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Information gathering and cost of living differences among searchers
Economics Letters North-Holland
247
28 (1988) 247-250
INFORMATION
GATHERING
AND COST OF LIVING DIFFERENCES
AMONG SEARCHERS
Paul M. ANGLIN University
of Windsor, Windsor, Ont., Canada N9B 3P4
Michael R. BAYE Texas A&M University, College Station, TX 77843.4228, USA
Received Accepted
17 May 1988 23 June 1988
This note extends recent work on cost of living index numbers for a population of individuals who face imperfect and costly information about prices. It is known that individuals with identical preferences but different incomes will require different cost of living adjustments unless preferences are homothetic. However, this result in based on a model of consumer behavior under costless information. We analyze differences in the cost of living among searchers with different incomes, and show that even if preferences are homothetic, expected cost of living indices will generally vary across searchers with different references standards of living. The reason for the differences is that, under homothetic preferences. reservation prices are a decreasing function of the searcher’s standard of living.
1. Introduction The introduction of the economics of information into the analysis of cost of living indices has raised several important questions regarding their use and interpretation. For example, Baye (1985) has argued that consumers with identical preferences will likely experience different cost of living indices because they may pay different prices for commodities. This, of course, raises the question ‘whose cost of living is to be measured?’ In Anglin and Baye (1987) we extend the analysis of cost of living indices to situations where consumers search for lower prices in a multigood framework, and show that consumers’ reservation price for one good can change as new information about prices for other goods changes during the search process. One implication of this is that the distribution of ‘acceptable prices’ will generally be stochastically dependent even if the distributions of offer prices are stochastically independent. One question that has yet to satisfactorily be answered is how cost of living indices vary across searchers who seek to maintain different standards of living. Empirical studies [Balk (1986) and Finke and Lu (1984)] reveal that individuals with different incomes have substantially different living costs. It is well known that under perfect information [Samuelson and Swamy (1974)] or spatial price dispersion [Baye (1985)] homotheticity of preferences is both necessary and sufficient for cost of living indices to be independent of the standard of living. This paper demonstrates that this fundamental result fails to hold when consumers search. 0165-1765/88/$3.50
0 1988, Elsevier Science Publishers
B.V. (North-Holland)
248 2.
P. M. Anglin, M. R. Buye / Information gathering, cost of living differences
Theory and results
Following Anglin and Baye (1987) define the expected expenditure required by a searcher to buy utility level u, given the vector P of known prices, as E(P, CY,u, 0) =min where
for k= 1, 2 ,..., n, and where Q = (P ,,..., Pk_,, s, Pk+l ,..., P,). In eq. (1) Fk is the cumulative probability density function 1 of offered prices for good k[k = 1, 2,. . . , n]; s2 is the probability space of the n prices; cx denotes a vector of search costs for each good k, that is exogenous and denominated in dollars; and e( .) represents a neoclassical (minimum) expenditure function that is homogeneous of degree one and non-decreasing in accepted prices. It is assumed that all goods are essential and that the first observation for the price of each good is obtained by randomly choosing a store selling that good. ’ Given the above formulation of the search process, the expected factor by which the searcher’s base expenditure level 3 [ y-e( PO, u) + N. a] must be adjusted in order to maintain utility level, u, is given by
ns( W’, a’)-
[j
E(z, a’,
u, ti’) dF’(z)
1
/y.
(2)
As noted above, agents with different incomes (and hence different reference levels of u) will, on average, require different cost of living adjustments even if they are identical in all other respects. In the standard framework for constructing cost of living indices, which ignores search behavior, homotheticity of preferences is necessary and sufficient to eliminate this problem. In the presence of search behavior and price dispersion, it turns out that homotheticity is no longer sufficient for the searcher’s expected cost of living index to be independent of the base standard of living. As we shall demonstrate below, the reason is that reservation prices generally depend on the searcher’s standard of living. Individuals with different incomes will differ in their willingness to accept or reject price quotes obtained during search. The first thing to note is that the index in eq. (2) reduces to the standard Konus cost of living index whenever prices are not dispersed and search costs are zero. Hence, homotheticity of preferences is a necessary condition for a searcher’s cost of living index to be independent of the standard of living. However, the following counter-example demonstrates that homotheticity is not a sufficient condition. Suppose there are only two goods and that search is costless in the period ‘0’. Further suppose prices are not dispersed in the first period. Under these circumstances, CX~ = 0 for both goods, the law of one price also holds, and the total expenditure required by the standard of living associated with u is e(PO, 24). ’ It is assumed the consumer knows these distributions, and that the consumer samples with replacement and free recall. * For further details on this search model see Anglin and Baye (1987). 3 This is inclusive of any search costs, which we denote by (Y. N.
