Consumer Search Markets with Costly Re-Visits∗

Maarten C.W. Janssen†and Alexei Parakhonyak‡

April 29, 2013

Abstract

This paper characterizes equilibrium outcomes in consumer search markets taking

the cost of going back to stores already searched explicitly into account. We show

that the optimal sequential search rule under costly re-visits is very different from the

traditional reservation price rule in that it is non-stationary and not independent of

previously sampled prices. We explore the implications of costly re-visits on market

equilibrium in two celebrated search models. In the Wolinsky model some consumers

search beyond the first firm. In this class of models costly re-visits do make a substan-

tive difference and their impact can be of the same order of magnitude as the initial

search cost. In the Stahl oligopoly search model where consumers do not search beyond

the first firm, there remains a unique symmetric equilibrium that has firms use pricing

strategies that are identical to the perfect recall case.

JEL-codes: D11, D40, D83, L13

Key Words: Search, Costly Re-Visits, Oligopoly, Competition

∗We thank two anonymous referees, Mark Armstrong, Jan Brinkhuis, Vladimir Karamychev, ManfredNermuth, Vladimir Pavlov, Andrew Rhodes, Jidong Zhou and seminar participants at Tinbergen Institute,the University of Vienna, the World Congress of the Econometric Society (Shanghai, 2010), 2nd Workshopon Search and Switching Costs, University of Edinburgh and the 2011 Midwest Economic Theory Conferencefor their valuable comments. This paper replaces the paper ’Consumer search with costly recall’ where weonly characterize the optimal search strategy. Support from the Basic Research Program of the NationalResearch University Higher School of Economics is gratefully acknowledged. Janssen acknowledges financialsupport from the Vienna Science and Technology Fund (WWTF) under project fund MA 09-017.

†University of Vienna and National Research University Higher School of Economics. E-mail:maarten.janssen@univie.ac.at

‡National Research University Higher School of Economics. E-mail: parakhonyak@hse.ru.

1

1 Introduction

The main focus of the literature on consumer search is to analyze how market outcomes

are affected by explicitly taking into account the cost consumers have to incur to obtain

information about prices and/or product characteristics. The basic results of the extensive

literature are that firms have some market power that they can exploit even if there are

many firms in the market, and that price dispersion emerges as a consequence of the fact

that some firms aim at selling to many consumers at low prices, while others make higher

margins over fewer customers (see, e.g., Stigler (1961) and Reinganum (1979)).

Most, if not all, of the consumer search literature implicitly or explicitly makes the

assumption of perfect or free recall: consumers can always come back to previously sampled

firms without incurring a cost.1 One of the important consequences of this assumption is that

the optimal sequential search rule is characterized by a reservation price that is constant over

time (Kohn and Shavell (1974)): for any observed price sequence, consumers stop searching

and buy at the firm from which they received a price quote if that price is not larger than

this reservation price; otherwise they continue searching.2

As we argue in this paper, the assumption of perfect recall is at odds with the general

philosophy of the consumer search literature which has search frictions at its core. If con-

sumers have to incur a cost to go to a shop in the first place, then in almost any natural

environment it is also costly (in terms of time, effort, or money) to return to that shop.

Even while searching on the internet, where the cost of search is arguably lower than in

non-electronic markets, it takes some mouse clicks and time to go back to previously visited

web-sites. In other words, in consumer search it is not only important to remember the offers

previously received, but also one has to incur a cost to activate these offers again.

1See, e.g., Reinganum (1979), Morgan and Manning (1985), Stahl (1989) and Stahl (1996) for earlypapers. An extensive survey of this literature has recently been given by Baye et al. (2006). Recent papersthat make this assumption include Armstrong et al. (2009), Haan and Moraga-Gonzales (2011), Janssenet al. (2011) and Tappata (2009).

2An alternative setting is studied by Weitzman (1979). He considers the interesting case where alternativesdiffer in the cost of inspection as well as in the distribution of revenues and he asks the question in whichorder the alternatives should be explored.

2

In this paper, we replace the perfect recall assumption by the more natural assumption

of costly re-visits, where the cost of going back to stores previously sampled is explicitly

modeled. When re-visits are costly, we have three sets of results. The first set of results

relates to the optimal sequential search strategy for consumers. The second set of results

relates to the implications for the equilibrium pricing strategies of firms in the Wolinsky

(1986) model. The third set of results concerns the equilibrium pricing strategies of firms

in the Stahl (1989) model. We consider the models of Wolinsky (1986) and Stahl (1989)

as they are generally conceived to be the two seminal papers in consumer search theory on

which the recent literature builds. Another reason for looking at the implications in these

two models is that these two models differ in the extent to which consumers really do search.

In the equilibrium of the Wolinsky model some consumers search beyond the first firm, while

in the equilibrium of the Stahl model consumers stop searching at the first alternative they

observe. We are interested in the extent to which the impact of costly re-visits differs in

these two environments.

Concerning the optimal search rule under costly re-visits, we show that the reservation

price is not constant over time, but rather that at any moment in time it depends on (i)

the number of firms that are not yet sampled and (ii) the lowest price sampled so far. In

particular, for a given lowest price in the sample the reservation price is (weakly) decreasing

in the number of firms that are not yet sampled (increasing over time) and increasing in

the minimum price in the sample if this minimum price is not too large. Thus, under costly

re-visits it may happen that if consumers observe as part of a price sequence two prices pt

and pt+1, with pt < pt+1, they will rationally decide to accept to buy at pt+1 and not at

pt. Only when there are infinitely many prices to sample, stationarity re-appears and the

reservation price in that case coincides with the reservation price under perfect recall. Thus,

one implication of this part of our paper is that competitive search models are robust to

introducing costly re-visits. We also show that when the cost of re-visiting firms is above a

critical threshold, the re-visit option is never used, and the model essentially coincides with

the no recall model (see, e.g., Lippman and McCall (1976)).

3

In Wolinsky (1986), each of N firms produces a single distinct brand. Each consumer

has a particular value for a brand, which is called the match value. These match values are

stochastic variables and when searching a particular firm, a consumer is informed about the

price of that firm and its match value. When we introduce costly re-visits in the Wolinsky

(1986) model, it turns out that firms’ pricing behaviour is affected when we go beyond perfect

recall. Although it is difficult to derive an analytic solution for the general model with costly

recall, we show that one can still derive useful predictions for the case where N = 2 and the

match values are uniformly distributed. Since consumers do search beyond the first firm,

costly re-visits do influence equilibrium pricing behaviour as they affect expected demand

both because consumer reservation prices change and because the chance that consumers

come back to the previously visited store is affected. We characterize the new equilibrium

price and show that when the initial search and re-visit costs are small, the impact of the

return cost on the equilibrium price is of the same order of magnitude as the impact of the

search cost related to the first search. Moreover, we show that the effect of the impact of the

return cost on the equilibrium price can be non-monotone, whereas the equilibrium price is

always increasing in the cost of the first visit.

Stahl (1989) analyzes a homogeneous goods oligopoly market where firms compete for

two types of consumers, informed and uninformed. Informed consumers have zero search

cost and always buy the product at the lowest price in the market. Uninformed consumers

have positive search cost and engage in optimal sequential search. The surprising result we

obtain for the Stahl (1989) model is that even though the search strategy is different and

more complicated, the market equilibrium does not involve firms choosing different pricing

strategies. We have two types of results that underline this general conclusion.

First, the symmetric equilibrium that is found by Stahl (1989) remains an equilibrium. In

this equilibrium, the price distribution is such that the uninformed consumers buy immedi-

ately in the first store they visit and their reservation price is the same as under perfect recall.

This first result is quite intuitive: at the reservation price (which is the upper bound of the

price distribution) consumers are indifferent between buying immediately and continuing to

4

search and buy (with probability one) at the next store and thus consumers never seriously

consider going back to previously visited stores. The second result is less intuitive: we show

there are no other types of symmetric equilibria. With costly re-visits, firms may hypothet-

ically benefit from setting prices above the reservation price of the first search round. We

show, however, that the structure of the profit function is such that if firms charge prices

above the first round reservation price, they can never compensate the loss of demand with

higher margins per consumer.

The drawback of studying costly second visits is that the optimal search rule of consumers

becomes more complicated which, despite being more realistic, may limit the use in practical

applications. However, even for more practical applications, our paper has three important

conclusions. First, it is important to know when one can approximate costly re-visits by the

easier assumptions of costless recall or no recall. Our paper shows that if (i) the cost of re-

visits is small relative to the search cost, or (ii) the number of firms is large, the costless recall

assumption is a good approximation, whereas if (iii) the cost of second visits is larger than a

critical value, the no recall assumption can be used without any adjustments to the results.

Second, in certain models, particularly in models based on Stahl (1989), our paper shows

that the equilibrium analysis remains unaffected when costly second visits are introduced

even when the optimal search rule is affected. Third, in other models, even when the search

rule becomes more complicated and the equilibrium analysis is affected, like in the Wolinsky

(1986) model, our Section 3 shows that one can perform an analytic equilibrium analysis of

the model. As the search rule is more complicated, there is a trade-off between introducing

costly re-visits and introducing other features of the model environment, but it is not the

case that any model with costly re-visits is inherently intractable.

