Mechanisms of Patent Licensing. Sibo Wang

Transcription

1 Mechanisms of Patent Licensing Sibo Wang May 12, 201

2 ACKNOWLEDGEMENT I would like to thank my advisor, Prof. William Rogerson, for sparking my interest in game theory, leading me to the field of industrial organization, guiding me through obstacles that an inexperienced researcher usually encounters, and encouraging me to challenge myself. I would like to thank Prof. Eddie Dekel and Prof. Marciano Siniscalchi for training me so well in formal microeconomic theory that I can read through the abstruse theory literature without too many problems. I would also like to thank my friend Will Doyoon Kim who took his time peer reviewing my ideas with very constructive feedbacks. Last but not least, I would like to thank my family for their consistent support.

3 1. INTRODUCTION AND SET UP Technology has been the main engine of economic development during the last century. Innovation has played a crucial role in development of technology, and patent, protection to intellectual property, has given the creators economic incentives to invent. Fascinated by how patent boosted innovation, I started to research about how much incentive a patent can profit its inventor, and then noticed the importance of selling mechanism. Unlike a piece of chocolate or a painting, patent is usually used to compete in some market, so a buyer may agree to pay more than their valuation of the patent to prevent their competitors from getting the technological advantage. In addition, a good mechanism may coordinate production and raise the total profit of the industry, so the innovator can extract more profit from his buyer. Observing those interesting phenomenon, I decide to survey through the common mechanisms of patent licensing, and conduct an examination for the optimal licensing mechanism. To talk about the mechanisms of patent selling, we need to think about how could a patent benefits a buyer. On one hand, the buyer may use the patent to add some new features to its product to differentiate with competitors. On the other thand, the buyer may use the patent to boost its productivity, or decrease the cost of production. Regarding two ways of

4 1. Introduction and Set Up modeling innovation, I have two concerns. First, the cost reduction aspect of innovation depicts how inventions boost the economic productivity very well. Second, to model competition with heterogeneous goods requires more mathematical details than homogeneous good, and the majority of literature follows this definition of innovation. Therefore, I will adopt the later notion that the innovation helps with cost reduction. Then, I need to describe what the competition structure is like in the downstream market. I am modeling the market as a Cournot duopoly, because it captures the cost aspect of competition well. In addition, the inventor could be either in the market, or out of the market before the innovation. [Wang, 2002] developed a simple framework using Cournot duopoly to compare basic mechanisms such as linear royalty and fixed fee to capture the patent sharing phenomenon in various industries. However, I will follow most of the literature assuming that the innovator does not compete in the market for simplicity, and concentrate on discussions about different mechanisms. I assume that the innovation will reduce the marginal cost of its user by ɛ in this paper. In addition, I will denote the total profit of the innovator as π s Formally, I model the downstream competition as a two-player game. Two identical firms, firm 1 and firm 2, are producing homogenous good and compete in a Cournot duopoly by choosing the quantity to produce. I assume the market demand is linear, or p = a (q 1 + q 2 ). Both firm has a marginal cost of c without innovation. In this case, the Nash equilibrium quantity of both firms are q i respectively (See Apendix I). = a c for i = 1, 2, and their profits are π i = (a c)

5 1. Introduction and Set Up 5 I will first compare the basic pricing schemes, including fixed fee, royalty, and two-part tariff, in Chapter II. [Kamien and Tauman, 186] conducted a comparison on fixed fee and royalty. The main goal of this chapter is a review to their result, and analyze the draw back of those mechanisms. Then, I will discuss auctions and some variation of them in Chapter III. [Katz and Shapiro, 186] and [Kamien et al., 12] discussed the advantage of patent over fixed fee and royalty, and I will formally review their derivation. In addition, I will compare auction and two-part tariff, because they have distinct advantages in profiting. [Sen and Tauman, 2007] attempted to use a mechanism they named Auction with Royalty (AR) to combine those two features, and I will review how this mechanism is better than both auction and two-part tariff. But the question remaining is: could we do better than the AR mechanism? To explore this question, I employ the framework [Segal, 1] to help me with the analysis. I will introduce this framework in Chapter IV, and apply this framework to the specific case in Chapter V. I will analyze why some mechanisms do not achieve the optimal profit using the framework, and calculate the range that AR mechanism could achieve optimal under a non-rebate assumption. In the concluding Chapter VI, I will remark on issues such as asymmetric costs and rebate, and describe the implication of this assumptions.

6 2. FEE AND ROYALTY To sell a patent, the simplest mechanism for the inventor to use is to set a fixed fee price, and the downstream firms decide if they want to purchase the patent. In addition, the seller can also set the price at a per unit royalty as the price, and then let the downstream firms decide if they want to purchase. It is natural to ask the following question: can the inventor make more profit by fixed fee or royalty? In this chapter, I will first discuss the optimal pricing using fixed fee and royalty, and then compare the profit of them. 2.1 Optimal Fixed Fee Pricing To discuss the optimal pricing of the patent, I would like to specify the game under this pricing scheme. I formulate this game as a three-stage game. In the first stage, the inventor announces the price of the patent, denoted t. Then, both firms decide if they want to purchase the patent simultaneously in the second stage. Last, the downstream competition occurs and firms that bought the patent pay the inventor. We want to solve for the sub-game perfect Nash equilibrium of the game, in which the inventor would play the optimal pricing strategy, so then we can find the maximized profit using fixed fee. We solve this game using backward induction. In stage three, the firms

