Suppose we had some Sraffian model of an economy, 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?
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.
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).
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 As one might expect, the links to various external web-pages are out of date or completely broken...so just fixing the references might be a full time job.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.
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.
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."
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.
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
where 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.,
This is half the condition for linearity. We also expect
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 function.
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 in the equation
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
where is some real number. This describes a change in value for units of commodity .
5.1. Remark. We don't measure this variation directly. We gauge it from how the value at time for units equates to other goods, units of at time , etc. Then consider the value of units in terms of units of at time . We suppose the ratio describes the change in value of .
This should be viewed as problematical, since the values for every commodity fluctuates over time. So it may not be practical to consider 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.
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 .]
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...
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
and where , 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
So, suppose we have for our toy example (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:
For simplicity, the production of 'a' and 'b' are blackbox
functions which takes "some vector" of inputs.
We have combined . The relative quantities of 'a' and
'b' are, literally, one-to-one. Observe the surplus is 3 c...and
we had contribute.
But if we change how we produce things, say use only the first
process, then we have
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 to
.
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
where A is the input-output matrix, is the price-vector,
w wage, the labor vector, and r the rate of
profits. Then we have
where I is the identity matrix.
This gives us
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:
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
or if we introduce new symbols to stress the similarity to the
previous case:
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 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.
Stock Simulation in Clojure, a basic introduction to modeling using software. Fairly mainstream, but I work with clojure professionally, so there it is.
This is a review of the concept of Joan Robinon's "metaphysics" in economics...I suppose we might call it "economic metaphysics" or something like that. The interested reader may peruse her original book on the subject:
Joan Robinson, Economic Philosophy. Penguin books, 1962. Freely available online at archive.org.
My page numbers will refer to the Penguin edition.
Definition of Metaphysics
Robinson begins with the definition of metaphysics, saying:
The hallmark of a metaphysical proposition is that it is incapable of
being tested. We cannot say in what respect the world would be different
if it were not true. The world would be just the same except that we
would be making different noises about it. It can never be proved wrong,
for it will roll out of every argument on its own circularity; it claims
to be true by definition of its own terms. It purports to say something
about real life, but we can learn nothing from it. Adopting Professor
Popper's [fn: See The Logic of Scientific Discovery] criterion
for propositions that belong to the empirical sciences, that they are
incapable of being falsified by evidence, it is not a scientific
proposition. (pp. 8.75--9.1)
Question 1.Is this a good definition of a "metaphysical proposition"?
But look, we really have two different criteria for a metaphysical proposition given:
Popperian Critera: "...it is incapable of being tested. [...] they are incapable of being falsified by evidence [...]."
Logical-Positivist Criteria: "It can never be proved wrong,
for it will roll out of every argument on its own circularity; it claims
to be true by definition of its own terms. It purports to say something
about real life, but we can learn nothing from it."
Well, okay, so we have two different criterion...is this really a bad thing? It is if they are inconsistent, i.e., we have a proposition be metaphysical in one criterion but not in the other.
We have to dissect the Popperian critera a bit further before answering our first question.
Question 2.Is "Testability" Well-Defined?
Robinson continues, asking about the Classical Economist's notion of 'value' "What is it? where shall we find it? Like all metaphysical concepts,
when you try to pin it down it turns out to be just a word" (p. 29.66).
The objection raised is we cannot observe value, therefore it cannot be falsifiable or testable. (This is the subject of the second chapter.)
This seems too naive stating "observable = testable = scientific". After all, hard sciences don't use this criteria (otherwise renormalization in QFT would be "metaphysical", as well as natural selection in biology, among many other scientific concepts).
Do we "observe" the business cycle or its symptoms? Well, it's the latter (with unemployment, inflation, and other indicators reflecting the state of the cycle).
So the business cycle is not testable, according to this strict criteria. It's "metaphysics".
After all, we observe only "symptoms"...does that mean we should designate the disease as "metaphysics"?
I don't think so...
Back to Question 1:Consistency of Criteria?
Well, the business cycle isn't "observable" although its symptoms are observed. The Popperian criterion qualifies it as a "metaphysical concept".
