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Selling patent rights and the incentive to innovate
Economics Letters 114 (2012) 241–244
Contents lists available at SciVerse ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Selling patent rights and the incentive to innovate Yair Tauman a,b,∗ , Ming-Hung Weng c a
Department of Economics, Stony Brook University, NY 11794-4384, USA
b
Arison School of Business, Interdisciplinary Center (IDC) Herzliya, Herzliya, Israel
c
Department of Economics, National Cheng Kung University, Tainan, Taiwan
article
info
Article history: Received 3 March 2011 Received in revised form 2 October 2011 Accepted 20 October 2011 Available online 26 October 2011
abstract We show that an outside innovator has a higher incentive to innovate than an incumbent innovator, by auctioning off his patent rights exclusively to an incumbent firm. For significant innovations this is also superior to selling licenses directly. © 2011 Elsevier B.V. All rights reserved.
JEL classification: D45 L13 O32 O33 Keywords: Patent licensing Cournot oligopoly Process innovation Patent rights Incentive to innovate
1. Introduction The literature on various licensing methods of a process innovation is quite extensive. Kamien and Tauman (1984, 1986) and independently Katz and Shapiro (1985, 1986) used strategic models to analyze the private value of a patent of an outside innovator (a research lab) who can sell licenses by means of an upfront fee, royalty, auction, or a mixture of the above.1 See also the survey of Kamien (1992). Kamien et al. (1992) studied the market structure resulting from the three common licensing strategies of selling licenses by an auction, by an upfront fee or by a per-unit royalty, but not by the combinations of these three strategies.2 Recently Sen and Tauman (2007) (ST hereafter)
∗ Corresponding author at: Department of Economics, Stony Brook University, NY 11794-4384, USA. E-mail addresses: [email protected] (Y. Tauman), [email protected] (M.-H. Weng). 1 Kamien and Tauman (1984) studied licensing by means of a combination of upfront fee and royalty only for sufficiently competitive markets. 2 Wang (1998) studied a process innovation by an incumbent firm in a duopoly market and showed that a per-unit royalty is a better licensing strategy for the innovator than an upfront fee strategy. The extension of his result to oligopoly of a general size is in Kamien and Tauman (2002). 0165-1765/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2011.10.023
analyzed the optimal combination of an upfront fee and royalty, its impact on the market structure and the diffusion of a cost-reducing innovation in a Cournot oligopoly industry by both inside or outside innovators. They show that irrespective of the magnitude of innovation, an outside innovator (e.g. a research lab) has a higher incentive to innovate than an inside one (an incumbent firm). This result is true for Cournot oligopoly competition and for linear demand. In this paper we extend this result to a general competitive environment (not necessarily Cournot) and to any demand or cost structure. The only assumption we make is, if at least one firm produces with the new technology, the profit of a non-licensee firm who produces with the old technology is smaller than its pre-innovation profit level (where all firms produced with the old technology). First notice that an outside innovator can sell the property rights of his innovation to an incumbent firm. We refer to this strategy as the SPR strategy. To the best of our knowledge this strategy is not analyzed in the literature. The buyer firm of the property rights can then license the new technology to other firms and with any licensing strategy. Consider next an industry with an arbitrary number of firms (greater than or equal to two). An outside innovator who puts on auction to sell the property rights of his innovation to the highest bidder will obtain the difference between the payoff of
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Y. Tauman, M.-H. Weng / Economics Letters 114 (2012) 241–244
the winner (the sum of his operating profit and the revenue he extracts from licensee firms, using his optimal licensing strategy) and the operating profit of a non-licensee firm (the profit he obtains if some other firm wins the auction). On the other hand the incremental profit of an incumbent innovator is the difference between his optimal payoff (which coincides with the winning bid of the previous case) and his pre-innovation profit level (where all firms produced with the old technology). By our assumption, the winning bid paid to the outside innovator is higher than the incremental profit of an inside innovator. The motivation of an outside innovator to use the SPR strategy is based not only on practical grounds but also on a theoretical one. On the practical side, an external research lab may incur a significantly higher licensing cost. The implementation of a new technology often requires service facilities that a research lab may not have, while an incumbent firm can offer more efficiently. Turning to the theoretical side, recall that an outside innovator when using the SPR strategy guarantees a higher payoff than an inside innovator. Still it is not clear whether or not the SPR strategy of the outside innovator dominates the direct licensing strategy, the DL strategy. For oligopoly Cournot competition with linear demand we show that even if the licensing costs are the same for both inside and outside innovators, for a significant innovation the SPR strategy dominates the DL strategy. 2. The model Consider a homogeneous industry with n firms who are engaged in a certain type of competition (Cournot or other). Let N = {1, . . . , n} be the set of firms. Suppose that all firms incur the same constant marginal cost of production c. Let π (N , c ) be their equilibrium profit level. Suppose next that a subset S ⊆ N, S ̸= φ , of firms have access to a superior technology with a marginal cost of production c ′ < c. Let πL (S , c ′ ) and πNL (S , c ′ ) be the equilibrium profit levels of each firm in S and each firm in N \S, respectively (later on the firms with the access to the superior technology will be the ones who purchase a license to use that technology. Hence L stands for licensee firms and NL for non-licensee firms). Assumption 1. Suppose that c ′ < c. Then π (N , c ) > πNL (S , c ′ ). This assumption asserts that firms with the old technology become worse off when some other firms use a new and superior technology.3 Let us show that this assumption holds in an oligopoly Cournot model with an inverse demand P (Q ) satisfying the following assumptions. Assumption 2. P (Q ) is strictly decreasing in Q and differentiable for Q > 0. Assumption 3. The total revenue function, QP (Q ), is strictly concave in Q . Proposition 1. In a Cournot competition π (N , c ) > πNL (S , c ′ ) whenever c ′ < c and S ⊆ N, S ̸= φ . Proof. Consider first the case where all firms in N produce with the less efficient technology. The first order condition of a firm i producing qi units is P (Q (N , c )) + qi P (Q (N , c )) = c . ′
This implies that nP (Q (N , c )) + Q (N , c )P ′ (Q (N , c )) = nc .