P. M. Anglin,
M.R.
Ba_ve / Informorion
gathering,
casi
of lioing
249
difference.?
Suppose that in the comparison period (period ‘l’), the price of good two remains constant, but the price of good one is dispersed according to a non-degenerate distribution of prices, Fr( p,). Further suppose that CY~ > 0, but that CY\remains unchanged at zero. Under these circumstances the searcher’s reservation price in period ‘1’ is given by pi
/0 {
e P:,P2, (
or equivalently
u) - e(s, P2, u)} dF,(s) = al,
(3)
by
(4) Therefore,
the expected
I E(z,
ZI, Q’)
the
expenditure
may be written
as
u),
z) = expected
Tlr(S2’, d)=e(P;,
when searching
of
index
situations
and
is
P2,u).
P2, u)/e(Pr,
(5)
On the basis of eq. (5), it is tempting conclude that the searcher’s cost of living index is independent of u if and only if preferences are homothetic. After all, homothet~ci~y implies e( P,u) = #(u)ti/( P) for some functions # and 4, so that it appears that the standard result of Samuelson and Swamy can be applied to ‘cancel out’ the functions of utility. This reasoning is flawed, however, as the reservation price generally depends on M. To see this, differentiate eq. (3) with respect to the standard of living. The marginal cost of search remains unchanged, but the expected benefits of search change by P;, Pz, u),/‘~uApplying
ae(s,
P2,u)/au)
the mean value theorem
for some 17E [O, Pi]. Applying
dF,(s).
for ~emann-Stieltjes
the mean value theorem
integrals,
for derivatives
we may write
for some q E [T, P;]. Application of Shephard’s lemma reveals that this equation is positive (negative) depending upon whether good one is a normal (an inferior) good. This implies that the benefits to searching for a lower price of good 1 are an increasing function of the reference standard of living whenever good 1 is a normal good. The economic reason for this is that, under normality, the greater the desired standard of living, the greater the desired quantity of good 1, and hence the greater the gains to searching for a lower price. This, of course, implies that the reservation price for good one is a decreasing function of the standard of living.
250
P. M. Anglin, M.R. Baye / Information
gathering, cost of living differences
The upshot of this analysis is as follows. If preferences are homothetic, the expenditure function may be written as e( P, u) = Cp(u)$( P). But if preferences are homothetic, goods are normal goods, and the reservation price for good 1 is a decreasing function of the standard of living: Pi = P;(u). Using these results, along with eq. (5) we see that the searcher’s expected cost of living index under homotheticity is
which depends on the standard of living because the reservation price of good 1 is a decreasing function of u. Because the reservation price is a decreasing function of u, and 4 is an increasing function of the price good 1, individuals with greater incomes will have lower expected cost of living indices. In essence, there are economies of scale in information gathering; for a given expenditure on search, individuals seeking to maintain higher standards of living spread the search cost over a greater quantity consumed.
3. Conclusions We have shown that, under search behavior, expected cost of living indices are independent of the standard of living if reservation prices are independent of the standard of living and preferences are homothetic. There are two situations in which reservation prices will be independent of the standard of living. The first situation is when search costs are zero and prices are not dispersed, for in this instance all consumers will pay identical prices for goods regardless of their incomes. So long as search costs are zero and prices are not dispersed, homotheticity is a necessary and sufficient condition for a cost of living index to be invariant to the standard of living. Whenever prices are dispersed and search costs are non-zero, reservation prices are independent of the standard of living whenever the demands for the subset of goods for which prices are dispersed are independent of income. This is, of course, not consistent with the assumption that preferences are homothetic. Hence, whenever prices are dispersed and search costs are nonzero, expected cost of living indices of searchers will generally depend on the standard of living even if preferences are homothetic.
References Anglin, Paul M. and Michael R. Baye, 1987, Information, multiprice search, and cost-of-living index theory, Journal of Political Economy 95, 117991195. Balk, B.M., 1986, The behavior of the cost of living at different levels of real income: A correction, Economics Letters 20. 301-302. Baye, Michael R., 1985, Price dispersion and functional price indices, Econometrica 53, 217-224. Diewert, Erwin, 1982, Duality approaches to microeconomic theory, in: K. Arrow and M. Intriligator, eds., Handbook of mathematical economics, Vol. II (North-Holland, Amsterdam) 535-599. Finke, R. and W. Lu, 1984, The behavior of the cost of living at different levels of real income, Economics Letters 15. 223-228. Samuelson, P. and S. Swamy, 1974, Invariant economic index numbers and canonical duality: Survey and synthesis?, American Economic Review. 5666593.
