) g> 3 A sufficient condition for stability to be met is that g>4/3, which also satisfies the second-order condition. q C = 3(a c)g 9g 4 2λ +2λ 2 and π C = (a c)2 g(9g 8+8λ 2λ 2 ) (9g 4 2λ +2λ 2 ) 2. The results follow by substitution. (iv) Show that consumer surplus and welfare are given, respectively, by CS C = 18(a c) 2 g 2 (9g 4 2λ +2λ 2 ) 2 and W C = 4(a c)2 g(9g 4+4λ λ 2 ) (9g 4 2λ +2λ 2 ) 2. Recall that CS C = ( a p C) Q C /2 and that W C = CS C +2π C. 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 x J = 2(a c)(1 + λ) 9g 2(1 + λ) 2. Hint: Maximize 2 i=1 π i(x i,x j ) and focus on the symmetric equilibrium. The firms solve [ 2 (a ) ] c + xi (2 λ)+x j (2λ 1) 2 max = g x 1,x x2 i. i=1 Taking first derivatives and focusing on the symmetric equilibrium, the result follows. 25

26 (ii) Show that q J = 3(a c)g 9g 2(1 + λ) 2 and π J = (a c) 2 g 9g 2(1 + λ) 2. The results follow by substitution. (iii) Show that consumer surplus and welfare are given, respectively, by CS J = 18(a c) 2 g 2 (9g 2 4λ 2λ 2 ) 2 and W J = 4(a c)2 (9g 1 2λ λ 2 ) (9g 2 4λ 2λ 2 ) 2. Recall that CS J = ( a p J) Q J /2 and that W J = CS J +2π J. 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) x J > x C, (ii) q J > q C, and (iii) W J > W C. Provide an intuitive explanation for these results. See d Aspremont and Jacquemin (1988). 26

27 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 firms colluding, government intervention, or labour unions), there is an innovator and a number of imitators 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 their product as perfect substitutes for the patented technology and the price would be lower than the monopoly price but higher than the competitive price (p m > p >p c ) since entry is costly. Furthermore, assume that the market demand is given by x(p) and that marginal production costs are normalized to zero. (a) Let K be the cost of entering the protected market and assume that there are n active firms in the market (including the innovator). Given discounted patent length T,find the 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 firm produces the same quantity. Therefore, the profit of each entrant is the aggregate profit, x(p(n))p(n), divided by the number of firms, n, minus entry cost, K. Entry stops when firms are making positive profits and one additional entrant 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 be duplication cost for inventing around the protected technology or, alternatively, as a licensing fee charged by the patentholder to allow entry. Does it make a difference which interpretation is used? Discuss this issue from the point of view of the patentholder. The patent holding firm would prefer licensing, because by charging a license fee of K, it can at least earn (n 1)K and compensate part of the loss from giving up its monopoly position. (c) Put yourself in the social planner s shoes and assume that K is large enough to cover the patentees cost. Let τ T = e rt dt 0 denote discounted patent length from time 0 to τ, and suppose that there are two policies to choose from: (T m,k) is a short enough patent, so that no single competitor enters the market, but long enough to cover the patentholder s cost. (T c,k) is a long lived patent that allows entry in the protected market. Let the discounted profit of the patentholder be π m = p m x(p m )T m under the policy (T m,k) and π = px( p)t c under the policy (T c,k). Graphically illustrate the per-period profit of the patentholder and the deadweight loss caused by each policy. 27

28 The profit of the patentholder and the associated deadweight loss can be seen below (filled for the short lived patent): p p m p x(p m ) x( p) x (d) Which policy will consumers prefer? Hint: Use the Ratio Test and proceed along Technical Note (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 for the firm. Consumers are better off with the policy that generates a larger consumer surplus. They will prefer policy (T c,k) to (T m,k) if ( ) ( ) 1 1 T c s( p)+ r T c s(0) >T m s(p m )+ r T m s(0). (S1) Rearranging, we have or, equivalently, T c (s( p) s(0)) >T m (s(p m ) s(0)), T c (s(0) s( p)) <T m (s(0) s(p m )). In exercise (c), we saw that the sum of the profit and the deadweight loss is equal to the loss in consumer surplus. Thus, we can write s(0) s(p) = px(p)+l(p) for p [0,p m ]. Inserting this above gives us: T c [ px( p)+l( p)] <T m [p m x(p m )+l(p m )]. Making use of the fact that both policies are equally profitable if p m x(p m )T m = px( p)t c, we can rewrite the above condition as T c [ px( p)+l( p)] < px( p)t c p m x(p m ) [pm x(p m )+l(p m )]. 28

