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# Blackwell Publishers Ltd and the Board of Trustees of the Bulletin of Economic Research 2001. Published by Blackwell Publishers, 108 Cowley Road, Oxford OX4 1JF, UK and 350 Main Street, Malden, MA 02148, USA. Bulletin of Economic Research 53:2, 2001, 0307± 3378
THE RELEVANCE OF BARGAINING FOR THE LICENSING OF A COST-REDUCING INNOVATION* Jose J. Sempere Monerris and Vincent J. Vannetelbosch University of Valencia and Universite Catholique de Louvain
ABSTRACT
In the context of a Cournot duopoly, this paper studies the licensing of a cost-reducing innovation by means of three possible allocation mechanisms: auction, fixed fee, and direct negotiation. Once the use of an arbitrary reserve price (which is not credible) has been excluded, it is no longer true that auction always yields higher profit to the patentee than a fixed fee. However, the authors propose a direct negotiation mechanism which restores the patentee's profit to the level of an auction with an arbitrary reserve price (which is unimplementable). Direct negotiation is superior to both an auction with a nonarbitrary reserve price and a fixed fee. From the social point of view, however, licensing with a fixed fee is the best option. I.
INTRODUCTION
Our aim is to compare bargaining with auctioning or setting a fixed fee for the licensing of a cost-reducing innovation. The dominant stream of the patent licensing literature hinges on the patentee's ability to exploit the licensees' competition for a licence. Kamien and Tauman (1986), Katz and Shapiro (1986) and Kamien et al. (1992) use a game-theoretic framework to study the strategic interdependence * Vincent Vannetelbosch is Charge de Recherches at the Fonds National de la Recherche Scientifique. We wish to thank an anonymous referee and Xavier Wauthy for helpful comments. Financial support from the research projects PI-98-48 (Basque Country government), G17-99 (UPV-EHU) and TMR Network FMRX CT 960055 `Cooperation and Information' (THEMA, Universite de Cergy Pontoise) is also gratefully acknowledged.
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between the patentee and an oligopolistic industry, the potential licensees. Two trading mechanisms of patent licensing have dominated the literature: the patentee either (a) sets a per-unit royalty and=or a fixed fee, or (b) auctions a fixed number of licences. In a standard licence auction game, a non-cooperative three-stage game, the patentee decides first the number of licences to be auctioned. In the second stage, all the interested firms decide simultaneously and independently how much to bid for a licence. Finally, in the third stage, licensed and unlicensed firms choose outputs to maximize profits. Licences are awarded to the highest bidders and ties are broken arbitrarily. However, when the number of licences announced is the same as the number of oligopolistic firms, the patentee must also state a reserve price (slightly below the benefit to a firm if all are licensed) below which he will not sell a licence. The reason for setting a reserve price is to prevent any firm from offering nothing for a licence, because it knows it will get one anyway. The questionable point lies in the patentee's ability or commitment to set any particular reserve price different from the continuation value, and stick to it. It is not a credible commitment. However, the patentee may look for alternative trading mechanisms, which allow him to overcome the credibility problem associated with the reserve price. Also, the case where the number of licences sold equals the number of firms may be considered as a bargaining problem, and auction theory predictions are quite sensitive to the bargaining model used. Indeed, the terms of trade between the patentee and the buyer are determined by negotiation, the course of which is influenced by each agent's opportunities for matching and trading with other partners. When there are fewer licences than firms, auction theory is insensitive to the bargaining theory used. Alternative methods of licensing by means of bargaining may then be of interest. One result in the licensing literature when firms are Cournot competitors in the third stage is that a fixed-fee mechanism of licensing is superior to a royalty one in terms of patentee profits and consumer surplus (Kamien and Tauman, 1986). Also, in general, auctioning licences yields the patentee higher profits than selling licences by means of a fixed fee or a royalty (Katz and Shapiro, 1986; Kamien, 1992; Kamien et al., 1992). The intuition behind this is as follows. In the case of an auction, a licensee rejection decision does not reduce the number of licences, while in the case of a fixed fee it is reduced by one. Therefore, the profits of unlicensed firms are lower in the case of an auction than in the case of a fixed fee. As the most a firm will pay for a licence is the difference between its profits as a licensed firm versus as an unlicensed firm, it will pay more when licensing is by means of an auction. In a two-period Cournot duopoly, we consider the licensing by an independent laboratory of a cost-reducing innovation by means of three possible allocation mechanisms: auction, fixed fee, and direct negotiation. # Blackwell Publishers Ltd and the Board of Trustees of the Bulletin of Economic Research 2001.
