## Models of Innovation I: The Patent Race

I’ve been going through some old literature on innovation again as part of a current project, so I figured I ought put up a little review of this literature. I’ll cover five strands: the patent race, the partial equilibrium/auction, the quality ladder, sequential innovation a la Scotchmer and Green, and bandit experimentation. Roughly speaking, all economic models of invention builds on these models (though in the best of all possible worlds, the model we are currently developing will provide a sixth member of the canon!) The patent race I discuss below is the model of Loury (1979) and Lee/Wilde (1980), but if you are interested in this model with feedback during the invention process, Jennifer Reinganum’s 1982 Econometrica (gated copy) and Ken Judd’s 1985 Kellogg working paper are the places to look.

Glenn Loury (an old professor of mine back at the alma mater) not only gets some great applied results in his famous patent race paper, but he also proves the importance of two things you’ve ignored since your micro qualifying exam: general equilibrium effects and the value of implicit differentiation. Let there be a set of homogeneous R&D firms who can buy random Poisson variables which produce an invention. The random R&D variable costs x and produces with Poisson rate h(x), where h is increasing and single-peaked, meaning possibly increasing returns to scale when x is low but decreasing returns to scale as x goes to infinity. All invention activity across firms is independent. The first inventor wins a prize V and everyone else earns 0. Is there too much, too little, or the right amount of research?

Let’s first try solving this problem in partial equilibrium. That is, take the sum of other firms’ R&D efforts as a given constant called A. Taking the derivative of my own effort with respect to opponent efforts, you will find that dx/da is positive iff h(x) is greater than A plus the interest rate. It is also (intuitively) obvious that as total opponent effort goes to infinity, my own effort goes to zero: my probability of inventing first goes to zero, but I still have to pay a constant for the random R&D variable. We know from those two properties, then, that my own R&D effort is increasing when other firms’ effort is low, then decreasing monotonically to zero after we pass a cutoff. It seems that the “optimal” level of competition for eliciting R&D is somewhere between monopoly and perfect competition, a very Schumpeterian notion.

But solving this problem in partial equilibrium is odd: as I increase my own R&D, other firms will react to this change. A natural modification is to solve for a Cournot-style Nash equilibrium. Since firms are identical, a natural restriction is to symmetric equilibria. Recall that we can solve analytically for dx/da. Setting this equation to 0, substituting (n-1)h(x) for A using the symmetry assumption, and then solving for x would give us the equilibrium level of effort. It turns out that solving for x analytically is rather difficult in this problem. But no worries: we don’t actually care what the optimal level of x (call this x*) is, but only about what properties this equilibrium contains. So let’s set the derivative to 0 implicitly: let x*=f(a,p) where p is the other parameters like the payoff, the interest rate, etc. By symmetry, x*=f((n-1)h(x*),p).

How is this implicit function useful? Let’s see what happens to equilibrium effort when the number of firms increases. That is, what is dx*/dn? Taking the total derivative of both sides of the function for x* above gives

dx*/dn=df/da*da/dn+(n-1)dx*/dn

df/da is, by construction, equal to dx/da, and since own effort is not greater than total effort by other firms in a symmetric equilibrium, we can use the result two paragraphs back to see that the derivative dx/da is negative. Solving the total derivative for dx*/dn and using the property that dx/da is negative, we have that dx*/dn is always negative. That is, more firms decreases effort for each firm, a very different result from what we got in partial equilibrium. Note that we found this property without ever having to solve for what each firms’ equilibrium effort was!

From here, it is straightforward to get two more important results: firms in competitive equilibrium spend more on R&D than a social planner would, and there is inefficiently too much entry into R&D in the long run. The first result basically just comes from the fact that each firm, when choosing how much R&D to invest in, does not take into account that an increase in its R&D effort decreases the probability that all other firms will create an invention given their level of effort. The second result is very similar to a Chamberlain entry game: if there are positive profits in the R&D industry when each firm is operating at its most efficient scale, then other firms will enter. They will continue to enter as long as they can gain any profits, even if this pushes the scale of every firm to the left of its most efficient point. We can regain the socially efficient outcome by decreasing the payoff to an invention. By, for instance, decreasing patent length and charging firms to enter R&D in an industry, the payoff to invention decreases. Artificially increasing the cost of entry and decreasing the payoff to invention can regain the socially optimal level of investment.

One clarification is necessary, due to Lee and Wilde (1980). In Lee and Wilde, instead of paying a fixed amount x to buy a random R&D variable, firms pay a flow cost x when they are working on an invention. Changing the fixed cost to a variable cost does not change most of the welfare results above, but it does completely flip the effect of having more firms on my effort choice. The more firms there are, the more effort I will exert. Why? Profit in Loury is just the integral from 0 to infinity of the discount rate at time t times the probability no one has made a discovery by t times the instantaneous probability of a discovery by my firm times the payoff, minus the fixed cost x. In Lee and Wilde, the fixed cost x is replaced by a flow cost x, meaning I pay the flow cost until the first firm makes a discovery. Simplifying those integrals gives profit functions for Loury of

Vh(x)/(A+r+h(x))-x

and for Lee and Wilde

(Vh(x)-x)/(A+r+h(x))

That is, the more effort from other firms, the lower the cost I have to pay to work on R&D, since the invention will be found (by someone) sooner. This is enough to cause me to inefficiently “race” harder no matter how many firms are working on the project.

“Market Structure and Innovation (final QJE version and IDEAS link) by Glenn Loury was in QJE in 1979. The clarification by Lee and Wilde (final QJE version and IDEAS link) appeared in the QJE in 1980. The major earlier papers on patent races in patent equilibrium are a series by Kamien and Schwartz, and a 1971 paper by Bob Lucas.