There are several papers studying optimal search rules without perfect and costless re-

call. Karni and Schwartz (1977) and Landsberger and Peleg (1977) study situations where a

decision maker has “uncertain recall”, where there is a positive probability that past obser-

vations can be successfully retrieved. They motivate their analysis by interpreting the large

part of consumer search and labour search theory as making specific assumptions on the

5

probability with which past observations can be successfully retrieved: in consumer search,

the probability of successful retrieval is one, whereas in job market search, this probability

is zero. Karni and Schwartz (1977) and Landsberger and Peleg (1977) study the intermedi-

ate case where there is a positive probability previous observations remain valid that is less

than one and larger than zero. In the operations research literature Kang (1999) studies an

optimal stopping problem where there is a cost for re-visits, but the cost (linearly) depends

on the value of the observation. Compared to Karni and Schwartz (1977) and Landsberger

and Peleg (1977) we interpret the difference between consumer search and job market search

differently, namely in terms of the cost one has to incur to reactivate the offer. In our inter-

pretation, most of the literature assumes this cost to be either zero or prohibitively high. We

study the intermediate case where the cost is positive, and characterize the optimal stopping

rule when consumers consider the option of going back to previously sampled firms.3 We

also show that the no recall search rule for a finite number of firms, defined in Lippman and

McCall (1976), obtains when the re-visit cost is larger than a threshold value the value of

which depends on whether or not consumers were searching optimally on previous search

rounds. Compared to Kang (1999), we see the cost as being embodied in search frictions,

and therefore independent of the value of past observations. Despite these differences in

interpretation, some of the more technical results of Landsberger and Peleg (1977) and Kang

(1999) are similar in nature to ours. Most notably, they also characterize a time-dependent

reservation price and they show for their set-up that this price is constant in case of perfect

recall and when there are infinitely many firms. None of these authors investigates, however,

the implications of these optimal search rules for firms’ strategic behaviour.

Armstrong and Zhou (2011) give a particular interpretation of costly re-visits. They show

that costly re-visits can be re-interpreted as buy-now discounts, i.e. as discounts consumers

only get when they visit a firm for the first time: as soon as they walk out of the store without

buying, the possibility to receive the discount disappears. The main difference between their

3As far as we are aware, there is no paper studying this most relevant case. Kohn and Shavell (1974) saythat some of their results continue to hold if there is no possibility of recall, but they also do not analyzethe situation of costly recall.

6

paper and ours is that the buy-now discount in Armstrong and Zhou (2011) is a strategic

variable chosen by firms, whereas in our model the cost of a second visit to a firm is an

exogenous feature of the search technology. This means that our analysis may have various

other applications, such as in a search theoretic explanation for the existence of shopping

malls (see Non (2010)).

The structure of the rest of the paper is as follows. Section 2 analyzes the optimal sequen-

tial search rule in a setting where consumers have to make a choice between a finite number

of objects, where there is a cost of investigating an object for the first time and another cost

for coming back to objects already visited. Section 3 investigates the implications of this

optimal search rule for the model studied in Wolinsky (1986), while section 4 presents the

results for the model presented in Stahl (1989). Section 5 concludes. All technical proofs

are in the appendix.

2 Optimal Sequential Consumer Search

The environment we discuss in this section and that will be relevant in the market setting

discussed in the next two sections is one where consumers have a choice whether or not to buy

one alternative out of a finite number of alternatives. The utility each alternative delivers

is unknown before consumers investigate the properties of the alternative. Before inspection

all alternatives look the same, but ex post they are likely to be different. The notation we

use in this section is based on the idea that the alternatives only differ in price p, but this

is not in any way essential: anything can also be expressed in terms of the utility received

by an agent. Thus, we concentrate on an environment where alternative i has a price pi

that is distributed according to the distribution function Fi(p). For now, these distribution

functions are exogenous, but in later sections they are endogenous. As in later sections,

we concentrate on symmetric equilibria where in equilibrium all firms are identical before

visiting them, we analyze the case where Fi(p) = F (p) for all i. We assume that F (p) is a

7

continuous function (i.e., it has no mass points) and has a bounded support.4 We define p

and p to be the lower and upper bound of the support of the distribution. Consumers engage

in sequential search after firms have made their choices (and cannot change them anymore).

Consumers get their first price quotation for free (following most of the literature),5 but any

subsequent price quotation comes at a search cost c. Consumers have unit demand and an

identical valuation for the good which we denote by v and v > c. If the consumer decides to

go back to the store she already visited, she incurs costs b > 0.

The main issue we address in this section is how the presence of costly re-visits (b > 0)

affects the optimal search rule when F (p) is known. Since the expected value of continuing

to search depends on future period expected values we use backward induction to analyze

the optimal stopping rule. To this end, psk−1 is defined as the smallest price in a sample of

k − 1 prices previously sampled. We will argue that for each value of psk−1 there is a unique

value of pk such that an individual consumer is indifferent between buying at pk and either

going back to one of the previously sampled firms and buying there or continuing to search.

We denote this (reservation) price by ρk(psk−1). If pk ≤ ρk(p

sk−1), the consumer decides to

buy at pk. Otherwise, she either buys at psk−1 (if this price is relatively small) or continues

to search.

We introduce the reservation price under perfect recall, where b = 0, as a benchmark

case. It is well known, that if the return costs equal to zero the reservation price ρpr is

defined by the following equation (see, e.g., Stahl (1989)):

c =

∫ ρpr

p

F (p)dp.

Before we start with the formal analysis under costly re-visits we introduce the following

notation. Let

4In the Wolinsky model analyzed in section 3, continuity is exogenously imposed by the distributionfunction describing the match value. In the Stahl Model analyzed in section 4, it is a standard result thatthe price distribution is described by a continuous function.

5Alternatively, following Janssen et al. (2005) we can easily incorporate the case where the first searchcomes at a cost as well.

8

C(x) = c+ F (x+ b)E(y|y < x+ b) + (1− F (x+ b))(x+ b) = c+ x+ b−∫ x+b

p

F (p)dp.

We refer to C(x) as the continuation cost function as it represents the costs of continuing

to search for one more round. The expression can be understood as follows. If a consumer

continues to search, she has to pay the search cost and either finds then, with probability

F (x + b), an alternative that is better than returning to the alternative x, or she does not,

and then returns to get the previously visited alternative. The second expression is obtained

by integrating in parts.

The construction of the sequence of reservation price functions is by induction starting

at the last search round. The following lemma introduces the base of induction.

Lemma 1. Let F (p) be a distribution of prices. Then for k = N − 1 the reservation price

ρN−1 is uniquely defined as a function of psN−2 ∈ [p, p] by

ρN−1(psN−2) = min

(ps

N−2 + b, C(psN−2), p

∗N−1

)where p∗N−1 satisfies the equation

p∗N−1 = C(p∗N−1)

Moreover, if the consumer decides to continue searching, the continuation cost of the

search, defined as the additional net expected cost of continuing to search conditional on

optimal behaviour after the search is made, equals C(psN−1).

Figure 1 illustrates the lemma for sufficiently small values of b. The reservation price as

a function of psN−2 is presented by the bold curves. It is easy to see that this line consists of

three parts:

9

Figure 1: Reservation Price ρN−1 as a function of psN−2, the smallest price sampled prior to

period N − 1.

psN−2

pN−1

CN−1

p∗N−1p

b

Return

Buy now

Search

(i) for psN−2 < p,6 the best alternative to buying at pN−1 is to go back to the lowest-priced

firm in the sample so far. Thus, the reservation price is determined by ρN−1 = psN−2 + b.

(ii) for p ≤ psN−2 < p∗N−1 the option to continue searching is always preferred to the option

of going back to the lowest-priced firm in the sample so far. Thus, the consumer’s optimal

choice is based on a comparison between the current price and the expected continuation

costs of the search;

(iii) for the region psN−2 ≥ p∗N−1 the situation is similar to the previous case, except that

the previously sampled prices are too high to be considered, implying that the continuation

cost does not depend on psN−2. Therefore, the reservation price is independent of ps

N−2 in

6Here p is a price with 0 < p < p∗N−1 such that if that price was the lowest price observed so far andthe current observed price is p + b, then the consumer would be indifferent between continuing to search,buying now and going back to the lowest price observed so far. In the proof of proposition 1 we show thatp = ρpr − b.

10

this case.

Along the bold curve the consumer is indifferent between buying now at the shop she is

currently visiting and either continuing to search or going back to the lowest-priced firm in

the sample so far.

Since optimal search behaviour is completely determined by the pair (pN−1, psN−2) we can

characterize it in the same figure. Indeed, in the upper left region, which is bounded from

below by ρN−1 and from the right by p, the consumer always goes back and buys at the

lowest-priced firm in the sample so far. In the lower, unshaded region, which is bounded

from above by the reservation price, the consumer always buys at the current shop. Finally,

in the upper right (dotted) region, which is bounded from below by the reservation price

and for which psN−2 > p, the consumer always continues to search.

Given the lemma, we are now ready to state and prove the main result of this section. The

theorem gives a recursive definition of the reservation price and shows that the reservation

price is a monotone function of psk−1.