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Two potential licensees are in the market. This small market allows us to focus on the case where the number of licences sold equals the number of potential licensees. As already mentioned, this particular case questions the auctioning mechanism with an arbitrary reserve price. Our main result is that, once we exclude the use of an arbitrary reserve price (which is not credible), it is no longer true that the auction always yields higher profit to the patentee than a fixed fee. In fact, the auction with a nonarbitrary reserve price implies exclusive licensing. That is, only one license will be sold at equilibrium. In contrast, the fixed-fee mechanism implies complete diffusion of the innovation: two licences are always sold at equilibrium. However, we propose a direct negotiation mechanism which restores the patentee's profit to the level of an auction with an arbitrary reserve price. This direct negotiation is weakly superior to both an auction with a nonarbitrary reserve price and payment of a fixed fee. Firstly, the patentee strictly prefers direct negotiation to auction with a nonarbitrary reserve price when the nondrastic innovation is small; otherwise, he is indifferent. Secondly, the patentee strictly prefers direct negotiation to the fixed fee when the nondrastic innovation is more significant; otherwise, he is indifferent. From the social point of view, licensing through a fixed fee is the best option. Finally, our main results are qualitatively robust to an alternative specification where the selling of one licence brings to the buyer not only advantages in terms of costs but also the advantage of becoming the Stackelberg leader in the market. In Section II we give a description of the market for the cost-reducing innovation and we consider two classic modes of licensing: a licence auction game and a fixed-fee licensing game. Section III is devoted to an alternative mode of licensing where trading is carried out through a direct negotiation mechanism. Section IV undertakes a social welfare appraisal.
II.
DESCRIPTION OF THE MARKET
We consider a duopolistic industry consisting of two identical firms. A patentee with a cost-reducing innovation seeks to license the patent to both firms, to one or to none so as to maximize her profit. We assume that the patentee is an independent research laboratory and cannot enter the market of the final good directly.1 A licensing game is a noncooperative game between the patentee and the duopolists, which 1 After the Second World War a class of specialized process design and engineering firms appeared which played an important role in the developing and diffusion of process innovations. See Arora (1997) for the case of the chemical industry.
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consists of three stages. In the first stage, either the patentee chooses the number of licences to sell, k, belonging to {0; 1; 2} K, or he settles a fixed fee. In the second stage the licence(s) is (are) traded following the trading rules in force. Finally, in the third stage the Cournot competition takes place.2 Without the new technology, the duopolists are producing the same good with a linear cost function f (qi ) cqi , where qi is the quantity produced by firm i (i 1; 2), and c > 0 is the constant marginal cost of production. Both firms face a homogenous linear inverse demand for the good given by P(Q) a Q, where a > c, and Q q1 q2 is the aggregate quantity demanded and produced. Firm i's production profits are given by i (q1 ; q2 ) (a q1 q2 )qi cqi . The patentee owns a costreducing innovation that reduces the marginal cost of production from c to c ", c > " > 0. Both firms' technologies are common knowledge when the duopolists are choosing their quantities to produce. Let i (i ) be firm i's Cournot± Nash equilibrium production profits when both firms produce with the old (new) technology. Let Ai (i ) be firm i's Cournot ± Nash equilibrium production profits when firm i produces with the new (old) technology while firm j produces with the old (new) technology; j 6 i. Analytically, we have that: i 19 (a
c) 2
Ai 19 (a
c 2") 2
i 19 (a
c
i 19 (a
c ") 2
") 2
where Ai > i > i > i . It should be noted that if only one firm owns the innovation then the other one is worse off, since there exists a negative externality due to the market interdependence between the duopolists. We restrict our study to the case of nondrastic innovations. A nondrastic innovation is such that the nonpurchasing firm would produce a positive quantity at the Cournot± Nash equilibrium. We also assume that all innovations are nondrastic, so that " Æ a c. Another situation that might be interesting to explore is what happens when the selling of one licence brings to the buyer not only advantages in terms of costs but also the advantage to become the Stackelberg leader in the market. In such a situation, the profits of the buyer and the nonbuyer 1 (a c 2") 2 , become, respectively, A Li 18 (a c 2") 2 and Fi 16 F L where A i > i > i > i . The superscripts `L' and `F' identify the leader and the follower. However, the range of " that implies a nondrastic innovation shrinks and becomes " Æ 12 (a c). Although the Stackelberg 2 It is assumed that, at the beginning of each stage, everything that happened at previous stages is common knowledge among the agents.
