Aggregating non-linear consumer demands: A maximum entropy approach

The analysis of consumer demands is limited by data availability. Typically, aggregate consumption and expenditure data are all that is available to draw conclusions based on the theory of individual consumer behavior. Such limitations are problematic as some demand systems possess non-linear Engel effects. Non-linear Engel effects imply differences in demand responses across heterogeneous individuals. These effects may invalidate arguments justifying estimation of per capita demands based on aggregate data. Given that disaggregate data are frequently not available, assumptions are employed to restrict individual demands to obtain empirically tractable aggregate demand functions. However, these restrictions often exclude the possibility of non-linear Engel effects. The objective of this study is to develop a numerical scheme for dealing with the aggregation of consumer demands. Rather than directly estimating an aggregate demand model whose form and restrictions are inherited from the form and restrictions of individual demands, unobservable disaggregate demands and expenditure distributions are recovered in a manner consistent with observable information. This is accomplished using maximum entropy and traditional econometric methods to simultaneously estimate parameters of a demand system, and recover the unobserved expenditure distributions and disaggregate demands. Key to the recovery process are conditions ensuring the recovered distributions reflect known information regarding the actual distribution, and that recovered disaggregate demands add up to the known level of aggregate demand. This scheme is used to model international consumption patterns with an implicit, directly additive demand system (AIDADS). AIDADS is desirable as it has potentially non-linear Engel effects. However, this non-linearity limits it usefulness in directly modeling aggregate demands. Nevertheless, when account is taken of the distribution of expenditure via the numerical aggregation scheme, results are more plausible than when demands are modeled using aggregate data alone. Modeling disaggregate demand rather aggregate demand allows for estimation of more meaningful subsistence levels for consumption of individual goods such as food, clothing and shelter. Consequences of estimating AIDADS based on country-level cross sectional data versus a disaggregate approach that recognizes there is a distribution of expenditure within each country are evaluated via a demand projections exercise