How the deadweight loss of a tax depends on the size of the tax and on the elasticities of demand and supply

What the deadweight loss is. A tax drives a wedge between the price buyers pay and the price sellers receive. Some buyers who valued the good above the old price, but not above the new one, stop buying, and the quantity traded falls. The surplus that buyers and sellers would have gained on those lost trades is not collected by anyone as tax revenue either: it simply disappears. That fall in total surplus is the deadweight loss of the tax. Taxes discourage production and consumption, which is why they create market inefficiency.

The loss as a triangle. On a supply and demand diagram the deadweight loss is the triangle between the demand and supply curves over the units no longer traded. With straight-line curves:

deadweight loss = 1/2 × (tax per unit) × (fall in quantity traded).

Both relations asked about follow from this. The height of the triangle is the tax itself; the base, the number of trades lost, depends on how strongly buyers and sellers respond to the price changes the tax causes, which is what the elasticities measure.

The size of the tax. A larger tax enlarges the triangle in both directions at once: it is taller (the wedge is larger) and wider (more trades are lost). With straight-line curves the lost quantity grows in proportion to the tax, so the deadweight loss grows with the square of the tax: doubling the tax makes the loss four times as large, and tripling it makes it nine times as large. Each additional rupee of tax drives out trades that were worth more to buyers and sellers than the ones it drove out before, so the loss rises faster than the tax.

Tax revenue behaves differently. Revenue is the tax per unit times the quantity still traded, and the quantity shrinks as the tax rises, so revenue first rises less than in proportion and, beyond some rate, actually falls. This is the idea of the Laffer curve: past an optimal rate, raising the tax rate reduces revenue. A high tax can therefore produce a very large deadweight loss for little extra revenue.

The elasticities of demand and supply. The more elastic demand or supply is, the larger the deadweight loss of a given tax:

• Elasticity is the percentage change in quantity for a percentage change in price. When buyers respond strongly to the higher price they pay, or sellers to the lower price they receive, many trades are abandoned, so the base of the triangle is wide.
• When both demand and supply are inelastic, buyers and sellers keep trading nearly the same quantity despite the tax; few trades are lost, and the deadweight loss is small.
• In the extreme case of perfectly inelastic demand, the quantity bought does not change at all whatever the price; buyers bear the whole tax, no trades are lost, and there is no deadweight loss.

Illustration (our own figures). Quantity is in thousand units and price in rupees. In each market the curves cross at P = 50, Q = 50, and supply is Q = P (elasticity 1) unless stated.

• Size of the tax (first two rows): doubling the tax from 10 to 20 doubles the lost quantity (5 to 10) and raises the deadweight loss four times (25 to 100), while revenue rises from 10 × 45 = 450 to 20 × 40 = 800, less than double.
• Elasticity of demand (rows 1, 3 and 4): the same tax of 10 costs 16.67 when demand is inelastic, 25 when its elasticity is 1, and 33.33 when it is elastic.
• Elasticity of supply (last row): a more elastic supply raises the loss in the same way, to 33.33.

What follows for tax policy. Since the deadweight loss rises with the square of the tax, a modest tax costs society little in lost trades while a very high one costs a great deal; and since it rises with the elasticities, a tax on goods whose demand and supply are inelastic causes a smaller loss than the same tax on goods whose demand or supply is elastic. The second lesson has a cost in fairness: inelastic demand means that buyers carry most of the tax.

Conclusion. The deadweight loss of a tax is the triangle of lost trades between the demand and supply curves. It rises more than in proportion to the tax (with its square, for straight-line curves), because a larger tax both widens the wedge and destroys more trades; and it is larger the more elastic demand and supply are, because responsive buyers and sellers abandon more trades. With perfectly inelastic demand or supply there is no deadweight loss at all.