Optimal licensing of technology in the face of (asymmetric) competition

Accepted Manuscript

Optimal Licensing of Technology in the Face of (Asymmetric) Competition Cuihong Fan, Byoung Heon Jun, Elmar G. Wolfstetter PII: DOI: Reference:

S0167-7187(18)30068-7 https://doi.org/10.1016/j.ijindorg.2018.07.009 INDOR 2466

To appear in:

International Journal of Industrial Organization

Received date: Revised date: Accepted date:

8 November 2017 27 April 2018 27 July 2018

Please cite this article as: Cuihong Fan, Byoung Heon Jun, Elmar G. Wolfstetter, Optimal Licensing of Technology in the Face of (Asymmetric) Competition, International Journal of Industrial Organization (2018), doi: https://doi.org/10.1016/j.ijindorg.2018.07.009

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Highlights • We consider licensing of technology by an incumbent firm in a Cournot oligopoly. • Focusing on how competition among licensees shapes optimal license contracts. • Relaxing standard assumption about demand, cost profiles, and the number of licenses. • We compare license contracts with standard and menu license auctions.

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• We design a dynamic mechanism that extracts the maximum surplus.

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Optimal Licensing of Technology in the Face of (Asymmetric) Competition* Byoung Heon Jun

Shanghai University of Finance and Economics [email protected]

Department of Economics, Korea University [email protected]

Elmar G. Wolfstetter†

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Cuihong Fan

Humboldt-University at Berlin and Korea University [email protected]

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August 3, 2018

Abstract

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We reconsider the optimal technology licensing by an incumbent firm facing multiple competitors. First, we cover the case of one license and show that competition has a drastic effect on optimal two-part tariffs. We also consider license auctions and design a more profitable dynamic mechanism. Next, we allow the licensor to award multiple licenses and design a dynamic mechanism that extracts the maximum industry profit. It awards licenses to all firms, prescribes maximum permitted royalty rates and positive fixed fees, and is more profitable than other dynamic mechanisms. Finally we show that a slight modification of that mechanism is also optimal for outside patent holders.

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K EYWORDS : Patent licensing, dynamic mechanisms, menu auctions. JEL C LASSIFICATIONS : D21, D43, D44, D45.

1 Introduction

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In the present paper we examine optimal patent licensing by an incumbent firm that interacts with potential licensees in a downstream oligopoly market. We relax the standard assumptions that there is only one potential licensee, the licensor issues only one license, potential licensees have the same unit cost, demand is linear, and that license contracts are simple take-it-or-leave-it offers. Prior to licensing firms have different unit costs that are common knowledge. The licensor owns a more efficient technology; its transfer reduces licensees’ unit cost to the licensor’s cost level. Firms compete in a downstream Cournot oligopoly with homogeneous goods. Licensing mechanisms are either simple two-part-tariffs that prescribe a combination of per-unit royalty rates and fixed fees or more complex dynamic contracts or license auctions. * We would like to thank an anonymous referee for his comments that helped us to improve our paper. Research support by Korea University (Grant: K1701281) and the National Natural Science Foundation of China (Grant: 71371116) is gratefully acknowledged. † Corresponding author. Humboldt University at Berlin, Institut f. Wirtschaftstheorie I, Spandauer Str. 1, 10178 Berlin, Germany, Tel.: +49 30 8100 1770, Fax: +49 30 8100 1770.

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In a first step, we analyze the case when only one license is issued, which plays a prominent role in the literature. We show that the optimal two-part tariff changes drastically when the number of potential licensees is increased from one to two or more. We also consider alternative mechanisms such as standard license auctions and more sophisticated menu auctions and design a two-stage dynamic mechanism. We show that the dynamic mechanism outperforms all other mechanisms and generates the highest payoff of the licensor.

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In a second step we relax the assumption that the licensor issues only one license. We design a dynamic mechanism that allows the licensor to extract the maximum industry profit that can be achieved by two-part tariffs, minus the minimum payoffs that licensees can assure themselves by not acquiring a license. This mechanism is also more profitable than other dynamic mechanism like the “chutzpah” mechanism.

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The literature on patent licensing by an incumbent or inside firm was initiated by Shapiro (1985) and Wang (1998). They showed that there is a sharp distinction between the licensing by an outside patent holder, who is not a player in the potential licensee’s market game, and an incumbent patent holder who is a competitor in the product market of potential licensees. Whereas outside innovators are advised to auction patent licenses to a limited number of licensees, inside patent holders are advised to employ output based royalty contracts without fixed fees.1 However, their analysis was carried out in the framework of a duopoly with one potential licensee and linear demand.

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The case of licensing by an inside licensor in duopoly was further pursued in numerous contributions. Among the more recent contributions Niu (2013) considers profit-share licensing and proves the equivalence of profit-share and per-unit royalty licensing, allowing for non-linear demand, and Fan, Jun, and Wolfstetter (2017) consider a general model of incomplete information that allows for all possible cost profiles and a continuum of types and show that it is generally optimal to employ two-part tariffs rather than pure royalty contracts.

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The papers that are most closely related to the present inquiry are Kamien and Tauman (2002), Sen and Tauman (2007), Kamien, Tauman, and Zamir (1990), Kamien, Oren, and Tauman (1992), Katz and Shapiro (1986), and Creane, Ko, and Konishi (2013) who pioneered the analysis of the role of competition between potential licensees.

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Kamien and Tauman (2002) generalize Wang (1998) to multiple potential licensees and multiple licenses and show that pure royalty licensing is superior for an incumbent inventor to both auctions or fixed-fee licensing if the magnitude of the invention is not too small. Sen and Tauman (2007) consider optimal two-part tariffs in the presence of multiple potential licensees and multiple licenses. They assume that all potential licensees have the same unit cost and linear demand and find that it is generally optimal to employ both royalty rates and fixed fees and award licenses to almost all firms. Creane, Ko, and Konishi (2013) assume that the licensor awards one license to one among several potential licensees. They focus on the question whether the optimal selection of licensee maximizes the joint profit of licensee and licensor and whether it maximizes welfare, generalizing earlier contributions by Katz and Shapiro (1985) and La Manna (1993), and explore whether awarding the license through a standard or a menu license auction may improve the efficiency of the allocation, without assessing its relative profitability for the licensor. Katz and Shapiro (1986), Kamien, Tauman, and Zamir (1990), and Kamien, Oren, and Tauman (1992) introduce dynamic licensing mechanisms known as “chutzpah mechanisms” to the licensing 1 However, Giebe and Wolfstetter (2008) and Fan, Jun, and Wolfstetter (2013) show that an outside innovator can increase his payoff by supplementing license auctions with per-unit royalty contracts for those who lose the auction.

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problem of an outside patent holder. These mechanisms are essentially license auctions with entry fees, supplemented by a threat to expose those who fail to pay the entry fee to an unfavorable allocation of licenses. In Katz and Shapiro (1986) that threat takes the extreme form to award free licenses to those who paid the entry fee, which raises a commitment problem. That problem is resolved by Kamien, Oren, and Tauman (1992) who replace the threat of awarding free licenses by the optimal time-consistent threat in the class of fixed fee licenses.