29 Rearranging terms, the condition boils down to p m x(p m ) l(p m ) < px( p) l( p) (S2) The policy that generates lower prices p is (T c,k) and it is preferred by the consumers 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

30 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 that the patentee obtains during patent protection. Assume that a broader patent increases the patentee s market power and therefore the associated deadweight loss. In other words, if we let 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 and the interest rate is r. The social planner s goal is to maximize total social welfare Ω(T,π) by optimally choosing T and π subject to achieving a cost-effective reward for the innovation with value V,thatis V (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 profits as functions of T and π. Hint: Ω(T,π) is the discounted social welfare during patent protection plus the discounted social welfare after the patent expires. The present value of the patentee s profits V (T,π) is the sum of the discounted profits obtained during patent protection and the discounted profits after patent expiration. Discounted social welfare and the present value of the patentee s profit are given by T Ω(T,π)= W (π)e rt dt + We rt dt and respectively. 0 V (T,π)= T 0 πe rt dt + T T πe rt dt, (b) Define π = φ(t ) as the flow of profits needed to obtain V. For a given T>0 we can then write V as: V T 0 φ(t )e rt dt + T πe rt dt. Solve the integral and differentiate with respect to T to find an expression for φ (T ). T Solving the integrals in V φ(t )e rt dt + πe rt dt, wefind 0 T V = φ(t ) 1 e rt r Differentiating with respect to T yields + π e rt r. 0=e rt (φ(t ) π)+φ (T ) 1 e rt r 30,

31 which can be rewritten as φ (T )= e rt r (φ(t ) π). (S3) 1 e rt (c) Show that social welfare is given by Ω(T,π)= W (π) e rt r Derive the first-order conditions. Total welfare is given by + W (π) 1 r + W e rt r. Ω(T,π)= T 0 W (π)e rt dt + = W (π) e rt r The first-order conditions read: Ω T + W (π) 1 r We rt dt + W e rt r T =W (π)e rt We rt = ( W (π) W ) e rt Ω π = W (π) e rt r + W (π) 1 r = W (π) 1 e rt r. (S4). (S5) (d) Show that dω/dt > 0 when the flow of profits is set optimally π = φ(t ). If welfare increases 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 π φ (T ). Then make use of the results obtained in (b) to substitute for the expression φ (T ). Finally, assume that patent breadth is increasingly costly in terms of social welfare, i.e. W (π) < 0 and W (π) < 0 on [ π, φ(t )]. Graph the welfare function to 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 to see the argument) W (φ(t )) W >W (φ(t )) (φ(t ) π) or equivalently: W W (φ(t )) < W (φ(t ))(φ(t ) π). 31

32 Thus, dω dt > 0 and increasing T always has a positive effect on welfare, so the optimal patent length is infinite. The idea is that increasing patent breadth raises market power and hence deadweight loss. Therefore, increasing the reward 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 environment. If the environment is not predictable and there is uncertainty about future demand and costs, risk averse firms might prefer shorter and broader patents in order to share risk. In an non-stationary environment, Cumulative innovation plays an important role. Since inventions build on each other, patents with infinite length is likely to have deterrent effects on firms incentives to invest in related research. 32

33 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 for a 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-period increment to market value generated by the application is y. Letc 1 and c 2 represent the cost of undertaking basic research and to develop the second-generation product, respectively. The interest rate is r and T denotes the discounted patent length. Further let π and l denote fractions of the total per-period value to end users. That is, π(x + y) is the per-period profit at 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 finances the innovations through a lump-sum tax and offers the product for free. Write down the social value obtained from both innovations, W. If both innovations were public and sold at price p =0, the social value is maximal (as there is no deadweight loss). In that case: W = x + y r c 1 c 2. (b) Write down the social value, W p, and consumer surplus, S p, obtained when the innovations are performed by the two firms under the patent regime. The social value when both innovations are performed by the two firms is the discounted total value to end users less the deadweight loss due to patent protection, and less invention cost. Therefore, W p (x + y) = r The consumer surplus is given by (x + y)lt c 1 c 2. S p = (x + y) r (x + y)(l + π)t c 1 c 2. (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 innovations during patent protection less the cost of turning both ideas into innovations. Hence, Π=(x + y)πt c 1 c 2. 9 Assume that most of the profit is due to the second-generation product and both firms have blocking patents on the application. 10 Thus, 1 π l is the fraction of (x + y) defining consumer surplus. 33

