Those familiar with the Austrian School of economics will remember the Economic Calculation Problem, how to decide what to produce with what resources. While this is usually presented as a problem of what goods are best for consumers there is another problem, how people would prefer to use their resources, particularly their labor.
For example Paul Armstrong was a high level health administrator in the New South Wales governement. He was good, so good other sections of the NSW health administration headhunted him for his skills in getting diverse people to work together. His high pay represented the benefits others saw in his organisation and people skills (and the lack of competing people who could provide the same skills for less of course). In other words his effort was thought to be able to provide the same or better results than large amounts of capital, land and other people's labor. So why then did he quit and become an actor, a profession notorious for not being financially stable? Put simply his valuation of doing the job he wanted to do had to be compared to other people's valuation of doing the job they wanted him to do. This fundamental problem of how to balance the preferences of those who want services with the preferences of those who want to provide them I call "The Paul Armstrong problem". It is a subset of the Economic Calculation Problem and I want to determine if can be solved without prices or without a free market*.
Obviously if there is an opportunity to do a job that is highly valued by others but highly disvalued by the person doing it then someone has to make a decision. Whose desire predominate? Since the abolition of slavery nobody can be coerced into doing a particular job if they haven't agreed to do it. The only exception is "National Service" that is to say "Government Slavery" i.e. conscription or conscription-like institutions. So given that people can veto what other people think they should do how do we persuade them to do what would suit us?
Several methods suggest themselves. Social pressure/guilt can work in some cases, but it isn't the sort of thing that can or should be applied on societal scales. Attempting to do so would result in those with the least active consciences benefitting most and those with the most active getting the shaft. This in the long run is not good. It also encourages people to be bad at unpopular jobs (or pretend to be) which is not what we want. In any case it does not really measure the cost/benefit of a particular person doing a job. The disutility of a person doing a job (i.e. how much he would prefer not to do it, for whatever reasons) is a cost of them doing a job. If it is not expressed in a monetary form, it's still a cost. There might be people who would do the job almost as well and dislike it much less, thus having almost as much of a benefit and much more of a cost. This is not measurable using social pressure.
Consciption, which is fundamentally no different from slavery with a paycheck could also be tried, but it suffers from a central defect. Under slavery there is no incentive to actually do a good job. Under selective slavery, where only those who seem good at the job are enslaved, there is even less incentive. The worse job you do, the sooner people stop forcing you to do it. Any of you guys out there used the "Drop a few plates and get out of drying up" trick? Same deal, society wide. It also doesn't allow cost/benefit analysis in the same way social pressure did not.
The current solution is some form of recompense. That is provision of more goods and services in return for the more valuable labor. Labor can be more valuable because there are a limited number of people who can do it, because those who can do it can do other valuable labor or because nobody wants to do it. With some jobs the value difference between them being done well and not as well is very great. For instance the aforementioned Mr. Armstrong had a job that involved various different health services, including drug & alcohol, pyschiatric hospitals, regular hospitals etc. working together. Anyone could try to do that, but doing it well meant high value in terms of better patient outcomes, less time shuttling them around, less bureacratic effort and more time actually spent helping patients etc. So his labor was highly valued in terms of goods and services. It was cheaper to provide him with large amounts of goods and services (via the medium of a paycheck since double coincidence of wants is a thing) than to try and accomplish goals without using his labor. In other words his labor substituted for large amounts of capital and other people's labor.
So could people be given recompense without prices? Fundamentally no. All that would happen is a de facto barter system where, for instance, doing a particular job gets you particular packages of goods and services and you trade them for the goods and services you want. This in effect gives you a price system but with the inefficiencies of a barter system. While it is possible to provide part recompense in goods and services rather than money (e.g. free housing for some workers, especially those who need to be a particular place) that is only a modification to the idea of prices. For some organisations in some cases it is cheaper to provide a benefit by providing a good or service rather than the cash necessary to provide an equivalent value to the worker. For instance mining companies provide free accommodation in remote areas to many workers. This is cheaper than providing the worker money which he then uses to rent accomodation which would be difficult if the company hadn't built said accommodation.
* Yes his job wasn't in the free market, but it was subject to competition from other employers who were in the free market and so had some free market discipline imposed.