Patent Licensing and Capacity in a Cournot Model

We consider the problem of patent licensing in a Cournot duopoly in which the innovator (the patentee) is one of the competing firms and it is capacity constrained. When the capacity constraint is maximum (that is, the innovator cannot produce), the model coincides with the case of an outside patentee; when the capacity constraint is not binding, the model coincides with the case of an unconstrained inside patentee. Therefore, our model provides a bridge between the two cases usually considered in the literature: outside innovator and unconstrained inside innovator.

A capacity constraint is often relevant for the innovator. Pavitt et al. (1987), Acs and Audretsch (1990), OECD (2004), Marx et al. (2014), and Scholz (2017) provide evidence of the importance of small firms in generating technological innovations that are diffused by means of licenses. For example, Marx et al. (2014) explore the commercialization strategy in which a start-up temporarily enters the product market in order to establish the value of its innovation; ultimately, the entrant may switch to a strategy of cooperating with incumbents. Scholz (2017) emphasizes that—due to the increasing scarcity of raw materials that posit severe capacity constraints, especially for small firms—licensing agreements that delegate production (or part of production) to other firms are becoming widespread.¹ In many cases the innovator is a small firm with limited production possibilities, which licenses its innovation to other firms.

The theoretical literature has rarely considered the role of a capacity constraint in determining the licensing choice of the patentee. Scholz (2017) analyses a vertical model where the upstream firms are capacity constrained, while the patentee is an outside innovator. Alderighi (2008) proposes a licensing method that involves a maximum authorized production for the licensee. However, to the best of our knowledge, the case of a capacity-constrained patentee has not been considered yet.

In a different context, Mukherjee (2001) examines the possibility of technology transfer when firms have pre-commitment strategies such as a capacity commitment or an incentive delegation under the assumption of fixed-fee licensing. Filippini (2005) suggests that restrictions on licensing contracts reduce the viability of technology transfers. Bagchi and Mukherjee (2011) show that an incumbent firm may hold excess capacity: not to deter entry but to extract a larger licensing fee.²

While the literature has shown that in the case of an outside (inside) innovator the fixed fee (unit royalty) is preferred by the patentee (Kamien & Tauman, 1986; Sen & Tauman, 2007; Wang, 1998), we show that a fixed fee is preferred by the patentee even if the patentee competes with the licensee—provided that the patentee is able to produce only a relatively small quantity. Therefore, a fixed fee might be preferred to a unit royalty even if the patentee is an insider. This happens both in the case of drastic and non-drastic innovation. We show that the results are driven by two parameters: k, the maximum quantity that can be produced by the capacity-constrained firm; and c, the cost reduction after the innovation (which is a measure of the innovation size).

Furthermore, we show that when the patentee can set two-part tariffs in the form of combinations of fixed fees and unit royalties, it charges a positive fixed fee if and only if the patentee can produce only a relatively small quantity due to the capacity constraint. We also show that with combinations of fixed fees and royalties, the royalty rate is lower than is true for the standard case.

The reason is the following: When the quantity that can be produced by the patentee is relatively small, it is better for the patentee to let the licensee expand its production, and then extract the extra profit of the licensee by means of a fixed fee. By contrast, when the patentee can produce a relatively large quantity, it benefits from expanding production and earns additional revenues from the royalties that it charges on the licensee’s output. Therefore, the patentee chooses a per-unit royalty.

The rest of the paper proceeds as follows: In Sect. 2 we introduce the model. In Sect. 3 we derive some preliminary results with regard to a constrained Cournot duopoly with asymmetric firms. In Sect. 4 we derive the equilibrium profits under licensing. In Sect. 5 we compare different licensing mechanisms. Section 6 discusses several extensions. Section 7 concludes.

Consider a Cournot duopoly with two firms—1 and 2—with inverse demand: p = 1 − q₁ − q₂. Firm 1 (the patentee) has a cost-reducing innovation. The marginal cost of a firm is 0 with the innovation and c > 0 without the innovation where 0 < c < 1. Since firm 1 has the innovation, its marginal cost is 0. Firm 1 is constrained by capacity k ≥ 0, which is the maximum quantity that can be produced in a certain period of time, whereas firm 2 has no capacity constraint.

