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ieaH
working paper
department
of economics
THE PRODUCTION-THEORETIC MEASUREMENT
OF INPUT PRICE AND QUANTITY INDICES
Franklin M. Fisher
No.
575
April 1991
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
THE PRODUCTION-THEORETIC MEASUREMENT
OF INPUT PRICE AND QUANTITY INDICES
Franklin M. Fisher
No.
575
April 1991
JUL
1
8
^
TBB-BB0DHCIIDBrTBBDBBTIC-asaSHB£[3BSI-0E
IHEBT-BBICB-aMD-QUaHTIKf-IUDICSS
Franklin M. Fisher
^
Massachusetts Institute of Technology
Paper prepared for Conference in Honor of Zvi
*
Jerusalem,
May 1991.
forthcoming
tually)
Griliches,
This paper adapts material from my (even-
with Karl Shell
book
on
the
production-
theoretic approach to price and quantity measurement.
la_lD£loguc£iQD
has long been understood that conventional
It
cost
the
living such as the consumer price
of
approximations to
a
measures
index
but
are
"true" index based on consumer theory.
It is
well accepted that measurement of prices and quantities
less
production
are
or ought to be focussed on approximating
indices oased on production theory.
gregates
for production,
properties.
Indeed,
of
in
"true"
in calculating ag-
concentration has oeen on arithmetical
In a real sense,
and practice has dominated.
practice has come before
theory,
Yet the question of what it is chat
we are trying to measure is of oovious importance.
In the form considered here,
in
Fisher and Shell
(1972)
.
that question was firsz raised
Snell and
I
considered the case of
the measurement of output and, especially, output prices when the
production
frontier
oauer
system
(P?F)
(1972)
is descrioed by
of a closed economy.
the
production
possibility
Following that, John Muell-
pointed oun that an isomorphic theory can oe
for the case of input deflation and proceeded to do so,
built
deriving
many interesting results.
Unfortunately,
not
however, Kuellbauer's isomorphic theory does
economy characterization of
its
That is oecause the closed-
seem of great direct interest.
PPF oy fixed factor supplies has as
a
parallel the characterization of output as
vector.
fixed
a
Tnis
means that the index numoer question oeing asked has to
with
the relative input usage required to produce a fixed output
vector under two different regimes.
narrow
treatment.
a
(Ti\e
production
Since firms and
systems usually do not face fixed demand vectors,
do
this seems too
single-
exception is the case of the
output competitive firm.)
Fortunately,
low
widening taat treatment does enaoie us to fol-
up on Muellbauer's insight that input deflation and measure-
ment requires
case,
production-theoretic treatment.
a
conditions that
demand
the
given are not represented oy
out
by
deflation
a
tiiis
at
quantities
vector of fixed output
a
as
firm
can
oe
to
assumption.
treatment does not end the matter, however, for input
and
measurement is seldom done at the
individual competitive firm.
sold
takes
input deflation and usage proolem ought tnus
analyzed using
Such
competitive firm
vector of fixed output prices at which the
a
Tne
sell.
a
In the simplest
v7hile tne same
level
the
of
treatment of outputs
fixed prices will serve for the case of
a
small
fully
open economy trading outputs on world markets at fixed prices,
will
not
cover
the
case of a large economy or of
industry fating downward sloping demand curves.
a
firm
it
or
As it turns out,
such cases can involve substantial complexities and cast doubt on
tne use of Paasciie and Lasoevres indices as appropriate bounds.
monetary magnitude
A
It
change into a price effect and
that
the money value of
example,
we
changes.
wish to speak of
(or deflationary)
a
.,.__;..ut
real
change.
Thus,
for
is observed to increase,
and
a
real output change versus an
effect.
separate
desired to
is
inflationary
Similarly, when money expenditures on
we may wish to speak of the extent to which this is
inputs rise,
simply due to changes in factor prices and the extent to which it
represents
In
real increase
a
case
the
income
(or decrease)
of the consumer,
a
in tne use of resources.
change in the value
to De separated into a price effect and a
is
money
of
change
in
real income.
One important thing that all such examples have in common is
that
is natural to think
it
changes
the side of the markets being studied faced
on
under analysis
unit
deflation;
supply
price index.
action
—
from
by
the
supply prices for factors in the case
prices for goods in the case of the
of
output
demand prices for goods in the case of
deflation;
input
of the price effect as coming
consumer
Then any change in real magnitudes must involve an
on the part of the unit that is different from that which
would have been observed had the unit,
other circumstances
with
unchanging, been faced with the price changes.
Consider
a
competitive firm producing
change in costs is to be divided into
real change in input usage.
single output.
A
factor-price effect and
a
If we consider factor price changes
as coming from outside the firm,
would
a
a
then it is natural to ask
have happened to costs had the otner circumstances of
what
cne
.
effect
had
this amounts to asking
what
changes in factor prices would have had if the
firm
remained constant.
firm
the
Clearly,
remained on the same isoquant as in the base
change
any further change
the pure price effect;
is
represents
a
That
period.
costs
in
real change.
There is another way to think about this.
Any
solution to
the problem of measuring real aggregate input use must oegin with
an answer to the following question:
vectors
input
ootain
to
Since we shall be reducing
index-numser scalars,
complete ordering for the resulting
a
vectors to wnich they correspond,
input
and since we
\;e
scalars
and
must choose
classes for input vectors such that all
equivalence
wisn
a
to
the
set of
vectors
in
the same equivalent class will be said to involve the same aggre-
How is this to be done?
gate input.
