Searching for a bargain

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Searching for a bargain

M.J.W. van Megen

University of Groningen

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Research Master thesis

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Searching for a bargain

By M.J.W. van Megen

Abstract

In this paper, we consider bargaining off posted prices in retail stores. We model infinitely many firms selling differentiated products. Some-times, when consumers refuse to pay the posted price, they obtain a second offer based on the information their refusal reveals. With the prospect of a better deal, refusing the posted price becomes a more at-tractive strategy for many consumers. Consequently, firms are forced to lower prices and profits drop. Consumers, on the other hand, do benefit with, aside from lower prices, a better allocation of goods. Due to the increase in search, however, welfare overall turns out lower.

1 Introduction

Although a common occurrence in non-Western countries, bargaining in stores is rarely observed in Western countries. This difference may be put aside as a cultural difference. However, public commentators regularly refer to evidence that bargaining off the posted price is in fact possible in stores such as jewellers, shoe shops, travel agents, furniture stores and electronics stores (see e.g. The Guardian, 2008, Time, 2013, The New York Times, 2013). One explanation as to why haggling in stores is still a rarely seen sight seems to be that consumers are largely unaware that there is even a chance they might succeed.

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with the simple instruction to ask for a discount. A discount was offered approximately 40% of the time on products ranging from a backpack to a bottle of perfume to a surveillance camera for babies. Conditional on getting a discount, the average size of the discount was noted to be approximately 25 EUR, and the average discount percentage almost 10%. The experiment clearly shows that retail stores do not necessarily commit to their prices.

In this paper, we model markets in which firms are unable to fully commit to their posted prices. Consumers engage in sequential search and firms sell differentiated products which closely relates the model to that of Anderson and Renault (1999). Firms set posted prices, only observed by consumers upon visitation. The lack of commitment to that price features in the fact that consumers who reject the posted price will sometimes be made a sec-ond offer by the seller. Consumers incur search costs in order to learn about posted prices and their personal valuations for the products. Moreover, con-sumers are aware to what extent firms are committed to their posted price. Rejecting the posted price can therefore be seen as a bargaining strategy. We derive the effects of the integration of bargaining in this framework on prices, profits, consumer welfare, and welfare overall.

We find in this paper that with the firms’ lack of commitment to their prices, posted prices are lower than if firms would have take-it-or-leave-it prices. The intuition behind this result is that the bargaining process is favourable for consumers. It is more favourable in the sense that it al-lows them to be more critical either to find a product they highly value or to obtain a second offer. Indeed, on average, goods are better allocated among consumers which, in combination with lower prices, results in higher consumer surplus. The increase of search, however, outweighs the better allocation of goods and therefore total welfare turns out lower.

In the past, researchers have in the context of bargaining in consumer markets mainly considered the choice between committing to posted prices and allowing for bargaining without posted prices (e.g., Bester, 1993, Wang, 1995, Arnold and Lippman, 1998, Camera and Delacroix, 2004). More re-cently, a small but growing literature considers bargaining off posted prices which is where this paper clearly fits in.

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2009, Gill and Thanassoulis, 2016); a(n) (exogenous) probability of suc-cessful bargaining (King and Patras, 2014, Gill and Thanassoulis, 2016); consumers who non-sequentially search for second quotes (Gill and Thanas-soulis, 2009). In our model, consumers can credibly convey their rejection of the posted price and some do so strategically in the hope of obtaining a second offer.

The focus of these papers also differs from ours. Desai and Purohit (2004) ask the question whether retailers can gain a strategic advantage by permitting bargaining. They find that, depending on the parameters and in particular the proportion of haggling consumers, retailers find themselves in a prisoners’ dilemma either where they end up with a fixed price policy where symmetric haggling policies would have earned higher profits, or vice versa. Different from Desai and Purohit (2004), we consider in this paper a non-binary bargaining policy. That is, firms deal with a frequency of how often they engage in bargaining with consumers rather than whether to permit it or not.