7 2. Fee and Royalty 7 are playing a Cournot duopoly game. As a result, they know the optimal amount they should produce at this stage if the marginal costs are known. We can use the general Cournot duopoly result to find the payoffs of the firms based on the maginal costs decided in the previous stages. Now, let s consider the second stage. Given any price t offered by the inventor and magnitude of innovation ɛ, both firms can choose weither to buy the patent or not. Based on the choices of, there are in total eight kinds of terminal payoffs. To discuss the outcomes, I first define the concept of drastic innovation: Definition 1 (Drastic). An innovation is drastic if ɛ > a c, in which case a firm can monopolize the market with the patent if none of its opponent has the technology. If the innovation is drastic, then we have the payoffs for (π 1, π 2 ) as: Firm 2 Has Patent No Patent Firm 1 Has Patent No Patent (a c+ɛ) 2 t, (a c+ɛ)2 t 0, (a c+ɛ)2 t (a c+ɛ) 2 t, 0 (a c) 2, (a c)2 (See Appendix I for the general derivation of Cournot duopoly). Proposition 2.1. If the innovation is drastic, the fixed fee pricings yields a payoff of π f s = (a c+ɛ)2 (a c)2 for the patentee. Proof. For the case of drastic innovation, t (a c+ɛ)2 will induces both firms to buy the patent. The innovator s greatest profit therefore can be achieved by setting t = (a c+ɛ)2 in this situation, with π f s = 2(a c+ɛ)2.

8 2. Fee and Royalty 8 Further, (a c+ɛ)2 < t (a c+ɛ)2 (a c)2 will make the subgame has two NE that either the buyers buy the patent while the other decide not to. In this situation, the seller can achieve a maximized profit of πs f = (a c+ɛ)2 (a c)2 by choosing t equals to the same number. If t > (a c+ɛ)2 (a c)2, the only downstream NE in this subgame is both players choose not to purchase, and therefore the inventor yield π f s = 0. Therefore, the seller will set the price as t = (a c+ɛ)2 (a c)2 in the subgame perfect equilibrium, and yield a profit of π f s = (a c+ɛ)2 (a c)2. Intuitively, the innovator observe that his invention can help either firm to monopolize the market, so he wants to set the price to help either firm to do so, and extract the profit.as a result, one firm is forced out of the market by the innovation. Now let s consider the case that the innovation is not drastic. The payoffs for the firms given they are optimizing in the third stage are: Firm 2 Has Patent No Patent Firm 1 Has Patent No Patent (a c+ɛ) 2 t, (a c+ɛ)2 t (a c ɛ) 2, (a c+2ɛ)2 t (a c+2ɛ) 2 t, (a c ɛ)2 (a c) 2, (a c)2 Proposition 2.2. If the innovation is not drastic, the optimal profit is obtained by selling two pieces of patent at price t = (a c)ɛ, and the innovator gains a profit of π f s = 8(a c)ɛ. Proof. Suppose the patentee wants to sell two copy of licenses. Then, he would like to set the highest price which results in a NE that both firms would

9 2. Fee and Royalty like to purchase the patent. In this case, the highest t = (a c+ɛ) (a c ɛ)2, and π f s = 2(a c+ɛ) 2(a c ɛ)2 = 8(a c)ɛ. Suppose the patentee wants to sell one copy of license, then he would like to set the highest price which results in a NE that one firm rejects the purchase and the other firm accepts. In this case, the highest t that obtains equilibrium is (a c+2ɛ)2 (a c)2, and the total profit for the seller is πs f = (a c+2ɛ)2 (a c)2 = (a c+ɛ)ɛ The patentee never wants to sell no copy of license, so it cannot be the optimal outcome. Since the innovation is not drastic, we have a c+ɛ < 2(a c). Therefore, the innovator gains more profit by selling two pieces of patent, so the subgame perfect Nash equilibrium is the innovator set teh price at t = (a c+ɛ) (a c ɛ) 2 = 8(a c)ɛ, and the buyers maximize their profit. In this situation, the seller could not help any of the firm monopolize the market, so he decides to sell his invention to both firms, and extract profits from them. 2.2 Optimal Royalty In last section, I have calculated the optimal lump sum price for the seller and their corresponding profit. In this section, I am calculating the optimal royalty. The game of royalty pricing is similar to the game of lump sum price, except in the first stage, the inventor announces a royalty rate δ. The third stage of the game is also a Cournot competition, which is discussed in Appendix I, so we concentrate our discussion on the second stage

10 2. Fee and Royalty 10 first as we did in the lump sum pricing. A subtle difference between royalty and fixed fee is that the royalty may distort the value of innovation, so some drastic innovation may not be able to help a downstream firm to monopolize the market because of the royalty. Therefore, we make the following definition: Definition 2 (drastic policy). A royalty δ is said to be a drastic policy if ɛ δ a c, or a firm can monopolize the market with the patent and the royalty policy δ. So the payoff matrix for both firms if δ is a drastic policy is: Firm 2 Has Patent No Patent Firm 1 Has Patent No Patent (a c+ɛ δ) 2, (a c+ɛ δ)2 0, (a c+ɛ δ)2 (a c+ɛ δ) 2, 0 (a c) 2, (a c)2 And the payoff matrix if δ is not a drastic policy is: Firm 2 Has Patent No Patent Firm 1 Has Patent No Patent (a c+ɛ δ) 2, (a c+ɛ δ)2 (a c ɛ+δ) 2, (a c+2ɛ 2δ)2 (a c+2ɛ 2δ) 2, (a c ɛ+δ)2 (a c) 2, (a c)2 In both cases, if δ ɛ, both firms would choose to purchase at the equilibrium, and δ > ɛ will result in that both firm not purchasing the patent. Therefore, the inventor is choosing a royalty rate δ to maximize his profit. Since both firms will buy the license, the payoff of the inventor is: π s = 2δ(a c + ɛ δ)

