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Adam Smith's macrodynamic conception of the natural wage.

Abstract: Adam Smith argued that the 'natural' wage is an increasing function of the rate of accumulation; and only in the stationary state would it be at the 'subsistence' (ZPG) level. In a world with no capital goods, no technical progress and no scale effects, Smith's macrodynamic conception would constitute a theory of the natural wage in the steady state. But when these complications are allowed for, no steady state is possible in general. Moreover, even without these complications, Smith's reasoning is shown to rest crucially on historically contingent behavioural assumptions. If the lower orders make decisions that affect the rate of population growth, Smith's conclusions are turned upside down. Given the masters' degree of parsimony, high wages are then associated with a stationary or declining economy and vice versa.

The characteristic feature of Smith's analysis is the role accorded to the rate of capital accumulation as an 'independent' variable governing the demand for labour, upon which depends the (long-run or secular) real wage rate and the growth of population; to each growth rate of labour demand there corresponds a long-run real wage rate which assures an equivalent rate of growth of population, and therefore the work force.

Samuel Hollander (1973, p. 157)

1 Introduction

Every reader of chapters vii, viii and ix in Book I of Wealth of Nations (WN) has been struck by repeated reminders of the taxonomic importance of the 'advancing, stationary or declining condition' of society in the determination of the 'natural' rates of wages and profit (WN I.vii.l, 33, 34, 36; viii.41, 43, 52; ix.l). Chapter viii presents a loosely constructed macrodynamic account of a positive relation between the natural wage and the rate of growth of 'the funds destined for the maintenance of labour' which underlies that famous aphorism which almost everyone knows:

The progressive state is in reality the chearful and hearty state to all the different orders of the society. The stationary is dull; the declining melancholy. (WN I.viii.43)

Wages are determined by supply of and demand for labour. The demand for labour is given by the stock of circulating capital (or by circulating and fixed capital if each productive worker is assumed to be equipped with his share of capital goods). When circulating capital grows, wages start to go up and, as a result, population grows too.

Smith assumed that the whole of last period's output is owned and controlled by masters. Their circulating capital in the current period is that portion of their revenue 'destined for the maintenance of productive labour' (WN II.iii.11). It is renewed by the parsimony of masters, who dedicate a portion of their revenue at the beginning of each production period as advances for the employment of 'servants' (in this case productive workmen). Because productive workmen produce more than it costs to employ them, a social surplus is created which is available to support unproductive workmen who provide not only menial service and low entertainment but also government, defence, the arts, religion and culture: 'everything, indeed, that distinguishes the civilized, from the savage state' as Robert Malthus (1798, pp. 286-7) put it, having clergymen and college fellows in mind. (2) If the parsimony of masters were such as to assign from last period's output exactly the inputs needed to maintain the current level of productive employment, then production, the surplus and the stock of capital would remain constant. If parsimony is greater than this, the unproductive sector is squeezed, and productive employment, output and capital all grow, and vice versa (WN II.iii.1, 6, 13-20, 30-32).

The 'demand for men, like that for any other commodity, necessarily regulates the production of men' (WN I.viii.40). Rising demand caused by capital accumulation raises wages; and since 'every species of animals naturally multiplies in proportion to the means of their subsistence' (WN I.viii.39), population growth follows capital accumulation. If the latter is constant long enough for the former to catch up, wages stop rising and a steady-state growth equilibrium may exist in which supply of and demand for labour are equal, and the wage is constant at that level required to induce a rate of population growth equal to the rate of accumulation.

It would seem from all this that Smith wished his readers to understand that the natural wage is at the 'subsistence', or zero-population-growth level when the economy is stationary, above this level when it is growing, and below it when it is declining. Does this macrodynamic conception constitute a theory of the natural wage? Most commentators have doubted and some have simply ignored it. Joseph Schumpeter (1954, p. 190) had no very high opinion of Adam Smith as a theorist, and remarked that WN I.viii contains elements of at least three different theories of wages. Elementary textbooks (for example, Blaug 1996, p. 43; Landreth and Colander 2002, p. 99; O'Brien 2004, p. 127; Staley 1989, p. 47) have dutifully recycled this unflattering opinion. The most powerful analysis of Smith's value theory is Paul Samuelson's (1977) 'Vindication of Adam Smith' from the criticisms 'of Ricardo and Marx and from the general supercilious discounting of Smith as an unoriginal theorist who is logically fuzzy and eclectically empty'. (Could he have meant to refer to his old Harvard supervisor, Joseph Schumpeter?) But it contains no account of the level of wages in a steadily growing economy. For since it was necessary to bring rents into the determination of product prices, Samuelson was not interested in the special case of free land. (3) Only Takashi Negishi (1989, pp. 83-9) has addressed the matter formally. His elegant von Neumann growth model produces the desired result: 'the natural rate of wage is higher than the subsistence level in a growing economy' (p. 86). But, for reasons which will appear below (see Appendix 3), I believe it is preferable to use an alternative, though less polished, formulation of Smith's growth theory: one which takes seriously the distinction between productive and unproductive labour that the Neumann approach obliterates.

