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

Producer Price Index Manual
(16.87)
n
n t
t b
b i
∑ pq
i i ∑ si
⎜ b
0 1
i 1 p
t b i=
= i
Lo( , , ) ≡ =
⎝
n n 0
0 b b
⎛
pq
pk
∑ k k ∑ sk ⎜ b
k = 1 k = 1 pk
P p p q
⎛ p ⎞
⎟
⎠
.
⎞
⎟
⎝ ⎠
16.135 Drawing on those that have been listed in
Sections C and E, we highlight 12 desirable axioms
for price indices of the form P(p 0 ,p 1 ). The period
0 and t price vectors, p 0 and p t , are presumed
to have strictly positive components.
T1—PositivityTtest: P(p 0 ,p t ) > 0 if all prices are
positive.
T2—Continuity Test: P(p 0 ,p t ) is a continuous function
of prices.
T3—Identity Test: P(p 0 ,p 0 ) = 1.
T4—Homogeneity Test for Period
P(p 0 ,λp t ) = λP(p 0 ,p t ) for all λ > 0.
t Prices:
T5—Homogeneity Test for Period 0 Prices:
P(λp 0 ,p t ) = λ −1 P(p 0 ,p t ) for all λ > 0.
T6—Commodity Reversal Test: P(p t ,p 0 ) =
P(p 0 *,p t *) where p 0 * and p t * denote the same permutation
of the components of the price vectors p 0
and p t . 75
T7—Invariance to Changes in the Units of Measurement
or the Commensurability Test.
P(α 1 p 1 0 ,...,α n p n 0 ; α 1 p 1 t ,...,α n p n t ) = P(p 1 0 ,...,p n 0 ;
p 1 t ,...,p n t ) for all α 1 > 0, …, α n > 0.
T8—Time Reversal Test: P(p t ,p 0 ) = 1/P(p 0 ,p t ).
T9—Circularity or Transitivity Test: P(p 0 ,p 2 ) =
P(p 0 ,p 1 )P(p 1 ,p 2 ).
T10—Mean Value Test: min{p i t /p i 0 : i = 1,…,n} ≤
P(p t ,p 0 ) ≤ max{p i t /p i 0 : i = 1,…,n}.
T11—Monotonicity Test with Respect to Period t
Prices: P(p 0 ,p t ) < P(p 0 ,p t* ) if p t < p t* .
T12—Monotonicity Test with Respect to Period 0
Prices: P(p 0 ,p t ) > P(p 0* ,p t ) if p 0 < p 0* .
16.136 It is straightforward to show that the Lowe
index defined by equation 16.87) satisfies all 12 of
the axioms or tests listed above. Hence, the Lowe
index has very good axiomatic properties with respect
to its price variables. 76
16.137 It is straightforward to show that the
Young index defined by equation (16.86) satisfies
10 of the 12 axioms, failing T8, the time reversal
test, and T9, the circularity test. Thus, the axiomatic
properties of the Young index are definitely
inferior to those of the Lowe index.
Appendix 16.1: Proof of Optimality
of Törnqvist-Theil Price Index
in Second Bilateral Test Approach
16.138 Define r i ≡ p i 1 /p i 0 for i = 1,…,n. Using T1,
T9, and equation (16.66), P(p 0 ,p 1 ,v 0 ,v 1 ) =
P*(r, v 0 ,v 1 ). Using T6, T7, and equation (16.63):
(A16.1)
0 1 0 1 ∗ 0 1
P( p , p , v , v ) = P ( r, s , s ),
where s t is the period t revenue share vector for t
= 0,1.
16.139 Let x ≡ (x 1 ,…,x n ) and y ≡ (y 1 ,…,y n ) be
strictly positive vectors. The transitivity test T11
and equation (A16.1) imply that the function P*
has the following property:
(A16.2)
P xs s P ys s
0 1
= P ( xy,..., x y; s, s)
.
∗ 0 1 ∗ 0 1
( ; , ) ( ; , )
∗
1 1 n n
16.140 Using T1, P*(r,s 0 ,s 1 ) > 0 and using T14,
P*(r, s 0 ,s 1 ) is strictly increasing in the components
of r. The identity test T3 implies that
(A16.3)
P s s
∗ 0 1
(1
n, , ) 1
= ,
75 In applying this test to the Lowe and Young indices, it
is assumed that the base year quantity vector q b and the
base year share vector s b are subject to the same permutation.
76 From the discussion in Chapter 15, it will be recalled
that the main problem with the Lowe index occurs if the
quantity weight vector q b is not representative of the quantities
that were purchased during the time interval between
periods 0 and 1.
432