In
question
the case of the single-output firm,
immediately suggests itself
—
the use of the isoquants
of the production function to define equivalence classes.
given technology,
this
the answer to
we will say thai: two input vectors
With
correspon-
ding to the same output involve the same aggregate input use.
factor
prices change and the
firra
remains on the same
a
If
isoquant,
then any cnange in money input costs will oe considered as a pure
price pnenomenon.
Ooviously,
price index
.
tiiis
is
i_,o..,orpnic
to the theory of the consumer
There the consumer faces prices as given from out-
side and the equivalence classes are provided by the indifference
curves
In the case of production,
however, matters need not always
oe quite so simple.
For one thing, the production unit involved
have monopoly or monopsony power,
may
taken as given from outside.
the principles just described.
deflation and
Thus,
taken
A change in the money
.
place had the production unit
responded
—
—
real
to
change.
—
we separate the world into price effects
all cases,
—
would
merely
caused oy the ocher side of the markets being studied
effects
input
in the case of
altered supply conditions and the remaining
In
generalization
input can then be separated into the change that
of
nave
a
firm with monopsony power, input prices cannot be
a
taken as given, but supply conditions can
value
be
shall treat such cases in detail
I
Suffice it to say now that they involve
later.
of
so that prices cannot
—
and real
the actions of the productive unit being studied that
are not merely reactions to the price effects.
But the case
of
monopoly or monopsony points up the somewhat artificial nature of
such a division.
After all, when we scudy
a
unit as large as the
entire productive system, the prices at which outputs can be sold
not oe independent of the actions of the production
will
system
itself, even if no monopoly power is involved.
The
answer
monetary
single
here is simple.
The division of
change
a
into a single measurement of price change and
value
measurement of quantity change is necessarily
In the true world of general equilibrium effects,
is truly possible.
In particular,
sector,
can
production
conditions).
sector
ask
no such change
if we wish to examine the
wnat -would have happened if
had faced different input prices
Answering
such
questions provides
a
arbitrary.
all we can do is to ask "what
Instead,
questions.
we
in
if"
production
zhe
original
(or
insight
supply
into
whether real input usage has increased or decreased,
even though
actual change has consequences beyond the production
any
sector
itself.
3*_SiG)£l£_lDBU£_D£flaliOD_iD_IJQrsJ2e£ail
now concentrate on the easy case of simple input deflation
I
—
deflation of inputs in the case of
the
Tnis
firm.
case
a
is quite instructive,
single
and
the
competitive
propositions
will be seen to be of quite general applicability,
discussed
so
chat a gooo deal more than an example is involved.
[FIGURE 3.1 HERE]
the base-period isoquant is drawn with base-
In Figure 3.1,
period factor prices,
input
use
notation,
is v,
I
w,
and total input costs C =
wv
.
(For
on the context to make clear what is intended.)
current
period,
wv*
.
the
oeen
ease
of
omit transposition signs when writing inner products,
relying
lines.
Optimal
represented by the solid line.
factor
prices,
w,
In
are denoted by the
the
dashed
Actual factor usage is v* and actual money costs are C* =
Had factor prices been w instead of w in the base
period,
output corresponding to the base-period isoquant would
efficiently produced with inputs v rather than v and
have
money
UN
costs would have been C rather than C.
tion
The view of input-defla-
taken here is that the change in costs from C to C*
should
be thougnt of as:
(3.1)
C*/C
=
(C*/C) (C/C)
,
with the first factor the increase in real aggregate input
and the second reflecting price changes.
usage
more
In
general
where
C(w, x),
base-period
let the firm's cost
output,
x,
function
The construction just given
is output.
x
input costs as C(w,
the
terms,
and calculates the deflator
x)/C(w,
x)
.
be
uses
money
for
This deflator is divided into
relative change in money input costs to give the measure
of
real aggregate input usage relative to the base period.
It
bounded
the
not harci co see that the deflator just
is
above by
deflator
oounded
Laspeyres index of input prices.
a
oased
below
oy
technology and
a
produced
on the current period's
Paasche index.
a
nomothetic isoquant map,
Similarly,
isoquanc
In the case
is
will
be
unchanging
of
ooth deflators will oe
the same and both bounds will apply.
Evidently,
phic
Fisher-Shell
approach leads to
a
of the cost-of-living index
that
to
this
theory of output deflation)
.
theory largely isomor(as well
In considering
sible objections to the theory oeing advanced,
well
what those same objections
consider
to
to
as
therefore,
imply
the
posit is
about
the
relatively well-established theory of the cost-of-living index.
1.
Space
objections here.
does
not
permit consideration of
In particular, Diewert
the parallel case of output deflation)
(1983)
all
possible
points out (using
that the method here given
will often not lead to a measure of real input usage that doubles
if
the
firm exactly doubles its usage
matters are taken up in Fisher (1988)
One
of
every
input.
Such
.
such objection is as follows.
The procedure just
de-
scrioed treats input vectors as identical if they can produce the
output and treats
same
as
an input increase.
a
movement to a higher-numbered isoquant
But we are trying to build a
input aggregation and measurement.
theory
Is it not odd that levels of
outputs become central to the theory?
Moreover, different firms
different technologies facing the same set of input
with
of
prices
will have different: factor price deflators constructed for them.
