Compensating Variation

Compensating Variation as the Hicksian Money Measure Behind Willingness to Pay

Compensating variation is the welfare-economic quantity that a willingness-to-pay question for a gain, or a willingness-to-accept question for a loss, tries to measure. When a cost-benefit analysis of a health programme adds up what people would pay for a health gain, or would need to be paid to bear a loss, the theoretical object being summed is each person's compensating variation. The measure has a formal definition, applies differently to price changes and to changes in health itself, and has to be kept distinct from equivalent variation and Marshallian consumer surplus. Willingness to pay and willingness to accept can diverge, and in applied work compensating variation is recovered from contingent valuation and discrete choice experiments, as the illustrative example later shows. Here CV means compensating variation, not contingent valuation or the coefficient of variation; elicitation formats, survey biases and the value of a QALY are covered on the willingness to pay page.

From a fall in the price of oranges to health programmes

The idea comes from Hicks's attempt to put Marshall's consumer surplus on a firmer footing. In Value and Capital, Hicks described the gain from a fall in price as the compensating variation in income, the loss of income that would just offset the price fall and leave the consumer no better off than before. In his example a person buys 6 oranges at 2d. each and the price falls to 1d.; the compensating variation cannot be less than 6d., because after losing 6d. the person could still buy the original bundle and would usually do better by substituting towards oranges.

After Henderson argued that there were four alternative measures of consumer surplus, Hicks accepted the point and set out price and quantity versions of both compensating and equivalent variations. The distinction matters because the changes valued in health cost-benefit analysis are of two kinds. Some are price changes, such as a new user charge or a change in an insurance premium. Many more are changes in things the person cannot buy in a market, such as their own health state, a shorter wait or access to a new treatment.

Defining the measure with expenditure and indirect utility

Compensating variation fixes the reference point at the person's utility before the change. It asks how much income must be taken away (for a gain) or given (for a loss) after the change, so that the person ends up exactly as well off as before. Using an indirect utility function, it is defined as follows.

$$v(p^1, q^1, Y - CV) = v(p^0, q^0, Y)$$

where $v$ is the indirect utility function giving the highest utility attainable at given prices, non-market quantities and income, $p^0$ and $p^1$ are prices before and after the change, $q^0$ and $q^1$ are non-market quantities or qualities (such as a health state) before and after, $Y$ is income and $CV$ is the compensating variation. The same quantity in terms of the expenditure function is given below.

$$CV = e(p^0, q^0, u^0) - e(p^1, q^1, u^0) = Y - e(p^1, q^1, u^0)$$

where $e$ is the expenditure function, the least income needed to reach a given utility at given prices and quantities, and $u^0$ is utility before the change, so that $e(p^0, q^0, u^0) = Y$. Under this convention CV is positive for an improvement, when it is the most the person would pay, and negative for a deterioration, when its absolute value is the least compensation the person would accept. Some texts use the opposite sign, so a report should state its convention.

When a prescription charge or copayment changes, the person can adjust how much they buy, and the measure allows for that adjustment. When a programme improves health, the person cannot choose how much of the new health state to consume. This quantity version is often called the compensating surplus, and the name is sometimes used as a synonym for compensating variation, although strictly it is the case in which the changed quantity is fixed for the person. HM Treasury's wellbeing guidance, which supplements the Green Book, uses compensating surplus for the money value of a change in an outcome, defined as the amount paid or received that leaves the person in their initial welfare position after a change from the status quo.

Compensating variation, equivalent variation and consumer surplus compared

The three measures answer different questions about the same change and agree only in special cases. Equivalent variation holds utility at its level after the change and asks what change in income, without the policy, would have the same effect. Consumer surplus is the area to the left of the ordinary (Marshallian) demand curve between the old and new prices, which mixes substitution and income effects; it is defined for price changes only, since a change in a person's health has no ordinary demand curve. In the same notation equivalent variation is written as follows.

$$EV = e(p^0, q^0, u^1) - Y$$

where $EV$ is the equivalent variation and $u^1$ is utility after the change.

