Understanding the Action Axiom
The action axiom can be stated as follows (Cf. Rothbard [1962, 1970] 2004, pp. 1–2, 7–8, 19–20; Rothbard, 2011, pp. 113, 290; Mises,  1998, pp. 14–16): “Human beings engage in purposive behavior—i.e., they choose which scarce means are to be more fruitfully (or economically, or rationally) employed in order to satisfy their most preferred ends. This behavior—stemming from human free will—is what we call action. As long as means are scarce and wants are not fully satisfied, human beings will keep on intentionally (or purposefully) acting.”
Why is this an axiom? Because you cannot disprove it without either conceding its truth or incurring a self-contradiction (Cf. Rothbard 2011, pp. 6, 10; Rothbard  2002, p. 32 and 32n6). In fact, anyone trying to disprove the action axiom would indeed engage in purposive behavior—i.e., he would be employing scarce means (his time, his intellectual labor, etc.) in order to achieve a preferred end (trying to disprove the action axiom instead of, say, watching TV or reading Rothbard). Therefore, the action axiom denier would either contradict himself—claiming the untruth of a statement he is instead performatively proving to be true—or be forced to concede the truth of the axiom itself—because otherwise he could not maintain to be acting in the sense we defined above.
The Action Axiom and “Microeconomics”
Generally, microeconomics textbooks frame the main tenets of consumer (and, analogously, producer) theory as follows: “Consumers are supposed to be rational—i.e., they employ scarce means to attain desired ends. Moreover, consumers’ preferences are assumed to feature (at least) the following properties: completeness, transitivity, and non-satiation.”
As I will briefly explain, it is very easy to derive these properties from the action axiom itself. There are many other properties of human behavior that we could derive from this axiom—such as, e.g., time preferences, the law of diminishing marginal utility, and the law of optimum returns—but those will not be examined here.
First, consider “completeness”—i.e., human beings’ capability of always ranking alternative ends. Since action requires having preferred ends, it also entails that human beings will act if and only if they indeed have ends they want to achieve—i.e., they are willing to act in order to substitute a state of the world with a more desired one. In other words, were human beings not always capable of deciding between at least two possible ends and ranking them (e.g., B is preferred to A, or vice versa), they simply could not act (Cf. Rothbard 2011, pp. 305–07). But, as we saw above, action is an axiomatic truth of human nature—we cannot conceive of nonacting human beings. Therefore, action itself implies complete preferences—i.e., human beings need to be always capable of ranking the various potential states of the world (A, B, C, etc.) they can attain while engaging in action.
Second, consider “transitivity”—i.e., if I prefer A over B and B over C, then I must also prefer A over C. Again, it is straightforward that absent transitivity human beings would not be capable of conclusively ranking preferences—hence, they would not have precisely definable desired ends to struggle for. Consider, for instance, the simplest possible case: Fabrizio is faced with the possibility of employing some means (say, one hour of his labor) in order to attain one among three alternative ends—A, B, and C. However, suppose also that he prefers A to B, B to C, and then C to A; the question now arises: How could he act? In fact, it is obvious that he would face a paradox—not being able to decide which end is to be pursued and which ones are to be foregone. Again, this proves that transitivity is directly—and easily—derived from the action axiom: men with nontransitive preferences would not be able to act—but this would contradict their nature as human beings!
Third, consider nonsatiation. Here, suffice it to say that “a man perfectly content with the state of his affairs [i.e., satiated] would have…neither wishes nor desires; he would be perfectly happy. He would not act; he would simply live free from care” (Mises  1998, p. 13). Again, denying nonsatiation of preferences would be tantamount to denying the acting nature of human beings. Thus, we proved again (via reductio ad absurdum) that nonsatiation is consistently derived from a true axiom—and must thus be itself true.
There are two other properties that conventional microeconomics textbooks ascribe to economic agents’ preferences—namely, continuity and convexity (i.e., consumers are assumed to love goods’ variety). Suffice it to say that while convexity can be postulated (but need not be, since it is not directly derived from the action axiom), the assumption of continuity is simply wrong—because human action does always involve choices among discrete, noncontinuous units (pounds of bread, gallons of milk, haircuts per year, etc.) (Cf. Rothbard [1962, 1970] 2004, pp. 130n27, 305–07).
The Action Axiom and Ethics
If the fundamental feature of human nature is that human beings act, then what does this (a priori) truth imply for other branches of human sciences? Take, for instance, the task of building a rational (i.e., a priori true) ethics of human liberty.
Starting from the action axiom, we can, for instance, build a rational and cogent argument for human beings’ natural and absolute self-ownership—i.e., any human being shall be the sole owner of his body, his labor, and his mind (namely, his choices, his values, his free will, etc.). In fact, let’s assume the simplest possible case: there are two persons—A and B. Now, there are three possible arrangements (Cf. Rothbard 2011, pp. 353–54).
First, we can assume that A is the owner of B (or vice versa). However, if A is the owner of B, then B is A’s slave—and slaves, by definition, are not free to act as they will. But if B is no longer capable of free-willed action, then he would no longer be a human being as we defined this concept (i.e., acting man) above. But this would imply a paradox—B being a human being in our hypothesis but no longer being one in our conclusion. Hence, this first arrangement is to be discarded as self-contradictory.
Second, we can assume that A owns a share of himself and a share of B (and vice versa). However, this would entail that A cannot act without B’s approval (and vice versa). But, for B to approve A’s action, B is required to purposefully cast a vote approving such an action (and vice versa). But then we reach again a paradox: A cannot act without B’s consent, but B’s consent is itself an action requiring A’s approval, but A’s approval is itself an action requiring B’s consent, et cetera—an infinite regress. But if both A and B’s action is stalled, then neither A nor B can act—and, if they cannot act, they are at odds with their nature as human beings (i.e., acting men). Hence, again, this second arrangement is to be discarded as paradoxical.
Lastly, we are left with the third option, the only consistent—i.e., noncontradictory—solution to our problem of human beings’ ownership: both A and B are to be self-owners. No other arrangement is compatible with human nature—i.e., action as deliberate choices employing scarce means in order to achieve preferred ends.
Economics has conspicuously departed from its Mengerian-Misesian-Rothbardian framework as a purely deductive science. By doing so, mainstream economists have forgotten the epistemological groundwork of economics, abandoning proper praxeological analysis, neglecting the apodictical (a priori) truths of praxeology, and imitating empirical (a posteriori) sciences. In the end, this departure from the Mengerian-Misesian-Rothbardian framework has proved detrimental for (at least) two reasons.
First, it prompted mainstream economists to justify their assumptions as “operational” (or “heuristic”) hypotheses—like empirical scientists do—instead of acknowledging the a priori truth of praxeological tenets. Second, it undermined sound and rational analysis even in other branches of human sciences—such as ethics—thus leaving intellectual room for moral relativism, postmodernism, and collectivistic ideologies.