The answer lies in consideration of the object of the enter-
prise.
are treating the firm as the object of interest with
We
factor prices given from outside.
of
Any production-theoretic view
input deflation must involve the production function
as tne cost-of-living index describes price changes
Just
firm.
the
of
from the point of view of the individual consumer, so the produc-
tion-theoretic
input price index describes factor price
The
the point of view of the individual firm.
from
different
not
firms have different points of view,
valid objection.
a
prices.
input
To
a
fact
that
so to speak,
The aggregation problem
points cannot be solved by choosing
changes
to
is
which
it
firm-independent measure of
do tnat is merely to impose on all
firms
a
measure not relevant to any one of them.
Two
more comments on this before proceeding.
the point of view of the input-producing sector
be
made up of firms if inputs are
construction
intermediate
of an index of prices for
(which could also
products),
sector.
Here,
to
But output
is a different enterprise with price changes taken
originating in the purchasing sector.
deflation,
the
its cu£pu£§ ought not
depend on the production functions of its customers.
deflation
from
First,
in tne case of
as
input
price cnanges are taken as originating in the selling
Tnis
is
consonant with tne general view of price
and
quantity indexation taken above.
Second,
index
the fact that the production-theoretic input
will
different for firms
be
with
price
production
different
functions does not mean that nothing can be said about the nature
of such dependence.
to
On the contrary, interest certainly attaches
the way in which such indices change as the production
tion varies over firms or over time.
with
func-
Such matters can be treated
comparative static analysis (altnough space does not permit
doing so in the present paper).
When
we examine more general
(and more
cases
interesting)
than that of the single-output competitive firm, the same princiIn each case, we shall examine what would have
ples will apply.
happened had the productive unit in question faced the new set of
input prices
will
(or,
more generally,
input supply conditions).
restrict the productive unit to an isoquant or to
a
We
genera-
lization thereof.
Before proceeding to the details of that analysis,
two
They
can be discussed at the
subjects
present,
however,
general
level.
are the treatment of corner solutions and the treatment
of
quality changes in inputs.
3*_CarD£r..Selu£iSDS
The
issues
resolved.
goods
in
The
the
involved in corner solutions here
isomorphic case to that of new
or
analysis of the cost-of-living index
production-theoretic
output
price
index is that of
are
readily
disappearing
or
a
the
of
new
or
This does not seem to be
disappearing
factor of production.
particularly
interesting case unless we are considering as
a
fac-
not primary factors but inputs of goods or materials
tors
If we do consider
one sector or country buys from another.
cases,
then
which,
in
is not hard to imagine
it
produces
effect,
technical
a
such
discovery
previously
good that serves as a
a
which
unknown factor of production from the point of view of the sector
or economy that purchases it.
Despite
analysis
—
tioned.
It
(1972,
plain
is
99-105)
for
the
there
is no need to give
from consideration of Fisher
is
reservation
a
problem for
Paasche index
a
bounding
The
between
em-
intercept on
the
considered would be just indifferent
factor
it
is
c»i^
demand
productive
(This is the price at which the
ploying the factor and not doing so;
base-period
not
and
period's
base
any price at or above its
price.
oeing
Shell
and
Paasche index will be preserved if the factor price
a
chosen
men-
already
isomorphisms
production-theoretic input price index.
property of
lengthy
a
that the problem of what factor price to use
new factor is only
a
sector
cases,
part because of the
in
pp.
for
so
such
demand curve.) Of all prices in that
range,
Paascne
the demand reservation price produces the most efficient
lower bound.
Further, that price can also be interpreted as the
shadow
of
price
the constraint involved in not being
able
to
employ the factor in question in the base period.
It is
which
important to realize,
however,
that corner solutions
arise through the appearance of new goods or the disappea-
rance of old ones do not raise similar problems.
the
isoquant.
relative to which the
production-theoretic
price deflator is calculated is defined for
10
This is because
a
input
given set of output
Just how those conditions are properly defined is a
conditions.
matter we consider below,
be
but however they are defined they will
the same for both periods.
question;
change,
New goods do therefore
raise
a
the appearance of a new good, like any other technical
will
generally alter the isoquant which determines
production-theoretic
input price index.
the
This is quite a diffe-
rent class of effects from those involved in corner solutions
in
£ac£or. space.
To
take
the simplest example,
describes
isoquant
suppose that
the efficient factor combinations
given set of goods in specified quantities.
ducing
a
goods
either includes the new one or it does not.
is
relevant
the
pro-
for
That set of
What the set
certainly affects the shape of the isoquant and the value
of
the resulting input price deflator but the fact that a
different
of goods would produce a different isoquant and a
different
set
raises no problem for the construction of the
deflator
deflator
using the given isoquant corresponding to a given set of goods.
analysis of quality change
The
presents
somewhat
deeper problems.
In the theory of the cost-of-living index,
question
is
a
fairly natural
that of the treatment of quality change in
more of the goods consumed.
one
or
While such changes can be handled in
principle as the disappearance of one good (with the old quality)
and the appearance of a new good
not
done in practice and,
theory
of
(with the new quality)
indeed,
the cost-of-living index,
11
,
is rather awkward.
quality improvement
this is
the
In
in
a
good is equivalent to an improvement in the opportunity
consumed
set
facing consumers and tnerefore equivalent to a fall
cost
(1972,
one
living
of
pp.
can
and
analyzed such effects and considered
26-37)
treat
Fisher
prices remain constant.
if
quality improvement as equivalent to
a
rally
done,
oe
change
independent
involved
consumed
dependent
(e.g.
