Comparing stochastic price regimes

Economics

Letters

14 (1984)

173

173-178

North-Holland

COMPARING The Limitations

Received

21 March

Marshallian Hicksian ing

1983

consumer’s compensating

variation

cannot

STOCHASTIC PRICE REGIMES of Expected Surplus Measures

is not

he Jutified

surplus variation.

a generally

can

often

Under valid

as an approximation

price

welfare

he

defended

as

uncertainty. measure.

to a rigorous

3 useful

however. and

approximation

expected

expected

to

compensat-

consumer‘s

surplus

construct.

1. Introduction

Although consumer’s surplus is not generally an exact representation of utility loss or gain, Willig (1976) has shown that it often closely approximates Hicksian compensating (or equivalent) variation: it is therefore a valuable practical tool for welfare analysis under conditions of certainty. When variability of supply generates price instability, many authors have used the expected value of consumer’s surplus as a measure of the benefits of some form of government intervention in the market. ’ We seek to determine whether, as has been commonly supposed, the rationale of consumer’s surplus approximating a rigorous Hicksian measure can be extended to expected consumer’s surplus in this case of stochastic prices.

’ See. e.g.. Waugh (1944) Samuelson (1977), and Wright (1979). 0165-1765/84/$3.00

1’: 1984,

Elsevier

(1972).

Science

Turnovaky

Publishers

(1976).

Bradford

B.V. (North-Holland)

and

KeleJ&m

L-l.

174

2. Conditions

Helm

/

Comparrng

stochustrc

for the validity of expected

prrce regtme.,

compensating

variation

Consider an individual with income >;I making consumption choices over good x and n - 1 other goods. Under the usual assumptions, ’ if only the price, p, of good x is variable, the utility maximization problem generates demand, expenditure, and indirect utility functions which may be written as x = _x( p. .v,,), y = e( p, u). and u = c( p. .r(,), respectively. If the price of good x changes from y top. the Marshallian consumer’s surplus

from

that

price

change

is

It is well known that A( 4, p. y(,) is a rigorous change in utility only if the marginal utility respect

to the

price

of X. Nonetheless,

monetary of income

even

when

equivalent of the is constant with

P,,, f 0. the

use

of

consumer’s surplus as a practical tool has been justified by Willig’s result that A is approximately equal to Hicksian measures such as compensating variation.

under

typical

circumstances.

If uncertainty on variation in the price

’

the supply side of the market of good X, the prices before and

induces random after the change

will be random variables. 4 In this case it is common to use the expectation of consumer’s surplus, EA(q, p. >i,). as a welfare measure. Rogerson (1980) has shown that the necessary and sufficient conditions for EA to represent

consumer are

- EP(~,J;,)]] underlying

the consumer’s

It is therefore _a We

assume

’ ,411 results

tempting

that

differentiable,

the

strtctly and

[i.e.. to have the same sign as [ Ev(q, .r;,) same in this case as the assumptions

preferences exactly the surplus

measure

to conclude

consumer

maxunizes

increasing

van

rcmarkr

below

that

under

conditions

a strictly

quasiconcave,

Neumann-Morgenstern

are equally

of certainty. of A as an

the justification

applicable

utility to Hickbian

twcc

contlnuoubl)

function. equivalent

variation.

nllltutl~ nzutumh. 4 These

random

government he related average’.

variables, interventmn

q and

p.

might

to (partialI>)

to q by the requirement

represent

stabilize that

price

the price

the government’s

distributions of good sale\

hefore

.x. In that and

purchaw

and

after

C:IX. p would cancel

‘on

175

L.J. Helms / Comparing stochrrstlc prrce regimes

approximation to the rigorous measure C also carries over when expectations are taken in the stochastic case. Unfortunately, the conditions under which expected compensating variation. (3) represents consumer preferences when prices are random are quite restrictive. We begin by noting that (3) depends on the joint distribution of q and p. It is conceptually troublesome that the covariance of the results of two mutually exclusive policy alternatives, or any information beyond the marginal distributions, would affect the welfare comparison. But even in the absence of such correlations, EC can produce paradoxical results, as exemplified by the following Lemma: Lenmu. Suppose thut the prices for all but the first good ure fixed und thut income is constant at Jo,. Consider tM!o non-degenerate identicul!v und itldependently distributed rundom vuriubles q und p representing ulternutive price distributions for the first good. Then, if thut good is strictly normul Msithprobability one. EC(q.P,.h,)>O

Proof. random

Let F( q, p) be the joint cumulative variables q and p. Then

Ec(q,p,:,,)=/

function

for the

is zero for p = q, we can break the integral

up into

{e(p,

v(q.y,,))-e(q,

distribution

~‘(q..h)))dF(q~P).

