Limited Time Offer as an Invitation to Search∗

Zheng Gong† and Jin Huang‡

†University of Toronto

‡NYU Shanghai

Abstract

We present a price-directed search model to explore the idea that sellers promote lim-

ited time offer (LTO), a discount that lasts a short period of time, to invite early search.

We show that LTO leads to a socially optimal search order and higher total welfare rel-

ative to uniform pricing. For a monopolist, LTO is the most profitable pricing scheme

under mild conditions when consumers have uncertain alternatives. Competitive sellers

all use LTO in equilibrium, but they can make less profit than under uniform pricing. A

seller with a better product may charge a lower price and earn less than the competitor.

JEL classification: D43, D83, L13

Keywords: consumer search; limited time offer; price advertisements; sales techniques

∗Zheng Gong: zheng.gong16@rotman.utoronto.ca. Jin Huang: jinhuang@nyu.edu. We are grateful forhelpful comments from Heski Bar-Isaac, Guillermo Caruana, April Franco, Ignatius Horstmann, Gerard Llobet,Matthew Mitchell, Gabor Virag and Jidong Zhou. All errors are our own.

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1 Introduction

A limited time offer or flash sale is any kind of discount, gift or reward buyers can get if

they make a purchase during a short time period. Traditional media advertising (e.g., on

television or newspaper) usually accompanies a limited-time price reduction. Nowadays

with big data, online sellers can target potential consumers by sending personally tailored

limited-time coupons via email, or by showing display ads with flash discounts for products

that consumers have just been looking at. Online platforms like Amazon and Taobao hold

“lightning deals” and advertise those deals through email subscriptions and mobile apps.

Websites such as Groupon aggregate deals of the day from different local business and on-

line retailers. Black Friday is also a notable example of sellers advertising and competing in

limited time offers.

A common feature of the ubiquitous limited time offers is that they usually last a very

short period of time. It can be as short as one day or just a couple of hours. For example, in

2013, Kmart’s Black Friday advertisement states that its sale starts from 6am and lasts until

12pm in the noon. In 2014, Forever 21’s “early bird” sale on Black Friday ends at 9am for

online shoppers and 2pm for in-store purchases.1 Lightning sales on Amazon usually last

two to six hours, and on Taobao only one hour. Questions naturally arise: how do consumers

respond to the limited time offers and why do firms set such a short time for consumers to

consider the offers?

Responses to limited time offers have been documented in previous psychological and

behavioral literature. For example, Inman et al. (1997) present evidence suggesting that “re-

strictions (i.e., purchase limit, purchase precondition, or time limit) serve to accentuate deal

value and act as ‘promoters’ of promotions.” Most of the studies consider regret theory (Si-

monson, 1992) or unavailability theory (Inman et al., 1997) as a behavioral theoretical frame-

1 Sources: Kmart: https://www.nerdwallet.com/blog/shopping/black-friday/kmart-ad/. Forever 21:https://www.blackerfriday.com/forever-21/. The advertisement posters of Kmart and Forever 21 can befound in Appendix A.

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work. Some other empirical studies examine rational consumers’ searching and purchasing

behavior facing limited time offers. Using individual clickstream data on Groupon, a daily

deal website, Hu et al. (2019) find evidence supporting the postulation that subscribers visit

the webpage of the deal and invest time and effort to learn about deal’s quality, after receiv-

ing daily newsletter which covers only the price and some limited information about that

day’s featured deal.

In this paper, we provide an economic rationale behind this sales tactic and examine

whether a limited time offer could be the optimal pricing strategy for a seller in a consumer

search market. We point out that adding the time limit to discounts not only increases a

consumer’s purchase intent by creating endogenous search frictions, but also it restructures

a consumer’s search order in a manner that favors the seller who uses the tactic.

The main intuition of our paper is as follows. Before making a purchase decision, a typi-

cal consumer needs to sequentially visit sellers and discover her idiosyncratic match utility

for each product at some cost. Price information is observable to the consumer, and there-

fore she can optimally design her search strategy accordingly. As prominence is valuable

(Armstrong et al., 2009), a seller can advertise price discounts to earn a consumer’s first

visit. Besides, the seller can also announce a high buy-later price, which implies high fu-

ture search frictions on the seller’s product if the consumer takes time to investigate other

options first. A limited-time price discount and a high buy-later price together constitute a

limited time offer, inducing the consumer to investigate the product early on, and also be

more willing to purchase compared with a situation where prices are constant over time.

