Equivalent Variation

Equivalent Variation and Valuing a Change at the Utility It Delivers

Any money valuation of a health change has to fix a reference point, and equivalent variation is the measure that takes the person's utility after the change as that reference. The choice decides which question a valuation answers. For a health gain, equivalent variation is the least the person would accept to forgo the gain, and for a health loss it is the most they would pay to avoid the loss. The most a person would pay for a gain is the compensating variation. Because equivalent variation measures every option in money at the same status quo, it also ranks several options in the same order as the person's own preferences. The measure is also used in cost-benefit analysis, where the gap between willingness to accept and willingness to pay has to be taken into account. Here EV means equivalent variation, not expected value.

Hicks's second measure, equivalent rather than compensating

Hicks first described the gain from a fall in price as a compensating variation in income, the loss of income that would just offset the price fall and leave the consumer no better off than before. After Henderson argued that this analysis implied four measures of consumer surplus, Hicks set out price and quantity versions of both the compensating and the equivalent variation. The price equivalent variation asks how much money income the person would need, without the price fall, to reach the level of satisfaction that the price fall would have given.

Mishan later narrowed the acceptable measures to two, compensating and equivalent variation, and traced the difference between them to the diminishing marginal utility of money. One measures what the person would pay and the other what the person must be paid, and the sum needed to express the same difference in utility is larger when it is added to income than when it is taken away. In modern terms this is the income-effect explanation of the gap discussed below.

Defining equivalent variation at the post-change utility level

Equivalent variation fixes the reference point at the utility the person reaches after the change. The money, however, is measured at the prices and non-market quantities that held before it. Using an indirect utility function, it is defined as follows.

$$v(p^0, q^0, Y + EV) = v(p^1, q^1, Y)$$

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 $EV$ is the equivalent variation. In terms of the expenditure function, the definition is as follows.

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

where $e$ is the expenditure function, the least income needed to reach a given utility at given prices and quantities, $u^0$ is utility before the change and $u^1$ is utility after it, so that $e(p^0, q^0, u^0) = Y$. Under this convention EV is positive for an improvement and negative for a deterioration, the same convention as the compensating variation page; some texts reverse the sign, so a report should state which it uses.

For a fall in a single price from $p^0$ to $p^1$, with everything else unchanged, the expenditure function can be differentiated with respect to price to give the compensated (Hicksian) demand, so equivalent variation is the area to the left of that demand curve, evaluated at the final utility, between the two prices, as follows.

$$EV = \int_{p^1}^{p^0} h(p, u^1) \, dp$$

where $h(p, u^1)$ is the Hicksian demand for the good at price $p$ when utility is held at $u^1$. For a change in something the person cannot buy, such as their own health state, there is no market demand curve to integrate, the measure comes directly from the utility equation and it is often called the equivalent surplus. HM Treasury's wellbeing guidance, which supplements the Green Book, defines equivalent surplus as the money, paid or received, that would leave the person in their subsequent welfare position without the change from the status quo.

Willingness to accept for a gain, willingness to pay to avoid a loss

The direction of the money flow follows from the reference point. For a gain, the person is placed at the higher post-change utility, so the question is how much extra income they would need in the old situation to be as well off as with the gain, which is the least they would accept to forgo it. For a loss, the question is how much income could be taken away in the old situation to leave them as badly off as the loss would, which is the most they would pay to avoid it.

Weber states the same distinction formally. Compensating variation corresponds to willingness to pay to bring a welfare change about, while equivalent variation measures willingness to accept to reverse the change once it has taken place. HM Treasury's wellbeing guidance tabulates equivalent surplus as willingness to accept to forgo a gain and willingness to pay to avoid a loss.

Why equivalent variation ranks several options consistently

Many health decisions involve more than one alternative to the status quo, such as several screening intervals or several models of service delivery. Equivalent variation measures every option in money at the same status quo prices and quantities, so it works as a money-metric of each option's final utility. The difference between two options' equivalent variations is given below.

$$EV_k - EV_j = e(p^0, q^0, u^k) - e(p^0, q^0, u^j)$$

where $EV_k$ and $EV_j$ are the equivalent variations of options $k$ and $j$ measured from the same status quo, and $u^k$ and $u^j$ are the utilities the person reaches under each option. Because the expenditure function rises with utility at fixed prices and quantities, this difference is positive exactly when option $k$ gives higher utility than option $j$.

Compensating variation lacks this property, because each option's compensating variation is measured at that option's own prices and quantities. Hammond notes that equivalent variation, unlike compensating variation, is higher for better changes from an existing status quo and so can itself be used as a welfare indicator. McKenzie and Pearce argued that equivalent variation, not compensating variation, is a money-metric of utility, and Baqaee and Burstein adopt equivalent variation as their welfare measure in recent work for that reason. With a single move away from the status quo the two measures always agree in sign, and the choice between them is a judgement about the reference point; with several options, equivalent variation is the consistent basis for ranking.

Equivalent variation, compensating variation and consumer surplus

The three measures answer different questions about the same change and agree only when income effects are absent. Marshallian consumer surplus, the area under the ordinary demand curve, mixes substitution and income effects and is defined only for price changes. For a fall in the price of a normal good, the equivalent variation is the largest of the three, because demand for a normal good rises with real income. When utility is linear in income the two Hicksian measures coincide, and Willig's result that consumer surplus is close to either of them when the income elasticity of demand or the price change is small does not carry over to large health changes, as the compensating variation page explains.

