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

Producer Price Index Manual
16.84 Theil (1967, pp. 136–137) proposed a solution
to the lack of weighting in the Jevons index,
P J defined by equation (16.47). He argued as follows.
Suppose we draw price relatives at random
in such a way that each dollar of revenue in the
base period has an equal chance of being selected.
Then the probability that we will draw the ith price
n
i i i k k
k = 1
relative is equal to s 0 ≡ pq 0 0 ∑ pq
0 0 , the period
0 revenue share for product i. Then the overall
mean (period 0 weighted) logarithmic price change
n
is ∑ s 0 ln ( 1 0
i
pi pi
)
i=
1
. 53 Now repeat the above mental
experiment and draw price relatives at random in
such a way that each dollar of revenue in period 1
has an equal probability of being selected. This
leads to the overall mean (period 1 weighted) loga-
n
rithmic price change of ∑ s 1 ln ( 1 0
i
pi pi
)
i=
1
. 54 Each of
these measures of overall logarithmic price change
seems equally valid, so we could argue for taking a
symmetric average of the two measures in order to
obtain a final single measure of overall logarithmic
price change. Theil 55 argued that a nice, symmetric
index number formula can be obtained if the probability
of selection for the nth price relative is
made equal to the arithmetic average of the period
0 and 1 revenue shares for product n. Using these
probabilities of selection, Theil’s final measure of
overall logarithmic price change was
(16.48)
n
1
0 1 0 1 0 1 i
ln PT( p , p , q , q ) ≡ ∑ ( si + si)ln( ).
0
i=
1 2 pi
53 In Chapter 19, this index will be called the geometric
Laspeyres index, P GL . Vartia (1978, p. 272) referred to this
index as the logarithmic Laspeyres index. Yet another
name for the index is the base weighted geometric index.
54 In Chapter 19, this index will be called the geometric
Paasche index, P GP . Vartia (1978, p. 272) referred to this
index as the logarithmic Paasche index. Yet another name
for the index is the current period weighted geometric index.
55 “The price index number defined in (1.8) and (1.9)
uses the n individual logarithmic price differences as the
basic ingredients. They are combined linearly by means of
a two stage random selection procedure: First, we give each
region the same chance ½ of being selected, and second,
we give each dollar spent in the selected region the same
chance (1/m a or 1/m b ) of being drawn. (Henri Theil, 1967,
p. 138).
1
p
Note that the index P T defined by equation (16.48)
is equal to the Törnqvist index defined by equation
(15.81) in Chapter 15.
16.85 A statistical interpretation of the righthand
side of equation (16.48) can be given. Define
the ith logarithmic price ratio r i by:
(16.49)
p
ri
≡ ln( ) for i = 1,..., n.
p
1
i
0
i
Now define the discrete random variable—we will
call it R—as the random variable that can take on
the values r i with probabilities ρ i ≡ (1/2)[ s i 0 + s i 1 ]
for i = 1,…,n. Note that since each set of revenue
shares, s i 0 and s i 1 , sums to one over i, the probabilities
ρ i will also sum to one. It can be seen that the
expected value of the discrete random variable R is
(16.50) [ ]
1 p
E R ≡ ρ r = ( s + s )ln( )
n
n
1
0 1 i
∑ i i ∑ i i 0
i= 1 i=
1 2 pi
0 1 0 1
PT
p p q q
= ln ( , , , ) .
Thus, the logarithm of the index P T can be interpreted
as the expected value of the distribution of
the logarithmic price ratios in the domain of definition
under consideration, where the n discrete
price ratios in this domain of definition are
weighted according to Theil’s probability weights,
ρ i ≡ (1/2)[ s i 0 + s i 1 ] for i = 1,…,n.
16.86 Taking antilogs of both sides of equation
(16.48), the Törnqvist (1936, 1937) Theil price index,
P T , is obtained. 56 This index number formula
has a number of good properties. In particular, P T
satisfies the proportionality in current prices test
(T5) and the time reversal test (T11) discussed in
Section C. These two tests can be used to justify
Theil’s (arithmetic) method of forming an average
of the two sets of revenue shares in order to obtain
his probability weights, ρ i ≡ (1/2)[ s i 0 + s i 1 ] for i =
1,…,n. Consider the following symmetric mean
class of logarithmic index number formulas:
56 The sampling bias problem studied by Greenlees (1999)
does not occur in the present context because there is no
sampling involved in equation (16.50): the sum of the p i t q i
t
over i for each period t is assumed to equal the value aggregate
V t for period t.
420