Design Patent Damages under Sequential Innovation

Transcription

1 Design Patent Damages under Sequential Innovation Yongmin Chen and David Sappington University of Colorado and University of Florida February / 32

2 1. Introduction Patent policy: patent protection and infringement damages. Extensive literature on patents and patent protection. Economic analysis of patent damages is sparse and focuses exclusively on deterring imitation. Schankerman-Scotchmer (2001), Anton-Yao (2006), Henry-Turner (2010) Patent damages from the U.S. patent law (35 U.S.C. Section 284). lost profit (LP): infringer is required to pay the patent holder the reduction in profit due to the infringement. reasonable royalty (RR): no less than the royalty rate that would have been agreed upon under an ex ante licensing contract Patent damages are highly controversial (e.g., Apple vs. Samsung). 2 / 32

3 1. Introduction Many patent litigations occur under sequential innovation, where a follow-on discovery may infringe an earlier patent. Patent damages may affect competition and the incentives for the initial and follow-on innovations. Anticipation of potential damages may soften price competition. higher price by the infringer can reduce the "lost profit" of the patent holder higher price by the patent holder can raise the royalty payment from the infringer Damage policy can affect not only the ex post profit divisions, but also the size of industry profit. The profit effects in turn impact the innovation incentives of the initial and follow-on innovators. 3 / 32

4 1. Introduction This research: Design patent damages to increase total welfare under sequential innovation. A simple model with an initial and a follow-on innovator. costly innovations may occur stochastically the follower may infringe the leader s patent with some probability potential distortions: ex ante ineffi cient innovation incentives; ex post ineffi cient output allocations Consider a general linear policy with LP and RR as special cases. characterize the optimal policy how does it compare to either LP or RR? (when) can it maximize total welfare achieving the first best? when the first best cannot be attained, how does the policy balance the trade off? 4 / 32

5 2. The Model A variant of the Hotelling model: A mass of Q consumers uniformly distributed on a line of length Q. Firm 1 is the first potential innovator at cost k 1 [ 0, k 1 ]. Q = 1 if firm 1 is the only firm in the market, located at point 0 Firm 2 can innovate at cost k 2 [ 0, k 2 ] iff firm 1 has innovated. Q = L 1 if firm 2 also innovates, located at point L v i is consumers value for a unit of firm i s product, i = 1, 2. = v 2 v 1 firm 2 s product can expand the market and has a higher quality k 1 and k 2 are random draws from distributions F (k 1 ) and G (k 2 ). Each consumer has unit transportation cost t = 1. Production costs are normalized to zero. 5 / 32

6 2. The Model λ (0, 1] is the probability that product 2 will be found to have infringed firm 1 s patent. λ measures the strength of patent protection λ will be considered as a pre-determined parameter If found infringement, firm 2 is obligated to pay firm 1 D m + d 1 R 1 + d 2 π 2 d 1 0, d 2 0, and m are policy parameters R 1 is the reduction of firm 1 s profit due to firm 2 s operation π 2 is firm 2 s profit: the royalty base is profit, not output m is a monetary transfer that can be negative This damage policy generalizes LP and RR. simple, and has other desirable properties (to be shown) 6 / 32

7 2. The Model The timing of the game: Given λ, damage policy (d 1, d 2, m) is set. Firm 1 learns the realization of k 1 and decides whether to innovate product 1. If firm 1 decides not to innovate, the game ends. If firm 1 innovates, firm 2 learns the realization of k 2 and decides whether to pursue the follow-on innovation. If firm 2 innovates, it competes with firm 1; otherwise firm 1 acts as a monopolist. If both firms produce, they set prices simultaneously and non-cooperatively, after which consumers make purchases. It then becomes known whether 2 has infringed 1 s patent. In case of infringement, 2 makes the damage payment to 1. 7 / 32

8 3. Market Equilibrium We consider market equilibrium in which at least one firm innovates, given (d 1, d 2, m). Assumption 1. (i) min {v 1, v 2 } > 2 L; (iii) < L. Lemma 1. In the subgame where only firm 1 innovates, it will set p 1 = v 1 1 and earn profit π M v 1 1. Firm 1 s demand is q 1 = v 1 p 1 for p 1 [v 1 1, v 1 ]. p 1 (v 1 p 1 ) is decreasing on [v 1 1, v 1 ] because v 1 > 2 by assumption. optimal for firm 1 to sell to all consumers at price p 1 = v / 32

