# 16. Axiomatic and Stochastic Approaches to Index Number Theory

- International Monetary Fund
- Published Date:
- September 2004

## A. Introduction

**16.1** As Chapter 15 demonstrated, it is useful to be able to evaluate various index number formulas that have been proposed in terms of their properties. If a formula turns out to have rather undesirable properties, then doubt is cast on its suitability as a target index that could be used by a statistical agency. Looking at the mathematical properties of index number formulas leads to the *test* or *axiomatic approach to index number theory*. In this approach, desirable properties for an index number formula are proposed; then it is determined whether any formula is consistent with these properties or tests. An ideal outcome is that the proposed tests are desirable and completely determine the functional form for the formula.

**16.2** The axiomatic approach to index number theory is not completely straightforward, since choices have to be made in two dimensions:

The index number framework must be determined; and

Once the framework has been decided upon, the tests or properties that should be imposed on the index number need to be determined.

The second point is straightforward: different price statisticians may have different ideas about what tests are important, and alternative sets of axioms can lead to alternative best index number functional forms. This point must be kept in mind while reading this chapter, since there is no universal agreement on what is the best set of reasonable axioms. Hence, the axiomatic approach can lead to more than one best index number formula.

**16.3** The first point about choices listed above requires further discussion. In the previous chapter, for the most part, the focus was on *bilateral index number theory*; that is, it was assumed that prices and quantities for the same *n* commodities were given for two periods, and the object of the index number formula was to compare the overall level of prices in one period with that of the other period. In this framework, both sets of price and quantity vectors were regarded as variables that could be *independently varied*, so that, for example, variations in the prices of one period did not affect the prices of the other period or the quantities in either period. The emphasis was on comparing the overall cost of a fixed basket of quantities in the two periods or taking averages of such fixed-basket indices. This is an example of an index number framework.

**16.4** But other index number frameworks are possible. For example, instead of decomposing a value ratio into a term that represents price change between the two periods times another term that represents quantity change, one could attempt to decompose a value aggregate for one period into a single number that represents the price level in the period times another number that represents the quantity level in the period. In the first variant of this approach, the price index number is supposed to be a function of the *n* product prices pertaining to that aggregate in the period under consideration, and the quantity index number is supposed to be a function of the *n* product quantities pertaining to the aggregate in the period. The resulting price index function was called an *absolute index number* by Frisch (1930, p. 397), a *price level* by Eichhorn (1978, p. 141), and a *unilateral price index* by Anderson, Jones, and Nesmith (1997, p. 75). In a second variant of this approach, the price and quantity functions are allowed to depend on *both* the price *and* quantity vectors pertaining to the period under consideration.^{1} These two variants of unilateral index number theory will be considered in Section B.^{2}

**16.5** The remaining approaches in this chapter are largely bilateral approaches; that is, the prices and quantities in an aggregate are compared for two periods. In Sections C and E, the value ratio decomposition approach is taken.^{3} In Section C, the bilateral price and quantity indices, *P(p ^{0}, p^{1}, q^{0}, q^{1})* and

*Q(p*, are regarded as functions of the price vectors pertaining to the two periods,

^{0}, p^{1}, q^{0}, q^{1})*p*and

^{0}*p*, and the two quantity vectors,

^{1}*q*and

^{0}*q*. Not only do the axioms or tests that are placed on the price index

^{1}*P(p*reflect reasonable price index properties, some of them have their origin as reasonable tests on the quantity index

^{0}, p^{1}, q^{0}, q^{1})*Q(p*. The approach in Section C simultaneously determines the best price and quantity indices.

^{0}, p^{1}, q^{0}, q^{1})**16.6** In Section D, attention is shifted to the *price ratios* for the *n* commodities between periods 0 and 1, *r _{i}≡ p_{i}^{1}/p_{i}^{0}* for

*i*= 1,…,

*n*. In the

*unweighted stochastic approach to index number theory*, the price index is regarded as an evenly weighted average of the

*n*price relatives or ratios,

*r*

_{i}. Carli (1804; originally published in 1764) and Jevons (1863, 1865) were the early pioneers in this approach to index number theory, with Carli using the arithmetic average of the price relatives and Jevons endorsing the geometric average (but also considering the harmonic average). This approach to index number theory will be covered in Section D.1. This approach is consistent with a statistical approach that regards each price ratio

*r*as a random variable with mean equal to the underlying price index.

_{i}**16.7** A major problem with the unweighted average of price relatives approach to index number theory is that it does not take into account the economic importance of the individual commodities in the aggregate. Arthur Young (1812) did advocate some form of rough weighting of the price relatives according to their relative value over the period being considered, but the precise form of the required value weighting was not indicated.^{4} However, it was Walsh (1901, pp 83–121; 1921a, pp 81–90) who stressed the importance of weighting the individual price ratios, where the weights are functions of the associated values for the commodities in each period, and each period is to be treated symmetrically in the resulting formula:

What we are seeking is to average the variations in the exchange value of one given total sum of money in relation to the several classes of goods, to which several variations [price ratios] must be assigned weights proportional to the relative sizes of the classes. Hence the relative sizes of the classes at both the periods must be considered. (Correa Moylan Walsh, 1901, p. 104)

Commodities are to be weighted according to their importance, or their full values. But the problem of axiometry always involves at least two periods. There is a first period and there is a second period which is compared with it. Price variations

^{5}have taken place between the two, and these are to be averaged to get the amount of their variation as a whole. But the weights of the commodities at the second period are apt to be different from their weights at the first period. Which weights, then, are the right ones—those of the first period or those of the second? Or should there be a combination of the two sets? There is no reason for preferring either the first or the second. Then the combination of both would seem to be the proper answer. And this combination itself involves an averaging of the weights of the two periods. (Correa Moylan Walsh, 1921a, p. 90)

**16.8** Thus, Walsh was the first to examine in some detail the rather intricate problems^{6} in deciding how to weight the price relatives pertaining to an aggregate, taking into account the economic importance of the commodities in the two periods being considered. Note that the type of index number formulas that he was considering was of the form *P(r, v ^{0}, v^{1})*, where

*r*is the vector of price relatives that has

*i*th component

*r*and v

_{i}= p_{i}^{1}/p_{i}^{0}^{t}is the period

*t*value vector that has ith component

*v*for

_{i}^{t}= p_{i}^{t}q_{i}^{t}*t*= 0,1. His suggested solution to this weighting problem was not completely satisfactory, but he did at least suggest a useful framework for a price index as a value-weighted average of the

*n*price relatives. The first satisfactory solution to the weighting problem was obtained by Theil (1967, pp 136–37), and his solution will be explained in Section D.2.

**16.9** It can be seen that one of Walsh’s approaches to index number theory^{7} was an attempt to determine the best weighted average of the price relatives, *r _{i}*. This is equivalent to using an axiomatic approach to try to determine the best index of the form

*P(r, v*. This approach will be considered in Section E below.

^{0}, v^{1})^{8}

**16.10** Recall that in Chapter 15, the Young and Lowe indices were introduced. These indices do not fit precisely into the bilateral framework because the value or quantity weights used in these indices do not necessarily correspond to the values or quantities that pertain to either of the periods that correspond to the price vectors *p ^{0}* and

*p*. In Section F, the axiomatic properties of these two indices with respect to their price variables will be studied.

^{1}## B. The Levels Approach to Index Number Theory

### B.1 Axiomatic approach to unilateral price indices

**16.11** Denote the price and quantity of product *n* in period *t* by *p _{i}^{t} and q_{i}^{t}*, and

*q*, respectively, for

_{i}^{t}*i*= 1,2,…,

*n*and

*t*= 0,1,…,

*T*. The variable

*q*is interpreted as the total amount of product

_{i}^{t}*i*transacted within period

*t*. In order to conserve the value of transactions, it is necessary that

*p*be defined as a unit value; that is,

_{i}^{t}*q*must be equal to the value of transactions in product

_{i}^{t}*i*for period

*t*divided by the total quantity transacted,

*q*. In principle, the period of time should be chosen so that variations in product prices within a period are quite small compared with their variations between periods.

_{i}^{t}^{9}

For *t* = 0,1,…,*T*, and *i* = 1,…,*n*, define the value of transactions in product *i* as *v _{i}^{t} ≡ p_{i}^{t}q_{i}^{t}* and define the

*total value of transactions in period t*as

**16.12** Using the notation above, the following *levels version of the index number problem* is defined as follows: for *t* = 0,1,…,*T*, find scalar numbers *P ^{t}* and

*Q*such that

^{t}**16.13** The number *P ^{t}* is interpreted as an aggregate period

*t*price level, while the number

*Q*is interpreted as an aggregate period

^{t}*t*quantity level. The aggregate price level

*P*is allowed to be a function of the period

^{t}*t*price vector,

*p*, while the aggregate period

^{t}*t*quantity level

*Q*is allowed to be a function of the period

^{t}*t*quantity vector,

*q*. As a result we have the following:

^{t}**16.14** The functions *c* and *f* are to be determined somehow. Note that equation (16.3) requires that the functional forms for the price aggregation function *c* and for the quantity aggregation function *f* be independent of time. This is a reasonable requirement, since there is no reason to change the method of aggregation as time changes.

**16.15** Substituting equations (16.3) and (16.2) into equation (16.1) and dropping the superscript *t* means that *c* and *f* must satisfy the following functional equation for all strictly positive price and quantity vectors:

for all *p _{i}* > 0 and for all

*q*

_{i}> 0.

**16.16** It is natural to assume that the functions *c(p)* and *f(q)* are positive if all prices and quantities are positive:

if all *pi* > 0 and for all *qi* > 0.

**16.17** Let 1_{n} denote an *n* dimensional vector of ones. Then equation (16.5) implies that when *p = 1 _{n}*,

*c(1*is

_{n})*a*positive number, a for example, and when

*q = 1*, then

_{n}*f(1*is also a positive number,

_{n})*b*for example; that is, equation (16.5) implies that

*c*and

*f*satisfy

**16.18** Let *p = 1 _{n}* and substitute the first expression in equation (16.6) into (16.4) in order to obtain the following equation:

**16.19** Now let *q = 1 _{n}* and substitute the second part of equation (16.6) into (16.4) in order to obtain the following equation:

**16.20** Finally substitute equations (16.7) and (16.8) into the left-hand side of equation (16.4) and the following equation is obtained:

for all *p _{i}* > 0 and for all

*q*> 0. If

_{i}*n*is greater than 1, it is obvious that equation (16.9) cannot be satisfied for all strictly positive

*p*and

*q*vectors. Thus, if the number of commodities

*n*exceeds 1, then there are no functions

*c*and

*f*that satisfy equations (16.4) and (16.5).

^{10}

**16.21** Thus, this levels test approach to index number theory comes to an abrupt halt; it is fruitless to look for price- and quantity-level functions, *P ^{t} = c(p^{t})* and

*Q*, that satisfy equations (16.2) or (16.4) and also satisfy the very reasonable positivity requirements in equation (16.5).

^{t}= f(q^{t})**16.22** Note that the levels price index function, *c(p ^{t})*, did not depend on the corresponding quantity vector

*q*, and the levels quantity index function,

^{t}*f(q*, did not depend on the price vector

^{t})*p*. Perhaps this is the reason for the rather negative result obtained above. As a result, in the next section, the price and quantity functions are allowed to be functions of both

^{t}*p*and

^{t}*q*.

^{t}### B.2 A second axiomatic approach to unilateral price indices

**16.23** In this section, the goal is to find functions of *2n* variables, *c(p, q)* and *f(p, q)* such that the following counterpart to equation (16.4) holds:

for all *p _{i}* > 0 and for all

*q*> 0.

_{i}**16.24** Again, it is natural to assume that the functions *c(p, q)* and *f(p, q)* are positive if all prices and quantities are positive:

if all *p _{i}* > 0 and for all

*q*> 0.

_{i}**16.25** The present framework does not distinguish between the functions *c* and *f*, so it is necessary to require that these functions satisfy some reasonable properties. The first property imposed on *c* is that this function be homogeneous of degree 1 in its price components:

Thus, if all prices are multiplied by the positive number λ, then the resulting price index is λ times the initial price index. A similar linear homogeneity property is imposed on the quantity index *f*; that is, *f* is to be homogeneous of degree 1 in its quantity components:

**16.26** Note that the properties in equations (16.10), (16.11), and (16.13) imply that the price index *c(p, q)* has the following homogeneity property with respect to the components of *q*:

Thus *c(p, q)* is homogeneous of degree 0 in its *q* components.

**16.27** A final property that is imposed on the levels price index *c(p, q)* is the following: Let the positive numbers *d _{i}* be given. Then it is asked that the price index be invariant to changes in the units of measurement for the

*n*commodities, so that the function

*c(p, q)*has the following property:

**16.28** It is now possible to show that the properties in equations (16.10), (16.11), (16.12), (16.14), and (16.15) on the price-levels function *c(p, q)* are inconsistent; that is, there is no function of *2n* variables *c(p, q)* that satisfies these quite reasonable properties.^{11}

**16.29** To see why this is so, apply equation (16.15), setting *d _{i} = q_{i}* for each

*i*, to obtain the following equation:

If *c(p, q)* satisfies the linear homogeneity property in equation (16.12) so that *c(λp, q) = λc(p, q)*, then equation (16.16) implies that *c(p, q)* is also linearly homogeneous in *q*, so that *c(p,λq) = λc(p, q)*. But this last equation contradicts equation (16.14), which establishes the impossibility result.

**16.30** The rather negative results obtained in Section B.1 and this section indicate that it is fruitless to pursue the axiomatic approach to the determination of price and quantity levels, where both the price and quantity vector are regarded as independent variables.^{12} Therefore, in the following sections of this chapter, the axiomatic approach to the determination of a *bilateral price index* of the form *P(p ^{0}, p^{1}, q^{0}, q^{1})* will be pursued.

