“Self-financing” tax reforms: a simple formula

There is much talk these days about tax reforms, both in Austria and around world. Most political parties seem to agree that taxes on labor are too high and that cuts should be made. There is disagreement as to whether these tax cuts should be accompanied by cuts in government spending or increases in other taxes.

One recurrent issue in this debate is the extent to which tax cuts are “self-financing”. This usually comes from a vague notion that reducing tax rates has a “stimulating” effect on “growth” and “job creation”. Such “stimulus” makes the tax revenue increase thus offsetting some of the revenue loss due to the reduction in tax rates.

Although I usually take great pleasure in brutally debunking popular myths with my profound knowledge of Economic Science (insert resounding laughter here), let me say that I think that in this matter the vague notion of the layman is broadly correct.

Economics being a hard quantitative science, the careful economist always strives to replace broadly correct but vague notions with mathematically exact but only vaguely correct formulas. In this spirit, I offer a formula for calculating to which degree a cut in the marginal labor tax rate is “self-refinancing”.

We start from a definition: total tax revenue (T) is the tax rate (t) times income (Y):

 \displaystyle T = t\times Y. 

We treat t as both the average and marginal tax rate. In fancy language: income taxes are assumed to be linear. Not true, but (one hopes) true enough.

We want to know how T changes if t is reduced by a small amount dt. There are two effects, one direct, one indirect. The direct effect is to reduce T by an amount  \displaystyle Y dt . The indirect effect comes from realizing that Y depends on labor input L which, in turn, depends on the tax rate. So therefore, if we reduce the tax rate by dt, labor supply rises by  \displaystyle n dt , where n is the elasticity of labor supply. The increase in labor input raises output and thus income. Suppose the elasticity of output with respect to labor input is a. Then the total change in income is:  \displaystyle dY = (\alpha\times n)dt. 

The indirect effect is where “self-financing” comes from. Let us measure the self-refinancing effect of the tax cut by  \displaystyle X = t\times dY/Y, which is the indirect change in revenue measured in percent of income.

 \displaystyle X = (t\times\alpha\times n)dt.  *

The self-financing share X is larger, the higher the initial tax rate, and the higher the two elasticities  \displaystyle \alpha and n.

How big is  \displaystyle \alpha ? Well, consider a Cobb-Douglas production function  \displaystyle Y=K^{1-\alpha}\times L^{\alpha} , where K stands for other factors of production which we hold fixed for purposes of this exercise. The labor elasticity of output is  \displaystyle \alpha. It is well-known that under competitive conditions a is equal to the labor share of income. In Austria, as well as in most developed countries, this share is about 2/3. So let’s take that as our answer.

How big is n? That’s a tough one to measure. Theoretically, it depends on the labor-leisure preferences of households as well as on other „deep” parameters of the economy. The empirical evidence I have seen suggests that a 1 percent decrease in t increases L by less than 1, but more than 1/3 of a percent. Let’s take 1/2 as a guess.

Finally, what is t? In Austria the marginal income tax rate is close to 50%, the average rate is in the area of 30%.

Feeding these numbers to our formula we arrive at the following conclusion. The self-financing share of a tax cut is in the range between 10 and 17 percent. This means that a tax cut of 1 billion euros indirectly creates additional revenues between 100 and 170 million euros. That still leaves a hole in the public budget of at least 830 million euros, though.

*) The General Formula is:

 \displaystyle dT = Ydt + t\times\frac{dY}{dL}\frac{L}{Y}\times\frac{dL}{L}\frac{1}{dt}\times Y dt 

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