This document summarizes research on licensing non-linear technologies. It finds that when technologies exhibit increasing returns to scale (sub-additivity), only small innovations will be licensed, while decreasing returns to scale (super-additivity) means all innovations will be licensed. Royalty rates are higher under increasing returns. For drastic innovations, sub-additivity leads to monopoly while super-additivity leads to duopoly. For non-drastic innovations, licensing occurs when the elasticity of the innovation function is below 1. Under increasing returns, only royalties are optimal, while under decreasing returns royalties alone may be optimal for small innovations but fees plus royalties are optimal for large innovations.

1. Licensing non-linear technologies
Debapriya Sen
Ryerson University
Giorgos Stamatopoulos
University of Crete
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2. Literature status
• Kamien & Tauman (1984, 1986), Katz
& Shapiro (1985, 1986): seminal works
in strategic patent licensing
• vast expansion (product differentiation,
asymmetric inform, location choices, delegation, Stackelberg, etc)
• however, all works build on linear technologies (exceptions: Sen & Stamatopoulos 2008, Mukherjee 2010)
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3. 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
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4. Snapshot of the model
• cost-reducing innovation
• Cournot duopoly
• incumbent innovator
• super-additive or sub-additive cost functions
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5. • super-additivity: weaker notion than convexity (decreasing returns to scale)
• sub-additivity: weaker notion than concavity (increasing returns to scale)
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6. Main findings
• super-additivity: all innovations are licensed
• sub-additivity: only ”small” innovations
are licensed
• royalties are higher under concavity/subadditivity
• interplay between super-additivity and
royalties produces a paradox
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7. 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)
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8. 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)
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9. 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
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10. • 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
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11. • analyze both drastic and non-drastic innovations
• drastic innovation: firm 2 cannot survive in the market without new technology
•non-drastic innovation: firm 2 survives
without new technology
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12. VI. Drastic innovations
Proposition 1 Consider a drastic innovation. If the cost function is sub-additive,
licensing does not occur.
Proposition 2 Consider a drastic innovation. If the cost function is super-additive,
licensing occurs. The optimal policy has
positive royalty and fee.
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13. 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´ (2002): drasıs
tic innovation + product differentiation
+constant returns lead to duopoly too
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14. VI. Non-drastic innovations (diffusion)
• F (q) ≡ C0(q)−Cε(q) innovation function
• H(q) =
F (q)
elasticity of innovation
F (q)/q
function at q.
Proposition 3a Consider a non-drastic
innovation. Assume that H(q2) ≤ 1. Then
licensing occurs.
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15. Remark on Proposition 3a
Condition H(q) ≤ 1 can hold under either
super-additive or sub-additivity
• C0(q) = cq + bq 2
• Cε(q) = (c − ε)q + bq 2
• H(q) = 1, for positive and negative b
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16. 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
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17. Proposition 3c If Cε is convex, the optimal 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
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18. VIII. The linear-quadratic case
• Cε(q) = (c − ε)q + bq 2/2
• b > 0 super-additivity
• p=a−Q
• licensing always occurs
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19. Observation 1 The optimal royalty, r(b, ε),
is decreasing in b.
• high b ⇒ high marginal cost
• by charging a lower royalty, firm 2 produces more
• hence firm 1 stays in more efficient production zone
• inverse relation between r and b has an
interesting implication
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20. 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?
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21. Intuition
• Q = Q(b, r(b, ε)) industry output
dQ
∂Q ∂Q ∂r(b, ε)
+
=
db
∂b
∂r
∂b
<0
<0
<0
• in certain ranges, the positive effect dominates
• in these ranges price falls when marginal
cost increases
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