Note that when k = 0, the model coincides with the case of an outside innovator, whose production is zero (Kamien & Tauman, 1986). In this case, the non-producing inventor sets a fixed fee that absorbs all of the licensee’s profits: The inventor wants to maximize the licensee’s profits (and grab them via the fixed fee), and it does not want to distort production or sales (and the resulting profits) by means of a per-unit royalty (Kamien & Tauman, 1986).

On the other hand, when k is sufficiently high such that the capacity constraint is never binding in equilibrium (namely, k > 1), the model coincides with the case of an inside innovator with no capacity constraint (Wang, 1998). In this case, setting a per-unit royalty is preferred by the innovator, as it re-establishes cost asymmetry between the firms and, in addition, allows the patentee to get the license revenues (Wang, 1998).

Next, we introduce the distinction between drastic and non-drastic innovation (Arrow, 1962) Consider a monopolist that faces demand p = 1 − Q. The monopolist is not capacity constrained and has the cost-reducing innovation, so its marginal cost is 0. The monopoly price under marginal cost 0 is p_M ≡ 1/2. A cost-reducing innovation is drastic if the monopoly price p_M under the reduced cost (= 0) does not exceed c (the marginal cost without innovation); otherwise the innovation is non-drastic. Thus an innovation (which yields a marginal cost of 0) is drastic if c ≥ 1/2 and is non-drastic if c < 1/2.

We consider three licensing policies:

Since firm 1 has the cost-reducing innovation, its marginal cost is 0. If firm 2 does not have the innovation, its marginal cost is c. Therefore, c can be interpreted as the innovation size. If firm 2 has the innovation under a fixed fee policy, its marginal cost is 0. If firm 2 has the innovation under a policy that has royalty r (either a royalty policy or a policy that is a combination of royalty and fee), then the effective marginal cost of firm 2 is r.

The strategic interaction between firms 1 and 2 is modeled as the three-stage licensing game G: In stage 1 of G, firm 1 decides whether to licenses its innovation to firm 2 or not and offers a licensing policy to firm 2; in stage 2 firm 2 decides whether to accept the policy or not; in stage 3, firms 1 and 2 compete in the Cournot duopoly, and payments are made according to the policy.

For 0 ≤ r ≤ c and k ≥ 0, denote by D^k(r) the Cournot duopoly in which firm 1 has marginal cost 0 and capacity constraint k; firm 2 has marginal cost r and no capacity constraint. In particular note that with respect to the marginal cost of firm 2, r = c corresponds to the situation where firm 2 does not have the innovation.

Thus, if firm 2 has the innovation under a fixed fee policy, the resulting Cournot duopoly is D^k(0). If firm 2 has the innovation under a policy that has royalty r, it is D^k(r). If firm 2 does not have the innovation, the resulting Cournot duopoly is D^k(c).

