Signature
EV = Y * (exp(b * Delta_h) - 1)
| Inputs | Definition | Unit |
|---|---|---|
Y | Income of the person without the change for the period valued | money per period, above zero |
b | Utility weight on the health index relative to log income: the coefficient on health divided by the coefficient on log income | per unit of the health index |
Delta_h | Change in the health index, h_1 less h_0, positive for a gain | units of the health index |
EV | Least the person would accept to forgo a gain (positive), or minus the most the person would pay to avoid a loss (negative) | money per person for the period valued |
|---|
Function
Equivalent variation function for health and income changes measured at the status quo
Maps a change from the status quo to the equivalent variation: the change in income, without the change, that would leave the person as well off as the change would. For a gain it is the least the person would accept to forgo the gain, and for a loss the most the person would pay to avoid it. Because every option is valued at the same status quo prices and quantities, equivalent variation ranks several options in the same order as the person's utility. The general expenditure-function definition is given on the Welfare Economics formula page; these records give the closed forms under a log-income utility function, and the log-sum formula of a logit model, HE-FM-CVAR-002, gives both Hicksian measures at once. The records follow the notation of the Equivalent Variation article.
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Implementations
Excel
Equivalent variation from named income, coefficient and health change cells
Excel multiplies the named cell holding income by the exponential of the health coefficient times the health change, less one.
=Income*(EXP(HealthCoef*HealthChange)-1)
Assumptions
Utility additive in log income and health, valued at the status quo
Utility is ln Y plus b times h, and the money is measured at the status quo, without the change, with utility held at its level after the change. Under this utility the money value rises with income, so the equivalent variation is at least the compensating variation, as Weber shows for any change whose money value does not fall as income rises.
Health state not purchasable by the person
The person cannot buy the new health state, so the measure is the quantity case often called the equivalent surplus, valued for one period at the stated income.
Worked examples
Equivalent variation of a gain from 0.60 to 0.75 in a health index
With an annual income of £30,000 and b equal to 1, a programme that raises the health index from 0.60 to 0.75 has an equivalent variation of £4,855.03, the least the person would accept to forgo the gain and £676.27 more than its compensating variation.
Y = 30000; b = 1; Delta_h = 0.15; EV = 4855.03
Equivalent variation of a loss from 0.60 to 0.45 in a health index
A loss of the same size has an equivalent variation of -£4,178.76: the person would pay up to £4,178.76 to avoid it.
Y = 30000; b = 1; Delta_h = -0.15; EV = -4178.76
Common errors
Describing the equivalent variation of a gain as willingness to pay
For the gain of 0.15, £4,855.03 is the least the person would accept to forgo it. The most the person would pay for it is the compensating variation, £4,178.76; equivalent variation is willingness to pay only for a loss.
Using the linear value in place of the equivalent variation of a large change
Valuing the gain of 0.15 at income times b times the change gives £4,500, £355.03 or about 7.3% below the equivalent variation of £4,855.03 (computed here for illustration).
Expecting a logit log-sum value to show the equivalent variation gap
A log-sum figure from a discrete choice model assumes a constant marginal utility of income, so it is the equivalent and compensating variation at once and cannot reproduce a gap such as the £676.27 in this example.
Sources
Treasury ln(income) formula for the equivalent surplus of a large change
HM Treasury. Wellbeing Guidance for Appraisal: Supplementary Green Book Guidance. London: HM Treasury; July 2021, last updated November 2022. Annex 2, ln(income) approach: the equivalent surplus of a change in an outcome as average income multiplied by the exponential of the outcome coefficient times the change divided by the coefficient of log income, less one; for an increase the equivalent surplus always exceeds the compensating surplus, and an equivalent surplus of a loss is a willingness to pay to avoid it.
Weber on equivalent variation exceeding compensating variation
Weber TA. Hicksian welfare measures and the normative endowment effect. American Economic Journal: Microeconomics. 2010;2(4):171-194. The equivalent variation measures willingness to accept to reverse a welfare change; income monotonicity implies that the equivalent variation exceeds the compensating variation.
Canonical Identity
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