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The theory of search for several goods
JOURNAL
OF ECONOMIC
THEORY
24, 362-316
(1981)
The Theory of Search for Several Goods KENNETH Department
of Economics,
BURDETT*
University
of Wisconsin,
Madison,
Wisconsin
53706
AND DAVID Department
of Economics,
A. MALUEG
Northwestern
University,
Evanston,
Illinois
60201
The problem analyzed in this study can be illustrated by the following example. Before starting out to buy groceries for the week, suppose an individual makes a list of the n goods required (n > 2). The list also states the amount required of each good. There are many grocery stores the individual can visit. Each of these stores sells all the goods desired. The individual can visit any store at a cost and observe the n vector of prices offered, one price for each of the n goods. The individual can purchase any number of the goods required from this store and then continue to search for the remaining goods. Given each pric,e vector observed can be envisaged as a random draw from a known nondegenerate distribution of price vectors, the problem is to determine the search strategy that minimizes the individual’s expected total cost of purchasing the n goods. The nature of the above problem is similar to that posed in the literature on consumer search for the lowest price for a single good.’ In the literature on single good search a consumer is assumed to be shopping for a given amount of a good. On the payment of a fixed cost the consumer may visit a store and observe the price offered, which is assumed to be a random draw from a known nondegenerate distribution of prices.’ Many studies have assumed that the consumer can return to any store preciously visited without incurring a cost; the other studies have assumed that the consumer cannot * Supported in part by an N.S.F. grant. ’ Lippman and McCall [6] provide an excellent survey of this literature and the related literature on worker job search. Stigler [ 111 presents an early example of work in this area. Pratt et al. ]8] have presented evidence that the distribution of prices for many goods is nondegenerate, at least in Boston. * Rothschild [9] considers a model of consumer search where the consumer does not know the distribution of prices in the market. In this tase the well-known reservation price property may not hold. 362
0022-0531/81/030362-15$02.00/O Copyright C: 1981 by Academic Press, Inc. All rights of reproduction in any form reserved.
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return to any store.3 Which of these two assumptions is used is of no great importance in consumer search for a single good, however, as it has been shown many times that the search strategy that minimizes the consumer’s expected cost of purchasing the good is the same in either case. Specifically, it has been shown that the optimal strategy can be characterized by a reservation price, p*. The consumer should purchase the first time a price no greater than p* is observed, where
c is the cost of a price observation, and F(-) is the distribution function associated with any price observation.4 In the present study a model similar in most respects to those presented in the single good search literature will be used. There are, however, two important differences. First, it will be assumed that the consumer wants to purchase n goods, where n is any positive integer. Second, the consumer can visit a store and observe the price quotation for each one of the n goods desired. Unlike the single good consumer search models, the assumption made about the consumer’s ability to return to any store is critical in consumer search models for several goods. In Section 1 it will be assumed that the consumer may return to any store without incurring a cost. In Section 2 the consumer is assumed to be unable to return to any store. It will be shown’that the strategies that minimize the consumer’s expected total cost of purchasing differ significantly in the two situations considered, at least in the case where n = 2. Suppose the consumer can freely return to any store previously visited. It follows immediately that in this case the consumer need not purchase any of the goods until he or she selects to by all the it goods. When the decision to purchase all the goods has been made, the consumer can freely return to the store that offered the lowest price observed for good i and purchase good i from that store, i = 1,2 ,..., n. Using the results presented in the single good consumer search literature, it appears reasonable to conjecture that the consumer’s best strategy can be characterized by what may be loosely termed a reservation frontier. In Section 1 this conjecture is shown to be true. Specifically, it is shown that the cost minimizing search strategy implies the consumer should stop searching and purchase all n goods if and only if the vector of lowest prices observed to date is no greater than some point on the reservation frontier. Most of Section 1 is taken up characterizing the reser’ To the authors’ knowledge, all but one of the studies presented to date have utilized one of these two restrictions. Landsberger and Peled 15 1 provide the exception. ’ See Lippman and McCall [6] for a derivation of this result. Note that p* is independent of the recall assumption.
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vation frontier and hence the set of acceptable price vectors. It will be demonstrated that the set of acceptable price vectors is convex. Further, it will be shown that the consumer will never purchase good i at a price higher than the reservation price for good i in a single good search for good i, i = 1, 2,..., n. In the final part of Section 1 we investigate the consequences of changes in the parameters of the model of consumer search. For example, it is shown that an increase in the cost of search will shift out the relevant reservation frontier and hence enlarge the set of acceptable price vectors. The effect of a change in the distribution of prices for one of the goods is also investigated. First, it is shown that if the distribution of one of the n prices is translated up, the set of acceptable price vectors increases. Second, if the distribution of one of the prices becomes riskier in the sense specified by Rothschild and Stiglitz [lo], the set of acceptable price vectors decreases. In Section 2 the case where the consumer cannot return to any store is considered. In this case if the consumer decides to leave a store and visit another one without purchasing, then the consumer loses the opportunity to purchase from that store at the offered price. Some of the prices offered. however, may be too good to miss. In Section 2 we consider a model of this type where the consumer is searching for only two goods, i.e., n = 2. In this case it is shown that there exists a reservation frontier as in the recall case. The nature of this frontier, however, is different. The strategy that minimizes the expected cost of purchasing when the consumer cannot return to stores can be briefly summerized as follows. First, the consumer will not purchase good i from a store that offers a price for good i that is greater than p;, where pT is the reservation price the consumer would use in a single good search for good i, i = 1, 2. Second, given good i has not been purchased to date, the consumer will always purchase good i from a store that offers a price for good i no greater than fii, jli < pT,i = 1, 2. Finally, suppose the consumer has purchased neither of the two goods to date. If the consumer observes a price for good i, say pi, such that fii < pi < pT, then both of the goods will be purchased if the price of the other good is no greater than some price hi(pi), where hi(pi) is a decreasing function, i = 1. 2. It should be stressed that in either of the models considered in this study the consumer need not purchase all the goods desired from the same store. Suppose, for the moment, that the consumer must purchase all the n goods from the same store. In this case the consumer search for several goods is formally equivalent to a single good search model. Specifically, the consumer will select a reservation “price,” z*. and then purchase from the first store visited which offers a price vector (pl, pzr..., p,) such that Cy=, pi < z*.
