Signature
EV_k = Y_k * exp(b * Delta_h_k) - Y_0
| Inputs | Definition | Unit |
|---|---|---|
Y_k | Income of the person under option k, after any payment the option requires | money per period, above zero |
b | Coefficient on health divided by the coefficient on log income, the same for every option | per unit of the health index |
Delta_h_k | Change in the health index under option k against the status quo, h_k less h_0 | units of the health index |
Y_0 | Income at the status quo, the same for every option | money per period, above zero |
EV_k | Equivalent variation of option k, measured at the status quo | 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 of an option from named income and health cells
Excel multiplies the named cell holding the option's income by the exponential of the health coefficient times the option's health change, and subtracts the status quo income.
=OptionIncome*EXP(HealthCoef*OptionHealthChange)-StatusQuoIncome
Assumptions
Copayment counted as part of the option's change
Any payment the option requires reduces the person's income under that option. In a social appraisal a copayment is a transfer to the health system rather than a resource cost, so the ranking describes one person's preferences, not a full cost-benefit result.
Same status quo for every option ranked
Every option is valued from the same income and health index, so their equivalent variations are comparable and differences between them follow the differences in utility.
Worked examples
Treatment raising the health index by 0.30 with a 3,900 pound copayment
In the article's example treatment B raises the health index from 0.60 to 0.90 and its £3,900 copayment cuts income from £30,000 to £26,100. Its equivalent variation is £5,231.31, above the £4,855.03 of treatment A, which raises the index by 0.15 at no cost, matching the person's preference for B.
Y_k = 26100; Y_0 = 30000; b = 1; Delta_h_k = 0.3; EV_k = 5231.31
Treatment raising the health index by 0.15 with no copayment
Treatment A leaves income at £30,000 and raises the health index by 0.15, so its equivalent variation is £4,855.03, the same as for the health gain alone.
Y_k = 30000; Y_0 = 30000; b = 1; Delta_h_k = 0.15; EV_k = 4855.03
Common errors
Ranking options by compensating variation instead of equivalent variation
Compensating variation puts treatment A (£4,178.76) above treatment B (£3,875.45), contradicting the person's preference: a direct comparison shows the person would pay up to a further £278.76 to receive B rather than A.
Leaving the copayment out of the option's income
Valuing treatment B at the status quo income of £30,000 gives an equivalent variation of about £10,495.76 instead of £5,231.31, overstating it by the £3,900 copayment multiplied by the exponential of 0.30 (computed here for illustration).
Sources
Hammond on equivalent variation as a welfare indicator
Hammond PJ. Money metric measures of individual and social welfare allowing for environmental externalities. In: Eichhorn W, editor. Models and Measurement of Welfare and Inequality. Berlin: Springer-Verlag; 1994. p. 694-724. In contrast to the compensating variation, the equivalent variation is higher for better changes from an existing status quo, so it can itself be used as a welfare indicator.
Baqaee and Burstein on equivalent variation as a money-metric
Baqaee DR, Burstein A. Welfare and output with income effects and taste shocks. NBER Working Paper 28754. Cambridge, MA: National Bureau of Economic Research; 2021, revised January 2022. Equivalent variation is adopted as the welfare measure because, unlike the compensating variation, it is a money-metric of utility under a common price system.
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