BLS WORKING PAPERS

U.S. DEPARTMENT OF LABOR

Bureau of Labor Statistics

 

OFFICE OF PRICES AND LIVING

CONDITIONS

A Comparison of the Substitution Effects for Input and Output Price Indexes

Keith Waehrer, U.S. Bureau of Labor Statistics

Working Paper 327

April 2000

 

The views expressed are those of the author and do not reflect the policies of the U.S. Bureau of Labor Statistics or the

views of other staff members. The paper benefited from the helpful comments of Dennis Fixler, Jack Galvin, John Greenlees,

and Ken Stewart.
1

A Comparison of the Substitution Effects for Input

and Output Price Indexes

KEITH WAEHRER*

Division of Price and Index Number Research

U.S. Bureau of Labor Statistics

Current Version: February, 1999

ABSTRACT

The substitution effect is different for output and input indexes like the consumer price

index and the producer price index respectively. The conceptual index for both output

and input indexes are defined and compared. Then the conceptual indexes are compared

with some common index number formulas. Some results relating to which formulas are

closer to the conceptual index are reviewed.

 

*

 The paper benefited from the helpful comments of Dennis Fixler, Jack Galvin, John Greenlees, and Ken

Stewart.
2

A Comparison of the Substitution Effects for Input

and Output Price Indexes

The recent report to the Senate Finance Committee by the Advisory Committee on

the Consumer Price Index highlighted a number of sources of bias in the CPI; one of these

being substitution bias. While the Boskin Report only discusses the CPI, some have

suggested that its recommendations may also apply to the producer price index.

However, conceptual differences between CPI and PPI imply that substitution bias has

different effects on the two indexes and, thus, will likely require different remedies.

In this paper I describe the conceptual target indexes for input and output price

indexes.1 Conceptually the CPI is an input index and the PPI is an output index. Once

defined, it is straight forward to show that the Laspeyres is a lower bound on the

conceptual output price index and an upper bound on the conceptual input price index.

The geometric mean index has been proposed for use in the CPI, but a similar use in the

PPI may be inappropriate. The geometric mean index is always lower than the Laspeyres.

Hence, the Laspeyres must lie between the geometric mean and the conceptual output

price index. Therefore, the Laspeyres is always closer to the conceptual target output

price index than the geometric mean. I also give conditions that imply that the Laspeyres

is a better estimate of the conceptual output price index than the Fisher Ideal index.

Output Price Indexes:

The important indexes produced as part of the PPI are net output indexes. Net

output indexes are calculated for different industries. These indexes are used to deflate

the value of goods and services produced by these industries in order to determine

changes in real output. For instance, if the value of the output of the mining industry rises,

the PPI net output index for mining is used to determine how much of that rise is due to

an increase in the price of the goods produced by the mining industry and how much is

due to an increase in production.

Conceptually output indexes depend on how one defines an increase or decrease in

aggregate output. If each industry produced only one good, this problem has a

straightforward solution. Suppose that the mining industry’s only product is copper.

Then clearly the mining industry’s output increases if and only if the physical quantity of

copper produced increases. However, an individual industry such as mining produces a

 

1

 The concept of an output price index has been well developed. Much of the analysis of this paper has

been drawn from this literature. For example, see Robert Archibald (1975) “On the Theory of Industrial

Price Measurement: Output Price Indexes,” BLS Working Paper #44, W.E. Diewert (1983) “The Theory

of the Output Price Index and the Measurement of Real Output Change,” in Price Level Measurement,

W.E. Diewert and C. Montmarquette (eds.), Ottawa: Statistics Canada, pp. 1049-1113, and Franklin M.

Fisher and Karl Shell (1981) “Output Price Indices,” BLS Working Paper #120.
3

whole array of products. Therefore, it is more difficult to define increases and decreases

in aggregate output. While two output combinations may be different, they may be

considered the same level of output in an aggregate sense. When an index is used as a

deflator for aggregate output measures, whether intentionally or not, that index makes a

particular assumption about which output combinations are taken as equivalent in terms of

aggregate output.

