Food Prices and Overweight Patterns in Italy

of the paper empirically and can support policy interventions selectively. The rest
of the paper is devoted to applying this framework to one of the major demand
systems proposed in the literature, the AI model of Deaton and Muellbauer (1980).

3 Theoretical background

We briefly review the static AI model. The specification of this demand system
arises from a class of preferences in the logarithm of total expenditure, known as
’price independent generalized logarithmic’. It satisfies the necessary and sufficient
conditions for consistent aggregation across consumers (Muellbauer, 1976; Deaton
and Muellbauer, 1980) and allows the estimation of demand elasticities with limited
restrictions (Deaton, 1986).

It is assumed that there are n goods which can be purchased by consumers
and which can potentially be included in the demand system. We index these
categories of goods (and services) by i = 1.., n. We note that choosing the goods to
be included in the demand system depends on the purpose of the paper. Because we
are interested in the reaction of relative consumption between two macrocategories of
foods, i.e., healthy and unhealthy, this classification determines what sub-categories
of goods are included in the AI ; its dimensionality is of negligible interest.

Analytically, the budget share of a certain good is equal to the expenditure
generated by the good divided by the total expenditure of the categories of goods
included in the demand system. We use w_i to represent the budget share of a good
i, i = 1, ..., n. Under the AI specification, w_i takes the form of:

n

wi = αi +   γij log pj + βi log (Y/P) + νi                  (1)

j=1

where pj are j = 1, ..., n, are the prices of goods, Y is the total expenditure in the
demand system, P is an overall price index for the goods, ν_i is a stochastic error,
and α_i, β_i, and γ_ij are parameters to be estimated. Note that the last term of (1) is
based on the real expenditure (Y/P = y*) devoted to category w_i. The budget share
of product i increases as the total real expenditure of the category increases if βi is
positive, and decreases if βi is negative. The second term is based on the price effects
of the various goods. We will return to this point later, after introducing all the
ingredients of this flexible demand system to estimate the (cross) price elasticities
of demand and to identify the patterns of substitution between goods.

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