Price Dispersion and Cost-Of-Living Indexes

Robert McClelland

PSB, Room 3105

Washington, D.C. 20212

MCCLELLAND_R@BLS.GOV

November 5, 1996

The author would like to thank Mike Baye, Tim Erickson, Brent Moulton, Marshall Reinsdorf

and participants of seminars at the 1996 American Economic Association meetings and the

Bureau of Labor Statistics for helpful comments and suggestions. All remaining errors are my

own. The views expressed here do not reflect the policy of the Bureau of Labor Statistics or the

views of other BLS staff members.
Abstract

Recently, several Konüs cost-of-living indexes that allow stochastic prices have been described.

However, under some conditions these indexes may violate an identity axiom and allow utility to

increase with the measured cost of living. As an alternative I use literature on consumer surplus

to describe a new Konüs index that does not violate the identity axiom and ranks price regimes in

the opposite order as indirect utility functions. In addition, it provides a natural way to introduce

risk aversion into cost-of-living indexes. This is demonstrated by examining the implication of

risk aversion on the cost-of-living from fixed sample size searches.
1

I. Introduction

Standard cost-of-living indexes assume that prices in each of two time periods are

nonstochastic. Research, however, suggests that prices are in fact dispersed across outlets, even

after accounting for quality differences.1

 In a world of zero search costs this dispersion is

irrelevant because every household will locate the minimum available price. More realistically,

when search costs are positive, each household faces a distribution of offered prices, and the

prices actually paid will vary across households. Therefore, an index based on nonstochastic

prices has few applications in the real world and isn’t necessarily related to indexes calculated

with stochastic prices.

Baye (1985), however, describes a Konüs index for random prices by taking expectations

over the indexes of individual agents. He also shows that his index obeys several appealing

axioms. Reinsdorf (1994a) notes a limitation with Baye’s index and suggests an alternative in

which expectations are taken separately over the expenditure functions in each time period. He

then uses his index to examine the cost-of-living when households use a fixed sample size search

rule.

This paper draws upon research on consumer surplus to describe two additional

complications with these indexes. These new complications have not been previously described

because the level of utility is left undetermined. When the utility level is set by an indirect utility

function the above indexes can collapse into essentially the same index that can violate an identity

 

1

 See, for example, Van Hoomissen (1988).
2

axiom. Failure to obey this axiom implies that such an index can indicate an increase in the cost

of living when the distribution of prices is actually stationary. In addition, utility may increase

when the cost of living index increases. Given this possibility, the index is arguably of little value

because households may prefer a higher level of the index to a lower one.

As an alternative, I use the expenditure functional concept of Helms (1985), developed for

a measure of expected compensating variation, to create a new Konüs index. This index describes

the amount of money in a later time period necessary to equate expected indirect utility in both

periods, which offers several advantages. First, it does not violate the above-mentioned identity

axiom. Second, it ranks price regimes in the opposite order as expected indirect utility functions.

Therefore, it is a useful measure of welfare changes. Finally, because it uses expected indirect

utility, it provides a natural way to introduce risk aversion into cost-of-living indexes. For

example, the gains or loses from increasing the spread of prices will depend upon household riskloving behavior in prices.

To show some of the implications of introducing risk aversion into a cost-of-living index,

the fixed sample size search cases in Reinsdorf (1994a) are considered. The two major results

are: First, that for the index introduced here, risk aversion, not just search, determines whether or

not a consumer benefits from increased price dispersion. Second, Reinsdorf’s intuition that a high

elasticity of substitution decreases the relative gains from search is given a precise meaning.

II. The Standard Konüs Index

A cost-of-living index is a useful way to consider welfare changes caused by changees in

factors exogenous to the individual household, such as the monetary and trade policies of

governments. By describing the effect of changes in terms of the percent of income necessary to
3

leave the household indifferent, it provides a unit-free measure of the change in welfare. Price

indexes, such as the U.S. Consumer Price Index, are approximations of these indexes.

To describe a cost-of-living index, first we need to assume that an individual household

has a well-behaved utility function and a budget constraint. These generate a Marshallian demand

function x(p,y), a Hicksian (compensated) demand function h(p,u) and an expenditure function

e(p,u), where p is an m-dimensional vector of nonstochastic prices, m > 1, y is a fixed level of

income, and u is some level of utility. The standard definition of a Konüs cost-of-living index is

the ratio:

(1) K(p,q,u) = e(p,u)/e(q,u)

where q represents prices in the earlier reference period and p represents prices in the later

comparison period. The index K shows the percentage change in income in the comparison

period necessary to return the individual to the same level of utility as in the reference period.

This use of utility information is the crucial feature distinguishing cost-of-living indexes from

‘mechanical’ price indexes.

As a consequence of this distinction, any specific price index formula is an exact cost-ofliving index only for a proper subset of utility functions. This implies that no price index formula

is ideal for all utility functions. Desirable axioms for price indexes to follow have been created,

however, and they can help researchers sort meaningfully among price indexes. Baye (1985)

suggests some analogous axioms for cost-of-living indexes when prices are random. The

nonstochastic versions for a cost-of-living index I(p,q,u) are:
4

Monotonicity

For an m-dimensional vector qλ with element j equal to qj for all j ≠ i and element i equal to qiλ,

with λ>1, I(qλ,q,u)>1 and I(q, qλ,u)<1.