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1. THE MODEL
Suppose a consumer wants to purchase n different goods (n > 2). One unit of each of these n goods is desired. There are many stores that sell all of these goods. By paying c (c > 0) the consumer can visit a store and observe the vector of prices offered, one price for each one of the n goods. The stores the consumer can visit, however, do not all offer the same price vector. Specifically, assume there is a known nondegenerate distribution function, G, that describes the vectors offered by these stores. Assume that each price vector observed can be envisaged as a random draw from this distribution. After visiting a store the consumer may purchase some number of the goods desired and/or visit another store at cost c. Although the consumer can purchase only from a store visited, all of the n goods need not be purchased from the same store. Given the environment specified above, the problem is to determine and characterize the search strategy that minimizes the consumer’s expected total cost of purchasing the II goods. Throughout this section it will be assumed that the consumer can return to any store previously visited without incurring a cost. In the next section the consequences of utilizing an alternative to this free recall assumption will be considered. Let pi denote the lowest price for good i observed by the consumer to date, i = 1, 2,..., n; and let p = (p, , pz ,..., p,) be the vector of these prices. If the consumer has observed at least one price vector, there are three options open. Option
1.
Purchase all n goods and thus end the search process.
Option 2. vector.
Purchase none of the goods and then observe another price
Option 3. vector.
Purchase fewer than n goods and observe another
price
If the consumer is starting the search process and has not observed a price vector, then Option 2, must be selected. The free recall assumption guarantees that the expected cost of purchasing when Option 3 is selected is at least as great as the expected cost if Option 2 were selected instead. Hence, without loss of generality, assume that Option 1 or 2 only will be chosen. Specifically, Option 2 will be chosen at any time until Option 1 is chosen. The search process then ends. The free recall assumption implies that p . 1 = Cr=r pi denotes the minimum cost of purchasing when Option 1 is chosen and p is the vector of lowest prices observed to date. The consumer will choose Option 1 at any time if and only if p . 1 is no greater than the expected cost of purchasing if Option 2 were selected instead. When the consumer selects Option 2,
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however, the expected cost of purchasing depends on the future search strategy to be used. At any time the consumer is assumed to know the vector of lowest prices observed to date and the probability laws generating the price vectors that may be observed in the future. In what follows it will be shown that any search strategy can be characterized by a stopping rule. To construct any such rule some notation needs to be developed. Let R = Xi”, R: denote the set of infinite sequences of n vectors consisting of positive reals. Since the distribution function G is assumed to be known, the probability the consumer would observe any measurable set in R if he or she were to search indefinitely is well defined. Let (0, /3,p) denote the probability measure space, where p is the probability measure induced by G, and b is the a-field. The u-field generated by the first k searches is denoted by Pkr k = l-2,3,... . Using the above notation, a stopping rule is an integer valued function s:B+IsuchthatskE/Ikforallk=1,2,3 ,..., where sk = {w E Q 1S(W) = k}. It is straightforward to check that any search strategy based on the information known to the consumer can be described by a stopping rule. For example, suppose the consumer utilizes the following simple search strategy: observe four price vectors and then purchase. The stopping rule S that describes this strategy is such that ? = R, and ? = 0 for all k # 4. Let .Y denote the set of all possible stopping rules. Suppose the consumer has decided to observe at least one more price vector when p denotes the vector of lowest prices observed to date. If the consumer utilizes the search strategy that can be described by stopping rule s, the expected total cost by purchasing, ty(p, s), can be written as ’
min(pi,
W(k) + ck
,e,
where Wik is the random variable describing the lowest price observed for good i in k observations.5 In (1) Wik is conditioned by w E sk. The expected cost of purchasing when any search strategy is used depends on the probability measure p which is induced by G. Throughout the study it will be assumed that G has a compact support. Further, let V= {p E R: ) p < p < ~7) denote the smallest closed n rectangle enclosing the support, i.e., G(P) = 0
0 < G(p) < 1 G(P) = 1
for all p < p,
forallp
p,
for all p > p.6
’ Specifically, Wi, = min(P,, , Pi, ,..., P,,), where P, denotes the random variable describing the price the consumer may observe for good i in the jth search, i = 1,2,..., n. 6 Throughout this study it will be assumed that (p,, &,..., p,) > (p{, pi,.... p:) if and only if pi >p;, i = 1, 2 ,..., n. Further, the inequality is strict of pi > p: for i = 1, 2 ,..., n and pi > p; for somej= I,2 ,..., n.
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Terming pi (pi) the lowest possible (highest possible) price for good i, although formally incorrect, should lead to no confusion. It is possible to weaken this restriction on G without disturbing the basic results to be presented. The restriction used, however, appears natural in the situation envisaged. Below an important result is stated. As the proof of this claim has been presented many times before (see, for example, DeGroot [2, Chap. 13, Sec. 8]), none will be presented. Claim 1. Given G has a compact support, for any fixed p E R: there exists an s,* such that v/(p, s,*) < ty(p, s) for any s E 5“. The stopping rule s,” is termed the optimal stopping rule when p is the vector of lowest prices observed to date. Utilizing Claim I we may define
4(P) = $3 w(P. s>
(2)
to be the expected cost of purchasing when Option 2 is chosen. Above it has been shown that the expected cost of purchasing is well defined when either Option 1 or Option 2 is chosen. In either case the expected cost of purchasing depends on the vector of lowest prices observed to date. When will Option 1 be chosen, i.e., when will the consumer decide to purchase the n goods? In what follows the answer to this question will be presented. Specifically, the set of acceptable prices A = (p E R: 1p . 1 < 4(p)} will be characterized. To achieve such a goal three lemmas are first stated and proved. LEMMA
1. #(a) is a concave nondecreasing function.
Proof:
To establish concavity it is shown that #(aP + (1 - a)fl-4(p)
- (1 -a)
4(F) > 0,
for any p, ~7,and a, where 0 < a < 1. Let s^be the optimal stopping rule when (ap t (1 - a)p”) denotes the vector of lowest prices observed to date, i.e., Q(ap+(l-a)fl=v(apt(l-a)j,i). Since (2) guarantees that yl(p, f) > 4(p) and ~(8, s^)> d(j), it follows that 4(aP + (1 - a)F> - 4(p)
- (1 - a) YUI
> v(ap + (1 - a>A i) - w(P, s1)- (1 - a) w(j, $1
3
k&
ip iel+
- a min(p,,
(min(ap, + (1 - a)ji,
Wik)
Wik) - (1 - a) min(p’,, Wik)} dp(w).
(3)
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Clearly, min(crp, + (1 - cf)p(, Wik) -a min(p,, Wik) - (1 - cz)min(j,.
