Notes on Sraffa's Production, Chapter 2

Ch. 2. Production with a Surplus

4. The rate of Profits

• If the economy produces more than the bare minimum necessary for replacement, i.e., we have some surplus to be distributed, then our model becomes self-contradictory. Why? Because the "Gross National Product" (the right hand side) will contain the sum of the columns on the left hand side plus bonus parts. We cannot use the basic linear algebra one might naively use.
• We allot the surplus simultaneously as when the prices are determined.
  • We cannot allot the surplus before the prices are determined. The surplus must be distributed in proportion to the means of production advanced in each industry. Such a proportion between two aggregates of heterogeneous goods ("the rate of profits") cannot be determined before we know the prices of goods.
  • OTOH, we cannot defer the allotment of the surplus till after the prices are known, since the prices cannot be determined before knowing the rate of profits.
  • The distribution of the surplus must be determined through the same mechanism and at the same time as the prices of commodities.
• We add the rate of profits ("which must be uniform for all industries") as an unknown, r, and the system becomes

  (A_ap_a + B_ap_b + ... + K_ap_k)(1 + r) = Ap_a
  (A_bp_a + B_bp_b + ... + K_bp_k)(1 + r) = Bp_b
  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
  (A_kp_a + B_kp_b + ... + K_kp_k)(1 + r) = Kp_k
  

  where, since we have a self-replacing state, we need A_a + A_b + ... + A_k ≤ A, B_a + B_b + ... + B_k ≤ B, and so on. In other words: the quantity produced of each commodity is at least equal to the quantity of it used in other sectors' production together.

• The system has k independent equations, which determines the k - 1 unknowns plus the rate of profit.

5. Example of Rate of Profits

• Lets revise our example from §1 to have a surplus:

  280 qr wheat + 12 t iron → 575 qr. wheat
  120 qr wheat + 8 t iron → 20 t. iron
  

  The answer is 15 qr wheat = 1 t. iron will restore the initial condition, and the rate of profit is r = 25%
  • The trick is to rewrite this as producing one unit output in each sector:

    (56/115) qr wheat + 3/5 t iron → 1 qr. wheat
    (24/115) qr wheat + 2/5 t iron → 1 t. iron
    

    Then we let p_w be the price of 1 qr wheat, p_i be the price of 1 t iron, r the rate of profits.
  • We have an eigen-problem of the form: Ax = λx where λ = 1/(1+r), A is the matrix we deduced, and x is the vector (p_w, p_i).
  • Note that the matrix has eigenvalues λ = 4/5, 2/23.
  • The rate of profit r = λ^-1 - 1 could be either 1/4 or 23/2. Since the rate of profit must satisfy 0 ≤ r ≤ 1, we see r = 1/4 is the solution.
    
    Correction: my reasoning for r = 1/4 is incorrect here: it's because the exchange corresponding to r = 23/2 gives us a negative quantity of iron may be exchanged for a positive quantity of wheat, which is absurd. Hence we throw it away...and that is the correct reasoning here.
  • Now that we have our solution for r, we plug it into one of the sectors at random and solve for the prices. Since 1 + r = 5/4, we see:

    (280 p_w + 12 p_i)(5/4) = 575 p_w
    

    implies 15 p_i = 225 p_w, or equivalently 1 t iron may be traded for 15 qr wheat.

6. Basic and Non-Basic Products

• Definition. Notice before, without surplus, all commodities produced must be used in the production of other commodities. But now, with surplus, we may have commodities which are not needed in the production process. These commodities are called luxury goods.
  • You should really convince yourself this must be the case, since we affirmed before the sum of the columns for the production matrix must be equal to the output. That is to say: the total inputs must be equal to the total outputs. Otherwise, by definition, there is surplus.
• Notice also the luxury goods do not affect the rate of profits.
• If the production of a given luxury good doubled with constant inputs, the price per unit of given luxury good would halve. The price relations of all other goods would remain the same, however.
• The price of a luxury good is not an unknown we are trying to solve for. But the prices of non-luxury goods are unknowns which we need to determine. With the non-luxury good prices determined, we may deduce the luxury goods prices.
• Definition. The criteria is: does a commodity enter (either directly or indirectly) the production of all commodities. Those that do, we shall call basic and those which do not are non-basic commodities.

