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Random Observations for Students of Economics

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I usually refrain from commenting on all the silliness found over at Paul Krugman's blog. But in a post a couple days ago, Paul is especially dyspeptic and calls me out by name. Let me offer a few comments.

1. Paul says I have never admitted to making a math error. Well, I would if I thought I made such an error. I make them all the time. But in this case I am not convinced. Neither is University of Chicago professor Casey Mulligan, who thinks Paul made a math error. I spoke with several other economists (some of whom share Paul's politics) and they don't see Paul's point either.*

2. Paul says that economists like me have not been sufficiently critical of President Trump and his policies. Let me point out

4. Paul seems to take the position that unless you agree with him about the tax bill, you are unprincipled. In the world as I see it, reasonable people can disagree, and progress is best made when people do not question the moral rectitude of others simply because they hold different opinions.

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**Update*: Robert Waldmann looks at the issue, concluding "no one has made an algebra mistake. Taxes on capital and capital income are different."

1. Paul says I have never admitted to making a math error. Well, I would if I thought I made such an error. I make them all the time. But in this case I am not convinced. Neither is University of Chicago professor Casey Mulligan, who thinks Paul made a math error. I spoke with several other economists (some of whom share Paul's politics) and they don't see Paul's point either.*

2. Paul says that economists like me have not been sufficiently critical of President Trump and his policies. Let me point out

A . I said during the election that I would not vote for him.3. Paul thinks that economists like me should be more vocal about how horrible the tax bill is. I might be if I thought it was completely horrible, but despite its many flaws, there are parts that I like, including the lower corporate tax rate, the move to a territorial tax system, the reduced deduction for state and local taxes, and the scaled back mortgage interest deduction (in the House bill). Overall, the tax bill is a mixed bag, with some bad features and some good features.

B. I criticized his obsession with the trade deficit.

C. I encouraged tax reform to be revenue neutral.

D. I called the tax bill an "unworkable mess."

E. I lamented the new tax on university endowments.

4. Paul seems to take the position that unless you agree with him about the tax bill, you are unprincipled. In the world as I see it, reasonable people can disagree, and progress is best made when people do not question the moral rectitude of others simply because they hold different opinions.

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This came out last year, but I only heard about it recently: The Night Manager, a six-part miniseries based on the John le Carre novel. Compelling story and great acting. Available now via Amazon.

I don't agree with all of it, but these slides from a recent talk by Jason Furman make a lot of good points.

One of the most surprising parts of the proposed tax bill is its tax on university endowments. If my rough calculations are correct, the tax would cost schools like Harvard between $1,000 and $2,000 per student every year.

Is there a good argument for this policy? Not that I can see.

Is there a good argument for this policy? Not that I can see.

Click here to read my column in Sunday's *NY Times* on the Trump tax plan.

There has been a lot (maybe too much) commentary on my simple pedagogical exercise. To move the discussion forward, let me offer a challenge to readers: What is the dynamic Furman ratio in such a model--that is, the ratio the wage increase to the dynamic revenue loss, which I will call dw/dz?

Note that

dz = - d[t*f '(k)*k]

but now we take into account that k and f '(k) will change with t.

Furman was the first to point out to me that, for Cobb-Douglas, the correct answer is:

dw/dz = (1-α) / (1- α - t).

For a capital share and tax rate of 1/3, we get dw/dz = 2. Each dollar of a capital tax cut (dynamicly scored) raises wages by two dollars. See the derivation at Cochrane's blog. (Note that John and I have a slightly different notation, so don't be misled by the minus sign.)

What is the general case?

I have not worked it out, but I will offer a conjecture: You can write dw/dz as a function of the tax rate, the capital share, and the elasticity of substitution between capital and labor.

You can find a helpful hint in footnote 19 of this paper.

Note that

dz = - d[t*f '(k)*k]

but now we take into account that k and f '(k) will change with t.

Furman was the first to point out to me that, for Cobb-Douglas, the correct answer is:

dw/dz = (1-α) / (1- α - t).

For a capital share and tax rate of 1/3, we get dw/dz = 2. Each dollar of a capital tax cut (dynamicly scored) raises wages by two dollars. See the derivation at Cochrane's blog. (Note that John and I have a slightly different notation, so don't be misled by the minus sign.)

What is the general case?

I have not worked it out, but I will offer a conjecture: You can write dw/dz as a function of the tax rate, the capital share, and the elasticity of substitution between capital and labor.

You can find a helpful hint in footnote 19 of this paper.

Click here to read "Friedman’s Presidential Address in the Evolution of Macroeconomic Thought," an essay I just finished writing with Ricardo Reis for the *Journal of Economic Perspectives*.

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.

An 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)*k*dt.

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 on my blog.

By the way, 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.

Bonus question: If there are positive externalities to capital accumulation, as suggested by DeLong and Summers, would the effect of the tax cut on wages be larger or smaller than in the standard neoclassical model above?

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*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.

And if DeLong and Summers are right that there are positive externalities to capital, the effect will be larger than $1.50.

*Update 2*: A friend asks to see the proof. Here goes. Start with my second equation

w = f(k) -f '(k)*k.

Take the total differential of this equation to get

dw = -k*f "(k)*dk.

