Post on 14-Apr-2018
7/30/2019 Neoclassical Classical Growth
1/24
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 TheWealth of Nations and probably even Smith
had predecessors. (R. M. Solow 1988, 307)
A recent Supplementto this journal (Boianovsky and Hoover 2009) took Robert Solows 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 WNII.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 Swans 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.
7/30/2019 Neoclassical Classical Growth
2/24
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 (WNI.xi.b.2). Labor employed in agriculture isproductive. Thesurplus
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 (WNII.iii.1, note *). It was Adam Smiths achievement (WNII.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 foodbut 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 years produce will be greater or smaller
accordingly; the whole annual produce . . . being the effect of productive labour. (WN
II.iii.3)
2
7/30/2019 Neoclassical Classical Growth
3/24
The aggregate of masters decisions as to the disposal of last periods total product is therefore
crucial in determining the rate of growth. These decisions are governed by a psychological
propensity of masters which Smith calledparsimony.
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. (WNIII.iii.16, 17).
The more parsimonious each master, the greater the proportion of last years 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 (WNIII.iii.28). It is important to note that it is parsimony and
not the rate of profitwhich 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? (WNIV.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 Samuelsons 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 where 0 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
3
7/30/2019 Neoclassical Classical Growth
4/24
comes furnished with the requisiteper 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 Smiths analysis
in WNII.iii. Therefore I abstract from fixed capital here, and follow Smith in specifying the
capital stockKt, 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 (1)
It is this lag between last years output and this years capital which makes the classical model
inherently dynamic.
Let the production of foodstuff in the current period be
Ft = Np
t , (2)
where is a technical parameter, andNp 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 inFunits) we may regardNp 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 toNp.
Employment of productive workers in period tmade possible byKt 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:
(KtKt-1)/Kt-1 = /w1. (4)
Define a growth-rate operatorgsuch that for any continuous, differentiable function of timeX(t),
gX(t) d/dt(lnX). Then for small proportionate changes inK, (4) is approximated as
4
7/30/2019 Neoclassical Classical Growth
5/24
gK= /w1, (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. =w/, employment of
productive labor is the same as in the previous period and therefore capital stock remains the
same. Given , 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 WNII.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, sil ont le moen 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 (WNI.viii.39): which is obviously the source of Malthuss geometrical
ratio. LetNnow stands for total population, assumed to be equal to (productive +
unproductive) work force, m > 0the speed of adjustment of population to excess subsistence,
and > 0 the ZPG wage rate, culturally determined in human populations. Then
gN = m(w ). (5)
The market wage-rate w is determined by supply of and demand for productive labor. If
Kincreases the demand for labor rises, bidding up w. If the increase inKis once-for-all, w will
return to its initial level. But if it is sustained at a constant exponential rategK, higherw will
induce an increase inNaccording to (5); and asKcontinues 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 ofWN: 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).
5
7/30/2019 Neoclassical Classical Growth
6/24
Given that =Np/N, thengNp = gNfor any given degree of parsimony. Then upon the
assumption that remains constant asNvaries, (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 m)w* = , (6)
g*2 + (1 + m)g* = m(). (7)
These results could be obtained graphically by plotting (4a) and (5) in w,gspace. 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
Smiths 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 thequadratic 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, ron 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 sinceF = Np and in this simple, Smithian caseK =wNp, then what Samuel Hollander
(1987, 108-12) calls the fundamental theorem on distribution appears as
r = /w 1 (9)
By solving (9) forw = /(1 + r) and substituting in (4a) and (5) we obtain
gK = (1) + r (10)
6
7/30/2019 Neoclassical Classical Growth
7/24
gN = m + m/(1 + r). (11)
Equations (10) and (11) afford simultaneous (quadratic) solutions forg* 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 Samuelsons equation (5) when my m =/. The wage-profit
relation, equation (9), is equivalent to Samuelsons equation (4) when my =f(V), which will
be the case ifdoes not vary asNp. 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,gspace its curve is a rectangular hyperbola with asymptotes
gN= mand r =1, and interceptsgN = m( ) and r = (/ 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 ofr
7/30/2019 Neoclassical Classical Growth
8/24
It was Harrods strategy to investigate the conditions under which both flow and stock
conditions could be continuously satisfied as YandKgrew; the stock condition being understood
as V[ K/Y] = V* [ K*/Y], whereK* 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.
