# 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.

## Model

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

${\displaystyle \log(c(u,p))=\alpha _{0}+\sum _{k}\alpha _{k}\log(p_{k})+{\frac {1}{2}}\sum _{k}\sum _{j}\gamma _{kj}^{*}\log(p_{k})\log(p_{j})+u\beta _{0}\prod _{k}p_{k}^{\beta _{k}}}$

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:

${\displaystyle w_{i}=\alpha _{i}+\sum _{j}\gamma _{ij}\log(p_{j})+\beta _{i}\log\{x/P\}}$

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

${\displaystyle \log(P)\equiv \alpha _{0}+\sum _{k}\alpha _{k}\log(p_{k})+{\frac {1}{2}}\sum _{k}\sum _{j}\gamma _{kj}\log(p_{k})\log(p_{j})}$.

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

• homogoeneous of degree 0 in prices and total expenditure
• sum of budget shares add up to 1 (i.e., ${\displaystyle \sum w_{i}=1}$)
• satisfy the symmetry of the Slutsky matrix

## Origin

The AIDS system is derived from the piglog model,[1] which allows researchers to treat aggregate consumer behavior as if it were the outcome of a single maximizing consumer.

## Applications

Many studies have used the AIDS system to determine the optimal allocation of expenditure among broad commodity groups, i.e., at high levels of commodity aggregation.

In addition, the AIDS system has been used as a brand demand system to determine optimal consumption rates for each brand using product category spending and brand prices alone. [2] Assuming weak separability of consumer preferences, the optimal allocation of expenditure among the brands of a given product category can be determined independently of the allocation of expenditure within other product categories.[3]

## References

1. ^ The Piglog Model USDA Web site
2. ^ Baltas, George (2002). "An Applied Analysis of Brand Demand Structure". Applied Economics. 34 (9): 1171–1175. doi:10.1080/00036840110085996.
3. ^ Thomas, R.L. (1987). Applied Demand Analysis. Essex: Longman.
• An Almost Ideal Demand System, Angus Deaton, John Muellbauer, The American Economic Review, Vol. 70, No. 3. (Jun., 1980), pp. 312–326. [1]