Supply-Chain Collapse Puzzle

Suppose we had some Sraffian model of an economy, $(1+R)A\stackrel{\to <!--\; \to \; -->}{p}=\stackrel{\to <!--\; \to \; -->}{p}$ with only basic commodities. There is a cycle of production at time t, then the goods go to market at time t + 1, then production starts again at t + 2, and so on.

What happens if the supply chain for one sector collapses completely? In the sense that the product cannot get to market after production. How do prices of commodities change in response to this shortage? What about the rate of profits: how does it change?

What happens if we extend the system to allow non-basic commodities? How do their prices alter?

Recommended Economics Writing: Link Exchange

Sraffa's archival material is a gift to the science of economics. Scott Carter has been working in the Sraffa archives for a while, and this is an update about his efforts (among others) in getting the Sraffa archives published digitally.

The History of Economic Thought Website, I'm excited to say this has been brought back from the dead. The website's author has returned, so this is the official version.

The State of Macro Is Sad (Wonkish); Paul Krugman manifesting what looks like cognitive dissonance about the state of Macroeconomics.

How to Think about Own Rates of Interest, Version 2.0. When Hayek wrote Prices and Production, Sraffa wrote a scathing review about it. In the review, Sraffa used the concept of "own rates of interest" in a unique manner. Unfortunately it seems that Keynes had a very similar but distinct notion of "own rates of interest" in chapter 17 of his General Theory. I've recently become curious about the subject, so this blog-post has piqued my interest (even though I vehemently disagree with just about everything the author has written).

CEPA's "History of Economic Thought"

So, the History of Economic Thought site appears to be back and stable at its new location, but I'm not so sure it will last.

I've backed it up using wget. I will probably translate the site to Markdown, then put the result up on Github. As one might expect, the links to various external web-pages are out of date or completely broken...so just fixing the references might be a full time job.

Addendum (August 23, 2016): it seems the author of the History of Economic Thought has returned from...wherever he was, and has hosted the site elsewhere. It seems back up for good now.

Logical Structure of Austrian Economics

1. Introduction. We will attempt to reconstruct the Austrian approach to economics using first-order logic.

We observe (in section 3) Austrian economists confuse deduction with introducing logically independent propositions. The general reasoning is "This doesn't directly contradict our foundational axiom, therefore it must logically follow from it" promoting the non-sequitur from fallacy to rule of inference.

Nevertheless, we bravely continue, and in section 4 discover it's impossible to deduce marginal utility from the action axiom. This spells disaster for any marginal analysis in the Austrian school.

Definition and Explication

2.1. Axiom ("Action Axiom"). Murray Rothbard's The Logic of Action One: Method, Money, and the Austrian School (1997) describes the "action axiom" as:

Praxeology rests on the fundamental axiom that individual human beings act, that is, on the primordial fact that individuals engage in conscious actions toward chosen goals. This concept of action contrasts to purely reflexive, or knee-jerk, behavior, which is not directed toward goals. The praxeological method spins out by verbal deduction the logical implications of that primordial fact. In short, praxeological economics is the structure of logical implications of the fact that individuals act. This structure is built on the fundamental axiom of action, and has a few subsidiary axioms, such as that individuals vary and that human beings regard leisure as a valuable good. Any skeptic about deducing from such a simple base an entire system of economics, I refer to Mises's Human Action. Furthermore, since praxeology begins with a true axiom, A, all the propositions that can be deduced from this axiom must also be true. For if A implies B, and A is true, then B must also be true. (58--59)

This outlines the Austrian methodology fairly faithfully (I hope).

In order to make heads or tails out of it, lets first refine the meaning of "action" (since "humans act" is ambiguous at the moment).

2.2. Definition (Action). Ludwig Mises' Human Action itself defines "action" rather vaguely:

Human action is purposeful behavior. Or we may say: Action is will put into operation and transformed into an agency, is aiming at ends and goals, is the ego's meaningful response to stimuli and to the conditions of its environment, is a person's conscious adjustment to the state of the universe that determines his life. Such paraphrases may clarify the definition given and prevent possible misinterpretations. But the definition itself is adequate and does not need complement of commentary.

Personally, I find this unsatisfactory, but I will resign myself to accept the definition of "action" as "physical and psychological processes which render a specific state". (Even then, I'm nervous.)

If it makes much of a difference, Rothbard insists that All action in the real world, furthermore, must take place through time; all action takes place in some present and is directed toward the future (immediate or remote) attainment of an end (59). I thought this went without saying, but it is good to be explicit.

Immediate "Deductions"

3.1. Corollary. Rothbard continues:

Let us consider some of the immediate implications of the action axiom. Action implies that the individual's behavior is purposive, in short, that it is directed toward goals. Furthermore, the fact of his action implies that he has consciously chosen certain means to reach his goals. (59)

Well, is "the ego's meaningful response to stimuli" necessarily "consciously chosen"? Wasn't that the point of Pavlov's dogs?

