Revenue royalties: comment
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Revenue royalties: comment Marta San Martı´n1,2 • Ana I. Saracho1,2 Received: 20 August 2019 / Accepted: 6 September 2020 Ó Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract We show that the comparison of the two-part licensing mechanisms considered in the San Martı´n and Saracho (2015) context is exactly identical to that in Colombo and Filippini (J Econ 118:47–76, 2016) context. This means that the results obtained in Colombo and Filippini (J Econ 118:47–76, 2016) when the patentee chooses between per-unit and ad valorem royalties are incorrect. We conclude that when the convexity of the cost function is not too low, the patentee prefers to license by means of a royalty on the value of sales. Keywords Cournot oligopoly Product differentiation Decreasing returns to scale Royalties
JEL Classification D45
It is well known that when we consider ‘‘a n-good industry with representative consumer with utility function for the differentiated goods given by UðqÞ ¼ P ð ni¼1 qbi Þh ; in the Cournot case a change of units transforms the Cournot differentiated market with constant marginal costs into a Cournot homogeneous Financial support from Spanish government, Grants ECO 2015-66027-P and PID2019-106146GB-I00 and from the Departamento de Educacio´n, Universidades e Investigacio´n del Gobierno Vasco IT-869-13 and IT-1367-19 is gratefully acknowledged. Any errors are ours alone. & Ana I. Saracho [email protected] Marta San Martı´n [email protected] 1
Dpto. de Ana´lisis Econo´mico, University of the Basque Country UPV/EHU, Avda. Lehendakari Agirre 83, 48015, Bilbao, Spain
2
BRiDGE Group, University of the Basque Country UPV/EHU, Bilbao, Spain
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M. San Martín, A. I. Saracho
product market with strictly convex costs’’ (Vives, page 181). Similarly, from a quadratic utility function UðqÞ ¼ a1 q1 þ a2 q2 1=2ðb1 q21 þ 2cq1 q2 þ b2 q22 Þ, with all the parameters positive and b1 b2 c2 [ 0, the inverse demand function for good i is given by pi ¼ a bqi cqj when a1 ¼ a2 ¼ a and b1 ¼ b2 ¼ b: It is then straigthforward to transform the Cournot differentiated market with constant marginal costs into a Cournot homogeneous product market with decreasing returns to scale, and vice versa. As a result, the comparison of the two-part licensing mechanisms (fixed fee, F, plus a per-unit royalty, r, or ad valorem royalty, d, in the San Martı´n and Saracho (2015) context (SMS henceforth) and in the Colombo and Filippini (2016) context (CF henceforth) is exactly identical. Consider a duopolistic industry producing a homogeneous good. The inverse demand function for this good is p ¼ a ðqi þ qj Þ, with a [ 0; where qi and qj represent the quantities produced by firms i and j. Firms produce with convex costs, in particular Ci ðqi Þ ¼ cqi þ k
q2i 2
where c \a and k [ 0, and consequently the q2
profit function is pi ðqi ; qj Þ ¼ ða qi qj Þqi cqi k 2i . Rewriting this expression, we get pi ðqi ; qj Þ ¼ ða ð1 þ 2kÞqi qj Þqi cqi . As we can se
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