Neoclassical and Classical Growth Theory Compared



Neoclassical and Classical Growth Theory Compared

A. M. C. Waterman*

Growth theory did not begin with my articles of 1956 and 1957, and it certainly did not end there. Maybe it began with The Wealth of Nations and probably even Smith had predecessors. (R. M. Solow 1988, 307)

A recent Supplement to this journal (Boianovsky and Hoover 2009) took Robert Solow’s “key papers from the 1950s as its anchor” and “addressed the intellectual currents that formed the background of that work . . .” (1). It is the purpose of this article to add to that discussion by identifying those features of what we may, with hindsight, think of as “classical” growth theory –

which did indeed begin with The Wealth of Nations (WN) – in order to compare them with the characteristic features of “neoclassical” growth theory as constructed by Solow and Trevor Swan.

In the first part of what follows I formalize the growth theory in WN II.iii and I.viii, and summarize it in a diagram with the rate of profit on the ordinate and the growth-rate on the abscissa. In the second part I rearrange the material in Swan’s version of the basic neoclassical model to represent it graphically with the same magnitudes on the axes, so to facilitate a direct comparison between the models. The third part discusses similarities and differences between classical and neoclassical growth theory, takes note of some complications, and considers whether there has been progress in this branch of economic theory.

1. A Classical Growth Model

Eighteenth-century growth theory emerged from the commonplace insight that “land . . . produces a greater quantity of food than what is sufficient to maintain all the labour necessary for bringing it to market” (WN I.xi.b.2). Labor employed in agriculture is productive. The surplus

of produce over what is needed to feed the labor needed to obtain it may be spent on unproductive labor – employed in personal services, luxury goods, government, defence, education, religion and the arts – thereby sustaining “everything that distinguishes the civilized, from the savage state” (Malthus 1798, 287). But part of that surplus may instead be used to feed additional productive labor in the next period and so to increase total output and income. “Some French authors of great learning and ingenuity” [i. e. the “Physiocrats”] had thoroughly grasped and developed this point (WN II.iii.1, note *). It was Adam Smith’s achievement (WN II.iii.1; IV.ix.29-39) to generalize “his predecessors’” conception of productive labor (Chernomas 1990) and therefore of the surplus, and so to formulate the first complete theory of economic growth.

Productive labor, for Smith, affords not only food but any goods which may be used as inputs into subsequent periods’ production. Given the state of technique, a certain proportion of the total work-force employed in productive labor in one period can produce exactly what was produced by the same fraction of the work force in the previous period. Smith had in mind an economy of small masters, each of whom provides wages, raw materials etc. in advance, and who in aggregate own the total product at the end of each period. Some portion of this they destine for the replacement of their capitals used up in the previous period, the remainder may either be added to capital or spent on unproductive labour. Thus “the annual produce of the land and labour of the country” maintains all who labor together with “those who do not labour at all.” And

According . . . as a smaller or greater proportion of it is in any one year employed in maintaining unproductive hands, the more in one case and the less in the other will remain for the productive, and the next year’s produce will be greater or smaller accordingly; the whole annual produce . . . being the effect of productive labour. (WN II.iii.3)

The aggregate of masters’ decisions as to the disposal of last period’s total product is therefore crucial in determining the rate of growth. These decisions are governed by a psychological propensity of masters which Smith called parsimony.

Parsimony, and not industry, is the immediate cause of the increase of capital

. . . Parsimony, by increasing the fund which is destined for the maintenance of productive hands . . . tends to increase the exchangeable value of the annual produce of the land and labour of the country. (WN III.iii.16, 17).

The more parsimonious each master, the greater the proportion of last year’s income will he spend on productive labor, and the less on domestic servants, fine china and fashionable clothes for his wife and daughters.

The incentive to parsimony is emulation: “the principle which prompts us to save, is the desire of bettering our condition, a desire which . . . comes with us from the womb, and never leaves us till we go into the grave” (WN III.iii.28). It is important to note that it is parsimony and not the rate of profit which governs the saving-and-investment decisions of masters. Indeed Smith believed that a high rate of profit might have an adverse effect on accumulation.

