Technis slides for webseminar by Dr. George Stamatopoulos

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, del-
egation, Stackelberg, etc)
• however, all works build on linear tech-
nologies (exceptions: Sen & Stamatopou-
los 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 func-
tions
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5. • super-additivity: weaker notion than con-
vexity (decreasing returns to scale)
• sub-additivity: weaker notion than con-
cavity (increasing returns to scale)
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6. 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
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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 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
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12. 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.
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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´ıs (2002): dras-
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)
F(q)/q
elasticity of innovation
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 + bq2
• Cε(q) = (c − ε)q + bq2
• 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 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
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18. VIII. The linear-quadratic case
• Cε(q) = (c − ε)q + bq2/2
• 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 pro-
duces more
• hence firm 1 stays in more efficient pro-
duction 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
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
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