This equation relates the change in wages to the change in capital. To find dk, use my first equation

r = (1-t)f '(k).

Take the total differential and solve for dk to obtain

dk = {f '(k)/[(1-t)*f "(k)]}*dt

This equation relates the change in capital to the change in the tax rate. Substitute this expression into the dw equation to obtain

dw = -[k*f '(k)/(1-t)]*dt.

This equation relates the change in wages to the change in the tax rate. The third equation in the model can be rewritten as

dt = dx/[-f '(k)*k].

This equation relates the change in the tax rate to the static revenue loss. Substitute this expression into the preceding equation to yield the result

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

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.

By the way, this derivative (dw/dx) is slightly different from what Casey calls the Furman ratio in his post. Casey looks at the ratio of the wage change to the dynamic revenue loss, whereas dw/dx is the ratio of the wage change to the static revenue loss. We might call dw/dx the*static Furman ratio*. The dynamic Furman ratio is typically larger.

*Update 3*: Alan Auerbach emails me the following comment:

Just to place this result in context, it's a combination of (1) the standard result that in a small open economy labor bears 100% of a small capital income tax; and (2) the fact that starting at a positive tax rate, the burden of a tax increase exceeds revenue collection due to the first-order deadweight loss.

An 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)*k*dt.

How much will the tax cut increase wages? In particular,

By the way, 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.

Bonus question: If there are positive externalities to capital accumulation, as suggested by DeLong and Summers, would the effect of the tax cut on wages be larger or smaller than in the standard neoclassical model above?

-------------

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.

And if DeLong and Summers are right that there are positive externalities to capital, the effect will be larger than $1.50.

w = f(k) -f '(k)*k.

Take the total differential of this equation to get

dw = -k*f "(k)*dk.

This equation relates the change in wages to the change in capital. To find dk, use my first equation

r = (1-t)f '(k).

Take the total differential and solve for dk to obtain

dk = {f '(k)/[(1-t)*f "(k)]}*dt

This equation relates the change in capital to the change in the tax rate. Substitute this expression into the dw equation to obtain

dw = -[k*f '(k)/(1-t)]*dt.

This equation relates the change in wages to the change in the tax rate. The third equation in the model can be rewritten as

dt = dx/[-f '(k)*k].

This equation relates the change in the tax rate to the static revenue loss. Substitute this expression into the preceding equation to yield the result

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

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.

By the way, this derivative (dw/dx) is slightly different from what Casey calls the Furman ratio in his post. Casey looks at the ratio of the wage change to the dynamic revenue loss, whereas dw/dx is the ratio of the wage change to the static revenue loss. We might call dw/dx the

Just to place this result in context, it's a combination of (1) the standard result that in a small open economy labor bears 100% of a small capital income tax; and (2) the fact that starting at a positive tax rate, the burden of a tax increase exceeds revenue collection due to the first-order deadweight loss.

Most people forget about the second point when arguing about where between 0 and 100% of a tax cut goes to labor vs. capital, and this is exacerbated by the fact that distribution tables assume revenue change = burden change, except in special cases (such as where a cut in capital gains taxes is presumed to lose little or no revenue).

*Update 4*: John Cochrane weighs in.

*Update 5*: Steven Landsburg weighs in.

Two of my favorite young macroeconomists (and former students) have a new essay on Identification in Macroeconomics.

Back in July, I wrote a *NY Times* column about the economics of healthcare. Yesterday, my friend John Cochrane posted a lengthy response. I won't take the time to reply to all of John's points, but like everything John writes, his post is provocative and thoughtful. So I would encourage people to read it and decide for themselves.

John is certainly correct when he speculates about my motivation in writing the column:

*Times* column is just one outlet.

I wrote this particular column around the same time I was writing about the economics of healthcare in a longer piece, which is designed to be an optional chapter for users of my favorite textbook. You can read the longer piece here.

John is certainly correct when he speculates about my motivation in writing the column:

It sounded like a good column idea, "I'll just run down the econ 101 list of potential problems with health care and insurance and do my job as an economic educator."I have always thought of my job as first and foremost being an economics educator, and my

I wrote this particular column around the same time I was writing about the economics of healthcare in a longer piece, which is designed to be an optional chapter for users of my favorite textbook. You can read the longer piece here.

Every few years, I teach (in addition to ec 10) a freshman seminar for about a dozen students. The seminar is essentially a book group for students who are taking introductory economics concurrently or who have advanced placement credit in economics. Here is a list of this year's books:

- The Worldly Philosophers, by Robert Heilbroner
- On Liberty, by John Stuart Mill
- Capitalism and Freedom, by Milton Friedman
- Equality and Efficiency: The Big Tradeoff, by Arthur Okun
- The Economics of Inequality, by Thomas Piketty
- Fair Play, by Steven Landsburg
- Finance and the Good Society, by Robert Shiller
- Scarcity, by Sendhil Mullainathan and Eldar Shafir
- The Moral Economy, by Samuel Bowles
- The Myth of the Rational Voter, by Bryan Caplan

I much appreciated today's column by David Brooks, though he seems to be describing center-right moderates more than center-left moderates (or is that my own bias showing up?).

David also taught me a new word:*syncretistic*. It refers to combining different forms of belief.

David also taught me a new word: