This post continues a series of notes on the main theoretical models of innovation. The first post covered the patent race literature. Here I’ll cover the sequential innovation literature most associated with Suzanne Scotchmer, particularly in her 1991 JEP and her 1995 RAND with Jerry Green.

Let there be two inventions instead of one, where the second builds upon the first. Let invention 1 cost c1, and invention 2 cost c2, with firm 1 having the ability to invent invention 1, and firm 2 invention 2. If only invention 1 exists, the inventing firm earns v1 (where v1 is a function of patent length T). If both invention 1 and 2 exist, and compete for sales in a market, then they earn v1c and v2c, where c stands for “compete”. If both invention 1 and 2 exist, but are sold by a monopolist, they earn v12>=v1c+v2c. With probability p, 2 will infringe on 1, and hence inventor 2 will need a license to sell product 2.

With one invention, it’s intuitive that the length of the patent should be just long enough to allow the inventor to cover the cost of that invention. This logic does not hold when inventions build on each other. Invention 1 makes invention 2 possible, so it seems we should give some of the social surplus created by invention 2 to the inventor of 1. But doing so makes it impossible to give all of the surplus created by invention 2 to inventor 2. This is a standard problem in the theory of complementary goods: if left shoe has social value 0, and right shoe by itself has social value 0, but the two together have value 1, then the “marginal value” created by each shoe is 1. Summing the marginal values created gives us 2, but the total social value of the pair of shoes is only 1. This wedge between the partial equilibrium concept of marginal value and our intuition about general equilibrium actually comes up quite a bit: willingness-to-pay, by definition, is *only* meaningful in a partial equilibrium sense despite frequent misuse to the contrary.

So how should efficiently give patent rights with sequential inventions? First assume there is no possibility to form an ex-ante license between the two firms, though of course firms can sell inventions to each other once the product is invented. Also assume that profits are divided using Nash bargaining when firms sell a patent to each other: in this case, each firm garners half of the profit earned using the patents minus the threat point representing what each firm earns in the absence of an agreement. Consider our logic from the one invention case, where we get incentives correct by making patent length just long enough to cover costs: v12(T)=c1+c2, where v12 is the revenue earned by having both products 1 and 2 in the same monopoly firm given patent length T, c1 is the cost of developing product 1, and c2 is the cost to develop product 2. Setting v12(T) just equal to c1+c2 will, in general, provide insufficient incentives for both products to be developed. That is, making patent length long enough that inventor 1 can afford to cover her costs, and the costs of inventor 2, while making precisely zero profit, is insufficient for inducing the invention of both 1 and 2. Why? One reason is that once 2 is invented, the development costs of 2 are sunk. Therefore, once 2 is invented, the licensing agreement will not take into account inventor 2’s costs. Inventor 2, knowing this, may be reluctant to invest in product 2 in the first place.

How might I fix this? Allow ex-ante joint ventures. That is, let firm 1 and firm 2 form a joint venture before the costs of creating invention 2 are sunk. If ex-ante joint ventures is allowed, the optimal patent breadth is p=1: the second invention always infringes. The reason is simply that longer patent length diminishes the bargaining power of firm 2 at the stage in the game where the joint venture is created, 2 knows that he will be required to get an ex-post license after inventing if no joint venture is formed. The Nash bargaining share given to 2 in an ex-post license is always higher if there is a chance that 2 does not infringe because 2’s Nash threat point is higher. Therefore, the share of monopoly profit that needs to be given in an ex-ante joint venture to 2 is higher, because this share is determined in Nash bargaining by the “threat” 2 has of not signing the joint venture agreement, developing product 2, and then signing an ex-post license agreement. Since the total surplus when both products are sold by a monopoly is a fixed amount, giving more profit to 2 means giving less profit to 1. By construction, this increased profit for 2 does change the probability that firm 2 invests in invention 2; rather, the distortion is that less profit to 1 means less incentive for firm 1 to invest in invention 1. So optimal patent breadth is always p=1: follow-up inventions should always infringe.

The intuition above has been modified in many papers. Scotchmer and Green themselves note that that if the value v2c of the second invention is stochastic, and only realizes after firm 2 invests, less than perfectly broad patents can be optimal. Bessen and Maskin’s 2009 RAND, discussed previously on this site, notes that imperfect information across firms about research costs can make patents strictly worse than no patents, because with patents I will only offer joint ventures that are acceptable to low-cost researchers even when social welfare maximization would require both low and high-cost researchers to work on the next invention. A coauthor here at Northwestern and I have a result, which I’ll write up here at some point, that broad patents are not optimal when we allow for multiple paths toward future inventions. Without giving away the whole plot, the basic point is that broad patents cause distortions early on – as firms race inefficiently to get the broad patent – whereas narrow patents cause distortions later – as firms inefficiently try to invent around the patent. The second problem can be fixed with licenses granted by the patentholder, but the first cannot as the distortion occurs before there is anything to license.

The main papers discussed here are Scotchmer and Green’s 1995 RAND (Final RAND copy, IDEAS) and Scotchmer’s 1991 JEP (Final JEP copy, IDEAS). Despite the dates, the JEP was written after the RAND’s original working paper.