It is therefore my purpose in this article to combine Smith's macrodynamic reasoning in Book I, chapter viii, with that in Book I1, chapter iii, of Wealth of Nations, and to specify the conditions under which the result could fairly be regarded as a theory of the natural wage. First, 1 present a simple model of wages and growth for an economy in which there is no fixed, only circulating, capital. With one important exception considered below, my simple model bears a family resemblance to that of Walter Eltis (2000, chapter 3), in particular in its Smithian assumption that it is the psychological characteristic of 'parsimony', rather than the rate of profit, that motivates masters to accumulate (see Appendix 3). In the following sections, I consider some of the more serious complications which have to be acknowledged in any attempt to be faithful to Smith's text; I construct a more complicated model with fixed capital; and I attempt to show what assumptions have to be made in the light of these complications in order to regard Smith's macrodynamic conception of the natural wage as a genuine theory. The final section contains conclusions, some of which are unintended consequences of the previous analysis. Two appendices elucidate the mathematics of my model specifications; and a third explains why the rate of profit is irrelevant to any account of Adam Smith's growth theory.

The most novel and challenging of my conclusions is the discovery that Smith's macrodynamics of the real wage rests upon historically contingent, class-based behavioural assumptions; and that when it is servants rather than masters who make autonomous decisions--which is what Smith's immediate successors were interested in--Smith's results are turned upside down. High wages are then correlated with a declining (or at any rate stationary) economy and vice versa.

2 A Simple Model

Let the capital stock in period t, consisting wholly of circulating capital, be

[K.sub.t] = [pi. [F.sub.t-1], (1)

where F is the output of a single homogeneous product ('foodstuff'), 0 < [pi] < 1 is the degree of parsimony, a behavioural parameter, and the subscript (t - 1) refers to values in the preceding period. Let production in current period be

[F.sub.t] = [alpha][N.sup.p]t, (2)

where a is a technical parameter and [N.sup.p] is the population of productive workers, fully employed at all times. Let

[N.sup.p]t = [K.sub.t]/[w.sub.t], (3)

where w is the real market wage-rate measured in units of 'foodstuff'. Then, by (3) into (2), and (2) into (1), the annual growth rate of capital is, ([K.sub.t] - [K.sub.t-1])/[K.sub.t-1] = [alpha][pi]/w - 1. (4)

For small differences we may regard (4) as a discrete approximation to the logarithmic growth rate, noting that in general the operator g is interpreted such that gY(t) [equivalent to] d/dt(lnY) where Y(t) is any continuous, differentiable function of time, t. Then

gK [approximately equal to] [([K.sub.t]/[K.sub.t-1]) - 1] - [alpha][pi]/w - 1; (5)

or, alternatively,

w = [alpha][pi]/(1 + gK). (5a)

If the degree of parsimony [pi] is greater than the ratio of the real wage to average product, the economy will grow and vice versa. If [pi] = w/[alpha] the entire per-period surplus (1 - [pi])[F.sub.t-1] is spent on unproductive labour, therefore per-period production and the end-of-period capital stock remain stationary. Note that the rate of profit plays no part in Adam Smith's growth theory (see Appendix 3).

Let the population and workforce (assumed to be the same) 'naturally' multiply qn proportion to the means of their subsistence', such that

[N.sub.t+1] - [N.sub.t][1 + m([w.sub.t] - s)], m > 0, (6)

or in continuous terms, since [([N.sub.t+1]/Nt) - 1] = m([w.sub.t] - s)],

gN = m(w - s); (7)

and alternatively,

w = s + gN/m; (7a)

where s is the socially determined 'subsistence', or zero-population-growth, wage; a behavioural parameter, but which might become endogenous if workers learned to expect higher living standards after a prolonged period of rapid growth. The parameter m measures the speed of population response to above-subsistence per capita income. Since [pi] measures the fraction of productive labour in the total workforce then gN = g[N.sup.p] for any given [pi].

If (5a) and (7a) are graphed in (w, g) space, the negatively sloped w(gK) locus intersects with the positively sloped w(gN) locus to determine a balanced growth rate g* for capital, population and productive employment, and a natural wage-rate w*. However, because the w(gK) function is a rectangular hyperbola, there will exist a second point of intersection with the w(gN) function in the third quadrant. But since negative wage-rates are not economically meaningful this may be ignored (see Appendix 1). Moreover because the vertical asymptote of the w(gK) function is g = -1, and the curve approaches the horizontal axis as g [right arrow] +[infinity], the function may be approximated for graphical purposes as a straight line over that narrow range of values of g, say from -0.1 to +0.1, for which the analysis is intended to apply. From (5a), the slope of the w(gK) function in general is

[??]w/[??]gK = -[alpha][pi]/[(1 + gK).sup.2] (8)

= - [alpha][pi] in the special case when gK = 0, that is to say on the vertical axis. A linear approximation to (5a) may therefore be written, and graphed in figure 1, as:

w = [alpha][pi] - [alpha][pi].gK. (9)

[FIGURE 1 OMITTED]

Figure 1 bears a family resemblance to diagrams used by Hollander (1984) and Eltis (2000, chapter 4) to analyse the growth theories of Malthus and Ricardo.

