It should be noted the minimum wage isn’t the only intervention that messes with the labor markets.
But for this example let us consider an unhampered market. We shall define the following terms:
We shall call the infimum rate of an employment activity the least monetary level at which the utility gained by the laborer through the wages is greater than the disutility of performing the job. Like all value judgements this is subjective and differs between person, employment activity, and time.
Likewise, we shall call the supremum rate of the employment activity the greatest monetary level where total utility gained by the employer when that activity is performed for him is greater than the utility of the wages given.
It is obvious that an employment relationship can only occur where the wage given is above the infimum and below the supremum. In both cases both parties are gaining more than they are losing. Given an employer and employee, an employment activity, and a time, we may call the set of wage rates above the infimum and below the supremum the acceptable range. We say that a worker is unemployable to a given employer for a given activity at a known time if and only if the supremum is below the infimum, and thus the acceptable range is empty. To say that a worker is unemployable is to say that the relationship would be only wasteful.
The law of association proves that a possible relationship within the acceptable range must always exist, and indeed always exists between every person at all times. This leaves the only variable the employment activity. Thus a person cannot be unemployable for every activity: A person cannot completely be a debt to society.
We may call an acceptable employment between an employer and employee at a given time an employment activity for which the acceptable range is populated, that is, not empty. Given a set of people and a set of known acceptable employments, it is obvious the worker will choose the employment where the supremum is highest.
The problem is, it is not obvious what the acceptable employments are. A person cannot trade with someone they do not have any way to contact, or any known things to trade. It is perfectly reasonable to conject, and indeed happens very often, that two people could profit enormously by cooperating yet the profit does not occur because they do not know of this opportunity, or even that the other person exists at all. We may thus define unemployment as the situation where a person does not know of any acceptable employments available to them. However, it is also known that acceptable employments always exist. Thus the unemployment problem is one of information.
I don’t really like the use of “involuntary employment” on the unhampered market as it really twists around the definition of voluntary. If it starts raining when I’m out for a walk, am I suffering from “involuntary weather?”
As for “the unemployment is voluntary, they can lower their reserve wage if they want to” it is perfectly compatible with my definitions without having to mess around with fuzzy concepts. As a person waits longer, their savings deplete, and they become more desperate, the infimum lowers and the acceptable range widens, increasing the amount of acceptable employments, making one easier to discover.