Figure 1 diagramitically describes the difficulty when there are two goods

produced. Suppose that the mining industry produces only copper and iron. Each point

on the diagram represents a quantity of output for both iron and copper. It is clear that

points b and c represent a higher level of output than point a and point d depicts a lower

level of aggregate output than point a. More generally an output combination to the

northeast of point a would depict more aggregate output and an output combination to

the southwest of point a would represent less aggregate output. What is less clear is how

point e should be compared to point a: more, less, or equal in aggregate output.

Copper

Iron

l l

l

l

l

a

b

c

d

e

 Figure 1

Economists have resolved this issue by making use of the concept of a production

possibility frontier. A production possibility set is the set of all output combinations that

can be produced from a particular set of inputs and a particular technology. A production

possibility frontier (PPF) is the upper boundary of a production possibility set. Consider

Figure 2. The area enclosed by the curved line and the axes is an example of a production

possibility set. Since points a, d, and e are within the set, any of those combinations of

copper and iron can be produced with the same set of inputs and technology. (A

production possibilities set always includes its boundary.) Points a and e are on the
4

boundary of the production possibility set and thus are on the production possibility

frontier.

Once the production possibility frontier has been defined, it is easy to describe

which output combinations constitute more, less, or the same level of aggregate output for

the industry. Points on the same PPF as point a are considered the same level of

aggregate output. Thus, while two output combinations may be different, they are said to

represent the same level of aggregate output if they can be produced from the same initial

set of inputs and technology. Output combinations above point a’s PPF are considered

higher levels of aggregate output. And points below a’s PPF correspond to lower levels

of aggregate output for this industry. Therefore, for the purpose of calculating an output

price index, holding aggregate output constant can be thought of as staying on the same

PPF.

The conceptual index underlying the PPI is the Fixed-input Output Price Index

(FIOPI). Let pib and pic denote the price of good i in the base and current periods.

Define qib and qic as the quantity produced of good i in the base and current periods.

Then the FIOPI is defined as

{ ( ) }

{ q p (q q ) PPF}

q p q q PPF

I p p PPF

n iq i ib

n iq i ic

O b c

∈

∈

=

∑

∑

max | ,...,

max | ,...,

( , , )

1

1

This notation requires some explanation. The FIOPI is the ratio of revenues:

revenue in the current period divided by revenue in the base while staying on the same

production possibility frontier. Recall that we defined holding aggregate output constant

as staying on the same PPF or in other words holding inputs and technology constant.

The numerator in the expression above is the maximum revenue that can be generated

from output combinations on the production possibility frontier PPF when facing current

period prices. The denominator of the expression is the maximum revenue that can be

achieved from output combinations on the same production possibility frontier PPF when

faced with base period prices.
5

Copper

Iron

l l

l

l

l

a b

c

d

e

PPF

Figure 2

There are actually many possible production possibility frontiers that could be used in the

calculation of output indexes. As a result of the inputs and technology in use at any given

time, an industry faces a particular production possibility frontier. Over time the

production possibility frontier faced by an industry changes due to changes in the inputs

and technology used. In general, the choice of production possibility frontiers to use in

the FIOPI formula has an impact on the resulting index. Therefore, it seems difficult to

argue that the choice of production possibility frontier is completely arbitrary. Recall that

the production possibility frontier is used in the FIOPI essentially defines the definition of

aggregate output. Arguing that the choice of production possibility frontiers to use in a

FIOPI is arbitrary implies that the definition of aggregate output and analogously the price

index are to some extent arbitrary.

Two natural choices of production possibility frontier for use in the FIOPI are the

production possibility frontiers in the base and current periods. The base period

production possibility frontier PPFb is the boundary of the set of output combinations that

could be produced by the same technology and inputs used to produce the base period

outputs ( ,..., )

1 b nb q q . The current period production possibility frontier PPFc is the

boundary of the set of output combinations that could be produced by the same

technology and inputs used to produce the current period outputs ( ,..., )

1c nc

q q . There is

no reason to believe that PPFb and PPFc are the same or equivalent for the purposes of

calculating a FIOPI. I say that PPFb and PPFc are equivalent for the purposes of

calculating a FIOPI if ( , ', ) ( , ', ) O b O PPFcI p p PPF = I p p no matter what prices p and p’

are plugged into the formula.2

 

2

Production possibility frontiers with this relationship has also been described as parallel. This property is

also related to homothetic production possibility maps. A more detailed description of this property and
6