(2)
Similarly in a Cournot industry where firms in S produce with a marginal cost c ′ and firms in N \S produce with a marginal cost c nP (Q (S , c ′ )) + Q (S , c ′ )P ′ (Q (S , c ′ ))
= |S |c ′ + (n − |S |)c < nc .
(3)
By Assumption 3, P (Q ) + QP ′ (Q ) is decreasing in Q and by Assumption 2, nP (Q ) + QP ′ (Q ) is also decreasing in Q. Hence by (2) and (3) Q (S , c ′ ) > Q (N , c ).
(4)
Next observe that
π (N , c ) = −(P (Q (N , c )) − c )2 /P ′ (Q (N , c )), and
πNL (S , c ′ ) = −(P (Q (S , c ′ )) − c )2 /P ′ (Q (S , c ′ )). The result now follows from (4) and from the fact that (P (Q ) − c )2 /P ′ (Q ) is increasing in Q .4 Consider next a cost-reducing innovation that reduces the marginal cost from c to c ′ = c − ϵ , 0 < ϵ < c. The innovator can license his innovation to any number of firms in the industry. The next proposition compares the incentive to innovate of an outside innovator with an inside innovator and holds true independently of the licensing strategy. It makes use of the following assumption. Assumption 4. For the same investment level the probability of a successful innovation is the same for both an outside innovator and an incumbent firm. Proposition 2. Suppose that Assumption 4 holds and the industry consists of at least two firms. The incentive to innovate is higher for an outside innovator than it is for an inside one, irrespective of the licensing strategy. Proof. Suppose that both the outside lab and the incumbent firm invest I and their probability of having a successful innovation is λ. Suppose that conditional on success the outside innovator puts on auction to sell his patent rights to an incumbent firm, say Firm 1. Suppose next that Firm 1 chooses a licensing strategy σ (which includes royalty payments, upfront fees, and the number of licenses sold). Then the innovator obtains in equilibrium
π1 (S (σ ), c (σ )) − πNL (S (σ ), c (σ )) ≡ b1 , where S (σ ) is the set of licensee firms under the licensing strategy σ , c (σ ) is the marginal cost of every licensee firm, and π1 (S (σ ), c (σ )) is the payoff of Firm 1, the licensor, which consists of its operational profit plus the revenue it extracts from the licensee firms in S (σ ). Also, πNL (S (σ ), c (σ )) is the profit level of a non-licensee firm. Next suppose that an incumbent firm invents the same innovation. Now the equilibrium incremental profit of the innovator is
π1 (S (σ ), c (σ )) − π (N , c ) ≡ b1 ′ .
(1)
3 The inequality π(N , c ) > π (S , c ′ ) in Assumption 1 holds as equality for NL a Bertrand competition (with unlimited capacity). The same is true if an outside innovator charges a per-unit royalty r = c − c ′ , in which case every licensee firm actually produces with the pre-innovation marginal cost.
4 It is easy to verify that (P (Q ) − c )2 /P ′ (Q ) is increasing in Q iff P ′′ (Q ) < 2(P ′ (Q ))2 /(P (Q )− c ). By Assumption 3 2P ′ (Q )+ P ′′ (Q )Q < 0. Hence it is sufficient to show that Q > −(P (Q ) − c )/P ′ (Q ). By the F.O.C. if all firms produce with a marginal cost c we have P (Q ) − c + qi P ′ (Q ) = 0 or qi = −(P (Q ) − c )/P ′ (Q ). Since qi < Q , (P (Q ) − c )2 /P ′ (Q ) is strictly increasing.
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The expected (incremental) profit of the lab is λb1 − I and the expected incremental profit of the incumbent innovator is
incumbent firm has no incentive to invest in an outside lab, at least as long as the SPR strategy is better for the lab than the DL strategy.8
λπ1 (S (σ ), c (σ )) + (1 − λ)π (N , c ) − π (N , c ) − I = λb1 ′ − I .
