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working paper
department
of economics
PRODUCTION-THEORETIC INPUT PRICE INDICES AND THE
MEASUREMENT OF REAL AGGREGATE INPUT USE
Franklin M, Fisher
Number 384
July 1985
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 021 39
PRODUCTION-THEORETIC INPUT PRICE INDICES AND THE
MEASUREMENT OF REAL AGGREGATE INPUT USE
Franklin M. Fisher
Number 384
July 1985
I
JAN 2 5
f^f^o
PROD UCTION-THEORE TIC TN PIIT PRTCE INDICES
AND THE MEASUREME NT OF REAL AGGREGATE INPUT USE
Franklin M. Fisher
Massachusetts Institute of Technology*
*
Paper
prepared for the Fourth Karlsruhe Seminar on
surement in Economics (Theory and Application of
1985.
I
Indices)
MeaJuly
,
am indebted to Karl Shell for helpful conversations but
retain responsibility for error.
1
Index
number
.
theory,
Introduction
and the theory of price
indices
particular, tends to run in terms of arithmetic properties.
questions as whether
ty
or
a
however,
The
particular index satisfies
reversal test are well discussed and
understood.
are,
a
Important
relatively
well
discussions
they are only one side of the index number story.
V7hat are
I
shall term the economic theory
we trying to measure?
if we had all the information possible?
we
Such
chain proper-
as such properties and such
other side lies in what
index numbers:
a
in
V7hat
of
would we do
if anything,
are
attempting to approximate with the index numbers computed
in
What,
practice?
In the case of the theory of the individual consumer,
sure,
the economic theory of index numbers is v/ell
to be
established.
We are accustomed to regarding computed cost-of-living indices as
to the ratio of expenditures required to attain a
approximations
unchanging indifference curve at base and current
given
Where the analytic questions are properly asked,
be
prices.
the theory
can
extended to cast light on what to do in the presence of taste
and quality change.
(See,
for example,
Fisher and Shell 1972,
While the aggregation problems accompanying an
Essay I.)
exten-
sion to many consumers are formidable, they present no problem of
principle.
The
situation
is
somewhat different when we turn
the
to
production side of the economy and the measurement of real output
and
input.
there is relatively little agreement
Here
analytic basis for index numbers.
Indeed,
the
on
the notion that what
one means by aggregate real output is a Laspeyres output index is
^^irly widespread.
meaning for words,
own
production
light
established
knew
theory can cast on the
the
usage,
questions
that
index
Putting aside the question
of
what index numbers would one compute if
one
prices
and
technology perfectly but had to summarize
on the one hand or factor prices and inputs on the other
outputs
pair of aggregate measures.
a
what
it is still appropriate to inquire
are supposed to answer.
numbers
in
Yet while anyone is free to choose his or her
problem
Only by thinking
about
in this way can one investigate the strengths and
the
weak-
nesses of actually computable indices as approximations to analy-
tically satisfactory constructs.
More
than
along these lines
a
decade ago,
Karl Shell and
(Fisher and Shell 1972,
I
began
Essay II)
.
thinking
We consi-
the production-theoretic foundation for
dered
presented
and
to which we still adhere)
(and
is that real output must
to a given technology,
relative
fined
deflation
The position that we
number of results.
a
output
took
de-
be
with points on the
same
production possibility frontier (PPF) considered as involving the
same real output. This leads to a theory of output deflation that
is
isomorphic
to the theory of the
cost-of-living
index.
We
developed that theory on the assumption that the money output
be
is that of an entire closed economy but have
deflated
since
gone on to consider the more complex case in which inputs can
purchased
by
the
productive unit involved
(Fisher
to
be
Shell
and
1981).
shall not discuss that work in detail but shall follow
I
a closely related subject.
Upon reading
a
the output-deflation section of our book,
pointed
of
deflation
use
preliminary version of
John Muellbauer
(1972)
case
out that an isomorphic theory can be built for the
input
—
up
That topic
deflation and proceeded to do so.
—
of inputs and the measurement of real aggregate
Dealing
is the subject of the present paper.
with
the
input
the
conceptual problems involved casts light on the parallel problems
of
output
deflation;
further,
there are some interesting
new
results to present.
2.
I
cing
Ne .a5uring-Re.aI_Aggregi
begin with the simplest case, that of
one
output from several inputs with
a
a
single firm produ-
well-behaved
neo-
classical production function.
Any
solution
to the problem of measuring
real
aggregate
.
use must begin with an answer to the
input
following
question:
Since we shall be reducing input vectors to index-number scalars,
and since we wish to obtain a complete ordering for the resulting
and the input vectors to which they correspond,
scalars
choose
a
set of equivalence classes for input vectors such
all vectors in the same equivalent class will be said to
the same aggregate input.
question
this
isoquants
classes.
immediately
of
the
With
a
suggests itself
production
function
given technology,
involve
to
the answer to
—
use
the
define
the
In
a
Figure
equivalence
input
we will say that two
pure price phenomenon.
1,
the base-period isoquant is drawn with
w,
represented by the solid line.
input use is v, and total input costs C
For
aggre-
then any change in money input costs will be
period factor prices,
1.
the
If factor prices change and the firm remains on
same isoquant,
considered as
this
of
vectors corresponding to the same output involve the same
gate input use.
that
How is this to be done?
the case of the single-output firm,
In
we must
ease of notation,
I
baseOptimal
In the current
= wv.
when
omit transposition signs
writing inner products, relying on the context to make clear what
is intended.
period,
Actual
Had
factor
prices,
w,
are
dashed
denoted by the
factor usage is v* and actual money costs are C*
factor
prices been w instead of w in the base
lines.
=
period,
output corresponding to the base-period isoquant would have
efficiently
would
v;v*
the
been
produced with inputs v rather than v and money costs
have been C rather than C.
The view
of
inout-def lation
Figure
taken
is that the change in costs from C to C*
here
should
be
thought of as:
C*/C = {C*/C) (C/C)
(2.1)
with the first factor the increase in real aggregate input
usage
and the second reflecting price changes.
[Figure
general
more
In
where
C{w, x),
base-period
x
x,
input costs as C(w,
money
here]
let the firm's cost
terms,
and calculates the deflator
x)/C(w,
be
uses
money
for
This deflator is divided into
x).
aggregate
costs to give the measure of real
input
function
The construction just given
is output.
output,
1
input
usage relative to the base period.
