Increasing Returns and Long-Run Growth

Romer replaces the standard assumption of diminishing returns to capital with an assumption that knowledge, treated as a capital good, has increasing marginal productivity, which reverses the standard model’s predictions of convergence and vanishing per capita growth. The Ramsey-Cass-Koopmans model implies that the rate of return on investment and the growth rate of per capita output are decreasing functions of the capital stock, that wage rates and capital-labor ratios across countries should converge, and that initial conditions or current disturbances have no long-run effect on the level of output (Introduction, p. 1002). Romer’s model instead allows per capita output to “grow without bound, possibly at a rate that is monotonically increasing over time,” allows the level of per capita output in different countries to fail to converge, and does so “without depend[ing] on any kind of exogenously specified technical change or differences between countries” – preferences and technology are stationary and identical, and even population size can be held constant (pp. 1002-1003).

Three elements combine to produce a well-specified model: an externality (knowledge cannot be perfectly patented or kept secret, so one firm’s research raises other firms’ production possibilities), increasing returns in the production of consumption output as a function of the aggregate stock of knowledge, and decreasing returns in the production of new knowledge itself (p. 1003). The externality is essential for a competitive equilibrium to exist at all despite increasing returns; the diminishing returns in research are required to ensure that consumption and utility do not grow “too fast” for a finite-valued optimization problem to have a solution (p. 1004).

The paper traces the idea to Adam Smith’s pin factory and Alfred Marshall’s internal/external economies distinction, credits Allyn Young’s 1928 address with the key insight, but notes that Frank Knight (a student of Young’s) criticized externalities based on specialization as an “empty economic box,” since increased specialization “opens new markets and introduces new goods” rather than creating a pure technological externality (Sec. II, pp. 1004-1005). Arrow’s (1962) learning-by-doing model, and its elaborations by Levhari and Sheshinski, avoided the finiteness problem by assuming output is increasing-returns-to-scale in capital and labor jointly but that the marginal product of capital is still diminishing given a fixed labor supply – which implies the empirically questionable prediction that per capita output growth is monotonically increasing in population growth, and that growth must go to zero with zero population growth. Romer’s model instead assumes knowledge itself, not the joint capital-labor aggregate, has increasing marginal product, and relies on diminishing returns in the research technology rather than a bound on labor growth to keep the optimization problem well-defined (pp. 1005-1007).

Romer presents four kinds of evidence consistent with increasing rather than diminishing returns, while explicitly cautioning that none of it is decisive. First, using Maddison’s (1982) estimates for the three successive productivity “leader” countries since 1700 (the Netherlands, the United Kingdom, and the United States), the annual compound growth rate of GDP per man-hour rises from essentially zero in eighteenth-century Netherlands to 2.3 percent per year in the United States since 1890 (Table 1, p. 1009). Second, decade-by-decade U.S. per capita GDP growth from 1800 to 1978 shows a rising rather than falling pattern (Tables 2-3, Fig. 1, pp. 1009-1011). Third, a nonparametric test for trend across eleven countries’ growth-rate histories rejects a nonpositive trend at the 5 percent level for five of eleven countries and at the 10 percent level for eight of eleven (Table 3, p. 1011), and there is no clear evidence of convergence in the dispersion of per capita income across all countries, only within already-industrialized groups (pp. 1012-1013, citing Streissler 1979 and Baumol 1985). Fourth, growth-accounting exercises (citing Kendrick 1976) repeatedly find that measured input growth cannot account for measured output growth over 1929-69, with output growing 1.06 to 1.30 times faster than a weighted measure of input growth – evidence Romer describes as “subject to substantial, unquantified uncertainty” and not “decisive support,” but consistent with ruling out constant-returns production functions (p. 1013).

Romer characterizes the competitive equilibrium not by solving a social planning problem but by defining, for each candidate aggregate knowledge level K, an ordinary concave “restricted” maximization problem P(K) that a representative firm and consumer would solve taking K as given, and then finding a fixed point K* at which the aggregate outcome implied by solving P(K*) equals K* itself. In the two-period setup, each of S identical consumers chooses first-period consumption and an investment in knowledge k to maximize utility over two periods, taking the aggregate stock of knowledge K = sum of all firms’ ki as given; because knowledge cannot be patented, individual firm output depends on both firm-specific knowledge ki and the economywide aggregate K (Sec. IV, pp. 1014-1016). Since collusion among firms to jointly overinvest in research “cannot be supported for the same reasons that collusive agreements fail in models without externalities” (each firm has an incentive to free-ride), the resulting competitive equilibrium is suboptimal (p. 1016). The paper shows formally that any fixed point of the mapping from K to the solution of P(K) can be supported as a competitive equilibrium with externalities, using standard Kuhn-Tucker shadow prices from the concave problem P(K) as market prices (pp. 1016-1018).

The paper assumes that a firm’s rate of growth of its own knowledge, k-dot = G(I,k), is homogeneous of degree one and can be rewritten as a proportional growth rate k-dot/k = g(I/k), where g is concave, normalized so that Dg(0) = 1, and – crucially – bounded above by a constant α no matter how large the investment-to-knowledge ratio I/k becomes (Sec. V.A, pp. 1018-1019). This “strong form of diminishing returns in research” is what bounds the maximal technologically feasible rate of growth of knowledge and hence of per capita output; Romer argues this is a plausible feature of actual research even though the level of production exhibits increasing, not diminishing, returns to knowledge – illustrated by the remark that even devoting a large research effort to nuclear fusion “could not be produced by next year regardless of the size of the current research effort” (p. 1020).