Wik) > 0 (4)
for any i and k. Inspection of (3) and (4) establishes the concavity of $(.). To demonstrate that d(.) is nondecreasing note that for any s E Y’ v/(p, s) > w(p’, s), if p > p’. This fact and (2) imply 4(p) > #(p’), if p > p’. This completes the proof of the lemma. Using Lemma 1 it follows immediately that (g(p) - p . 1) is a concave function of p and thus the set of acceptable prices A is convex. To facilitate the exposition let G, denote the marginal distribution for the price of good k, k = 1, 2, 3,..., n, i.e., Gk(Pk) q” I..... Pk-l.Pk,Pk+ I...., En)dG(p) @I,....Pk-l.ek,Pk+I.....en)
for any pk. LEMMA 2. Suppose p E R: is such that pi < pi for i = 1,2 ,..., n, i f k, where k is an integer such that 1
Proof: If p satisfies the hypothesis of the lemma, then with probability one no lower price will be observed for good i, i # k. Hence, letting II(p) = min(p . 1,$(p)), we have 4(P) = c + j’*n(P, Pk
,...v pk-l,s,
pkf l,..., P,) dG,(s)
if the kth element of p is pk and pi
+ [ 1 - GktPk)]
n(P),
i f k. Rearranging terms yields
Since 4(p) -L!(p) = c > 0 when pi <
-n(P)
=
c -
=
0,
p*(pk ktk
-
s> dGk(s)
This completes the proof of the lemma.
>
O,
ifpk
if
Pk
<
pt,
>
Pz.
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Lemma 2 links the search for several goods with the theory of search for one good. Specifically, if the consumer has observed the lowest possible price for n - 1 goods, then the reservation price used for the other good is precisely that which would be used in a single good search for that good; i.e., pz is the reservation price the consumer would use if searching for a low price for good k only, k = 1, 2 ,..., n. The next lemma demonstrates that if the consumer prefers to purchase all n goods when p is the vector of lowest prices observed to date, then the consumer would also prefer to purchase if p’ < p is the vector of lowest prices observed to date. LEMMA
3.
. 1, then
If#(p)=p
(i) p’ < p implies #(p’) > p’ . 1, with inequality strict ifpi > pi and Gi(pi) > 0 fir some i, i = 1, 2 ,..., n, and (ii) p2 > p implies #(p’) < p2 . 1, with the inequality strict if pf + pi and G,(pt) > 0 for some i, i = 1, 2 ,..., n. ProoJ To establish the claims of the lemma it is shown that for any two vectors p1 and p2 such that p2 > p’, #(p’) - #(p’) < p2 . 1 - p1 . 1, with the inequality strict if pi # p’ and G,(pf) > 0 for some i = 1, 2,..., n. Let s1 denote the optimal stopping rule when p’ is the vector of lowest prices observed to date, i.e., 4(p’) = w(pl, s,). From (2) it follows that #(P’) < v(p2, s,). Hence, #(P’) - #(PI) < W(P2, s,) - W(PIYs1)
< 2 (pf - Pi>. i=l
Note that the last inequality is strict if pf > pi and G,(pf) > 0 for some i, i = 1, 2,..., n. The two claims made in the lemma now follow directly from the result established above. The lemmas established above are now used to prove the main result of this section. The theorem provides a complete characterization of the optimal strategy of a consumer searching for n goods with free recall. THEOREM 1. (i) The set of acceptable prices A = {p E RI 1$(p) > p . 1 } is convex. (ii) The set A is a strict subset of the set T= {p E R: 1pi
i = 1, 2 ,..., n}.
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BURDETT
(iii)
If@)
AND
MALUEG
=p . 1, then c = ,$ jll (Pi - s, dGi(‘)*
(6)
Further,
dPj
=
dPi
Gi(Pi) -q&r
for i # j, i = 1, 2,..., n and j = 1, 2 ,..., n.
-fl(p)=p.l
Proof. (i) From Lemma 1 it follows that Q(p) -p . 1) is a concave function of p. Hence, A is a convex set. (ii) Lemma 2 and Lemma 3 establish that if p E A, then pi < pf, i = 1, 2 ,..., 12. This implies A c T. Lemmas 2 and 3 also imply that (PT, Pz*...., p;) @?/I. Thus T&A. (iii) Assume that if p denotes the vector of lowest prices observed to date, then q+(p) = p - 1. Using Lemma 3 it follows that
q7 min(p,,
g(p) = c + E
Wil)
,Tl
c min(p,, i= 1 + PrW,,
< P,)
[
WC,
I K,
< P,)
n-1
+ E { ?‘ min(pi, ,r1
n-l
=c+E{
L‘ min(pi, lrl
wil) I w,, > P,
@‘,I> 1 W,I < Pn Will
II - s) dG,(s) + P,.
Continuing in the same fashion we obtain the following relationship for any k, k = I, 2 ,..., n: k-l 4(p)
=
c +
E
2,
min(p,,
H’il)
-
I
5 i=k
‘pi(pi
-
S) dGi(s)
J&
+
[gk
Pi*
Since 4(p) = p . 1 by assumption, $(p) = x1= 1 pi. This establishes the first part of claim (iii). To prove the second part of claim (iii) note that integrating (6) by parts yields i i=l
f’(pi ei
- s) dG,(s) = i lpiGi(s) ds. i=l ef
(7)
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1
To illustrate the results presented above we consider the case where the consumer is searching for two goods, i.e., n = 2. This case is illustrated in Fig. 1. The boundary of the acceptance set where #(PI, PJ = p, +p2 is given by the curve p:XYpf. Note that this curve in Fig. 1 is horizontal in the section PTX and vertical in the section Ypf. The curve is drawn this way to represent the fact #(pT, pJ = pT + p2 if pz p . 1 when the cost of search is c’ > c. The result is illustrated in Fig. 2a for the case where n = 2. In Fig. 2a an increase in the cost of search shifts out the reservation frontier from the curve PTPT to the curve j$PT. In the final part of this section the consequences of a change in the distribution of one of the prices is discussed. There are many ways this distribution can change. Two special cases are considered. First, assume there is an upward translation in the mean of the distribution of prices of good k, 1 < k < n; i.e., the new distribution of prices for good k, Gb(.), is such that for some E > 0, Gk(pk) = G;(p, + E) for any pk. Second, we consider the case where the new distribution of prices for good k is less risky than the old one. Following Rothschild and Stiglitz [lo], we define the distribution function G;(.) to be less risky than Gk(.), which has the same mean. if
(’ (G&j-
G;(s)) ds > 0
‘The frontier of the acceptance set in Figure 1 has been drawn as if the function G(‘) is strictly increasing on its support. If G(.) is not strictly increasing on its support, the frontier will have “flat” sections even if pi > pi, i = 1,2.
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a P; -p;’ P2
P2
pl
I
b p; FIGURE
p2
2
for any z, and as z + co the above integral tends to zero. Using (7) it follows immediately that either of the two changes in the distribution function, Gk(.), will increase the set of acceptable price vectors. Specifically, given either of the two changes, if 4(p) =p . 1 when p is the vector of lowest prices observed to date, then an increase in the mean or a reduction in the riskiness of the distribution function Gk(.) will imply g(p) > p . 1, with the inequality strict if and only if pk > fk. An increase in the mean of the distribution of prices of good 1 is illustrated in Fig. 2b for the case where n = 2. In Fig. 2b the upward translation in the mean has increased the lowest possible price from p, to pi. The old reservation frontier is illustrated by the curve prpf, the new one by the curve pT’pf.