OK, lets overlook this and continue analyzing the consequences of the action axiom. (I mean, real and meaningful consequences, not tautological statements.)

3.2. Corollary. Rothbard tries to pull a fast one, insisting

Furthermore, that a man acts implies that he believes action will make a difference; in other words, that he will prefer the state of affairs resulting from action to that from no action. (59)

How does this logically follow at all? The actor's belief in his success seems irrelevant to the supposition the actor "acts" (in the Austrian sense). It seems Rothbard assumes "conscious actions toward chosen goals" implies that choosing a goal requires first belief in succeeding at accomplishing that goal. So without that prior belief in success, we would have no action?

So, if I had doubt or no belief whatsoever in my success to bring about a desired state, and I resign myself to this fate, am I still "acting"?

This is so stupid a point to make, because this has no bearing on anything at all in Austrian economics. But Rothbard insists on making it! So, I should say Rothbard will say two things: first, that I am not acting (otherwise he immediately contradicts himself); and second, I am acting, because I have belief in my success in my resignation.

My own personal belief is that this point should be disregarded, as it has no bearing on Austrian economics...nor does it illuminate the action axiom (or any other proposition "shown").

Fine, I'm willing to expand the definition of "action" to include the condition "The actor consciously believes in his or her own success".

3.3. Corollary (Uncertainty). Rothbard continues in his analysis, suggesting:

Action therefore implies that man does not have omniscient knowledge of the future; for if he had such knowledge, no action of his would make any difference. Hence, action implies that we live in a world of an uncertain, or not fully certain, future. Accordingly, we may amend our analysis of action to say that a man chooses to employ means according to a technological plan in the present because he expects to arrive at his goals at some future time. (59)

The proposition "Humans live in a world of uncertain future" is compatible with the Action axiom, but in no way does it logically follow. That is, there are no rules of inference which gets us from the Action axiom to this Uncertainty proposition. (Why? Because they're independent propositions!)

At the same time, there is no rule of inference denying this Uncertainty proposition. The two (the Action axiom and this uncertainty proposition) are compatible, like the Continuum hypothesis and ZFC set theory.

But the term "technological plan" here (introduced for the first time) Rothbard does not define.

3.4. Corollary (Scarcity). Rothbard's fifth conclusion:

The fact that people act necessarily implies that the means employed are scarce in relation to the desired ends; for, if all means were not scarce but superabundant, the ends would already have been attained, and there would be no need for action.

So if I want to read Mises' Human Action, that is only possible provided there is scarcity? This does not logically follow from anything stated thus far.

The proof Rothbard gives is a proof by contradiction, which is worse than useless.

Rothbard attempts clarifying this proposition, Stated another way, resources that are superabundant no longer function as means, because they are no longer objects of action (60). So the argument basically boils down to "Because the current state of the world is not the end-state desired by an action, there must be scarcity." This is a non-sequitur.

3.5. Observation. The "logic" Rothbard uses appears to be "Here's a proposition B. It's logically compatible with the action axiom. (But in no way does the action axiom logically imply B or its negation.) Therefore we deduce B must be true."

This is an invalid rule of inference. Why? Because you're not proving anything! You don't have a statement "If A, then B."

Instead you have a statement "We have A. And here's an independent proposition B. Therefore A implies B."

Marginal Utility

4.1. Scarcity. Thorsten Polleit "deduces"

Human action implies employing means to the fulfillment of ends, and the axiom of human action implies that means are scarce. For if they were not scarce, means would not serve as objects of human action; and if means were not scarce, there would be no action — and that is unthinkable.

But nothing in the definition of "action" necessitates the existence for any "means to the fulfillment of ends". Having such "means" exist is not necessary for the definition of "action".

If we change the definition for "action" to "employing some 'means' to achieve some 'end'", then we have problems: we have introduced two undefined terms. We can handwave 'end' as "The state of the world after the action is done" to arbitrary precision (specify how long afterwards, etc. etc. etc.).

But the term "means" here is completely ambiguous. If we take it as "physical objects", then the axiom of action collapses on itself: the argument "Trying to refute the action axiom is a contradiction" becomes false, and all the preceding "deductions" in section 3 become false.

If we weaken the meaning for "means" as both physical objects and mental processes, then we still have problems: the claim for scarcity has a metaphysical statement that needs to be shown (namely, "Mental processes are scarce").

If we ignore everything except the proposition if means were not scarce, there would be no action, then...this claim still needs to be demonstrated. Why? Because it is the contrapositive of the claim "If there is action, then the means are scarce" which has not been shown.

