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

18. Aggregation Issues
(18.4) R 0 (p 0 ,v 0 ) =
and R 1 (p 1 ,v 1 ) =
E
N
∑∑
e= 1 n=
1
p q
e0 e0
n n
E N
e1 e1
∑∑ pn
qn
.
e= 1 n=
1
Under these revenue maximizing assumptions,
adapting the arguments of F. M. Fisher and Shell
(1972, pp. 57–58) and Archibald (1977, p. 66),
Diewert (2001) showed that the two theoretical indices,
P 0 (p 0 ,p 1 ,v 0 ) and P 1 (p 0 ,p 1 ,v 1 ), described in (i)
and (ii) above, satisfy the following inequalities of
equations (18.5) and (18.6):
(18.5) P 0 (p 0 ,p 1 ,v 0 ) R 0 ( p 1 , v 0 ) R 0 ( p 0 , v
0
)
using equation (18.3)
using equation (18.4)
≡ ,
E N
= ( , ) ∑∑
R p v p q
0 1 0 e 0 e 0
n n
e= 1 n=
1
E N E N
e1 e0 e0 e0
pn qn pn qn
e= 1 n= 1 e= 1 n=
1
≥ ∑∑ ∑∑
since q e0 is feasible for the maximization problem
that defines R e0 (p e1 ,v e0 ), and so
( , )
N
e 0 e 1 e 0 e 1 e 0
n n
n=
1
R p v ≥ ∑ p q for e = 1,...,E
≡ P L (p 0 ,p 1 ,q 0 ,q 1 ),
where P L is the Laspeyres output price index,
which treats each commodity produced by each establishment
as a separate commodity. Similarly,:
(18.6) P 1 (p 0 ,p 1 ,v 1 ) ≡ R 1 ( p 1 , v 1 ) R 1 ( p 0 , v
1
)
using equation (18.3)
using equation (18.4)
E
N
= ∑∑
e= 1 n=
1
( , )
p q R p v
e 1 e 1 1 0 1
n n
E N E N
e1 e1 e0 e1
pn qn pn qn
e= 1 n= 1 e= 1 n=
1
≤ ∑∑ ∑∑
since q e1 is feasible for the maximization problem
that defines R e1 (p e0 ,v e1 ),and so
( , )
N
e 1 e 0 e 1 e 0 e 1
n n
n=
1
R p v ≥ ∑ p q for e = 1,...,E
≡ P P (p 0 ,p 1 ,q 0 ,q 1 ),
where P P is the Paasche output price index, which
treats each commodity produced by each establishment
as a separate commodity. Thus, equation
(18.5) says that the observable Laspeyres index of
output prices P L is a lower bound to the theoretical
national output price index P 0 (p 0 ,p 1 ,v 0 ) and equation
(18.6) says that the observable Paasche index
of output prices P P is an upper bound to the theoretical
national output price index P 1 (p 0 ,p 1 ,v 1 ).
18.9 It is possible to relate the Laspeyres-type
national output price index P 0 (p 0 ,p 1 ,v 0 ) to the individual
establishment Laspeyres-type output price
indices P e0 (p e0 ,p e1 ,v e0 ), defined as follows:
(18.7) P e0 (p e0 ,p e1 ,v e0 )
≡ R p , v R p , v
for e = 1,...,E
( ) ( )
e0 e1 e0 e0 e0 e0
( , )
e 0 e 1 e 0 N
e 0 e
∑
0
n n
n=
1
= R p v p q ≥
where the establishment period 0 technology revenue
functions R e0 were defined above by equation
(18.1) and assumptions in equation (18.4) were
used to establish the second set of equalities; that
is, the assumption that each establishment’s observed
period 0 revenues,
N
∑
n=
1
p q
e0 e0
n n
, are equal to
the optimal revenues, R e0 (p e0 ,v e0 ). Now define the
revenue share of establishment e in national revenue
for period 0 as
(18.8) S e
0
N E N
e0 e0 e0 e0
pn qn pn qn
n= 1 e= 1 n=
1
≡ ∑ ∑∑ ; e = 1,...,E.
Using the definition of the Laspeyres type national
output price index P 0 (p 0 ,p 1 ,v 0 ), equation (18.3), for
(t,v) = (0,v 0 ), and using also equation (18.2):
(18.9) P 0 (p 0 ,p 1 ,v 0 )
E
0 ( 1 0 E
e e e e
, ) 0 e
( 0 e
≡ ∑R p v ∑ R p , v
0
)
e= 1 e=
1
E
e=
1
( ,
⎝
) E
∑
= ∑ R p v
e0 e0 e0
e=
1
( , )
( , )
⎛ R p v
⎜R p v
e0 e1 e0
e0 e0 e0
( , )
R p v
e0 e0 e0
⎞
⎟
⎠
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