Homogeneity

For λ>0, λI(p,q,u)=I(λp,q,u)

Dimensionality

For λ>0, I(λp,λq,u)=I(p,q,u)

Identity

I(q,q,u) = 1.

The standard Konüs satisfies all of these properties. The first axiom states that, if all

prices but one stay constant, then if that price increases/decreases, the index must be greater/less

than unity. This property forces indexes to show an increase in the cost of living if any price

increases. Without it, prices could rise, while the index indicates a decrease in the cost of living.

(Linear ) homogeneity just states that increasing all prices in the second period by the same

proportion should increase the total index by that proportion. Dimensionality forces indexes with

proportional prices to equal each other. The identity axiom forces indexes with identical prices to

show no change in the cost of living. Without this feature, the cost of living can appear to rise or

fall even if prices and utility do not changes.

Note that, while utility u is explicitly used, the source of utility is not specified in either the

index or in the axioms. Nevertheless, because the expenditure function e(p,u) is the inverse of the

indirect utility function v(p,y), the Konüs also satisfies an additional useful axiom:

Preference Monotonicity
5

sgn(I(p,q,u) - 1) = sgn(v(p,y) - v(q,y)).

Without this axiom, it is difficult to make normative statements about the cost of living. For

example, one cannot conclude that a higher cost of living is undesirable because an individual may

prefer it.

III. Konüs Indexes Under Price Dispersion

Since the assumptions usually made to create a Konüs index (e.g., nonstochastic prices)

are highly restrictive, it is not surprising that the standard Konüs satisfies the above axioms. One

possible weakening of these assumptions is to allow for the random dispersion of prices. This

could represent an array of retail outlets offering different prices to a set of otherwise identical

consumers with limited information who search over prices at some cost or who search a fixed

number of times. At least two distributions are of interest here. First, the distribution of prices

that the consumer could possibly encounter i.e., the offered prices. This is the relevant

distribution when the Bureau of Labor Statistics collects price quotes from outlets to form the

consumer price index. A second distribution of interest would be a distribution over prices

located by consumers, i.e., the distribution of accepted, or transacted, prices.

Any index constructed from prices with these distributions will have to aggregate across

both prices and goods. This implies that the axioms above could not be applied because they are

designed solely in terms of goods aggregation. But without any underlying theory as guidance,

the construction of an index over stochastic prices is necessarily ad hoc. In addition, intuition

developed for indexes such as that in equation (1) may be useless.

As a step to developing a cost-of-living index methodology when prices are stochastic,

Baye (1985) proposes an index that can be used with the distribution of transacted prices. Rather
6

than simply use the expectation of prices in place of fixed prices, Baye takes expectations over the

whole index. This Expected Konüs (EK) index is defined by

(2) EK u E epu

p q equ ( ,) (

~, )

(

~, ) Ω = , 

 .

where the expectation is taken over the random price vectors p ̃ and q ̃ associated with the

probability space Ωp,q. One interpretation of EK is that it minimizes the expected squared

deviation from a randomly selected consumer’s Konüs index of transacted prices. It also equals

the expected index of offered prices under the assumption that the consumer does not change

outlets in the second period. Baye states that EK obeys useful axioms adapted for cost-of-living

indexes with random prices.

These axioms, and the random price version of preference monotonicity are:

Monotonicity (random) Define Ωp,qλ as the probability space caused by the random price vectors p ̃

and q ̃

 λ, where element i of q ̃ λ equals λ⋅qi, with λ being a constant greater than unity. Then for

some index EI of random prices, EI(Ωqλ,q,u) > EI(Ωq,q,u) and EI(Ωq,q,u) > EI(Ωq,qλ,u).

Homogeneity (random)

Define Ωλp,q as the probability space of q ̃, λ⋅p ̃. For λ>0 λEI(Ωp,q,u)=EI(Ωλp,q,u).

Dimensionality (random)

For λ>0, EI(Ωλp, λq,u) = EI(Ωp,q,u)

Identity (random)

EI(Ωq,q,u)=1.

Preference Monotonicity (random)

sgn(EI(Ωp,q,u) - 1) = sgn(E[v(p,y)] - E[v(q,y)]).
7

For a given household, let the Konüs index be K(λiq,q,u), where λi is the realization of the

random variable λ ̃ for household i. Baye shows that, if E[λ ̃]=1 so that E[p ̃] equals q, then EK is

less than one. This follows from the concavity of the expenditure function with respect to prices

and from Jensen’s inequality.

As a result, the cost of living always falls with the introduction of price dispersion, which

raises the issues of risk aversion generated by Waugh (1944), although it has received little

attention in the literature on cost-of-living indexes. In the present context it seems

counterintuitive that the cost of living always falls with the introduction of price dispersion

because this seems to say that households would always prefer greater dispersion to less

dispersion. On the contrary, one might expect that there is a level of risk aversion that makes an

increase in dispersion undesirable.

Reinsdorf (1994a) touches indirectly on this issue by pointing out that although EK

satisfies the identity axiom above for accepted prices, it may not satisfy the identity axiom for the

distribution of offered prices if the household searches in both periods. He gives the following

example:

[S]uppose that a single good is consumed and that offered prices in both period 0

and period 1 are distributed such that after searching consumers have equal

probabilities of paying $1 or $2. Then EK will equal 0.5⋅(0.25)+1⋅(0.5) +2⋅(0.25),

or 1.125. In order for EK to satisfy the identity property, it must be the case that

whenever offered prices have the same distribution all consumers locate exactly the

same price that they did in the reference period.