IfS(Y) can be assumed to besYwhere the saving ratios 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 Fishers 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, thengY = g*Yis the warranted rate of growth at which stock and flow conditions are
simultaneously satisfied and all expectations continuously fulfilled. Harrod (1948, 85ff.) argued
that ifgY > g*Y, and if boths and v* remained constant, then Y(t) would diverge increasingly
from, and above, the warranted growth path Y*(t); and vice versa ifgY < 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 progressgN = n, the natural rate
of growth, an exogenously given constant. IfgY < n unemployment will grow until some vague
floor is reached at which v* may change so as to induce a faster rate of growth. IfgY > n a
hard ceiling will eventually be reached at which Y(t) is constrained by labor shortage.
8
7/30/2019 Neoclassical Classical Growth
9/24
Therefore even ifgY = g*Ybefore this point, when it is reachedgY = n must fall short ofg*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 offixed
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)
wherey Y/Nand k K/N, then the desired (or intended, or profit-maximizing) capital-output
ratio, V* = k*/y, is determined at that point on they(k) function at which the marginal product of
capitaly(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 Solows achievement to construct the first complete neoclassical theory of economicgrowth 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 sincegk = gK n and
gK = I/K = sY/K, then
9
7/30/2019 Neoclassical Classical Growth
10/24
dk/dt = k(gK n) = K/N.(sY/K n), whence
dk/dt = sy(k) nk: (16)
which is Solows famous equation (6), a differential equation involving the capital-labor ratio
alone (1956, 69). Now since
d/dk[dk/dt] = sy(k) n (17)
wherey(k) is the slope of the production function (15), then kwill increase assy(k) > n and vice
versa. Note thaty(k)= v-1, the incremental output-capitalratio. 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 ofkimplies 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, whateverv, 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)
ory = kain labor-intensive form, is evidently sufficient for the existence, uniqueness and
stability ofk* since it is well-behaved in Uzawas sense (Hahn and Matthews 1965, 10, n.1):
that is to say,y(0)= andy()= 0. Some low-valued range ofy(k) must exist at whichsy(k) >
n, and some higher-valued range at whichsy(k) < n.
It was with this production function that Trevor Swan (1956) constructed his owncontribution to neoclassical growth theory, published some months after Solows, but perhaps
excogitated months or even years before (Dimand and Spencer 2009, 112-20).
It follows from (18) and the assumption thatgK = s(Y/K), that
gY = as(Y/K) + (1 a)n; (19)
10
7/30/2019 Neoclassical Classical Growth
11/24
and therefore, by subtractinggKfrom 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/Kratio (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 Swans diagram into one in which rappears 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 ofgK = s(Y/K) = sr/a is plotted as a ray from the origin of slope
a/s. The locus ofgN = n is plotted as a vertical line intercepting thegaxis at n. By substitution of
r/a forY/Kin (19) we see that the plot ofgYmust lie between thegKandgNcurves, with a slope
of 1/s and an intercept ofgY = (1 a)n. WhengK = gN = g*Y, the steady-state rate of profit,
r* = aV*, is determined. It is clear from the diagram that whengY < gK, rand hence Y/K, will
increase and vice versa.
INSERT FIGURES 1 AND 2 HERE
_____________________________________________________________________________
_
11
7/30/2019 Neoclassical Classical Growth
12/24
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 slopedgKcurve with a
gNcurve. The slope of thegKcurve is in each case a decreasing function of the saving ratio, s,
or its classical analogue, . SincegK = gYin the classical case, it turns out indeed that the slopes
of the twogYcurves are exactly the same insofar as we can allow that 1/= 1/s. In each case
the equilibrium r*,g* pair is dynamically stable.
The most obvious difference is that whereas in the classical casegK = gYbecause of the
(seeming) assumption of fixed factor proportions, the neoclassical model is more general in
permittinggYto diverge fromgKas the output-capital (or capital-labor) ratio varies. Except in
steady state,gY gK. Another seeming difference is that exogeneity ofgNin 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. . . (WNII.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 Yeither by wage
flexibility or by government stabilization policy (Solow 1956, 93).
