The propensity to patent with differentiated products.

I. Introduction

It has been noted in the literature [5; 8; 9], and observation readily confirms, that not all inventions are patented. There are, of course, several explanations for why a firm might not actually patent all of its innovations including the ideas that a patent would give information on profitability to non-inventors, that some inventions might not satisfy the novelty requirement [6; 10], or that a patent would give information to non-inventors on the production of the invention. Horstmann, MacDonald, and Slivinski [3] utilized the first explanation in providing a theoretical framework for why the propensity to patent--the proportion of innovations which actually are patented--is somewhere between zero and one. This paper will utilize the last explanation in offering a complementary framework.

Economically, a patent system is justified for three basic reasons: it rewards the inventor, creating an incentive to invent [7]; it allows for the development of inventions into marketable innovations by defining the property rights over the inventions, lessening the need for secrecy in outside contracting [4]; and it entails the full disclosure of an invention such that someone "skilled in the art" could understand and duplicate it. It is this disclosure which is intended to be one of the main positives that counters the negative of artificially-created market power for the inventor since it allows for a faster diffusion of the new idea. This implies that a non-innovating firm would have little or no need of reverse-engineering the product, making imitation easier. In exchange for this information, the government guarantees the inventor certain property rights over that invention. It is then up to the inventor to decide if gaining those rights is worth letting that information become known. This paper tries to model this trade-off using a differentiated-products model. It is found that the firm will patent if the follower will not use that information, and that the firm might or might not patent if the follower will use the information in order to imitate. Simply put, some inventors will patent, and some will not. Thus, a propensity to patent is obtained between zero and one.

It is assumed that innovation has already occurred for one of the firms. The question at hand is whether or not the inventor will patent, and the answer will depend on the non-innovator's intentions. Except for indifference, the inventing firm will know for certain if it will patent. However, neither firm knew a priori which would be the innovator and which firm would not (whether a patent is applied for also depends on the equilibrium being played if there are multiple equilibria).

The purpose of this paper is to examine the patenting decision of a firm when a patent would allow non-inventors to learn more about how the product is made, potentially hurting the profits of the inventor. While this is a sub-game of a larger, differentiated-products game which would include the choices of product varieties by the firms, the patent issue must first be addressed and is an interesting topic on its own. The result here is a framework for explaining a propensity to patent between zero and one which acts as a complement to Horstmann, MacDonald, and Slivinski.

The rest of the paper is organized as follows: section II describes the basic model; the patenting decision of the innovating firm is discussed in section III; and section IV concludes.

II. Basic Model

The model is an adaptation of that found in Harter [2]. This paper, however, adds the option of imitation to the non-innovator's choices and includes a patent system. A patent will block off a segment of the market around the invention from further entry, but will make imitation less costly. Thus, this is a fencepost patent system, not a signpost system [11].

The two risk-neutral, profit-maximizing firms, A and B, have simultaneously chosen product varieties which are described by locations on the unit interval and have begun R&D. The product variety chosen by firm A is described by the location a, and that chosen by B is described by b.

The R&D process is location-specific, and is represented by a known, Poisson discovery process with a hazard rate denoted by h. Each firm faces a fixed cost, F, and a flow cost, z, to R&D, both of which are assumed to be strictly positive. Once a firm invents, it innovates immediately, producing the invented good at a constant marginal cost which, for simplicity, is assumed to be zero. The firm does not innovate again. The original innovator is a monopolist in this class of goods unless and until the non-innovator has also innovated.

The innovating firm has the option of patenting its invention immediately after the successful completion of R&D. The patent is assumed to be of infinite length and to force any subsequent innovations to be farther than w from the location characterizing the patented invention (thus, if w = 1/3, then a patented innovation located at b = 3/4 would imply that no other firm could produce a product located at a if a [is an element of] [5/12, 1]). Assume that the innovator will patent if it is indifferent between patenting and not patenting.

If the innovation is not patented, then the follower (firm A, say) has four options: discontinue R&D, exiting the market (referred to as an exiting strategy)--no additional costs are involved; continue R&D as originally planned (referred to as a staying strategy)--the flow cost to R&D, z, is still incurred; re-focus R&D toward a different location than was originally chosen (referred to as a moving strategy)--the flow cost to R&D is still incurred, and the fixed cost, F, must be paid again; or reverse-engineer and imitate the innovation (referred to as an imitating strategy)--with cost [Mu][(b - [Alpha]).sup.2] + f, where [Alpha] is firm A's imitation location and [Mu] is some positive coefficient on the cost of differentiating the two products. There is a fixed cost, f, of reverse-engineering the innovation.

