The Welfare Effects of Licensing Product-Differentiating Technology in a Commodity Market

Abstract

An innovation that creates horizontal product differentiation is offered to downstream producers that face price competition in a homogeneous commodity market. Innovations can be single-product or multi-product technologies. Adopters of single-product technologies produce identical products, but their products are differentiated from the commodity product. Adopters of multi-product technologies differentiate their products from other adopters and non-adopters. Model results indicate that widespread licensing improves consumer and social welfare; however, the number of licensees that maximizes the patent owner’s profits exceeds the number of licensees that maximizes consumer and social welfare.

Appendices

Appendix: Proposition Proofs

This appendix provides formal proofs of the propositions.

Proof of Proposition 1

First note that the patent owner will not be able to license to both firms because both firms would then produce identical products. To maintain product differentiation, one firm will choose not to adopt and consequently earn positive profits. It follows that the patent owner will license to at most one firm.

Suppose that the patent owner offers a contract $(r\ge 0,F\ge 0)$ to firm one. Without loss of generality, assume that firms have zero marginal costs of production. Then each firm will solve the following maximization problems:

$$\begin{aligned} \max _{p_1} \pi _1^l = \max _{p_1} (p_1-r)q_1 -F = \max _{p_1} (p_1-r) \left( \frac{p_1-1-\sigma p_2+\sigma }{\sigma ^2-1} \right) -F \end{aligned}$$

(7)

and

$$\begin{aligned} \max _{p_2} \pi _2^o = \max _{p_2} p_2q_2 = \max _{p_2} p_2 \left( \frac{p_2-1-\sigma p_1+\sigma }{\sigma ^2-1} \right) \end{aligned}$$

(8)

The Nash equilibrium is $(p_1^*,p_2^*) = \left( \frac{-2+\sigma +\sigma ^2-2r}{\sigma ^2-4},\frac{-2+\sigma +\sigma ^2-r\sigma }{\sigma ^2-4}\right)$.

The patent owner solves the following problem:

$$\begin{aligned} \max _{r,F}\quad rq_1 + F \end{aligned}$$

(9)

$$\begin{aligned} \text{ s.t., }\quad \pi _1^l \ge 0 \end{aligned}$$

(10)

Equation 10 is the participation constraint of Firm 1 which states that its profits after adopting the technology are non-negative. If we assume that the participation constraint is binding, then the patent owner’s problem simplifies to the following:

$$\begin{aligned} \max _{r,F}\quad p_1q_1 \end{aligned}$$

(11)

$$\begin{aligned} \text{ s.t., }\quad \pi _1^l \ge 0 \end{aligned}$$

(12)

If we solve the equation with the use of the Nash equilibrium that is found above, the optimal value of the royalty is $r^* = \frac{1}{4}\frac{\sigma ^2(1-\sigma )(\sigma +2)}{2-\sigma ^2} > 0$ when $0<\sigma <1$. The fixed fee becomes $F^*=\frac{1}{16}\frac{(1-\sigma )(\sigma +2)^2}{\sigma +1}$. The patent owner earns licensing profits of $\pi ^T=\frac{1}{8}\frac{(1-\sigma )(\sigma +2)^2}{(\sigma +1)(2-\sigma ^2)}$. Note that substituting $r^*$ into the Nash equilibrium price from above gives $(p_1^*,p_2^*) = \left( \frac{1}{2}\frac{(1-\sigma )(\sigma +2)}{2-\sigma ^2},\frac{1}{4}\frac{(1-\sigma )(\sigma ^2-2\sigma -4)}{\sigma ^2-2}\right)$. $\square$

Proof of Proposition 2

Note that the patent owner will not be able to license to all N firms because then all firms would produce identical products. At least one firm will refuse to adopt, thereby maintaining a differentiated product and earn positive profits.

Now examine the case where the patent owner chooses to license to multiple firms and leaves multiple firms without a license: The number of adopters satisfies the condition $2\le N_l < N-1$. Then all downstream producers would price at marginal cost. If the patent owner charges a royalty fee, then the Nash equilibrium product prices are $(p_1^*,p_2^*)=(r,0)$ where $p_1^*$ is the Nash equilibrium price for adopters and $p_2^*$ is the Nash equilibrium price for non-adopters. If we solve Eq. (11), then $r^* = \frac{1-\sigma }{2}$. The patent owner earns profits of $\pi ^T=\frac{1}{4}\left( \frac{1-\sigma }{1+\sigma }\right)$.