It is evident from I Sa) that the negatively sloped w(gK) function will be shifted upwards by an increase in either [alpha] or [pi].

Because the balanced-growth rate of accumulation is (gK = gN = g*) when the balanced-growth wage-rate is w = w*, a pair of simultaneous, linear equations,

w* = [alpha][pi] - [alpha][pi].g* (9*)

w* = s + g*/m, (7a)

determines the solutions:

w* = [alpha][pi](1 + ms)/(1 + [alpha][pi]m) (10)

g* = m([alpha][pi] - s)/(1 + [alpha][pi]m). (11)

It can be seen from (10) and (11) that for any given values of m and s:

g* > 0 when [alpha][pi] > s, and hence that w* > s; the 'chearful and hearty state';

g* = 0 when [alpha][pi] - s, and hence that w* = s; the stationary, 'dull' state;

g* < 0 when [alpha][pi] < s, and hence that w* < s; the 'declining, melancholy' state.

Adam Smith's macrodynamic conception of the natural wage is illustrated in figure 2.

[FIGURE 2 OMITTED]

That the market wage-rate w will 'gravitate' to the natural wage-rate w*, for reasons similar to those that Smith considered for product prices in WN I.vii.7-15, is easily seen. For when w < w* capital accumulation is faster than growth in the workforce, hence demand for labour will rise in relation to supply and force up the market wage--and vice versa.

Let the out-of-steady-state rate of wage adjustment be

dw/dt = H[gK - gN] = H[(1 - w/[alpha][pi]) - m(w - s)];H' > 0. (12)

Then d/dw(dw/dt) = -H'(m + 1/[alpha][pi]) < 0, (13)

which is sufficient for convergence of w upon w*.

With the stability of balanced growth assured, we may proceed to comparative dynamics. The effect upon w* and g* of a once-for-all change in each of the four parameters (a, x, m and s,) may be seen from figure 1. An increase in [alpha] or [pi] shifts the w(gK) curve upwards; an increase in s shifts the w(gN) curve upwards; and an increase in m causes the w(gN) curve to rotate clockwise about its vertical intercept. The outcomes of these shifts may be verified by partial differentiation of the balanced-growth equations: which it is preferable to use in their general, quadratic forms obtained from (5a) and (7a) by substitution (see Appendix 1), setting gK = gN = g* which implies that w = w*:

[mw*.sup.2] + (1 - ms)w* = [alpha][pi] (14)

[g*.sup.2] + (l + ms)g* = m([alpha][pi] - s) (15)

The results are summarised in table 1.

The effect of the population parameter m upon w* and g* requires elucidation.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

In the 'progressive' state an increase in the speed of population response - say because of reduced child mortality - will reduce the natural wage, and vice versa in the 'declining' state. It will have no effect in the 'stationary' state.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (17)

exactly so in the stationary state, approximately so as g* [not equal to] 0 for the relatively small values of g* considered.

These results show that in the 'progressive' state an increase in the speed of population response will increase the rate of accumulation but lower the natural wage, and vice versa in the 'declining' state. It has no effect in the 'stationary' state. Together with the opposite signs on [partial derivative]w*/[partial derivative]s and [partial derivative]g*/[partial derivative]s (table 1), it appears that a shift in the w(gN) curve--for any given values of [alpha] and [pi]--will produce effects upon w* and g* that are contrary to those assumed by Smith in his macrodynamic conception of the natural wage. Figure 3 illustrates this seeming anomaly with a family of three w(gN) curves intersecting a given w(gK) curve. The significance of this for Smith's account of the natural wage will be considered below.

[FIGURE 3 OMITTED]

3 Some Complications

The simple model cannot capture all that Smith attempted to say in chapters vii, viii and ix of Book I of WN.

The most serious complication is the possibility, taken seriously by Eltis (2000, pp. 88-9, 92-3), that Smith really supposed that at any given rate of accumulation both population and real wages would rise. Since [N.sup.p.sub.t] = [K.sub.t]/[w.sub.t] in the notation of my model, gK = gN + gw. Eltis argues that gw is normally greater than zero, and hence that gN < gK. It is obvious that Eltis' interpretation, if correct, would completely subvert my attempt at modelling by ruling out any possibility of steady-state growth equilibrium. More seriously, it would also seem to challenge what appears, at least from some parts of the text (for example, I, viiii, pp. 40-3, 52), Smith's own conception of the relation between the rate of growth and the level of wages. Intellectual historians have been warned by Quentin Skinner (1969) against falling into a 'mythology of coherence'. We need not assume that a great thinker like Adam Smith could never have been confused or mixed up. In what follows, however, I shall ignore this possibility, and assume that Hollander's interpretation, cited at the head of this article, is an accurate summary of Smith's own thinking on the matter.