The prin-
gene-
virtual
price
prices
of all other
only on the physical
amounts
and
nature
the
of
The necessary
quality change) under very special circumstances.
and
virtual
while this could
could only be done with the
it
Shell
whether
a
price decrease in the good whose quality had changed.
cipal result was the demonstration that,
the
in
sufficient conditions for such treatment are that the consu-
mer view the new quality of good as exactly equivalent to a fixed
number of units of the old quality, i.e., that the quality change
enter the utility function as
a
shift in
meter
.
>:
(as b
in U(bx,
±
ties
consume^
case
"repackaging"
good
as
a
"
c.;ic
,
x of
.
.
i.
U(.)
n
)
,
a
good-augmenting
where the
;:
.
are the quanti-
1
is the utility function)
We called
since the consumer regards the
repackaged" version of the old one,
Obviously, this is
a
para-
quality
new
so
this
to
speak.
very special case, and the result calls
into question a whole class of methods used to adjust for quality
change,
a
class
of
whic/i
.i^acnics is the
most
sophisticated
representative.
The case of output price deflation has an isomorphic problem
(considered
turns out
equivalent
used
in Fisher and Shell 1972,
pp.
105-7)).
There
it
chat one would scili wish to treat a quality change as
to a price decrease.
This is because the
resources
to produce the new quality of good could have been used
12
to
produce
different amounts of the original quality had
desired it.
In effect, quality change is interpreted as simply a
in demand.
shift
The isomorphic statement to
repackaging
the
is that quality improvement can be treated as a
theorem
decrease
price of the good whose
the
in
independent
for
frontier
quality
'
virtual
changed
has
of prices and quantities of all goods produced
under special circumstances.
tions
consumers
such
(PPF)
only
The necessary and sufficient condi-
treatment are that the
production
possibility
with the new quality good differ from the PPF with
tne old one by a shift in a good-augmenting parameter.
This
is
equivalent to requiring that the production functions for the new
and
old quality good differ by
so that,
of
the
a
in terms of resource use
Kicks-neutral technical
change
(rather than utility), one unit
new quality of good is equivalent to
a
fixed num,ber
of
units of the old.
Leaving the repackaging theorem aside for the moment,
important
quality
to recognize the difference between the
constructed
technical
deflator.
treatment
is
of
change and that of technological change in the theory of
The production-theoretic output price deflator
output deflation.
is
it
from the point of view of
a
specific
will shift that PPF and (usually)
change
However,
PPF.
change
A
the
the PPF used to construct the deflator will
be constant when comparing base and current period prices even if
the
technical change occurred oetween the two periods.
This is
because the question to be answered concerns the responses
an
sets
economy with
of prices.
a
which
given PPF would have made when faced with
A shift in the PPF may increase real output
13
cwo
—
if
it leads to a greater value of production than would have been
achieved
—
prices
the base period's PPF and
with
period's
but this is do£ because the change is equivalent to
With base-period and current-period prices
price decrease.
a
the
deflator will be unity whichever. PPF is used to
the
same,
current
the
con-
struct it.
Quality
guous.
It
on the other hand,
change,
is treated as
unambi-
is treated as though the choice between new and
quality of goods reflected not
a
£emaDy conditions.
quality change requiring more
Hence,
a
change in supply but
change in
a
re-
to produce the new quality than to proauce the old leads
sources
to an unambiguous increase in real output if the number of
of
old
units
the new quality of good produced in the current period is the
as the number of units of the old quality of good
same
in the base period,
The
decrease.
Indeed,
because
with the production of all other goods
quality
constant.
produced
change
prices
is thus equivalent to
price
a
will be treated as declining
of the quality change,
held
solely
even if money prices remain
the
same in the two periods.
Obviously,
whether
a
it
is
a
matter of some importance
theory of the cost-of-living index,
the
decide whether
or
a
2.
and Shell
This
demand.
it
given change is a taste change,
quality change due to supply.
a
decide
given change should be treated as technological and due
to supply or as a quality change and due to
in
to
2
)
is important
due to
7-8).
14
to
demand,
The basic question to
is not always as simple as it looks.
(1972, pp.
(Similarly,
be
See Fisher
asked is always whether,
if prices do not change,
one wishes to
consider the value of the deflator as necessarily altered by
change under consideration.
Now,
If so,
then it is a quality change.
question of how a given change sno^id be
the
arises again when we consider input deflation.
to ask is whether,
say
treated
Here the question
with money input prices the same,
that input prices have gone down as
we wish to
result of the
a
then the change will be treated as
If so,
the
quality
a
change.
change;
if
not, then it will be treated as a technical change.
inside
important to realize here that
is
It
change that
unit whose input prices are to be deflated
the
treated as
a
a
technical change (just as such
a
occurs
change in the
of households is treated as a shift in taste).
be
will
case
In general,
the
of a more efficient process within the production unit
discovery
itself will not be considered as a decrease in input prices.
price
input
index is to measure the cost
usefulness of inputs once they are used.
other
hand,
means
a
which
Changes
in
we
inputs,
not
the
A quality change on the
decrease in input costs for §DY
technology
Thus, the kind of quality change
used by the production sector.
with
of
An
are here concerned is change in
eu£py£ quality will simply ae treated
changes for purposes of input deflation.
treat an su£pu£ quality improvement as
a
iopy£
as
quality.
technical
It is not reasonable to
virtual decline in iBPy£
prices.
3.
Uote the reversal of roles from the case of output price
deflation.
