Grasping Reality on TypePad, by Brad DeLong

So I surfed over to the source on Greg Mankiw's blog, and found some correct math:

A [small] open economy has the production function y=f(k), where y is output per worker and k is capital per worker. The capital stock adjusts so that the after-tax marginal product of capital equals the exogenously given world interest rate r: r=(1-t)f'(k). Wages are set by the marginal product of labor, which (by Euler's theorem) equals: w=f(k)-f'(k)k. 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)kdt.

How much will the tax cut increase wages? In particular, what is dw/dx? The first person to email me the correct answer will get a shout-out.... The same calculation would apply to the steady-state of a Ramsey model of a closed economy, where r would be interpreted as the rate of time preference.... Update: Casey Mulligan, who has been thinking along similar lines, was the first to email me the correct answer:

dw/dx = 1/(1 - t).

So if the tax rate is one third, then every dollar of tax cut to capital (on a static basis) raises wages by 1.50...

It seemed to be intended as some kind of backup for Kevin Hassett's beyond ludicrous and unprofessional claims about the consequences of corporate tax cuts for real wage levels:

There has been a lot of discussion lately about how much a cut in the tax on capital will increase wages. So I thought I would pose a relevant exercise for my readers...

But Greg ended his post with a plea for help:

I must confess that I am amazed at how simply this turns out. In particular, I do not have much intuition for why, for example, the answer does not depend on the production function...

And then I vanished down the rabbit hole for four hours, getting more and more confused. I have finally emerged. And I think I understand what is going on—and it has absolutely nothing to do with any sort of boost to economic growth from cutting "taxes on capital".

But I could be wrong: additional insights would be welcome. And corrections would be especially welcome.

Let us make Greg's example even simpler.

Let us suppose an economy is composed of typical workers who can produce 50,000 dollar/year in useful marketed goods and services as long as they are able to rent 100,000 dollars worth of capital goods to assist them. More than 100,000 dollars worth of capital goods per worker does not boost their productivity at all. Less than 100,000 dollars worth of capital goods per worker, and workers cannot produce at all.

Let us follow Greg and say that the required rate of return that must be paid to the corporations that are the owners of capital is fixed: say, 10%/year. And let us say that there is a corporate income tax at a potentially variable rate t.

What can we say about this model?

First, if the corporate tax rate is more than 80%, the economy collapses: At an 80% corporate tax rate, the corporation must collect 50,000 dollars/year per worker in order to pay its 40,000 dollars/year in taxes and still earn the 10,000 dollars/year to make its 10% hurdle rate. But at a tax rate below 80%, there is an equilibrium with production.

In that equilibrium, for each possible tax rate t in [0%, 100%]:

• 100,000 worth of capital is used by each worker.
• 50,000/year of output is produced by each worker.
• 10,000/(1-t) is taken from that output as pre-tax corporate earnings.
• 10,000[t/(1-t)] of pre-tax corporate earnings is paid to the government as taxes.
• 10,000 per worker remains for the corporation as after-tax profits
• This typical worker earns 50,000 - 10,000/(1-t) per year in wages.

Let's start with the tax rate at 60% at a 60% tax rate:

• Production is 50,000/year per worker
• Pre-tax corporate earnings are 25,000/year per worker
• After-tax corporate earnings are 10,000/year per worker
• Tax revenues are 15,000/year per worker
• Wages are 25,000/year per worker

Now let's cut the tax rate to 50%. At a 50% tax rate:

• Production is 50,000/year per worker
• Pre-tax corporate earnings are 20,000/year per worker
• After-tax corporate earnings are 10,000/year per worker
• Tax revenues are 10,000/year per worker
• Wages are 30,000/year per worker

The shift from a 60% to a 50% tax rate has lowered tax revenues by 5,000/year per worker, and raised wages by 5,000/year per worker. There is no factor 1/(1-t) anywhere—no greater increase in wages than reduction in tax revenue.

But take a look back at how Greg Mankiw defined his "static cost of the tax cut": it was by multiplying the tax base—at a 60% tax rate, the 25,000/year per worker pre-tax corporate earnings—by the 10%-point change in the tax rate: 25,000 x 10% = 2,500. There is the 1/(1-t) factor: 2,500 is indeed half of the 5,000 wage increase.

But is this tax cut good for economic growth? No. Production is the same.

Let's look at the taxes more closely:

• At a 60% tax rate, we have 25,000 tax base times 60% tax rate = 15,000 in government revenue
• If we were to cut the tax rate from 60% to 50% while the tax base remained unchanged, we would have 25,000 tax base times 50% tax rate = 12,500 in government revenue—the 2,500 difference, half the gain in wages.
• But when we cut the tax rate the tax base falls as well. With a 50% tax rate, the tax base—the pre-tax earnings of the corporation—is not 25,000 per worker per year, but rather 20,000
• And 20,000 x 50% = 10,000, which is indeed the same as the 5,000 difference in wages.

More generally, if you have a tax rate t and if you cut it by an amount Δt, you will find that you are:

1. collecting less revenue per unit of pre-tax profit: a (tax base) x (Δt) term
2. collecting off of a lower tax base: a (change in required after-tax profits)/(1-t) term
3. In Greg Mankiw's notation, these two terms are:
   1. (TR)Δt
   2. [(TR)/(1-t))Δt]t term
   3. Both terms are money that ceases to flow to the government and flows to workers instead.
4. And, sure enough the second term is t/(1-t) times the first, so if you add the two terms together, you get that the boost to wages is 1/(1-t) times the first term.

But this has nothing to do with economic growth spurred by tax cuts. And this has nothing to do with "static" revenue estimation as performed by Treasury OTA, JTC, or OMB: the revenue estimators, in my knowledge, do not assume that asset prices and returns are unaffected by changes in tax law. They would, I think, include both (1) and (2). They have to estimate changes in valuations in order to do their job. (Ask Charlie Steindel about the fresh hell that is revenue forecasting for the state of New Jersey sometime.) (They do assume, in "static" estimates, that levels of economy-wide production and income are unaffected.)

And so I think I have the answer to Greg's question: the intuition for the reason that "the answer does not depend on the production function" is that the 1/(1-t) fact has absolutely nothing to do with changes in production. It's just as much there when there are no changes in production at all.

But I could be wrong...

I do, however, wish Greg were out there saying: "Encourage productive investment, yes! Further enrich successful past rent-seekers no!", rather than trying to provide some sort of backup for the ethics-free Kevin Hassett...