A Contribution to the Theory of Economic Growth

Solow argues that the Harrod-Domar model’s instability conclusion – that even in the long run “the economic system is at best balanced on a knife-edge of equilibrium growth” – flows specifically from its “crucial assumption that production takes place under conditions of fixed proportions,” with “no possibility of substituting labor for capital in production” (Introduction, p. 65-66). He is explicit about what counts as a crucial versus an incidental assumption: “a ‘crucial’ assumption is one on which the conclusions do depend sensitively, and it is important that crucial assumptions be reasonably realistic” (p. 65). Solow’s diagnostic strategy is to keep every other Harrod-Domar premise – a single composite commodity, a constant savings ratio, an exogenously growing labor force – and change only the production technology, to see whether the instability survives.

The model has one composite output Y(t), a constant savings ratio s (so sY is the flow of investment, K-dot = sY), an exogenously growing labor force L(t) = L0e^(nt), and a constant-returns-to-scale neoclassical production function Y = F(K,L) (Section II, pp. 66-68). Substituting the labor path into the accumulation identity gives K-dot = sF(K, L0e^(nt)), one differential equation in the single unknown K(t). Converting to the capital-labor ratio r = K/L and using constant returns to divide through by L, Solow derives what becomes the paper’s central tool: r-dot = sF(r,1) - nr (equation 6, Section III, p. 69) – the rate of change of the capital-labor ratio equals the (scaled) contribution of saving/investment to capital per worker, minus the dilution of existing capital per worker caused by labor-force growth at rate n.

In Figure I, the function sF(r,1) is drawn convex and rising from the origin (as with a Cobb-Douglas function), against the ray nr; where they intersect (r), r-dot = 0 and capital, labor, and output all grow at the common rate n.* Away from r*, the mechanism is self-correcting: “to the right of the intersection point, when r > r*, nr > sF(r,1) … r will decrease toward r*. Conversely if initially r < r*… r will increase toward r*” (p. 70). “Whatever the initial value of the capital-labor ratio, the system will develop toward a state of balanced growth at the natural rate” – the opposite of Harrod-Domar’s knife-edge, in which any deviation from exact parameter balance is not self-correcting but self-reinforcing.

No – Solow is explicit that “the strong stability shown in Figure I is not inevitable” (p. 71). Figure II depicts a production function generating three intersections with the ray nr: r1 and r3 are stable, but the middle equilibrium r2 is unstable, so that “any accidental disturbance will be magnified over time” and which of the two stable paths the economy reaches depends on whether the initial capital-labor ratio starts above or below r2 (p. 71). Figure III goes further, showing production functions for which no balanced-growth equilibrium exists at all: one curve lies wholly above nr (capital-labor ratio and output per head grow without limit under full employment), the other lies wholly below (income per capita falls without limit even though aggregate income still rises) (p. 72). The paper’s basic conclusion is thus narrower than “growth is always stable” – it is that once variable proportions are allowed, a knife-edge is no longer a logical necessity, though pathological production functions can still produce multiple or absent equilibria.

Example 1 recovers the Harrod-Domar fixed-proportions case itself (Y = min(K/a, L/b)) and works out three sub-cases depending on whether the natural rate n is greater than, equal to, or less than the warranted rate s/a: when n exceeds s/a, capital is the bottleneck and unemployment develops and grows over time; when n is below s/a, capital becomes redundant with a growing absolute excess of capacity even though the marginal product of capital in equilibrium is exactly zero (pp. 73-76). Example 2, the Cobb-Douglas function Y = K^a*L^(1-a), always produces a unique stable equilibrium r* = (s/n)^(1/(1-a)) regardless of parameter values, and the capital coefficient K/Y converges to exactly s/n – so the natural and warranted rates become equal “not as an odd piece of luck but as a consequence of demand-supply adjustments” (pp. 76-77). Example 3, the CES-type function Y = (a1*K^(1/2) + L^(1/2))^2, shows a genuine either/or: if sa1^2 > n there is no equilibrium and the capital-labor ratio (and output per head) grows without limit; if sa1^2 < n a stable equilibrium exists (pp. 77-78).

