Licensing Schemes in Endogenous Entry

3 Result

We solve this game by backward induction. Given a license scheme (w, f ), the licensees
compete in quantities in the third stage. The profit maximization problem of each
licensee i (= 1, ..., n) is as follows:

max q_i(P(Q) - w - c) - f.                          (1)

Qi

The first-order condition of each licensees i (= 1, ..., n) is given by

P(Q) - c - w + P^,q_i = 0.                           (2)

Note that since we assume that P'(Q) + QP"(Q) < 0, the second-order condition is
satisfied.

In the second stage, licensees enter the market as long as they can obtain positive
profits. Thus, we have the following zero-profit condition for each licensee i (= 1, ..., n):

qi(P(Q) - w - c) - f = 0.                           (3)

In what follows, we focus on the symmetric equilibrium where all the licensees choose
the same strategies, i.e., q_i = q for all i (= 1,...,n). Now we have the equilibrium
conditions in the second and third stages as follows:

P(Q) - c - w + P'q = 0,                        (4)

q^{(P(Q) - c - w) - f} = ⁰.                         ⁽⁵⁾

Given w and f, let q(w, f ) and n(w, f ) be the solutions to (4) and (5). Therefore, from
equations (4) and (5), the implicit function theorem implies the following⁷:

⁷Since the determinant is (P - c - w)(P' + wP") - q(P')² < 0(≠ 0), the implicit function theorem
can be applied.

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