The high rate of profit seems everywhere to destroy that parsimony which in other circumstances is natural to the merchant. When profits are high, that sober virtue seems to be superfluous. . . Have the exorbitant profits of the merchants of Cadiz and Lisbon augmented the capital of Spain and Portugal? (WN IV.vii.c.61)

The following model, in which parsimony is the motor of economic growth, is similar to those originally formulated by Leif Johansen (1967) but seemingly unknown to his successors, and Walter Eltis (1975); and in most respects it can be assimilated to Paul Samuelson’s “Canonical Classical Model” (1978). I have expounded its properties in two recent articles (Waterman 2009; forthcoming) and there will be some unavoidable self-plagiarism in this section.

Let the degree of parsimony, understood as the fraction of their total proceeds per production period that masters decide to spend on productive employment in the following period, be π where 0 ≤ π ≤ 1. Output consists of a single, homogeneous subsistence good F which we may label “foodstuff”. Workers need more than food, and we must assume that each comes furnished with the requisite per capita share of necessary equipment: tools, wagons, barns, horses, cottages etc. which require some fraction of the productive work force to maintain at the desired level. In principle the cost of these goods could be represented as flow magnitudes by means of their depreciation rates, which was the strategy of Karl Marx (1954, vol. I, chap. VIII et passim). But though fixed capital goods must exist they play no part in Smith’s analysis in WN II.iii. Therefore I abstract from fixed capital here, and follow Smith in specifying the capital stock Kt, as “the funds destined for the maintenance of productive labour” in period t (WN II.iii.11), that is to say, advance wages measured in “foodstuff” units. Then

Kt = π.Ft - 1 (1)

It is this lag between last year’s output and this year’s capital which makes the classical model inherently dynamic.

Let the production of “foodstuff” in the current period be

Ft = αNpt , (2)

where α is a technical parameter, and Np is the population of productive workers, fully employed at all times. Since productive workers must come with their unit share of capital (in this simple case wage per period, wt, measured in F units) we may regard Np as the number of what Samuelson (1978, 1416) called “doses” of a joint “labor-cum-capital” variable factor applied to production. Then α is the average product of the joint factor, given for any state of technique when there are constant returns to scale (CRS) and no diminishing returns to Np.

Employment of productive workers in period t made possible by Kt is

Npt = Kt/wt. (3)

Then from (1), (2) and (3) it appears that the rate of capital accumulation is an increasing function of the degree of parsimony and a decreasing function of the real wage:

(Kt – Kt-1)/Kt-1 = απ/w – 1. (4)

Define a growth-rate operator g such that for any continuous, differentiable function of time X(t), gX(t) ≡ d/dt(lnX). Then for small proportionate changes in K, (4) is approximated as

gK = απ/w – 1, (4a)

which is identical to equation 3.9 in Eltis (2000: 94). When the degree of parsimony is exactly equal to the wage-rate divided by the average product of labor, i.e. π =w/α, employment of productive labor is the same as in the previous period and therefore capital stock remains the same. Given π, a lower wage implies a faster growth-rate because more productive labor can be employed with any given capital π.Ft – 1. Equations (1) to (4a) are intended to summarize the implicit macrodynamic analysis in WN II.iii.1-18.

However, there is more to classical growth theory than capital accumulation, since the supply of labor is endogenous. It was universally supposed by eighteenth-century economic thinkers that “Les hommes se multiplient comme des Souris dans une grange, s’il ont le moïen de subsister sans limitation” (Cantillon 1931: 82), or as Smith put it more generally, “every species of animals naturally multiplies in proportion to the means of their subsistence, and no species can ever multiply beyond it” (WN I.viii.39): which is obviously the source of Malthus’s “geometrical ratio.” Let N now stands for total population, assumed to be equal to (“productive” + “unproductive”) work force, m > 0 the speed of adjustment of population to excess subsistence, and σ > 0 the ZPG wage rate, culturally determined in human populations. Then

gN = m(w – σ). (5)

The market wage-rate w is determined by supply of and demand for productive labor. If K increases the demand for labor rises, bidding up w. If the increase in K is once-for-all, w will return to its initial level. But if it is sustained at a constant exponential rate gK, higher w will induce an increase in N according to (5); and as K continues to grow w will rise until it reaches that level at which supply and demand curves are shifting to the right at the same rate, and gN = gK. Hence in steady state there will be some equilibrium or “natural” wage rate corresponding to each rate of accumulation, positive, negative or zero. This is the message of Book I, chapter viii of WN: e.g.