Gill and Thanassoulis (2009) study an oligopoly competing in Cournot-fashion to sell a homogeneous good. Consumers have individual valuations for the good. Moreover, the group of consumers is split into two groups: price takers and bargainers. Bargainers attempt to extract multiple second quotes from a number of firms while price takers submit to the Cournot market list price. The authors find that an increase in the proportion of bargainers increases both the list price and the lowest price offered to bar-gainers. The part of the model that concerns itself with the bargaining con-sumers closely relates to Burdett and Judd’s (1983) model of non-sequential search. Our framework, on the other hand, is largely based on Anderson and Renault’s (1999) model of sequential search. Furthermore, Gill and Thanas-soulis (2009) are interested in the effects of an increase in the proportion of bargainers in the market on prices, while we focus on effects on search and welfare when all consumers have a chance of getting a second quote.

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Gill and Thanassoulis (2016), similar to the bargaining process we study, include a probability of successful bargaining. The process differs in that consumers do not have to abandon the list price first. When the two firms, each on one end of the Hotelling line (Hotelling, 1929), compete, first in list prices, then in discount prices, the authors find that a misallocation of goods causes welfare to increasingly decline with an increasing proportion of bargainers relative to price takers. In addition, the authors find that a higher proportion of bargainers leads to both higher list prices as well as higher discount prices. An important difference with our paper is that we consider consumers who each time they visit a firm weigh the option in front of them with an uncertain outside option, while Gill and Thanassoulis (2016) consider completely informed consumers.

The remainder of the paper is organized as follows. In Section 2, we set out the model. In Section 3, we derive results for the benchmark model where bargaining is excluded. We then analyze the full model in Section 4. In Section 5, we derive comparative statics. Section 6 considers an endogenous probability of obtaining a second offer. Robustness is discussed in Section 7. Finally, Section 8 concludes.

2 The model

The framework in which we operate is largely based on Anderson and Renault (1999). There are infinitely many† single-product firms whose marginal production costs equal zero. Products are horizontally differen-tiated and sold to a unit mass of consumers. A consumer incurs search costs s when a firm is visited. Search is sequential. Consumer j buying a product at firm i at price pi obtains utility

uij = v + εij − pi.

v is the baseline valuation for any consumer buying any firm’s product. It is assumed, for ease of analysis, to be sufficiently high such that the consumer will always buy in equilibrium. εij is consumer j’s additional valuation for

product i and can be interpreted as his match value for that product. This match value is consumer j’s private information. It is common knowledge that εij is independently and identically distributed across consumers and

firms with distribution function F (ε). In particular, unless explicitly stated otherwise, we will assume match values are uniformly distributed on the

†

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[0, 1] interval. The robustness of the results to this simplifying assumption will be discussed in Section 7.1.

In our model, we assume that consumers are able to credibly convey their rejection of the posted price p they observe upon visiting a firm. With an exogenously given probability γ, sellers pick up on rejecting consumers and make them a second offer; bargaining price pb. Both the consumers

and the firms are aware of this probability γ. Consumers who either reject the posted price and do not obtain a second offer or reject the bargaining price will visit the next firm, incurring search costs s. The credibility of rejection of the posted price, we assume, stems from the psychological costs associated with the shame of buying at the posted price after walking away but not obtaining a second offer. Without this assumption, consumers would always first attempt to obtain a second offer and only then consider the posted price. Thus, firms would gain no additional information about a consumer’s match value upon observing his rejection and the second offer would therefore not be lower than the first offer. Note that we have not included the aforementioned psychological costs in the model as it is merely a justification for the assumption and it plays no role beyond that.

The timing of the game as played by a single consumer is as follows: t=1: The exogenously given probability γ becomes common knowledge. t=2: Firms set their posted prices, though not yet observed by the

con-sumer.

t=3: The consumer randomly visits one of the firms allowing him to observe his personal match value at that firm and the posted price the firm has set.

t=4: The consumer can either accept or reject the posted price. If he ac-cepts, the game ends.

t=5: If the consumer rejects, then with probability 1 − γ the consumer does not obtain a second offer he pays search costs s and visits the next firm. With probability γ, the firm makes the consumer a second offer. t=6: In case a second offer was obtained, the consumer can choose to either

accept or reject the bargaining price. If he accepts, the game ends. t=7: If the consumer rejects the bargaining price as well, he pays search

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Note that after paying search costs s, effectively, the game resumes again from time t=3 onwards. As such, the game iterates until it ends.