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assumption is an interesting extension, our analysis will focus mainly on the Cournot assumption. The Cournot assumption makes sense once we interpret the leader position in a market as the ability of a firm to precommit to a given action regardless of its cost structure or product design. Nevertheless, we will comment about the implications of the alternative Stackelberg situation. The only available mode of patent licensing are lump-sum fees, Fi (we do not exclude, a-priori, Fi 6 Fj ), which are independent of the quantity produced. Two trading mechanisms have been studied in the literature: a licence auction game and a fixed-fee licensing game. II.1. The licence auction game The licence auction game considered here is the one developed by Katz and Shapiro (1986) and Kamien (1992), except that we have a duopolistic industry and we exclude the use of arbitrary reserve prices. In the first stage the patentee decides how many licences k 2 {0; 1; 2} K to auction. Let 0 be the monetary value of the innovation to the patentee. We focus on the case where the patentee cannot choose a reserve price different from below which she will not sell a licence. The second stage sees the sealed-bid first price auction.3 Both firms decide independently and simultaneously how much to bid for a licence. Licences are sold to the highest bidder at their bid price, and in the event of a tie licensees are chosen randomly. In the third stage, each firm, licensed or unlicensed, competes on the good market and chooses its Cournot profit-maximizing level of output. We denote by A(K) the licence auction game with a nonarbitrary reserve price. The subgame-perfect equilibrium (SPE) in pure strategies is the solution concept used. Thus, the licence auction game A(K) is solved backwards. Let A(k) be the auction game where the patentee offers k licences in the first stage. That is, A(1) is the licence auction game with exclusive licensing. At the SPE of the license auction game A(1), both firms make a bid equal to: F*i [A(1)] [Ai
i ] 19 [6(a
c) 3"]";
i 1; 2
and the licensee is chosen randomly. But at the SPE of the licence auction game A(2), both firms make a bid equal to F*i [A(2)] 0 (for i 1; 2), and obtain the licence. Proposition 1: Consider the licence auction game A(K) with nonarbitrary reserve price. At the SPE it is optimal for the patentee to auction only one licence if the innovation is nondrastic. 3 The interested reader on auction theory is referred to Wilson (1992) or Wolfstetter (1996).
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At equilibrium, the agents' (patentee=firm i=firm j) payoffs of A(K) are, respectively: [A(K)] 19 [6(a
c) 3"]"
(1)
i [A(K)] 19 [a
c
"] 2
(2)
j [A(K)] 19 [a
c
"] 2 :
(3)
Proposition 1 is due to the absence of an arbitrary reserve price. Let A*(K) be the licence auction game where the patentee chooses an arbitrary reserve price in the first stage. At the SPE of the licence auction game A*(1), both firms make a bid equal to F*i [A(1)], for i 1; 2, and the licensee is chosen randomly. The reserve price matters only for A*(2) Indeed, at the SPE of the licence auction game A*(2), both firms make a bid equal to (i i ) 49 (a c)" and obtain the licence. To sustain such an SPE the patentee must, along with her announcement that two licences will be auctioned, fix a reserve price slightly below the benefit to a firm if both firms are licensed. The use of the reserve price prevents a firm from offering nothing for a licence because it knows it will get one anyway. Therefore, at the SPE of the licence auction game A*(K) with an arbitrary reserve price, the patentee will auction one licence if " 2 (23 (a c); a c] and two licenses if " 2 (0; 23 (a c)). That is, if the nondrastic innovation is relatively small, it is optimal for the patentee to sell licences to both firms. At equilibrium, the agents' (patentee=firm i=firm j) payoffs of A*(K) are, respectively: 8 < 1 [6(a c) 3"]" if " 2 (2 (a c); a c] 9 3 [A*(K)] (4) 8 : (a c)" if " 2 (0; 23 (a c)) 9 i [A*(K)] 19 [a
c
"] 2
(5)
j [A*(K)] 19 [a
c
"] 2 :
(6)
But the patentee's ability or commitment to fix any particular reserve price and stick to it may be questionable. If at the end of the auction game the patentee has to choose between selling the licences at the highest bids or withdrawing the licences from the auction, then fixing a reserve price different from the continuation value is not a credible commitment. II.2
The fixed fee licensing game
The fixed-fee licensing game was introduced by Kamien and Tauman (1986); see also Kamien (1992). This game is similar to the licence auction game except that now in the first stage the patentee only chooses # Blackwell Publishers Ltd and the Board of Trustees of the Bulletin of Economic Research 2001.
