Notes on Sraffa's Production, Chapter 1

Sraffa wrote his Production of Commodities in a very "mathematician's style"...i.e., it was a grocery list of propositions chunked together. Consequently, I analyze each "chunk" in my series of notes. The section titles are Sraffa's.

Ch. 1. Production for Subsistence

1. Two Commodities

• We consider an economy with two commodities, producing just enough to maintain itself.
• Commodities are produced by separate industries and are exchanged at a market held after the harvest.
• Wheat and iron are produced, representing sustenance for the workers and the means of production (respectively).
• We suppose that 280 quarters of wheat and 12 tons of iron are used to produce 400 quarters of wheat, whereas 120 quarters of wheat and 8 tons of iron are used to produce 20 tons of iron. We represent a year's operations as:
  

  280 qr. wheat + 12 t. iron → 400 qr. wheat
  120 qr. wheat + 8 t. iron → 20 t. iron.
  

  • Note that when we add up the columns, it's equal to the output. This is what we mean when we say "There is no surplus."
  • These relations Sraffa calls "the methods of production and productive consumption" or the methods of production.

2. Three or More

• The same applies to any number of commodities. For example, if we add a third sector (pigs):
  

  240 qr. wheat + 12 t. iron + 18 pigs → 450 qr. wheat
  90 qr. wheat + 6 t. iron + 12 pigs → 21 t. iron
  120 qr. wheat + 3 t. iron + 30 pigs → 60 pigs
  

• The exchange values which ensure replacement all around are 10 qr wheat = 1 t iron = 2 pigs.
• The general procedure for calculating this is to normalize the output, so we consider

  (8/15) qr. wheat + (4/7) t. iron + (3/10) pigs → 1 qr. wheat
  (3/15) qr. wheat + (2/7) t. iron + (2/10) pigs → 1 t. iron
  (4/15) qr. wheat + (1/7) t. iron + (5/10) pigs → 1 pig
  

• Then we just use linear algebra to solve this as usual.

3. General Case

• We have commodities of different "species" (e.g., iron, wheat, etc.). Lower case variables will keep track of each species a, b, c, ..., k. NB: the term "species" is mine, not Sraffa's.
  • Caution: Sraffa abuses notation and mixes up A and a when referring to the species.
  • Abuse of Notation: We will write k to indicate the number of sectors, as well as its proper meaning. It will depend on the context, but it should be clear when used.
• We denote A the quantity annually produced of a, B the quantity annually produced of b, etc.
• We will call A_a, B_a, ..., K_a the quantities of a, b, ..., k annually used in the industry producing a, and so on.
• The prices are the unknowns p_a, p_b, ..., p_k, which represent the values of 1 unit of the commodities a, b, ..., k (respectively) necessary to restore the initial position.
• We can now cast the conditions of production as:
  

  A_ap_a + B_ap_b + ... + K_ap_k = Ap_a
  A_bp_a + B_bp_b + ... + K_bp_k = Bp_b
  . . . . . . . . . . . . . . . . . . . . . . . . . .
  A_kp_a + B_kp_b + ... + K_kp_k = Kp_k
  

• Note that we assume the system to be in a self-replacing state, so

  A_a + A_b + ... + A_k = A,
  B_a + B_b + ... + B_k = B,

  and so on. That is to say, the sum of the first column equals the output of the first row, etc.
  • Again, the approach one should take is to work with modified quantities A'_a = A_a/A and so on, working with primed quantities as inputs and output quantities (NOT PRICES) set to 1.
• It is unnecessary to assume every commodity enters directly into the production of every other. We could have pigs enter into production of wheat but not iron, and no iron enter into the production of pigs.
• One commodity is taken as the standard measure of value, and its price is set to 1. This leaves k - 1 unknowns. It doesn't really matter which commodity we pick, since the relative exchange-value doesn't change. This leaves k - 1 independent linear equations which uniquely determine the k - 1 prices. (This is similar to Marx's "general form of value" discussed in Das Kapital, Vol. 1, Ch. 1.)