## C. First Axiomatic Approach to Bilateral Price Indices

### C.1 Bilateral indices and some early tests

**16.31** In this section, the strategy will be to assume that the bilateral price index formula, *P(p ^{0}, p^{1}, q^{0}, q^{1})*, satisfies a sufficient number of reasonable tests or properties so that the functional form for

*p*is determined.

^{13}The word bilateral

^{14}refers to the assumption that the function

*p*depends only on the data pertaining to the two situations or periods being compared; that is,

*p*is regarded as a function of the two sets of price and quantity vectors,

*(p*, that are to be aggregated into a single number that summarizes the overall change in the

^{0}, p^{1/}, q^{0}, q^{1})*n*price ratios,

*p*.

_{1}^{1}/p_{1}^{0},…, p_{n}^{1}/p_{n}^{0}**16.32** The value ratio decomposition approach to index number theory will be taken; that is, along with the price index *P(p ^{0}, p^{1}, q^{0}, q^{1})*, there is a companion quantity index

*Q(p*such that the product of these two indices equals the value ratio between the two pe- riods.

^{0}, p^{1}, q^{0}, q^{1})^{15}Thus, throughout this section, it is assumed that

*p*and

*q*satisfy the following

*product test*:

The period *t* values,*v _{t}*, for

*t*= 0,1 are defined by equation (16.1). Equation (16.17) means that as soon as the functional form for the price index

*p*is determined, then equation (16.17) can be used to determine the functional form for the quantity index

*Q*. However, a further advantage of assuming that the product test holds is that if a reasonable test is imposed on the quantity index

*Q*, then equation (16.17) can be used to translate this test on the quantity index into a corresponding test on the price index

*P*.

^{16}

**16.33** If *n* = 1, so that there is only one price and quantity to be aggregated, then a natural candidate for *p* is *p _{1}^{1}/p_{1}^{0}*, the single-price ratio, and a natural candidate for

*q*is

*q*, the single-quantity ratio. When the number of products or items to be aggregated is greater than 1, index number theorists have proposed over the years properties or tests that the price index

_{1}^{1}/q_{1}^{0}*p*should satisfy. These properties are generally multidimensional analogues to the one good price index formula,

*p*. In sections C.2through C.6, 20 tests are listed that turn out to characterize the Fisher ideal price index.

_{1}^{1}/p_{1}^{0}**16.34** It will be assumed that every component of each price and quantity vector is positive; that is, *p ^{t} > > 0_{n} and q^{t} > > 0_{n}*

^{17}for

*t*= 0,1. If it is desired to set

*q*, the common quantity vector is denoted by

^{0}= q^{1}*q*; if it is desired to set

*p*, the common price vector is denoted by

^{0}= p^{1}*p*.

**16.35** The first two tests are not very controversial, so they will not be discussed in detail.

T1—*Positivity*:^{18}*P(p ^{0}, p^{1}, q^{0}, q^{1})* > 0.

T2—*Continuity*:^{19}*P(p ^{0}, p^{1}, q^{0}, q^{1})* is a continuous function of its arguments.

**16.36** The next two tests are somewhat more controversial.

T3—*Identity or Constant Prices Test*:^{20}

*P(p, p, q ^{0}, q^{1})* = 1.

**16.37** That is, if the price of every good is identical during the two periods, then the price index should equal unity, no matter what the quantity vectors are. The controversial part of this test is that the two quantity vectors are allowed to be different. ^{21}

T4—*Fixed-Basket or Constant Quantities Test*:^{22}

That is, if quantities are constant during the two periods so that *q ^{0} = q^{1} ≡ q*, then the price index should equal the revenue in the constant basket in period 1,

**16.38** If the price index *p* satisfies test T4 and *p* and q jointly satisfy the product test, equation (16.17), then it is easy to show^{23} that *q* must satisfy the identity test *Q(p ^{0}, p^{1}, q, q)* = 1 for all strictly positive vectors

*p*. This constant quantities test for

^{0}, p^{1}, q*q*is also somewhat controversial, since

*p*and

^{0}*p*are allowed to be different.

^{1}### C.2 Homogeneity tests

**16.39** The following four tests restrict the behavior of the price index *p* as the scale of any one of the four vectors *p ^{0}, p^{1}, q^{0}, q^{1}* changes.

T5—*Proportionality in Current Prices*:^{24}

*P(p ^{0},λp^{1}, q^{0}, q^{1}) = λP(p^{0}, p^{1}, q^{0}, q^{1})* for λ > 0.

**16.40** That is, if all period 1 prices are multiplied by the positive number λ, then the new price index is λ times the old price index. Put another way, the price index function *P(p ^{0}, p^{1}, q^{0}, q^{1})* is (positively) homogeneous of degree 1 in the components of the period 1 price vector

*p*. Most index number theorists regard this property as a fundamental one that the index number formula should satisfy.

^{1}**16.41**Walsh (1901) and Fisher (1911, p. 418; 1922, p. 420) proposed the related proportionality test *P(p,λp, q ^{0}, q^{1}) = λ*. This last test is a combination of T3 and T5; in fact, Walsh (1901, p. 385) noted that this last test implies the identity test T3.

**16.42** In the next test, instead of multiplying all period 1 prices by the same number, all period 0 prices are multiplied by the number λ.

T6—*Inverse Proportionality in Base-Period Prices*:^{25}

*P(λp ^{0}, p^{1}, q^{0}, q1) = λ^{1}P(p^{0}, p^{1}, q^{0}, q^{1})* for λ > 0.

That is, if all period 0 prices are multiplied by the positive number λ, then the new price index is 1/λ times the old price index. Put another way, the price index function *P(p ^{0}, p^{1}, q^{0}, q^{1})* is (positively) homogeneous of degree minus 1 in the components of the period 0 price vector

*p*.

^{0}**16.43** The following two homogeneity tests can also be regarded as invariance tests.

T7—*Invariance to Proportional Changes in Current Quantities*:

*P(p ^{0}, p^{1}, q^{0},λq^{1}) = P(p^{0}, p^{1}, q^{0/}, q^{1})* for all λ > 0.

That is, if current-period quantities are all multiplied by the number λ, then the price index remains unchanged. Put another way, the price index function *P(p ^{0}, p^{1}, q^{0}, q^{1})* is (positively) homogeneous of degree zero in the components of the period 1 quantity vector

*q*. Vogt (1980, p. 70) was the first to propose this test,

^{1}^{26}and his derivation of the test is of some interest. Suppose the quantity index

*q*satisfies the quantity analogue to the price test T5; that is, suppose

*q*satisfies

*Q(p*for λ > 0. Then using the product test in equation (16.17), it can be seen that

^{0}, p^{1}, q^{0},λq^{1}) = λQ(p^{0}, p^{1}, q^{0}, q^{1})*p*must satisfy T7.

T8—*Invariance to Proportional Changes in Base Quantities*:^{27}

*P(p ^{0}, p^{1},λq^{0}, q^{1}) = P(p^{0}, p^{1}, q^{0}, q^{1})* for all λ > 0.

That is, if base-period quantities are all multiplied by the number λ, then the price index remains unchanged. Put another way, the price index function *P(p ^{0}, p^{1}, q^{0}, q^{1})* is (positively) homogeneous of degree 0 in the components of the period 0 quantity vector

*q*. If the quantity index

^{0}*q*satisfies the following counterpart to T8:

*Q(p*for all λ > 0, then using equation (16.17), the corresponding price index

^{0}, p^{1},λq^{0}, q^{1}) = λ^{-1}Q(p^{0}, p^{1}, q^{0}, q^{1})*p*must satisfy T8. This argument provides some additional justification for assuming the validity of T8 for the price index function

*P*.

**16.44** T7 and T8 together impose the property that the price index *p* does not depend on the absolute magnitudes of the quantity vectors *q ^{0}* and

*q*.

^{1}### C.3 Invariance and symmetry tests

**16.45** The next five tests are invariance or symmetry tests. Fisher (1922, pp 62–63, 458–60) and Walsh (1901, p. 105; 1921b, p. 542) seem to have been the first researchers to appreciate the significance of these kinds of tests. Fisher (1922, pp 62– 63) spoke of fairness, but it is clear that he had symmetry properties in mind. It is perhaps unfortunate that he did not realize that there were more symmetry and invariance properties than the ones he proposed; if he had realized this, it is likely that he would have been able to provide an axiomatic characterization for his ideal price index, as will be done in Section C.6. The first invariance test is that the price index should remain unchanged if the *ordering* of the commodities is changed:

T9—*Commodity Reversal Test (or invariance to changes in the ordering of commodities)*:

*P(p ^{0}*, p^{1}*, q^{0}*, q^{1}*) = P(p^{0}, p^{1}, q^{0}, q^{1})*,

where *p ^{t}** denotes a permutation of the components of the vector

*p*, and

^{t}*q** denotes the same permutation of the components of

^{t}*q*for

^{t}*t*= 0,1. This test is due to Irving Fisher (1922, p. 63);

^{28}it is one of his three famous reversal tests. The other two are the time reversal test and the factor reversal test, which will be considered below.

**16.46** The next test asks that the index be invariant to changes in the units of measurement.

T10—*Invariance to Changes in the Units of Measurement* (commensurability test):

_{1} > 0, …, α_{n} > 0.

That is, the price index does not change if the units of measurement for each product are changed. The concept of this test comes from Jevons (1863, p. 23) and the Dutch economist Pierson (1896, p. 131), who criticized several index number formulas for not satisfying this fundamental test. Fisher (1911, p. 411) first called this test the *change of units test*, and later (Fisher, 1922, p. 420) he called it the *commensurability test*.

**16.47** The next test asks that the formula be invariant to the period chosen as the base period.

T11—*Time Reversal Test*: *P(p ^{0}, p^{1}, q^{0}, q^{1}) = 1/P(p^{1}, p^{0}, q^{1}, q^{0})*.

That is, if the data for periods 0 and 1 are interchanged, then the resulting price index should equal the reciprocal of the original price index. In the one good case when the price index is simply the single-price ratio, this test will be satisfied (as are all of the other tests listed in this section). When the number of goods is greater than one, many commonly used price indices fail this test; for example, the Laspeyres (1871) price index, *P _{L}*, defined by equation (15.5) in Chapter 15, and the Paasche (1874) price index,

*P*, defined by equation (15.6) in Chapter 15, both fail this fundamental test. The concept of the test comes from Pierson (1896, p. 128), who was so upset with the fact that many of the commonly used index number formulas did not satisfy this test that he proposed that the entire concept of an index number should be abandoned. More formal statements of the test were made by Walsh (1901, p. 368; 1921b, p. 541) and Fisher (1911, p. 534; 1922, p. 64).

_{P}**16.48** The next two tests are more controversial, since they are not necessarily consistent with the economic approach to index number theory. However, these tests are quite consistent with the weighted stochastic approach to index number theory to be discussed later in this chapter.

T12—*Quantity Reversal Test* (quantity weights symmetry test): *P(p ^{0}, p^{1}, q^{0}, q^{1}) = P(p^{0}, p^{1}, q^{1}, q^{0})*.

That is, if the quantity vectors for the two periods are interchanged, then the price index remains invariant. This property means that if quantities are used to weight the prices in the index number formula, then the period 0 quantities *q ^{0}* and the period 1 quantities

*q*must enter the formula in a symmetric or evenhanded manner. Funke and Voeller (1978, p. 3) introduced this test; they called it the

^{1}*weight property*.

**16.49** The next test is the analogue to T12 applied to quantity indices:

T13—*Price Reversal Test* (price weights symmetry test):^{29}

Thus, if we use equation (16.17) to define the quantity index *q* in terms of the price index *P*, then it can be seen that T13 is equivalent to the following property for the associated quantity index *Q*:

That is, if the price vectors for the two periods are interchanged, then the quantity index remains invariant. Thus, if prices for the same good in the two periods are used to weight quantities in the construction of the quantity index, then property T13 implies that these prices enter the quantity index in a symmetric manner.

### C.4 Mean value tests

**16.50** The next three tests are mean value tests.

T14—*Mean Value Test for Prices*:^{30}

That is, the price index lies between the minimum price ratio and the maximum price ratio. Since the price index is supposed to be interpreted as a kind of average of the *n* price ratios, *p _{i}^{1}/p_{i}^{0}*, it seems essential that the price index

*p*satisfy this test.

**16.51** The next test is the analogue to T14 applied to quantity indices:

T15—*Mean Value Test for Quantities*:^{31}

where *v _{t}* is the period

*t*value for the aggregate defined by equation (16.1) above. Using the product test in equation (16.17) to define the quantity index

*q*in terms of the price index

*P*, it can be seen that T15 is equivalent to the following property for the associated quantity index

*Q*:

That is, the implicit quantity index *q* defined by *p* lies between the minimum and maximum rates of growth *q _{i}^{1}/q_{i}^{0}* of the individual quantities.

**16.52** In Section C of Chapter 15, it was argued that it was reasonable to take an average of the Laspeyres and Paasche price indices as a single best measure of overall price change. This point of view can be turned into a test:

T16—*Paasche and Laspeyres Bounding Test*:^{32} The price index *p* lies between the Laspeyres and Paasche indices, *P _{L}* and

*P*, defined by equations (15.5) and (15.6) in Chapter 15.

_{P}A test could be proposed where the implicit quantity index q that corresponds to *p* via equation (16.17) is to lie between the Laspeyres and Paasche quantity indices, *Q _{P}* and

*Q*, defined by equations (15.10) and (15.11) in Chapter 15. However, the resulting test turns out to be equivalent to test T16.

