Demographic Challenges to Regional Development - Regional Science Matters - page 186

A modified AIDS cost function can be written as:

2 γ ii ln P i ln P i þγ ii ln P i ln P i þγ ii ln P i ln P i þ

1

ln CU ð ; P Þ α 0 þα i ln P i þα i ln P i þ

ʳ ii ln P i ln P i þʲ 0 P ʲ i

i P ʲ i

i

ð 11 : 3 Þ

*
Applying Shepherd's lemma to this expenditure function, the function can be
described as:

γ ii ¼γ i * ,

*

where γ i i ¼γ i i ¼γ ii

ln CU ð ; P Þ P i Q i

CU

w i ¼ ∂ ʲ 0 P ʲ i P ʲ i

¼

Þ ¼ α i þ ʳ ii ln P i þ ʳ ii ln P iþ ʲ i U

ð 11 : 4 Þ

i

ln P i ð ; P

∂

For a utility-maximizing consumer, total expenditure X is equal to C(U, P) and
this equality can be inverted to give U as a function of P and X , the indirect utility.
Equation ( 11.4 ) can be rewritten as the AIDS demand functions in budget share
form:

w i ¼ α i þ ʳ ii ln P i þ ʳ ii ln P i þ ʲ i ln X ð = P Þ ð 11 : 5 Þ

where w i is the budget share of the good i for the household, P i is the price of good i ,
and (X/P) is the total expenditure on all goods and services in real terms. A price
index P is defined by

2 γ ii ln P i ln P i þ γ ii ln P i ln P i þ γ ii ln P i ln P i þ

1

ln P ¼ α 0 þ α i ln P i þ α i ln P i þ

þγ ii ln P i ln P i

ð 11 : 6 Þ

1
2

γ ii þ γ ii

where ʳ ii ¼

Since the price index P is defined as (Eq. 11.6 ), the AIDS model is non-linear.
Deaton and Muellbauer ( 1980 ) suggested the real price index P can be replaced by
the Stone price index P to transform the AIDS into a linear one.

= P w i ¼ α i þ ʳ ii ln P i þ ʳ ii ln P i þ ʲ i ln X ð 11 : 7 Þ

ln P ¼ w i ln P i þ w i ln P i ð 11 : 8 Þ

The AIDS with the Stone price index is called 'Linear Approximate Almost
Ideal Demand' (LA/AIDS), employed by many researchers because of it is more
convenient to estimate. To be consistent with consumption theory, the model
should require the following conditions.