11 2. Fee and Royalty 11 Proposition 2.. If the innovation is not drastic, the seller want to set the royalty rate equal to ɛ, gaining an optimal profit of π r s = 2(a c)ɛ. If the innovation is drastic, the seller s optimal profit using royalty is π r s = (a c+ɛ)2 6, by setting the royalty rate of a c+ɛ 2. Proof. We take the FOC of π r s with respect to δ, we have: π s = 2 (a c + ɛ 2δ) As a result, the δ that satisfies FOC is a c+ɛ 2. This expression satsisfies δ ɛ if ɛ a c. Therefore, if the innovation is drastic, the seller want to set the royalty rate at ɛ, yielding a profit of π r s = 2(a c)ɛ. If the innovation is drastic, the inventor want to sell at a rate of a c+ɛ 2, gaining a total profit of π r s = (a c+ɛ) Fee vs. Royalty After we calculated the optimal profit for both fixed fee and royalty pricing, we want to conduct a comparison between them. In the Cournot duopoly case, the fixed fee is a more profitable pricing scheme. Proposition 2.. Fixed fee yields more profit than royalty. Proof. When the innovation is drastic, fixed fee yields a profit of π f s = (a c+ɛ) 2 (a c)2, and royalty yields a profit of π r s = (a c+ɛ)2 6. Denote z = a c.

12 2. Fee and Royalty 12 By the innovation is drastic, we have ɛ z. Therefore, we have: π f s π r s = = (z + ɛ)2 (z + ɛ)2 12 z2 z2 (z + ɛ)2 6 z2 12 z2 = 2z2 > 0 So the fixed fee policy yields more profit in this case. If the innovation is not drastic, denoting z = a c, we have: πs f πs r = 8zɛ 2zɛ = 2 zɛ > 0 Also fixed fee is more profitable. The result coincides with [Kamien and Tauman, 186]. However, this does not indicate a pervasion of fixed fee. First, [Sen, 2005] points out if the number of firms increases, royalty may surpass both auction and fixed fee because the optimal number of licenses to sell is not closed to any integers, so the rounding of this number will make fixed fee a suboptimal option. So what s wrong with royalty? The case of drastic innovation shows the issue clearly. In the case of drastic innovation, the fixed fee policy would drive one of the firm out of the market to realize the monopoly profit for the industry, but a royalty policy cannot optimally achieve that. The monopoly yields a higher industry profit than oligopoly, so the seller can gain more profit

13 2. Fee and Royalty 1 by inducing monopoly. The problem for royalty here is that it cannot extract all the surplus made from the innovation. As a result, the seller chooses not to induce a monopoly because he can only extract a small portion of the monopoly profit. 2. Two-part Tariff Despite the drawback mentioned in the last section, royalty is useful in coordinating downstream production. From the industry planner s perspective, the downstream firms are overproducing from the optimal amount. Therefore, the seller should be able to obtain more profit if he extracts surplus with fixed fee and constrains the number of production with royalty in some cases. In this section, I will show that there are cases that positive royalty improves the seller s profit. The three-stages game is similar to previous ones, except the seller announces a positive pair of tariff δ and t at the same time in the first stage. As a result, we still try to use backward induction to solve the game, and the firms are still competing in a downward Cournot game in the third stage. Knowing the result of Cournot duopoly, we can figure out what the terminal payoffs for actions of each firm in stage 2. To help ourselves with the algebra, consider the case where one firm gets the patent while the other does not. Denote w(δ) be the optimal quantity for the firm with th patent under Cournot competition where the royalty rate is δ. And denote l(δ) be the optimal quantity of the opponent. According to the general Cournot result,

14 2. Fee and Royalty 1 we have: w(δ) = a c+ɛ δ 2 δ ɛ (a c) a c+2ɛ 2δ δ > ɛ (a c) and 0 δ ɛ (a c) l(δ) = a c ɛ+δ δ > ɛ (a c) Therefore, then the payoff matrix is: Firm 2 Firm 1 Has Patent No Patent Has Patent No Patent (a c+ɛ δ) 2 t, (a c+ɛ δ)2 t w 2 (δ) t, l 2 (δ) l 2 (δ), w 2 (δ) t (a c) 2, (a c)2 To maximize the profit, the seller can either choose to sell two licenses or one. First, let s consider the case that the seller sells one license. Lemma 2.1. The seller cannot do better employing two-part tariff than fixed fee if he insists to sell one license only. Proof. The seller have to maintain the downstream equlibrium, so we have the following constraints on t: (a c + ɛ δ) 2 t l 2 (δ) w 2 (a c)2 (δ) t

