A Contribution to the Theory of Economic Growth

Robert Solow’s 1956 paper argues that the Harrod-Domar model’s famous conclusion – that steady economic growth is only ever balanced on a “knife-edge,” liable to tip into growing unemployment or prolonged inflation – follows specifically from its assumption that capital and labor must be combined in fixed proportions, with no possibility of substitution between them. Solow keeps every other Harrod-Domar assumption (a single composite commodity, a constant savings ratio s applied to output Y, an exogenously growing labor force L(t) = L0e^(nt)) but replaces fixed-coefficient technology with a standard neoclassical production function Y = F(K,L), homogeneous of degree one. Substituting the labor-force path into the savings identity and converting to the capital-labor ratio r = K/L yields a single first-order differential equation, r-dot = sF(r,1) - nr, whose qualitative behavior can be studied graphically by comparing the curve sF(r,1) against the ray nr. Under the “normal” case illustrated in Figure I (essentially the Cobb-Douglas case), this equation has a unique, globally stable equilibrium capital-labor ratio r*: starting from any positive initial ratio, the economy converges to balanced growth in which capital, labor, and output all expand at the labor force’s natural rate n, so that “no simple opposition between natural and warranted rates of growth is possible.” Solow is careful to show this stability is not automatic for every conceivable production function – Figure II exhibits a case with three equilibria (two stable, one unstable, so initial conditions determine which stable path is reached) and Figure III exhibits cases with no equilibrium at all, where the capital-labor ratio either grows or shrinks without bound. Three worked examples (fixed-proportions/Harrod-Domar, Cobb-Douglas, and a two-parameter constant-elasticity-of-substitution family) make the algebra explicit, including the specific redundant-labor and redundant-capital sub-cases that arise in the fixed-proportions case when the natural and warranted rates diverge. Section V shows that competitive factor prices (real wage and real rental) adjust smoothly along the way to equilibrium – directly contradicting a claim by Harrod that a perpetually falling interest rate would be needed to sustain balance. Section VI extends the model to neutral technical progress (which raises the asymptotic growth rate above n), a wage-elastic labor supply, a capital-yield-dependent savings ratio, income taxation, and endogenous (income-dependent) population growth, the last of which can produce a low unstable threshold capital-labor ratio separating permanent stagnation from self-sustaining growth. Section VII explicitly limits the claim to a frictionless, full-employment “neoclassical side of the coin,” noting that rigid real wages, liquidity-trap-like asset preferences, and the general absence of perfect foresight can each still generate unemployment or excess capacity through familiar Keynesian channels.

Technical Change and the Aggregate Production Function

Robert Solow’s 1957 paper proposes a simple method for separating shifts in the aggregate production function (“technical change,” broadly defined) from movements along it caused by capital accumulation, and applies it to U.S. private non-farm output from 1909-1949, finding that seven-eighths of the doubling in output per worker-hour is attributable to technical change and only one-eighth to increased capital per worker. Starting from an aggregate production function Q=F(K,L;t), Solow specializes to the case of neutral technical change, Q=A(t)f(K,L), where neutrality means the shift “leaves marginal rates of substitution untouched” and simply scales output at any given capital-labor ratio; under the standard (and, he argues, practically unavoidable) assumption of constant returns to scale and competitive factor markets paying marginal products, this yields a simple decomposition of the growth rate of output per worker, q-dot/q, into a technical-change term A-dot/A and capital’s income share times the growth rate of capital per worker, w_k*(k-dot/k) – requiring, to estimate it, only time series of output per worker, capital per worker, and capital’s share of income, and one new assumption (competitive factor pricing), without needing to specify the exact functional form of the production function. Applying this to U.S. private non-farm GNP per man-hour, an estimate of the capital stock (Goldsmith’s data, crudely corrected for unemployment but not for wartime multi-shift operation) and factor-share data for 1909-1949, Solow reconstructs the cumulative shift factor A(t) year by year; a scatter of the year-to-year technical-change term against the capital-labor ratio shows essentially no relationship, so he concludes technical change over the period was, on average, neutral, though the average annual rate of shift roughly doubled between the first and second halves of the sample (about 1 to 1.2 percent per year before 1929 versus roughly 2 percent per year after 1930). The paper’s headline growth-accounting result compares the near-doubling of output per man-hour ($0.623 to $1.275) against the roughly 80 percent cumulative rise in A(t): correcting the 1949 output figure for the estimated technical-change factor implies that about one-eighth of the 40-year increase in output per hour is attributable to increased capital intensity and the remaining seven-eighths to technical change broadly defined. Dividing the resulting technical-change-corrected output-per-worker series by A(t) and plotting it against capital per worker (Chart 4) reveals a scatter with a distinct, though not violent, curvature consistent with diminishing returns; several two-parameter curves (including Cobb-Douglas, semi-logarithmic, and others with upper asymptotes) fit this corrected scatter about equally well, with the linear specification performing noticeably worse, and the data show no sign of approaching capital saturation within the observed range. Solow flags a cluster of wartime and postwar observations (1943-1949) as anomalously high relative to the rest of the scatter, likely reflecting underestimated capital utilization from unmeasured multi-shift wartime operation, and, after experimentation, excludes these years from the regressions reported in the paper’s tables.

Robert M. Solow | Macro Paper Warehouse

A Contribution to the Theory of Economic Growth

Technical Change and the Aggregate Production Function