We will analyze a benchmark model, the full model as described above, and an extension in which γ is assumed to be a decision variable in the upcoming sections. Note that in these analyses, consumer index j is omitted for ease of exposition so long as it causes no confusion.

3 Benchmark: no second offers

We first consider the benchmark case where firms never make a second offer. The timing of this game is as follows

t=1: Firms set their posted prices, though not yet observed by the con-sumer.

t=2: The consumer randomly visits one of the firms allowing him to observe his personal match value at that firm and the posted price the firm has set.

t=3: The consumer can either accept or reject the posted price. If he ac-cepts, the game ends.

t=4: If the consumer rejects, he pays search costs s and visits the next firm. Here, after paying search costs s, the game effectively resumes from time t=2 onwards. We solve by first deducing the consumer’s actions given the prices set by the firms and then infer the price that firms will actually set given consumer’s response. We look for a pure strategy equilibrium in posted prices.

Suppose we are in an equilibrium where all firms charge the same price p∗, where the asterisk henceforth indicates an equilibrium value.‡ Let any consumer’s expected utility of playing the above game be denoted by ∆. Note that because there infinitely many firms, ∆ does not depend on the number of firms the consumer has already visited. Suppose the consumer visits firm i, observes utility§ εi − p, and weighs his options, knowing that

walking away yields ∆−s. The consumer is indifferent when the two options

‡

The asterisks will be suppressed in the remainder for ease of exposition as long as it does not cause any confusion.

§

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yield an equal amount of (expected) utility and we will call the corresponding match value the consumer’s reservation value, denoted by ˆε. We thus have

ˆ

ε = p + ∆ − s. (1)

With this reservation value, the consumer, a priori, visiting the first firm, expects utility ∆ = Z εˆ 0 (∆ − s)dF (ε) + Z 1 ˆ ε (ε − p)dF (ε),

easily solved (by assumption of standard uniformly distributed ε) by ∆ = 1 + ˆε

2 − p −

ˆ ε

1 − ˆεs. (2)

Substituting (2) into (1), we find ˆ

ε = 1 −√2s.

Now suppose all firms but firm i set price p. With δ ≡ pi− p, given that

i is visited k-th, the probability the consumer buys from i is F (ˆε)k−1(1 − F (ˆε + δ)),

i.e. the probability the consumer has not bought at any of the earlier visited firms times the probability that he will buy at firm i. Firm i’s expected profit function is thus given by

Πi(pi) = lim N →∞ 1 N N X k=0 F (ˆε)k(1 − F (ˆε + δ))pi = lim N →∞ 1 N 1 − F (ˆε + δ) 1 − F (ˆε) pi.

Taking the first-order condition and imposing symmetry yields equilibrium price and profit (now denoted with clear reference to this section)

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4 Introducing second offers

We now consider the full model described in Section 2. We solve by first deducing the bargaining price given γ and the posted price using the in-formation about a consumer’s match value revealed by his rejection of the posted price. Then, we continue by deducing the consumer’s actions given his match value, γ, the posted price, and the inferred bargaining price. Fi-nally, we infer the posted price given γ and the consumer’s response. We look for a pure strategy equilibrium in posted prices.

Suppose again we are in an equilibrium where all firms charge the same price p∗. Suppose the consumer visits firm i and rejects the posted price. The firm can infer from the rejection that the consumer’s match value is below a reservation value, say ¯ε. Hence, firm i now knows that εi < ¯ε.

Furthermore, suppose the seller has come into a position where he can make a second offer to the consumer. The consumer will accept if his utility from buying at the bargaining price, εi − pb, exceeds his expected utility from

going to the next firm, ∆ − s. (Note that ∆ still denotes the consumer’s expected utility of playing the game, but obviously, the game now differs from the previous section which, in turn, changes the ‘value’ of ∆.) Thus, from the seller’s point of view, the probability that the consumer will accept the second offer equals

P r(εi > pb+ ∆ − s|εi< ¯ε) = (¯ε − pb− ∆ + s)/¯ε.

Firm i’s expected profit from this consumer is thus pb(¯ε − pb− ∆ + s)/¯ε.

