### 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,

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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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.

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 equationw = 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.