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*parameters are equal to one; thus implying that all household have similar preferences.

_{h}**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_{i}α_{0}) + Σ_{j}γ_{ij}log p_{j}+ β_{i}*{log x – Σ_{k}α_{k}*log p_{k}– .5*Σ_{k}Σ_{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:

*Adding up restriction*: Σ_{i}α_{i}=1; Σ_{i}γ_{ij}=0; Σ_{i}β_{i}=0;*Homogeneity restriction*: Σ_{j}γ_{ij}=0;*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*. If

_{k}(w_{k}*log p_{k})*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_{i}log φ) + Σ_{j}γ_{ij}log p_{j}+ β_{i}log(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_{i}log(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 = α
_{0}+ Σ_{k}α_{k}*log p_{k}+ 0.5*Σ_{k}Σ_{j}(γ_{kj}*log p_{k}*log p_{j})}

In order to calculate this “correct” price index, one must choose an appropriate value for α_{0}.

*Source*:

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

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