In order to utilize scarce goods the most efficiently, people need to make economic calculations via the price mechanism. Its been said that private property is essential to perform viable economic calculation. Why is this so?
Because if there is not private property in the capital markets there is no way to decide compare the relative scarcity of various heterogenous inputs and their alternative means of deployment. Without someone owning the capital factors of production any price applied to them is not a real price (an exchange ratio) but is instead a meaningless number which in no way reflects the actual scarcity.
There is an entire field of combinatorial optimisation in mathematics and computer science, dedicated to solving such problems in kind, yet even the most complex in this category are always defined with regard to some objective function in view from which optimisation of the bundle of determining variables takes place. This, as Edmund points out is not available for acentral planner trying to solve an economic calculation problem, There has to be a rational basis through which it must be decided how prouction is optimised, but where it should be optimised further and where less, given that most factors of production are substitutabe to an extent, but not completely along the many stages of production of an industrial economy. This is not to say, a central planner, akin to Robinson Crusoe, might not try to produce an extremely botched structure of production, with immense wastage and chaos in the process of trying to “solve” the calculation problem.
The profit-loss system is the only system which allows these decisions to be made via a non-arbitrary cardinal number basis, through which success in changing and expanding various domains of the structure of production, and thereby the remuneration of the capitalist entrepreneur, is proportional to the extent to which the rest of society evaluates this as beneficial.
Objectively comparing opportunity costs of the use of various means(with alternative uses) to the ends they achieve with the use of cardinal numbers can only be achieved via a system where means of production(capital) is privately owned, since only then can there be exchange, and subsequently prices for these factors of production. only with a pricing system for factors can entrepreneurs(note of course entrepreneurs do not evaluate present prices in their decisions but estimate future prices) objectively compare the opportunity costs of means and ends from an entire plethora of processes they could engage in, and guide themselves in picking what they perceive the best. Those who make incorrect valuations are subsequently weeded out, as their estimations of future prices failed to come to fruition.