Tilburg University

Consumer's welfare and change in stochastic partial-equilibrium price

Stennek, M.J.

Publication date:1995

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Citation for published version (APA):Stennek, M. J. (1995). Consumer's welfare and change in stochastic partial-equilibrium price. (CentERDiscussion Paper; Vol. 1995-72). CentER.

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Tilburg University

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Centerfor

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~-i y ~ r' 7 No. 9572

CONSU~fER'S ~~'ELFARE AND CHANGE ItiSTOCHASTIC PARTIAL-EQUILIBRIUI~f PRICE

B~ Johan Stennek

August 199i

j~'i~Lt

(,UeC~Ctit; ~ Ct't~C~tittt: ~

CJt pCG~ CZ ~`t iC P~~Cc` cce S

ISSN 0924-7815

Consumer's t?~'elfare and Change in

Stochastic Partial-Equilibrium Price

b~'

Johan Stennek'

June 23, 199~

Abstract

First, I show that the expected consumer's surplus is equivalent to er

ante compensating variation if and only if the consumer is risk neutral,

and the consumer's income elasticitc of demand for the commoditv is zero.

~Ioreo~.er, the conditions are equivalent to the von ~euman - tilorgenstern

utilitc function being quasi-linear. Second, I show that the expected con-

sumer's surplus is an approximation for the consumer's welfare, measured

bv expected utilin, also if the expenditure share is sma1L Third, I pro-

pose a(ormula to evaluate approximatelc the consumer's welfare. measured

both by~ expected utility and by~ er ante compensating variation, when the

above conditions are not met.

'~1ail Institute (or International Economic Studies, Stockholm tiniversity, 5-106 91 Stock-

holm. SNeden Phone t46 8 162000, Fax: t46 8.161443. E-mail: Johan.Stennekaiies.su.se. 1

am grateful to Jan Bouckaert, Eric van Damme, Jos Jansen, Hatald Lang, Stefan Lundgren,

and Frank Vetboven for helpful discussions Finally, I thank CentER for Economic Research,

Tilburg l'niversity, The lethetlands, for an enjoyable visit during which this tesearch was done.

~

1 Introduction

In applied micro, for example industrial economics, it is frequentlv necessary

to evaluate the effect of a change in industry structure on consumer welfare.

The change may be a policy proposal that will affect the industry, or it may

concern some predicted change in firm behavior such as a merger. Often the

context is stochastic, so that one needs to evaluate the effect on consumer's

welfare associated with the move from one stochastic price regime to another.

~lore precisel}'. the present paper is concerned with the evaluation of the effect

on consumer's welfare when there is a change in the stochastic partial-equilibtium

price of a single commodity.' The purpose is to find practical tools that can be

easily used in applied work.

It is usefu] to distinguish between two different, but exact, measures of the

effect of a change in industry structure on consumer welfare, in a stochastic set-

ting. The first measure of the consumer's welfare is the expected indirect von

`eumann -`lorgenstern utility. The utilitv-theoretic íramework pro~-idcs impor-

tant qualitati~~e results (by definition it ranks different price regimes according to

the consumer's preferences), but it does not produce a quantitative measure of

the benefits or costs associated with the move from one price regime to another.

.-~ quantitative measure is nevertheless desirable in man}' applications, because

it could be compared ~eith the polic}~ makers' costs of program implementation

and~or changes in the producers' profits. To find a building block that is ron-

venient for socia! welfare judgments, Helms (1985) proposes a second approach

- the ez ante compensating cariation. This is the amount of income which. if

provided to the consumer in the new price regime, would restore expected utilitv

1Through out this paper, the variability in prices is assumed to stem exclusively from thesupply side. ~toreover, the price may be stochastic for two reasons: First, there may bevaria[ions in the exogenous conditions of the situation, for example the firms' cost condiuons.that generate variations in the market price. Second, the firms on a market may be requiredto use mixed strategies in equilibrium, so that the source of the variability of the prices isendogenous

to the le~-el attained under the old price regime. Helms shows that this mea-

sure generates rigorous qualitati~-e and quantitative assessments of price regimes,

~~ithout imposing any restrictions on preferences. (Helms (1984) shows that the

expected compensating variation is generally not a good measure of welfare.)

A third (but inexact) approach. which is the approach frequently taken in

applied work (see for example Baron and 11}-erson 1982, Shapiro 1986, Riordan

and Sappington 198 ï, and Kirby 1988) is to use the expected consumer's surplus

as a measure of the consumer's ~~~elfare. As an example, Shapiro (1988, p.43-1)

moti~~ates his use of the expected consumer's surplus by ignoring income effects.~

However, even when income effects are small, the expected consumer's surplus is

generally not a good measure of welfare, in a stochastic setting, since it is only

based on information about the consumer's demand (ordinal utility information)

and hence does not take into account the consumer's preferences towards risk.

To illustrate the problem, consider a consumer who has unit demand for the

commodity and willingness to pa}- a. Hence, there are no income (or price)

effects. Assume that the price is stochastic, but that we always havep G a(C m).

The residual income m- p is spent on a composite commodity with unit price.

The consumer's surplus is defined as the area under the (~farshallian) demand

function over the price, that is o-p. The expected consumer's surplus is Q- Ep,

which is independent of price dispersion. Since the consumer alwa~-s consumes

one unit of the good, his utilit}' can be indexed solel}' b}' his consumption of

the composite good, that is m- p. If the consumer dislikes variations in the

consumption of the compositegood (then his utility function is a concavefunction

of m- p), then the consumer is risk-averse with respect to variations in residual

income. Hence, the consumer's welfare decreases if (a mean preserving) increase

in the ~-ariance of the price occurs. However, the expected consumer's surplus

21C the mcome elasticity of demand is small, then the consumer's surplus is a good ap-proximation of compensating variation, in a non-stochastic setting, as shown by K'illig (1976)-However. when [he price is stochastic, then ~ti'illig's results are not applitable.

