Producer Price Index Manual: Theory and Practice ... - METAC

22. Treatment of Seasonal Products
proximations to them can be constructed. 11 For
each month m = 1, 2,...,12, let S(m) denote the set
of products that are available for purchase in each
year t = 0, 1,...,T. For t = 0, 1,...,T and m = 1,
2,...,12, let p t,m n and q t,m n denote the price and
quantity of product n that is available in month m
of year t for n belongs to S(m). Let p t,m and q t,m
denote the month m and year t price and quantity
vectors, respectively. Then the year-over-year
monthly Laspeyres, Paasche, and Fisher indices
going from month m of year t to month m of year
t+1 can be defined as follows:
tm , t+
1, m tm ,
(22.1) PL
( p , p , q )
m = 1, 2,...12;
t, m t+ 1, m t+
1, m
(22.2) P ( p , p , q )
m = 1,,2,...12;
=
∑
n∈
S( m)
=
∑
n∈
S( m)
∑
p
q
t+
1, m t,
m
n n
p
n∈
S( m)
∑
n∈
S( m)
q
tm , tm ,
n n
p
q
;
t+ 1, m t+
1, m
n n
p
q
tm , t+
1, m
n n
tm , t+ 1, m tm , t+
1, m
(22.3) PF
( p , p , q , q )
t, m t+ 1, m t, m t, m t+ 1, m t+
1, m
PL( p , p , q ) P ( p , p , q )
≡ ;
m = 1, 2,...,12.
22.20 The above formulas can be rewritten in
price relative and monthly revenue share form as
follows:
tm , t+ 1, m tm , tm , t+
1, m tm ,
(22.4) PL( p , p , s ) sn ( pn pn
)
m = 1, 2,...12;
t, m t+ 1, m t+
1, m
(22.5) P ( p , p , s )
m = 1, 2,...12;
= ∑ ;
n∈S( m)
−1
∑
t 1, m t 1, m t,
m
−1⎤
+ +
sn ( pn pn
) ;
n∈S( m)
⎡
= ⎢ ⎥
⎢⎣
⎥⎦
11 Diewert (1996b, pp. 17–19; 1999a, p. 50) noted various
separability restrictions on purchaser preferences that
would justify these year-over-year monthly indices from
the viewpoint of the economic approach to index number
theory.
;
tm , t+ 1, m tm , t+
1, m
(22.6) PF
( p , p , s , s )
tm , t+
1, m tm ,
t, m t+ 1, m t+
1, m
≡ PL
( p , p , s ) P ( p , p , s )
m = 1, 2,...,12
;
−1
−1
⎡
⎤
tm , t+ 1, m tm , tm , t+
1, m tm ,
= ∑ sn ( pn pn ) ⎢ ∑ sn ( pn pn
) ⎥
n∈S( m)
⎢⎣n∈S ( m)
⎥⎦
where the monthly revenue share for product
n∈S(m) for month m in year t is defined as:
(22.7)
s
p q
= ; ; m = 1, 2,...,12 ;
∑
tm , tm ,
tm ,
n n
n tm , tm ,
pi
qi
i∈S( m)
n∈S(m) ; t = 0,1,...,T;
and s t,m denotes the vector of month m expenditure
shares in year t, [s n t,m ] for n∈S(m).
22.21 Current period revenue shares s n t,m are not
likely to be available. As a consequence, it will be
necessary to approximate these shares using the
corresponding revenue shares from a base year 0.
22.22 Use the base period monthly revenue
share vectors s 0,m in place of the vector of month m
and year t expenditure shares s t,m in equation (22.4)
and use the base period monthly expenditure share
vectors s 0,m in place of the vector of month m and
year t + 1 revenue shares s t+1,m in equation (22.5).
Similarly, replace the share vectors s t,m and s t+1,m in
equation (22.6) with the base period expenditure
share vector for month m, s 0,m . The resulting approximate
year-over-year monthly Laspeyres,
Paasche, and Fisher indices are defined by equations
(22.8)–(22.10) below: 12
12 If the monthly revenue shares for the base year, s n 0,m ,
are all equal, then the approximate Fisher index defined by
(22.10) reduces to Fisher’s (1922, p. 472) formula 101.
Fisher (1922, p. 211) observed that this index was empirically
very close to the unweighted geometric mean of the
price relatives, while Dalén (1992a, p. 143) and Diewert
(1995a, p. 29) showed analytically that these two indices
approximated each other to the second order. The equally
weighted version of equation (22.10) was recommended as
an elementary index by Carruthers, Sellwood, and Ward
(1980, p. 25) and Dalén (1992a p. 140).
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