P-q

Because the integrand two components:

Ec(q.p,_v,,)=J {e(P,v(q,~~:,,))-e(q.V(q,.~,,))}dF(q,p) P(Y

-tj

{e(p,u(q,.y,y,,))--e(q,v(q.~~,))}dF(q,p). Y (P

Now because q and p are identically and independently distributed. F(q, p) is identical to F( p, q). Rewriting the second integral by making

176

L.J.

this substitution the same integral

and then leaves

Helms

+/ PC4

/

Compurrng

stochastrr

interchanging

price regimes

the variables

of integration

in

b+b d~d’o?o))-e(~. hu,))}dF(q.p).

Using the property that Hicksian demand at price 5 and utility level 11can be written as A([. rl) = e,,({. u) and recombining the integrals, the expression is equivalent to

Since h({.

h(.t,

u(q,

,v(,))

=

x({,

e(l,

u(q.

.k;,)))

<

-Y(l.

e(l,

rl( p.

>;,))I

square

=

for a strictly normal good when p < q. the integrand brackets in (4) is strictly negative and so the integral in braces

u( p, ytLi)))

strictly positive for p < q. Moreover, since p and q are identically independently distributed and non-degenerate. the set of values p c q has positive measure, establishing the result. Q.E.D.

The Lemma shows that, for a normal good. tion above an identical. unrelated distribution, yield the same expected utility. Moreover, always possible to find two independently and p, such that q stochastically dominates

EC ranks despite

in is

and with

a price distributhe fact that both

continuity assures that it is distributed price variables. q p and yet EC ranks q below

P. These results suggest that EC is a valid welfare extremely limited circumstances. That this is indeed the following Theorem: Theorrnl .

thut income equivalent:

criterion only under the case is proven in

Suppose thut the prices for ull hut the first c~ood are fised is constant ut .r;,. Then the following two propositions

utd are

L.J. Helms / Comparing stochastic prm

117

regmes

(i) Given any two alternative price distributions, good,

q and p, for the first

(ii) v,, =x, = 0, and v,,,,= 0 whenever demand for

the first good is

positive Proof. (i) + (ii): Suppose x,.(s, y,)) > 0 for some price s. Then by continuity x,. is strictly positive within some neighborhood of s for fixed _r,). Let q and p be non-denegerate identically and independently distributed random variables taking on positive values within this neighborhood with probability one. Then from the Lemma EC(q. p, y,)) > 0. but since Eu(q, yO) = Eu( p, yo), this contradicts (i). Similarly, x, (s. _J+)< 0 leads to a contradiction, establishing that x,. = 0. Since x,. = 0 we know from the Slutsky equation that Hicksian and Marshallian demand curves coincide, so EC(q, p, yo) = EA(q, p, y,)). Rogerson’s (1980) Theorem 1 shows that EA, and hence EC, represents consumer preferences in this case only if differentiating Roy’s identity, x(s, _r,,) = - ~;~(.s,.1;,)/ u,,,, = 0. Finally, I), (s, yU). with respect to y yields L’,./,= -xv,, - xp, which. together with x = 11,,, = 0, establishes the result. ’ (ii) + (i): R o ge rson’s Theorem demonstrates that u,.,, = 0 is sufficient to establish that EA represents consumer preferences. This, together with the equivalence of EC to EA implied by x,. = 0 and the Slutsky equation, completes the proof. Q.E.D. In order to ensure that EC provides the same ranking of two alternative price regimes as does expected utility, we must assume not only that the marginal utility of income is constant with respect to changes in the price of good x (as required for the validity of EA), but also that the marginal utility of income is invariant to changes in income (i.e., that the consumer is risk neutral). Thus only in these extremely restrictive circumstances, when income effects are nil and EC coincides with EA, is EC a valid welfare measure. ’

’

If

we consider

questions relaxed Helms welfare

of

only the

somewhat. (198321) indicator.

cases

welfare See

indicate

where effects

Helms that

p 1s subject

to stochastic

of complete

price

(1983b). EC

(as well

However. as EA)

variation

stabilization). even

in these

can be qwte

but these

cases, misleading

q is comtant restrictions

numerical

(i.e.. can

results

as a quantitative

be in

L.J. Helms / Comparrng stochcrstlc price regimes

178

3. Conclusion

In a non-stochastic setting, compensating variation provides a rigorous measure of the willingness to pay for a change in price. As a result. Marshallian consumer’s surplus, which is approximately equal to compensating variation in many cases, may be useful as an approximately correct welfare measure. When prices are uncertain, however. no such reassurances can be made. The expectation of compensating variation can rank alternative price distributions perversely and will always rank some distributions above others which provide greater expected utility except when the marginal utility of income is constant and income effects are nil. Because expected compensating variation is a valid welfare measure only under conditions which are even more restrictive than those required for the validity of expected consumer’s surplus. the usefulness of the latter in more general cases cannot be justified by arguments that it is an approximation to the former.

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