The demand for labour, according as it happens to be increasing, stationary, or declining, or to require an increasing, stationary or declining population, determines the quantity of the necessaries and conveniences of life which must be given to the labourer (WN I.viii.52).

Given that π ’ Np/N, then gNp = gN for any given degree of parsimony. Then upon the assumption that α remains constant as N varies, (4a) and (5) afford simultaneous solutions for the steady-state rate of balanced growth, g* = gK = gN, and the equilibrium wage rate, w*:

mw*2 + (1 – mσ)w* = απ, (6)

g*2 + (1 + mσ)g* = m(απ –σ). (7)

These results could be obtained graphically by plotting (4a) and (5) in w,g space. Because (4a) is a rectangular hyperbola there will be two solutions, corresponding to the quadratics in (6) and (7). An economically meaningful solution appears in the first or fourth quadrants, illustrating Smith’s argument that the natural wage depends upon the rate of capital accumulation (Johansen 1967, fig. 1; Waterman 2009, figs. 1, 2). It can be shown (Waterman 2009, appendix 1) that the quadratic in (7) is identical in form to the characteristic equation of the second-order, discrete system obtained from (1) – (3) plus a discrete version of (5). Its dominant root generates the economically meaningful solution of (6) and (7).

Since it is the purpose of this section to produce a diagram not with w but with the rate of profit, r on the ordinate, some further manipulation is required.

Under competitive conditions the joint labour-cum-capital factor is paid the value of its marginal product, which must be divided between wages and profits. When labor is in strong demand wages are high and profits low, and vice versa. Define the rate of profit (gross of depreciation, if any), as

r ≡ (F – wNp)/K. (8)

Then since F = αNp and in this simple, Smithian case K = wNp, then what Samuel Hollander (1987, 108-12) calls “the fundamental theorem on distribution” appears as

r = α/w – 1 (9)

By solving (9) for w = α/(1 + r) and substituting in (4a) and (5) we obtain

gK = (π – 1) + πr (10)

gN = – mσ + αm/(1 + r). (11)

Equations (10) and (11) afford simultaneous (quadratic) solutions for g* and r* corresponding to those in (6) and (7) above. By modelling accumulation as an increasing function of the profit rate, (10) is made comparable with equation (6) in (Samuelson 1978, 1421). When (11) is changed back to (5) it is identical to Samuelson’s equation (5) when my m = λ/ε. The wage-profit relation, equation (9), is equivalent to Samuelson’s equation (4) when my α = f′(V), which will be the case if α does not vary as Np. These three equations have been the stuff of most subsequent expositions of classical growth theory (e.g. Eltis 1980, 20-21; Hollander 1984, figs. I-VII).

If (11) is plotted in r,g space its curve is a rectangular hyperbola with asymptotes gN = – mσ and r = – 1, and intercepts gN = m(α – σ) and r = (α/σ – 1). For ease of exposition it will be assumed that the line segment between the intercepts can be approximated as a straight line. The other branch of the hyperbola with values of r < – 1 (i.e. less than minus 100%) can be ignored. When (10) is also plotted in the same space we obtain figure 1, which closely resembles figure 2 in Eltis (1980). In drawing (10) as a straight line we are implicitly ignoring Smith’s fears about the adverse effect upon parsimony of a high rate of profit. It is evident that the gN curve can only stay put if α remains constant whatever is happening to Np. These matters will be considered further in section 3 below.

2. A Neoclassical Growth Model

Twentieth-century growth theory emerged from the commonplace insight that “Positive saving, which plays such a great rôle in the General Theory, is essentially a dynamic concept” (Harrod 1948, 11). For if the product-market flow condition in Keynesian macroeconomics is satisfied when I = S(Y) ≠ 0, the equilibrium value of Y can only be momentary, since I = dK/dt ≠ 0, and K should be an argument of both the I and the S functions. Thus if the I-curve lies above or below the horizontal axis in the “Keynesian cross” with which Samuelson (1948) adorned the front cover of early editions of his textbook, neither curve will stay put. It was therefore necessary to deliver Keynesian macroeconomics from incoherence.