15 2. Fee and Royalty 15 Therefore we have: (a c + ɛ δ) 2 l 2 (δ) t w 2 (δ) (a c)2 The objective function of the seller is π t s = δw(δ) + t. Since the biggest t the seller can choose is w 2 (δ) (a c)2, the seller wants to maximize π t s = w 2 (δ) + δw(δ) (a c)2. If the innovation is drastic, the optimal δ to choose is 0. The firm with patent would maximize the industry surplus, and therefore the seller can take all of that except (a c)2 to induce the purchase of patent. In this case, δ = 0, π t s = t = (a c+ɛ)2 (a c)2. If the innovation is not drastic, we have δ 0 > ɛ (a c), so w(δ) = a c+2ɛ 2δ. Therefore, the FOC is: 2w(δ)w (δ) + w(δ) + δw (δ) = (a c + 2ɛ 2δ) + 1 2δ (a c + 2ɛ 2δ) = 1 δ (a c + 2ɛ) < 0 Therefore, the seller want to choose the smallest possible δ in this case, which is 0 by our assumption, yielding a maximized profit of (a c+2ɛ)2 (a c)2 = ɛ(a c+ɛ) with the same amount of fixed fee price. Although royalty can potentially benefit the industry surplus in a duopoly, the coordination effect does not exist if applied to one firm only. In this case, royalty only distorts the amount of output if the inventor only sells one license, and decreases the potential profit that the inventor can extract, so

16 2. Fee and Royalty 16 this result is not surprising. Then, let s consider the case that the seller sells two licenses. Proposition 2.5. When the innovation is not drastic, there exists a level of innovation such that the optimal two part tariff has a positive royalty, particularly when ɛ > 1 (a c). Proof. According to Lemma 2.1, the best of selling one license with two part tariff is selling at 0 royalty. We have also proved in Proposition 2.2, selling two licenses is more profitable than selling one with fixed fee if the innovation is not drastic. As a result, the optimal two-part tariff needs to sell two pieces given ɛ < a c. When the seller wishes that both firms buy the patent, his choice of t and δ should satisfy (a c+ɛ δ)2 t l 2 (δ) in order to induce both firms buy the patent. The objective function of the seller in this case is: π t s = 2δ(a c + ɛ δ) + 2t To maximize π t s, the equality of constraints on t should always hold. Plugging in, we have: π t s = 2δ(a c + ɛ δ) + 2(a c + ɛ δ)2 2l 2 (δ) Since we have assumed δ 0, so l(δ) = a c ɛ+δ. Therefore, the payoff function for the inventor becomes: π t s = 2δ(a c + ɛ δ) + 2(a c + ɛ δ)2 2(a c ɛ + δ)2

17 2. Fee and Royalty 17 Take the FOC, we have: 6(a c + ɛ δ) 6δ (a c + ɛ δ) (a c ɛ + δ) = 0 Solve it, the optimal δ should satisifies: δ = ɛ 1 (a c) In the case that ɛ > 1 (a c), the optimal royalty rate is positive, so the statement is proven. Proposition 2.6. The innovator can extract the monopoly industry profit with two-part tariff if and only if ɛ 5 (a c). Proof. We prove the if direction first. Suppose ɛ 5 (a c), the innovator can set a royalty rate of δ = a c+ɛ and t = a c+ɛ 16 to achieve the monopoly profit. In this case, the equilibrium quantity for both firms are a c+ɛ because of the royalty, and the total quantity is the monopoly quantity for the market. In addition, since l(δ) = 0 as δ ɛ (a c), so the equilibrium can be sustained with the inventor takin all the remaining profit from the industry by setting t = a c+ɛ 16 for both firms. Therefore, two-part tariff can achieve monopoly profit if ɛ 5 (a c) by selling to both of the firms. Now we prove the only if direction. First, δ has to be a drastic policy for the inventor to extract the monopoly profit. Suppose not, then l(δ) > 0. Since each firm will retain at least a profit of l(δ), the inventor therefore could not extract the monopoly profit. Second, if δ is drastic policy, then δ ɛ (a c) < a c+ɛ by ɛ < 5 (a c). So the quantity that each firm

18 2. Fee and Royalty 18 would produce is more than a c+ɛ a c+ɛ = a c+ɛ. As a result, the industry is overproducing from the monopoly quantity, so the innovator could not extract the monopoly profit. From the propositions above, we can see that a combination of both royalty and fixed fee yields a better result than using fixed fee solely. The phenomenon captures the coordination effect of royalty well. Especially when ɛ 5 (a c), the coordination via royalty returns the inventor a monopoly profit.

19 . AUCTIONS In addition to two-part tariff, another popular way of selling something is auction. A unique advantage of auction is that it threatens a firm by selling the patent to its opponent. Such an advantage deserves some detailed analysis. I will first discuss an auction without royalty, and then write about auction with royalty..1 Auction without Royalty The most common versions of auction would be the first price seal bid auction or the second price seal bid auction. By the revenue equivalence theorem, we only need to consider the first price seal bid auction. The trading game is set up as a three-stage game. In the first stage, the seller announces number of licenses for sale, denoted as k, and a reservation price r. Then, the firms bid in the second stage. The top k bidders win the auction paying their bid and getting the patent. We denote b i as the bid for firm i. Last, the downstream Cournot competition occurs. Since there are only two firms in the downstream market, k can only be either 1 or 2. First, let s consider the situation that k = 2. Lemma.1. The optimal profit for k = 2 is obtained by set r = t, where

