Notes on Sraffa's Production, Chapter 9

66. Quantity of labour embodied in two commodities jointly produced by two processes

• So what results from single-product systems generalize over to joint-product systems?
• One rule we should study: when the rate of profits is zero, the relative value of each commodity is proportional to the quantity of labor which (directly and indirectly) has gone to producing them (§14).
  • For joint-products, there is no obvious criterion apportioning the labor among individual products. It seems doubtful whether it makes any sense to speak of a "separate" quantity of labor as having gone to produce one among many jointly produced commodities.
  • We get no help from the "Reduction" approach, where we sum the various dated labor inputs weighted by the product of rate of profits. (This is further discussed in §68)
• With the system of single-product industries, we had an alternative (if less intuitive) approach using the method of "Sub-systems" (Sraffa discusses this in his Appendix A).

  It was possible to determine — for each of the commodities composing the net product — the share of aggregate labor which could be regarded as directly or indirectly entering its production.

  • This method (with appropriate adaptation) extends to joint-products, so the conclusion about the quantity of labor "contained" in a commodity and its proportionality to value (at zero profits) can be generalized to joint products.
• Consider two commodities jointly produced through each of two processes in different proportions.

  Instead of looking separately at the two processes and their products, lets consider the system as a whole and suppose quantities of both commodities are included in the net product for the system.

• We further assume the system is in a self-replacing state, and whenever the net product is changed...the self-replacing state is preserved (i.e., immediately restored through means of suitable adjustments in the proportions of the processes composing it).
• We also note: it is possible to change (within certain limits) the proportions in which two commodities are produced if we alter the relative sizes of the two processes producing them.
• If we wish to increase the quantity which a commodity enters the net product of the system (while leaving all other components unchanged), we normally must increase the total labor employed by society.

  It's natural to conclude we must increase labor for producing the commodity in question. This may go directly (i.e., directly into the process in question) or indirectly (i.e., producing the means of production).

  • The commodity added will (at the prices corresponding to zero rates of profits) be equal in value to the additional quantity of labor.
• This conclusion seems to hold for commodities jointly produced, as it holds for single-product systems.
  • The conclusion appears to hold even when we change the quantities of the means of production, since any additional labor needed to produce the latter is included as indirect labor in the quantity producing the addition to the net product.
  • Footnote. Since joint-products are present, the contraction for some processes might occur, and thus we might fall into the awkward "negative industries" scenario again...but even then, the adjustments noted include them!

    This can be avoided, provided the initial increase for the commodity in question is supposed to be "sufficiently small", and the net product for the system is assumed to comprise at the start "sufficiently large quantities" of all products...so any necessary contraction may be absorbed by existing processes, without the need for any of them having to receive a negative coefficient.

67. Quantity of labour embodied in two commodities jointly produced by only one process

• Similar reasoning holds for the case when two commodities ('a' and 'b') are jointly produced through only one process...but are used as means of production (in different relative quantities) through two processes, each produes singly the same commodity 'c'.
  • So we have two processes of the form $q_{1,a}a+q_{1,b}b\to <!--\; \to \; -->q_{1,c}c$ and $q_{2,a}a+q_{2,b}b\to <!--\; \to \; -->q_{2,c}c$ where $q_{1,a}/q_{1,b}\ne <!--\; \ne \; -->q_{2,a}/q_{2,b}$, and none of the coefficients vanish.
• We can't change the proportions which 'a' and 'b' appear in the output of their production processes (i.e., the processes producing them). But we can (through altering the relative size of the two processes using them) vary the relative quantities in which they enter as means for producing a given quantity of 'c'.

  We can vary the relative quantities of 'a' and 'b' this way, and this by itself alters the relative quantities in which they enter the net social product. (The relative quantities in which the two enter the gross product are fixed.)

  • Remark. As a childish example, we could have $$\begin{array}{r}a+2b\to <!--\; \to \; -->c\\ 3a+b\to <!--\; \to \; -->c\end{array}$$ So, suppose we have for our toy example $q_a=q_b$ (there is a one-to-one ratio between the quantity of 'a' and 'b' produced).

    The relative quantities of 'a' and 'b' seems like a strange term to me. We could consider enlarging the first process and keep the second process constant: $$\begin{array}{rl}f(\stackrel{⃗<!--\; ⃗\; -->}{x})+L_a& \to <!--\; \to \; -->5a\\ g(\stackrel{⃗<!--\; ⃗\; -->}{y})+L_b& \to <!--\; \to \; -->5b\\ 2(a+2b)& \to <!--\; \to \; -->2c\\ 3a+b& \to <!--\; \to \; -->c\end{array}$$ For simplicity, the production of 'a' and 'b' are blackbox functions which takes "some vector" of inputs. We have combined $5a+5b\to <!--\; \to \; -->3c$. The relative quantities of 'a' and 'b' are, literally, one-to-one. Observe the surplus is 3 c...and we had $L_a+L_b$ contribute.

