Functions & Formulae

Each applied formula has its own function page, with a signature, implementations, and tests.

Compensating variation function for health and choice-set changes

v(p_1, q_1, Y - CV) = v(p_0, q_0, Y); CV = Y * (1 - exp(-b * Delta_h)); CV = (log(sum exp(V1_j)) - log(sum exp(V0_j))) / alpha

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.

  • Compensating variation for a health change under log-income utility

    CV = Y * (1 - exp(-b * Delta_h))

    With utility equal to the natural log of income plus b times a health index h, the compensating variation of a change in h solves ln(Y less CV) plus b times h_1 equals ln Y plus b times h_0. Because utility is concave in income, the money value is not proportional to the health change: for a gain it is below the linear value Y times b times Delta_h, and for a loss its absolute value is above it. HM Treasury's wellbeing guidance gives the same form for the compensating surplus of a large change in life satisfaction, with b equal to the coefficient on the outcome divided by the coefficient on log income, and presents it as a sensitivity test for the high end of the range, not the central value.

  • Compensating variation from a multinomial logit model by the log-sum formula

    CV = (log(sum_(j=1)^J1 [exp(1)^V1_j]) - log(sum_(j=1)^J0 [exp(1)^V0_j])) / alpha

    Values a change in the attributes of the alternatives, or in the set of alternatives, in a discrete choice experiment or another logit model. Expected utility from the choice is the log of the sum of the exponentiated representative utilities, and the change in it is converted to money by dividing by the marginal utility of income, the negative of the coefficient on the cost attribute. Train attributes the closed form to Williams and to Small and Rosen; it holds for independent extreme value errors and utility linear in income, so it gives a single figure that is both the compensating and the equivalent variation, per person per choice occasion.