20 . Auctions 20 t is the optimal fixed fee pricing, and obtain a profit of π a s = 8(a c)ɛ if the innovation is not drastic, or π a s = (a c+ɛ)2 (a c)2 if the innovation is drastic. Proof. (Trivial) When the patentee is selling two licenses, both firms choose if they buy the patent at the price r independently, because the supply for licenses is abundant. Therefore, the profit and optimal setting of r is the same as fixed fee. Then, we want to find the optimal profit for k = 1. Lemma.2. The optimal profit for k = 1 is obtained by setting r = 0, gaining the monopoly profit (a c+ɛ)2 if the innovation is drastic and a if the innovation is not drastic. Proof. In the second stage, both firms are bidding for a license of patent, so we want to find out their willingness to pay for the license. If b 1 > b 2, then firm 1 gets the patent, yielding a payoff of (a c+ɛ)2 b 1 if the patent is drastic and (a c+2ɛ)2 b 1 if the patent is not drastic. Firm 2 loses the auction, getting a payoff of 0 if the innovation is drastic, and (a c ɛ) 2 if the innovation is not drastic. If b 1 = b 2, since the tie breaks at random, both firms have a expected payoff of 1 2 ( (a c+ɛ)2 b 1 if the innovation is drastic and 1 2 ( (a c+2ɛ)2 b 1 + (a c ɛ) 2 ) if the innovation is not drastic. For the case b 1 < b 2, it is similar to b 1 > b 2 by symmetry. The winning and losing firms are getting the same payoff for winning/losing. As a result, the downstream equilibrium is b 1 = b 2 = (a c+ɛ)2 if the innovation is drastic. (A deviation upward will result in a loss because firm bids

21 . Auctions 21 too much, and a deviation downward is still 0 profit.) And if the innovation is not drastic, the dowanstream equilibrium is b 1 = b 2 = (a c+2ɛ)2 (a c ɛ)2 = 2ɛ(a c)+ɛ 2. (Deviation will not improve payoff similarily to the drastic case.) An r less than equilibrium bid is not affecting the result, and r greater than that will yield a 0 profit for the innovator since no one buys the license. As a result, r = 0 is optimal in this case. Therefore, the profit of auctioning 1 license is the monopoly profit (a c+ɛ)2 if the innovation is drastic, and 2ɛ(a c)+ɛ2 if the innovation is not drastic. Based on the calculation of profits for issuing different numbers of licenses, we can now derive the optimal number of license to auction. Proposition.1. It is optimal for the innovator to license 1 patent if ɛ 2 (a c)ɛ (a c), and 2 copies of patent with a reservation price r = if ɛ < 2(a c). Proof. In the case of drastic innovation, k = 1 is the optimal choice since it extracts the monopoly profit for the seller. Therefore, we only needs to consider the non-drastic case. π k=1 s π k=2 s = 2ɛ(a c) + ɛ2 = ɛ2 2(a c)ɛ 8(a c)ɛ = ɛ (ɛ 2(a c)) As a result, issuing 1 license is more profitable if ɛ 2 (a c), and 2 copies of licenses otherwise. In addition to the optimal choice of k, I would also like to discuss the industry output after auctioning the patent.

22 . Auctions 22 Proposition.2. If the innovation is drastic, auction yields an monopoly output in downstream market. If the innovation is not drastic, auction will not yield a monopoly output. Proof. If the innovation is drastic, the downstream output is at the monopoly quantity since the firm with the patent is monopolying the market. If the innovation is not drastic, the total output of the downstream market is either 2(a c+ɛ) if two patents are licensed or 2(a c)+ɛ, both are not the monopoly quantity after innovation..2 Auction with Royalty The profitability of auction is different from that of two-part tariff. On one hand, when 5 (a c) > ɛ a c, auction yields monopoly profit, but a two-part tariff cannot, by Proposition 2.6. On the other hand, when 2 (a c) > ɛ > 1 (a c), a two-part tariff is doing striclty better than the fixed fee by Proposition 2.5, and therefore better than auction by Lemma.1. Therefore, a nature question is that can we develop a mechanism combining the advantage of both two-part tariff and auction to achieve a bigger profit? [Sen and Tauman, 2007] proposed a mechanism named Auction Plus Royalty (AR), where the seller announces a royalty rate δ 0 in addition to the number of license he wants to auction and the reservation price. The profit of this mechanism can be characterized by the following proposition: Proposition.. The optimal profit of AR mechanism is the maximum of two-part tariff and auction, or π AR s = max π a s, π t s.

23 . Auctions 2 Proof. If the innovation is drastic, the AR mechanism reaches the monopoly profit by setting the royalty at 0, reserve price at 0, and auction 1 license. In this case, π AR s = π a s, which is the monopoly profit. If the innovation is not drastic, consider the case that the patentee sells two licenses. In this situation, the firms trivally play the same game as pricing with two-part tariff. Therefore, the profit yield with the AR mechanism is the same as two-part tariff. Then, consider the case that the patentee intends to sell one license. Given the royalty rate δ, the firms are willing to bid: (a c + 2ɛ 2δ) 2 (a c ɛ + δ)2 = (a c + ɛ δ)(ɛ δ) In addition, the income from royalty for the seller is δ(a c+2ɛ 2δ), so the total payoff is: π A s (δ) = δ(a c + 2ɛ 2δ) + (a c + ɛ δ)(ɛ δ) Taking FOC, we have: a c + 2ɛ 2δ 2δ (a c + ɛ δ) ɛ + δ = 2δ = 0 Therefore, the seller is gaining the maximize profit with a 0 royalty if auctioning off one license, so π AR s = π a s if only one license. In brief, when k = 1, the optimal π AR s = π a s. And when k = 2, the optimal π AR s = πs. t Therfore, the seller will choose k to maximize πs AR, and the maximizd value is either π t s or π a s.