There are a number of reasons for the differences between PPFb and PPFc. Over

time advancing technology allows producers to produce the same level of output with

fewer inputs. Industries may also adjust some of the inputs that are used. Between the

base period and current period, the prices of the inputs used by the industry are likely to

have changed. Such a change will most likely cause a change in the combination of inputs

used by the industry and hence a change in the PPF associated with the two periods. Our

use of a FIOPI does not require us to assume that the inputs and technology used in the

actual economy remain fixed. However, the calculation of such an index requires us to

calculate revenues as if the PPF stays the same while output prices change. (This is

analogous to the calculation of a cost of living index where it is necessary to calculate a

consumers cost as if they were kept on the same indifference curve.)

Having settled on either PPFb and PPFc as natural choices for inclusion in the

FIOPI, let me define the following notation in order to simplify the expressions in this

paper. First, I define what is referred to as the dot product.

∑=

⋅ =

n

i

q p qi pi 1

Define max qxy as the output combination that maximizes revenue when faced with period x

prices while remaining on the PPF for period y. That is,

{ ( n) y } i

qxy⋅ px = max q ∑qipix | q1

,..., q ∈ PPF max

.

Therefore, the formula for the FIOPI can be written more compactly as

bc b

cc c

O b c c

bb b

cb c

O b c b

q p

q p

I p p PPF

q p

q p

I p p PPF

⋅

⋅

=

⋅

⋅

=

max

max

max

max

( , , )

( , , )

Under the assumption that utilized an industry’s output combination is selected to

maximize revenue conditional on the inputs and technology, we have

b b bb b

q ⋅ p = q ⋅ p

max

.

Therefore, the expression for FIOPI using the base period PPF can be written as

 

its relation to homotheticity can be found in Franklin M. Fisher and Karl Shell, Economic Analysis of

Production Price Indexes (Cambridge, Cambridge University Press, 1998).
7

b b

cb c

O b c b

q p

q p

I p p PPF

⋅

⋅

=

max

( , , )

While calculating an exact FIOPI for an industry might be ideal, it is impractical since

PPFb is not generally observable or known by statistical agencies, and thus, the actual

calculation of cb cq ⋅ p

max is impossible in the absence of strong assumptions. Being unable

to construct an index over fixed inputs and technology, the BLS instead constructs an

index based on fixed outputs. That is, the Bureau aims to calculate a Laspeyres index as

an estimate of the conceptual target. The Laspeyres index is defined as

b b

b c

L b c

q p

q p

I p p

⋅

⋅

( , ) =

However, notice that b c cb cq ⋅ p ≤ q ⋅ p

max since while b

q is on PPFb, we would not

necessarily expect bq to continue to be the revenue maximizing output combination when

faced with current period prices. We would expect the industry to adjust its output

combination to capture more revenue.

To fix ideas geometrically, consider Figure 3. Suppose that point a corresponds to

the base period output combination. Assuming that this output combination maximizes

the industry's revenue conditional on its inputs, there is a line Rb that runs through point a

and with slope equal to minus the price of copper divided by the price of iron in the base

period. All of the output combinations on Rb generate the same revenue as point a under

base period prices. All output combinations above Rb generate more revenue than Rb and

all output combinations below Rb generate less revenue than Rb . It is important to see

that in terms of the notation defined above Rb qb pb = ⋅ .
8

Copper

Iron

l

l

a

e

PPF

Rc

Rb

Rf

Figure 3

Now consider a price change that corresponds to a relative increase in the price of

copper. If the industry does not change its output combination, then its new revenue will

be Rf qb pc = ⋅ . However, notice that the industry could earn higher revenue while still

remaining on the base period production possibility frontier. Specifically, if the industry

produced output combination e, it could earn higher revenue. That is,

Rc qcb pc ≥ Rf = ⋅

max . Notice that when the relative price of copper rises, in order to

maximize revenue the industry should change its output combination to produce more

copper and less iron. That is, substitute in the direction of the good whose relative price

has risen.

It is now possible to define substitution bias for output indexes. The discussion

above can be summarized by the following chain of inequalities.

