Economic Growth and Capital Accumulation

Swan states his aim as being “to illustrate with two diagrams a theme common to Adam Smith, Mill, and Lewis… namely, the connexion between capital accumulation and the growth of the productive labour force” (Section 1, p. 334), explicitly situating the exercise within the classical tradition even though “our illustration takes a neo-classical form.” In a footnote to the growth diagram, Swan notes that “a similar device is used by R. M. Solow (‘A Contribution to the Theory of Economic Growth,’ Quarterly Journal of Economics, February, 1956)” (p. 337, fn. 6) – the two papers were developed independently and close together in time, which is why the resulting framework is now conventionally called the Solow-Swan model.

With capital and labour as the only two factors under a constant-elasticity production function Y = K^a*N^b (a+b=1), logarithmic differentiation gives the basic growth-accounting formula y = as(Y/K) + bn, where y is output’s relative growth rate, s the saving ratio, and n labour’s relative growth rate (Section 2, p. 335). In Figure 1, the growth line of capital is a ray of slope s through the origin (since capital’s growth rate is s times the output-capital ratio); the growth line of labour is horizontal at height n; and the growth line of output lies between them, dividing the vertical distance in the proportion a:b. Because a+b=1, all three lines must intersect at a single point, where growth in each case equals n – and, crucially, the position of this intersection along the horizontal (output-capital-ratio) axis, not the common growth rate, is what a higher saving ratio shifts (p. 336).

Only the level. Swan traces the transition from a 5 percent to a 10 percent saving ratio: the growth line of output shifts, and “output per head begins to improve” as the economy moves to a new equilibrium with a permanently higher output-capital ratio and wage rate, but the improvement occurs “at an ever-slackening pace” during the transition and the long-run growth rate returns to the same value, n, as before – “this conclusion is not really surprising. It is in fact the counterpart in our present unclassical model of the classical proposition that capital accumulation leads ultimately to the stationary state” (p. 338). Swan flags the numerical result as “at first sight disconcerting”: in his example, doubling the saving ratio at a point where capital yields 8 percent adds only 0.4 percentage points to the growth rate at the moment of the change (p. 338).

A constant rate of “neutral” technical progress (m percent per year) shifts the growth line of output upward by m, producing a new equilibrium point at which the output-capital ratio and profit rate are permanently higher, and – the key qualitative change – output per head is not just permanently higher but perpetually rising (p. 337). Its rate of rise actually exceeds m, “because the contribution of capital is also sustained by technical progress at a higher level” (p. 337). Swan also works out the case where population growth itself responds to a demand for progressively rising living standards (a target improvement rate q in output per head): this is formally equivalent to reducing the effective rate of technical progress by q, and the resulting family of growth lines (indexed by q) can be read as “a grid that divides the economic map into characteristic zones of improvement or determination in output per head” rather than literal time paths (pp. 341-342). A higher saving ratio, in this reading, does not change growth rates at the equilibrium points but shifts the whole grid so as to enlarge the region of the map corresponding to rising output per head (p. 342).

With land as a third factor of fixed supply and production elasticity gamma (so a+b+gamma=1, a+b<1), land drops out of the growth formula itself but reduces the sum a+b below unity, so the growth line of output now cuts the growth line of capital below the horizontal growth line of labour (p. 340). Swan defines the Ricardian line as the locus of (labour-growth-rate, output-capital-ratio) pairs consistent with constant output per head, obtained by setting y=n in the growth formula. “Except at the origin, the growth line of capital lies above the Ricardian line, for capital must always grow faster than labour in order to sustain output per head in the face of continually diminishing returns on the land” – but since capital is then also growing faster than output, the output-capital ratio continually falls, and “the natural progress of society continues indefinitely towards the origin,” where the growth line of capital and the Ricardian line finally meet in “a stationary state” (p. 340). Swan explicitly reads this as the diagrammatic counterpart of the classical prediction that a very low rate of profit eventually “arrests all accumulation” (p. 341).

A constant rate of technical progress can permanently lift the Ricardian line above the level that would otherwise gravitate to the origin, producing a stable equilibrium with a positive, constant growth rate of output per head instead of stagnation – “so instead of gravitating towards the origin, the economy if necessary levitates to this stable equilibrium point” (p. 341). Swan immediately flags the tension this creates with the underlying diminishing-returns story: if population is assumed to grow forever at whatever rate is implied (in his example, about 2.9 percent), “it is very likely that at some point diminishing returns will set in with a violence not allowed for in our production function” – i.e., the model’s own assumption of a constant land-elasticity gamma cannot literally hold forever under a Malthusian population response (p. 341).