Theorem 1. The reservation price ρk(psk−1) is defined for any k and any ps

k−1 from the

support of F (p) by

ρk(psk−1) = min(ps

k−1 + b, Ck(psk−1), p

∗k),

where p∗k is defined as a solution of p = Ck(p) and Ck is defined recursively by

Ck(psk) = c+ P(pk+1 < min(ps

k + b, Ck+1(psk+1)))·

E(pk+1|pk+1 < min(psk + b, Ck+1(p

sk+1)))+

P(pk+1 ≥ min(psk + b, Ck+1(p

sk+1)))·

E(min(psk + b, Ck+1(p

sk+1))|pk+1 ≥ min(ps

k + b, Ck+1(psk+1))).

and CN−1 = C(psN−1).

Moreover, the time- and history-dependent reservation prices ρk(psk−1) are nondecreasing

in psk−1.

11

The proof of the theorem basically shows that the function ρk+1(psk) is defined over three

separate intervals and essentially looks like the reservation price for the last step (see Figure

1). When psk−1 is relatively small ρk(p

sk−1) = ps

k−1 + b. Then for intermediate values of psk−1,

the reservation price is defined by the indifference between continuing to search and buying

now, but where ρk(psk−1) crucially depends on ps

k−1 as the expected pay-off of continuing to

search depends on the lowest price observed so far (as this is the price the consumer will

go back to if a future search will only yield high prices). Finally, for higher values of psk−1,

ρk(psk−1) is independent of ps

k−1.

Next, we show that reservation prices are non-decreasing over time. In particular, if

a price smaller than p = ρpr − b is sampled before, then the reservation price is simply

ρk(psk−1) = ps

k−1 + b and therefore if psk = ps

k−1, then ρk+1(psk) = ρk(p

sk−1). However, if a price

strictly larger than p = ρpr − b is the lowest price in the sample so far, then ρk+1(psk) >

ρk(psk−1). Thus, under costly re-visits reservation prices are essentially non-stationary.7

Proposition 1. If psk = ps

k−1, then ρk+1(psk) ≥ ρk(p

sk−1), i.e., reservation prices are non-

decreasing over time. Moreover, ρk+1(psk) > ρk(p

sk−1) for all ps

k and psk−1 such that ps

k =

psk−1 > p = ρpr − b.

The following numerical example illustrates some of the aforementioned features of the

sequence of reservation prices and the quantitative impact of b on the reservation prices.

Consider that prices are uniformly distributed on [0, 1] and that there are four firms in the

market. We first present for fixed values of c and b the reservation prices after visiting zero,

one and two firms as well as the reservation price under perfect recall. Figure 2 considers the

case where c equals 0.05 and b equals 0.04. In this case, the reservation price under perfect

recall equals approximately 0.316, while the reservation price before visiting any shop under

costly re-visits equals 0.333, whereas the reservation price after observing two high prices is

0.382 (which is 20% larger than the reservation price under perfect recall). The figure also

clearly shows the non-stationarity of reservation prices.

7Note that when during the search process consumers learn more about the underlying price distributionfrom which prices are sampled, the reservation prices can also be increasing over time. See, e.g., Chou andTalmain (1993) and Janssen et al. (2011).

12

Figure 2: Reservation prices when N = 4 and prices are uniformly distributed over [0, 1].

N-1

N-2

N-3 - first step

perfect recall

0.30 0.35 0.40 0.45

0.30

0.32

0.34

0.36

0.38

ρk

psk−1

Parameters: N = 4, b = 0.04, c = 0.05.

The figure also illustrates that all reservation price functions are non-decreasing in psk

(theorem 1) , and that the sequence of reservation prices is non-decreasing in the number of

firms left and strictly increasing for all prices above p.

In Table 1, we show for the same uniform distribution of prices and for four firms how the

reservation prices after observing high prices before depend on c and b. Reservation prices

increase in both costs. The impact of the re-visit cost b is largest when c is small. Moreover,

one can observe that the difference to the perfect recall case can be substantial.

Table 1: Reservation Prices for Different Values of c and b, p ∼ U [0, 1], N = 4.

ρ1(∞) ρ2(∞) ρ3(∞) ρpr

c / b 0.02 0.05 0.1 0.02 0.05 0.1 0.02 0.05 0.1 –0.02 0.219 0.240 0.267 0.231 0.264 0.304 0.263 0.324 0.390 0.2000.05 0.325 0.336 0.351 0.332 0.352 0.378 0.354 0.397 0.448 0.3160.1 0.451 0.456 0.463 0.456 0.466 0.481 0.470 0.498 0.532 0.447

13

Next, we show two limit results, namely for the case where the cost of re-visits equals

zero and for the case where there are infinitely many firms.

Proposition 2. For any k and psk−1:

limb→0

ρk(psk−1) = min(ps

k−1, ρpr).

If the cost of revisits becomes arbitrarily small, there is essentially no difference for the

consumer between buying at the current price under review and buying at the smallest price

sampled so far (as previous offers can be retrieved incurring only an arbitrarily small cost).

Proposition 2 implies that if psk−1 < ρpr the consumer will always stop searching (as she has

already observed a low enough price) and will only buy at the current price if pk < psk−1. The

first term in the minimum operator only has an effect if consumer’s behaviour in previous

search rounds was suboptimal. If the consumer behaved optimally in all previous search

rounds, all the observed prices are larger than ρpr and, thus, she always has to compare the

current offer with ρpr.

Consider then the second limiting case where there are infinitely many prices to sample.

Here, the time dependency of the reservation prices disappears due to the fact that now the

cost of continuing to search is independent of time so that ρk(psk−1) = ρk+1(p

sk).

Proposition 3. Let K ∈ N. Then for any p ≥ p limN→∞ ρK(p) = ρpr.

Thus, with infinitely many firms the reservation price under costly re-visits is exactly

identical to the case where consumers have perfect recall (see, also e.g., Landsberger and

Peleg (1977), who obtain a similar result in their analysis of the case where a decision maker

has “uncertain recall”).

We next show that rational consumers never use the option of going back to previously

sampled stores, unless they have visited every store available. This result is used in Proposi-

tion 5 where threshold values for the return costs are determined for which the costly revisit

optimal search rule coincides with the no recall search rule. The result is also of use in

14

characterizing the demand function in the analysis of the Wolinsky model if more than two

firms are considered (which is not the case in the next section).

Proposition 4. Assume the consumer behaved optimally on all steps 1 ≤ k ≤ K. Then if

K < N, it is never optimal for this consumer to go back.

The numerical example presented above already incorporated the feature discussed in

Proposition 4: it can be rational to accept a price in a future period even if a lower price has

been observed in the past, but the reverse cannot be true. Thus, in the numerical example,

if a consumer faces, a price of 0.34 in the first period she decides to continue searching, but

if the third price the consumer encounters is 0.35 it is optimal for her to stop and buy at

that price (instead of going back to a previously observed price).

This proposition also has an implication for the empirical results reported in De Los San-

tos et al. (2011). Using data on web browsing, they find that actual search behaviour is not

consistent with the standard sequential search model as consumers do go back to previously

sampled firms even though they have not yet sampled all firms. The above proposition im-

plies that allowing for costly re-visits does not “rescue” the sequential search model as the

observed behaviour remains inconsistent with rational sequential search.

Proposition 4 helps us to compare our result with the results obtained in the no recall

literature (see, for example Lippman and McCall (1976)). Similar to the no recall case,

Proposition 1 already showed that the reservation prices are increasing over time, although

they are not dependent on the search history as they do in our case. We now compare our

results in more detail to the no recall scenario. To make the comparison tight, we consider

a scenario where consumers search optimally in each past search round. The following

proposition holds.

Proposition 5. Suppose, a consumer has searched optimally in all past search rounds up to

round k. The optimal search rule in round k is then identical to the no recall search rule, if

and only if, b ≥ b∗∗ ≡ p−maxpsN−2

ρN−1(psN−2).

Thus, if the re-visit cost b is larger than a threshold value, the optimal search rule under

15

costly recall coincides with the no recall case. If b is smaller than the threshold value, there

is a search history that is consistent with optimal past search behaviour until round k and

a price observation pk in round k, such that he buys in round k when there is no recall,

whereas he continues to search when there is costly recall.

3 The Wolinsky model with costly re-visits

After having defined the optimal search behaviour of consumers it is natural to look at the

equilibrium implications of this behaviour and ask whether costly re-visits imply different

firm behaviour in equilibrium. In this section, we study the Wolinsky (1986) model as a

prominent example of a model with true search where some consumers search beyond the

first firm. The next section deals with the Stahl model as an example of a search model

where no consumer with positive search cost searches beyond the first firm.

We make several simplifications to the model of Wolinsky (1986) in order to focus on

our main point – the analytic analysis of the implications of costly re-visits on equilibrium

outcomes. We analyze a duopoly market where firms have identical production costs which

are normalized to zero. There is a unit mass of consumers and each consumer buys at most

one unit of the product.8 Consumers maximize their surplus vi− pi, where vi is the (match)

value of a consumer to brand i. The vi’s are realizations of independent uniformly distributed

variables with support [0, 1]. Consumers are not informed about the prices and match values

before they search a particular firm. As explained in the previous section, a new search

costs c and it costs b to return to a previously sampled firm. We look for the symmetric

equilibrium of the model, where all firms charge price p∗. Note that if c is large enough, no

consumer will ever want to continue to search beyond the first firm and therefore the option

of coming back also does not play a role. In this case firms have monopoly power and charge

a price p = 1/2. The expected utility of searching in this case equals 1/8 implying that we

have to consider c < 1/8 for the option to come back at some cost having an influence on

8Derivations for the case of a general demand function are available upon request.

