Deadweight Loss

Deadweight Loss from Insurance, Taxes and Subsidies

Deadweight loss is the net surplus lost on trades that a price distortion prevents, or the amount by which cost exceeds value on trades that it adds. In health care the distortion can come from a tax, a subsidy, a price set above cost or insurance that makes care cheaper to the patient than to the health system. In health economics the measure is used to value overuse induced by insurance, the efficiency effects of taxes and subsidies, and the extra cost of raising tax revenue for public programmes. The loss is usually drawn as a triangle with an elasticity form traced to Harberger, and an illustrative coinsurance example with linear demand shows how it is measured, while US, Norwegian and UK appraisal guidance treat the excess burden of taxation in different ways. The term here carries its welfare-economic meaning, not the separate use of "deadweight" in UK appraisal for outcomes that would have happened without an intervention.

A wedge between marginal value and marginal cost

Total surplus is largest when output stands where the demand curve, read as the marginal value of each unit to its users, meets the marginal cost of supplying it. A distortion that restricts output stops trades that users value above their cost; one that expands output adds units that cost more than they are worth to the people who use them. In both cases the deadweight loss is the net loss of consumer surplus and producer surplus left after money that only changes hands, such as tax revenue or an insurer's payments, has been netted out. In general form the loss is given by the expression below.

$$DWL = \left| \int_{q^{*}}^{q^{d}} \big( P(q) - MC(q) \big)\, dq \right|$$

where $DWL$ is the deadweight loss, $q^{*}$ is the efficient quantity at which demand price equals marginal cost, $q^{d}$ is the quantity under the distortion, $P(q)$ is the inverse demand curve giving the marginal value of the $q$-th unit, and $MC(q)$ is the marginal cost of that unit. With straight-line demand and supply the area is a triangle.

$$DWL = \tfrac{1}{2}\, w\, |\Delta q|$$

where $w$ is the wedge between the demand price and the supply price at the distorted quantity, such as a tax per unit or the gap between the cost of care and the patient's price, and $|\Delta q|$ is the change in quantity the wedge causes. Hines describes the excise tax case in these terms. The height of the triangle is the tax rate, its base is the fall in sales, and its area is one measure of the efficiency cost, also called the excess burden of the tax. The rectangle of tax revenue beside the triangle is a transfer to the government, not a loss to society.

Harberger triangles and how they entered applied work

The idea is older than its name. Hines notes that Jules Dupuit, writing in 1844, is generally credited with first using demand schedules to infer the welfare effect of a price change, and records that Dupuit saw that the loss from a tax generally grows with the square of the tax rate. Hines also records that Arnold Harberger's two papers of 1964 derived the triangle method and applied it to the deadweight loss of US income taxes, and that in 1954 Harberger had put the distortion from monopoly in US industry at about 0.1 per cent of gross national product. Hines notes that economists very rarely estimated deadweight losses before Harberger, and that the triangles took his name because of his influence.

Why the loss rises with elasticity and with the square of the wedge

The size of the triangle depends on how far quantity responds to the wedge. That response is measured by the price elasticity of demand, which therefore does most of the work. For a small ad valorem tax on a good supplied at constant marginal cost, the triangle can be written in terms of initial spending.

$$DWL \approx \tfrac{1}{2}\, \varepsilon\, \tau^{2}\, p\, q$$

where $\varepsilon$ is the absolute value of the price elasticity of demand at the starting point, $\tau$ is the tax as a proportion of the pre-tax price $p$, and $q$ is the quantity bought before the tax. The expression follows from the triangle, because the wedge is $\tau p$ and the fall in quantity is approximately $\varepsilon \tau q$. When demand does not respond at all, the wedge changes who pays but removes no transactions, so the loss is zero; when it does respond, doubling the wedge roughly quadruples the loss.

