Food Prices and Overweight Patterns in Italy

We then define wi* as the ’’optimal” level of the observable expenditure budget
share wi for commodity i and log P = αo+∑n=₁ αk logpk+2 ∑,'n=_l ∑n=₁ Ykj logPk logPj.
As commonly done in empirical papers, we also employ a linear approximation in
this price index, i.e., Stone’s price index, defined as log P = ∑n=₁ wi log pi.

As discussed in the review by Barnett and Serletis (2008), the AI flexible model
has a number of desirable properties. It derives an expenditure function from a
second-order approximation to any expenditure function and provides the possibility
of including the theoretical restrictions of adding up, homogeneity and symmetry
in order to respect the predictions of the demand theory. Because the expenditure
function must be linearly homogeneous and strictly increasing in P, adding up and
homogeneity can be explained as ∑n=₁ αi = 1 and ∑n=₁ γ_ij∙ = ∑ⁿ=₁ Yij = ∑X₁ βi = 0,
respectively, while symmetry requires γij = γji for all i, j .

Another property of a robust expenditure function (and demand system) is that
it must be concave in prices. This means that the matrix of the second-cross partial
derivatives must be negative semi-definite. In turn, this property gives rise to the
matrix of the substitution effects of Slustky, Sij = ∂hi(p,u)/∂pj, with non-positive
own-price effects , where h_i (.) is Hicksian demand. Formally, it can be shown that
the Slutsky substitution coefficients of model (1) are given as:

y

Sij =----[γij + WiWj - δijwi]                           (2)

PiPj

where δ_ij is the Kronecker parameter (δ_ij = 1 if i = j, and δ_ij = 0 if i = j).
The matrix of substitution effects for the AI model varying with data determines
that negativity conditions must be evaluated (and eventually imposed) locally at
a specific point in the sample⁵ . That is, by scaling the data at a representative
point (e.g. the mean of the sample) in which P = y* = 1, we can obtain the local
substitution term θ_ij∙ = Sij(P = y* = 1)6.

Equation (1) is singular by construction, as the expenditure shares sum to 1. A
frequently employed procedure to avoid econometric problems consists of dropping
one equation from the system and, although the budget share demand system with
a n - 1 rank must be empirically confirmed, it provides complete characterisation
of consumer preferences. Consequently, it can be used to estimate the income, own-
and cross-price elasticities as well as the elasticities of substitution.

⁵ See Cranfield and Pellow (2004) for a more thorough discussion of the role of global and local negativity in
functional form selection.

⁶ In order to remark the properties of the demand system, the AI provide a reasonablly accurate approximation
at any set of prices not too far from the point of approximation.

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