2. THE No
RECALL
MODEL
In the previous section it was assumed that the any store previously visited without incurring a restriction is dropped. Instead, it will be assumed return to any store previously visited. It will
consumer could return to cost. In this section this that the consumer cannot be shown that this new
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assumption leads to a drastic change in the set of acceptable price vectors at least in the case where n = 2. It should be noted that the “free recall” assumption used in the previous section and the “no recall” assumption used in this section do not exhaust all possibilities. For example, it could be assumed that the consumer can return freely to the last four stores visited only. Nevertheless, the vast majority of contributions to the search literature have used one of the two restrictions used in the present study. Suppose the consumer visits a store and observes the vector of prices offered. Since this consumer cannot return to this store in the future, some of the prices offered may be too good to miss and therefore the consumer may purchase some subset of the goods and then continue to search for the other goods. As there are 2” subsets of a set of cardinality n, there are 2” options open to the consumer when visiting a store. Computing the expected total cost of purchasing the n goods when each of these options is selected is tedious when n is large, and leads to no new insights. Hence, to illustrate the essentials, the special case where n = 2 will be considered in this section. Suppose the consumer visits a store and observes price pI for good 1 and price pz for good 2. There are four options open to the consumer. Option A.
Buy both goods and thus end the search process.
Option B.
Buy good 2 and then continue to search for a low price for
good 1. Option C.
Buy good 1 and then continue to search for a low price for
good 2. Option D.
Buy neither of the goods and thus continue to search for both
goods. If Option A is chosen, then the cost of purchasing is p1 + p2. If Option B is chosen, then p, + V, is the expected cost of purchasing both goods, where V, is the minimum expected cost of purchasing good 2 in a single good search. Similarly, p2 + V, is the expected cost of purchasing both goods when Option C is chosen. Let W denote the expected cost of purchasing when Option D is selected. Note that the no recall assumption implies W, V, , and V, are independent of the prices observed to date. The consumer will choose the option that minimizes the expected cost of purchasing. Hence, the minimum expected cost of purchasing when prices p, and p2 have just been observed can be written as
It is well known that the expected cost of purchasing good i in a single
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search when at least one more price observation is to be taken, Vi, can be expressed as
*
Vi =
c
dGi(s) + (1 - Gi(pT)) Vi,
+
(9)
where p;” = Vi, i = 1,2 (Lippman and McCall [6] provide a derivation of this result). Note that the reservation price, pf, in the no recall case is the same as the reservation price for good i in the free recall case, other things being equal. Assume the consumer selects Option D. The expected cost of purchasing will depend on the future search strategy to be used. Suppose the consumer uses the following strategy: visit another store and purchase good i if and only if the price observed is no greater than pf. i = 1,2. If the consumer has not purchased good i after this observation, then he or she will search in an optimal manner for good i in a single good search. When this strategy is used the expected cost of purchasing, r, can be written as @= c + Pr(p, < PT and p2 < PT)E{ P, + la2 I pI < PT and p2 G PT I + PO,
G PT and p2 > P~JIE~P, I pI < PT and p2 > PTI + v21
+ Pr(p, > PT and p2 G PT)IE( p2 I p1 > PT and p2 < PT I+ v, 1 + PO, .
> PT and p2 > PTNV, + v2> t
= c + p’s dG,(s) + p’
sdG,(s)+
(1
-G,@:))
v,
+
(I--G,(P:))
v2.
Pz
!?I
Using (9) it follows immediately that w < I’, + V,. Hence, as r> W, we have W < V, + V,. This implies the expected total cost of purchasing both goods is lower in a search for both the goods at once than single good search for these goods. It is straightforward to demonstrate that the consumer will always choose to purchase good i when a price no greater than Pi is observed, i = 1, 2. Using this result it follows that p1 + V2 < W and g2 + I’, < W, and therefore Vi < W, i = 1,2, with the inequality strict if pi > 0. From the above arguments we may state the following result: v, + v2
>
w>
This result is important in characterizing no recall case. Given the consumer has visited a store when will the consumer purchase good l? purchase good 1 from the store visited if min(p,+
p2,
pl+
i= 1,2.
vi7
(10)
the set of acceptable prices in the and observed the prices p, and pz, By virtue of (8), the consumer will and only if
V,)
v,,
W.
(11)
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FIGURE
375
3
Using (10) and (11) it follows that the consumer will purchase good 1, independent of the price offered for good 2, if p, < a,, where 8, + pf = W. Further, inspection of (IO) and (11) establishes that the consumer will never purchase good 1 from a store that offers a price p, > PT. If the two prices observed are such that pi < pi < pT, i = I, 2, the consumer will purcase both goods if and only if p, + pz < W, otherwise the consumer will continue to search for both goods. This result leads to a complete characterization of the set of acceptable prices in the no recall case when )2= 2. The conclusions are summarized in Theorem 2 and illustrated in Fig. 3. THEOREM 2. Given the consumer has purchased neither of the two goods and has just observed prices p, and p2, the following strategy minimizes the consumer’s expected cost of purchasing both goods.
(a)
Select Option A f and only ifp, + p2 < W and pi < pT, i = 1,2.
(b)
Select Option B if and only ifp, < 6, and p2 > pf.
(c) (d) satisfied.
Select Option C if and only ifp, ,< $I and p, > pf. Select Option D if and only of none of the above conditions are
If only one of the goods has been purchased, the consumer should continue to search for the other good as in an optimal single good search. Let A* denote the set of acceptable price vectors in the no recall case. A* is illustrated by the area OpTRSpT in Fig. 3. Using Theorems 1 and 2 it follows that, ceteris paribus, the set of acceptable price vectors in the free recall case, A, is a strict subset of A *. Thus, there are advantages to the consumer having the right of free recall. This is not true with consumer search for a single good.
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BURDETTANDMALUEG ACKNOWLEDGMENTS
We would like to thank C. Wilson, M. Rothschild, D. Mortensen, and the anonymous referee for helpful comments. They are, of course, not responsible for any remaining errors.
