I think they are all wrong and that none of the four made a mistake.

update: I also now think that I was wrong about what Brad wrote when I wrote the silly post below. Like Mankiw, he was considering the ratio of the long term effect of a tax cut on wages divided by the short term effect on tax revenues. The difference is entirely that DeLong and Krugman consider a tax on capital and Mankiw and Mulligan consider a tax on capital income. Short run revenue effects changes in such taxes differ only by a constant (the initial marginal product of capital). Long run changes in tax revenue per unit of capital and of wages differ by an further factor 1-t explaining the different results. end update: Mankiw considers a reduction in the tax on capital income in a small open economy. He assumes that the after tax return is equal to a constant world rate of return r* (in the long run although he doesn't clearly state that he doesn't think this holds in the short run). He looks at the "static" cost to the Treasury of a tax cut. Here he assumes that the pre-tax return doesn't change quickly, so he assumes that, in the short run, the after tax return is greater than r*. Then he looks a the long run increase in total wages paid (the wage bill).

He notes that the ratio (long run)/short run = (1/(1-t)) where t is the initial tax rate.

DeLong scolds Mankiw very harshly for using the word "static" with a different definition that the JCT. I personally wonder why Mankiw thinks anyone should be interested in a (long run)/(short run) ratio.

First I think I understand the communication problem (update I didn't understand it end update). Mankiw is no more able than I to write the symbol for a partial derivative on the web.

He wrote "We cut the tax rate t. Because f '(k)*k is the tax base, the static cost of the tax cut (per worker) is

dx = -f '(k)*k*dt."

he means partial x/partial t = -f'(k)k. by "static" he means "holding k constant" that is taking a partial derivative. Now if k were constant, then wages and production would be constant so profits gross of taxes would be constant and the return on capital would be greater than r*. In Mankiw's example, the only thing which changes (other than taxes once) is k. You can't change t, keep k the same and keep (1-t)f'(k) = r* constant.

update 3: All that follows is my confusion. I can get to a model in which there is a short run wage gain equal to the short run revenue loss. However, it isn't Brad's model at all. Like Mankiw his is looking at long run wage gains vs short run revenue losses dw/dtau/(partial x/partial tau). The difference is that Brad considers a tax on capital not on capital income.

Everything that follows is irrelevant to the discussion and just an example of how one can get any result one wants out of an economic model by fiddling the assumptions.

end update 3

Brad *insists* on another definition of static -- one which he knows is used by the JCT to score tax reforms and generate the ultra important $ 1.5 trillion. In this defintion, prices may change (and accounting tricks definitely change) but actual production doesn't.

So in Brad's static calculation, k stays the same but the pre-tax return on capital falls so (1-t)(pretaxreturnoncapital) = r* stays the same. This can only happen if wages go up. The net of tax income of investors is (by assumption) fixed so the gain to workers is exactly equal to the loss to the Treasury.

Brad's static analysis is a bit odd. He assumes k is fixed *and* that wages and the pre tax return on capital change. He writes that it is very important to defer to the JCT. I agree with him about that as a matter of political economy. But I want at least a story for how w and pretaxreturnoncapital can change without k changing.

The story follows. Capital is like clay. Once it is assembled, the production function is Leontief so there is no way to substitute capital and labor. Output is firms choose a technology with a given capital labor ratio from a menut that looks like an ordinary production function, but, once chosen, the ratio is fixed.

In contrast w is not determined by the technology. It is determined so the after tax return on capital is r*.

If w is too low the return is higher than r* and foreigners send in capital and hire a worker (taking w as given). There would be excess capital so the return would be zero. Uh oh. if w is too high domestic investors send all their savings abroad. Then One tiny bit of capital deprciates and there is surplus labor and wages fall to zero.

So wages and pretaxreturnoncapital adjust instantly.

New capital is installed with a higher capital labor ratio (because wages are suddenly high in the USA). So as the old capial is replaced by new capital, demand for labour slowly changes.

Capital as clay makes it possible for prices to change quickly and quantities to change slowly. This is what Brad assumes, presumably following the JCT.

Mankis is assuming a smooth production function in which substitution of capital and labor is alway possible. His short term calculation is in the short term, k is the same so w = f(k)-kf'(K) is the same so the ratio of gain to workers to loss to the treasury is 0. not 1/(1-t) not 1, but exactly 0.

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