_{L}### C.5 Monotonicity tests

**16.53** The final four tests are monotonicity tests; that is, how should the price index *P(p ^{0}, p^{1}, q^{0}, q^{1})* change as any component of the two price vectors

*p*and

^{0}*p*increases or as any component of the two quantity vectors

^{1}*q*and

^{0}*q*increases?

^{1}T17—*Monotonicity in Current Prices*:

*P(p ^{0}, p^{1}, q^{0}, q^{1}) < P(p^{0}, p^{2}, q^{0}, q^{1}) if p^{1} < p^{2}*.

That is, if some period 1 price increases, then the price index must increase, so that *P(p ^{0}, p^{1}, q^{0}, q^{1})* is increasing in the components of p

^{1}. This property was proposed by Eichhorn and Voeller (1976, p. 23), and it is a reasonable property for a price index to satisfy.

T18—*Monotonicity in Base Prices*: *P(p ^{0}, p^{1}, q^{0}, q^{1}) < P(p^{0}, p^{2}, q^{0}, q^{1}) if p^{1} < p^{2}*. if p

^{0}p

^{2}.

That is, if any period 0 price increases, then the price index must decrease, so that *P(p ^{0}, p^{1}, q^{0}, q^{1})* is decreasing in the components of p

^{0}. This quite reasonable property was also proposed by Eichhorn and Voeller (1976, p. 23).

T19—*Monotonicity in Current Quantities*: If *q ^{1} q^{2}*, then

T20—*Monotonicity in Base Quantities*: If *q ^{0} q^{2}*, then

**16.54** Let *q* be the implicit quantity index that corresponds to *p* using equation (16.17). Then it is found that T19 translates into the following inequality involving *Q*:

if q_{1} < q^{2}.

That is, if any period 1 quantity increases, then the implicit quantity index *q* that corresponds to the price index *p* must increase. Similarly, we find that T20 translates into:

q^{0} < q^{2}.

That is, if any period 0 quantity increases, then the implicit quantity index *q* must decrease. Tests T19 and T20 are due to Vogt (1980, p. 70).

**16.55** This concludes the listing of tests. In the next section, it is asked whether any index number formula *P(p ^{0}, p^{1}, q^{0}, q^{1}*) exists that can satisfy all 20 tests.

### C.6 Fisher ideal index and test approach

**16.56** It can be shown that the only index number formula *P(p ^{0}, p^{1}, q^{0}, q^{1})* that satisfies tests T1—T20 is the Fisher ideal price index

*P*, defined as the geometric mean of the Laspeyres and Paasche indices:

_{F}^{33}

To prove this assertion, it is relatively straightforward to show that the Fisher index satisfies all 20 tests.

**16.57** The more difficult part of the proof is showing that it is the *only* index number formula that satisfies these tests. This part of the proof follows from the fact that if *p* satisfies the positivity test T1 and the three reversal tests, T11–T13, then p must equal *P _{F}*. To see this, rearrange the terms in the statement of test T13 into the following equation:

using T12, the quantity reversal test

using T11, the time reversal test.

Now take positive square roots on both sides of equation (16.28), and it can be seen that the lefthand side of the equation is the Fisher index *P _{F}(p^{0}, p^{1}, q^{0}, q^{1})* defined by equation (16.27) and the right-hand side is

*P(p*. Thus, if

^{0}, p^{1}, q^{0}, q^{1})*p*satisfies T1, T11, T12, and T13, it must equal the Fisher ideal index

*P*.

_{F}**16.58** The quantity index that corresponds to the Fisher price index using the product test in equation (16.17) is *Q _{F}*, the Fisher quantity index, defined by equation (15.14) in Chapter 15.

**16.59** It turns out that *P _{F}* satisfies yet another test, T21, which was Irving Fisher’s (1921, p. 534; 1922.pp 72–81) third reversal test (the other two being T9 and T11):

T21—*Factor Reversal Test* (functional form symmetry test):

(16.29)

A justification for this test is the following: assume *P(p ^{0}, p^{1}, q^{0}, q^{1})* is a good functional form for the price index; then if the roles of prices and quantities are reversed,

*P(q*ought to be a good functional form for a quantity index (which seems to be a correct argument). The product, therefore, of the price index

^{0}, q^{1}, p^{0}, p^{1})*P(q*and the quantity index

^{0}, q^{1}, p^{0}, p^{1})*Q(q*=

^{0}, q^{1}, p^{0}, p^{1})*P(q*ought to equal the value ratio,

^{0}, q^{1}, p^{0}, p^{1})*V*. The second part of this argument does not seem to be valid; consequently, many researchers over the years have objected to the factor reversal test. However, if one is willing to embrace T21 as a basic test, Funke and Voeller (1978, p. 180) showed that the only index number function

^{1}/V^{0}*P(q*that satisfies T1 (positivity), T11 (time reversal test), T12 (quantity reversal test), and T21 (factor reversal test) is the Fisher ideal index

^{0}, q^{1}, p^{0}, p^{1})*P*defined by equation (16.27). Thus, the price reversal test T13 can be replaced by the factor reversal test in order to obtain a minimal set of four tests that lead to the Fisher price index.

_{F}^{34}

### C.7 Test performance of other indices

**16.60** The Fisher price index*P _{F}* satisfies all 20 of the tests listed in Sections C.1–C.5. Which tests do other commonly used price indices satisfy? Recall the Laspeyres index

*P*, equation (15.5); the Paasche index

_{L}*P*, equation (15.6); the Walsh index

_{P}*P*, equation (15.19); and the Törnqvist index

_{W}*P*equation (15.81) in Chapter 15.

_{T}**16.61** Straightforward computations show that the Paasche and Laspeyres price indices, *P _{L}*and

*P*, fail only the three reversal tests, T11, T12, and T13. Since the quantity and price reversal tests, T12 and T13, are somewhat controversial and can be discounted, the test performance of

_{P}*P*and

_{L}*P*seems at first glance to be quite good. However, the failure of the time reversal test, T11, is a severe limitation associated with the use of these indices.

_{P}**16.62** The Walsh price index,*P _{W}*, fails four tests: T13, the price reversal test; T16, the Paasche and Laspeyres bounding test; T19, the monotonicity in current quantities test; and T20, the monotonicity in base quantities test.

**16.63** Finally, the Törnqvist price index *P _{T}* fails nine tests: T4, the fixed-basket test; T12 and T13, the quantity and price reversal tests; T15, the mean value test for quantities; T16, the Paasche and Laspeyres bounding test; and T17–T20, the four monotonicity tests. Thus, the Törnqvist index is subject to a rather high failure rate from the viewpoint of this axiomatic approach to index number theory.

^{35}

**16.64** The tentative conclusion that can be drawn from these results is that from the viewpoint of this particular bilateral test approach to index numbers, the Fisher ideal price index*P _{F}* appears to be best because it satisfies all 20 tests.

^{36}The Paasche and Laspeyres indices are next best if we treat each test as being equally important. However, both of these indices fail the very important time reversal test. The remaining two indices, the Walsh and Törnqvist price indices, both satisfy the time reversal test, but the Walsh index emerges as the better one because it passes 16 of the 20 tests, whereas the Törnqvist satisfies only 11 tests.

### C.8 Additivity test

**16.65** There is an additional test that many national income accountants regard as very important: the *additivity test*. This is a test or property that is placed on the implicit quantity index *Q(q ^{0}, q^{1}, p^{0}, p^{1})* that corresponds to the price index

*P(q*using the product test in equation (16.17). This test states that the implicit quantity index has the following form:

^{0}, q^{1}, p^{0}, p^{1})where the common across-periods price for product *i*, *p _{i}** for

*i*= 1,…,

*n*, can be a function of all

*4n*prices and quantities pertaining to the two periods or situations under consideration,

*p*. In the literature on making multilateral comparisons (that is, comparisons among more than two situations), it is quite common to assume that the quantity comparison between any two regions can be made using the two regional quantity vectors, q

^{0}, p^{1}, q^{0}, q^{1}^{0}and q

^{1}, and a common reference price vector, p* ≡(p1*,…, pn*)

^{37}

**16.66** Different versions of the additivity test can be obtained if further restrictions are placed on precisely which variables each reference price *P _{i}** depends. The simplest such restriction is to assume that each

*P** depends only on the product

_{i}*i*prices pertaining to each of the two situations under consideration,

*p*. If it is further assumed that the functional form for the weighting function is the same for each product, so that

_{i}^{0}and p_{i}^{1}*p*for

_{i}* = m(p_{i}^{0}, p_{i}^{1})*i*= 1,…, n, then we are led to the unequivocal quantity index postulated by Knibbs (1924, p. 44).

**16.67** The theory of the *unequivocal quantity index (or the pure quantity index)*^{38} parallels the theory of the pure price index outlined in Section C.2 of Chapter 15. An outline of this theory is now given. Let the pure quantity index *Q _{K}*have the following functional form:

It is assumed that the price vectors p^{0} and p^{1} are strictly positive, and the quantity vectors q0 and q1 are nonnegative but have at least one positive component.^{39} The problem is to determine the functional form for the averaging function *m* if possible. To do this, it is necessary to impose some tests or properties on the pure quantity index QK. As was the case with the pure price index, it is reasonable to ask that the quantity index satisfy the *time reversal test*:

**16.68** As was the case with the theory of the unequivocal price index, it can be seen that if the unequivocal quantity index *Q _{K}*is to satisfy the time reversal test of equation (16.32), the mean function in equation (16.31) must be

*symmetric*. It is also asked that

*Q*satisfy the following

_{K}*invariance to proportional changes in current prices test*.

for all p^{0}, p^{1}, q^{0}, q^{1} and all λ > 0.

**16.69** The idea behind this invariance test is this: the quantity index *Q _{K}(p^{0}, p^{1}, q^{0}, q^{1})* should depend only on the

*relative*prices in each period. It should not depend on the amount of inflation between the two periods. Another way to interpret equation (16.33) is to look at what the test implies for the corresponding implicit price index,

*P*, defined using the product test of equation (16.17). It can be shown that if

_{IK}*Q*satisfies equation (16.33), then the corresponding implicit price index

_{K}*P*will satisfy test T5, the

_{IK}*proportionality in current prices test*. The two tests in equations (16.32) and (16.33) determine the precise functional form for the pure quantity index

*Q*defined by equation (16.31): the

_{K}*pure quantity index or Knibbs’ unequivocal quantity index Q*must be the Walsh quantity index

_{K}*Q*

_{W}^{40}defined by

**16.70** Thus, with the addition of two tests, the pure price index*P _{K}* must be the Walsh price index

*P*defined by equation (15.19) in Chapter 15. With the addition of the same two tests (but applied to quantity indices instead of price indices), the pure quantity index

_{W}*Q*must be the Walsh quantity index

_{K}*Q*defined by equation (16.34). However, note that the product of the Walsh price and quantity indices is

_{W}*not*equal to the revenue ratio,

*V*. Thus, believers in the pure or unequivocal price and quantity index concepts

^{1}/V^{0}*have to choose one of these two concepts*; they cannot apply both simultaneously.

^{41}

**16.71** If the quantity index *Q(q ^{0}, q^{1}, p^{0}, p^{1})* satisfies the additivity test in equation (16.30) for some price weights

*P**, then the percentage change in the quantity aggregate,

_{i}*Q(q*− 1, can be rewritten as follows:

^{0}, q^{1}, p^{0}, p^{1})where the weight for product *i*, *w _{i}*, is defined as

Note that the change in product *i*going from situation 0 to situation 1 is *q _{i}^{1}—q_{i}^{0}*. Thus, the ith term on the right-hand side of equation (16.35) is the

*contribution of the change in product*0 to 1. Business analysts often want statistical agencies to provide decompositions like equation (16.35) so they can decompose the overall change in an aggregate into sector-specific components of change.

*i*to the overall percentage change in the aggregate going from period^{42}Thus, there is a demand on the part of users for additive quantity indices.

**16.72** For the Walsh quantity index defined by equation (16.34), the ith weight is

Thus, the Walsh quantity index *Q _{W}* has a percentage decomposition into component changes of the form in equation (16.35), where the weights are defined byequation (16.37).

**16.73** It turns out that the Fisher quantity index *Q _{F}*defined byequation (15.14) in Chapter 15 also has an additive percentage change decomposition of the form given byequation(16.35).

^{43}The ith weight

*W*for this Fisher decomposition is rather complicated and depends on the Fisher quantity index

_{Fi}*QF(p*as follows:

^{0}, p^{1}, q^{0}, q^{1})^{44}

where *Q _{F}* is the value of the Fisher quantity index,

*Q*, and the period

_{F}(p^{0}, p^{1}, q^{0}, q^{1})*t*normalized price for product

*i*,

*v*is defined as the period

_{i}^{t}*i*price

*q*divided by the period

_{i}^{t}*t*revenue on the aggregate:

**16.74** Using the weights *W _{Fi}* defined byequations (16.38) and (16.39), the following exact decomposition is obtained for the Fisher ideal quantity index:

^{45}

Thus, the Fisher quantity index has an additive percentage change decomposition.

**16.75** Due to the symmetric nature of the Fisher price and quantity indices, it can be seen that the Fisher price index*P _{F}* defined byequation (16.27) also has the following additive percentage change decomposition:

where the product *i* weight *v _{Fi}* is defined as

where*P _{F}* is the value of the Fisher price index,

*PF(p*, and the period

_{0}, p_{1}, q_{0}, q_{1})*t*normalized quantity for product

*i*,

*v*, is defined as the period

_{i}^{t}*i*quantity

*q*divided by the period

_{i}^{t}*t*revenue on the aggregate:

**16.76** The above results show that the Fisher price and quantity indices have exact additive decompositions into components that give the contribution to the overall change in the price (or quantity) index of the change in each price (or quantity).