( , ) ( , , ) O b c b

b

c

b

f

L b c

I p p PPF

R

R

R

R

I p p = ≤ = (since Rf ≤ Rc

)

Therefore, assuming that industries maximize revenue conditional on the inputs it uses, the

Laspeyres index is a lower bound on the FIOPI. The difference between IL and IO being

the so called substitution bias. Hence, one would expect that a price index based on the

Laspeyres concept to be on average below the conceptual fixed-input output price index.

It is also possible to relate the relative size of a Paasche to a FIOPI in a way similar

to tat just presented. The Paasche index is defined as
9

c b

c c

P b c

q p

q p

I p p

⋅

⋅

( , ) =

Assuming that firms seek to maximize revenue conditional on their input utilization notice

that c b bc b

q ⋅ p ≤ q ⋅ p

max which follows from the fact that the current period

output combination is one of the feasible choices in PPFc and 

cc c c c

q ⋅ p = q ⋅ p

max which

follows as long as an industry maximizes revenue subject to its inputs. Therefore,

( , ) ( , , )

max O b c c

bc b

c c

c b

c c

P b c

I p p PPF

q p

q p

q p

q p

I p p =

⋅

⋅

≥

⋅

⋅

= .

To summarize what has been demonstrated thus far, we have

( , , ) ( , )

( , ) ( , , )

O b c b L b c

P b c O b c c

I p p PPF I p p

I p p I p p PPF

≥

≥

That is, the Paasche index is greater than a FIOPI based on the current period PPF, and

the Laspeyres index is less than a FIOPI based on the base period PPF. What is not

known is the relationship between ( , , ) O b c PPFcI p p and ( , , ) O b c PPFbI p p . If PPFb

and PPFc are “equivalent”, then we have ( , , ) ( , , ) O b c c O b c PPFbI p p PPF = I p p and thus,

P L

I ≥ I . However, as I have argued above there is no reason to believe that PPFb and

PPFc are “equivalent”. In fact, there is nothing in the theory that suggests that the

Paasche should necessarily be greater than the Laspeyres.

Consider the following simple two product example. Suppose that each good is

produced independently with different inputs and the goods’ demands are independent.

Let the market for good 1 remain unchanged between the base and current periods, but

the price of an input used to produce good 2 rises in these periods. This change will cause

the marginal cost of producing good 2 to rise and in a perfectly competitive equilibrium

the output price of good 2 must also rise to equal the new marginal cost, decreasing the

quantity demanded. Therefore, in this simple example we have q1b = q1c, p1b = p1c, q2b >

q2c, and p2b < p2c. It is not difficult to show that in such a simple case the Laspeyres will

exceed the Paasche. In fact, in a companion paper, Galvin and Stewart find empirical

evidence using PPI data that the Laspeyres index is greater in general than the Paasche

index.3 A table from their analysis is reproduced below.

 

3

J. Galvin and K. Stewart (1998) "Alternative Measures of Price Change for the U.S. Producer Price

Index," mimeo, Bureau of Labor Statistics.
10

Table 1. Laspeyres and Paasche PPIs for 2-digit product groups, total percentage

change, 1987-92

Producer Price Index

2-digit Product Group Category

Laspeyres Paasche Difference,

Laspeyres minus

Paasche

 Farm products (01) 13.55 9.40 4.15

 Processed foods and feeds (02) 15.41 14.57 0.84

 Textile products and apparel (03) 11.98 11.47 0.51

 Hides, skins, leather, and related products (04) 19.00 18.23 0.77

 Fuels and related products and power (05) 7.40 7.36 0.04

 Chemicals and allied products (06) 15.20 16.27 -1.07

 Rubber and plastic products (07) 10.28 10.20 0.08

 Lumber and wood products (08) 29.85 26.20 3.65

 Pulp, paper, and allied products (09) 17.23 15.08 2.15

 Metals and metal products (10) 12.78 10.17 2.61

 Machinery and equipment (11) 12.26 9.86 2.40

 Furniture and household durables (12) 10.77 9.91 0.86

 Nonmetallic mineral products (13) 6.75 6.90 -0.15

 Transportation equipment (14) 15.13 14.38 0.75

 Miscellaneous products (15) 26.13 24.08 2.05

Reproduced from J. Galvin and K. Stewart (1998) "Alternative Measures of Price Change for the U.S.

Producer Price Index," mimeo, Bureau of Labor Statistics.