Samuelson (1978, p. 1416) has suggested that the 'classical' authors generally conceived of production as taking place by the employment of a variable 'labor-cum-capital' composite factor which may be applied to a third factor, 'land'. Then the parameter [alpha] in equation (1) is the marginal and average product of that composite factor. Now, as Hollander (1973, pp. 157-8), Eltis (2000, pp. 86-9) and others have noted, some part of total production is devoted to capital goods (horses, ploughs, wagons and so on) which--if growth is positive--may increase the capital intensity of the composite factor as time passes, thereby raising productivity and increasing the value of [alpha]. Moreover, if increasing capital intensity lowers the price of manufactures in relation to foodstuff ('corn'), the wage will rise when measured in units of manufactured goods, even if the corn wage were to remain constant (Eltis 2000, p. 89). Therefore the results of the simple model--even ignoring, for the present, the anomaly illustrated in figure 3--cannot 'constitute a true "theory"' of the natural wage 'because Smith failed to explain the allocation of total capital between wage-goods and other items' (Hollander 1973, p. 159).

The best-known feature of Smith's account of economic growth is the division of labour, which produces increasing returns to scale (IRS). In Eltis' growth model IRS plays an important part, and is captured by the exponent Z, such that

[F.sub.t] = [alpha][([N.sup.p.sub.t]).sup.Z] (18)

in the notation of this paper; thus with IRS, Z > 1. Negishi (1989, pp. 89-102) discusses IRS at some length, but does not incorporate its effect formally into his von-Neumann-type growth model.

All Smith's immediate successors, strongly influenced by Robert Malthus' first Essay on Population (1798), analysed the effect upon economic growth of land scarcity. As population and capital increase in relation to land, the average product of the labour-cum-capital factor falls, bringing down wages and profits to the point at which all accumulation ceases. There is some textual evidence for the view that Smith recognised land scarcity (for example, WN I.ix.11), and Samuelson (1978) subsumed Smith's analysis under the 'Canonical, Classical Model' of Malthus, Ricardo and their contemporaries. But Hollander (1980) contested this interpretation, and on other grounds (Waterman 2004, chapter 7) it seems implausible. For though Smith was aware of land scarcity he did not incorporate its effect in his analysis of growth, which for the most part proceeds as if land were a free good. Nevertheless, we should regard this theoretical possibility as a complication to be considered.

The remaining complications to be considered are the possibility of technical progress, and the effect upon Smith's argument of exogenous demographic changes. Each of these is discussed below. In order to deal with some of them it will be helpful to respecify the w(gK) function so as to take account of fixed capital; and to note that the marginal and average product, [alpha], should now be treated as a function of capital intensity, the state of technique and scale.

4 A More Complicated Model

Let the stock of fixed capital be Ct, and circulating capital as before be [V.sub.t] [equivalent to] [N.sub.t][w.sub.t]. Then the capital stock is now

[K.sub.t] = [C.sub.t] + [V.sub.t] = [C.sub.t] + [N.sub.t][w.sub.t]. (19)

Let the annual change in fixed capital (neglecting depreciation) be

[C.sub.t] - [C.sub.t-1] = (1 - [beta])[gamma][N.sub.t-1], (20)

where [beta] is the fraction of the workforce that masters assign to wage-good production, and [gamma] is the marginal and average product in fixed-capital production.

The parameter (1 - [beta]) is an index of the demand for fixed capital. Suppose that masters seek to maintain an optimum per-capita stock of ploughs, horses, cottages, barns and so on: k* - ([C.sub.t]/[N.sub.t])*. Then

[C.sub.t] - k* [N.sub.t]. (21

By (21) into (20) and the usual approximation of discrete differences as differentials,

(1 - [beta])* - (k*/[gamma])gN (22)

Given the technical parameter y and the growth rate of workforce/employment gN, the proportion of the workforce masters chose to assign to capital-goods production is determined by the optimum fixed-capital/labour ratio.