## D. Stochastic Approach to Price Indices

### D.1 Early unweighted stochastic approach

**16.77** The stochastic approach to the determination of the price index can be traced back to the work of Jevons (1863, 1865) and Edgeworth (1888) over a hundred years ago.^{46} The basic idea behind the (unweighted) stochastic approach is that each price relative, *p _{i}^{1}/p_{i}^{0}* for

*i*= 1,2,…,

*n*can be regarded as an estimate of a common inflation rate α between periods 0 and 1;

^{47}that is, it is assumed that

where α is the common inflation rate and the ε_{i} are random variables with mean 0 and variance σ^{2}. The least squares or maximum likelihood estimator for α is the *Carli* (1804) *price index P _{C}* defined as

A drawback of the Carli price index is that it does not satisfy the time reversal test, that is, *P _{C}(p^{1}, p^{0}) ≠ 1/ P_{C}(p^{0}, p^{1})*.

^{48}

**16.78** Now change the stochastic specification and assume that the logarithm of each price relative, *ln p _{i}^{1}/p_{i}^{0}*, is an unbiased estimate of the logarithm of the inflation rate between periods 0 and 1, β, say. The counterpart toequation (16.44) is:

where β ≡ ln α and the εi are independently distributed random variables with mean 0 and variance σ^{2}. The least-squares or maximum-likelihood estimator for β is the logarithm of the geometric mean of the price relatives. Hence, the corresponding estimate for the common inflation rate α^{49} is the *Jevons (1865) price index**P _{J}*defined as follows:

**16.79** The Jevons price index*P _{J}*does satisfy the time reversal test and thus is much more satisfactory than the Carli index

*P*. However, both the Jevons and Carli price indices suffer from a fatal flaw: each price relative

_{C}*p*is regarded as

_{i}^{1}/p_{i}^{0}*being equally important*and is given an equal weight in the index number equations (16.45) and (16.47). Keynes was particularly critical of this

*unweighted stochastic approach*to index number theory.

^{50}He directed the following criticism toward this approach, which was vigorously advocated by Edgeworth (1923):

Nevertheless

*i*venture to maintain that such ideas, which*i*have endeavoured to expound above as fairly and as*P*ausibly as_{L}*i*can, are rootand- branch erroneous. The “errors of observation”, the “faulty shots aimed at a single bull’s eye” conception of the index number of prices, Edgeworth’s “objective mean variation of general prices”, is the result of confusion of thought. There is no bull’s eye. There is no moving but unique centre, to be called the general price level or the objective mean variation of general prices, round which are scattered the moving price levels of individual things. There are all the various, quite definite, conceptions of price levels of composite commodities appropriate for various purposes and inquiries which have been scheduled above, and many others too. There is nothing else. Jevons was pursuing a mirage.What is the flaw in the argument? In the first place it assumed that the fluctuations of individual prices round the “mean” are “random” in the sense required by the theory of the combination of independent observations. In this theory the divergence of one “observation” from the true position is assumed to have no influence on the divergences of other “observations”. But in the case of prices, a movement in the price of one product necessarily influences the movement in the prices of other commodities, whilst the magnitudes of these compensatory movements depend on the magnitude of the change in revenue on the first product as compared with the importance of the revenue on the commodities secondarily affected. Thus, instead of “independence”, there is between the “errors” in the successive “observations” what some writers on probability have called “connexity”, or, as Lexis expressed it, there is “sub-normal dispersion”.

We cannot, therefore, proceed further until we have enunciated the appropriate law of connexity. But the law of connexity cannot be enunciated without reference to the relative importance of the commodities affected—which brings us back to the problem that we have been trying to avoid, of weighting the items of a composite commodity. (John Maynard Keynes, 1930, pp. 76-77)

The main point Keynes seemed to be making in the quotation above is that prices in the economy are not independently distributed from each other and from quantities. In current macroeconomic terminology, Keynes can be interpreted as saying that a macroeconomic shock will be distributed across all prices and quantities in the economy through the normal interaction between supply and demand; that is, through the workings of the general equilibrium system. Thus, Keynes seemed to be leaning toward the economic approach to index number theory (even before it was developed to any great extent), where quantity movements are functionally related to price movements. A second point that Keynes made in the above quotation is that there is no such thing as *the* inflation rate; there are only price changes that pertain to wellspecified sets of commodities or transactions; that is, the domain of definition of the price index must be carefully specified.^{51} A final point that Keynes made is that price movements must be weighted by their economic importance; that is, by quantities or revenues.

**16.80** In addition to the above theoretical criticisms, Keynes also made the following strong empirical attack on Edgeworth’s unweighted stochastic approach:

The Jevons-Edgeworth “objective mean variation of general prices,” or “indefinite” standard, has generally been identified, by those who were not as alive as Edgeworth himself was to the subtleties of the case, with the purchasing power of money—if only for the excellent reason that it was difficult to visualise it as anything else. And since any respectable index number, however weighted, which covered a fairly large number of commodities could, in accordance with the argument, be regarded as a fair approximation to the indefinite standard, it seemed natural to regard any such index as a fair approximation to the purchasing power of money also.

Finally, the conclusion that all the standards “come to much the same thing in the end” has been reinforced “inductively” by the fact that rival index numbers (all of them, however, of the wholesale type) have shown a considerable measure of agreement with one another in spite of their different compositions. … On the contrary, the tables given above (pp. 53, 55) supply strong presumptive evidence that over long period as well as over short periods the movements of the wholesale and of the consumption standards respectively are capable of being widely divergent. (John Maynard Keynes, 1930, pp 8– 81)

In the quotation above, Keynes noted that the proponents of the unweighted stochastic approach to price change measurement were comforted by the fact that all of the then existing (unweighted) indices of wholesale prices showed broadly similar movements. However, Keynes showed empirically that his wholesale price indices moved quite differently than his consumer price indices.

**16.81** In order to overcome these criticisms of the unweighted stochastic approach to index numbers, it is necessary to

Have a definite domain of definition for the index number; and

Weight the price relatives by their economic importance.

^{52}

**16.82** In the following section, alternative methods of weighting will be discussed.

### D.2 Weighted stochastic approach

**16.83**Walsh (1901, pp 88-89) seems to have been the first index number theorist to point out that a sensible stochastic approach to measuring price change means that individual price relatives should be weighted according to their economic importance or *their transactions’ value* in the two periods under consideration:

It might seem at first sight as if simply every price quotation were a single item, and since every commodity (any kind of commodity) has one price-quotation attached to it, it would seem as if price-variations of every kind of commodity were the single item in question. This is the way the question struck the first inquirers into pricevariations, wherefore they used simple averaging with even weighting. But a price-quotation is the quotation of the price of a generic name for many articles; and one such generic name covers a few articles, and another covers many. … A single price-quotation, therefore, may be the quotation of the price of a hundred, a thousand, or a million dollars’ worth, of the articles that make up the commodity named. Its weight in the averaging, therefore, ought to be according to these money-unit’s worth. (Correa Moylan Walsh, 1921a, pp 82–83)

However, Walsh did not give a convincing argument on exactly how these economic weights should be determined.

**16.84**Theil (1967, pp 136-37) proposed a solution to the lack of weighting in the Jevons index, *P _{J}*, defined by equation (16.47). He argued as follows. Suppose we draw price relatives at random in such a way that each dollar of revenue in the base period has an equal chance of being selected. Then the probability that we will draw the ith price relative is equal to

*i*. Then the overall mean (period 0 weighted) logarithmic price change is

^{53}Now repeat the above mental experiment and draw price relatives at random in such a way that each dollar of revenue in period 1 has an equal probability of being selected. This leads to the overall mean (period 1 weighted) logarithmic price change of

^{54}Each of these measures of overall logarithmic price change seems equally valid, so we could argue for taking a symmetric average of the two measures in order to obtain a final single measure of overall logarithmic price change. Theil

^{55}argued that a nice, symmetric index number formula can be obtained if the probability of selection for the

*n*th price relative is made equal to the arithmetic average of the period 0 and 1 revenue shares for product

*n*. Using these probabilities of selection, Theil’s final measure of overall logarithmic price change was

Note that the index *P _{T}* defined byequation (16.48) is equal to the Törnqvist index defined by equation (15.81) in Chapter 15.

**16.85** A statistical interpretation of the righthand side ofequation (16.48) can be given. Define the ith logarithmic price ratio *r _{i}* by:

Now define the discrete random variable—we will call it R—as the random variable that can take on the values *r _{i}* with probabilities

*ρ*. …,

_{i}≡ (½)[ s_{i}^{0}+ s_{i}^{1}] for i = 1,…, n*n*. Note that since each set of revenue shares,

*s*and

_{i}^{0}*s*, sums to 1 over

_{i}^{1}*i*, the probabilities ρi will also sum to 1. It can be seen that the expected value of the discrete random variable R is

Thus, the logarithm of the index *P _{T}* can be interpreted as

*the expected value of the distribution of the logarithmic price ratios*in the domain of definition under consideration, where the

*n*discrete price ratios in this domain of definition are weighted according to Theil’s probability weights,

*ρ*.

_{i}≡ (½)[ s_{i}^{0}+ s_{i}^{1}] for i = 1,…, n**16.86** Taking antilogs of both sides of equation (16.48), the Törnqvist- (1936, 1937) Theil price index, *P _{T}*is obtained.

^{56}This index number formula has a number of good properties. In particular, PT satisfies the proportionality in current prices test (T5) and the time reversal test (T11) discussed in Section C. These two tests can be used to justify Theil’s (arithmetic) method of forming an average of the two sets of revenue shares in order to obtain his probability weights,

*ρ*:

_{i}≡ (1/2)[ s_{i}^{0}+ s_{i}^{1}] for i = 1,…, n. Consider the following symmetric mean class of logarithmic index number formulaswhere *m(s _{i}^{0}, s_{i}^{0})* is a positive function of the period 0 and 1 revenue shares on product

*i*,

*s*and

_{i}^{0}*s*, respectively. In order for

_{i}^{1}*P*to satisfy the time reversal test, it is necessary for the function

_{S}*m*to be symmetric. Then it can be shown

^{57}that for

*P*to satisfy test T5,

_{S}*m*must be the arithmetic mean. This provides a reasonably strong justification for Theil’s choice of the mean function.

**16.87** The stochastic approach of Theil has another advantageous symmetry property. Instead of considering the distribution of the price ratios *r _{i} = ln (p_{i}^{1}/p_{i}^{0})*, we could also consider the distribution of the

*reciprocals*of these price ratios, say,

The symmetric probability, *ρ _{i} ≡ (½)[ s_{i}^{0} + s_{i}^{1}]*, can still be associated with the ith reciprocal logarithmic price ratio

*t*for

_{i}*i*= 1,…, n. Now define the discrete random variable,

*t*, say, as the random variable that can take on the values

*t*with probabilities

_{i}*ρ*, for

_{i}≡ (½)[ s_{i}^{0}+ s_{i}^{1}]*i*= 1,…, n. It can be seen that the expected value of the discrete random variable

*t*is

Thus, it can be seen that the distribution of the random variable *t* is equal to minus the distribution of the random variable R. Hence, it does not matter whether the distribution of the original logarithmic price ratios, *r _{i} =ln (p_{i}^{1}/p_{i}^{0})* is considered or the distribution of their reciprocals,

*t*, is considered: essentially the same stochastic theory is obtained.

_{i}=ln (p_{i}^{1}/p_{i}^{0})**16.88** It is possible to consider weighted stochastic approaches to index number theory where the distribution of the price ratios, *p _{i}^{1}/p_{i}^{0}*, is considered rather than the distribution of the logarithmic price ratios, ln (

*p*). Thus, again following in the footsteps of Theil, suppose that price relatives are drawn at random in such a way that each dollar of revenue in the

_{i}^{1}/p_{i}^{0}*base period*has an equal chance of being selected. Then the probability that the

*i*th price relative will be drawn is equal to

*s*, the period 0 revenue share for product

_{i}^{0}*i*. Thus, the overall mean (period 0 weighted) price change is

which turns out to be the Laspeyres price index,*P _{L}*. This stochastic approach is the natural one for studying

*sampling problems*associated with implementing a Laspeyres price index.

**16.89** Take the same hypothetical situation and draw price relatives at random in such a way that each dollar of revenue in period 1 has an equal probability of being selected. This leads to the overall mean (period 1 weighted) price change equal to

This is known as the Palgrave (1886) index number formula.^{58}

**16.90** It can be verified that neither the Laspeyres nor the Palgrave price indices satisfy the time reversal test, T11. Thus, again following in the footsteps of Theil, it might be attempted to obtain a formula that satisfied the time reversal test by taking a symmetric average of the two sets of shares. Thus, consider the following class of *symmetric mean index number formulas*:

where *m(s _{i}^{0}, s_{i}^{1})* is a symmetric function of the period 0 and 1 revenue shares for product

*i*,

*s*and

_{i}^{0}*s*, respectively. In order to interpret the right-hand side ofequation (16.56) as an expected value of the price ratios

_{i}^{1}*p*, it is necessary that

_{i}^{1}/p_{i}^{0}However, in order to satisfy equation (16.57), *m* must be the arithmetic mean.^{59} With this choice of *m*,equation (16.56) becomes the following (unnamed) index number formula, *Pu*:

Unfortunately, the unnamed index *P _{u}* does not satisfy the time reversal test either.