Taking the first-order condition yields

pb= (¯ε − ∆ + s)/2. (3)

It follows that the consumer will accept this bargaining price if εi > (¯ε +

∆ − s)/2. He thus also has a reservation value in case he obtains a second offer which we denote by

¯ε = (¯ε + ∆ − s)/2. (4)

Suppose now that the consumer has not yet abandoned the posted price. His options are to either accept and obtain εi− p or to reject yielding an

expected utility of γ(εi− pb) + (1 − γ)(∆ − s). Hence, the match value for

which he is indifferent is defined by the equation ¯

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Plugging in (3), we derive ¯ ε = 2p

2 − γ + ∆ − s. (5)

Furthermore, a priori, visiting the first firm, the consumer’s expected utility ∆ = Z 1 ¯ ε (ε − p)dF (ε) + γ Z ε¯ ¯ε (ε − pb)dF (ε) + (1 − γ) Z ε¯ ¯ε (∆ − s)dF (ε) + Z ¯ε 0 (∆ − s)dF (ε). (6)

Using (3), (4), (5), and (6), we can derive

∆ = 1 + s − p − s 2s − γ(1 − γ)p 2 (2 − γ)2 . Subsequently, we have pb = p 2 − γ, (7) ¯ ε = 1 + γp 2 − γ − s 2s −γ(1 − γ)p 2 (2 − γ)2 , (8) ¯ε = 1 − (1 − γ)p 2 − γ − s 2s − γ(1 − γ)p 2 (2 − γ)2 . (9)

Now suppose all firms but firm i set price p∗. Given that i would be visited k-th, the probability the consumer arrives at firm i is

[(1 − γ)(F (¯ε) − F (

¯ε)) + F (¯ε)]

k−1_{= [(1 − γ)F (¯}_{ε) + γF (}

¯ε)]

k−1_.

Since ∆, the expected utility from playing the game, is unchanged by firm i’s defection, the reservation values at firm i simply become

¯ εi= 2pi 2 − γ + ∆ − s, (10) ¯εi = (¯εi+ ∆ − s)/2 = pi 2 − γ + ∆ − s. (11)

Moreover, it is easily shown, following very similar steps as before, that pb,i =

pi

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Firm i’s profit function is now given by Πi(pi) = lim N →∞ 1 N N X k=0 [(1 − γ)F (¯ε) + γF ( ¯ε)] k × [(1 − F (¯εi))pi+ γ(F (¯εi) − F ( ¯εi))pb,i].

Using (10), (11), (12), taking the first-order condition, and imposing sym-metry, we are able to derive equilibrium price and, subsequently, profit

psec= s 2(2 − γ)4_s (4 − 3γ)(4 − 3γ2₎, (13) Πsec= lim N →∞ 1 N "s 2(2 − γ)4_{(4 − 3γ)s} (4 − 2γ − γ2₎2_{(4 − 3γ}2₎ # . (14)

5 Comparative statics

Theorem 1. (a)

psec ≤ pbench ∀γ ∈ [0, 1]; psec < pbench ∀γ ∈ (0, 1). (b)

Πsec ≤ Πbench ∀γ ∈ [0, 1]; Πsec < Πbench ∀γ ∈ (0, 1). Proof. See Appendix.

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sure they will secure a second offer. There is thus a trade-off by firms in setting their prices between a focus on two groups of consumers, separated by their respective range of match values: one that buys at the posted price and one that would only buy at the bargaining price. Clearly, this intuition also fits the result that firms earn lower profits when they are unable to commit to their prices.

What we have not yet explicitly highlighted are the properties of the boundary cases where γ = 0 or γ = 1. In either case, there is no difference in prices between the benchmark and the model with second offers. With γ = 0, this is obvious as there is then zero probability of getting a second offer and the bargaining price becomes irrelevant. With γ = 1, all consumers always go for the bargaining price and therefore, the posted price becomes irrelevant. Thus, in the boundary cases, there is just one relevant price and therefore no actual difference between the benchmark and the full model.

We now turn to the comparison of search, consumer surplus, and total welfare. Defining Λ as total expected search costs, we find for the benchmark case Λbench = lim N →∞ N X k=0 F (ˆε)k(1 − F (ˆε))ks =ps/2 − s.

