Cost-of-living index

A cost-of-living index is entirely analogous to an input-price index, except that it measures the cost to a representative consumer associated with achieving a fixed level of utility from consumption—how does a consumer’s expenditure change from a change in price to keep them as well off as they were in period 0? The theory is exactly the same as an input-price index, except that outputs \(y_0\) are replaced with a measure of utility from consumption in period 0 in the expenditure function. That is, a cost of living index is \[\begin{align*} I^{C} = \frac{e(p_{t}, u_{0})}{e(p_{0}, u_{0})}, \end{align*}\] where \(u_{0}\) is the consumer’s utility from consumption in period 0. Other than a change in interpretation, the mechanics of a cost-of-living index are identical to an input-price index, and will not be repeated.