^{60}

**16.91** Instead of considering the distribution of the price ratios, *p _{i}^{1}/p_{i}^{0}*, the distribution of the reciprocals of these price ratios could be considered. The counterparts to the asymmetric indices defined earlier by equations (16.54) and (16.55) are now

*backward*from period 1 to 0. In order to make these indices comparable with other previous forwardlooking indices, take the reciprocals of these indices (which lead to harmonic averages) and the following two indices are obtained:

using equation (15.9) in Chapter 15. Thus, the reciprocal stochastic price index defined by equation (16.60) turns out to equal the fixed-basket Paasche price index, pp This stochastic approach is the natural one for studying *sampling problems* associated with implementing a Paasche price index. The other asymmetrically weighted reciprocal stochastic price index defined by equation (16.59) has no author’s name associated with it, but it was noted by Irving Fisher (1922, p. 467) as his index number formula 13. Vartia (1978, p. 272) called this index the *harmonic Laspeyres index*, and his terminology will be used.

**16.92** Now consider the class of symmetrically weighted reciprocal price indices defined as

where, as usual, *m(s _{i}^{0}, s_{i}^{0})* is a homogeneous symmetric mean of the period 0 and 1 revenue shares on product

*i*. However, none of the indices defined by equations (16.59)–(16.61) satisfy the time reversal test.

**16.93** The fact that Theil’s index number formula*P _{T}* satisfies the time reversal test leads to a preference for Theil’s index as the best weighted stochastic approach.

**16.94** The main features of the weighted stochastic approach to index number theory can be summarized as follows. It is first necessary to pick two periods and a transaction’s domain of definition. As usual, each value transaction for each of the *n* commodities in the domain of definition is split up into price and quantity components. Then, assuming there are no new commodities or no disappearing commodities, there are *n* price relatives *p _{i}^{1}/p_{i}^{0}* pertaining to the two situations under consideration along with the corresponding

*2n*revenue shares. The weighted stochastic approach just assumes that these

*n*relative prices, or some transformation of these price relatives,

*f(p*, have a discrete statistical distribution, where the ith probability,

_{i}^{1}/p_{i}^{0})*ρ*, is a function of the revenue shares pertaining to product

_{i}= m(s_{i}^{0}, s_{i}^{0})*i*in the two situations under consideration,

*s*and

_{i}^{0}*s*. Different price indices result, depending on how one chooses the functions

_{i}^{1}*f*and

*m*. In Theil’s approach, the transformation function

*f*was the natural logarithm, and the mean function

*m*was the simple unweighted arithmetic mean.

**16.95** There is a third aspect to the weighted stochastic approach to index number theory: one must decided what *single number* best summarizes the distribution of the *n* (possibly transformed) price relatives. In the analysis above, the *mean* of the discrete distribution was chosen as the best summary measure for the distribution of the (possibly transformed) price relatives, but other measures are possible. In particular, the *weighted median* or various *trimmed means* are often suggested as the best measure of central tendency because these measures minimize the influence of outliers. However, a detailed discussion of these alternative measures of central tendency is beyond the scope of this chapter. Additional material on stochastic approaches to index number theory and references to the literature can be found in Clements and Izan (1981, 1987), Selvanathan and Rao (1994), Diewert (1995b), Cecchetti (1997), and Wynne (1997, 1999).

**16.96** Instead of taking the above stochastic approach to index number theory, it is possible to take the same raw data that are used in this approach but use them with an axiomatic approach. Thus, in the following section, the price index is regarded as a value-weighted function of the n price relatives, and the test approach to index number theory is used in order to determine the functional form for the price index. Put another way, the axiomatic approach in the next section looks at the *properties* of alternative descriptive statistics that aggregate the individual price relatives (weighted by their economic importance) into summary measures of price change in an attempt to find the best summary measure of price change. Thus, the axiomatic approach pursued in Section E can be viewed as a branch of the theory of descriptive statistics.

## E. Second Axiomatic Approach to Bilateral Price Indices

### E.1 Basic framework and some preliminary tests

**16.97** As was mentioned in Section A, one of Walsh’s approaches to index number theory was an attempt to determine the best weighted average of the price relatives,*P _{i}*.

^{61}This is equivalent to using an axiomatic approach to try to determine the best index of the form

*P(r, v*, where

^{0}, v^{1}*v*and

^{0}*v*are the vectors of revenues on the

^{1}*n*commodities during periods 0 and 1.

^{62}However, rather than starting off with indices of the form

*P(r, v*, indices of the form

^{0}, v^{1}*P(p*will be considered, since this framework will be more comparable to the first bilateral axiomatic framework taken in Section C. If the invariance to changes in the units of measurement test is imposed on an index of the form

^{0}, p^{1}, v^{0}, v^{1})*P(p*, then

^{0}, p^{1}, v^{0}, v^{1})*P(p*can be written in the form

^{0}, p^{1}, v^{0}, v^{1})*P(r, v*.

^{0}, v^{1})**16.98** Recall that the product test, equation (16.17), was used in order to define the quantity index, *Q(p ^{0}, p^{1}, q^{0}, q^{1}) ≡ V^{1}/[V^{0}P(p^{0}, p^{1}, q^{0}, q^{1}]*, that corresponded to the bilateral price index P(p

^{0}, p

^{1}, q

^{0}, q

^{1}). A similar product test holds in the present framework; that is, given that the functional form for the price index

*P(p*has been determined, then the corresponding

^{0}, p^{1}, v^{0}, v^{1})*implicit quantity index*can be defined in terms of

*p*as follows:

**16.99** In Section C, the price and quantity indices P(p^{0}, p^{1}, q^{0}, q^{1}) and *Q(p ^{0}, p^{1}, q^{0}, q^{1})* were determined jointly; that is, not only were axioms imposed on

*P(p*, but they were also imposed on

^{0}, p^{1}, q^{0}, q^{1})*Q(p*, and the product test in equation (16.17) was used to translate these tests on

^{0}, p^{1}, q^{0}, q^{1})*q*into tests on

*P*. In Section E, this approach will not be followed: only tests on

*P(p*will be used in order to determine the best price index of this form. Thus, there is a parallel theory for quantity indices of the form

^{0}, p^{1}, v^{0}, v^{1})*Q(q*where it is attempted to find the best valueweighted average of the quantity relatives,

^{0}, q^{1}, v^{0}, v^{1})*.*

*q*_{i}^{1}/q_{i}^{0}^{63}

**16.100** For the most part, the tests that will be imposed on the price index *P(p ^{0}, p^{1}, v^{0}, v^{1})* in this section are counterparts to the tests that were imposed on the price index

*P(p*in Section C. It will be assumed that every component of each price and value vector is positive; that is,

^{0}, p^{1}, v^{0}, v^{1})*P*> > 0n and

_{T}*v*> > 0n for

_{t}*t*= 0,1. If it is desired to set

*v*, the common revenue vector is denoted by v; if it is desired to set p

^{0}= v^{1}^{0}= p

^{1}, the common price vector is denoted by

*p*.

**16.101** The first two tests are straightforward counterparts to the corresponding tests in Section C.

T1—*Positivity*: *P(p ^{0}, p^{1}, v^{0}, v^{1})* > 0.

T2—*Continuity*: *P(p ^{0}, p^{1}, v^{0}, v^{1})* is a continuous function of its arguments.

T3—*Identity or Constant Prices Test*:

P(p, p, v^{0}, v^{1})= 1.

That is, if the price of every good is identical during the two periods, then the price index should equal unity, no matter what the value vectors are. Note that the two value vectors are allowed to be different in the above test.

### E.2 Homogeneity tests

**16.102** The following four tests restrict the behavior of the price index *p* as the scale of any one of the four vectors p^{0}, p^{1}, v^{0}, v^{1} changes.

T4—*Proportionality in Current Prices*:

*P(p ^{0},λp^{1}, v^{0}, v^{1}) = λP(p^{0}, p^{1}, v^{0}, v^{1}) for λ > 0*.

That is, if all period 1 prices are multiplied by the positive number λ, then the new price index is λ times the old price index. Put another way, the price index function *P(p ^{0}, p^{1}, v^{0}, v^{1})* is (positively) homogeneous of degree 1 in the components of the period 1 price vector p

^{1}. This test is the counterpart to test T5 in Section C.

**16.103** In the next test, instead of multiplying all period 1 prices by the same number, all period 0 prices are multiplied by the number λ.

T5—*Inverse Proportionality in Base-Period Prices*:

*P(p ^{0}, p^{1}, q^{0}, q^{1}) < P(p^{0}, p^{2}, q^{0}, q^{1}) if p^{1} < p^{2}*

That is, if all period 0 prices are multiplied by the positive number λ, then the new price index is 1/λ times the old price index. Put another way, the price index function *P(p ^{0}, p^{1}, v^{0}, v^{1})* is (positively) homogeneous of degree minus 1 in the components of the period 0 price vector p

^{0}. This test is the counterpart to test T6 in Section C.

**16.104** The following two homogeneity tests can also be regarded as invariance tests.

T6—*Invariance to Proportional Changes in Current-Period Values*:

*P(p ^{0}, p^{1}, q^{0}, q^{1}) < P(p^{0}, p^{2}, q^{0}, q^{1}) if p^{1} < p^{2}*

That is, if current-period values are all multiplied by the number λ, then the price index remains unchanged. Put another way, the price index function *P(p ^{0}, p^{1}, v^{0}, v^{1})* is (positively) homogeneous of degree 0 in the components of the period 1 value vector

*v*.

^{1}T7—*Invariance to Proportional Changes in Base- Period Values*:

*P(p ^{0}, p^{1},λv^{0}, v^{1}) < P(p^{0}, p^{2}, v^{0}, v^{1}) for all λ > 0*.

That is, if base-period values are all multiplied by the number λ, then the price index remains unchanged. Put another way, the price index function *P(p ^{0}, p^{1}, v^{0}, v^{1})* is (positively) homogeneous of degree 0 in the components of the period 0 value vector

*v*.

^{0}**16.105** T6 and T7 together impose the property that the price index *p* does not depend on the absolute magnitudes of the value vectors *v ^{0}* and

*v*. Using test T6 with

^{1}*p*has the following property:

where *s ^{0}* and

*s*are the vectors of revenue shares for periods 0 and 1; that is, the

^{1}*i*th component of s

^{t}is

*t*= 0,1. Thus, the tests T6 and T7 imply that the price index function

*p*is a function of the two price vectors p

^{0}and p

^{1}and the two vectors of revenue shares,

*s*and

^{0}*s*.

^{1}**16.106** Walsh suggested the spirit of tests T6 and T7 as the following quotation indicates:

What we are seeking is to average the variations in the exchange value of one given total sum of money in relation to the several classes of goods, to which several variations [i.e., the price relatives] must be assigned weights proportional to the relative sizes of the classes. Hence the relative sizes of the classes at both the periods must be considered. (Correa Moylan Walsh, 1901, p. 104)

**16.107** Walsh also realized that weighting the *i*th price relative*P _{i}*by the arithmetic mean of the value weights in the two periods under consideration, (½)[v

_{i}

^{0}+ v

_{i}

^{1}], would give too much weight to the revenues of the period that had the highest level of prices:

At first sight it might be thought sufficient to add up the weights of every class at the two periods and to divide by two. This would give the (arithmetic) mean size of every class over the two periods together. But such an operation is manifestly wrong. In the first place, the sizes of the classes at each period are reckoned in the money of the period, and if it happens that the exchange value of money has fallen, or prices in general have risen, greater influence upon the result would be given to the weighting of the second period; or if prices in general have fallen, greater influence would be given to the weighting of the first period. Or in a comparison between two countries, greater influence would be given to the weighting of the country with the higher level of prices. But it is

*P*ain that_{L}*the one period, or the one country, is as important, in our comparison between them, as the other, and the weighting in the averaging of their weights should really be even*. (Correa Moylan Walsh, 1901, pp 104-05)

**16.108** As a solution to the above weighting problem, Walsh (1901, p. 202; 1921a, p. 97) proposed the following *geometric price index*:

where the *i*th weight in the above formula was defined as

The second part ofequation (16.65) shows that Walsh’s geometric price index*P _{GW}(p^{0}, p^{1}, v^{0}, v^{1})* can also be written as a function of the revenue share vectors,

*s*and

^{0}*s*; that is,

^{1}*P*is homogeneous of degree 0 in the components of the value vectors

_{GW}(p^{0}, p^{1}, v^{0}, v^{1})*v*and

^{0}*v*, and so

^{1}*P*=

_{GW}(p^{0}, p^{1}, v^{0}, v^{1})*P*. Thus, Walsh came very close to deriving the Törnqvist-Theil index defined earlier byequation (16.48).

_{GW}(p^{0}, p^{1}, s^{0}, s^{1})^{64}

### E.3 Invariance and symmetry tests

**16.109** The next five tests are *invariance or symmetry tests*, and four of them are direct counterparts to similar tests in Section C. The first invariance test is that the price index should remain unchanged if the *ordering* of the commodities is changed.

T8—*Commodity Reversal Test* (or invariance to changes in the ordering of commodities):

*P(p ^{0}*, p^{1}*, v^{0}*, v^{1}*)* =

*P(p*,

^{0}, p^{1}, v^{0}, v^{1})where p^{t}* denotes a permutation of the components of the vector *P _{T}* and

*v** denotes the same permutation of the components of

_{t}*v*for

_{t}*t*= 0,1.

**16.110** The next test asks that the index be invariant to changes in the units of measurement.

T9—*Invariance to Changes in the Units of Measurement* (commensurability test):

That is, the price index does not change if the units of measurement for each product are changed. Note that the revenue on product *i*during period t, v_{i}t, does not change if the unit by which product i is measured changes.

**16.111** Test T9 has a very important implication. Let *α _{1} =1/p_{1}^{0}, …, α_{n} = 1/p_{n}^{0}*and substitute these values for the α

*i*into the definition of the test. The following equation is obtained:

where 1_{n} is a vector of ones of dimension n, and r is a vector of the price relatives; that is, the ith component of r is*P _{i}≡ p_{i}^{0}/p_{i}^{1}*.Thus, if the commensurability test T9 is satisfied, then the price index

*P(p*, which is a function of 4n variables, can be written as a function of 3

^{0}, p^{1}, v^{0}, v^{1})*n*variables,

*P*(r, v*, where

^{0}, v^{1})*r*is the vector of price relatives and

*P*(r, v*) is defined as

^{0}, v^{1}*P(1*.