Should the innovation be patented, the follower (still A) may instead imitate at a cost of [Mu][(b - [Alpha]).sup.2] + [Phi], where [Phi] [is less than or equal to] f, so long as [Alpha] is not within w of b. Note that the imitation location, [Alpha], is independent of how the follower gained the information on the innovation--whether by reverse-engineering or through the patent application--unless [Alpha] violates the area of protection given by the patent grant. Also, since the follower will not violate the area of protection, but instead differentiates its product from the original invention, then no royalties need to be paid.

The consumers are assumed to have unit demands for this class of good up to a reservation price, s, which is sufficiently high so that all consumers buy (i.e., s [is greater than] 3--this guarantees that a monopolist located anywhere on the interval will price so that all consumers would be willing to buy). Each consumer has a most-preferred good, x, and the values of x are distributed uniformly over the unit interval. The disutility of consuming a good which is not located at x is quadratic in the distance from x.

For the purpose of this paper, it is assumed that both firms have begun R&D and that one of the firms (firm B) has successfully innovated a variety described by the location b. Without loss of generality, assume that b [is greater than or equal to] 1/2. Since there is a unique set of prices for any pair of locations [1], the patenting decision (together with the non-innovator's decision) is sufficient to describe the subgame-perfect, Nash equilibrium in pure strategies.

III. The Patenting Decision

Waterson [11, 860] said "that the main impact of a product patent is not to create a monopoly but rather to affect the variety choices that rivals make." Here, the inventor has to decide whether or not to "affect the variety choices" made by the non-innovator based on whether the patent will limit (to the innovator's advantage) the profitable choices for the follower. The following three lemmas combine to show that when the non-innovator does not take advantage of the information in the patent grant which would allow it to imitate, then the innovator loses nothing by patenting.

LEMMA 1. If exiting is the non-innovator's response when a patent is granted, then the innovator will patent the innovation.

Proof. Show that the innovator (firm B) will prefer the smallest monopoly profits to even the highest duopoly profits:

The lowest monopoly flow profits are given by (b = 1):

[Pi][m.sub.B] = s - 1, [is greater than] 3 - 1 since s [is greater than] 3, = 2.

The highest duopoly flow profits are given by (a = 0, b = 1):

[Pi][d.sub.B] = 1/2.

Since 2 [is greater than] 1/2, monopoly profits are always preferred by the innovator to duopoly profits, and the innovator will, therefore, patent in order to secure the monopoly. Q.E.D.

The best outcome the innovator could hope for is that the follower exit the market. Obviously, if the innovator could achieve a monopoly by patenting, it will. In the absence of exiting, however, the innovator prefers that the follower re-focus R&D for the endpoint which is farther from the innovation--i.e., that the follower play a moving strategy. Of the three alternatives left to the non-innovator (moving, staying, or imitating), the innovator prefers that the follower move because it allows time for some monopoly profits and leads to the maximum possible duopoly flow profits for the given innovation.

LEMMA 2. If a moving strategy is the non-innovator's response when a patent is granted, then the innovator will patent the innovation.

Proof. If the follower (firm A) will play a moving strategy when the innovation is patented, then the moving strategy must yield to firm A an expected profit that is positive. Since moving is available in the absence of a patent grant and is profitable, exiting will not occur regardless of whether or not the innovation is patented.

Because the expected time until firm A has completed R&D is independent of the location firm A has chosen, firm B's expected monopoly profits are the same if A stays or moves. The duopoly flow profits, however, are potentially different. Since A locates in the moving response so as to maximize the distance between the products produced by the two firms, the moving response yields duopoly flow profits to B which are the highest possible for the given variety, b. Thus, firm B will never prefer firm A staying to firm A moving.

Similarly, the duopoly flow profits when the follower moves are always at least as great as those when the follower imitates. In addition, imitation would decrease to zero the expected time that firm B could enjoy a monopoly. Since monopoly profits are always preferred to duopoly profits by the firm, firm B will prefer moving to imitation.

Thus, a moving response by firm A is preferred by firm B over either a staying or an imitating response. Since exiting will not be a response, firm B will patent when moving is firm A's response to the patent. Q.E.D.