For a fall in the price of a normal good, the compensating variation is smaller than the Marshallian area and the equivalent variation is larger, because demand for a normal good rises with real income. Willig showed that the error from using consumer surplus in place of either Hicksian measure is small when the income elasticity of demand is small or the price change is small. That justification is weaker for changes in health, which are not market price changes and can be large relative to income.

The choice also matters when several options are ranked. Every option's equivalent variation is measured at the same status quo prices and quantities, so equivalent variation ranks options in the same order as their utilities, whereas each compensating variation is measured at that option's own prices and quantities and a ranking by it can disagree with the utility ranking when there are income effects. Compensating variation suits a single move away from the status quo, which is the usual appraisal question.

Why willingness to accept can exceed willingness to pay

For a health gain, the compensating variation is the willingness to pay to obtain it and the equivalent variation is the willingness to accept to forgo it. The two coincide only when the reference point does not matter, which requires the absence of income effects. With utility linear in income, for example, the two are equal.

$$v = \alpha Y + V(q) \implies CV = EV = \frac{V(q^1) - V(q^0)}{\alpha}$$

where $\alpha$ is the constant marginal utility of income and $V(q)$ is the part of utility that depends on the non-market quantity $q$.

Weber shows that if the two Hicksian measures are to coincide at every income, both must be independent of income and indirect utility must be additively separable in income. He also shows that when the money value of the change does not fall as income rises, as expected if health behaves as a normal good, equivalent variation is at least as large as compensating variation, so in theory willingness to accept for a gain is at least willingness to pay. Hanemann found that, for a change in a non-market good, the gap can be arbitrarily large, depending on how easily the good can be substituted by ordinary market goods. A person's own health, or access to the only effective treatment, has few substitutes, so a large gap is consistent with economic theory and is not by itself evidence of survey error. Behavioural explanations based on the reference point offer a competing account, discussed on the willingness to pay page.

How surveys and choice models recover compensating variation

Stated preference methods estimate compensating variation by building the change into the question. Asking the most a person would pay for a programme they do not yet have targets compensating variation for a gain, and asking the least compensation they would accept to lose an existing service targets compensating variation for a loss; the payment vehicle and the description of the status quo fix the reference point.

Discrete choice experiments recover the measure from an estimated random utility model. For a multinomial logit model with utility linear in income, Small and Rosen showed that expected consumer surplus takes a closed form, so the welfare effect of a change in the choice set or in the attributes of the alternatives is given by the following expression.

$$CV = \frac{1}{\alpha}\left[\ln \sum_{j=1}^{J^1} e^{V_j^1} - \ln \sum_{j=1}^{J^0} e^{V_j^0}\right]$$

where $\alpha$ is the marginal utility of income, estimated as the negative of the coefficient on the cost attribute, $V_j^0$ and $V_j^1$ are the representative utilities of alternative $j$ before and after the change, and $J^0$ and $J^1$ are the numbers of alternatives before and after. Because the formula assumes a constant marginal utility of income, it cannot separate compensating from equivalent variation, and it is most defensible for changes that are small relative to income.

Lancsar and Savage argued that the methods then used in health economics to derive welfare measures from discrete choice experiments were not consistent with random utility theory or welfare theory, and proposed a compensating variation measure derived from welfare theory. Ryan replied that the Small and Rosen formula reduces to the usual health economics method for state of the world models, which turns the question into when state of the world models, as opposed to multiple alternative models, are appropriate. The exchange continued in a second comment and a reply from Lancsar and Savage in the same issue, so the choice of model, and with it the formula, remains a judgement for each study. The method is used in applied work. An Australian study of genomic newborn screening used it to value a publicly funded programme, reporting about 4,600 to 5,700 Australian dollars per newborn screened for a restrictive model as the additional diagnostic yield rose from 10 to 50 per 1,000 newborns.

Compensating variation in cost-benefit analysis and UK appraisal

In a cost-benefit analysis of a health programme, the benefit side is the sum of compensating variations across everyone affected, with losses counted as negative. The net benefit is that sum minus the opportunity cost of the resources used. A positive result is conventionally read as a potential Pareto improvement, meaning the gainers could in principle compensate the losers and still be better off, but the compensation is not paid and an unweighted sum gives more weight to people with more income.

