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.

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