As a solution to this problem Reinsdorf describes an index which he titles the ratio of

expectations index and will be referred to as the REK index here, for the case where prices are

random variables due  for example  to a consumer facing dispersed prices in retail markets.
8

This index describes the ratio of expected expenditures over offered prices in both periods. The

REK index is defined as:

(3) REK u Eepu

Eequ p q ( ,) [ (~, )][ (~, )]

Ω = ,

where the expectation is taken over the prices in the first argument of the expenditure function.

IV. New Complications

While the REK index is designed to measure the changes in offered prices, there are

potential hazards when using an expectation operator on expenditure functions. For example,

under price dispersion, the expenditure minimization problem implied by the REK has an

unsettling dual: a utility maximization problem in which prices and income are random variables,

with income varying in such a way as to hold utility constant.

This difficulty exists because the source of the base utility level u isn’t specified. If instead

one assumes that this level is the outcome of a standard utility maximization program, then utility

will itself vary with different prices. Income can then be thought of as fixed, which implies that

expenditures in the base period are also fixed. Now u can be replaced with the indirect utility

function v(b,y), where b represents prices in some base period. Substituting the indirect utility

function into the expenditure function, the Konüs cost-of-living index becomes

K*(p,q,b,y)= e(p,v(b,y))/e(q,v(b,y))

=a(p,b,y)/a(q,b,y).

The index K*(p,q,y) can be interpreted as the percent change between p and q in the cost of

maintaining the level of utility attained with base prices b and income y. The function a(p,b,y) is

the amount of income necessary to maintain the same level of utility with prices p, as is possible
9

with prices b and income y. Appendix A describes several useful relationships involving a(p,b,y)

and v(b,y).

Although in principle base prices can be any period, a logical selection is the reference

period associated with prices q. Diewert (1993) defines a Konüs index using this base period as a

Laspeyres-Konüs index, and it will be the focus of this paper. Setting b equal to q immediately

yields

 (4) K*(p,q,y)=e(p,v(q,y))/e(q,v(q,y))

=a(p,q,y)/y.

This shows that the calculation of the Konüs index involves only a single expenditure and utility

function. The second equality in (4) holds by P1 and P5 of Appendix A. Because q and y are the

same in both the numerator and denominator, utility is the same in both time periods. The index

K*(p,q,y) is then equal to the percent of expenditures with prices p necessary to achieve the utility

achieved with prices q and income y.

To form the random price analog of K*, transform EK by the substitution of v(q,y) with

u.

2

 The expected Konüs index is then:

(5) EK y E apqy

aqqy

E apqy

p q y *( , ) (

~,~, )

(

~,~, )

(

~,~, ) Ω = , 

  =  

 

2

 As with K*, there is no requirement that the base period for utility be the same as the period

with prices q. The generalization of REK* and EK* to an arbitrary base period are

EK y E pby

qby pqb *( , ) (

~,

~

, )

(

~,

~

, )

Ω = , ,





 



 μ

μ

and

REK y E pby

E qby

*( , ) pqb

[ (~,

~

, )]

[ (~,

~

, )]

Ω = , ,

μ

μ .
10

and the ratio of expectations index is:

(6) REK y Eapqy

Eaqqy

Ea pqy

y

*( , ) p q

[ (~,

~, )]

[ (~,

~, )]

[ (~,

~, )] Ω= = ,

Because y is a constant and base period prices are q ̃, the only distinction between REK* and EK*

lies in interpreting the probability measure as applying to offered or transacted prices. Given the

probability measure, the calculation of the two statistics is otherwise identical. One can then

consider many characteristics of the expected expenditure income ratio EIR(Ωp,q,y) =

E[a(p ̃, q ̃ ,y)]/y , without having to specify whether the index is REK* or EK*.

For example, one can consider how EIR satisfies the cost-of-living axioms:

Monotonicity (random)*

Define Ωp,qλ as before. Then for some index EI*(Ωp,q,y), EI*(Ω qλ,q,y) > EI*(Ωq,q,y) and

EI*(Ωq,q,y)> EI*(Ωq,qλ,y).

Homogeneity (random)*

Define Ωλp,q as before. For λ>0, λEI*(Ωp,q,y)=EI*(Ωλp,q,y).

Dimensionality (random)*

For λ>0, EI*(Ωλp,λq,y) = EI*(Ωp,q,y)

Identity (random)*

EI*(Ωq,q,y)=1.

Preference Monotonicity (random)*

sgn(EI*(Ωp,q,y) - 1) = sgn(E[v(p,y)] - E[v(q,y)]).

The following theorem shows that the identity axiom fails to hold for EIR, regardless of the

probability measure of the index:
11

Proposition 1) If the prices for a normal good are random draws from a nondegenerate IID

distribution within both periods then EIR(Ωp,q,y) >1.

Proof: See Appendix B

By specifying REK* and EK* as EIR(Ωp,q,y), we can also use a theorem in Helms (1985)

to examine preference monotonicity. Because this theorem describes the conditions under which

E[e(p ̃,v(q,y)]-y has the same sign as v(q,y)- E[v(p ̃,y)], it also gives the conditions under which

EIR - 1 has the same sign as v(q,y)- E[v(p ̃,y)]. If these conditions fail to hold, then EIR > 1 may

accompany an increase in expected utility and households actually prefer the cost of living, as

measured by EIR, to rise.

