1 Introduction
Article 28.2 of the Agreement on Trade Related Aspects of Intellectual Property Rights (TRIPs, 1994) states that “patent owners shall also have the right to assign, or transfer by succession, the patent and to conclude licensing contracts”. Licensing a patent enables an innovator simultaneously to appropriate the knowledge that she or he creates while also guaranteeing its access to firms that generate value from exploiting that knowledge.1
This article investigates the economic incentives that drive the patent holder’s choice of the optimal number of licensing contracts. It does so by assuming an external patent holder who offers a superior, cost-reducing technology to downstream Cournot-competing companies that are technologically asymmetric with each other.
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I focus on pure royalty licensing and show that the patent owner faces a trade-off between the price of the technology and the volumes of royalties that the owner collects. On the one hand, licensing a technology to fewer inefficient firms guarantees the patent owner larger revenues per sale (the intensive margin), as she can set a higher royalty rate; on the other hand, including more firms at a lower price maximizes market penetration of the patented technology, which increases the amount of royalties that are paid to the innovator (the extensive margin).
Incentives that increase the intensive margin are defined as the price effect (PE), and the incentives that increase the extensive margin are defined as the marker share effect (MSE). Under the assumption of ex-ante symmetric firms, PE does not play a role. Technically, the royalty rate is constrained above by the value of the cost-saving effect that is enabled by the new technology, which, by construction, is the same for all adopting firms. Hence, the innovator can leverage only on the volumes of royalties that are collected to increase her revenues. In this case, from the innovator’s standpoint, the only rationale choice is to maximize market penetration. The innovator accomplishes this by licensing the technology to all firms in the market.
In contrast, accounting for firms’ heterogeneity shows that PE is an essential driver of the innovator’s choice. In this article, I show that licensing outcomes with a per-unit royalty scheme are not trivial and can result in the innovator’s choice to limit the diffusion of her superior technology.
In more detail, I show that the innovator must compromise on the price of the technology to include more efficient firms in the licensing agreement. To increase the price, the innovator must limit the number of firms to which she offers licenses.
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This analysis extends beyond the limited domain of patent licensing. Indeed, it applies also to software and digital services providers. Many companies in labor-intensive sectors (such as call centers, logistics, and healthcare) introduce ad-hoc software solutions to limit production inefficiencies and to organize the workforce in a standard and measurable fashion: They move from their ex-ante particular production process (and associated costs) and converge towards a standard procedure; they, eventually, match the cost structures of similar companies.2
Moreover, the analysis applies also to sellers that compete in the same product category in digital marketplaces—e.g., Amazon. Despite sharing the same cost of producing the homogeneous good, merchants face different costs of handling and delivering the parcel to final consumers. To merchants, the digital platform offers a subscription-based service (e.g., Fulfillment by Amazon) that standardize costs. By subscribing, the retailers outsource these final stages of the supply chain to a more efficient economic agent.3
The following sub-paragraph summarizes the relevant literature. Section 2 provides a description and a detailed analysis of the model. Section 3 presents the main results and derives sufficient conditions for the main trade-off between price and market penetration to emerge. Section 3.1 applies the analysis to a market structure with n heterogeneous firms under the assumption of linear demand function. In Sect. 4, the welfare implications of the results derived are analyzed. Section 5 presents various extensions of the main model. In 5.1, I extend the baseline model to price competition with capacity constraints. In Sect. 5.2, I assume that licensing occurs via ad-valorem fees. Fixed fee licensing is briefly described in Sect. 5.3. Finally, Sect. 6 concludes.
Review of the literature Licensing under imperfect competition was first analysed by Kamien and Tauman (1986), Kamien et al. (1992), and Katz and Shapiro (1985). These early contributions seek to identify the most efficient licensing scheme. The studies suggest upfront fees dominate royalties from the innovator’s perspective, while an auction is the most efficient licensing scheme for an outside innovator,4
Building on these results, a large body of scholarship on optimal licensing scheme has developed in more recent years (see Gallini & Wright, 1990; Sen, 2005; Erutku & Richelle, 2007; Sen & Tauman, 2007, 2018; Parra, 2019; Marshall & Parra, 2019; Ferreira et al., 2021, among others).5
Focusing on royalty licensing, Llobet and Padilla (2016) find that equilibrium prices under ad-valorem royalties are lower than under per-unit royalties. Moreover, ad-valorem royalties benefit upstream innovators and do not necessarily hurt downstream producers. Hence, most licensing contracts include royalties that are based on the product’s value.
Using a Cournot duopoly model, Hsu et al. (2019) argue that ad-valorem royalty licensing is superior to per-unit royalty licensing from the patent holder’s perspective if the cost-reducing effect of the innovation is modest. Similarly, Fan et al. (2018) show that the profitability of different royalty licensing schemes depends on how effectively a patent holder can employ the new technology compared to the potential licensee. If the former is more efficient, then per-unit royalty licensing is superior to ad-valorem royalty licensing. Otherwise, ad-valorem royalty licensing is more advantageous.6
Most of these studies analyze the outcomes of a cost-reducing innovation when firms share a homogeneous technology that an invention can be improved by an invention under a licensing agreement. Under this assumption, royalty licensing by an external patent holder has been associated with the full diffusion of the licensed technology (Sen & Tauman, 2018). Studies with exceptions to this result include Lapan and Moschini (2000) and Sandrini (2022).
Lapan and Moschini (2000) demonstrate that if producing the final good requires more than one input, partial royalty licensing may emerge endogenously. More specifically, if the innovation generates negative pressure on the prices of the other inputs by altering the demand for the targeted input, some firms may decide to keep the obsolete technology, which has become cheaper.