It follows from (19) that the growth rate of capital,

gK = (1 - v)gC + v.gV. (23)

where v [equivalent to] V/K. Dividing (20) by [C.sub.t-1] we obtain the growth rate of fixed capital ([C.sub.t] - [C.sub.t-1])/[C.sub.t-1], or

gC = (1 - [beta])[gamma]N/C = (1 - [beta])[gamma]/k. (24)

The growth rate of circulating capital is determined as in the Simple Model, except that equation (1) is now

[V.sub.t] = [pi].[F.sub.t-1], (25)

and the wage-goods production function, equation (2), must capture the effect of division of the productive workforce between fixed capital and wage-goods, and become

[F.sub.t] = [alpha][beta].[N.sup.p.sub.t]. (26)

By (26) into (25) and the definition of V

[V.sub.t] = ([pi][alpha][beta]/w)[V.sub.t-1] or

gV = [pi][alpha][beta]/w - 1. (27)

Thus from (23), (24) and (27) the growth rate of capital is

gK = (1 - v)(1 - [beta])[gamma]/k + v([pi][alpha][beta]/w - 1). (28)

However, upon the assumption that masters are successful in maintaining their desired per capita stocks of fixed capital (k = k *), it follows from (21) that gC = gN, and thus from (23) that gK = gC = gN for any w. Therefore by substitution of gK for gC on RHS (23),

gK = [pi][alpha][beta]/w - 1. (29)

It is apparent from (22) that [beta] is a decreasing function of k and an increasing function of [gamma]. Therefore the effect of these parameters of the fixed capital growth-rate function (24) are incorporated into (29) although it is formally identical with (27), the growth rate of circulating capital only. When there is no fixed capital, [beta] = 1 and (29) reduces to (5) of the Simple Model.

The population/workforce growth-rate function is unchanged in the more complicated model. Hence equations (29) and (7) together determine g* and w* as in the Simple Model, and could be represented graphically as in figure 1, noting only that the intercept of the w(gK) locus would now become [pi][alpha][beta]. Slightly more complicated quadratic equations for g* and w* (see Appendix 2) now follow from (29) and (7). But since the new w(gK) curve closely resembles the old and will shift in the same way for the same reasons, we may refer to figure 1 for diagrammatical illustrations of what follows.

5 Conditions of Steady State

If the intersection of w(gK) and w(gN) curves produces an equilibrium (w*, g*) pair, the question that is crucial to an appraisal of Adam Smith's conception of wage determination is whether balanced growth at that rate can persist without exogenous disturbance? If it can, Smith has a theory of the natural wage; if it cannot, he has only a vague and suggestive conception.

In order to answer this question we must consider the factors which may cause [alpha] and/or [beta] to vary endogenously if accumulation is non-zero. As discussed above, these are fixed-capital intensity, k; scale, proxied by productive employment, [N.sup.p], and land scarcity; and the state of technique, A. Let

[alpha] = [alpha](k, [N.sup.p], A); [[alpha].sub.1] > 0; [[alpha].sub.2] [greater than or equal to] 0 or [[alpha].sub.2] [less than or equal to] 0; [[alpha].sub.3] > 0. (30)

When increasing scale leads to more division of labour and higher productivity, [[alpha].sub.2] > 0, which captures the same effect as Eltis' Z > 1. But if land scarcity causes diminishing returns to the variable composite factor--and if this effect dominates that of IRS--then [[alpha].sub.2] < 0. If the two exactly offset one another, [[alpha].sub.2] = 0.

The factors affecting marginal and average product in capital-goods production, [gamma], may be expected to resemble (30) in all save land scarcity: [[gamma].sub.2] > 0 only, for economies of scale will be probable and land scarcity irrelevant. Since [beta] is an increasing function of [gamma], it will vary in the same direction as [alpha] with respect to changes in [N.sup.p] and A. But the effects of changes in k are ambiguous. Increasing capital intensity may increase [gamma] and thus [beta] on the one hand, but on the other will lower [beta] by requiring a larger proportion of the workforce in capital-goods production. For economy of exposition it will be assumed that the effect on a of changes in k dominates the effect on [beta], and hence that we can confine our attention to a alone.

(a) Capital Intensity

The positive sign of a1 follows from the assumption--which Smith certainly made--that masters are rational and self-interested. If masters assign productive workers between capital-goods and wage-goods production according to equation (22), k can remain constant and steady state can persist, ceteris paribus. But if for any reason, say because actual [beta] > [beta]* is too high, or because masters seek to equip their servants with larger amounts of fixed capital in order to raise productivity [k*.sub.1] < [k*.sub.1+n], capital grows faster than the productive workforce, [alpha], w and g begin to rise and steady state can no longer exist. Hence the rate of accumulation can no longer explain the natural wage.

(b) Division of Labour and Increasing Returns to Scale

If [[alpha].sub.2] > 0, continued growth in population, productive employment and capital stock will cause [alpha] and therefore gK to increase. In a diagram like that of figure 1, in which the w(gK) curve is now described by equation (29), there will be continual upward shift of the w(gK) curve, causing a continual increase in g* and w*. Though balanced growth might still exist, steady state cannot. Accumulation can no longer explain the natural wage. Note that this objection applies as much to the simple model as to one in which there is fixed capital, since division of labour and IRS may take place even when there is only circulating capital.