16

the analysis.

Even though our results of the previous section are formulated in terms of costs and

prices, they can be easily interpreted in terms of values and utilities. Let us denote by

w = v − p the reservation surplus. Using lemma 1, the consumers’ reservation price can be

defined by the following equation.

w =− c+ P(v − p∗ > max(w − b, 0))E(v − p∗|v − p∗ > max(w − b, 0))+

+ P(v − p∗ < max(w − b, 0)) max(w − b, 0).

This equation can be understood as follows. While being at the shop where the consumer

receives a utility v − p (the LHS of the expression), he should consider whether to buy now

or to continue searching at a cost c. Continue searching and coming back to the current

shop is only interesting if the current utility is larger than b. Hence, if continuing to search

the consumer should consider max(w − b, 0). Continuing to search gives then rise to two

possibilities. Either, the utility that can be obtained is larger than what the consumer can

get if he goes back to the previous shop (v − p∗ > max(w − b, 0)), or it is not. If it is, the

consumer buys at the next shop and gets an expected utility given that it is not beneficial to

go back or not to buy at all. If it is not, the consumer either goes back, or does not buy at

all. To make our analysis interesting, the cost of re-visits should not be prohibitively high.

We thus focus on the case where b < w.

We will consider Nash equilibrium as solution concept, and assume that consumers’ belief

that the second (unsampled) firm’s price equals the equilibrium price if they observe an off-

the-equlibrium path offer from the first sampled firm. After characterizing the equilibrium

we will show for which exogenous values of b and c this restriction holds, and what are the

equilibrium prices for b > w. If b < w, the optimal stopping rule can be written as

w =− c+(1− p∗ − w + b)(1− p∗ + w − b)

2

+ (p∗ + w − b)(w − b).

This quadratic equation has one economically sensible solution, namely

17

w = 1− p∗ + b−√

2b+ 2c, (3.1)

so that the reservation utility v(p) is given by:

v(p) = 1 + p− p∗ + b−√

2b+ 2c. (3.2)

We can now proceed with identifying the demand of a firm that charges a price p, while

the equilibrium price is p∗. We first write the demand functions for local deviations, namely

for p− p∗ ≤√

2c+ 2b− b.9 After we have derived the equilibrium candidate value for p∗ we

verify that it is not optimal to deviate to larger price differences.

The first part of the demand stems from consumers who visit a firm at their first search

and immediately buy there. This is the case if v1 − p1 ≥ w, or, equivalently if v1 > v(p):

D1 = 1− w − p = p∗ − p− b+√

2b+ 2c. (3.3)

A second source of demand stems from consumers who first visited the competitor, de-

cided to continue to search and then found an acceptable offer. For these consumers it

has to be the case that the utility the consumer gets at the competitor is smaller than the

reservation value, that the match value at the firm under consideration is larger than the

price it charges and that it does not pay to go back to the competitor whom the consumer

first visited, i.e., the following three inequalities have to hold: v2 < v(p∗), v1 > p and

v2 − p∗ − b < v1 − p. The left panel in Figure 3 illustrates these three restrictions.

Note, that from (3.1) it follows that p∗ + w < 1, so v2 is bounded by v(p∗) and not by

one (see Figure 3).

This gives us the following expression:

D2 = (1− p)(1 + b−√

2b+ 2c)− 1

2(1− p∗ −

√2b+ 2c)2. (3.4)

9Since max(b, c) ≤ 1, the RHS of this inequality is always positive.

18

Figure 3: Demand

D2: come from competitor D3: return from competitor

p p+ bu1 p+ w u1

p∗ + b

p∗ + w

u2

p∗

p∗ + w − b

u2

Finally, we compute demand from consumers who decided to leave the firm at the first

search, but come back after having visited the second firm. For this third source of demand

it has to be the case that v1 − p < w, v1 − p > b and v1 − p− b > v2 − p∗. The right panel

in Figure 3 illustrates these restrictions and this demand is then given by

D3 =1

2(1 + p∗ −

√2b+ 2c)(1− p∗ −

√2b+ 2c). (3.5)

Thus, total profit of the firm is given by

π = p(1 +

(−2− b+

√2b+ 2c

)p− p∗

(−2 +

√2b+ 2c+ p∗

)). (3.6)

Clearly the profit function has a unique maximum. Solving for the first-order condition

and substituting p = p∗ gives

p∗(b, c) =1

2

(−2 + x+

√4 + (2− x)2

), (3.7)

where x = −2b+√

2b+ 2c, as the only economically sensible solution. The next proposi-

19

tion shows that there are also no profitable global deviations. Thus, (3.7) gives the symmetric

equilibrium price for the duopoly model with costly re-visits.

Proposition 6. For all b, c such that

2 + b− 3√b+ c√2

− 1

2

√4 +

(2 + 2b−

√2√b+ c

)2

> 0 (3.8)

the symmetric equilibrium price of the Wolinsky duopoly model with costly re-visits is given

by (3.7).

Condition (3.8) guarantees that w > b and is satisfied for sufficiently small values of b

and c (see Figure 4 for a graphical interpretation of condition (3.8)). To understand why b

and c have to be small enough for the proposition to hold, recall that it only makes sense to

consider returning to a firm that is already visited if the cost b of doing so is smaller than the

reservation utility w. We already explained that also c has to be small enough for the re-visit

option to be considered, as a consumer first has to continue to search (i.e., leave the firm)

before she can come back. As the continuation pay-off of searching is negatively affected

by the return cost, the critical boundary of b below which a consumer may consider the

return option is negatively dependent on the search cost c. If condition (3.8) is not satisfied,

consumers never use the revisit option (“no recall” in equilibrium). It is straightforward to

verify that in this case p∗ is defined as the solution to the equation

1− 2(1− c)p∗ − p∗3 = 0.

Depending on the parameter values, the impact of b on the equilibrium price can be of a

similar magnitude as the impact of c. Note, that

∂p∗/∂b

∂p∗/∂c= 1− 2

√2b+ 2c. (3.9)

Thus, if both b and c are small, the impact of re-visit costs is of the same order of

20

Figure 4: Region of parameters b, c for which re-visits are sometimes optimal (dark gray)and a necessary condition for which “buy and search” option may be optimal (light gray).

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

0.00

0.05

0.10

0.15

0.20

c

b

magnitude as the impact of search costs. If both costs become large, the impact of return

costs becomes small.

There is, however, also an interesting qualitative difference between the impact of b and c

on the equilibrium price. It is well-known that the equilibrium price in the Wolinsky model

is increasing in the search cost parameter c. It is relatively easy to see that this result

continues to hold in our model for a given re-visit cost.10 Figure 5 shows, however, that

the equilibrium price is non-monotone in b. Equation (3.9) implies that for small b, the

equilibrium price is increasing in b. This follows from the fact that consumers only consider

10See the previous footnote and note that the derivative of p∗(b, c) with respect to x and the derivative ofx with respect to c are both positive.

21

re-visiting a firm if c < 1/8, and ∂p∗/∂c > 0. The reason is that if b increases, from initially

low levels, consumers are more willing to buy at the first firm they visit as the option value

of continuing to search becomes lower. For larger values of b, but still such that (3.8) holds,

the reverse is true. From (3.9) it is clear that ∂p∗/∂b < 0 if, and only if, 2b + 2c > 0.25.

Intuitively, the reason is the following. If a firm would only serve demand according to D1

and D2 stemming from first searchers and those that left the competitor, the optimal price

reaction to a given price of the competitor is increasing in b up to the level of b where re-

visits do not play any role. Under a uniform distribution of the match value, the demand D3

from consumers that return to the firm they searched first does not depend on price. The

reason is that their decision to leave the firm in the first instance is driven by the same price

consideration, and for those that first left, it is only match value and not price that makes

them decide to come back. This insensitivity of returning consumers gives firms an incentive

to increase their prices relative to the situation where these consumers do not play a role,

whenever b < w. When b is just equal to the critical value where recall is too expensive

and not considered anymore, this demand D3 equals 0, and this “overshooting” does not

arise anymore. As the marginal change in the overshooting effect can be quite strong (due

to the way b impacts on D3), for re-visit cost just below w, the equilibrium price negatively

depends on b.

Figure 5: Equilibrium price depending on b, c = 0.0675

0.04 0.05 0.06 0.07 0.08

0.4773

0.4774

0.4775

0.4776

0.4777

0.4778

b

p∗

22

So far, we have considered the standard setting where consumers either buy the product

or they continue to search, but they do not consider to combine these two options in which

case they “buy and search”.11 Potentially, this “buy and search” option could be interesting

for a consumer, if the price she encounters at a firm is relatively low, but the match value

could be much better, but is not low either (and the re-visit cost is relatively high). The

question that may then arise is whether the equilibrium we have analyzed so far is robust to

this enlargement of the search options of consumers.12 The answer is yes, and the reason is

the following. Suppose that at the first firm, the utility of the buying consumer equals w. If

the consumer continues to search after buying, her continuation pay-off is

−c+ P(v − p∗ > w + p)E(v − p∗ − p|v − p∗ > w + p) + (1− P(v − p > w + p))w. (3.10)

It is important to note that the price at the first firm p is a sunk cost when considering

buying another product (as the consumer only values one unit), and that therefore, the

utility at the newly visited firm v − p∗ should be compared to w+ p (v obtained at the first

firm).

