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:

$$\begin{array}{rl}(A_1{p}_a+B_1{p}_b+\dots <!--\; \dots \; -->+K_1{p}_k)(1+r)+L_{1}w& =A_{(1)}{p}_a+B_{(1)}{p}_b+\dots <!--\; \dots \; -->+K_{(1)}{p}_k\\ (A_2{p}_a+B_2{p}_b+\dots <!--\; \dots \; -->+K_2{p}_k)(1+r)+L_{2}w& =A_{(2)}{p}_a+B_{(2)}{p}_b+\dots <!--\; \dots \; -->+K_{(2)}{p}_k\\ \dots <!--\; \dots \; -->& =\dots <!--\; \dots \; -->\\ (A_k{p}_a+B_k{p}_b+\dots <!--\; \dots \; -->+K_k{p}_k)(1+r)+L_{k}w& =A_{(k)}{p}_a+B_{(k)}{p}_b+\dots <!--\; \dots \; -->+K_{(k)}{p}_k\end{array}$$

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.