Signature
CV = (log(sum_(j=1)^J1 [exp(1)^V1_j]) - log(sum_(j=1)^J0 [exp(1)^V0_j])) / alpha
| Inputs | Definition | Unit |
|---|---|---|
V1_j | Representative utility of alternative j after the change | utility |
V0_j | Representative (observed) utility of alternative j before the change, including any opt-out | utility |
alpha | Marginal utility of income, the negative of the coefficient on the cost attribute | utility per unit of money, above zero |
CV | Money value of the change per person per choice occasion | money, positive for an improvement |
|---|
J0Number of alternatives available before the change (count)J1Number of alternatives available after the change, which may differ from J0 when an alternative is added or removed (count)
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
Log-sum compensating variation from named utility ranges
With the representative utilities before and after the change in ranges named UtilitiesBefore and UtilitiesAfter and alpha, entered as a positive number, in a cell named MargUtilIncome, Excel takes the difference in the logs of the sums of exponentials and divides by alpha. SUMPRODUCT makes the exponentials evaluate as arrays in any version.
=(LN(SUMPRODUCT(EXP(UtilitiesAfter)))-LN(SUMPRODUCT(EXP(UtilitiesBefore))))/MargUtilIncome
Assumptions
Logit errors and utility linear in income for the log-sum formula
The unobserved parts of utility are independent and identically distributed extreme value, and the marginal utility of income is constant, so the formula cannot separate compensating from equivalent variation.
Change small relative to income for the log-sum formula
Train notes that the formula can be used for changes that alter consumer surplus by small amounts per person relative to income, even though in reality the marginal utility of income varies with income. Lancsar and Savage and Ryan disagreed over when health studies should apply it, so the choice of model is a judgement for each study.
Worked examples
Log-sum value of improving a service against an opt-out
A choice between opting out (utility 0) and a service (utility 0.5) changes when the service improves to utility 0.9. With a cost coefficient of -0.01 per pound, alpha is 0.01 and the log-sum compensating variation is about £26.71 per person (computed here for illustration).
J0 = 2; V0_j = [0,0.5]; J1 = 2; V1_j = [0,0.9]; alpha = 0.01; CV = 26.71
Log-sum value of adding a third alternative to a choice set
Adding a new alternative with utility 0.3 to the original opt-out and service, with nothing else changed, gives about £41.19 per person (computed here for illustration).
J0 = 2; V0_j = [0,0.5]; J1 = 3; V1_j = [0,0.5,0.3]; alpha = 0.01; CV = 41.19
Common errors
Dividing the improved alternative's utility gain by alpha
In the first example the service's utility rises by 0.4, which divided by alpha gives £40 per person, about 50% above the log-sum value of £26.71, because about 38% of choosers took the opt-out before the change (computed here for illustration).
Using the cost coefficient itself as alpha in the log-sum formula
With the cost coefficient of -0.01 in the denominator the first example gives -£26.71, a loss for an improvement (computed here for illustration). Alpha is the negative of the cost coefficient.
Reading a log-sum result as a pure compensating variation
Because alpha is constant, the log-sum figure is the compensating and the equivalent variation at once and cannot show any gap between willingness to pay and willingness to accept.
Sources
Train on the log-sum formula for consumer surplus in logit models
Train KE. Discrete Choice Methods with Simulation. 2nd ed. Cambridge: Cambridge University Press; 2009. Chapter 3, section 3.5 (author's online edition): with independent extreme value errors and utility linear in income (Williams 1977; Small and Rosen 1981), the change in expected consumer surplus is one over alpha times the change in the log of the sum of exponentiated representative utilities; alpha is the negative of a price or cost coefficient; the number of alternatives can change; usable for changes small relative to income.
Lancsar and Savage on compensating variation from discrete choice experiments
Lancsar E, Savage E. Deriving welfare measures from discrete choice experiments: inconsistency between current methods and random utility and welfare theory. Health Economics. 2004;13(9):901-907. Abstract: methods then used in health economics were not consistent with random utility and welfare theory, and a compensating variation measure consistent with both is described.
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