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a price at which any firm wishing to can buy a licence. The licence price is independent of the number of units produced with the new technology and therefore is a fixed cost just as in the auction case. In the second stage, both firms choose simultaneously whether or not to buy a licence at the fixed price.4 The production stage does not change. We denote by FF the fixed-fee licensing game and we solve it backwards. For all nondrastic innovations, the SPE licensing policy of the patentee is to choose a fee equal to: 4 F* i [FF ] 9 (a
c)";
i 1; 2
such that both firms buy a licence. Proposition 2: Consider the fixed-fee licensing game FF. At the SPE two licences are sold by the patentee if the innovation is nondrastic. At equilibrium, the agents' (patentee=firm i=firm j) payoffs of FF are, respectively: [FF ] 89 (a
c)"
i [FF ] 19 [a
c
"] 2
(8)
j [FF ] 19 [a
c
"] 2 :
(9)
(7)
Comparing the expressions (4) and (7), we recover Kamien's (1992) result that the patentee's licensing profits are, in general, lower under fixed-fee licensing than under licence auctioning. This result is no more valid once we exclude the use of arbitrary reserve prices (which are not credible). Comparing expressions (1) and (7), we obtain [A(K)] > [FF ] if " 2 (23 (a c); a c], and [A(K)] < [FF ] if " 2 (0; 23 (a c)). That is, the optimal choice for the patentee becomes to choose either the fixed fee (when the innovation is relatively small) or the auction with nonarbitrary reserve price (when the innovation is more significant). In the next section we propose a single mechanism which restores (whatever the size of the nondrastic innovation) the patentee's profit to the level of the auction with arbitrary reserve price. However, instead of proposing another mechanism as we do next, one could also consider some options that would make a reserve price different from zero a credible commitment. One option (as suggested by a referee) is to allow a potential entrant into the market. When the 4 Choosing a fixed fee, such that it is optimal only for one firm to buy the innovation, implies a coordination problem since both identical firms decide simultaneously whether or not to buy a licence at the settled price. To avoid coordination problems in the fixed-fee licensing game, we could assume that both firms choose sequentially: firstly, firm i decides whether or not to buy a licence; secondly, firm j decides after having observed firm i's decision.
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patentee decides to auction two licences to the incumbents, the credibility problem of setting an arbitrary reserve price arises. But by having a third option, or the threat of a third option, both incumbents would bid according to A*(2); i.e. both would prefer to buy the costreducing innovation if the threat of a potential entrant exists. They would know that if they did not bid according to A*(2), the patentee could always try to sell it to a third firm, which would then enter the market and affect the incumbent's profits. This would be a credible threat from the patentee, which would surely make the auction A*(2) implementable. But free entry would only partially solve the problem, since now the patentee could consider to auction three or more licences. III.
LICENSING THROUGH DIRECT NEGOTIATION
The licensing game with direct negotiation we consider is a simple takeit-or-leave-it bargaining with voluntary matching. In the first stage the patentee decides how many licences k 2 K to sell. In the second stage the agents are matched and negotiate the terms of trade. In the third stage, each firm, licensed or unlicensed, competes on the good market and chooses its Cournot profit-maximizing level of output. We denote by T(K) the licensing game with direct negotiation. So, T(k) is the licensing game with direct negotiation where the patentee offers k licences. Formally, the negotiation proceeds as follows. *
*
Direct-negotiation licensing game T(1). In the second stage, the patentee voluntary matches with one of the firms and makes an offer. The matched firm either accepts or rejects the offer. If it accepts, then the negotiation ends and it starts to produce in the third stage with the new technology. If it rejects, then the patentee again voluntary matches with one of the firms (not necessarily the same firm as previously) and makes an offer which is accepted or rejected. Direct-negotiation licensing game T(2). In the second stage, the patentee makes an offer (F1 ; F2 ). Both firms, simultaneously, accept or reject the offer. If both firms accept the offer, then the negotiation ends and both firms start producing with the new technology. If at least one firm rejects the offer, then the negotiation will proceed. The patentee makes again an offer to each firm which has not bought the new technology, and the offers are (simultaneously) accepted or rejected.5
5 Allowing the patentee to match and to make an offer to the potential licensees more than twice would not change the SPE offers and outcomes. The reader interested in more sophisticated bargaining theory is referred to Osborne and Rubinstein (1990) or Binmore et al. (1992).