Obviously,
this
approach
will
lead to a
largely
theory
isomorphic to that of the cost-of-living index (as well as to the
Fisher-Shell
theory of output deflation)
In considering
.
therefore,
sible objections to the theory being advanced,
well
consider
to
what those same objections
imply
Consideration
it is
about
well-established theory of the cost-of-living
relatively
pos-
the
index.
of such objections leads to insight as to what
is
involved in the present analysis.
The first such objection is conceptual.
The procedure just
described
treats input vectors as identical if they can
the
output
same
isoquant
as
and treats
a
an input increase.
movement
to
of
outputs
higher-numbered
But we are trying to
theory of input aggregation and measurement.
levels
a
become central to
the
produce
build
a
Is it not odd that
theory?
Moreover,
.
firms with different technologies facing the same
different
of
set
input prices will have different factor price deflators
con-
structed for them.
The answer lies in consideration of the object of the enter-
We
prise.
are treating the firm as the object of interest with
factor prices given from outside.
2
Any production-theoretic
Cases of monopsony can also be treated but are irrelevant
2.
to the present discussion.
of input deflation must involve the production function
view
as the cost-of-living
Just
firm.
the
index
describes
of
price
changes from the point of view of the individual consumer, so the
production-theoretic
input
price index describes
factor
changes from the point of view of the individual firm.
that different firms have different points of view,
is
not a valid objection.
price
The fact
so to speak,
The aggregation problem to which
it
points cannot be solved by choosing a firm-independent measure of
input
prices.
To
do that is merely to impose on all
firms
a
measure not relevant to any one of them."
3.
interest
To say this is not,
in
of course,
to say that there is no
the variation of the production-theoretic index
the production function varies over firms
(or
as
changes over time)
Some results on such variation are presented below.
D.QC[.££ieity..
and- Rela te d.. Pr o p^jtj. £s
The second objection to the approach taken here is
more troublesome,
at least at first glance.
somewhat
Suppose that input
V,
Figure
2
prices change but that it just so happens that the usage of every
changes in the same proportion,
factor
so that the actual input
point in the current period is on the ray through v in Figure
To
that the usage of every
does
not lead to
a
exactly
aggregate
measure of increase in
usage which doubles.
input
input
It is evident that the production-theoretic view taken
doubles.
here
suppose
ideas,
fix
In terms of Figure
usage will be said to increase by
>
input
real
2,
factor of C*/C
a
real
2,
and it is
easy to see that the sense of the inequality is no accident.
Because
4.
2.
duction-theoretic
a
Laspeyres
price index must bound
input-price deflator from
above,
aggregate
C*/(C*/2),
the
The Paasche quantity
input index from below.
pro-
the
corre-
production-theoretic
Paasche quantity index bounds the
sponding
4
index,
is obviously 2, however.
the production-theoretic index of aggregate input usage is
Hence
not homogeneous of degree one in the inputs being aggregated.
[Figure
2
here]
This objection was made by Diewert (1983,
parallel
glance,
theory
of the measurement of
it seems a telling one.
real
pp. 16-26)
output.
to the
At
first
Careful consideration, however,
reveals that its force is far less than first appears.
Suppose first that the technology is one of constant returns
and that there has been no technological change between the
and the current periods.
must
be
however,
Then the isoquant through the point 2v
parallel along rays to that through v.
the
base
situation pictured in Figure
2
In that
cannot
occur.
case,
In
such circumstances,
the only way that 2v can be the input
point
for the current period is if relative input prices do not change.
But
if relative input prices do not change,
see
that
sensible
then it is easy
production-theoretic approach (like
the
every
approach in such circumstances) will lead to an
usage index of
other
input-
2.
Now suppose (still with no technical change) that the
nology does not exhibit constant returns.
homothetic,
the
situation
pictured
tech-
If the isoquant map is
in Figure 2
in more general circumstances, however,
occur;
to
cannot
still
Once one
it can.
leaves constant returns, however, it ceases to be obvious that it
is
desirable
geneous
of
for an index of aggregate input use
degree
one in the individual
homo-
be
to
Inputs
inputs.
are
The measure-
important because they are employed in production.
ment of aggregate input usage properly ought to be from the point
of view of that employment.
under-
Without homotheticity in the
lying technology it is not at all clear why movements along a ray
should play any special role in aggregate input measurement.
If this view that doubling does not mean doubling seems hard
to swallow, consider the parallel issue that arises in the theory
of
the
cost-of-living index.
could
that
There is nothing about Figure
not apply to that theory with
isoquants
being replaced by indifference curves and expenditures.
in
the
little
returns,
case of the cost-of-living index,
interest
is
in homothetic maps and none at all in
so situations such as Figure
the exception.
there
In such a context,
2
and
2
costs
Indeed,
relatively
constant
are the rule rather than
one surely hesitates to insist
that
doubling of the consumption of every commodity musi mean
a
doubling of real income.
Yet the apparent appeal of the homogen-
property is every bit as strong in the consumer context
eity
—
a
as
precisely because that appeal rests
in
the production one
no
consideration of the context involved or the
question
on
being
asked.
This is, of course, not to say that the homogeneity property
may not be an interesting one.
an
One can perfectly well
construct
input index by asking by what factor the current-period input
must
vector
quant.
5
be multiplied to place it on the base
period
iso-
Such a construction guarantees the homogeneity property,
Diewert
(1983,
approach.
case of real output measurement this
the
In
5.
He
18)
p.
terms the
is
what
"Malmquist-Bergson-Moorsteen"
gives an extensive bibliography and discussion of
these matters in that context.
but
concentrates
homothetic
on movements along a
isoquant
maps,
Once
ray.
it is not clear why
leaves
one
such
movements
should be of special interest.
One
can
different
period
way.
input
think
of such a ray-centered
Instead
construction
of asking by v/hat factor
the
point must be multiplied to place it on
in
a
current
the
base
period isoquant, ask the equivalent question: By what factor must
the
base
period isoquant be expanded (or
contracted)
along rays to pass through the current-period input
parallel
point.
Ob-
viously, this amounts to the same thing.
Thinking about matters in this way,
however,
points up the
difference between such an approach and the one taken here.
