Monday, June 3, 2013

Notes on Sraffa's Production, Chapter 8

Ch. 8. The Standard System with Joint Products

53. Negative Multipliers: I. Proportions of Production Incompatible with Proportions of use

  • When we consider in detail how we construct a Standard system with joint products, it becomes obvious some of the multipliers may be negative.
  • Consider two products jointly produced, each through two different methods.
    • The possibility that varying the extent to which one or the other method is used ensures a certain range of variation in the proportions in which the two goods may be produced in aggregate.
    • For each commodity, its two methods limits the range of proportionality. The limits are reached as soon as one or the other method is exclusively employed.
  • Now suppose in all cases which two joint products 'a' and 'b' are used as means of production, the proportion in which 'a' is employed relatively to 'b' is invariably higher than the highest of the proportions in which it is produced.
    • In such circumstances we may say some process must enter the Standard system with a negative multiplier: but whether such a multiplier will have to be applied to the low producer or high user of commodity 'a' cannot be determined a priori—it can only be discovered through the solution of the system.

54. Negative Multipliers: II. Basic and non-basic jointly produced

  • Non-basic products are "the most fertile ground" for negative multipliers.
    • (NB: non-basic goods needs a new definition under these new circumstances…but we may say that the main class, i.e. products altogether excluded from the means of production, will still be non-basic; see §60)
  • Consider again the case of two commodities (jointly produced in different proportions by two processes). One is to be included in the Standard product while the other — not entering the means of production for any industry — must be excluded from the Standard product.
    • This will be effected by giving a negative multiplier to the process which produces relatively more of the second commodity, and a positive one to the other process.

      The two multipliers being so proportioned when the two equations are added up to the two quantities produced of the non-basic exactly cancel out…while a positive balance of its companion product is retained as a component of the Standard commodity.

55. Negative Multipliers: III. Special raw material

  • Once negative multipliers have been admitted for some processes, others (which shine with a reflected light) are liable to appear.
  • Hence, suppose we have a raw material be directly used in only one process. Suppose that process has a negative multiplier. Then the industry which produces the raw material will itself follow suit and enter the Standard system with a negative multiplier.

56. Interpretation of negative components of the Standard commodity

  • Since no meaning could be attached to "negative industries" which such multipliers entail, it becomes impossible to visualize the Standard system as a conceivable rearrangement of the actual processes.
  • We must therefore (in the case of joint-products) be content with the system of abstract equations, transformed by appropriate multipliers, without trying to think of it as having a bodily existence.
    • Remark. I'm sure many marginalist economists howl out in frustration over this, which is amusingly ironic.
  • The Standard system's purpose is to provide a Standard commodity. When it has negative components, there is no difficulty interpreting them: they are liabilities or debts. This is analogous to accounting (negative numbers = liabilities/debts; positive numbers = assets).
  • Hence a Standard commodity which includes both negative and positive quantities may be adopted as money of account without straining the imagination, provided the unit represents a fraction of each asset and each liability (like a share in a company)…with the liability in the shape of an obligation to deliver without payment certain quantities of particular commodities.

57. Basics and non-basics, new definition required

  • We have another difficulty we must tackle before constructing the Standard commodity: the criterion distinguishing basic and non-basic goods fail…since it's ambiguous whether a product entering the means of production for only one industry producing a given commodity should or should not be regarded as entering directly the means of production for that product.
    • Footnote: The trouble lies deeper, and as we shall see presently there would be uncertainty even if the commodity entered directly the means of production of all the processes in the system! See §59.
  • And the uncertainty would naturally extend to the question whether it did or did not enter "indirectly" the production of commodities, into which the latter entered as means of production.

58. Three types of non-basics

  • All three distinct types of non-basics are met in the single-product system will find their equivalents in the case of multiple-product industries.

    Taking advantage of this circumstance, we begin defining for the latter case the three types of non-basics, each as the extension of the corresponding single-product type (cf. §35).

    1. Products which do not enter the means of production for any industry. This type can be immediately extended to the multiple-product system without modifying anything.
    2. Products each of which enters only its own means of production. The equivalent would be a commodity which enters the means of production for each of the processes by which it is itself produced, and no others — but enters them to such an extent that the ratio of its quantity among the means of production to its quantity among the products is exactly the same in each of the processes concerned.
    3. Products which only enter the means of production for an interconnected group of non-basics; in other words, products which (as a group) behave in the same way as a non-basic of the second type does individually.
  • In order to define (in the multiple system of k processes) the type which corresponds to the third case (with the interconnected group consisting of 'a', 'b', and 'c'), we arrange the quantities in which these commodities enter any one process, as means of production, and as products, in a row. We shall thus obtain k rows ordered in 2×3 columns as follows: A 1 B 1 C 1 A ( 1 ) B ( 1 ) C ( 1 ) A 2 B 2 C 2 A ( 2 ) B ( 2 ) C ( 2 ) A k B k C k A ( k ) B ( k ) C ( k )
    • Footnote: Some of these quantities may be zero, of course.
  • The condition for the three products being non-basic: not more than three of the rows should be independent, and the others should be a linear combination of those three. (For the general definition, see §60.)