Our paper builds on the literature that study how firms choose observable prices to be-

come prominent in search market (Armstrong and Zhou, 2011; Armstrong, 2017; Choi et al.,

2018; Ding and Zhang, 2018; Haan et al., 2018). While most of the models in the literature

assume that each firm chooses a uniform price, the new feature of limited time offers is that

pricing is dynamic. A seller strategically announces a price plan, and consumers take both

the current price and future prices into account to optimize their search behavior. A seem-

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ingly similar sales tactic with a dynamic pricing setting has been studied by Armstrong and

Zhou (2015). They model that consumers randomly choose a firm to visit first and discover

the product’s price and match value at a cost. In contrast, on the basis of real-life situations

where consumers are exposed to massive advertisements containing price discount infor-

mation, consumers in our model can observe product prices and strategically decide on the

search order among firms. Due to different assumptions on observables, firms in our model

use limited time offers to attract new consumers to visit them first (hereafter “search invi-

tation”), while firms in Armstrong and Zhou (2015) use buy-now discounts to retain the

consumers who have randomly sampled their products (hereafter “search deterrence”).

In this paper, we consider two settings that represent different real-world examples of

limited time offers. We start with a monopoly model as our benchmark, in which a repre-

sentative buyer chooses between a monopolist’s product and an outside option. The match

values of both choices are unknown to the buyer and the buyer incurs a search cost each time

she inspects an option. The monopolist advertises his price plan which could be designed as

a limited time offer to the buyer. One such example in real life could be: after reading some

travel blogs, a consumer receives a promotional email from a travel agency, encouraging

her to check out a one-week trip to New York and book it within a day to get a discount.

Alternatively, the consumer could also plan a trip on her own, to any city she prefers. In the

benchmark model, we find that a limited time offer could induce the consumer to investi-

gate the monopolist’s product first, and such an offer strictly dominates any uniform pricing

strategy that also yields search prominence. However, it decreases consumer surplus be-

cause the discounted buy-early price is still higher than the price that a monopolist would

set under uniform pricing. From the perspective of total welfare, the equilibrium search or-

der is socially optimal: a consumer first investigates the monopolist’s product if the product

has a higher expected search value relative to the outside option, where the search value

depends approximately on the ex-ante likelihood that the product is a good match and the

value of a good match.

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We also introduce competition and consider a setting in which sellers advertise to con-

sumers at the same time. This is often the case during traditional shopping holidays like

Black Friday and Boxing Day, when multiple retailers advertise discounts which last hours

or a few days to consumers. We model a duopoly market with two sellers, each of whom

sells a different product. In the dynamic pricing framework, we find that all pure-strategy

market equilibria exhibit a unique search order: to investigate first the product with a higher

expected search value. Similar to the monopoly market, the duopoly market also achieves

the socially optimal search order.

In the duopoly model, there exist multiple pure-strategy equilibria under dynamic pric-

ing. We focus on the profit-maximizing equilibrium in which both sellers obtain their highest

profits among all possible equilibria. In that equilibrium, the prices chosen by both sellers

display a low-to-high pattern across periods, implying a market outcome of competing in

limited time offers. We find a counter-intuitive result: the seller whose product has a higher

search value earns search prominence in the competition, however, he might charge a lower

price and profit less than the non-prominent seller in equilibrium. It is interesting to find that

the search advantage does not necessarily yields a higher profit. The non-prominent seller

is essentially a monopolist to the consumer who fails to find a good match in the prominent

product, and thus the seller could possibly profit more.

To gain policy insights, we compare the profit-maximizing limited-time-offer equilib-

rium with the equilibrium under uniform pricing. When both sellers use uniform pricing,

allowing for price observability does not admit pure-strategy equilibria. Mixed-strategy

equilibria exist, in which consumer search order is not definite. We show that limited time

offers can increase total welfare because it induces a socially efficient search order: consumer

always starts investigation from the product with a higher search value in equilibrium. We

also show that under dynamic pricing, it is never the case that only the seller with a worse

product is better off while others are worse off. All other situations can arise under different

parameter regions, including a situation where all agents are (weakly) better off, or only the

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consumer is better off but both sellers profit less.