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The present paper complements these contributions. The main value added is that we consider alternative licensing mechanisms, including menu license auctions, and assess their relative profitability, allow the licensor to award multiple licenses, and design a globally optimal licensing mechanism that awards licenses to all licensees and prescribes maximum royalty rates equal to firms’ cost reductions together with positive fixed fees. That mechanism yields a higher payoff to the licensor than all licensing mechanisms considered in the literature, including the chutzpah mechanism that controls outputs by restricting the number of licenses rather than by using two-part tariffs.

Model

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The plan of the paper is as follows: In Section 2 we introduce the model and explain the constraints imposed by antitrust rules. In Section 3 we consider the case when one license is awarded to one of several potential licensees. We consider alternative mechanisms, ranging from optimal take-it-orleave-it contracts to first-price license auctions and the more sophisticated menu auctions, design a dynamic mechanism, and rank their profitability. In Section 4 we allow for multiple licenses and design a globally optimal license mechanism. In Section 5 we compare that proposed mechanism with other dynamic mechanisms such as the chutzpah mechanism and rank their profitability. In Section 6 we adapt our proposed mechanism to a world where the licensor cannot use contract clauses as commitment devices. In Section 7 we close with a brief Discussion where we point out that a slight modification of our proposed mechanism is also optimal for an outside patent holder which indicates that there is no sharp distinction between optimal licensing by an inside and an outside licensor.

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Consider a dynamic licensing game played between an incumbent firm that owns a superior technology and n competitors that operate in the same product market. In the first-stage the licensor offers two-part tariff contracts that prescribe royalty rates per output unit, ri , and fixed fees, fi , or more complex dynamic mechanisms, which licensees either accept or reject. After observing their rivals’ acceptance decisions and contracts, in the second-stage firms play a Cournot market game.

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Firms are indexed by i ∈ N+ := {0, 1, . . . , n}, where firm 0 is the licensor and firms N := {1, . . . , n} the potential licensees. Firms are ordered by their prior efficiency. Prior to licensing, firms’ unit cost profile is c0 = (c0 , c1 , . . . , cn ) with c0 < c1 ≤ c2 ≤ . . . ≤ cn . Licensing to firm i ∈ N reduces that firm’s unit cost from ci to c0 and its effective unit cost to ci (ri ) := c0 + ri .

The payoff functions of the oligopoly subgames are: πi (qi , q−i ) = (P(Q) − ci (ri )) qi , if firm i ∈ N is a licensee, πi (qi , q−i ) = (P(Q) − ci ) qi , if firm i ∈ N does not obtain a license, and π0 (q0 , q−0 ) = (P(Q) − c0 ) q0 , with Q := ∑i∈N+ qi . There qi denotes the output of firm i, q−i the output vector of all firms other than i, and P the inverse demand function which is decreasing and concave. Also, we denote the equilibrium outputs induced by licensing with the effective unit cost profile c(r) := (c0 , c1 + r1 , . . . , cn + rn ) by qi (c(r)), Q(c(r)) := ∑i∈N+ qi (c(r)), and, with slight abuse of notation, the associated equilibrium profits by πi (c(r)). Licensing mechanisms are regulated by antitrust authorities that interfere if they suspect collusive schemes that are geared to raise the equilibrium price above the level pN that prevails without 4

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licensing: P(Q) ≤ pN . That constraint corresponds to the “upward pricing pressure” (UPP) methodology in merger policy that advises to approve mergers only if the upward pressure on price due to the change in market structure is compensated by a downward pressure due to efficiency gains (see Farrell and Shapiro, 2010, Willig, 2011, Niu, 2013). Therefore, this constraint assumes that antitrust authorities apply the same principle to review license contracts which they apply to review mergers. In the case of linear per unit royalties, P(Q) ≤ pN is equivalent to the usual requirement that royalty rates cannot exceed the cost reduction:2 (antitrust constraints).

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ri ∈ Ri := {ri ∈ R+ | ri ≤ ci − c0 }

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The cost profiles are such that no firm is ever crowded out of the market, before and after licensing. This assumption is relaxed in Section 4.

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The licensor maximizes his payoff, which consists of his own profit plus license income. He offers either take-it-or-leave-it contracts, which individual licensees either accept or reject, or more complex dynamic contracts that make the terms of the contract contingent on the acceptance decisions of all licensees, or first-price license auctions. The licensor observes outputs and commits not to renege license contracts.

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Optimal licensing if one license is offered

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The bulk of the literature on licensing technology by an inside patent holder assumes that the licensor offers one license to one competitor with whom he interacts in a duopoly market. In that case, the optimal license contract is a pure royalty contract without fixed fee that prescribes the maximal permitted royalty rate equal to the licensee’s cost reduction (see Shapiro (1985) and Wang (1998)).3

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However, this result is not robust. As a first test, in this section, we increase the number of potential licensees to n ≥ 2, while maintaining the assumption that only one license is offered. This variation has a drastic impact on the optimal licensing scheme.

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For convenience, denote the cost profile that applies if the license is awarded to firm i ∈ N by ci := (c0 , c1 , . . . , ci−1 , c0 , ci+1 , . . . , cn ) and indicate the allocation of the license by the induced cost profile. With slight abuse of notation, we also define the associated effective cost profile ci (ri ) := (c0 , c1 , . . . , ci−1 , c0 + ri , ci+1 , . . . , cn ), and q j (ri ) := q j (ci (ri )), Q(ri ) := ∑ j∈N+ q j (ri ). Take-it-or-leave-it mechanisms

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In line with the literature, we (first) consider the class of static take-it-or-leave-it mechanisms that prescribe a two-part tariff contract, (ri , fi ), to one designated licensee, i. The optimal mechanism determines the optimal contract and allocation of license that maximize the licensor’s payoff: Π0 = π0 (ci (ri )) + ri qi (ci (ri )) + fi .

Proposition 1 (Optimal take-it-or-leave-it mechanism). The optimal take-it-or-leave-it mechanism is the pure fixed-fee contract: (ri , fi ) = (0, πi (ci )−πi (c0 )), to be awarded to the firm i that maximizes 2 For a detailed discussion of antitrust issues see Fan, Jun, and Wolfstetter (2017). There we also point out that the lower bound on permitted royalty rates does not completely rule out negative royalty rates. Without loss of generality we ignore this possibility in the present framework. 3 For a proof that does not assume linearity of demand (which is usually assumed in the literature) see Footnote 5 below.

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the increment of the joint profit of licensee and licensor. The fixed fee fi makes the licensee indifferent between the allocations ci and c0 . Proof. 1) Suppose the license is offered to firm i. No matter what royalty rate is set, it is optimal to set the highest acceptable fixed fee, i.e., fi = πi (ci (ri )) − πi (c0 ). Therefore, one can write the total payoff of the licensor, Π0 , as a function of ri : Π0 (ri ) = π0 (ci (ri )) + ri qi (ri ) + πi (ci (ri )) − πi (c0 )

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= (P(Q(ri )) − c0 ) (q0 (ri ) + qi (ri )) − πi (c0 ).

It follows that the optimal ri maximizes the joint profit of licensor and licensee.

