Relevant Quantitative Analysis

Microeconomic Analysis

An important contribution to demand analysis was Stone’s work on the estimation of the linear expenditure system (LES) (Stone 1954b). One interesting feature of the LES is that it takes account of the fact that in allocating their expenditure consumers have prior commitments to meet, such as paying the mortgage, buying food, and paying utility bills, before responding to prices and allocating the rest of their incomes. Stone represented this situation by the equation

where p and q represent price and quantity vectors, respectively. In Equation (5.1), q corresponds to the quantities that consumers have to buy, b is a vector of constants that sum to unity, and m ° p'q is total expenditure. Equation (5.1) may be rewritten as

where i is the unit vector, I is the unit matrix, and c = —q .

Deaton (1998: 600) gave the following evaluation of Stone’s work:

Stone’s achievement lay not in deriving the demand functions, but in thinking to estimate them. The demand functions [Equation 5.1 above], even if fitted to the data by least-squares, require non-linear optimization, and Stone invented a simple and not very efficient scheme, but one that allowed him to obtain parameter estimates and a good fit to interwar British data for a six commodity disaggregation of expenditures. This was a major breakthrough, not only in demand analysis, but also in applied econometrics in general. Indeed, much of demand analysis for a decade or so after Stone’s paper consisted of applying better algorithms and faster computers to the fitting of Stone’s model to different data sets.

Stone’s solution to the non-linearity problem was to derive some parameter estimates from a simpler, restricted version of the model and then substitute these estimates into the full version of the model to estimate the remaining parameters and to iterate this process until stable values of the parameter estimates were obtained.