2.1. The SPR versus DL strategy
Since by Assumption 1 π (N , c ) > πNL (S (σ ), c (σ )) we have λb1 − I > λb1 ′ − I as claimed.5 Proposition 2 raises a natural question. Why in practice we do not observe firms investing in outside R&D firms (labs), even though they have higher incentive to innovate? One answer is that Proposition 2 holds under Assumption 4 which asserts that for the same level of investment the two entities are equally likely to invent. If an incumbent firm believes that it has a significantly higher probability to invent it will not invest in an outside lab. But we claim that even if Assumption 4 holds and it is commonly known that the probability to invent is the same for both entities the incumbent firm can not benefit from investing in outside labs. This claim however is true for the case where it is more profitable for the outside innovator to sell its property rights to an incumbent firm (the SPR strategy) rather than to license the innovation directly (the DL strategy) to firms in the industry. In the next section we provide sufficient conditions for the SPR strategy to dominate the DL strategy.6 To verify our claim, suppose that the two potential innovators invest I and have the same probability λ of success. Suppose that the incumbent firm, say Firm 1, owns the outside lab and it acts as to maximize its own profit, and not that of Firm 1. The lab, conditional on success, auctions off its property rights to an incumbent firm, say Firm j (not necessarily Firm 1). Let σ be the optimal licensing strategy of Firm j. Denote by S (σ ) the set of licensee firms and let c (σ ) be their marginal cost. In equilibrium every firm in the industry (whether licensee or not) obtains a net payoff of πNL (S (σ ), c (σ )).7 Hence the net payoff of Firm 1, the owner of the outside lab (whether or not it has access to the new technology) is the sum of the profit of a non-licensee firm, πNL (S (σ ), c (σ )), and the winning bid, b(σ ), submitted by Firm j. Thus the incremental profit of Firm 1 conditional on the lab’s success is
∆ = πNL (S (σ ), c (σ )) + b(σ ) − π (N , c ) − I . But ∆ is also the incremental profit of Firm 1 if it invests the same amount I in-house and succeeds to innovate, since it will implement the same optimal licensing strategy σ as Firm j. Since the probability of success is the same in these two scenarios, the expected incremental profits are also equal. We conclude that even when the probability of success is the same for the two entities, an
5 If the licensor charges a per-unit royalty r ≤ ϵ then all licensee firms produce with marginal cost c − ϵ + r ≤ c while Firm 1 produces with c − ϵ . Note that we use the same notation for the profit of the firms even though we may have now three different marginal costs: the old technology of a non-licensee firm with marginal cost c, the superior technology of the innovator with marginal cost c − ϵ and of every licensee firm with the marginal cost c − ϵ + r. Also note that in a Cournot competition π(N , c ) > πNL (S (σ ), c (σ )) even if r = ϵ . In this case every firm except Firm 1 produces with marginal cost c but Firm 1 produces with marginal cost c − ϵ . 6 One can argue that the SPR strategy is better than the DL strategy for an outside innovator if it is significantly more costly for an outside innovator to transfer its technology to firms in the industry since it requires a training force that usually is limited for a research lab. 7 Except when S (σ ) = N \{j}. Namely the case when Firm j sells licenses to every other firm in the industry. In equilibrium for this case every firm will end up with a net payoff of πNL (N \{j}, c (σ )) for any j ∈ N. The reason is that a deviant firm will lose the license and it will compete with N − 1 other firms with the superior technology. Note that in equilibrium when Firm j auctions off |S (σ )| licenses at least |S (σ )| + 1 firms will bid b = πL (S (σ ), c (σ )) − πNL (S (σ ), c (σ )). Then |S (σ )| firms are randomly selected from the set of firms that bid b and each of them will be awarded a license. Now a deviant licensee firm can be sure that it will be replaced by another firm that submitted a bid b and was not selected.
In the previous section, we proved that an outside innovator who uses the SPR strategy extracts a higher revenue than the incremental profit of an incumbent innovator with the same innovation. This result does not necessarily imply that the SPR strategy dominates the DL strategy for an outside innovator. ST provides the analysis for the DL strategy for two-part-tariff licensing which is a combination of a per-unit royalty r and an upfront fee determined in an auction, where r and the number of licenses offered, k, are pre-announced. Using ST (which applies to a Cournot model with a linear demand) we show that the SPR strategy is better for an outside innovator than the DL strategy if the magnitude of the innovation is sufficiently significant but not drastic. An outside innovator of a drastic innovation obtains the same revenue whether he uses the SPR strategy or the DL strategy. Consider two possible interactions (games) of an outside innovator with the firms in the industry. In the first one, GDL , the innovator licenses his innovation directly to some firms in the industry and in the second one, GSPR , licenses are sold by an incumbent firm who first purchases the property rights from the innovator. Let us describe first the game GSPR . In the first stage, the innovator auctions off the property rights to the highest bidder firm in a first price sealed bid auction. (Ties are resolved at random.) In the second stage, the winning firm, say Firm 1, licenses the new technology to the firms in a subset S1 of N \{1} (including the possibility that S1 = φ ). In the third stage, all licensee and nonlicensee firms compete à la Cournot. Consider next the game GDL . It has two stages. The innovator in the first stage licenses his technology to a subset S2 ⊆ N, S2 ̸= φ , of firms. All firms compete à la Cournot in the second stage. It is assumed in both GSPR and GDL that the licensing strategy of the innovator is commonly known (i.e., the royalty level and the set of licensee firms) before the firms choose their quantities. The class of licensing schemes considered here is a combination of an upfront fee and a per-unit royalty. The upfront fee is determined by a first price auction. The licensor selects a pair (k, r ) where k is the number of licenses to be sold and r, the per-unit royalty that every licensee firm pays the licensor for every unit sold. The pair (k, r ) is announced and firms bid for the license. The k highest bidders win a license and pay their bids upfront in addition to future royalty payments. Ties are resolved at random. In practice the licensing cost is not negligible and it provides an outside innovator with more incentive to use the SPR strategy. Nevertheless, for a significant innovation and for linear demand we show that even with zero licensing cost the use of the SPR strategy is better for an outside innovator than the DL strategy. Proposition 3. Consider an outside innovator in a Cournot industry of size n and suppose that the inverse demand function is P (Q ) = max(a − Q , 0). Then there exists ϵ¯ , 0 < ϵ¯ < min(c , a − c ), such that
8 If it is optimal for the lab to use the DL strategy then it is not clear whether or not an incumbent firm should invest in an outside lab even in the equal probability of success case. Suppose that the optimal licensing strategy of the lab is σ ′ and suppose that an incumbent firm, say Firm 1, succeeds in innovating in-house and adopts the same strategy σ ′ . If σ ′ contains a per unit royalty component r, r > 0, then the operating profit of every licensee firm is smaller compared with the case where the lab is the innovator. The reason is that in the former case the licensee firms who produce with a marginal cost of c − ϵ + r compete with Firm 1 who produces with a smaller marginal cost of c −ϵ . For the same reason, the net profit of a non-licensee firm is also smaller when the innovator is an incumbent firm and it is not clear if the willingness to pay of a licensee firm is higher or lower when the innovator is an incumbent firm compared with the case where the innovator is an outside lab.