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Using the characterization of the equilibrium outputs in the oligopoly subgames spelled out in Appendix A, the derivative of the licensor’s payoff with respect to ri is equal to:
∂ri Π0 (ri ) = (q0 (ri ) + qi (ri )) P0 (Q(ri ))Q0 (ri ) + (P(Q(ri )) − c0 ) q00 (ri ) + q0i (ri ) α1 + α2 = 0 2 P (Q(ri )) ((n + 2)P0 (Q(ri )) + Q(ri )P00 (Q(ri ))) where α1 = ((n − 2) (P(Q(ri )) − c0 ) + ri ) P0 (Q(ri ))2 > 0, if ri > 0 ! α2 = − (P(Q(ri )) − c0 ) (n − 1)P(Q(ri )) −

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Because the denominator is negative and α1 + α2 > 0 for ri > 0, it follows that ∂ri Π0 (ri ) < 0 for ri > 0. Together with the antitrust constraint (1) this implies that it is optimal to set ri = 0.

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2) Suppose the allocation ci maximizes the increment of the joint profit (relative to no licensing). Then, it also maximizes the licensor’s profit and vice versa, because:4


0 < π0 (ci ) + πi (ci ) − π0 (c0 ) + πi (c0 ) − π0 (c j ) + π j (c j ) − π0 (c0 ) + π j (c0 )
= π0 (ci ) + πi (ci ) − πi (c0 ) − π0 (c j ) + π j (c j ) − π j (c0 )
= π0 (ci ) + fi − π0 (c j ) + f j .

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Evidently, it makes a drastic difference if the licensor face one or more competitors.

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If the coalition of licensee and licensor face no competition (i.e, if n = 1), it is in their best interest to restrict the licensee’s output as much as possible to bring aggregate output as close as possible to monopoly output. Because output can only be restricted indirectly by charging the licensee a high royalty rate, this is achieved by setting the highest permitted royalty rate equal to the licensee’s cost reduction, i.e., r1 = c1 − c0 .5

If the coalition of licensor and licensee face at least one competitor, it is in their best interest to weaken the competitor(s) by charging the licensee the minimal permitted royalty rate, ri = 0. The

4 Assuming fixed-fee licensing, Creane, Ko, and Konishi (2013) already observed that it is optimal to award the license to the firm that maximizes the increment of the joint profit of licensor and licensee, without checking whether fixed-fee licensing is optimal. 5 Formal proof: If n = 1, the licensor’s profit is equal to Π (r ) = (P(Q(r )) − c ) Q(r ) − π (c0 ). Therefore, i 0 1 1 0 1 evaluated at r1 = c1 − c0 , Π00 (r1 ) = Q0 (r1 )(P(Q(r1 )) − c0 + P0 (Q(r1 ))Q(r1 )) > 0, because there Q0 (r1 ) < 0 and, by the assumption of a non-drastic technology, P(Q(r1 )) − c0 + P0 (Q(r1 ))Q(r1 ) < 0.

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licensor can fully extract the licensee’s gain by means of the fixed fee. Therefore, what is in the best interest of the coalition is also in the best interest of the licensor. However, “take-it-or-leave-it” mechanisms are too restrictive. One can easily design dynamic mechanisms that are more profitable for the licensor. 3.2

Dynamic mechanisms

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Consider two-stage “dynamic mechanisms”, (i, j, ri , fi , r0j , f j0 ), i 6= j, where: 1) In the first-stage, the licensor makes the take-it-or-leave-it offer (ri , fi ) to firm i. If firm i accepts, it gets the license and pays fi + ri qi . 2) If firm i rejects, then in the second-stage the licensor makes the optimal take-it-or-leave-it default offer (r0j , f j0 ) to firm j; if that offer is also rejected, no licensing occurs. Proposition 2 (Optimal dynamic mechanism). The optimal dynamic mechanism prescribes pure fixed-fee contracts: (ri , fi ) = (0, πi (ci ) − πi (c j )) (first-stage offer) and (r0j , f j0 ) = (0, π j (c j ) − π j (c0 )) (second-stage offer).

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These offers are made to firms (i, j) = (i∗ , j∗ ) where (i∗ , j∗ ) maximizes π0 (ci ) + πi (ci ) − πi (c j ) over all (i, j) ∈ N 2 , i 6= j. If i∗ < n then j∗ = n and if i∗ = n then j∗ = n − 1.

That mechanism is more profitable for the licensor than the optimal take-it-or-leave-it mechanism.

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Proof. In three steps: 1) We characterize the optimal contracts if the first offer is made to firm i and the second to firm j; 2) we characterize the optimal selection of (i, j), and 3) we show that the optimal dynamic mechanism is more profitable for the licensor than the optimal take-it-or-leave-it mechanism.

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1) Using backward induction we first consider the subgame after firm i has rejected the offer and an optimal offer is made to firm j. That default contract can only be a take-it-or-leave-it contract. By Proposition 1 the optimal default contract is a fixed-fee contract with f j0 = π j (c j ) − π j (c0 ).

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Using this result, the first offer is accepted by firm i if and only if πi (ci (ri )) − fi ≥ πi (c j ). Given the royalty rate, it is optimal for the licensor to set the highest possible fixed fee. Therefore, Π0 (ri ) = (P(Q(ri )) − c0 ) (q0 (ri ) + qi (ri )) − πi (c j ).

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This function differs from (2) only by a constant. Therefore, the optimal royalty rate ri is the same as that in the optimal take-it-or-leave-it contract and the optimal first-stage contract is a fixed-fee contract with fi = πi (ci ) − πi (c j ).

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2) From these results it follows that the licensor’s payoff when the first offer is made to firm i and the second to firm j is equal to ˆ 0 (i, j) = π0 (ci ) + fi = π0 (ci ) + πi (ci ) − πi (c j ). Π

It follows immediately that it is optimal to make the second stage default offer to the firm j with the highest unit cost, which minimizes πi (c j ) and thus maximizes the implicit threat to firm i. Therefore, one has either i∗ < n, j∗ = n or i∗ = n, j∗ = n − 1 and the remainder follows immediately. 3) Let k be the firm that obtains the license in the optimal take-it-or-leave-it mechanism. The ˆ 0 (i∗ , j∗ ), is greater than that of the optimal take-it-or-leave-it licensor’s equilibrium payoff, Π

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mechanism, Π0 , because, using the definition of (i∗ , j∗ ):   ˆ 0 (i∗ , j∗ ) − Π0 = π0 (ci∗ ) + πi∗ (ci∗ ) − πi∗ (c j∗ ) − π0 (ck ) + πk (ck ) − πk (c0 ) Π   ≥ π0 (ck ) + πk (ck ) − πk (c j ) − π0 (ck ) + πk (ck ) − πk (c0 ) for all j 6= k = πk (c0 ) − πk (c j ) > 0.

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We mention that the license is not necessarily allocated either to the most efficient or the least efficient licensee. Allocating it to the least efficient firm raises the profit of that firm more than any alternative allocation, but also lowers the profits of the other firms more. Therefore, the allocation depends on the relative strength of these effects. For example, if n = 2 and P(Q) = 1 − Q, allocating to firm 1 is optimal if c0 = (0, 1/11, 3/11), and allocating to firm 2 is optimal if c0 = (0, 1/33, 2/33).