34 (d) Now suppose the firms do not coordinate their research activity but instead the firms can either sign a license agreement ex-ante, before the second innovator invests c 2 (but after the first innovation has been made), or ex-post, after the second innovator invests c 2. Illustrate the bargaining situation. An illustration of the bargaining situation is seen below (including payoffs that will be explained in the following exercises): Ex-ante license? Yes No Payoffs with licence: πtx c πty, 1 2 πty c 2 or πtx c (πty c 2), 1 2 (πty c 2) Yes Firm 2: invest? No Ex-post license? πtx c 1, 0 Yes No πtx c πty, 1 2 πty c 2 πtx c 1, c 2 (e) In the game, each firm s threat point is defined as the expected profit it can guarantee itself if it does not license. Find the threat points and bargaining outcome for each firm at the ex-post licensing node under the assumption that the bargaining surplus is split equally. Indicate the pay-off s in the diagram. Ex-post, the producer of the basic innovation has a threat point of πtx c 1 and the second generation producer has a threat point of c 2. The bargaining surplus is πty and it is shared equally between the firms. Payoffs can be seen at the bottom of the diagram in exercise (d). (f) Find the threat point and bargaining outcome for each firm at the ex-ante licensing node under the assumption that the bargaining surplus is split equally. Indicate the payoffs in the diagram. Following failed ex-ante licensing negotiations, firm 2 is better off by investing than not investing if 1 2 πty c 2 > 0. The thread points at the ex-ante bargaining node are therefore πtx c πyt and 1 2 πyt c 2 for firms 1 and 2 respectively. Since the firms are bargaining for πtx c 1 + πyt c 2 there is nothing left to divide and the bargaining surplus is zero. The ex-ante licensing agreement payoffs are simply the thread points as indicated in the first line of the upper-left branch on the diagram in exercise (d). 34

35 On the other hand, the costs c 2 couldbesohighthat 1 2 πty c 2 < 0. Firm 2 then anticipates that it will be held up ex-post and that it will not be able to cover costs. Firm 2 will therefore not invest following failed negotiations for ex-ante licensing. The thread point are then given by πtx c 1 and 0 for firms 1 and 2 respectively. The firms are bargaining for πtx c 1 + πyt c 2 which gives a bargaining surplus of πyt c 2. Splitting the bargaining surplus equally gives the ex-ante licensing agreement payoff indicated in the second line of the upper-left branch on the diagram in exercise (d). 35

36 Exercise 8.2: Anti Competitive Cross Licensing (Motta, 2004) Consider two firms that play the following game. In the first stage, they jointly decide whether they want to cross-license their technologies. The technologies are assumed to be perfect substitutes and are used to produce the same homogenous good. At this stage, if crosslicensing is agreed upon, they also jointly decide the same per-unit of output royalty c L for the cross-license. In the following stage, they compete in quantities. Assume for simplicity that the only unit cost, if any, is given by c L. Assume linear demand p =1 Q, whereq is total output. (a) Find each firm s equilibrium output and the associated per-firm profit if no cross-licensing is agreed upon. When no cross-license is agreed upon, each firm solves max π i =(1 q i q j )q i. q i The reaction functions are given by q i (q j )= 1 q j. 2 In a symmetric equilibrium, each firm s output is q NL =1/3 and the associated per-firm profit is π NL =1/9. (b) Assuming that cross-licensing is agreed upon, write down the maximization problem of each firm and find equilibrium outputs and per-firm profits. Firm i solves max π i =(1 q i q j c L )q i + c L q j. q i The term c L appears both as cost and as revenue, because each firm has to pay the other a unit royalty. The reaction functions are q i (q j )= 1 q j c L 2 and Equilibrium outputs and per-firm profits are, q j (q i )= 1 q i c L. 2 q L (c L )= 1 c L 3 and π L (c L )= (1 + 2c L)(1 c L ), 9 respectively. Note that here is an example where profit is not equal to quantity squared for all c L. (c) In the first stage, firms not only decide whether to cross-license or not but also they determine the level of c L. Find the equilibrium level of the unit royalty. To find the equilibrium level of c L, we need to find the value at which the function π L (c L ) reaches its maximum. Solving dπ L = 2(1 c L) (1 + 2c L ) dc L 9 36! =0,