In economics, compensating variation (CV) is a measure of utility change introduced by John Hicks (1939). Compensating variation is the amount of additional money an agent would need to reach its initial utility. Compensating variation can be used to find the effect of a price change on an agent's net welfare. CV reflects new prices and the old utility level. It is often written using an expenditure function, e(p,u): This article or section does not cite its references or sources. ... For other persons named John Hicks, see John Hicks (disambiguation). ... In microeconomics, a consumers expenditure function describes how much money a consumer needs to be happy. ...

CV = e(p₁,u₀) − e(p₁,u₁)

= e(p₁,u₀) − w

= e(p₁,u₀) − e(p₀,u₀)

where w is the wealth level, p₀ and p₁ are the old and new prices respectively, and u₀ and u₁ 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):

v(p₁,w + CV) = u₀

e(p₁,v(p1,w + CV)) = e(p₁,u₀)

w + CV = e(p₁,u₀)

CV = e(p₁,u₀) − 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. Kaldor-Hicks efficiency is a type of economic efficiency that occurs only if the economic value of social resources is maximized. ...

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. For quasi-linear preference EV=CV=Consumer surplus, but this is not always true. Equivalent variation (EV) is a measure of how much more money a consumer would need before a price increase to be just as well off after the price increase. ...

Example of Adding a New Product

Assume a log-linear demand function for a product given by x(p,y) = Ap^αy^δ.

The compensating variation resulting from the introduction of this new product is

Assuming no income effect δ = 0 and no sales of the product prior to introduction p_n0x₀ = 0, 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.

References

Hicks, J.R. Value and capital: An inquiry into some fundamental principles of economic theory Oxford: Clarendon Press, 1939

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.

• Graphical explanation