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Comparison with license auctions

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The optimal dynamic mechanism threatens to award the optimal take-it-or-leave-it offer (ri0 , fi0 ), stated in Proposition 1, to the least efficient firm j∗ = max {N\{i∗ }} if the designated winner, i∗ , has rejected the first-stage offer. This is time consistent, provided the licensor makes a legally binding pledge to award a license to firm j∗ if the first-stage offer has been rejected. Of course, if all firms in N are identical, as it is assumed in the bulk of the literature on patent licensing, no such pledge is needed because then it does not matter to whom the second-stage offer is made.6

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The optimal dynamic mechanism is more profitable than the optimal take-it-or-leave-it mechanism for two reasons: because, by threatening the first responder to award the license to the competitor (rather than not allocate the license), it reduces the default payoff of the first responder to πi (c j ) rather than πi (c0 ) and implements a more profitable allocation.

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The first feature is similar to a standard license auction. There, bidders know that if they do not win a license, someone else wins, and this reflects in bidding.

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This begs the question whether the proposed dynamic mechanism is more profitable than a standard license auction. We now show that it is.

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Proposition 3. The optimal dynamic mechanism is more profitable for the licensor than a standard first- or second-price license auction.

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Proof. Without loss of generality consider a first-price auction. There each potential licensee, i, simultaneously submits a sealed-bid, bi . The licensor awards the license to the bidder iA = arg maxi∈N π0 (ci ) + bi , the winner iA pays his bid, biA , and all other bidders pay nothing.

Suppose this auction game has an equilibrium in pure strategies, in which case it has actually multiple equilibria.7 The equilibrium that yields the highest payoff of the licensor is the one in which the winning bidder, iA , bids πiA (ciA ) − πiA (c jA ), where jA = arg max j∈N−iA π j (c j ) − π j (ciA ).8 6 In

Section 6 we explain how to proceed if firms are asymmetric and contract clauses cannot be used as commitment devices. 7 To derive the equilibrium and prove existence is not trivial, essentially because bidders’ maximum willingness to pay for the license depends on who wins the auction if that bidder loses it. Creane, Ko, and Konishi (2013) establish existence if demand is linear. 8 Note, in this equilibrium one bidder spitefully “overbids” to such an extent that the winner is driven to bid his gain from winning. To support this equilibrium one needs an appropriate tie rule or think of a smallest currency unit and let it vanish in the limit.

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In that equilibrium, the licensor’s payoff, ΠA0 , is lower than in the proposed dynamic mechanism, ˆ 0 (i∗ , j∗ ), for the following reasons. Π By Proposition 2 the optimal dynamic mechanism makes the first offer to firm i∗ and the second to firm j∗ , where (i∗ , j∗ ) = arg max(i, j)∈N 2 ,i6= j π0 (ci ) + πi (ci ) − πi (c j ). Therefore, ˆ 0 (i∗ , j∗ ) = Π

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(i, j)∈N 2 ,i6= j

π0 (ci ) + πi (ci ) − πi (c j )

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≥ π0 (ciA ) + πiA (ciA ) − πiA (c jA ) = ΠA0 .

Like the optimal dynamic mechanism, standard license auctions implicitly threaten bidders to award the license to another bidder if they fail to win the auction. However, unlike the optimal dynamic mechanism, the auction does not control the best possible threat. This explains why it is less profitable for the licensor.

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A standard license auction does not perform well in terms of efficiency because it fails to internalize the externality that licensing inflicts upon those who do not win the license. As Creane, Ko, and Konishi (2013) point out, using a menu auction in lieu of a standard auction remedies this problem. Indeed, as Bernheim and Whinston (1986, Theorem 2) show in their seminal paper, a first-price menu auction implements the efficient allocation that maximizes the industry profit, provided one employs a plausible equilibrium refinement (which we explain below).9 This begs the question whether a first-price menu auction is also more profitable for the licensor, and, in particular, whether it can be more profitable than the optimal dynamic mechanism. We now briefly address this issue.

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In a first-price menu license auction the licensor chooses an allocation of the license to one of the firms k ∈ N+ , where k = 0 denotes the special case when the licensor retains the license. Each potential licensee, i ∈ N, submits a vector of bids, bi := (b0i , b1i , . . . , bni ), where bki denotes his bid on allocation k. Bids cannot be negative. If the licensor picks an allocation, say k, each bidder pays bki , and the licensor collects the auction revenue ∑i∈N bki . The licensor chooses the allocation, k, that maximizes his payoff, k = arg max j∈N+ π0 (c j ) + ∑i∈N bij .

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For convenience, in the remainder of this section we denote the profit earned by firm i ∈ N+ in allocation k ∈ N+ by πik := πi (ck ), firms’ payoffs by Πk0 := π0k + ∑i∈N bki , Πki := πik − bki , i ∈ N, and the industry profit in allocation k by I k := ∑i∈N+ πik .

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From Bernheim and Whinston (1986, Propositions 2 and 3) it is known that menu auctions have multiple Nash equilibria. Yet there is always an equilibrium in which the licensor chooses the allocation that maximizes the industry profit. This is the “truthful Nash equilibrium” (TNE). That equilibrium is the only one that is not susceptible to cheap-talk renegotiations; i.e., it is the only one that is coalition-proof (as defined in Bernheim, Peleg, and Whinston, 1987). Therefore, it is reasonable to use TNE to predict equilibrium bidding.

Stated formally, following Bernheim and Whinston (1986): Definition. bi is a “truthful strategy relative to allocation k” if and only if for all allocations j ∈ N+ bidder i is either indifferent between allocations j and k, i.e., Πij = Πki , or allocation j is inferior, Πij < Πki , and bij = 0. 9 Menu

auctions should not be confused with package auctions. In package auctions bidders bid on all possible bundles of auctioned goods, whereas in menu auctions they bid on all possible allocations of auctioned goods. In the presence of externalities, when bidders’ payoffs depend not only on what they get but also on who gets the good(s) allocated to others, menu and package auctions differ.

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¯ is a TNE if and only if it is a Nash equilibrium and b¯ := (b¯ 1 , . . . , b¯ n ) are truthful ( b¯ 1 , . . . , b¯ n , k) ¯ strategies relative to allocation k. ¯ k) ¯ is a TNE, bidders offer “subsidies”, b¯ j , for each allocation j 6= k¯ that reflect their Evidently, if (b, i maximum willingness to pay for switching from the equilibrium allocation k¯ to j, with the caveat that for some allocations the subsidy is too high because bids cannot be negative.