3

does not depend on the variance. and does not "detect'~ the change. Hence. the

expected consumer~s surplus cannot represent the consumer's preferences.

To iliustrate that risk preferences toward price risk ha~~e important conse-

quenses in the econom}., interpret the above discussion in terms of the market for

mortgages. In the short run, the consumer is stuck with his house and hence the

amount borrowed is fixed to the consumer, so that there are no price and income

effects. Jforeover, mortgage payments normally constitutes a large expenditure

share, so that variations in the interest rate creates substantial variations in the

consumer's real (or residual) income. It is, in this light, not surprising that the

consumer's risk-aversion toward income risk implies aversion toward price risk,

as manifested b}' the regular use of fixed rate mortgages 3

The facts that the expected consumer's surplus is a practical tool that is often

used, and that it is based solel}' on obser~.ables (the consumer's demand func-

tion), makes it important to establish the conditions under which it generates

rigorous assessments of consumer welfare. This question is discussed in Section

3. Obser~~ation 1 is a(simplified) restatement of Rogerson's (1980) Proposition 1,

and it shows that the consumer's surplus is a rigorous equivalent of the change in

utilit}~ if onlv if the marginal utilit}. of income is constant with respect to the price

oC the commodity-. Stated more operationall}~, the necessar}. and sufTicient condi-

tíon is that the consumer's relative risk aversion is equal to his income elasticit}-

of demand for the commodity. Obser~~ation 2 extends Rogerson's anal}.sis and

shows that the expected consumer's surplus is equivalent to ex arzte compensating

~.ariation if and only if (1) the consumer is risk neutral. and (2) the consumer's

income elasticit~- of demand for the commoditv is zero. It is also shown that

the conditions are equícalent to the von `eumann -`forgenstern utility function

3It ~s difficult to find othet markets where consumers insure against price risk However,if the expenditure share is smalL then the price risk does not create much real-income risk.1loreover, i( the quantity is not fixed. so that the consumer buys more when the price is lowand vice versa, then the consumer may actually benefit from price variability Also the supplyof price insurante may be limited due to mora] "hazard problems," if not the quantity is alsofixed ez ante

being quasi-linear.

Section 4 is concerned with sufficient conditions for the change in expected

consumcr~s surplus to approrimate the change in welfare, measured b}~ expected

utilit~- (Observation 3). In the context of approximations, the expected con-

sumer's surplus ma}~ be motivated also b}~ a small expenditure share on the com-

modity.

The expected consumer's surplus is a good quantitative measure of welfare

only if the consumer is risk-neutral with respect to income-risk. However, stan-

dard estimates indicate that the coefficient of relative risk-aversion is at least

one (~lehra and Prescott 1985). Aioreover, the expected consumer's surplus is a

good qualitative measure of welfare only if the consumer's relative risk-aversion

with respect to income-risk is equal to his income-elasticity of demand. How-

ever. standard estimates of the income-elasticity of demand indicates dispersion

between .5 and 1.5 for different goods (see for example the review b}' Deaton and

?~luellbauer 1986). It is consequentl}~ important to design tools that can be used

also in the case the consumer is not risk-neutral, and when his income elasticity

of demand is different from relative risk-aversion. Section 5 is concerned with

this case and suggests simple formulas to approximately evaluate the effect on

consumer welfare, measured both b}~ expected utility (Obsen~ation -1) and b}~ ez

ante compensating ~~ariation (Observation 5). In the formula, the change in the

price regime is described b}- onl}~ two variables: (i) the change in the expected

price, and (ii) the change of the variabilit}' of the price. All other aspects of

the changed price regime are abstracted from. The parameters of the formula

are given b}' the consumer's preferences, namel}' (i) the consumer's relative risk

aversion with respect to income risk, (ii) the price elasticity of demand, (iii) the

income elasticit~. of demand, and (iv) the commodity~s expenditure share. .All

parameters are evaluated in the old price regime, and may hence (apart from the

risk aversion) be estimated by observation of the consumers demand in the old

price regime.

1

J

2 Price Risk and Welfare Measures

Consider a consumer who allocates budget mo 1 0 between purchases of a ho-

mogeneous commodity with price p and a composite good which serves as a

numeraire. The price is distributed according to cumulative distribution func-

tion F on (O.fx). The consumer makes his purchases at known prices after

the uncertainty is resolced. ~Iaximization of a quasi-concave, twice continu-

ously differentiable, strictl}~ increasing von ~eumann - 17orgenstern utility func-

tion subject to the budget constraint generates the demand function D(p, m)

and the indirect von `eumann - 1lorgenstern utility W(p,m). It is assumed

that D(p, mo) ~ 0 for all prices p. Consequently, Wp (p, m) G 0 for all p. Let

s- p- D(p, m) ~m be the share of consumer's budget allocated to the commod-

ity. Let e - DP(p.m)(p~D(p,m)) be the own uncompensated price elasticity

of demand for the commodity, and rr - Dm(p,m)(m~D(p,m)) be the income

elastic'ity of demand for the commodity. Sub-indices indicate partial derivati~.es.

The consumer's risk-preferences toward income risk are represented by p-

(-6V m(p, m) . m) ~ Wm (p, m) which is the income elasticity of the marginal util-

ity of income, which in turn is the ~rrow (19 i 1) - Pratt (1964) measure of relative

risk aversion with respect to income risk. In analogy, define a price risk averter

as a consumer w~ho is ~cilling to take an insurance that stabilizes the price at its

expected ~~alue. Hence. lï~ (p,m) ~(1~2) G6' (p - h, m) f(1~2) ib' (p t h, m) or

W' (p, m) - 6i~ (p - h, m) , W' (p t h, m) - bi' (p, m), that is the utility difference

corresponding to equal changes in the price are decreasing as the price increases.

