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).

## 10 comments:

One correction: I did not accuse anyone of making an algebra mistake. I only said that I didn't. Thank you for confirming this.

Sorry. I will update. Yes now I remember, you said that Mulligan thought someone had made a mistake.

I think you are wrong here, Robert. What do you 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:

>r/(1-t(0))

and total pretax income from capital was initally:

>k(0)(r/(1-t(0)))

and total government revenue was initially:

>T = t(0)k(0)(r/(1-t(0)))

As I see it, Greg maintains, when calculating the tax revenue loss, that, after the tax cut, the pretax rate of profit is still r/(1-t(0)), so that the amount of revenue collected is:

>T - ΔT = (t(0)-Δt)k(0)(r/(1-t(0)))

and the change in revenue is:

>ΔT = (Δt)k(0)(r/(1-t(0)))

But, as I see it, Greg also maintains, when calculating the wage gain, that, after the tax cut, the pretax rate of profit is r/(1-t(0)+Δt), so that there is an extra cash flow per unit of capital ΔC of:

>ΔC = r/(1-t(0)) - r/(1-t(0)+Δt) = r[[(1-t(0)+Δt) - (1-t(0)]/[(1-t(0))(1-t(0)+Δt)]]

>ΔC = r[Δt/[(1-t(0))(1-t(0)+Δt)]]

The total extra cash flow available for wages is that times k(0):

>ΔW = (Δt)k(0)(r/(1-t(0))(1-t(0)+Δt))

These two are not equal: there is an extra factor 1-t(0) (+Δt).

This inequality does not arise because of the difference between a tax on capital and capital income, it arises because Mankiw has calculated the tax loss assuming that the pretax rate of profit is not immediately impacted by the tax cut, while calculating the cash flow available to pay wages assuming that the pretax rate of profit is immediately impacted by the tax cut.

I don't think we really disagree except for one word. Mankiw assumes that the pre tax rate of profit is gradually (not immediately) affected by the tax cut when calculating the cash flow which will gradually become available to pay wages.

The immidiate impact of the tax cut on wages is 0.

The 1/(1-t) is a ratio of a long term effect on wages to a short term effect on revenues.

The ratio of the long term effect on wages to the long term effect on revenue collected on income of old capital is 1 . Both are the same rectangle in your figure.

The 1/(1-t) is all about long term vs short term. It is the ratio of the long term effect on tax revenues from old capital to the short term effect on tax revenues.

The reason that taxing capital Tau K rather than revenue from capital

t f'(k)K matters is that, if the tax is a tax on capital, the long term effect of the tax on revues from old capital is equal to the short term effect.

Stepping back, one might consider a (long term)/(short term) ratio an odd thing to calculate. One might consider it a mistake to suggest it is a policy relevant ratio. But it isn't an algebra mistake

immediately/immediately = 1

eventually/immediately !=1 maybe also != anything we should care about, but definitely !=1.

So your view is:

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 (Δt)(k)(r/(1-t))

3.In the short run in which the pretax rate of capital does not fall, profits rise by (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 (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 (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 (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 (Δt)(k)(r/[(1-t)(1-t+Δt)]) - {rtΔk[1/(1-t+Δt) -1]}

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 (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...

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

thx...

I too think the 1/(1-t) is seven divided by 2.

The amount of valuable time spent on this has become absurd. At the moment, I have insomnia and am trying to bore myself back to sleep. However, I do strongly recommend that extremely smart economists not waste their extremely valuable time reading this.

The only useful point is that some of my overlong posts were specifically about the Leontief case in whick k never changes. Only in that case is it necessary to introduce nominal stickiness to get a short run and a medium run.

So really there is a fork not a series. With Brad's numbering things can go 1,2,3,4,5,6 for the Leontief case with nominal stickiness or

1,2,3,7,8,10 for a smooth production function with wage and price flexibility.

To set the tone of utterly pointless algebra.

OK so above I wrote 1/(1-t) = 7/2 so 1-t = 2/7 so t = 5/7 ...

Semi seriously, I have been wondering why I started typing about sticky wages and prices. I suspected I had just made a conceptual error. In fact, ther are calculations which end up with something/(something else) = 1/(1-t) works with flexible prices. I think it is 1/(1-t) = seven/two with no need for a medium run.

Looking back, I see why I was typing about sticky prices and wages. Brad asked what happens if production is Leontief so K doesn't change. How can there be a 1/(1-t) factor when output doesn't ever change and all tax reforms are just transfers ? Actually interesting question.

In that case only, wage and price stickiness matter and there is a difference between the extremely short run and Brad's medium run (which in that very special case is the long run, because K never changes). Leontief is special, because wages and profits are not determined by supply and demand.

For a standard smooth production function, with standard flexible price assumptions, the wages are a function of k w=f(k)-kf'(k). This means that there is no medium run (no 4, 5 or 6).

For a Leontief production function, there is no increase in k, so the medium run (4, 5 and 6) are the long run.

Only in the Leontief case is there any interest in wage and price stickiness. If there isn't any stickiness, the real wage jumps so the pretax return on capital falls instantly, so investors get the same income, the Treasury gets less and workers get more and the ratio dw/d(revenue) = 1.

But with sticky wages and prices, in the short run, the pretax rate of profit stays at r*/(1-t) , investors get briefly extraordinary returns

Kr*(1-t+delta)/(1-t). workers get the same old wage. The Treasury gets

(delta)r*/(1-t) less.

There is excess demand for labor as everyone would love to make that return building new capital, but they need workers & all the workers are already employed. The frustrated potential new entrants slowly bid up wages through some sort of realistic model of wage an dprice setting with nominal stickiness which you guys have been seeking since 1985 at the latest.

In the Leontief long run investors get kr*. Total production = f(k). Workers get more and the Treasury gets less. the ratio is one to one. The long run loss of revenue is even greater than the short run loss of revenue, because the pre-tax rate of profit declines as wages increase. I don't have to introduce the ugly concept of tax_revenue_collected_on_old_capital because, in this case only, there is no new capital. All of my pointless algebra works. It gives

(long run revenue loss)/(short run revenue loss) = 1/(1-t) =

(long run wage gain)/(short run revenue loss)

So in the middle of this national debate about taxes, Greg Mankiw, poses this question: "How much will the tax cut increase wages?". Now I think he might say that his question has nothing to do with the national debate and is just an academic exercise. But as a probably average reader of his blog, I assume he is knowledgeable and I assume he is honest. And since I cannot follow the details of the economics, what I get from the blog is that yes, the proposed tax cuts can greatly increase wages! Now Greg says he is not trying to say that, he is just proposing an academic exercise. But that is still what his blog says to me. Without the interference of Krugman and DeLong I would never know that Mankiw was trying to post just an academic exercise. Whether Greg likes it or not, his academic exercise becomes part of the national debate.

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