Almost ideal demand system

The Almost Ideal Demand System (AIDS) (1980) is a consumer demand model used primarily by economists to study consumer behavior. The AIDS model gives an arbitrary first-order approximation to any demand system and has many desirable qualities of demand systems. For instance it satisfies the axioms of order, aggregates over consumers without invoking parallel linear Engel curves, is consistent with budget constraints, and is simple to estimate.

The AIDS model is based on a first specification of a cost/expenditure function c(u,p):

where p stands for the prices of the L goods, and u the utility level. This specification satisfies homogeneity of order 1 in prices, and is a second order approximation of any cost function.

From this, demand equations are derived (using Shephard's lemma), but are however simpler to put in term of budget shares:

with x total expenditures, ${\displaystyle \gamma _{ij}=1/2(\gamma _{ij}^{*}+\gamma _{ji}^{*})}$, and P is a price index:

Under relevant constraints on the parameters ${\displaystyle \alpha ,\beta ,\gamma }$, These budget shares equations share the properties of a demand function:

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