Growth and Ideas

The chapter states the chain of reasoning as “Ideas ⇒ Nonrivalry ⇒ IRS ⇒ Problems with CE,” distinguishing ideas (codifiable instructions, “things that can be codified in a bitstring”) from objects (rivalrous goods such as capital, labor, and the atoms that make them up) following Romer (1993) (Sec. 1, pp. 1065-1066). For a production function Y = F(A, X) where A is the stock of knowledge and X is a vector of rivalrous inputs, the chapter shows that “there are constant returns to scale to the standard rivalrous inputs X and, therefore, increasing returns to scale to these inputs and A taken together,” formalized as F(A, λX) = λY alongside F(λA, λX) > λY for λ > 1, because doubling production of a good requires replicating the rivalrous inputs but not the nonrivalrous knowledge needed to produce it (Sec. 1, pp. 1066-1067, Eqs. 1-3).

The chapter shows that constant returns in the rivalrous inputs X alone guarantee that paying each such factor its marginal product exactly exhausts output (F_X·X = Y), so that “if the stock of knowledge is also paid its marginal product, then the firm would make negative profits[; t]his means that the standard competitive equilibrium will run into problems in a model that includes ideas” (Sec. 1, pp. 1067-1068, Eqs. 4-5). The chapter is explicit that this is not true of human capital: a production function constant-returns in physical and human capital (Y = K^α H^(1-α)) remains constant-returns even when rewritten in per-worker terms, “a simple change of variables cannot change the underlying convexity of a production function” – the increasing-returns property is specific to treating the stock of ideas, not human capital, as a third argument of the production function (Sec. 1, pp. 1067-1068).

With output Y_t = A_t^σ L_Yt and an idea production function Ȁ_t = νL_At A_t^φ, the chapter derives that along a balanced growth path “the growth rate of the stock of ideas, in the long-run, is proportional to the rate of population growth” (g_A = n/(1-φ)), so that “the growth rate of per capita output is proportional to the rate of population growth, where the factor of proportionality depends on the degree of increasing returns in the two sectors” (g_y = σn/(1-φ)) (Sec. 3.2, pp. 1072-1073, Eqs. 12-13). The chapter explains the mechanism in two steps: “more researchers produce more new ideas,” and because of nonrivalry, “per capita output depends on the total stock of ideas, not on ideas per person” – so that a larger population means more researchers discovering ideas that benefit everyone without being “degraded by the presence of a larger population” (Sec. 3.3, p. 1073).

The chapter states that Result 1 – that output per capita along a balanced growth path is proportional to the total stock of ideas, which in turn is proportional to the effective amount of research and hence to population size, with growth rates proportional to the population growth rate (times the degree of increasing returns, γ) – “continue[s] to hold” under the optimal allocation (Result 2a) and under the decentralized equilibrium with monopolistic competition in capital-goods production (Result 3a), “provided the allocations are interpreted” accordingly (Sec. 4.3-4.4, pp. 1080, 1085-1086). This robustness across very different institutional assumptions about how resources get allocated is what makes the scale prediction, in the chapter’s view, a structural feature of idea-based growth models rather than an artifact of any one allocation mechanism.

The chapter shows that comparing the equilibrium and optimal expressions for the share of human capital allocated to research reveals “three differences”: first, “the true marginal product of human capital in research is lower by a factor of λ<1 than the equilibrium economy recognizes because of the congestion/duplication externality, which tends to lead the equilibrium to overinvest in research”; second, the equilibrium “ignores the fact that the discovery of new ideas may raise the future productivity of research if φ>0,” which could cause underinvestment; and third, “the research sector appropriates only the fraction θ<1” of a new idea’s marginal product in final-goods production because of the monopoly markup, which “works to cause the equilibrium allocation of human capital to research to be too low.” The chapter concludes that “these three distortions do not all work in the same direction, so that theory cannot tell us whether the equilibrium allocation to research is too high or too low” (Sec. 4.4, pp. 1085-1086, Result 3d). Separately, the equilibrium investment rate in physical capital is unambiguously too low, because capital is paid less than its marginal product under the monopoly markup (Sec. 4.5, p. 1086).

The chapter defines models with “strong” scale effects as those in which “the growth rate of the economy is an increasing function of scale,” associated with the first-generation models’ assumption that φ=1 in the idea production function, versus models with “weak” scale effects in which only “the level of per capita income in the long run is an increasing function of the size of the economy” (Sec. 5, p. 1089). The chapter argues the strong form is “inconsistent with the basic trends in aggregate data”: U.S. per-capita GDP growth has been remarkably stable (fit well by a straight line on a log scale back to 1870), while “resources devoted to research have exhibited a tremendous amount of growth in the post-war period” – an index of researchers in the G-5 countries “rose by more than a factor of eight” between 1950 and 1993 – so that a model in which the growth rate itself should rise with research effort “is a bug,” not a feature, of the first-generation models (Sec. 5.1, pp. 1090-1093).

