Licensing non-linear technologies

Debapriya Sen

Ryerson University

Giorgos Stamatopoulos

University of Crete

1

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

• Faulı-Oller and Sandonıs (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)/qelasticity 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