There
technical change
—
quality change in an iDPut was treated as
a
a
change in the PPF
15
.
Here it is treated as
a
a
On the other hand, a quality change in an cu£2u£
quality change.
—
—
there treated as such
is here treated as a technical change
change in the isoquant.
a
To
suppose that the change in question concerns
fix ideas,
quality
the
more
productivity is due to
a
productive.
change and not as
inputs.
education of labor,
increase
in
as
a
decrease in the effective cost
a
of
the increase is due to oetter
then one may very well wish to treat it as
in input costs and a reduction
decrease
virtual
that
If
then this will be treated
the other hand,
on
If,
factors
(other
discovery on the part of firms as to how
use labor more efficiently,
tecnnical
labor
labor, so that
of
constant) becomes
to
—
input
the
in
a
price deflator.
The
issue is not
workers
the
simple one,
suppose
being
that the production sector whose
input
inputs
—
For
quality
a
more efficient way of using
"raw" labor.
those
cases in which
theorem
are
a
they
Or do
given
set
applies
a
a
given change is
treated
as
result isomorphic to that of
and is very
natural.
A
in the price of that
a
the
quality
improvement in an input can always be treated as equivalent to
decrease
of
Either answer is possible.
change in an input,
repackaging
costs
Do better edu-
cated workers represent lower virtual input prices?
reflect
to
training.
on-the-job
considered is that of the entire economy.
simply
prices
this is not so (or at least not obvi-
if the education comes through
so)
educated
Better
however.
may represent an effective decrease in input
firm that hires them;
ously
Now
a
a
input with input quality constant.
16
price
That
decrease
will be independent of
purchases only under special conditions,
input
however.
prices
Necessary and
sufficient conditions for such independence are that the
change
quality
can be represented within the productive technology as
shift
in
parameter augmenting the factor whose
a
In other words,
changed.
unit
one
and
'
a
has
quality
the quality change must be such
that
of the new quality of factor is exactly equivalent
in
old
quality
of
the
repackaging
production
to
fixed numbetf of units of the
a
factor.
may
It
condition
thought that this version
be
seems more likely to oe satisfied in the present
quality changes in factors than in either the case
of
augmenting
changes
function
the utility
in
—
in production functions
changes
spectively.
considered
change
not
Once
as
a
too
augmenting.
the conditions that apply
strictive
quality change in
a
than
factor rather
stringent to suppose that tne cnange
But is this really so?
index
tion
a
is
worker,
a
then
—
it
—
perhaps
deflation
is not obvious that factor
totally natural assumption.
for example,
is
be
a
may
factor-
quality change already supposes
in the case of output price
than
re-
as
it
to
Aside from the fact that the
restrictive set of circumstances
cost-of-living
cated
deflator,
one has decided that a given change is to
decision to treat the change as
fairly
good-
of
in technology affecting how that factor is used,
seem
case
Hicks-neutral
or
cost-of-living index and to the output price
the
a
of
Can
an
more
re-
or
the
augmentaeducated
really do eysiyibiog oetter than an unedu-
one and better in the same proportion no matter
17
what
the
task?
not
ging
If not,
then the quality change involved in education is
merely factor-augmenting in the way required by the repackatheorem.
simple adjustment in wages will
No
suffice
to
production-theoretic
account for the effect of the change on the
input price deflator or on the corresponding production-theoretic
index of real input use.
5 J _Tbe - Eully_QpeD-£§§e
As already indicated,
the easiest case for the analysis
economy-wide input price deflation is that of
my.
In
fully open econo-
a
the present circumstances this amounts to assuming that
the productive sector
sell
of
(like a multiproduct competitive firm)
all the outputs it wants at fixed output prices.
can
The iso-
quant relative to which the input price deflator is defined
then
becomes
the locus of all efficient input combinations which will
produce
output
The
plans
bundles of equal value.
economic
F(x,v)
x
production
efficient
can be summarized in terms of the production
(5.1)
where
unit's technologically
relation
= 0,
We are
is the output vector and v is the input vector.
also given a vector of output prices p =
(p,
,
.
.
. ,
p
)
and a
money value of output, y.
The
tneoretic
isoquant
to
be used in constructing
the
production-
input price deflator is defined in the following
Choose any output vector, x, whose value at prices p is y.
consider the set of v corresponding to that x and the
function
F(x,v)
=
0.
Next
production
This is an isoquant for the production
18
way:
of
.
the
given vector x.
Consider the family of isoquants generated
in this way by varying x over all vectors satisfying px = y.
The
isoquant that will be used for input price deflation is the lower
envelope of this family.
Each point,
on that envelope also
v,
on some isoquant defined for x fixed for some x with
lies
total
value of output at the given output prices is equal to y.
Formally, we have:
I__ = {v| v is minimal subject to F(x,v)
(5.2)
Given
the isoquant
I
pn
the construction of the correspon-
r
ding production-theoretic input price index is
application
given two input price vectors, w A and w
C
(5.3)
B
We
above.
First
.
straightf orward
a
the general approach discussed
of
and px = y}.
=
are
define
v/e
A =
A
min w v subject to v lying on I^q.
v
Let
be the minimizing value of v in Problem (5.3);
v
h,—
ave C
t\
»*i
= w v
=
C
we
c\
.
Similarly,
(5.4)
thus,
we define
min w v subject to v lying on
v
I
po
.
