Signature
DWL = 0.5 * w * abs(q_d - q_star)
| Inputs | Definition | Unit |
|---|---|---|
w | Gap between the demand price and the supply price at the distorted quantity, such as a tax per unit or the cost of care minus the patient's price, zero or above | currency per unit |
q_d | Quantity traded with the wedge in place | units per period |
q_star | Quantity at which the demand price equals marginal cost | units per period |
DWL | Net loss of consumer and producer surplus after transfers are netted out | 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
Deadweight loss triangle from named wedge and quantities
With the wedge in a cell named Wedge and the quantities in DistortedQty and EfficientQty, the formula returns the deadweight loss, held in a cell named DeadweightLoss.
=0.5*Wedge*ABS(DistortedQty-EfficientQty)
Assumptions
Straight-line demand and cost between the two quantities
Demand and marginal cost are linear between q_star and q_d, so the area between them is a triangle; with curved demand the triangle is an approximation. Marginal cost is constant in the insurance case, as Feldstein and Gruber assume.
Demand curve measures the marginal value of each unit
The demand price is read as what the marginal unit is worth to its users. Where patients misjudge the value of care, or where use has external effects, the demand curve does not measure social value and the triangle misstates the loss.
Only one market distorted for the deadweight triangle
Related markets are undistorted. Hines shows that when a tax pushes demand into a market that is already distorted, the loss becomes a trapezoid rather than a triangle.
Worked examples
Physiotherapy at a 20 per cent coinsurance rate on linear demand
Sessions cost 50 pounds and demand is q = 10000 minus 100p. Full-price use is 5,000 sessions; at the insured price of 10 pounds use is 9,000. The wedge of 40 pounds over 4,000 extra sessions gives a deadweight loss of 80,000 pounds a year, as in the article.
w = 40; q_d = 9000; q_star = 5000; DWL = 80000
Same demand curve at a 40 per cent coinsurance rate
At a patient price of 20 pounds use falls to 8,000 sessions, the wedge is 30 pounds and the loss 45,000 pounds, 0.5625 of the previous loss, as the square-of-the-wedge rule predicts.
w = 30; q_d = 8000; q_star = 5000; DWL = 45000
Less responsive demand at a 20 per cent coinsurance rate
With demand q = 6000 minus 20p, use at 10 pounds rises only to 5,800, and the same wedge of 40 pounds gives a loss of 16,000 pounds instead of 80,000, as in the article.
w = 40; q_d = 5800; q_star = 5000; DWL = 16000
Common errors
Counting a transfer as part of the deadweight loss
In the first example the insurer pays 40 pounds on each of 9,000 sessions, 360,000 pounds, and the extra 4,000 sessions cost 200,000 pounds, but the deadweight loss is 80,000 pounds. Tax revenue and insurer payments are transfers; only the triangle is a loss to society.
Ignoring a distortion in a related market
If a charge in one health care setting shifts use to another subsidised setting, the extra use there carries its own wedge, and the triangle in the first market understates the total loss.
Sources
Harberger triangle with the tax as height and the fall in sales as base
Hines JR Jr. Three sides of Harberger triangles. Journal of Economic Perspectives. 1999;13(2):167-188 (read as NBER Working Paper 6852, 1998). For an excise tax, the height of the Harberger triangle is the tax rate, its base the amount by which sales fall, and its area one measure of the efficiency cost, deadweight loss or excess burden; revenue is a transfer to the state; a tax pushing demand into a distorted market turns the triangle into a trapezoid.
Deadweight loss triangle of insurance-induced health spending
Feldstein M, Gruber J. A major risk approach to health insurance reform. In: Poterba JM, editor. Tax Policy and the Economy. Vol 9. Cambridge, MA: MIT Press; 1995. p. 103-130. Section 4.1 and Figure 1: with care supplied at constant cost and measured in units whose uninsured price is 1, the deadweight loss caused by the induced increase in health spending is the triangle ACD.
Canonical Identity
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