24 . Auctions 2 The attempt to combine the features here by AR yields a weakly better profit than both two-part tariff and auction by this proposition. However, the mechanism indeed basically says using the better one of auction and two-part tariff. The threat to give to competitor imposed by auction and the coordination effect of two-part tariff do not operate simultaneously in AR. As a result, an investigation about how to combine those features for a greater profit is still needed.

25 . CONTRACTING WITH EXTERNALITIES In last chapter, I asked a question that how to combine the coordination of production and threat to give the opponents well into a mechansim. Further, we could also consider if there are other features that the seller needs to achieve an even greater profit. As a result, a general theory about optimal mechanism suiting out situation is needed. Therefore, I review the theoratic framework proposed by [Segal, 1] on contracting with externalities in this chapter. In [Segal, 1], the contracting game is a three-stage game with n + 1 players, among which there is one seller, or principal, and n buyers, or agents. The seller announces a mechanism in the first stage, and the buyers response in the second stage. In the third stage, the contracts are enforced according to announced mechanism and the actions of the buyers in the second stage. Now, let s define mechanism formally: Definition (Mechanism). A mechanism is a function Γ : S 1 S 2... S n (x 1, x 2... x n, t 1, t 2... t n ) where S i is the signal space designed for agent i, x i represents the contract offered to agent i (no contract if x i = 0), and t i represents the lump sum payment that agent i needs to pay to the principal. In the second stage, the response of agent i must be the element of S i.

26 . Contracting with Externalities 26 The payoff of the principal is: π p = f(x 1, x 2,..., x n ) + n i=1 t i where f is some payoff for the principal by offering contracts x 1, x 2,..., x n, and the sum is the payment received from the agents. The payoff of the agents are: π i = u i (x 1, x 2,..., x n ) t i where u i is the payoff depending on contracts of all the players and t i is the lump sum transfer that agent i has to make to contract. We should notice that u i does not only depend on the contract obtained by agent i, but also depends on the others offers. To discuss the optimal mechanism, we first qualify our discussion into a subset of all mechanisms: Proposition.1. [Segal, 1] Suppose Γ : n i=1s i R 2n is a mechansim where for each agent i there exists 0 S i s.t. playing 0 in the second stage would receive an offer (x i, t i ) = (0, 0). It yields an outcome {(x i, t i )} n i=1 in equilibrium if and only if there exists a mechanism Γ : n i=1s i R 2n in which S i = {0, 1} i {1, 2... n} yields the same outcome of {(x i, t i )} n i=1 in equilibrium. We name such a mechanism with signal spaces {0, 1} for all players a direct mechansim. In addition, the outcome should be reached when all agents decide to accept the trade by sending a response 1. Proof. See [Segal, 1] Proposition 7.

27 . Contracting with Externalities 27 By the proposition above, we then only need to consider an optimal mechanism for the principal with S i = {0, 1}, because more complicated mechanisms have to yield the same outcome as such a mechanism if they allow agents to reject contracting with the principal. In the specific case of patent licensing, the downstream firms can always choose not to purchase the license, so we can apply the result safely. Proposition.2. Denote s i as the vector of signals send by agents other than i such that all the other agents reply 1, and s 0 = (1, 1,..., 1) For any s as a signal profile,denote Γ(s) = (χ(s), τ(s)), where χ(s) is the contract profile and τ(s) is the payment profile given signal s. A direct mechanism maximizes the payoff of principal if and only if: a. χ(s 0 ) solves max χ f(χ) + n i=1 u i(χ) b. χ(0, s i ) solves min χ u i (χ) for any i c. τ i (s 0 ) = u i (χ(s 0 )) u i (χ(0, s i )) Proof. The principal is maximizing his payoff: π p = f(χ(s 0 )) + n τ i (s 0 ) i=1 subject to: τ i (s 0 ) u i (χ(s 0 )) u i (χ(0, s i )) The payoff is maximized if and only if the equalities of constraints hold,

28 . Contracting with Externalities 28 yielding result c. Plug the constraints in, we have: n π p = f(χ(s 0 )) + (u i (χ(s 0 )) u i (χ(0, s i ))) = [f(χ(s 0 )) + i=1 n n u i (χ(s 0 ))] [ u i (χ(0, s i ))] i=1 i=1 The expression is maximized if and only if the term in the first square bracket is maximized and the term in the second square bracket is minimized.