Real factor prices are determined by ordinary marginal-productivity conditions (partial-F/partial-L = w/p, partial-F/partial-K = q/p) and move smoothly, not catastrophically, toward their equilibrium values as r moves toward r* (Section V, pp. 78-84). Solow shows the real wage and real rental adjust “in general … not indefinite,” and explicitly credits John Chipman with the observation that this result “directly contradicts Harrod’s position that a perpetually falling rate of interest would be needed to maintain equilibrium” (p. 83) – catastrophic factor-price movements, Solow argues, are a special consequence of the fixed-proportions assumption (visible starkly in the Harrod-Domar Figure IV, where factor prices are indeterminate except at exactly one capital-labor ratio), not a general feature of growth.

Neutral technical change is modeled as a multiplicative scale factor, Y = A(t)F(K,L), defined as a shift that “leaves marginal rates of substitution untouched” while raising attainable output at every capital-labor ratio (Section VI, p. 85). In the Cobb-Douglas case with A(t) = e^(gt), the long-run growth rate of capital rises from n to n + g/(1-a), and of output to n + ag/(1-a) – both faster than population growth alone would generate, and the capital-labor ratio “never reaches an equilibrium value but grows forever,” eventually at rate g/(1-a) (pp. 85-86). The relative share of labor stays constant at 1-a even as technical progress proceeds, a special property of the Cobb-Douglas form (p. 86).

No – Solow shows each of these extensions preserves a stable balanced-growth path, though it may shift where the equilibrium capital-labor ratio sits. With a wage-elastic labor supply of the form L = L0e^(nt)(w/w0)^h, the Cobb-Douglas equilibrium capital-labor ratio turns out to be unaffected, “a consequence of the special supply-of-labor schedule,” though this need not hold for other specifications (pp. 86-87). Making the savings ratio s(r) a decreasing function of the yield on capital “tends to be stabilizing: when the capital-labor ratio is high, saving is cut down; when it is low, saving is stimulated,” though “there is still no possibility of a stationary state: should r get so high as to choke off saving and net capital formation, the continual growth of the labor force must eventually reduce it” (pp. 87-89). A proportional income tax simply rescales the effective savings ratio depending on how the government spends or invests the proceeds (p. 89-90).

When the rate of population growth is itself made a function of the capital-labor ratio (via per capita income), the ray nr in the diagram is twisted into a curve, and multiple equilibria can appear even with a perfectly ordinary, single-valued production function (Section VI, “Variable Population Growth,” pp. 90-91). In Solow’s illustrative Figure IX, a low equilibrium r1 is stable, and a higher r2 is unstable: “if the initial ratio could somehow be boosted above the critical level r2, a self-sustaining process of increasing per capita income would be set off … small-scale capital accumulation only leads back to stagnation but a major burst of investment can lift the system into a self-generating expansion of income and capital per head” (p. 91) – a threshold effect arising purely from the population-growth mechanism, without any indivisibility or increasing returns in production itself.

Solow is explicit that “everything above is the neoclassical side of the coin … full employment economics,” and that the paper does not deny the importance of Keynesian causes of unemployment or excess demand – only that these should be attributed “less readily to any deviation from a narrow ‘balance’” (Section VII, p. 91). He briefly sketches how rigid real wages can produce sustained unemployment or labor shortage (depending on whether the implied growth rate falls short of or exceeds n), how a liquidity-trap-like infinitely elastic demand for idle balances can rigidify the interest rate and cause capital underutilization, and notes that the whole apparatus assumes away risk and uncertainty, concluding that “no credible theory of investment can be built on the assumption of perfect foresight and arbitrage over time,” a simplification he considers “perhaps justifiable” only “in the context” of this exercise (pp. 91-94).