DeLong & Krugman vs Mankiw and Mulligan III

I thought it was all so absurdly simple after all that I could explain to the many readers over at AngryBear. But I am so bad at expressing myself that even my very trivial explanation of the controversy was a mess. so I brought it here. Now I have to add paragraph breaks by hand

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).

DeLong & Krugman Vs Mulligan & Mankiw II

Below, I tried to understand why Brad DeLong and Greg Mankiw were having so much trouble understanding each other. The story so far: Delong and Paul Krugman think that Mankiw and Casey Mulligan made an elementary algebra mistake. Mankiw and Mulligan think that DeLong and Krugman made a math mistake.

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.

DeLong, Mankiw, Krugman, Mulligan and Cochrane Argue About Elementary Economics

I think you should read this post by Brad DeLong to understand the issue and the very grave condition of the discussion in which academic economists try to contribute to the policy debate.

The TL:DR version is that Greg Mankiw blogged a little exercize in which he asked the interested reader to calculate the ratio of two effects of cutting the tax on profits. The ratio was the long run increase in wages divided by a very short run loss of revenues to the Treasury.

The point was that this ratio is 1/(1-t) where t is the initial tax rate. I have no doubt that, as a partisan Republican, Mankiw was eager to lead people to a ratio greater than 1.

Brad DeLong objected that Mankiw incorrectly called his extremely short run analysis a static analysis. The exact definition of "static" matters, because it appears in the rules of the Senate which determine if a bill can be filibustered.

In Mankiw's extremely short run, the capital stock is fixed and so are wages and prices. This is a perfectly standard Keynesian short run. In static analysis as conducted by the CBO, the OTA and the JCT, wages and prices are assumed to adjust (and all accounting tricks are used).

Astonishingly, there is a heated debate about this. I think it can be resolved if Mankiw says he didn't use static in its Senatorial sense and should have written "extremely short run". I also think he should, but definitely won't, note that his calculation is just a calculation with no policy relevance at all (it would have none even if the super simple modeling assumptions were the truth, the whole truth, and nothing but the truth).

Oh crap my summary for those who find DeLong's post TL is Too Long too. Just click the link.

My interest is in totally pointless theory. (no JCT no CBO). Why, in the model, does the long run take a long time to arrive ? What assumption is made which prevents K from jumping ?

I can think of 3

1. What Mankiw really has in mind. The economy is a closed economy. higher after tax interest implies higher saving and capital accumulation (there is a substitution effect but Ricardian equivalence means there is no income effect). The economy converges to a new steady state with after tax interest equal to the rate of time preference (1-t)f'(k) = rho. But this is hard, so (like the Tax Foundation as denounced by Krugman) he semi shifts to an open economy, but just to say that the after tax interest rate reaches a constant in the long run.

But then, if there are no installation costs and domestic and foreign goods are perfect substitutes, then domestic K will jump. Oooops. One needs one or the other. Krugman has very wonkishly done imperfect substitutes here.

so I will whip out Q. To avoid Krugman's insanely wonkish math (and replace it with other insanely and pointlessly wonkish math) I assume that domestic and foreign goods are perfect subsitutes (with no transportation costs either). This means that there is alway perfect purchasing power parity and current account deficits can jump up and down. This good can be consumed or assembled to make capital. I use its price as numeraire. It really just means I am setting the after tax rate of interest to a constant r*.

I will assume that labor input is constant and L=1. So I can write production as f(K) = F(K,1) and talk about derivatives. Capital income gross of taxes is Kf'(k), investors get (1-t)Kf'(K), the IRS gets tKf'(K) and workers get f(K)-Kf'(K). Here notice that I assume that reinvested profits are taxed -- no expensing investment here.

Now I will intoduce an installation cost. The cost of increasing K by dK is dk+dk^2 . The second term is called an installation cost. This means that the value of a unit of capital is not necessarily one unit of the final product. The ratio of the prices is called Q.

The convention is to call the dk increase I (investment) and not explain where installation costs appear on profit and loss statements. I assume that the installation costs are counted as investment not expenses for tax purposes (this is also conventional). I am just insisting that the tax collected is equal to tKf'(K) no matter how much or little firms invest.

The Standard results now are that

1) Q = 1+2I

2) r*Q = (1-t)f'(K) + dQ/dt

so in steady state r*=(1-t)f'(K) . There is math behind the equations, but they make sense. The marginal cost of capital is 1+2I so equation 1 just means that there is no arbitrage opportunity based on building new capital and selling it. Equation 2 says the return on ownership of capital is equal to r*. In other words, there is no arbitrage opportunity based on borrowing, buying some capital, operating it for a while collecting after tax revenues then selling it for a capital gain or loss.

Now what happens quickly if t is suddenly cut by dt ?

K can't jump. production can't jump. The real wage doesn't jump. real profits gross of taxes don't jump. This short term is Mankiw's extremely short term. There is no need for wage or price stickiness.

The variable Q jumps up (owners of capital are richer -- that is the actual point of the whole operation even if Republicans won't admit it).

OK I haven't proven this (and have no intention of doing so) but the transfersaility condition and the budget constraint imply that K will converge to a new steady state where r* ( 1-t+dt)f'(K), dK/dt = I = 0 and Q = 1. So Q has to head back down (K,Q) moves down a saddle path.

This means that the dQ/dt term is negative. This means that Q jumps up to a level lower than (1-t+dt)/(1-t).

Well that was almost exactly pointless. The only tiny point is that I have a model which has been fully worked out (I didn't here -- it's in the literature google [Q theory hayashi]) in which the very short run is exactly Mankiw's very short run.

Twitter AI Fail

I just got 2 new twitter followers.

Asteroid day is "Raising awareness to protect Earth from asteroid impacts and inspire the next generation." kay mccull's avatar tells people to vote.

I have trouble doubting that there is some connection with the poll I recently posted in which I asked if Clinton weren't on the ballot would her supporters have voted for the Sweet Meteor O'Death. A.I. is getting scary, but still confused support for a life destrying meteor with opposition. I assure my (few) blog readers that only one of my (few) twitter followers voted for the Sweet Meteor O'Death

Republicans Reject the NFL, the CIA and the FBI

I forget who said she never expected that, after the national divorce, liberals would get custody of the NFL. But it's beyond that. Now Republicans reject the CIA and the FBI too. Donald Trump sometimes sounds like a paranoid 60s hippy claiming he is being persecuted by the evil FBI (except some of them really were persecuted).

So what else can they reject and abandon ? Hmm the flag. Some of it is blue like blue states -- they can't have that, and some is commie red. So they will probably decide to purify it.

I can't wait to see Republicans wave their white flag.