9 3. Market Equilibrium Next consider the subgame in which both firms innovate. Given (p 1, p 2 ), demand functions facing firms 1 and 2 are: q 1 (p 1, p 2 ) = L + p 2 p 1 2 ; q 2 (p 2, p 1 ) = L + + p 1 p 2. (1) 2 In the absence of infringement, profits (exclusive of innovation costs) are: π N 1 = p 1 q 1 (p 1, p 2 ) and π N 2 = p 2 q 2 (p 2, p 1 ). The firms profits in the event of infringement (excluding m) are: ) ( ) π I 1 = π N 1 + d 1 (π M π N 1 + d 2 π N 2 ; π I 2 = π N 2 d 1 π M π N 1 d 2 π N 2. 9 / 32

10 3. Market Equilibrium The ex ante expected profits of firms 1 and 2 (exclusive of innovation costs and m) when they both innovate are, respectively: [ ] π e 1 = (1 λ d 1 ) p 1 q 1 (p 1, p 2 ) + λ d 1 π M + d 2 p 2 q 2 (p 2, p 1 ) and [ ] π e 2 = [ 1 λ d 2 ] p 2 q 2 (p 2, p 1 ) λ d 1 π M p 1 q 1 (p 1, p 2 ). (2) When p i and q i are equilibrium values, the equilibrium profits in (2) will be denoted as π i. 10 / 32

11 3. Market Equilibrium Lemma 2. Given (d 1, d 2 ), the equilibrium prices and quantities when all consumers purchase and both firms produce positive outputs are: p1 = (1 λ d 2 ) L [3 λ (3 d 1 d 2 )] [1 λ (d 1 + d 2 )] ; [1 λ (d 1 + d 2 )] [3 λ (d 1 + d 2 )] p2 = (1 λ d 1 ) L [3 λ (3 d 2 d 1 )] + [1 λ (d 1 + d 2 )] ; [1 λ (d 1 + d 2 )] [3 λ (d 1 + d 2 )] (4) q1 = L (3 2 λ d 2) 2 [3 λ (d 1 + d 2 )] ; q 2 = L (3 2 λ d 1) + 2 [3 λ (d 1 + d 2 )]. (5) The existence of this equilibrium requires λ (d 1 + d 2 ) < 1, and if d 1 and d 2 are suffi ciently small, then this equilibrium indeed exists. Furthermore, for any p1 and p 2, there exist d 1 0 and d 2 0 that induce these prices in equilibrium. We can confine our analysis to this equilibrium. (3) 11 / 32

12 3. Market Equilibrium Lemma 3. (i) As d 1 increases, q 1 and p 2 p 1 p 1 p 2 decrease; (ii) As d 2 increases, q 1 and p 2 p 1 q 2 and p 1 p 2 increase. increase whereas q 2 and decrease whereas d 1 firm 2 expects to pay a higher portion of R 1 with prob. λ. Thus p 2 to reduce this loss, q 1 and p 2 p 1 d 2 firm 1 expects to receive a higher portion of π N 2 λ. Thus p 1 to raise π N 2 q 2 and p 1 p 2. with prob. 12 / 32

13 3. Market Equilibrium Proposition 1. When both firms innovate, p1 and p 2 both increase as d 1 or d 2 increases. Furthermore, Π π1 + π 2 increases with d 1 if [ 2 λ (d 1 + d 2 ) ] L λ [ d 2 d 1 ]. Π increases with d 2 if [ 2 λ (d 1 + d 2 ) ] L λ [ d 2 d 1 ]. prices are strategic complements. d 1 p 2 p 1 and p 2. d 2 p 1 p 1 and p 2. Damage policy has a strategic effect on competition. By raising either d 1 or d 2, policy can increase industry profit. q i may decrease as d j increases 13 / 32

14 3. Market Equilibrium Total welfare when only product 1 is produced is: W 1 = W 1 k 1 where W 1 = v x dx = v (6) Total welfare following innovation by both firms is W 12 = W 12 k 1 k 2, where: q W 12 = v 1 q1 1 q y dy + v 2 q2 2 y dy. (7) 0 0 Ex ante expected welfare is: W = F ( ˆk 1 ) G ( ˆk 2 ) W 12 + F ( ˆk 1 ) [ 1 G ( ˆk 2 ) ] W 1 ˆk 2 ˆk 1 F ( ˆk 1 ) k 2 dg (k 2 ) k 1 df (k 1 ). (8) / 32

