Schumpeter famously argued for the economic importance of market power. Even though large firms cause static inefficiency, they had dynamic benefits in that large firms demand more invention since they can extract more revenue from each new product. Further, they supply more invention, Schumpeter hypothesized, since the rate of invention has increasing returns to scale in the number of inventors, and in the number of other employees at the firm. (Axioms A and B). The second part of that statement may be for many reasons; for instance, if the output of a research project could be many potential products, a larger firm has the ability to capitalize on many of those new projects, whereas a small firm might have more limited complementary capabilities. Often, this hypothesis has been tested by checking whether larger firms are more research intensive, meaning that larger firms have a higher percentage of their workforce doing research (Hypothesis 1). Alternatively, a direct reading of Schumpeter is that a 1% increase in the non-research staff of a firm leads to a more than 1% increase in total R&D output of a firm, where output is just the number of research workers times each worker’s average output as a function of firm size (Hypothesis 2).

And here is where theory comes into play. Are axioms A and B necessary or sufficient for either hypothesis 1 or 2? If they don’t imply hypothesis 1, then the idea of testing the Schumpeterian axioms about increasing returns to scale by examining researcher employment is wrong-headed. If they don’t imply hypothesis 2, then Schumpeter’s qualitative argument is incomplete in the first place. Fisher and Temin (that’s *Franklin Fisher* and *Peter Temin*, two guys who, it goes without saying, have had quite some careers since they wrote this paper in the early 70s!) show that, in fact, for both hypotheses the axioms are neither necessary nor sufficient.

An even more basic problem wasn’t noticed by Fisher and Temin, but instead was pointed out by Carlos Rodriguez in a 1979 comment. If Axiom 1 holds, and the average product per researcher is increasing in the number of researchers, then marginal product always exceeds average product. If market equilibrium means I pay all research workers their marginal product, then I will be making a loss if I operate at the “optimal” quantity. Hence I will hire no research workers at all. So step one to interpreting Schumpeter, then, is to restate his two axioms. A weaker condition might be that if the number of research and the number of nonresearch workers increase at the same rate, then average product per research worker is increasing. This is implied by Axioms A and B, but doesn’t rely on always-increasing average product per research worker (Axiom C). This is good for checking our two hypotheses, since anything that would have been implied by Axioms A and B is still implied by our more theoretically-grounded axiom C.

So what does our axiom imply about the link between research staff size and firm size? Unsurprisingly, nothing at all! Surely the optimal quantity of research workers depends on the *marginal* product of more research workers as firm size grows, and not on the average product of those workers. Let’s prove it. Let F(R,S) is the average product per research worker as a function of R, the number of researchers, and S, the number of other employees at the firm. I hire research workers as long as their marginal product exceeds the researcher wage rate. The marginal product of total research output is the derivative of R*F(R,S) with respect to R, or F+R*dF/dR. As S increases, this marginal product goes up if and only if dF/dS+R*dF^2/dRdS>0. That is, I hire more research workers in equilibrium if my non-research staff is bigger according to a function that depends on the second derivative of the average output per researcher. But my axioms had only to do with the first derivative! Further, if dF/dS+R*dF^2/dRdS>0, then larger firms have a larger absolute number of scientists than smaller firms, but this implication is completely independent of the Schumpeterian axioms. What’s worse, even that stronger assumption involving the second derivative does not imply anything about the *share* of research workers on the staff.

The moral is the same one you were probably taught you first day of economics class: using reasoning about averages to talk about equilibrium behavior, so dependent on marginals, can lead you astray very quickly!

1971 working paper; the final version was published in JPE 1973 (IDEAS). Related to the comment by Rodriguez, Fisher and Temin point out here that the problem with increasing returns to scale does not ruin their general intuition, for the reasons I stated above. What about the empirics of Schumpeter’s prediction? Broadly, there is not much support for a link between firm size and research intensity, though the literature on this is quite contentious. Perhaps I will cover it in another post.