    But if we change how we produce things, say use only the first process, then we have $$\begin{array}{rl}f(0.4\stackrel{⃗<!--\; ⃗\; -->}{x})+0.4L_a& \to <!--\; \to \; -->2a\\ g(0.8\stackrel{⃗<!--\; ⃗\; -->}{y})+0.8L_b& \to <!--\; \to \; -->4b\\ 2a+4b& \to <!--\; \to \; -->2c\end{array}$$ and hence we have the surplus be 2c. The relative proportion which 'a' and 'b' enter production change; is this what Sraffa means? We varied the size of the processes producing 'a' and 'b', without deforming the processes (i.e., changing the proportions of the coefficients, just reduced the ratio to produce a lesser amount).

    The amount of labor also changed from $L_a+L_b$ to $0.4L_a+0.8L_b$.

• It is thus possible (through an addition to total labor) to arrive at a new self-reproducing state, where a quantity for one of the two products (say 'a') is added to the net product, while all other components of the latter remain unchanged.

  We can conclude the addition to labor is the quantity which directly and indirectly is required to produce the additional amount of 'a'.

68. Reduction to dated quantities of labour not generally possible

• Sraffa claims there is no equivalent (in the case of joint-products) to the "alternative method", i.e., Reduction to a series of dated labor terms. Sraffa explains the "essence" of Reduction is that each commodity should be (a) produced separately, (b) by only one industry, and (c) the whole operation consists in tracing back the successive stages of a single-track production process.
  • Remark. I am very suspicious of this claim, and I don't follow the reasoning given. After all, consider the system given as $$(1+r)A\stackrel{⃗<!--\; ⃗\; -->}{p}+w\stackrel{⃗<!--\; ⃗\; -->}{L}=\stackrel{⃗<!--\; ⃗\; -->}{p}$$ where A is the input-output matrix, $\stackrel{⃗<!--\; ⃗\; -->}{p}$ is the price-vector, w wage, $\stackrel{⃗<!--\; ⃗\; -->}{L}$ the labor vector, and r the rate of profits. Then we have $$(I-<!--\; -\; -->(1+r)A)\stackrel{⃗<!--\; ⃗\; -->}{p}=w\stackrel{⃗<!--\; ⃗\; -->}{L}$$ where I is the identity matrix. This gives us $$\stackrel{⃗<!--\; ⃗\; -->}{p}=(I-<!--\; -\; -->(1+r)A)\cdot <!--\; \cdot \; -->w\stackrel{⃗<!--\; ⃗\; -->}{L}=\left(\underset{0}{\overset{\mathrm{\infty <!--\; \infty \; -->}}{\sum <!--\; \sum \; -->}}(1+r{)}^n{A}^n\right)w\stackrel{⃗<!--\; ⃗\; -->}{L}$$ Isn't this a Reduction-type equation?

    If so, it could be suitably generalized in the straightforward way for a joint-product. Provided the joint-product system satisfies the conditions Sraffa gives (basically, the general linear algebraic conditions that a solution exists).

  • Remark (Cont'd). Now, we are dealing with a slightly more general situation, specifically: $$(1+r)A\stackrel{⃗<!--\; ⃗\; -->}{p}+w\stackrel{⃗<!--\; ⃗\; -->}{L}=B\stackrel{⃗<!--\; ⃗\; -->}{p}$$ where the matrix B is necessary for joint-products. Without loss of generality, we may assume it is an invertible matrix. Thus we re-write this system as $$(1+r)B^{-<!--\; -\; -->1}A\stackrel{⃗<!--\; ⃗\; -->}{p}+w{B}^{-<!--\; -\; -->1}\stackrel{⃗<!--\; ⃗\; -->}{L}=\stackrel{⃗<!--\; ⃗\; -->}{p}$$ or if we introduce new symbols to stress the similarity to the previous case: $$(1+r)\stackrel{˜<!--\; ˜\; -->}{A}\stackrel{⃗<!--\; ⃗\; -->}{p}+w{\stackrel{⃗<!--\; ⃗\; -->}{L}}^{\prime }=\stackrel{⃗<!--\; ⃗\; -->}{p}.$$ We should observe this becomes the previous situation.
• Sraffa suggests we should have to give a negative coefficient to one of the two joint-production equations and a positive coefficient to the other, thus eliminating one of the products while retaining the other in isolation.

  Some of the terms in the Reduction equation would represent negative quantities of labor, which Sraffa insists "no reasonable interpretation could be suggested."

  • Sraffa insists the series would contain both positive and negative terms, so the "commodity residue" wouldn't necessarily be decreasing at successive stages of approximation. Instead, it might show steady or even widening fluctuations — the series might not converge!
  • Sraffa will investigate this in §79 ("Different depreciation of similar instruments in different uses").
• Reduction could not be attempted if the products were jointly produced by a single process, or by two processes in the same proportions, since the apportioning of the value and of the quantities of labor between the two products would depend entirely on the way the products were used as means of production for other commodities.

69. No certainty that all prices will remain positive as the wage varies

• Sraffa urges us to reconsider another proposition considered earlier: if the prices of all commodities are positive at any one value of the wage between 1 and 0, no price could become negative as a result of varying the wage within those limits (§39).
  • Sraffa denies the possibility we could generalize this proposition to joint-product systems.
• Recall, the premise underpinning this proposition: the price of a commodity could only become negative if the price for some other commodity (one of its means of production) had become negative first — so no commodity could ever be the first to do so.
  • But for joint-products, there is a way around and the price for one of them may become negative...provided the balance was restored by a rise in the price of its companion product sufficient to maintain the aggregate value of the two products above that of their means of production by the requisite margin.