Theorem 1 (Helms 1985): Assume that v(p,y) is an indirect utility function of two goods,

one with price q in the base and random price p ̃ in the second period and the second a composite

numeraire good, and income y. If

sgn(E[e(p ̃,v(q1,y))]-y)= sgn(v(q1,y)-E(v(p ̃,y))

then for any p and y with x(p,y)>0

i) vyp=xyvy

ii) ρy=2ηy

iii) ρp=-ε

h

,

where

ρy yy

y

v

v = y − , the relative level of income risk aversion

ρp pp

p

v

v = p − , the relative level of price risk aversion
12

ηy x py y

y

x(p y = (,) , ) , the income elasticity of demand

ε h h pu p

p

hpu = (,) (,) , the compensated price elasticity of demand

with the subscripts on functions denoting partial derivatives.

For expected expenditures to always decrease while expected utility increases, households must

be risk averse in income in a very specific manner, while being risk loving in prices in another

specific manner. For all other preferences sets, there will exist p ̃ and q such that the change in

expected utility will move in the same direction as the change in the EIR index. By continuity,

one can also apply this argument to random q as well.

The intuition for the result is straightforward: while the expenditure function makes use of

ordinal information about utility, risk aversion uses cardinal properties. This means that

agreement between the expenditure function and the indirect utility function imposes restrictions

on the allowable types of risk aversion.

Such lack of welfare properties makes the cost-of-living index of little value in this case.

An EIR index is still generally useful for examining the cost of living increase moving from a fixed

price regime to a variable price regime where the mean prices are the same as the fixed prices in

the initial regime. It can be shown that in this case the condition necessary for a decrease in the

EIR index to always accompany an increase in expected indirect utility is that ρp

 be greater than

zero. Turnovsky, Shalit and Schmitz (1980) show that this is not unusually restrictive because a

sufficient condition for ρp

 to be positive is that 2ηy-ρy > 0. For example, this holds in constant

elasticity of substitution utility functions because ηy is equal to one while ρy equals zero.
13

V. A New Konüs Index Under Price Dispersion

The previous section describes problems when explicitly setting base period utility in the

expenditure function when prices are random. This section suggests a possible solution that also

allows the incorporation of risk aversion into cost-of-living indexes. Once done, the large body of

literature on expected consumer surplus measures can then be applied to consumer price indexes.

As an example of the implications of introducing risk aversion, let us examine two cases described

in Reinsdorf (1994a).

Before addressing risk aversion an additional idea is needed. This concept, as described in

Helms (1985), is the ex ante expenditure functional. Denoted as ê(Ωq,v), it is the solution to the

following problem:

Minimize y

subject to E[v(q ̃,y)] > u.

This functional is the amount of money required to achieve a given expected level of utility and

can be calculated by inverting E[v(p ̃,y)]. It is ‘ex ante’ in the sense that the random price is

realized before the expenditure function is formed. For a cost of living index one might still wish

to know how much money is necessary to restore utility to this level if the price distribution

changes. That is, it is of interest to ask what level of income ÿ is necessary to satisfy E[v(p ̃,ÿ)] =

E[v(q ̃,y)]. Solving for ÿ yields â(Ωp,q, y). The implied ex ante Konüs index is

(7) EAK y a y

y p q

p q ( ,) $( ,)

,

, Ω

Ω = .

To show that EAK satisfies the modified Eichhorn-Voeller axioms of monotonicity,

homogeneity, dimensionality and identity, note that by considering the price of the same good in
14

different states as separate prices on separate goods, E[v(q ̃,y)] can be considered to be an indirect

utility function itself. Therefore, EAK satisfies these axioms for the same reason that the standard

Konüs satisfies them.

While clearly in the spirit of the REK index, EAK offers several advantages. First, the

maximization problem generating ê(Ωp,v) and â(Ωp,q, y) is sensible. Second, the identity property

is satisfied because we are only taking expectations with respect to the indirect utility function.

Third, the index follows the preference monotonicity axiom. Helms (1985) shows this for the

compensating variation case:

Theorem 2 (Helms 1985): Given two random price vectors p ̃ and q ̃ fixed utility u and income y

sgn(ê(Ωp,v) - ê(Ωq,v))=sgn(E[v(q ̃,y)]- E[v(p ̃,y)]).

From this theorem it therefore holds that

sgn(EAK(Ωp,q,y)-1)=sgn(E[v(q ̃,y)]- E[v(p ̃,y)]).

The final advantage of this index is that it incorporates risk aversion. The change in the index

depend upon how the household views risk because the amount of income necessary to raise the

expected indirect utility level will depend upon the level of risk aversion imbedded in the indirect

utility function.

As an example, consider the cases of increased in price dispersion in Reinsdorf (1994a).

Following Reinsdorf, assume that the cumulative distribution function Φ(p;z) of offered prices is

twice continuously differentiable. Also, assume that for all z the function Φ(p;z) has a positive

support (B,T), B < ∞ and for at least one point in the support Φz ≠ 0. Define

Q r z dp z

B

r

(;) ≡ ∫Φ
15

Then the Rothschild-Stiglitz conditions for an increase in z to cause a mean-and-supportpreserving (MASP) increase in the dispersion of p are that (i) Q(r;z) > 0 for r ∈ (B,T) and (ii)

Q(B;z) = 0.

