Assume a utility function U(P) (P is take home pay) which is downward sloping (i.e., dU(x)/dW < dU(y)/dW whenever x < y). This is just the law of diminishing marginal returns.

Negative income tax: take home pay is P = A x W, where A > 1 and W = wages. Marginal utility = dU(AW)/dW = A dU(AW)/dW. If A dU(AW)/dW > dU(W)/dW, negative income tax creates an incentive for work.

Basic minimal income: take home pay is P = BI + W (BI + basic income). dU(BI+W)/dW < dU(W)/dW. This is always a disincentive for work.

I'd really be curious, do you have more information on Hayek's support for minimum income?

In his version, there's an allowance calculated by family size, dependents, etc., and a "subsidy rate", which is the proportion of any excess allowance that's refundable. He proposed a 50% subsidy rate. If the allowance for a given taxpayer is $20,000, and the taxpayer earns $15,000, that leaves $5,000 in unused allowance, of which 50% is refunded, so the taxpayer gets a $2,500 transfer payment. If the taxpayer earns no income at all, the entire $20,000 is unused, of which 50% is refunded, so the taxpayer gets a $10,000 transfer payment.

So, if A is allowance, S is subsidy rate, and W is wages, the taxpayer earns W if W >= A, or W + S * (A-W) otherwise. The guaranteed minimum income is when W=0, and equal to S * A ($10,000 in the above example).

For Hayek, here's one of several places he discusses his rationale: http://books.google.com/books?id=nclLLOfnGqAC&pg=PA55. One reason is that, unlike many libertarians, he's strongly against private-sector safety nets through e.g. church charities, because he feels those inhibit human freedom by making people scared to leave their ethnic/religious/racial/social group for fear of losing its safety net, which he views as a variety of collectivism. So he sees a guaranteed minimum income as a way of promoting individual freedom and undermining the power of tribalist collectives.

But I get Friedman's rationale - he wanted to replace the the "ragbag" of assorted other welfare programs with this. It does have the advantage that dP/dW is always positive (just reduced by (1-S) if you are poor), which is not necessarily the case for the existing system.

So I'm not sure it's fair to say Friedman advocated it - he just proposed replacing a much worse system with it. Do you know if he considered it good policy, as opposed to merely better policy?

First, if the utility function was "downward sloping" nobody would ever work. It's the marginal utility that is assumed to slope downwards (I assume you know this, because you later argue as if you did). But this is the opposite as in your example: It should read dU(x)/dW < dU(y)/dW whenever x > y (not x < y).

More importantly, your negative income tax definition is faulty. A NIT has never the form proposed by you. It wouldn't make much sense. Also, if A > 1 (in fact, if A > 0) you always have an incentive to work in your model, assuming dU/dW > 0.

And by the way, I don't know where you received your math education, but dU(AW)/dW = A dU(AW)/dW does not follow. For example, let U(P) be a simple concave utility function U(P) = ln(P) = ln(AW). Then dU(P)/dW = d(ln(A)+ln(W))/dW = 1/W != A 1/W.

This is all not so bad. Everybody makes mistakes (maybe I made one above). What bothers me is that some hacker news readers have apparently upvoted you without ever checking the results.

Aggh, I miswrote. I meant concave down, which is actually what I used in my calculation. Thanks for the correction. In defense of my readers, they probably glossed over my incorrect definitions, and didn't notice any problem because my calculations didn't match my definition.

But I will defend my calculus: dU(AW)/dW = A dU(AW)/dW follows from the chain rule. Another way to do your calculation: d(ln(AW))/dW = A x 1/(AW) = 1/W.

Also, the NIT I described has been proposed (that's how it was described in my intro to econ class), and it does not always create an incentive to work. The relevant criteria is A dU(AW)/dW > dU(W)/dW. Since dU(AW)/dW decreases with A, there the marginal utility of work can be negative with respect to NIT.

Also if you apply ln(P) as utility function then the partial derivate for W yields the same result to U(P) = U(AW) as to U(P) = U(W), namely 1/W. So in that case your inequality does not hold.

    [U(G(W))]' = U'(G(W))G'(W)
    [ln(AW)] ' = ln'(AW) [AW]' = (1/(AW)) A = 1/W

So in that case [U(P)=ln(P)] your inequality does not hold.

The inequality I gave is a criteria, not a theorem. I didn't claim it was always satisfied, I just claimed it was possible.

If it is satisfied (e.g., U(P)=sqrt(P)), a NIT creates an incentive for work. If equality holds (U(P)=ln(P)), a NIT has no marginal effect on work. If the reverse inequality holds (U(P)=1-1/P), a NIT creates a disincentive for work.

Not necessarily - simply provide everyone, no matter their income level with the basic payment; that also decreases the paperwork and the power of the bureaucracy to bully the recipients. Then, since everyone has the minimum to live on, have a fairly high, fixed percentage, income tax on all income. There would be no effects on the marginal utility of work, and shouldn't be too much on the average utility, provided the minimum really is only a minimum subsistence level.