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.