While the calculation of a FIOPI requires the hypothetical output combination

max

cb q , its use does not necessarily imply that we expect supply conditions such as the

quantity of inputs or technology to actually remain unchanged from period to period.

Input Price Indexes:

For input price indexes like the CPI, the substitution bias works in the opposite

direction. Consumers substitute away from goods whose relative price have risen, and a

Laspeyres index is an upper bound of the “true” price index concept. The concept

underlying input indexes like the CPI can be referred to as Fixed-output Input Price Index

(FOIPI). Fixed-output Input Price Indexes are defined as
11

{ ( ) }

{ q p (q q ) IO}

q p q q IO

I p p BF

n iq i ib

n iq i ic

I b c

∈

∈

=

∑

∑

min | ,...,

min | ,...,

( , , )

1

1

where IO is an iso-output curve denoting the input combinations that produce a given

output combination while utilizing a particular level of technology. The iso-output curve

would be exactly analogous to an indifference curve in the theory of cost of living indexes.

Therefore, the following analysis also applies to cost of living indexes.

The iso-output curve is represented graphically in Figure 4. In the diagram input

combinations a and e can produce the same output combination. Combinations b and c

can produce more output than combination a, and combination d cannot produce as much

as input combination a.

It is now possible to interpret the expression for Fixed-Output Input Price Index.

The numerator is the least cost way of reaching a certain output combination given the

prices in the current period, and the denominator is the least cost way of producing the

same output combination given the prices in the base period.

Let min qxy denote the input combination that minimizes the cost of producing the

same output as in period y while facing the prices from period x. That is,

{ ( n) y } i

xy x q i ix q ⋅ p = min ∑ q p | q1

,..., q ∈ IO min

Input 2

Input 1

l l

l

l

l

a

b

c

d

e

IO

Figure 4
12

Let IOb and IOc denote the iso-output curves associated with the output

combinations in the base and current periods. Assuming that producers attempt to achieve

a particular output combination at minimum cost, we have

c c cc c

c b bc b

b c cb c

b b bb b

q p q p

q p q p

q p q p

q p q p

⋅ = ⋅

⋅ ≥ ⋅

⋅ ≥ ⋅

⋅ = ⋅

min

min

min

min

The explanations for these equalities and inequalities is similar to the arguments

presented in the previous section on output price indexes. If behavior is optimal, then the

input combination selected in the base period will be the least cost way of producing the

base output combination, and similarly, the input combination selected in the current

period will be the least cost way of producing the current period output combination. The

inequalities follow from the facts that the base and current period input combinations can

produce the base and current period output combinations respectively, but they do not

necessarily produce those output combinations at least cost when faced with different

prices.

A graphic representation of these facts is straightforward to derive using consumer

theory. Consider Figure 5. Let the straight line labeled Cb represents the set of input

combinations that cost the consumer the same amount of money as the base period input

combination point a when facing base period prices. That is, b b b C = q ⋅ p .

Suppose that between the base period and the current period the relative price of

input 1 rises. Let Cc’ denote the set of input combinations that cost the consumer the

same amount of money as the base period input combination under the current period

prices. Notice that it is possible to find a different input combination that produces the

same output combinations at a lower cost. That is, input combination e represents the

input combination on IOb that costs the least. Therefore, we have

c b c cb c Cc C′ = q ⋅ p ≥ q ⋅ p =

min

Notice that when the price of input 1 rises the consumer wishes to substitute some of

input 1 for input 2, effectively substituting away from the more expensive good.
13

Input 2

Input 1

l

l

a

e

BFb

Cc

Cc’

Cb

Figure 5

By making use of the expressions above we can establish the following

relationships between the Laspeyres, Paasche, and FOIPI.

( , , ) ( , , )

min

I b c b

b b

cb c

b b

b c

L b c b

I p p IO

q p

q p

q p

q p

I p p q =

⋅

⋅

≥

⋅

⋅

=

( , , ) ( , , )

min L b c c

c b

c c

bc b

c c

I b c c

I p p q

q p

q p

q p

q p

I p p IO =

⋅

⋅

≥

⋅

⋅

=

The Laspeyres is necessarily larger than the FOIPI when the base period iso-output curve

is used, and similarly the Paasche is necessarily smaller than the FOIPI when the current

period iso-output curve is used. However, as in the theory of output indexes, there is

nothing in the arguments made thus far that allows us to draw any conclusion regarding

the relative sizes of ( , , )

I b c b

I p p IO and ( , , )

I b c c

I p p IO .