70. Negative quantities of labour

• Sraffa suggests his conclusion is "not in itself very startling". He interprets the situation quite simple. Sraffa notes in fact all prices are positive...but a change in the wages may create a situation which necessarily requires prices to become negative. Since this is unacceptable, those methods giving negative prices would be discarded in favor of those giving positive prices.
• When we consider this with the previous section (concerning the quantity of labor entering a commodity), the combined effect requires some explaining...
  • What's involved is not merely something like "In the remote contingency of the rate of profits falling to zero, the price of such a commodity would (if other things remain equal) have to become negative"...but we conclude in the actual situation, with profits at the perfectly normal rate of (say) 6%, that particular commodity is in fact produced by a negative quantity of labor.
  • Caution: We will work supposing 6% is the "normal rate of profits" throughout this section, so bear that in mind...
• Sraffa says "This looks at first as if it were a freak result of abstraction-mongering that can have no correspondence in reality." He has such a way with words, sometimes!
  • If we apply it to the test employed for the general case in §66, where — under the supposed conditions — the quantity of such a commodity entering the system's net product is increased (the other components remaining constant), we shall find as a result the aggregate quantity of labor society employs has diminished.
• Nevertheless! Since the change in production occurs while the "ruling rate of profits" is 6%, and the system of prices is the one appropriate to that rate, Sraffa argues "nothing abnormal will be noticeable".

  In effect the diminution in the expense for labor will be more than balanced by an increased charge for profits, the addition to net output will entail a positive addition to the cost of production.

• So, what happened? In order bring about the required change in the net product, one of the two joint-production processes must be expanded while the other contracted.

  In the case under consideration, the expansion of the former employs (either directly or through "other processes as it carries in its train the ensure full replacement") a quantity of labor which is smaller...but means of production which at the prices appropriate to the given rate of profits are of greater value — and thus attracts a heavier charge for profits — than the contraction of the latter process "under a similar proviso".

• Sraffa concludes "It seems unnecessary to show in detail that what has been said in this section concerning negative quantities of labor can be extended (on the same lines as was done for positive quantities in §67) to the case in which two commodities are jointly produced by only one process, but are used as means of production by two distinct processes both producing a third commodity."

71. Rate of fall of prices no longer limited by rate of fall of wages

• Sraffa has one further proposition about prices which needs reconsideration for the case of joint products.
• We have seen (§49) for single-product industries, when the wage falls in terms of the Standard commodity that no product can fall in price at a higher rate than does the wage.
  • The premise underpinning this: were a product able to do so, it must be owing to one of its means of production falling in price at a still higher rate.

    Since this could not apply to the product that fell at the highest rate of all, that product itself could not fall at a higher rate than wage.

• With one of a group of joint products, there is the alternative possibility the other commodities jointly produced with it should rise in price (or suffer only a "moderate" fall) with the fall of wage so as to make up — in the aggregate product of the industry — for any excessive fall of the first commodity's price.

  To such a rise, there is no limit...and thus there is none to the rate at which one of the several joint products may fall in price.

• But as soon as it is admitted the price of one (out of two or more joint products) can fall at a higher rate than does the wage, it follows even a singly produced commodity can do so...provided it employs — as one of its means of production, and to a sufficient extent — the joint product so falling.

72. Implication of this

• This possibility — price may fall faster than the wage — has some noteworthy consequences...
• First we have an exception to the rule "The fall of wage in any Standard involves a rise in the rate of profits."
• Suppose a 10% fall in the Standard wage entails (at a certain level) a larger proportionate fall — say 11% — in the price of 'a' as measured in the Standard product.
  • This means labor has risen in value by about 1% relative to the commodity 'a'.
    • Remark. I think the ratio would be $90/89\approx <!--\; \approx \; -->1.01123595505$ or the rise of value of labor relative to 'a' is about 1.12%.
  • If we were to express the wage in terms of commodity 'a', a fall in such a wage over the same range would involve a rise in the Standard wage and consequently a fall in the rate of profits.
• Moral. We can't speak of a rise or fall in the wage unless we specify the standard, for what is a rise in one standard may be a fall in another.
• For the same reasons, it becomes possible for the wage-line and price-line of a commodity 'a' to intersect more than once as the rate of profits varies
  • Figure 5: Several intersections are possible in a system of multiple-product industries.
    

• As a result, to any one level of the wage in terms of commodity 'a', there may correspond several alternative rates of profits.
  • In figure 5, the several points intersective the solid black curve — representing the price of 'a' — with the dashed wages curve represent equality in value between a unit of labor and a unit of commodity of 'a'...i.e., the same wage in terms of 'a'.

    Of course, they represent different levels of wage in terms of the Standard commodity.

  • On the other hand, as in the case of the single-products system, to any one level of the rate of profits there can only correspond one wage, whatever the standard in which the wage is expressed.