12
7/30/2019 Neoclassical Classical Growth
13/24
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 interpretKt in equation (1) as the current addition to the capital stock (which we
are entitled to do since by assumption last periods stock was completely used up) and interpret
as equivalent to the Keynesians, then from (1) the incremental capital-output ratio is
v = (.Ft 1)/(Ft Ft1) = /gF; (23)
and in steady state, whengF = gNand v has the value that masters desire,
s/v* = g*N. (14a)
This will be brought about, as in Solows model, by flexibility of the capital-labor ratio. For
when all capital is simply the wages fundNpw, then k = w. And asgK > gN, kwill 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 WNI.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 depend on
each:
= (Np, A); 1 > 0,or1 = 0,or1 < 0; 2 > 0. (24)
If there are increasing returns to scale (IRS), then 1 > 0. If there are diminishing returns, 1 < 0.
If there are neither, or if their effects cancel out, then 1 = 0. IfA is an index of the state of
13
7/30/2019 Neoclassical Classical Growth
14/24
technique and 2 > 0, then when there is technical progress such as crop rotation, horse-hoe
husbandry or the draining of the Fens 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, thegNcurve in
figure 1 would shift continually upward and rightward, increasing r* andg* without bound. Note
that from (9) both rand w may increase in this case, notwithstanding the inverse relation
between the two when is constant; and it may be seen from (6) that
dw*/d = /[1 + (2w* )], (25)
which will be positive, since for the economically meaningful, positive root of (6),
w* = /[1+ m(w* )] > 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
WNI.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). Smiths other
successors simply ignored IRS, and this obvious truth about the real world was forgotten for acentury.
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
14
7/30/2019 Neoclassical Classical Growth
15/24
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
1 < 0 then as growth proceeds and population/work force increases, the vertical intercept of the
gNcurve in figure 1will fall until thegNcurve intersects thegKcurve on the raxis, and a
stationary state will exist at which = . Both wages and profits will fall until w = and
r = (1/1). 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
(WNI.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 induced by rising living standards.
In Swans version of the neoclassical model the Cobb-Douglas production function
permits scarce land to be added to the story with ostensibly classical results. LetL stand for the
supply of land, then
Y = KaNbLc, a + b + c = 1. (26)
SincegL = 0 by assumption,gY = agK + bgN. Hence whengY = gKin figure 2
gY = [b/(b + c)]gN < gN. (27)
Output per head must therefore fall untily is just sufficient to inducegNat the rategY, 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)
15
7/30/2019 Neoclassical Classical Growth
16/24
In figure 2 thegYandgNloci should therefore replaced by a ray from the origin of slope
(1 b)/s along whichgN = gY(not drawn), and which would lie above the locus ofgKsince
(1 b) > a when there are three factors. Hence at any r > 0,gY < gK: which would cause (Y/K),
rand (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/Kratio 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 WNI.viii.26. More serious, the relative shares of factors remain
constant in Swans 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
gNlocus, 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 aground-breaking empirical study of technical progress in the US economy, 1909-49. Figure 2 can
illustrate the effect of technical progress in Swans version of the model. For if (18) is now
written as
Y = A(t)KaN1-a (18a)
16
7/30/2019 Neoclassical Classical Growth
17/24
whereA(t) is an index of the state of technique, increasing at the proportionate annual rategA,
then
gY = gA + asY/K + (1 a)gN: (19a)
and all that is necessary is to add a new gYlocus, parallel to the original, lyinggA to the right.
ThengYandgKwill intersect to the right ofgNwith a higher rate of profit, illustrating the fact
that output per head will now rise in steady state at a rate exceedinggA 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
Smiths 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.
17
7/30/2019 Neoclassical Classical Growth
18/24
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 Thnen 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 Ricardos
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.
Solows 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 novelfacts, 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.
18
7/30/2019 Neoclassical Classical Growth
19/24
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: Solows 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 Says
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 Johns College, Winnipeg R3T 2M5, Canada. The author is grateful to Robert Solow for
valuable criticism.
References
WNstands for Smith, Adam.An Inquiry into the Nature and Causes of the Wealth of Nations
(1976) [1776]. Oxford: Oxford University Press, 2 vols. Exact photographic reproduction of the
foregoing in Liberty Classics, Indianapolis: Liberty Fund, 1981. All references to this
source list book number, chapter number and paragraph number.
Bacon, R. W. and Eltis,Walter. 1976.Britains Economic Problem: Too Few Producers.
London: Macmillan.
19
7/30/2019 Neoclassical Classical Growth
20/24
Boianovsky, Mauro and Hoover, Kevin D. (eds) 2009.Robert Solow and the Development of
Growth Economics.Annual Supplement to Volume 41, History of Political Economy.
