Compensating variation in the context of "Cost-of-living index"

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 their initial utility after a change in prices, 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):

where ${\displaystyle w_{1}}$ is the new wealth level, ${\displaystyle p_{0}}$ and ${\displaystyle p_{1}}$ are the old and new prices respectively, and ${\displaystyle u_{0}}$ and ${\displaystyle u_{1}}$ 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 a substraction of CV in exchange for allowing the change to occur.