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Computing TNE is tedious even if the number of possible allocations is small. Therefore, it seems to be difficult to make general statements about the relative profitability of first-price menu auctions. However, one can unambiguously rank their profitability without using the explicit solution of bidding strategies, as follows.10 ¯ by K: ¯ Denote the set of allocations that maximize the licensor’s payoff at TNE bids, b, ¯ K¯ := {k ∈ N+ | k = arg max Π0j , where b = b}. j∈N+

(3)

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¯ k) ¯ is a TNE. Then for each Lemma 1 (Bernheim and Whinston (1986), Lemma 2). Suppose (b, j ¯ ¯ bidder i ∈ N there exists an allocation j ∈ K such that bi = 0. Lemma 2. Suppose bi is a truthful strategy relative to some allocation k. If bij = 0, then bhi = 0 for all h > j, h 6= i. Proof. Suppose bhi > 0 for some h > j, h 6= i. If this were the case, one would have Πhi = πih − bhi < πih ≤ πij = Πij ≤ Πki , which contradicts truthfulness. ∗

M

¯ k) ¯ is a TNE. Then b¯ h ≤ π h − π j (where j∗ = n if i 6= n and j∗ = n − 1 if Lemma 3. Suppose (b, i i i i = n). ¯ Proof. Suppose b¯ hi > πih − πij ≥ 0. If this were the case, then Πki = Πhi = πih − b¯ hi < πij . If i ¯ deviates and bids 0 for all allocations his payoff will be πik for some k, which is larger than Πki . ¯ k) ¯ cannot be a Nash equilibrium. Thus (b, ∗

ED

∗

PT

Denote the largest j ∈ K¯ by k∗ .

CE

¯ k) ¯ is a TNE and k¯ 6= 0. Then Πk¯ = π k∗ + bk∗∗ . Proposition 4. Suppose (b, 0 0 k ¯

¯ k). ¯ Then Πk = Πk . By Lemmas 1 and 2 one must have b¯ k = 0 for all Proof. Consider a TNE, (b, i 0 0 ∗ ∗ ∗ i 6= k∗ . Thus Πk0 = π0k + bkk∗ . ∗

∗

AC

Proposition 5. The optimal dynamic mechanism is more profitable for the licensor than a first-price menu auction. ∗

∗

∗

Proof. First, consider the case where k∗ 6= 0. From Proposition 4 it follows that Πk0 = π0k + bkk∗ . ∗ ∗ ∗ By Lemma 3, bkk∗ ≤ πkk∗ − πkj∗ , where j∗ = n if k∗ 6= n and j∗ = n − 1 if k∗ = n. Therefore ¯

∗

∗

Πk0 = Πk0 ≤ π0k + πkk∗ − πkj∗ ≤ max π0k + πkk − πkj . ∗

∗

∗

k∈N

10 One may wonder why we consider a complicated first-price menu auction in lieu of a Clark-Groves mechanism which

has an equilibrium in (weakly) dominant strategies. One reason is that, under complete information, the equilibrium of the Clark-Groves mechanism is vulnerable to cheap-talk renegotiations.

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∗ Next, consider the case where k∗ = 0. In this case Lemma 1 implies b¯ 0i = 0 for all i. Thus Πk0 = π00 . Because the licensor has the option to offer no license, the optimal dynamic mechanism does not yield a lower payoff to the licensor.

4

Generalization: Globally optimal licensing

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If there are n ≥ 2 potential licensees, the licensor may award more than one license. This raises the questions: can one improve the above dynamic licensing mechanism by issuing more than one license, what is the optimal number of licenses, and how much payoff can the licensor extract by awarding multiple licenses?

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A simple way to increase the profitability for the licensor is to replace the threat of awarding the license to another competitor if the designated licensee i rejects the offer by the stronger threat of awarding pure fixed fee licenses to all other potential licensees, N\{i}. This way, the licensor can increase the first-stage fixed fee from πi (ci ) − πi (c j ) to πi (ci ) − πi (c−i ) and thus increase his payoff. There c−i := (c0 , . . . , c0 , ci , c0 , . . . , c0 ) denotes the cost profile in which each except the i-th component is equal to c0 and the i-th component is equal to ci . This cost profile is the least profitable for firm i ∈ N, among all subsidy free cost profiles.

In this particular mechanism more licenses are issued only off the equilibrium path, if the designated licensee has rejected the first-stage offer. However, one can further increase the licensor’s payoff by making a first-stage offer to all potential licensees. In fact, one can design a dynamic dominant strategy mechanism that extracts the maximum surplus.

ED

M

At the outset we define a mechanism as “globally optimal” if it allows the licensor to appropriate the maximum industry profit through the use of two-part tariffs with linear royalties, while reducing the potential licensees payoff to the minimum level that they can assure themselves. That minimum level is the profit a firm earns in the worst case, when it is the only one that did not get a license, while all rival firms operate at the lowest effective unit cost c0 . Stated formally:

PT

Definition. A mechanism is “globally optimal” (in the class of two-part tariff mechanisms that influence outputs only indirectly, by charging output dependent royalties), if the licensor’s payoff, Π0 , is equal to Π∗0 = max (P(Q(c(r))) − c0 ) Q(c(r)) − ∑ πi (c−i ), r∈R

R := R1 × . . . × Rn .

(4)

CE

i∈N

Consider the following two-stage dynamic mechanisms, M(r, ε):

AC

1) in the first-stage the licensor offers license contracts (ri , fi ) to all n potential licensees, with 0 ≤ ri ≤ ci − c0 , fi = πi (c(r)) − πi (c−i ) − ε, and ε > 0. These contracts are applied if and only if all licensees accept. If no one accepts, no licensing occurs. 2) If a true subset A $ {1, . . . , n} of potential licensees accepts, the licensor makes a new, take-it
or-leave-it default offer (ri0 , fi0 ) = 0, πi (cA ) − πi (c0 ) − ε to all i ∈ A and no offer to all those who rejected the first-stage offer. For convenience, for each subset S ⊆ N define cS as the cost vector with: ( c0 if i ∈ S S ci = ci if i ∈ N\S. 11

(5)

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Proposition 6 (Dominant strategy dynamic mechanisms). Consider a dynamic mechanism M(r, ε). Eliminating dominated strategies in the second-stage subgames it is a strictly dominant strategy for each licensee to accept the first-stage offer. Proof. First, consider the second-stage subgames. There, a true subset of licensees, A, has accepted the first-stage offer. We show that accepting the second-stage offer is a dominant strategy for each i ∈ A. Consider an i ∈ A and let B denote the set of players other than i who accept the second-stage offer. If i accepts the offer, his payoff is equal to

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Πi = πi (cB∪{i} ) − πi (cA ) + πi (c0 ) + ε, whereas if he rejects, his payoff is equal to Π0i = πi (cB ). Evidently,

Πi − Π0i = πi (cB∪{i} ) − πi (cA ) + πi (c0 ) − πi (cB ) + ε > 0, because πi (cB∪{i} ) − πi (cA ) ≥ 0 and πi (c0 ) − πi (cB ) ≥ 0.

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Second, consider the first-stage, assuming that the licensees play the dominant strategies in the second-stage subgames. Consider a licensee i and let B denote the set of players other than i who accept the first-stage offer. Suppose B 6= N\{i}. If i accepts the offer, his payoff is equal to   Πi = πi (cB∪{i} ) − πi (cB∪{i} ) − πi (c0 ) − ε = πi (c0 ) + ε,

whereas if he rejects, his payoff is equal to Π0i = πi (cB ). Evidently,

M

Πi − Π0i = πi (c0 ) − πi (cB ) + ε > 0,

ED

because πi (c0 ) ≥ πi (cB ). Next, suppose B = N\{i}. If i accepts the offer, his payoff is equal to
Πi = πi (c(r)) − πi (c(r)) − πi (c−i ) − ε = πi (c−i ) + ε,

PT

whereas if he rejects, his payoff is equal to Π0i = πi (c−i ). Evidently, Πi − Π0i = ε > 0.

