which we assume strictly positive for p ∈ [p, pm ), and
∏00 = 1 [(P — D0 + 2D-2 ( D ) C0
Applying Propositions 1 and 2 to this case, the following result follows:
Proposition 3 A sufficient condition for WJ > WS is that
(3) 2P0 + qP00 <0,
that is, that marginal revenue be decreasing in q.
Proof: Since in the relevant range we have S0 < 0 and π0 > 0, it follows from Proposition 2 that it suffices
to show that (3) ensures that
(4) S00π0 < S0π00
for all p ∈ [p, pm). Some straightforward calculations show that (4) is equivalent to:
(5) (p - C)[D00D - 2(D0)2] < -2π0D0 + 2 (D0)2Dc00.
Lemma 2 implies that the left hand side of (5) is negative for all p ∈ [p, pm ] if marginal revenue is decreasing
in q. On the other hand, the right hand side of (5) is positive, because C0 ≥ 0 and π0 > 0 for all p ∈ [p,pm).
I
3.2 Royalties
Consider a licensing agreement where the licensee pays the principal a royalty t per unit produced and sold,
but no fixed fee.14 For example, this is the case when a patent holder licences the right to manufacture the
good, but does not participate in the product market.15 In this case the principal is worried about downstream
double marginalization, and, given t, would like the licensee to sell as much as possible. The principal’s sur-
plus is S(p) = tD(p) (where now D is market demand for the good), and π(p) = 1 (p -1)D(p) - c(D(p)/2)
is the surplus of each licensee.
We then have that Proposition 3 also applies, i.e. 2P0 + qP00 < 0 is sufficient for WJ > WS. We postpone
the proof until the next subsection.
As is well known, a disadvantage of licensing through royalties is that any market power exercised
downstream reduces industry profits (this is the double marginalization problem). One solution is to auction
14Calvert (1964) and Taylor and Silberston (1973) observe that about 50% of all licensing contracts specify royalties only. Also,
Lafontaine and Shaw (1999) show that, on average, franchise fees amount to no more than 8% of actual payouts from franchise
holders to franchisees.
15See Tirole (1988, ch. 10.8) for a review of the literature on licensing.
10