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First we consider the direct-negotiation licensing game T(1). At the SPE the patentee voluntarily matches with firm i and makes the offer Fi such that firm i is indifferent between accepting and rejecting. If firm i rejects the offer, then firm i will obtain the payoff of the unmatched buyer: i . Indeed, the patentee can credibly threaten firm i to match next with firm j if firm i rejects the patentee's current offer. Therefore, at equilibrium, the patentee makes an offer: F* i [T(1)] Ai
i 19 [6(a
c) 3"]":
This offer is immediately accepted by the matched firm i. At equilibrium, the agents' (patentee=firm i=firm j) payoffs of T(1) are, respectively: [T(1)] Ai
i 19 [6(a
c) 3"]"
(10)
i [T(1)] i 19 [a
c
"] 2
(11)
j [T(1)] j 19 [a
c
"] 2 :
(12)
The main difference between T(1) and FF is that, in the direct negotiation T(1), the patentee can credibly threaten firm i to match next with firm j 6 i if firm i rejects the patentee's current offer. Using this credible threat the patentee will obtain a higher revenue in T(1). Next we consider the direct-negotiation licensing game T(2). We show that to offer Fi Fj 49 (a c)" at any subgame is the SPE for the direct negotiation licensing game T(2). Moreover, both firms will accept this offer immediately. At any subgame where no agreement was reached before, the patentee offers two contracts. Both firms, simultaneously, either accept it (buy) or reject it (don't buy). We compute the pair (Fi ; Fj ) that maximizes the patentee's revenue. Two Nash equilibria are analysed: (buy, buy) and (buy, don't buy). The following conditions, on Fi and Fj , are necessary and sufficient for (buy, buy) to be a Nash equilibrium: i Fi æ i and j Fj æ j . That is, Fi Æ 49 (a c)" and Fj Æ 49 (a c)". Therefore, when (buy, buy) is the Nash equilibrium outcome the patentee's revenue is 89 (a c)". The following conditions, on Fi and Fj , are necessary and sufficient for (buy, don't buy) to be the unique Nash equilibrium: Ai Fi æ i and j æ Aj Fj . That is, Fi Æ 49 (a c ")" and Fj > 49 (a c ")". Therefore, when (buy, don't buy) is the unique Nash equilibrium the patentee's revenue is 4 c ")". Comparing the patentee's revenue for (buy, buy) and 9 (a (buy, don't buy) and given that the innovation is nondrastic, the patentee's optimal decision is to offer the licence at the price 49 (a c)" to both firms. At any subgame where one licence was sold before, the nonbuyer will accept the offer Fi i i 49 (a c)". Therefore, at equilibrium, the patentee makes (at the first subgame) an offer: F*i [T(2)] 49 (a
c)";
i 1; 2
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to both firms. This offer is immediately accepted by both firms. At equilibrium, the agents' (patentee=firm i=firm j) payoffs of T(2) are, respectively: [T(2)] 2(i
i ) 89 (a
c)"
(13)
i [T(2)] i 19 [a
c
"] 2
(14)
j [T(2)] j 19 [a
c
"] 2 :
(15)
On comparing the expressions (10) and (13), we can state the following proposition. Proposition 3: Consider the direct negotiation licensing game T(K). The SPE licensing policy of the patentee is to sell one licence if " 2 (23 (a c); a c] and two licences if " 2 (0; 23 (a c)). Which mode of licensing does the patentee prefer? Before comparing the various modes of licensing, it is worthwhile to notice the following results. Firstly, exclusive licensing is assigned by the direct-negotiation mechanism when the nondrastic innovation is not too small. Secondly, there is always exclusive licensing with the auction mechanism with a nonarbitrary reserve price. Finally, there is never exclusive licensing with the fixed fee. From Propositions 1, 2 and 3, and expressions (1), (7), (10) and (13), we obtain the patentee's preferences at equilibrium: * *
If " 2 (23 (a c); a c] then [A*(K)] [A(K)] [T(K)] > [FF ]. If " 2 (0; 23 (a c)) then [A*(K)] [T(K)] [FF] > [A(K)]:
Firstly, we compare the direct-negotiation mechanism with the fixed fee. Direct negotiation yields greater profit whenever only one licence is sold by means of the direct negotiation; i.e. when the nondrastic innovation is more significant. Otherwise, the direct negotiation provides the same profit as the fixed-fee mechanism; i.e. when the nondrastic innovation is small. The reason is that the threat of matching with the other firm, if the offer is not accepted, allows the patentee to obtain greater profits using the direct negotiation and selling one licence than using the fixed fee (and selling two licences). But the credible threat the patentee has in T(1) disappears in T(2). As a consequence, the patentee is indifferent between the direct negotiation and the fixed fee whenever it is optimal for her to sell two licenses by means of both mechanisms. Secondly, comparing auction with a nonarbitrary reserve price and direct negotiation, we observe that both mechanisms yield the same profit to the patentee when one licence is granted using the direct negotiation (i.e. when the innovation is not too small). But, the patentee prefers direct negotiation to auction with a nonarbitrary reserve price when the innovation is small; i.e. the case where had the auction with an # Blackwell Publishers Ltd and the Board of Trustees of the Bulletin of Economic Research 2001.