To separate the “search invitiation” mechanism from “search deterrence”, and also be

able to admit tractable characterization of the mixed-strategy equilibria, in the main mod-

els we assume that the match value of a product or the outside option is drawn according

to a certain two-point distribution.2 To relax the assumption on the distribution function,

we further extend the monopoly model to two variations. In the first extension, we still

assume that the net surplus from outside option is uncertain for the buyer and it follows

a two-point distribution, but it is private information and not observable by the monop-

olist. In the second extension, we extend the two-point distribution to a general form of

distribution functions. The two extensions show that limited time offer strategy under mild

conditions strictly dominates (and always weakly dominates) uniform pricing in inducing

search prominence, and it is also the most profitable sale tactic when the seller’s product is

good enough relative to a consumer’s outside option.

The paper unfolds as follows. In the next section, we review the related literature. Section

3 describes the monopoly model, solves for the optimal pricing scheme for the monopolist

and also compares the monopoly market equilibrium with the first-best scenario. Section 4

first introduces the duopoly model, then characterizes the market equilibrium, and at the

end of the section, we compare welfare under dynamic pricing with welfare under uniform

pricing in the classic setting. We provide two model extensions in Section 5 and conclude in

Section 6. Proofs are in the Appendix.

2 Literature Review

This paper builds on the literature on sequential consumer search, which focuses on study-

ing the impact of consumer search on price competition and other market outcomes. Seminal2 “Search deterrence” mechanism in Armstrong and Zhou (2015) essentially works through adding search

frictions into the free-recall regime. Under our setting, a consumer does not have the incentive to recall regard-less of the sampling result, which ensures that the “search deterrence” mechanism is not the driving force ofthe “limited time offer” pricing scheme in our equilibrium.

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literature including Wolinsky (1986) and Anderson and Renault (1999) develop a standard

framework in which a consumer incurs a search cost to learn the price charged by a firm as

well as her match value for the product sold by that firm. In their framework, the search pro-

cess does not follow a specific order, and thus is called random search. Different from those

papers with random consumer search, some other papers assume that firms are not visited

at random, and the order of search can be exogenous or endogenous. Among exogenous

search order papers, Arbatskaya (2007) shows that there is price dispersion among firms

selling homogeneous goods when consumer search order is exogenously determined, and

prices and profits decline in the order of search. Armstrong et al. (2009) study a search mar-

ket with prominence in which all consumers sample the prominent firm first and randomly

search among remaining firms if the prominent product is not satisfactory. Zhou (2011)

generalizes Armstrong et al. (2009) by considering a completely ordered search model with

differentiated products and obtains similar results that the prominent firms charge lower

prices but earn higher profits due to their larger market shares.

In addition to models with exogenous search order, several papers endogenize the search

order using different approaches. The seminal work Weitzman (1979) and other papers like

Doval (2018) study a general problem where an agent faces heterogeneous alternatives and

finds the optimal search sequence. Garcia and Shelegia (2018) assume that the search order

of a consumer is guided by the observational learning of previous consumers’ purchases.

Both Chen and He (2011) and Haan and Moraga-Gonzalez (2011) assume that the order in

which firms are visited is influenced by advertising. Chen and He (2011) consider online

search advertising on a search engine. At a separating equilibrium of their model, a firm

submits a higher bid for prominent placement if its product is more relevant for a given key-

word. In Haan and Moraga-Gonzalez (2011), if a firm spends more on advertising expenses,

it is more likely to be visited early on. When search cost increases, the incentive of gaining

search prominence gets stronger and thus equilibrium advertising expenses increase, which

may result in less firm profits. Athey and Ellison (2011) studies the auction game of adver-

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tisers on search platforms like Google. The advertisers are ranked according to the auction

outcome and consumers sequentially explore the advertisers until their need is met. See also

Armstrong (2017) for a comprehensive introduction of the literature on ordered search.

A particular strand of the ordered-search models focus on the price-directed search. The

key insight is that if prices are observable, firms can strategically influence the search order

through prices as it is beneficial for a firm to visited early on. As a result, firms may engage

in fierce price competition. A common and intuitive feature in such setting is that the search

cost is procompetitive, that is, a higher search cost leads to lower prices. There is a well-

recognized modelling difficulty though: allowing for price observability normally does not

yield pure-strategy equilibrium and the mixed-strategy equilibrium is hard to characterise.

Besides, there are many potential search paths that consumers can follow, and a small change

in the price of one firm may result in a very different consumer search order and market

outcomes. To tackle the problem, Armstrong and Zhou (2011) present a duopoly model in

which match values are negatively correlated across two firms, Haan et al. (2018) and Choi

et al. (2018) build in ex-ante product differentiation such that consumers observe part of

their idiosyncratic match values before searching, and Ding and Zhang (2018) assume that

match values follow a two-point probability distribution.

