Compensated Demand Curve

Compensated Demand Curve in Health Economics Welfare Analysis

A compensated demand curve, also called a Hicksian demand curve, is a demand curve drawn at one fixed level of utility, so that every change in price comes with whatever change in income keeps the person on the same indifference curve. Moving along the curve therefore shows only the substitution effect of a price change, without the income effect in the ordinary, or Marshallian, demand curve. In health economics the curve matters because the areas under it are the exact money measures of welfare that a cost-benefit analysis adds up when a copayment, a health tax or an insurance subsidy changes the price people face. The same distinction runs through the argument about how large the welfare cost of moral hazard in health insurance really is.

Holding utility fixed instead of income

The ordinary demand curve comes from utility maximisation with income fixed. The compensated curve comes from the mirror-image problem, in which the person reaches a target utility at the least cost. Quint's lecture notes call the solution Hicksian demand, or compensated demand, and explain the name by the thought experiment of raising a price while giving the consumer enough money to afford the same utility as before.

$$h(p, u) = \arg\min_{x} \; p \cdot x \quad \text{subject to} \quad U(x) \ge u, \qquad h_i(p, u) = \frac{\partial e(p, u)}{\partial p_i}$$

where $h(p, u)$ is the vector of compensated demands at prices $p$ and target utility $u$, $x$ is a bundle of goods, $U(x)$ is the utility function, $e(p, u)$ is the expenditure function, the least income needed to reach $u$ at prices $p$, and $h_i$ and $p_i$ are the demand for and price of good $i$. The second identity is Shephard's lemma as Quint states it.

Quint shows that the two demands match at corresponding levels of income and utility, so the curves cross at the price actually faced. Every point on a compensated curve lies on one indifference curve, so its height at each quantity is the person's marginal rate of substitution between the good and money at that utility. Applied to a new prescription charge, the curve gives the items a patient would buy if the charge came with a lump-sum payment just large enough to leave the patient as well off as before.

Substitution and income effects in the Slutsky equation

Compensated demand separates the two reasons why a price change alters what people buy. A higher prescription charge makes the medicine dearer relative to other goods, the substitution effect, and leaves the person poorer, the income effect. Quint derives the Slutsky equation by differentiating the identity between the two demands.

$$\frac{\partial x_i(p, Y)}{\partial p_i} = \frac{\partial h_i(p, u)}{\partial p_i} - x_i(p, Y) \frac{\partial x_i(p, Y)}{\partial Y}, \qquad \varepsilon^{h}_{i} = \varepsilon_{i} + s_i \eta_i$$

where $x_i(p, Y)$ is ordinary demand at income $Y$, the first term on the right of the first equation is the substitution effect and the second is the income effect, $\varepsilon^{h}_{i}$ is the compensated own-price elasticity, $\varepsilon_{i}$ is the ordinary price elasticity of demand, $s_i$ is the budget share of good $i$ and $\eta_i$ is its income elasticity of demand.

The compensated curve never slopes upwards. Quint shows that the matrix of compensated price responses is symmetric and negative semidefinite, so the own-price response cannot be positive, which his notes call the law of compensated demand. An ordinary curve can slope upwards in the Giffen case, which Quint explains as an inferior good whose income effect overwhelms the substitution effect. For a normal good the two effects work together, so Hines notes that the compensated curve is steeper than the Marshallian curve in the usual diagram with price on the vertical axis.

Nobody actually compensates patients when a charge rises, so market data trace ordinary demand. Quint notes that the Slutsky equation lets compensated responses be inferred from ordinary demand, and Hines reports a sizeable literature on doing so.

Areas under the curve as exact money measures of welfare

For a change in one price, the compensating variation and the equivalent variation are areas under compensated curves drawn at two utility levels. Quint writes both for a fall in the price of good $i$.

$$CV = \int_{p_i^1}^{p_i^0} h_i(p, u^0) \, dp, \qquad EV = \int_{p_i^1}^{p_i^0} h_i(p, u^1) \, dp$$

where $p_i^0$ and $p_i^1$ are the old and new prices, $u^0$ and $u^1$ are utility before and after the change, and $CV$ and $EV$ are both positive for a price fall under this convention. Hines records that Hicks named the area between such a curve and the initial price line the compensating variation, and the corresponding area at the post-change utility the equivalent variation.

