of other COLIs, each associated with a different level of living. Just as with basket price indexes for which, in principle, one can think about using any basket as the base, so too can one use any level of living to construct the COLI. This multiplicity of possible COLIs is often inconvenient, so that it is natural to ask in what circumstances the multiple indexes are the same. This turns out to be the case if the consumer behaves in accord with what are known as homothetic preferences. This is also the condition that is necessary for the Laspeyres to be at least as large as the Paasche whatever the prices may be, a result first established by Frisch (1936). When preferences are not homothetic, there will always be at least one level of living, somewhere between the reference and comparison levels, for which the COLI lies between the Paasche and the Laspeyres (Konüs, 1924). Homotheticity in preferences implies that the way the consumer ranks different bundles of goods is the same no matter what her level of living so that, for example, the rate at which a person is prepared to trade food for tobacco, or baseball tickets for opera tickets, is the same whether the person is rich or poor. Homotheticity also implies that, as people become better off, they simply scale up their purchases without changing the pattern of consumption. However, such behavior is inconsistent with more than a century of empirical evidence dating back to Engel, who showed that the share of food in the budget diminishes at higher levels of income. Because homothetic preferences are not a reasonable description of reality, one must acknowledge a multiplicity of cost-of-living indexes.
So far, we have introduced the concept of a COLI and presented the classic results about the relationship between the Laspeyres and Paasche indexes and the associated COLIs. By themselves, these arguments are of limited practical application. Although they explain the limits of basket price indexes for thinking about cost-of-living indexes or compensation, they tell nothing about how to calculate a cost-of-living index more accurately. For example, one might argue that compensating social security recipients according to a Laspeyres-based CPI ignores their ability to substitute in response to changes in relative prices and therefore overcompensates them. But, from the discussion so far, it is not clear that it is possible to do better without a direct way of observing the standard of living.
One approach to constructing better cost-of-living indexes is to find out more about how consumers respond to changes in prices and income, something that in principle is directly observable. Between the 1950s and the late 1970s, economists worked out theoretical and empirical procedures for measuring the standard of living, given a knowledge of consumer demand functions, the relationships that tell us how purchases depend on prices and income. In particular, if the demand functions are known, cost-of-living indexes can be calculated exactly. Here then is a possible procedure. Econometric methods can be used to estimate the demand functions from market data on each individual’s purchases, prices, and income and the results used to calculate any cost-of-living index numbers that one wants. While it is useful to know that this is possible, there are serious drawbacks to recommending such procedures for routine use in national