Notes on Sraffa's Production, Chapter 6

45. Cost of production aspect

• Sraffa considers prices from their "cost of production" aspect, and examines the way they "resolve themselves" into wages and profits.
• Sraffa would have introduced the argument earlier "had it not been for the necessity of following one line of argument at a time".

46. "Reduction" defined

• Definition. We call Reduction to Dated Quantities of Labor (or "Reduction" for short) an operation where the equation for a commodity, the different means of production used are replaced with a series of quantities of labor, each with its appropriate "date".
• Consider the equation representing the production for commodity 'a' (where wage and prices are expressed in terms of the Standard commodity): $$(A_a{p}_a+B_a{p}_b+...+K_a{p}_k)(1+r)+L_{a}w=A{p}_a$$
  • We start with replacing the commodities forming the means of production for A with their own means of production and quantities of Labor.

    In other words: we replace them with the commodities and labor which (as appears from their own respective equations) must be employed to reproduce those means of production; and they, having been expended a year earlier (§9), will be multiplied by a profit factor at a compound rate for the appropriate period…namely, the means of production by (1 + r)² and labor by (1 + r).

  • It may be noted that A_a — the quantity of commodity a itself used in the production of A — is to be treated like any other means of production…i.e., replaced by its own means of production and labor.
  • Remark. Here we are "almost dynamic" but "still quite static"! We are taking into account time, kind of, but we are really…not.
• We next replace these latter means of production with their own means of production and labor, and to these will be applied a profit factor for one more year. Or to the means of production (1 + r)³ and to the labor (1 + r)².
• We can carry this operation on as far as we like. If next to the direct labor L_a we place the successive aggregate quantities of labor which we collect at each step and we call respectively $L_{a_1}$, $L_{a_2}$, …, $L_{a_n}$, …, we shall obtain the Reduction Equation for the product in the form of an infinite series $$L_{a}w+L_{a_1}w(1+r)+...+L_{a_n}w(1+r{)}^n+...=A{p}_a.$$
• How far reduction needs to be pushed in order to obtain a given degree of approximation depends on the level of the rate of profits: the nearer the latter is to its maximum, the further must the reduction be carried.
• Beside the labor terms, there will always be a "commodity residue" consisting of minute fractions of every basic production; but it is always possible, by carrying the reduction sufficiently far, to render the residue so small as to have a negligible effect on price (at any prefixed rate of profits short of R).
  • Remark. I object to this supposition. If we carry this operation "infinitely far back", then we carry it back to a time predating humans. From a strictly historical perspective, humans began with labor alone and constructed simple tools…then constructed complex tools. Sraffa, I believe, errs suggesting "things were as they are" — a common sin among economists!
• Sraffa notes only at r = R the residue becomes all-important as the sole determinant of the price of the product.
  • Mathematically, this makes sense since w = 0 when r = R. Hence the infinite series sums infinitely many zeroes.

47. Pattern of movement of individual terms with changes in distribution

• As the rate of profits rises, the value for each of the labor terms is pulled in the opposite direction by the rate of profits and by the wage…and it moves up or down as the one or the other prevails.
• The relative weight of these two factors varies at different levels of distribution. Besides, it varies differently in the case of terms of different "date", as we shall see.
• We have seen (§30) that — if wage is expressed in terms of the Standard net product — when the rate of profits r changes, the wage w moves as $$w=1-<!--\; -\; -->\frac{r}{R}$$ where R is the maximum rate of profits.
• Substituting this expression for the wage in each term in the reduction equation, the general form of any n^th labor term becomes $$L_{a_n}\left(1-<!--\; -\; -->\frac{r}{R}\right)(1+r{)}^n$$
• Consider the values (for this expression) as r moves from 0 to its maximum R.
  • At r = 0, the value for a labor term depends exclusively on its size regardless of date.
  • With a rise in the rate of profits, terms fall into two groups:
    1. those that correspond to labor done in more recent past (which begin at once to fall in value and fall steadily throughout);
    2. those representing labor more remote in time (which rise at first, then as each of them reaches its maximum value, turn and begin downward movement).
  • In the end, at r = R, the wage vanishes and with it vanishes the value of each labor term.
  • This is best shown by a selection of curves, representing terms of widely different dates n and different quantities of labor. Lets doodle this when R is 25%.

    Variation in value of "Reduction terms" of different periods $[L_{n}w(1+r{)}^n]$ relative to the Standard commodity as the rate of profits varies between zero and R (assumed to be 25%).