Thus, the utility function of a price risk averter is characterized by the condition

that Wp (p, m) is strictly decreasing as p increases, or that Yi~yy (p, m) G 0. Price

risk neutrality and price risk love are defined in a similar way. To measure price

risk preferences quantitatively, Turnovsky, Shalit, and Schmitz (1980) propose

the use of the coefficient of relative price risk aversion, O -(-WpP (p,m) . p)

~WP (p,m), defined in analogy with p. tiote that the measure is invariant under

s

positive affine transformations of the utility function. lote that o GO) 0 if

61~ar G(1)0 that is if the consumer is risk avert. (To avoid corifusion note that

o c 0 implies price risk acersion, but that p 1 0 implies income risk aversion.)

~foreover, Turnovsky. Shalit, and Schmitz (1980. expression (23)) proposes an

identity that links the consumer's relative risk-aversion with respect to price risk

to more familiar concepts:

o-s'(0-P)-E. (I)

Consequently, the net risk-aversion with respect to price risk can be decomposed

into three terms: the "price-elasticit}~ effect," the "income-elasticity effect," and

the "income-risk effect." tiormally, the "price-elasticity effect" makes the con-

sumer positive toward price risk (for a good with DP c 0). Intuitively, the

reason is that the consumer buys a large quantity when the price is low and

small quantity when the price is high, hence he may in expectation bu}' the same

amount as he would under stahle prices. but at a lower expected expenditure

(price times quantity). `ormally, the "income-risk effect" makes the consumer

negative toward price risk (for a consumer with p~ 0). Intuitively, the reason

is that variations in the price creates variations in real income, in proportion to

the expenditure share. and that such income ~.ariations are evaluated according

to income risk preferences. ~ormallv, the "income-elasticity effect" makes the

consumer positi~e toward price risk (for a good with p~ 0). This effect is related

to the effect on other markets.

The change in industry structure is represented by a move from cumula-

tive distribution Fo to cumulative distribution F1. The consumer's preferences

over price regimes are represented by expected indirect utility EW (F, m) -

j W(p, m) dF, so that ELb` (F',m) ~ EW (Fo, m) if and only if the consumer

weakly prefers F' to Fo. Since the discussion concerns change, define the change

in expected consumer's welfareas ~EW (Fo, F',m) - EW (F', m)-EW (Fo, m).

Jloreover, any function G(Fo, F', m) represents the consumer's preferences if it

'f

pro~ ides the same ranking, that is if G( Fo, F',m) ? 0 t~ :,E6i~- (Fo, F',m) ~ 0.

or equi~'alently if

,Eli' ~Fo. F', m~ - k- G~Fa. F'. m~ , for some k~ 0. (2)

The er ante compensating variation, denoted b}' C(Fo, F', mo), is defined as

the amount of income which. if provided to the consumer in the new price regime

F', would restore expected utility to the level attained under the old price regime

Fo. Hence, C(Fo. F', mo) is defined implicitlv by

I 4i" ~p. mo f C ~Fo, F', mo~~ dF' - 1 6i' ~p,mo~ dFo. (3)

The prescript ea ante indicates that C(Fo, F',mo) is not contingent on the re-

alization of the price and may hence be given to the consumer before the price is

known. Helms (1985) shows that the ez anfe compensating variation represents

the consumer~s preferences, without imposing an}' restrictions on preferences. To

see this, first note that C(Fo, F',mo) G 0 implies

~ Ib' ~p, mo~ dF' ~~ lti ~p, mo f C lFo' F'' mo~ ~ dF' -~ W lp' mo~ dFo

since li' is increasing in income. Second, note that by the same argument

C(Fo, F',mo) ~ 0 implies J 6V (p, mo) dF' G f W( p. mo) dFo. bloreover, the

er ante compensating variation has a quantitative, willingness-to-pay, interpreta-

tion. that follows directly from the definition.4 Helms ( 198~) also defines ei ante

equicalent variation, as the amount oí income which, if taken from the consumer

in the old price regime Fo, would give him the expected utility associated with

F'. The e.r ante equivalent variation hence measures the consumer's willingness

to pa}~ to avoid the change from Fo to F'. Although this measure also represents

the consumer's welfare and has a willingness to pay interpretation, I confine at-

tention to ez ante compensating variation since it is a natural building block for

social welfare assessments using the compensation principle.

"The er antc compensating variation was also defined by Schmalensee (19ï2) to assess thevalue of a price change under uncertainty about income and ptefetences.

Consumer~s surplus is defined as the area under the ~farshallian demand

function over the price, that is

CS ~p. mo~ - f x D~x, mo) dx. (-f )

Expected consumer~s surplus, w-hen price is distributed accordíng to F, is

ECS ~F, mo~ - f CS ~p, ma~ dF, (5)

and the change in consumer's surplus resulting from the move from Fo to F' is

gi~-en by

~ECS ~Fa, F',mo~ - ECS ~F', ma~ - ECS ~Fo, ma~ . (6)

3 Expected Consumer's Surplus as an Exact

Measure of Welfare

Rogerson (1980) has shown that, under the assumptions of the present paper, the

necessan~ and sufficient conditions for the expected consumer's surplus to repre-

sent the consumer~s preferences is that the marginal utility of income is constant

w~ith respect to the price of the commodity, that is lYmp (p. m) - 0. Stated more

operationallc. the necessart~ and sufficient condition is that the consumer's rela-

ti~-e risk a~.ersion is equal to the income elasticity of demand for the commodity,

that is p- p(see Claim 1 in .appendix A). To summarize,

Observation 1.~Eli' (Fo, F', m) - k.~ECS(Fo. F',m) iJ and only iJ

(1~ P-rl.

u~here k~ 0 is an arbitrary constant.

