The fact that two commodities exchange for each other in some proportion does not mean that they are therefore 'equal' in value and can be 'represented by an equation.' As we have learned ever since Buridan and the scholastics, two things exchange for each other only because they are unequal in value to the two participants in the exchange. A gives up x to B in exchange for y, because A prefers y to x, and B, on the contrary, prefers x to y. An equal sign falsifies the essential picture. And if the two commodities, x and y, were really equal in value in the sight of the two exchangers, why in the world did either of them take the time and trouble to make the exchange? --Murray N. Rothbard, Classical Economics: An Austrian Perspective on the History of Economic Thought, Volume 2, 410, emphasis mineRothbard's opening quotation was written as a response to the objective theory of value supported by Marx. One should note that Marx is not the originator of the objective theory of value; the objective theory has its origins going back to the writings of Aristotle. Rothbard summarizes the objective theory of value by citing Marx's rendition of it as follows:
Let us take two commodities, e.g., corn and iron. The proportions in which they are exchangeable, whatever these proportions may be, can always be represented by an equation in which a given quantity of corn is equated to some quantity of iron: e.g., 1 quarter corn = x cwt. iron. What does this equation tell us? It tells us that in two different things--in 1 quarter of corn and x cwt. of iron, there exists in equal quantities something common of both. (Murray N. Rothbard, Classical Economics, 409-410)The significance of these two quotations is simply that they represent one of the fundamental differences between laissez-faire and Marxism. Rothbard's introductory quotation is based purely on the subjective-value theory, which supports the laissez-faire doctrine, and Marx's quotation is based purely on the objective-value theory, which supports the communist doctrine.
The quotation from Marx, reminded me of the same thought process that appeared in my 2nd year undergraduate macroeconomics textbook by Blanchard and Melino. They used a similar line of reasoning when they constructed their "real exchange rates" and made statements such as 2 Canadian goods are exchanged for 1 American good and then said 2 Canadian goods = 1 American good. This is completely analogous with Marx's quotation above with the corn and the iron because both try to stick in an equal sign between things of unequal value. Moreover, in the famous Keynesian aggregate expenditure function (being equal to real output), Y = C + I + G + X - Q, they noted that the Q, which stands for imports, is measured in foreign goods, but the other variables are measured in domestic variables. This creates an "apples and oranges" problem, which they propose to solve by multiplying Q by the real exchange rate. By doing so, they argue that E*Q (i.e., the real exchange rate times the quantity of imports of foreign goods) is now measured in terms of domestic goods. At this point they claim that all the variables in the modified equation Y = C + I + G + X - E*Q are measured in terms of domestic goods; consequently they have solved the "apples and oranges" problem that was present in the initial equation. Personally, I found the whole idea of "a Canadian good" and "an American good" to be nonsensical. What exactly is a Canadian good if the sub-components that went into manufacturing the final consumer good were imported from 40 different countries? To construct the "real exchange rate," they compare the overall price level in Canada to that of the United States. This raises another question about the suitability of trying to construct overall price levels. Consumers who engage in a trade do not look at overall price levels; they look at the individual prices of the goods that they are considering buying.
Rothbard's critique of the Marxian position stresses the subjective nature of value and the fact that exchange takes place because people have different preferences. Moreover, Rothbard raises a similar argument that Mises (see Theory and History, 16) also raises concerning human action. If the two goods exchanged really were of equal value, then "his judgment of value expresses indifference. No action can result from such a neutral disposition" (16, emphasis mine). In other words, people won't engage in exchange if the actor views both goods as equal in value. People exchange in order to benefit from receiving something that they value more than the thing that they give up.