The same arithmetic governs the welfare cost of insurance. Feldstein and Gruber modelled a shift from existing US group insurance to a plan with 50 per cent coinsurance and an out-of-pocket limit of 10 per cent of income. They found no gain from reduced distortion at zero elasticity, a fall in deadweight loss of USD 534 per insurance unit (about 13 per cent of initial spending) at an elasticity of 0.33, and USD 902 at 0.5, in 1995 US dollars. They noted that the RAND experiment had estimated an elasticity of only 0.2 and argued that a community-wide change in coinsurance would draw a larger response, so the result rests on an input that is itself uncertain.

Insurance-induced overuse and the coinsurance wedge

Insurance that pays part of the price at the point of use works, for the patient, like a subsidy on care. Einav and Finkelstein credit Pauly's 1968 comment with first showing that if use responds to the out-of-pocket price, the cost of insurance rises and people may no longer be willing to pay the break-even price of full cover. The incentive problem is covered on the moral hazard page, and the concern here is how the loss is measured.

Feldstein and Gruber state that their method follows the approach Feldstein developed in 1973. They assume care is supplied at constant cost and measure it in units whose price without insurance is one. A person with a 20 per cent coinsurance rate buys care until the last dollar of services is worth 20 cents to them, an 80 cent welfare loss on that dollar. Raising the coinsurance rate reduces the loss by a rectangle and a triangle.

$$\Delta DWL = (E_0 - E_1)(1 - P_1) + \tfrac{1}{2}(P_1 - P_0)(E_0 - E_1)$$

where $\Delta DWL$ is the reduction in deadweight loss, $E_0$ and $E_1$ are the quantities of care at the initial and higher coinsurance rates $P_0$ and $P_1$, and the price of care without insurance is normalised to one. With a constant price elasticity $\eta$, in absolute value, $E_1 = E_0 (P_0 / P_1)^{\eta}$. The triangle term treats demand as a straight line between $E_1$ and $E_0$, so under constant elasticity it is an approximation. Feldstein and Gruber treated the fall in deadweight loss as an unambiguous benefit and the change in the risk people bear as a separate effect of ambiguous sign, and added the two. They also noted that US tax rules make much more complete cover personally optimal, a tax subsidy to insurance that deepens the first wedge.

Worked example with linear demand and a 20 per cent coinsurance rate

The figures are illustrative. An insurer covers physiotherapy sessions that cost £50 each to provide, at constant cost. Annual demand in the covered population is $q = 10000 - 100p$, where $q$ is sessions per year and $p$ is the price the patient pays in pounds.

1. Find use at the full price and at the insured price. With a 20 per cent coinsurance rate the patient pays £10 a session, which gives the following.

$$q_0 = 10000 - 100 \times 50 = 5000, \qquad q_1 = 10000 - 100 \times 10 = 9000$$

where $q_0$ is use if patients paid the full £50 cost and $q_1$ is use at the £10 insured price.

2. Measure the triangle. The wedge is £40 and insurance adds 4,000 sessions, so the triangle is as follows.

$$DWL = \tfrac{1}{2} \times (50 - 10) \times (9000 - 5000) = 80000$$

where $DWL$ is the deadweight loss in pounds per year. As a check, users value the extra sessions at an average of £30 each, £120,000 in all, against a cost of £200,000, which leaves the same £80,000.

3. Raise the coinsurance rate to 40 per cent. The patient now pays £20 and use falls to 8,000 sessions, which gives the following.

$$DWL_2 = \tfrac{1}{2} \times (50 - 20) \times (8000 - 5000) = 45000$$

where $DWL_2$ is the deadweight loss at the 40 per cent rate. The fall of £35,000 matches the Feldstein and Gruber formula in money terms.

$$(9000 - 8000) \times (50 - 20) + \tfrac{1}{2} \times (20 - 10) \times (9000 - 8000) = 35000$$

where the first product is the rectangle between cost and the new patient price over the 1,000 forgone sessions and the second is the triangle above it. The wedge fell from £40 to £30 and the loss fell to $(30/40)^2 = 0.5625$ of its previous value, as the square-of-the-wedge rule predicts.

