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

Producer Price Index Manual
=
N
∑
i=
1
using equation (A17.5)
N
∑
{ ⎤}
() () ⎡⎣
1() ,
2() ,..., ()
Qtr () *()
t
p t Q t r p t p t p t
i i N
(), (),..., () ' *()
= r ⎡⎣p t p t p t ⎤⎦p r t
i 1 2
N i
i=
1
( ) ( )
=⎡⎣1 r* t ⎤⎦
dr*
t dt
using equation (A17.4)
≡ r*' t r*
t .
( ) ( )
Thus, under the above continuous time revenuemaximizing
assumptions, the Divisia price level,
P(t), is essentially equal to the unit revenue function
evaluated at the time t prices, r*(t) ≡
r[p 1 (t),p 2 (t),…,p N (t)].
17.104 If the Divisia price level P(t) is set equal
to the unit revenue function r*(t) ≡
r[p 1 (t),p 2 (t),…,p N (t)], then from equation (A17.2) it
⎦
follows that the Divisia quantity level Q(t) defined
in Chapter 15 by equation (15.30) will equal the
producer’s output aggregator function regarded as
a function of time, f*(t) ≡ f[q 1 (t),…,q N (t)]. Thus,
under the assumption that the producer is continuously
maximizing the revenue that can be achieved
given an aggregate output target where the output
aggregator function is linearly homogeneous, it has
been shown that the Divisia price and quantity levels
P(t) and Q(t), defined implicitly by the differential
equations (15.29) and (15.30) in Chapter 15,
are essentially equal to the producer’s unit revenue
function r*(t) and output aggregator function f*(t),
respectively. 45 These are rather remarkable equalities
since, in principle, given the functions of time,
p i (t) and q i (t), the differential equations can be
solved numerically, 46 and hence P(t) and Q(t) are
in principle observable (up to some normalizing
constants).
17.105 For more on the Divisia approach to index
number theory, see Vogt (1977; 1978) and Balk
(2000).
45 The scale of the output aggregator and unit revenue
functions are not uniquely determined by the differential
equations (15.29) and (15.30); that is, given f(q) and r(p),
one can replace these functions by αf(q) and (1/α)r(p) respectively,
and still satisfy equations (15.29) and (15.30) in
Chapter 15.
46 See Vartia (1983).
462