Taking ( , , )

I b c b

I p p IO as the conceptual target, the Laspeyres index must over

estimate the target input index because it does not take into account that the

consumer/producer will reoptimize the input combination in response to the change in

prices. Since in the case of input price indexes, the consumer wishes to minimize its cost

of achieving a certain level of utility the Laspeyres which does not account for this

reaction over estimates the effect of the price increase.

Using Fisher and Geometric Means Indexes for Input and Output Price Indexes:
14

The actual calculation of price indexes is usually limited to formulas that only

require observed price and quantity data. In general this limitation makes the calculation

of FIOPI and FOIPI impossible since each would require hypothetical data. That is, the

numerator of the FIOPI using a base period production possibility frontier is the maximum

revenue that can be achieved under current period prices while remaining on the base

period PPF. Such a revenue calculation is purely hypothetical since the actual revenue

observed in the current period is likely to be associated with a PPF other than the base

period PPF. A similar problem arises in the case of input price indexes.

Various strategies have been proposed to circumvent these difficulties. I consider

two here: the Fisher index that requires prices and quantities from both periods and the

geometric mean index that requires prices from both periods and quantities only from the

base period. They are defined as follows.

( )

∑ ∏

∑

∑

∑

∑

=

















=

qib pib j q jb p jb

i G b c ic ib

i

ic ib

i

ic ic

i

ib ib

i

ib ic

F b c

I p p p p

q p

q p

q p

q p

I p p

( , )

( , )

2

1

Notice that the Fisher index, IF, is equal to the geometric average of the Laspeyres and the

Paasche indexes. Therefore, mathematically the Fisher must lie between the Laspeyres

and Paasche indexes. It is straight forward prove that the geometric mean index must

never be larger than the Laspeyres index. Note that,

ln ( , , )

ln

ln ( , ) ln

L b c c

i

ib

ic

j jb jb

ib ib

i

ib

ic

j jb jb

ib ib

G b c

I p p q

p

p

q p

q p

p

p

q p

q p

I p p

=

















∑

≤

















∑

=

∑

∑

The inequality in the expression above follows from Jensen's inequality.4 Thus,

( , ) ( , ) G b c L pb pcI p p ≤ I since for any x, y > 0 , ln x ≤ ln y implies x ≤ y.

The Fisher index has a number of properties that make it attractive as an index

number. The property most relevant for the discussion here is its status as a superlative

index. The Fisher index is equal to the conceptual index when the production function

takes the form of a generalized quadratic function. It has also been shown that under

more general conditions the Fisher index closely tracks the conceptual index bases on a

PPF that is an "average" of PPFb and PPFc .

 

4

 For a discription of Jensen's inequality see for example William Noveshek, Mathematics for Economists

(San Diego, Academic Press, 1993).
15

Before the Fisher and geometric mean indexes can be evaluated as possible input

or output price indexes it is important to settle on the conceptual target for each. I argue

in the sections above that the conceptual output and input price indexes are the FIOPI and

the FOIPI respectively. However, it remains to be determined what production possibility

frontier and iso-output curve to use as a basis in each of the formula. As I argue above,

natural choices appear to be the base or current period production possibility frontiers,

PPFb or PPFc and the base or current period iso-output curves, IOb or IOc. Its seems

natural to fix the PPF and IO in the base period. Therefore, for the purpose of discussion

I settle on ( , , ) O b c PPFbI p p and ( , , )

I b c b

I p p IO as the conceptual targets for output

and input price indexes.

First consider the case of the geometric mean. If it can be shown that the

Laspeyres always falls between the geometric mean and the target index, then it must be

the case that the Laspeyres index is closer to the target than the geometric mean index.