The gap between willingness to accept and willingness to pay

For a health gain, the equivalent variation is the willingness to accept to forgo it and the compensating variation is the willingness to pay to obtain it, so the gap between the two measures is the gap between WTA and WTP. Economic theory predicts a gap whenever there are income effects.

Weber shows that when the money value of a 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, and that the two can be equal at every income only if indirect utility is additively separable in income. The gap can also be arbitrarily large for a good with few market substitutes, as Hanemann showed, which fits a person's own health, so a large gap is consistent with theory. Behavioural accounts based on the reference point, often called the endowment effect, offer a competing explanation; Weber argues that studies which label a positive difference between the two measures as such an anomaly implicitly assume there are no income effects.

Equivalent variation also has a limit for very large losses. Hammond notes that money-metric measures such as equivalent variation can become undefined when a deterioration is severe enough, a similar problem first noticed by Jones-Lee for large increases in the probability of death, since a person may be willing to give up everything they own to avoid such a risk.

Equivalent variation in cost-benefit analysis and choice experiments

In a cost-benefit analysis of a health programme, the benefit side can be measured as the sum of equivalent variations across everyone affected, with losses counted as negative, and the net benefit is that sum minus the opportunity cost of resources. Using equivalent variation treats each person as entitled to the post-change position, which is a value judgement. HM Treasury's wellbeing guidance takes the opposite starting point. It states that compensating surplus is in general the preferred measure for appraisal because people are taken to have a right to the status quo, while it gives formulae for both measures based on the logarithm of income and notes that, under those formulae, equivalent surplus will always exceed compensating surplus for a gain.

In discrete choice experiments a standard welfare measure is the log-sum formula of Small and Rosen, set out on the compensating variation page. It assumes a constant marginal utility of income, so its result is both the compensating and the equivalent variation and cannot show the gap between them. When the marginal utility of income varies with income, more complex procedures are needed, and the two Hicksian measures then generally differ.

Worked example of the same health change seen from the other side

The figures below are illustrative and match the health example on the compensating variation page. A person has an annual income of £30,000 and utility $u = \ln Y + b h$, where $h$ is a health index and $b = 1$; the change is valued for one year.

1. A health gain. A programme raises $h$ from 0.60 to 0.75. 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 30,000 is income in pounds, 0.15 is $b$ multiplied by the change in the health index, and 4,855.03 is the least the person would accept, in pounds, to forgo the gain. The compensating variation for the same gain is £4,178.76, so willingness to accept exceeds willingness to pay by £676.27.

2. The same size of health loss. Now $h$ falls from 0.60 to 0.45. Solving $\ln(Y + EV) + b h^0 = \ln Y + b h^1$ again gives the following.

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

where the negative sign marks a loss and 4,178.76 is the most the person would pay, in pounds, to avoid it. The compensating variation of the loss is -£4,855.03, the least compensation the person would accept to bear it. The two measures swap values between a gain and a loss of the same size, which shows that equivalent variation is willingness to pay only when the change is a loss.

3. Ranking two treatments. Treatment A raises $h$ from 0.60 to 0.75 at no cost to the person. Treatment B raises $h$ from 0.60 to 0.90 but carries a copayment of £3,900, which is part of the change and reduces the person's income to £26,100. The utility gain over the status quo ($\ln Y + 0.60$) is 0.15 under A and $\ln(26100/30000) + 0.30 = -0.139262 + 0.30 = 0.160738$ under B, so the person prefers B. The equivalent variation of B is calculated as follows.

$$EV_B = 26100 \times e^{0.30} - 30000 = 26100 \times 1.3498588 - 30000 = 35231.31 - 30000 = 5231.31$$

where 26,100 is income in pounds after the copayment and 5,231.31 is the equivalent variation of B in pounds. The compensating variation of B is calculated as follows.

$$CV_B = 26100 - 30000 \times e^{-0.30} = 26100 - 30000 \times 0.7408182 = 26100 - 22224.55 = 3875.45$$

where 3,875.45 is the most, in pounds, the person would pay for B on top of the copayment.

4. Interpretation. Equivalent variation ranks B (£5,231.31) above A (£4,855.03), which matches the person's preference. Ranking this person's options by compensating variation contradicts their own preference, since it puts A (£4,178.76) above B (£3,875.45), even though a direct comparison shows the person would pay up to a further £278.76 (26,100 minus 30,000 × 0.8607080) to receive B rather than A. In a social appraisal the copayment would be a transfer to the health system rather than a resource cost, so the example shows the ranking property for one person, not a full cost-benefit result.

Common misreadings of equivalent variation

Frameworks that value health outcomes in QALYs do not use equivalent variation directly, although it rests on the same welfare-economic foundations as the money valuation of health in cost-benefit analysis. Where it is used, the same errors recur.

Equivalent variation for a gain is described as the most a person would pay for it, which is compensating variation. A discrete choice estimate is read as an equivalent variation distinct from the compensating variation, when the model's constant marginal utility of income makes them identical by assumption.

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