If the patent owner decides to license only one firm, then the remaining $N-1$ firms that do not adopt the technology are direct competitors; and, if we assume again that firms do not have marginal costs of production, they will price at $p^*=0$. Then the patent owner solves the equivalent problem from Eqs. 9 and 11 but where the non-adopters are equivalent to Firm 2 from Eq. 8, and the Nash equilibrium from Eqs. 7 and 8 is $(p_1^*,p_2^*)=\left( \frac{1}{2}(1-\sigma +r),0\right)$. Then $r^* = 0$ and $F^*=\frac{1}{4}\left( \frac{1-\sigma }{\sigma +1}\right)$. Then the patent owner’s profits are $\pi ^T=F^*$.

Now compare the patent owner’s profits if the patent owner decides to license to all but one firm, or $N-1$ firms, where $N>2$. Now the remaining firm that does not adopt the technology serves as the equivalent of Firm 2 from Eq. 8. Since all technology adopters now become direct competitors, they will price where $p^*=r$, where r is the royalty rate. The patent owner again solves the equivalent problem from Eqs. 9 and 11 but where the Nash equilibrium from Eqs. 7 and 8 is $(p_1^*,p_2^*)=\left( r, \frac{1}{2}(1+\sigma r - \sigma )\right)$. Then $r^* = \frac{1}{2}\frac{(1-\sigma )(\sigma +2)}{2-\sigma ^2}$. The patent owner’s profits are $\pi ^T=\frac{1}{8}\frac{(1-\sigma )(\sigma +2)^2}{(\sigma +1)(2-\sigma ^2)}$ (and $p_1^*\ge p_2^*$) which exceed the profits it earns from licensing to only one firm.¹⁹$\square$

Proof of Proposition 3

I first verify this in the case of two downstream producers $(N=2)$. In the proof, the technology owner is always the first-mover and the non-adopters are the second-movers. However, the results depend merely on who is the first- versus the second-mover and the Stackelberg–Bertrand–Nash equilibrium prices would be reversed if the technology owner was the second mover and the non-adopter was the first mover. The technology owner owns the technology and therefore there are no licensing fees.

Assume Firm 1 owns the technology (and therefore there are no licensing fees) and is able to commit to a price before Firm 2 chooses its price. From Eq. 8, Firm 2’s best response function is $p_2^* = \frac{1}{2}(1+\sigma p_1 - \sigma )$. Firm 1 solves the problem:

$$\begin{aligned} \max _{p_1} \pi _1^l = \max _{p_1} p_1q_1 = \max _{p_1} p_1 \left( \frac{p_1-1-\sigma p_2+\sigma }{\sigma ^2-1} \right) \end{aligned}$$

(13)

Substituting Firm 2’s best response function into Firm 1’s maximization problem above and solving for $p_1^*$, then $(p_1^*,p_2^*) = \left( \frac{1}{2}\frac{(1-\sigma )(\sigma +2)}{2-\sigma ^2}, \frac{1}{4}\frac{(1-\sigma )(\sigma ^2-2\sigma -4)}{\sigma ^2-2}\right)$. These are the same Nash equilibrium prices from Propositions 2 and 3. If there are $N>2$ firms, the Stackelberg–Bertrand–Nash equilibrium prices with no licensing results in lower market prices than when there are $N=2$ firms.²⁰$\square$

Proof of Proposition 4

The proof of Proposition 4 follows the proof of Proposition 2. First note that the patent owner will not be able to license to all N firms. This is because the Nth firm will choose not to adopt the technology since if $N-1$ firms adopt, the Nth firm will already be differentiated from the other $N-1$ downstream producers.

Now to show that the patent owner will license to all but one firm, or $N-1$ firms, it must be shown that the patent owner’s profits that are earned from licensing to all but one firm exceed the profits that can be earned from licensing to fewer than $N-1$ firms. Begin by supposing that the patent owner licenses to only one firm. Then the remaining $N-1$ non-adopting firms will produce the commodity product. Since the non-adopting firms are direct competitors producing identical products, they will price where $p^*=0$. The analogous results from Proposition 2 follow, and the patent owner’s profits are $\pi ^T=F^*=\frac{1}{4}\left( \frac{1-\sigma }{\sigma +1}\right)$.

Now suppose that the patent owner licenses to $N-1$ firms. It follows that since the potential adopters are identical, the patent owner will offer identical licensing contracts to all potential adopters. Therefore suppose that the patent owner offers a contract (r, F) to firm i. Following Eqs. 7 and 8 but for N firms, each technology-adopting firm $i=1,\ldots ,N-1$ solves the following maximization problem:

$$\begin{aligned} \max _{p_i} \pi _i^l&= \max _{p_i} (p_i-r)q_i -F \nonumber \\&= \max _{p_i} (p_i-r) \left( \frac{(1+(N-2)\sigma )p_i-1+\sigma -\sigma (\sum _{j=1,j\ne i}^N p_j)}{((N-1)\sigma +1)(\sigma -1)} \right) -F \end{aligned}$$