Assume that the consumer engages in a fixed sample size (FSS) search where the search

over prices pi occurs ni times. Nonsearchers for this good are defined as those for whom ni equals

one. Reinsdorf states that the cumulative distribution function for the minimum prices after ni

searches of prices with cumulative distribution function Φ(p;z) is 1-[1-Φ(p;z)]ni. The probability

density function is then niφ(p;z)[1-Φ(p;z)] ni -1, where φ(p;z) is the probability density function of

p. Reinsdorf shows that, for a single good, FSS search is a sufficient condition for a MASP

increase in price dispersion to imply REK <1. The equivalent proposition for multiple goods and

EAK is:

Proposition 2) Assume the distribution of the price of good one is independent of prices

p2,...,pm ≡p_, that good one has positive demand and that all prices pi have positive support

[Bi,Ti]. Then a sufficient condition for a MASP increase in the dispersion of p1 to cause

EAK(Ωp,q,y) <1 for searchers and non-searchers is ρp

1 > 0. FSS search is itself neither necessary

nor sufficient. If ρp

1 < 0, then there exists some distribution function and MASP increase in

price dispersion such that EAK(Ωp,q,y) >1 for searchers.

Proof: See Appendix B

In this case FSS search does not necessarily cause EAK(Ωp,q,y) < 1. The more important

condition is that ρp

1>0, i.e., that the searcher is risk-loving in prices. While ρp1>0 is sufficient, it is

not necessary. A weaker condition for an increase in dispersion to cause EAK(Ωp,q,y) < 1 is that -

E(v1|p1) is everywhere greater than 1/(1-Φ(p;z))n

1

 -1, where v1 is the partial derivative of the

indirect utility function with respect to the price of good one. This limit allows searchers to be

risk-averse in the price p1 but only up to some point determined by the specific distribution
16

function. Hanoch (1977) shows that, if all income elasticities are equal to unity and ρy>2, then

expected utility increases from a MASP increase in all prices.

Reinsdorf also shows that in the multi-good case a MASP increase in dispersion may

reduce the REK index for nonsearchers more than searchers, even for search samples as small as

two. He points out that this occurs because high levels of commodity substitution reduce the

distinction between a consumer visiting markets for multiple goods and visiting multiple markets

for the same good. Comparisons of nonsearchers with FSS searchers is then similar to

comparisons between FSS searchers with different numbers of searches. In these comparisons the

searchers with large ni do not necessarily reduce, on the margin, expected expenditures more than

nonsearchers when price dispersion increases. While the same intuition holds for EAK(Ωp,q,y), a

more precise connection between substitutability and the gains from price dispersion can be made.

Proposition 3) Consider the case in proposition 2, and let si be the expenditure share for good i.

If either |ε

h

| > s1(2ηy - ρy) or 2ηy - ρy > 0 then a MASP increase in dispersion benefits

nonsearchers more than searchers. If searchers benefit more than nonsearchers, then either |ε

h

|

< s1(2ηy

-ρy) or 2ηy-ρy < 0 over some price range.

Proof: See Appendix B.

This proposition explicitly lays out the connection among search, elasticities and risk

aversion. For example the first sufficient condition for dispersion to relatively benefit

nonsearchers implies that the greater the level of risk aversion in income the lower the

compensated price elasticity of substitution necessary for nonsearchers to relatively benefit. This

occurs because the marginal benefits of search are reduced with greater risk aversion. Greater

risk aversion must then be accompanied by greater elasticity or searchers will be the relative
17

winners. The second sufficient condition states that, beyond a sufficiently high level of risk

aversion (ρy=2ηy), nonsearchers are relative winners regardless of the compensated elasticity of

substitution. Also, note that income elasticity works in the opposite direction as compensated

price elasticity. Finally, if preferences are Leontief, then a MASP increase in price dispersion

benefits nonsearchers only if 2ηy>ρy.

VI. Neoclassical Utility: A Useful Special Case

Diewert (1993) defines a useful set of utility functions whose elements U(x) are

neoclassical if U(x) is (i) continuous, (ii) positive, i.e., U(x) > 0 if each element of x is positive,

and (iii) linearly homogeneous, i.e., U(λx) = λU(x) if λ>0. This class of utility functions is useful

in clarifying the relation between the EAK index and the REK index by using the fact that the

indirect utility function has the form v(p,y) = y⋅v(p,1). In this case, the expenditure function

e(p,u) is equal to u/v(p,1), a(p ̃,q ̃,y) = y⋅v(p ̃,1)/v(q ̃,1) and â(Ωp, Ωq, y) = y⋅E[v(p ̃,1)]/E[v(q ̃,1)].

The indexes EAK and REK* can then be expressed as

(8) EAK y

y Evq

Evp

y

Evq

Evp p q ( ,)

[ (~, )]

[ (~, )] [ (~, )]

[ (~, )]

Ω = ,

⋅

=

1

1 1

1

and

(9) REK y

y E v q

v p

y

E v q

v p

*( , ) p q

(

~, )

(

~, ) (

~, )

(

~, )

Ω = ,

⋅ 

  = 



 





1

1 1

1 .

