What is the importance of elasticity in government revenue collection?

The effect of income elasticity on taxation is quite large, and quite analogous to the effect of demand elasticity on optimal pricing for a monopoly.

When you're setting a price as a monopoly, you have two effects to consider: Raising the price will make you more profit for each good sold; but it will also reduce the number of goods sold. The amount by which quantity sold changes due to a change in price is the elasticity of demand. The optimal price is the one at which these two effects are balanced, so that either raising or lowering from that price level would reduce profit.

Similarly, as a government setting tax rates you have two effects to consider: A higher tax rate will get you more revenue per dollar of GDP, but it will also reduce total GDP. The amount by which a change in tax rates reduces GDP is the taxable income elasticity.

If the taxable income elasticity is small---as it probably is, at least under current tax rates---then raising taxes will increase revenue, because it will have a large effect on revenue per dollar of GDP and a small effect on total GDP.

But if the taxable income elasticity is large, we get what is called the Laffer effect, in which an increase in tax rates can actually cause a decrease in tax revenue, because a small change in revenue per dollar of GDP causes a large loss of total GDP.

The revenue-maximizing tax rate is the one that balances these two effects, much like a monopoly would balance demand elasticity; empirically the revenue-maximizing rate appears to be about 70%. Below this, you get less revenue per dollar of GDP; above this, you get more revenue per dollar of GDP, but you lose so much in GDP that you get less revenue overall.

Mathematically, let r be the tax rate, T be tax revenue, and Y be total GDP. In this very simple model we'll pretend the taxes are flat, so:

T = r Y

If Y were constant, obviously T would be maximized by setting r = 1. But Y is in fact dependent on r, so to find the maximum we need to take the derivative of this function:

dT/dr = 0 = Y + r dY/dr

Y = - r dY/dr

r/Y * dY/dr = -1

r/Y * dY/dr is simply the taxable income elasticity, so what we're saying is that revenue is maximized when the elasticity is -1.

There are reasons you might want to choose tax rates below the revenue-maximizing rate: Even at much lower rates, higher taxes are reducing overall GDP, and that is undesirable. You may not need or want such high tax revenues because they are in excess of what you need for government spending. You may decide as a matter of public policy to leave more of the economy under private spending rather than government spending.

There is rarely any reason to choose tax rates above the revenue-maximizing rate: At that high a rate, the government actually loses money by raising taxes. If you wanted that same amount of revenue, you should choose a tax rate below revenue-maximizing, in order to improve overall economic efficiency. Certainly you would not want to raise taxes above revenue-maximizing in order to raise revenue for public services; that makes no sense. But one case that might make sense is suggested by Thomas Piketty, which is that as a matter of public policy you may choose to forcibly equalize the distribution of wealth even at the cost of less tax revenue. Even this would only make sense if you had strong reasons to equalize the income distribution (which is true---a lot of socioeconomic problems are linked to high wealth inequality) and the loss of GDP and tax revenue was not too large (that part is not as clear).