Tuesday, October 24, 2017

The Dynamic Furman Ratio

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.