Signature
CV = Y * (1 - exp(-b * Delta_h))
| Inputs | Definition | Unit |
|---|---|---|
Y | Income of the person at the status quo 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 |
CV | Most the person would pay for a gain (positive), or minus the least compensation the person would accept for a loss (negative) | money per person for the period valued |
|---|
Function
Compensating variation function for health and choice-set changes
Maps a change from the status quo, in prices, in a non-market quantity such as a health state, or in the alternatives a person can choose from, to the compensating variation: the income that must be taken away after a gain, or given after a loss, to leave the person exactly as well off as before the change. Under the article's sign convention it is positive for a gain, when it is the most the person would pay, and negative for a loss, when its absolute value is the least compensation the person would accept. The general expenditure-function definition is given on the Welfare Economics formula page; these records give the closed forms used with a log-income utility function and with a multinomial logit choice model. The records follow the notation of the Compensating Variation article.
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Implementations
Excel
Compensating variation from named income, coefficient and health change cells
Excel multiplies the named cell holding income by one less the exponential of minus the health coefficient times the health change.
=Income*(1-EXP(-HealthCoef*HealthChange))
Assumptions
Utility additive in log income and health
Utility is ln Y plus b times h, so the marginal utility of income falls as income rises and the compensating and equivalent variations differ. Weber shows that the two Hicksian measures coincide at every income only when indirect utility is additively separable in income, as with utility linear in income.
Health state fixed for the person after the change
The person cannot buy more or less of the new health state, the quantity case often called the compensating surplus, and the valuation covers one period at the stated income.
Worked examples
Compensating 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 a compensating variation of £4,178.76, the most the person would pay for the gain.
Y = 30000; b = 1; Delta_h = 0.15; CV = 4178.76
Compensating variation of a loss from 0.60 to 0.45 in a health index
A loss of the same size has a compensating variation of -£4,855.03 (as in the Equivalent Variation article's loss example): the person would need at least £4,855.03 to accept it.
Y = 30000; b = 1; Delta_h = -0.15; CV = -4855.03
Common errors
Reporting willingness to pay to avoid a loss as its compensating variation
For the loss of 0.15 the compensating variation is -£4,855.03 (as in the Equivalent Variation article's loss example), the least compensation the person would accept. Reporting £4,178.76, the most the person would pay to avoid the loss, as its compensating variation mixes in the equivalent variation.
Using the linear value for a large health change
Valuing the gain of 0.15 at income times b times the change gives £4,500, about 7.7% above the compensating variation of £4,178.76 and about 7.3% below the equivalent variation of £4,855.03 (computed here for illustration).
Ranking several options by compensating variation
In the Equivalent Variation article's two-treatment example, compensating variation puts treatment A (£4,178.76) above treatment B with a £3,900 copayment (£3,875.45), although the person prefers B. Each compensating variation is measured from its own option's position; equivalent variation (HE-FM-EVAR-002) ranks the options as the person does.
Sources
Treasury ln(income) formula for the compensating 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 compensating surplus of a change in an outcome as average income multiplied by one minus the exponential of minus the outcome coefficient times the change divided by the coefficient of log income; compensating surplus as the preferred measure for appraisal, with the formula as a sensitivity test for the high end of the range.
Weber on compensating variation, willingness to pay and income effects
Weber TA. Hicksian welfare measures and the normative endowment effect. American Economic Journal: Microeconomics. 2010;2(4):171-194. The compensating variation corresponds to willingness to pay to bring a welfare change about; both Hicksian measures are independent of income only when indirect utility is additively separable in income.
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