values are assigned to u. More important is the concept of an indifference curve or indifference surface; this is a collection of q’s all of which yield the same value of the utility function. They are therefore bundles between which the consumer is indifferent. Higher indifference curves are those with a higher value of u and correspond to a higher standard of living.
The most useful concept for cost-of-living theory is the cost or expenditure function, which measures the least amount of money that the consumer would have to pay at specified prices to reach a specified indifference curve. We write this function as c(u,p) where, as before, p is the vector of prices, and u is some arbitrary label that identifies the indifference curve. Given that the consumer has a total x to spend, and given the assumption that she spends that money to do as well as possible, we can write
x= c(u,p ).
Note that this function also can be thought of as defining u, the standard of living, in terms of the prices p, and total expenditure x.
Cost-of-living index numbers are defined directly from the cost function. Suppose that the base period level of living is u0. The cost-of-living index number using base period level of living is the ratio of the costs of reaching the indifference curve u0 at the two sets of prices, p0 and p1. Hence,
is the cost-of-living analog to the Laspeyres index (1). Both indexes compare the current prices in the numerator with the base prices in the denominator. Because c(u0,p0) = x0 = p0·q0, the denominators of (1) and (18) are the same. However, the numerator of the Laspeyres is the cost of the base basket q0 evaluated at period t prices pt, while the numerator of the cost-of-living index is the minimum cost of obtaining the base period indifference curve at prices pt. If instead of the base indifference curve in (18), we use the current indifference curve, we get the cost-of-living index corresponding to the Paasche index, which is
Each of the two cost-of-living indexes (18) and (19) involves a counterfactual cost; in (18) it is the minimum cost of reaching u0 at prices pt, while in (19) it is the minimum cost of reaching ut at p0. Although we do not immediately know what these counterfactuals are, we can set limits on them. In particular, since one way of reaching u0 is to buy the original bundle q0, the minimum cost of reaching u0 at pt can be no larger than the cost of that bundle at the current prices, which is q0. pt. Similarly, one way of reaching ut at the original prices is to buy the bundle qt, so that the minimum cost of ut at p0 can be no larger than qt. p0. Hence, if we go back to the definition of the base period cost-of-living index (18) and note that the minimum cost of u0 at prices p0 is the actual expenditure q0. p0, we have