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

Producer Price Index Manual
value vectors v 0 and v 1 and so P GW (p 0 ,p 1 ,v 0 ,v 1 ) =
P GW (p 0 ,p 1 ,s 0 ,s 1 ). Thus, Walsh came very close to
deriving the Törnqvist-Theil index defined earlier
by equation (16.48). 64
E.3 Invariance and symmetry tests
16.109 The next five tests are invariance or symmetry
tests, and four of them are direct counterparts
to similar tests in Section C. The first invariance
test is that the price index should remain unchanged
if the ordering of the commodities is
changed.
T8—Commodity Reversal Test (or invariance to
changes in the ordering of commodities):
P(p 0 *,p 1 *,v 0 *,v 1 *) = P(p 0 ,p 1 ,v 0 ,v 1 ).
where p t * denotes a permutation of the components
of the vector p t and v t * denotes the same
permutation of the components of v t for t = 0,1.
16.110 The next test asks that the index be invariant
to changes in the units of measurement.
T9—Invariance to Changes in the Units of Measurement
(commensurability test):
P(α 1 p 1 0 ,...,α n p n 0 ; α 1 p 1 1 ,...,α n p n 1 ; v 1 0 ,...,v n 0 ;
v 1 1 ,...,v n 1 )
= P(p 1 0 ,...,p n 0 ; p 1 1 ,...,p n 1 ; v 1 0 ,...,v n 0 ; v 1 1 ,...,v n 1 )
for all α 1 > 0, …, α n > 0.
That is, the price index does not change if the units
of measurement for each product are changed.
Note that the revenue on product i during period t,
v i t , does not change if the unit by which product i
is measured changes.
16.111 Test T9 has a very important implication.
Let α 1 =1/p 1 0 , … , α n = 1/p n 0 and substitute these
64 One could derive Walsh’s index using the same arguments
as Theil except that the geometric average of the
revenue shares (s i 0 s i 1 ) 1/2 could be taken as a preliminary
probability weight for the ith logarithmic price relative, ln
r i . These preliminary weights are then normalized to add up
to unity by dividing by their sum. It is evident that Walsh’s
geometric price index will closely approximate Theil’s index
using normal time-series data. More formally, regarding
both indices as functions of p 0 ,p 1 ,v 0 ,v 1 , it can be
shown that P W (p 0 ,p 1 ,v 0 ,v 1 ) approximates
P T (p 0 ,p 1 ,v 0 ,v 1 ) to the second order around an
equal price (that is, p 0 = p 1 ) and quantity (that is,
q 0 = q 1 ) point.
values for the α i into the definition of the test. The
following equation is obtained:
(16.66)
P p p v v P r v v
≡ P (, rv, v)
,
0 1 0 1 0 1
( , , , ) = (1
n, , , )
∗ 0 1
where 1 n is a vector of ones of dimension n and r
is a vector of the price relatives; that is, the ith
component of r is r i ≡ p i 1 /p i 0 . Thus, if the commensurability
test T9 is satisfied, then the price index
P(p 0 ,p 1 ,v 0 ,v 1 ), which is a function of 4n variables,
can be written as a function of 3n variables, P*(r,
v 0 ,v 1 ), where r is the vector of price relatives and
P*(r, v 0 ,v 1 ) is defined as P(1 n ,r,v 0 ,v 1 ).
16.112 The next test asks that the formula be invariant
to the period chosen as the base period.
T10—Time Reversal Test: P(p 0 ,p 1 ,v 0 ,v 1 ) =
1/ P(p 1 ,p 0 ,v 1 ,v 0 ).
That is, if the data for periods 0 and 1 are interchanged,
then the resulting price index should
equal the reciprocal of the original price index.
Obviously, in the one good case when the price index
is simply the single price ratio, this test will be
satisfied (as are all of the other tests listed in this
section).
16.113 The next test is a variant of the circularity
test that was introduced in Section F of Chapter
15. 65
T11—Transitivity in Prices for Fixed Value
Weights:
P(p 0 ,p 1 ,v r ,v s )P(p 1 ,p 2 ,v r ,v s ) = P(p 0 ,p 2 ,v r ,v s ).
In this test, the revenue-weighting vectors, v r and
v s , are held constant while making all price comparisons.
However, given that these weights are
held constant, then the test asks that the product of
the index going from period 0 to 1, P(p 0 ,p 1 ,v r ,v s ),
times the index going from period 1 to 2,
P(p 1 ,p 2 ,v r ,v s ), should equal the direct index that
compares the prices of period 2 with those of period
0, P(p 0 ,p 2 ,v r ,v s ). Clearly, this test is a manyproduct
counterpart to a property that holds for a
single price relative.
65 See equation (15.77) in Chapter 15.
426