Proof: Rogerson (19t30).

lote that JEbi~ is only proportional to ,,ECS, as indicated by the constant k.

However. proportionality gi~-es all essential information about utility differences.

9

In particular, .,ECS and .~E6b" w.ill rank different price-regimes in the same

wa}., regardless of k.

Ob~er~.ation 2 extends Rogerson's anal}~sis and shows that the change ex-

pected consumer's surplus not only represents the consumer's preferences, but

also is equivalent to (the negative of) ez ante compensating variation and hence

has a w~illingness-to-pay interpretation. if and only if ( 1) the consumer is risk-

neutral, and ( 2) the income elasticity oí demand for the commodíty is zero.

Observation 2 .)ECS (Fo, Fl, mo) --C (Fo, Ft, mo) if and only if

(1~ p - 0, and

(~~ rl - 0.

Proof: See Appendix A.

The positive side of this result is that ez ante compensating variation can be

computed as the consumer's surplus under D(p, mo) which can be observed. The

negative side of the result is that the conditions under which this is possible are

cer~- restrictive. This issue is further discussed in Section 5.

For modeling purposes it is interesting to note that p- rl - 0 is equivalent

to the indirect utility function being of the form

6~'(P.m)-ti~(P)~m. (i)

and positive affine transformations thereof (see Claim 7 in Appendix A). ~1ore-

over, this is equivalent to the (direct) utilit}~ function C(q, y), where q denotes

the commodit~~ under discussion and y denotes the composite good, being of the

fórm

f' (9. y) - C' (9) ~ y, (8)

and a(fine transformations thereof. Hence, the requirement is equivalent to the

utilit}~ function being "quasi-linear."

10

4 Expected Consumer's Surplus as an Approx-

imation of Welfare

Suppose that s. (p - p) is small, and that we are willing to abstract from the sum

of the `'income-elasticity effect" and the "income-risk effects when we evaluate

the consumer~s preferences, then expression (1) reduces to o:-e, and by the

definitions of a and "

Hwa (P~ m) ,.. Dn ( P, m)(9)

Wvl~P,m) ~ D(P.m).

Rewriting the expression as (8~8p) ( ln M'a (p, m)) - ( 8~8p) (ln D ( p, m)) and in-

tegrating both sides with respect to p, gives D ( p, m) .~ -k-~ - Li'o (p, m), where k

is an arbitrary positive constant. Integrating both sides with respect to p again,

and using the definition oï the consumer's surplus yields

CS(p,m).~k-'.i~'(p,m)-k-l.ht'(x.m). (10)

Consequently, if s (rt - p) ti 0 then the consumer's surplus is approximately a

positive affine transformation of utility, and hence an approximate representa-

tion oí preferences. Observation 3 is an approximation counterpart to Rogerson

(1980).

Observation 3 .1EW (Fo, F',mo) ti k ..~ECS (Fo, F',mo) if

(Il s (rl - P) - 0~

u~here k~ 0 is an aróitrary constant.

Proof: Take expectations of expression ( 10) and form the differences.

Again, note that JE6i' is only approximately proportional to .,ECS, as indicated

by the constant k. However, proportionality gives all essential information about

utility differences. In particular, :~ECS and ,,EW will rank different price-

regimes in approximately the same way, regardless of k.

.As expected, also in the case of approximations the condition is that income

elasticity of demand should be (approximately) equal to the relative risk aversion.

11

Howe~.er, in the context of approximations, then a small expenditure share is also

a sufficient condition for using the expected consumer's surplus as a qualitative

measure of welfare.

.~n important difference between the approximations suggested in the present

paper and the approximations suggested by ~i'illig ( 1976) is that he derives ob-

servable bounds on the error made by using the approximation. tio such bounds

are offered here.

5 Direct Approximations of Welfare

The expected consumer~s surplus is a good quantitative measure of welfare only

if the consumer is risk-neutral with respect to income-risk. However, b4ehra and

Prescott (1985) survey a variety of studies that conclude that the coefi`icient of

relative risk-aversion is at least one. ~loreover, the expected consumer's surplus is

a guod qualitative measure of welfare only if the consumer's relative risk-aversion

with respect to income-risk is equal to his income-elasticity of demand. How-

ever, standard estimates of the income-elasticity of demand indicates dispersion

between .5 and 1.~ for different goods (see for example the review by Deaton and

Jluellbauer 1986). It is consequently important to design tools that can be used

also in the case the consumer is not risk-neutral, and when his income elasticity

of demand is different from relative risk-a~-ersion.

To derive the approximation results of this section, consumer welfare is ap-

proximated by a second degree Taylor expansion of W(p, m) around (Eo, mo) .

:~loreover. the price regime F` is characterized by the mean E' - J p dF` and the

i?

variance t'' - J(p - E')~ dF'.

t1'(p.m) .ti tt'(Eo.mo)

f[ty(Eo~mo)'(P-Eo)t2t~~nv(Eo,mo)'(P-Eo)~ (11)

~-hL~m (Eo. mo) . (m - mo) f 2 t4~m,~ (Eo, mo) . (m - mo)s

ttl'my (Eo- ma) ~ (m - mo) (P - Ea) .

The expected utility oí situation (F'. m') is approximated by taking expectations

Eli~ (F', m') ti t4' ( Ea. ma)

rliF(Eo,mo)'(E'-Eo)~2t'~.vn(Eo,mo) ~l''t(E'-Eo)~~

fti'm (Eo, Tna) '(m' - mo) -1. 2 bi-'mm (Ea. ma) -( m' - mo)~

ttt~mv (Eo, mo) ' (Tn~ - mo) (E' - Eo) .(12)

Consequently, the expected welfare is written as a function of income and only

the first two moments (mean and variance) of the price regime.

