where is the wealth level, and are the old and new prices respectively, and and are the old and new utility levels respectively. The first equation can be interpreted as saying that, under the new price regime, the consumer would accept CV in exchange for allowing the change to occur.
More intuitively, the equation can be written using the value function, v(p,w):
one of the equivalent definitions of the CV.
Compensating variation is the metric behind Kaldor-Hicks efficiency; if the winners from a particular policy change can compensate the losers it is Kaldor-Hicks efficient, even if the compensation is not made.
Equivalent variation (EV) is a closely related measure that uses old prices and the new utility level. It measures the amount of money a consumer would pay to avoid a price change, before it happens. When the good is neither normal nor inferior, or when there are no income effects for the good, then EV (Equivalent variation) = CV (Compensating Variation) = CS (Consumer Surplus)
Assume a log-linear demand function for a product given by .
The compensating variation resulting from the introduction of this new product is
Assuming no income effect and no sales of the product prior to introduction , this simplifies to
For no income effect but previous products on the market at a different price,
In the case of online book sellers, Brynjolfsson, Hu, and Smith find that the compensating variation is quite large and mostly the result of a wider assortment of books being offered.
Brynjolfsson, E., Y. Hu, and M. Smith. "Consumer Surplus in the Digital Economy: Estimating the Value of Increased Product Variety at Online Booksellers," Management Science: 49, No. 1, November, pp. 1580-1596. 2003.