[video; combinatorial innovation overview] New ideas are often combinations of existing goods or ideas, a point emphasized by Romer1993 and Weitzman1998. A separate literature highlights the links between exponential growth and Pareto distributions: Gabaix1999 shows how exponential growth generates Pareto distributions, while Kortum1997 shows how Pareto distributions generate exponential growth. But this raises a “chicken and egg” problem: which came first, the exponential growth or the Pareto distribution? And regardless, what happened to the Romer and Weitzman insight that combinatorics should be important?

This paper answers these questions by demonstrating that combinatorial growth in the number of draws from standard thin-tailed distributions leads to exponential economic growth; no Pareto assumption is required. More generally, it provides a theorem linking the behavior of the max extreme-value to the number of draws and the shape of the tail for any continuous probability distribution.

…the number of combinations we can create from existing ingredients is so astronomically large as to be essentially infinite, and we are limited by our ability to process these combinations. Let N_t denote the number of ingredients whose recipes have been evaluated as of date t. In other words, our “cookbook” includes all the possible recipes that can be formed from N_t ingredients: if each ingredient can either be included or excluded from a recipe, a total of 2^N_t recipes are in the cookbook. Finally, research consists of adding new recipes to the cookbook—i.e. evaluating them and learning their productivities. In particular, suppose that researchers evaluate the recipes that can be made from new ingredients in such a way that N_t grows exponentially. We call a setup with 2^N_t recipes with exponential growth in the number of ingredients combinatorial growth.

One key result in the paper is this: combinatorial expansion is so fast that drawing from a conventional thin-tailed distribution—such as the normal distribution–generates exponential growth in the productivity of the best recipe in the cookbook. Combinatorics and thin tails lead to exponential growth.

The way we derive this result leads to additional insights. For example, let K denote the cumulative number of draws (eg. the number of recipes in the cookbook) and let Z_K be max of the K draws. Let F̄(10) denote the probability that a draw has a productivity higher than x—the complement of the cdf—so that it characterizes the search distribution. Then a key condition derived below relates the rise in Z_K to the number of draws and to the search distribution: Z_K increases asymptotically so as to stabilize ZF̄(Z_K). That is, given a time path for the number of draws K_t, the maximum productivity marches down the upper tail of the distribution so as to make K_tF̄(Z_Kt) stationary. Kortum1997 can be viewed in this context: exponential growth in the max Z_K is achieved by an exponentially growing number of draws K from a Pareto tail in F̄(·). Alternatively, with thinner tailed distributions like the normal or the exponential, combinatorial growth in K is required to get exponential growth in the max. Even the Romer1990 model can be viewed in this light: linear growth in K requires a log-Pareto tail for the search distribution. This same logic can essentially be applied to any setup: if you want exponential growth in Z_K from a particular search distribution F̄(·), then you need the rate at which you take draws from the distribution to stabilize KF̄(Z_K).

…Theorem 1 (a simple extreme value result). Let Z_K denote the maximum value from K > 0 independent draws from a continuous distribution F(10), with F̄(10) ≡ 1—F(10) strictly decreasing on its support. Then for m ≥ 0:

…Results related to Theorem 1 are of course known in the mathematical statistics literature. It is closely related to Proposition 3.1.1 in Embrechts et al 1997. Galambos1978 (Chapter 4) develops a “weak law of large numbers” and a “strong law of large numbers” for extreme values; some of the results below will fit this characterization.² However, the tight link between the number of draws, the shape of the tail, and the way the maximum increases is not emphasized in these treatments.

…In §5, we see that the combinatorial case has an important empirical prediction that distinguishes it from other cases: in the combinatorial setup, the number of “good” new ideas grows exponentially over time. By contrast, Kortum1997 predicts that the flow of superior new ideas should be constant, even as the number of researchers grows.

Empirically, the flow of annual US patents exhibits rapid growth in recent decades, supporting the prediction of the combinatorial model.

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