•p
be the minimizing value of v in Problem (5.4);
Let v
B B
= w v
nave „B
C
The
.
production-theoretic
money costs at input prices w
A
input price index
quant,
Ir,^,
which
comparing
.
value is held constant at y is then
This index is,
for
.
to money costs at input prices
outputs can be freely sold at fixed prices,
when
thus, we
A
(C
/C
R
)
is determined by technology
19
B
and output
p,
defined relative to
of course,
w
a
given iso-
(4.1),
output
prices,
output
logy,
period
bound
prices,
indexed as B)
,
and output value of the base period
(the
then a Laspeyres input price
will
If they are the technology,
output prices,
above.
and output value
of
current
period (indexed as A),
then a Paasche input
price
will
bound the corresponding
production-theoretic
index
index
from below.
If the two isoquants are
production-theoretic
two
the
index
corresponding production-theoretic index from
the
the
If these are the actual techno-
and output value, y.
p,
parallel along rays,
indices will be
equal
then
botn
and
oounds will apply.
will such isoquants be parallel in this
When
conditions are:
cient
(i)
common to both periods;
and
(ii)
Suffi-
way?
The production technology F(.,.)
F(.,.)
is
is constant returns to scale;
the output prices, p, are proportional in both periods.
(iii)
These conditions
—
—
particularly that of constant returns
are
far stronger than necessary.
When
such
parallelism along rays is not present,
production-theoretic
input
differ
price indices will
and Laspeyres bounds need not both hold
Paasche
the
two
and
the
simultaneously.
Sa.CeDgfsl-SsBSD^^CsDdiiioDsi-Tbs-UflDeppiisiic^gase
The
are
where
fully open case just analyzed in which
output
demands
perfectly elastic is the appropriate one for input deflation
the
productive sector involved is small
—
a
firm
or
a
small group of firms in competition, or a small country in inter-
national trade.
Larger units or aggregates,
so simply treated.
however, cannot be
We must therefore analyze the case of decli-
ning demand curves.
20
It turns out to matter a good deal whether or not the agents
in
the production sector realize that they face declining demand
curves and take that fact into account in their decision
If
they
—
do
simpler
—
than if they do not
behaves
firm
the case of monopoly
present
—
the analysis is
open
section takes up the monopoly case;
each
situation.
The
the more
difficult
competitive case is treated later.
production plans are
efficient
before,
As
somewhat
where
the competitive case
as if it were in the fully
(and more interesting)
making.
given
by
the
relation
F(x,v) =
(6.1)
,
where
x
Let
p be the corresponding r-vector of output
is an r-vector of outputs and v is an m-vector of inputs.
prices;
outputs
are sold according to the demand schedule
(6.2)
x
= x
D
(p)
If demand for some good is perfectly elastic at a constant price,
then
the
corresponding component of
price, zero above it, and any value in
x
is infinite
[0,
+oo]
below
that
at the exogenous-
ly given price.
that we observe the economic unit in question
Suppose
convenience,
prices
revenue,
the
"economy")
producing an output vector
p* so that its total revenue is y* = p*x*.
y*
,
Fixing
(for
x*
at
total
what are the output combinations which the economy
could have sold?
The answer depends on what is assumed about the
output demand conditions which the economy faces.
21
In Figure 6.1,
the
point
For the fully
x* is indicated.
economy,
open
line shows the output combinations consistent with
dashed
revenue, y*
i.e.
,
{x
|
p*x* = y*
} .
the
total
Points to the northeast of the
dashed line produce more revenue for the fully open economy
The solid curve represents the output combi-
it receives at x*.
nations
consistent
demand schedules,
x
(p*).
than
i.e.
economy
y* for the
with
{x
|
x
= x
(p)
declining
facing
and px = y*
where x*
} ,
=
The solid curve lies to the northeast of the dashed line
reflecting
fact that increased outputs can only be sold
the
at
lower prices.
From Figure 6.1, we see that the fully open model provides
more
"optimistic" isoquant than does the declining demand
dule
model.
showing
This
is reflected in the fact that
a
sche-
the
isoquant
to
generate
the efficient factor combinations required
the given revenue y* shows greater required inputs with declining
demand than in the fully open case.
It
theoretic
not hard to see,
is
price
input
demand-curve
therefore,
that
deflator constructed in
case will be less than that for
production-
the
declining-
the
corresponding
the
fully open case because the fully open economic unit with a given
isoquant
responds
more fully to factor ojI^l. changes than
the unit facing declining demand curves.
Hence
a
does
given change in
the money value of costs will be considered more of a real change
and
less
of
a
monetary one when declining
demand
curves
are
present than when they are not.
The
isoquant
for the general case of
unit facing declining demand curves,
22
IM
,
is
a
monopoly
economic
defined as follows:
}
(6.3)
=
I
"K
{v
is minimal subject to F(x,v)
v
|
= x
x
It
and px = y
,
important to realize that this isoquant is based
is
assumption that the economic unit "sees" and acts
the
entire demand schedule.
but
px
(p)
= 0,
minimizes
rather
D
(p)
its
It does not take output prices as given
subject
cost
its
In other words,
= y.
upon
on
in deriving
constraint
the
to
we have assumed the
I..,
economic unit to have monopoly power in its product markets.
The
case of competition with declining demand is studied later.
Wow, assume that
I.,
is derived from technological and output
market conditions actually prevailing in the base period.
by the superscript B,
conditions
these
that
so
Denote
isoquant
the
becomes
3
(6.4)
I
=
.