The
Figure
3
real input index constructed in the theory here
aggregate
to
By what factor must
the
base
isoquant be expanded (or contracted) parallel along
rays
espoused
period
the question:
answers
become tangent to the isocost line at the new prices
the new input point?
unchanging)
this construction gives the
(and
answer
does the ray-centered one.
as
,
lacking (or the isoquant map changes)
[Figure
3
,
Where
here]
ray-cen-
The
where
ray through the current-period input point crosses the
period
isoquant.
In effect,
base
it begins by asking the question:
What would inputs have been in the base period had the firm
restricted
is
the answers are different.
construction takes as its reference point the point
tered
same
homotheticity
difference can be seen in another way.
That
through
Where the isoquant map is
(See Figure 3.)
homothetic
the
being
to the current period's input proportions?
By
been
con-
trast, the production-theoretic index takes as its base point the
point on the base-period isoquant where that isoquant is
to
an
prices.
isocost line corresponding to the current period's
In effect,
it begins by asking the question:
inputs
have been in the base period had the firm
period
input
interesting,
prices?
but
I
input
What would
faced
current
Both ways of looking at the problem
believe the approach taken here is much
of an economics-oriented one.
(as
tangent
are
more
Certainly it lends itself readily
the ray-centered approach does not)
to an accompanying theory
of input-price deflation.
Having
said all this,
I
should point out that much of
10
the
comparative-static
analysis given below is of interest primarily
the case of homothetic technologies.
in
under homotheticity
As
already
observed,
if the technology does not change, both the
,
ray-centered approach and the one used here give the same answers
and
the real aggregate input use index here constructed does
in
fact have the homogeneity property.
what if technology does change between the base
But
and
the current period?
tion-theoretic
surprisingly,
there
is
a
approach
despite
Here the ray-centered and the
will give different
answers.
period
producPerhaps
the failure of homogeneity of degree one,
powerful case to be made that the
answers
the
of
production-theoretic approach are superior.
The
reason
for
every
this lies in the fact that
way
of
constructing a real input-usage index implies the construction of
a
corresponding input-price deflator and conversely.
considering
must
the properties of any approach to the
Hence,
problem,
in
one
consider the properties of both price and quantity indices,
not merely the properties of only one of them.
Implicit
input
vector
in the idea that homogeneity of degree one in
the
is a desirable property for a real input index
is
the view that a weaker property is even more naturally desirable:
The
measure
vector
of aggregate input should not change if
Parallel to this is
itself does not change.
statement about the input-price deflator:
the
a
input
similar
The input price defla-
tor should not change if input prices do not change
(and,
we may
add, should be homogeneous of degree one in those prices).
In
addition,
one
wants the two indices
natural consistency properties.
In particular,
11
to
have
certain
it is natural to
the "circle property".
require
price
the
should have the property that the change in relative input
index
prices
in
In terms of prices,
from situation A to situation B multiplied by the
change
relative prices from situation B to situation C should
equal
the change in relative prices from situation A to situation C.
A
similar property should hold for the index of real input usage.
Unfortunately,
one
cannot have all these desirable properIn particular:
ties at the same time.
Theorem 3.1: A. The production-theoretic input-price deflator and
its
associated
index
.
of real input use are
the
only
indices
having the circle property and also the following properties:
1.
relative change in the index of factor
The
prices
multiplied by the relative change in the index of real input
usage equals the relative change in expenditures on inputs.
2.
base
With an unchanging technology,
a
movement along the
period isoquant leaves the index of real
usage
input
unchanged.
3.
The index of factor prices does not change if factor
prices remain constant.
In
addition,
the
production-theoretic input-price deflator
is
homogeneous of degree one in the input prices.
B.
There
exists
no way of constructing a pair of
input-
price and real input-usage indices that have properties 1-3
just
given, the circle property, and also the following property:
4.
inputs
The
index of real input usage does not
remain constant.
change
which lead to those amounts
chosen.
12
if
being
,
Proof.
It
A.
obvious that the indices resulting
is
production-theoretic
the resulting input-price
Figure
1
once again.
isoquant,
period
property
from C to C* is entirely
shown
(as
property
by
3
two points.
from
costs
is
Consi-
Since v and v both lie on the
base-
implies that the movement in
costs
2
price phenomenon.
a
pro-
deflator
obviously homogeneous of degree one in the input prices.
der
the
approach have the circle property and
Further,
1-3.
perties
from
the dashed lines)
Since input prices
are the same at
v
as
at
implies that there is no price change between
v*
those
Hence, by the circle property, the movement of money
C to C*
is entirely a change in
real
usage.
input
This division of the movement from C to C* into monetary and real
changes,
however,
(See equation
approach.
B.
that
is
In
that
just
of
the
production-theoretic
(2.1).)
view of part A of the theorem,
it suffices to
the production-theoretic approach does not have property 4.
Consider Figure
4.
Here, the technology changes between the base
and the current periods,
and it just so happens that input usage
remains at v despite the fact that relative input prices
(Obviously,
this
requires
a
however,
will
The
still
the change in money costs from C to C a pure price phenome-
non and the change from C to C* an increase in real input
It
change.
change in the isoquant map.)
production-theoretic input-price deflator,
call
show
follows
that real input usage will- be said to
from V to V*, violating property
[Figure
13
4.
4
here]
have
usage.
changed
soquant
Figure 4
Lest
violation of property
the
4
just exemplified
be considered particularly damaging,
proof
examine the situation in Figure
property
between
corresponding
4
4
it is instructive to
In Figure
input usage at v is the same as at v.
real
approach (the ray-centered one,
perty
and
4
prices
the
4,
costs C and C* are the same.
to
contradiction
again to see the
and property 3.
the
in
By property
It follows that
2,
any
that satisfies pro-
for example)
has real input usage the same at v* as at
must
v
the change in costs from C to C* as purely monetary
describe
property
satisfy
property.
to
such
1
and the circle
approach will show
a
change in input prices between the situation
corresponding
C and that corresponding to C* even though
to
Hence
any
no
input price changes.
Thus,
and
4
the presence of properties
in
are contradictory.
1
properties
and 2,
One cannot have both an
price
input
that depends only on input prices and an input-usage index
index
depends
that
obvious way,
only on input usage,
have them
multiply
retain the natural circle property,
in
along an isoquant represent no change in input
Something
has
indices,
give.