Swan states that his model “differs from Harrod’s model of economic growth only in that it systematizes the relations between the ‘warranted’ and ’natural’ rates of growth, and introduces land as a fixed factor” (p. 342). The growth line of capital, s(Y/K), corresponds point-for-point to Harrod’s warranted rate (since the output-capital ratio is the reciprocal of Harrod’s capital coefficient Cr); the growth line of output corresponds to Harrod’s natural rate; and their intersection is the point where warranted and natural rates coincide. Swan uses this correspondence to correct what he describes as a misreading of Harrod by some readers, who took Harrod to mean that equality of the warranted and natural rates could occur only “by a fluke”: Swan quotes Harrod’s own statement that policy should aim at “a progressive reduction in the rate of interest” via a “deepening” factor d until the output-capital ratio adjusts, arguing that “the mechanism of Figures 1 and 2 merely makes explicit what this statement implies” (p. 343).

Robinson’s objection, as Swan quotes it, is that the standard neo-classical notation “O = f(L,C)” conceals an unresolved question of what unit C (capital) is measured in: measuring capital “in terms of product” is natural for discussing accumulation, but “when we consider what addition to productive resources a given amount of accumulation makes, we must measure capital in labour units,” and “one symbol, C, cannot stand both for a quantity of product and a quantity of labour time” (p. 344, quoting Robinson). Swan’s response constructs a “scarecrow” economy of homogeneous labour and land plus capital in the form of durable, freely-reassemblable “meccano sets,” in which the basic model of the text “could be rigorously established in a form that would deceive nobody” (p. 344) – and then argues that Robinson conflates two distinct senses of “capital measured in terms of product”: (1) the current value of the pre-existing capital stock, which is revalued whenever relative prices between capital goods and product change, versus (2) the cumulated historical cost of past saving and investment, which is the neo-classical tradition’s “perpetual inventory” measure; “the two measures may in fact diverge very widely” away from a stationary state, but “in a stationary equilibrium, the two measures coincide,” and for marginal comparative-statics analysis around a single equilibrium, capital can validly be treated as “an equilibrium dollar’s worth” regardless of its physical composition (pp. 348-350).

No – he concedes an important limitation. The “equilibrium dollar’s worth” argument, resting on what Swan calls “the familiar… Wong-Viner-Harrod envelope theorem,” is valid only for marginal (“virtual”) displacements around a single equilibrium point, not for comparing two different stationary states “in the large” with different factor endowments: “comparative statics” is, on these terms, “a misnomer: not different situations, but only ‘virtual’ displacements at the margin of one situation, can be considered” (p. 351). For genuine structural comparisons between two stationary states, Swan concludes that either capital must be measurable in a naturally homogeneous technical unit (as with meccano sets), or an artificial device is needed – he points to Champernowne’s proposed “chain index” of capital, and dedicates the rest of the appendix to showing how such a chain index “emerges naturally” from Wicksell’s analysis of the specific problems Robinson raised (p. 351).

Reworking Wicksell’s “point-input, point-output” model – labour N applied at a point in time yields, after a “period of production” t, a final output Q=Nf(t), with the real wage as the discounted product per unit of labour and the interest rate as the marginal productivity of waiting – Swan shows that the proportional change in the value of capital splits into a “productive” component and a “financial” (revaluation) component (Part II, pp. 352-357). The Wicksell effect is Wicksell’s finding that an increase in the social capital stock is “partly absorbed by increased wages and rent,” so that only the residual is “really effective as far as a rise in production is concerned” – in this model, increasing the value of capital always means a rising wage rate and falling interest rate, and the net effect is necessarily a rise in the value of a unit of capital in terms of product, which Swan characterizes as “an apparent absorption of capital” (Part IV, p. 359) rather than a change in physically productive capacity.

Part III reworks Wicksell’s solution to Gustaf Akerman’s problem of durable capital equipment (“axes” of a chosen optimal service life n years, forming in equilibrium a “balanced equipment” of uniform age distribution from 0 to n), deriving the Champernowne-Kahn formula for the value of such equipment as a proportion of its replacement cost, and showing the same neoclassical marginal-productivity apparatus applies once capital is measured in a “standard axe” (p. 358-359). Part IV shows that in this durable-equipment setting, unlike the earlier point-input model, the “financial” (interest-rate) component of a capital revaluation can outweigh the “productive” (wage) component, so that the value of a standard axe can actually fall even as wages rise and the interest rate falls – the Wicksell effect in reverse (p. 359-360). Swan notes this reversal left Wicksell himself “very puzzled” and forced him to concede his earlier absorption explanation was “not general” (p. 360), but argues that once the effect is understood purely as a revaluation phenomenon, there is “nothing perverse about it,” and “in general there is no presumption either way” as to its direction (p. 360) – which Swan closes by using to question Joan Robinson’s claim that Wicksell’s finding is “the key to the whole theory of accumulation and of the determination of wages and profits” (p. 361).