15 3. Market Equilibrium Lemma 4. The industry output allocation maximizes W 12 when In this case, d 2 d 1 d 2 = d2 w (d 1 ) d [ 1 λ d 1 ]. (9) λ [ L + ] p 1 = p 2 = (1 λ d 1) ( L 2 2) L (1 2 λ d 1 ) as 0, equilibrium prices and quantities are ; q 1 = L 2, q2 = L +, (10) 2 and equilibrium industry profit is Π = p 1 L, which rises as d 1 increases. The maximum Π is Π w = d 1 = d w 1 1 λ [ v 1 12 (L ) ] L, which obtains when (11) [ ] 1 L 2 p 1 = p 2 = p w v 1 L 2 2 (v 1 L) L 2 (v 1 L) L < 1 2λ, and. (12) 15 / 32

16 3. Market Equilibrium When p1 = p2, welfare when both firms produce ( W 12 ) is maximized. all consumers will then purchase the product with the higher net utility p 1 = p 2 can be induced by some d 1 and d w 2 (d 1) ; p i increases in d 1. Under a higher λ, the same p i can be induced by a lower d i. stronger patent protection can partially substitute for higher damages Industry profit Π increases in d 1 until d 1 = d w 1, where Π = Π w. further increase in Π is either not possible if = 0 or is achieved by distorting the output allocation if = 0 16 / 32

17 3. Market Equilibrium Lemma 5. The maximum industry profit is: Π = L 2 [ v 1 + v 2 L ] = Πw + 2, under (13) 8 p 1 v 1 L 2 ; p 2 2 = v 2 L + 2. (14) 2 These equilibrium prices are obtained when d 1 and d 2 are given by: d 1 = (L 2 ) (6L + 7 4v 1) 4λ ( 3 2 2Lv 1 + 2L 2 ), d 2 = (L + 2 ) (6L 7 4v 1) 4λ ( 3 2 2Lv 1 + 2L 2 ). (15) The marginal consumer has zero surplus: p 2 = p 2 (p 1 ) v 1 + v 2 p 1 L. if d 2 = d2 w (dw 1 ), then p 1 = p w and Π = Π w if v 1 = v, Π may rise above Π w as p 1 departs p w until Π = Π if v 1 = v 2 = v, Π = Π w = L 2 [ 2 v L ], p i = p w 17 / 32

18 4. Innovation Incentives and Welfare Let ˆk i denote the realization of k i for which firm i secures the same expected profit with or without innovation. ˆk 1 = G ( ˆk 2 ) ( π 1 + λ m) + [ 1 G ( ˆk 2 ) ] π M 1 and ˆk 2 = π 2 λ m. (16) Lemma 6. Given innovation by firm 1, innovation by firm 2 increases W if and only if k 2 < W 12 W 1 k2 w. (17) Moreover, if d 2 = d2 w (d 1), then k2 w = L 2 [ v 1 + v 2 ] + 1 [ 2 L 2 ] v kw 2. (18) k w 2 is the effi cient critical innovation cost for firm 2 when d 2 is set (given d 1 ) to ensure the effi cient output allocation. 18 / 32

19 4. Innovation Incentives and Welfare Lemma 7. Suppose firm 2 innovates if and only if k 2 ˆk 2 = π 2 λ m. Then innovation by firm 1 increases total welfare if and only if: k 1 < G ( ˆk 2 ) W 12 + [ 1 G ( ˆk 2 ) ] W 1 Moreover, if d 2 = d w 2 k w 1 ˆk 2 (d 1) and ˆk 2 = k2 w, then = G ( k o 2 ) W 12 + [ 1 G ( k 2 ) ] W 1 0 k 2 w 0 k 2 dg (k 2 ) k w 1 ( ˆk 2 ). (19) k 2 dg (k 2 ) k w 1. (20) k1 w is the effi cient critical innovation cost for firm 1 when ˆk 2 = k2 w and d 2 is set (given d 1 ) to ensure effi cient allocation of industry output. 19 / 32