To determine the optimal licensing policies for firm 1, it is therefore useful to determine the equilibrium outcomes of D^k(r) for all 0 ≤ r ≤ c and k ≥ 0. When there is no capacity constraint, the quantities that are produced by firms 1,2 in the unique (Cournot-Nash) equilibrium are \(\overline{q}_{1} (r) = \left\{ {\begin{array}{*{20}l} {{{(1 + r)} \mathord{\left/ {\vphantom {{(1 + r)} 3}} \right. \kern-\nulldelimiterspace} 3}} \hfill \\ {{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \hfill \\ \end{array} } \right.\) \(\begin{array}{*{20}l} {if} \hfill & {0 \le r < {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \hfill \\ {if} \hfill & {r \ge {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \hfill \\ \end{array}\) and \(\overline{q}_{2} (r) = \left\{ {\begin{array}{*{20}l} {{{(1 - 2r)} \mathord{\left/ {\vphantom {{(1 - 2r)} 3}} \right. \kern-\nulldelimiterspace} 3}} \hfill \\ 0 \hfill \\ \end{array} } \right.\) \(\begin{gathered} \begin{array}{*{20}c} {if} & {0 \le r < {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \\ \end{array} \hfill \\ \begin{array}{*{20}c} {if} & {r \ge {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \\ \end{array} \hfill \\ \end{gathered}\); the equilibrium price is \(\overline{p}(r) = \left\{ \begin{gathered} {{(1 + r)} \mathord{\left/ {\vphantom {{(1 + r)} 3}} \right. \kern-\nulldelimiterspace} 3} \hfill \\ {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} \hfill \\ \end{gathered} \right.\) \(\begin{gathered} \begin{array}{*{20}c} {if} & {0 \le r < {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \\ \end{array} \hfill \\ \begin{array}{*{20}c} {if} & {r \ge {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \\ \end{array} \hfill \\ \end{gathered}\); and the equilibrium profits are \(\overline{\pi }_{1} (r) = \left\{ \begin{gathered} {{(1 + r)^{2} } \mathord{\left/ {\vphantom {{(1 + r)^{2} } 9}} \right. \kern-\nulldelimiterspace} 9} \hfill \\ {1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4} \hfill \\ \end{gathered} \right.\) \(\begin{array}{*{20}l} {if} \hfill & {0 \le r < {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \hfill \\ {if} \hfill & {r \ge {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \hfill \\ \end{array}\) and \(\overline{\pi }_{2} (r) = \left\{ {\begin{array}{*{20}l} {{{(1 - 2r)^{2} } \mathord{\left/ {\vphantom {{(1 - 2r)^{2} } 9}} \right. \kern-\nulldelimiterspace} 9}} \hfill \\ 0 \hfill \\ \end{array} } \right.\) \(\begin{array}{*{20}l} {if} \hfill & {0 \le r < {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \hfill \\ {if} \hfill & {r \ge {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \hfill \\ \end{array}\).

For the Cournot duopoly D^k(r), denote the equilibrium price by p^k(r); quantities of firms 1,2 by q₁^k(r), q₂^k(r); and profits by φ₁^k(r), φ₂^k(r).

No license. When firm 1 does not license the innovation, the resulting Cournot duopoly is D^k(c), where firm 1 obtains Cournot profit φ₁^k(c).

Unit royalty policy. When firm 1 licenses the innovation to firm 2 with a unit royalty r ≥ 0, the Cournot duopoly game D^k(r) is played where the Cournot quantity of firm 2 is q₂^k(r). So for firm 1, the licensing revenue from royalty is rq₂^k(r). The payoff of firm 1 is the sum of its Cournot profit and licensing revenue, which is given by:

Recall that no royalty with r > c is acceptable to firm 2. So under the unit royalty policy, the problem of firm 1 is to choose r (0 ≤ r ≤ c) so as to maximize π^k_R(r) given in (1). We also need to compare the payoff from optimal royalty policy with φ₁^k(c) to see whether licensing by means of a royalty is superior than not licensing.

Fixed-fee policy. When firm 1 licenses the innovation to firm 2 with fixed fee f ≥ 0, the resulting Cournot duopoly is D^k(0) in which firm 2 obtains the Cournot profit φ₂^k(0). If firm 2 refuses to have a license, the resulting Cournot duopoly is D^k(c) in which firm 2 obtains the Cournot profit φ₂^k(c). Therefore the maximum fixed fee that firm 1 can set is φ₂^k(0) − φ₂^k(c) (provided that this is non-negative), which makes firm 2 just indifferent between accepting and rejecting. So the payoff that firm 1 has under the fixed-fee policy has two parts: (i) firm 1's Cournot profit φ₁^k(0); and (ii) fixed fee φ₂^k(0) − φ₂^k(c). This payoff is:

We need to compare this payoff with φ₁^k(c) to see whether licensing by means of a fixed fee is superior than not licensing.

