Almost Ideal Demand System (AIDS)

How does one model a demand system? In general, researchers only observe the equilibrium prices and quantities of goods over time. Changing prices or quantities could be due to shifts in either the demand or supply curve. Thus, modeling demand systems is difficult.

Deaton and Muellbauer (1980) propose one method: the Almost Ideal Demand System (AIDS). Today I will review this demand estimation strategy.

Origin

The origin of the AIDS system comes from the piglog model. Piglog models allow researchers to treat treat aggregate consumer behavior as if it were the outcome of a single maximizing
consumer. One must assume that in equilibrium, the marginal propensity to consume is the same across households.

One could add sophistication to the model by including parameters–k_h in the paper–which measure household size, age composition, and other household characteristics. In general, Deaton and Muellbauer assume that all the k_h parameters are equal to one; thus implying that all household have similar preferences.

Estimation

The AIDS estimations strategy is attractive because it is simple to estimate and–under certain assumptions–avoids the need for non-linear estimation. Further, it can also test homogeneity and symmetry restrictions through assumptions on the parameter values in the estimation.

Deaton and Muellbauer (1980) describe how one can begin with primitive, individual utility functions and aggregate them to form the following estimation framework:

• w_i=(α_i+β_iα₀) + Σ_j γ_ijlog p_j + β_i*{log x – Σ_kα_k *log p_k – .5*Σ_kΣ_j (γ_kj*log p_k*log p_j)}

In this equation, w_i represents the budget share of good i. The index j indexes the good. Prices are represented by p, total expenditures are represented by x, and w represents budget shares. One can estimate all parameters using a maximum likelihood methodology. In general, the following three restrictions are imposed to simplify estimation:

1. Adding up restriction: Σ_i α_i=1; Σ_i γ_ij=0; Σ_i β_i=0;
2. Homogeneity restriction: Σ_j γ_ij=0;
3. Slutsky Symmetry restriction: γ_ij=γ_ji

One can simplify this estimation in situations where prices are closely collinear. Instead of using an exact price index, P, one could calculate an approximate price index P^*. One candidate recommended by Deaton and Muellbauer is Stone’s (1953) index: log P^*=Σ_k (w_k*log p_k). If P^* is a good approximation, then one can use the following equation as an approximation of the full estimation above:

• w_i=(α_i+β_ilog φ) + Σ_j γ_ijlog p_j + β_ilog(x/P^*)

Testing Restrictions

In order to test the homogeneity restriction, one can leaves out a single p_j term and instead focus use relative prices (p_j/p_n).

• w_i = α_i^* +Σ_j=1^n-1 {γ_ij*log(p_j/p_n)} + β_ilog(x/P^*)

An F-ratios are calculated for each of the i equations to determine if the homogeneity restriction holds. Next the paper also gives the steps needed to test the symmetry restriction as well. In order to conduct the symmetry test, one must calculate the price index as follows:

• log P = α₀ + Σ_k α_k*log p_k + 0.5*Σ_kΣ_j (γ_kj*log p_k*log p_j)}

In order to calculate this “correct” price index, one must choose an appropriate value for α₀.

Source:

• Deaton and Muellbauer (1980) “An Almost Ideal Demand System” American Economic Review, v70(3):312- 326.