For the full model, we find

Λsec= lim N →∞ N X k=0 [(1 − γ)F (¯ε) + γF ( ¯ε)] k_{[1 − F (¯}_{ε) + γ(F (¯}_{ε) − F (} ¯ε))]ks = s (4 − 3γ)(4 − 3γ2)s 2(4 − 2γ − γ2₎2 − s,

where we have used that plugging in equilibrium prices into (8), (9) yields

¯ ε = 1 − 4(1 − γ) s 2s (4 − 3γ)(4 − 3γ2₎, ¯ε = 1 − (6 − 5γ) s 2s (4 − 3γ)(4 − 3γ2₎.

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that is paid and the total expected search costs. Note that the average price paid by a unit mass of consumers is equal to expected producer surplus. For the benchmark, we have

CSbench= v + lim N →∞ N X k=0 F (ˆε)k× [(1 − F (ˆε))(E(ε|ε > ˆε) − p − ks)] = v + 1 − (2√2s − s).

Similarly for the case with second offers,

CSsec= v + lim N →∞ N X k=0 [(1 − γ)F (¯ε) + γF ( ¯ε)] k × [(1 − F (¯ε))(E(ε|ε > ¯ε) − p − ks) + γ(F (¯ε) − F ( ¯ε))(E(ε|¯ε < ε < ¯ε) − pb− ks)] = v + 1 −     s (4 − 3γ)(4 − 3γ2) (4 − 2γ − γ2₎2 + s (2 − γ)4(4 − 3γ) (4 − 2γ − γ2₎2_{(4 − 3γ}2₎  × √ 2s − s   (15) Finally, we can compute total welfare for the two models which is the sum of consumer and producer surplus. Note that total welfare is not affected by the difference in the average price that is paid by consumers as that is merely a shift between consumer and producer surplus. It quickly follows that total welfare in the benchmark case

T Wbench= v + 1 − (√2s − s), and in the model with second offers

T Wsec= v + 1 − s 2(4 − 3γ)(4 − 3γ2_)s (4 − 2γ − γ2₎2 − s ! . Theorem 2. (a)

Λsec ≥ Λbench ∀γ ∈ [0, 1]; Λsec > Λbench ∀γ ∈ (0, 1). (b)

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T Wsec ≤ T Wbench ∀γ ∈ [0, 1]; T Wsec< T Wbench ∀γ ∈ (0, 1). Proof. See Appendix.

The intuition behind these results is as follows. Since, as we explained earlier, rejecting the posted price has become cheaper, consumers can be more critical. This leads to more search and hence higher expected search costs, but also a higher match value on average. Due to the additional effect of lower prices, consumer surplus turns out higher. As consumers do not merely search for better match values but also for better prices, it is possible and it turns out that total welfare is lower: the higher expected search costs outweigh the on average higher match values.

Finally, we will shortly comment on the effects of changes in search costs s. As s increases, consumers become less critical. In turn, average prices paid increase, while average match values (of the goods bought) decrease. Thus, producers gain while consumers lose. Overall, welfare increases as s increases.

6 Extension: endogenous γ

In this section, we consider γ as a decision variable in the model. One possi-ble interpretation for an exogenous γ is that firms only sometimes pick up on consumers rejecting the posted price. We can then interpret an endogenous parameter as a way for firms to pick up on as many rejecting consumers as they want. The timing of the game as played by a single consumer is as follows:

t=1: Firms set a probability γ with which they make a second offer to consumers who reject the posted price.

t=2: Firms set their posted prices, though not yet observed by the con-sumer.

t=3: The consumer randomly visits one of the firms allowing him to observe his personal match value at that firm, the posted price the firm has set, and the probability that the consumer obtains a second offer if he decides to reject the posted price.

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t=5: If the consumer rejects, then with probability 1 − γ the consumer does not obtain a second offer he pays search costs s and visits the next firm. With probability γ, the firm makes the consumer a second offer. t=6: In case a second offer was obtained, the consumer can choose to either

accept or reject the bargaining price. If he accepts, the game ends. t=7: If the consumer rejects the bargaining price as well, he pays search

costs s and visits the next firm.

