Licensing Schemes in Endogenous Entry

2 The Model

Consider a license holder who has innovative technology and n of identical licensees that
produce and sell goods in the retail market. The license holder offers a licensing scheme
(w, f ) to all licensees where w ∈ R is a unit royalty, and f ∈ R ∣ is a fixed fee. Only
a positive fixed fee is assumed in this study. If w = 0 in an optimal contract, a license
holder selects a fixed fee scheme. On the other hand, if in an optimal contract contains
both w and f, a license holder selects a two-part tariff scheme.⁶ Licensee i (= 1, ..., n)
requires the technology of a license holder to produce its product. We assume that all
licensees are identical and produce homogenous goods. If a licensee buys a license, then
it can produce a product with marginal cost c. These identical licensees compete in
quantity.

Given a total output Q, let P(Q) denote market price. We assume that the function
P is differentiable and that P'(Q) < 0 and P'(Q) + QP''(Q) < 0 for all Q. These are
standard assumptions and the latter one guarantees the stability of equilibrium. The
cost function of licensee i is given by

(c + w)q_i + f.

where q_i ∈ R₊ is the output level of licensee i.

The profit of the license holder is given by

π^b = nf + wQ.

The profit of licensee i (= 1, ..., n) is given by:

¾ = qi(P(Q) - w - c) - f.

As a benchmark, consider a case where a license holder does not sell its technology
and produces goods with his own technology. Obviously, such a license holder will
obtain monopoly profit, which is a solution to:

max q_L(P(qL) - c),

QL

where q_L ∈ R₊ is the license holder’s output. The first-order condition is P(q_L) — c +
P'q_L = 0 and the license holder obtains a profit that is denoted by π^ . Hereafter, the
profit equal to π^ is termed monopoly profit.

This game runs as follows. In the first stage, the license holder offers terms (w, f )
to all its potential licensees. We assume that (w, f ) is identical for all i. In the second
stage, each licensee decides whether to buy a license and to enter the market. In the
third stage, licensees compete in quantities.

⁶Note that a negative unit royalty is allowed in this model. A negative unit royalty implies a subsidy
for production. See Liao and Sen (2005) for further discussion on negative unit royalty.

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