There is an even more serious objection to the Smithian conception of wages and growth, noted by no previous analyst as far as I know. For if growth is negative, scale decreases and [alpha] falls, other things being equal. Only in the stationary state, where w(gK) intersects w(gN) on the vertical axis in figure 1 and w* = [alpha][beta][pi] = s, can steady state exist in the presence of IRS: and that stationary steady state is dynamically unstable. Any positive or negative displacement must lead to cumulative departures from stationarity in either direction. Some analysts (for example, Barkai 1969; Eltis 2000, chapter 3) have modelled IRS in Smith's growth theory upon the implicit assumption, certainly not entertained by Smith himself, that growth is always non-negative. Others (for example, Samuelson 1977; Negishi 1989, chapter 3) have prudently evaded the issue.

(c) Land Scarcity and Diminishing Returns

This is the 'Canonical Classical Model of Political Economy' that Samuelson (1978) correctly attributed to Malthus, Ricardo and their contemporaries but more questionably to Adam Smith. If [[alpha].sub.2] < 0 with gK positive, continued growth in population, productive employment and capital stock will cause [alpha] and therefore gK to decrease. In figure 1 the w(gK) curve will shift downwards until it intersects gN on the vertical axis and the stationary state is reached. In this case the stationary state is dynamically stable, since negative growth would increase the marginal product [alpha] and thus shift the gK curve back upwards. As in the case of IRS, however, steady state can only exist when g* = 0, and hence accumulation can no longer explain the natural wage. Recognising this fact, Smith's successors (for example, Ricardo 1817 [1951], p. 93) took the natural wage to be simply the subsistence wage, s. Note that this objection too applies as much to the simple model as to one in which there is fixed capital.

(d) Technical Progress

A once-for-all increase in A will simply shift the w(gK) curve upwards in figure 1, raising w* and g* in the new steady state. It would seem that Adam Smith, like his successors in the English School with the possible exception of Malthus (Eltis 2000, pp. 169-70), generally conceived of technical change in terms of occasional, discrete increases in A. But if technical progress is continuous or even continual, w* and g* will always be rising, steady state cannot exist, and hence the natural wage cannot be explained by the rate of accumulation.

(e) The Demographic Parameters

Any change, ceteris paribus, in either the speed of population response to changes in wages, m, or the zero-population-growth wage, s, will affect w* and g* in a way opposite to that assumed by Smith in his macrodynamic conception of the natural wage. The explanation offered for the simple model above is equally applicable to the general model with fixed capital goods. That the possibility of such changes is not far-fetched is illustrated by the sustained campaign waged by Smith's immediate successors, especially Malthus and Thomas Chalmers, to persuade the lower orders to aim at higher standard of living by a voluntary (and virtuous) reduction in childbearing (Waterman 2004, chapter 9). The 'moral and religious education' they promoted, if successful, would raise s and thus the gN curve, lower g* and increase w*. It would also tend to lower m. A generation later J. R. McCulloch (1832; see Waterman 1991, p. 237) explained how s could become endogenous after a prolonged period of high wages. In terms of figure 1, the gN curve gradually shifts upwards for as long as w* > s, until it intersects gK on the vertical axis. Throughout this process the rate of accumulation falls and the dynamic-equilibrium wage (which we can no longer call the 'natural wage') rises. This objection also applies to the Simple Model as to the More Complicated Model.

6 Conclusions

1 If a steady-state growth equilibrium can persist without endogenous change in which the rate of accumulation is equal to the rate of population and workforce growth, then Adam Smith's macrodynamic conception of the relation between wages and growth may be regarded as a genuine theory of the natural wage. However, the conditions required for steady state are stringent, and some of these would seem to be at variance with other things Smith wanted to say about growth.

(i) When capital and employment of productive labour grow, the equilibrium wage must remain constant.

(ii) Capital intensity must remain constant, which can only be the case if the proportion of the productive workforce assigned to capital goods production is such that fixed capital grows at the same rate as circulating capital and population.

(iii) There can be no division of labour, or at any rate no consequent increasing returns to scale.

(iv) Land must be a free good.

(v) There can be no technical progress.

(vi) The speed at which population responds to any difference between the actual market wage and the 'subsistence' wage must remain constant.

(vii) The 'subsistence' wage must not be endogenous and must remain constant.

2 Even if steady-state growth could exist, however, we have seen that Smith's famous recipe for high wages rests crucially upon the assumption that the factors determining population growth in relation to the real wage remain constant, and that changes in the rate of accumulation are brought about by changes in the degree of parsimony: that is to say by changes in the behaviour of masters. But if parsimony remains constant and wages and accumulation are affected instead by autonomous changes in the zero-population-growth real wage, that is to say by changes in the behaviour of servants, then Smith's prediction is turned upside down. It is now the 'declining' state that is associated with a wage which is high but yet below 'subsistence' and therefore hardly likely to be 'chearful'; and the 'progressive' state in which wages are low though still above 'subsistence' and therefore not exactly 'melancholy'.