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arbitrary reserve price been implementable, the patentee would have sold two licences. Finally, the direct negotiation mechanism restores the patentee's profit to the level obtained with the unimplementable auction with an arbitrary reserve price. This result is true whatever the size of the nondrastic innovation. The next proposition summarizes our main results. Proposition 4: At the SPE, the patentee weakly prefers the directnegotiation mechanism to both the fixed fee and the auction with nonarbitrary reserve price. More precisely: (1) The patentee strictly prefers direct negotiation to auction with a nonarbitrary reserve price when the nondrastic innovation is small; i.e. " 2 (0; 23 (a c)). Otherwise, the patentee is indifferent. (2) The patentee strictly prefers direct negotiation to fixed fee when the nondrastic innovation is more significant; i.e. " 2 ( 23 (a c); (a c)]. Otherwise, the patentee is indifferent. (3) The direct-negotiation mechanism restores the patentee's profit to the level of the unimplementable auction with an arbitrary reserve price. We have learned that, for the case of exclusive licensing, auction with a nonarbitrary reserve price and direct negotiation are the best mechanisms to extract rents from the duopolistic industry. This is because both mechanisms are able to make credible the threat to the potential licensee that there will be one licensee in the product market regardless of the potential licensee's decision. However, the auction suffers from a credibility problem for the case of complete diffusion. In the latter case, either fixed fee or direct negotiation can be used by the patentee. Therefore, except for direct negotiation, the patentee's choice of the licensing mechanism is linked to the choice between exclusive licensing and complete diffusion. Consequently, it is important to understand why exclusive licensing is not always the best option. The patentee's choice between exclusive licensing and complete diffusion is based on which amount, either 2F*i [T(2)] 2(i i ) or F*i [A(1)] Ai i , is greater. This comparison can be expressed as 2i (Ai i ); i.e. whether total industry profits, gross of licensing costs, under the complete diffusion of the technology are greater or smaller than total industry profits under exclusive licensing. One should note that total industry profits are increasing and convex with the size of the innovation in the above two cases. However, the increasing rate is higher in the case of complete diffusion if the innovation is small, and the opposite otherwise. Therefore, if the innovation is small enough, it is better to completely diffuse the technology. But, as the innovation becomes more significant, the exclusive licensing policy tends to dominate the complete diffusion in terms of industry profits. # Blackwell Publishers Ltd and the Board of Trustees of the Bulletin of Economic Research 2001.