Combinations of unit royalties and fixed fees policy. Suppose that firm 1 licenses the innovation to firm 2 with the use of a licensing policy (r, f) where r (0 ≤ r ≤ c) is the unit royalty and f ≥ 0 is the fixed fee that firm 2 has to pay to firm 1. If firm 2 accepts this policy, it obtains the Cournot profit φ₂^k(r). If it rejects, it operates with marginal cost c and obtains the Cournot profit φ₂^k(c). So for any r, the maximum fixed fee that firm 1 can set is:

Under the licensing policy (r, f), the payoff of firm 1 has three parts: (i) its Cournot profit φ₁^k(r); (ii) its royalty revenue rq₂^k(r); and (iii) the fixed fee f that is given by (3). When f is chosen optimally as in (1), the payoff of firm 1 as function of r is:

As the fixed fee f is chosen optimally for any r, under combinations of unit royalties and fees the problem of firm 1 is to choose r (0 ≤ r ≤ c) so as to maximize π^k_RF(r) that is given in (4). We also need to compare the payoff from the optimal combination with φ₁^k(c) to see whether such a policy is superior than not licensing.

For the analysis it will be convenient first to characterize the optimal combinations of unit royalties and fees:

Optimal combinations of unit royalties and fixed fees When the unit royalty is r, the resulting Cournot duopoly is D^k(r) in which firm 1 has marginal cost 0 and firm 2 has (effective) marginal cost r. Therefore Cournot profits are: φ₁^k(r) = p^k(r)q₁^k(r) and φ₂^k(r) = [p^k(r) − r]q₂^k(r).

Using this in (4) and denoting Q^k(r) = q₁^k(r) + q₂^k(r) (the total quantity), we have:

For any price p (0 ≤ p ≤ 1), let φ_M(p) = p(1 − p) be the profit of the monopolist at price p which has marginal cost 0. Observe from (5) that:

Since φ₂^k(c) is a constant that is not affected by r, by (4) the problem of firm 1 is to choose r so as to maximize φ_M(p^k(r)). Note that φ_M(p) is increasing for p < 1/2 and is decreasing for p > 1/2; and its unique maximum is attained when p equals p_M ≡ 1/2: the monopoly price with marginal cost 0. Let φ_M^* ≡ φ_M(p_M) = 1/4 (the monopoly profit at marginal cost 0). From (4), the maximum possible payoff that firm 1 can obtain is φ_M^* − φ₂^k(c): the monopoly profit with marginal cost 0 by leaving firm 2 its reservation profit φ₂^k(c).

Proposition 1 characterizes the optimal combinations of unit royalties and fixed fees:

The optimal licensing policy for the innovator—and the equilibrium price, quantity, and profits—depends on the model parameters’ values: the innovation size/original unit cost, c; and the maximum quantity of output that the innovator can produce, k (see Fig. 1). In particular, a positive fixed fee emerges only in the parameter area where k is low enough—where the patentee can produce only a relatively small quantity—for both the case of drastic innovation and of non-drastic innovation. Moreover, we show that with combinations of fixed fees and royalties, the royalty rate is lower than the standard case.

For example, consider the case of non-drastic innovation: It is well-know that, without a capacity constraint, the optimal royalty would be r = c. However, with a capacity constraint, we observe r = k < c. Therefore, the capacity constraint modifies the optimal combinations of royalty and fee.

In particular, differently from the case of optimal two-part tariffs with no-capacity constraint (see for example Faulì-Oller & Sandonìs, 2002, and Colombo, 2012), there are situations where the optimal royalty rate is smaller than the marginal cost reduction that is due to the process innovation. Furthermore, in these situations, the fixed fee is positive, rather than zero. This happens when k is small enough: The patentee is able to produce only a relatively small quantity.

The explanation is the following: When the patentee’s output is limited and relatively small (k is low), it is better for the patentee to allow the licensee to expand production (by means of a lower royalty, which is equal to k⁵), and then extract the licensee’s extra profit by means of a positive fixed fee. Note that this explanation is similar to the argument in Kamien and Tauman (1986). Indeed, when k is relatively small, our model is similar to a model with an outside patentee.

In contrast, when the patentee is not strongly constrained by the capacity constraint (k is high), it is better for the patentee to expand its own production and also receive additional revenues from the royalty on the licensee’s output.⁶ Therefore, the per-unit royalty is maximized—r = c—and the fixed fee is zero.⁷ The intuition in this case is similar to Wang (1998). Indeed, when k is high, our model is similar to a model with an inside patentee with no capacity constraint.