It should be clear that the EAK index in (8) is inversely related to the indirect utility function so

that the EAK index only indicates an increase in the cost of living if the expected indirect utility

function decreases. Then EAK-1 varies inversely with E[v(p ̃,1)] - E[v(q ̃,1)] and Theorem 2 holds.
18

Theorem 1, however, still constrains an equivalent relationship for REK*. This applies if the

expectation operator generates the arithmetic mean of the random variable. In many cases,

however, we are interested in generating the geometric mean instead. For example, Flemming et.

al. (1977) show that the geometric mean avoids potential problems with the selection of a

numeraire good.

Proposition 4) Assume that utility is neoclassica and define the geometric mean of x as

exp(E[ln(x)]). Then if expectations are taken using a geometric mean, EAK=REK*.

To see this note that E[1/χ]=1/E[χ] for the geometric mean operator. In general, one cannot sign

the difference between REK* and EAK. For the neoclassical case, however, it is positive by

Jensen’s inequality.

While both risk aversion and risk loving in prices is possible with neoclassical preferences,

this attitude doesn’t change with income. Therefore, if an individual with a low income benefits

from a MASP increase in dispersion, he or she will still benefit even with an extremely high

income. In addition, the conditions in proposition (3) necessary for nonsearchers to benefit more

from a MASP increase in the price dispersion of good 1 can be simplified to |ε

h

|>s1. Again, if this

condition holds, it will do so for all income levels.

Finally, one can use the neoclassical utility function and the properties of a(p,q,y) to

describe the cost-of-living index version of the formula bias problem in the U.S. consumer price

index.3

 If we use base period prices b that differ from q, REK* =

E[v(b,1)/v(p,1)]/E[v(b,1)/v(q,1)].

 

3

 See Erickson (1995), McClelland (1996) and Reinsdorf (1994b).
19

Assume that the base prices are unavailable. For example, we may randomly select outlets

in the base period without recording prices. We may then wish to impute them using prices q.

The bias of this imputation is

(10) [ ]

[ ]

[ ]

[ ] [ ] [ ][ ] β= − = −

E v q

v p

E v q

v q

E v b

v p

E v b

v q

E v q

v p

E v b

v p

E v b

v q

( ,)

( ,)

( ,)

( ,)

( ,) ( ,)

( ,)

( ,)

( ,)

( ,)

( ,) ( ,)

( ,)

( ,)

1

1

1

1

1

1

1

1

1

1

1

1

1

1

 .

Cross-multiplying and using the definition of covariance, the bias can be written as

[ ] β =

− 



 



 cov

( ,)

( ,), ( ,)( ,)

( ,)

( ,)

v q

v p

v b

v q

E v b

v q

1

1

1

1

1

1

.

This is analogous to the example of formula bias given in Erickson (1995). As in that case the

bias is likely to be positive because increases in v(q,1) will cause v(q,1)/v(p,1) to rise and

v(b,1)/v(q,1) to fall.

VII. Conclusions

Cost-of-living index theory assumes that prices are nonstochastic. Recent work on costof-living indexes, however, has developed indexes for the case when prices are randomly

distributed. This paper uses consumer surplus theory to show that if utility is set by an indirect

utility function in the reference period then two of these indexes are calculated in an identical

manner. In this case, prices with stationary distributions can result in these indexes showing an

increase in the cost of living. In addition, the index may show an increase in the cost of living

even if households are better off than before.

Both of these problems are solved if the index is formed by comparing the level of income

with the income necessary to compensate the household for the change in expected utility. This
20

procedure not only resolves these problems, but also creates a context within which researchers

may discuss the use of risk aversion in cost-of-living indexes.
21

References

Baye, M.R. (1985), Price Dispersion and Functional Price Indexes, Econometrica 53, pp. 217-

223.

Diewert, WE (1993), Index Numbers, in The New Palgrave: A Dictionary of Economics The

Macmillan Press

Eichhorn, W. & J. Voeller (1970), Theory of the Price Index, Springer-Verlag, Berlin.

Erickson, T. (1995), Effects of Mismeasuring Base Period Prices When Estimating the Laspeyres

Index: some Idealized Cases, memo, November 1995

Flemming, J., Turnovsky, S. & M.C. Kemp (1977), On the Choice of Numeraire and Certain

Price in General Equilibrium Models of Price Uncertainty, Review of Economic Studies, 44, pp.

573-583.

Hanoch, G. (1977), Risk Aversion and Consumer Preferences, Econometrica 45, pp. 413-426

Helms, L.J. (1984), Comparing Stochastic Price Regimes: The Limitations of Expected Surplus

Measures, Economics Letters 14, pp. 173-178

Helms, L.J. (1985), Expected Consumer’s Surplus and the Welfare Effects of Price Stabilization,

International Economic Review 26, pp. 603-617

McClelland, R. (1996), Evaluating Formula Bias in Various Indexes Using Simulations, BLS

Working Paper 289

Reinsdorf, M. (1994a), The Effect of Price Dispersion on Cost-of-Living Indexes, International

Economic Review 35, pp. 137-149

Reinsdorf, M. (1994b), Price Dispersion, Seller Substitution and the U.S. CPI, BLS Working

Paper 252

Turnovsky, S., Shalit, H., & A. Schmitz (1980), Consumer’s Surplus, Price Instability, and

Consumer Welfare, Econometrica 48, pp. 135-152

Van Hoomissen, T. (1988), Price Dispersion and Inflation: Evidence from Israel, Journal of

Political Economy 96, pp. 1303-1314

Waugh, F. (1944), Does the Consumer Benefit from Price Instability?, Quarterly Journal of

Economics 58, pp. 602-614
22

Appendix A

P.1 a(p,q,y)=e(p,v(q,y)) is the expenditure function on prices p and utility level u and v(q,y) is

the indirect utility function with prices q and income level y. It also implies that for a fixed y and

prices p-= q-, sign(a(p1,q1,y)-y)=sign(p1-q1), where p-=[p2,p3,...,pn], q-=[q2,q3,...,qn].

