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
$$\mathrm{value}:\mathbf{Commodities}\to \mathbb{R}$$
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.,
$$\mathrm{value}(2\phantom{\rule{thickmathspace}{0ex}}\mathrm{tons}\text{}\mathrm{steel})=2\mathrm{value}(1\phantom{\rule{thickmathspace}{0ex}}\mathrm{ton}\text{}\mathrm{steel}).$$
This is half the condition for linearity. We also expect
$$\mathrm{value}(1\phantom{\rule{thickmathspace}{0ex}}\mathrm{ton}\text{}\mathrm{steel}+1\phantom{\rule{thickmathspace}{0ex}}\mathrm{quarter}\text{}\mathrm{wheat})=\mathrm{value}(1\phantom{\rule{thickmathspace}{0ex}}\mathrm{ton}\text{}\mathrm{steel})+\mathrm{value}(1\phantom{\rule{thickmathspace}{0ex}}\mathrm{quarter}\text{}\mathrm{wheat})$$
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
$$\mathrm{value}(1\phantom{\rule{thickmathspace}{0ex}}\mathrm{ton}\text{}\mathrm{steel})=x\phantom{\rule{thinmathspace}{0ex}}\mathrm{value}(\mathrm{quarter}\text{}\mathrm{wheat}).$$
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
$${\mathrm{value}}_{t+dt}(x\phantom{\rule{thickmathspace}{0ex}}\mathrm{units}\phantom{\rule{thickmathspace}{0ex}}A)=c\phantom{\rule{thinmathspace}{0ex}}{\mathrm{value}}_{t}(x\phantom{\rule{thickmathspace}{0ex}}\mathrm{units}\phantom{\rule{thickmathspace}{0ex}}A)$$
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.