REFERENCES 1. L. BRIEMAN, Stopping rule problems, in “Applied Combinatorial Mathematics” (E. F. Beckenbach, Ed.), Univ. of California Press, Berkeley 1964. 2. M. H. DEGROOT, “Optimal Statistical Decisions,” McGraw-Hill, New York, 1970. 3. J. L. GASTWIRTH, On probabilistic models of consumer search for information, Quart .I. Econ. 90 (1) (February 1976), 38-50. 4. M. G. KOHN AND S. SHAVELL, The theory of search, J. Econ. Theory 9 (2) (October 1974), 93-123. 5. M. LANDSBERGER AND D. PELED, Duration of offers and price realization, J. Econ. Theory 16 (January 1977), 17-37. 6. S. A. LIPPMAN AND J. J. MCCALL, The economics of job search: A survey, Econ. Inquiry 14 (June 1976), 155-189, and (Sept. 1976), 347-368. 7. J. J. MCCALL, The economics of information and optimal stopping rules, J. Business 38 (July 1965), 30@-317. “Price Variations in Almost Competitive 8. J. PRATT, D. WISE, AND R. ZECKHAUSER, Markets,” Discussion Paper 370, Kennedy School of Government, Harvard U., December 1975. 9. M. ROTHSCHILD, Searching for the lowest price when the distribution of prices is unknown, J. Pal. Econ. (July/August 1974), 689-7 11. 10. M. ROTHSCHILD AND J. STIGLITZ, Increasing risk I: A definition, J. Econ. Theory 2 (1970), 225-243. 11. G. J. STIGLER, The economics of information, J. Pol. Econ. (June 1961), 213-225.
OF ECONOMIC
THEORY
24, 362-316
(1981)
The Theory of Search for Several Goods KENNETH Department
of Economics,
BURDETT*
University
of Wisconsin,
Madison,
Wisconsin
53706
AND DAVID Department
of Economics,
A. MALUEG
Northwestern
University,
Evanston,
Illinois
60201
The problem analyzed in this study can be illustrated by the following example. Before starting out to buy groceries for the week, suppose an individual makes a list of the n goods required (n > 2). The list also states the amount required of each good. There are many grocery stores the individual can visit. Each of these stores sells all the goods desired. The individual can visit any store at a cost and observe the n vector of prices offered, one price for each of the n goods. The individual can purchase any number of the goods required from this store and then continue to search for the remaining goods. Given each pric,e vector observed can be envisaged as a random draw from a known nondegenerate distribution of price vectors, the problem is to determine the search strategy that minimizes the individual’s expected total cost of purchasing the n goods. The nature of the above problem is similar to that posed in the literature on consumer search for the lowest price for a single good.’ In the literature on single good search a consumer is assumed to be shopping for a given amount of a good. On the payment of a fixed cost the consumer may visit a store and observe the price offered, which is assumed to be a random draw from a known nondegenerate distribution of prices.’ Many studies have assumed that the consumer can return to any store preciously visited without incurring a cost; the other studies have assumed that the consumer cannot * Supported in part by an N.S.F. grant. ’ Lippman and McCall [6] provide an excellent survey of this literature and the related literature on worker job search. Stigler [ 111 presents an early example of work in this area. Pratt et al. ]8] have presented evidence that the distribution of prices for many goods is nondegenerate, at least in Boston. * Rothschild [9] considers a model of consumer search where the consumer does not know the distribution of prices in the market. In this tase the well-known reservation price property may not hold. 362
0022-0531/81/030362-15$02.00/O Copyright C: 1981 by Academic Press, Inc. All rights of reproduction in any form reserved.
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return to any store.3 Which of these two assumptions is used is of no great importance in consumer search for a single good, however, as it has been shown many times that the search strategy that minimizes the consumer’s expected cost of purchasing the good is the same in either case. Specifically, it has been shown that the optimal strategy can be characterized by a reservation price, p*. The consumer should purchase the first time a price no greater than p* is observed, where
c is the cost of a price observation, and F(-) is the distribution function associated with any price observation.4 In the present study a model similar in most respects to those presented in the single good search literature will be used. There are, however, two important differences. First, it will be assumed that the consumer wants to purchase n goods, where n is any positive integer. Second, the consumer can visit a store and observe the price quotation for each one of the n goods desired. Unlike the single good consumer search models, the assumption made about the consumer’s ability to return to any store is critical in consumer search models for several goods. In Section 1 it will be assumed that the consumer may return to any store without incurring a cost. In Section 2 the consumer is assumed to be unable to return to any store. It will be shown’that the strategies that minimize the consumer’s expected total cost of purchasing differ significantly in the two situations considered, at least in the case where n = 2. Suppose the consumer can freely return to any store previously visited. It follows immediately that in this case the consumer need not purchase any of the goods until he or she selects to by all the it goods. When the decision to purchase all the goods has been made, the consumer can freely return to the store that offered the lowest price observed for good i and purchase good i from that store, i = 1,2 ,..., n. Using the results presented in the single good consumer search literature, it appears reasonable to conjecture that the consumer’s best strategy can be characterized by what may be loosely termed a reservation frontier. In Section 1 this conjecture is shown to be true. Specifically, it is shown that the cost minimizing search strategy implies the consumer should stop searching and purchase all n goods if and only if the vector of lowest prices observed to date is no greater than some point on the reservation frontier. Most of Section 1 is taken up characterizing the reser’ To the authors’ knowledge, all but one of the studies presented to date have utilized one of these two restrictions. Landsberger and Peled 15 1 provide the exception. ’ See Lippman and McCall [6] for a derivation of this result. Note that p* is independent of the recall assumption.
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vation frontier and hence the set of acceptable price vectors. It will be demonstrated that the set of acceptable price vectors is convex. Further, it will be shown that the consumer will never purchase good i at a price higher than the reservation price for good i in a single good search for good i, i = 1, 2,..., n. In the final part of Section 1 we investigate the consequences of changes in the parameters of the model of consumer search. For example, it is shown that an increase in the cost of search will shift out the relevant reservation frontier and hence enlarge the set of acceptable price vectors. The effect of a change in the distribution of prices for one of the goods is also investigated. First, it is shown that if the distribution of one of the n prices is translated up, the set of acceptable price vectors increases. Second, if the distribution of one of the prices becomes riskier in the sense specified by Rothschild and Stiglitz [lo], the set of acceptable price vectors decreases. In Section 2 the case where the consumer cannot return to any store is considered. In this case if the consumer decides to leave a store and visit another one without purchasing, then the consumer loses the opportunity to purchase from that store at the offered price. Some of the prices offered. however, may be too good to miss. In Section 2 we consider a model of this type where the consumer is searching for only two goods, i.e., n = 2. In this case it is shown that there exists a reservation frontier as in the recall case. The nature of this frontier, however, is different. The strategy that minimizes the expected cost of purchasing when the consumer cannot return to stores can be briefly summerized as follows. First, the consumer will not purchase good i from a store that offers a price for good i that is greater than p;, where pT is the reservation price the consumer would use in a single good search for good i, i = 1, 2. Second, given good i has not been purchased to date, the consumer will always purchase good i from a store that offers a price for good i no greater than fii, jli < pT,i = 1, 2. Finally, suppose the consumer has purchased neither of the two goods to date. If the consumer observes a price for good i, say pi, such that fii < pi < pT, then both of the goods will be purchased if the price of the other good is no greater than some price hi(pi), where hi(pi) is a decreasing function, i = 1. 2. It should be stressed that in either of the models considered in this study the consumer need not purchase all the goods desired from the same store. Suppose, for the moment, that the consumer must purchase all the n goods from the same store. In this case the consumer search for several goods is formally equivalent to a single good search model. Specifically, the consumer will select a reservation “price,” z*. and then purchase from the first store visited which offers a price vector (pl, pzr..., p,) such that Cy=, pi < z*.
