Cost-of-living Index - Economic Theory

Economic Theory

The basis for the theory behind the cost of living index is attributed to Russian economist A.A. Konüs. The economic theory behind the cost of living index assumes that consumers are optimizers and get as much utility as possible from the money that they have to spend. These assumptions can be shown to lead to a "consumer's cost function", C(u,p), the cost of achieving utility level u given a set of prices p. Assuming that the cost function holds across time (i.e., people get the same amount of utility from one set of purchases in year as they would have buying the same set in a different year) leads to a "true cost of living index." The general form for Konüs's true cost of living index compares the consumer's cost function given the prices in one year with the consumer's cost function given the prices in a different year:

Since u can be defined as the utility received from a set of goods measured in quantity, q, u can be replaced with f(q) to produce a version of the true cost of living index that is based on price and quantities like most other price indices:

In simpler terms, the true cost of living index is the cost of achieving a certain level of utility (or standard of living) in one year relative to the cost of achieving the same level the next year.

Utility is not directly measurable, so the true cost of living index only serves as a theoretical ideal, not a practical price index formula. However, more practical formulas can be evaluated based on their relationship to the true cost of living index. One of the most commonly used formulas for consumer price indices, the Laspeyres price index, compares the cost of what a consumer bought in one time period (q0) with how much it would have cost to buy the same set of goods and services in a later period. Since the utility from q0 in the first year should be equal to the utility from q0 in the next year, Laspeyres gives the upper bound for the true cost of living index. Laspeyres only serves as an upper bound, because consumers could turn to substitute goods for those goods that have gotten more expensive and achieved the same level of utility from q0 for a lower cost. In contrast, a Paasche price index uses the cost of a set of goods purchased in one time period with the cost it would have taken to buy the same set of goods in an earlier time period. It can be shown that the Paasche is a lower bound for true cost of living index. Since upper and lower bounds of the true cost of living index can be found, respectively, through the Laspeyres and Paasche indices, the geometric average of the two, known as the Fisher price index, is a close approximation of the true cost of living index if the upper and lower bounds are not too far apart.