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As already mentioned, another situation that might be interesting to explore is what happens when the selling of one licence brings to the buyer not only advantages in terms of costs but also the advantage of becoming the Stackelberg leader in the market. Before considering the welfare we briefly comment on the implications of the alternative Stackelberg situation. In such a situation (see the Appendix for the details), one can show that, at equilibrium, the optimal policy for the patentee is to sell always two licences and to use either a fixed fee or direct negotiation when the innovation is nondrastic. Indeed, both mechanisms provide the same profit to the patentee and dominate the auction with a nonarbitrary reserve price, which leads to exclusive licensing. Not surprisingly, the direct-negotiation mechanism still restores the patentee's profit to the level of the unimplementable auction with an arbitrary reserve price. To explain the above result we have to emphasize that the effect of the Stackelberg is twofold. On one side, there are innovations that under the Cournot setting are nondrastic, but become drastic under the Stackelberg setting (i.e. 12 (a c) < " < a c). In the latter case, to sell two licences is not an option for the patentee. Therefore, in order to draw comparisons, we concentrate our discussion on nondrastic innovations irrespective of the setting. Turning back to the effect of the Stackelberg setting on the potential licensees' bidding behavior, it is true that: first, the buyer of an exclusive licence bids more aggressively to become a leader with cost advantages; and second, the buyers of the two licences also bid more aggressively to avoid the possibility of becoming a follower (i.e. F*i [T(2)] C i i under the Cournot setting, while S Fi under the Stackelberg setting, and F*i [T(2)] S > F* i [T(2)] i C F* i [T(2)] ). The conclusion is that the patentee, irrespective of the size of the nondrastic innovation, prefers to sell two licences under the Stackelberg setting and using a fixed fee or direct negotiation as selling mechanism. Finally, the contrast between auction with an arbitrary reserve price (which is equivalent to direct negotiation) and auction with a nonarbitrary reserve price is now greater. Indeed, the former mechanism implies that there is always complete diffusion of the innovation, while there is always exclusive licensing with the later.
IV.
WELFARE CONSIDERATIONS
In this section, we investigate the desirability of licensing modes when a public agency cares either for domestic welfare (DW) or for world welfare (SW). Indeed, the patentee we have considered is a foreign independent laboratory. Expressions for DW and SW are, respectively: DW CS i j ;
SW CS i j
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where CS is the consumer surplus, i is firm i's profits, and is the patentee's revenue from licensing. Expressions for the consumer surplus are given below; CS[k] denotes the consumer surplus when k licences have been sold: 1 [4(a CS[1] 18
CS[2] 19 [2(a
c) 2 4(a c) 2 4(a
c)" " 2 ] c)" 2" 2 ]:
It is obvious that CS[2] > C[1], and consumers prefer complete diffusion of the new technology. The expressions of DW and SW, at equilibrium, for the different modes of licensing are: 1 [8(a DW [T(1)] 18
DW [T(2)] 49 [(a 1 [8(a SW [T(1)] 18
SW [T(2)] 19 [4(a
c) 2
4(a
c)" 5" 2 ] DW [A*(1)] DW [A(1)]
c) 2 " 2 ] DW [A*(2)] DW [FF ] c) 2 8(a c) 2 8(a
c)" 11" 2 ] SW [A*(1)] SW [A(1)] c)" 4" 2 ] SW [A*(2)] SW [FF ]:
Comparing these different expressions, we obtain: *
*
If " 2 (0; 23 (a c)), DW [FF ] DW [T(K)] DW [A*(K)] > DW [A(K)] and SW [FF] SW [T(K)] SW [A*(K)] > SW [A(K)]. If " 2 (23 (a c); (a c)], DW [FF] > DW [T(K)] DW [A*(K)] DW [A(K)] and SW [FF] > SW [T(K)] SW [A*(K)] SW [A(K)].
We can now state the following proposition about the welfare recommendations. Proposition 5: A public agency maximizing either the domestic or the world welfare will recommend licensing through the fixed-fee mechanism. For world welfare ± that is, adding the profit earned by the patentee to the consumer surplus and the firms' profits ± what is relevant is the number of licences sold and not the price paid for the technology. Since it is easily shown that the best outcome is the selling of two licences, then the fixed fee ± the licensing mechanism that always implies complete technology diffusion regardless of the size of the nondrastic innovation ± is the mechanism recommended by the social agency. For domestic welfare, the relevant aspects in order to choose a licensing mechanism are the number of licences sold and the price paid by the firms to the patentee. Again a fixed-fee mechanism is recommended because it allows complete diffusion of the technology at the lowest price. # Blackwell Publishers Ltd and the Board of Trustees of the Bulletin of Economic Research 2001.