SEARCHFORSEVERALGOODS
365
1. THE MODEL
Suppose a consumer wants to purchase n different goods (n > 2). One unit of each of these n goods is desired. There are many stores that sell all of these goods. By paying c (c > 0) the consumer can visit a store and observe the vector of prices offered, one price for each one of the n goods. The stores the consumer can visit, however, do not all offer the same price vector. Specifically, assume there is a known nondegenerate distribution function, G, that describes the vectors offered by these stores. Assume that each price vector observed can be envisaged as a random draw from this distribution. After visiting a store the consumer may purchase some number of the goods desired and/or visit another store at cost c. Although the consumer can purchase only from a store visited, all of the n goods need not be purchased from the same store. Given the environment specified above, the problem is to determine and characterize the search strategy that minimizes the consumer’s expected total cost of purchasing the II goods. Throughout this section it will be assumed that the consumer can return to any store previously visited without incurring a cost. In the next section the consequences of utilizing an alternative to this free recall assumption will be considered. Let pi denote the lowest price for good i observed by the consumer to date, i = 1, 2,..., n; and let p = (p, , pz ,..., p,) be the vector of these prices. If the consumer has observed at least one price vector, there are three options open. Option
1.
Purchase all n goods and thus end the search process.
Option 2. vector.
Purchase none of the goods and then observe another price
Option 3. vector.
Purchase fewer than n goods and observe another
price
If the consumer is starting the search process and has not observed a price vector, then Option 2, must be selected. The free recall assumption guarantees that the expected cost of purchasing when Option 3 is selected is at least as great as the expected cost if Option 2 were selected instead. Hence, without loss of generality, assume that Option 1 or 2 only will be chosen. Specifically, Option 2 will be chosen at any time until Option 1 is chosen. The search process then ends. The free recall assumption implies that p . 1 = Cr=r pi denotes the minimum cost of purchasing when Option 1 is chosen and p is the vector of lowest prices observed to date. The consumer will choose Option 1 at any time if and only if p . 1 is no greater than the expected cost of purchasing if Option 2 were selected instead. When the consumer selects Option 2,
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BURDETT AND MALUEG
however, the expected cost of purchasing depends on the future search strategy to be used. At any time the consumer is assumed to know the vector of lowest prices observed to date and the probability laws generating the price vectors that may be observed in the future. In what follows it will be shown that any search strategy can be characterized by a stopping rule. To construct any such rule some notation needs to be developed. Let R = Xi”, R: denote the set of infinite sequences of n vectors consisting of positive reals. Since the distribution function G is assumed to be known, the probability the consumer would observe any measurable set in R if he or she were to search indefinitely is well defined. Let (0, /3,p) denote the probability measure space, where p is the probability measure induced by G, and b is the a-field. The u-field generated by the first k searches is denoted by Pkr k = l-2,3,... . Using the above notation, a stopping rule is an integer valued function s:B+IsuchthatskE/Ikforallk=1,2,3 ,..., where sk = {w E Q 1S(W) = k}. It is straightforward to check that any search strategy based on the information known to the consumer can be described by a stopping rule. For example, suppose the consumer utilizes the following simple search strategy: observe four price vectors and then purchase. The stopping rule S that describes this strategy is such that ? = R, and ? = 0 for all k # 4. Let .Y denote the set of all possible stopping rules. Suppose the consumer has decided to observe at least one more price vector when p denotes the vector of lowest prices observed to date. If the consumer utilizes the search strategy that can be described by stopping rule s, the expected total cost by purchasing, ty(p, s), can be written as ’
min(pi,
W(k) + ck
,e,
where Wik is the random variable describing the lowest price observed for good i in k observations.5 In (1) Wik is conditioned by w E sk. The expected cost of purchasing when any search strategy is used depends on the probability measure p which is induced by G. Throughout the study it will be assumed that G has a compact support. Further, let V= {p E R: ) p < p < ~7) denote the smallest closed n rectangle enclosing the support, i.e., G(P) = 0
0 < G(p) < 1 G(P) = 1
for all p < p,
forallp
p,
for all p > p.6
’ Specifically, Wi, = min(P,, , Pi, ,..., P,,), where P, denotes the random variable describing the price the consumer may observe for good i in the jth search, i = 1,2,..., n. 6 Throughout this study it will be assumed that (p,, &,..., p,) > (p{, pi,.... p:) if and only if pi >p;, i = 1, 2 ,..., n. Further, the inequality is strict of pi > p: for i = 1, 2 ,..., n and pi > p; for somej= I,2 ,..., n.
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Terming pi (pi) the lowest possible (highest possible) price for good i, although formally incorrect, should lead to no confusion. It is possible to weaken this restriction on G without disturbing the basic results to be presented. The restriction used, however, appears natural in the situation envisaged. Below an important result is stated. As the proof of this claim has been presented many times before (see, for example, DeGroot [2, Chap. 13, Sec. 8]), none will be presented. Claim 1. Given G has a compact support, for any fixed p E R: there exists an s,* such that v/(p, s,*) < ty(p, s) for any s E 5“. The stopping rule s,” is termed the optimal stopping rule when p is the vector of lowest prices observed to date. Utilizing Claim I we may define
4(P) = $3 w(P. s>
(2)
to be the expected cost of purchasing when Option 2 is chosen. Above it has been shown that the expected cost of purchasing is well defined when either Option 1 or Option 2 is chosen. In either case the expected cost of purchasing depends on the vector of lowest prices observed to date. When will Option 1 be chosen, i.e., when will the consumer decide to purchase the n goods? In what follows the answer to this question will be presented. Specifically, the set of acceptable prices A = (p E R: 1p . 1 < 4(p)} will be characterized. To achieve such a goal three lemmas are first stated and proved. LEMMA
1. #(a) is a concave nondecreasing function.