The innovator, therefore, prefers that the follower exit or, failing that, that the follower refocus R&D. Exiting by the follower would mean monopoly profits forever for the innovator, and moving would mean monopoly profits for a while followed by the maximum duopoly profits for that particular innovation. Because of the necessary distance between the firms' products in the event of a staying response to a patented innovation, a similar lemma can be formulated for the staying response.

LEMMA 3. If a staying strategy is the non-innovator's response when a patent is granted, then the innovator will patent the innovation.

Proof. Since exiting, moving, and staying are all available to the follower (firm A) with and without a patent (the moving response is always at least as far from b as the staying response is--see above), and since staying is the response with a patent, then neither moving nor exiting could have been the response without a patent. Therefore, either a staying or an imitating response would occur if firm B does not patent.

If A stays in the absence of a patent, then the patent makes no difference. Therefore, assume firm A imitates--with a product described by the location [Alpha]--if there is no patent. Since staying is an option in the absence of a patent and was not chosen, imitating at [Alpha] must not be available if there is a patent, else staying would not be chosen (recall that a patent determines whether f or [Phi] must be paid in order to imitate and not the choice of [Alpha] unless the expected profit-maximizing value of [Alpha] falls within the area of patent protection). This, however, implies that b - [Alpha] [is less than or equal to] w and that b - a [is greater than] w. The patent, therefore, increases the duopoly flow profits for firm B as well as increasing the expected time that firm B enjoys monopoly profits (making the expected time positive). Thus, firm B will prefer that firm A stays to imitate and will patent. Q.E.D.

THEOREM 1. If imitating is not the non-innovator's response when a patent is granted, then the innovator will patent the innovation.

The proof of Theorem 1 is simply the result of Lemmas 1, 2, and 3 showing the patenting decision for specific actions by the follower. The innovator will, therefore, patent if imitation does not occur after the patent. Essentially, the innovator will patent if the follower would neither produce a less differentiated product nor innovate sooner than if the innovator did not patent. Even if imitation is the follower's strategy when there is a patent, however, the innovator will sometimes patent.

Table I. The Patenting Decision

Follower's Action without a Patent    Follower's Action with a Patent
Patent?

Imitate                               Imitate                            Yes
Imitate                               Move                               Yes
Imitate                               Stay                               Yes
Imitate                               Exit                               Yes
Move                                  Imitate                            No
Move                                  Move                               Yes
Move                                  Stay
N.A.(b) Move                                  Exit
   Yes Stay                                  Imitate
  ?(a) Stay                                  Move
  Yes Stay                                  Stay
 Yes Stay                                  Exit
Yes Exit                                  Imitate
No Exit                                  Move
N.A.(b) Exit                                  Stay
   N.A.(b) Exit                                  Exit
      Yes

Notes:
a. The innovating finn might or might not patent here.
b. This combination of strategies with and without a patent is not possible
for a profit-maximizing firm.

LEMMA 4. If imitating is the non-innovator's response in the absence of a patent, then the innovator will patent the innovation.

Proof. If any response other than imitation would result after the patent, then Lemmas 1, 2, and 3 show that the innovation will be patented. Therefore, assume that the follower (firm A) will choose, in the absence of a patent, to imitate and produce the product variety described by location [Alpha]. Assume, also, that after a patent has been granted, firm A will imitate and produce the product variety described by location [Alpha][prime]. Since in both cases, the finns begin production immediately, the only way of decreasing firm B's profits in the case of a patent is for [Alpha][prime] to be closer to b than [Alpha] is. The variety, [Alpha][prime], however, is an option for firm A when there is no patent and [Alpha] is chosen. Therefore, the location [Alpha] must be at least as profitable for finn A as [Alpha][prime]. If [Alpha] = [Alpha][prime], then firms A and B are indifferent. If [Alpha] [is not equal to] [Alpha][prime], then, since [Alpha] is not chosen in the case of a patent, b - [Alpha] [is less than or equal to] w. Thus, b - [Alpha][prime] [is greater than] b - [Alpha], implying that firm B will receive higher duopoly profits from an imitation at [Alpha][prime] and will, therefore, patent. Q.E.D.

The above lemmas give sufficient conditions for patenting to occur. It is shown that patenting might not occur only when imitation would be the resulting response to the grant. Even then, the innovator will apply for patent protection if the follower would imitate regardless.

There are, however, situations in which patenting would not occur. Given that the follower imitates if there is a patent, the proofs of Lemmas 1 and 2 are sufficient to show that the innovator will not patent if the follower would exit or move in the absence of a patent. There is less of a clear-cut indication as to the patenting decision when the follower would stay when there is no patent grant and would imitate when there is a grant since examples can be constructed which show both patenting and non-patenting. The appendix gives examples of both.

