Chapter 4 The Theory of Market Demand
The theory developed and explained in the previous chapters was for individual households. If we want to use microeconomic theory for “aggregates of households”, then there have to be “necessary conditions under which it is possible to treat aggregate consumer behavior as if it were the outcome of the decisions of a single maximizing behavior”.
The authors note that there is also a view that microeconomic theory has greater relevance for aggregate data, arguing on largely intuitive grounds that the variations in circumstances of individual households average out to negligible proportions in aggregate, leaving only the systematic effetcs of variations in prices and budgets (Hicks (1956)).
The authors introduce some doubt on replicating microeconomic foundations with macroeconomic relations. They say that …For example, most goods are not bought by all consumers, although the proportion purchasing any one good can be expected to rise as its price falls. Consequently, a decrease in price not only causes individuals who already buy the good to buy more, but also causes new consumers to purchase for the first time. A correct treatment requires that both these effects be adequately modeled, but this is impossible if aggregate demand is treated as coming from a “representative” consumer who buys some of all of the goods.
I will follow Deaton and Muellbauer (1980, AER) for the technical details of elasticity calculation from aggregate demand.
The budget share equation explained in the previous chapter can be generalized to describe behavior of an individual household \(h\):
\[\begin{equation} w_{ih}=\alpha_{i} + \sum_{j}\gamma_{ij}logp_{j} + \beta_{i}log(x_{h}/k_{h}P). \end{equation}\]The authors interpret the parameter \(k_{h}\) “as a sophisticated measure of household size which, in principle, could take account of age composition, other household characteristics, and economies of household size; and which is used to deflate the budget \(x_{h}\) to bring it to a ‘needs corrected’ per capita level. This allows a limited amount of taste variation across hosueholds.”
Then, the share of aggregate expenditure on good \(i\) in the aggregate budget of all households, denoted \(\bar{w}_{i}\) is given by
\[\begin{equation} \sum_{h}p_{i}q_{ih}/\sum x_{h} \equiv \sum_{h}x_{h}w_{ih}/\sum x_{h}. \end{equation}\]Substituting this into above equation will enable us to have the equation at the average level of total expenditure \(x_{h}\):
\[\begin{equation} \bar{w}_{i}=\alpha_{i} + \sum_{j}\gamma_{ij}logp_{j} + \beta_{i}log(\bar{x}/kP). \end{equation}\]The authors states that “…under these assumptions aggregate budget shares correspond to the decisions of a rational representative household whose preferences are given by the AIDS cost function and whose budget is given by \(\bar{x}/k\), the ‘representative budget level’”.
For further note on \(k_{h}\), the authors state that “when \(k_{h}\) differs across households, for example, because of differences in household composition, the index \(k\) reflects not only the distribution of budgets but the demographic structure.”
“When the distribution of household budgets and household characteristics is invariant except for equiproportional changes in household budgets, \(k\) is constant. In this case, there is considerable extra scope for taste variations in the individual demand functions without altering the validity of the representative consumer hypothesis embodied in the above equation.”