29 5. OPTIMAL MECHANISM FOR PATENT LICENSING 5.1 Discussion on Previous Mechanisms under the Framework Now we can apply the framework in [Segal, 1] to find out the optimal mechanism for patent selling. The mechanisms that we have discussed, either two-part tariff or auction plus royalty (AR), could be represnted under the theoratic framework. In two-part tariff, signals that sent by the buyers are decisions of purchase or not. In AR, the signals sent by the buyers are the bids they submit. The resulting contract for firm i in all mechanisms can be denoted as (y i, δ i, t i ), where y i denotes if the buyer is getting the license (0 for no license, 1 for getting the patent), δ i denotes the royalty rate that person i received, and t i deontes the lump sum transfer. Note that (y i, δ i ) together forms a specfication of a contract instead of a real number x i. We can do this because there exists a bijection between R and R 2. The lump sum payment t i works exactly the same as in the theoratic framework. Now, we should discuss the payoff functions for both buyers and the seller. Denote qi (y 1, y 2, δ 1, δ 2 ) as the downstream equilibrium quantity for firm i. Then, we have: π i = qi 2 (y 1, y 2, δ 1, δ 2 ) t i

30 5. Optimal Mechanism for Patent Licensing 0 for the buyers. And: π s = 2 (y i δ i qi (y 1, y 2, δ 1, δ 2 ) + t i ) i=1 Denote (y s 1s 2 i, δ s 1s 2 i, t s 1s 2 i ) as the contract that player i gets when the signal sent by the buyers is (s 1, s 2 ). Then, according to Proposition., the optimal profit of π s could be written as: π s = [ 2 i=1 ] (y i δ i qi (y1 11, y2 11, δ1 11, δ2 11 ) + qi 2 (y1 11, y2 11, δ1 11, δ2 11 )) [ ] (q1 2 (y1 01, y2 01, δ1 01, δ2 01 ) + q2 2 (y1 10, y2 10, δ1 10, δ2 10 )) The expression in the first square bracket is in fact the industry profit for the downstream, and the expression in the second square bracket is the profit for each firm if they deviate. Two-part tariff cannot coordinate for a maximum industry profit and minimize the deviating profit at the same time in most cases. The biggest penalty that the innovator can give for the deviator is to sell the patent to its competitor without royalty. If the one license is sold, the deviation of the buyer would result in both firms not getting the patent, so the two-part tariff cannot minimizing the deviating profit. If two licenses are sold, a positive royalty rate is needed to coordinate the production, so the punishment to deviator optimal only when the royalty rate is a drastic policy. Therefore, the only case that two-part tariff is optimal is that the royalty rate yields the monopoly quantity while selling to two firms is a drastic policy, coinciding with our previous result.

31 5. Optimal Mechanism for Patent Licensing 1 AR only achieves optimal when the innovation is drastic, where the patentee can threat with a 0 profit and extract the monopoly profit with an auction. If the innovation is not drastic, selling one license that could not drive the firm without it out of the market by auction cannot coordinate the quantity to produce. Selling two licenses, same as two-part tariff, cannot produce a big enough punishment for the deviator. As a result, AR mechansim only yields the optimal outcome when the innovation is drastic. 5.2 Optimal Mechanism for Patent Licensing As I pointed out in last section, the mechanisms we discussed before all have some drawbacks. Naturally, we want to know more about what the optimal mechanism in the specific case of patent licensing. Proposition 5.1. The optimal profit can always be achieved by a direct mechanism Γ s.t. y 11 i = 1 for all i. Proof. Suppose not, then without loss of generality, we have: Γ(1, 1) = (1, 0, δ 1, δ 2, t 1, t 2 ) (5.1) or for the optimal mechanism. Γ(1, 1) = (0, 0, δ 1, δ 2, t 1, t 2 ) (5.2) If equation 5.1 holds, then consider a mechanism Γ that is the same as Γ, except Γ(1, 1) = (1, 1, δ 1, δ 2 + ɛ, t 1, t 2 ). Notice that under both contracts, the marginal costs for production of each buyers are the same, so both quantities

32 5. Optimal Mechanism for Patent Licensing 2 and the price are the same. Consequently, using the mechanism Γ, no buyers make less profit, so they would choose to accept the trade still. However, in Γ, the profit of the seller is increased by q2ɛ. The weakly increment is 0 if and only if q 2 = 0, a.k.a. the innovation is drastic. As a result, Γ can achieve weakly more profit than Γ. Similar derivation if the equation 5.2 holds. As a result, we can always find an optimal mechanism Γ where Γ(1, 1) = (1, 1, δ 1, δ 2, t 1, t 2 ). The intuition here is that an additional patent in the market always creates additional total profit for the industry. If the seller can always extract this piece of profit, he would like to do so. The problem for both AR and two-part tariff are that both agents will be offered the same rate of royalty regardless of how many licenses are sold or who gets the license. As a result, selling to one of the firm may seem more profitable in some cases because royalty, if set at incorrect level, will decrease the punishment to deviation. Proposition 5.2. If δ i and t i could be drawn from any real number, the seller always gets the monopoly profit. Even if we constraint δ i 0, the monopoly quantity is always reached if we allow t i to be chosen along the real line. Proof. To maximize industry profit, the seller would set the royalty rate at a c+ɛ for both firms when they accept the deal. The optimal quantity for each firm becomes a c+ɛ if accepts the trade, and the total output becomes a monopoly quantity. In this case, δ i 0, but t i might be negative to extract all the profit. Suppose in addition, δ i can be drawn from any real number. If any of the firm deviates from accepting the trade, the profit they can obtained could

33 5. Optimal Mechanism for Patent Licensing be driven to 0 by giving a δ i = ɛ (a c) to its competitor. Therefore, the minimized profit from deviation is 0 for both firms. Considering the industry will have an output at the monopoly quantity, the seller can extract the monopoly profit with the 0 profit threat to the deviator. A natural question following this proposition is that what if we constrain t i 0 as well. On one hand, there are still some cases such that a monopoly output is reached if the innovation is big enough. On the other hand, the principal could also set up a non-linear royalty mechanism to coordinate the market. For example δ(q i ) = 0 if q i a c+ɛ, and δ(q i ) = a c if q i a c+ɛ would coordinate the downstream production to produce the monopoly quantity. As a result, the optimal profit for the seller can always result in a monopoly quantity of downstream market.

34 6. CONCLUDING DISCUSSIONS In this paper, I first compared fixed fee and royalty. The patent is sold to both downstream firms in Cournot duopoly if the innovation is not drastic under fixed fee pricing, but sold to one firm only if the innovation is drastic. The patent is sold to all the buyers no matter the degree of innovation. In addition, under Cournot duopoly, fixed fee is always a more profitable pricing mechanism. Further, two-part tariff does better than both of them, and it obtains the monopoly output if the innovation is big enough, or ɛ a c. Then, I discussed about auction. It is optimal for the seller to sell one license to the buyers when the innovation is big, and to sell two licenses if the innovation is relatively small. The seller obtains a monopoly profit if the innovation is drastic under first price seal bid auction. Two-part tariff and auction have different advantages for profitting. Somtimes the two-part tariff makes more profit, and somtimes it is the opposite. A greater profit can be obtained with auction plus royalty (AR), which tried to combine desirable features of both mechanisms. We can still do better than AR. I reviewed [Segal, 1] s theory about optimal mechanism, it shows that the optimal mechanism must maximize the industry profit at equilibrium, minimize the deviating profit for each buyers, and extract the proper profit by a lump sum transaction. Every

35 6. Concluding Discussions 5 simple mechanism has some of the good property. Fixed fee allows the seller extracts all the surplus that created by patents from the buyers. Royalty has a coordinating effect for production so the firms can achieve a better downstream industry profit. Auction provides threat for not purchasing. However, each traditional mechanism has some drawbacks as discussed in Chapter V. The optimal configuration of mechanism will yield monopoly industry profit if a per-unit/lump sum rebate, or negative royalty/payment is allowed, and yield a monopoly industry output if only the lump sum transfer is allowed to be negative. However, the result that the optimal mechanism yields the monopoly output should not be observed in practice. On one hand, an important assumption in this paper is that both firms have the same cost before innovation. Without this assumption, the seller has to force the firm with higher cost out of the market in order to achieve the industry optimal output for the greatest profit. It would be hard to do so if the innovation is not drastic. As a result, the optimal profit is hard to reach in the case of asymmetric cost. However, the seller may not care about the small efficiency loss if the differences in cost is small, and the innovation can be drastic much easier if the cost is big. Therefore, we can ignore the issue of asymmetric costs. On the other hand, coordination to produce the monopoly quantity via patent can still be regarded as collusion by anti-trust authority. Therefore, bearing the regulatory constriants, the seller would not carry out the theoratical optimal mechanism. Still, the results of this paper provide intuition for innovators to maximize their profit under the legal constraints.

36 7. APPENDIX I: COURNOT DUOPOLY In this section, I am writing the derivation to Nash equilibrium for the Cournot duopoly game, since I have used the conclusions extensively in my paper. Now here is the formal description of the game: Players: {1, 2} Actions: q i [0, + ), i {1, 2} Payoffs: π i (q i, q i ) = p(q i, q i )q i c i q i, where p = a (q i + q i ) In this Cournot duopoly game, the demand is assumed to be linear, and firm i has a per unit cost c i for production. And without loss of generality, we here assume that c 1 c 2 Now given q i, we take the FOC of π i respect to q i to find the best response, so we have: π i(q i, q i ) = p (q i, q i )q i + p(q i, q i ) c i = q i + a q i q i c i = 2q i q i + a c i = 0 Notice that the SOC is 2, so the solution to FOC yields the global maximizer. Combine equations for both i = 1 and i = 2, since the two players

37 7. Appendix I: Cournot Duopoly 7 are best responding each other in Nash equilibrium, we have: 2q 1 + q 2 = a c 1 2q 2 + q 1 = a c 2 to yield an interior solution. Solve them, we have: q 1 = a 2c 1 + c 2 q 2 = a 2c 2 + c 1 Therefore, if a 2c 2 c 1, we yield an interior solution for Nash equilibrium, where player i chooses to produce a quantity of a 2c i+c i. In this case, the profit of firm π i is (a 2c i+c i ) 2. If c 1 < a < 2c 2 c 1, then there is no interior solution since firm 2 could not respond to firm 1 with a positive quantity. In this case, firm 1 will produce at the monopoly quantity, observing that firm 2 cannot produce at positive quantity without a loss. Therefore, q 1 = a c 1 2 and q 2 = 0. In this case, the equilibrium profit would be π 1 = (a c 1) 2 and π 2 = 0. If c 1 a, then none of the firm can make a positive profit since the max possible price is less than the per unit cost for both of them. Therefore, they will both produce 0 and get 0 profit at equilibrium. According to the discussion above, there is a very useful observation I want to highlight: Proposition 7.1. In Cournot duopoly, π i = q 2 i for both firms. Proof. Observing the profit and quantity at equilibrium that we dervied, we

38 can make easily draw this conclusion. 7. Appendix I: Cournot Duopoly 8

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