Endogenous Technological Change

The three premises are that technological change – improved instructions for combining raw materials – lies at the heart of economic growth; that this change arises largely from intentional actions by people responding to market incentives; and that instructions for working with raw materials, once created, can be reused at no additional cost, unlike other economic goods (Introduction, pp. S71-S72). Section II shows that once all three premises are granted, an equilibrium with price-taking behavior cannot be supported: if a nonrival input A is productive, the production function F(A,X) cannot be concave, since a standard replication argument implies F(λA,λX) > λF(A,X) for λ>1, and a firm paying every input its value marginal product would run losses (Sec. II, pp. S73-S76).

Rivalry is a purely technological attribute – a purely rival good’s use by one person precludes its use by another, while a purely nonrival good’s use by one does not limit another’s use at all – whereas excludability depends on both technology and the legal system, since a good is excludable if its owner can prevent others from using it (Sec. II, p. S73). A design for a new good is nonrival because, once created, it can be used in as many productive activities as desired, in contrast to a piece of human capital (such as the ability to add), which is rivalrous because it is tied to a single physical body that can be in only one place at a time. A design is only partially excludable: a patent lets the inventor exclude others from manufacturing the specific patented good, but the inventor “has no ability to stop the inventor of a wodget from learning from the design of a widget” – so the benefit of a design to future researchers is completely nonexcludable even though its direct productive use is excludable (Sec. II, pp. S74-S85).

Solow (1956) and Shell (1966, 1967) treated the technology index as an exogenous or government-provided public input that is nonexcludable and receives no compensation, which is inconsistent with the premise that technological change results from intentional, profit-motivated investment; Arrow’s (1962) learning-by-doing model made knowledge accumulation an unintentional side effect of physical-capital accumulation, forcing a strict, unexplained proportionality between the growth of knowledge and of capital. Romer’s own 1986 model made the same kind of assumption to keep the dynamic analysis to a single state variable, so it too ends up with the growth of knowledge moving in lockstep with capital (Sec. II, pp. S76-S78). The model in this paper instead makes the accumulation of designs an intentional investment decision, decoupled from physical-capital accumulation, and resolves the resulting nonconvexity through monopolistic competition rather than through an unexplained public-good assumption (Sec. II, p. S78).

The economy has a research sector that combines human capital devoted to research with the existing stock of designs A to produce new designs; an intermediate-goods sector in which each firm that owns a design converts forgone output into units of a specific, patented producer durable and rents them out; and a final-goods sector that combines labor, human capital, and the available range of producer durables to produce output that can be consumed or saved as capital (Sec. III, pp. S78-S80). Final output is Cobb-Douglas in labor and human capital devoted to final-goods production, and additively separable across the (continuum of) available durables, so that – unlike the conventional model, in which all capital goods are perfect substitutes – one additional unit of one type of durable has no effect on the marginal productivity of another type (Sec. III, eq. 1, pp. S79-S81).

Once a firm owns a design, it is the sole supplier of that durable and faces a downward-sloping demand curve; given a constant elasticity of demand determined by the parameters of the final-output production function, the firm sets a constant markup over marginal cost, and free entry into the business of producing designs bids the price of a new design up to exactly the present discounted value of the monopoly profit stream it will generate (Sec. III, eqq. 4-6, pp. S85-S87). Differentiating the resulting zero-profit condition shows that at every instant the flow of monopoly profit net of the interest cost on the initial design investment must be exactly zero in present-value terms (Sec. III, eq. 6’, p. S87).

On the balanced growth path, the common growth rate of output, capital, and the stock of designs equals the productivity of research (delta) times the amount of human capital devoted to research, and this in turn is pinned down by the equation g = (delta)H - Ar, where A is a constant depending on the technology and markup parameters and r is the interest rate – notably, the size of the labor force L and the unit cost of manufacturing durables do not appear (Sec. V-VI, eq. 11, pp. S90-S93). Although an increase in L or a fall in the unit capital cost both raise the flow of monopoly profit a new design can earn, they simultaneously raise the return to human capital in manufacturing by exactly the same amount, so the two effects “exactly cancel” for the functional forms used and the allocation of human capital between research and manufacturing – and hence the growth rate – is unaffected (Sec. VI, pp. S93-S94). Romer explicitly notes this exact cancellation “is not a robust feature of the model” and that in a companion model (Romer 1990, a different paper) an increase in L can even reduce research effort (p. S94).