The chapter states that “the only difference between the two models… is essentially the strength of the knowledge spillover parameter,” but that this has a large consequence for policy: with weak scale effects (φ<1), “policy changes such as a subsidy to R&D or a tax on capital will have no [e]ffect on the long-run growth rate,” only on its level and transition path, whereas with strong scale effects a research subsidy permanently raises long-run growth (Sec. 5.2, pp. 1093-1094). Reviewing the Young/Peretto/Dinopoulos-Thompson/Howitt line of models that add a product-variety dimension to recover policy potency without strong scale effects, the chapter shows their key result “depends crucially on a knife-edge case” (the variety-creation parameter β exactly equal to 1), and that, even at that knife edge, “the model still exhibits the weak form of scale effects” despite many of these papers being titled “growth without scale effects” (Sec. 5.2, pp. 1094-1095).

The chapter reports that studies controlling explicitly for international trade openness – Backus-Kehoe-Kehoe (1992), Frankel-Romer (1999), and especially Alcala-Ciccone (2002), who also instrument for trade and institutional quality – “all find an important role for scale,” with Alcala-Ciccone estimating “a long-run elasticity of GDP per worker with respect to the size of the workforce… equal to 0.20” (Sec. 5.3, pp. 1096-1097). The chapter also presents Kremer’s (1993) natural experiment of five continents isolated from one another for roughly 12,000 years by rising sea levels after the last ice age, noting that “the technological rank of these regions more than 10,000 years later matches up exactly with their initial population ranks at the end of the last ice age” (Sec. 5.3, pp. 1095-1096). On the other side, the chapter argues that the negative coefficient on population growth found in cross-country growth regressions such as Mankiw-Romer-Weil (1992) is “not inconsistent with models in which weak scale effects play a role,” because it is capturing ordinary Solow-style investment-dilution transition dynamics that operate in any growth model, idea-based or not (Sec. 5.3, p. 1097).

Using Lee’s (1988) log-linear model – in which output depends on land, labor, and ideas, idea production depends on (the log of) population, and population growth is Malthusian in the gap between output per worker and a subsistence level – the chapter shows the system generates “double exponential” growth, i.e. a growth rate that itself grows exponentially over very long historical time, which “mathematically… allows the model to deliver a graph that looks approximately like” the actual history of world per-capita income (Sec. 5.4, pp. 1098-1100). The chapter credits scale effects as the mechanism that “gets growth started,” with the intuition that “thousands and thousands of years ago, the population was relatively small and the productivity of the population at producing ideas was relatively low,” so that each new idea raised population further via higher income, which then produced further ideas sooner, in “a virtuous circle” that accelerates until a demographic transition sets in and growth rates level off (Sec. 5.4, p. 1100).

Citing Jones (2002b), the chapter reports that “approximately 80 percent of U.S. growth in the post-war period is due to transition dynamics associated in roughly equal parts with increases in educational attainment and with increases in world R&D intensity[;] only about 20 percent of U.S. growth is attributed to the scale effect associated with population growth in the idea-generating countries” (Sec. 6.1, p. 1102). This finding is reported as surprising precisely because U.S. growth rates have been so stable, which might have suggested the economy is already near its balanced growth path; the chapter’s resolution is that the Solow-model analogy of “an investment rate… grow[ing] exponentially” can sustain a constant elevated growth rate for an extended period even while transition dynamics, not the balanced-growth-path scale effect, are doing most of the work (Sec. 6.1, pp. 1102-1103).

The chapter explains that “all growth models that exhibit steady-state growth ultimately rest on an assumption that some differential equation takes the form Ẋ = X” – i.e., a knife-edge linearity requirement – whether that equation governs capital (AK models), human capital (Lucas 1988), the stock of ideas (first-generation idea-based models), or population (Sec. 6.2, pp. 1103-1104). Rather than treating this as fatal, the chapter argues “one can see the linearity critique as an opportunity… [to] look for an economic explanation for why linearity should hold,” and concludes that “population growth is the least objectionable place to locate a linear differential equation in a growth model,” since holding individual fertility behavior fixed, doubling the population quite naturally doubles the number of offspring – unlike the comparable claims required of a capital-share-of-one AK model or of human-capital returns that must stay proportional to the existing stock (Sec. 6.2, pp. 1105-1106).

The chapter poses three open questions: first, “What is the shape of the idea production function?” – the chapter is candid that the standard Cobb-Douglas specification “is elegant, but at this point we have little reason to believe that it is correct,” since research productivity need not move smoothly with the existing stock of ideas. Second, “What is the long-run elasticity of output per worker with respect to population?” – the parameter the chapter labels γ, “crucially related to the long-run rate of growth of the economy,” which the chapter says has not yet been pinned down precisely. Third, a policy question: “What are better institutions and policies for encouraging the efficient amount of research?” – noting that “none of these individual studies [of social returns to R&D] is especially compelling, and more accurate estimates of these gaps would be valuable” (Sec. 7, pp. 1106-1107).