Now, to answer our question "Are the two criteria for metaphysics consistent?" We will examine...whether the business cycle qualifies in the "Logical Positivist" sense as "metaphysics".
Really, we have two different notions here: (a) the model explaining the business cycle, and (b) the phenomenon of the business cycle [i.e., the actual process itself].
The logical positivist criteria would consider (a).
I will state here my confessed belief that the model for the business cycle is not "circular" (no more so than any other model). If pushed, I don't know if I could defend this position in a short time --- we would have to specify which model!
But we have inconsistent results from the two criteria: Popper says the business cycle is metaphysics, whereas the Logical Positivist approach disagrees.
Answer to question 1: the criteria given is inconsistent.
This dawned on me after posting: Wittgenstein's "language games" and "rule-following" are important and relevant concepts when discussing "metaphysics".
Question 3.Can we classify a proposition as neither "scientific"
nor "metaphysical"?
After all, the Popperian criterion boils down to "analytic proposition =
metaphysical, and synthetic proposition = scientific".
This tacitly assumes that language reflects reality. (Yep, time
to drop the W-bomb!)
After all, doesn't this alleviate our anxieties Popper raised concerning the distinction between model and symptoms?
I should probably flesh this out a bit more, but I'm certain anyone could write a thesis on "Robinson's 'metaphysics' in the context of Wittgenstein's 'language games'"...
Question 4.Is the concept of a "metaphysical proposition" itself a metaphysical proposition?
I am a bit sloppy here. I should specify: isn't the proposition "A
metaphysical proposition exists" a metaphysical proposition?
What do I mean by "metaphysical proposition"? It doesn't matter.
Lets consider both criterion for a "metaphysical proposition".
Using Popperian Criterion. We can't observe a
"metaphysical proposition", since it's entirely a human construct. So, I
guess that means it is metaphysical.
I fear I am using a caricature of the Popperian criterion, so critical
comments would be welcome!
Using Logical-Positivist Criterion. "It can never be proved wrong,
for it will roll out of every argument on its own circularity; it claims
to be true by definition of its own terms. It purports to say something
about real life, but we can learn nothing from it."
Isn't this vacuous?
After all, regarding the proposition "A metaphysical proposition exists"
— well, we "cannot say in what respect the world would be different
if it were not true."
Hence, we conclude the existence of metaphysical propositions...is a
metaphysical notion.
Question 5.Does "metaphysics" contribute to economics? More precisely, does is this a useful dichotomy [a proposition is "scientific" or "metaphysical"] or not?
I wonder about this, because what are we left with when we
examine only "scientific propositions" in economics?
We're left with what may be observed. Isn't this merely econometrics?
Whenever discussing econometrics, someone ivariably invokes
the Lucas Critique.
Tangential Question: Is the Lucas Critique a metaphysical
proposition? [We'll have to consider this another time...]
Categorizing a proposition as either "scientific" or "metaphysics"
implicitly gives it a positive or negative connotation (respectively).
Denoting propositions as "metaphysics" boils down to using loaded
terms.
What's worse, Robinson notes "Yet metaphysical statements are not
without content" and "Metaphysical propositions also provide a quarry
from which hypotheses can be drawn" (9.1--9.5).
The only positive twist I could put is if Robinson meant "metaphysical
propositions" belong to an Althusserian problematic...a background
"idea-logical" [ideological] framework an economist brings to the game.
Is that even useful to know?
Question 6.If we weaken "testability", can we get a good notion of "metaphysics"?
I think we're being too strict with our notion of "metaphysics". We
should use a slightly different criterion, a little weaker than either given.
It seems "testability" is too strict for economics. We should be a
little weaker...if the concept produces "accurate predictions", we
should consider it scientific.
Joan Robinson says the notion of "Value" [in Classical Economics] is
metaphysical since we cannot "observe" it...we don't find it living
under a rock.
BUT we can predict prices using "values". If the predicted
prices are "wrong", then the concept of "value" would be incorrect.
Although this is a more realistic criterion for "testability", it would
violate the understanding that metaphysical propositions constitute the
ideological framework an economist uses to analyze phenomena.