So, in short, this nice-sounding couple of sentences is ambiguous.

4.2. Can Scarcity Be Deduced? The statement concerning scarcity's existence (or non-existence) is necessarily an a postereori claim, since it is an empirical statement.

If we buy into this Kantian taxonomy of propositions (a priori vs a postereori, analytic vs synthetic), then there is no way to deduce an a postereori proposition from an a priori one...otherwise it would be, by definition, a priori.

Consequently, by definition, it is impossible to "deduce" anything about scarcity's existence.

4.3. Scholium. What's the consequence of this? Any proposition in Austrian economics dependent on scarcity's existence has no logical grounding. So, basically, all of Austrian economics has no logical grounding.

Conclusion

We have examined the action axiom and the definition of "action". We found it mildly ambiguous, but workable.

We have seen the "immediate consequences" are really just independent propositions that are not logically linked to the action axiom.

We tried to reconstruct the inference "action implies scarcity", and found this to be impossible (trying to deduce a postereori from the a priori is always impossible). Consequently, all Austrian economics depending on scarcity has no logical grounding.

Future research might include analyzing the Austrian business cycle, or other macroeconomic theories.

Addenda

12 May 2014, 8:23AM (PST). It dawns on me the "Action axiom" isn't a priori --- it's based on the observation that people "act", and the observation attempts to refute it are "actions". No one really cares about this in Austrian circles nowadays, it seems, as no one seriously defends Mises peculiar Kantian inclinations.

I wonder about the "synthetic-ness" of the "Action axiom", too.

NB: the fact that the "Action axiom" is a proposition that's neither a priori nor synthetic doesn't seriously alter anything in Austrian economics. Fundamentally, it's an "axiom" in the modern mathematical sense rather than the Kantian sense: a specification we expect to hold while making "deductions" (in some vague sense).

12 May 2014, 8:48AM (PST). After thinking deeply about a priori synthetic statements (in the Kantian sense), it dawns on me that Kant used Aristotlean logic. Theoretically, Austrian economics cannot use first-order logic because of their Kantian underpinnings. I suppose it would be an interesting philosophical project to re-cast Austrian economics in rigorous first-order logic, and see what happens. A project, I hope, I will not commit myself to...

But it does mean the proposition "Man acts" is not a valid proposition for Aristotlean logic. Rothbard's proposition individuals engage in conscious actions toward chosen goals is invalid within Aristotlean logic. Mises' Human action is purposeful behavior. likewise is invalid. Hence it's invalid to consider it either a priori or a postereori, analytic or synthetic. Being charitable, perhaps a better form of the action axiom would be "All humans are 'actors'". But this only confirms the previous point: this is clearly not a priori.

The Definition of Value

So, what is a theory of value? In this post, these are just my notes defining "value" in some suitably abstract way, such that every paradigm has its own theory of value. In this way, we can meaningfully discuss theories of value from different paradigms on equal terms...or so I hope will be the case (eventually)!

1. Definition. In one sense, value is a mapping from commodities to numbers. That is to say, value is some mapping $$\mathrm{value}:<!--\; :\; -->\mathbf{Commodities}\to <!--\; \to \; -->\mathbb{R}$$ where $\mathbf{Commodities}$ is the module of commodities over the integers (we interpret a negative quantity of commodities as a debt to be repaid), or perhaps a vector space over the rationals[1]. The basis is formed by the different "species" of commodities (e.g., iron, corn, wheat, tobacco, computers, cars, etc.).

2. Remark. Value has to be linear. Why? Because we expect, e.g., $$\mathrm{value}(2\mathrm{tons}\mathrm{steel})=2\mathrm{value}(1\mathrm{ton}\mathrm{steel}).$$ This is half the condition for linearity. We also expect $$\mathrm{value}(1\mathrm{ton}\mathrm{steel}+1\mathrm{quarter}\mathrm{wheat})=\mathrm{value}(1\mathrm{ton}\mathrm{steel})+\mathrm{value}(1\mathrm{quarter}\mathrm{wheat})$$ or more generally, the value of any linear combination of commodities is precisely the sum of the constituents of that basket of goods. This would be sufficient to imply linearity. (End of Remark)

3. Remark (Theories of Value). The main contention between different paradigms in economics (notably the Neoclassical, Ricardian & Neo-Ricardian, and I think post-Keynesian paradigms) has to do with how we determine the $\mathrm{value}$ function.

Adam Smith writes:

I.5.1. Labour, therefore, is the real measure of the exchangeable value of all commodities.

I.5.2. The real price of every thing, what every thing really costs to the man who wants to acquire it, is the toil and trouble of acquiring it. What every thing is really worth to the man who has acquired it, and who wants to dispose of it or exchange it for something else, is the toil and trouble which it can save to himself, and which it can impose upon other people.