It was Harrod’s strategy to investigate the conditions under which both flow and stock conditions could be continuously satisfied as Y and K grew; the stock condition being understood as V [≡ K/Y] = V* [≡ K*/Y], where K* is the desired (or expected, or equilibrium) capital stock and V* the desired (etc.) capital-output ratio. In steady-state, V = v [≡ dK/dY], the incremental capital-output ratio with which Harrod worked.

If S(Y) can be assumed to be sY where the saving ratio s is a constant, then when the product market is in flow equilibrium, and when we abstract from interaction with all other markets,

s = (dK/dt)/Y, (12)

from which, by manipulation,

s = [(dY/dt)/Y].(dK/dY), (13)

or the actual rate of growth,

gY = s/v. (13a)

(13a) is a tautology like Fisher’s equation of exchange, and its heuristic function – not unimportant in the early stages of a new research program – is merely taxonomic. But if v = v*, that is if the current increment to capital in relation to output growth is what entrepreneurs expect and desire, then gY = g*Y is the warranted rate of growth at which stock and flow conditions are simultaneously satisfied and all expectations continuously fulfilled. Harrod (1948, 85ff.) argued that if gY > g*Y, and if both s and v* remained constant, then Y(t) would diverge increasingly from, and above, the warranted growth path Y*(t); and vice versa if gY < g*Y. The warranted growth-path is thus a “knife edge.” (Harrod denied it. See Hagemann 2009, 84; Dimand and Spencer 2009, 115.)

Both capital and labor are required for production, and in twentieth-century growth theory it is generally assumed that in the absence of technical progress gN = n, the natural rate of growth, an exogenously given constant. If gY < n unemployment will grow until some vague “floor” is reached at which v* may change so as to induce a faster rate of growth. If gY > n a hard “ceiling” will eventually be reached at which Y(t) is constrained by labor shortage. Therefore even if gY = g*Y before this point, when it is reached gY = n must fall short of g*Y: hence Y(t) will slide off the warranted growth path. It is therefore necessary for steady-state equilibrium growth that

s/v* = n, (14)

which Solow (1970, 8-12) later called the “Harrod-Domar consistency condition.”

Solow (1956, 65) noted that the “opposition of warranted and natural rates turns out in the end to flow from the crucial assumption that production takes place under conditions of fixed proportions.” If instead it takes place by means of a CRS production function with continuous substitutability of capital and labor, written in labor-intensive form (Hahn and Matthews 1965, 10-11), as

y = y(k), y(0) = 0, y' > 0, y" < 0 (15)

where y ≡ Y/N and k ≡ K/N, then the desired (or intended, or profit-maximizing) capital-output ratio, V* = k*/y, is determined at that point on the y(k) function at which the marginal product of capital y′(k), which is also the rate of profit r, is equal to the current real rate of interest. Harrod was well aware of this, but believing that the rate of interest is determined by monetary factors feared that it, and hence V*, might get “stuck” (Hahn and Matthews 1965, 11-15). Perhaps with this in mind, Solow (1956, 78-84) made a detailed analysis of the “price-wage-interest reactions” necessary for the neoclassical adjustment process to occur. He found, among other things, that

within the narrow confines of our model (in particular, absence of risk, a fixed average propensity to save, no monetary complications) the money rate of interest and the return to holders of capital will stand in just the relation required to induce the community to hold the capital stock in existence. (Solow 1956, 81, my italics)

It was Solow’s achievement to construct the first complete neoclassical theory of economic growth on the basis of these assumptions – together with the assumption of continuous flow equilibrium at full employment which evades the knife-edge problem. His model shows that market forces can reconcile natural and warranted rates of growth. For since gk = gK – n and gK = I/K = sY/K, then

dk/dt = k(gK – n) = K/N.(sY/K – n), whence

dk/dt = sy(k) – nk: (16)

which is Solow’s famous equation (6), “a differential equation involving the capital-labor ratio alone” (1956, 69). Now since

d/dk[dk/dt] = sy((k) – n (17)

where y((k) is the slope of the production function (15), then k will increase as sy((k) > n and vice versa. Note that y((k) = v-1, the incremental output-capital ratio. The capital-labor ratio will therefore be stationary when RHS (17) = 0, that is when

s/v* = n; (14)

for if entrepreneurs are rational, stationarity of k implies that v = v*. Equation (17) thus shows how flexibility of the capital-labor ratio can ensure that the Harrod-Domar consistency condition can always be satisfied in steady state, whatever v, provided that (17) can afford a unique, stable solution.