Theorem 1 establishes that, given the bound on the research technology (g bounded by α) and a bound on the growth of the socially feasible production function (relating the exponent of that bound, denoted ρ, to the discount rate δ via αρ < δ), both the social planning problem and the family of restricted problems P(K) have finite-valued solutions (Sec. V.B, p. 1021). Theorem 2 adds a further asymptotic condition – using the “asymptotic exponent” concept of Brock and Gale (1969) – under which a suboptimal competitive equilibrium with externalities exists in which both consumption and knowledge grow without bound, provided the initial stock of knowledge exceeds a threshold at which the private marginal product of knowledge exceeds the discount rate (Sec. V.C, pp. 1023-1025). The proofs of both theorems are given in Romer’s 1983 dissertation and a companion 1986 Econometrica paper rather than in full in this article (pp. 1021, 1024).

Because each firm perceives only its own private marginal product of knowledge and ignores the positive externality it confers on other firms, the competitive equilibrium has too much consumption and too little research investment relative to the social optimum; any government intervention that shifts current resources from consumption toward research is welfare-improving (Sec. V.D, pp. 1025-1026). With access to lump-sum taxation, the government can restore the optimum with a subsidy on knowledge holdings equal to the externality term the private sector neglects, or – in the special production form f(k,K) = kᵛKᵞ – with a constant subsidy rate on research investment equal to γ/(v+γ) (p. 1026). Romer also notes the subtler point that interest rates need not be higher in the socially optimal path than in the suboptimal one: under linear utility the two paths can have identical interest rates despite different marginal products of knowledge, because capital gains on knowledge holdings adjust to compensate; but with any curvature in utility, interest rates in the optimum will typically be higher, implying cost-benefit analysis in a suboptimal economy should use a social discount rate above the market interest rate (pp. 1026-1027).

With logarithmic utility, Romer shows geometrically (using a phase-plane argument) that a competitive equilibrium trajectory exists that stays trapped between the k-dot=0 and lambda-dot=0 loci and converges toward what behaves like “a saddle point… moved infinitely far to the right,” so that both knowledge and consumption grow without bound, though the exact asymptotic growth rate cannot be pinned down in closed form because of the algebraic complexity (Sec. VI.A, pp. 1027-1029). This example establishes existence of an unbounded-growth equilibrium concretely but does not, in this specific case, allow Romer to show that the growth rate itself is monotonically increasing.

With linear utility, the equilibrium trajectory is upward sloping and behaves asymptotically like k^(v+γ-1), so that consumption, investment, and the investment-to-knowledge ratio all go to infinity, driving the growth rate of knowledge toward its technological upper bound α(v+γ) – meaning the percentage growth rate of output and consumption is itself increasing over time toward this asymptote. (Sec. VI.B, pp. 1029-1031). Romer then perturbs this equilibrium with a foreseen future exogenous increase in aggregate knowledge at a future date T: because the shadow price of knowledge must be continuous through time under arbitrage, the anticipated future increase causes an immediate jump in the value of knowledge today and a corresponding immediate increase in current research investment, and “because of the increasing returns, the private response to an aggregate increase in the stock of knowledge will be to reinforce its effects rather than to dampen them” – so that small, even purely anticipated, disturbances can have permanent, growing aggregate consequences, a result Romer shows depends on marginal utility not falling too quickly (it would not hold under sufficiently curved bounded utility) (pp. 1031-1032).

Extending Example 2 to two closed economies with no trade, Romer shows that even if both start with identical initial knowledge stocks, the country with the larger initial stock grows persistently faster in percentage terms, so that the level gap between the two economies’ knowledge and consumption paths goes to infinity even as both growth rates converge to the same asymptotic bound α. Relaxing the assumption of closed economies, Romer considers two countries linked by a freely mobile composite knowledge/capital good, where each country’s production depends more strongly on its own domestic aggregate knowledge than on the foreign aggregate (exponent a on the domestic aggregate exceeds exponent b on the foreign aggregate); equalizing the private marginal product of the composite good across borders admits, in addition to the symmetric solution, an asymmetric equilibrium in which the composite good flows steadily toward the initially more developed country – a form of ongoing “capital flight” or “brain drain” that requires no fundamental difference in technology between the two countries and that can be triggered purely by agents’ beliefs about which country is destined to be the slow-growing one (Sec. VI.C, pp. 1032-1034).

Romer is explicit that the model is deliberately restricted to a single state variable (knowledge), with population and physical capital held fixed, chosen specifically because this permits a geometric phase-plane analysis that would be “impossible with more than one state variable,” where numerical methods would instead be required (Sec. VII, p. 1034). He frames the model as filling “part way” a theoretical gap left by earlier incomplete treatments that took technological change as exogenous or addressed it only descriptively, but stresses that incorporating both increasing marginal productivity of knowledge and decreasing marginal productivity of physical capital simultaneously is a natural and tractable extension left for future work, and that the specific functional forms used in Section VI’s examples (particularly the choice between logarithmic and linear utility) determine which qualitative behaviors – accelerating growth, amplified disturbances, cross-country divergence – actually appear (pp. 1034-1035).