In contrast to Romer’s earlier (1987) model, where an increase in scale measured by population L increased the growth rate, in this model an increase in L has no effect on growth, while an increase in the total stock of human capital H unambiguously speeds growth, because the research technology exhibits increasing returns in human capital specifically (a doubling of both human capital and the existing stock of knowledge more than doubles the marginal product of human capital in research) (Sec. VI, pp. S94-S96, Fig. 2). If the total stock of human capital H is too small, the nonnegativity constraint on human capital devoted to research binds and growth does not take place at all, since every feasible growth rate for designs is too low relative to the discount rate to justify sacrificing current output – an outcome Romer likens to explanations offered by historians for the absence of growth in prehistoric times, when no human capital could be spared from production for immediate consumption (Sec. VI, p. S96).

Two independent reasons make equilibrium human capital devoted to research too low: first, an additional design raises the productivity of all future researchers, a benefit that is completely nonexcludable and therefore uncompensated by the market price of designs; second, a design is purchased by a manufacturing sector that engages in monopoly pricing, so the price a researcher’s design commands captures only a fraction (1 - alpha - beta of the relevant markup parameters) of the design’s true social marginal product (Sec. VI, pp. S96-S97). Comparing the market equilibrium’s balanced growth rate to the solution of an explicit social planning problem (worked out in the Appendix) shows that correcting both distortions – replacing the coefficient that reflects the monopoly markup with one reflecting only the true externality, and replacing 1 with a smaller constant reflecting foregone monopoly rents – raises the socially optimal allocation of human capital to research and hence the socially optimal growth rate above the equilibrium rate (Sec. VI, eqq. 13-14, pp. S97-S98).

Because this model uncouples the decision to invest in physical capital from the decision to invest in research – unlike Arrow’s (1962) learning-by-doing model and Romer’s own 1986 model, in which the growth rate of knowledge was forced by assumption to equal the growth rate of physical capital, so that any policy (such as an investment tax credit) that raised capital accumulation necessarily raised knowledge accumulation too – a subsidy to physical capital here has no such automatic effect on research. Romer states plainly that “if the fundamental policy problem is that we have too many lawyers and MBAs and not enough engineers, a subsidy to physical capital accumulation is a weak, and possibly counterproductive, policy response,” and that the model’s most robust welfare conclusion instead is that the rate of technological change is sensitive to the interest rate: any policy change that lowers the interest rate (more patience, or a higher intertemporal elasticity of substitution) speeds growth by raising the present value of a research investment’s future payoff (Sec. VI, pp. S93-S95, S98).

Comparing two identical closed economies operating in isolation with what would happen if they were fully integrated, the model implies that integration raises the effective stock of human capital available for research (replacing each country’s H with the combined worldwide total), raising both the share of human capital devoted to research and the common growth rate – which is why a populous but poorly integrated economy such as China or India cannot simply substitute a large domestic population for participation in world markets (Sec. VII, pp. S98-S99). As supporting historical evidence, Romer cites Sokoloff’s (1988) finding that U.S. counties with access to navigable waterways in the early nineteenth century had higher rates of patenting, and that the construction of a new canal or dredging of a river was followed by a sharp increase in patenting in adjacent counties, together with Sokoloff and Khan’s (1989) finding of a fairly elastic short-run supply of people moving in and out of research activity in response to aggregate disturbances (Sec. VII, p. S99).

For a fixed level of the technology index A, the model behaves almost identically to the Solow model with labor- and human-capital-augmenting technological change, exhibiting the usual diminishing returns to capital accumulation and converging to a steady state (or, with exogenous growth in A, to a balanced path in which capital grows at the same rate as A); the intuition that A can grow at a constant endogenous rate instead comes from the Uzawa model, in which the growth of technology depends on how resources are allocated between a research sector and a final-goods sector (Sec. IV, pp. S88-S90). Romer’s contribution is to show that a balanced growth equilibrium of this Solow/Uzawa hybrid type exists once the market for designs is modeled with monopolistic competition rather than with an exogenous or price-taking treatment of technology (Sec. IV-V, pp. S89-S92).

The model holds population, the total supply of labor, and the total stock of human capital fixed; assumes research uses only human capital and existing knowledge (not labor or capital); and assumes the production of new designs is linear in the existing stock of knowledge A (Sec. III, pp. S79-S80). Romer is explicit that this last assumption – linearity in A – “is what makes unbounded growth possible, and in this sense, unbounded growth is more like an assumption than a result of the model”: if the productivity of research depended on some concave (rather than linear) function of A, the marginal product of human capital in research would eventually fall relative to manufacturing, human capital would shift out of research, and growth would slow. Romer defends the linear specification not as an empirical finding but because “there is no evidence from recent history to support the belief that opportunities for research are diminishing,” while acknowledging that whether research opportunities will eventually peter out is “an empirical question that this kind of theory cannot resolve” (Sec. III, p. S84).