_{n}, r, v^{0}, v^{1})**16.112** The next test asks that the formula be invariant to the period chosen as the base period.

T10—*Time Reversal Test: P(p ^{0}, p^{1}, v^{0}, v^{1}) =1/ P(p^{1}, p^{0}, v^{1}, v^{0})*.

That is, if the data for periods 0 and 1 are interchanged, then the resulting price index should equal the reciprocal of the original price index. Obviously, in the one good case when the price index is simply the single-price ratio, this test will be satisfied (as are all of the other tests listed in this section).

**16.113** The next test is a variant of the circularity test that was introduced in Section *f* of Chapter 15.^{65}

T11—*Transitivity in Prices for Fixed-Value Weights*:

*P(p ^{0}, p^{1}, v^{r}, v^{s})P(p^{1}, p^{2}, v^{r}, v^{s}) = P(p^{0}, p^{2}, v^{r}, v^{s})*.

In this test, the revenue-weighting vectors, *v ^{r}* and

*v*, are held constant while making all price comparisons. However, given that these weights are held constant, then the test asks that the product of the index going from period 0 to 1,

^{s}*P(p*, times the index going from period 1 to 2,

^{0}, p^{1}, v^{r}, v^{s})*P(p*, should equal the direct index that compares the prices of period 2 with those of period 0,

^{1}, p^{2}, v^{r}, v^{s})*P(p*. Clearly, this test is a manyproduct counterpart to a property that holds for a single price relative.

^{0}, p^{2}, v^{r}, v^{s})**16.114** The next test in this section captures the idea that the value weights should enter the index number formula in a symmetric manner.

T12—*Quantity Weights Symmetry Test*:

*P(p ^{0}, p^{1}, v^{0}, v^{1})* =

*P(p*.

^{0}, p^{1}, v^{0}, v^{1})That is, if the revenue vectors for the two periods are interchanged, then the price index remains invariant. This property means that if values are used to weight the prices in the index number formula, then the period 0 values *v ^{0}* and the period 1 values

*v*must enter the formula in a symmetric or evenhanded manner.

^{1}### E.4 Mean value test

**16.115** The next test is a mean value test.

T13—*Mean Value Test for Prices*:

That is, the price index lies between the minimum price ratio and the maximum price ratio. Since the price index is to be interpreted as an average of the *n* price ratios, *p _{i}^{1}/p_{i}^{0}*, it seems essential that the price index

*p*satisfy this test.

### E.5 Monotonicity tests

**16.116** The next two tests in this section are *monotonicity tests*; that is, how should the price index *P(p ^{0}, p^{1}, v^{0}, v^{1})* change as any component of the two price vectors p

^{0}and p

^{1}increases?

T14—*Monotonicity in Current Prices*:

*P(p ^{0}, p^{1}, q^{0}, q^{1}) < P(p^{0}, p^{2}, q^{0}, q^{1}) if p^{1} < p^{2}*.

That is, if some period 1 price increases, then the price index must increase (holding the value vectors fixed), so that *P(p ^{0}, p^{1}, v^{0}, v^{1})* is increasing in the components of p

^{1}for fixed p

^{0}, v0, and

*v*.

^{1}T15—Monotonicity in Base Prices:

*P(p ^{0}, p^{1}, q^{0}, q^{1}) < P(p^{0}, p^{2}, q^{0}, q^{1}) if p^{0} < p^{2}*.

That is, if any period 0 price increases, then the price index must decrease, so that *P(p ^{0}, p^{1}, v^{0}, v^{1})* is decreasing in the components of p

^{0}for fixed p

^{1}, v

^{0}and

*v*.

^{1}### E.6 Weighting tests

**16.117** The preceding tests are not sufficient to determine the functional form of the price index; for example, it can be shown that both Walsh’s geometric price index*P _{GW}(p^{0}, p^{1}, v^{0}, v^{1})* defined by equation (16.65) and the Törnqvist-Theil index

*P*defined byequation (16.48) satisfy all of the above axioms. At least one more test, therefore, will be required in order to determine the functional form for the price index

_{T}(p^{0}, p^{1}, v^{0}, v^{1})*P(p*.

^{0}, p^{1}, v^{0}, v^{1})**16.118** The tests proposed thus far do not specify exactly how the revenue share vectors *s ^{0}* and

*s*are to be used in order to weight, for example, the first price relative,

^{1}*p*. The next test says that only the revenue shares

_{1}^{1}/p_{1}^{0}*s*0 and

^{1}*s*1 pertaining to the first product are to be used in order to weight the prices that correspond to product 1, p

^{1}_{1}

^{1}and p

_{1}

^{0}.

T16—*Own-Share Price Weighting*:

Note that *s _{1}^{t}*, the revenue share for product 1 in period

*t*. This test says that if all of the prices are set equal to 1 except the prices for product 1 in the two periods, but the revenues in the two periods are arbitrarily given, then the index depends only on the two prices for product 1 and the two revenue shares for product 1. The axiom says that a function of

*2 + 2n*variables is actually only a function of four variables.

^{66}

**16.119** If test T16 is combined with test T8, the commodity reversal test, then it can be seen that *p* has the following property:

Equation (16.69) says that if all of the prices are set equal to 1 except the prices for product *i*in the two periods, but the revenues in the two periods are arbitrarily given, then the index depends only on the two prices for product *i*and the two revenue shares for product *i*.

**16.120** The final test that also involves the weighting of prices is the following:

T17—*Irrelevance of Price Change with Tiny Value Weights*:

The test T17 says that if all of the prices are set equal to 1 except the prices for product 1 in the two periods, and the revenues on product 1 are 0 in the two periods but the revenues on the other commodities are arbitrarily given, then the index is equal to 1.^{67} Roughly speaking, if the value weights for product 1 are tiny, then it does not matter what the price of product 1 is during the two periods.

**16.121** If test T17 is combined with test T8, the product reversal test, then it can be seen that *p* has the following property: for *i*= 1,…, n:

Equation (16.71) says that if all of the prices are set equal to 1 except the prices for product *i*in the two periods, and the revenues on product *i*are 0 during the two periods but the other revenues in the two periods are arbitrarily given, then the index is equal to 1.

**16.122** This completes the listing of tests for the weighted average of price relatives approach to bilateral index number theory. It turns out that these tests are sufficient to imply a specific functional form for the price index as will be seen in the next section.

### E.7 Törnqvist-Theil price index and second test approach to bilateral indices

**16.123** In Appendix 16.1, it is shown that if the number of commodities *n* exceeds two and the bilateral price index function *P(p ^{0}, p^{1}, v^{0}, v^{1})* satisfies the 17 axioms listed above, then

*p*must be the Törnqvist-Theil price index

*P*defined byequation (16.48).

_{T}(p^{0}, p^{1}, v^{0}, v^{1})^{68}Thus, the 17 properties or tests listed in Section E provide an axiomatic characterization of the Törnqvist-Theil price index, just as the 20 tests listed in Section C provided an axiomatic characterization of the Fisher ideal price index.

**16.124** There is a parallel axiomatic theory for quantity indices of the form *Q(p ^{0}, p^{1}, v^{0}, v^{1})* that depend on the two quantity vectors for periods 0 and 1, q0 and q1, as well as on the corresponding two revenue vectors,

*v*and

^{0}*v*. Thus, if Q(p0, p1, v0, v1) satisfies the quantity counterparts to tests T1–T17, then q must be equal to the Törnqvist-Theil quantity index

^{1}*Q*, defined as follows:

_{T}(q^{0}, q^{1}, v^{0}, v^{1})where, as usual, the period *t* revenue share on product i, *s _{i}* t, is defined as

*i*= 1,…, n and

*t*= 0,1.

**16.125** Unfortunately, the implicit Törnqvist- Theil price index *P _{IT(}q^{0}, q^{1}, v^{0}, v^{1})*, which corresponds to the Törnqvist-Theil quantity index QT defined by equation (16.72) using the product test, is not equal to the direct Törnqvist-Theil price index

*P*defined byequation (16.48). The product test equation that defines PIT in the present context is given by the following equation:

_{T}(p^{0}, p^{1}, v^{0}, v^{1})The fact that the direct Törnqvist-Theil price index PT is not in general equal to the implicit Törnqvist- Theil price index *P _{IT}* defined by equation (16.73) is a bit of a disadvantage compared with the axiomatic approach outlined in Section C, which led to the Fisher ideal price and quantity indices as being best. Using the Fisher approach meant that it was not necessary to decide whether one wanted a best price index or a best quantity index: the theory outlined in Section C determined both indices simultaneously. However, in the Törnqvist-Theil approach outlined in this section, it is necessary to

*choose*whether one wants a best price index or a best quantity index.

^{69}

**16.126** Other tests are, of course, possible. A counterpart to test T16 in Section C, the Paasche and Laspeyres bounding test, is the following *geometric Paasche and Laspeyres bounding test*:

where the logarithms of the geometric Laspeyres and geometric Paasche price indices, *P _{GL}* and

*P*, are defined as follows:

_{GP}As usual, the period *t* revenue share on product *i*, *s _{i} t*, is defined as

*i*= 1,..., n and

*t*= 0,1. It can be shown that the Törnqvist-Theil price index

*P*defined byequation (16.48) satisfies this test, but the geometric Walsh price index

_{T}(p^{0}, p^{1}, v^{0}, v^{1})*P*defined byequation (16.65) does not satisfy it. The geometric Paasche and Laspeyres bounding test was not included as a primary test in Section E because, a priori, it was not known what form of averaging of the price relatives (for example, geometric, arithmetic, or harmonic) would turn out to be appropriate in this test framework. The test equation (16.74) is an appropriate one if it has been decided that geometric averaging of the price relatives is the appropriate framework. The geometric Paasche and Laspeyres indices correspond to extreme forms of value weighting in the context of geometric averaging, and it is natural to require that the best price index lie between these extreme indices.

_{GW}(p^{0}, p^{1}, v^{0}, v^{1})**16.127**Walsh (1901, p. 408) pointed out a problem with his geometric price index *P _{GW}* defined by equation (16.65), which also applies to the Törnqvist- Theil price index

*P*defined by equation (16.48): these geometric-type indices do not give the right answer when the quantity vectors are constant (or proportional) over the two periods. In this case, Walsh thought that the right answer must be the Lowe index, which is the ratio of the costs of purchasing the constant basket during the two periods. Put another way, the geometric indices PGW and

_{T}(p^{0}, p^{1}, v^{0}, v^{1})*P*do not satisfy T4, the fixed-basket test, in Section C above. What, then, was the argument that led Walsh to define his geometric average type index

_{T}*P*? It turns out that he was led to this type of index by considering another test, which will now be explained.

_{GW}**16.128**Walsh (1901, pp 228-31) derived his test by considering the following simple framework. Let there be only two commodities in the index, and suppose that the revenue share on each product is equal in each of the two periods under consideration. The price index under these conditions is equal to *P(p _{1}^{0}, p_{2}^{0};p_{1}^{1}, p_{2}^{1};v_{1}^{0}, v_{2}^{0};v_{1}^{1}, v_{2}^{1}) =P*(r_{1}, r_{2};½,½;½,½) ≡ m(r_{1}, r_{2})*, where m(r

_{1}, r

_{2}) is a symmetric mean of the two price relatives,

*r*.

_{1}≡ p_{1}_{1}/p_{1}_{0}and r_{2}≡ p_{2}_{1}/p_{2}_{0}^{70}In this framework, Walsh then proposed the following price-relative reciprocal test:

Thus, if the value weighting for the two commodities is equal over the two periods, and the second price relative is the reciprocal of the first price relative *I _{1}*, then Walsh (1901, p. 230) argued that the overall price index under these circumstances ought to equal 1, since the relative fall in one price is exactly counterbalanced by a rise in the other, and both commodities have the same revenues in each period. He found that the geometric mean satisfied this test perfectly, but the arithmetic mean led to index values greater than 1 (provided that r1 was not equal to 1), and the harmonic mean led to index values that were less than 1, a situation that was not at all satisfactory.

^{71}Thus, he was led to some form of geometric averaging of the price relatives in one of his approaches to index number theory.

**16.129** A generalization of Walsh’s result is easy to obtain. Suppose that the mean function, *m(r _{1}, r_{2})*, satisfies Walsh’s reciprocal test,equation (16.77), and, in addition,

*m*is a homogeneous mean, so that it satisfies the following property for all

*r*, and λ > 0:

_{1}> 0, r_{2}> 0Let *r _{1} > 0, r_{2} > 0*. then

where the function of one (positive) variable f(z) is defined as

Usingequation (16.77):

using equation (16.78) with λ=1

Using equation (16.80), equation (16.81) can be rearranged in the following form:

Letting *z ≡ r _{1}^{-2} so that z_{1}^{-2} = r_{1}^{-1}*, equation (16.82) becomes

Now substitute equation (16.83) into equation (16.79) and the functional form for the mean function *m(r _{1}, r_{2})* is determined:

Thus, the geometric mean of the two price relatives is the only homogeneous mean that will satisfy Walsh’s price-relative reciprocal test.