In contrast, Sandrini (2022) shows how a licensor of a drastic innovation may strategically reduce the royalty rate to restrict the number of contracts and generate an asymmetry in the market. As a consequence, only some firms become licensees in equilibrium, whereas non-adopters exit the market.
In studies that consider only asymmetric firms, research has mostly focused on product differentiation. In this respect, my paper closely relates to the analysis in Hernández-Murillo and Llobet (2006). Specifically, Hernández-Murillo and Llobet (2006) investigate process innovations with homogeneous firms that offer differentiated goods. The authors assume that the novel technology is mostly fit to produce a specific product but can also be employed by other firms with a loss of efficiency. Moreover, conditions are derived for an optimal two-part tariff to be implemented as a combination of per-unit royalties and a fixed fee. In contrast, I do not focus on evaluating the optimal licensing scheme. Instead, I show that the patent holder’s problem of allocating royalty licensing contracts is not trivial if firms are asymmetric.
Some articles have investigated licensing outcomes with technologically asymmetric firms. In their seminal paper, Gallini and Winter (1985) analyze the effect of licensing on duopolistic firms’ incentives to invest in innovation. When the ex-ante difference in the cost structures of the two firms is small (large), the authors find that licensing stimulates (discourages) innovation.
Creane et al. (2013) analyze technology transfer between heterogeneous firms. The authors investigate the welfare effects of technology transfer when a relatively more efficient firm offers its technology to a less efficient rival. Importantly, they focus on technology transfer between firms that compete in the market for the final good.7
Poddar et al. (2021) design a model of spatial price competition with two rivals that are endowed with different technology. The authors show that a pure royalty contract is the most efficient licensing policy from the outside innovator’s perspective and that there is full diffusion of the technology.
In a recent contribution, Colombo et al. (2023) analyze optimal licensing contracts under the assumption of a capacity-constrained innovator. The authors show that royalty licensing is preferred when the inventor can produce relatively large quantities of output, whereas, fixed fee is preferred otherwise.
In this article, I contribute to the literature by highlighting the underlying forces—the price effect and the market share effect—that drive the inventor’s licensing decision. Moreover, I show that partial diffusion of the technology with a pure royalty contract is a possible equilibrium outcome if the new technology is not excessively more efficient than the best available one. The result is robust to both per-unit and ad-valorem royalty licensing schemes and extends to the case of price competition with capacity constrained companies.
2 The Model
Setup and assumptions Let \({\mathcal {I}} = \{1,...,n\}\) denote the set of active firms in a market. Each firm \(i \in {\mathcal {I}}\) produces the same (homogeneous) good with a different technology. With a slight abuse of terminology, I define technologies as the marginal cost of production \(c_i\) that is implied by adopting the identified technology. Firms are technologically asymmetric. Moreover, I assume there are n different technologies. Each technology is allocated to one firm only; thus it is possible to rank firms in the technology domain. If firms \(i,j \in {\mathcal {I}}\) produce the good with technologies \(c_i\) and \(c_j\), respectively, then i is more efficient than j if \(c_i < c_j\). Hence, the most efficient firm (firm 1) would be the one that produces with technology \(c_1\), whereas the least efficient one produces with technology \(c_n\). The ranking of technologies can be written as \(0<c_1<c_2<...<c_i<...<c_n\).
Firm i’s production level is indicated by \(q_i\), and it is assumed to be a function that decreases in its own costs \(c_i\) and increases in the costs of any other rival \(c_{-i}\). Formally, \(q'_{i,c_{i}} < 0\) and \(q'_{i,c_{-i}} > 0\). Accordingly, the final output can be ordered on (i) as follows: \(q_1> q_2>...>q_i>...>q_n > 0\). The last inequality in this ranking implies that all firms are active in the market. The total production level is \(Q = \Sigma _{i=1}^{n} q_i\).
As a result, the most efficient firm has the largest share of the market, while the least efficient one has the smallest (but positive) market share. In that goods are assumed to be homogeneous, the above assumption requires a restriction on the market structure. More in detail, competition can not be in prices.8 Therefore, in what follows, I assume that firms compete in quantities and face an inverse demand function p(Q). The demand is decreasing in total output and sufficiently large to allow the least efficient firm to earn non-negative profits (\(p(Q) > c_n\)).
The firms’ profit function is \(\pi _i = q_i (p(Q) - c_i)\), which leads to the following equilibrium output absent licensing:where \(p'(Q) = \partial p(Q)/\partial q_i\). I assume strategic substitutability for all \(i \in {\mathcal {I}}\):Consequently, increasing the marginal cost of any of the rival \(c_{-i}\) generates a positive effect on the output of firm i.
$$\begin{aligned} q_i^* = \frac{p(Q)- c_i}{-p'(Q)} \end{aligned}$$
$$\begin{aligned} p''(Q)q_i + p'(Q) \le 0 \end{aligned}$$
Lemma 1
Under the strategic substitutability condition, the equilibrium is unique. Moreover, keeping other firms’ marginal costs intact, an increase in \(c_i\) decreases the equilibrium aggregate output Q if \(c_i < P(Q)\), and has no effect otherwise.
Proof
See Creane et al. (2013), Lemma 1. \(\square\)
I assume that there is an outside innovator (she) who owns a patent for a technology that allows the firms to produce at a cost \(c^* < c_1\). For the sake of exposition and without loss, I assume \(c^* = 0\). All firms have a strictly positive willingness to pay for superior technology, as that is more efficient than the best one that is available in the market. The licensing scheme is the per-unit royalties. In Sect. 5.2, I extend the analysis and show that the main results hold, with some minor differences, in the case of an ad-valorem royalty scheme. Finally, I briefly describe fixed fee licensing in Sect. 5.3.
