Almost Ideal Demand System

Definition

The Almost Ideal Demand System (AIDS, Deaton-Muellbauer 1980) models budget shares $w_{it}$ — the fraction of total expenditure $x_t$ devoted to good $i$ — as a function of log prices and log real income:

$w_{it} = a_i + \sum_{j=1}^n b_{ij}\ln p_{jt} + c_i\ln\!\left(\frac{x_t}{P_t}\right) + u_{it}$

where $P_t = \sum_i w_{it}\ln p_{it}$ is the aggregate Stone price index (in the linear approximation, LA/AIDS), and $n-1$ equations are estimated (one is dropped by the adding-up restriction). The model arises from a flexible expenditure function and provides first-order approximations to any demand system consistent with utility maximization.

Key Ideas

Demand restrictions: Three conditions must hold for consistency with utility theory:
- Adding-up: $\sum_i a_i=1$ , $\sum_i b_{ij}=0$ , $\sum_i c_i=0$ — automatically satisfied by construction; use $n-1$ equations.
- Homogeneity: $\sum_j b_{ij}=0$ for each $i$ — price parameters in each equation sum to zero (no money illusion).
- Symmetry: $b_{ij}=b_{ji}$ for all $i\neq j$ — Slutsky symmetry.
SUR structure: The AIDS model is a Seemingly Unrelated Regression (SUR) with the same regressors in every equation, so the Zellner (1962) same-regressor result applies: Generalized Least Squares (GLS) and Ordinary Least Squares (OLS) coincide for $\beta$ estimation (though not for $\Sigma$ ). This makes conjugate Gibbs sampling straightforward.
Elasticities: Income elasticity $\eta_i=1+c_i/w_i$ ; own-price elasticity $\varepsilon_{ii}=-\delta_{ii}+b_{ii}/w_i-(c_i/w_i)[\sum_k w_k\ln p_k(\varepsilon_{ki}+\delta_{ki})]$ . Normal goods have $\eta_i>0$ ; necessities have $\eta_i<1$ ; luxuries have $\eta_i>1$ . Own-price elasticities should be negative (demand curves downward-sloping).
LA/AIDS linearization: The nonlinear translog price index $\ln P_t^*=\alpha_0+\sum_i\alpha_i\ln p_{it}+\tfrac{1}{2}\sum_i\sum_j b_{ij}\ln p_{it}\ln p_{jt}$ is replaced by the Stone index $\ln P_t=\sum_i w_{it}\ln p_{it}$ , saving parameters at the cost of potential inconsistency (Stone index involves $w_{it}$ which is also the dependent variable, introducing endogeneity).
Bayesian testing of restrictions: The Markov Chain Monte Carlo (MCMC) posterior directly yields credible intervals for elasticities and marginal likelihoods for restricted vs. unrestricted models. Bayes factors for homogeneity can be computed via Newton-Raftery harmonic mean, though this estimator has poor numerical properties in practice.

How It Works

Specify the system: Choose goods ( $n$ commodities); compute Stone price index; form the $n-1$ SUR design matrix $Z=\text{diag}(X^{(1)},\ldots,X^{(n-1)})$ with $X^{(i)}=[1,\ln p_{1t},\ldots,\ln p_{nt},\ln(x_t/P_t)]$ .
Estimate by SUR Gibbs sampler (Percy 1992; Chib-Greenberg 1995b): Since regressors are identical across equations, the Gibbs cycle alternates $\boldsymbol{\beta}|\Sigma\sim N$ and $\Sigma|\boldsymbol{\beta}\sim \mathrm{IW}$ (Inverse-Wishart).
Recover dropped equation: Use adding-up to derive coefficients for good $n$ as negatives of the sums over goods $1,\ldots,n-1$ .
Compute elasticities: Evaluate income and own/cross-price elasticities at sample mean budget shares $\bar{w}_i$ .
Test homogeneity: Estimate restricted model ( $\sum_j b_{ij}=0$ imposed) and unrestricted; compare via Likelihood Ratio (LR) statistic (frequentist) or Bayes factor (Bayesian).

Why It Matters

The AIDS model is the workhorse of demand system estimation in microeconomics and applied policy analysis. It is flexible enough to approximate any demand system while remaining tractable.
Casting AIDS as a SUR allows direct use of well-developed Bayesian SUR samplers (Percy 1992, Chib-Greenberg 1995b), providing credible intervals for elasticities — something frequentist SUR estimates do not readily deliver.
Testing homogeneity and symmetry restrictions is central to welfare and policy analysis (e.g., tax incidence, equivalence scales, trade policy). Bayesian Bayes factors provide a coherent alternative to classical LR tests.

Open Questions