The ordinary curve lies between the two compensated curves. Quint concludes that for a normal good $EV \ge \Delta CS \ge CV$, where $\Delta CS$ is the change in Marshallian consumer surplus, and that with quasilinear preferences, which have no income effects, all three coincide. Hines adds that compensated curves satisfy Hotelling's integrability conditions, so the welfare effect of several prices changing at once does not depend on the order in which the changes are taken.

Hicks doubted that the distinction would usually matter. Hines explains that the two elasticities differ only by the marginal propensity to spend on the good, which is small for a cheap medicine but not for a large bill for care.

Worked example with an illustrative tax on sugar-sweetened drinks

All numbers in this example are illustrative. A household has GBP 1,000 a month of discretionary spending and Cobb-Douglas preferences that give sugar-sweetened drinks a budget share of $a = 0.05$, with other spending priced at 1. A tax of GBP 0.50 a litre, passed on in full, raises the price from GBP 2.00 to GBP 2.50. Ordinary demand is $x = aY/p$, and the compensated demand through the starting point is $h(p, u^0) = a \, e(p, u^0) / p$ with $e(p, u^0) = Y (p / p^0)^a$.

1. Ordinary demand. The household buys the following litres a month before and after the tax.

$$x^0 = \frac{0.05 \times 1000}{2.00} = 25, \qquad x^1 = \frac{0.05 \times 1000}{2.50} = 20$$

where $x^0$ and $x^1$ are quantities at the old and new prices.

2. Compensated demand at the old utility. Keeping the household as well off as before costs the income below, at which it would buy the quantity shown.

$$e(p^1, u^0) = 1000 \times 1.25^{0.05} = 1011.22, \qquad h(p^1, u^0) = \frac{0.05 \times 1011.22}{2.50} = 20.22$$

where 1.25 is the ratio of new price to old. Of the 5 litre fall, 4.78 litres is the substitution effect and the remaining 0.22 litres is the income effect.

3. Slopes at the starting point. The ordinary slope is $-aY/p^2 = -12.5$ litres per pound. The income term is $x \, \partial x / \partial Y = 25 \times 0.025 = 0.625$, so the compensated slope is $-12.5 + 0.625 = -11.875$. In elasticities the ordinary value is $-1$ and the compensated value is $-1 + 0.05 \times 1 = -0.95$.

4. Three money measures of the loss. The three measures are as follows.

$$|CV| = 1011.22 - 1000 = 11.22, \qquad |EV| = 1000 - 1000 \times 0.8^{0.05} = 1000 - 988.90 = 11.10, \qquad |\Delta CS| = 50 \times 0.22314 = 11.16$$

where $|CV|$ is the compensation needed to restore the old utility, the area under $h(p, u^0)$, $|EV|$ is the most the household would pay to avoid the tax, and $|\Delta CS| = aY \ln 1.25$ is the Marshallian loss, all in pounds. The Marshallian figure sits between the two Hicksian ones.

5. Deadweight loss. The tax raises GBP 0.50 on each of 20 litres, or GBP 10. Subtracting it gives a deadweight loss of GBP 1.16 on the Marshallian measure and GBP 1.10 on the equivalent variation measure. At the old utility the compensated household would buy 20.22 litres and pay GBP 10.11 in tax, so the loss is GBP 1.11.

6. Interpretation. On unrounded figures the Marshallian loss of 1.157 exceeds the equivalent variation loss of 1.095 by about 6 per cent, matching Hines's account that, when a single normal good is taxed, the equivalent variation measure is the smaller. The gap is small because drinks take 5 per cent of the budget, which is Hicks's point. Where a household pays for a large course of care, the budget share and the gap are far larger.

Moral hazard and the welfare cost of insurance that lowers the price of care

Insurance that pays part of the bill at the point of use lowers the price of care. Nyman describes Pauly's analysis as setting Marshallian consumer surplus for the extra care against its cost, under an assumption that medical care use is not related to income. In Nyman's illustration of the conventional calculation, use rises from 10,000 units at USD 1 a unit to 20,000 units at a price of zero, and the extra units cost USD 10,000 but are worth USD 5,000, a welfare loss of USD 5,000.

