Information about Licensing non linear technologies

Technis slides for webseminar by Dr. George Stamatopoulos

Literature status • Kamien & Tauman (1984, 1986), Katz & Shapiro (1985, 1986): seminal works in strategic patent licensing • vast expansion (product diﬀerentiation, asymmetric inform, location choices, del- egation, Stackelberg, etc) • however, all works build on linear tech- nologies (exceptions: Sen & Stamatopou- los 2008, Mukherjee 2010) 2

Aim of current work • analyze optimal licensing under (more) general cost functions • derive optimal two-part tariﬀ policies • identify impact of non-constant returns on royalties/diﬀusion 3

Snapshot of the model • cost-reducing innovation • Cournot duopoly • incumbent innovator • super-additive or sub-additive cost func- tions 4

• super-additivity: weaker notion than con- vexity (decreasing returns to scale) • sub-additivity: weaker notion than con- cavity (increasing returns to scale) 5

Main ﬁndings • super-additivity: all innovations are li- censed • sub-additivity: only ”small” innovations are licensed • royalties are higher under concavity/sub- additivity • interplay between super-additivity and royalties produces a paradox 6

I. Market • N = {1, 2} set of ﬁrms • qi quantity of ﬁrm i, q1 + q2 = Q • p = p(Q) price function • C0(q) initial technology (for both ﬁrms) 7

II. Post-innovation • ﬁrm 1 innovates (not part of the model) • Cε(q) post-innovation cost funct, ε > 0 • Cε(q) < C0(q), any q > 0 • either exclusive use of new technology or also sell to ﬁrm 2 • two-part tariﬀ policy (r, α): ﬁrm 2 pays rq2 + α (royalties and fee) 8

IV. Three-stage game stage 1: ﬁrm 1 decides whether to sell new technology or not. If it sells, it oﬀers a policy (r, α) stage 2: ﬁrm 2 accepts or rejects the oﬀer stage 3: ﬁrms compete in the market we look for sub-game perfect equilibrium outcome of this game 9

• focus on super-additive and sub-additive cost functions Deﬁnition Cε is super-additive if Cε(q + q ) > Cε(q) + Cε(q ) If inequality reverses, Cε is sub-additive. • convexity ⇒ super-additivity • concavity ⇒ sub-additivity 10

• analyze both drastic and non-drastic in- novations • drastic innovation: ﬁrm 2 cannot sur- vive in the market without new technol- ogy •non-drastic innovation: ﬁrm 2 survives without new technology 11

VI. Drastic innovations Proposition 1 Consider a drastic innova- tion. If the cost function is sub-additive, licensing does not occur. Proposition 2 Consider a drastic innova- tion. If the cost function is super-additive, licensing occurs. The optimal policy has positive royalty and fee. 12

Remarks on Propositions 1 and 2 • drastic innovation+sub-additivity lead to monopoly • drastic innovation+super-additivity lead to duopoly • Faul´ı-Oller and Sandon´ıs (2002): dras- tic innovation + product diﬀerentiation +constant returns lead to duopoly too 13

VI. Non-drastic innovations (diﬀusion) • F(q) ≡ C0(q)−Cε(q) innovation function • H(q) = F (q) F(q)/q elasticity of innovation function at q. Proposition 3a Consider a non-drastic innovation. Assume that H(q2) ≤ 1. Then licensing occurs. 14

Remark on Proposition 3a Condition H(q) ≤ 1 can hold under either super-additive or sub-additivity • C0(q) = cq + bq2 • Cε(q) = (c − ε)q + bq2 • H(q) = 1, for positive and negative b 15

VII. Non-drastic innovations (optimal mechan.) Proposition 3b Consider a non-drastic innovation. If Cε is concave, the optimal policy has only royalty. • in order to exploit increasing returns, ﬁrm 1 needs to produce high quantity • charge the highest royalty, so that rival’s quantity is low and own quantity is high 16

Proposition 3c If Cε is convex, the opti- mal policy has: (i) only royalty, if ε suﬃciently low (ii) both royalty and fee, if ε suﬃciently high (⇒ not a complete characterization) • high royalty raises ﬁrm 1’s output and its marginal cost • lower incentive to charge high royalty 17

VIII. The linear-quadratic case • Cε(q) = (c − ε)q + bq2/2 • b > 0 super-additivity • p = a − Q • licensing always occurs 18

Observation 1 The optimal royalty, r(b, ε), is decreasing in b. • high b ⇒ high marginal cost • by charging a lower royalty, ﬁrm 2 pro- duces more • hence ﬁrm 1 stays in more eﬃcient pro- duction zone • inverse relation between r and b has an interesting implication 19

Observation 2 There exist ranges of ε and b such that: • industry output increases when marginal cost (expressed by b) increases • market price decreases when marginal cost increases • surprising/interesting result? 20

Intuition • Q = Q(b, r(b, ε)) industry output dQ db = ∂Q ∂b <0 + ∂Q ∂r <0 ∂r(b, ε) ∂b <0 • in certain ranges, the positive eﬀect dom- inates • in these ranges price falls when marginal cost increases 21

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