tilburg university consumer's welfare and change in ... · on consumer's welfare when...
TRANSCRIPT
Tilburg University
Consumer's welfare and change in stochastic partial-equilibrium price
Stennek, M.J.
Publication date:1995
Link to publication
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.
General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.
- Users may download and print one copy of any publication from the public portal for the purpose of private study or research - You may not further distribute the material or use it for any profit-making activity or commercial gain - You may freely distribute the URL identifying the publication in the public portal
Take down policyIf you believe that this document breaches copyright, please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.
Download date: 17. Feb. 2020
~ ~ra~~~-~ Discussion8414199572
mic Research a ermii i iiu i i iu i i i ii ii i i i uu i mi i i iui iui ui i
Tilburg University
,,,
Centerfor
Economic Research
~.~ l
~-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
References
ARRO~ti, K. (1971). "The Theory of Risk Aversion," in Essays in the Theory
of Rtsk-Bearing, ed: K. .4rrow, `orth-Holland, Amsterdam.
BARO1, D., A~D ti1YERSOV, R. Í 1982). '`Regulating a:~lonopolist with L'n-
known Costs." Econometrica 50, 911-930.
DEATON, A., A~D ~SCELLBAI;ER, J. (1986). Economics and Consumer Be-
hacior, Cambridge Cniversit~ Press, Cambridge.
HELtifs, J. (1984). "Comparing Stochastic Price Regimes - The Limitations of
Expected Surplus !~ieasures," Economics Letters 19, 173-178.
HEL~is, J. (198~). "Expected Consumer~s Surplus and the ~~'elfare Effects of
Price Stabilization,"' International Economic Reziew 26, 603-61 ï.
KIRBI', A. (1988). "Trade Associations as Information Exchange tilecha-
nisms," Rand Journal of Economics 19, 138-46.
~-fENRA, R., AvD PRESCOTT, E. (1985). '`The Equity Premium: A Puzzle,"
Journal of .bfonetary Economics 15, 145-61.
PR,~TT. J. (1964). "Risk .A~~ersion in the Small and in the Large," Economet-
rica 32. 122-36.
RtoRDA~, ~1., .atiD SAPpI:~GTOti, D. (1987). ".Awarding ` lonopoly Fran-
chises," American Economic Review 77, 375-87.
ROGERSO~, ~~'. (1980). "Aggregate Expected Consumer Surplus as a ~i~'elfare
Index with an `lpplication to Price Stabilization," Econometrica 48. 423-
436.
SCH~fALE`SEE, R. (1972). "Option Demand and Consumer's Surplus: Valuing
Price Changes under lincertainty," American Economic Reviewó2, 813-24.
23
SH.aP[RO. C. (1986). '`Exchange of Cost Information in Oligopoly," Reviea~ of
Economic Studies LIII. 433-43.
TrR~o~sx~', S., SHALiT, H., AvD SCH~f[TZ, A. (1980). "Consumer~s Sur-
plus. Price Instability, and Consumer ~~'elfare," Economefrica 48, 135-1~2.
~CILLIG, R. (1976). "Consumer's Surplus ~~'ithout Apology," American Eco-
nomic Reriew 66, 589-~97.
tio. Author(s)
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
An Empirical Investigation of the Factors that Detetirtine thePricing of Du[ch Index Warrants
A Globally and Universally Stable Quantity AdjustmentProcess for an Exchange Economy with Price Rigidities
Noisy Communication and the Fast Evolution ofCooperation
S.L.I.M. - A Small Linear Interdependent Model of EightEU-Member States, the USA and Japan
Identifying the Important Factors in Simulation Models withMany Factors
[ncreasíng the Capital Income Tax Leads to Faster Growth
Monetary and Fiscal Policy Interaction and GovernmentDebt Stabilization
Currency Substitution in Eastern Europe
~o. Author(s)
9503 Z. Yang
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
9519 R.F. Hartl and P.M. Kort
9520 A. Lejour
Title
A Constructive Proof of a Unirnodular TransformationTheorem for Símplices
Sensitivity Analysis and Optimization of Sys[em DynamicsModels: Regression Analysis and S[atistical Design ofExperiments
The UI[imate Determinants of Central Bank Independence
A Production and Maintenance Planning Model for theProcess Industry
Computer Integrated Manufacturing in the ChemicalIndustry: Theory 8c Practice
Can a Brain Drain be Good for Growth?