That is, the difference between the geometric mean and the target index must be larger

than the difference between the Laspeyres and the target. Notice this relationship always

holds for output indexes. We have shown

( , ) ( , ) ( , , ) G b c L b c O b c PPFbI p p ≤ I p p ≤ I p p

Hence, for the case of output indexes, the geometric mean must be farther from the target

index than the Laspeyres. Therefore, using closeness to the target index as the sole

criteria in judging an index formula, the Laspeyres is a better formula than the geometric

mean for output indexes. Intuitively, the geometric mean implies that firms would

decrease their production of products whose relative price increases when its inputs stay

the same. This is clearly not consistent with the hypothesis that firm maximize revenue

conditional in their inputs.

The same is not true for input indexes. Recall that we have shown that

( , , ) ( , )

I b c b L b c

I p p IO ≤ I p p ;

the relative quantities of the geometric mean and the target input price index cannot be

definitively established without further assumptions. Therefore, it cannot be the case that

the Laspeyres falls between the target input index and the geometric mean. However,

with the assumptions made thus far, the geometric mean is not necessarily closer to the

target index than the Laspeyres.

Now let us consider the case of the Fisher index. As stated above, the Fisher index

will always fall between the Paasche and Laspeyres indexes. Therefore, just as we did in

the case of the geometric mean, we can establish definitively that the Laspeyres is closer to

the target index than the Fisher if the Paasche is less than the Laspeyres for output indexes

and if the Paasche is greater than the Laspeyres for input indexes. That is,
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( , ) ( , ) ( , , ) ( , ) ( , ).

 and

( , ) ( , ) ( , ) ( , ) ( , , )

L b c P b c I b c b L b c F b c

P b c L b c F b c L b c O b c b

I p p I p p I p p IO I p p I p p

I p p I p p I p p I p p I p p PPF

≤ ⇒ ≤ ≤

≤ ⇒ ≤ ≤

It is possible to make additional assumptions that make the Fisher index closely

approximate the target index. One such assumption is that the production possibility

frontiers or the iso-output curves are parallel (also know as the homothetic case). When

production possibility frontiers are parallel, it must be the case that

( , ) ( , , ) ( , , ) ( , ) L b c O b c b O b c c P pb pcI p p ≤ I p p PPF = I p p PPF ≤ I .

Therefore, anytime ( , ) ( , ) L b c P pb pcI p p > I , the production possibility frontiers cannot

be parallel.

In fact, Galvin and Stewart, in their paper (see Table 1), provide empirical

evidence that for PPI data the Laspeyres is often larger than the Paasche index. In such

cases the Laspeyres index will actually be closer to a FIOPI based on the base period PPF

than a Fisher index. However, basing the FIOPI on an "average" of base period and

current period PPFs will imply that Fisher index is a better estimate than the Laspeyres.

Summary:

The theory of input and output price indexes indicate that substitution effects in

the two are fundamentally different. For input price indexes like the CPI, consumers

would like to consume less of goods whose relative price increases. For output price

indexes like the PPI, the theory indicates the firms would want to produce a larger

quantity of those goods whose relative price increases. Hence, the two substitution effects

are in fact in opposite direction.

The conceptual input and output price indexes require hypothetical calculations

that are often impractical to perform. Therefore, other index formulas are used in their

place. If an index formula is to be judged by its proximity to the conceptual target, then

regardless of the assumptions made the geometric mean index is never a better candidate

output index than the Laspeyres index. On the other hand, under certain assumptions the

geometric mean index is exactly equal to the conceptual input price index. Therefore,

while the geometric mean index may be appropriate for the CPI, it is not likely to be

appropriate for implementation in the PPI.

The Fisher index will definitely be farther from the conceptual output price index

than the Laspeyres index when the Laspeyres index is larger than the Paasche. While for

input price indexes, the Fisher is a worse approximation of the conceptual input price

index than the Laspeyres when the Laspeyres is less than the Paasche.
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Price data reflects the interaction of supply and demand in the real world. If price

fluctuations are predominantly caused by the supply curve moving up and down the

demand schedule, then we should expect to see price and quantity demanded moving in

opposite directions and, thus, the Laspeyres greater than the Paasche. This type of effect

would result if, for instance, the price or supply of productive inputs change over time.

On the other hand, if price fluctuations are predominantly caused by the demand curve

moving up and down the supply curve, then we should expect to see price and quantity

moving in the same direction and, thus, the Laspeyres less than the Paasche. Changes in

consumer preferences or income can result in demand curve fluctuations.