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Why are diamonds, which are little more than decorative baubles, so much more valuable than water, without which we would die? The answer is that value is determined at the margin, meaning that we value not "diamonds" as a category compared to "water" as a category, but one more diamond compared to one more unit of water.

Water is super-abundant while diamonds are not; this is one reason why a diamond is so expensive while water is not. This also illustrates an important point about decision making. Instead of "setting priorities" and viewing things in terms of all-or-nothing decisions, we should look at trade-offs.

One of the most important principles of economics is that decisions are made at the margin, and one of the key problems in classical economics concerned the source of value. The law of diminishing marginal utility is a fundamental tenet of economics, and it is every bit as much a scientific law as the law of gravity (more so, perhaps, as it can be deduced from an axiom — man acts — that is self-evidently true). Marginal utility is not decreasing just because we assume it is. The law of marginal utility is an implication of the action axiom, not merely an ad hoc assumption.

The "utility" one derives from consuming a good or undertaking an activity is best understood as a set of wants that can be satisfied by employing means, not as the output of a function mapping from an agent's consumption set into the real number line. Following this definition, the "marginal utility" of employing another unit of a homogeneous supply of goods or services must be understood as the additional set of wants that can be satisfied by employing that marginal unit. From the fundamental axiom of praxeology — that "human action is the use of means to arrive at preferred ends"[1] — we can see that the marginal utility of employing unit n is preferred to the marginal utility of employing unit n+1; in the language of mainstream economics, marginal utility must be decreasing.

In his classic article "Toward a Reconstruction of Utility and Welfare Economics" and in his treatise Man, Economy, and State, Murray Rothbard asks us to consider eggs as an example. (I alter the example slightly here.) Consider Joe, who has a wife, a daughter, a dog, and the following value scale:

Feed his family with cake Feed his daughter with scrambled egg Feed his wife with scrambled egg Feed himself with scrambled egg Feed his dog with scrambled egg

Suppose he needs four eggs to bake a cake. With his first egg, he will feed his daughter because he prefers this to all other sets of wants he can satisfy with one egg. With his second egg, he will feed his wife, and with his third egg, he will feed himself.

Now, suppose Joe purchases a fourth egg. This leads us to a possible misconception: one might be tempted to look at the situation and exclaim, "Aha! With the fourth egg, Joe can feed his family with cake, which he clearly prefers to feeding them with scrambled eggs! Clearly, then, the marginal utility of the fourth egg is higher than the marginal utility of the third egg; therefore, marginal utility is increasing!"

This line of reasoning neglects a crucial point: the fourth egg can only be used to bake a cake in the presence of the first three eggs. Since "marginal utility" is a concept that can only be applied to homogeneous units of a given supply, "one egg" is no longer the relevant unit of analysis. The homogeneity of units is determined by the set of wants that can be satisfied with a unit of a good; in this case, the relevant unit of analysis becomes "1 unit=a set of four eggs." Thus, Joe's value scale may now be

Feed his family with cake Feed his family with scrambled eggs

He will clearly choose to feed his family with cake and, should he happen upon a second set of four eggs, scrambled eggs as well.

The astute reader will notice that the value scales listed above were listed according to the wants satisfied by the marginal unit of a given good, not by the good itself after the fashion of Rothbard in "Toward A Reconstruction." Our hero Joe didn't prefer the first egg to the second in and of itself; he preferred feeding his daughter to feeding his wife. If only one egg is available, he must choose between competing ends, and the end that satisfies him most is feeding his daughter.

It should be evident that the law of marginal utility should be accorded just that epistemological status: a law. As Rothbard explains (and as Carl Menger and others showed before him), this theorem, which can be deduced from the action axiom, is more than merely empirically demonstrable: it is irrefutably true.