There has been a very odd debate among very smart economists in which Brad DeLong and Paul Krugman are convinced that Greg Mankiw made a silly algebra mistake and Greg Mankiw is not convincedade a silly algebra mistake [update oh my Prof Mankiw appeared in my comments noting that he didn't say anyone made a mistake & just wrote that he hadn't. Sorry about that].

I have struggled to understand the disagreement, which, again is elementary algebra and geometry. There is no point in trying to make sense of my efforts to understand. I am now quite sure I understand the disagreement. I am also quite sure that none of the three made a silly algebra mistake.

Mankiw's question is here

He assumes a small open economy (with something making adjustment gradual) so the after tax return on capital must be equal to the world interest rate r*. then he asks a very odd question: what is the ratio of the long term gain in wages due to a (small) reduction in the capital income tax to the short term loss of revenue. There is no particular reason to ask this question, except that it has an oddly elegant answer. That ratio is 1/(1-t) where t is the initial tax on capital income.

Brad's latest effort to explain is here

Just click the links. I finally understand that Brad too is asking a very similarly odd question. The only difference is that Brad considers a tax on capital (tau)k not on capital income (t)f'(k)k. This makes the difference.

The reason is that changing t by delta t (delta t <0 so this is a cut) has three effects on revenues. First there is the immediate loss (delta t)f'(k)k (this is what Mankiw calls the static cost and I think that's standard terminology). Second there is the additional revenue because the tax cut will cause higher investment (t+delta t)(delta k). Third and critically there is a gradual reduction in tax revenue per unit of k due to the decline in f'(k) equal to (t+delta t) f''(k) (delta k) so this causes a loss of revenue equal to (t+delta t) f''(k) (delta k)(k+delta k) or, to first order

tf''(k)(delta k)k

This means that the change in revenue per unit of capital is (to first order) (delta t)f'(k) + t f''(k)(delta k). Now imagine that new capital is due to entry of new firms, so I can talk about revenue collected from old capital. that changes by

(delta t)f'(k)k + t f''(k)(delta k)k

if delta t is negative, delta k is positive. f''(k) is negative so the second term is an additional cost to the treasury of cutting t. It taxes at a lower rate and the profits earned by the old firms are lower bcause of the competition from the new firms.

wages paid equal f(k)-f'(k)k so the change in total wages is (always to first order)

f''(k)(delta k) k.

OK as noted by Brad, the after tax returns on the old capital are always kr* so the reduction in revenue collected on old capital must be equal to the gain in wages (to first order in delta t)

(delta t)f'(k)l + t f''(k)(delta k)k = f''(k)(delta k)k

so

(delta t)f'(k)l = (1-t)f''(k)(delta k)k

Oh look that's Mankiw's short term loss in revenue equals (1-t) times the long term gain in wages. The long term loss of revenue from taxes on income of old capital is equal (to first order) to the long term gain in wages.

Now consider a tax on capital Tau if it is changed by delta Tau then there are only two effects on revenue. A short term loss of (delta tau)k and a gain of (tau +delta tau)(delta k). the long term effect on revenues from taxing old capital is just (delta tau)k.

The long term effect on after tax income from old capital is zero again, so the long term effect on wages is, to first order (delta tau)k. So again the ratio of the long term gains to wages and the long term reduction in revenue from old capital is 1.

But now the long term reduction in revenue from old capital is equal to the short term reduction in revenue from capital. So now the ratio of long term wage gains to short term revenue losses is 1 not 1-t.

Now I think the actual lesson here is that it makes no sense to look at a long term change divided by a short term change.

But no one has made an algebra mistake. Taxes on capital and capital income are different. The effect of changing them on revenue collected from old capital is different if the change in the taxes affects the pre-tax return on capital.

Now something is gained by drawing the figure (see Brad's figure). It makes it very clear that the gain to workers is equal to the loss of revenues collected on old capital (plus the little triangle which is second order in the changes in taxes).