One century ago, Mises began the socialist calculation debate, publishing his essay Economic Calculation in the Socialist Commonwealth (1920) and his subsequent treatise Socialism: An Economic and Sociological Analysis (1922). Later, Mises included his antisocialist arguments in Human Action  (1999) , his magnum opus, especially in Sections III (about economic calculation) and V (about the economic impossibility of socialism).
Mises question on socialism is straightforward and simple: Can a socialist economy allocate resources efficiently as the free market does (cf. Mises  1999, p. 691)? In order to answer, we need to understand (1) how does a free market economy work, (2) the importance of economic calculation and entrepreneurship, and (3) the reason why socialism is intrinsically incompatible with the very idea of economy.
How does a free market economy work? It’s a system of human interactions wherein human beings make their choices of consumption and production—efficiently allocating different privately owned means (scarce resources with alternative uses) to satisfy different ends (consumptive wants). Since human ends (consumptive wants and desires) are subjectively valued, the means conducive to their satisfaction (production goods) are subjectively valued as well—according with the ends they satisfy, i.e., the consumptive goods and services they produce. Of course, a free market economy features human beings freely exchanging both consumptive and productive goods and services. Such exchanges occur at freely agreed ratios (prices), which express the essence of economy: satisfy (directly—via consumption—or indirectly—via production) chosen ends while giving up other less preferred ones.
It’s therefore clear that the concept of economy is linked with the idea of exchange—thus economics, the science concerned with economy, is more aptly labeled catallactics, i.e., the science of exchanges, from the Greek verb katallassein, meaning “to exchange” (cf. Mises  1990, pp. 15–16). But exchange requires previous estimation and calculation of pros and cons, assessing whether what we give up is actually worth less than what we gain (cf. Mises,  1999, p. 230).
In a free market economy, productive choices are governed by the profit and loss mechanism, whereby sovereign consumers signal—through their consumptive choices—which entrepreneurs they are willing to “reward” and which ones they are willing to “punish” (cf. Mises,  1999, pp. 295–97). When entrepreneurs supply consumers with desired consumptive goods (ends) at affordable prices (i.e., when they employ scarce productive means effectively and efficiently), they are rewarded by consumers with entrepreneurial profits—thus increasing entrepreneurs’ net worth, their capital (cf. Mises  1999, p. 231). Otherwise, consumers “punish” entrepreneurs through losses—decreasing entrepreneurs’ net worth, their capital, and turning their investments into malinvestments.
Thus, we understand the pivotal importance of entrepreneurship within a free market economy. Entrepreneurs, indeed, are the transmission belt between consumers’ wants (consumptive goods and services) and the means conducive to their satisfaction (production goods). Hence, entrepreneurs are the central cog of the economic choice mechanism. They (1) forecast, or speculate, which wants consumers are eager to satisfy, (2) perform the economic calculation establishing whether such wants can be efficiently satisfied, and (3) employ their own savings—skin in the game—while investing and buying production goods.
It’s hence evident that entrepreneurs are both speculators (they envisage future possible scenarios) and savers-capitalists (they save and accumulate the capital they later invest).
But speculation requires calculating tools—the price system. How does it emerge? Prices can emerge only while exchanging, buying, selling, purchasing, etc. (cf. Mises  1999, p. 202). Prices are, indeed, the ultimate expression of economic action—gaining something (say, a T-shirt) while foregoing something else in exchange (say, twenty dollars). Absent exchange, prices cannot originate: they would be not only impossible, but even inconceivable. Prices are, in fact, ratios (or tradeoffs) at which given exchanges are performed—if exchanges are abolished, prices will follow suit. Thus, were exchanges for particular goods (say, production goods) to be abolished, these same goods would cease to feature market prices.
Saving and capital accumulation , on the other hand, require private ownership to be in place: it’s indeed thanks to private ownership over their own capital—i.e., production goods—that entrepreneurs enjoy profits and suffer losses (cf. Mises  1999, pp. 254, 302–04, 704–05; and Mises  1999, p. 37), thus allowing the profit and loss mechanism to function properly and to steer productive activities on consumers’ behalf.
Here comes socialism: collectivizing production goods’ ownership, socialism abolishes entrepreneurship via two logical steps. First, entrepreneurs are “directly” abolished as capitalists, since they are legally forbidden from privately own and accumulate capital—i.e., production goods. Second, being all production goods now owned collectively, they can no longer be exchanged, bought, sold, etc.; hence, prices cannot emerge any more for these goods, and entrepreneurs can no longer compute costs of production while choosing what to produce and how to produce it—thus, entrepreneurs are “indirectly” abolished as speculators.
What about the fashionable critique of socialism proffered by Hayek and Robbins, i.e., that socialism is impossible because the central planner would lack (1) the knowledge and/or (2) the intelligence necessary to plan production? Hayek and Robbins, indeed, ground their critique of socialism on the central planner’s incapability of (1) obtaining all the relevant information necessary to plan production and/or (2) computing and calculating the optimal level of production (cf. Salerno 1990, pp. 57–64).
But central planner’s knowledge and intelligence are not the relevant argument for Mises. By means of abolition of entrepreneurs (the pivots of free market production choices), socialism itself gets incompatible with the very idea of economy—economize available means to attain desired ends.
If, indeed, production goods are collectively owned by a single entity (government, State, folk, etc.), how would it be possible to trade, to exchange, to sell and purchase them? It would be impossible—hence, there wouldn’t exist a market for them. But, without a market, how could prices emerge? Of course, they couldn’t (cf. Mises,  1990, p. 4). But again, without prices for production goods, how to compute costs of production? And profits and losses? Of course, it would be impossible as well (cf. Mises,  1999, p. 701). And, without profits and losses, a socialist economy has no tool conducive to efficient allocations of production goods.
Thus, without knowing whether revenues are higher or lower than costs (because costs cannot be computed), how would the socialist central planner know whether production is being carried out according to consumers’ desires? Of course, that would be impossible to know (cf. Mises,  1999, p. 209). A socialist central planner, in fact, even knowing which consumptive goods are desired most, would know neither which ones could be profitably produced, nor how to efficiently produce them (cf. Mises,  1990, p. 21). Absent market prices for production goods, no profit nor loss can be computed—hence, producers have no “compass” guiding them through production choices. Economic calculation is impossible for goods with no market (cf. Mises,  1999, pp. 215, 230).
So, why does socialism fail? It fails because it’s the very negation of the idea of economy—economizing man. Abolishing private ownership for production goods, socialism abolishes the market for them and makes it impossible for market prices to emerge and for costs of production to be computed—thus impairing the profit and loss mechanism. Socialism does not necessarily fail, as Hayek and Robbins thought, because the central planner lacks the knowledge and/or the intelligence needed to plan production; it fails, instead, because it abolishes entrepreneurship and economic calculation.