IV. Conclusion

This paper uses a differentiated-products model in an attempt to give a theoretical basis for the empirical observation that not all inventions are patented. It is shown that, if the non-innovator will not use the information contained in the patent grant in order to imitate, then the innovator will patent. It is also shown, however, that situations exist when the innovator might not wish to patent. The decision on whether or not to patent will depend on the intended actions of the non-innovator in the particular equilibrium being played. This is true regardless of the magnitude of such things as the patent width, w, and the cost savings to an imitator of a patent, f - [Phi]. These do affect the results indirectly by determining the follower's intended actions, however.

The general conclusions in this paper would appear to hold up well when there are multiple non-innovators. If no other firm would utilize the information in the patent grant to imitate, then the innovating firm would patent in order to prevent production of close substitutes. If the non-innovators do imitate, then the firm might or might not patent.

Perhaps the most obvious extension to the model is to see how this patent system affects the original choices of product variety. This paper assumed an invention and asked about the patenting decision. This is, however, just a sub-game of the bigger game which begins with product-variety choices. This would also be an extension to the Waterson paper which addressed this issue, but which did not allow price competition. The effects on the market of certain patent policies--the length and width, for instance--could then be studied.

An additional extension would be to allow the innovator to license the product. In this paper, the non-innovator, regardless of its choice of action, does not produce a product which is within the protected area around the patented good. Thus, an imitator need pay no royalties. It is possible that the innovating finn would allow a follower to produce, for a fee, within the protected area, but the results would depend critically on the bargaining to choose the level of the royalties.

Appendix

Example (the patenting decision when the follower would stay in the absence of a patent grant and would imitate when there is a grant):

Assume firm B has innovated at b = 1 and firm A has not innovated at its original variety choice, [Alpha] = .9. The flow costs to R&D are zero and the fixed costs are given by F = h(r/h + 1 - [Epsilon])/2r(r + h), where [Epsilon] = 1/9000 and r is used to denote the rate of interest. The costs to imitation are given by [Mu] = 0 (implying that [Alpha] = 0) and f = [Phi] = h(r/h + 1 - [Epsilon])/2r(r + h). It is given that r/h = 317/9000. The width of the patent is assumed to be w = .25.

The follower has, in the absence of a patent, four options, whose expected profits are given by:

[Mathematical Expression Omitted]

[E[Pi].sub.B][where]moving = [integral of][e.sup.-rt][Pr([Tau] [is less than or equal to] t)[Pi][d.sub.A][where]moving + Pr([Tau] [is greater than] t)z]dt - F between limits [infinity] and 0,

= h/2000r(r + h); and

[E[Pi].sub.A][where]imitating = [integral of][e.sup.-rt][[Pi][d.sub.A][where]imitating]dt - [Mu][(1 - [Alpha]).sup.2] - [Phi] between limits [infinity] and 0,

= h/1000r(r + h).

Thus, the follower prefers staying to imitating to moving to exiting. The follower will stay if there is no patent, but, since 1 - a [is less than] w, it will imitate if there is a patent.

The innovator, firm B, must choose, then, between the situations when firm A imitates and stays. Firm B's expected profits are given by:

[E[Pi].sub.B][where]staying = [integral of][e.sup.-rt][Pr([Tau] [is less than or equal to] t)[Pi][d.sub.B][where]staying + Pr([Tau] [is greater than] t)[Pi][m.sub.B]]dt between limits [infinity] and 0,

= h([Pi][d.sub.B][where]staying)/r(r + h) + r[Pi][m.sub.B]/r(r + h),

= h(49/1000 + 2(r/h)(s - 1))/2r(r + h); and [E[Pi].sub.B][where]imitating = [integral of][e.sup.-rt][[Pi][d.sub.B][where]imitating]dt between limits [infinity] and 0,

= h((r/h) + 1)/2r(r + h).

Thus, if E[[Pi].sub.B][where]staying - [E[Pi].sub.B][where]imitating [is greater than] 0, then the firm will not patent. In comparing these, the firm need only look at v [ie equivalent to] 2(r/h)s - 3(r/h) - 951/1000 = (634s - 9150)/9000. If s [is greater than] 15, then v [is greater than] 0, and B does not patent. If, however, s [is less than] 15, B does patent. Thus, the innovator might or might not patent here (under these assumptions), depending on the consumers' reservation price.

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