Whether this can be so depends on the shape of the production function, and Solow investigated a number of possibilities. The most tractable of these, the Cobb-Douglas function

Y = KaN1-a (18)

or y = ka in labor-intensive form, is evidently sufficient for the existence, uniqueness and stability of k* since it is “well-behaved” in Uzawa’s sense (Hahn and Matthews 1965, 10, n.1): that is to say, y((0)= ( and y((()= 0. Some low-valued range of y(k) must exist at which sy((k) > n, and some higher-valued range at which sy((k) < n.

It was with this production function that Trevor Swan (1956) constructed his own contribution to neoclassical growth theory, published some months after Solow’s, but perhaps excogitated months or even years before (Dimand and Spencer 2009, 112-20).

It follows from (18) and the assumption that gK = s(Y/K), that

gY = as(Y/K) + (1 – a)n; (19)

and therefore, by subtracting gK from both sides, that

g(Y/K) = (a – 1)s(Y/K) + (1 – a)n, (20)

which is a first-order (logarithmic) differential equation in Y/K = V-1, with a stable solution for

V* = s/n, or n = s/V*. (21)

Once again, the Harrod-Domar consistency condition is seen to be satisfied in steady state (in which V*= v*) by flexibility in factor proportions, implied from (15) by flexibility in the output-capital ratio.

Swan illustrated his story with a diagram in which growth-rates of capital, labor and output are plotted against the Y/K ratio (Dimand and Spencer 2009, 117, fig. 1). But since, among many other convenient properties of the Cobb-Douglas function, the rate of profit

r = ∂Y/∂K = a(Y/K), (22)

(Swan 1956, 335 equation 2) we can transform Swan’s diagram into one in which r appears on the ordinate and growth-rates on the abscissa, thus enabling an exact comparison to be made with the classical model illustrated in figure 1.

In figure 2, the locus of gK = s(Y/K) = sr/a is plotted as a ray from the origin of slope a/s. The locus of gN = n is plotted as a vertical line intercepting the g axis at n. By substitution of r/a for Y/K in (19) we see that the plot of gY must lie between the gK and gN curves, with a slope of 1/s and an intercept of gY = (1 – a)n. When gK = gN = g*Y, the steady-state rate of profit, r* = aV*, is determined. It is clear from the diagram that when gY < gK, r and hence Y/K, will increase and vice versa.

INSERT FIGURES 1 AND 2 HERE

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3. Discussion

(a) Comparisons

There are some obvious similarities between the classical and neoclassical models as caricatured in figures 1 and 2. In each case a rate of profit exists at which steady-state growth can take place, and in each case that rate is determined by the intersection of a positively sloped gK curve with a gN curve. The slope of the gK curve is in each case a decreasing function of the saving ratio, s, or its classical analogue, π. Since gK = gY in the classical case, it turns out indeed that the slopes of the two gY curves are exactly the same – insofar as we can allow that 1/π = 1/s. In each case the equilibrium r*,g* pair is dynamically stable.

The most obvious difference is that whereas in the classical case gK = gY because of the (seeming) assumption of fixed factor proportions, the neoclassical model is more general in permitting gY to diverge from gK as the output-capital (or capital-labor) ratio varies. Except in steady state, gY ≠ gK. Another seeming difference is that exogeneity of gN in the neoclassical model leads to the counter-intuitive result that changes in the saving ratio can have no effect on the steady-state rate of growth. In this respect, therefore, the neoclassical model would appear to be less general. However as both Solow (1956, 90-91) and Swan (1956, 340-41) showed in their original articles, it is a small matter to generalise their simple model to accommodate endogenous population growth.