In the case of Paasche
the missing property is property
isoquant.
either
to
property
In
3
more
the case of
or property
2
—
or
usage.
Laspeyres
equivalence along
sophisticated
approaches,
must be abandoned once we
4
the
and still have
movements
an
3
leave
homotheticity and an unchanging technology.
Which property should be retained?
is
that one is trying to do.
I
That depends on v/hat it
take the view that input
price
deflation means looking at input prices from the point of view of
14
the
input-using unit
prices
—
the firm in the
simplest
are among the givens of that unit's problem,
quantities
that should be retained.
then,
it is property
3
the fact that the corresponding
usage index will depend both on prices and
reflects
we
If
The resulting input-price index should
depend only on the input prices;
merely
Those
whereas the
of input used are functions of those prices.
wish to form aggregates in this context,
input
case.
on
the fact that quantities themselves
quantities
depend
on
prices.
I
add two points in this connection that may serve to
the argument more convincing.
14A
make
First, we have already seen that
tension between properties
the
3
and
does not arise
4
until
leave the case of an unchanging homothetic isoquant map.
homotheticity behind,
leave
we
When we
the case in favor of an input-usage
index homogeneous of degree one in the input vector stops being a
convincing
or without a homogeneous
With
one.
isoquant
map,
however, the case in favor of an input-price index homogeneous of
in the input prices remains
one
degree
convincing.
cost
The
function continues to have that property even when the underlying
isoquant map is not homothetic.
consider again the case of the cost-of-living index
Second,
interpret
and
property
4
Figure
4
in that context leads to a case in
consumer prices (the dashed lines)
living index.
Insistence
as an indifference map.
which
on
unchanging
imply a change in the cost-of-
Retention of property
3,
on the other hand, does
not do this but does lead to the proposition that the change from
C to C*
involves an increase in real income despite the fact that
V and V* appear to be on the same base-period indifference curve.
indiffe-
one realizes that v* must be on a higher-numbered
Once
rence curve than v according to the current period's indifference
property does not seem so
that
miap,
must apply to input
argument
(or
odd.
Plainly,
same
the
measure-
output) deflation and
ment.
while
Thus,
property
and 2.
4,
it
might be nice to have both property
this is impossible in the presence of properties
Certainly,
the implications
proach that does so are well worth investigating;
fatal
1
Given that, it seems sensible to me to retain property
and abandon property 4.
a
and
3
o^f
an
ap-
it is far from
objection that such an approach fails to make the
15
3
mea-
surement of real input usage depend only on the input vector
be homogeneous of degree one in the elements
to
fails
and
that
of
vector.
4.
continue
now
I
Ge neralizati o n to Man y Outputs
with the development
input-price deflator.
theoretic
the case of
generalize
a
of
To do so,
production-
the
it is necessary
single firm producing a single
output.
While
the generalization to many outputs is an easy one,
shall
see,
the
straightforward.
is because of the
either output or input prices.
the present paper,
unit
This
as
we
always
so
possibility
that
of firms involved is large enough to have an
aggregation
on
generalization to many firms is not
I
the
effect
shall avoid this issue
in
however, and shall assume that the productive
involved is small enough so that it takes prices as
In view of this,
to
we may as well keep on thinking of the
given.
produc-
tive unit as a competitive firm.
6.
problems
See
Fisher and Shell,
involved
in
1981,
for a discussion
more general situations in
the
of
case
the
of
output deflation.
Even
the case of the competitive
firm,
however,
requires
generalization to allow several outputs, and that generalization,
while not hard, has some interesting features.
In particular, we
must ask how the underlying isoquant map is to be constructed.
In the isomorphic case of output deflation, this question is
16
.
that of how to construct the production possibility frontier.
case,
that
—
inputs as fixed
possibility is to take the
obvious
one
vector
at least up to scalar multiplication.
the approach taken in Fisher and Shell,
In
of
(This is
1972, Essay II.
Such
)
is natural when dealing with output price deflation in
choice
closed economy,
however.
but it is not inevitable,
a
a
If the pro-
ductive unit being studied purchases inputs at fixed prices,
for
example, then it becomes natural to draw the production possibility
frontier
constant
as the locus of outputs that can
input
cost.
(This
produced
be
is studied in Fisher
and
at
Shell,
1981.)
In the present case of input deflation and measurement,
the
isomorphic choice to that of the closed economy with fixed inputs
is to take the output vector of the firm as fixed
to scalar multiplication.
sumption
given,
is no interesting problem in which output
there
firm,
makes sense to analyze matters from
it
Naturally,
cular
changes
makes
a
same
combinations
input
given value of output.
this makes the isoquant map depend on the parti-
output prices used.
in
the
This means fixing output prices
drawing an isoquant as the locus of all
that can produce
propor-
Rather, given that we are dealing with a compe-
point of view as does the firm.
and
Even apart from the as-
that the productive unit being analyzed takes prices as
tions are fixed.
titive
at least up
This is the choice made in Muellbauer
but it is not an appealing one.
(1972),
—
Since
v;e
the isoquant map on the
shall study the
constructed
effect
indices,
of
this
it important to study the effect of changing output prices
in particular
7
17
Note that the case of unchanged output prices is no differ-
7.
ent from that of a single output or of fixed output
so
that
the only difference in terms of exposition
of actual index construction)
terms
lies in whether
proportions,
(but not
comparative
exercises are done with changing prices or with
static
in
changing
output proportions.
5.
I
Thg Inpjjt-Price Deflator; Foxipal Stateipgflt
now give a formal description of the
The givens of the problem are: the vector
input-price deflator.
of
base-period factor prices,
factor prices,
output
prices,
w;
p,
production-theoretic
the vector of current
w;
period
total base-period costs, C; and the vector of
the same in both periods.
Production takes
place according to the production function:
F(x, V)
(5.1)
where
x
vector
=
is the vector of outputs produced by the firm and v
of inputs it uses.
Hats will be used to describe
period values.
Consider the solution to the following problem:
(5.2)
Maximize y h px
subject to F(x,
v)
=
and wv = C
Call the resulting value of y, y.
Next, solve the following problem:
18
the
base-
,
Minimize C
(5.3)
=
wv
subject to F(x,
v)
=
and px = y
production-theoretic
The
input-price deflator is then
the
ratio, J = C/C.
comments are in order.