247
28 (1988) 247-250
INFORMATION
GATHERING
AND COST OF LIVING DIFFERENCES
AMONG SEARCHERS
Paul M. ANGLIN University
of Windsor, Windsor, Ont., Canada N9B 3P4
Michael R. BAYE Texas A&M University, College Station, TX 77843.4228, USA
Received Accepted
17 May 1988 23 June 1988
This note extends recent work on cost of living index numbers for a population of individuals who face imperfect and costly information about prices. It is known that individuals with identical preferences but different incomes will require different cost of living adjustments unless preferences are homothetic. However, this result in based on a model of consumer behavior under costless information. We analyze differences in the cost of living among searchers with different incomes, and show that even if preferences are homothetic, expected cost of living indices will generally vary across searchers with different references standards of living. The reason for the differences is that, under homothetic preferences. reservation prices are a decreasing function of the searcher’s standard of living.
1. Introduction The introduction of the economics of information into the analysis of cost of living indices has raised several important questions regarding their use and interpretation. For example, Baye (1985) has argued that consumers with identical preferences will likely experience different cost of living indices because they may pay different prices for commodities. This, of course, raises the question ‘whose cost of living is to be measured?’ In Anglin and Baye (1987) we extend the analysis of cost of living indices to situations where consumers search for lower prices in a multigood framework, and show that consumers’ reservation price for one good can change as new information about prices for other goods changes during the search process. One implication of this is that the distribution of ‘acceptable prices’ will generally be stochastically dependent even if the distributions of offer prices are stochastically independent. One question that has yet to satisfactorily be answered is how cost of living indices vary across searchers who seek to maintain different standards of living. Empirical studies [Balk (1986) and Finke and Lu (1984)] reveal that individuals with different incomes have substantially different living costs. It is well known that under perfect information [Samuelson and Swamy (1974)] or spatial price dispersion [Baye (1985)] homotheticity of preferences is both necessary and sufficient for cost of living indices to be independent of the standard of living. This paper demonstrates that this fundamental result fails to hold when consumers search. 0165-1765/88/$3.50
0 1988, Elsevier Science Publishers
B.V. (North-Holland)
248 2.
P. M. Anglin, M. R. Buye / Information gathering, cost of living differences
Theory and results
Following Anglin and Baye (1987) define the expected expenditure required by a searcher to buy utility level u, given the vector P of known prices, as E(P, CY,u, 0) =min where
for k= 1, 2 ,..., n, and where Q = (P ,,..., Pk_,, s, Pk+l ,..., P,). In eq. (1) Fk is the cumulative probability density function 1 of offered prices for good k[k = 1, 2,. . . , n]; s2 is the probability space of the n prices; cx denotes a vector of search costs for each good k, that is exogenous and denominated in dollars; and e( .) represents a neoclassical (minimum) expenditure function that is homogeneous of degree one and non-decreasing in accepted prices. It is assumed that all goods are essential and that the first observation for the price of each good is obtained by randomly choosing a store selling that good. ’ Given the above formulation of the search process, the expected factor by which the searcher’s base expenditure level 3 [ y-e( PO, u) + N. a] must be adjusted in order to maintain utility level, u, is given by
ns( W’, a’)-
[j
E(z, a’,
u, ti’) dF’(z)
1
/y.
(2)
As noted above, agents with different incomes (and hence different reference levels of u) will, on average, require different cost of living adjustments even if they are identical in all other respects. In the standard framework for constructing cost of living indices, which ignores search behavior, homotheticity of preferences is necessary and sufficient to eliminate this problem. In the presence of search behavior and price dispersion, it turns out that homotheticity is no longer sufficient for the searcher’s expected cost of living index to be independent of the base standard of living. As we shall demonstrate below, the reason is that reservation prices generally depend on the searcher’s standard of living. Individuals with different incomes will differ in their willingness to accept or reject price quotes obtained during search. The first thing to note is that the index in eq. (2) reduces to the standard Konus cost of living index whenever prices are not dispersed and search costs are zero. Hence, homotheticity of preferences is a necessary condition for a searcher’s cost of living index to be independent of the standard of living. However, the following counter-example demonstrates that homotheticity is not a sufficient condition. Suppose there are only two goods and that search is costless in the period ‘0’. Further suppose prices are not dispersed in the first period. Under these circumstances, CX~ = 0 for both goods, the law of one price also holds, and the total expenditure required by the standard of living associated with u is e(PO, 24). ’ It is assumed the consumer knows these distributions, and that the consumer samples with replacement and free recall. * For further details on this search model see Anglin and Baye (1987). 3 This is inclusive of any search costs, which we denote by (Y. N.