**16.130** There is one additional test that should be mentioned. Fisher (1911, p. 401) introduced this test in his first book that dealt with the test approach to index number theory. He called it the *test of determinateness as to prices* and described it as follows:

A price index should not be rendered zero, infinity, or indeterminate by an individual price becoming zero. Thus, if any product should in 1910 be a glut on the market, becoming a “free good, “that fact ought not to render the index number for 1910 zero. (Irving Fisher, 1911, p. 401)

In the present context, this test could be interpreted to mean the following: if any single price *p _{i}^{0}* or

*p*tends to zero, then the price index P(p0, p, v0, v1) should not tend to zero or plus infinity. However, with this interpretation of the test, which regards the values

_{i}^{1}*v*as remaining constant as the

_{i}^{t}*p*or

_{i}^{0}*p*tends to zero, none of the commonly used index number formulas would satisfy this test. As a result, this test should be interpreted as a test that applies to price indices

_{i}^{1}*P(p*of the type that were studied in Section C, which is how Fisher intended the test to apply. Thus, Fisher’s price determinateness test should be interpreted as follows: if any single price

^{0}, p^{1}, q^{0}, q^{1})*p*or

_{i}^{0}*p*tends to zero, then the price index

_{i}^{1}*P(p*should not tend to zero or plus infinity. With this interpretation of the test, it can be verified that Laspeyres, Paasche, and Fisher indices satisfy this test, but the Törnqvist-Theil price index will not satisfy this test. Thus, when using the Törnqvist-Theil price index,

^{0}, p^{1}, q^{0}, q^{1})*care must be taken to bound the prices away from zero in order to avoid a meaningless index number value*.

**16.131** Walsh was aware that geometric average type indices like the Törnqvist-Theil price index PT or Walsh’s geometric price index *P _{GW}* defined by equation (16.64) become somewhat unstable

^{72}as individual price relatives become very large or small:

Hence in practice the geometric average is not likely to depart much from the truth. Still, we have seen that when the classes [that is, revenues] are very unequal and the price variations are very great, this average may deflect considerably. (Correa Moylan Walsh, 1901, p. 373)

In the cases of moderate inequality in the sizes of the classes and of excessive variation in one of the prices, there seems to be a tendency on the part of the geometric method to deviate by itself, becoming untrustworthy, while the other two methods keep fairly close together. (Correa Moylan Walsh, 1901, p. 404)

**16.132** Weighing all of the arguments and tests presented in Sections C and E of this chapter, it seems that there may be a slight preference for the use of the Fisher ideal price index as a suitable target index for a statistical agency, but opinions can differ on which set of axioms is the most appropriate to use in practice.

## F. Test Properties of Lowe and Young Indices

**16.133** In Chapter 15, the Young and Lowe indices were defined. In the present section, the axiomatic properties of these indices with respect to their price arguments will be developed.^{73}

**16.134** Let*q ^{b} ≡ [q_{1}^{b},..., q_{n}^{b}] and p^{b} ≡ [p_{1}^{b},..., p_{n}^{b}]* denote the quantity and price vectors pertaining to some base year. The corresponding base-year revenue shares can be defined in the usual way as

Let *s ^{b} ≡ [s_{1}^{b},..., s_{n}^{b}]*denote the vector of base-year revenue shares. The Young (1812) price index between periods 0 and

*t*is defined as follows:

The Lowe (1823, p. 316) price index^{74} between periods 0 and *t* is defined as follows:

**16.135** Drawing on those that have been listed in Sections C and E, we highlight 12 desirable axioms for price indices of the form *P(p ^{0}, p^{1})*. The period 0 and

*t*price vectors, p

^{0}and

*P*are presumed to have strictly positive components.

_{T}T1—*Positivity Test*: *P(p ^{0}, p^{t})* > 0 if all prices are positive.

T2—*Continuity Test*: *P(p ^{0}, p^{t})* is a continuous function of prices.

T3—*Identity Test*: *P(p ^{0}, p^{0})* = 1.

T4—*Homogeneity Test for Period t Prices*:

T5—*Homogeneity Test for Period 0 Prices*:

T6—*Commodity Reversal Test*: *P(p ^{t}, p^{0})* =

*P(p*, where p

^{0}*, p^{t}*)^{0}* and pt* denote the same permutation of the components of the price vectors p

^{0}and pt.

^{75}

T7—*Invariance to Changes in the Units of Measurement or the Commensurability Test*:

T8—*Time Reversal Test: P(p ^{t}, p^{0}) = 1/P(p^{0}, p^{t})*.

T9—*Circularity or Transitivity Test: P(p ^{0}, p^{2}) =P(p^{0}, p^{1})P(p^{1}, p^{2})*.

T10—*Mean Value Test: min{pi ^{t}/pi^{0} : i = 1,…, n} ≤P(p^{t}, p^{0}) ≤ max{pit/pi^{0} : i = 1,…, n}*.

T11—*Monotonicity Test with Respect to Period t Prices: P(p ^{0}, p^{t}) < P(p^{0}, p^{t}*) if p^{t} < p^{t}**.

T12—*Monotonicity Test with Respect to Period 0 Prices: P(p ^{0}, p^{t}) > P(p^{0}*, p^{t}) if p^{0} < p^{0}**.

**16.136** It is straightforward to show that the Lowe index defined by equation (16.87) satisfies all 12 of the axioms or tests listed above. Hence, the Lowe index has very good axiomatic properties with respect to its price variables.^{76}

**16.137** It is straightforward to show that the Young index defined by equation (16.86) satisfies 10 of the 12 axioms, failing T8, the time reversal test, and T9, the circularity test. Thus, the axiomatic properties of the Young index are definitely inferior to those of the Lowe index.

Eichhorn (1978 p. 144) and Diewert (1993d, p. 9) considered this approach.

In these unilateral index number approaches, the price and quantity vectors are allowed to vary independently. In yet another index number framework, prices are allowed to vary freely, but quantities are regarded as functions of the prices. This leads to the *economic approach to index number theory*, which will be considered in more depth in Chapters 17 and 18.

Recall Section B in Chapter 15 for an explanation of this approach.

Walsh (1901, p. 84) refers to Young’s contributions as follows: “Still, although few of the practical investigators decidhave actually employed anything but even weighting, they have almost always recognized the theoretical need of allowing for the relative importance of the different classes ever since this need was first pointed out, near the commencement of the century just ended, by Arthur Young. … Arthur Young advised simply that the classes should be weighted according to their importance.”

A price variation is a price ratio or price relative in Walsh’s terminology.

Walsh (1901, pp 104-105) realized that it would not do to simply take the arithmetic average of the values in the two periods, *[v _{i}^{0} + v_{i}^{0}]/2*, as the correct weight for the

*i*th price relative

*r*since, in a period of rapid inflation, this would give too much importance to the period that had the highest prices, and he wanted to treat each period symmetrically: ?But such an operation is manifestly wrong. In the first place, the sizes of the classes at each period are reckoned in the money of the period, and if it happens that the exchange value of money has fallen, or prices in general have risen, greater influence upon the result would be given to the weighting of the second period; or if prices in general have fallen, greater influence would be given to the weighting of the second period. Or in a comparison between two countries greater influence would be given to the weighting of the country with the higher level of prices. But it is

_{i}*P*ain that

_{L}*the one period, or the one country, is as important, in our comparison between them, as the other, and the weighting in the averaging of their weights should really be even*.” However, Walsh was unable to come up with Theil’s (1967) solution to the weighting problem, which was to use the average revenue share [

*s*]/2, as the correct weight for the ith price relative in the context of using a weighted geometric mean of the price relatives.

_{i}^{0}+ s_{i}^{1}Walsh also considered basket-type approaches to index number theory, as was seen in Chapter 15.

In Section E, rather than starting with indices of the form *P(r, v ^{0}, v^{1})*, indices of the form

*P(p*are considered. However, if the invariance to changes in the units of measurement test is imposed on this index, it is equivalent to studying indices of the form

^{0}, p^{1}, v^{0}, v^{1})*P(r, v*. Vartia (1976a) also used a variation of this approach to index number theory.

^{0}, v^{1}This treatment of prices as unit values over time follows Walsh (1901, p. 96; 1921a, p. 88) and Fisher (1922, p. 318). Fisher and Hicks both had the idea that the length of the period should be short enough so that variations in price within the period could be ignored as the following quotations indicate: “Throughout this book ‘the price’ of any commodity or ‘the quantity’ of it for any one year was assumed given. But what is such a price or quantity? Sometimes it is a single quotation for January 1 or July 1, but usually it is an average of several quotations scattered throughout the year. The question arises: On what principle should this average be constructed? The *practical* answer is *any* kind of average since, ordinarily, the variation during a year, so far, at least, as prices are concerned, are too little to make any perceptible difference in the result, whatever kind of average is used. Otherwise, there would be ground for subdividing the year into quarters or months until we reach a small enough period to be considered practically a point. The quantities sold will, of course, vary widely. What is needed is their sum for the year (which, of course, is the same thing as the simple arithmetic average of the per annum rates for the separate months or other subdivisions). In short, the simple arithmetic average, both of prices and of quantities, may be used. Or, if it is worth while to put any finer point on it, we may take the weighted arithmetic average for the prices, the weights being the quantities sold” (Irving Fisher, 1922, p. 318). “I shall define a week as that period of time during which variations in prices can be neglected. For theoretical purposes this means that prices will be supposed to change, not continuously, but at short intervals. The calendar length of the week is of course quite arbitrary; by taking it to be very short, our theoretical scheme can be fitted as closely as we like to that ceaseless oscillation which is a characteristic of prices in certain markets” (John Hicks, 1946, p. 122).

Eichhorn (1978, p. 144) established this result.

This proposition is due to Diewert (1993d, p. 9), but his proof is an adaptation of a closely related result due to Eichhorn (1978, pp 144–45).

Recall that in the economic approach, the price vector *p* is allowed to vary independently, but the corresponding quantity vector *q* is regarded as being determined by *p*.

Much of the material in this section is drawn from Sections 2 and 3 of Diewert (1992a). For more recent surveys of the axiomatic approach, see Balk (1995) and Auer (2001).

Multilateral index number theory refers to the case where there are more than two situations whose prices and quantities need to be aggregated.

See Section B of Chapter 15 for more on this approach, which was initially due to I. Fisher (1911, p. 403; 1922).

This observation was first made by Fisher (1911, pp 400-406). See also Vogt (1980) and Diewert (1992a).

Notation: *q >> 0 _{n}* means that each component of the vector q is positive;

*q ≥ 0*means each component of q is nonnegative; and

_{n}*q > 0*means

_{n}*q ≥ 0*and

_{n}*q ⧣ 0*.

_{n}Eichhorn and Voeller (1976, p. 23) suggested this test.

Fisher (1922, pp 207–15) informally suggested this.

Laspeyres (1871, p. 308), Walsh (1901, p. 308), and Eichhorn and Voeller (1976, p. 24) have all suggested this test. Laspeyres came up with this test or property to discredit the ratio of unit-values index of Drobisch (1871a), which does not satisfy this test. This test is also a special case of Fisher’s (1911, pp 409–10) price proportionality test.

Usually, economists assume that given a price vector*p* the corresponding quantity vector *q* is uniquely determined. Here, the same price vector is used, but the corresponding quantity vectors are allowed to be different.

The origins of this test go back at least 200 years to the Massachusetts legislature, which used a constant basket of goods to index the pay of Massachusetts soldiers fighting in the American Revolution; see Willard Fisher (1913). Other researchers who have suggested the test over the years include Lowe (1823, Appendix, p. 95), Scrope (1883, p. 406), Jevons (1865), Sidgwick (1883, pp 67-68), Edgeworth (1925, p. 215; originally published in 1887), Marshall (1887, p. 363), Pierson (1895, p. 332), Walsh (1901, p. 540; 1921b, pp 543-44), and Bowley (1901, p. 227). Vogt and Barta (1997, p. 49) correctly observe that this test is a special case of Fisher’s (1911, p. 411) proportionality test for quantity indices, which Fisher (1911, p. 405) translated into a test for the price index using the product test in equation (15.3).

See Vogt (1980, p. 70).

This test was proposed by Walsh (1901, p. 385), Eichhorn and Voeller (1976, p. 24), and Vogt (1980, p. 68).

Eichhorn and Voeller (1976, p. 28) suggested this test.

Fisher (1911, p. 405) proposed the related test

This test was proposed by Diewert (1992a, p. 216).

“This [test] is so simple as never to have been formulated. It is merely taken for granted and observed instinctively. Any rule for averaging the commodities must be so general as to apply interchangeably to all of the terms averaged” (Irving Fisher, 1922, p. 63).

This test was proposed by Diewert (1992a, p. 218).

This test seems to have been first proposed by Eichhorn and Voeller (1976, p. 10).

This test was proposed by Diewert (1992a, p. 219).

Bowley (1901, p. 227) and Fisher (1922, p. 403) both endorsed this property for a price index.

See Diewert (1992a, p. 221).

Other characterizations of the Fisher price index can be found in Funke and Voeller (1978) and Balk (1985, 1995).

However, it will be shown later in Chapter 19 that the Törnqvist index approximates the Fisher index quite closely using normal time-series data that are subject to relatively smooth trends. Under these circumstances, the Törnqvist index can be regarded as passing the 20 tests to a reasonably high degree of approximation.

This assertion needs to be qualified: there are many other tests that we have not discussed, and price statisticians could differ on the importance of satisfying various sets of tests. Some references that discuss other tests are Auer (2001; 2002), Eichhorn and Voeller (1976), Balk (1995), and Vogt and Barta (1997). In Section E, it is shown that the Törnqvist index is ideal for a different set of axioms.p. 19) used the term additive multilateral system. For axiomatic approaches to multilateral index number theory, see Balk (1996a, 2001) and Diewert (1999b).

Hill (1993, pp 395–97) termed such multilateral methods *the block approach*, while Diewert (1996a, pp 250–51) used the term *average price approaches*. Diewert (1999b, p. 19) used the term *additive multilateral system*. For axiomatic approaches to multilateral index number theory, see Balk (1996a, 2001) and Diewert (1999b).