Smith appears to have thought of masters as the only active and innovating class in society, and of their 'servants' (that is, 'employees') as responding passively and merely biologically to circumstances created by their betters. This may have born some resemblance to reality during the mid-eighteenth century. But in the new, post-revolutionary century, Smith's successors in the English School of political economy (Waterman 2008) took a very different view. As Malthus and Chalmers never tired of insisting, 'the comfort and independence of the working classes are in their own hands' (Chalmers 1844, p. 32). By setting themselves a higher target income and by postponing marriage and child-bearing until this was in prospect, economic growth could be retarded and wages permanently increased.

Smith's macrodynamic conception of the real wage therefore rests upon a set of contingent, class-based behavioural assumptions that were becoming less plausible by the end of the eighteenth century and were rejected in the nineteenth.

3 It would appear that increasing returns to scale produced by the division of labour are as much of an embarrassment to classical, macroeconomic theory as they are to neoclassical, microeconomic theory. In the latter case, as is well-known, IRS undermine the possibility of general competitive equilibrium. In the former case--it would appear from the argument of this paper--by making the stationary state dynamically unstable IRS undermine growth theory.

4 When carefully shielded from IRS, 'classical' growth theory reveals that it is more general than the yon Neumann growth theory in at least one not unimportant respect. For when the degree of parsimony is unitary, that is to say when the unproductive sector is suppressed, we obtain the neo-classical and von Neumann equality of growth rate with the rate of profit. In general however, g* < r because even the most parsimonious master will spend some part of his revenue on unproductive labour (see Appendix 3).

Appendices

1 The Simple Model

Equations (1), (2), (3) and (6) imply a second-order difference equation in N(t):

[N.sub.t+1] - (1 - ms)[N.sub.t] - m[pi][alpha][N.sub.t-1] = 0. (A1)

When the standard procedures are applied (for example, Chiang 1984, chapter 17.1), the characteristic equation of (A1) appears as

[b.sup.2] - (1 - ms)b - m[pi][alpha] = 0. (A2)

There are thus two roots and the larger (dominant) root will determine the time-path of the system [N(t), K(t), w(t)]. But in terms of my exposition in terms of growth rates,

b = (1 + g*), (A3)

therefore by substitution of(A3) in (A2), we obtain

[g*.sup.2] + (1 + ms)g* - m([alpha][pi] - s) = 0. (A4)

This is identical to the quadratic in g*, equation (15) above, which arises from the fact that the w(gK) function is a rectangular hyperbola. My model specification in terms of two simultaneous equations in w(gK) and w(gN), selected for its convenience in graphical analysis, short-circuits the more formal procedure and arrives at the same result. It should be noted that since (A4) is expressed in terms of g*, then gN = gK, and hence the difference equation (A1) in N(t) has the same form as that difference equation in K(t) which could be extracted from the system with more effort.

2 The More Complicated Model

Equations (19), (20), (25), (26) and (6) in K(t), C(t), N(t), F(t) and w(t) imply a second-order difference equation in N(t):

[N.sub.t+1] - (1 - ms)[N.sub.t] - m[pi][alpha][beta][N.sub.t-1] = 0 (A5)

which is only slightly different from (A1). Actually (25), (26) and (6) alone are sufficient for (A5). But since it can be shown, by reasoning similar to that of Appendix 1, that (AS) corresponds to the quadratic that could be obtained for g* = gK = gN from equations (29) and (7), then it must also describe the time-path of K(t)although this variable does not enter into equations (25), (26) and (6).

3 The Irrelevance of Profits and of the Rate of Profit

Some authors who have modelled Smith's growth theory (for example, Barkai 1969; Samuelson 1978) have made the growth of capital an increasing function of the rate of profit. It is easy to relate the wage rate to the profit rate when the latter is defined for the economy as a whole as r = (F - [N.sup.p]w)/K, and hence that

r = ([alpha][beta] - w)/k, (A6)

which is a form of Hollander's (for example, 1979, p. 11) 'inverse wage-profit theorem'. In the Simple Model in which w = k and [beta] = 1, (A6) reduces to

r = [alpha]/w - 1. (A7)

Figures 1, 2 and 3 could therefore be redrawn with r on the vertical axis and with positively sloped r(gK) curves and negatively sloped r(gN) curves, somewhat similar to that used by Eltis (2000, chapter 5) to analyse Malthus' theory of oscillations.

It is evident from (5) and (A7) that if [pi] = 1, that is to say, if the whole of last period's production were used by masters to employ productive labour in the current period, the rate of profit would be equal to the rate of accumulation in steady state. This well-known result--which appears in the von Neumann growth model (Negishi 1989, p. 88) and also in the neoclassical growth model when it is assumed that workers spend all their wages and capitalists invest all their profits--is obviously dependent upon the absence of any 'surplus' in the classical (macroeconomic) sense, and therefore the absence of any 'unproductive' sector. Not a penny is frittered away on 'churchmen, lawyers, physicians' or 'men of letters' like Adam Smith himself--not to mention 'players, buffoons, musicians, opera singers' and so on. (WN II.iii.2).

