so that the base period cost-of-living index is always no larger than the Laspeyres. For the current period true cost-of-living index, the hypothetical cost is in the denominator, so that replacing it by something larger will make the result smaller. Hence, using (20) and this inequality, we have
so that the current period cost-of-living index is always at least as large as the Paasche price index .
There is an immediate link between each of these cost-of-living indexes and a measure of compensation. The amount of money that the consumer needs to reach the base level of living at the current prices is simply c(u0,pt) so that the (possibly negative) compensation that the consumer requires to make up for the price change from p0 to pt is given by
CV=c(u0,pt) - c(u0,p0).
This quantity is known as the compensating variation. It is the difference between the same two costs whose ratio is the cost-of-living index for the base level of living . We can also construct the equivalent variation, defined as the maximum amount of money that the consumer would have been prepared to pay in the base situation to avoid the price change from p0 to pt. It is
EV = c(ut,pt) - c(ut, p0)
and bears the same relationship to the cost-of-living index for the current period level of living ut as does the compensating variation to the cost-of-living index for the base period level of living u0. To illustrate how these measures work with the cost-of-living indexes, suppose that a consumer’s base level of total expenditures is x0 = c(u0,p0) and that we escalate this by the base period cost-of-living index (20). The new escalated total expenditure will be c(u0,pt), so that the escalation pays the compensating variation (22) and exactly compensates the consumer for the change in prices. If in the absence of the cost-of-living index, we escalate by the Laspeyres price index , the consumer will have at least as much as needed to remain as well off. If the object of policy is to ensure that compensation is adequate, and if it is better to compensate too much than to compensate too little, this would be an argument for the use of the Laspeyres price index for escalation.
In principle, we can construct a cost-of-living index around any level of living. We might write this arbitrarily based cost-of-living index in the form