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for any n ≥ 4 and for any ϵ , ϵ¯ < ϵ < min(c , a − c ), the payoff of the innovator in GSPR is higher than that in GDL .9 The condition ϵ < a − c is the definition of a non-drastic innovation as introduced first by Arrow (1962). This is equivalent to the condition that the monopoly price under the new technology is higher than the marginal cost c of the old technology.10 Proof. Using ST, there exists ϵ¯ , 0 < ϵ¯ < min(c , a − c ), such that for every ϵ , ϵ¯ < ϵ < min(c , a − c ) and for any industry size n ≥ 4, (i) the payoff in GDL of an outside innovator who uses an optimal combination (kO , rO ) is ϵ(a − c ). (ii) the payoff of an incumbent innovator who uses an optimal combination (kI , rI ) is higher than ϵ(a − c ). The unique optimal policy (kI , rI ) satisfies 0 < rI < ϵ , and kI = n − 1, namely the new technology is used by every firm and each licensee firm pays a per-unit royalty smaller than ϵ . Consider next the game GSPR . Suppose that the outside innovator auctions off the patent rights to the highest bidding firm, say Firm 1. Given the optimal licensing strategy described in (ii), the equilibrium winning bid is
π1 − πN , where π1 is the equilibrium payoff of Firm 1 (the sum of its net profit and the revenue it extracts from licensing the new technology) and πN is the profit of a non-licensee firm when there are n − 1 firms who use the new technology (Firm 1 and the n − 2 other licensee firms). To complete the proof we will show that if ϵ¯ < ϵ < min(c , a − c ) then π1 − πN > ϵ(a − c ). Consider the following non-optimal strategy, σ , of an incumbent innovator: he licenses the new technology to any firm that agrees to pay the per-unit royalty r = ϵ (in this case the willingness to pay of a firm for a license is zero since the firms are indifferent between having or not having the license). In equilibrium all the n − 1 firms, other than the innovator, become licensees (otherwise, the innovator is better off slightly reducing the royalty level below ϵ ). Effectively, every firm except the innovator produces with a marginal cost c. Given σ , denote by π1′ the payoff of the innovator (the sum of his Cournot profit and the licensing revenue) and by πN′ the profit of any other firm. By (ii), rI < ϵ is the optimal royalty level and hence the actual marginal cost of production is c − ϵ + rI which is strictly smaller than c. That is, except for Firm 1 who in both cases produces with marginal cost c − ϵ the other n-1 firms produce with marginal cost c − ϵ + rI < c under the optimal licensing strategy and with marginal cost c under the licensing strategy r = ϵ . Therefore, the profit of a firm other than Firm 1 is smaller in the latter case. Namely πN < πN′ . With a similar argument we have π1 > π1′ . Consequently
9 It can also be shown that there exists ϵ , 0 < ϵ < ϵ¯ < min(c , a − c ), such that for n = 3 and for any ϵ , 0 < ϵ < ϵ , the payoff of the innovator in GDL is higher than that in GSPR . 10 If the innovation is drastic, namely c ≥ ϵ > a − c, then the optimal licensing strategy is to auction off an exclusive license for zero royalty. The licensee firm obtains a monopoly profit and all other firms are driven out of the market. In this case the licensor obtains the entire monopoly profit in both games GDL and GSPR .
π1 − πN > π1′ − πN′ . It is therefore sufficient to prove that π1′ − πN′ = ϵ(a − c ). Let Q ′ be the Cournot equilibrium quantity where there are n−1 firms producing with the marginal cost c and Firm 1 is producing with the marginal cost c − ϵ . Let q′1 and q′i be the Cournot output levels of Firm 1 and of any Firm i, i ̸= 1, respectively. Clearly Q ′ = q′1 + (n − 1)q′i and it is easy to verify that q′1 − q′i = ϵ . Then
π1′ = ϵ(n − 1)q′i + (P (Q ′ ) − c + ϵ)q′1 = ϵ Q ′ + (P (Q ′ ) − c )q′1 , and
πN′ = (P (Q ′ ) − c )q′i . Hence,
π1′ − πN′ = = = = =
ϵ Q ′ + (P (Q ′ ) − c )(q′1 − q′i ) ϵ Q ′ + (P (Q ′ ) − c )ϵ ϵ(P (Q ′ ) − c + Q ′ ) ϵ(a − Q ′ − c + Q ′ ) ϵ(a − c ).
