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?

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.

Monday, October 09, 2017

A Talk from the CEA Chair