{Journals}boer/53_2/z239/makeup/z239.3d
114
BULLETIN OF ECONOMIC RESEARCH
APPENDIX: THE STACKELBERG CASE
Here we give the expressions of the equilibrium fees (and the patentee's profit) for the auction, the fixed and the direct negotiation mechanism, under the Stackelberg assumption. Remember that under the Stackelberg assumption an innovation is nondrastic if and only if " Æ 12 (a c). Auction with a nonarbitrary reserve price At the SPE of the licence auction game A(1), both firms make a bid equal to: F*i [A(1)] S [A Li
1 Fi ] 16 [(a
c) 2 12(a
c)" 4" 2 ];
i 1; 2
S and the licensee is chosen randomly. Obviously, F* i [A(1)] > F* i [A(1)]. But at the SPE of the licence auction game A(2), both firms make a bid S equal to F* i [A(2)] 0, i 1; 2, and obtain the licence. Therefore, at the SPE of the auction game A(K) with a nonarbitrary reserve price, it is optimal for the patentee to auction only one licence if the innovation is nondrastic. At equilibrium, the patentee's payoff of A(K) is [A(K)] S 1 c) 2 12(a c)" 4" 2 ]. 16 [(a
Auction with an arbitrary reserve price At the SPE of the licence auction game A*(1), both firms make a bid equal to F*i [A(1)] S , i 1; 2, and the licensee is chosen randomly. The reserve price matters only for A*(2). At the SPE of the licence auction game A*(2), both firms make a bid equal to: F*i [A*(2)] S [i
1 Fi ] 144 [7(a
c) 2 68(a
c)"
20" 2 ];
i 1; 2
S
and obtain the licence; where F* Since i [A*(2)] > F* i [A*(2)]. S S is always true for nondrastic innovations, 2F* i [A*(2)] > F* i [A*(1)] the SPE policy is to auction two licences in A*(K). At equilibrium, 2 [7(a c) 2 68(a c)" 20" 2 ]. [A*(K)] S 144 Fixed-fee licensing For all nondrastic innovations, the SPE licensing policy of the patentee is to choose a fee equal to: S 1 F* i [FF ] 144 [7(a
c) 2 68(a
c)"
20" 2 ];
i 1; 2
such that both firms buy a licence. At the SPE two licences are sold by the patentee if the innovation is nondrastic by means of the fixed fee FF. 2 [7(a c) 2 At equilibrium, the patentee's payoff of FF is [FF ] S 144 2 68(a c)" 20" ]. # Blackwell Publishers Ltd and the Board of Trustees of the Bulletin of Economic Research 2001.
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LICENSING OF A COST-REDUCING INNOVATION
115
Direct negotiation At the SPE of T(1), the patentee makes an offer: F*i [T(1)] S A Li
1 Fi 16 [(a
c) 2 12(a
c)" 4" 2 ]:
This offer is immediately accepted by the matched firm i. Obviously, S F* i [T(1)] > F* i [T(1)]. In the direct-negotiation licensing game T(2), at equilibrium, the patentee makes an offer: F*i [T(2)] S i
1 Fi 144 [7(a
c) 2 68(a
c)"
20" 2 ];
i 1; 2
to both firms. This offer is immediately accepted by both firms. Consider the direct negotiation licensing game T(K). Given that " Æ 12 (a c), the optimal licensing policy for the patentee is to sell always two licences, 2 [7(a c) 2 and the patentee SPE payoff of T(K) is [T(K)] S 144 68(a c)" 20" 2 ]. REFERENCES
Arora, A. (1997). `Patents, licensing, and market structure in the chemical industry', Research Policy, vol. 26, pp. 391±403. Binmore, K., Osborne, M. J. and Rubinstein, A. (1992). `Noncooperative models of bargaining', in: Aumann, R. J. and Hart, S. (eds), Handbook of Game Theory, vol. 1. North-Holland, Amsterdam, pp. 179±225. Kamien, M. I. (1992). `Patent licensing', in: Aumann, R. J. and Hart, S. (eds), Handbook of Game Theory, vol. 1. North-Holland, Amsterdam, pp. 331±54. Kamien, M. I. and Tauman, Y. (1986). `Fees versus royalties and the private value of a patent', Quarterly Journal of Economics, vol. 101, pp. 471±92. Kamien, M. I., Oren, S. S. and Tauman, Y. (1992). `Optimal licensing of costreducing innovation', Journal of Mathematical Economics, vol. 21, pp. 483± 508. Katz, M. and Shapiro, C. (1986). `How to license intangible property', Quarterly Journal of Economics, vol. 101, pp. 567±89. Osborne, M. J. and Rubinstein, A. (1990). Bargaining and Markets. Academic Press, London. Wilson, R. (1992). `Strategic analysis of auctions', in: Aumann, R. J. and Hart, S. (eds), Handbook of Game Theory, vol. 1. North-Holland, Amsterdam, pp. 227± 79. Wolfstetter, E. (1996). `Auctions: an introduction', Journal of Economic Surveys, vol. 10, pp. 367±420.
# Blackwell Publishers Ltd and the Board of Trustees of the Bulletin of Economic Research 2001.