There are two other ways, not captured in the diagrams, in which these two simplest possible models may be compared and found similar.

In the first place each assumes that saving and investment are equal at full employment. In the classical case, as equation (1) illustrates, an act of saving is ipso facto an act of investment: “A man must be perfectly crazy, who, where there is tolerable security, does not employ all the stock he commands. . .” (WN II.i.30). Full employment always obtains because any redundant labor will “die like flies” (Samuelson 1978, 1423). In the neoclassical case entrepreneurs’ investment is kept equal to the saving determined by full employment Y either by wage flexibility or by government stabilization policy (Solow 1956, 93).

Secondly, as we might expect if it really is the case that “within every classical economist there is to be discerned a modern economist trying to be born” (Samuelson 1978, 1415), the classical model, like the neoclassical, satisfies the Harrod-Domar consistency condition in steady state. For if we interpret Kt in equation (1) as the current addition to the capital stock (which we are entitled to do since by assumption last period’s stock was completely used up) and interpret π as equivalent to the Keynesian s, then from (1) the incremental capital-output ratio is

v = (π.Ft –1)/(Ft – Ft–1) = π/gF; (23)

and in steady state, when gF = gN and v has the value that masters desire,

s/v* = g*N. (14a)

This will be brought about, as in Solow’s model, by flexibility of the capital-labor ratio. For when all capital is simply the wages fund Npw, then k = w. And as gK > gN, k will rise and vice versa. The classical model is only a “fixed proportions” model in the sense that each worker is assumed to require the same equipment of capital goods. But as WN I.viii.52, quoted above, makes clear, the “quantity of the necessaries and conveniences of life which must be given to the labourer,” proxied in this case by the real wage w, depends on the steady-state rate of growth.

(b) Complications

Three complications of the model in part 1 must be considered, both in order to do justice to those who originally worked with it, and also to compare it more fruitfully with the neoclassical model: returns to scale, diminishing returns to the labor-cum-capital variable factor, and technical progress. The effects of these can be captured by making the parameter α depend on each:

α = α(Np, A); α1 > 0, or α1 = 0, or α1 < 0; α2 > 0. (24)

If there are increasing returns to scale (IRS), then α1 > 0. If there are diminishing returns, α1 < 0. If there are neither, or if their effects cancel out, then α1 = 0. If A is an index of the state of technique and α2 > 0, then when there is technical progress – such as crop rotation, horse-hoe husbandry or the draining of the Fens – α will increase.

Increasing returns to scale

As all the world knows, The Wealth of Nations begins in a pin factory, used as an example of the division of labor and economies of scale. If this were all that was happening, the gN curve in figure 1 would shift continually upward and rightward, increasing r* and g* without bound. Note that from (9) both r and w may increase in this case, notwithstanding the inverse relation between the two when α is constant; and it may be seen from (6) that

dw*/dα = π/[1 + (2w* – σ)], (25)

which will be positive, since for the economically meaningful, positive root of (6), w* = απ/[1+ m(w* – σ)] > 0. As the denominator is positive, the denominator of RHS (25) must also be positive. Hence wages too will rise without limit. Balanced growth might still be possible, but not steady state. Yet this contradicts the detailed analysis of the “natural wage” in WN I.viii, according to which a stationary wage rate is associated with each rate of steady-state growth. Either Smith must be assuming that IRS are offset by diminishing returns (as Eltis 2000, 91-100 seems to think possible) or IRS are not integrated into his analysis, which seems more likely. If they were, moreover, they would present an anomaly that Smith never considered, for the stationary state would be dynamically unstable. Any displacement from stationarity in either direction would lead to cumulative departures into never-ending growth or never-ending decay. There are a few scattered references to the division of labor in Malthus but he made no analytical use of the concept, and in his testimony to the Parliamentary Select Committee on Artizans and Machinery he expressed reservations about the principle (Malthus 1989 I: li). Smith’s other successors simply ignored IRS, and this obvious truth about the real world was forgotten for a century.

As for neoclassical growth theory, both Solow (1956) and Swan (1956) assumed constant returns to scale, which is necessary – unless all economies are external – to preserve perfect competition, part of the “hard core” of neoclassical general equilibrium theory. A predilection for CRS is therefore another similarity of classical and neoclassical growth theory.