A necessary condition for this to be larger than the current utility w is that P(v − p∗ >

w + p) > 0. This can be rewritten as 1− w − p− p∗ > 0. After plugging in values of w and

p∗ we get:

1 +

√b+ c√

2− 1

2

√4 +

(2 + 2b−

√2√b+ c

)2

> 0

This condition is also represented in Figure 4, and one can easily see that if the re-visit

option is considered, the option of buying and searching is not attractive to consumers.

11The interesting suggestion to consider this alternative search option was given to us by one of the referees.12There may be another question to ask, namely whether in this alternative search environment there

exists other equilibria than the equilibrium we have analyzed here. This may be an interesting question forfuture research.

23

4 The Stahl model with costly re-visits

We now turn attention to the question whether costly re-visits imply different equilibrium

behaviour of firms in the model analyzed by Stahl (1989). Recall that Stahl (1989) considers

a market where N firms produce a homogenous good and have identical production costs,

which we normalize to zero. Each firm decides upon the price at which it is going to sell the

good in the market. There are two types of consumers in the market. A fraction λ of all

consumers are “shoppers”, i.e. these consumers like shopping or have zero search costs for

other reasons. These consumers know all prices in the market and buy at the firm with the

lowest price. The remaining fraction 1 − λ of consumers is uninformed. These consumers

engage in sequential search and get their first price quotation for free. The timing in the

model is simple: first, firms simultaneously decide on price, where the strategy of firm i is

described by the cumulative distribution function Fi(p). After firms have set their prices,

uninformed consumers randomly visit one of the firms, observe the price that is set at that

firm, and then decide whether or not to continue to search to the next firm, observe the price

there and then decide whether to buy there, go back to the first firm visited and buy there

or continue to search to the third firm, and so on. Firms’ prices are fixed during this search

process, but unknown to consumers. These uninformed consumers will search optimally in

the way analyzed in section 2. Stahl (and we) concentrate on symmetric equilibria where

Fi(p) = F (p) for all i.

We start with the question whether a “Stahl-type” of pricing strategy, i.e. one where

all firms play mixed strategies with the support up to first round reservation price is indeed

an equilibrium in the model with costly re-visits. Then we proceed with the investigation

whether other types of equilibria are possible.

Our first result states that the “Stahl-type” of equilibrium is also an equilibrium in the

model with costly re-visits.

Proposition 7. There is a mixed strategy equilibrium where all firms charge prices below

the first-round reservation price, which equals the reservation price under perfect recall ρpr.

24

This result can be explained as follows. Since nobody searched beyond the first firm with

perfect recall, the upper bound of the price distribution (the worst option for consumers)

was not worse than the reservation price (value) under perfect recall. Thus, provided that

firms stick to the same strategy, all the reservation prices under costly re-visits are equal

to the reservation price with perfect recall. Therefore, none of the firms individually has a

profitable deviation.

Now, using the results from section 2 we can formally prove the idea that there are

no symmetric equilibria other than the Stahl equilibrium when we consider the natural

restriction to b < c. For other equilibria to exist it must be the case that the upper bound of

the price distribution is strictly larger than ρpr. Imposing the condition b < c, we show that

this upper bound cannot be larger than two times the upper bound under perfect recall.

Lemma 2. If b < c, then ρN−1(p)+b

ρpr < 2 for any p in the support of F (p).

As it follows from proposition 1 that ρk(p) ≤ ρN−1(p) and as ρ1 ≥ ρpr we have that

ρk(p)+bρ1

< ρN−1(p)+b

ρpr and therefore that ρk(p)+bρ1

< 2 for all k ∈ (2, N).

The claim that there are no other symmetric equilibria is proved in three consecutive

steps that are stated and proved in the Appendix. We first show that there are no equilibria

where the upper bound of the support is smaller than to rN−1; next we rule out upper

bounds in between rN−1 and rN−1 +b, and, finally, we can rule out upper bounds larger than

rN−1 + b. By ruling out all these candidates we prove the main result of this section.

Theorem 2. The unique symmetric equilibrium in the model with costly re-visits is the

equilibrium characterized in proposition 7.

5 Conclusions

Consumer search models have assumed that consumers have to pay a search cost to visit the

store in the first place, but have costless access to prices in stores they already visited. We

have argued that this assumption is often not justified and that when there are search costs

25

for visiting a store in the first place, there are also (typically smaller) costs of going back

to a store (re-visits). We have shown that without the assumption of costless re-visits, the

optimal sequential search rule is no longer characterized by a unique, stationary reservation

price. Instead, the reservation price in a particular search round is a function of the number

of firms that are not yet visited and the lowest price sampled so far.

We have studied the implications of costly re-visits for two strands of literature, one

where (under perfect recall) in the market equilibrium firms price in such a way that some

consumers do search beyond the first firm and another class where no consumer does so. In

the first class of search models, inspired by Wolinsky (1986), costly re-visits imply a change

in the equilibrium behavior of firms that is of the same order of magnitude as the impact of

a change in the search cost. We also showed that despite the more complicated search rule,

the Wolinsky (1986) model can still be analytically solved for certain specifications and that

equilibrium prices can be non-monotone in the cost associated with re-visits.

Stahl (1989) is an example of a model “without true search” in equilibrium. For this

model, we have shown that the equilibrium analysis is robust to the assumption of costly

re-visits. Our analysis shows that the equilibrium analyzed by Stahl remains an equilibrium

under the alternative assumption of costly re-visits and that, in addition, other possible

symmetric equilibrium outcomes do not exist. Even though the optimal search behaviour of

the consumers can be very complicated, firms behave in such a way that they do not charge

prices above the first search round reservation price.

Interesting next steps for future research could be the following. First, at different parts

of the analysis, we have made some assumptions and it would be interesting to relax them.

For example, it would be interesting to analyze the Wolinsky model for markets with more

than two firms. Second, it would be interesting to see whether the many applications in

industrial organization that recently have been studied from a search theoretic perspective,

such as the issue of prominence (see, e.g., Armstrong et al. (2009), Haan and Moraga-

Gonzales (2011)), asymmetric price adjustments (see, e.g., Tappata (2009)) or minimum

price guarantees (Janssen and Parakhonyak (2013)) are robust to the introduction of costly

26

re-visits.

27

Appendix: Proofs

Lemma 1. Let F (p) be a distribution of prices. Then for k = N − 1 the reservation price

ρN−1 is uniquely defined as a function of psN−2 ∈ [p, p] by

ρN−1(psN−2) = min

(ps

N−2 + b, C(psN−2), p

∗N−1

)where p∗N−1 satisfies the equation

p∗N−1 = C(p∗N−1)

Moreover, if the consumer decides to continue searching, the continuation cost of the

search, defined as the additional net expected cost of continuing to search conditional on

optimal behaviour after the search is made, equals to C(psN−1).

Proof. We consider the situation where N − 2 firms have been sampled and the consumer

has decided to make one more search. In this case, the consumer has three options: to buy

now at the newly observed price pN−1, to buy now at lowest price among the previously

sampled prices psN−2, or to continue searching. Knowing the value of min(pN−1, p

sN−2), the

last option gives an expected value of

v − C(min(pN−1, psN−2))

Let us first concentrate on the case where pN−1 ≥ psN−2. In this case the pay-off of continuing

to search does not depend on pN−1 so that the reservation price is given by the point where

the consumer is either (i) indifferent between buying now at pN−1 or buying at psN−2 (and

paying the additional cost of going back b) or (ii) indifferent between buying now at pN−1

and continue searching. In the first case ρN−1(psN−2) = ps

N−2 + b; in the second case

ρN−1(psN−2) = C(ps

N−2)

28

It is easily seen that 0 < ∂C(x)∂x

< 1. Moreover, it is easily seen that at psN−2 = p, C(ps

N−2) >

psN−2 + c > ps

N−2 + b. Hence, by continuity, for small values of psN−2 the reservation price is

given by ρN−1(psN−2) = ps

N−2 + b. For larger values of psN−2 it is ρN−1(p

sN−2) = C(ps

N−2) at

least when ρN−1(psN−2) is still larger than ps

N−2.

Let us next concentrate on the case where pN−1 ≤ psN−2. In this case the consumer

will never go back to previously sampled prices and thus the reservation price is implicitly

characterized by the price that solves

pN−1 = C(pN−1).

Because of continuity at pN−1 = psN−2, the fact that when ps

N−2 < ρN−1(psN−2) < ps

N−2 +

b, the derivative of the reservation price is strictly smaller than 1, and the fact that left

differentiability holds at pN−1 = psN−2, we should have that there is exactly one pN−1that

solves the above equation. This implies that in the region where pN−1 ≤ psN−2, ρN−1(p

sN−2)

is independent of psN−2. Thus, also in this case ρN−1(p

sN−2) is uniquely defined and non-

decreasing in psN−2.

Once price pN−1 is observed the continuation cost by definition equals to C(psN−1).

