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X
i ʱ i ¼
1, X
i
0, X
ʳ ij ¼
i ʲ i ¼
0 adding up
ð
Þ
ð
11
:
9
Þ
X
j γ ij ¼
0 homogeneity
ð
Þ
ð
11
:
10
Þ
γ ij ¼ γ ji symmetry
ð
Þ
ð
:
Þ
11
11
Since homogeneity implies
ʳ ii ¼ʳ i i , (Eq. 11.7 ) can be re-written as
= P
w i ¼ α i þ ʳ ii ln P i =
ð
P i
Þ ʲ i ln X
ð
11
:
12
Þ
Since the modified AIDS is flexible, it is not guaranteed to satisfy the homoge-
neity and symmetry conditions. So these conditions are introduced as parameter
restrictions in the estimation process. The modified AIDS assumes that the size of
the family affects budget share and N , the number of household members, was
introduced as a shift parameter of
α i . Then the model can be expressed as:
þ
w it ¼ α i þ
e i N t þ ʳ ii ln P it =
Þ þ ʲ i ln X t = P t
e i
ð
P it
ð
11
:
13
Þ
where e i k is the error term and i and k refer to ten consumption types and six ages of
reference person.
w it ¼ α i þ
e i N t þ ʳ ii ln P it =
Þ þ ʲ i ln X t = P t
e i
ð
P it
þ
ð
11
:
14
Þ
where e i r is the error term and i and r refer to ten consumption types and five
quintiles of income.
The empirical results of (Eq. 11.13 ) show life cycle changes in consumption
behavior (a sample is provided in Fig. 11.4 ). From the estimation results (Eq. 11.13 )
and (Eq. 11.14 ), the price elasticity
ʵ it j and expenditure elasticity
ʷ it j are:
it ¼ ʳ ii
j
w it
ʵ
w it þ
1 for j
¼
k age of reference person
ð
Þ
, r quintiles of income
ð
Þ
ð
11
:
15
Þ
it ¼ ʲ i
j
ʷ
w it þ
1 for j
¼
k age of reference person
ð
Þ
, r quintiles of income
ð
Þ
ð
11
:
16
Þ
11.2.2 The AIDS-Type Chicago Region Econometric Input-Output
Model
To predict the demographic changes in the Chicago region up to 2030, this research
extends the Chicago Region Econometric Input-Output Model (CREIM). The
regional econometric input-output model (REIM), initially designed by Conway
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