Some
First,
the solution
the
to
problem in (5.2) amounts to finding the isoquant (for the production
value,
of
base-period
in
(5.3)
tangent to the plane
input prices and cost.
corresponding
period's factor prices,
w.
This construction
readily seen to be equivalent to that in Figure
1
at
is
above.
for beginning with the maximization problem
in
instead of directly with the base period's isoquant has
to
reason
The
(5.2)
to
The solution to the problem
then takes that isoquant and minimizes cost along it
current
the
y = px)
do
with the analysis of comparative statics given below.
Since
we
shall wish to ask how the deflator
index
of
real input usage) would differ if the isoquant map were
ferent,
picks
it
out
structed.
is
necessary to have
a
(and its associated
method
which
the
con-
isoquant map is the actual base-period
one,
then this will be the actual base-period isoquant.
will
the isoquant tangent to the
be
unambiguously
be
the isoquant with which the deflator is
If
dif-
period factor prices and costs.
plane
to
Otherwise, it
representing
base-
This approach is consonant with
the general view that the givens of the problem are factor prices
and
base-period costs
(current-period costs are to be deflated)
not actual base-period inputs.
Next,
tions.
the
deflator
can be defined in terms of cost
func-
Remem.bering that "output" here is really y, we can think
19
of the firms cost function as C(w, y)
revenue,
period
C(w, y)/C(w, y)
deflator
the
This
.
static analysis,
is
readily
be
to
comparative
as is the more extensive
description
above because it presupposes that the
given
seen
base
is not so helpful for
form
however,
Letting y be actual
.
"output",
will
y,
remain the same when the isoquant map changes.
It
is
obvious that,
as in the case of the
index, the input-price deflator here defined
the Laspeyres price index,
by
both J and L are the same
(C)
L = wv/wv.
is the cost of doing so by using
glance at Figure
line
1
index
The denominators
of
however, is
while the numerator of
base-period
inputs,
v.
(A
will confirm that the cost of v at the dashedIt follows that the production-
prices is greater than C.)
theoretic
is bounded above
The numerator of J,
.
the minimum cost of producing "output" y,
L
(J)
cost-of-living
of
real input usage is bounded
below
the
by
Paasche quantity index, wv/wv.
Now,
the
whole exposition so far has used the
isoquant to make comparisons.
made
to
Equally valid comparisons can
using the current-period isoquant.
that
base-period
just given shows that
the
be
An analysis isomorphic
production-theoretic
input
price deflator formed using the current-period isoquant is bounded below by the Paasche input-price index,
P = wv/wv,
while the
corresponding production-theoretic index of input usage is bounded above by the Laspeyres quantity index, wv/wv.
If
the isoquant map is homothetic and unchanging,
production-theoretic
and
indices that use the
base-period
then the
isoquant
those that use the current-period isoquant will be the same.
20
the production-theoretic input-price deflator will
In that case,
be
bounded below by L and above by P,
with
a
similar
statement
holding for the index of input usage.
While
the
assumption of homotheticity is
however,
interest,
the assumption of an unchanging isoquant map
Merely a change in relative output prices will alter the
is not.
isoquant
map
and may destroy the relation between
index and the deflator constructed using
price
considerable
of
isoquant
(and similarly for other inequalities)
Paasche
the
base-period
the
One important
.
way of looking at the comparative static analysis given below
is
thus as an analysis of the ways in which the production-theoretic
indices using the current-period isoquant differ from those using
the
be
base-period isoquant and of the changes that must
therefore
made in the Paasche input-price index to restore the bounding
inequality.
8.
should
course,
the
same analysis gives the
changes
be made in a Laspeyres input-price index to
relation
the
Of
restore
to the production-theoretic deflator constructed
current-period isoquant.
things like this, and
Notice,
I
It is tedious to
keep
that
its
using
repeating
shall not do so henceforth.
however, that such an interpretation of comparative
statics requires that the inequalities in question apply if there
is
no
change in the isoquant map.
This means that either
the
homothe-
base
period or the current-period isoquant map must be
tic.
Since such homotheticity is the interesting leading case,
I
shall assume it for the rest of this paper, pointing out where it
is
needed explicitly for purposes other than the
21
interpretation
of results just discussed.
9.
ty
9
The assumption involved is weaker than that of homogeneiof the production function
(of any degree)
little point in going into details here, however.
Shell
(1981,
80-81)
pp.
There seems
(5.1).
See Fisher and
for a discussion of the parallel issues
in the case of output price deflation.
the comparative static analysis given below
Of course,
be
viewed as interesting in itself.
deflator
can
It shows the ways in which
constructed
from
the
point of view of one firm differs from that constructed from
the
the
(and the input-usage index)
point of view of another.
For this purpose, homotheticity is not
required except where stated explicitly below.
Comparative Statics: Changing Output Prices
6.
now turn to
I
leading case of comparative static analysis
a
to exemplify what is involved therein.
exhibit the Lagrrangians for
and
(5.2)
It will be convenient to
(5.3),
respectively.
They
are:
A
px + 7f{x, V)
L =
(6.1)
-
(1/V) (wv
- C)
and
L
(6.2)
= wv +
(^F(x, V)
-
kipx
- y)
A
Here,
one
(\
'
being
K
'
M-
'
written
^^^
{1/lc
)
are Lagrange multipliers, the last
as a reciprocal for reasons of
interpretation.
22
symmetry
of
already remarked,
As
the obvious first case to examine
that of a change in one of the output prices,
p.
lO.c.l, pp.
82-83)
.
A.
'^
Thsozsm-^^lj-
c/^
Pj^
=
y^i\
~
We prove the
.
following theorem (isomorphic to Fisher and Shell
unchanged),
prices
would
Theorem
1981,
^k^
If a rise in the price of the ith factor, w.
B.
is
(with output
increase (reduce) output of
the
kth
good, given the base-period isoquant, then a rise in the price of
the
good will reduce (increase) the importance of
kth
the
ith
factor price in the production-theoretic input-price deflator.
Proof:
(6.1)
A.
Apply the Envelope Theorem first to (6.2) and then to
obtaining:
,
9
(6.3)
C/3
p^ =
9
L/9
P;^
=7^(^k
= --/4x^
B.
-
V
L/9
Pj^)
=
Pk^
y<^x^
-
Xj^)
.