20 4. Innovation Incentives and Welfare Definition. The first-best outcome is the outcome where industry output is always allocated to maximize welfare and each firm innovates if and only if the associated increase in expected welfare exceeds the realized innovation cost. That is, at the first best: q1 = 1 2 [ L ] and q 2 = 2 1 [ L + ] when both firms innovate firm 2 innovates if and only if k 2 k2 o min {kw 2, k 2 } firm 1 innovates if and only if k 1 k1 o min {kw 1, k 1 } Firm 1 s innovation incentive is ineffi ciently low (i.e., ˆk 1 < k w 1 ( ˆk 2 ) ) if m = 0, because its innovation creates consumer surplus that it does not fully capture, regardless of whether firm 2 innovates. Firm 2 s innovation incentive can be excessive when v 2 and L are low, due to business stealing. 20 / 32

21 4. Innovation Incentives and Welfare For k o 2 min { kw 2, k 2 } and consider the following condition: G (k o 2 ) [ Π w k o 2 ] + [ 1 G (k o 2 ) ] π M 1 k 1. (E1) (E1) holds if: The maximum industry profit, when industry output always maximizes welfare and firm 2 s innovation incentive is always effi cient, is large relative to innovation costs. Theorem 1. Suppose (E1) holds. Then d 1, d 2 = d2 w (d 1), and m can be chosen to induce the first-best outcome, with π 2 λ m = k o 2 and π 1 + λ m = k 1 π M 1 G (k o 2 ) + π M 1. Under this (optimal) policy, d 2 d 1 as / 32

22 4. Innovation Incentives and Welfare Nature of the optimal damage payment: It generally entails d 1 > 0 and d 2 > 0, combining LP and RR. If v 1 > v 2, then d 1 > d 2, so the initial innovator s lost profit receives more weight in the optimal damage payment. shifting consumption toward the product that consumers value more, because q 1 increases as d 1 increases. If v 1 < v 2, then d 2 > d 1, so the follower s profit receives more weight in the optimal damage payment. It is desirable to allow part of the damage payment (m) to be independent of either firm s profit. This facilitates the provision of effi cient innovation incentives while using d 1 and d 2 to ensure effi cient output allocation 22 / 32

23 4. Innovation Incentives and Welfare Corollary 1. Suppose v 1, v 2, and L are suffi ciently large to ensure k2 o = k 2. Then the first-best outcome is feasible if: k 1 + k 2 L 2 [ v 1 + v 2 L ]. (21) The first best is likely to attain if k 1 and k 2 are small while v 1, v 2, and L are large. that is, if innovation costs are relatively low compared to product values and market size 23 / 32

24 4. Innovation Incentives and Welfare Corollary 2. If (E1) does not hold, then it is not possible to ensure d 2 = d2 w (d 1), ˆk 1 = k1 o, and ˆk 2 = k2 o, so the first-best outcome cannot be secured. This is the situation where, when output is allocated effi ciently ex post, Π w is not high enough to ensure effi cient innovation incentives. The optimal choice of (d 1, d 2, m) will then balance the need to: (i) allocate output effi ciently when both products are produced (ii) induce effi cient innovation decisions The marginal consumer will receive zero surplus in order to maximize industry profit. if = 0, then d 2 = d w 2 (d 1) so as to increase industry profit beyond Π w 24 / 32

25 4. Innovation Incentives and Welfare Theorem 2. Suppose (E1) does not hold. Let p 1 and k 2 denote the values of p 1 and ˆk 2 that solve: [ ] k w F ( ˆk 1 ) 1 ( ˆk 2 ) ˆk 1 + F ( ˆk 1 ) [ k w ] G ( ˆk 2 ) 2 ˆk 2 + F ( ˆk 1 ) G ( ˆk 2 ) W 12 = 0, p 1 p 1 p 1 (22) [ ] k w F ( ˆk 1 ) 1 ( ˆk 2 ) ˆk 1 + F ( ˆk 1 ) [ k ˆk w ] G ( ˆk 2 ) 2 ˆk 2 = 0. 2 ˆk 2 (23) (i) If Π = p 1 (v 1 p 1 ) + p 2 (L + p 1 v 1 ) < Π, then the optimal policy is ( d 1, d 2, m ), where d 1 and d 2 induce p 1 and p 2 = p 2 ( p 1 ), with m = 1 ( ) λ π2 k 2, and π2 is obtained from (2). (ii) If Π Π, then the optimal policy is ( d 1, d 2, m ), where d 1 and d 2 are as specified in (15) (so equilibrium ) prices are p 1 and p 2 ), k 2 solves (23), and m = 1 λ (π 2 k / 32

