>What does inefficiency matter if there is not enough work for everyone anyway?

I am using the term "efficiency" in a technical sense. An inefficiency means that there exists an alternative way of distributing resources (including time) such that no one is worse off, and at least one person is better off. By this definition if their is an inefficient, then we would have a strictly better world if we could remove that inefficiency without causing any secondary effects.

Suppose we have a widget X that can be produced by person A in 8 hours, or by persons A and B in 10 man-hours (at 5 hours each). Further, assume that both A and B value their time at a constant $100/hr [0].

Suppose that we live in the second world, where A and B each spend 5 hours working on X. At the end of the day, A and B are both down by 5 hours of their time, and up by $500, and society is up by 1 X.

Suppose that, the next day, B hire A to do his share of the work. We have defined that A can do the entire job 8 hours, which means that this transaction only add 3 hours of work to A's day, meaning that A would be willing to do this for only $300. At the end of this day, A is down by 8 hours, and up by $800, B is up by $200, and society is up by 1 X. Again, A breaks even in this exchange, and society still has just as many Xs, however by B got $200 out of thin air.

We can modify the above example to where B pays A $400, in which case both A and B can $100 worth of value out of this exchange. This extra wealth is not just fictitious money. In doing this exchange, we saved 2 hours of human time, which is a real, limited, resource.

Of course, in the above situation, there is no way of deciding how exactly how much B should pay A, all we know is that any value between $300 and $500 produces an efficient system. However, we may, as a society prefer, the situation where the profit is split, and both A and B walk away with a surplus $100. In fact, we may prefer this possibility so much, that we are willing to destroy $2 of wealth somewhere else in the system to assure that A and B split the surplus. The argument for a BI is that a minimum wage achieves the desired result by destroying wealth (in the form of exchanges that never get made), whereas a BI can achieve the same desired goals while destroying less wealth. If that is the case, then a BI is strictly better than a minimum wage.

Of course, a BI itself introduces a market distortion (in the form of tax). The questions that need to be answered is how much wealth does the BI tax destroy, how much wealth does the minimum wage destroy, and how effective are both the BI and minimum wage at achieving the social goal of providing resources for all citizens.

>I am not sure what you want to say here, seems like it does no harm and provides no benefits.

In a society of perfectly rational actors, this would be correct. However, having multiple contributors to the effective marginal tax rate obscures the issue, and is how we get into situations where we have unreasonably high (in some cases >100%) rates.

[0] This is clearly a simplification, but it should not affect the analysis. The important thing is that there exists an "indifference" curve, whereby for any quantity of time, there exists a dollar value at which the worker is equally happy getting paid for doing the work as he is with doing nothing and not getting paid. The only potentially important assumption about this curve that I can think of is that the derivative is non-decreasing. This means that if I am willing to sell my first hour of labor at $100, I will sell my next hour for no less than $100, although I may need more than an extra $100 to be willing to spend an extra hour.