Technology Licensing under Successive Monopoly

Abstract

Assume that there is an outside innovator who owns a cost-reducing innovation and the market structure of the industry in question is that of successive monopoly. It is found that, an innovation that is aimed at an upstream firm will tend to be accompanied by a fixed fee license, while an innovation that is aimed at a downstream firm will tend to be accompanied by a per-unit royalty license. But the former is reversed if the market structure of the final goods becomes duopolistic: The optimal licensing contract could never be that of fixed fee when licensing occurs at the upmost production stage. Moreover, the industry profit, consumer surplus and social welfare are all maximized when the licensing occurs at the upmost production stage.

Appendix A

We use n = 1 as an example to check if the results that we derived with the linear demand curve and constant marginal costs are still robust with more general demand and marginal cost functions. It is found that the main results still hold if the demand function is not too convex.

The proof is as follows:

Assume n = 1: There are only two production stages. When the innovation is aimed at firms at the final production stage–k = 0–the non-input cost of the final good producer before licensing is assumed to be $c(q_{0} )$, where $c(q_{0} )$ is the general marginal cost function with the properties $c^{\prime}(q_{0} ) > 0$ and $c^{\prime\prime}(q_{0} ) > 0$. It changes from $c(q_{0} )$ to $c(q_{0} ) + r_{0} q_{0} - \varepsilon q_{0}$ after licensing. Moreover, the inverse final demand function is assumed to be $p_{0} (q_{0} )$ with $p_{0}^{\prime } (q_{0} ) < 0$. The profit function of the final good producer–firm 0–is therefore specified as follows:

$$\pi_{0} (q_{0} ) = [p_{0} (q_{0} ) - p_{1} ]q_{0} - c(q_{0} ) - r_{0} q_{0} + \varepsilon q_{0}$$

(A1)

The first-order condition for profit maximization of the above objective function is derivable as follows:

$$\frac{{d\pi_{0} }}{{dq_{0} }} = p^{\prime}_{0} q_{0} + p_{0} - c^{\prime}(q_{0} ) - r_{0} + \varepsilon - p_{1} = 0$$

(A2)

and the second-order condition requires: $\frac{{d^{2} \pi_{0} }}{{dq_{0}^{2} }} = 2p^{\prime}_{0} + p_{0}^{\prime \prime } q_{0} - c^{\prime\prime}(q_{0} ) < 0$. We can also derive the following comparative statics: $\frac{{dq_{0} }}{{dp_{1} }} = \frac{{dq_{0} }}{{dr_{0} }} = \frac{1}{{2p^{\prime}_{0} + p_{0}^{\prime \prime } q_{0} - c^{\prime\prime}(q_{0} )}} < 0$, the value of which depends on the concavity of the demand and the cost curves.

In the second stage of the game, firm 1–the upmost-stage producer–has the following profit function:

$$\pi_{1} (p_{1} ) = p_{1} q_{1} (p_{1} ) - c(q_{1} (p_{1} )).$$

where $c(q_{1} )$ is the marginal cost function with $c^{\prime}(q_{1} ) > 0$ and $c^{\prime\prime}(q_{1} ) > 0$. The first-order condition for profit maximization of the above objective function is derivable as follows:

$$\frac{{d\pi_{1} }}{{dp_{1} }} = q_{1} + (p_{1} - c^{\prime})\frac{{dq_{1} }}{{dp_{1} }} = 0.$$

Note that $q_{1} = q_{0} = q$ because we assume that the production technology exhibits a one-to-one relationship. The comparative statics effect becomes: $\frac{{dp_{1} }}{{dr_{0} }} = \frac{{ - ({{d^{2} \pi_{1} } \mathord{\left/ {\vphantom {{d^{2} \pi_{1} } {dp_{1} dr_{0} }}} \right. \kern-0pt} {dp_{1} dr_{0} }})}}{{{{d^{2} \pi_{1} } \mathord{\left/ {\vphantom {{d^{2} \pi_{1} } {dp_{1}^{2} }}} \right. \kern-0pt} {dp_{1}^{2} }}}}$, where $\frac{{d^{2} \pi_{1} }}{{dp_{1} dr_{0} }} = \frac{dq}{{dp_{1} }}(1 + 3qp^{\prime\prime}_{0} \frac{dq}{{dp_{1} }} - c^{\prime\prime}\frac{dq}{{dr_{0} }})$ and $\frac{{d^{2} \pi_{1} }}{{dp_{1}^{2} }} < 0$ by the second-order condition. It is found that if the demand function is not too convex, $\frac{{d^{2} \pi_{1} }}{{dp_{1} dr_{0} }}$ and $\frac{{dp_{1} }}{{dr_{0} }}$ are negative.

At the downstream stage, the objective function of the licensor k–k = 0–is the same as Eq. (6). The first-order condition of the licensor k is as follows:

$$\frac{{d\Omega_{0} (r_{0} )}}{{dr_{0} }} = q + \mathop {r_{0} \frac{dq}{{dr_{0} }}}\limits_{{}} + \mathop {\frac{{\partial \pi_{0}^{{}} }}{{\partial p_{1} }}\frac{{\partial p_{1} }}{{\partial r_{0} }}}\limits_{{}} - q = 0$$

(A3)

From (A3), we find that with a non-linear demand and a non-constant marginal cost, our results still generally hold, and the optimal licensing contract is generally a pure per-unit royalty or a two-part tariff if the innovation is aimed at the downstream firm. This is because the input price effect is generally negative, which is the same as Eq. (7). However, the input price effect becomes ambiguous if the demand function is too convex. From the above discussions, we conclude that if the innovation is aimed at firms at the final goods production stage, the licensor still charges a positive royalty rate as long as the demand is not too convex.

Now consider the case that the innovation is aimed at firms at the upmost production stage:$k=1$. The non-input cost of the upstream producer is $c(q_{1} )$. After licensing, it changes to $c(q_{1} ) + r_{1} q_{1} - \varepsilon q_{1}$. The last stage is the same as in the previous paragraph. In the second stage of the game, the profit function of the upmost stage producer–firm 1–is as follows:

$$\pi_{1} (p_{1} ) = p_{1} q_{1} (p_{1} ) - c(q_{1} (p_{1} )) - r_{1} q_{1} + \varepsilon q_{1}$$

(A4)

In the first stage, the objective function of the licensor k, k = 1, is the same as that in Eq. (6). The first-order condition for profit maximization of the licensor k is as follows:

$$\frac{{d\Omega_{1} (r_{1} )}}{{dr_{1} }} = q + r_{1} \frac{dq}{{dr_{1} }} + 0 - q = 0$$

(A5)

From (A5), we find that with a non-linear demand and non-linear marginal cost, the equilibrium $r_{1}$ set by the outside licensor is still 0 when the innovation is aimed at the upmost stage producer: Regardless of whether the demand and cost curves are linear or non-linear, an innovation that is aimed at an upstream firm will be accompanied by a fixed-fee license because there is no input price effect.