59. Example of the third type

  • This third type gives us "curiously intricate patterns". Sraffa demonstrates this with an example.
  • Given a system of four processes and four products, two commodities ('b' and 'c') are jointly produced by one process and are produced by no other.

    But while 'b' does not enter the means of production for any process, 'c' enters the means of all four processes.

    Supposing the process producing 'b' and 'c' corresponds to the equation ( A 1 p a + C 1 p c + K 1 p k ) ( 1 + r ) + L 1 w = A ( 1 ) p a + B ( 1 ) p b + C ( 1 ) p c + K ( 1 ) p k the "rows" for the two commodities will be C 1 B ( 1 ) C ( 1 ) C 2 C 3 C 4 Only the first row and any other are independent, the remaining two rows are linear combinations of the first pair. So both 'b' and 'c' are non-basic.

  • If we look at the matter from constructing the Standard system, we see: (a) it's obvious 'b' can't enter the Standard commodity, (b) 'c' looks like it could be a suitable component.
    • However, since 'b' occurs only in one process, the only way to eliminate 'b' is omitting that process altogether.
    • But that process was the exclusive producer of 'c', so it only appears as means of productions…not as a produced commodity. So 'c' cannot possibly enter the Standard commodity, and must be dropped.

60. General Definition

  • The formal definition given is not as satisfying as the intuition "entering (or not entering) the means of production for all commodities", which it supercedes. BUT it has the advantage of greater generality.
  • Lets observe the first two types of non-basics may be considered as special cases of the third.
  • The definition covers the three types of the single-product system.
    • (It is quite general, and as the example in §59 suggests, it includes a final type of non-basic, which is introduced subsequently…namely commodities which enter the means of production but are not produced — a type which land is the outstanding example.)
  • We can give this general formulation between the distinction between basic and non-basic goods:
  • Critera. In a system of k productive processes and k commodities (no matter whether produced singly or jointly), we say that a commodity — or more generally a group of n linked commodities (where 1≤ n< k) — are "Non-Basic" if:

    of the k rows (formed by the 2 n quantities in which they appear in each process) not more than n rows are independent, the others being linear combinations of these.

    Or, in linear algebraic terms, the matrix of k rows and 2 n columns is of rank less than or equal to n.

  • All commodities which do not satisfy this condition are "Basic" (Note that, as has been stated in §6, every system is assumed to include at least one basic product.)

61. Elimination of non-basics

  • It follows we can (through linear transformations) entirely eliminate non-basic commodities from the system…both on the side of the means of production and the products.
  • This operation achieves the same result as we obtained in the single-products system by the much simpler method of crossing out equations of industries producing non-basics (§35).

62. The system of Basic equations

  • If the number of basic products is j, the system thus obtained will consist of j equations: these may be described as Basic equations.
  • Supposing the j basic commodities are a, b, …, j we shall denote the net quantities in which they appear using the "barred-quantities" A ˉ , B ˉ , …, J ˉ to distinguish them from the quantities in the original processes.
  • The Basic equations will accordingly be as follows:
( A ˉ 1 p a + B ˉ 1 p b + + J ˉ 1 p j ) ( 1 + r ) + L ˉ 1 w = A ˉ ( 1 ) p a + B ˉ ( 1 ) p b + + J ˉ ( 1 ) p j ( A ˉ 2 p a + B ˉ 2 p b + + J ˉ 2 p j ) ( 1 + r ) + L ˉ 2 w = A ˉ ( 2 ) p a + B ˉ ( 2 ) p b + + J ˉ ( 2 ) p j ( A ˉ j p a + B ˉ j p b + + J ˉ j p j ) ( 1 + r ) + L ˉ j w = A ˉ ( j ) p a + B ˉ ( j ) p b + + J ˉ ( j ) p j
  • This system is equivalent to the original one inasmuch as the values it determines for R and the prices will also be solutions of that system.
  • It differs from the original system (aside from obviously excluding non-basics):

    (a) A basic equation does not represent a productive process — it merely is the result of combining the equations of a number of processes.

    (b) It may contain negative quantities as well as positive ones.