4. Change the elasticity. If demand is instead $q = 6000 - 20p$, which also gives 5,000 sessions at £50, use at £10 rises only to 5,800 and the loss is as follows.

$$DWL = \tfrac{1}{2} \times (50 - 10) \times (5800 - 5000) = 16000$$

where $DWL$ is the deadweight loss under the less responsive curve. At the full price the point elasticity is $-100 \times 50 / 5000 = -1.0$ on the first curve and $-20 \times 50 / 5000 = -0.2$ on the second, so the same coinsurance rate costs £80,000 or £16,000.

5. Interpret. On the first curve, the move to 40 per cent coinsurance cuts spending by £50,000, the cost of the 1,000 forgone sessions, which patients valued at £15,000, leaving the £35,000 gain. Patients' out-of-pocket payments rise from £90,000 to £160,000, offset in expectation by lower premiums; the loss of risk protection is a cost the triangle does not count. The gain is real only if the demand curve measures what the sessions are worth to patients.

The marginal excess burden of tax finance in appraisal

When a public health programme is paid for from taxation, each extra pound raised through taxes that change behaviour carries a deadweight loss of its own. The marginal excess burden is that loss per unit of revenue at the margin, and the marginal cost of public funds is one plus it. An appraisal that applies it scales each option's call on public budgets in the following way.

$$C^{adj} = (1 + \lambda)\, G$$

where $C^{adj}$ is the social cost of the public funding, $G$ is the option's net effect on public budgets, and $\lambda$ is the marginal excess burden per unit of revenue.

National guidance differs. The 1992 version of US Office of Management and Budget Circular A-94, which OMB reinstated in April 2025 when it revoked a 2023 revision, takes 25 cents per dollar as a reasonable estimate and asks for a supplementary analysis of public investments not justified on cost savings, with public expenditure multiplied by 1.25; it says this is not required for cost-effectiveness analyses and that investments funded by user charges that work like market prices carry no excess burden. Norway's Ministry of Finance circular R-109/2021, as set out by the Norwegian Agency for Public and Financial Management (DFØ), requires a tax financing cost of 20 øre per krone of net public financing for measures in every sector. DFØ explains that the cost covers the efficiency loss from taxation and also the administrative costs of collecting tax, and adds that it applies even within an allocated budget. The UK Green Book (2026) accepts that taxes create distortions but tells practitioners not generally to include the costs of raising public funds, because most proposals are funded from pre-determined departmental budgets.

Within a fixed health budget the factor matters less than it first appears. If every cost in the comparison falls on the same tax-financed budget, scaling all of them by the same factor leaves the ranking of options by incremental cost per QALY unchanged, although each ratio rises against a threshold expressed in unadjusted money. The adjustment changes decisions when options are financed differently, through user charges or private premiums, or in a cost-benefit analysis that sets public cost against money benefits. An ISPOR Special Task Force report on US value assessment frameworks discusses, for the pluralistic US health care system, the excess burden of tax-financed public programmes compared with private ones.

Deadweight loss beside transfers, corrective taxes and additionality

Several nearby ideas are easily confused with deadweight loss. They differ in what they count as a cost to society.

Transfers. The Green Book treats tax revenues, grants and subsidies as economic transfers that cost the payer what they give the recipient, and excludes taxes and subsidies from the market prices used in appraisal. Deadweight loss is the part of a tax's cost that is not a transfer.

Corrective taxes. Where use of a good harms others through an externality, the private demand curve overstates its social value, and a Pigouvian tax set at the external cost removes a deadweight loss instead of creating one. This is the efficiency argument made for taxes on tobacco, alcohol and sugar-sweetened drinks, whose harms fall partly on others.

Several distorted markets. Hines shows that a triangle becomes a trapezoid when a tax pushes demand into a market that is already distorted. Where a charge in one health care setting shifts use to another subsidised one, this links deadweight loss to cross-price elasticity.

Market power. Monopoly pricing of a patented medicine above its marginal production cost leaves out uses worth more than that cost but less than the price. Patent systems accept this static loss in return for the incentive to invest in research and development. Where the medicine is insured, patients face only their share of the price, so the loss arises mainly at the payer or coverage level, when restricted coverage at the high price excludes patients whom it would be efficient to treat at marginal cost.

