Competition In or For the Field: Which is Better

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.

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