3 Main Results and Analysis
Exclusive dealing First, let us consider the case in which the innovator (patent holder) sells the technology to one firm only, which means that the licensing contract includes an exclusivity agreement. For simplicity, I assume that technology adoption by any of the firms in the market will drive the least efficient firm out of business: The innovation is strictly non-drastic.
The innovator decides which firm is most profitable to sell the technology to and at what price. I assume that there is complete and perfect information: The innovator and the firms know exactly the ranking and allocation of all of the technologies.
The maximum price that a firm i is willing to pay is \(c_i - c^* = c_i\). The price of the new technology (the royalty rate) \(r_i\) charged to a firm \(i \in {\mathcal {I}}\) is constrained by the cost-saving effect that it yields. Hence, the participation constraint of firm i is \(r_i \le c_i\). Otherwise, the adoption represents a pure cost, and the firm i rejects the offer. To sum up: Suppose that firm i gets the license for the new technology. Its new cost function becomes \(c_i = c^* + r_i = r_i\)
Given the output of the firms in stage 2, for any firm i, the innovator maximization problem is:From the f.o.c.,The element in the f.o.c. that is marked as the Market Share Effect (MSE) represents the indirect effect that an increase in the royalty rate has on the innovator licensing revenues via the output of the licensee. As royalties enter the marginal costs of the firm, increasing the royalty rate (and keeping all rivals’ costs constant) imposes a negative pressure on that firm’s output. Conversely, the element that is marked as the Price Effect (PE) represents the direct effect that increasing the royalty rate has on the patent holder’s revenues, which is positive. When \(r_i = 0\), the patent holder earns no revenues, whereas the output of the licensee is at its maximum. As the royalty rate increases, the patent holder starts collecting positive licensing revenues due to the positive price, but the volume of sales decreases. Formally, when evaluated at \(r_i=0\), the derivative in the f.o.c. is strictly positive. Eventually, if \(r_i\) is sufficiently high, the MSE becomes sizable enough to offset the PE, and, ultimately, dominate it.
$$\begin{aligned} \max _{r_i} R^{ph} = \underbrace{\frac{p(Q) - r_i}{-p'(Q)}}_{q_i} \times r_i \qquad s.t. \qquad r_i \le c_i \end{aligned}$$
(1)
$$\begin{aligned} \frac{\partial R^{ph}}{\partial r_i} = \underbrace{\frac{\partial q_i}{\partial r_i} r_i}_{MSE} + \underbrace{q_i}_{PE} = 0 \end{aligned}$$
It must be recalled that the royalty rate is bound by the participation constraint of the downstream firms \(r_i \le c_i\). Hence, for any given firm \(i \in {\mathcal {I}}\), the revenue-maximizing royalty rate isBecause goods are strategic substitutes, the monopoly rate is firm-dependent. When the innovator chooses a firm, she is also choosing the set of rivals that the candidate licensee will face. Choosing a relatively efficient firm implies that there are going to be fewer efficient rivals and more inefficient ones. In turn, this means that an ex-ante efficient licensee produces more than ex-ante inefficient firms (\(q_i \uparrow\)). For this reason, if the innovator was unconstrained, she would set the monopoly rate that applies to the most efficient firm in \({\mathcal {I}}\): \(r^* = r^m\)
$$\begin{aligned} r^* = \min \{c_i, r^m\} \quad \text {with} \quad r^m = \left\{ r_i > 0 \parallel \frac{\partial q_i}{\partial r_i} r_i + q_i = 0\right\} \end{aligned}$$
However, efficient firms face participation constraints that are stricter than the constraints that are faced by inefficient firms and are more likely to bind the innovator maximization problem. Putting all of these results together, it is possible to derive the following two Propositions.
Proposition 1
Assume that \(c_1 \ge r^m\). Then, any firm \(i \in {\mathcal {I}}\) is a candidate licensee. The innovator chooses the most efficient firm \(i=1\) and sets the royalty rate \(r^m\).
Proof
See the Appendix. \(\square\)
Proposition 2
Assume that \(c_1 < r^m\). Then, the innovator offers a licensing contract to a firm \(i \in {\mathcal {I}}\) at a price \(r_i = \min \{c_i, r^m\}\) such that
$$\begin{aligned} \frac{p(Q) - \min \{c_i, r^m\}}{-p'(Q)}\times \min \{c_i, r^m\} \ge \frac{p(Q) - \min \{c_{-i}, r^m\}}{-p'(Q)}\times \min \{c_{-i}, r^m\}\end{aligned}$$
(2)
Proof
See the "Appendix". \(\square\)
Proposition 1 states that an unconstrained monopolist would set the profit-maximizing royalty rate and license the most efficient firm in the market. Instead, Proposition 2 nuances this result and shows that the problem of allocating licensing contracts is nontrivial. Moreover, a constrained innovator has to balance the maximum price she can charge to a given firm with the volume of royalties that are collected by doing so. The next Corollary complements Proposition 2:
Corollary 1
Define the technological gap between the least and the most efficient firms as \(\Delta c \equiv c_n - c_1\). If \(\Delta _c \rightarrow 0\) (the PE is weak), the innovator is more likely to sign a licensing agreement with the most efficient firm. On the contrary, if \(\Delta _c \rightarrow c_n\) (the PE is strong), there exists a threshold \({\bar{c}} \in \left( 0, P(Q)\right)\) such that if \(c_n < \bar{c}\) (the MSE is weak), the innovator licenses the least efficient firm, whereas she licenses an interior firm \(i \in (1,n)\) if \(c_n>{\bar{c}}\) (the MSE is strong).
Proof
The proof follows immediately from (2). See the appendix. \(\square\)
Non-exclusive dealing The assumption that the innovator could sign only exclusive deals with downstream firms helped understand the main trade-off and the economic forces that govern it. In what follows, I relax this assumption and allow all firms that are willing to pay the price of the technology to become licensees. Consequently, the problem of the innovator is modified.