The second degree Taylor polynomial is a good approximation of the indirect

utilitv function, close to ( Eo, mo), in the sense that all parameters s, e, r), p, and v

are unaffected by the transformation at (Eo, mo).5 Jforeover, it is an improvement

over the expected consumer's surplus, since it is flexible enough to incorporate the

consumer's attitudes to~~~ards risk. However, a straight forward use oí quadratic

utility functions has been criticized, by for example .~rrow (19 i 1), on two related

grounds: (1) for incomes (prices) above (below) a certain level, additional incomes

w.ill decrease (increase) utility; and (2) it violates the "principle" of decreasing

absolute risk-aversion. Howe~er, in the present context the quadratic form is

used as a local approximation of the true utility function. Consequently, two

identical consumers, but with different incomes, are indeed both described by

some quadratic utility, however the (estimated) parameters will differ between

5`ote that T (Eo. mo) - lt' (Eo, mo), and Ty (Eo, mo) - Wp (Eo, mo), Tvv (Eo, mo) -

wnv (Eo, mo). m(Eo, mo) - W .,~ (Eo, mo), ~,.~ (Eo. mo) - K',.~m (Eo, mo). andTy,., (E", m~) - 4b'ym (Eo, mo). bloreover, define pr --(T ,,, - m)~ m and similarly for57~, ET~ 7r. and oT. Then pT - p (Eo, mo) and so on.

13

the two consumers.

For convenience. I consider the percentage change in expected price, denoted

b}~ E-(E' - Eo] ~Eo. and a relati~-e measure of the change in variance Y' -

(I'' -['o) ~(Eo)~. Consequenth., all changes are measured without dimension

(S~~ and S'~SZ respectively), and are hence comparable.

Observation 4 provides a simple approximation of the change in welfare.

Observation 4:~EF4' (Fo, F', mo) : I-E f 2~E2 f V)J k,

a~here o is et~aluated at (Eo,mo), and kl1 0 is an arbitrary constant.

Proof: See Appendix B.

Again, due to the constant k~ 0, DEW is only approximately proportional to

-E -~ 2~E~ -~ V~. However, only the sign of ~EW is of interest, and since the

constant is strictly positive, the sign of .,EW' is independent of k's value.

A potential problem with the formula is that we have very vague ideas about

consumer's preferences towards price risk, as measured by o. However, the

Turnovsky et al. identity, expression (1) above, helps to evaluate consumer's

risk preferences with respect to price risk, in terms of more familiar concepts.

`ote also that the expenditure share s, the income-elasticity of demand rl, and

the price-elasticit} of demand :. are all obsen-able. Jloreover, the approximations

used in Observation -t are made around price Eo and income mo. Consequently,

all the parameters are e~aluated at the price Eo and income mo. This is conve-

nient since then s. q, and e can then be estimated by observation of the agents

consumption beha~~ior in the old price regime Fo. However, the consumers rel-

atice risk acersion with respect to income risk p can not be estimated just by

using such ordinal information, but must be provided by other means.

By inspection of Observation 4. and using expression (1), the following com-

parative statics results are immediate. The consumer is made worse off if the

expected price increases E' 1 Eo. (The formula is an approximation so the

change must not be too large. For example, if o-.1, then the change in ex-

pected price should not exceed one thousand percent, in order for the effect to

have thc "right" sign.) The consumer's preferences towards price variability are

ambiguous. Suppose that the commodity is a normal good, so that r) ~ 0 and

hence ~ c 0. If risk aversion p and the expenditure share s are not too high.

then the "price-elasticity effect" determines the net effect, and then the con-

sumer prefers price variabilit}'. According to Turnovsky et al. (1980) we may

take e- -.2, rt - 0.6, p- 1, and s-.3 as typical estimates of a composite

commoditv such as food. For a more extensive survev, see Deaton and ~luell-

bauer (1986). Hence, for such a commodity v-.32 and, consumers prefer risky

prices. On the other hand, if the expenditure-share of the commodity is high

and the consumer is very risk-averse, then the consumer benefits from price sta-

bilization. Finally, note that Observation 4 suggests that the relative importance

of the change in expected price and the change in price variability is determined

solely by the consumer's relative risk aversion towards price risk o.

Obser~.ation ~ provides an approximation of the ez ante compensating varia-

tion.

Observation 5 The ex ante compensating rariation, in relation to income,

C(Po, Pi. mo) - C(Po, Pi, mo) ~ma is approrimated b y

C: ~~I-s(rl-P)E~-~P~~1-s(rl-P)E~~f2~s~-Ef-2~E~-~6'~~.

u.here s. rt and p are evaluated at (Eo, mo) .

Proof: See .~ppendix B.

It is possible to derive a much "cleaner" expression for the ex ante compensat-

ing variation if one is willing to make the Taylor approximation around price

(Eo -{- E' ) ~2 and income (mo ~ ml) ~2. However, this means that the parame-

ters cannot be directly estimated by observation of the consumer's consumption

pattern in regime (Fo, mo).

ls

6 Concluding Remarks

The paper suggest simple tools to evaluate the welfare-effect of a change in the

stochastic partial-equilibrium price of a single commodity. The paper provides

conditions under which the expected consumer's surplus provides an exact or

approximate index of consumer welfare. measured both by expected indirect von

`eumann - ~forgenstern utility and by ez ante compensating variation. Moreover,

approximation-formulas are provided for the case when these conditions are not

satisfied.

The discussion in Section 3 pointed out that not only the income-elasticity

of demand, but also the consumer's attitudes towards income-risk is an impor-

tant determinant of how a consumer ranks different price-regimes. It turns out

that risk-neutrality with respect to income risk is a necessary condition for the

expected consumer's surplus to be equi~.alent to the er ante compensating vari-

ation.