Announcemen[ Effec[s of Convenible Bond Loans VersusWarrant-Bond Loans: An Empirical Analysis for the DutchMarket
Testing Nes[ed and Non-Nested Periodically Integra[edAutoregressive Models
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
On Con.sistency of Reward Allocation Rules in 5equencingSituations
Optimal [nput Substitution of a Firm Facing anEnvironmental Constraint
Cooperative and Competitive Policies in the EU: TheEuropean Siamese Twin?
tio. Author(s) Title
9521 H.A. Keuzenkamp The Econometrics of the Holy Grail: A Critique
9522 E. van der Heijden Opinions conceming Pension Systems. An Analysis ofDutch Sun~ey Data
9523 P. Bossaerts and Local Parametric Analysis of Hedging in Discrete TimeP. Hillion
9524 S. Hochgiirtel, Household Portfolio Allocation in the Netherlands: SavingR. Alessie and A. van Soest Accounts versus Stocks and Bonds
9525 C. Fernandez. Inference Robustness in Multivariate Models with a ScaleJ. Osiewalski and ParameterM.F.1. S[eel
9526 G.-1. Otten, P Borm,T. Storcken and S. Tijs
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
On the Concavity of Delivery Games
On the Generic Instability of Mixed Strategies inAsymmetric Contests
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
The Instability of Fixed Exchange Rate Systems whenRaising the Nominal Interest Rate is Costly
Altruism and Faimess in a Public Pension System
Aging and Public Pensions in an Overlapping-GenerationsModel
International Trade and Migration in the Presence of Sector-Specific Labor Quality Pricing Distortions
`o. Author(s) Title
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
A Comment on Holmlund 8t Lindén's "Job Matching,Temporary Public Employment, and Unemployment"
Taxation and the Transfer of Technology by Multina[ionalFirms
Statistical Validation of Simulation Models: A Case Study
Relative Convergence in a Dual Economy with Tradeableand Non-Tradeable Goods
Generalized Switching Regression Analysis of Private andPublic Sector Wage Structures in Germany
Supervision and Performance: The Case of World BankProjects
Marketing Cooperatives and Financial Structure
Designing Fiscal and Monetary Institutíons in a Second-BestWorld
Collusion and Renegotiation in a Principal-Supervisor-AgentRelationship
Localized Competition, Multimarket Operation andCollusive Behavior
Propenies of N-person Axiomatic Bargaining Solutions if thePare[o Frontier is Twice Differentiable and Stricdv Concave
The Open-Loop Nash Equilibrium in LQ-Games Revisited
Expected and Realized Income Changes: Evidence from theDutch Socio-Economic Panel
On Incentive Compatibility and Budget Balancedness inPublic Decision Making
Can Habi[ Formation be Reconciled with Business CycleFacts?
The Partnered Core of an Economy
Competition Reduces X-Inefficiency. A Note on a LimitedLiabiliry Mechanism
9557 K. Aardal and S. van Hoesel Polyhedral Techniques in Combinatorial Optimization
9558 R.M.W.J. Beetsma and Designing Fiscal and Monetary Institutions for a EuropeanA.L. Bovenberg Monetary Union
No. Author(s)
9559 R.M.W.J. Beetsma andA.L. Bovenberg
9560 R. Strausz
9561 A. Lejour
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
9568 M.F.J. Steel
9569 M.P. Berg
9570 F. Verboven
9571 B. Melenberg andA. van Soest
9572 J. Stennek
Title
Monetary Union without Fiscal Coordination May DisciplinePolicymakers
Delegation of Monitoring in a Principal-Agent Relationship
Social Insurance and the Completion of the Interrtal Market
Monopolistic Competition with a Mail Order Business
Household Decisions and Equilibrium Efficiency
The Division of Profit in Sequential InnovationReconsidered
Learning, Experimentation, and Long-Run Behavior inGames
Transition and Financial Collapse
Optimal Design of Simulation Experiments with NearlySaturated Queues
Posterior Analysis of Stochastic Volatility Models withFlexible Taíls
Age-Dependent Failure Modelling: A Ha7ard-FunctionApproach
Testing for Monopoly Power when Products areDifferentiated in Quality
Semiparametric Estimation of Equivalence Scales UsingSubjective Information
Consumer's Welfare and Change in S[ochas[ic Partial-Equilibrium Price
P.O. BOX 90153, 5000 LE TILBURG, THE NETHERLANDSBibliotheek K. U. Brabant~ in~n~m~ ~~ i~~i~