53. Negative Multipliers: I. Proportions of Production Incompatible with Proportions of use
When we consider in detail how we construct a Standard system with
joint products, it becomes obvious some of the multipliers may be
negative.
Consider two products jointly produced, each through two different
methods.
The possibility that varying the extent to which one or the other
method is used ensures a certain range of variation in the
proportions in which the two goods may be produced in aggregate.
For each commodity, its two methods limits the range of proportionality.
The limits are reached as soon as one or the other method is
exclusively employed.
Now suppose in all cases which two joint products 'a' and 'b' are
used as means of production, the proportion in which 'a' is
employed relatively to 'b' is invariably higher than the highest
of the proportions in which it is produced.
In such circumstances we may say some process must enter the
Standard system with a negative multiplier: but whether such a
multiplier will have to be applied to the low producer or high user
of commodity 'a' cannot be determined a priori—it can only be
discovered through the solution of the system.
54. Negative Multipliers: II. Basic and non-basic jointly produced
Non-basic products are "the most fertile ground" for negative multipliers.
(NB: non-basic goods needs a new definition under these new
circumstances…but we may say that the main class, i.e. products
altogether excluded from the means of production, will still be
non-basic; see §60)
Consider again the case of two commodities (jointly produced in
different proportions by two processes). One is to be included in the
Standard product while the other — not entering the means of
production for any industry — must be excluded from the Standard product.
This will be effected by giving a negative multiplier to the process
which produces relatively more of the second commodity, and a
positive one to the other process.
The two multipliers being so proportioned when the two equations are
added up to the two quantities produced of the non-basic exactly
cancel out…while a positive balance of its companion product is
retained as a component of the Standard commodity.
55. Negative Multipliers: III. Special raw material
Once negative multipliers have been admitted for some processes,
others (which shine with a reflected light) are liable to appear.
Hence, suppose we have a raw material be directly used in only one
process. Suppose that process has a negative multiplier. Then the
industry which produces the raw material will itself follow suit and
enter the Standard system with a negative multiplier.
56. Interpretation of negative components of the Standard commodity
Since no meaning could be attached to "negative industries" which such
multipliers entail, it becomes impossible to visualize the Standard
system as a conceivable rearrangement of the actual processes.
We must therefore (in the case of joint-products) be content with the
system of abstract equations, transformed by appropriate multipliers,
without trying to think of it as having a bodily existence.
Remark. I'm sure many marginalist economists howl out in
frustration over this, which is amusingly ironic.
The Standard system's purpose is to provide a Standard commodity. When
it has negative components, there is no difficulty interpreting them:
they are liabilities or debts. This is analogous to accounting
(negative numbers = liabilities/debts; positive numbers = assets).
Hence a Standard commodity which includes both negative and positive
quantities may be adopted as money of account without straining the
imagination, provided the unit represents a fraction of each asset and
each liability (like a share in a company)…with the liability in the
shape of an obligation to deliver without payment certain quantities
of particular commodities.
57. Basics and non-basics, new definition required
We have another difficulty we must tackle before constructing the
Standard commodity: the criterion distinguishing basic and non-basic
goods fail…since it's ambiguous whether a product entering the means
of production for only one industry producing a given commodity should
or should not be regarded as entering directly the means of production
for that product.
Footnote: The trouble lies deeper, and as we shall see presently
there would be uncertainty even if the commodity entered directly
the means of production of all the processes in the system! See §59.
And the uncertainty would naturally extend to the question whether it
did or did not enter "indirectly" the production of commodities, into
which the latter entered as means of production.
58. Three types of non-basics
All three distinct types of non-basics are met in the single-product
system will find their equivalents in the case of multiple-product
industries.
Taking advantage of this circumstance, we begin defining for the
latter case the three types of non-basics, each as the extension of
the corresponding single-product type (cf. §35).
Products which do not enter the means of production for any
industry. This type can be immediately extended to the
multiple-product system without modifying anything.