CE

Therefore, in either case it is a dominant strategy to accept the first-stage offer.

AC

Proposition 7 (Globally optimal mechanism). For each given ε, the optimal mechanism prescribes ri = ci − c0 for all i ∈ N. By choosing ε arbitrarily small, in the limit the mechanism becomes globally optimal. Proof. By Proposition 6 the first-stage offers are accepted by all licensees, if 0 ≤ ri ≤ ci − c0 , for all i ∈ N. Substituting the fixed fees one finds that the licensor’s payoff is equal to the industry profit minus a constant: Π0 (r) = (P(Q(c(r))) − c0 ) Q(c(r)) − ∑ πi (c−i ) − nε. i∈N

It only remains to be shown that the royalty rates ri = ci − c0 maximize Π0 (r) over the set R.

12

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Using a procedure similar to the proof of Proposition 1 and Appendix A, one finds: (n + 1) (P(Q(c(r))) − c0 ) − ∑i∈N ri −P0 (Q(c(r))) 1 ∂ri Q(c(r)) = < 0. (n + 2)P0 (Q(c(r))) + P00 (Q(c(r)))Q(c(r))

(7)

Q(c(r)) ≡

(8)

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Using these results we now show that ∂ri Π0 (r) > 0, for all r that satisfy the constraints ri ≤ ci − c0 :
∂ri Π0 (r) = ∂ri Q(c(r)) P(Q(c(r))) + P0 (Q(c(r)))Q(c(r)) − c0 ! = −∂ri Q(c(r)) nP(Q(c(r))) − ∑ (c0 + ri ) i∈N

(by (7))

= −∂ri Q(c(r)) ∑ (P(Q(c(r))) − (c0 + ri )) i∈N

> 0 (by (8)).

(9)

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Therefore, the optimal solution is ri = ci − c0 . We conclude that the stated mechanism is globally optimal in the limit as ε approaches zero.

ED

M

If this mechanism is applied, each potential licensee is charged the maximum permitted royalty rate equal to its cost reduction together with a positive fixed fee. Charging the maximum royalty rates reduces the industry output as much as possible under the constraint of antitrust law. Charging fixed fees together with maximum royalty rates makes all potential licensees worse off than prior to licensing. This is compatible with voluntary participation because the first offer is contingent upon its acceptance by all potential licensees, and the second-stage license contract penalizes all those who rejected the first-stage contract by awarding pure fixed fee licenses that entail maximum cost reductions to all those who accepted the first offer. One may wonder whether the proposed mechanism could be improved by restricting the number of licenses, prescribing some royalty rates lower than the maximum. The answer is no:

CE

PT

Proposition 8. Suppose it is possible to drive out some firm(s) by awarding licenses to a subset of firms prescribing some royalty rates lower than the cost reduction. It is nevertheless optimal to award licenses to all firms and apply the “globally optimal” mechanism stated in Proposition 7.

AC

Proof. Suppose, per absurdum, that one can improve the above stated globally optimal mechanism by awarding licenses to a subset of firms S ⊂ N and crowd out some firm(s). Denote the set of firms that are active by T ⊂ N. Then a firm, i ∈ / T , is also driven out in the worst case when all other firms are awarded the technology at zero royalty rates. Therefore, πi (c−i ) = 0, for all i ∈ / T . If the alternative mechanism is applied, the licensor’s payoff is bounded from above, as follows:

∑

i∈S∪{0}

(P(Q) − c0 ) qi − ∑ πi (c−i ) ≤ i∈S

∑

i∈T ∪{0}

(P(Q) − c0 ) qi − ∑ πi (c−i ) i∈T

= (P(Q) − c0 ) Q − ∑ πi (c−i )


i∈N

≤ P(Q(c )) − c0 Q(c ) − ∑ πi (c ) 0

=

Π∗0

13

(by (4) ).

0

−i

i∈N

(10)

ACCEPTED MANUSCRIPT

There, the first inequality follows from the fact that (P(Q) − c0 ) qi ≥ πi ≥ πi (c−i ) for all i ∈ T \S; the equality in the second line follows because πi (c−i ) = 0 for all i ∈ / T ; and the last inequality follows from the fact that (P(Q) − c0 )) Q is decreasing in Q (because Q(c0 ) exceeds the monopoly output11 ) and Q > Q(c0 ). Therefore, the globally optimal mechanism cannot be improved even if the technology is such that one can drive out some firms by awarding licenses only to a subset of potential licensees.

Comparison with other dynamic mechanisms

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5

How does the proposed mechanism compare with other dynamic mechanisms such as the “chutzpah mechanism”? That mechanism was introduced by Katz and Shapiro (1986), Kamien, Tauman, and Zamir (1990), and Kamien, Oren, and Tauman (1992), who consider an outside innovator and symmetric licensees with identical unit costs.

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In a symmetric framework, the chutzpah mechanism is essentially a license auction with an entry fee, supplemented by a threat to expose those who fail to pay that fee to an unfavorable allocation of licenses. In Katz and Shapiro (1986) it takes the following form: The licensor commits to auction a (limited) number of licenses, k ≤ n, and asks each potential licensee to pay an “entry fee”. If all potential licensees pay that fee, they play a standard auction and bid for one of the k licenses. If at least one of them fails to pay, all those who did pay obtain a license for free and a full refund of the entry fee. Choosing the optimal k and entry fee, Katz and Shapiro (1986) show that this mechanism induces all firms to pay the entry fee, and extracts the maximum surplus, maximizing over the number of licenses, k ≤ n.

PT

ED

M

In their discussion of the chutzpah mechanism Katz and Shapiro (1986) express doubts about its effectiveness, on three grounds: 1) the prescribed entry fees and the associated restriction of the number of auctioned licenses tend to “. . . reduce the extent to which the innovation is disseminated”, which may provoke antitrust authorities; 2) “. . . there is a serious question of whether the innovator can credibly commit himself to . . . ” the threat of offering free licenses; 3) that threat exposes the licensor to the risk that his “. . . profits drop to zero if even two of the buyers can cooperate in making their participation decision” (Katz and Shapiro, 1986, p. 587).

AC

CE

In response to these concerns, Kamien, Oren, and Tauman (1992, Sect. 7) show that one can make the chutzpah mechanism less extreme and resolve the commitment issue by replacing the threat of awarding free licenses by the threat to award optimal take-it-or-leave-it pure fixed fee license contracts. With this modification it remains an equilibrium in dominant strategies to pay the requested entry fee and maximum surplus extraction is preserved without giving rise to the commitment problem nor the risk of suffering a zero profit if firms collude. However, the antitrust concern remains an issue because it is generally optimal to auction fewer than n licenses.

