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

Producer Price Index Manual
H.1 Short-run comparisons: illustration
of some quality adjustment
methods
7.200 A comparable replacement C may be
found. In the previous example, the focus was on
the use of the Jevons index at the elementary level
since it is shown in Chapter 20 that this has much
to commend it. The example here uses the Dutot
index, the ratio of arithmetic means. This is not to
advocate it but only to provide an example using a
different formulation. The Dutot index also has
much to commend it on axiomatic grounds but
fails the commensurability (units of measurement)
test and should only be used for relatively homogeneous
items. The long-run Dutot index for April
compared with January is
P
D
⎡
⎢
≡ ⎢
⎢
⎢
⎣
N
∑
i=
1
N
∑
i=
1
Apr ⎤
p
i
/N ⎥
⎥ ,
Jan
p ⎥
i
/N
⎥
⎦
which is 8/5 = 1.30, a 30 percent increase. The
short-run equivalent is the product of a long-run
index up to the immediately preceding period and
an index for the preceding to the current period,
that is, for period t + 4 compared with period t:
(7.35)
⎡ ⎤ ⎡ ⎤
⎢ p /N ⎥ ⎢ p /N ⎥
⎢ ⎥ ⎢ ⎥,
⎢ ⎥ ⎢ ⎥
⎢
p /N
⎣
⎥
⎦
⎢
p /N
⎣
⎥
⎦
N
N
t+ 3 t+
4
∑ i ∑ i
i= 1 i=
1
D
≡ ×
N N
t t+
3
∑ i ∑ i
i= 1 i=
1
P
or, for example, using a comparison of January
with April:
⎡ ⎤ ⎡ Apr ⎤
⎢ p / N ⎥ ⎢ pi
/ N ⎥
⎢ ⎥ ⎢ ⎥ ,
⎢ Mar
p / N
⎥ ⎢
pi
/ N
⎥
⎢
⎣
⎥
⎦
⎢
⎣
⎥
⎦
N
N
Mar
∑ i ∑
i= 1 i=
1
D
≡ ×
N N
Jan
∑ i ∑
i= 1 i=
1
P
6 8
which is of course × = 1.30 as before.
5 6
7.201 Consider a noncomparable replacement
with an explicit quality adjustment: say C’s value
of 6 in April is quality-adjusted to be considered to
be worth only 5 when compared to the quality of
B. The quality adjustment to prices may have
arisen from an option cost estimate, a quantity adjustment,
a subjective estimate or a hedonic coefficient
as outlined above. Suppose the long-run
comparison uses an adjusted January price for C,
which is B’s price of 3 multiplied by 6/5 to upgrade
it to the quality of C, that is 6/5 x 3 = 3.6.
From April onward the prices of the replacement
product C can be readily compared to its January
reference period price. Alternatively, the prices of
C in April onward might have been adjusted by
multiplying them by 5/6 to downgrade them to the
quality of B and enable comparisons to take place
with product B’s price in January: for April the adjusted
price is 5/6 x 6 = 5; for May, the adjusted
price is 5.8; and for June, it is 6.67 (see Table 7.5).
Both procedures yield the same results for longrun
price comparisons. The results from both
methods (rounding errors aside) are the same for
product B.
7.202 However, for the overall Dutot index, the
results will differ because the Dutot index weights
price changes by their price in the initial period as
a proportion of total price (Chapter 21, equation
(21. )). The two quality-adjustment methods will
have the same price changes but different implicit
weights. The Dutot index in May is 9/5.6 = 1.607
using an adjustment to the initial period, January’s
price and 7.8/5 = 1.56 using an adjustment to the
current period, May’s price. The short-run indices
give the same results for each adjustment:
8 9
× = 1.607 using an adjustment to the initial
period, January’s price, and
5.6 8
7 7.8
× = 1.56 using an adjustment to the current
period, May’s
5 7
price.
7.203 The overlap method may also take the
short-run form. In Table 7.5, there is a price for C
in March of 5 that overlaps with B in March. The
ratio of these prices is an estimate of their quality
difference. A long-run comparison between January
and April would be ⎜6× + 2⎟/5
= 1.36. The
⎛ 4 ⎞
⎝ 5 ⎠
short-run comparison would be based on the product
of the January to March and March to April
link: 6.8 × 6 = 1.36 .
6 5
192