Comment of the Day: But, alas, I think Robert Waldmann has the completely wrong. Draw the graph, Robert! What areas on what graphs does Greg's calculation correspond to? Please tell me!: Robert Waldmann: DeLong & Krugman vs Mankiw and Mulligan III: "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...

No, I do not think that it does.

What does Mankiw think the pretax rate of profit is in the short run after you cut the tax rate from t0 by Δt? The pretax rate of profit was initially:

$\frac{r}{(1-t_{0})}$

and total pretax income from capital was initally:

$\frac{k_{0}r}{(1-t_{0})}$

and total government revenue was initially:

$T_{0} = \frac{t_{0}k_{0}r}{(1-t_{0})}$

As I see it, Mankiw maintains, when calculating the tax revenue loss, that, after the tax cut, the pretax rate of profit is still:

$\frac{r}{(1-t_{0})}$

So that the amount of revenue collected is:

$T - ΔT = \frac{(t_{0}-Δt)k_{0}r}{(1-t_{0})}$

and the change in revenue is:

$ΔT = \frac{Δt(k_{0})r}{(1-t_{0})}$

But, as I see it, Mankiw also maintains, when calculating the wage gain, that, after the tax cut, the pretax rate of profit is:

$\frac{r}{(1-t_{0}+Δt)}$

so that there is an extra cash flow ΔC per unit of capital arising from the reduced pretax cost of renting capital equal to:

$ΔC = \frac{r}{(1-t_{0})} - \frac{r}{(1-t_{0}+Δt)}$

$ΔC = \frac{1-t_{0}+Δt - 1+t_{0}}{(1-t_{0})(1-t_{0}+Δt)}$

$ΔC = \frac{Δt}{(1-t_{0})(1-t_{0}+Δt)}$

The total extra cash flow available to pay higher wage is then that times the initial capital stock:

$Δw = \frac{k_{0}Δt}{(1-t_{0})(1-t_{0}+Δt)}$

Δw is not equal to ΔT: there is the extra factor in Δw of:

$\frac{1}{1-t_{0} (+Δt)}$

But this factor does not arise because of the difference between a tax on capital and capital income. It, rather, arises because Mankiw has calculated:

• the tax loss assuming that the pretax rate of profit is not immediately impacted by the tax cut,

• the cash flow available to pay wages assuming that the pretax rate of profit is immediately impacted by the tax cut.

What if he assumed in making both calculations that the pretax rate of profit did not fall? Then the calculation of the tax revenue loss would be correct. But with no decline in the pretax rate of profit, there is no extra cash flow to pay wages, and so the ratio of wage gain to tax revenue loss is zero.

What if he assumed in making both calculations that the pretax rate of profit did fall to:

$\frac{r}{(1-t_{0}+Δt)}$

?

Then the calculation of the wage gain is correct, but your calculation of the tax revenue loss is in error. Tax revenue falls both because the government is levying a lower rate on pretax profits, and because pretax profits are now lower.

In the long run, as k expands to k(0)+Δk, the ratio is greater than one of course—but production function parameters do not drop out... Comment of the Day: Brad DeLong: Robert Waldmann: DeLong & Krugman vs Mankiw and Mulligan III:

1. In the short run in which the pretax rate of capital does not fall, wages don't rise.
2. In the short run in which the pretax rate of capital does not fall, tax revenue falls by: $\frac{(Δt)(k)r}{(1-t)}$
3. In the short run in which the pretax rate of capital does not fall, profits rise by: $\frac{(Δt)(k)r}{(1-t)}$
4. In the medium run in which the pretax rate of capital does fall but investment ordered is not yet installed, wages rise by: $\frac{(Δt)(k)r}{(1-t)(1-t+Δt)}$
5. In the medium run in which the pretax rate of capital does fall but investment ordered is not yet installed, tax revenue falls by: $\frac{(Δt)(k)r}{(1-t)(1-t+Δt)}$
6. In the medium run in which the pretax rate of capital does fall but investment ordered is not yet installed, profits revert to their initial level.
7. In the long run in which the capital stock reaches its equilibrium, wages rise by $\frac{(Δt)(k)r}{(1-t)(1-t+Δt)}$ + {Harberger triangle term}
8. In the long run in which the capital stock reaches its equilibrium, tax revenues fall by $\frac{(Δt)(k)r}{(1-t)(1-t+Δt)} - t(Δk)r\left(\frac{1}{(1-t+Δt)} -1\right)$
9. In the long run in which the capital stock reaches its equilibrium, total profits rise by $rΔk$
10. In the long run in which the capital stock reaches its equilibrium, those extra profits flow to foreigners, and the difference between GDP and GNP grows...
11. In the "fully phased in static" calculation by JCT, OTA, CBO, and company, in which $Δk = 0$ in order to eliminate political noise and get a favorable bias-variance tradeoff, wages rise and tax revenues fall by $\frac{(Δt)(k)r}{(1-t)(1-t+Δt)}$

And Mankiw is dividing (4) by (2) and claiming it is a "static" analysis.

Which nobody has ever done before, and which makes no sense. The "static" analysis is (11)

My assessment is still: 95% chance this is retconned...

Greg Mankiw: Using Brad's breakdown above, I believe it is more accurate to say that my exercise was dividing (7) by (2).