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).

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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 = .Ft - 1

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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 = Npt , (2)

where 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 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 .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

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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 thatNp/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* =

g*2 + (1 + mσ)g* = m(–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

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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 = Np and in this simple, Smithian case K = wNp, then what Samuel Hollander

(1987, 108-12) calls “the fundamental theorem on distribution” appears as

r = /w – 1 (9)

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

gK = (– 1) + r (10)

gN = – mσ + m/(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 =f′(V), which will

be the case if 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).

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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(– σ) and r = (/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 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)

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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

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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

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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.

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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

______________________________________________________________________________

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

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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

as equivalent to the Keynesian s, then from (1) the incremental capital-output ratio is

v = (.Ft –1)/(Ft – Ft–1) = gF; (23)

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

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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

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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,

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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

15

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

16

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

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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

18

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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