{v
v is minimal subject to F
|
x = x
Assume
that
v
,
DB
,
P
(x,v)
0,
,
and px = y
,
(p)
=
,
,
}
.
period
the actual vector of base
inputs
a
minimizes
constraint, F
3
(x,v)
= 0,
the demand constraint
B
revenue constraint px = y
in
and I„^,
ru
(6.4)
share
(5.2))
B
of I„~.
4
I
isoquant
market
B
.
DB
I.,
3
defined
I,.,
isoquant
the corresponding fully open
but
and the
(p)
(see
lies above and to the right
denote the production-theoretic input price
37.
would lie below and to the left of
defined
basket
= x
x
Then the two isoquants,
.
common point,
a
Let
4.
subject to the technological
costs at factor prices w
of
v/ith
perfectly inelastic
outputs
—
a
demands
but the latter is of
interest.
23
closed"
"fully
—
very
a
fixed
little
deflator defined relative to
input price deflator defined
L 2
(6.5)
jJJ
I
,.
and J pn the production-theoretic
relative to
I,,,-.
Then we have:
i Jp
where L denotes the Laspeyres input price index.
Similarly, for
the indices derived from current-period conditions,
(superscrip-
ted A) we have:
P * J
(6.6)
A
<
[v;
j£ Q
,
where P denotes the Paasche input price index.
Note
flator
that because the production-theoretic input price
de-
for the monopoly case is always greater than or equal
corresponding
a
given
real
input
usage in the fully open case than in the case of monopoly.
This
the
increase
in
deflator for the fully
to
open
money costs will be attributed more to
corresponds to the isoquants drawn in Figure 6.1.
case
has a flatter isoquant,
a
case.
Cost
The fully open
so that factor price changes
will
the
monopoly
changes resulting from movements along an
isoquant
greater movement in factor usage than in
induce
case,
are counted as monetary only.
now turn to the important case of
I
unit
(an industry,
There
taking
competitive
The economic unit faces
demand coniditions as in the preceding section
not
know it.
output
prices
economic
facing declining output demand schedules.
is no monopoly power.
general
does
say)
a
The firms which make up the
as given as
24
in the fully
same
the
—
but it
unit
optimize
open
case
of
Section
above.
5
In fact, however,
not in a fully open environment,
taken all together, they are
and output prices do depend on
the sum of their decisions.
This
view
fact raises a new problem.
production-theoretic
input deflation asks what the economic unit would
of
spent
The
inputs at the new input prices holding
on
output constant.
have
value
the
of
But now holding output value constant is not a
simple matter.
It cannot be done,
as in the fully open case, by
restricting the output vector to an isovalue line at fixed output
because output prices are not fixed.
prices,
it cannot be done,
economic
unit
On the other hand,
as in the monopoly case, by assuming that the
minimizes
cost subject to outputs
lying
on
an
isorevnue curve, because the economic unit does not in fact solve
such
problem.
a
theoretic
input
Indeed,
price
the construction of the
deflator in this case is
problem in constrained optimization;
a
demand
in
that construction also
cases so far considered.
and
statics
far more difficult than in the
Further,
a
in-
supply
This makes comparative
all output markets.
(not treated in this paper)
merely
not
fixed-point argument ensuring equality of
volves
production-
other
Paasche and Laspeyres bounds
need no longer apply.
As in the preceding section, technology is summarized by
F(x,v)
(7.1)
5.
industry.
=0
5
Note that (7.1) gives the production technology for
In the absence of constant returns,
25
the
this will general-
:
}
.
ly not also be the tachnology for the individual firm.
This is a
matter of no consequence here, however.
demand by
and output
(7.2)
= x
x
D
(p)
Firms in our competitive industry face a given output
vector,
p,
perceive
Section
god ac£
as.
Thus, the firms
ds£ affsci i£.
£}p
as operating in a fully open economy as
themselves
5.
if they
price
If that perception were correct,
industry facing output prices,
in
the isoquant for the
and earning total revenue, y,
p,
would be
I„_(p)
t u
(7.3)
=
v is minimal subject to F(x,v)
{v|
and px
=
= y
By fixing total revenue, y, and varying output prices, p, we
derive from (7.3) the implied industry supply
can
outputs
schedule
for
(parametric on y, of course)
(7.4)
= x S (p)
x
How, we cannot case the analysis of our competitive industry
on
the isoquant,
I^Cp),
defined in (7.3).
derived for fixed output prices,
at the industry level.
the
Second,
First,
The
fact
that
x
D
,
(p)
is
and output prices are not fixed
I
(p)
does not take into account
industry-wide constraint that supply and demand for
must be equal, i.e., that
Ic
(p)
= x
S
(?)
this constraint is not
recognized
competitive industry makes the analysis complex.
26
outputs
by
the
Tnis is because
equilibrium output prices which equate output
the
demands
and
themselves depend on factor prices since output supplies
so depend.
Thus, while the competitive industry, given y, mini-
mizes costs while remaining on I^-Jp)
depends on factor prices,
is
supplies
for some p,
that
whj,£h p
Were w different, p would also
w.
be different, and the competitive economy would solve a different
problem.
must therefore take this into account and (in principle)
We
use the p that corresponds to equilibrium in output markets given
the factor prices involved.
defining the production-theoretic input price index
In
the competitive general case,
relation
(7.1),
J n r-r
total revenue,
First,
production
we are given the
the output demand schedules
y,
(7.2), and two input price vectors, w
for
A
and w
take the output price vector,
B
.
as a parameter and
p,
let
C
A
(p)
=
A
min w v
x,v
(7.5)
subject to F(x,v) =
(C
A
is
the
(p)
is thus money cost at factor prices,
on I„_(p).)
r O
w
and px = y
A
,
given that input
Let the minimizing input vector be v
resulting vector of optimal output supplies be
x
SA
(p)
find that value of p (for convenience assumed to be unique)
A
D
SA
that x
(p) = x (p); call it p
.