Nyman argued that part of the extra use is caused by the income that insurance transfers from people who stay well to people who fall ill, and that this part should be excluded from the welfare loss. As a first approximation he applied the Slutsky equation to RAND Health Insurance Experiment estimates reported by Manning and Marquis, a price elasticity of 0.18 in absolute value and an income elasticity of 0.22, with a budget share for medical care of 0.15, which he took from the share of GDP spent on personal medical care in 1997. The compensated elasticity is then $-0.18 + 0.15 \times 0.22 = -0.147$, which Nyman rounds to 0.15 in absolute value, putting the welfare loss at about 83 per cent of the Marshallian estimate. He also noted that RAND's cost-sharing groups were paid an amount equal to their highest possible out-of-pocket spending, so some might read the RAND price elasticity as a compensated elasticity already.

Nyman went beyond the Hicksian curve, proposing a decomposition that holds the person to the original budget constraint, with a demand curve below the Hicksian one and a welfare loss smaller than on either Pauly's measure or a Hicksian one. The compensated curve is thus the benchmark in this debate, and the dispute is over which counterfactual income the insured person's demand should be measured at.

Compensated demand in cost-benefit and cost-effectiveness analysis

Compensated demand belongs to the welfarist tradition of welfare economics. Brouwer and colleagues quote Hurley's summary that welfarist tenets require any policy to be judged solely by the resulting utilities achieved by individuals, as assessed by those individuals. Money measures read from compensated curves express those utility changes, and in a cost-benefit analysis of a user charge, a health tax or a subsidy they are the gains and losses summed across people. Because they depend on ability to pay, an unweighted sum gives more weight to the gains and losses of richer people.

Most health technology assessment does not use demand curves. NICE's technology appraisal manual, PMG36, states that health effects should be expressed in QALYs and justifies cost-effectiveness analysis by NICE's focus on maximising health gains from a fixed NHS and personal social services budget. Brouwer and colleagues describe cost-utility analysis as using something like need instead of effective demand and willingness to pay, and many health economists as treating health itself as the maximand. In this extra-welfarist framing a compensated demand curve has no direct role, although it remains the right tool when a health policy works through prices, as taxes, charges and insurance design do.

Compensated demand beside neighbouring demand concepts

The general demand curve page deals with observed demand for care and its estimation, and the Marshallian demand curve is that ordinary curve written as a function of prices and income. The compensated curve is never observed directly and is defined by a utility level, not an income. The Slutsky decomposition that Nyman discusses, which lets the person just afford the original bundle, gives yet another curve.

Misreadings and limits of compensated demand

Most errors come from treating the compensated curve as an observed curve or from forgetting which utility level it is drawn at. The common ones in health applications are these.

1. Treating an estimated demand curve as compensated. Elasticities from claims data or trials are ordinary elasticities unless the design or model removes the income effect, and the RAND hold-harmless payments show how hard the line can be to draw.
2. Assuming the distinction never matters. Hicks's argument relies on small budget shares, so it is weakest for large out-of-pocket health spending, for low incomes and for insurance payouts that transfer large sums to people who fall ill.
3. Using the curve for changes in health itself. A gain in health or survival has no market price to vary, so its money value comes out of stated or revealed willingness to pay, not out of integrating a demand curve.

Sources

• Brouwer WBF, Culyer AJ, van Exel NJA, Rutten FFH. Welfarism vs. extra-welfarism. Journal of Health Economics. 2008;27(2):325-338. https://doi.org/10.1016/j.jhealeco.2007.07.003
• Hines JR Jr. Three sides of Harberger triangles. Journal of Economic Perspectives. 1999;13(2):167-188. https://doi.org/10.1257/jep.13.2.167 (read as NBER Working Paper 6852)
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual. PMG36. 2022, last updated 13 October 2026. Chapter 4. https://www.nice.org.uk/process/pmg36/chapter/economic-evaluation
• Nyman JA. The welfare economics of insurance contracts that pay off by reducing price. Discussion Paper No. 308. Center for Economic Research, University of Minnesota; 1999. https://conservancy.umn.edu/server/api/core/bitstreams/a5bf0296-3317-4e72-a0ed-753b517c443d/content
• Quint D. Lecture 11. Expenditure minimization and Slutsky. Econ 711 lecture notes. University of Wisconsin-Madison; 2021. https://users.ssc.wisc.edu/~dquint/econ711%20fall%202021/lecture%2011.pdf
• Quint D. Lecture 12. Testing via Slutsky, GARP, aggregating demand. Econ 711 lecture notes. University of Wisconsin-Madison; 2021. https://users.ssc.wisc.edu/~dquint/econ711%20fall%202021/lecture%2012.pdf