There are three main differences between the chutzpah mechanisms and our mechanism: 1) whereas the chutzpah mechanisms controls industry output by restricting the number of licenses, our mechanism awards licenses to all firms and controls output by prescribing output dependent royalty rates and thus achieves maximum dissemination of technology; 2) whereas the chutzpah mechanism assumes an outside innovator facing symmetric firms, we consider an inside innovator and allow 11 Proof:

Denote the “inclusive reaction functions” by φi (Q) and φ (Q) := ∑i∈N+ φi (Q) (see Selten 1970, Ch. 9 or Wolfstetter 1999, p. 91-93). Then, the monopoly output is the fixed point of φ0 and the Nash equilibrium aggregate output is the fixed point of φ , and the assertion follows immediately from the fact that φ (Q) > φ0 (Q).

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for asymmetries across firms; 3) whereas the chutzpah mechanism requires all firms to pay an entry fee, in our mechanism firms pay only in exchange for a license. One can easily adapt the chutzpah mechanisms to allow for asymmetries and an inside patent holder. Instead of restricting the number of licenses, the mechanism must then specify the set of firms, say M ⊆ N, to which licenses will be awarded if all firms pay the prescribed entry fees, and add the licensor’s profit to his payoff. However, in this generalization the mechanism can no longer be interpreted as an auction.

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It may be disturbing that the firms N\M are requested to pay entry fees even though they know beforehand that they will not receive a license. However, these firms are willing to pay the requested fees because otherwise they are exposed to the most unfavorable cost profile c−i . Altogether, the thus generalized chutzpah mechanism employs the same pure fixed fee licenses in the second stage as our proposed mechanism and extracts the maximum surplus, maximizing by choice of A ⊆ N rather than by choosing the number of licenses. With these extensions the chutzpah mechanism is comparable to our proposed mechanism and we are now ready to rank their profitability.

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Proposition 9. The proposed mechanism is more profitable for the licensor than the “chutzpah” mechanism. Proof. We consider the proposed mechanism in the limit, as ε goes to zero. To characterize the payoff that the licensor earns in the generalized chutzpah mechanism, let M ⊆ N denote the subset of firms that are given access to the technology with cost c0 .

M⊆N i∈N

M

That mechanism yields the following payoff of the licensor:
M −i max ∑ P(Q(cM )) − cM i qi (c ) − ∑ πi (c ).

(11)

i∈N

ED

Now, suppose M is the optimal subset in the chutzpah mechanism. Consider the particular vector of royalty rates r0 with ri0 = 0 for i ∈ M and ri0 = ci − c0 for i ∈ / M. Then, max Π0 (r) ≥ Π0 (r0 )


= P(Q(c(r0 ))) − c0 Q(c(r0 )) − ∑ πi (c−i )

CE

PT

r∈R

≥

∑

i∈N

i∈N

(12)


0 −i P(Q(c(r0 ))) − cM i qi (c(r )) − ∑ πi (c ). i∈N

AC

If M = N the first inequality is strict by (9) and if M 6= N the second inequality is strict. Therefore, at least one of the inequalities is strict and we conclude that the proposed mechanism is more profitable for the licensor than the generalized chutzpah mechanism. Essentially, the proposed mechanism is superior because it awards licenses to all potential licensees and employs royalty rates to control outputs and extract surplus whereas the chutzpah mechanism controls outputs by restricting the number of licensees, or in the present more general case of asymmetric firms, the set of firms that will be awarded a license. The use of royalty rates is obviously more powerful for creating and extracting surplus and, if these are employed, it never pays to adopt restrictive licensing.12 12 Kamien,

Oren, and Tauman (1992) consider output dependent royalties when they compare pure royalty contracts and license auction; however, when they discuss the chutzpah mechanisms they assume that outputs cannot be observed by the licensor.

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6

Extension: Dynamic licensing with two-part tariffs as default contracts

Katz and Shapiro (1986) express concern that their own version of the chutzpah mechanism may not be credible because it relies on the licensor’s binding commitment to award free licenses as default contracts.13 In response, Kamien, Oren, and Tauman (1992) resolve that commitment problem by replacing the threat to offer free licenses by the threat to offer ex post optimal and thus time consistent fixed fee licenses as default contracts.

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Our proposed globally optimal dynamic mechanism employs the very same default contracts as the chutzpah mechanism in Kamien, Oren, and Tauman (1992), i.e., we include the optimal license contracts in the class of fixed fee license contracts as default contracts. However, as we allow for two-part tariffs in first-stage contracts, one may wonder whether the default contracts that we employ are also time consistent in the larger class of two-part tariffs, i.e., if the licensor may replace fixed fee contracts by two-part tariffs when the second stage has been reached.

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If only one license is awarded, Proposition 2 implies that our default contracts are also ex post optimal in the class of two-part tariff contracts. However, if more than one license is offered, the licensor can in some cases benefit from replacing fixed fee default contracts by two-part tariffs if the second stage has been reached. As an example, consider linear demand, n = 3, and the cost profile (c0 , c1 , c2 , c3 ) = (0, 1/9, 1/8, 1/4). Suppose firm 1 has unilaterally rejected the first-stage offer. Then, the ex post optimal take-it-or-leave-it contracts offered to firms 2 and 3 prescribe r2 = 1/8, r3 = 0; hence, in this case, our default contracts are not time consistent.

M

There are two ways to resolve this issue: One way is to use a contract clause as commitment device; the other way, which is imperative if such contract clauses are not feasible, is to replace the ex post optimal fixed fee default contracts by ex post optimal two-part tariff contracts.

ED

A contract clause may serve as commitment device if it states a legally enforceable pledge to award pure fixed-fee licenses as default contracts. Licensing is governed by contracts and contracts are, in principle, enforceable. Therefore, the use of contract clauses as commitment device is not a far-fetched proposition.

CE

PT

If, instead, one replaces the optimal fixed-fee default contracts by ex post optimal two-part tariff contracts, our proposed dynamic mechanism is easily adapted, and it is still the case that it awards licenses to all firms and extracts the maximum industry profit through the use of output dependent royalties together with fixed fees, while reducing licensees’ payoff to the minimum that they assure themselves. The only difference is that this minimum payoff is changed to the profit that licensees earn if they do not get a license while all other firms obtain optimal two-part tariff default contracts.

AC

Similarly, if one applies the same ex post optimal two-part tariffs as default contracts to the chutzpah mechanism, one finds that the chutzpah mechanism extracts the maximum industry profit through restricting the set of firms that will be awarded a license, while reducing firms’ payoffs to the minimum they can assure themselves. Using a proof that is similar to the proof of Proposition 9, it follows again that our optimal dynamic mechanism is more profitable for the licensor than the optimal chutzpah mechanism. The details are spelled out in Appendix B, without reiterating proofs that are essentially unchanged.

Altogether we conclude that if one applies the same ex post optimal default contracts in the chutzpah mechanism as in our proposed dynamic mechanism, our mechanism is more profitable for the licensor and assures greater, and indeed maximum, dissemination of the best available technology. 13 Incidentally,

if the licensor had strong commitment power, he could threaten to apply negative rather than zero royalty rates, thereby reduce the minimum payoff that firms can assure themselves, and hence further increase his profit.

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It is optimal to award licenses to all firms and to control their outputs by prescribing maximum permitted royalty rates equal to firms’ cost reductions. Restricting the number (or set) of licenses, as in the chutzpah mechanism, is never optimal.

