>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.

I obviously have to disagree with your analysis, let me offer an alternative. The difference between the two scenarios is the difference between value(X) + value(8 hours leisure time) and value(X) + 2 * value(3 hours leisure time). Therefore in case the value of 3 hours of leisure time exceeds half of the value of 8 hours of leisure time, i.e. the value of additional leisure time diminishes sufficiently quickly, the second scenario provides the better total value.

By restricting the entire world to the two actors wanting to produce one X one can keep money out of the equation. I argue your analysis fails or at least may fail because you don't assign a value to the leisure time. Also the assignment of monetary values obscures things, where is this money coming from? From selling the X to the society? In that case you will have to take into account that producing one X may take either 8 or 10 hours - will this raise the price or decrease the monetary value of an hour of work?

Because value(X) is constant, your observation boils down to a comparison of value(8 hours leisure time) and value(X) + 2 * value(3 hours leisure time).

I did consider the value of leisure time, which I arbitrarily set to $100/hr (I just called this time, but unsold time can be exchanged 1:1 for leisure time). You seem to be objecting to my assumption that the marginal value of time is constant for the workers. I agree with this objection (in fact, I raised it myself in the footnote). However, even without this simplifing assumption, there are still situations where selling ones time is a net positive.

>By restricting the entire world to the two actors wanting to produce one X one can keep money out of the equation.

My entire argument can be made without invoking money. Indeed, I believe the non monetary form is more compelling, but also more abstract. Again, I will invoke the idea indifference. We say that things X and Y have equal value to person A if she is just as happy receiving 1 X as she is receiving 1 Y. With this concept we can replace "money" with any object of value and have an equivalent analysis. Of course, the selected object would itself have diminishing returns, but this is equivalent to an increasing marginal value of ones time, which I have already addressed. Ultimately, to properly explore this route, I would formalize an abstract notion of "utility" as an equivalence class over indifference. I hope this explanation makes sense, doing it properly would take far to much effort to put into an internet comment.

>where is this money coming from, From selling the X to the society?

The above framework provides a nice way to sidestep this issue. Let U be the utility function. So U(5 hours) is the value of 5 hours of time. [0]. For simplicity, assume that A and B share a utility function.

Suppose that A and B are not selling X, but rather using it themselves (for simplicity, suppose that X is a widget that A and B can share without conflict). We know that X is worth at least 2 * U(5 hours). Suppose that 2 * U(5 hours) > U(8 hours). In this case, by my previous argument, it would be preferable for A to do all of the work, and B to barter something worth between U(8 hours) - U(5 hours) and U(5 hours) for his share of X.

Ultimately, this argument is just a rediscovery of the value of trade and specialization. To translate a minimum wage into this framework, we would define some artificial lower bound for the compensation A needs to be provided in a trade such as this. If this minimum value is less than U(8 hours) - U(5 hours) or U(5 hours), then we are just influencing how wealth gets distributed between A and B. However, if the minimum value to big, than the above trade is prohibited, even though we have seen that it is strictly beneficial.

>will this raise the price or decrease the monetary value of an hour of work?

What do you mean by "monetary value of an hour of work". Returning to a more realistic employer/employee example, there are two important monetary values: the value the employee gives to an hour of work, and the value the employer gives to an hour of work.

From the employer perspective, the value of an hour of work is V(X)/T, where V(X) is the value of X [1], and T is the time spent making it. V(X) is independent of how long it took to make, which means that as the work takes longer, the value of an hour goes down to the employer.

From the employee perspective, the issue is a question of their utility function. Assuming a reasonable utility function, I would expect the marginal value of an hour to increase to the employee. That is to say that the employee needs more money to be indifferent to working the last 2 hours than she does for any of the prior 2 hours. If this is the case, than the value of an hour to the employee goes up with the time.

[0] More formally, U(5 hours) is the equivalance class of objects for which the person would be indifferent to spending 5 hours of time to obtain.

[1] Or rather, V(X) is the labor-excluded profit of X.

Okay, after rethinking it once a gain I will move a bit towards your position. By increasing the required work for one X from 8 to 10 hours we are definitely introducing an inefficiency so we are losing something somewhere, we are doing 2 hours of unnecessary work. Assuming both agents can share the X or get to use half the X in both scenarios, we can just leave it out of the equation as mentioned before. If one gets the same value from X in both scenarios, then working 5 hours is strictly preferable over working 8 hours and not working is strictly preferable over working for 5 hours.

So this looks somewhat like the prisoner's dilemma. Let A and B be the scenario that agent A respectively agent B works 8 hours and AB the scenario that both agents work 5 hours each. Then the preferences for agent A are B > AB > A and for agent B they are A > AB > B. But that doesn't help much, no matter which transition you consider, one agent will be better off and one will be worse off.

So there is still a conflict. I acknowledge that an inefficiency gets introduced when the work is split but also want to maintain that 3 hours of leisure time twice is more valuable than 8 hours of leisure time once. Surly value can not increase and decrease at the same time. I am more certain that the inefficiency is real than what the shape of the value function for leisure time is so I will pick the former one and conclude that 8 hours of leisure time are - contrary to my earlier comment - more valuable than 3 hours of leisure time twice.

So do I have to say that maximizing the total value is the wrong metric? Do I have to try to assign a value to the fairness of work distribution? Why wouldn't that already show up in the individual value functions? I am a bit stuck here. But I may have a way around that, I am just not sure how good the argument is. One obvious problem is that in case agent A does all the work, then agent B has no means to transfer some of the additional value from his additional leisure time to A. Maybe he could give away some of his share of X.

But now I can no longer ignore the value function for X and that hopefully saves my argument and I avoid the conflict between decreasing and increasing total value without the need to explicitly appeal to the value of fair work work distribution because it is in the two value functions for X and leisure time. But I am not sure, I did not think this fully through.

I think in the end it boils down to nonlinearities between the individual and the global view, i.e. we can globally waste 2 hours of work but still increase the total value because we have to take the global set of things, split it up for all individuals, push them through the nonlinear value functions and then sum it up again. Now it is no longer necessary that a global loss of 2 hours results in a global loss of value even if all the individual value functions are monotone.