The following corollary is immediate from Proposition 1:

Optimal unit royalty policies In view of Corollary 1, to completely characterize optimal pure royalty policies, we need to find optimal pure royalty policies for the parameter space where k < 1/2 and k < c. The next proposition presents the result:

Figure 2 presents optimal pure royalty policies for firm 1 in the (c, k) plane. Note that this result coincides with Wang (1998) when the innovation is non-drastic: When the fixed fee is zero, it is optimal for the patentee to set the highest per-unit royalty—r = c—and to license to the rival. Therefore, the capacity constraint plays no role when the innovation is non-drastic.

By contrast, when the innovation is drastic and k is smaller than 1/2, the patentee cannot produce the monopoly output due to the capacity constraint. In this case, it is better for the patentee to produce as much as possible and to get extra revenues from the per-unit royalty that is charged to the licensee. Note that this result is different from Wang (1998), and highlights the role that is played by the capacity constraint. Finally, when k is greater than 1/2, the patentee produces the monopoly outcome, and it does not license, as in Wang (1998).

Optimal fixed-fee policies By Corollary 1, if c ≥ 1/2 (drastic innovations) and k ≥ 1/2, not licensing is superior to combinations of unit royalties and fixed fees, so not licensing is also superior to fixed-fee policies. To characterize optimal fixed-fee policies completely, we examine the rest of the regions:

Figure 3 presents optimal fixed fee royalty policies for firm 1 in the (c, k) plane. If c < 2/5, the fixed fee is superior to not licensing for any k. If c ≥ 2/5, the fixed fee is superior to not licensing if k is below k₀(c), and not licensing is superior if k is above k₀(c).

It is interesting to observe that when c < 2/5, we are back to the result in Wang (1998, Proposition 1): Licensing by means of a fixed fee is always better than no licensing. However, for greater values of c, the capacity constraint starts to play a role in determining the optimal choice of the patentee between fixed fee and no licensing. Indeed, when k is low, the patentee’s output is relatively small. Therefore, firm 1 cannot benefit so much from the marginal cost differential: It is better for firm 1 to allow firm 2 to expand its production (by licensing the cost-reducing innovation) and extract the extra profits of firm 2 by means of the fixed fee. By contrast, when k is high, firm 1’s output is large. Therefore, firm 1 keeps the marginal costs difference by not licensing the innovation to the rival.

We can now compare the optimal royalty and fixed-fee policies.

Figure 4 summarizes the above discussion, by indicating the parameter spaces where each licensing mechanism emerges in equilibrium. Figure 4 shows that a fixed fee is superior to unit royalties (and to no licensing), provided that the patentee is able to produce only a relatively small quantity—both for the case of drastic innovation and for non-drastic innovation.

As was argued above, when the quantity that can be produced by the patentee is relatively small, it is better for the patentee to let the licensee expand the latter’s production and then extract the extra profit of the licensee by means of a fixed fee. This result is similar to Kamien and Tauman (1986). Indeed, when k is relatively low, our model is similar to a licensing game with an outside innovator.

By contrast, when the patentee can produce a relatively large amount of output, it gets additional revenues from the royalties that are charged on the licensees’ output. In this case, a per-unit royalty is better than a fixed fee from the patentee’s perspective. This result resembles that in Wang (1998). Indeed, when k is relatively large, our model is similar to a licensing game with an inside unconstrained innovator.

In this section, we briefly discuss some extensions of the basic model. The mathematical derivations are relegated in the Technical Appendix.

We introduce a capacity constraint for an innovator and we discuss optimal licensing in a Cournot duopoly. Our model links the two models that are usually considered in the literature: an outside innovator, and an unconstrained inside innovator. We consider unit royalties, fixed fees, and combinations of unit royalties and fixed fees.¹³ We show that a fixed fee is used if and only if the patentee is able to produce only a relatively small quantity (because of a binding capacity constraint). Therefore, a fixed fee might be preferred by the patentee even if the innovator competes with the licensee. When the patentee sets a two-part tariff by combining fixed fees and unit royalties, it charges a positive fixed fee if and only if the patentee is able to produce only a relatively small quantity. Furthermore, the optimal royalty rate in the optimal two-part tariff is lower than the standard case.

The results are driven by two key parameters: i) the maximum quantity that can be produced by the capacity-constrained firm; and ii) the innovation size.¹⁴