Acknowledgments We would like to thank Debapriya Sen for useful discussions. Also, the paper benefitted considerably from the insightful remarks of an annonymous referee. References Arrow, K.J., 1962. Economic welfare and the allocation of resources for invention. In: Nelson, P.R. (Ed.), The Rate and Direction of Inventive Activity. Princeton University Press, pp. 609–626. Kamien, M.I., 1992. Patent licensing. In: Aumann, R.J., Hart, S. (Eds.), Handbook of Game Theory, vol. 1. North Holland, Elsevier, pp. 331–354. Kamien, M.I., Oren, S., Tauman, Y., 1992. Optimal licensing of cost-reducing innovation. Journal of Mathematical Economics 21, 483–508. Kamien, M.I., Tauman, Y., 1984. The private value of a patent: A game theoretic analysis. Journal of Economics (Supplement) 4, 93–118. Kamien, M.I., Tauman, Y., 1986. Fee versus royalties and the private value of a patent. Quarterly Journal of Economics 101, 471–491. Kamien, M.I., Tauman, Y., 2002. Patent licensing: The inside story. The Manchester School 70 (1), 7–15. Katz, M.L., Shapiro, C., 1985. On the licensing of innovation. Rand Journal of Economics 16, 504–520. Katz, M.L., Shapiro, C., 1986. How to license intangible property. Quarterly Journal of Economics 101, 567–590. Sen, D., Tauman, Y., 2007. General licensing schemes for a cost-reducing innovation. Games and Economic Behavior 59, 163–186. Wang, X.H., 1998. Fee versus royalty licensing in a Cournot duopoly model. Economics Letters 60, 55–62.
Contents lists available at SciVerse ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Selling patent rights and the incentive to innovate Yair Tauman a,b,∗ , Ming-Hung Weng c a
Department of Economics, Stony Brook University, NY 11794-4384, USA
b
Arison School of Business, Interdisciplinary Center (IDC) Herzliya, Herzliya, Israel
c
Department of Economics, National Cheng Kung University, Tainan, Taiwan
article
info
Article history: Received 3 March 2011 Received in revised form 2 October 2011 Accepted 20 October 2011 Available online 26 October 2011
abstract We show that an outside innovator has a higher incentive to innovate than an incumbent innovator, by auctioning off his patent rights exclusively to an incumbent firm. For significant innovations this is also superior to selling licenses directly. © 2011 Elsevier B.V. All rights reserved.
JEL classification: D45 L13 O32 O33 Keywords: Patent licensing Cournot oligopoly Process innovation Patent rights Incentive to innovate
1. Introduction The literature on various licensing methods of a process innovation is quite extensive. Kamien and Tauman (1984, 1986) and independently Katz and Shapiro (1985, 1986) used strategic models to analyze the private value of a patent of an outside innovator (a research lab) who can sell licenses by means of an upfront fee, royalty, auction, or a mixture of the above.1 See also the survey of Kamien (1992). Kamien et al. (1992) studied the market structure resulting from the three common licensing strategies of selling licenses by an auction, by an upfront fee or by a per-unit royalty, but not by the combinations of these three strategies.2 Recently Sen and Tauman (2007) (ST hereafter)
∗ Corresponding author at: Department of Economics, Stony Brook University, NY 11794-4384, USA. E-mail addresses: [email protected] (Y. Tauman), [email protected] (M.-H. Weng). 1 Kamien and Tauman (1984) studied licensing by means of a combination of upfront fee and royalty only for sufficiently competitive markets. 2 Wang (1998) studied a process innovation by an incumbent firm in a duopoly market and showed that a per-unit royalty is a better licensing strategy for the innovator than an upfront fee strategy. The extension of his result to oligopoly of a general size is in Kamien and Tauman (2002). 0165-1765/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.econlet.2011.10.023
analyzed the optimal combination of an upfront fee and royalty, its impact on the market structure and the diffusion of a cost-reducing innovation in a Cournot oligopoly industry by both inside or outside innovators. They show that irrespective of the magnitude of innovation, an outside innovator (e.g. a research lab) has a higher incentive to innovate than an inside one (an incumbent firm). This result is true for Cournot oligopoly competition and for linear demand. In this paper we extend this result to a general competitive environment (not necessarily Cournot) and to any demand or cost structure. The only assumption we make is, if at least one firm produces with the new technology, the profit of a non-licensee firm who produces with the old technology is smaller than its pre-innovation profit level (where all firms produced with the old technology). First notice that an outside innovator can sell the property rights of his innovation to an incumbent firm. We refer to this strategy as the SPR strategy. To the best of our knowledge this strategy is not analyzed in the literature. The buyer firm of the property rights can then license the new technology to other firms and with any licensing strategy. Consider next an industry with an arbitrary number of firms (greater than or equal to two). An outside innovator who puts on auction to sell the property rights of his innovation to the highest bidder will obtain the difference between the payoff of
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the winner (the sum of his operating profit and the revenue he extracts from licensee firms, using his optimal licensing strategy) and the operating profit of a non-licensee firm (the profit he obtains if some other firm wins the auction). On the other hand the incremental profit of an incumbent innovator is the difference between his optimal payoff (which coincides with the winning bid of the previous case) and his pre-innovation profit level (where all firms produced with the old technology). By our assumption, the winning bid paid to the outside innovator is higher than the incremental profit of an inside innovator. The motivation of an outside innovator to use the SPR strategy is based not only on practical grounds but also on a theoretical one. On the practical side, an external research lab may incur a significantly higher licensing cost. The implementation of a new technology often requires service facilities that a research lab may not have, while an incumbent firm can offer more efficiently. Turning to the theoretical side, recall that an outside innovator when using the SPR strategy guarantees a higher payoff than an inside innovator. Still it is not clear whether or not the SPR strategy of the outside innovator dominates the direct licensing strategy, the DL strategy. For oligopoly Cournot competition with linear demand we show that even if the licensing costs are the same for both inside and outside innovators, for a significant innovation the SPR strategy dominates the DL strategy. 2. The model Consider a homogeneous industry with n firms who are engaged in a certain type of competition (Cournot or other). Let N = {1, . . . , n} be the set of firms. Suppose that all firms incur the same constant marginal cost of production c. Let π (N , c ) be their equilibrium profit level. Suppose next that a subset S ⊆ N, S ̸= φ , of firms have access to a superior technology with a marginal cost of production c ′ < c. Let πL (S , c ′ ) and πNL (S , c ′ ) be the equilibrium profit levels of each firm in S and each firm in N \S, respectively (later on the firms with the access to the superior technology will be the ones who purchase a license to use that technology. Hence L stands for licensee firms and NL for non-licensee firms). Assumption 1. Suppose that c ′ < c. Then π (N , c ) > πNL (S , c ′ ). This assumption asserts that firms with the old technology become worse off when some other firms use a new and superior technology.3 Let us show that this assumption holds in an oligopoly Cournot model with an inverse demand P (Q ) satisfying the following assumptions. Assumption 2. P (Q ) is strictly decreasing in Q and differentiable for Q > 0. Assumption 3. The total revenue function, QP (Q ), is strictly concave in Q . Proposition 1. In a Cournot competition π (N , c ) > πNL (S , c ′ ) whenever c ′ < c and S ⊆ N, S ̸= φ . Proof. Consider first the case where all firms in N produce with the less efficient technology. The first order condition of a firm i producing qi units is P (Q (N , c )) + qi P (Q (N , c )) = c . ′
This implies that nP (Q (N , c )) + Q (N , c )P ′ (Q (N , c )) = nc .