Proof:
To establish concavity it is shown that #(aP + (1 - a)fl-4(p)
- (1 -a)
4(F) > 0,
for any p, ~7,and a, where 0 < a < 1. Let s^be the optimal stopping rule when (ap t (1 - a)p”) denotes the vector of lowest prices observed to date, i.e., Q(ap+(l-a)fl=v(apt(l-a)j,i). Since (2) guarantees that yl(p, f) > 4(p) and ~(8, s^)> d(j), it follows that 4(aP + (1 - a)F> - 4(p)
- (1 - a) YUI
> v(ap + (1 - a>A i) - w(P, s1)- (1 - a) w(j, $1
3
k&
ip iel+
- a min(p,,
(min(ap, + (1 - a)ji,
Wik)
Wik) - (1 - a) min(p’,, Wik)} dp(w).
(3)
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BURDETTANDMALUEG
Clearly, min(crp, + (1 - cf)p(, Wik) -a min(p,, Wik) - (1 - cz)min(j,.
Wik) > 0 (4)
for any i and k. Inspection of (3) and (4) establishes the concavity of $(.). To demonstrate that d(.) is nondecreasing note that for any s E Y’ v/(p, s) > w(p’, s), if p > p’. This fact and (2) imply 4(p) > #(p’), if p > p’. This completes the proof of the lemma. Using Lemma 1 it follows immediately that (g(p) - p . 1) is a concave function of p and thus the set of acceptable prices A is convex. To facilitate the exposition let G, denote the marginal distribution for the price of good k, k = 1, 2, 3,..., n, i.e., Gk(Pk) q” I..... Pk-l.Pk,Pk+ I...., En)dG(p) @I,....Pk-l.ek,Pk+I.....en)
for any pk. LEMMA 2. Suppose p E R: is such that pi < pi for i = 1,2 ,..., n, i f k, where k is an integer such that 1
Proof: If p satisfies the hypothesis of the lemma, then with probability one no lower price will be observed for good i, i # k. Hence, letting II(p) = min(p . 1,$(p)), we have 4(P) = c + j’*n(P, Pk
,...v pk-l,s,
pkf l,..., P,) dG,(s)
if the kth element of p is pk and pi
+ [ 1 - GktPk)]
n(P),
i f k. Rearranging terms yields
Since 4(p) -L!(p) = c > 0 when pi <
-n(P)
=
c -
=
0,
p*(pk ktk
-
s> dGk(s)
This completes the proof of the lemma.
>
O,
ifpk
if
Pk
<
pt,
>
Pz.
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Lemma 2 links the search for several goods with the theory of search for one good. Specifically, if the consumer has observed the lowest possible price for n - 1 goods, then the reservation price used for the other good is precisely that which would be used in a single good search for that good; i.e., pz is the reservation price the consumer would use if searching for a low price for good k only, k = 1, 2 ,..., n. The next lemma demonstrates that if the consumer prefers to purchase all n goods when p is the vector of lowest prices observed to date, then the consumer would also prefer to purchase if p’ < p is the vector of lowest prices observed to date. LEMMA
3.
. 1, then
If#(p)=p
(i) p’ < p implies #(p’) > p’ . 1, with inequality strict ifpi > pi and Gi(pi) > 0 fir some i, i = 1, 2 ,..., n, and (ii) p2 > p implies #(p’) < p2 . 1, with the inequality strict if pf + pi and G,(pt) > 0 for some i, i = 1, 2 ,..., n. ProoJ To establish the claims of the lemma it is shown that for any two vectors p1 and p2 such that p2 > p’, #(p’) - #(p’) < p2 . 1 - p1 . 1, with the inequality strict if pi # p’ and G,(pf) > 0 for some i = 1, 2,..., n. Let s1 denote the optimal stopping rule when p’ is the vector of lowest prices observed to date, i.e., 4(p’) = w(pl, s,). From (2) it follows that #(P’) < v(p2, s,). Hence, #(P’) - #(PI) < W(P2, s,) - W(PIYs1)
< 2 (pf - Pi>. i=l
Note that the last inequality is strict if pf > pi and G,(pf) > 0 for some i, i = 1, 2,..., n. The two claims made in the lemma now follow directly from the result established above. The lemmas established above are now used to prove the main result of this section. The theorem provides a complete characterization of the optimal strategy of a consumer searching for n goods with free recall. THEOREM 1. (i) The set of acceptable prices A = {p E RI 1$(p) > p . 1 } is convex. (ii) The set A is a strict subset of the set T= {p E R: 1pi
i = 1, 2 ,..., n}.
370
BURDETT
(iii)
If@)
AND
MALUEG
=p . 1, then c = ,$ jll (Pi - s, dGi(‘)*
(6)
Further,
dPj
=
dPi
Gi(Pi) -q&r
for i # j, i = 1, 2,..., n and j = 1, 2 ,..., n.
-fl(p)=p.l
Proof. (i) From Lemma 1 it follows that Q(p) -p . 1) is a concave function of p. Hence, A is a convex set. (ii) Lemma 2 and Lemma 3 establish that if p E A, then pi < pf, i = 1, 2 ,..., 12. This implies A c T. Lemmas 2 and 3 also imply that (PT, Pz*...., p;) @?/I. Thus T&A. (iii) Assume that if p denotes the vector of lowest prices observed to date, then q+(p) = p - 1. Using Lemma 3 it follows that
q7 min(p,,
g(p) = c + E
Wil)
,Tl
c min(p,, i= 1 + PrW,,
< P,)
[
WC,
I K,
< P,)
n-1
+ E { ?‘ min(pi, ,r1
n-l
=c+E{
L‘ min(pi, lrl
wil) I w,, > P,
@‘,I> 1 W,I < Pn Will
II - s) dG,(s) + P,.
Continuing in the same fashion we obtain the following relationship for any k, k = I, 2 ,..., n: k-l 4(p)
=
c +
E
2,
min(p,,
H’il)
-
I
5 i=k
‘pi(pi
-
S) dGi(s)
J&
+
[gk
Pi*
Since 4(p) = p . 1 by assumption, $(p) = x1= 1 pi. This establishes the first part of claim (iii). To prove the second part of claim (iii) note that integrating (6) by parts yields i i=l
f’(pi ei
- s) dG,(s) = i lpiGi(s) ds. i=l ef
(7)
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FIGURE
GOODS
371
1
To illustrate the results presented above we consider the case where the consumer is searching for two goods, i.e., n = 2. This case is illustrated in Fig. 1. The boundary of the acceptance set where #(PI, PJ = p, +p2 is given by the curve p:XYpf. Note that this curve in Fig. 1 is horizontal in the section PTX and vertical in the section Ypf. The curve is drawn this way to represent the fact #(pT, pJ = pT + p2 if pz
(’ (G&j-
G;(s)) ds > 0
‘The frontier of the acceptance set in Figure 1 has been drawn as if the function G(‘) is strictly increasing on its support. If G(.) is not strictly increasing on its support, the frontier will have “flat” sections even if pi > pi, i = 1,2.
