The Alrost Ideal Demand System

An alternative specification can avoid some of the problems

associated with the loglinear form, and usefully serve as a test of the sensitivity of the obtained results to demand specification. One reasonably flexible functional form is Deaton and Muellbauer's (1980b) Almost Ideal Demand System, or AIDS. This is the second model to be estimated.

In this model, the dependent variable takes the form of the budget share. Underlying the AIDS functional form is the Working-Leser semi- logarithmic Engel curve which has the value share of good i in total expenditure linearly related to the log of total expenditure. Thus on introducing a stochastic term the share of expenditure devoted to good i by household h (S^j^) is given by

for i=l,..,n, where a, P are parameters. It is easy to see that here, and consequently

in the AIDS demand equations, P->0 denotes a luxury, while points to a necessity or possibly an inferior good. The latter must also have the property that

Deaton and Muellbauer allow prices to shift the Engel curve, and manipulate the underlying cost function. The latter is a member of the PIGLOG (price independent generalized linear log) class of cost functions which permits exact aggregation over households without implying parallel linear Engel curves (see Muellbauer, 1975b, 1976). The following AIDS demand functions are generated:

where PIj^ is interpreted as a general price index (interpreted as the cost of zero utility^) and given by:

logPIj^ = ®o EfljlogP.^ + E E7^.1ogP^^logP.j^/2 (3.12) j ^ i j ^^ ^ ^

and all variables are defined as before. Note that any of the parameters a, p, 7 may themselves be functions of the demographic variables or other relevant variables.

AIDS is made fully consistent with demand theory through a series of linear restrictions on the parameters and the concavity restriction on the cost function. Namely,

Adding up: Homogeneity: Symmetry: Negativity: 1 = 0 Ea. = 1, ip^ i i

ll.. = 0

Slutsky matrix must be negative semi definite; i.e. all H nxn matrices with elements,

have only negative eigenvalues (where is the Kronecker delta).

In the present case of two goods with only the price of rice allowed to vary across the sample, the model collapses to:

SRj^ = a + 7logPRj^ + /?log(TEXj^/PIj^) + /i^ (3.13)

where SRj^ is the value of S^^ for i=rice, and similarly PRj^ is the price of rice faced by household h. The price index PI, is now defined to include the vector of other household characteristics giving:

where 6 = (5 , 5,,..., 5„) and Z, = (Z Note that in the

" -i- n on Ih nh

AIDS model, the demographics (and other household characteristics variables) denoted by the vector Z^^, could enter through any of the model's parameters (though consistency with aggregation may then require further parameter restrictions). For this study, Z^^ is assumed to only shift the intercept of the consumer's cost function in which case it enters into the price index as in (3.14).

Together, equations (3.13) and (3.14) provide the following equation with which to estimate demands,

SRj^ = a - pz^6 + ^logTEX^ + clogPR^ + d(logPRj^)^ + /ij^ (3.15)

where the estimated parameters are defined in terms of the original ones according to,

c = 7 - a/? d = -^7/2

In the two goods case the model can be estimated by linear

regression methods, which allows all parameters to be identified (i.e. from knowing the regression coefficients one can uniquely retrieve the a, P , 7 , and 6). This will not be true when more goods are accommodated unless a simplified price index is imposed in the place of (3.12) (see Deaton and Muellbauer, 1980). This will be more fully discussed in Chapter 6 when a three goods AIDS model is estimated.

Further, it is exactly integrable to an explicit indirect utility

function for many goods, such that the equivalent income function can be computed from such a demand system. Finally, the AIDS functional form allows the elasticities to vary across the sample as functions of the shares and prices. Differentiation of equation (3.15) easily confirms that the income elasticity of demand for rice is 1+/J/SR, while the own price elasticity of demand is -1+{c+2dlogPR)/SR. In the two goods case, the demand theory restriction which must be satisfied takes the form of the negativity condition of the compensated own price response. This elasticity is given by SR+^-l+(c+2dlogPR)/SR and must be negative over the range of the data.

3.5.4 Linear Demaind

As a test of whether the true rice demand relationship is indeed non linear as is assumed in both previous models, a linear demand model is also estimated. Linear demand can be specified as follows:

X.^ = 2. c + Ea. .PR.^ + b.TEX. + e.^ (3.16) ih h 3h 1 h ih

in obvious notation. This form provides the simplest possible

representation of the rice demand relationship. As for the AIDS but unlike the loglinear model, the elasticities vary according to their data evaluation points. The own price elasticity of demand is given by

aPR/X, while the income elasticity of demand for rice is bTEX/X. I will not, however, go further into the properties of the linear model as it is later rejected in favour of the alternatives discussed above.