P. 2 ap(p,q,y) = h(p,v(q,y)) where h(p,u) is a Hicksian (compensated) demand function. This

means that ap(p,q,y) is increasing in p if demand is positive.

P. 3 h(p,u) is increasing in u for a normal good and v(q,y) is increasing in y.

P. 4 v(q,y) is non-increasing in q and strictly decreasing in q when prices are positive, the

individual is unsatiated and demand is positive.

P. 5 a(q, q, y) = y.

By P.1 a(p, q, y) = e(p, v(q, y)), so that a(q, q, y) = e(q, v(q, y)), which is equal to y.
23

Appendix B

Proposition 1) If the prices for a normal good are random draws from a nondegenerate IID

distribution within both periods then EIR(Ωp,q,y) > 1.

Proof:

This proof is similar to that of Helms (1984). Without loss of generality let good 1 have random

prices in both periods. Let Φ(p) be the nondegenerate cumulative distribution function for the

random variables p and q. Then:

Eaqqy ( )

y

apqyd pd q

y

B

T

B

T

[ ~,

~, ] (,,) () ()

=

∫ ∫ Φ Φ

Adding and subtracting y, the numerator becomes:

y apqy yd pd q

B

T

B

T

+ − ∫ ∫[(, , ) ] () () Φ Φ

or

y apqy yd pd q apqy yd pd q

B

q

B

T

q

T

B

T

+ − ∫ ∫[(, , ) ] () () [(, , ) ] () () ΦΦ ΦΦ + − ∫ ∫ 1

1

.

By P1., when p < q then y ≡a(q,q,y) < a(p,q,y) and y ≡ a(q,q,y) > a(p,q,y). This implies that the

first integral is negative and the second is positive. Whether the numerator is greater or less then

y depends upon whether the absolute magnitude of the second integral exceeds the first. That

being said, we can cancel out y to get:

apqyd pd q apqyd pd q

B

q

B

T

q

T

B

T

(,,) () () (,,) () () ΦΦ ΦΦ 1

1

∫ ∫ + ∫ ∫

In this case, the numerator will be greater than y if the second integral exceeds the first. Because
24

p and q are identically distributed we can reverse the order of integration in the second integral.

Doing this and changing the variables of integration, the numerator becomes:

apqyd pd q aqpyd pd q

B

q

B

T

p

T

B

T

(,,) () () (,,) () () ΦΦ ΦΦ 1

1

∫ ∫ + ∫ ∫

Both integrals are being taken over the region p<q, and they are equal at the point p=q, so the

relative magnitudes are a function of the slopes of a(χ, ξ, y) with respect to χ. This depends upon

the sign of aχξ(χ,ξ,y) = dh(χ,v(ξ,y))/dξ. We can write this out as:

dh(χ,v(ξ,y))/dξ = hu(χ,v(ξ,y))vξ(ξ,y).

By P3., hu(χ,v(ξ,y)) is positive for a normal good and by P4. vξ(ξ,y) is negative. The slope of

a(χ,p,y) is therefore greater than the slope of a(χ,q,y) for p<q. This implies that the second

integral exceeds the first, so that the numerator exceeds y and REK* > 1.
25

Proposition 2) Assume the distribution of the price of good 1 is independent of prices

p2,...,pm ≡p_ , that good one has positive demand and that all prices pi have positive support

[Bi,Ti]. Then a sufficient condition for a MASP increase in the dispersion of pi to cause

EAK(Ωp,q,y) < 1 for searchers and non-searchers is ρp

1>0. FSS search is itself neither necessary

for sufficient). If ρp1>0, then there exists some distribution function and MASP increase in price

dispersion such that EAK(Ωp,q,y)>1 for searchers.

Proof:

Because EAK(Ωp,q,y) = â(Ωp,q, y)/y, ∂EAK(Ωp,q,y)/ ∂z = âz(Ωp,q, y), where z increases the

dispersion of the distribution of p, which is equivalent to êz(Ωp,v) when the distribution of q is

held fixed. Because ê(Ωp,v) varies inversely with the expected indirect utility function, it

sufficient to examine the change in expected indirect utility with respect to z. It is also useful to

note that, if for all z a function H(p) is positive and strictly decreasing in p over (B,T), then

H p dp z

B

T

∫ ( )Φ > 0 .

If instead H(p) is strictly increasing in p over (B,T) for all z, then:

H p dp z

B

T

∫ ( )Φ < 0 ,

where Φ is the cumulative distribution function.