HM Treasury's Green Book places stated preference, in which people are asked what they would be willing to pay or willing to accept, below market prices, generic prices and revealed preference in its hierarchy of valuation methods, and says it should be used only if robust revealed preference data are not available. The supplementary wellbeing guidance states that compensating surplus is in general the preferred measure for appraisal, because it reflects the idea that people have a right to the status quo, and gives a formula for large changes based on the logarithm of income with the same form as the health example below. The guidance presents that formula as an optional consideration, a sensitivity test that gives values for the high end of the range, not the central value.

Worked example of a health gain and a price cut for a health product

The figures below are illustrative. A person has an annual income of £30,000, and two separate changes are valued for one year, each with its own utility function, to show when the three measures diverge.

1. A health gain. Utility is $u = \ln Y + b h$, where $h$ is a health index and $b = 1$. A programme raises $h$ from 0.60 to 0.75. Solving $\ln(Y - CV) + b h^1 = \ln Y + b h^0$ gives the following.

$$CV = 30000 \times (1 - e^{-0.15}) = 30000 \times 0.1392920 = 4178.76$$

where 30,000 is income in pounds, 0.15 is $b$ multiplied by the change in the health index, and 4,178.76 is the most the person would pay for the gain, in pounds. Solving $\ln(Y + EV) + b h^0 = \ln Y + b h^1$ gives the following.

$$EV = 30000 \times (e^{0.15} - 1) = 30000 \times 0.1618342 = 4855.03$$

where 4,855.03 is the least the person would accept to forgo the gain, in pounds. Willingness to accept exceeds willingness to pay by £676.27, about 16% of the compensating variation.

2. A price cut. Utility is Cobb-Douglas with a budget share $a = 0.004$ on a health product bought at the person's discretion, such as an over-the-counter remedy; the Cobb-Douglas form, which implies a price elasticity of demand of -1, is chosen for tractability. At a price of £10 the person buys 12 items, spending £120; the price falls to £5 and demand rises to 24 items. Hicks's argument gives a lower bound of 12 × 5 = 60, so the compensating variation is at least £60. The three measures are calculated below.

$$CV = 30000 \times (1 - 0.5^{0.004}) = 30000 \times 0.0027687 = 83.06$$

where 0.5 is the ratio of the new price to the old and 83.06 is the compensating variation in pounds.

$$CS = 120 \times \ln 2 = 120 \times 0.693147 = 83.18$$

where $CS$ is the Marshallian consumer surplus in pounds and 120 is spending at the original price, which with Cobb-Douglas demand equals $aY$.

$$EV = 30000 \times (2^{0.004} - 1) = 30000 \times 0.0027764 = 83.29$$

where 83.29 is the equivalent variation in pounds.

3. Interpretation. For the price cut, CV is below consumer surplus, which is below EV, as theory predicts for a normal good, and all three lie within £0.23 of each other, so consumer surplus is an adequate approximation. For the health gain, which is large relative to income, the Hicksian measures differ by about 16%. An analysis that asked about paying for the gain would record £4,178.76, while one that treated the person as entitled to it and asked about giving it up would record £4,855.03. Which figure is right depends on whether the person is taken to be entitled to the status quo or to the gain, and that is a value judgement for the analyst to state.

Assumptions and common misreadings

Compensating variation treats the individual as the judge of their own welfare, depends on income, so the same health gain is worth more in money to a richer person, and is defined only for a specified change from a specified status quo. Frameworks that value health outcomes in QALYs do not use it directly.

Four errors recur. Sign conventions are mixed, so a loss is reported as a positive figure without comment. Compensating variation, equivalent variation and consumer surplus are treated as interchangeable for health changes, although the close agreement seen for small price changes does not carry over to large health gains. A discrete choice estimate is read as a pure compensating variation when the model's constant marginal utility of income means it cannot distinguish the two Hicksian measures. Many options are ranked by compensating variation, where equivalent variation is the consistent measure.

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