The Conditions for a Falling Rate of Profit

C. The Conditions for a Falling Rate of Profit

The “standard” formula for the Marxian profit rate is:

s /v

R = 100 · s /(c + v) = 100 (c /v) + 1

where s, c, and v refer respectively to surplus, “constant” capital, and “variable” capital, all in “value” terms. 3 This formula implies that both constant and variable capital are turned over once per period, i.e., that (c + v) is the stock as well as the “flow” of capital. Marx himself justified the simplification in the chapter on the formation of a general rate of profit: “The magnitude of the actual value of [the] product depends on the magnitude of the fixed part of the constant capital, and on the portion which passes from it through wear and tear into the product. But since this circumstance has absolutely no bearing on the rate of profit, and hence, [on] the present analysis, we shall assume, for the sake of simplicity, that the constant capital is everywhere uniformly and entirely transferred to the annual product of

the capitals” (MECW 37: 153). 4 In the discussion of the falling profit rate, Marx also proceeds in this manner, obliging Engels to spell out the implicit assumption that “capital is turned over in exactly one year” (225; cf. 54 regarding the period of turnover of variable capital). 5

The formula is more revealing when rewritten to distinguish explicitly between the physical capital-labor ratio and the prices (values) of capital-goods and wage-goods. Moreover, we shall treat the variables in average terms (per unit of

3 See the formula for the general profit rate, Chapter 1, p. 18. 4 In proceeding with his formal Transformation analysis, Marx does include depreciation in his calculation of cost price, thus allowing for a difference between total and “used up” capital. “But,” he repeats, “this is immaterial to the rate of profit” (MECW 37: 155). In any event, cases existed where “the entire constant capital [goes] into the annual product” (154). 5 The chapter dealing explicitly with the effect of the turnover on the rate of profit (Capital 3, Chapter 4) is an insertion by Engels.

C. The Conditions for a Falling Rate of Profit 115 labor – say, the workday) rather than as totals. Selecting a day’s labor as value unit,

we have: L · (surplus per workday)

R = 100 · s /(c + v) = 100 L · (constant capital per workday)

+ L · (variable capital per workday)

+ P L where L is the number of workdays; P L < 1 the value of the (gives) commodity wage

L k + LP L

per day, the condition that labor embodied in the daily wage falls short of one day assuring a positive surplus; (1 – P L ), the surplus generated per day; and (P k K)/L

the value of constant capital per day. 6 Dividing through by P L we obtain:

100 · (1 − P L )/P L

R=

(P K / P L )(K /L) + 1 100/P L

· (1 − P L )

(P K / P L )(K /L) + 1

The profit rate is thus the product of the number of workdays employed per $100 of capital invested 7 and surplus per day, which is intuitively convincing. Technical progress largely takes the form of raising K /L – say by a constant rate of increase α. Let the effect of such progress be to reduce the values of capital-goods (for simplicity without lag) by β K , and those of wage-goods by β L . We then have a time path for R(n),

R(n) = L 100 (n) · [1 − P L (1 − β L ) n ]

P L (1−β L ) = n

n · [1 − P L (1 − β n L ) 1+ ]

P K (1−β k ) n

P L (1−β L ) n · L (1 + α)

Evidently, a necessary (though insufficient) condition for falling R over time is falling L per $100. For surplus per day is rising continuously towards 1, so that the product L 100 (1 – P L ) can fall only if L 100 is falling.

We proceed to summarize the results for the limit to R(n) – the “ultimate” trend in the profit rate – with reference to three sets of situations: (1) β L = β K ; (2)

6 Our choice of value unit is arbitrary and one that Marx himself sometimes makes: “If . . . e.g. £ 1 = 1 working day (no matter whether you think in terms of a day or a week, etc.), the working day = 12 hours, and the necessary labour (i.e., reproductive of the pay) = 8 hours, then the wage of 30 workers (or working days) = £ 20 and the value of their labour = £ 30, the variable capital per worker (daily or weekly) = £ 2/3 and the value he creates = £ 1” (letter to Engels, 2 August 1862; in MECW 41: 395).

7 The left-hand expression can be derived directly from the budget constraint: P L L+P K K= 100, i.e., L per $100 investment (written L 100 ) = (100/P L ) – (P K /P L )K. But K = gL where g

is the technologically-determined “capital-labour” ratio. Therefore L 100 = (100/P L ) − (P K / P L )gL 100 =

100/P L 1+(P K / P L )(K /L)

Economic Growth and the Falling Rate of Profit

β L <β K , implying some check to the impact of new technology on the wage- goods sector relative to that on the capital-goods sector; 8 and (3) β L >β K . Since lim [1 – P L (1 – β L ) n ] is 1, it does not enter into the calculation of lim R(n) – only L 100 is relevant. (The details of the analysis are given in Hollander 1991, Appendix A.)