(2)
Similarly in a Cournot industry where firms in S produce with a marginal cost c ′ and firms in N \S produce with a marginal cost c nP (Q (S , c ′ )) + Q (S , c ′ )P ′ (Q (S , c ′ ))
= |S |c ′ + (n − |S |)c < nc .
(3)
By Assumption 3, P (Q ) + QP ′ (Q ) is decreasing in Q and by Assumption 2, nP (Q ) + QP ′ (Q ) is also decreasing in Q. Hence by (2) and (3) Q (S , c ′ ) > Q (N , c ).
(4)
Next observe that
π (N , c ) = −(P (Q (N , c )) − c )2 /P ′ (Q (N , c )), and
πNL (S , c ′ ) = −(P (Q (S , c ′ )) − c )2 /P ′ (Q (S , c ′ )). The result now follows from (4) and from the fact that (P (Q ) − c )2 /P ′ (Q ) is increasing in Q .4 Consider next a cost-reducing innovation that reduces the marginal cost from c to c ′ = c − ϵ , 0 < ϵ < c. The innovator can license his innovation to any number of firms in the industry. The next proposition compares the incentive to innovate of an outside innovator with an inside innovator and holds true independently of the licensing strategy. It makes use of the following assumption. Assumption 4. For the same investment level the probability of a successful innovation is the same for both an outside innovator and an incumbent firm. Proposition 2. Suppose that Assumption 4 holds and the industry consists of at least two firms. The incentive to innovate is higher for an outside innovator than it is for an inside one, irrespective of the licensing strategy. Proof. Suppose that both the outside lab and the incumbent firm invest I and their probability of having a successful innovation is λ. Suppose that conditional on success the outside innovator puts on auction to sell his patent rights to an incumbent firm, say Firm 1. Suppose next that Firm 1 chooses a licensing strategy σ (which includes royalty payments, upfront fees, and the number of licenses sold). Then the innovator obtains in equilibrium
π1 (S (σ ), c (σ )) − πNL (S (σ ), c (σ )) ≡ b1 , where S (σ ) is the set of licensee firms under the licensing strategy σ , c (σ ) is the marginal cost of every licensee firm, and π1 (S (σ ), c (σ )) is the payoff of Firm 1, the licensor, which consists of its operational profit plus the revenue it extracts from the licensee firms in S (σ ). Also, πNL (S (σ ), c (σ )) is the profit level of a non-licensee firm. Next suppose that an incumbent firm invents the same innovation. Now the equilibrium incremental profit of the innovator is
π1 (S (σ ), c (σ )) − π (N , c ) ≡ b1 ′ .
(1)
3 The inequality π(N , c ) > π (S , c ′ ) in Assumption 1 holds as equality for NL a Bertrand competition (with unlimited capacity). The same is true if an outside innovator charges a per-unit royalty r = c − c ′ , in which case every licensee firm actually produces with the pre-innovation marginal cost.
4 It is easy to verify that (P (Q ) − c )2 /P ′ (Q ) is increasing in Q iff P ′′ (Q ) < 2(P ′ (Q ))2 /(P (Q )− c ). By Assumption 3 2P ′ (Q )+ P ′′ (Q )Q < 0. Hence it is sufficient to show that Q > −(P (Q ) − c )/P ′ (Q ). By the F.O.C. if all firms produce with a marginal cost c we have P (Q ) − c + qi P ′ (Q ) = 0 or qi = −(P (Q ) − c )/P ′ (Q ). Since qi < Q , (P (Q ) − c )2 /P ′ (Q ) is strictly increasing.
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The expected (incremental) profit of the lab is λb1 − I and the expected incremental profit of the incumbent innovator is
incumbent firm has no incentive to invest in an outside lab, at least as long as the SPR strategy is better for the lab than the DL strategy.8
λπ1 (S (σ ), c (σ )) + (1 − λ)π (N , c ) − π (N , c ) − I = λb1 ′ − I .