7. Terminological Note

• Why do we call the ratios satisfying conditions of production "values" or "prices" rather than "costs of production"?
• The latter would be adequate so far as non-basic products were concerned, since their exchange ratio is merely a reflection of what must be paid for their means of production, labour, and profits in order to produce—there is no mutual dependence.
• Basic products have another dimension
  • Its exchange-ratio depends on its use in the production of other basic commodities, as much as on the extent to which those commodities enter its own production.
  • One might be tempted to say "it depends as much on the Demand side as on the Supply side", but one would be wrong
• The price of non-basic products depends on the prices of its means of production, but these (the prices of its means of production) do not depend on it
• A basic product has the prices of its means of production depend on its own price no less than the latter depends on them
• Sraffa argues a "less one-sided description than cost of production seems therefore required".
  • Classical terms meeting the case include:
    1. "necessary price" (e.g., the physiocrats;
       Thomas Hodgskin's Popular Political Economy, IX.1 "...the natural and necessary price of money being determined...by the quantity of labour required to produce it"),
    2. "natural price" (e.g., Adam Smith's Inquiry Ch. 4, Ch. 7 "The natural price...is...the prices of all commodities are continually gravitating";
       David Ricardo's Principles Ch. 5;
       J.S. Mill's Principles III.4 "value...proportional to its cost of production, [is] its Natural Value (or its Natural Price)";
       NB: John Locke appears to be the first(?) English economist to use the terms "natural price" and "market price" in Some Considerations), or
    3. "price of production" (e.g., Marx?)
  • But value and price have been preferred, because (a) it's shorter; and (b) in the present context — which has no reference to "market prices" — it's no more ambiguous.
• In general Sraffa avoids the term "cost of production", as well as the term "capital" (in its quantitative connotation), even at the expense of tiresome circumlocution
  • These terms have become inseparably linked with the supposition they stand for quantities which can be measured independently of — and prior to — the determination of the prices of the products.
  • Consider the "real costs" of Marshall ("But now we have to take account of the fact that the production of a commodity generally requires many different kinds of labour and the use of capital in many forms. The exertions of all the different kinds of labour that are directly or indirectly involved in making it; together with the abstinences or rather the waitings required for saving the capital used in making it: all these efforts and sacrifices together will be called the real cost of production of the commodity." Alfred Marshall's Principles of Economics, Book 5, Chapter 3) and the "quantity of capital" which is implied in the marginal productivity theory.
  • Sraffa avoids suppositions which such terms connote, since he's trying to critique the marginalist paradigm.

8. Subsistence-Wage and Surplus-Wage

• We have regarded wages as consisting of the necessary subsistence of the workers, and enters the system on equal footing as fuel for engines or feed for cattle.
• Sraffa takes into account the "other aspect of wages" since, besides the ever-present element of subsistence, they may include a share of the "surplus product".
• We separate the wages into two components: one is the subsistence, which we keep as inputs on equal footing as fuel or feed; the other is the "division of the surplus", which we should as variable.
  • Working with tradition, we will refrain from parting with tradition, and shall follow the usual practice treating the whole wage as variable.
• Drawback: This approach relegates the necessaries of consumption to the "limbo" of non-basic products.
  • This is because the necessaries of consumption no longer appear alongside the other means of production, i.e., they don't appear on the left hand side of the equations.
  • An improvement in the methods of production for necessaries of life will no longer directly affect rates of profits and the prices of other products.
  • Necessaries are essentially basic, and if they are prevented from exerting influence on prices and profits under that label, they do so in devious ways (Sraffa suggests, e.g., "by setting a limit below which the wage cannot fall", a limit which would itself fall with any improvement in the methods of production for necessaries, "carrying with it a rise in the rates of profits and a change in the prices of other products".)
• The discussion Sraffa entertains can "easily be adapted to the more appropriate, if unconventional, interpretation of the wage suggested above".

9. Wages paid out of the product

• We shall hereafter assume the wage is paid post factum as a share of the annual product. Thus we abandon the classical economists' idea of a wage "advanced" from capital.
• We retain the supposition of an annual cycle of production with an annual market.

10. Quantity and Quality of Labor

• The quantity of labor employed in each industry should now be represented explicitly, taking the place of the corresponding quantities of subsistence.
• We suppose labor to be uniform in quality or (what amounts to the same thing) we assume any difference in quality to have been previously reduced to equivalent differences in quantity, so each unit of labor receives the same wage.
• We call L_a, L_b, ..., L_k the annual quantities of labor respectively employed in the industries producing a, b, ..., k and we define them as fractions of the total annual labor of society, which we take as unity. So:

  L_a + L_b + ... + L_k = 1
  

  • Remark. It seems labor is treated differently than other commodities. For example, it doesn't have its own "sector" (equation). And it's already normalized!
• We call w the wage per unit of labor, which like the prices will be expressed in terms of the chosen standard
  • (See also, on the choice of a standard, in §12)

11. Equations of Production

• So, with these additional assumptions, the equations take the form:

  (A_ap_a + B_ap_b + ... + K_ap_k)(1 + r) + L_aw = Ap_a
  (A_bp_a + B_bp_b + ... + K_bp_k)(1 + r) + L_bw = Bp_b
  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
  (A_kp_a + B_kp_b + ... + K_kp_k)(1 + r) + L_kw = Kp_k
  

• We assume, as before, the system is in a self-replacing state, so A_a + A_b + ... + A_k ≤ A, B_a + B_b + ... + B_k ≤ B, etc.

12. National Income in a Self-Replacing System

• Definition. The National Income of a system in a self-replacing state consists of the set of commodities which are "left over" after the articles replacing the means of production are used up.
  • In other words, if we denote ΔA = A - (A_a + A_b + ... + A_k) and so on for all other industries, we have (ΔA)p_a + (ΔB)p_b + ... + (ΔK)p_k be the national income.
• The value of this set of commodities, or "composite commodities" as it may be called, which forms the national income...we set to 1.
• Thus the national income becomes the standard in terms of which the wage and k prices are expressed (taking the place of the arbitrarily chosen single commodity in terms of which the k - 1 prices, besides the wage, were expressed).
• We have the additional equation:

  (ΔA)p_a + (ΔB)p_b + ... + (ΔK)p_k = 1.

• It is impossible for the aggregate quantity of any commodity represented in this expression to be negative, otherwise we contradict the assumption the economy is in a self-replacing state!
• This gives k + 1 equations as compared to k + 2 variables (the k prices, the wage w, and the rate of profits r).
• The result of adding the wage as one of the variables is that the number of these now exceeds the number of equations by one. The system has "one degree of freedom". If one of the variables is fixed, the others will be too.