(14)

Let the non-adopting producer be the Nth firm. Then it solves the maximization problem:

$$\begin{aligned} \max _{p_{N}^o} \pi _{N}&= \max _{p_{N}} p_{N}q_{N} \nonumber \\&= \max _{p_{N}} p_{N} \left( \frac{(1+(N-2)\sigma )p_{N}-1+\sigma -\sigma (\sum _{i=1}^{N-1}p_i)}{((N-1)\sigma +1)(\sigma -1)} \right) \end{aligned}$$

(15)

Then the Nash equilibrium is

$$\begin{aligned} p_i^*&= \frac{2r\sigma ^2N^2-2\sigma ^2 N+2\sigma N-8r\sigma ^2 N+4r\sigma N+2r-5\sigma +8r\sigma ^2-8r\sigma +2+3\sigma ^2}{-9\sigma ^2 N+9\sigma ^2+4+6\sigma N-12\sigma +2\sigma ^2 N^2} \nonumber \\ p_{N}^*&= \frac{2\sigma N-2\sigma ^2 N-5\sigma +r\sigma N+r\sigma ^2 N^2-3r\sigma ^2 N+2r\sigma ^2+3\sigma ^2-r\sigma +2}{-9\sigma ^2 N+9\sigma ^2+4+6\sigma N-12\sigma +2\sigma ^2 N^2} \end{aligned}$$

(16)

The patent owner solves the maximization problem

$$\begin{aligned} \max _{r,F}\quad r(q_1+q_2+\cdots q_{N-1}) + (N-1)F \end{aligned}$$

(17)

$$\begin{aligned} \text{ s.t., }\quad \pi _i^l \ge 0 \quad \forall i=1,\ldots ,N-1 \end{aligned}$$

(18)

With the use of the Nash equilibrium from above, the profit-maximizing contract is $(r^*,F^*)= \left( \frac{1}{4}\frac{(2\sigma N^2-7\sigma N+2N+7\sigma -4)\sigma (2\sigma N-3\sigma +2) (-1+\sigma )}{(-2-2\sigma N+4\sigma +\sigma ^2 N-\sigma ^2)(1+\sigma N-2\sigma )^2}, \frac{1}{16}\frac{(2\sigma ^2N-2\sigma N-3\sigma ^2+5\sigma -2)(2\sigma N-3\sigma +2)}{(1-\sigma N+\sigma )(1+\sigma N-2\sigma )^3}\right)$. The patent owner earns profits of $\pi ^T=\frac{N-1}{8}\frac{(2\sigma ^2 N-2\sigma N-3\sigma ^2+5\sigma -2)(2\sigma N-3\sigma +2)}{(-2-2\sigma N+4\sigma +\sigma ^2 N-\sigma ^2)(\sigma N-\sigma +1)(1+\sigma N-2\sigma )}$. Technology adopting firms will charge price $p_i^*=\frac{1}{2}\left( \frac{(2\sigma N-3\sigma +2)(-1+\sigma )}{-2-2\sigma N+4\sigma +\sigma ^2 N-\sigma ^2}\right)$. The remaining firm will charge $p_N^*=\frac{1}{4}\left( \frac{(2\sigma ^3 N^2-2\sigma ^2 N^2-7\sigma ^3 N+13\sigma ^2 N-6\sigma N+5\sigma ^3-15\sigma ^2+14\sigma -4)}{(1+\sigma N-2\sigma )(-2-2\sigma N+4\sigma +\sigma ^2 N-\sigma ^2)}\right)$.

Now compare this to the licensing profits that are earned by the patent owner if it licenses to more than one firm but less than $N-1$ firms: $1< N_l< N-1$. Assume that $N\ge 4$.²¹ Then the remaining non-adopters will charge a Nash equilibrium price of $p_k^*=0$ since they are producing the commodity product. Then in the Nash equilibrium, the optimal royalty fee is $r^*=\frac{1}{2}\left( \frac{\sigma (1-\sigma )(N_l-1)}{N_l\sigma -\sigma +1}\right)$ and the fixed fee is $F^*=\frac{1}{4}\left( \frac{1-\sigma }{(\sigma N_l -\sigma + 1)(1+\sigma N_l)} \right)$. The technology adopters will charge the price $p_j^* = \frac{1}{2}(1-\sigma )$ and non-adopting firms will charge the price $p_k^*=0$. The patent owner’s profit is $\pi ^T=\frac{N_l}{4}\left( \frac{1-\sigma }{\sigma N_l + 1} \right)$. One can verify that the patent owner’s profits are greatest when it licenses to all but one firm, or $N_l=N-1$.²²$\square$

Appendix: Monsanto Licensing Examples

Biotechnology innovation for the U.S. seed market began in the 1980s. The innovations created genetically modified plant varieties that were significantly differentiated from conventional seed varieties. Prior to biotechnology, new varieties were created by cross-breeding existing plant varieties to produce desirable traits. Biotechnology enabled new varieties to be created by directly inserting genetic material—some of which did not originate from plants, into seed varieties to produce desirable traits. Regulatory guidelines were created to evaluate agricultural biotechnology’s impact on public safety and the environment. At the same time, the landmark case Diamond v. Chakrabarty, 447 U.S. 303 (1980), established the patentability of newly created hybrid plants. The novel creation of “frankenfoods” combined with stronger property rights and new regulatory guidelines created genetically modified varieties that were distinct from conventional hybrids.