Diminishing returns

Diminishing returns are the finger-print, or DNA test of the “Canonical Classical Model.” If α1 < 0 then as growth proceeds and population/work force increases, the vertical intercept of the gN curve in figure 1will fall until the gN curve intersects the gK curve on the r axis, and a stationary state will exist at which απ = σ. Both wages and profits will fall until w = σ and r = (1/π – 1). Land rent is simply [F – w(1 + r)Np] and rises to a maximum in the stationary state. Whether or not Adam Smith was aware of all this, as Samuelson (1980) insisted against Hollander (1980) that he was, there can be no doubt that by 1815 at the latest Malthus, West, Torrens and Ricardo most certainly were. In his macrodynamic conception of the natural wage (WN I.viii) Smith requires steady-state growth to make sense of his argument (Waterman 2009). But for Malthus and Ricardo, and the entire English School down to and including J. S. Mill, steady-state growth is only possible in an agricultural economy with fixed land if the effect of diminishing returns is exactly offset by technical progress; and also if there is no endogenous increase in σ induced by rising living standards.

In Swan’s version of the neoclassical model the Cobb-Douglas production function permits scarce land to be added to the story with ostensibly classical results. Let L stand for the supply of land, then

Y = KaNbLc, a + b + c = 1. (26)

Since gL = 0 by assumption, gY = agK + bgN. Hence when gY = gK in figure 2

gY = [b/(b + c)]gN < gN. (27)

Output per head must therefore fall until y is just sufficient to induce gN at the rate gY, upon which further population growth must be constrained to the rate of output growth:

(gN = gY) = [s/(1 – b)]a(Y/K) = [s/(1 – b)]r. (28)

In figure 2 the gY and gN loci should therefore replaced by a ray from the origin of slope (1 – b)/s along which gN = gY (not drawn), and which would lie above the locus of gK since (1 – b) > a when there are three factors. Hence at any r > 0, gY < gK: which would cause (Y/K), r and (gN = gY) to fall continuously to zero.

Swan (1956, 341, fig. 2) labelled the locus of (28) “The Ricardian Line” and called his story “A Classical Case.” Yet there are some obvious differences from the classical model as expounded above. The stationary state is reached only when the Y/K ratio and the rate of profit have fallen to zero. There is no room in his model for negative growth, which is explicitly recognized and considered in WN I.viii.26. More serious, the relative shares of factors remain constant in Swan’s model, whereas in the classical model the relative share of rent rises continuously at the expense of capital and labor until the stationary state is reached. This is because only the labor-cum-capital factor, which operates in competitive conditions, is paid the value of its marginal product. But land receives an ever-growing surplus because of the monopoly power of each landlord as land becomes scarcer in relation to labor and capital.

Technical Progress

Though Malthus seems to have believed that technical progress might be or become endogenous (Eltis 2000, 169-70), most of his contemporaries tended to think of it as intermittent series of random “inventions.” In figure 1 there might be occasional once-for-all, rightward shifts of the gN locus, but soon to be reversed by ever-present diminishing returns.

No doubt because of its omnipresence in modern industrial society, technical progress is far more prominent in neoclassical growth theory. Both Solow (1956, 85-6) and Swan (1956, 337) incorporated a constant annual rate of neutral technical progress in their models; and the following year Solow (1957) used equation (13) of his 1956 paper as the starting point of a ground-breaking empirical study of technical progress in the US economy, 1909-49. Figure 2 can illustrate the effect of technical progress in Swan’s version of the model. For if (18) is now written as

Y = A(t)KaN1-a (18a)

where A(t) is an index of the state of technique, increasing at the proportionate annual rate gA, then

gY = gA + asY/K + (1– a)gN: (19a)

and all that is necessary is to add a new gY locus, parallel to the original, lying gA to the right. Then gY and gK will intersect to the right of gN with a higher rate of profit, illustrating the fact that output per head will now rise in steady state – at a rate exceeding gA by the added effect of continually increasing capital per head.