Similarly, let
27
and
(p)
.
Now
such
C
B
=
(p)
B
min w v
x,v
(7.6)
subject to F(x,v)
Let the minimizing input vector be v
of
optimal output supplies be
x
SB
(p)
CO
(assumed unique) such that x
The
and px = y
=
(p)
and the resulting vector
.
Now find the value of
= x
(p)
p
D
V)
(p)
call it p
;
production-theoretic input price index
.
appropriate
to
the competitive general case is then defined by
J
(7.7)
The
=
CG
C
(p
)/C
B
B
(p
)
fact that the firms making up the competitive
unit
behave
unit
as
the
difficulties
for
whole does not creates
a
First,
becomes difficult.
parameter
practical
comparative static analysis (not here
The effect of
a
shift in
given
a
does not only involve changes in the solution
now
the unit's optimizing proolem given that shift
handled
economic
as though they face flat demand curves whereas
further analysis.
treated)
A
A
by the Envelope Theorem)
.
Such
a
to
(which are readily
shift
involves
also
shifts in the other parameters of the unit's optimization problem
through
changes
markets.
in
the position of equilibrium in
output
all
Without more information as to demand schedules,
such
shifts cannot be studied.
Second,
and
more important for the practice of price index
construction,
Laspeyres and Paasche bounds are no longer guaran-
teed to hold.
To see this, Let B denote actual base-period and A
3
actual current-period conditions, respectively.
Then C
denominator
Laspeyres
of J ,,_ and the denominator
28
of the
(p
)
,
the
input
.
price
index
will be the same.
establishing
J
C
CG ,
which
A
the
A
(p
situation
(p
)
bound would be to show that the
This is no longer guaranteed.
in which the value of v
(just for good measure),
by
Plainly,
.
B
numerator
of
w
A
(repre-
is lower than
C
A
A
(p
at prices
the value of v
the slopes of the solid lines)
Laspeyres
period)
Figure 7.1 shows a
at factor prices
the slope of the dashed lines)
by
(represented
C
of
is the value of the solution to a minimum problem in
)
was feasible.
Further
method
the usual
(the actual input combination used in the base
v
sented
However,
w L
lower
is
)
than
and Paasche bounds are inapplicable
here.
Except where demand curves can in f§£i
The moral is clear.
be
taken as approximately flat because the unit is
Paasche
Laspeyres
and
for output price deflation.
than
enough,
another case
generating
in
There,
—
if the unit
inappropriate
to
assume
Here,
laige
is
fixed,
as
economy
that of tne fully colsed
which Paasche and Laspeyres bounds apply.
Fisher and Shell 1972.)
the
The case is even worse
appropriate to treat factor supplies
is
it
bound
input price indices will not
production-theoretic input price indices.
small,
very
(See Essay II
no matter how big the unit,
that the demand curves
it is
faces
it
of
are
perfectly inelastic.
and Laspeyres input price indices will thus give
Paasche
picture except for very small
misleading
other
cases,
general,
of
detailed
economic
technology as well) will be required.
(and,
One
assume that changes in input prices leave output prices
ted, and this creates a serious proolem.
29
For
units.
knowledge of demand schedules
a
in
cannot
unaffec-
course,
Of
dices
this difficulty with Paasche and Laspeyres
It comes about because
is a form of aggregation problem.
the situation facing an entire industry is not that perceived
the firms that make it up.
of
If
at
prices from the point of view of
Laspeyres
weights would have to be
quantity
sponding to an individual firm's purchases).
input
narrow
used
an
quantity
measurement,
however,
is
on an industry or economy-wide basis,
production theory is not
those
Paasche
corre-
Input deflation and
seldom done
from
When Paasche or Laspeyres indices
a point of view.
a
strong one.
30
to
individual
such problems would not arise (although even then
firm,
and
input
by
(parallel to the case of the cost
living index and an individual household) we were content
look
in-
their
grounding
so
are
in
.
Diewert,
W. E.
(1983),
"The Theory of the Output Price Index and
the Measurement of Real Output Change."
University of
Bri-
tish Columbia, Department of Economics, Discussion Paper No.
83-10.
Fisher,
and
P.M.
(1988),
"Production-Theoretic Input Price Indices
Measurement
the
DS3SUX£ffi£Qt_iD_5fiflD2IDiS5
Aggregate
Real
of
(W.
Eichhorn,
Input
ed
.
)
,
Use,"
in
Heidelberg:
Physica-Verlag
Fisher,
F.
M.
and K. Shell
(1972), TQQ„E£QnsRi£^Tb£QL}l-QL*.'ELi££
lB$iS§§r New York: Academic Press.
Muellbauer,
J.
Indices."
N.
J.
(1972),
"The Theory of True Input Price
University of Warwick,
search Paper 17.
31
Revision of Economic
Re-
V,
Figure
3.1
Economy with
declining demand
schedules
Fully open
economy
Figure
6.1
1
V-
C B (p B
)
WB V A
(Ap*)
WA V B
Ifo(p
V
Figure
7
367
1339
7.
B
)
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