Macroeconomic

A Growth Cycle

by R. M. Goodwin

[Originally published in C. H. Feinstein (ed), Capitalism and Economic Growth, Cambridge University Press, 1967, pp 54-8. Reprinted, revised and enlarged, in E. K. Hunt and Jesse G. Schwartz (eds), A Critique of Economic !eory, Penguin Books, 1972, pp 442-449. )is edition reprints the latter, with corrections of typographical errors.]

Since its *rst appearance capitalism has been characterized by alternating ups and downs. )is paper attempts to give more precise form to an idea of Marx’s — that it can be explained by the dynamic interaction of pro*ts, wages and unemployment. My thesis is that the very structure of capitalism constitutes a homeostatic mechanism which functions by means of variations in distributive shares but does so in such a way as to keep them constant in the long run. If real wages go up, pro*ts go down: if pro*ts go down, saving and investment lag, thus slowing up the creation of new jobs. But the labour force is continually growing both through natural increase and through men ‘released’ by technological progress. )e reserve army of labour grows, wages lag behind the growth of productivity, pro*ts rise and accumulation is accelerated back up to a high level. )is in turn gradually reduces unemployment, wages rise, and so it goes on, inde*nitely.

)e structure of such a system is somewhat more complex than might appear and therefore it seemed advisable to use some mathematics to check that the quantitative logic does indeed con*rm the conclusions. Also it enables us to get some further results which are not quite obvious, for example, that the mechanism implies a long-run constancy of relative shares of wages and of pro*ts. )is suggests an explanation of the paradox that every trade unionist feels he can and is certain that he has, in fact. raised wages at the expense of pro*ts, whereas the scanty evidence suggests that this distribution has not changed signi*cantly during a century of growing trade union power.

Presented here is a starkly schematized and hence quite unrealistic model of cycles in growth rates. )is type of formulation now seems to me to have better prospects than the more usual treatment of growth theory or of cycle theory, separately or in combination. Many of the bits of reasoning are common to both, but in the present paper they are put together in a di+erent way.

)e following assumptions are made for convenience:

1. Steady technical progress (disembodied).

2. Steady growth in the labour force.

3. Only two factors of production, labour and ‘capital’ (plant and equipment), both ho- mogeneous and non-speci*c.

4. All quantities real and net.

5. All wages consumed, all pro*ts saved and invested.

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)ese assumptions are of a more empirical, and disputable, sort:

6. A constant capital-output ratio.

7. A real wage rate which rises in the neighbourhood of full employment.

Number 5 could be altered to constant proportional savings, thus changing the numbers but not the logic of the system. Number 6 could be so-ened but it would mean a serious compli- cating of the structure of the model.

)ese assumptions are too simple and too crude to represent reality; they are not, however, arbitrarily or frivolously chosen. )ey were chosen because they represent, in my opinion, the most essential dynamic aspects of capitalism; furthermore, they are factually based, to the order of accuracy implicit in such a model. Number 6 should be a result and not an assump- tion, in which case it need only be roughly true over time. Number 7 should run in terms of money, not real, wages, which, with allowance for in.ation, would achieve the same sort of result but at a cost of considerable complication. Any Marxist-inclined economist should ask: why analyse an unreal, idealized system? )e answer is that to show the logic and plausibility of a type of behaviour and of its analysis, it is essential to get it clearly and simply stated. If and when such an analysis *nds wider acceptance, then it is not too di/cult to make the model more realistic by incorporating additional, empirically valid assumptions.

Symbols used are:

q is output ;

k is capital ;

w is wage rate;

a = a0eαt is labour productivity, α constant;

σ is capital-output ratio (inverse of capital productivity);

w/a is workers’ share of product, (1 − w/a) capitalists’; Surplus - pro*t = savings = investment = (1 − w/a)q = k̇ ; Pro*t rate = k̇/k = q̇/q = (1 − w/a)/σ; n = n0eβt is labour supply, β constant;

l = q/a is employment.