P. M. Anglin,
M.R.
Ba_ve / Informorion
gathering,
casi
of lioing
249
difference.?
Suppose that in the comparison period (period ‘l’), the price of good two remains constant, but the price of good one is dispersed according to a non-degenerate distribution of prices, Fr( p,). Further suppose that CY~ > 0, but that CY\remains unchanged at zero. Under these circumstances the searcher’s reservation price in period ‘1’ is given by pi
/0 {
e P:,P2, (
or equivalently
u) - e(s, P2, u)} dF,(s) = al,
(3)
by
(4) Therefore,
the expected
I E(z,
ZI, Q’)
the
expenditure
may be written
as
u),
z) = expected
Tlr(S2’, d)=e(P;,
when searching
of
index
situations
and
is
P2,u).
P2, u)/e(Pr,
(5)
On the basis of eq. (5), it is tempting conclude that the searcher’s cost of living index is independent of u if and only if preferences are homothetic. After all, homothet~ci~y implies e( P,u) = #(u)ti/( P) for some functions # and 4, so that it appears that the standard result of Samuelson and Swamy can be applied to ‘cancel out’ the functions of utility. This reasoning is flawed, however, as the reservation price generally depends on M. To see this, differentiate eq. (3) with respect to the standard of living. The marginal cost of search remains unchanged, but the expected benefits of search change by P;, Pz, u),/‘~uApplying
ae(s,
P2,u)/au)
the mean value theorem
for some 17E [O, Pi]. Applying
dF,(s).
for ~emann-Stieltjes
the mean value theorem
integrals,
for derivatives
we may write
for some q E [T, P;]. Application of Shephard’s lemma reveals that this equation is positive (negative) depending upon whether good one is a normal (an inferior) good. This implies that the benefits to searching for a lower price of good 1 are an increasing function of the reference standard of living whenever good 1 is a normal good. The economic reason for this is that, under normality, the greater the desired standard of living, the greater the desired quantity of good 1, and hence the greater the gains to searching for a lower price. This, of course, implies that the reservation price for good one is a decreasing function of the standard of living.
250
P. M. Anglin, M.R. Baye / Information
gathering, cost of living differences
The upshot of this analysis is as follows. If preferences are homothetic, the expenditure function may be written as e( P, u) = Cp(u)$( P). But if preferences are homothetic, goods are normal goods, and the reservation price for good 1 is a decreasing function of the standard of living: Pi = P;(u). Using these results, along with eq. (5) we see that the searcher’s expected cost of living index under homotheticity is
which depends on the standard of living because the reservation price of good 1 is a decreasing function of u. Because the reservation price is a decreasing function of u, and 4 is an increasing function of the price good 1, individuals with greater incomes will have lower expected cost of living indices. In essence, there are economies of scale in information gathering; for a given expenditure on search, individuals seeking to maintain higher standards of living spread the search cost over a greater quantity consumed.
3. Conclusions We have shown that, under search behavior, expected cost of living indices are independent of the standard of living if reservation prices are independent of the standard of living and preferences are homothetic. There are two situations in which reservation prices will be independent of the standard of living. The first situation is when search costs are zero and prices are not dispersed, for in this instance all consumers will pay identical prices for goods regardless of their incomes. So long as search costs are zero and prices are not dispersed, homotheticity is a necessary and sufficient condition for a cost of living index to be invariant to the standard of living. Whenever prices are dispersed and search costs are non-zero, reservation prices are independent of the standard of living whenever the demands for the subset of goods for which prices are dispersed are independent of income. This is, of course, not consistent with the assumption that preferences are homothetic. Hence, whenever prices are dispersed and search costs are nonzero, expected cost of living indices of searchers will generally depend on the standard of living even if preferences are homothetic.
References Anglin, Paul M. and Michael R. Baye, 1987, Information, multiprice search, and cost-of-living index theory, Journal of Political Economy 95, 117991195. Balk, B.M., 1986, The behavior of the cost of living at different levels of real income: A correction, Economics Letters 20. 301-302. Baye, Michael R., 1985, Price dispersion and functional price indices, Econometrica 53, 217-224. Diewert, Erwin, 1982, Duality approaches to microeconomic theory, in: K. Arrow and M. Intriligator, eds., Handbook of mathematical economics, Vol. II (North-Holland, Amsterdam) 535-599. Finke, R. and W. Lu, 1984, The behavior of the cost of living at different levels of real income, Economics Letters 15. 223-228. Samuelson, P. and S. Swamy, 1974, Invariant economic index numbers and canonical duality: Survey and synthesis?, American Economic Review. 5666593.