However, the existence of a surplus, and the consequent distinction between productive and unproductive employment, is central to Smith's account of accumulation in WN II.iii. There is no textual evidence to support the view that saving and investment take place out of profits rather than out of last period's total output. And as Blaug (1998, p. 55) correctly observes, 'Smith never suggested that saving is a function of the rate of interest or the size of net revenue'. Moreover, as I have shown elsewhere (Waterman 1999), there is no positive relation in WN between the rate of profit and the rate of accumulation: indeed such evidence as there is runs the other way.

Yet if we solve (A6) for w and substitute in (29) we obtain an equation in which the propensity to accumulate, gK, is an increasing function of r. What is the explanation? It is simply that given [alpha] and [beta], the higher the rate of profit the lower will be the wage rate, and therefore the larger the productive work force which can be employed by any capital, hence the larger the output in the next period: and therefore the larger the creation of new capital at any given degree of parsimony. It would appear, therefore, that profits and the rate of profit are redundant in any account of Smith's growth theory and may be ignored.

References

Barkai, H. 1969. 'A formal outline of a Smithian growth model', Quarterly Journal of Economics, 83, pp. 396-414.

Blaug, Mark. 1998. Economic Theory in Retrospect, 5th edn. Cambridge, UK: Cambridge University Press.

[Chalmers, Thomas] 1844. 'The Political Economy of the Bible', The North British Review, November, pp. 1-52.

Chiang, Alpha C. 1984. Fundamental Methods in Mathematical Economics, 3rd edn. New York: McGraw-Hill.

Eltis, Walter. 2000. The Classical Theory of Economic Growth, 2nd edn. Basingstoke, UK: Palgrave.

Hollander, Samuel. 1973. The Economics of Adam Smith. Toronto: University of Toronto Press.

Hollander, Samuel. 1979. The Economics of David Ricardo. Toronto: University of Toronto Press.

Hollander, Samuel. 1980. 'On Professor Samuelson's canonical classical model of political economy', Journal of Economic Literature, 18(2), pp. 559-74.

Hollander, Samuel. 1984. 'The wage path in classical growth models: Ricardo, Malthus and Mill', Oxford Economic Papers, 36, pp. 200-12.

Landreth, H. and Colander, D.C. 2002. History of Economic Thought, 4th edn. Boston, MA: Houghton Mifflin.

[Malthus, T. R.] 1798. An Essay on the Principle of Population. London: Johnson.

[McCulloch, J.R.] 1832. 'Dr Chalmers on Political Economy', Edinburgh Review, October, pp. 52-72.

Negishi, Takashi. 1989. History of Economic Theory. Amsterdam: North Holland.

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Ricardo, David [1817] 1951. On the Principles of Political Economy and Taxation, volume I in The Works and Correspondence of David Ricardo (edited by P. Sraffa). Cambridge, UK: Cambridge University Press.

Samuelson, P.A. 1977. 'A modern theorist's vindication of Adam Smith', American Economic Review, 41, pp. 42-9.

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Smith, Adam. 1776 [1976]. An Inquiry into the Nature and Causes of the Wealth of Nations, edited by R. R. Campbell, A. S. Skinner and W. B. Todd. Oxford: Clarendon.

Staley, C.E. 1989. A History of Economic Thought: From Aristotle to Arrow. Cambridge, MA: Blackwell.

Waterman, A. M. C. 1991. Revolution, Economics and Religion: Christian Political Economy, 1798-1833. Cambridge, UK: Cambridge University Press.

Waterman, A.M.C. 1999. 'Hollander on the "Canonical Growth Model": a comment', Journal of the History of Economic Thought, 21, pp. 311-13.

Waterman, A. M. C. 2004. Political Economy and Christian Theology Since the Enlightenment. Basingstoke, UK: Palgrave Macmillan.

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Notes

(1) I am grateful to Paul Samuelson for helpful comments on a first draft; and very grateful to an anonymous referee for his cogent and justified criticism of my mathematical style, which I hope is now more lucid. Any infelicities which remain are my sole responsibility.

(2) It was 'to the established administration of property'--which Godwin had assailed--that Malthus attributed these benefits. But as I have explained elsewhere (Waterman 1991, pp. 45-57, appendices 1 and 2), this is because Malthus saw property rights as a necessary condition of any surplus.

(3) Samuelson's (1977) vindication of Smith's 'value-added' account of the natural price requires a determinate and positive value for rental costs of production. But Smith's macrodynamic conception of the real wage requires that land be free and therefore that rents be zero. As in his 1978 publication, Samuelson assumes that Smith really meant that land is scarce, hence that a stationary equilibrium a la Ricardo will exist and rents will be determined. In effect, Samuelson vindicates Smith's microeconomics at the expense of his macroeconomics.

A. M. C. Waterman, John's College, Winnipeg. Manitoba R3T 2N2, Canada. Email: watermn@cc.umanitoba.ca.
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