26 4. Innovation Incentives and Welfare Theorem 2 deals with the case where the first-best outcome is not feasible. The optimal policy is derived in two steps. First, p 1 and k 2 that maximize expected welfare, when p 2 = p 2 is chosen to leave the marginal consumer with zero surplus, are identified. This determines Π. Second, d 1 and d 2 that induce the p 1 and p 2 are identified. Doing so determines π 2 and m = 1 λ [ π 2 k 2 ]. the maximum industry profit constraint is not binding if Π < Π otherwise the maximum industry profit constraint ) is binding, then p i = p i, k 2 solves (23), and m = 1 λ (π 2 k 2 26 / 32

27 4. Innovation Incentives and Welfare Allowing p 1 to differ from p 2 can increase industry profit, but it distorts ex post output allocation. Condition (22) states that the optimal p 1 balances these two considerations. When Π Π under p i, profit can no longer be increased by changing p i, so the industry profit constraint binds. then the optimal prices maximize industry profit, i.e., p i = p i Condition (23) shows that the balance of incentives between the two innovations depends on the difference between ˆk i and k w i. 27 / 32

28 4. Innovation Incentives and Welfare Corollary 3. Suppose (E1) does not hold. Then under the optimal policy, the welfare-maximizing value of m solves: k w 1 ( ˆk 2 ) ˆk 1 k w 2 ˆk 2 = F ( ˆk 1 ) f ( ˆk 1 ). (24) G ( ˆk 2 ) g ( ˆk 2 ) π 1 λ m + πm Moreover, if v 1 = v 2, then at the optimal policy d 1 = d 2 = d. k1 w ˆk 1 > 0 measures the extent to which firm 1 has insuffi cient incentive to innovate. F ( ˆk 1 ) [ k2 w ] ˆk 2 measures the extent to which firm 2 is expected to have insuffi cient incentive to innovate. m is optimally increased to the point where the marginal reduction in firm 1 s innovation deficiency is equal to the increase in firm 2 s expected innovation deficiency. 28 / 32

29 5. Discussion If patent strength λ is endogenous: λ λ (L, v 2 ) may decrease in L and v 2. Patent policy may need to coordinate patent protection and infringement damages. If patent strength (λ) is too low, infringement damages will generally not be able to implement the first best. Once λ is above certain critical value, high damages can substitute for even stronger patent protection. 29 / 32

30 5. Discussion If the Royalty base is units of sales by firm 2, then: The damage policy would become D = m + d 1 R 1 + d 2 q 2 (p 1, p 2 ). the policy is then much less likely to achieve the first best. e.g., if v 1 = v 2, then p 1 = p 2 only when d 1 = 0 unlikely to induce effi ciency in both innovation and output allocation The royalty base in D should be the infringer s profit, not output. reducing the infringer s unjust enrichment That is: D = m + d 1 R 1 + d 2 π 2 is superior to D. 30 / 32

31 5. Discussion If the parties can sign ex ante licensing contracts, then: The royalty payment under the standard licensing contract has the form R = m + d 2 q 2 (p 1, p 2 ). This is equivalent to damage policy m + d 2 q 2 (p 1, p 2 ) with λ = 1. This policy is generally inferior to D, which is in turn inferior to D, provided that λ is not too low. Thus, when λ is not too low, our optimal damage policy is superior to ex ante effi cient licensing, even if the latter is feasible. 31 / 32

32 6. Concluding Comments Patent damages impact sequential innovation and output allocation. A policy based on a linear combination of LP and RR, with a fixed transfer payment, strictly dominates either LP or RR alone. If innovation costs are not too high, this simple policy achieves the first best: effi cient innovation incentives and output allocation. higher d 1 and d 2 when innovation costs are higher higher weight on LP (RR) when the first (second) innovation is more valuable desirable to allow a transfer payment independent of realized profits When the first best is not attained, the optimal policy balances output allocation effi ciency vs. total innovation incentive incentives for the initial vs. for the follow-on innovation 32 / 32