Although the approximation results of Section 5 are (by definition) not exact,

they do provide important additional insights about the crucial importance of

risk-preferences. An immediate consequence of Observation 4 is that all studies

that rely on the expected consumer's surplus to make welfare judgments should be

interpreted with much caution. First, the Turnovsky et al. identity shows that

b} ~aning the coefficient of relative risk-acersion with respect to income risk

nny preference towards price-risk can be generated, compatible with any given

demand functions. Second, Observation 4 shows that. by varying the relative

risk-aversion ~cith respect to price-risk any conclusion about the welfare-effect of

a change in the price-regime (if the variability of the price is changed) can be

generated.

16

A Equivalence Between ~ECS and -C

In all proofs it is assumed that m 1 0 and p~ 0. I sav that two functions f(p, m)

and g(p.;n) are identical. f( p,m) - g(p,m), if and only if f( p,m) - g(p,m)

for all m~ 0 and p 1 0. ~ote that [i'm(p.m) ~ 0 and Li'y(p,m) G 0 for all

m 1 0 and p~ 0. In some proofs a star sy-mbol ' indicates the steps in the proofs

w.here the conditions of the Claim are cruciaL If f(p, m) - f ( p', m) for all p and

p' then I write f(.,m).

Claim l p- r~ ea bi'mp (p, m) - 0.

Proof:

1 D(p.m)--Wv(P,m)Wm (P~m)

2~ Dm (n.m) --Bv,n(P~m) ~ 1L~n(P,m) W m(P.m)

3 r~ rl - D ([i'am ÍP, m) m I~'mm (P, m) m

4 t~ r1-bii~c(P-m) - l~m(P,m)

rl-~nm(P.m)m~PJ q

6iP(p.m)

6 t~ l~f nn` (P~ m) m-1'Iv(P,m)

-r1-p-

Step 1: Ro}-'s identity. 5tep 2: Differentiate Roy's identity with respect to m.

Step 3: Jlultipl}- both sides by m~D and use the definition r~ -(Dm m) ~D at

the left hand side and Ro}.'s identity. at the right hand side. Step ~1: Rearrange.

Step 5: L-se the definition p--(lt'mm . m) ~Ii~m. Step 6: Rearrange.

Finally-. since m~lb~ G 0 for all m 1 0 and p~ 0

11'ym(P,m) -0 a rl-p-0.

Claim 2 p- 0 ta it mm (p, m) - 0.

Proof: B~~ definition

lb'm (P,m) lbm (P,m) wín (P,m)m ) ~ tilnmlP-m) Wmm(P,m)~p m D(P~ m) 6Vy ( P, m)

- D(p, m) W (R m)

14~mm(P,m)m- l~~~m (P, m ) - P.

17

Since-m;ótmcOfora11m~0andp10

W~m.~(P~m)-0qp-0.

Claim 3 If and on(y if p- 0,

W~p, ml) - tL' ~p, mo) f lL~„ lP' mo~ ~m~ - mo~ '

Proof:

1 Ió W~m(P.z)dz- jó Wm(P.z)dz

2' q fmó ti'm (P. zl dz - LL'm (P. mo) f,no dz

3 q fó~ 1b'm ( P, z) dz - W~m ( P, mo) ( m` - mo)

4 q W(P,m~) - W ( P,mo) - W:n (P,mo) ( ml - mo)

5 q W(P,m')-W~(P.mo)tVi~,n(P~mo)(m'-mo)

Step 1: Self-evident. Step 2: First, note that t4'R, (p,z) can be "factored out" if

and only if it is constant in the relevant interval. Second, note that W'm (p,z) is

constant for all inten-als if and only if Wmm ( p,m) - 0, which is equivalent to

(Claim 2) p - 0. Step 3: Lse fó3 dz - (m' - mo). Step ~: The function W is

twice continuousl~. differentiable. Step 5: Rearrange.

Claim 4 If and only if p- 0 and ri - p,

C~Fa.F',mo) - li'm (., mo) ~ I I I4' ~p, mo) dFo - I bi' ~p, mo) dFIJ

Proof: lJ J

t f i~i (P, mo f C) dF~ - f W(P, mo) dFo

2' p f~W~ (P, mo) t W m(P, mo) C~ dF' - f W' (P, mo) dFo

3~ a f w~ (P, mo) dF' f Yi'm (', mo) C f dF' - f LF (P~ mo) dFo

~ q C- ti,n (', mo) ~~f t~ (P, mo) dFo - I w. (P, mo) dF~~ -

Step 1: Definition of C. Step 2: lise Claim 3. Step 3: First, note that Wm (p, mu)

can be "factored out" if and only if it takes on the same value at all p with positive

ls

mass. Second, note that bi~'m (p, mo) takes on the same value at all p with positive

mass, for all distribution functions F', if and only if Gi'mn(p,mo) - 0, which is

equivalcnt to (Claim 1) p - p. Step 4: Cse JdF' - 1 and rearrange.

Claim 5 If and only :f ri - p,

,,ECS ~Fo, F', mo~ --Lb'm ~., mo~ 1 LJ t~. ~p. ma~ dFo - f lL' ~p, mo~ dF1 J .