The model and its predictions provide a possible explanation for the licensing behavior of Monsanto’s patented traits (Howell et al., 2009; Jefferson et al., 2015; Deconinck 2020) .Monsanto²³ holds a significant share of the industry’s key trait technologies in its patent portfolio.²⁴ First-generation trait technologies include herbicide-tolerance and insect-resistance. Downstream seed firms licensed Monsanto’s traits to incorporate them into their seed varieties for field crops such as corn. These seed firms likely engaged in Bertrand price competition because commodity agriculture such as corn typically experiences low profit margins (Cahoon 2007), seeds are produced easily once the variety itself is created, and firms are not likely to face production capacity constraints.

Monsanto licensed its trait technologies in two different ways: first, to create mono-trait varieties, exhibiting only the single, licensed trait; and second, to create “stacked” trait varieties, which allowed the licensee to combine multiple trait technologies from different trait providers.²⁵ When Monsanto licensed a given trait for the creation of mono-trait varieties, the newly-created varieties were close substitutes to one another because they all exhibited the same single, licensed trait, which suggests single-product technology adoption. In contrast, when Monsanto licensed a given trait for a stacked variety, this enabled seed firms to combine Monsanto’s trait technology with other trait technologies—often from other trait providers—and create products that were differentiated from others that were licensing that particular Monsanto trait. Therefore, licensing for stacked varieties suggests multi-product technology adoption.

Two traits that Monsanto widely licensed to create single-trait corn varieties are its Round-up Ready (RR2) and YieldGard corn borer (YGCB) traits. RR2 confers herbicide tolerance to the herbicide Roundup Ready while YGCB confers resistance to the corn borer insect. The downstream seed firms that created mono-trait varieties with RR2 include Pioneer, Bayer, Stine, and Genective, and Monsanto’s Dekalb. Meanwhile Pioneer and Bayer licensed Monsanto’s YGCB technology to create mono-trait insect-resistant varieties.

Monsanto also licensed RR2 and YGCB traits to create stacked varieties. Examples of stacked traits with RR2 include Herculex I-Liberty Link-RR2, Herculex XTRA-Liberty Link-RR2, and its SmartStax. The first two combine Monsanto’s RR2 trait with Bayer’s Liberty Link herbicide tolerance trait and Pioneer’s Herculex insect resistance trait to create new corn hybrids. SmartStax combines Monsanto’s Yieldgard VT Triple and RR2 traits, Pioneer’s Herculex XTRA, and Bayer’s Liberty Link. Examples of stacked varieties with the YGCB trait include YGCB-Clearfield which combines Monsanto’s YGCB with BASF’s Clearfield herbicide tolerance trait; and YieldGard VT Triple combines Monsanto’s YGCB, YieldGard Rootworm, and RR2 traits.

If Monsanto’s behavior is consistent with the model, then the model predicts that as Monsanto increases the number of adopters and stops short of full adoption, it should be able to increase its licensing fee, which would lead to an increase in all corn prices (both genetically modified and conventional). Deconinck (2020) notes that past studies suggest that corn seed prices have risen.²⁶

The theory suggests that consolidation of licensing seed firms will reduce welfare only in the case of multi-product technologies. The industry itself had experienced mergers between biotechnology firms (such as Monsanto) and seed firms (such as Dekalb and Monsanto). Despite this, Chan (2011) notes that many major patent-holding firms do not create many new varieties. This activity of “specializing” in genetic technologies rather than also creating new varieties is also consistent with the idea that licensing technology to outside producers rather than manufacturing the product itself is consistent with profit maximization.²⁷

Other explanations may also support Monsanto’s widespread licensing behavior. For example, because Monsanto’s patent portfolio is so extensive, it may be entering into a multitude of cross-licensing agreements that would make it profitable to license its technology widely to its competitors. Wilson and Huso (2008) discuss licensing and acquisition strategy for agricultural biotechnology firms. Jefferson et al. (2015) explicitly detail the need for better publicly available information of patent ownership and cross-licensing agreements so as to gauge industry competitiveness and welfare effects more accurately.