Recently, I was asked by a commenter in one Internet discussion to clearly explain the essence of the calculation problem. Later, I got thankful for the opportunity to do that because it was actually not so easy but in the end I managed to come up with what I think is an adequate formulation.
1) There is no such thing as a social welfare function. There is no way for any agent to spell out the structure of preferences of all the society among various combinations of consumer goods.
2) Because of (1) the only thing even a socialist government can do to even try to respond to the needs of consumers is to retain the market for consumer goods. However, because we are talking about the calculation problem for socialism, there can be no markets and thus prices for producer goods.
3) Suppose that the government in the beginning of the first socialist production period distributes a certain amount of coupons among the population that are valid for the period for buying consumer goods. In parallel it implements a certain production plan for the period. What it gets in the end of the period is a set of quantities of various consumer goods and prices paid for them. In the next period it can distribute the same amount of coupons and make adjustments to the production plan for the period. In the end of the period it will get some other set of quantities and prices but the crucial thing is that there will be no basis for a rational comparison between those two sets to tell which of them is associated with better satisfaction of consumers’ wants. Thus, economic calculation is literally impossible even under market socialism.
4) In contrast, in the presence of markets for producer goods, the adjustment of the production structure towards better satisfaction of consumers’ wants happens through the process of arbitrage between the prices of consumer goods and producer goods. If an entrepreneur finds a way to use certain producer goods to create more value for consumers she will receive profit. If profit is high it will attract other producers. Also, if consumers still want the consumer goods produced with the relevant producer goods strongly enough, the prices for the relevant producer goods will tend to rise and push producers of the producer goods which are inputs into the production of the former into producing more of those. There will also be incentives over the whole chain to innovate in order to lower the costs of production. Thus, resources will tend to be continuously reallocated towards more productive uses over the whole production structure.