### Labor Supply and Distortionary Taxation

8/16/2012 02:30:00 PM
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Miles Kimball has an interesting post titled "The Flat Tax, The Head Tax and the Size of Government: A Tax Parable," but read it carefully, because it isn't necessarily about what it seems. At any rate, for those of you who are not familiar with mathematical economics but are curious enough, go check it out: it is a very simple, well explained example that I think should be quite accessible to outsiders yet still informative (only algebra is required to understand, although you may want to learn a little about convexity and concavity of functions).

Kimball's point, I think, is very similar to one I've been thinking a lot about in recent weeks. We often speak about labor supply being increasing in the wage and decreasing in the marginal income tax rate--this is only sort of true. The fact is that there is no particular reason for a household to distinguish between a low wage with zero income taxes versus a high wage with an offsetting income tax, they should behave the same in both cases. In the data we see that real wages have risen dramatically in the past century, yet the labor supply has barely budged (after all, there's only so much time in a day, it can't fluctuate much). That means that any reasonable model of labor supply must show that labor supply is basically orthogonal to real wages, save for some modest imperfections. That, in turn, means that labor supply is basically orthogonal to income tax rates as well. Those "imperfections" arise as a result of household wealth--if the household has amassed a source of wealth or income other than their income from labor, then their labor supply will be modestly sensitive to the tax rate. Alternatively, if the household has debts that must be honored, then we would also expect labor supply to be sensitive to tax rates, since a higher tax rate means the household must work harder to meet the debt obligation.

So the model that Kimball gives is one in which households have cobb-douglas preferences with equal weights on consumption C, liesure L, and government spending G. There is one unit of time divided between labor and leisure, C and G are perfect substitutes, and production is such that one unit of labor (1-L) produces one unit of output: C+G=1-L

Suppose that the government finances G through a combination of income tax at rate t and a lump-sum tax of amount T. Then the household budget constraint is C=(1-t)(1-L)-T, which gives us the labor supply equation
L=1/2-T/(1-t)
So to clarify my remarks above, just note that if T=0, then the income tax rate has absolutely no effect. If the government also has a lump-sum tax T>0, then the income tax rate matters somewhat. But wait! Labor supply is increasing as taxes go up! In other words, the household is forced to work harder in order to meet its binding obligation to pay the government at least T.

Now, in reality we don't have either lump-sum or flat taxes. Instead we have progressive tax rates that are locally flat rate for the interior of each tax bracket. As it turns out, we can model our progressive system pretty effectively by supposing that t is the household's marginal tax rate, and then giving the household a negative transfer T<0 that compensates the household for the fact that his income below his marginal tax bracket was taxed at a lower rate. Returning to the labor supply equation, note that with T<0, we have that
-T/(1-t)>0
which implies that labor supply is decreasing in the income tax rate. Hence you could argue that income taxes are only discretionary because they are progressive, not because they tax income. Moreover, what I like about Kimball's post is that it highlights a common bias that a lot of economists commit without realizing it--lump-sum taxation may allow us to reach a more efficient outcome than income taxes, but it doesn't really make sense to say that they aren't distortionary--they can and do have very large wealth effects, even if there is no substitution effect. And as far as labor is concerned, lump-sum taxes (or transfers) are far more relevant than the marginal tax rate.