Proof:

1 .~ECS - ECS (F'. mo) - ECS ( Fo, mo)

2 q .)ECS- JCS(p,mo)dF' - JCS(p,mo)dFo

3 t~ ~ECS - j Jy D(z, mo} dz dF' - j Jy D (z, mo) dz dFo0 0

4 e~ ~1ECS--JJa LW~(zm~dzdF'tjJy W~~i'~~dzdFo

5' q .~ECS -- J W'm (P, mo)-' JP li-y (z, mo) dz dF'-t-

f f W~,n (p, mo)-1 fo 4i'p (z, mo) dz dFo

6" a.7ECS --4i'm (' , rna)-1 [f fp l~Lp (z, mo) dz dF` - J Jy ti9(z, mo) dz dFol

ï q :)EC S - - l4'm (., mo) ~ [f [LL~ (x, mo) - w~ (P, mo)~ dF'-

- J[W ( x, mo) - W( P, mo)~ dFu~8 e~ JECS --li'm (.. ma)-1 [j W' (p. mo) dFo - I LL' (P, mo) dF`j

Step 1: Definition of ,EC S. Step 2: Definition of ECS. Step 3: Definition

of CS. Step ~: Roy~s identity. Step .3: First, note that W'm(p.m)-` can be

`'factored out" if and onh~ if 6i~m (z, m) is constant for all z E (p, x), where p

is fixed. Second, note that the operation can be done for all F1 and Fo if and

onl}~ 6t'm ( z. m) is constant for all z E(0. x), that is for all z E(p, x) where

p is variable. Third, note that 4lm ( z,m) is constant for all z E ( 0, x) if and

only if lL'my ( p, m) - 0, which is equivalent to (Claim 1) ~- p. Step 6: First,

note that 6L'm (p,m)-` can be "factored out" if and only if Wm (p,m) takes on

the same value at all p with positive mass. Second, note that LL'm (p, mo) takes

on the same value at al] p with positive mass, for all distribution functions F',

19

if and onl}~ if [i'mD (p, mo )- 0. ~~.hich is equivalent to (Claim 1) p- p. Step 7:

The function L[' is twice continuously differentiable. Step 8: Rearrange and use

f[~4'(:~,,m)dF-[Y'(x.m)JdF-[b"(x.m).

Claim 6 C(Fo.F',mo) --.JECS(Fo,F'.mo) a p- rt - 0.

Proof: First, show that C- -.~ECS ~ p- p: According to Helms (1985)

C represents the consumer~s preferences. B}~ assumption C --DECS, and

hence -L1ECS represents the consumer's preferences. But, according to Roger-

son (1980) expected consumer surplus represents the consumer's pteferences if

and only if 1Lmp (p,m) - 0. ( This was shown in a more general framework than

used here.) Hence Limy(p,m) - 0. Equivalently ( Claim 1) rl - p.

Second, show that C- -.,ECS ~ p - 0. or equivalently (Claim 2) C-

-.~ECS ~ [i'mm (p, m) - 0: Since rt - p we have

.1EC S ~Fo. F', mo~ - - [ti'm ~ , mo~ ~ I f [i' ~p. mo~ dFo - f [i' ~p. mo~ dF'

according to Claim ~. Then, b} assumpttion

C~Fa, F',mo) -[[',,, ~~, mo) -~ I f[i' ~p, mo~ dFo - f 6{~' ~p, mo~ dF'1 .

L'se Claim ~ to conclude p- 0.

Third, sho~c that p- q- 0 ~,,ECS (Fo, F',mo) - -C (Fo. F', mo).

.~ssume p- rt and p- 0 and combine Claim 5 and Claim 4.

Claim 7 If and only if p- p- 0, there exist a function L' such that

4["(P,rn)-a~V(P)fm]fb,

for some constants a ~ 0 and b.

(13)

20

Proof: First, it is eas}~ to see that if expression (13) holds then r~ - p- 0.

Second, consider the reverse.

1

2'3'4

6i~(P,m) - ff (P.m)

p 6~'(P~m)-I~ (P.C)~-li'm(P,C)(m-C)

t~ ib'(P,m)-H~'ÍP.~)-~bi'm(-,i~)(m-C)

~ LL' (P.m) - [V (P) t m] bL'm ('. ~) - ~Wm (', ~) ~

Step 1: Self evident. Step 2: Claim 3. Step 3: If and only if W P(p,m) - 0,

that is if and onlv if p - p, there exist a constant [ti'm (., ~) such that 6i"m (p, ~) -

li'm (~.~). Step 4: Let V(P) - ~L (P~~) ~II'm (~,~)-

B Direct Approximations

Observation 5 Form the equation .~E6t' (Fo. F',mo) - EW (F',mo)-EW (Fo, mo),

but using the approximation (12). Then

DEW (Fo, F',mo) ~

W P (Eo, mo) ' (E' - Eo) ~ r~ IVPP (Eo~ mo) ' ~(~~1 - V o) } (E' - Eo)~

Factor out I~'P (Eo, m) - Eo

(14)

.~Eli~ ( Fo. F', mo) ti

~ Il' (En.ma)EaE' - Eo } 1 6b~PP(Eo~molEo

lv'-vo } `E,-Eo~2JP Eo 2 l6y (Ea, ma) ~Eal' Eo ~-

(IJ)Hence

QE6i" ~Fo, F', mo~ ~ -6i y ~Eo. mo~ - Eo . ~-Ê f 2 . o . ~V f Ê~~ ~ . (16)

Remember that the multiplication of utility (difference) by a positive constant is

an unessential transformation of von `eumann - Jlorgenstern utility (c:ifference).

Proportionality gives all essential information. Let k- [WP (Eo, mo) - Eo] ~ 0,

then Observation 4 results.

21

Observation 6 Form the equation

Ebi' ~F', mo f C~ - Eli' ~Fa. mo~ (1 i)

but using the approximation (12). Then

2tf ~,mCZ t~6i m~- Li~'m, ~E~ - Eo~~ C-~pEa [-Ê f 2a ~Ê~ t V~] .~: 0(18)

Consequently

C2t2K,mmm [1 t wlwmeo (E's~)1 C--2~ wmm .~ mo í-Ê t z~ (Ê~ t~')1 ~ o

and0 1 r

C~-21 ltw' E Ê1 C-2spl-Êt2o~Ê~tV~,tiO. (20)P f{m L

Ste 6 in the roof of Claim 1 shows tfo~` m titPm p~m m-P P

tt p -~- p. Hence tit,m~i,o p-Li'my.p1

- - p - p. Consequently,H ~,., s

pC2-211-s(ri-p)Ê~C-2sf-Êf2Q~Ê2fV~~tiO. (21)

Finally. solve for C and use the root that gi`ves C- 0 when there is no change.