More specifically, by choosing the (uniform) price, the innovator uniquely determines the number of companies that become licensees. Firms can be ranked by their cost functions; thus, the inventor knows that by setting a price \(r_i \le c_i\), she will attract all firms that have marginal costs that are greater than \(c_i\). Rather than choosing i, with a slight abuse of notation, the inventor selects the set \({\mathcal {L}} = \{i,.., n\}\) of licensees, with \({\mathcal {L}} \subseteq {\mathcal {I}}\).
The problem of the innovator becomes:This implies the following revenue-maximizing royalty rate:In essence, the innovator chooses (i, \(r_i\)) to maximize licensing revenues. Naturally, the problem is interesting if the participation constraint is binding for at least some of the most efficient firms. Otherwise, the innovator simply sets the monopoly royalty rate \(r_1^m\) and licenses all the downstream firms. As a result:
$$\begin{aligned} \max _{i, r_i} R^{ph} = (n-i+1) \underbrace{\left( \frac{p(Q\big |{r=r_i}) - r_i}{-p'(Q\big |{r=r_i})}\right) }_{q_i\big |_{c_i=r_i}} \times r_i \qquad s.t. \qquad r_i \le c_i. \end{aligned}$$
$$\begin{aligned} r^* = \min \{c_i, r_i^m\} \quad \text {with} \quad r_i^m = \left\{ r > 0 \parallel \frac{\partial (n-i+1) q_i}{\partial r_i} r_i + q_i = 0\right\} \end{aligned}$$
Remark 1
Assume that \(r_1^m \le c_1\). Then the innovator sets price of technology \(r_1^m\), and there is complete adoption of the superior technology.
Instead, if the innovator is constrained, the problem of allocating licensing contracts is nontrivial. The innovator has to decide how many firms to include in the licensing agreement, knowing that: 1) all firms with a marginal cost greater than the selected royalty rate will pay for the technology; and 2) including an additional firm in the licensing agreement requires the innovator to reduce the royalty rate.
As before, the innovator anticipates that there is a trade-off between the number of contracts and the price \(r_i\) of the technology. By increasing the latter, fewer firms pay the price (MSE), but those who do guarantee a higher margin (PE). In the next proposition, I characterize the equilibrium outcome of the licensing game:
Proposition 3
Assume that \(c_1 < r_1^m\). Then, the innovator offers a licensing contract to \(n-i+1\) firms at a price \(r_i = \min \{c_i, r_i^m\}\) such thatwith \(i,j \in {\mathcal {I}}\), \(i \ne j\), and
$$\begin{aligned} \begin{aligned}&\frac{(n-i+1) p(Q\big |_{r=\min \{c_i, r_i^m\}}) - \min \{c_i, r_i^m\}}{-p'(Q\big |_{r=\min \{c_i, r_i^m\}})}\times \min \{c_i, r_i^m\} \\ {}&\quad \ge \frac{(n-{j}+1) p(Q\big |_{r=\min \{c_{j}, r_{j}^m\}}) - \min \{c_{j}, r_{j}^m\}}{-p'(Q\big |_{r=\min \{c_{j}, r_{j}^m\}})} \times \min \{c_{j}, r_{j}^m\} \end{aligned}\end{aligned}$$
$$\begin{aligned} Q\big |_{r=\min \{c_i, r_i^m\}} \equiv&\sum _{\ell =1}^{i-1} q_\ell \big |_{c=c_\ell } + (n-i+1) q_i\big |_{c=\min \{c_i, r_i^m\}};\\ Q\big |_{r=\min \{c_j, r_j^m\}} \equiv&\sum _{\ell =1}^{j-1} q_\ell \big |_{c=c_\ell } + (n-j+1) q_j\big |_{c=\min \{c_j, r_j^m\}}.\end{aligned}$$
Proof
See the appendix. \(\square\)
Finally, the innovator is weakly better off by offering non-exclusive deals because she is able to collect a weakly larger volume of royalties for any given price of the technology.
3.1 Example: Oligopoly with Linear Demand Function
In this subsection, I provide an example of the general results that were derived above by means of a discrete model with n oligopolistic firms that face a linear inverse demand function \(P=A - Q\), where \(Q=\sum _{i=1}^n q_i\). Assume that each company produces the homogeneous final good at a marginal cost c(i), such that \(c_1< c_2<...<c_i<...<c_n\).
Accordingly, the final output can be ordered on (i) as follows: \(q_1> q_2>...>q_i>...>q_n\), whereThe patent holder owns a technology that allows firms to produce the good at a marginal cost \(c^*<c_1\).
$$\begin{aligned} q_i = \frac{A - c_i + \sum _{j\ne i} (c_j-c_i)}{n+1}. \end{aligned}$$
To keep the analysis short, I focus on the scenario where the patent holder offers non-exclusive dealing, and all the participation constraints of the firms are binding: The patent holder cannot set the profit-maximizing royalty rate.
We know from the analysis above that for a given price of technology r, if r satisfies the participation constraint of firm i, it also satisfies the participation constraint of all the companies with a less efficient technology (higher marginal cost of production). That is, if the patent holder sets a price \(r(i) \le c_i - c^*\), all companies \(j\ne i\) such that \(c_j > c_i\) will also purchase the license. As before, I assume there is no scope for price discrimination.