2.1. The SPR versus DL strategy
Since by Assumption 1 π (N , c ) > πNL (S (σ ), c (σ )) we have λb1 − I > λb1 ′ − I as claimed.5 Proposition 2 raises a natural question. Why in practice we do not observe firms investing in outside R&D firms (labs), even though they have higher incentive to innovate? One answer is that Proposition 2 holds under Assumption 4 which asserts that for the same level of investment the two entities are equally likely to invent. If an incumbent firm believes that it has a significantly higher probability to invent it will not invest in an outside lab. But we claim that even if Assumption 4 holds and it is commonly known that the probability to invent is the same for both entities the incumbent firm can not benefit from investing in outside labs. This claim however is true for the case where it is more profitable for the outside innovator to sell its property rights to an incumbent firm (the SPR strategy) rather than to license the innovation directly (the DL strategy) to firms in the industry. In the next section we provide sufficient conditions for the SPR strategy to dominate the DL strategy.6 To verify our claim, suppose that the two potential innovators invest I and have the same probability λ of success. Suppose that the incumbent firm, say Firm 1, owns the outside lab and it acts as to maximize its own profit, and not that of Firm 1. The lab, conditional on success, auctions off its property rights to an incumbent firm, say Firm j (not necessarily Firm 1). Let σ be the optimal licensing strategy of Firm j. Denote by S (σ ) the set of licensee firms and let c (σ ) be their marginal cost. In equilibrium every firm in the industry (whether licensee or not) obtains a net payoff of πNL (S (σ ), c (σ )).7 Hence the net payoff of Firm 1, the owner of the outside lab (whether or not it has access to the new technology) is the sum of the profit of a non-licensee firm, πNL (S (σ ), c (σ )), and the winning bid, b(σ ), submitted by Firm j. Thus the incremental profit of Firm 1 conditional on the lab’s success is
∆ = πNL (S (σ ), c (σ )) + b(σ ) − π (N , c ) − I . But ∆ is also the incremental profit of Firm 1 if it invests the same amount I in-house and succeeds to innovate, since it will implement the same optimal licensing strategy σ as Firm j. Since the probability of success is the same in these two scenarios, the expected incremental profits are also equal. We conclude that even when the probability of success is the same for the two entities, an
5 If the licensor charges a per-unit royalty r ≤ ϵ then all licensee firms produce with marginal cost c − ϵ + r ≤ c while Firm 1 produces with c − ϵ . Note that we use the same notation for the profit of the firms even though we may have now three different marginal costs: the old technology of a non-licensee firm with marginal cost c, the superior technology of the innovator with marginal cost c − ϵ and of every licensee firm with the marginal cost c − ϵ + r. Also note that in a Cournot competition π(N , c ) > πNL (S (σ ), c (σ )) even if r = ϵ . In this case every firm except Firm 1 produces with marginal cost c but Firm 1 produces with marginal cost c − ϵ . 6 One can argue that the SPR strategy is better than the DL strategy for an outside innovator if it is significantly more costly for an outside innovator to transfer its technology to firms in the industry since it requires a training force that usually is limited for a research lab. 7 Except when S (σ ) = N \{j}. Namely the case when Firm j sells licenses to every other firm in the industry. In equilibrium for this case every firm will end up with a net payoff of πNL (N \{j}, c (σ )) for any j ∈ N. The reason is that a deviant firm will lose the license and it will compete with N − 1 other firms with the superior technology. Note that in equilibrium when Firm j auctions off |S (σ )| licenses at least |S (σ )| + 1 firms will bid b = πL (S (σ ), c (σ )) − πNL (S (σ ), c (σ )). Then |S (σ )| firms are randomly selected from the set of firms that bid b and each of them will be awarded a license. Now a deviant licensee firm can be sure that it will be replaced by another firm that submitted a bid b and was not selected.
In the previous section, we proved that an outside innovator who uses the SPR strategy extracts a higher revenue than the incremental profit of an incumbent innovator with the same innovation. This result does not necessarily imply that the SPR strategy dominates the DL strategy for an outside innovator. ST provides the analysis for the DL strategy for two-part-tariff licensing which is a combination of a per-unit royalty r and an upfront fee determined in an auction, where r and the number of licenses offered, k, are pre-announced. Using ST (which applies to a Cournot model with a linear demand) we show that the SPR strategy is better for an outside innovator than the DL strategy if the magnitude of the innovation is sufficiently significant but not drastic. An outside innovator of a drastic innovation obtains the same revenue whether he uses the SPR strategy or the DL strategy. Consider two possible interactions (games) of an outside innovator with the firms in the industry. In the first one, GDL , the innovator licenses his innovation directly to some firms in the industry and in the second one, GSPR , licenses are sold by an incumbent firm who first purchases the property rights from the innovator. Let us describe first the game GSPR . In the first stage, the innovator auctions off the property rights to the highest bidder firm in a first price sealed bid auction. (Ties are resolved at random.) In the second stage, the winning firm, say Firm 1, licenses the new technology to the firms in a subset S1 of N \{1} (including the possibility that S1 = φ ). In the third stage, all licensee and nonlicensee firms compete à la Cournot. Consider next the game GDL . It has two stages. The innovator in the first stage licenses his technology to a subset S2 ⊆ N, S2 ̸= φ , of firms. All firms compete à la Cournot in the second stage. It is assumed in both GSPR and GDL that the licensing strategy of the innovator is commonly known (i.e., the royalty level and the set of licensee firms) before the firms choose their quantities. The class of licensing schemes considered here is a combination of an upfront fee and a per-unit royalty. The upfront fee is determined by a first price auction. The licensor selects a pair (k, r ) where k is the number of licenses to be sold and r, the per-unit royalty that every licensee firm pays the licensor for every unit sold. The pair (k, r ) is announced and firms bid for the license. The k highest bidders win a license and pay their bids upfront in addition to future royalty payments. Ties are resolved at random. In practice the licensing cost is not negligible and it provides an outside innovator with more incentive to use the SPR strategy. Nevertheless, for a significant innovation and for linear demand we show that even with zero licensing cost the use of the SPR strategy is better for an outside innovator than the DL strategy. Proposition 3. Consider an outside innovator in a Cournot industry of size n and suppose that the inverse demand function is P (Q ) = max(a − Q , 0). Then there exists ϵ¯ , 0 < ϵ¯ < min(c , a − c ), such that
8 If it is optimal for the lab to use the DL strategy then it is not clear whether or not an incumbent firm should invest in an outside lab even in the equal probability of success case. Suppose that the optimal licensing strategy of the lab is σ ′ and suppose that an incumbent firm, say Firm 1, succeeds in innovating in-house and adopts the same strategy σ ′ . If σ ′ contains a per unit royalty component r, r > 0, then the operating profit of every licensee firm is smaller compared with the case where the lab is the innovator. The reason is that in the former case the licensee firms who produce with a marginal cost of c − ϵ + r compete with Firm 1 who produces with a smaller marginal cost of c −ϵ . For the same reason, the net profit of a non-licensee firm is also smaller when the innovator is an incumbent firm and it is not clear if the willingness to pay of a licensee firm is higher or lower when the innovator is an incumbent firm compared with the case where the innovator is an outside lab.