Note that after paying search costs s, effectively, the game resumes again from time t=3 onwards. Solving is done as in Section 4 except that we finish the analysis here with deducing what probability γ firms set in equilibrium. Suppose all firms but firm i set p∗ and γ∗. Instead, firm i defects to γi

and thus also reconsiders his price pi(γi). Firm i’s profit function is now

given by Πi(pi, γi) = lim N →∞ 1 N N X k=0 [(1 − γ)F (¯ε) + γF ( ¯ε)] k × [(1 − F (¯εi))pi+ γi(F (¯εi) − F ( ¯εi))pb,i], (16) where it is easily shown that, analogous to (10), (11), (12), we now have

¯ εi = 2pi 2 − γi + ∆ − s, (17) ¯εi = (¯εi+ ∆ − s)/2 = pi 2 − γi + ∆ − s, (18) pb,i = pi 2 − γi . (19)

Since we solve by backward induction, given the timing of the game, we first take the first-order condition with respect to pi. Using that all other

firms set the equilibrium price as in (13), we find

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Plugging this expression back into the profit function, one finds Πi(γi) = lim N →∞ 1 N N X k=0 [(1 − γ)F (¯ε) + γF ( ¯ε)] k × " (1 − ∆ + s) s 2(4 − 3γ)s 4 − 3γ2 − 2(4 − 3γ)s 4 − 3γ2 # (2 − γi)2 4 − 3γi = lim N →∞ 1 N N X k=0 [(1 − γ)F (¯ε) + γF ( ¯ε)] k_× 2(4 − 3γ)s 4 − 3γ2 (2 − γi)2 4 − 3γi ,

where the second equality follows when we plug in ∆. Taking the first-order condition with respect to γi and then imposing symmetry such that γi = γ

yields the condition

(2 − γ)(2 − 3γ) (4 − 3γ)2 = 0.

The condition is solved only by γ = 2/3. However, this point is a minimizer. Hence, the two boundary cases (which are effectively equal to each other) are profit maximizing and therefore indicate the equilibria:

γ∗= 0; γ∗= 1.

This result clearly reveals that firms also individually are better off com-mitting to their prices in this model regardless of what other firms choose to do. One theory is that the driving factor behind this result is that firms cannot differentiate between consumers that reject in the hope of a better offer and consumers that reject because their valuation for the product is too low. This makes the lack of commitment to prices here a tool not strong enough for firms to gain strategic advantage.

7 Discussion

In this section, we discuss robustness of the results to some key assumptions in our models.

7.1 Uniformly distributed match values

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still hold when a different distribution is assumed. Unfortunately, closed-form solutions could not be obtained when employing a general distribution. Claims about robustness of the results to this assumption can therefore not be defended.

7.2 Infinitely many firms

We have assumed in our analyses an infinite number of competing firms. Now consider the case of a limited number, say N , firms. When arriving at the N -th firm, when the consumer rejects the posted price, he either obtains a second offer or he gets nothing, since we assume that no recall is possible. At the (N − 1)-th firm, rejecting the posted price means either a second offer or another chance at playing the game. The point is that, when a consumer is at the N -th firm, he is more likely to take the posted price than when he is at the first firm as the risk of getting nothing is further away at that point. In other words, the outside option decreases as the consumer has visited more firms. The less firms there are, the lower the outside option becomes already at the first firm. Of course, the firm does not know how many firms the consumer has visited before him, which makes for a complicated analysis. Nonetheless, we conjecture that the less firms there are, the higher prices will be, and the sum of profits over all the firms per consumer would increase at the expense of consumer surplus.

We are inclined to say that for the benchmark model where there are no second offers, the mechanism is very similar and the comparatives stat-ics results derived in Section 5 sill hold. However, there is one important difference between the two models which may complicate those results. If a consumer arrives at the last firm without a chance of a second offer, he will always buy and so all consumers still buy in this model. With a chance of a second offer, some consumers with low match values may find it worthwhile to ‘gamble’ for a better offer still which may also lead them to end up with-out the product. Hence, at this point, the baseline valuation v becomes of some import. However, when v is high, no consumer would likely take the gamble. When v is low, some consumers ending up not buying the product has less of an impact on consumer welfare.

