The Economic Calculation Problem is a serious, possibly fatal, problem for claims that socialism can increase well-being. It posits that planners cannot make rational decisions about resource management without prices formed by profit maximisation by private owners of factors of production. So in other words you need a free market, at least in capital, land and labour to maximise the accomplishment of goals, even socialist goals like good houses for everyone.
The first false counterargument is to attack the word “rational” and assume it means a false assumption of what goals are rational. It does not. It merely means the ability to reason about how to achieve those goals. The original article was called “Economic Calculation in the Socialist Commonwealth” but it equally applies to a theocratic commonwealth, a feudal imperialistic state, or any other area where a planner seeks to maximise something through control of the economy. The question isn't “Is this a smart thing to try to do?”, it's “Can we find the best way to do it, given current resources?”. This means rationality is always possible and required, because trying to achieve your goals irrationally is stupid.
Similarly just because socialists aren't trying to maximise money but the provision of needs that doesn't mean you don't need to plan rationally. The ECP isn't about trying to be rich, it's about trying to achieve the goals planners are trying to achieve with the limited means available. It doesn't matter if the planner is trying to maximise yachts and private planes for his employers as possible or food and shelter for the hungry. Optimisation is still possible and desirable.
Imagine a world where everyone agrees everyone can have anything, and work is strictly voluntary but they recognise a social duty of the able to create more than they consume. How do you decide whether to use create more solar panels or increase transmission efficiency to get more power? Or should you discourage people from doing the activities that consumer power? All approaches reduce total utility, the first two by diverting resources from other useful endeavours the last by reducing the enjoyment of the people, that enjoyment being the goal of the whole thing. Without factor prices you can't tell what's more expensive OR what's more valuable, and they are different things. If you increased solar power when reducing transmission losses would be use less valuable resources or vice versa people have less access to goods. If you encouraged people to use less electricity when generating more is the most efficient way to produce more happiness again, you're failing to optimise what you're trying to optimise.
Using engineers and experts to try and overcome this problem won't work either. Yes, the optimisation of resources will involve engineers, agronomists, chemists etc. but they are not specialists in optimising resources. They all rely on outside price signals to tell them what tasks are worthy of their attention, what is worth doing and what isn't. Consider whether to build a mine to extract a potentially valuable ore. To know whether to do this one must know how much more valued things will happen if you have the ore. But to know this you must know what other means could be substituted for this ore, what those means would otherwise be used for and how valuable that is. It's true that engineers routinely include budget estimates for projects that allow planners to assess the viability of a project. But these budgets don't set their own prices, they observe prices established outside the project. Even if it were possible to establish correct prices in the absence of a private market in factors of production, engineers/chemists/designers don't have that skill. In addition you must know how valuable goals are relative to each other.
Another non-starter of an argument is that prices could be set by a central authority. Prices are there to express the desirability of not using the factors of production. They only work if they indicate the actual utility and difficulty of replacing the factor. The planner doesn't know this, and can't know this. To know enough to set the price of a resource he would have to know the relative scarciy of everyof every resource that could be substituted for it, partly or wholey and the interactions of that substitution with other resources and projects. This is impossible. You can't know everything about every transport route, every mine, every factory, let alone every worker or every technical subject. That's the whole problem.
Profit-maximisers however know enough to quote a price, which is all you need to assess whether to use this resource or another. Profit-maximisers will not price perfectly of course, that's why losses happen, but they have both the information and the incentive to make a good estimation of actual value.
Another thought was to make decisions on resource use without using prices and just account for each resource separately. So a project would use X amount of steel, Y amount of concrete, Z amount of skilled labor etc. But this involves tracking literally thousands of different resources (or even hundreds of thousands depending on how you define categories) and comparing the billions of ways they could be combined. Even if this could be done theoretically with perfect inputs, you don't have perfect inputs. You would have to know the disposition and condition of every useful resource in the economy, including workers, who are not fungible or identical and then run billions of simultaneous equations hoping nobody put in false information. They did.
Speaking of computers, no they don't help. The big problem is not the lack of information processing, it's the amount of information that would have be conveyed to the planner, much of it subjective, possessed by an enormous number of people and/or hard to communicate to a formal body in real time. Only by summarising the information in a price can it be made simple enough to communicate efficiently. The ECP could just as easily be called the economic communication problem. Imagine spending half your life typing answers into a computer, and then realizing they didn't ask the important question they had no idea even existed. That's how computer would “solve” this problem.
So how serious is the ECP? Some planning can occur under socialist systems. Just because you can't find a precise number for how much more resource approach A would cost versus approach B doesn't mean you can't make decisions. They just won't be very good ones. Resources will be expended to achieve goals that should have gone to other goals, and resources that went to the first goal go to the others. Now there will be cases where the correct choice is obvious. Nobody makes rail tracks out of titanium but the cost is enormous for the more marginal cases. It is not a financial cost but a cost in results. One hospital instead of two, 90 KPH trains instead of 120, cancer deaths reduced by 20% instead of 50%, that sort of thing. It is not trivial.