Writing (q̇/l) for d/dt(q/l), we have

(q̇/l) q/l

= q̇

q −

l̇

l = α

so that l̇

l =

1 − w/a σ

− α .

Call u = w

a , v =

l

n ,

3

so that v̇

v =

1 − u σ

− (α + β)

Assumption 7 may be written as

ẇ

w = f (v)

as shown in Figure 1.

0 +1 v

ẇ

w

f (v)

Figure 1

)e following analysis can be carried out using such an f (v), with a change in degree but not in kind of results. Instead, in the interest of lucidity and ease of analysis, I shall take a linear approximation (as shown in Figure 1),

ẇ

w = −∞ + ρv

and this does quite satisfactorily for moderate movements of v near the point +1. Both ∞ and ρ must be large. Since

u̇

u =

ẇ

w − α

u̇

u = −(α + ∞) + ρv

From this and the equation above for v, we have a convenient statement of our model.

v̇ = ∑Ω

1 σ

− (α + β) æ

− u

σ

∏ v(1)

u̇ = {−(α + ∞) + ρv}u(2)

In this form we recognize the Volterra (1931) case of prey and predator. To some extent the similarity is purely formal, but not entirely so. It has long seemed to me that Volterra’s problem of the symbiosis of two populations — partly complementary, partly hostile — is helpful in

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the understanding of the dynamical contradictions of capitalism, especially when stated in a more or less Marxian form.

)is Golden Goose Egg )eory of capitalism seems to me to *t actual working-class expe- rience and trade union strategy better than the straight Marxian one. )us it may help to explain some of the lack of success of Marxism in the unions. It also helps to explain, and in some measure to forgive, the fatuity and pusillanimity of social democracy.

Eliminating time and performing a *rst integration we get

1 σ

u + ρv − ∑

1 σ

− (α + β) ∏ log u − (∞ + α) log v = constant.

Letting θ1 = 1 σ ; η1 =

1 σ

− (α + β) ,

θ2 = ρ ; η2 = ∞ + α,

we can transform this into

ϕ(u) = uη1 e−θ1u = Hv−η2 eθ2v = H√(v),(3)

where H is an arbitrary constant, depending on initial conditions; since 1/σ > (α + β), all coe/cients are positive. By di+erentiating

dϕ

du = (−θ1 +

η1 u

)ϕ,

d√

dv = (θ2 +

η2 v

)√,

so that we can see that these functions have the sorts of slopes given in Figure 2.

ϕ

u

ϕ(u)

η1/θ1

√

v

√(v)

η2/θ2 Figure 2

Our problem as stated in 3 is to equate ϕ(u) to √(v) multiplied by a constant H. )is can be done neatly in the four quadrant positive diagram in Figure 3. We draw through the origin a straight line, A, with the slope ϕ/√ = H (arbitrary since dependent on the given initial condition). )en in symmetrical quadrants we place the two curves ϕ and √ and equating

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these two through the constant of proportionality gives a possible pair of values for u and v. All possible pairs of u and v constitute a solution, which may be plotted in the remaining quadrant. It can be shown, and indeed is quite obvious, that these solution points lie on a closed, positive curve, B, in u, v space. By going back to equations 1 and 2 we can *nd in what order the points succeed each other and hence in what direction we traverse curve B, as indicated by arrows in Figure 3. A second integration will yield u and v as functions of time, thus allowing us to determine the second arbitrary factor, the point on B at which we start. By varying the slope of A we can generate a family of closed curves broadly similar to B, thus yielding all the possible solutions. One initial condition selects the curve, a second *xes the starting point, and then we traverse some particular curve B in the direction of the arrows for ever, in the absence of given outside changes. )ere remains only to spell out the meaning of the motion.