Diewert (2001) used this term.

It is assumed that *m(a, b)* has the following two properties: *m(a, b)* is a positive and continuous function, defined for all positive numbers *a* and *b*; and *m(a, a)* = *a* for all *a* > 0.

This is the quantity index that corresponds to the price index 8 defined by Walsh (1921a, p. 101).

Knibbs (1924) did not notice this point!

Business and government analysts also often demand an analogous decomposition of the change in price aggregate into sector-specific components that add up.

The Fisher quantity index also has an additive decomposition of the type defined by equation (16.30) due to Van Ijzeren (1987, p. 6). The ith reference price *P _{i}** is defined as

*i*= 1,…, n and where

*P*is the Fisher price index. This decomposition was also independently derived by Dikhanov (1997). The Van Ijzeren decomposition for the Fisher quantity index is currently being used by the Bureau of Economic Analysis; see Moulton and Seskin (1999, p. 16) and Ehemann, Katz, and Moulton (2002).

_{F}This decomposition was obtained by Diewert (2002a) and Reinsdorf, Diewert, and Ehemann (2002). For an economic interpretation of this decomposition, see Diewert (2002a).

To verify the exactness of the decomposition, substitute equation (16.38) into equation (16.40) and solve the resulting equation for *Q _{F}*. It is found that the solution is equal to

*Q*defined byequation (15.14) in Chapter 15.

_{F}For references to the literature, see Diewert (1993a, pp 37–38; 1995a; 1995b).

“In drawing our averages the independent fluctuations will more or less destroy each other; the one required variation of gold will remain undiminished” (W. Stanley Jevons, 1863, p. 26).

In fact, Fisher (1922, p. 66) noted that *P _{C}(p^{0}, p^{1})PC(p^{1}, p^{0})* ≥ 1 unless the period 1 price vector p

^{1}is proportional to the period 0 price vector p

^{0}; that is, Fisher showed that the Carli index has a definite u

*P*ard bias. He urged statistical agencies not to use this formula. Walsh (1901, pp 331 and 530) also discovered this result for the case

_{W}*n*= 2.

Greenlees (1999) pointed out that although *P _{J}*defined by equation (16.47), will generally not be an unbiased estimator for α under our stochastic assumptions. To see this, let

*x*. Taking expectations, we have:

_{i}= ln (p_{i}^{1}/p_{i}^{0})*Exi = β = ln α*. Define the positive, convex function

*f*of one variable

*x*by

*f(x)*≡

*ex*. By Jensen’s (1906) inequality,

*Ef(x) ≥ f(Ex)*. Letting

*x*equal the random variable

*xi*, this inequality becomes

*E(p*, and it can be seen that the Jevonsprice index will generally have an u

_{i}^{1}/p_{i}^{0}) ≥ α*P*ard bias under the usual stochastic assumptions.

_{W}Walsh (1901, p. 83) also stressed the importance of proper weighting according to the economic importance of the commodities in the periods being compared: “But to assign uneven weighting with approximation to the relative sizes, either over a long series of years or for every period separately, would not require much additional trouble; and even a rough procedure of this sort would yield results far superior to those yielded by even weighting. It is especially absurd to refrain from using roughly reckoned uneven weighting on the ground that it is not accurate, and instead to use even weighting, which is much more inaccurate.”

See Section B in Chapter 15 for additional discussion on this point.

Walsh (1901, pp 82–90; 1921a, pp 82–83) also objected to the lack of weighting in the unweighted stochastic approach to index number theory.

In Chapter 19, this index will be called the *geometric Laspeyres index*, *P _{GL}*. Vartia (1978, p. 272) referred to this index as the

*logarithmic Laspeyres index*. Yet another name for the index is the

*base-weighted geometric index*.

In Chapter 19, this index will be called the *geometric Paasche index*, *P _{GP}*. Vartia (1978, p. 272) referred to this index as the

*logarithmic Paasche index*. Yet another name for the index is the

*current-period weighted geometric index*.

“The price index number defined in (1.8) and (1.9) uses the *n* individual logarithmic price differences as the basic ingredients. They are combined linearly by means of a two stage random selection procedure: First, we give each region the same chanceof being selected, and second, we give each dollar spent in the selected region the same chance (1/ma or 1/mb) of being drawn” (Henri Theil, 1967, p. 138).

The sampling bias problem studied by Greenlees (1999) does not occur in the present context because there is no sampling involved in equation (16.50): the sum of the *p _{i}^{t}q_{i}^{t}* over

*i*for each period

*t*is assumed to equal the value aggregate

*V*for period

^{t}*t*.

See Diewert (2000) and Balk and Diewert (2001).

It is formula number 9 in Fisher’s (1922, p. 466) listing of index number formulas.

For a proof of this assertion, see Balk and Diewert (2001).

In fact, this index suffers from the same u *P _{W}* ard bias as the Carli index in that

*P*≥ 1. To prove this, note that the previous inequality is equivalent to

_{u}(p^{0}, p^{1}, q^{0}, q^{1})P_{u}(p^{1}, p^{0}, q^{1}, q^{0})*[P*, and this inequality follows from the fact that a weighted harmonic mean of

_{u}(p^{1}, p^{0}, q^{1}, q^{0})]_{−1}≤ P_{u}(p^{0}, p^{1}, q^{0}, q^{1})*n*positive numbers is equal to or less than the corresponding weighted arithmetic mean; see Hardy, Littlewood, and Pólya(1934, p. 26).

Fisher also took this point of view when describing his approach to index number theory: “An index number of the prices of a number of commodities is an average of their price relatives. This definition has, for concreteness, been expressed in terms of prices. But in like manner, an index number can be calculated for wages, for quantities of goods imported or exported, and, in fact, for any subject matter involving divergent changes of a group of magnitudes. Again, this definition has been expressed in terms of time. But an index number can be applied with equal propriety to comparisons between two places or, in fact, to comparisons between the magnitudes of a group of elements under any one set of circumstances and their magnitudes under another set of circumstances” (Irving Fisher, 1922, p. 3). However, in setting up his axiomatic approach, Fisher imposed axioms on the price and quantity indices written as functions of the two price vectors, p^{0} and p^{1}, and the two quantity vectors, q0 and q1; that is, he did not write his price index in the form*P(r, v ^{0}, v^{1})* and impose axioms on indices of this type. Of course, in the end, his ideal price index turned out to be the geometric mean of the Laspeyres and Paasche price indices, and, as was seen in Chapter 15, each of these indices can be written as revenue share-weighted averages of the

*n*price relatives,

*r*.

_{i}≡ p_{i}^{0}/p_{i}^{1}Chapter 3 in Vartia (1976a) considered a variant of this axiomatic approach.

It turns out that the price index that corresponds to this best quantity index, defined as *P(p ^{0}, p^{1}, v^{0}, v^{1})*. Thus, the axiomatic approach in Section E generates separate best price and quantity indices whose product does not equal the value ratio in general. This is a disadvantage of the second axiomatic approach to bilateral indices compared with the first approach studied in Section C.

One could derive Walsh’s index using the same arguments as Theil except that the geometric average of the revenue shares (*s _{i}^{1}s_{i}^{1})^{1/2#x00BD;}* could be taken as a preliminary probability weight for the ith logarithmic price relative, ln ri. These preliminary weights are then normalized to add up to unity by dividing by their sum. It is evident that Walsh’s geometric price index will closely approximate Theil’s index using normal time-series data. More formally, regarding both indices as functions of

*p*, it can be shown that

^{0}, p^{1}, v^{0}, v^{1}*P*approximates

_{W}(p^{0}, p^{1}, v^{0}, v^{1})*P*to the second order around an equal price (that is,

_{T}(p^{0}, p^{1}, v^{0}, v^{1})*p*) and quantity (that is,

^{0}= p^{1}*q*point.

^{0}= q^{1})See equation (15.77) in Chapter 15.

In the economics literature, axioms of this type are known as separability axioms.

Strictly speaking, since all prices and values are required to be positive, the left-hand side of equation (16.70) should be replaced by the limit as the product 1 values, *v _{1}^{0}* and

*v*

_{1}^{1}, approach 0.

The Törnqvist-Theil price index satisfies all 17 tests, but the proof in Appendix 16.1 did not use all of these tests to establish the result in the opposite direction: tests T5, T13, T15, and either T10 or T12 were not required in order to show that an index satisfying the remaining tests must be the Törnqvist-Theil price index. For alternative characterizations of the Törnqvist-Theil price index, see Balk and Diewert (2001) and Hillinger (2002).

Hillinger (2002) suggested taking the geometric mean of the direct and implicit Törnqvist-Theil price indices in order to resolve this conflict. Unfortunately, the resulting index is not best for either set of axioms that were suggested in this section.

Walsh considered only the cases where *m* was the arithmetic, geometric, and harmonic means of *r ^{1} and r^{2}*.

“This tendency of the arithmetic and harmonic solutions to run into the ground or to fly into the air by their excessive demands is clear indication of their falsity” (Correa Moylan Walsh, 1901, p. 231).

That is, the index may approach zero or plus infinity.

Baldwin (1990, p. 255) worked out a few of the axiomatic properties of the Lowe index.

This index number formula is also precisely Bean and Stine’s (1924, p. 31) Type A index number formula. Walsh (1901, p. 539) initially mistakenly attributed Lowe’s formula to G. Poulett Scrope (1883), who wrote *Principles of Political Economy* in 1833 and suggested Lowe’s formula without acknowledging Lowe’s priority. But in his discussion of Fisher’s (1921) paper, Walsh (1921b, pp 543–44) corrects his mistake on assigning Lowe’s formula: “What index number should you then use? It should be this:

In applying this test to the Lowe and Young indices, it is assumed that the base-year quantity vector *q ^{b}* and the base-year share vector

*s*are subject to the same permutation.

^{b}From the discussion in Chapter 15, it will be recalled that the main problem with the Lowe index occurs if the quantity weight vector *q ^{b}* is not representative of the quantities that were purchased during the time interval between periods 0 and 1.

More explicitly, *β1(s _{1}^{0}, s_{1}^{1})*, the function

*α1(s*is used where all components of the vectors

_{1}^{0},1,…,1;s_{1}^{1},1,…,1)*s*and

^{0}*s*except the first are set equal to an arbitrary positive number like 1.

^{1}**16.138** Define*r _{i} ≡ p_{i}^{1}/p_{i}^{0}* for

*i*= 1,.., n. Using T1, T9, and equation (16.66),

*P(p*. Using T6, T7, and equation (16.63):

^{0}, p^{1}, v^{0}, v^{1}) =P*(r, v^{0}, v^{1})where st is the period *t* revenue share vector for *t* = 0,1.

**16.139** Let*x ≡ (x _{1},…, x_{n}) and y ≡ (y_{1},…, y_{n})* be strictly positive vectors. The transitivity test T11 and equation (A16.1) imply that the function P* has the following property:

**16.140** Using T1, *P*(r, s ^{0}, s^{1})* > 0 and using T14, P*(r, s0, s1) is strictly increasing in the components of r. The identity test T3 implies that

where 1_{n} is a vector of ones of dimension n. Using a result due to Eichhorn (1978, p. 66), it can be seen that these properties of *P** are sufficient to imply that there exist positive functions *αi(s ^{0}, s^{1})* for

*i*= 1,…, n such that

*P**has the following representation:

**16.141** The continuity test T2 implies that the positive functions *α i(s ^{0}, s^{1})* are continuous. For λ > 0, the linear homogeneity test T4 implies that

using equation (A16.4).

Equating the right-hand sides of the first and last lines in (A16.5) shows that the functions *α _{i}(s^{0}, s^{1})* must satisfy the following restriction:

for all strictly positive vectors *s ^{0}* and

*s*.

^{1}**16.142** Using the weighting test T16 and the commodity reversal test T8, equation (16.69) holds. Equation (16.69) combined with the commensurability test T9 implies that P* satisfies the following equation:

for all *r _{i}* > 0, where

*f*is the function defined in test T16.

**16.143** Substitute equation (A16.7) into equation (A16.4) in order to obtain the following system of equations:

But the first part of equation (A16.8) implies that the positive continuous function of 2*n* variables *α _{i}(s^{0}, s^{1})* is constant with respect to all of its arguments except

*s*and

_{i}^{0}*s*, and this property holds for each i. Thus, each

_{i}^{1}*α*can be replaced by the positive continuous function of two variables

_{i}(s^{0}, s^{1})*β*) for

_{i}(s_{i}^{0}, s_{i}^{1}*i*= 1,…, n.

^{77}Now replace the

*α*in equation (A16.4) with the

_{i}(s^{0}, s^{1})*β*) for

_{i}(s_{i}^{0}, s_{i}^{1}*i*= 1,…, n and the following representation for P* is obtained:

**16.144**Equation (A16.6) implies that the functions *β _{i}(s_{i}^{0}, s_{i}^{1}*) also satisfy the following restrictions:

**16.145** Assume that the weighting test T17 holds, and substitute equation (16.71) into (A16.9) in order to obtain the following equation:

Since the *p _{i}^{1}* and

*p*can be arbitrary positive numbers, it can be seen that equation (A16.11) implies

_{i}^{0}**16.146** Assume that the number of commodities *n* is equal to or greater than 3. Using equations (A16.10) and (A16.12), Theorem 2 in Aczél (1987, p. 8) can be applied and the following functional form for each of the *α _{i}(s_{i}^{0}, s_{i}^{1}*) is obtained:

where γ is a positive number satisfying 0 > γ >1.

**16.147** Finally, the time reversal test T10 *or* the quantity weights symmetry test T12 can be used to show that γ must equal. Substituting this value or γ back into equation (A16.13) and then substituting those equations back into equation (A16.9), the functional form for *P**, and hence *p*, is determined as