Theorem 1. The reservation price ρk(psk−1) is defined for any k and any ps

k−1 from the

support of F (p) by

ρk(psk−1) = min(ps

k−1 + b, Ck(psk−1), p

∗k)

where p∗k is defined as a solution of p = Ck(p) and Ck is defined recursively by

Ck(psk) = c+ P(pk+1 < min(ps

k + b, Ck+1(psk+1)))·

E(pk+1|pk+1 < min(psk + b, Ck+1(p

sk+1)))+

P(pk+1 ≥ min(psk + b, Ck+1(p

sk+1)))·

E(min(psk + b, Ck+1(p

sk+1))|pk+1 ≥ min(ps

k + b, Ck+1(psk+1)))

.

and CN−1 = C(psN−1).

29

Moreover, the time- and history-dependent reservation prices ρk(psk−1) are nondecreasing

in psk−1.

We show that on any step 1 < k < N −1 the reservation price as a function of the lowest

price in the sample is uniquely defined and has essentially the same shape on the step N −1.

The proof is by induction. Before we give the formal statement of the result and the proof,

we have to provide a technical result that turns out to be useful in making the induction

step. To this end, assume that y is a random variable with a continuous distribution function

F (y). Let for a given search and return cost c and b, the following function be defined

C∗(x) = P(y < min(x+ b, C∗∗(min(x, y))))·

E(y|y < min(x+ b, C∗∗(min(x, y))))+

P(y ≥ min(x+ b, C∗∗(min(x, y))))·

E(min(x+ b, C∗∗(min(x, y)))|y > min(x+ b, C∗∗(min(x, y)))) + c.

(A.1)

The function C∗(x) can be interpreted as a generalized continuation cost of additional search

given continuation cost on the next step. Then the following lemma holds.

Lemma A.1. If C∗∗(z) is a continuous function and for any z in the support of F (·) and

differentiable almost everywhere with 0 < ∂C∗∗(z)∂z

< 1, and C∗∗(p) ≥ p+c then 0 < ∂C∗(z)z

< 1

almost everywhere and C∗(p) ≥ p+ c.

Proof. Denote µ(y|x) = min(x + b, C∗∗(min(x, y))) and let α(x) be a solution of equation

α(x) = µ(α(x)|x). Note, that

∂µ(y|x)∂x

≤ 1

Thus, given that mu(y|p) ≥ p+ b, α(x) is well-defined and unique. Then,

C∗(x) = c+

∫ α(x)

p

ydF (y) +

∫ p

α(x)

µ(y|x)dx

30

Therefore,

dC∗(x)

dx= α(x)f(x)− µ(α(x)|x)f(x) +

∫ p

α(x)

∂µ(y|x)∂x

=

∫ p

α(x)

∂µ(y|x)∂x

which is less than 1, since ∂µ(y|x)∂x

≤ 1 and α(x) > p. Note, that C∗(x) is differentiable

almost everywhere, except the kink points α(x) and x0 +b = C∗∗(x0, x0), and at these points

both right and left derivatives are not greater than 1. Result C∗(p) ≥ p+ c follows from the

definition of C∗(·).

Now we are ready to prove theorem 1.

Proof. Let Ck(psk) be a continuation cost of additional search on the k-th step given realiza-

tions of (psk−1, pk) (recall that ps

k = min(psk−1, pk)). Then, given the optimal search behaviour

of the consumer, Ck(psk) is the expected payoff of two events: either the consumer buys at

the next firm to be searched (first event) or she continues to search onwards or goes back

(second event). Thus, we get that

Ck(psk) = c+ P(pk+1 < min(ps

k + b, Ck+1(psk+1)))·

· E(pk+1|pk+1 < min(psk + b, Ck+1(p

sk+1)))+

+ P(pk+1 ≥ min(psk + b, Ck+1(p

sk+1)))·

· E(min(psk + b, Ck+1(p

sk+1))|pk+1 ≥ min(ps

k + b, Ck+1(psk+1))),

(A.2)

which is a recursive definition of the cost function in the theorem.

We prove that 0 ≤ ∂Ck(psk)

∂psk

< 1. The proof is by backward induction. From lemma 1 it is

easy to see that 0 ≤ ∂CN−1(psN−1)

∂psN−1

< 1, and CN−1(p) ≥ c + p. Thus, the base of induction is

proven. We will now argue that this property also holds for any other period. For proving the

induction step we can apply lemma A.1 by substituting in equation (A.1) x = psk, y = pk+1,

C∗(x) = Ck(psk), C

∗∗(min(x, y)) = Ck+1(psk+1). Therefore, from 0 ≤ ∂Ck+1(ps

k+1)

∂psk+1

< 1 and

Ck+1(p) ≥ p + c it follows that 0 ≤ ∂Ck(psk)

∂psk

< 1 and Ck(p) ≥ p + c and thus, by induction it

follows that for any k it holds that 0 ≤ ∂Ck(psk)

∂psk

< 1 and Ck(p) ≥ p+ c.

31

The rest of the proof is straightforward. If pk ≥ psk−1, then ρk(p

sk−1) = min(ps

k−1 +

b, Ck(psk−1)), which is well-defined and unique. Moreover, it is non-decreasing in ps

k−1 since

both psk−1 + b and Ck(p

sk) are non-decreasing in ps

k−1. If, on the other hand, pk < psk−1, then

the reservation price is a solution to the equation pk = Ck(pk), which is unique since Ck(pk)

has a slope strictly smaller than 1. In this case, the reservation price does not depend on

psk−1 and is thus nondecreasing in ps

k−1.

Proposition 1. If psk = ps

k−1, then ρk+1(psk) ≥ ρk(p

sk−1), i.e., reservation prices are non-

decreasing over time. Moreover, ρk+1(psk) > ρk(p

sk−1) for all ps

k and psk−1 such that ps

k =

psk−1 > p = ρpr − b .

Proof. Note, that the reservation price essentially represents the cost of the next-best avail-

able alternative to buying now at the shop the consumer is currently visiting. If the next-

best available alternative is to go back to the lowest-priced firm in the sample before visiting

this shop, i.e., psk−1 < p the reservation price is simply independent of the periods, i.e.,

ρk+1(psk−1) = ρk(p

sk−1) = ps

k−1 + b.

Now consider the case where the next-best available alternative is to continue searching.

Let {ρk(psk−1)}N

k=1 be the sequence of the reservation price functions. Consider the following

suboptimal strategy. If on step k the consumer makes a decision to visit one more firm she

either buys at the firm she visits at step k+ 1 or continues her search but forgets about this

firm later on (thus, she never comes back to that firm). Let us denote a reservation price

under this suboptimal strategy by ρ′k(psk−1). Then ρk(p

sk−1) ≤ ρ′k(p

sk−1). On the other hand

for any psk−1 > p we get

ρ′k(psk−1) = F (ρk+1(p

sk−1))E(pk+1|pk+1 < ρk+1(p

sk−1) +

+ (1− F (ρk+1(psk−1)))ρk+1(p

sk−1) + c < ρk+1(p

sk−1)

which completes the proof.

32

Proposition 2. For any k and psk−1:

limb→0

ρk(psk−1) = min(ps

k−1, ρpr).

Proof. This proof uses geometrical logic represented in Figure 1. Consider the reservation

price

ρk(psk−1) = min(ps

k−1 + b, Ck(psk−1), p

∗k),

Note, that psk−1 + b and Ck(p

sk−1) always intersect at p = ρpr − b. Then, limb→0 p = ρpr. At

the same time since p∗k is the intersection point of Ck(psk−1) with the 45-degree line, we have

limb→0 p∗k = limb→0 p = ρpr. Thus,

limb→0

ρk(psk−1) = lim

b→0min(ps

k−1 + b, Ck(psk−1), p

∗k) = min(ps

k−1, ρpr).

Proposition 3. Let K ∈ N. Then for any p ≥ p limN→∞ ρK(p) = ρpr.

Proof. Note, that for any p ≥ p, CN−1(p) is fixed and does not depend on N . On the other

hand for any p ≥ p we have

Ck(p) = F (ρk+1(p))E(pk+1|pk+1 < ρk+1(p)) + c

+ (1− F (ρk+1(p))E(Ck+1(pk+1)|pk+1 ≥ ρk+1(p)) ≤

≤ C ′k(p) ≡ F (ρk+1(p))E(pk+1|pk+1 < ρk+1(p)) + (1− F (ρk+1(p))ρk+1(p) + c

Note, that C ′k(p) can be rewritten in the form:

C ′k(p) = C(ρk+1(p)− b)

33

Therefore, following our notation

∂C′

k

∂p=∂ρk+1

∂p(1− F (ρk+1(p))) ≤

∂C′

k+1

∂p(1− F (ρk+1(p)))

Then∂C

′

k

∂p≤

N−1∏i=K

∂C′i+1

∂p(1− F (ρi+1(p)))

As 1 − F (ρi+1(p)) < 1 for any p > p and i > K (note, that ρi+1(p) < ρi+2(p) ⇒

1− F (ρi+1(p)) > 1− F (ρi+2(p))) we get

limN→∞

∂C′K

∂p= 0.

.

Now note that from proposition 3 it follows that ρK(p) = ρpr and therefore CK(p) = ρpr.

Therefore, since C ′K(p) is a continuous function we get that for any p ≥ p,

limN→∞

C ′K(p) = ρpr.

Therefore

limN→∞

CK(p) = ρpr.

Proposition 4. Assume the consumer behaved optimally on all steps 1 ≤ k ≤ K. Then if

K < N, it is never optimal for this consumer to go back.