Suppose a rise in w. would increase production of the kth
given the base-period isoquant.
good,
^^/^
~
Consider such
rise from
a
the base to the current period, with all other factor prices held
constant.
Then the input-price deflator must be greater than one
the comparison is made using the isoquant map before the
whether
output price change or using the isoquant map after that
production of the kth good goes up,
Since
isoquant,
tive
x.
>
x,
(It is the
theorem
.
given the base-period
Since the Lagrange multiplier,
marginal cost of "output",
change.
y.)^
/<^
,
is posi-
part A of
the
lower
if
map after the output price change is used than
if
implies that the input-price deflator must be
the
isoquant
the
isoquant map before the output price change is.
23
Since
the
only factor price that changes is w.,
of
increase in
an
p.
input-price deflator.
which
a
it follows that the effect
is to reduce the importance of w.
(A
in
the
similar analysis applies to the case in
rise in w. decreases production of the kth
given
good,
the base-period isoquant.)
way
It
is illuminating to relate these results directly to
in
which the weights in
Paasche input-price
a
shift with changes in output prices.
(isomorphic,
lemmas
two
Proof
;
would
index
To do this, we first prove
respectively,
Lemmas
to
lO.c.l
and
84-85 of Fisher and Shell, 1981).
10. c. 2, pp.
I,emma 6.1:
the
Under homotheticity
'^/p
,
w^
i^(v^/C).
=
~
Differentiating (6.2) yields
7)L/'3^i=v.
(6.4)
pL/'Py =/^
;
.
Hence
(6.5)
?/</9wi
=
'>V9y'3w.
Now, we can evaluate
()
O ^ i/ u^'i that is, consider
^ / c) Y
v/hat
=
ii^
5v./pJ
.
two steps.
First, consider
happens to employment of the ith
factor as costs increase with factor prices constant.
theticity,
it
suffices
this is just v./C.
to
evaluate
To evaluate
^C/a
y.
(6.6)
^v./?y=
//{,
(^ v./pC) (^C/^y)
and the lemma now follows from
(6.4)
24
and
homo-
(yv./t)y, therefore,
From the
applied to (6.2), however, this is just
By
Envelope
Hence
.
=
/^(w./C),
(6.5).
Theorem
,
:
Under homotheticity
fi.2:
r.pTnTTia
Proof;
,
Differentiate (6.2) with respect to
treating C as
a
constant)
^ L/3w.
(6.7)
w.
and
p,
(this time
to obtain:
= V.
9l,/Pp^
;
-/^x^
=
Hence
(o
.
b)
^Pk^^i
Pk
^
9
^k
9
^i
_
jA^\Ci.
where the last step follows from Lemma 6.1.
We can now prove (isomorphic to Theorem 10. c.
2
of Fisher and
Shell 1981, p. 86)
Theorem 6.2
:
Under homotheticity
9
(V /C)
^
-
Q
^
(^/C)(^x^/^w.)
-
Proof:
9(v./C)
(6.9)
^
C(Pv./^p.)
k
1
''
?Pk
C
-^/^(?x^/^ w.)
-
-
V
1
(Pc/Pp,)
K
2
(/^x^v./C)
/^^i>^i
C2
25
.
=
-
U^/C)
{')'i^^/? w.)
using Lemma 6.2 and applying the Envelope Theorem to (6.2).
Combining this with Theorem 6.1,
p,
we see that an increase in
will increase the importance of w. in the input-price deflator
if and only if the "weight",
in
naturally associated with w.
of total costs is also increased.
index
an
v./C,
This
sort
of
duality is typical of comparative static results in this area.
Although
10.
restricted
it seems unlikely that such relationships are
to the case of homothetic technologies,
unable to find
a
I
have
been
proof of Theorem 6.2 that does not make explicit
use of homotheticity
A
great many other comparative results can be
particular,
case
given
the
In
results just given can be adapted to cover
the
of a Hicks-neutral technical change in the production of
The presentation of such particular results
output.
have to await
the
proved.
a
different occasion,
however,
for
I
will
want to
remaining space to return to ,ore general problems
—
a
use
those
of aggregation.
7.
There
considered.
are
Aggregation over Input Prices
two
types of aggregation problem
can
be
These are aggregation over input prices and aggrega-
tion over firms
(or
industries or sectors)
.
with the question of output-price deflation,
aggregation
that
When one is dealing
these two types
of
tend to coincide because it is natural to associate
26
particular kinds of output with particular firms.
with input-price deflation,
on the other hand,
When
dealing
the two types of
aggregation problem are less naturally associated, and it is best
to
take them up separately.
begin by considering aggregation
I
over input prices.
The production-theoretic input-price index can be written as
a
function of current-period factor prices,
holding base-period
factor prices fixed:
J = J(w^,
(7.1)
.
.
w^)
,
that we wish to form an aggregate of the first
Suppose
prices,
.
1
t
<
r,
<
so that J can be written as
J = H(A(w^,
(7.2)
where A(.)
.
.
.,
w^)
,
w^_^^,
theorem (Leontief,
.
.
.,
Wj.)
,
By Leontief's well-known
is a scalar-valued function.
aggregation
factor
t
this can be done if
1947),
and
only if
Q(J./J.)
^_^
(7.3)
^
where
J.
=
^
k
Wj^
/)j/Qw..
l,...,t;
=
(i,j
=
= t+1,
.
.
.
,r)
,
This means that the J-constant
rate of substitution between any pair of prices in the
marginal
aggregate
must be independent of any price not in the aggregate.
Now,
application
of
the Envelope Theorem to
(6.2)
above
shows that
(7.4)
J./Jj = v./v^
(i,j
Hence (7.3)
requires that changes in
in which v.
and v. are emploved.
27
w,
=
l,...,t).
leave unchanged the ratio
It follov/s that such ratios can
.
depend only on the first
t
factor prices, the ones to be included
Under homotheticity
in the aggregate.
that dependence will be
,
on the t-1 ratios of the factor prices to be aggregated.
Now such price ratios,
substitution
of
rates
in production among the
in the aggregate.
included
of course, will also be the marginal
factors
be
to
Since the employment of any
factor
not so included certainly depends on that factor's own price, the
aggregation condition just described is equivalent to the
the marginal rates of substituion among
that
tion
cluded
aggregate be independent of
the
in
A second application
not
Theorem
now shows that aggregation over the first
is
so included.
factors
in-
employment
the
factors
condi-
of
Leontief's
of
input prices
t
possible if and only if the efficient production surface
can
be written as:
= F(x,
(7.5)
where
v)
= G(x,
B(v^,
.