Uniform cost reductions (β) in capital-goods and wage-goods sectors

R tends to zero if (1 – β)(1 + α) > 1, i.e., if α > β/(1 – β) or β < α/(1 + α) where α is the percentage annual increase in the technical capital-labor ratio or, in this case, in the organic composition of capital.

Non-uniform cost reduction (β L

If K (1 + α) ≤ 1 (4.1)

1−β L

then R tends to infinity. If K (1 + α) > 1 (4.2)

1−β L

the outcome will vary: if (i) (1-β K )(1 + α) ≤ 1, R tends to infinity; but if (ii) (1 – β K )(1 + α) > 1, R tends to zero. Now the first condition for ultimately falling R (namely, 4.2) relates to the denominator in the expression

1/P L (1 − β L ) n

L 100 (n) = 100

n ; P L (1−β L ) n · (K /L)(1 + α)

1+ P K (1−β K ) n

and it implies that the value of capital-goods per labor unit relative to the value of the wage (P K K)(P L L) – the “value composition of capital” – is rising. 9 In the event β K >β L , the ratio P K /P L declines, and should this decline outweigh the increase in the physical capital-labor ratio, then L 100 (n) → ∞. However, with β L >β K the upward trend in the technical composition is reinforced by a rising price ratio. But the denominator may rise less than the numerator. To avoid this outcome the second condition (4.2ii) must be satisfied, namely (1– β K )(1 + α) > 1 or α > β K /(1– β K ), the latter constituting in fact the sufficient condition for (ultimately) falling L 100 (n). 10

8 Alternatively, let β L = β K , with α L >α K implying the need for more constant capital per man to assure a given level of cost reduction.

9 Where β K = β L this condition is necessarily satisfied since the only force acting on the value composition is the rise in the capital-labor ratio, K/L.

10 Note that β K /(1 – β K ) = 1/(1 – β K ) – 1 which is the percentage change:

1 1 (1−βK )n−1 (1−βK )n−1

(1−βK )n 1 −

C. The Conditions for a Falling Rate of Profit 117 For a specific example of the case of uniform price reductions (β K = β L ), we

consider values for β of .03, .05, .09, and .091 with α = .10:

n→∞ lim R(n) .03

In brief, with the physical capital-labor rates rising at α = 0.1 per period, the profit rate will (ultimately) decline for all β up to a value just short of .091; for higher β value, the trend is upwards. (For a detailed comparison between the cases, especially the speed of decline in the first three, see Hollander 1991, Appendix B, Figures 1.1(a)–(d).)

In the case β K >β L consider first β K = .09, β L = .045. Then K (1 + α) = 1.048 > 1

1−β L

and (1 – β K )(1 + α) = 1.001, so that R tends to zero. Needless to say, satisfaction of the second condition assures satisfaction of the first. But the opposite is not true. Thus with β K = 0.092 and β L = .046,

K (1 + α) = 1.047 > 1;

1−β L

but (1 – β K )(1 + α) = .999 < 1 so that R tends to infinity. (Hollander 1991, Appendix B, Figures 1.2(a)–(d).)

When we reverse the relative magnitudes of β K and β L there occurs a startling transformation: in the case of β L = .09 and β K = .045, the decline in R now sets in much earlier and markedly so; and a similar pattern now also emerges for β L = .092, β K = .046. (For (1 – β K ) (1 + α) = .954 (1.1) = 1.05 > 1.) Notice that β L >β K does not guarantee that L 100 (n) and, therefore, R(n) tend to zero. (Thus β K = .0909, β L = .091, entails an ultimate decline in L 100 and R; a slightly different configuration, β K = .09091, β L = .091, yields a continuous increase.) But β L >β K does add to the pressures reducing L n and R, and may transform a rising into a falling R trend. This result is not contingent upon the specific initial factor values or factor ratios selected. (For example: with K/L = 4 and P K /P L = 1, almost the same pattern emerges as with the values in Hollander 1991, Appendix B, Figure 1.2, namely K/L = .25, P K /P L = 20.) An ultimate decline in R(n) does not, however, preclude an initially rising trend. Initial conditions do play a strategic role as far as oncerns the entire course of R(n). On the other hand, an upward trend is incompatible with an initial decline (see Hollander 1991, Appendix A).

Economic Growth and the Falling Rate of Profit