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# Compensating variation

In economics, compensating variation (CV) is a measure of utility change introduced by John Hicks (1939). 'Compensating variation' refers to the amount of additional money an agent would need to reach its initial utility after a change in prices, or a change in product quality, or the introduction of new products. 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):

$CV = e\left(p_1, u_1\right) - e\left(p_1, u_0\right)$

$= w - e\left(p_1, u_0\right)$
$= e\left(p_0, u_0\right) - e\left(p_1, u_0\right)$

where $w$ is the wealth level, $p_0$ and $p_1$ are the old and new prices respectively, and $u_0$ and $u_1$ 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):

$v\left(p_1,w-CV\right)=u_0$

$e\left(p_1,v\left(p1,w-CV\right)\right)=e\left(p_1,u_0\right)$
$w-CV=e\left(p_1,u_0\right)$
$CV=w-e\left(p_1,u_0\right)$

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)

## Example of Adding a New Product

Assume a log-linear demand function for a product given by $x\left(p,y\right)=Ap^\left\{alpha\right\}y^\left\{delta\right\}$.

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

$CV = left\left[\left\{fracy^\left\{ - delta \right\} \left(p_\left\{n_\left\{0\right\}\right\} x_0 - p_\left\{n_\left\{1\right\}\right\} x_1 \right) + y^\left\{\left(1 - delta \right)\right\} \right\} right\right]^\left\{1/\left(1 - delta \right)\right\} - y.$

Assuming no income effect $\left\{ delta \right\} = 0$ and no sales of the product prior to introduction $p_\left\{n_\left\{0\right\}\right\}x_\left\{0\right\} = 0$, this simplifies to

$CV = -frac x_1 \right\}\right\}.$

For no income effect but previous products on the market at a different price,

$CV = -frac\left\{\left\{p_\left\{1\right\} x_1 -p_\left\{0\right\} x_0 \right\}\right\}.$

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.