Additionality. In the Green Book, deadweight means outcomes that would have taken place without any intervention. It is a different concept and is not measured from demand and cost curves.

Misreadings and limits of the triangle

The triangle rests on strong assumptions. Several errors recur when it is applied to health care.

A common error reads a fall in use after cost sharing as a welfare gain of the same size. At an elasticity of 0.5, Feldstein and Gruber found that a plan with a deductible of 15 per cent of income cut spending more than one with 50 per cent coinsurance up to the same limit (USD 1,629 against USD 1,426 per insurance unit) but reduced deadweight loss by less (USD 735 against USD 1,045), because it removed spending within a narrower range, some of which consumers valued. Where patients misjudge the value of care, the demand curve may not measure value at all, as the consumer surplus page explains.

A second error leaves out the value of risk protection, although a case for more cost sharing needs the change in financial risk alongside the triangle. A third ignores income effects. The triangle treats the whole rise in use as a response to price. Nyman argued that Pauly's analysis overstates the welfare loss because it includes the effect on use of income transferred from the healthy to the ill, and that the welfare loss can be seen as the transaction cost of insurance. Hines reports that Harberger triangles usually approximate compensated measures closely, but with examples where the two differ significantly.

A fourth error treats the excess burden rate as a constant, although OMB allows a different figure where specific information supports one. Unweighted triangles also add losses across people regardless of income; Hines notes that distributional weights can be attached, although Harberger doubted that agreed weights could usually be found.

Sources

• Danzon PM, Drummond MF, Towse A, Pauly MV. Objectives, budgets, thresholds, and opportunity costs: a health economics approach. An ISPOR Special Task Force report [4]. Value in Health. 2018;21(2):140-145. doi:10.1016/j.jval.2017.12.008
• Direktoratet for forvaltning og økonomistyring (DFØ). Veileder i samfunnsøkonomiske analyser. Oslo: DFØ; 2023. Section 3.4.8, reproducing Finansdepartementet circular R-109/2021, point 7.1. https://www.dfo.no/sites/default/files/2023-10/Veileder_samfunnsokonomiske_analyser_nov23.pdf
• Einav L, Finkelstein A. Moral hazard in health insurance: what we know and how we know it. Journal of the European Economic Association. 2018;16(4):957-982. doi:10.1093/jeea/jvy017
• Feldstein M, Gruber J. A major risk approach to health insurance reform. In: Poterba JM, ed. Tax Policy and the Economy. Vol 9. Cambridge, MA: MIT Press; 1995:103-130. https://www.nber.org/chapters/c10892
• Feldstein MS. The welfare loss of excess health insurance. Journal of Political Economy. 1973;81(2, Part 1):251-280. doi:10.1086/260027 (as reported in Feldstein and Gruber 1995)
• Harberger AC. The measurement of waste. American Economic Review. 1964;54(3):58-76. (as reported in Hines 1999)
• Hines JR Jr. Three sides of Harberger triangles. Journal of Economic Perspectives. 1999;13(2):167-188. doi:10.1257/jep.13.2.167 (NBER Working Paper 6852, 1998)
• HM Treasury. The Green Book (2026). London: HM Treasury; 2026. Chapters 4, 6 and 8. https://www.gov.uk/government/publications/the-green-book-appraisal-and-evaluation-in-central-government
• Nyman JA. The economics of moral hazard revisited. Journal of Health Economics. 1999;18(6):811-824. doi:10.1016/s0167-6296(99)00015-6
• Office of Management and Budget. Circular No. A-94: Guidelines and discount rates for benefit-cost analysis of federal programs. Washington, DC: OMB; 29 October 1992. Sections 6 and 11 and Appendix A. Reinstated by OMB Memorandum M-25-23, 8 April 2025. https://www.whitehouse.gov/wp-content/uploads/2025/12/a094.pdf
• Pauly MV. The economics of moral hazard: comment. American Economic Review. 1968;58(3):531-537. (as reported in Einav and Finkelstein 2018 and Nyman 1999)