372
BURDETT
AND
MALUEG
a P; -p;’ P2
P2
pl
I
b p; FIGURE
p2
2
for any z, and as z + co the above integral tends to zero. Using (7) it follows immediately that either of the two changes in the distribution function, Gk(.), will increase the set of acceptable price vectors. Specifically, given either of the two changes, if 4(p) =p . 1 when p is the vector of lowest prices observed to date, then an increase in the mean or a reduction in the riskiness of the distribution function Gk(.) will imply g(p) > p . 1, with the inequality strict if and only if pk > fk. An increase in the mean of the distribution of prices of good 1 is illustrated in Fig. 2b for the case where n = 2. In Fig. 2b the upward translation in the mean has increased the lowest possible price from p, to pi. The old reservation frontier is illustrated by the curve prpf, the new one by the curve pT’pf.
2. THE No
RECALL
MODEL
In the previous section it was assumed that the any store previously visited without incurring a restriction is dropped. Instead, it will be assumed return to any store previously visited. It will
consumer could return to cost. In this section this that the consumer cannot be shown that this new
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373
assumption leads to a drastic change in the set of acceptable price vectors at least in the case where n = 2. It should be noted that the “free recall” assumption used in the previous section and the “no recall” assumption used in this section do not exhaust all possibilities. For example, it could be assumed that the consumer can return freely to the last four stores visited only. Nevertheless, the vast majority of contributions to the search literature have used one of the two restrictions used in the present study. Suppose the consumer visits a store and observes the vector of prices offered. Since this consumer cannot return to this store in the future, some of the prices offered may be too good to miss and therefore the consumer may purchase some subset of the goods and then continue to search for the other goods. As there are 2” subsets of a set of cardinality n, there are 2” options open to the consumer when visiting a store. Computing the expected total cost of purchasing the n goods when each of these options is selected is tedious when n is large, and leads to no new insights. Hence, to illustrate the essentials, the special case where n = 2 will be considered in this section. Suppose the consumer visits a store and observes price pI for good 1 and price pz for good 2. There are four options open to the consumer. Option A.
Buy both goods and thus end the search process.
Option B.
Buy good 2 and then continue to search for a low price for
good 1. Option C.
Buy good 1 and then continue to search for a low price for
good 2. Option D.
Buy neither of the goods and thus continue to search for both
goods. If Option A is chosen, then the cost of purchasing is p1 + p2. If Option B is chosen, then p, + V, is the expected cost of purchasing both goods, where V, is the minimum expected cost of purchasing good 2 in a single good search. Similarly, p2 + V, is the expected cost of purchasing both goods when Option C is chosen. Let W denote the expected cost of purchasing when Option D is selected. Note that the no recall assumption implies W, V, , and V, are independent of the prices observed to date. The consumer will choose the option that minimizes the expected cost of purchasing. Hence, the minimum expected cost of purchasing when prices p, and p2 have just been observed can be written as
It is well known that the expected cost of purchasing good i in a single
314
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AND
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search when at least one more price observation is to be taken, Vi, can be expressed as
*
Vi =
c
dGi(s) + (1 - Gi(pT)) Vi,
+
(9)
where p;” = Vi, i = 1,2 (Lippman and McCall [6] provide a derivation of this result). Note that the reservation price, pf, in the no recall case is the same as the reservation price for good i in the free recall case, other things being equal. Assume the consumer selects Option D. The expected cost of purchasing will depend on the future search strategy to be used. Suppose the consumer uses the following strategy: visit another store and purchase good i if and only if the price observed is no greater than pf. i = 1,2. If the consumer has not purchased good i after this observation, then he or she will search in an optimal manner for good i in a single good search. When this strategy is used the expected cost of purchasing, r, can be written as @= c + Pr(p, < PT and p2 < PT)E{ P, + la2 I pI < PT and p2 G PT I + PO,
G PT and p2 > P~JIE~P, I pI < PT and p2 > PTI + v21
+ Pr(p, > PT and p2 G PT)IE( p2 I p1 > PT and p2 < PT I+ v, 1 + PO, .
> PT and p2 > PTNV, + v2> t
= c + p’s dG,(s) + p’
sdG,(s)+
(1
-G,@:))
v,
+
(I--G,(P:))
v2.
Pz
!?I
Using (9) it follows immediately that w < I’, + V,. Hence, as r> W, we have W < V, + V,. This implies the expected total cost of purchasing both goods is lower in a search for both the goods at once than single good search for these goods. It is straightforward to demonstrate that the consumer will always choose to purchase good i when a price no greater than Pi is observed, i = 1, 2. Using this result it follows that p1 + V2 < W and g2 + I’, < W, and therefore Vi < W, i = 1,2, with the inequality strict if pi > 0. From the above arguments we may state the following result: v, + v2
>
w>
This result is important in characterizing no recall case. Given the consumer has visited a store when will the consumer purchase good l? purchase good 1 from the store visited if min(p,+
p2,
pl+
i= 1,2.
vi7
(10)
the set of acceptable prices in the and observed the prices p, and pz, By virtue of (8), the consumer will and only if
V,)
v,,
W.
(11)
SEARCH FOR SEVERALGOODS
FIGURE
375
3
Using (10) and (11) it follows that the consumer will purchase good 1, independent of the price offered for good 2, if p, < a,, where 8, + pf = W. Further, inspection of (IO) and (11) establishes that the consumer will never purchase good 1 from a store that offers a price p, > PT. If the two prices observed are such that pi < pi < pT, i = I, 2, the consumer will purcase both goods if and only if p, + pz < W, otherwise the consumer will continue to search for both goods. This result leads to a complete characterization of the set of acceptable prices in the no recall case when )2= 2. The conclusions are summarized in Theorem 2 and illustrated in Fig. 3. THEOREM 2. Given the consumer has purchased neither of the two goods and has just observed prices p, and p2, the following strategy minimizes the consumer’s expected cost of purchasing both goods.
(a)
Select Option A f and only ifp, + p2 < W and pi < pT, i = 1,2.
(b)
Select Option B if and only ifp, < 6, and p2 > pf.
(c) (d) satisfied.
Select Option C if and only ifp, ,< $I and p, > pf. Select Option D if and only of none of the above conditions are
If only one of the goods has been purchased, the consumer should continue to search for the other good as in an optimal single good search. Let A* denote the set of acceptable price vectors in the no recall case. A* is illustrated by the area OpTRSpT in Fig. 3. Using Theorems 1 and 2 it follows that, ceteris paribus, the set of acceptable price vectors in the free recall case, A, is a strict subset of A *. Thus, there are advantages to the consumer having the right of free recall. This is not true with consumer search for a single good.
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BURDETTANDMALUEG ACKNOWLEDGMENTS
We would like to thank C. Wilson, M. Rothschild, D. Mortensen, and the anonymous referee for helpful comments. They are, of course, not responsible for any remaining errors.
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