22

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94100 A. Cukietrttan and S. Webb

94101 G.l. Almekinders andS.C.W. Eijffinger

94102 R. Aalbers

94103 H. Bester and W. Guth

94104 H. Huizinga

94105 F.C. Drost. T.E. Nijman,and B.J.h1. Werker

94106 V. Feltkamp, S. Tijs,and S. Muto

Title

Politícal Influence on the Central Bank - Interna[ionalEvidence

The Ineffec[iveness of Central Bank Intervention

Extinction of the Human Race: Doom-Mongering orReality?

Is Altruism Evolutionarily Stable?

Migration and Income Transfers in the Presence of LaborQuality Extemalities

Es[imation and Testing in Models Containing both Jumpsand Conditional Heteroskedasticity

On the Irreducible Core and the Equal RemainingObligations Rule of Minirnum Cost Spanning EztensionProblems

94107 D. Diamantaras, Efficiency and Separability in Economies with a TradeR.P. Gilles and P.H.M. Ruys Center

94108 R. Ray

94109 F.H. Page

94110 F. de Roon and C. Veld

94111 P.1.-J. Herings

94112 V. Bhaskar

94113 R.C. Douven andJ.E.J. Plasmans

94114 B. Bettom~il andJ.P.C. Kleijnen

94115 H. Uhlig and N. Yanagawa

9501 B. van Aarle,A.L. Bovenberg andM. Raith

9502 B. van Aarle andN. Budina

The Refotm and Design of Commodity Taxes in thePresence of Tax Evasion with Illustrative Evidence fromIndia

Optimal Auction Design with Risk Aversion and CorrelatedInformation

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A Globally and Universally Stable Quantity AdjustmentProcess for an Exchange Economy with Price Rigidities

Noisy Communication and the Fast Evolution ofCooperation

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[ncreasíng the Capital Income Tax Leads to Faster Growth

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9504 1.P.C. Kleijnen

9505 S. Eijffinger andE. Schaling

9506 1. Ashayeri, A. Teelenand W. Selen

9507 J. Ashaveri, A. Teelenand ~4'. Selen

9508 A. Mountford

9509 F de Roon and C. ~'eld

9510 P.H Franses andM. McAleer

9511 R.M.W.J. Beetsma

9512 V. Kriman andR.Y. Rubinstein

9513 J.P.C. Kleijnen,and R. Y'. Rubinstein

9514 R.D. van der htei

9515 M. Das

9516 P.W.J. De Bijl

9517 G. Koop, J. Osiewalskiand M.F.1. Steel

9518 l. Suijs, H. Hamers andS. Tijs

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9520 A. Lejour

Title

A Constructive Proof of a Unirnodular TransformationTheorem for Símplices

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The Political Economy of a Changing Population

Polynomial Time Algorithms for Estimation of Rare Eventsin Queueing Models

Optimization and Sensitiviry Analysis of ComputerSimula[ion Models by the Score Function Method

Polling Systems with Markovian Server Routing

Extensions of the Ordered Response Mode] Applied toConsumer Valuation of New Products

Entry Deterrence and Signaling in Markets for SearchGoods

The Components of Output Growth: A Cross-CountryAnalysis

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Optimal [nput Substitution of a Firm Facing anEnvironmental Constraint

Cooperative and Competitive Policies in the EU: TheEuropean Siamese Twin?

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9522 E. van der Heijden Opinions conceming Pension Systems. An Analysis ofDutch Sun~ey Data

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9527 M. Lettau and H. Uhlig

9528 F. van Megen, P. Borm,and S. Tijs

9529 H. Hamers

9530 V. Bhaskar

9531 E. Canton

9532 1.1.G. Lemmen andS.C.W. Eijffinger

9533 P.W.J. De Bijl

9534 F. de Jong and T. Nijman

9535 B. Dutta,A. van den Nouweland andS. Tijs

9536 B. Bensaid and O. leanne

9537 E.C.M. van der Heijden,1.H.M. Nelissen andH.A.A. Verbon

9538 L. Meijdam andH.A.A. Verbon

9539 H. Huizinga

Decomposable Effectivity Func[ions

Rule of Thumb and D}~namic Programming

A Perfectness Concept for Mul[icriteria Games

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Efficiency Wages and the Business Cycle

Financial Integration in Europe: Evidence from EulerEquation Tests

Stra[egic Delegation of Responsibility in Competing Firms

High Frequency Analysis of Lead-Lag RelationshipsBetween Financial Markets

Link Fotmation in Cooperative Situations

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Altruism and Faimess in a Public Pension System

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9540 J. Miller

9541 H. Huizinga

9542 J.P.C. Kleijnen

9543 H.L.F de GrootandA.B.T.M. van Schaik

9544 C. Dustmann andA. van Soest

9545 C. Kilby

9546 G.W.J. Hendrikse andC.P. Veetman

9547 R.M.W.1. Beetsma andA.L. Bovenberg

9548 R. Strausz

9549 F. Verboven

9550 R.C. Douven and1.C. Engwerda

9551 J.C. Engwerda andA.J.T.M. Weeren

9552 M. Das and A. van Soest

9553 J. Suijs

9554 M. Lettau and H. Uhlig

9555 F.H. Page andM.H. Wooders

9556 J. Stennek

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Supervision and Performance: The Case of World BankProjects

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Localized Competition, Multimarket Operation andCollusive Behavior

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9562 J. Bouckaert

9563 H. Haller

9564 T. Chou and H. Haller

9565 A. Blume

9566 H. Uhlig

9567 R.C.H. Cheng and1.P.C. Kleijnen

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9569 M.P. Berg

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9571 B. Melenberg andA. van Soest

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