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7.3 One second offer

Why would, by the same logic of making a second offer, a firm not (some-times) make a third or a fourth offer? This process would resemble a sort of one-sided bargaining procedure where at each refusal of the consumer, the firm learns more about his hidden match value. It seems, however, there is no reason to think this would end better for firms than it did when there is only a chance of a second offer. Therefore, we conjecture our results are robust to such an extension as discussed here.

8 Conclusion

In this paper, we studied the effects of the possibility of second offers. In a model where consumers search and bargain, firms do not commit to their prices but rather occasionally offer consumers that are leaving the store a better price. With the prospect of a better deal, consumers with not too low a valuation have a better outside option as it does not necessarily include search costs. Consumers thus gain a sort of bargaining power that forces firms to lower their prices. One might argue that a critical assumption that drives this result is the firms’ inability to differ between those leaving con-sumers that hope to get a bargain and those that simply value their product too little, though it remains for future research to actually show. Con-sumers not only benefit from their gained power through lower prices, they also obtain the luxury to be more critical which leads to a better allocation of goods in the sense that, on average, consumers value the product they end up with more. Although consumers pay on average more search costs, surplus is shown to be higher. Total welfare, on the other hand, is lower as the better allocation of goods does not outweigh the additional search undertaken for it. Finally, we have also shown in this paper that firms are, independent of the choice of other firms, individually better off to commit to their prices.

9 Appendix

9.1 Proof Theorem 1(a)

We compare equilibrium prices

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psec= s (2 − γ)4 (4 − 3γ)(4 − 3γ2₎× √ 2s. Let g1: [0, 1] → R be defined by g1(x) = s (2 − x)4 (4 − 3x)(4 − 3x2₎, such that psec= g1(γ) × pbench. To show: g1(x) ≤ 1 ∀x ∈ [0, 1]; g1(x) < 1 ∀x ∈ (0, 1). (20)

These inequalities hold if and only if the function h1 : [0, 1] → R defined by

h1(x) = (4 − 3x)(4 − 3x2) − (2 − x)4

satisfies the properties

h1(x) ≥ 0 ∀x ∈ [0, 1]; h1(x) > 0 ∀x ∈ (0, 1). (21)

The function h1 is a quartic function (fourth-degree polynomial) and

there-fore continuous. It can be rewritten as

h1(x) = 20x − 36x2+ 17x3− x4.

On its domain, we find two roots at x = 0 and x = 1. Furthermore, one can easily see that the derivative evaluated at zero

h0₁(0) = 20 > 0,

which implies that for any value of x strictly between zero and one, h1(x) >

0. We have now shown (21) holds, implying (20).

9.2 Proof Theorem 1(b)

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Let g2: [0, 1] → R be defined by g2(x) = s (2 − x)4_{(4 − 3x)} (4 − 2x − x2₎2_{(4 − 3x}2₎, such that Πsec= g2(γ) × Πbench. To show: g2(x) ≤ 1 ∀x ∈ [0, 1]; g2(x) < 1 ∀x ∈ (0, 1). (22)

h2(x) = (4 − 2x − x2)2(4 − 3x2) − (2 − x)4(4 − 3x)

h2(x) ≥ 0 ∀x ∈ [0, 1]; h2(x) > 0 ∀x ∈ (0, 1). (23)

The function h2 is an sixth-degree polynomial and therefore continuous. It

can be rewritten as

h2(x) = 112x − 256x2+ 168x3− 12x4− 9x5− 3x6.

h0₂(0) = 112 > 0,

which implies that for any value of x strictly between zero and one, h2(x) >

9.3 Proof Theorem 2(a)

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such that Λsec− Λbench = (g3(γ) − 1) p s/2. Hence, to show: g3(x) ≥ 1 ∀x ∈ [0, 1]; g3(x) > 1 ∀x ∈ (0, 1). (24)

h3(x) = (4 − 2x − x2)2− (4 − 3x)(4 − 3x2)

h3(x) ≤ 0 ∀x ∈ [0, 1]; h3(x) < 0 ∀x ∈ (0, 1). (25)

The function h3 is an fourth-degree polynomial and therefore continuous. It

can be rewritten as

h3(x) = −4x + 8x2− 5x3+ x4.

