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

Producer Price Index Manual
price index function P(p 0 ,p 1 ,q 0 ,q 1 ) is (positively)
homogeneous of degree minus one in the components
of the period 0 price vector p 0 .
16.43 The following two homogeneity tests can
also be regarded as invariance tests.
T7—Invariance to Proportional Changes in Current
Quantities:
P(p 0 ,p 1 ,q 0 ,λq 1 ) = P(p 0 ,p 1 ,q 0 ,q 1 ) for all λ > 0.
That is, if current period quantities are all multiplied
by the number λ, then the price index remains
unchanged. Put another way, the price index
function P(p 0 ,p 1 ,q 0 ,q 1 ) is (positively) homogeneous
of degree zero in the components of the period 1
quantity vector q 1 . Vogt (1980, p. 70) was the first
to propose this test, 26 and his derivation of the test
is of some interest. Suppose the quantity index q
satisfies the quantity analogue to the price test T5;
that is, suppose q satisfies Q(p 0 ,p 1 ,q 0 ,λq 1 ) =
λQ(p 0 ,p 1 ,q 0 ,q 1 ) for λ > 0. Then using the product
test in equation (16.17), it can be seen that p must
satisfy T7.
T8—Invariance to Proportional Changes in Base
Quantities: 27
P(p 0 ,p 1 ,λq 0 ,q 1 ) = P(p 0 ,p 1 ,q 0 ,q 1 ) for all λ > 0.
That is, if base period quantities are all multiplied
by the number λ, then the price index remains unchanged.
Put another way, the price index function
P(p 0 ,p 1 ,q 0 ,q 1 ) is (positively) homogeneous of degree
zero in the components of the period 0 quantity
vector q 0 . If the quantity index q satisfies the
following counterpart to T8: Q(p 0 ,p 1 ,λq 0 ,q 1 ) =
λ −1 Q(p 0 ,p 1 ,q 0 ,q 1 ) for all λ > 0, then using equation
(16.17), the corresponding price index p must satisfy
T8. This argument provides some additional
justification for assuming the validity of T8 for the
price index function P.
16.44 T7 and T8 together impose the property
that the price index p does not depend on the absolute
magnitudes of the quantity vectors q 0 and q 1 .
C.3 Invariance and symmetry tests
16.45 The next five tests are invariance or symmetry
tests. Fisher (1922, pp. 62–63, 458–60) and
Walsh (1901, p. 105; 1921b, p. 542) seem to have
been the first researchers to appreciate the significance
of these kinds of tests. Fisher (1922, pp. 62–
63) spoke of fairness, but it is clear that he had
symmetry properties in mind. It is perhaps unfortunate
that he did not realize that there were more
symmetry and invariance properties than the ones
he proposed; if he had realized this, it is likely that
he would have been able to provide an axiomatic
characterization for his ideal price index, as will be
done in Section C.6. The first invariance test is that
the price index should remain unchanged if the ordering
of the commodities is changed:
T9—Commodity Reversal Test (or invariance to
changes in the ordering of commodities):
P(p 0 *,p 1 *,q 0 *,q 1 *) = P(p 0 ,p 1 ,q 0 ,q 1 )
where p t * denotes a permutation of the components
of the vector p t , and q t * denotes the same
permutation of the components of q t for t = 0,1.
This test is due to Irving Fisher (1922, p. 63); 28 it
is one of his three famous reversal tests. The other
two are the time reversal test and the factor reversal
test, which will be considered below.
16.46 The next test asks that the index be invariant
to changes in the units of measurement.
T10—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 ; α 1 −1 q 1 0 ,...,α n −1 q n 0 ;
α 1 −1 q 1 1 ,...,α n −1 q n 1 ) =
P(p 1 0 ,...,p n 0 ; p 1 1 ,...,p n 1 ; q 1 0 ,...,q n 0 ;
q 1 1 ,...,q 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. The
concept of this test comes from Jevons (1863, p.
23) and the Dutch economist Pierson (1896, p.
131), who criticized several index number formulas
for not satisfying this fundamental test. Fisher
(1911, p. 411) first called this test the change of
26 Fisher (1911, p. 405) proposed the related test
P(p 0 ,p 1 ,q 0 ,λq 0 ) = P(p 0 ,p 1 ,q 0 ,q 0 ) =
n
n
1 0 0 0
piqi pi qi
i= 1 i=
1
∑ ∑ .
27 This test was proposed by Diewert (1992a, p. 216).
28 “This [test] is so simple as never to have been formulated.
It is merely taken for granted and observed instinctively.
Any rule for averaging the commodities must be so
general as to apply interchangeably to all of the terms averaged.”
Irving Fisher (1922, p. 63).
410