(c) Progress

In outline at any rate, neoclassical growth theory closely resembles the growth theory that Johansen (1967), Eltis (1975), Samuelson (1977, 1978), Negishi (1989) and others have reconstructed in present-day analytical terms from The Wealth of Nations and the works of Smith’s followers and successors, especially Malthus and Ricardo. It differs from classical growth theory chiefly in the more formal specification of its categories and conceptual relations and the greater generality of some of its theorems. Can this be regarded as progress? Perhaps, if more formal specification and greater generality have led to new knowledge, which I shall argue may have been the case.

Classical growth theory rested on the distinction between productive and unproductive labor, and the associated concept of the surplus. That distinction may still have some rough-and-ready use in commenting on such matters as the slow growth of the UK economy in the early post-war decades (Bacon and Eltis 1976), but for good reason it has been superseded in economic theory. The services of government, the church, the judiciary, the medical profession, even “players, buffoons, musicians etc.” may and almost certainly do have some positive effect on the production of material goods. In modern theory a “surplus” may exist, even when all factors receive their marginal product, if there are decreasing returns to scale (Darity 2009): steady-state growth could exist in this case if technical progress were exactly compensatory. But this is quite different from the classical conception, in which the surplus itself is what makes growth possible.

Classical thinkers discovered marginal-product pricing of competing factors and used it in their growth theory, but stopped short of applying it to all factors because of their fixation on productive labor. Moreover, by ignoring smooth substitutability between capital and labor they left unanalyzed the distribution of the joint marginal product between masters and laborers. When neoclassical thinkers (and Thünen much earlier) generalized marginal-product pricing to all factors of production, they abolished the surplus and allowed all workers and other factor-owners to be seen to play some part in producing the aggregate of what consumers as a whole want to be produced. And with the assumption of CRS they were able to deal with Ricardo’s problem of determining “the laws which regulate . . . distribution” about which “the classicists succeeded in saying little definite (and correct!)” (Samuelson 1978, 1421). It was precisely neoclassical distribution theory that allowed Solow (1957) to show that the growth of US output 1909-1949 could not be accounted for solely in terms of the increase in capital and labor. There was a significant unexplained “residual” which could be taken, and was taken, as the first attempt actually to measure the rate of technical progress in a market economy.

Solow’s article led to a vast and still increasing literature on technical progress, much of it empirical. And in a similar way the original contributions of Solow and Swan opened up expansive research programs in multi-sectoral growth, growth in open economies, linear models of general interdependence, and optimal growth (Hahn and Matthews 1967). Though much of this is “new knowledge” only in a formal, analytical sense (which is to say, that like mathematics, it is not really “knowledge” at all), some of it may in principle be empirically tested.

Imre Lakatos (1970, 116-20) has identified the conditions which must be met in order for it to be heuristically rational to replace an old scientific theory by a new one. (1) The new theory must “predict novel facts, that is facts improbable in the light of, or even forbidden, by” the older one; (2) The new theory must explain “the previous success” of the older one: it must contain “all the unrefuted content” of the latter; (3) “Some of the excess content” of the new theory must be corroborated. If (1) and (2) are satisfied, replacement of the old theory by the new is a “theoretically progressive problemshift.” If (3) is also satisfied we have an “empirically progressive problemshift.”

There would seem to be little doubt that neoclassical growth theory subsumes and contains the “unrefuted content” of classical growth theory. It tells the same story and tells it more fully, recognises that capital and labor are substitutable in production, recognises that land income could be determined by marginal product, and avoids the anomaly presented by the productive/unproductive labor distinction. At least one “new fact” is predicted by neoclassical theory: Solow’s “residual.” Moreover neoclassical growth theory grew out of, and is conceptually related to, Keynesian macroeconomics, which embodies new knowledge about market economies unknown to the classics (and indeed “forbidden” to all save Malthus by Say’s law.) It would therefore seem that there has been a “theoretically progressive” problemshift in growth theory. Whether the problemshift has also been “empirically progressive” is more difficult to say, since the “corroboration” of economic theories is always contestable. But at least we can say that since Solow (1957) there has been econometric investigation of economic growth.

Note

* St John’s College, Winnipeg R3T 2M5, Canada. The author is grateful to Robert Solow for valuable criticism.

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