Deadweight loss, tax revenue and the size of a tax, and taxing a good with inelastic demand

What a tax does to a market. A tax per unit drives a wedge between the price buyers pay and the price sellers keep, equal to the tax, and the quantity traded falls.

• Tax revenue is the tax per unit multiplied by the quantity still traded.
• Deadweight loss is the surplus lost on the units that are no longer traded: buyers valued those units above what they cost sellers, but the tax prevents the trade, and the lost surplus goes to no one, not even the government.

Both depend on the size of the tax, but in different ways.

Deadweight loss and the size of the tax. With straight-line demand and supply curves, the deadweight loss is a triangle whose height is the tax and whose base is the fall in quantity. A bigger tax makes both larger: the height grows with the tax, and the fall in quantity grows in proportion to the tax. So the deadweight loss grows with the square of the tax: doubling the tax makes the deadweight loss four times as large, and tripling it makes it nine times as large. Small taxes cause little inefficiency; large taxes cause a great deal.

Tax revenue and the size of the tax. Revenue is the tax times the quantity traded. A larger tax raises the amount collected on each unit but shrinks the number of units on which it is collected.

• When the tax is small, the first effect dominates and revenue rises with the tax.
• As the tax grows, the market shrinks so much that revenue stops rising and then falls; a tax high enough to stop all trade collects nothing.
• This rise and fall needs buyers or sellers to respond to price. If demand (or supply) is perfectly inelastic, the quantity traded does not fall at all, so revenue keeps rising with the tax and has no peak.

With the usual sloping demand and supply curves, this rise and fall is the Laffer curve: revenue is greatest at some tax rate T*, and raising the rate beyond T* lowers revenue, while any rate below T* leaves room to raise more. Where that rate lies is hard to measure in practice, so a cut in the rate is not certain to raise revenue.

A worked illustration. Our own figures: demand Q = 120 – P and supply Q = P, so the market clears at P = 60 and Q = 60. A tax t per unit cuts the quantity to 60 – t/2.

Each doubling of the tax (20 to 40, 40 to 80) multiplies the deadweight loss by four, while revenue rises only up to a tax of 60 and falls beyond it. At a tax of 80 the government collects no more than at 40, but the deadweight loss is four times as large.

Taxing a good with inelastic demand. Yes, the government can raise its revenue in this way, and it is the case in which a tax raises the most revenue for the least inefficiency.

• When demand is inelastic, buyers cut their purchases only a little as the price rises, so the quantity on which the tax is collected shrinks very little and revenue rises almost in step with the tax.
• Because few trades are lost, the deadweight loss, which arises from the lost production and consumption, is also small.
• Buyers bear most of the burden: when demand is relatively inelastic, the price buyers pay rises by most of the tax. In the extreme case of perfectly inelastic demand, the whole tax falls on buyers, quantity does not fall at all, revenue rises in full proportion to the tax, and there is no deadweight loss.
• The caveat is fairness: a tax on a necessity with inelastic demand falls heavily on the buyers who cannot do without it.

Conclusion. As a tax grows, its deadweight loss grows faster, with the square of the tax, while, with the usual sloping demand and supply curves, its revenue first rises and then falls, as the Laffer curve shows; only if demand or supply is perfectly inelastic does revenue go on rising. A tax on a good with inelastic demand raises revenue effectively, because the quantity taxed hardly falls; the same fact keeps its deadweight loss small but places most of its burden on buyers.