Wednesday, October 18, 2017

An Exercise for My Readers

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.


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.

Monday, October 09, 2017

A Talk from the CEA Chair

Saturday, September 30, 2017

What I am reading

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

Tuesday, September 26, 2017

More on the Economics of Healthcare

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:
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 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

Tuesday, September 19, 2017

Still #1

Friday, September 08, 2017

How to Get People to Get Along

Click here to read my column in Sunday's New York Times.

Monday, September 04, 2017

A Reading List

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

Sunday, September 03, 2017

What I did last night

Lady Gaga performed at Fenway Park yesterday (and the day before). Apparently, she is the first woman ever to headline an event at Fenway. It was a great concert.

Good people watching before the concert started. Ran into a few former ec 10 students.

Wednesday, August 30, 2017

News from Amazon

To users of my favorite textbooks: Thank you! 
Have a great semester.

Tuesday, August 22, 2017

What Moderates Believe

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.

Monday, August 14, 2017

Macro Musings

You can hear me interviewed by David Beckworth here.

Friday, July 28, 2017

Why Health Policy is Hard

Click here to read my column in Sunday's NY Times.

Does this make my Hamilton tickets a deductible business expense?

Economic Lessons from the Musical Hamilton, by Matthew C. Rousu and Courtney A. Conrad, discusses how the great musical can be used to teach economic principles in the classroom.

Wednesday, July 12, 2017

CEA Chairs on Steel Tariffs

Wednesday, June 21, 2017

Report from the NFF

It is now that time of year when I am enjoying the Nantucket Film Festival. My wife and I today saw The Big Sick. Despite the not very enticing title, we loved it. The film is based on the real-life romance of the two screen writers, emphasizing the difficulty of bridging cross-cultural expectations. It is more heartfelt than a standard rom-com, more comedic than a drama, more earnest than standard Hollywood fare. Most definitely recommended, especially for a date night.

Monday, June 12, 2017

Hamilton Tickets Redux

James Stewart takes a look at theater tickets on Broadway, a topic I discussed last year.  I love his ending:
Dynamic pricing and super-premium prices may be relatively new, but the scarcity of tickets for hit shows has a long tradition. Mr. Schumacher cited “My Fair Lady,” the “Hamilton” of the 1955-56 Broadway season. As Broadway lore has it, a man in the audience turned to his neighbor, an older woman, and asked why the fifth-row center seat next to her was empty. 
“My husband died,” she replied. 
“Didn’t anyone else want to come?” he asked. 
“No,” she answered. “They’re all at the funeral.”

Tuesday, June 06, 2017

Advice for Young Economists

Monday, June 05, 2017

Economists for Hassett

Saturday, June 03, 2017

On Taxes and Deficits

Click here to read my column in Sunday's New York Times.

Friday, May 26, 2017

A New Mankiw Publication

This one I am particularly proud of, though I cannot claim to fully understand it.