The patent holder selects (i) to maximize her objective function:where \((n-i+1)\) is the number of adopting firms; \(c_i-c^*\) is the maximum price for the license such that firm i is indifferent between buying or not; and \(q(c_i,c_j)=\left( A-c_i + \sum _{j=1}^{ i-1}(c_j-c_i)\right) /(n+1)\) is the associated output level of the adopting companies given their updated marginal costs \(c^* + r(i) = c^* + c_i - c^* = c_i\).
$$\begin{aligned} R^{ph} = (n-i+1) \, q(c_i,c_j) \, (c_i - c^*), \end{aligned}$$
(3)
In that \(c_i\) increases in i, the less efficient the targeted firm is, the larger the price of the technology that the patent holder can charge. However, a smaller number of firms will adopt the technology. This relationship illustrates the trade-off between the price effect and the market share effect that was explained above.
In equilibrium, firm i is chosen so that the innovator cannot increase her payoff by licensing any other group of companies. Formally, this condition requires:with \(k \ne i\) and \(q(c_k,c_j) = \left( A-c_k + \sum _{j=1}^{k-1}(c_j-c_k)\right) /(n+1)\). Hence, it is possible to derive the following proposition:
$$\begin{aligned} (n-i+1) \, q(c_i,c_j) \, (c_i - c^*) > (n-k+1) \, q(c_k,c_j) \, (c_k - c^*), \end{aligned}$$
Proposition 4
The patent holder decides to license the new technology to companies that are less or equally efficient than \(c_i\), where \(c_i\) solves \(\frac{c_i - c^*}{c_{k} - c^*} \ge \frac{(n-k+1)q(c_k,c_j)}{(n-i+1)q(c_i,c_j)}\) \(\forall\) \(k\ne i\).
Deriving an analytical solution is challenging without some restrictions on the number of firms in the market and their distribution of marginal costs. However, it is possible to numerically demonstrate the existence of an interior solution. Assume that \(A=20\), \(n=10\) and c(i) are ordered from \(c_1=1.1\) to \(c_{10}=2\) by a 0.1 increase (\(c_2=1.2, c_3=1.3,...\)). Table 1 illustrates the results in scenarios where the new technology enables the production of the final good at a cost that varies between \(c^*=1\) and \(c^*=0.5\).9 For low values of \(c^*\), the PE is weak, as the maximum price that is charged by the patent holder is always large regardless of the company selected. Thus, when \(c^*\) is low, everything else being equal, the patent holder licenses her technology to a larger number of companies. Vice versa, when \(c^*\) is large, the patent holder exploits the asymmetry of efficiency between companies to maximize licensing revenues. The patent holder may decide to limit the innovation diffusion and supply a smaller number of companies by charging a higher price.
Table 1
Optimal choice of the subset \(\{i,...,n\}\) of licensees by the patent holder. Entries represent the patent holder’s licensing revenues for any pair \((i, c^*)\) given the parameter values specified in the main text
\(i=1\) | \(i=2\) | \(i=3\) | \(i=4\) | \(i=5\) | \(i=6\) | \(i=7\) | \(i=8\) | \(i=9\) | \(i=10\) | |
|---|---|---|---|---|---|---|---|---|---|---|
\(c^*=1\) | 1.7181 | 3.0600 | 4.0146 | 4.5818 | *4.7727* | 4.6091 | 4.1236 | 3.3600 | 2.3727 | 1.2273 |
\(c^*=0.8\) | 5.1545 | 6.1200 | 6.6909 | *6.8727* | 6.6818 | 6.1454 | 5.3018 | 4.2000 | 2.9000 | 1.4727 |
\(c^*=0.6\) | 8.5909 | 9.1800 | *9.3673* | 9.1636 | 8.5909 | 7.6818 | 6.4800 | 5.0400 | 3.4273 | 1.7182 |
\(c^*=0.5\) | 10.3091 | *10.7100* | 10.7055 | 10.3091 | 9.5455 | 8.4500 | 7.0691 | 5.4600 | 3.6909 | 1.8409 |
4 Welfare Analysis
The previous section discusses the private incentives of a patent holder to license her innovation to a subset of firms that compete (à la Cournot) in the downstream sector. I focus here on the social welfare implications of the license rationing that is derived above. The literature has stressed the ambiguous effects of licensing on social welfare and consumer surplus. In particular, with the use of a per-unit linear royalty, licensing may represent a device through which companies coordinate and collude to increase their prices (Mukherjee, 2005; Faulí-Oller & Sandonís, 2002). Those results are derived from modeling licensing as a technology transfer from an innovative firm to its competing rivals. In other words, licensing is a horizontal market relationship.
When examining the problem from a vertical perspective—when an outside innovator licenses technology to firms competing in a downstream sector—the ambiguity shrinks. Licensing a process innovation is supposedly always (weakly) welfare-improving. The result of the present analysis is no exception. Nevertheless, some crucial differences are worth highlighting. With regard to licensing via per-unit linear royalties, the literature has focused mainly on licensing a process innovation to homogeneous companies (in terms of production efficiency). Hence, due to the constrained maximization problem of the patent holder, the price of the technology absorbs all of the cost-reducing effects of the innovation, which leaves the price unaltered. Welfare improves because some production costs disappear and are replaced by transfer payments (the royalties) to the patent holder, but consumers do not generally benefit from this introduction of a more efficient technology.10
However, if one allows downstream companies to be heterogeneous in their efficiency levels, then even if the patent holder has full bargaining power, the welfare effect of licensing tends to be more beneficial to consumers. To demonstrate this positive relationship, let us focus on non-exclusive licensing, such as the model that was presented in Sect. 2. Consider the case in which the innovation is adopted by a subset of firms \({\mathcal {L}} = \{j,...,n\}\) at a price \(r = c_j\) such that all firms that ranked worse or equal than j buy the license.
The new total output is \(Q=\sum _{i=1}^{j-1} q(c_i) + (n-j+1) q(c_j)\). Because \((n-j+1) q(c_j) >\sum _{i=j}^{n} q(c_i)dc\), total output expands as a consequence of licensing, and the output price falls.11 Hence, consumers benefit from the introduction of the innovation. From the social planner’s standpoint, a full diffusion of innovation would imply the largest expansion of consumers’ surplus.
Licensing fee discrimination In the analysis above, licensing fee discrimination has been ruled out by assumption. However, if the patent holder could practice perfect price discrimination vis-à-vis the downstream firms, she would offer each of them a royalty rate that matches their willingness to pay. The licensing game would result in full diffusion of the superior technology as the PE would not play any role. However, because the patent holder extracts all of the cost-saving effects of the innovation via the personalized royalty rate, the effect of innovation diffusion on consumer surplus would be zero. As in the case of homogeneous firms, welfare improves because some costs would become the surplus that is extracted by the patent holder, but the price of the final good would not fall. Hence, somewhat paradoxically, a consumer-oriented policy-maker should be indifferent between a market where the innovation is unlicensed and one with a full diffusion of innovation where the patent holder can practice perfect price discrimination vis-à-vis the downstream firms.
5 Extensions
In this section, I modify some of the main assumptions of the model that were presented in Sects. 2 and 3. First, I illustrate the results when the downstream firms are capacity-constrained so that demand cannot be satisfied by only one supplier (in the spirit of (Kreps & Scheinkman, 1983)). This framework implies that firms engage in a marginal pricing strategy, which is common in energy markets – in particular, the day-ahead market for electricity in European countries. Second, I analyze several licensing schemes: Pure ad-valorem royalties, and pure fixed fees. I show that the two economic forces that were described in the previous sections remain and regulate the strategic choice of the patent holder.
5.1 Price Competition with Capacity Constraints
Assume that there exists a non-empty set of technologically asymmetric firms \({\mathcal {I}} = \{1,...,n\}\); all of the firms are capacity constrained. Their capacity is correlated with their technology. I consider both the case where firms with better technologies have larger capacities (positive correlation), as well as the opposite case (negative correlation).12 Each firm \(i \in {\mathcal {I}}\) produces at most \(k(c_i)\) units of the final good.
Definition 1
Consider a subset \({\mathcal {A}} =\{1,...,\ell \} \subset {\mathcal {I}}\) of active firms that produce positive output, and the independent and complementary set \({\mathcal {O}}=\{\ell +1,...,n\} \subset {\mathcal {I}}\) that includes only inactive firms. I define the residual firm \(\ell\) as the least efficient active firm that produces at the marginal cost \(c_\ell \in (c_1, c_n)\).
A natural implication of Definition 1 is that firm \(\ell\) is the only active firm that may produce fewer goods than its capacity if the residual demand is limited.
Firms compete in prices to serve the market. The closing price of the final good equals the marginal cost of the residual firm \(P = c_\ell\). At this price, firms \(c_i \in [c_1, c_\ell )\) earn positive margins \(c_\ell - c_i >0\), whereas the residual firm makes zero profits. Inactive companies do not produce.13
The total quantity demanded in the market is fixed and equal towhere \(q(c_\ell ) \le k(c_\ell )\) is the residual demand covered by firm \(\ell\). As in the baseline model that was described in Sect. 2, an external innovator owns a patent for a technology that allows the production of the final good at \(c^* = 0 < c_1\), and licenses it with a pure per-unit royalty (r) contract. If adopted by a firm i, the novel technology would allow that firm to increase the margins that it earns per unit of the final good without altering its original capacity. For the sake of brevity, I focus only on the non-exclusive dealing scenario.
$$\begin{aligned} Q = \sum _{i=1}^{\ell -1} k(c_i) + q(c_\ell ) \end{aligned}$$
I generically denote active firms with \(i \in {\mathcal {A}}\) and inactive firms with \(j\in {\mathcal {O}}\). The patent owner can offer a licensing contract to firms in both groups. Intuitively, licensing the technology to inactive companies is profitable if and only if they become active.
Active firms’ maximum willingness to pay is simply given by \(c_i - c^*= c_i\), and from each firm, the patent owner earns \(R_i = k(c_i) \times c_i\).14
Inactive companies are willing to pay for the new technology if it allows them to become active. Consequently, they are willing to pay at most the cost difference between the residual firm’s technology and the innovative technology, namely \(c_\ell - c^* = c_\ell\).
One should notice that if an inactive firm becomes active, the residual firm \(\ell\) remains active if and only if the capacity of the newly activated firm is smaller than the residual demand: \(k(c_j) < q(c_\ell )\).
Licensing active firms The innovator chooses the price of the technology r and, consequently, the number of contracts that she offers. If she decides to license only active firms, the problem resembles the model that was analyzed in Sect. 3. The problem of the innovator can be written as follows:Intuitively, and consistently with the analysis in Sect. 3, choosing to license the novel technology to a firm i implies that all firms that are ranked worse than i will also become licensees. Hence, offering a licensing contract to an additional firm implies the usual trade-off that has been analyzed in this article: On the one hand, the innovator collects more royalties (MSE); and on the other hand, she needs to reduce the price \(r(c_i)\) of the technology (due to the firms’ participation constraints), which limits the licensing revenues per contract (PE).
$$\begin{aligned} \max _{r_i} \,\, R^{ph} =&r_i\, \left( \sum _{i}^{\ell -1} k(c_i) + q(c_\ell ) \right) = r_i\, \, K(c_i)\\ \text {subject to} \quad&r_i \le c_i \end{aligned}$$
Licensing inactive firms Suppose instead that the innovator decides to serve only inactive companies. In this case, the equilibrium price never exceeds the marginal costs of the residual firms even if all inactive firms face higher marginal costs.15
One should notice that the identity of the residual firm may change when inactive firms become active. Indeed, there is no demand expansion because the demand is given.
More specifically, two outcomes may occur: First, if the correlation between capacity and efficiency is positive, the number of active firms that would become inactive is smaller than the number of inactive firms that would become active. Conversely, if efficiency and capacity are negatively correlated, the number of active firms that exit the market is larger than the number of new entrants.
In both cases, the problem of the patent holder is to choose the number of licensees to maximize her licensing revenues. Formally:The problem of the patent holder is trivial only if the total capacity of inactive firms is less than the residual output that is produced by firm \(c_\ell\). In that case, all inactive firms enter the market, and the residual demand shrinks. The patent holder earns \(K(c_j) \cdot c_\ell\).
$$\begin{aligned} \max _{j} \,\, R^{ph} =&r(c_j)\, \left( \sum _{j}^{n-1} k(c_j) + q(c_n) \right) = r(c_j)\, K(c_j)\\ \text {subject to} \quad&r(c_j) \le c_\ell . \end{aligned}$$
Instead, when the aggregate capacity of inactive firms is larger than the residual demand, some active firms must leave the market if inactive firms start producing the final good. Thus, fixed demand implies that the amount of output that is produced in the market remains unaltered. Importantly, the identity of the residual firm changes as more inactive firms become active due to the innovative technology.
First, assume that capacities are positively correlated with firms’ technologies: Higher marginal costs imply smaller capacity. The equilibrium price of technology and the number of contracts are defined as follows: Define \({\hat{\ell }}\) with \(c_{{\hat{\ell }}} < c_\ell\) as the residual active firm when all inactive companies enter the market and produce at \(r(c_j)\). Then, we have the following proposition:
Proposition 5
If capacity and efficiency are ex-ante positively correlated, then there exist two firms \(\ell ^{'}\) and \(\ell ^{''}\) with \(c_{\ell ^{'}}\in [c_{{\hat{\ell }}}, c_\ell ]\) and \(c_{\ell ^{''}} \in (c_\ell , c_n]\) such thatThe patent holder sets price \(r = c_{\ell ^{'}}\) and offers \((\ell ^{''} - \ell + 1)\) contracts.
$$\begin{aligned} c_\ell -c_{\ell ^{'}} < c_{\ell ^{''}} - c_\ell , \qquad \text {and} \qquad \sum _{i=\ell ^{'}}^{\ell } k(c_i)\, = \sum _{j=\ell }^{{\ell }^{''}} k(c_j). \end{aligned}$$
Proof
See the appendix. \(\square\)
Consider now the opposite case where capacity and technology are negatively correlated. Then, we derive Proposition 6. Define the company \(\tilde{\ell }\) with \(c_{{\tilde{\ell }}} < c_\ell\) as the residual active firm when all inactive companies enter the market and produce at \(r(c_j)\). Then:
Proposition 6
If capacity and efficiency are ex-ante negatively correlated, then there exist two firms \(\ell ^{*}\) and \(\ell ^{**}\) with \(c_{\ell ^{*}} \in [c_{{\tilde{\ell }}}, c_\ell ]\) and \(c_{\ell ^{**}} \in (c_\ell , c_n]\) such thatThe patent holder sets price \(r = c_{\ell ^{*}}\) and offers \((\ell ^{**}-\ell +1)\) contracts.
$$\begin{aligned} c_\ell -c_{\ell ^{*}} > c_{\ell ^{**}} - c_\ell , \qquad \text {and} \qquad \sum _{i=\ell ^{*}}^{\ell } k(c_i) = \sum _{j=\ell }^{\ell ^{**}} k(c_j). \end{aligned}$$
Proof
See the appendix. \(\square\)
Condition \(k'_{c_{i}} > 0\) implies that every inactive firm that becomes a licensee exerts severe negative pressure on the price of the final good and royalty rate. Activating a new firm implies the shutdown of more than one firm; therefore, the PE is stronger, and we should expect less diffusion of the technology than occurs for the opposite case of \(k'_{c_{i}} < 0\).
Finally, the patent holder weakly prefers licensing inactive firms because the participation constraint of inactive firms is less strict than that of active firms. If all inactive firms are potentially included in the licensing agreement, some active firms may also start joining. In this case, we refer back to the analysis in Sect. 3.
5.2 Ad Valorem Royalty
We now focus on a different version of the royalty licensing scheme: assume that firms do not pay a price per unit of goods produced (and sold) with the new technology; instead, they pay a share of the gross revenues: For every unit of the final good sold, firms pay a share \(r \in (0,1)\) of the price to the patent holder. In this case, the profit-maximizing patent holder should consider the size of r, the output produced, and the final good’s price when choosing the targeted firm or group of firms. Assume that all of the other assumptions of the model that was described in Sect. 2 hold. Moreover, for simplicity, assume that the patent holder deals exclusively with one firm. Her maximization problem can then be written aswhere \(Q^{lav}(r(c_i)) \equiv q_i(c^*) + \sum _{j=1}^{n} q(c_j) - q(c_i)\) and \(^{lav}\) stands for licensing with ad-valorem fees.
$$\begin{aligned} \max _{r(c_i)} \quad R^{ph} = r(c_i) P\left( Q^{lav}(r(c_i))\right) q_i (c^*), \end{aligned}$$
For the sake of brevity, we define \(r_i \equiv r(c_i)\), \(P^{lav} \equiv P\left( Q^{lav}(r(c_i))\right)\), \(Q^{lav} \equiv Q^{lav}(r(c_i))\), and \(q_i^{lav} \equiv q_i (c^*)\). The maximization problem then yields the following first-order condition:The first component on the left-hand side of Eq. (4) represents the PE, while the second component is the MSE.