Products each of which enters only its own means of
production. The equivalent would be a commodity which enters the
means of production for each of the processes by which it is itself
produced, and no others — but enters them to such an extent that
the ratio of its quantity among the means of production to its
quantity among the products is exactly the same in each of the
processes concerned.
Products which only enter the means of production for an
interconnected group of non-basics; in other words, products which
(as a group) behave in the same way as a non-basic of the second
type does individually.
In order to define (in the multiple system of k processes) the type
which corresponds to the third case (with the interconnected group
consisting of 'a', 'b', and 'c'), we arrange the quantities in
which these commodities enter any one process, as means of production,
and as products, in a row. We shall thus obtain k rows ordered in
2×3 columns as follows:
Footnote: Some of these quantities may be zero, of course.
The condition for the three products being non-basic: not more than
three of the rows should be independent, and the others should be a
linear combination of those three. (For the general definition, see §60.)
59. Example of the third type
This third type gives us "curiously intricate patterns". Sraffa
demonstrates this with an example.
Given a system of four processes and four products, two commodities
('b' and 'c') are jointly produced by one process and are
produced by no other.
But while 'b' does not enter the means of production for any
process, 'c' enters the means of all four processes.
Supposing the process producing 'b' and 'c' corresponds to the
equation
the "rows" for the two commodities will be
Only the first row and any other are independent, the remaining two
rows are linear combinations of the first pair. So both 'b' and
'c' are non-basic.
If we look at the matter from constructing the Standard system, we
see: (a) it's obvious 'b' can't enter the Standard commodity, (b)
'c' looks like it could be a suitable component.
However, since 'b' occurs only in one process, the only way to
eliminate 'b' is omitting that process altogether.
But that process was the exclusive producer of 'c', so it only
appears as means of productions…not as a produced commodity. So
'c' cannot possibly enter the Standard commodity, and must be
dropped.
60. General Definition
The formal definition given is not as satisfying as the intuition
"entering (or not entering) the means of production for all
commodities", which it supercedes. BUT it has the advantage of
greater generality.
Lets observe the first two types of non-basics may be considered as
special cases of the third.
The definition covers the three types of the single-product system.
(It is quite general, and as the example in §59 suggests, it
includes a final type of non-basic, which is introduced
subsequently…namely commodities which enter the means of
production but are not produced — a type which land is the
outstanding example.)
We can give this general formulation between the distinction between
basic and non-basic goods:
Critera. In a system of k productive processes and k commodities
(no matter whether produced singly or jointly), we say that a
commodity — or more generally a group of n linked commodities
(where 1≤ n< k) — are "Non-Basic" if:
of the k rows (formed by the quantities in which they appear in
each process) not more than n rows are independent, the others being
linear combinations of these.
Or, in linear algebraic terms, the matrix of k rows and columns
is of rank less than or equal to n.
All commodities which do not satisfy this condition are "Basic"
(Note that, as has been stated in §6, every system is assumed to
include at least one basic product.)
61. Elimination of non-basics
It follows we can (through linear transformations) entirely eliminate
non-basic commodities from the system…both on the side of the means
of production and the products.
This operation achieves the same result as we obtained in the
single-products system by the much simpler method of crossing out
equations of industries producing non-basics (§35).
62. The system of Basic equations
If the number of basic products is j, the system thus obtained will
consist of j equations: these may be described as Basic equations.
Supposing the j basic commodities are a, b, …, j we shall
denote the net quantities in which they appear using the
"barred-quantities" , , …, to
distinguish them from the quantities in the original processes.
The Basic equations will accordingly be as follows:
This system is equivalent to the original one inasmuch as the values
it determines for R and the prices will also be solutions of that system.
It differs from the original system (aside from obviously excluding
non-basics):
(a) A basic equation does not represent a productive process — it
merely is the result of combining the equations of a number of
processes.
(b) It may contain negative quantities as well as positive ones.
63. Construction of the Standard system
The basic equations are designed for the construction of the Standard
product.
Footnote. It would be possible to construct the Standard product
directly from the original equations, and the final result would
have been the same. Why it has seemed simpler to go through the
intermediate step of the Basic equations, well, Sraffa explains it
in Appendix C.