h0₃(0) = −4 < 0,

which implies that for any value of x strictly between zero and one, h3(x) <

9.4 Proof Theorem 2(b)

We compare expected consumer surplus

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Hence, we need to show that the part between brackets is strictly positive for any γ except the boundary cases for which it should equal zero:

2 − s (4 − 3γ)(4 − 3γ2) (4 − 2γ − γ2₎2 − s (2 − γ)4(4 − 3γ) (4 − 2γ − γ2₎2_{(4 − 3γ}2₎ > 0 ⇐⇒    v u u t(4 − 3γ2) + s (2 − γ)4 (4 − 3γ2₎    s (4 − 3γ) (4 − 2γ − γ2₎2 < 2 ⇐⇒ 4 − 3γ2+ (2 − γ)4 < s 4(4 − 2γ − γ2₎2_{(4 − 3γ}2₎ 4 − 3γ ⇐⇒ 4(4 − 2γ − γ2₎2 _< 4(4 − 2γ − γ2)2(4 − 3γ2) 4 − 3γ ⇐⇒ 4 − 3γ 2 4 − 3γ > 1 ⇐⇒ 3γ(γ − 1) < 0 ⇐⇒ γ ∈ (0, 1).

The results for the boundary cases can be trivially shown to hold.

9.5 Proof Theorem 2(c)

We compare expected total welfare

T Wbench= v + 1 − ( √

2s − s), and in the model with second offers

T Wsec= v + 1 − s 2(4 − 3γ)(4 − 3γ2_)s (4 − 2γ − γ2₎ − s ! . Recall g3(x) = s (4 − 3x)(4 − 3x2) (4 − 2x − x2₎2 . We now have T Wsec− T Wbench= (1 − g3(γ)) √ 2s. As we have already shown

g3(x) ≥ 1 ∀x ∈ [0, 1]; g3(x) > 1 ∀x ∈ (0, 1), (26)

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References

Anderson, Simon P. and R´egis Renault (1999). Pricing, Product Diversity, and Search Costs: A Bertrand-Chamberlin-Diamond Model. The RAND Journal of Economics 30 (4), 719–735.

Arnold, Michael A. and Steven A. Lippman (1998, July). Posted Prices Versus Bargaining in Markets with Asymmetric Information. Economic Inquiry 36 (3), 450–457.

Bester, Helmut (1993). Bargaining versus Price Competition in Markets with Quality Uncertainty. The American Economic Review 83 (1), 278– 288.

Burdett, Kenneth and Kenneth L. Judd (1983). Equilibrium Price Disper-sion. Econometrica 51 (4), 955–969.

Camera, Gabriele and Alain Delacroix (2004, October). Trade mechanism selection in markets with frictions. Review of Economic Dynamics 7 (4), 851–868.

Desai, Preyas and Devavrat Purohit (2004, May). “Let Me Talk to My Man-ager”: Haggling in a Competitive Environment. Marketing Science 23 (2), 219–233.

Gill, David and John Thanassoulis (2009, August). The impact of bargain-ing on markets with price takers: Too many bargainers spoil the broth. European Economic Review 53 (6), 658–674.

Gill, David and John Thanassoulis (2016, April). Competition in Posted Prices with Stochastic Discounts. The Economic Journal .

Hotelling, Harold (1929). Stability in Competition. The Economic Jour-nal 39 (153), 41–57.

King, Stephen P. and Demitra Patras (2014). Posted prices and bargaining: the case of Monopoly. Monash Economics Working Paper 29-14, Monash University, Department of Economics.

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The Guardian (2008). ‘Can you knock a few quid off?’, 15 June. www. theguardian.com/money/2008/jun/15/shop.discounts; Accessed on 2 August 2016.

The New York Times (2013). More retailers see haggling as the price of do-ing business, 15 December. www.nytimes.com/2013/12/16/business/ more-retailers-see-haggling-as-a-price-of-doing-business. html?_r=0; Accessed on 2 August 2016.

Time (2013). 10 trick for haggling over price at any store,

18 December. http://business.time.com/2013/12/18/

10-tricks-for-haggling-over-price-at-any-store/; Accessed on 2 August 2016.