aggregation explicitly? Second, if the conditions are satisfied, and one computes a COLI for a representative agent, what relationship does this COLI bear to the COLIs of the underlying consumers?
We begin with a definition. The representative consumer of consumer theory is not defined to be representative of tastes or of levels of living but of behavior. The economic theory of consumer behavior works with a single person who is assumed to be greedy (always wants more), to make consistent choices, and to do the best to satisfy his or her desires within a fixed budget at a fixed set of prices. As a result of doing the best he or she can, each person will have demand functions that relate the amount demanded of each good to income and to the prices of all the goods. Because different people have different tastes and live in different socioeconomic and physical environments, these demand functions will generally differ from person to person, even if they have the same level of income and face the same prices. For the economy as a whole, one can sum (or average) the demand functions of each person to get the aggregate (or average) demand for each good in the economy as a function of prices and the incomes of each person in society. The representative consumer exists if this aggregate can be thought of as coming from a single consumer whose behavior replicates the aggregate of all consumers. Note again that this definition is in terms of behavior, not tastes or welfare.
In general, this construction cannot work, and for a very obvious reason. According to the theory of individual behavior, demand is a function of income and all the prices. But aggregate demand is a function of all the incomes and all the prices. The distribution of income between people matters for the aggregate but has no part in the theory of individual behavior. The representative consumer demands goods according to her representative income and the prices, and there is generally no way in which all the possible effects of the distribution of income can be captured through a single representative income. So the representative consumer cannot exist in general.
There are special cases where the distribution of income has no effect on aggregate demands: when each consumer spends his or her marginal dollar in exactly the same way. If so, taking a dollar from A will shrink A’s demands for goods in exactly the same way as giving the dollar to B will expand B’s demands for goods. As a result, the total demand for each good in the economy depends only on total income, not on who owns that income. Total income will work as representative income, and the representative agent exists.
How realistic is this condition? For it to hold exactly is clearly absurd; no one would seriously claim that everyone in the economy spends an additional dollar in exactly the same way so that, at the margin, all consumers are identical. A more serious question is whether, at the level of aggregation in the CPI (about