Monday, April 22, 2013

Notes on Sraffa's Production, Chapter 7

50. Two methods of production for two joint products: or, one method for producing them and two methods for using them in the production of a third commodity

  • So far we have worked with industries, each producing a single commodity. But we may now ask "What happens if a single industry produces multiple products?" For example, we have something like:

    150 q. wheat + 12 t. iron + 3 units Labor → 3 pigs + 50 q. wheat
    25 q. wheat + 3 t. iron + 25 units Labor → 4 pigs + 30 q. wheat

    Note we have two processes producing pigs and wheat. Having joint products usually has multiple different processes producing the joint-products.

  • The conditions for production can no longer determine the prices. There would be "more prices to be ascertained than there are processes" (and hence equations) to determine them.
    • The system of equations thus becomes under-determined, as the kids would say nowadays in linear algebra courses...
    • Remark. I realize now if we have a joint-process producing k different "species" of commodities, we need k distinct methods of production for the system to (mathematically) have a solution.

      So for my example above, we need two distinct processes (i.e., two processes which are linearly independent) for a solution to exist. But for the example given, we need an iron sector before we can solve it.

  • Sraffa suggests there will be a second, parallel process which will produce the two commodities by a different method...and in different proportions (Sraffa hints this may change later).
    • This is mathematically necessary to solve the system of equations.
    • Sraffa takes a step further and assumes (in such cases) a second process or industry exists.
      • Footnote. Incidentally, considering the proprtions which the two commodities are produced by any one method will (in general) differ from those required for use, the existence of two methods of producing them in different proportions will be necessary for obtaining the required proportion of the two producets through an appropriate combination of the two methods.
  • Problem: In every case, will there be a second (or third or n), distinct method of production?

    This is not immediately obvious to me.

    • Sraffa notes "this may appear an unreasonable assumption to make", implying for every process there exists a second, distinct, process which is neither more nor less productive.
    • But no such condition as to equal productiveness is implied! Nor would it have any meaning before prices were determined.

      With different proportions of products, a set of prices can generally be found where the two different methods are equally profitable.

    • Remark. This does not seem satisfactory, to me at least. Is there any reason why we should expect there to be multiple distinct methods of production for joint-products?

      How many different ways are there to raise sheep (which would then be turned into wool and mutton)?

  • Thus any other method of producing the two commodities will be compatible with the first, subject only to the general requirement: the resulting equations are mutually independent and have at least one system of real solutions.

    This rules of, e.g., proportionality of both products and means of production in the two processes.

    • The only economic restriction: while the equations may be formally satisfied by negative solutions for the unknowns, only those methods are practicable which do not involve other than positive prices in the conditions actually prevailing (i.e., at the given wage or rate of profits).
  • The same result could be achieved through the commodities being used as means of production in different proportions in various processes.
    • Remark. This is an important point that should not be overlooked. If we use the joint-products as means of production in other sectors (or production processes), then we can achieve the same result.

      I am having difficulty grasping this: is Sraffa trying to set up a system resembling:

      (1 + r) Ap + wL = (I + B)p

      where I is the identity matrix, and B is the "deformation" to take into account joint-production?

  • It could be achieved even if the two commodities were jointly produced by only one process, provided they were used as means of production to produce a third commodity by two distinct processses...and more generally provided that the number of independent processes in the system was equal to the number of commodities produced.
    • The assumption previously made of the existence of "a second process" could now be replaced by the more general assumption the number of processes should be equal to the number of commodities.

51. A System of Universal Joint Products

  • The possibility for an industry having more than one product makes it necessary to reconstruct – to some extent – the equations devised for the case of exclusively single-product industries.

    To do so in a perfectly general way we shall instead of regarding joint products as the exception, assume them to be universal and to apply all processes and all products.

  • We consider a system of k distinct processes each of which produces, in various proportions, the same k products.
  • This does not eliminate the possibility that some of the products have a zero coefficient (i.e., are not produced) in some of the processes: just as it has been admitted throughout it is not necessary for each of the basic products to be used directly as means of production by all the industries.
  • The system of single-product industries is thus subsumed as an extreme case in which each of the products, while having a positive coefficient in one of the processes, has a zero coefficient in all others.
  • An industry or production-process is characterized now by the proportions in which it uses and the proportions in which it produces the various commodities.
  • Notation Change! In the present chapter, and the next, processes will be distinguished by arbitrarily assigned numbers 1, 2, …, k (instead of their products 'a', 'b', …, 'k').
    • Thus A1, B1, …, K1 denote the quantities of the various goods 'a', 'b', …, 'k' which are used as means of production in the first process; A2, B2, …, K2 those in the second; etc.
    • The quantities produced will be distinguished with their indices in parenthetics: A ( 1 ) , B ( 1 ) , …, K ( 1 ) being the products of the first process; A ( 2 ) , B ( 2 ) , …, K ( 2 ) the products of the second process; etc.
    • In this notation, we have the joint-production equations:
( A 1 p a + B 1 p b + + K 1 p k ) ( 1 + r ) + L 1 w = A ( 1 ) p a + B ( 1 ) p b + + K ( 1 ) p k ( A 2 p a + B 2 p b + + K 2 p k ) ( 1 + r ) + L 2 w = A ( 2 ) p a + B ( 2 ) p b + + K ( 2 ) p k = ( A k p a + B k p b + + K k p k ) ( 1 + r ) + L k w = A ( k ) p a + B ( k ) p b + + K ( k ) p k

52. Complications in constructing the Standard system

  • We can also construct the Standard system in the same way as was done in the case of exclusively single-product industries (§33). How?

    Namely by finding a set of multipliers which — applied to the k production equations — will result in the quantity of each commodity in the aggregate means of production for the system bearing to the quantity of the same commodity in the aggregate product a ration which is equal for all commodities.
  • Before proceeding to do so, however, it is necessary to remove certain difficulties.

    These arise from the greater complexity of the interrelations, which results in the creeping in of negative quantities on the one hand, and the disappearance of the one-to-one relation between products and industries on the other.

Monday, April 1, 2013

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 Aa — 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 La 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 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 nth labor term becomes L a n 1 r R ( 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 = 1 + r R r
    • Conversely, the rate of profits at which any term of date n is at its maximum when r = R 1 + R n + 1
    • Accordingly, all terms for which nR-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 = 20 w ( 1 + r ) 8 ( 19 w + 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 = 20 w ( 1 + r ) 3 ( 19 w + 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 = 20 w ( 1 + r ) 8 ( 19 w + w ( 1 + r ) 25 ) This correction should produce the correct plot (with w = 1 - r/25%, of course).