.
.,
v^)
is scalar-valued and homothetic
B(.)
,
v^_^^
,
.
v^)
.,
.
,
property
(the latter
being guaranteed if the underlying technology is homothetic)
In other words,
theoretic
input-price
corresponding
factor
factor-price aggregation in the productiondeflator is possible if and only
aggregation is possible in the
if
the
production
function itself.
If
simply
to say.
the productive unit under study is a firm
given as its efficient technology,
If,
and
F(.,
.)
there is nothing more
however, the productive unit is an aggregate and its
efficient technology built up from the technologies of underlying
firms
by allocating a total stock of factors and assigning
28
out-
by allocating a total stock of factors and assigning
firms
puts to achieve efficient production,
more that can be said.
the literature,
and the conditions that permit such
See, for example, Fisher
Schworm (198^)
great deal
a
That case has been extensively studied in
12
shown to be extremely restrictive.
11.
then there is
out-
aggregation
It follows that aggregation
(1969,
1982)
and Blackorby and
.
of factor prices in the production-theoretic input-price deflator
is unlikely to be possible.
8.
Vertical Aagreaation over Productive Units
The other type of aggregation problem is that of aggregation
over
productive units.
over
units that do not buy or sell from each other
or
Here there are two
Horizontal
or
one
units
that
("vertical" aggregation).
conglomerate aggregation
that are beyond the scope of the present paper.
easy
("horizontal"
"conglomerate" aggregation) and aggregation over
trade directly with each other
as
aggregation
cases:
problems
presents
That is because
includes more and more productive units it becomes
less
indepen-
to maintain the assumption that output prices are
dent of the activities of the aggregate productive unit.
General
shall
not
for a discussion of
the
isomorphic problem of output deflation when input prices are
not
equilibrium
considerations
come to the fore,
and
I
discuss such considerations here. 12
12.
See
Fisher and Shell,
1981,
independent of the activities of the productive unit.
29
on the other hand,
Vertical aggregation,
raises some inte-
resting questions that can be discussed here.
suppose first that there are only two
fix ideas,
To
further that the first of these
Suppose
involved.
the
Seller,
suppose that the Buyer buys
Finally,
the "Buyer".
from
firms,
the
buys only primary factors and sells only to the second
"Seller",
firm,
firms
using no primary factors,
only
and sells only
to
consumers.
Obviously,
prices of the Buyer
—
output
the
prices
the
—
the Seller.
the
prices
of the Seller
—
which in
—
are
turn
in-
the input prices of
A sensible set of questions to ask is how to measure
relative contributions of the Seller and of primary
the
output
are influenced by the Buyer's input prices
by the prices of primary factors
fluenced
—
consumers
faced by
factors
to inflation as seen by the Buyer and how to measure the relative
contributions of primary factors,
and the Buyer
to
it is convenient to begin with
an
the Seller,
inflation as seen by consumers.
To study such questions,
even simpler case,
factors
stage
that in which a single firm buys only primary
and sells only to consumers,
of production.
Here,
so that there is only
ons
an obvious thought is to calculate
the single firm's production-theoretic output-price index and its
production-theoretic
these
It
input-price index and to take the ratio
two indices as the contribution of the firm to
is
important to understand that this
"obvious
of
inflation.
thought"
is
totally mistaken.
There
relatively
is
more than one way to see this.
formal way.
The
I
production-theoretic
30
begin
with
a
output-price
is constructed by assuming output prices to be given
index
the firm by demand conditions and having the firm
side
in
out-
optimize
various ways given those prices and the conditions of
factor
The construction of the production-theoretic input-price
supply.
assumes input prices to be given from
on the other hand,
index,
the
outside
firm
and performs
certain
optimization
those prices and the conditions of demand as
taking
compare
the
production-theoretic output-price
problems
given.
To
input-price
and
indices is to take as simultaneously valid two sets of conditions
can hold simultaneously only if £JA prices
that
outside
the firm,
determined
are
in which case the question of the firm's
own
contribution to inflation is vacuous.
There is more to this than the fact that we have been assumthe productive unit analyzed too small to influence
ing
The extension of the theory to relax that assumption
(whether
monopoly elements are involved) still constructs the
not
prices.
or
input-
price deflator by assuming input prices fixed outside the productive unit and the output-price deflator by assuming output prices
so
Comparison
fixed.
fixed
outside the unit,
of the two assumes both sets
and this is not
a
of
prices
useful assumption
in
the present context.
The
The
underlying reason for this problem is not hard to find.
method suggested by the "obvious thought" cannot provide
appropriate
has
to do
answer to the question being asked.
(in this example)
with contributions to
question
inflation
as
Inflation as seen by consumers,
however, is
by the cost-of-living index and not by the
production-
seen by consumers
measured
That
an
.
31
theoretic
output-price
The latter index
index.
output
takes
prices as demand determined, as reflective of the prices at which
firms can sell
It is the cost-of-living index that takes output
.
as production determined,
prices
which consumers can buy
as reflective of the prices at
but if
Each measure has its uses,
.
we
seek to evaluate contributions to inflation as seen by consumers,
it is the cost-of-living index that must be used.
Once this is realized, it is possible to see how to proceed.
As observed in Fisher and Shell
the
pp.
7-8)
,
as to where we take the interface between
opportunity set.
its
activities
in discussing
between taste and quality changes,
difference
choice
and
(1972,
there
the
Consumers buy goods and use them
inside the household to maximize
utility.
side
the
household
Some
preparation,
consumption
rather than directly as
for example)
.
a
household
activities can be considered as production activities
those
is
in
of
in(food
Now suppose that the productive sys-
tem were organized differently with the activities now carried on
in the productive unit under analysis being carried on inside the
household. 13
factors
Then the household would buy primary
and
aggregation
over
them in its productive activities to maximize utility.
The
As
13.
usual,
I
ignore difficulties of
households in the cost-of-living index.
use
prices
it
would face would be those of
primary
factors.
cost-of-living index would be computed using those prices.
parison of such
living
a
cost-of-living index with
the actual
index computed using the output prices of the
Its
Com-
cost-of-
productive
unit being studied thus measures the extent to v/hich that produc-
32
tive unit contributes to inflation as seen by the household.
an approach seems somewhat
such
If
Buyer and Seller.