Proof. Note that the price pk is defined such that after visiting k stores, the consumer is

indifferent between continuing searching and going back to the lowest-priced store in the

sample so far. Therefore, at pk the reservation price ρk(pk) = pk + b. The expected costs of

continuing to search are C(pk)

34

By equating it to the best current option (pk + b) and some simplifications we have also

used in previous proofs, we get

c =

∫ pk+b

p

F (p)dp.

It follows therefore that pk does not depend on k and that (by comparing this equation

to the definition of ρpr) it is actually just equal to ρpr − b.

Thus, the option of going back is preferred to continue searching or stopping only if

psK < p. On the first step any price p1 ≤ ρpr would be accepted immediately. So, if the

consumer continued her search it must be the case that p1 > ρpr. Given ps1 > ρpr on the

second step any price p2 ≤ ρpr also would be accepted immediately. Thus, if consumer

continued her search it must be the case that p2 > ρpr. Then by induction if customer

reached step K it must be the case that for any 1 ≤ k ≤ K it was the case that pk > ρpr.

Therefore psK > ρpr > p and it is never optimal to go back, except possibly at the last step.

Proposition 5. Suppose, a consumer has searched optimally in all past search rounds up to

round k. The optimal search rule in round k is then identical to the no recall search rule, if

and only if, b ≥ b∗∗ ≡ p−maxpsN−2

ρN−1(psN−2).

Proof. If re-visits are not possible, the continuation costs are defined recursively by:

CNRN−1 = c+ Ep = c+ p−

∫ p

p

F (p)dp

and

CNRk = c+ P(pk+1 < CNR

k+1)E(pk+1|pk+1 < CNRk+1) + (1− P(pk+1 < CNR

k+1))CNRk+1.

(i) “If” statement. Note that by the definition of the maximum reservation price in the

last search round, the consumer does not have a sampled offer to which she can return.

35

Denote ρ = maxpsN−2

ρN−1(psN−2). As,

CNRN−1 = c+

∫ p

p

pf(p)dp

and

CN−1(ρ) = c+ (1− F (min(ρ+ b, p)))(min(ρ+ b, p)) +

∫ min(ρ+b,p)

p

pf(p)dp,

we have that

CNRN−1 − CN−1(ρ) =

∫ p

min(ρ+b,p)

pf(p)dp− (1− F (min(ρ+ b, p)))(min(ρ+ b, p)) =

p−(min(ρ+b, p))F (min(ρ+b, p))−(1−F (min(ρ+b, p)))(min(ρ+b, p))−∫ p

min(ρ+b,p)

F (p)dp =

p−min(ρ+ b, p)−∫ p

min(ρ+b,p)

F (p)dp =

∫ p

min(ρ+b,p)

(1− F (p))dp.

Thus, if b > b∗∗ the continuation costs of search in the last search round coincide since

max(ρ+b, p) = p. The rest of the proof is by induction and is similar to the above last round

argument. Suppose that Ck+1(p) = CNRk+1. Note, that if the consumer behaved optimally in

all previous search rounds, then, due to Proposition 4 we have min(psk + b, Ck+1(p

sk+1)) =

Ck+1(psk+1) for all ps

k, psk+1 that are consistent with optimal past search behaviour. Again,

using (A.2) we get Ck(p) = CNRk , which implies that the optimal stopping rules in round k

coincide.

(ii) “Only if” statement. Suppose b < b∗∗, and suppose for all rounds up toN−1 consumer

observed prices from [p − b, p]. Since by the definition of b∗∗ we have ρ < p − b, there is an

ε > 0 such that if consumer observes p0 = ρ+ ε, CNRN−1−CN−1(p) =

∫ p

p0(1−F (p))dp > 0. As

it is possible to construct observations for the last search round such that the continuation

cost functions are different, it is also possible to do so for previous search rounds.

36

Proposition 6. For all b, c such that

2 + b− 3√b+ c√2

− 1

2

√4 +

(2 + 2b−

√2√b+ c

)2

> 0

the symmetric equilibrium price of the Wolinsky duopoly model with costly re-visits is given

by (3.7).

Proof. In the main body of the paper preceeding the statement of the proposition, we already

showed that it is not optimal to locally deviate from the equilibrium price. In order to

complete the proof we have to check for global deviations: i.e. cases when a firm deviates

to a price p such that p − p∗ >√

2c+ 2b − b. First we consider deviations to prices which

satisfy√

2c+ 2b ≥ p−p∗ >√

2c+ 2b− b. In this case different parts of the demand function

are given by:

D1 = 0, D2 = (1− p)(1 + b−√

2b+ 2c)− 1

2(1− p∗ −

√2b+ 2c)2

D3 =1

2(1− p− b)(1− p− b+ 2p∗)

Then, if we define the deviation x ≡ p − p∗ and plug in the equilibrium value of p∗ we

get the following expression for the profit function:

π = 14

(−2− 2b+

√2√b+ c+

√4 +

(2 + 2b−

√2√b+ c

)2+ 2x

)·(

− 4− 10b− b2 − 4c+ 4√

2√b+ c+ 3

√2b√b+ c

+2

√4 +

(2 + 2b−

√2√b+ c

)2+ b

√4 +

(2 + 2b−

√2√b+ c

)2−√

2√b+ c

√4 +

(2 + 2b−

√2√b+ c

)2 − 4x+ 2√

2√b+ cx+ x2

)This is a cubic polynomial with coefficient 1

2at x3. The local minimum and maximum

of this polynomial are given by:

x1,2 =1

6

(A±

√A2 +B

),

37

where

A = 10 + 2b− 5√

2√b+ c−

√4 +

(2 + 2b−

√2√b+ c

)2

B = 12

(4b+ b2 + 2c−

√2b√b+ c− b

√4 +

(2 + 2b−

√2√b+ c

)2).

In a technical note that is available upon request we show that for all relevant values of

b and c, A is larger than or equal to 3 and that B is positive.

It follows that one of the x-s is negative, while the other one is positive. In fact, x1 =

16

(A+

√A2 +B

)> 1 as x1 >

13A and A > 3.

Therefore, for all x ∈[√

2c+ 2b− b, 1], x2 < x < x1, and thus x lies in between the

local maximum and the local minimum. Thus, the profit function π(x) is decreasing and the

maximum is attained at x =√

2c+ 2b− b. However, for this deviation profit equals to profit

function derived in the main text (by continuity) and, thus is lower than the equilibrium

profit. That is, deviations to√

2c+ 2b ≥ p− p∗ >√

2c+ 2b− b are not profitable.

Next, we look at larger deviations. The demand functions are:

D1 = 0, D2 =1

2(1− p)(1 + 2p∗ + 2b− p)

D3 =1

2(1− p− b)(1− p+ b+ 2p∗)

Again, define x ≡ p−p∗ and now consider x >√

2c+ 2b. The profit function in this case

is given by:

38

π =1

4

(−2− 2b+

√2√b+ c+

√4 +

(2 + 2b−

√2√b+ c

)2

+ 2x

)(− 4− 8b− b2 − 2c+ 4

√2√b+ c+ 3

√2b√b+ c+

2

√4 +

(2 + 2b−

√2√b+ c

)2

+ b

√4 +

(2 + 2b−

√2√b+ c

)2

−

√2√b+ c

√4 +

(2 + 2b−

√2√b+ c

)2

− 4x+ 2x2

)Again, the profit function is a cubic polynomial with a positive coefficient for x3. The

local maximum and minimum are given by

x1,2 =1

6

(C ±

√C2 +D

),

where

C = 6 + 2b−√

2√b+ c−

√4 +

(2 + 2b−

√2√b+ c

)2

D =6(4b+ b2 + 2c− 2

√2√b+ c− 3

√2b√b+ c−

−(b−

√2√b+ c

)√4 +

(2 + 2b−

√2√b+ c

)2).

In the technical note we show that profit is again decreasing in the size of the deviation

and therefore by continuity of the profit function the deviation is not profitable.

Proposition 7. There is a mixed strategy equilibrium where all firms charge prices below

the first-round reservation price, which equals the reservation price under perfect recall ρpr.

Proof. If the upper bound of the support p = ρ1, then maxp ρ1(p) = . . . = maxp ρN−1(p) =

ρpr. Therefore, the equilibrium defined in Stahl (1989) is an equilibrium if none of the firms

has a profitable deviation. The only (potentially profitable) way for firms to deviate is to

39

charge prices above ρ1. However, then this firm has a zero demand both from informed and

uninformed consumers. Therefore, a profitable deviation does not exist, and the Stahl type

of equilibrium is indeed an equilibrium.

Lemma 2. If b < c, then ρN−1(p)+b

ρpr < 2 for any p in the support of F (p).

Proof. Lemma 1 states that

ρN−1(p) = min(p+ b, C(p), p∗N−1

),

where p∗N−1 satisfies the equation

p∗N−1 = C(p∗N−1).

Note, that first, ρN−1(p) ≤ p∗N−1, and, second, p∗N−1 satisfies the equation

c+ b =

∫ p∗N−1+b

p

F (p)dp. (A.3)

The reservation price under perfect recall is defined by:

c =

∫ ρpr

p

F (p)dp. (A.4)

From (A.3) and (A.4) it follows that p∗N−1 + b < 2ρpr. Indeed, if this were not true, by

subtracting one equation from the other we get:

b =

∫ p∗N−1+b

ρpr

F (p)dp >

∫ 2ρpr

ρpr

F (p)dp ≥∫ ρpr

0

F (p)dp > c, (A.5)

which contradicts the assumption b < c. The second inequality stems from the fact that

F (p) in a non-decreasing function, the last form the definition of ρpr. Therefore p∗N−1 + b <

2ρpr and since ρN−1(p) ≤ p∗N−1 the lemma is proved.