    The quantities of labor ($L_n$) in various "terms", which have been chosen so as to keep the curves within the page, are as follows: $L_0=1.04$ (dashed black line); $L_4=1$ (orange line); $L_8=.76$ (red line); $L_{15}=.29$ (green line); $L_{25}=.0525$ (blue line); $L_{50}=0.0004$ (solid black line).
• It is as if the rate of profits (when moving from 0 to R) generate a wave along the row of labor terms, the crest formed by successive terms, as one after another reach their maximum value.
  • At any value of the rate of profits, the term which reaches its maximum has the "date" $$n=\frac{1+r}{R-<!--\; -\; -->r}$$
  • Conversely, the rate of profits at which any term of date n is at its maximum when $$r=R-<!--\; -\; -->\frac{1+R}{n+1}$$
  • Accordingly, all terms for which n ≤ R^-1 have their maximum at r = 0 and thus form the group of "recent dates" mentioned above as falling in value for increasing r.

48. Movement of an aggregate of terms

• The labor terms may be regarded as the constituent elements of the price of a commodity, the combination of which may (with variation in the rate of profits) give rise to complicated patterns of price-movement, with several ups and downs.
• The simplest case, the "balanced commodity" (§21) or its equivalent, where the Standard commodity taken as an aggregate: its Reduction would result in a regular series, the quantity of labor for any term being (1 + R) times the quantity in the term immediately preceding it in date.
• Consider a complicated example: we suppose two products which differ in three of their labor terms, while being identical in all others.
  • One of them, a, has an excess of 20 units labor applied 8 years before, whereas the excess of the other (b) consists of 19 units employed in the current year and 1 unit bestowed 25 years prior.
  • (They are thus not unlike the familiar instances, respectively, of the wine aged in the cellars and of the old oak made into a chest.)
  • The difference between their Standard prices at various rates of profit, i.e. $$p_a-<!--\; -\; -->p_b=20w(1+r{)}^8-<!--\; -\; -->(19w+w(1+r{)}^{25})$$ is represented in the following figure:
• The price of "old wine" rises relative to the "oak chest" when the rate of profits move from 0 to 9%, then falls between 9% and 22% to rise again from 22% to 25%.
• (The reduction to dated labor has some bearing on the attempts to find in the "period of production" an independent measure of the quantity of capital which could be used — without circular reasoning — for determining prices and the shares in distribution.
  
  (But the case just considered seems conclusive showing the impossibility of aggregating the "periods" belonging to several quantity of labor into a single magnitude which could be regarded as representing the quantity of capital.
  
  (The reversals in direction of the movement of relative prices, in the face of unchanged methods of production, cannot be reconciled with any notion of capital as measurable quantity independent of distribution and prices.)
  • In this parenthetic remark, Sraffa just decimated the Neoclassical theory of production.

49. Rate of fall of prices cannot exceed rate of fall of wages

• Something restricts the movement of any product's price: if (as a result of a rise in the rate of profits) the price falls, its rate of fall cannot exceed the rate of fall of the wage.
• So, if we draw two lines showing how the price for product a and the wage (both expressed in terms of the Standard commodity) vary with the rise of the rate of profit, we see the price line cannot cut the wage line more than once…and even then, only in one direction: such that the price (from being lower) becomes higher than the wage with the rise of the rate of profits.
• How to see this? We may look at the Reduction series or the original production equation for a. Sraffa considers the former.
  • The only variables (besides the price for a) are the wage and rate of profits, which rises with the fall of the wage…the combined effect of the two can never fall in the price more than in proportion to that of the wage.
  • Sraffa next considers the production equation for commodity a. The prices for the means of production might upset the proposition if they were themselves capable of falling at a greater rate.

    But to see this is impossible, it is sufficient to turn our attention to the product whose rate of fall exceeds that of all others: this product (since it cannot have means of production capable of falling at a greater rate than it does) must itself fall less than wage.

• The conclusion is not affected if we take as measure of wages and prices any arbitrarily chosen product (instead of the Standard commodity), since what we are concerned with is the price-relation between labor and the given product…a relation which is independent of the medium adopted.
• It follows if wage is cut in terms of any commodity (no matter whether it is one that will rise or fall relative to the Standard commodity) the rate of profits will rise…and vice-versa for an increase of the wage.
• It also follows, if wage is cut in terms of one commodity, it is thereby cut in terms of all…and similarly for an increase. The direction of change is the same in relation to all commodities, however different may be the extent.

Addendum (5 March 2016): I realize I mistakenly wrote the equation $$p_a-<!--\; -\; -->p_b=20w(1+r{)}^3-<!--\; -\; -->(19w+w(1+r{)}^{25})$$ instead the exponent in the first term should not be a 3 but an 8, i.e., it should be: $$p_a-<!--\; -\; -->p_b=20w(1+r{)}^8-<!--\; -\; -->(19w+w(1+r{)}^{25})$$ This correction should produce the correct plot (with w = 1 - r/25%, of course).