7

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Having addressed the issue of time consistency, finally we mention that license contracts may also have to deal with a renegotiation issue.14 Whenever the licensor uses bilateral contracts to award multiple licenses with output dependent royalties, each coalition of licensor and licensee can increase its payoff, at the expense of other licensees, by secretly replacing royalties with fixed fees. This suggests that the licensor needs to commit not to renege the first-stage license contracts, for example by writing one multilateral rather than many bilateral contracts. Such a contract must promise to pay damages if there is direct or indirect evidence that some part of the agreement has been breached.

Discussion

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In the present paper we reconsider the optimal licensing by an inside patent holder, allowing for competition between potential licensees and asymmetric prior cost profiles. The main finding is that competition has a significant impact on optimal license contracts. In particular, we consider dynamic license contracts and design a globally optimal licensing mechanism that awards licenses to all firms and extracts the maximum industry profit while paying licensees the minimum payoff they can assure themselves. That optimal mechanism is superior to other dynamic mechanisms such as the chutzpah mechanism.

ED

M

While the chutzpah mechanism controls outputs by restricting the number of licenses, our proposed mechanism controls outputs by charging output dependent royalty rates. We show that restricting the number (or set) of licensees is never profitable, not even as a supplement to the use of royalties. These results hold true regardless of whether simple contract clauses are used as commitment device or default contracts are required to be time consistent, although the use of commitment devices is generally more profitable.

PT

The literature on patent licensing frequently draws a sharp distinction between the licensing problems of an inside and an outside patent holder. However, it turns out that our proposed globally optimal mechanism can also be adapted to licensing by an outside patent holder.15 The only difference is that one has to drop the profit of firm 0 from the licensor’s payoff. Therefore, it is optimal to apply the same kind of licensing mechanism in both settings.

AC

CE

Giebe and Wolfstetter (2008) show that an outside innovator can improve the profitability of a license auction by supplementing the auction with royalty licenses offered to those who lost the auction.16 Our results indicate that it is even better and indeed globally optimal to replace the auction (cum royalty licensing to losers) by royalty licensing (together with fixed fees) to all potential licensees in the form of our proposed dynamic mechanism. Finally, we emphasize that our analysis assumes that the licensor is able to perfectly monitor licensees’ outputs. If one considers imperfect observability, as in Gilbert and Kristiansen (2018), it may be optimal to rely to a lesser degree on output dependent royalties. 14 These issues figure prominently in the literature on the effectiveness of bilateral contracts in governing the relationship between a supplier and independent retailers (see, for example, O’Brien and Shaffer, 1992, Katz, 1991). 15 Provided awarding the technology to one firm with zero royalty rate does not crowd out any other firm. 16 In Fan, Jun, and Wolfstetter (2013, 2014) we show that this applies also if firms’ cost reductions are their private information or if cost reductions are common to all but firms privately observe imperfect signals, even though in these cases incentive compatibility implies that the royalty payments by the losers of the auction adversely affect auction revenue.

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A Appendix: Supplement to the proof of Proposition 1 Here we characterize the equilibrium outputs of the oligopoly subgames which are used in the proof of Proposition 1 and easily adapted to the proof of Proposition 7. Suppose the license is awarded to firm i ∈ N with royalty rate ri . The equilibrium outputs solve the following first-order conditions: P(Q(ri )) + P0 (Q(ri ))q0 (ri ) − c0 = 0

(A.1)

0

P(Q(ri )) + P (Q(ri ))qi (ri ) − c0 − ri = 0 0

(A.2)

Q(ri ) −

∑

qk (ri ) = 0.

k∈N+

These conditions yield: q00 (ri ) =

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(n + 1)P(Q(ri )) − (2c0 + ri + ∑ j∈{0,i} c j) / , −P0 (Q(ri ))

Q0 (ri ) =

∑

k∈N+

q0k (ri ).

(A.5) (A.6) (A.7)

(A.8)

M

B

(A.4)

Q0 (ri ) (P0 (Q(ri )) + q0 (ri )P00 (Q(ri ))) −P0 (Q(ri )) 0 0 Q (ri ) (P (Q(ri )) + qi (ri )P00 (Q(ri ))) − 1 q0i (ri ) = −P0 (Q(ri )) Q0 (ri ) (P0 (Q(ri )) + q j (ri )P00 (Q(ri ))) q0j (ri ) = −P0 (Q(ri ))

P(Q(ri )) − c0 , −P0 (Q(ri )) P(Q(ri )) − c0 − ri , qi (ri ) = −P0 (Q(ri )) P(Q(ri )) − c j , q j (ri ) = −P0 (Q(ri ))

q0 (ri ) =

Q(ri ) =

(A.3)

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P(Q(ri )) + P (Q(ri ))q j (ri ) − c j = 0

Appendix: Supplement to Section 6

CE

PT

ED

Here we outline in more detail the extension to globally optimal licensing with ex post optimal two-part tariffs as default contracts summarized in Section 6. In the first stage the licensor offers the same contracts as in Proposition 7. If these contracts are accepted only by a true subset of firms, A $ N, the firms in A are offered optimal two-part-tariff contracts (rA0 , fA0 ) as default contracts. Let ( ( ( 0 if i ∈ A 0 if i ∈ A r f c0 + ri0 if i ∈ A rA0 := i fA0 := i cA (rA0 ) = (B.1) 0 otherwise 0 otherwise ci otherwise.

AC

Then, the optimal default contracts, (rA0 , fA0 ), offered to the firms A solve the requirements:

s.t.

max π0 (cA (rA0 )) + ∑ ri0 qi (cA (rA0 )) + ∑ fi0 0 0 rA , fA

fi0

≤ πi (c

A

i∈A 0 0 (rA )) − πi (c ) − ε and ri0

i∈A

(B.2)

∈ Ri , ∀i ∈ A.

Using a procedure similar to the proofs of Propositions 6 and 7, we find that it is a dominant strategy to accept the second-stage default contracts and, using this fact, it is also a dominant strategy to accept the first-stage contracts. Choosing ε positive and arbitrarily small, it follows that by using the thus generalized optimal dynamic mechanism the licensor extracts the maximum surplus, maximizing over the first-stage royalty rates, r, while reducing potential licensees’ payoffs to the minimum they can assure themselves: −i 0 Π∗∗ 0 := max (P(Q(c(r)) − c0 ) Q(c(r)) − ∑ πi (c (r−i )), r∈R

i∈N

18

where − i := N\{i} .

(B.3)

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Evidently, the only difference between (B.3) and (4) is that the minimum payoff that each firm can assure itself is now equal to the profit that i can earn in the worst case, when it is the only one that 0 )), rather than π (c−i ). did not get the default two-part tariff contract, πi (c−i (r−i i

M⊆N i∈N

i∈N

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Next, consider the chutzpah mechanism by Kamien, Oren, and Tauman (1992) and replace the default contracts by the above stated optimal two-part tariff default contracts. Then one can easily show that this mechanism extracts the maximum surplus, maximizing over the set of firms that are offered licenses in the first-stage,
M −i 0 max ∑ P(Q(cM )) − cM (B.4) i qi (c ) − ∑ πi (c (r−i )). Using a proof that is similar to the proof of Proposition 9, it follows again that our proposed dynamic mechanism is more profitable for the licensor than the chutzpah mechanism.

References

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