The assumptions on the demand function and the first- and second-order conditions
indicate that dn∕dw < 0, dn∕df < 0 and dq∕df > 0. The sign of dq∕dw is indeterminate
and depends on the sign of P''. Moreover, the signs of dQ∕dw and dQ∕df are obtained
as follows:
|
dQ |
dq dn |
|
~r~ - |
= n---+ q—— < 0 |
|
dw |
dw aw |
|
dQ |
dq dn |
|
—7p- - |
= n— + q— < 0. |
,
Thus, the total output is decreasing in both w and f. Let q*, Q*, n*, w* and f * denote
the equilibrium output level of each licensee, equilibrium total output, equilibrium
number of licensees, equilibrium unit royalty and equilibrium fixed fee, respectively.
In the first stage, the license holder chooses a (w, f ) that maximizes its own profit.
Note that given (w, f, n, q), the license holder’s profit is kl = n(qw + f ). Substituting
the equilibrium condition (5), we can rewrite kl as follows:
kl = nq(P (nq) — c).
Therefore, the license holder’s profit maximization problem, denoted by (LP), is as
follows:
(LP) max nq(P(nq)—c).
{^,f }
Suppose that the unit royalty is given as w = w. As an auxiliary step, consider a
reduced problem, denoted by (LP’), as follows:
(LP’) max nq(P(nq)—c).
Using equations (8) and (9), the first-order condition of (LP’) is:
<Pl _ P,(P — c + n*q*P') _
~df~ = — P >(P — c — w — P 'q*) + P "q* (P — c — w) = '
Since it is assumed that P' < 0, we have P — c — w — qP' > 0. In addition, according
to the first- and second-order conditions in the third stage, the denominator, P'(P —
c — w — q*P') + P"q*(P — c — w), is negative. Thus, we have the following equilibrium
condition:
P (n*q*) — c + n*q*P ' = 0. (10)
Given any w, the system of equations (4), (5) and (10) determines q*, Q*, n*, and f *.8
8Note that the second-order condition in the first stage is satisfied. The second-order condition
with respect to f is as follows:
(2P' + QP'')(d^)2 + (P - c + QP') d^- < 0,
df df
where Q = nq. Since the first-order condition in the first stage implies P — c + QP' = 0, and
(dQ∕df )2 > 0, it can be rewritten as 2P' + QP'' < 0. Thus, since we assume P' + QP'' < 0 and P' < 0,
the second-order condition is satisfied.