The expected indirect utility from searching n times is:

E v p y z v p y n dp d

B

T n

B

T

n

n

( (~, ); ) ( , )( ) = ∫ ∫ − − K −

1

1

1 1 Φ φφ 1 ,
26

= − ∫ − E v p y p n dp n

B

T

( ( , ) )( ) 1

1

1 1

1

1

Φ φ

where φ- is the pdf of the prices of goods two to m, and n, Φ and φ are the numnber of searches,

cdf and pdf of good one. Integrating by parts:

E v p y z E v p y p E v p dp [ ] n

B

T n

B

T

( ( , ); ) ( ( , )| ) ( ) ( | )( ) = −− + − 1 11 1 1 1 ∫ 1

1

1

1

Φ Φ

where v1 is the partial derivative of v with respect to p1. Because Φ(T1)=1 and Φ(B1)=0, this may

be written as:

B.1) E v p y z E v p y p B E v p dp n

B

T

( ( , ); ) ( ( , )| ) ( | )( ) = =+ − 1 1 11 1 ∫ 1

1

1

Φ

=+ − C E v p dp ∫ n

B

T

( )( ) 11 1 1

1

1

Φ

Then:

B.2) E v p y z E v p n dp z

n

z

B

T

( ( , ); ) ( ) ( ) =− − ∫ −

1 1

1

1 1

1

1

Φ Φ ,

where Ez(v(p,y);z)= ∂E(v(p,y);z)/∂z. Because the good has positive demand and all prices are

positive, -E(v1|p1) is positive. The term n(1-Φ)

n-1 is also positive and decreasing. If ρp

1 >0 then -

E(v1|p1) is decreasing in p1 or is a positive constant. Then -E(v1|p1)n(1-Φ)

n-1 satisfies the

conditions necessary for Ez(v(p,y);z) to be positive so that expected utility increases with

dispersion and â(Ωp ,Ωq, y) falls. FSS search is clearly not sufficient. Nothing in the above relies

on n1>1. FSS search is therefore not necessary. If ρp

1 is negative, then -E(v1|p1) is increasing in

p1. If n(1-Φ)

n-1 falls more slowly than -E(v1|p1) rises then -E(v1|p1)n(1-Φ)n-1 rises. Then E

z(v(p,y);z) will be negative and an increase in dispersion raises â(Ωp,q, y).
27

 Proposition 3) Consider the case in proposition 2, and let si be the expenditure share for good

i. If either |ε

h

|>s1(2ηy - ρy) or 2ηy - ρy > 0 then a MASP increase in dispersion benefits

nonsearchers more than searchers. If searchers benefit more than nonsearchers, then either |ε

h

|

< s1(2ηy

-ρy) or 2ηy-ρy<0 over some price range.

Proof:

By the argument in proposition 2), we can consider the expected indirect utility function. By

equation (24) of Turnovsky, Shalit and Schmitz (1980), sgn(ρp

1)=sgn(s1(2ηy

-ρy) - ε

h

). Because s1

is positive and ε

h

 is negative, if 2ηy-ρy >0, then ρ p1>0. We therefore can consider the case where

ρp

1>0, which holds if and only if v11>0. First, we know that expected utility is higher for searchers

than nonsearchers. Using the notation in proposition 1):

B.3) E v n E v n E v p n dp E v p dp n

B

T

B

T

(| ) (| ) (| )[ ] (| ) >− == − − > ∫ ∫ − 11 1 0 1

1 Φφ φ 1 .

This holds by differentiating the righthand side of B.1) with respect to n:

[ ]

[ ][ ]

d E v p n dp

dn E v p n dp

n

B

T

n

B

T (|)

( | )ln

1 1

1 1

1

1 1 1

1

1

1

∫ −

= −− ∫

Φ

Φ Φ .

The terms E(v1|p1) and ln(1-Φ) are negative and the remaining term is positive, so the right-hand

side is positive. Integrating B.3) by parts and collecting terms:

E v n E v n E v p dp [ ][ ] ( )n

B

T

(| ) (| ) ( | ) >− == − − − ∫ − 1 1 11 1 1 1

1 Φ Φ

Differentiating with respect to z:

B.4) ( ) [ ] ( ) ∂

∂

Evn Evn

z

E v p n dp n

B

T

z

(| ) | (|)>− = `

= −− ∫ 1 1 − 1 1 1 1

1

1

1

Φ Φ
28

The term [1-n(1-Φ)

n-1] rises from 1-n to n as p increases from B1 to T1. If v11 > 0, then E(v1|p1)

rises and so the right-hand side of B.4) is negative. This means that, if a consumer is risk-loving,

then increased dispersion brings E(v|n>1) and E(v|n=1) relatively closer together. Because

E(v|n>1) is greater than E(v|n=1) and their difference decreases, increasing price dispersion

benefits nonsearchers more than searchers. Now suppose that E(v1|p1) falls very rapidly over the

region over which [1-n(1-Φ)

n-1 ] is negative and equals zero where [1-n(1-Φ)n-1 ] is non-negative.

Then the right-hand side of B.4 will be positive for some increase in dispersion. This shows that it

is possible for increases in dispersion to benefit searchers more than nonsearchers. Because it

cannot happen if v11 > 0 over the entire range of prices, it must be the case that v11 < 0 over

some range. This only occurs if either |ε

h

| < s1(2ηy-ρy) or 2ηy-ρy<0.