—
the
Consider inflation from the point of view
of
The relative contributions of the Seller and
factor
primary
asking
its
this is measured by the Buyer's production-theoretic
Buyer;
input-price index.
of
consider
to the case of two firms described earlier
generalization
the
forced,
prices can be analyzed
quite
naturally
how different the production-theoretic input-price
by
index
would be if the Seller did not exist and the Buyer were vertically integrated from purchase of primary factors to sale to
mers.
in
that the Seller's output-price index plays no role
(Note
This is the same procedure as that
this.)
consu-
house-
involving
holds but in a more familiar context.
It
easy to see that the same procedure applies to
is
restrictive
cases.
there is nothing in
For example,
requires the Buyer to use no primary inputs.
it
less
that
Whether or not such
inputs are used, the comparison to be made is that of the produc-
input-price index with and without vertical inte-
tion-theoretic
gration.
Note
Buyer's
that
and
a
similar procedure can be used to
consumers' relative contributions
inflation as seen by the Seller.
to
measure
the
output -price
This might be of some interest
when considering demand-pull rather than cost-push inflation.
I
leave the details to the reader.
Now,
indices
it is interesting to ask hov? such vertically aggregated
particular,
can
unaggregated
indices.
In
the same comparison of Seller-contribution
and
relate to the corresponding
33
e
primary-f actor-contribution
to inflation as seen by the Buyer be
made using the input-price index of the Buyer and the Inpiii -p r i c
case
the
from
which the Buyer buys only
in
Returning to
One would hope that it could.
index of the Seller?
Seller,
the
input-price index reflects inflation as the
Buyer's
it,
while
the
Seller
Buyer
sees
the Seller's input-price index reflects inflation
sees it
—
inflation as reflected in changes in
prices of primary factors.
the
It seems natural to compare the
as
the
two
to evaluate the Seller's contribution to inflation as seen by the
Buyer.
Perhaps construction of an input-price index for
a
verti-
cally integrated firm is not required.
Unfortunately,
given
those
for reasons similar
this will not v/ork,
earlier for the failure of the
to
thought."
"obvious
Seller's input-price index is calculated taking as given the
The
Seller's output prices
the
the
the prices at which the Seller sells to
But it is the change in those prices that
Buyer.
changes
—
in the Buyer's input-price index.
lead
to
The contribution
of
Seller to inflation as seen by the Buyer cannot be
assessed
using a construct that assumes there is no such inflation.
So long as we remain in the theoretical world in which there
is
sufficient
information
to construct all
theoretic indices, this presents no great problem.
ty
of constructing a vertically aggregated
production-
these
The difficul-
production-theoretic
price index is not analytically greater than that of
input
structing such an index without vertical integration.
ference
merely lies in what production processes are
The
con-
dif-
considered
to be under control of the productive unit being analyzed.
When
we
come
to the question of
34
approximations
used
in
.
practice,
the other hand,
on
the difficulties are considerably
Consider in particular the use of Laspeyres input-price
greater.
indices
above
already seen that (in the simple
have
We
14
a
)
Laspeyres input-price index for the Seller will bound
Where
14.
prices
input
productive unit is large enough
the
(1981,
59-68)
pp.
for
La-
firms)
See Fisher and
discussion of the parallel
a
affect
to
(even if it is made up of competitive
and Paasche bounds do not apply.
speyres
considered
case
Shell
case
of
output-price deflation.
the production-theoretic input-price index
above
from
Further,
Seller.
it
for
the
Laspeyres
is not hard to see that such a
index will also bound from above the production-theoretic
input-
price index that would apply under vertical integration.
So far
so good.
Unfortunately, the reason for constructing such
aggregated
vertically
input-price index was to compare it with the
tion-theoretic
In practice,
however, all
are likely to have for the Buyer is another Laspeyres
price
index,
this merely provides an upper
and
production-theoretic index we need.
input-price
indices
merely provides
produc-
input-price index of the Buyer and thus to assess
relative contributions to inflation.
we
a
a
bound
on
the
Comparison of the Laspeyres
for the Seller and
the
Buyer,
comparison of two upper bounds,
therefore,
That compari-
close approximation to the compari-
son
may or may not provide
son
of the production-theoretic input-price indices
a
input-
35
themselves.
.
and
it is obviously not possible to say anything general
as
to
the sign of the approximation error.
This is,
for
the
of course,
a
problem, but
regard it as
a
problem
use of Laspeyres indices and not for the theory of
production-theoretic input-price index.
of
I
As is true in the
index numbers and aggregation generally,
I
regard it as
the
area
im-
portant to ask the right questions and examine the defects of the
answers that can be given in practice.
able
That seems to me prefer-
to tailoring the questions to suit the currently
answers
36
available
.
Blackorby,
and W. Schworm
C.
(1989")
"Consistent Aggregation in
University of
Economies."
Competitive
f
British
Columbia,
Department of Economics (unpublished)
Diewert,
W.
E.
"The Theory of the Output Price Index and
(1983),
the Measurement of Real Output Change."
University of
Bri-
tish Columbia, Department of Economics, Discussion Paper No.
83-10.
Fisher,
F.
M.
Production
"The Existence of Aggregate
(1969),
Functions." Econometrica 37, pp. 553-577.
Fisher,
F.
"Aggregate Production Functions Revisited:
(1982),
M.
The Mobility of Capital and the Rigidity of Thought." Review
615-625.
of Economic Studies 49, pp.
Fisher,
F.
M.
Indices
Fisher,
(1972), The Econ omic The ory of Price
New York: Academic Press.
,
F.
and K. Shell
M.
and
Shell (1981),
K.
Indices."
Center for Analytic Research in
University of Pennsylvania,
Economics
"Output Price
CARESS Working Paper
and the Social Sciences,
#
81-05.
Leontief
,
W. W.
(1947)
,
"Introduction to
a
Theory of the Internal
Structure of Functional Relationships." Econometrica 15, pp.
361-373.
Muellbauer,
J.
Indices."
N.
J.
"The Theory of True Input Price
(1972),
University
of
Warwick,
Research Paper 17.
5 8 5
ii
u 6 7
37
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