40

Theorem 3. The unique symmetric equilibrium in the model with costly re-visits is the

equilibrium characterized in proposition 7.

Proof. To simplify notation the maximum possible reservation price in the k-th search round

is denoted by rk, i.e., rk = maxp ρk(p).

Lemma A.2 shows that there are no equilibria where the upper bound of the support is

smaller than to rN−1. Lemma A.3 shows that there are no equilibria where the upper bound

of the support is in between rN−1 and rN−1 + b. Finally, Lemma A.4 shows that there are

no equilibria where the upper bound of the support is above rN−1 + b. These three lemmas

together allow us to state that the “Stahl” equilibrium is the unique symmetric equilibrium

in the model with costly re-visits.

Lemma A.2. There is no equilibrium price distribution with r1 < p < rN−1.

Proof. It is easy to see that given the optimal search behavior all reservation prices are below

or equal to the upper bound of the support of the distribution. Indeed, suppose p < rN−1.

Recall, that

c+ b =

∫ rN−1+b

p

F (p)dp

then

c =

∫ rN−1

p

F (p)dp

and therefore rN−1 = ρpr, which is not possible.

Lemma A.3. There is no equilibrium price distribution with p ∈ [rN−1, rN−1 + b].

Proof. First, consider profits at r1 and at p:

π(r1) = λ(1− F (r1))N−1r1 +

1− λ

NSr1

41

and

π(p) = Y p

It is clear, that S ≥ 2−F (r1). If firm charges p > rN−1 it only sells something, if all other

firms set prices at least above r1 (otherwise all consumers stop on the first step). Therefore,

Y < (1− F (r1))N−1 < (1− F (r1)). Then it should be that

1− λ

N(2− F (r1))r1 <

1− λ

N(1− F (r1))p ≤

1− λ

N(1− F (r1))(rN−1 + b)

and therefore rN−1+b

r1> 2 which contradicts lemma 2.

Thus, the proposition is proved.

Lemma A.4. There is no equilibrium price distribution with p > rN−1 + b.

Proof. Let π0 be the equilibrium profits. First, consider the profits of a firm that charges

p. As, by construction, p is in the support of the equilibrium price distribution, equilibrium

profits are given by:

π0 =1− λ

N(1− F (p− b))N−1p (A.6)

As p > rN−1 + b, a firm charging p does not get any informed consumers and only those

uninformed consumers who have first visited all other firms, have observed these firms charge

prices above rN−1 and then do not want to go back to these stores because of the cost of a

second visit b. If a firm would charge p− b instead, its profits would be at least equal to

(λ(1− F (p− b))N−1 +

1− λ

N(1− F (p− 2b))N−1

)(p− b)

which is larger than or equal to

(λ(1− F (p− b))N−1 +

1− λ

N(1− F (p− b))N−1

)(p− b).

42

Whether or not p− b is in the support of the equilibrium price distribution, it should be

the case that π0 is larger than or equal to this expression, yielding

p ≤ 1− λ+ λN

λNb (A.7)

Therefore,

π0 < φ(λ,N) ≡ (1− λ)1− λ+ λN

λN2b (A.8)

this is the upper bound on the equilibrium profit. Next, we will construct a lower bound

on the equilibrium profit. To this end, consider profits at r1. It is easy to see that r1 should

be in the support of the equilibrium price distribution. Firstly, by definition of r1 it cannot

be the case that the whole price distribution lies above r1. Secondly, if there is part (or

whole) of probability distribution which lies below r1, then there is profitable deviation from

the largest price in this part of the support to r1, since demand does not change between

this to points. To simplify notation, let F (r1) = m. We then have that

π0 = λ(1−m)N−1r1 +1− λ

NSr1,

where S ≥ 1 is the total probability that a consumer buys from the firm, arising form all

possible search paths of consumers. The firm charging r1 gets at least 1/N consumers who

randomly arrive at its store in the first search round and N−1N

1N−1

(1−m) of consumers who

first visit another store, observe a price strictly larger than r1 and then randomly visits the

store under consideration. thus, it follows that S ≥ 2−m. Therefore, for any p ≤ r0 in the

equilibrium support:

π0 = λ(1− F (p))N−1p+1− λ

NSp,

which gives,

43

F (p) = 1−(π0

pλ− 1− λ

NλS

) 1N−1

and

p(r1) =Nλ(1−m)N−1 + (1− λ)S

Nλ+ (1− λ)Sr1.

Now consider a family of probability distributions:

F (p;K) = 1−(π0

pλ− 1− λ

NλS

) 1K−1

Then for M ≥ K F (p,M) ≤ F (p,K) for every p. Moreover, if we define r1(K) as

∫ r1(K)+b

p(r1(K))

F (p;K)dp = c+ b,

then we get that the solution of this equation r1(K) is an increasing function ofK, because

p(r1(K)) is linearly increasing in r1(K) with slope less than 1 and F (p,K) is decreasing in K.

Therefore, r1(2) ≤ r1(K) for any K. It is also clear that r1(K) is increasing in c, therefore,

r1(2)|c=b ≤ r1(2)|c>b. Let’s denote r∗ = r1(2)|c=b. It follows that r∗ is implicitly defined by

∫ r∗+b

p(r∗)

F (p, 2)dp = 2b

and therefore

∫ r∗

p(r∗)

F (p, 2)dp ≥ b

or

∫ r∗

p(r∗)

(1− π0

pλ+

1− λ

NλS

)dp =

(1 +

1− λ

NλS

)(r∗ − p(r∗))− π0

λln

r∗

p(r∗)≥ b.

As r∗ ≤ r1 for any N, b, c and fixed S,m it follows that

44

π0 ≥ λ(1−m)N−1r∗ +1− λ

NSr∗. (A.9)

By plugging in the expressions for p(r∗) and this lower bound on π0 we get

(1 +

1− λ

NλS

)(r∗ − p(r∗)) =

λN + (1− λ)S

λN

Nλ(1− (1−m)N−1)

λN + (1− λ)Sr∗ = (1− (1−m)N−1)r∗

π0

λln

r∗

p(r∗)≥ r∗

λ

(λ(1−m)N−1 +

1− λ

NS

)ln

Nλ+ (1− λ)S

(1−m)N−1Nλ+ (1− λ)S

which gives a lower bound for r∗:

r∗ ≥ λb

λ(1− (1−m)N−1)−(λ(1−m)N−1 + 1−λ

NS)ln Nλ+(1−λ)S

(1−m)N−1Nλ+(1−λ)S

.

Therefore π0 ≥ ψ0(λ,m,N, S) where

ψ0(λ,m,N, S) ≡λ(λ(1−m)N−1 + 1−λ

NS)b

λ(1− (1−m)N−1)−(λ(1−m)N−1 + 1−λ

NS)ln Nλ+(1−λ)S

(1−m)N−1Nλ+(1−λ)S

.

This is the lower bound on equilibrium profits. It is straightforward to verify that ∂∂Sψ0(λ,m,N, S) >

0 and because S ≥ 2−m we get that

π0 ≥ ψ0(λ,m,N, S) > ψ(λ,m,N) ≡ ψ0(λ,m,N, 2−m).

Now, since π0 < φ(λ,N) and π0 > ψ(λ,m,N) the equilibrium can only exist if the lower

bound on profits is smaller than the upper bound, or ξ(λ,m,N) ≡ φ(λ,N)−ψ(λ,m,N) > 0.

It is possible to verify that ψ(λ,m,N) is decreasing function of m and that

limm→1

1

(1− λ)b· ξ(λ,m,N) =

1− λ+ λN

λN2− λ

Nλ− (1− λ) ln 1−λ+Nλ1−λ

. (A.10)

45

Therefore ξ(λ,m,N) > 0 only if the denominator of the second fraction in (A.10) is

negative, which is equivalent to

lnNλ+ 1− λ

1− λ>

λN

1− λ, (A.11)

or, the denominator is positive, but the expression still holds, which is equivalent to

lnNλ+ 1− λ

1− λ<

λN

1− λ+Nλ. (A.12)

Let us start with (A.11). It is clear that at λ = 0 both the right hand side and the left

hand side of (A.11) are equal to 0. However,

∂

∂λ

(lnNλ+ 1− λ

1− λ− λN

1− λ

)= − λN2

(1− λ)2(1− λ+Nλ)< 0

Thus, the left hand side of (A.11) increases slower than the right hand side, and thus

(A.11) can never hold.

Now we proceed with (A.12). Again, at λ = 0 both the right hand side and the left hand

side of (A.12) equal to 0. If we take the derivative of the difference again we get

∂

∂λ

(lnNλ+ 1− λ

1− λ− λN

1− λ+Nλ

)=

λN2

(1− λ)(1− λ+Nλ)2> 0.

Therefore, the left hand side of (A.12) increases faster than the right hand side, and so

(A.12) cannot hold either. Therefore, there is no equilibrium with p > rN−1.

46

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