[...]

I.5.7. Labour alone, therefore, never varying in its own value, is alone the ultimate and real standard by which the value of all commodities can at all times and places be estimated and compared. It is their real price; money is their nominal price only.

David Ricardo refines this approach (Principles, Ch 1, Paragraphs 9–10), noting Smith's inconsistencies using corn as a standard of value at some times, then labor at other times:

“The real price of every thing,” says Adam Smith, “what every thing really costs to the man who wants to acquire it, is the toil and trouble of acquiring it. What every thing is really worth to the man who has acquired it, and who wants to dispose of it, or exchange it for something else, is the toil and trouble which it can save to himself, and which it can impose upon other people.” “Labour was the first price—the original purchase-money that was paid for all things.” Again, “in that early and rude state of society, which precedes both the accumulation of stock and the appropriation of land, the proportion between the quantities of labour necessary for acquiring different objects seems to be the only circumstance which can afford any rule for exchanging them for one another. If among a nation of hunters, for example, it usually cost twice the labour to kill a beaver which it does to kill a deer, one beaver should naturally exchange for, or be worth two deer. It is natural that what is usually the produce of two days’, or two hours’ labour, should be worth double of what is usually the produce of one day’s, or one hour’s labour.”*

That this is really the foundation of the exchangeable value of all things, excepting those which cannot be increased by human industry, is a doctrine of the utmost importance in political economy; for from no source do so many errors, and so much difference of opinion in that science proceed, as from the vague ideas which are attached to the word value.

I will refrain from reviewing the history of theories of value, as Dobb's Theories of Value and Distribution since Adam Smith does this in far better detail. But I will make note of a few other approaches.

The Neoclassical approach determines value from a microeconomic point of view using supply & demand curves.

I've discussed the Neo-Ricardian approach elsewhere (see, e.g., my notes on Sraffa's Production).

3.1. Questions to Self. David Ricardo notes how the price of a given commodity is expressing its value in terms of the money commodity. Does the Neoclassical approach do likewise?

In other words, is the concept of "value" an adequate abstraction such that each paradigm has their own theory of value? (Or, equivalently, no paradigm lacks a theory of value.)

4. Then from this mapping we induce an equivalence relation between commodities. That's the whole point of introducing value: to determine how much a given quantity of a given good will exchange for. We want to figure out $x$ in the equation $$\mathrm{value}(1\mathrm{ton}\mathrm{steel})=x\mathrm{value}(\mathrm{quarter}\mathrm{wheat}).$$ It tells us how much 1 ton of steel commands in the wheat market.

4.1. We will say that this is the expression of the value for steel in terms of wheat. When we express all commodities in terms of some "standard unit" (say, wheat), then we have some money-commodity (for us: wheat, since we chose it as the standard unit).

The function of money (how it gets value, etc.) is a completely different subject (why, it's the theory of money!). Each paradigm likewise has its own theory of money.

5. Value is a function of time (or parametrized by time). This is the difficulty with measuring value. When we see the value of a commodity change in time, we are uncertain if: (1) the value of a given commodity is fluctuating, (2) everything else is fluctuating, or (3) the value of money is fluctuating. (Or, worse, some combination of the three!)

More explicitly, we have $${\mathrm{value}}_{t+dt}(x\mathrm{units}A)=c{\mathrm{value}}_t(x\mathrm{units}A)$$ where $c>0$ is some real number. This describes a change in value for $x$ units of commodity $A$.

5.1. Remark. We don't measure this variation directly. We gauge it from how the value at time $t+\mathrm{d}t$ for $x$ units $A$ equates to other goods, $y^{\prime }$ units of $B$ at time $t+\mathrm{d}t$, etc. Then consider the value of $x$ units $A$ in terms of $y$ units of $B$ at time $t$. We suppose the ratio $y^{\prime }/y$ describes the change in value of $A$.

This should be viewed as problematical, since the values for every commodity fluctuates over time. So it may not be practical to consider $y^{\prime }/y$ as the defining factor for fluctuation.

Endnotes

[1] Technically, one could view it as a category - in the sense of category theory. This gets really complicated really quick if we try to interpret a negative quantity of commodities, since negative numbers haven't been adequately (vertically) categorified yet.

Revising Portions of my Notes on Sraffa

I realize, looking back through my notes on Sraffa's Production, I begin to get a little sloppy with chapter 7 or chapter 8. [Just updated chapter 7's notes August 8, 2013.]

Perhaps this is because I am having difficulty grasping various concepts, and I don't know which ones they are!

Consequently, I am going to pause, go back to these chapters, then revise them to a higher standard. I'll also revise chapter 9's notes, too, despite publishing them today!

I'm in the middle of moving, so this will take a while, but bear with me as I revise an otherwise incoherent summary...

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.