The multipliers , , …, which applied to the
Basic equations give the Standard system are determined by the
following equations:
The equations give an equation for R of the j-th degree, so there
may be up to j possible values of R and corresponding sets of
values of the q's. Each set will represent a Standard commodity of
different composition.
64. Only the lowest value of R economically significant
When deciding which (among the j possible sets of values) is the
relevant one, we can not rely on there being a value of R which
corresponds to an all-positive Standard commodity. Why? Because in a
system with joint-production all possible Standard commodities may
include negative quantities among their components.
If we reconsider the matter from the perspective of single-product
systems, we find while an all-positive Standard makes sense, its
superiority is due to it corresponding to the lowest possible value of
R (as we shown in §42).
We shall see the possession of this last property is — by itself
— sufficient to make the Standard net product, endowed with it,
the one eligible for adoption as unit of wages and prices.
This is regardless of whether the Standard commodity with this
crucial property consists of all positive quantities or otherwise.
Suppose that being the lowest possible value of , we adopted
as unit the Standard product corresponding to another value (say
).
As the wage measured in this Standard was gradually reduced from
1 it would (before vanishing) arrive at a level such that
when the rate of profits would be equal to .
If at such a level of , we suppose on the basis of , then the wage
must be zero (since the rate of profits is at its maximum). While on
the basis of the wage must be positive since the rate of
profits is below its maximum.
We reconcile this through the wage be a positive quantity of a
composite commodity who's exchange value is zero. This is because
(as we shown in §41) the exchange value for a Standard commodity
the composition corresponds to one solution of (in our case
) at the prices that correspond to another solution of (for
us, ) is zero.
This implies, under these circumstances, the prices of all
commodities would — in terms of the chosen Standard — be
infinite(!).
Economically, such a result is meaningless.
This anomaly, however, can be avoided if we adopt as unit the
Standard net product corresponding to the lowest value of .
This is the only Standard product in terms of which, at all the
levels of wage from 1 to 0 (and so at all the levels of the rate of
profits from 0 to its maximum), it is possible for the prices of
commodities to be finite.
65. Tax on non-basic product leaves rate of profits and prices of other products unaffected
The distinction between Basics and Non-Basics has become so abstract
in the multiple-product system, we may wonder if it has become void of
meaning.
The chief economic implication of the distinction was the basics have
an essential part in determining prices and the rate of profits, while
Non-Basics have none. And this remains true under the new definition.
For single-product systems this implies: if an improvement took place
in the method of production for a basic commodity, then the result
would necessitate a change in the rate of profits and the prices of
all commodities.
Whereas a similar improvement for a Non-Basic would affect only that
particular Non-Basic's price.
This cannot be directly extended to a system with multiple products,
where both basics and non-basics may be produced through the same
process.
We can find an equivalent in a tax (or subsidy) on the production
for a particular commodity.
Such a tax is best conceived as a tithe, which can be defined
independent of prices and has the same effect as a fall in the
output for the commodity in question all other things (viz., the
quantities of its mean of production and its companion products)
remaining unchanged.
A tax on a basic product will affect all prices and cause a fall in
the rate of profits corresponding to a given wage, while if imposed on
a Non-Basic…it will have no effect beyond the price of the taxed
commodity, and those other Non-Basics linked with it.
Footnote. The effect which tax has on the price for Non-basics
will vary with the type of Non-Basics.
If it does not enter any of the means of production, its price will
rise by the amount of the tax.
If it enters its own means of production, its price will change to
the extent required to maintain the original ratio of the value for
the aggregate product of the process (after deduction of the wage
and tax) to the value of its aggregate means of production.
If it belongs to a group of interconnected non-basics, the prices of
all or some of the components of the group will change s oas to
maintain that ratio.
(In the example of §59 if the production of commodity 'c' were
taxed, the price of 'c' itself would be unaffected and the brunt
would be borne by the price of 'b' which would have to rise to the
necessary extent.)
This is obvious if we consider the transformed system of Basic
equations (which by itself determines the rate of profits and prices
of basic products) cannot be affected by changes in the quantity or
prices of Non-basics which are not part of the system.