63. Construction of the Standard system

  • The basic equations are designed for the construction of the Standard product.
    • Footnote. It would be possible to construct the Standard product directly from the original equations, and the final result would have been the same. Why it has seemed simpler to go through the intermediate step of the Basic equations, well, Sraffa explains it in Appendix C.
  • The multipliers q 1 , q 2 , …, q j which applied to the j Basic equations give the Standard system are determined by the following equations:
( A ˉ 1 q a + B ˉ 1 q b + + J ˉ 1 q j ) ( 1 + r ) = A ˉ ( 1 ) q a + B ˉ ( 1 ) q b + + J ˉ ( 1 ) q j ( A ˉ 2 q a + B ˉ 2 q b + + J ˉ 2 q j ) ( 1 + r ) = A ˉ ( 2 ) q a + B ˉ ( 2 ) q b + + J ˉ ( 2 ) q j ( A ˉ j q a + B ˉ j q b + + J ˉ j q j ) ( 1 + r ) = A ˉ ( j ) q a + B ˉ ( j ) q b + + J ˉ ( j ) q j
  • The equations give an equation for R of the j-th degree, so there may be up to j possible values of R and corresponding sets of values of the q's. Each set will represent a Standard commodity of different composition.

64. Only the lowest value of R economically significant

  • When deciding which (among the j possible sets of values) is the relevant one, we can not rely on there being a value of R which corresponds to an all-positive Standard commodity. Why? Because in a system with joint-production all possible Standard commodities may include negative quantities among their components.
  • If we reconsider the matter from the perspective of single-product systems, we find while an all-positive Standard makes sense, its superiority is due to it corresponding to the lowest possible value of R (as we shown in §42).
    • We shall see the possession of this last property is — by itself — sufficient to make the Standard net product, endowed with it, the one eligible for adoption as unit of wages and prices.

      This is regardless of whether the Standard commodity with this crucial property consists of all positive quantities or otherwise.

  • Suppose that R being the lowest possible value of R , we adopted as unit the Standard product corresponding to another value (say R ′′ > R ).
    • As the wage w measured in this Standard was gradually reduced from 1 it would (before vanishing) arrive at a level w ′′ such that R ′′ ( 1 w ) = R when the rate of profits would be equal to R ′′ .
    • If at such a level of w , we suppose on the basis of R , then the wage must be zero (since the rate of profits is at its maximum). While on the basis of R ′′ the wage must be positive since the rate of profits is below its maximum.
    • We reconcile this through the wage w be a positive quantity of a composite commodity who's exchange value is zero. This is because (as we shown in §41) the exchange value for a Standard commodity the composition corresponds to one solution of R (in our case R ′′ ) at the prices that correspond to another solution of R (for us, R ) is zero.
    • This implies, under these circumstances, the prices of all commodities would — in terms of the chosen Standard — be infinite(!).

      Economically, such a result is meaningless.

      This anomaly, however, can be avoided if we adopt as unit the Standard net product corresponding to the lowest value of R .

      This is the only Standard product in terms of which, at all the levels of wage from 1 to 0 (and so at all the levels of the rate of profits from 0 to its maximum), it is possible for the prices of commodities to be finite.

65. Tax on non-basic product leaves rate of profits and prices of other products unaffected

  • The distinction between Basics and Non-Basics has become so abstract in the multiple-product system, we may wonder if it has become void of meaning.
  • The chief economic implication of the distinction was the basics have an essential part in determining prices and the rate of profits, while Non-Basics have none. And this remains true under the new definition.
  • For single-product systems this implies: if an improvement took place in the method of production for a basic commodity, then the result would necessitate a change in the rate of profits and the prices of all commodities.

    Whereas a similar improvement for a Non-Basic would affect only that particular Non-Basic's price.

    • This cannot be directly extended to a system with multiple products, where both basics and non-basics may be produced through the same process.
    • We can find an equivalent in a tax (or subsidy) on the production for a particular commodity.

      Such a tax is best conceived as a tithe, which can be defined independent of prices and has the same effect as a fall in the output for the commodity in question all other things (viz., the quantities of its mean of production and its companion products) remaining unchanged.

  • A tax on a basic product will affect all prices and cause a fall in the rate of profits corresponding to a given wage, while if imposed on a Non-Basic…it will have no effect beyond the price of the taxed commodity, and those other Non-Basics linked with it.
    • Footnote. The effect which tax has on the price for Non-basics will vary with the type of Non-Basics.

      If it does not enter any of the means of production, its price will rise by the amount of the tax.

      If it enters its own means of production, its price will change to the extent required to maintain the original ratio of the value for the aggregate product of the process (after deduction of the wage and tax) to the value of its aggregate means of production.

      If it belongs to a group of interconnected non-basics, the prices of all or some of the components of the group will change s oas to maintain that ratio.

      (In the example of §59 if the production of commodity 'c' were taxed, the price of 'c' itself would be unaffected and the brunt would be borne by the price of 'b' which would have to rise to the necessary extent.)

  • This is obvious if we consider the transformed system of Basic equations (which by itself determines the rate of profits and prices of basic products) cannot be affected by changes in the quantity or prices of Non-basics which are not part of the system.