s

s s

s s

s s

s s

s s s

° °

° °

° °

° °°

+1

+1 u

v

√ 0 ξ1 η1/θ1 ξ2

ϕ

≥1

≥2

η2/θ2

ϕ(u)

√(v)

B

A

❍❍❥

❍❍

Figure 3

Hence we may classify our model as a non-linear conservative oscillator of, fortunately, a sol- uble type. Since it is non-linear, the solution would not be essentially altered by replacing −∞ + ρv with f (v); it would still be a conservative (closed orbit) oscillator. However, more cumbersome, graphical methods become necessary in place of Volterra’s elegant analytic ones. As the representative point travels around the closed curve B, u vibrates between ξ1 and ξ2, and v between ≥1 and ≥2. Both u and v must be positive and v must, by de*nition, be less than unity; u normally will be also but may, exceptionally, be greater than unity (wages and con- sumption greater than total product by virtue of losses and disinvestment). Over the stretch 0 to +1 on the u axis, the point u indicates the distribution of income, workers’ share to the le-, capitalists’ to the right. )e capitalists’ share, multiplied by a constant, 1/σ, gives us the pro*t rate and the rate of growth in output, q̇/q. When pro*t is greatest, u = ξ1, employment

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is average, v = η2/θ2, and the high growth rate pushes employment to its maximum ≥2 which squeezes the pro*t rate to its average value η1/θ1. )e deceleration in growth lowers employ- ment (relative) to its average value again, where pro*t and growth are again at their nadir ξ2. )is low growth rate leads to a fall in output and employment to well below full employment, thus restoring pro*tability to its average value because productivity is now rising faster than wage rates. )is is, I believe, essentially what Marx meant by the contradiction of capitalism and its transitory resolution in booms and slumps. It is, however, un-Marxian in asserting that pro*tability is restored not (necessarily) by a fall in real wages but rather by their failing to rise with productivity. Real wages must fall in relation to productivity; they may fall absolutely as well, depending on the severity of the cycle. )e improved pro*tability carries the seed of its own destruction by engendering a too vigorous expansion of output and employment, thus destroying the reserve army of labour and strengthening labour’s bargaining power. )is in- herent con.ict and complementarity of workers and capitalists is typical of symbiosis.

An undisturbed system has constant average values η1/θ1 for u and η2/θ2 for v, hence a constant long-run average distribution of income and degree of unemployment. Much more remarkable is the fact that a disturbed system still has the same constant long-run values. )e time averages of u and of v are independent of initial conditions. We can see this from the fact that a rotation of A (an outside change) will only make the curve B larger or smaller but will not alter its central point. )erefore continual shocks will alter the shape of the cycle but not the long-run average values. Output and employment both will show alternating rates of growth. Whether they actually decrease or merely rise less rapidly will depend on the sever- ity of the cycle. For a mild cycle the growth rate may decrease but never become negative; in other cases there may be a sharp fall. However, the increases must predominate over the decreases, since the time average of 1 − u is positive and hence so also is that of q̇/q. Likewise employment grows in the long run at the same rate as labour supply, since the time average of v is constant. Similarly the equality of the growth rate in wages to that in productivity follows from the constancy of u. By contrast the pro*t rate is equal to 1 − u and therefore tends to constancy. We may look at this as standing Ricardo (and Marx) on his head. Progress *rst ac- crues as pro*ts but pro*ts lead to expansion and expansion forces wages up and pro*ts down. )erefore we have a Malthusian Iron Law of Pro*ts. )is is because of the tendency of capital, though not capitalists, to breed excessively. By contrast labour is something of a rent good since the supply, though variable, does not seem to be a function of wages. Hence it is the sole ultimate bene*ciary from technical progress. By now there would, I suppose, be considerable agreement that what happened in history is: wage rates went up; pro*t rates stayed down. It is to the explanation of this that the present paper is addressed.

Reference

VOLTERRA, V. (1931), !èorie mathèmatique de la lutte pour la vie, Gauthier-Villars, Paris