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for any n ≥ 4 and for any ϵ , ϵ¯ < ϵ < min(c , a − c ), the payoff of the innovator in GSPR is higher than that in GDL .9 The condition ϵ < a − c is the definition of a non-drastic innovation as introduced first by Arrow (1962). This is equivalent to the condition that the monopoly price under the new technology is higher than the marginal cost c of the old technology.10 Proof. Using ST, there exists ϵ¯ , 0 < ϵ¯ < min(c , a − c ), such that for every ϵ , ϵ¯ < ϵ < min(c , a − c ) and for any industry size n ≥ 4, (i) the payoff in GDL of an outside innovator who uses an optimal combination (kO , rO ) is ϵ(a − c ). (ii) the payoff of an incumbent innovator who uses an optimal combination (kI , rI ) is higher than ϵ(a − c ). The unique optimal policy (kI , rI ) satisfies 0 < rI < ϵ , and kI = n − 1, namely the new technology is used by every firm and each licensee firm pays a per-unit royalty smaller than ϵ . Consider next the game GSPR . Suppose that the outside innovator auctions off the patent rights to the highest bidding firm, say Firm 1. Given the optimal licensing strategy described in (ii), the equilibrium winning bid is
π1 − πN , where π1 is the equilibrium payoff of Firm 1 (the sum of its net profit and the revenue it extracts from licensing the new technology) and πN is the profit of a non-licensee firm when there are n − 1 firms who use the new technology (Firm 1 and the n − 2 other licensee firms). To complete the proof we will show that if ϵ¯ < ϵ < min(c , a − c ) then π1 − πN > ϵ(a − c ). Consider the following non-optimal strategy, σ , of an incumbent innovator: he licenses the new technology to any firm that agrees to pay the per-unit royalty r = ϵ (in this case the willingness to pay of a firm for a license is zero since the firms are indifferent between having or not having the license). In equilibrium all the n − 1 firms, other than the innovator, become licensees (otherwise, the innovator is better off slightly reducing the royalty level below ϵ ). Effectively, every firm except the innovator produces with a marginal cost c. Given σ , denote by π1′ the payoff of the innovator (the sum of his Cournot profit and the licensing revenue) and by πN′ the profit of any other firm. By (ii), rI < ϵ is the optimal royalty level and hence the actual marginal cost of production is c − ϵ + rI which is strictly smaller than c. That is, except for Firm 1 who in both cases produces with marginal cost c − ϵ the other n-1 firms produce with marginal cost c − ϵ + rI < c under the optimal licensing strategy and with marginal cost c under the licensing strategy r = ϵ . Therefore, the profit of a firm other than Firm 1 is smaller in the latter case. Namely πN < πN′ . With a similar argument we have π1 > π1′ . Consequently
9 It can also be shown that there exists ϵ , 0 < ϵ < ϵ¯ < min(c , a − c ), such that for n = 3 and for any ϵ , 0 < ϵ < ϵ , the payoff of the innovator in GDL is higher than that in GSPR . 10 If the innovation is drastic, namely c ≥ ϵ > a − c, then the optimal licensing strategy is to auction off an exclusive license for zero royalty. The licensee firm obtains a monopoly profit and all other firms are driven out of the market. In this case the licensor obtains the entire monopoly profit in both games GDL and GSPR .
π1 − πN > π1′ − πN′ . It is therefore sufficient to prove that π1′ − πN′ = ϵ(a − c ). Let Q ′ be the Cournot equilibrium quantity where there are n−1 firms producing with the marginal cost c and Firm 1 is producing with the marginal cost c − ϵ . Let q′1 and q′i be the Cournot output levels of Firm 1 and of any Firm i, i ̸= 1, respectively. Clearly Q ′ = q′1 + (n − 1)q′i and it is easy to verify that q′1 − q′i = ϵ . Then
π1′ = ϵ(n − 1)q′i + (P (Q ′ ) − c + ϵ)q′1 = ϵ Q ′ + (P (Q ′ ) − c )q′1 , and
πN′ = (P (Q ′ ) − c )q′i . Hence,
π1′ − πN′ = = = = =
ϵ Q ′ + (P (Q ′ ) − c )(q′1 − q′i ) ϵ Q ′ + (P (Q ′ ) − c )ϵ ϵ(P (Q ′ ) − c + Q ′ ) ϵ(a − Q ′ − c + Q ′ ) ϵ(a − c ).
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