$$\begin{aligned} \frac{d \,R^{ph}}{d\, r_i} = \underbrace{\left( P^{lav} + \frac{\partial \, P^{lav}}{\partial \,Q^{lav}}\, \frac{\partial \,Q^{lav}}{\partial \, r_i} \, r_i \, \right) q_i^{lav}}_{\text {Price Effect}} + \underbrace{\left( \frac{\partial \,q_i^{lav}}{\partial \, r_i} \right) \, r_i P^{lav}}_{\text {Market Share Effect}} = 0. \end{aligned}$$
(4)
To determine the existence of a trade-off, one needs to understand the sign of the components of Eq. 4. First, it is vital to assess the relationship between the royalty rate \(r(c_i)\) and the ex-ante cost of production \(c_i\). This relationship can be determined by examining the participation constraint of the candidate adopters. Define \(Q^{dev}\) as the total industry output if firm i decides to produce with the obsolete technology and not pay for the license. The deviation is profitable if:Hence, for any given level of royalty, the downstream firm accepts the offer if the above inequality is satisfied. The less efficient is the licensee ex-ante (\(c_i \,\uparrow )\), the less strict is the condition above, as \(\partial q(c_i) / \partial c_i < 0\) and \(\partial (P(Q^{dev}) -c_i) / \partial c_i < 0\). To see this, assume that a patent holder sets a price of technology \(r_i\) that is accepted by firm i. The same price would always be acceptable for less efficient firms, but not always for more efficient companies. Thus, it is possible to conclude that \(\partial r_i / \partial c_i \ge 0\).
$$\begin{aligned} \Delta \, \pi _i = \underbrace{q_i^{lav} \,P^{lav} - r_i q_i^{lav} P^{lav}}_{\text {profits with licensing}} - \underbrace{q(c_i) \left( P(Q^{dev}) - c_i\right) }_{\text {profits deviation}}&\,\ge 0 \\ \Longleftrightarrow \quad 1 - \frac{q(c_i)}{q^{lav}} \left( \frac{P(Q^{dev}) - c_i}{P^{lav}}\right)&\, \ge r_i. \end{aligned}$$
Given the relationship between the royalty rate and the ex-ante efficiency of the licensee, the MSE is non-positive: raising the royalty rate makes efficient firms less willing to accept the licensing contract. However, the analysis of the PE is a bit more complex. The first component is clearly positive. The second component depends on the sign of \(\partial Q^{tot} / \partial r_i\). In a linear demand model, the sign is negative, and the PE is unambiguously positive.
5.3 Fixed Fee
Fixed-fee licensing represents a far less interesting case. The patent holder sets a fee that is equal to the difference between the achievable payoff with the novel technology and the ex-ante payoff with the endowed technology. The payoff of a firm increases in the efficiency gap between itself and the less efficient rival. As a consequence, the larger is the number of adopters, the lower is the willingness to pay for the modern technology. This feature of fixed-fee contracts implies that even with homogeneous firms, the trade-off between PE and MSE drives the patent holder’s decision about the diffusion of the new technology.16 Recall that \(c^*=0\). Formally, the payoffs of the firms with and without the new technology arewhere \(Q^{fix}_{m}\) indicates the industry output when \(m \in [1,n]\) firms adopt the innovative technology and licensing occurs with a fixed-fee payment. The participation constraint is:Assume that firms are homogeneous: \(c_i = c_{-i} = c\). Then, \(\left( P(Q) - c\right) q(c)\) is common to all firms. In this case, it is the effect of adoption on the market price that alters the participation constraint: Industry output expands when many firms adopt the novel technology, and the market price falls. As a consequence, the markup that a firm can obtain with the new technology (\(P(Q^{fix}_{m})\)) declines with the number of adopters m. The patent holder must balance the trade-off between collecting more small fees (MSE) and fewer large fees (PE). The trade-off is more evident when firms are heterogeneous.
$$\begin{aligned} \pi (c_i) = \left( P(Q) - c_i\right) q(c_i); \qquad \pi (0) = P(Q^{fix}_{m}) \, q(0) - F, \end{aligned}$$
$$\begin{aligned} F \le \pi (0) -\pi (c_i) = P(Q^{fix}_{m}) \, q(0) -\left( P(Q) - c_i\right) q(c_i) \end{aligned}$$
6 Conclusion
In this article, I show that the allocation of licenses for a process innovation when the outside innovator uses a per-unit royalty scheme is nontrivial.
I disentangle two main economic drivers: a price effect, and a market share effect. The price effect identifies the firms’ higher willingness to pay for superior technology when their ex-ante technology is inefficient because their cost reduction effect would be more significant than would the cost reductions for efficient firms.
In contrast, the market share effect describes the positive effect on the licensee’s output when the innovator reduces the price of the technology. Choosing to license (or including in the licensing agreement) efficient firms allow the innovator to maximize the volume of royalties that she collects by expanding the aggregate share of total output that is produced with the licensed technology.
The dominance of these two forces determines which firms (and how many) obtain a license in equilibrium. Interestingly, this article shows that a full diffusion of innovation with royalty licensing is a particular result that depends on the closeness of firms’ originally embedded technologies. If the ex-ante technological asymmetry is significant, the patent holder may have a strong incentive to increase the price of the technology and licenses fewer firms. Similarly, if the novel technology is not excessively more productive than the best technology that is already available in the market, the price effect tends to dominate the market share effect, and not all companies become licensees.
From a welfare perspective, licensing increases consumer surplus. By reducing the price of the technology (royalty rate), the innovator limits her ability to extract the rent that is generated by the innovation, which is partially captured by firms. Consequently, because firms are, on average, more efficient, the price of the final good falls.
Declarations
Conflict of interest
The author has no relevant financial or non-financial interests to disclose.
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