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

Producer Price Index Manual
B.2 The Young index
1.42 Instead of holding constant the quantities
of period b, a statistical office may calculate a PPI
as a weighted arithmetic average of the individual
price relatives, holding constant the revenue shares
of period b. The resulting index is called a Young
index in this Manual, again after an another index
number pioneer. The Young index is defined in
Section D.3 of Chapter 15 as follows:
substitution is less than 1, the covariance will be
negative and the Young will be less than the
Laspeyres.
1.45 As explained later, the Young index fails
some critical index number tests discussed in Section
C of this chapter and in Chapter 16, Section C.
B.2.1 Geometric Young, Laspeyres,
and Paasche indices
n
b⎛
p ⎞
(1.8) P ≡∑
s ⎜ where s
0 ⎟
i=
1 ⎝ pi
⎠
pq
t b b
i b i i
Yo i i
≡
n
b b
∑ pi
qi
i=
1
.
1.46 In the geometric version of the Young index,
a weighted geometric average is taken of the
price relatives using the revenue shares of period b
as weights. It is defined as:
1.43 In the corresponding Lowe index, equation
(1.1), the weights are hybrid revenue shares that
value the quantities of b at the prices of 0. As already
explained, the price reference period 0 usually
is more current than the weight reference period
b because of the time needed to collect and
process and the revenue data. In that case, a statistical
office has the choice of assuming that either the
quantities of period b remain constant or the revenue
shares in period b remain constant. Both cannot
remain constant if prices change between b and
0. If the revenue shares actually remained constant
between periods b and 0, the quantities would have
had to change inversely in response to the price
changes. In this case the elasticity of substitution is
1, for example., the proportionate decline in quantity
is equal to the proportionate increase in prices.
1.44 Section D.3 of Chapter 15 shows that the
Young index is equal to the Laspeyres index plus
the covariance between the difference in annual
shares pertaining to year b and month 0 shares (s i b –
s i 0 ) and the deviations in relative prices from their
means (r – r i * ). Normally the weight reference period
b precedes the price reference period 0. In this
case, if the elasticity of substitution is larger than
one, for example, the proportionate decline in quantity
is greater than the proportionate increase in
prices, the covariance will be positive. Under these
circumstances the Young index will exceed the
Laspeyres index. 9 Alternatively, if the elasticity of
9 This occurs because products with the large relative price
increases (r – r i * is positive) would also experience declining
shares between periods b and 0 (s i b – s i 0 is positive), thus
having a positive influence on the covariance. In addition,
products with small relative price increases (r – r i *is negative)
would experience increasing shares between b and 0
(continued)
b
s i
n t
⎛ p ⎞
i
(1.9) PGYo
≡ ∏ ⎜ 0 ⎟
i=
1 ⎝ p
,
i ⎠
b
where s i is defined as above. The geometric
Laspeyres is the special case in which b = 0 : that
is, the revenue shares are those of the price reference
period 0. Similarly, the geometric Paasche
uses the revenue shares of period t. Note that these
geometric indices cannot be expressed as the ratios
of value aggregates in which the quantities are
fixed. They are not basket indices and there are no
counterpart Lowe indices.
1.47 It is worth recalling that for any set of positive
numbers the arithmetic average is greater than,
or equal to, the geometric average, which in turn is
greater than, or equal to, the harmonic average, the
equalities holding only when the numbers are all
equal. In the case of unitary cross elasticities of
demand and constant revenue shares, the geometric
Laspeyres and Paasche indices coincide. In this
case, the ranking of the indices must be:
the ordinary Laspeyres ≥ the geometric Laspeyres
and Paasche ≥ the ordinary Paasche.
1.48 The indices are, respectively, arithmetic,
geometric, and harmonic averages of the same price
relatives that all use the same set of weights.
1.49 The geometric Young and Laspeyres indices
have the same information requirements as their
ordinary arithmetic counterparts. They can be pro-
(s i b – s i 0 is negative), thus having a positive influence on the
covariance.
10