Durham NC: Duke University Press.
Cantillon, R. [1755] 1931.Essai sur la Nature du Commerce en Gnral, ed. and transl. H.
Higgs, London: Macmillan for the Royal Economic Society.
Chernomas, Robert. 1990. Productive and Unproductive Labor and the Rate of Profit in Malthus,
Ricardo and Marx.Journal of the History of Economic Thought12: 81-95.
Darity, William. 2009. More Cobwebs? Robert Solow, Uncertainty, and the Theory of
Distribution. In Boianovsky and Hoover.
Dimand, Robert W. and Spencer, Barbara A. 2009. Trevor Swan and the Neoclassical Growth
Model. In Boianovsky and Hoover.
Eltis, Walter, 1975. Adam Smiths Theory of Economic Growth. In Skinner, A. S. and Wilson,
T. (eds)Essays on Adam Smith. Oxford: Clarendon Press.
-----. 1980. Malthuss Theory of Effective Demand and Growth. Oxford Economic Papers 32:
19-56
-----. 2000. The Classical Theory of Economic Growth. 2nd edn. Basingstoke: Palgrave.
Hageman, Harald. 2009. Solows 1956 Contribution in the Context of the Harrod-Domar Model.
In Boianovsky and Hoover.
Hahn, F. H. and Matthews, R. C. O. 1965. The Theory of Economic Growth: a Survey. In
Surveys of Economic Theory, Growth and Development, Vol. II. London: Macmillan
Harrod, R. F. 1948. Towards a Dynamic Economics. London: Macmillan.
Hollander, Samuel.1980. On Professor Samuelsons Canonical Classical Model of Political
Economy.Journal of Economic Literature, 18: 559-74.
-----. 1987. Classical Economics. Oxford: Blackwell.
20
7/30/2019 Neoclassical Classical Growth
21/24
Johansen, Leif. 1967. A Classical Model of Economic Growth. In Feinstsein, C. H. (ed.),
Socialism, Capitalism and Economic Growth: Essays Presented to Maurice Dobb.
Cambridge: Cambridge University Press.
Lakatos, Imre. 1970. Falsification and the Methodology of Scientific Research Programmes. In
Lakatos, I. and Musgrave A. (eds.), Criticism and the Growth of Knowledge. Cambridge:
Cambridge University Press.
[Malthus, T. R.] 1798.An Essay on the Principle of Population as it affects The Future
Improvement of Society, with Remarks on the Speculations of Mr Godwin, M. Condorcet,
and Other Writers. London: Johnson. Facsimile reprint. London: Macmillan, 1966, for the
Royal Economic Society.
Malthus, T. R. 1989.Principles of Political Economy. Variorum edition, John Pullen ed. 2 vols.
Cambridge: Cambridge University Press.
Marx, Karl. [1867] 1954. Capital. A Critique of Political Economy, Volume I. Moscow: Progress
Publishers.
Negishi, Takashi. 1989.History of Economic Theory. Amsterdam: North Holland.
Samuelson, Paul A. 1948.Economics. An Introductory Analysis. New York: McGraw-Hill.
-----. 1978. The Canonical Classical Model of Political Economy.Journal of Economic
Literature,16: 1415-34.
-----. 1980. Noise and Signal in Debates Among Classical Economists: A Reply.Journal of
Economic Literature 18: 575-578.
Solow, Robert M. 1956. A Contribution to the Theory of Economic Growth. Quarterly Journal
of Economics 70:65-94.
-----. 1957. Technical Change and the Aggregate Production Function.Review of Economics and
Statistics 39:748-62.
-----. 1970. Growth Theory: An Exposition. Oxford: Oxford University Press.
-----. 1988. Growth Theory and After. The American Economic Review 78:307-17.
21
7/30/2019 Neoclassical Classical Growth
22/24
Swan, T. W. 1956. Economic Growth and Capital Accumulation.Economic Record32: 334-61.
Waterman, A. M. C. 2009. Adam Smiths Macrodynamic Conception of the Natural Wage.
History of Economics Review 49: 45-60.
-----. 2011. Adam Smith and Malthus on High Wages.European Journal of the History of
Economic Thought, forthcoming.
22
7/30/2019 Neoclassical Classical Growth
23/24
23
7/30/2019 Neoclassical Classical Growth
24/24
24