Signature
DWL = 0.5 * eps * tau^2 * p * q
| Inputs | Definition | Unit |
|---|---|---|
eps | Absolute value of the price elasticity of demand at the pre-tax price and quantity, compensated in principle | none |
tau | Tax or wedge as a proportion of the pre-tax price p, zero or above | proportion |
p | Price before the tax, equal to the constant marginal cost | currency per unit |
q | Quantity bought before the tax | units per period |
DWL | Approximate excess burden of the tax | currency per period |
|---|
Function
Deadweight loss of a wedge between marginal value and marginal cost
Maps a price distortion, such as a tax, a subsidy, a price above cost or insurance that lowers the price at the point of use, to the net surplus it destroys: the value of trades it prevents in excess of their cost, or the cost of trades it adds in excess of their value. Money that only changes hands, such as tax revenue or an insurer's payments, is netted out. With straight-line curves the area is a triangle whose height is the wedge and whose base is the change in quantity. The notation follows the Deadweight Loss article.
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Implementations
Excel
Harberger deadweight loss from named elasticity, tax rate and spending
With the absolute elasticity in Elasticity, the tax as a proportion in TaxRate and the pre-tax price and quantity in PreTaxPrice and PreTaxQty, the formula returns the approximate loss, held in a cell named HarbergerDWL.
=0.5*Elasticity*TaxRate^2*PreTaxPrice*PreTaxQty
Assumptions
Small tax on a good supplied at constant marginal cost
The producer price stays fixed, so the whole tax passes to buyers, and the tax is small enough for the second-order approximation to hold. On linear demand the expression is exact; on curved demand it is an approximation.
Compensated elasticity and no other distorted market
The elasticity is the compensated one; Hines reports that ordinary triangles usually approximate compensated measures closely, with exceptions. Other markets are undistorted, so no cross-market terms arise.
Worked examples
Twenty per cent tax at the article's full price
At the article's full price of 50 pounds and 5,000 sessions, demand q = 10000 minus 100p has a point elasticity of 1.0. A 20 per cent tax raises the price to 60 pounds and cuts use by 1,000, a loss of 5,000 pounds, which the approximation reproduces exactly because demand is linear (computed here for illustration).
eps = 1; tau = 0.2; p = 50; q = 5000; DWL = 5000
Doubling the tax rate quadruples the Harberger loss
At a 40 per cent rate on the same curve the loss is 20,000 pounds, four times the loss at 20 per cent (computed here for illustration).
eps = 1; tau = 0.4; p = 50; q = 5000; DWL = 20000
Elasticity of 0.2 on the less responsive curve
On the article's less responsive curve, q = 6000 minus 20p, the elasticity at 50 pounds is 0.2, and the same 20 per cent tax costs 1,000 pounds (computed here for illustration).
eps = 0.2; tau = 0.2; p = 50; q = 5000; DWL = 1000
Common errors
Applying the approximation to a large tax on curved demand
With a constant elasticity of 1 instead of linear demand, a 40 per cent tax on the same starting point gives a Marshallian loss of about 12,690 pounds, while the approximation gives 20,000 (computed here for illustration). For large wedges the area is computed from the demand curve itself.
Entering the tax per unit in place of the tax rate
Tau is a proportion of the price. Entering a tax of 10 pounds per session as tau = 10 instead of 0.2 multiplies the result by 2,500.
Sources
Second-order approximation to the excess burden of a single tax
Auerbach AJ, Hines JR Jr. Taxation and economic efficiency. In: Auerbach AJ, Feldstein M, editors. Handbook of Public Economics. Vol 3. Amsterdam: Elsevier; 2002. p. 1347-1421 (read as NBER Working Paper 8181, 2001). Section 2.2: the second-order approximation around the undistorted point approximates the total excess burden (Harberger 1964a); for a single tax with producer prices fixed, the excess burden is minus one half of the tax change times the compensated demand response to it times the tax change, so excess burden increases with the square of a tax. Section 2.3: Harberger derived the approximation used to measure deadweight loss.
Loss rising with the square of the tax rate and compensated measures
Hines JR Jr. Three sides of Harberger triangles. Journal of Economic Perspectives. 1999;13(2):167-188 (read as NBER Working Paper 6852, 1998). Dupuit observed that the welfare loss triangle is generally a function of the square of the tax rate; Harberger's 1964 papers derived and applied the triangle method; Harberger triangles usually approximate compensated measures closely.
Canonical Identity
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