Adapted, in essence, from Ludwig von Mises, Socialism, translated by J. Kahane (Indianapolis: Liberty Fund, 1981), second edition published in 1932.

Proposition: No rational economic action is possible in a collective economy (hereafter, "socialist" economy).

Demonstration:

1. Human beings can only make rational decisions (matching means efficient and proportionate to mediate ends) when the options and stakes are clearly represented to them, in a way that has meaning for their own situation. Only then can options be differently evaluated, to determine the choice. In short, rational choice must be meaningfully informed. [Axiom]
2. In a complex society, we must often make choices that affect others whose desires are not transparent to us, and whose knowledge of their own respective situations is not accessible to us (strangers and acquaintances). [Postulate]
3. In order for these choices to be rational, the effect of our actions on their options and stakes must be represented to us. Our respective situations must be put into communication or commensuration with one another, according to like terms. [By 2 & 1]
4. Commerce in a standard medium (money) is the process by which we develop representations of supply and demand (situation), cost and benefit (stakes), in such a way as to make this communication commensurate across anonymity. We inform one another's situations by offering or withholding incentives. [Definition #1]
5. A socialist state (in the old sense of of a collective economy, whether authoritarian or democratic) is one in which there is only one owner of capital or natural resources in the whole market. In short, there is no commerce. [Definition #2]
6. Let us suppose there to be a socialist state. [Declaration]
7. Since it is socialist, there is no commerce. [By 6 & 5]
8. No commerce, then no process by which these representations might be accomplished in money prices (ie., commensurable terms). [By 7 & 4]
9. No process by which representations might be made, then no way the options or stakes can be made meaningful to the decision-makers. [By 8 & 3]
10. Therefore no rational action is possible in a socialist economy. [By 9 & 1] — QED.

Two corollaries:

Mises's Corollary: A state that intervenes in its economy, will tend to suffer from the socialist calculation problem, the moreso the more often and more extensively it intervenes.

Rothbard's Corollary: A private firm that approaches monopoly will tend increasingly to suffer internal and external calculation problems within its industry, as if it were a socialist state. (For this reason, Rothbard says, we need not worry about true monopolies forming naturally in a free market.)