The biggest difference between our paper and the ones discussed above is that we explore

firm pricing and consumer search in a dynamic framework, where firms compete for promi-

nence in consumer search process not only through a low current price, but also through

high future prices. In this regard, our paper is most closely related to Armstrong and Zhou

(2015). They model a random-search process during which a firm uses a buy-now discount

or an exploding offer to retain consumers who have randomly visited it. Unlike their model,

our model features a price-directed search process in which a firm advertises limited time of-

fers to attract new consumers to visit it first. Whether or not consumers strategically choose

their search order based on observable prices constitutes the main difference between the

two models. Moreover, we obtain different results about total welfare compared with Arm-

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strong and Zhou (2015). We show a novel channel through which limited time offers can

increase total welfare because it induces a socially efficient search order, while Armstrong

and Zhou (2015) find that exploding offers and buy-now discounts lead to welfare losses.

3 A Monopoly Model

3.1 Model Settings

We consider a discrete-time model with a monopolist (seller, he) and a consumer (buyer,

she), t = 0, 1, 2 . . . .3 All agents are risk neutral and do not discount future payoffs.4 The

seller offers a product at a constant marginal cost normalized to zero. At t = 0, the seller

announces and commits to a dynamic pricing scheme over time, {pt}t=1,2,3....5 The consumer

has unit demand for the monopolist’s product (option A). The product’s match value to the

buyer is a random variable following a two-point distribution and takes the value vA > 0

(a good match) with probability θA ∈ (0, 1) and the value 0 (a bad match) with probability

1− θA. There is an outside option (option B) which yields uncertain net surplus to the buyer.

The net surplus also follows a two-point distribution: vB > 0 with probability θB ∈ (0, 1)

and 0 with probability 1− θB. The match value realizations of A and B are independent.

Starting from t = 1, at each period, the buyer can incur a search cost s to investigate

the value realization of either the seller’s product or of the outside option. She can only

investigate one alternative per period. The buyer can also freely wait until the next period

to investigate. The buyer must search before purchasing the product or taking the outside

option. Search is with free recall. The tie-breaking rule is assumed as follows: when the

3An equivalent interpretation is that the monopolist faces a unit mass of atomic consumers, whose matchvalues from the monopolist’s product and their outside options are i.i.d drawn from two independent two-point distributions.

4 We assume that the discount factor equals to one for the sake of simplicity. It is also considered to berealistic as most real-world limited time offers last a very short time. The results are qualitatively unchangedas long as the discount factor is within (0, 1).

5 We assume that the match value is privately learnt by the consumer after costly search, and therefore theseller cannot price discriminate contingent on the match value realization.

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buyer is indifferent between investigating or not investigating an option, she investigates;

when the buyer is indifferent between investigating the monopolist’s product or investigat-

ing the outside option first, she investigates the monopolist’s product first; when the buyer

is indifferent between buying and not buying, she buys.

If the buyer consumes neither the product nor the outside option, her payoff is zero. To

avoid trivial cases, we assume min{θAvA, θBvB} > s. This condition ensures that the other

option is worth investigating if the prominent option turns out to be a bad match.

3.2 The First Best

To being with, we look for the first-best search order which maximizes total welfare regard-

less of price (which is simply a transfer of surplus from consumers to firms).6 The optimal

search path could be characterized with two features: First, it is optimal for the buyer to con-

tinue searching if the previously investigated option is a bad match, due to the assumption

min{θAvA, θBvB} > s; Second, if the buyer investigates option i first and i turns out to be a

good match, the buyer will not proceed to option j, because otherwise, it is socially optimal

to start searching from option j.

Total welfare generated by the search order from A to B is:

WA→B = θAvA − s + (1− θA)(θBvB − s)

and total welfare generated by the search order from B to A is:

WB→A = θBvB − s + (1− θB)(θAvA − s)

It is immediate that WA→B > WB→A if and only if vA − sθA

> vB − sθB

. We hereafter call

vi − sθi(i ∈ {A, B}) as the expected search value of option i, and is denoted by SVi for the

6 Note that this is a special case of Weitzman (1979).

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sake of simplicity. It is socially optimal for the consumer to start searching from the option

with high expected search value to the option with low expected search value.

3.3 A Monopolist’s Optimal Pricing Scheme

In this setting, the buyer searches for at most two periods: one period for the product and

another period for the outside option. As the buyer is informed about the advertised prices,

she will only purchase the product at the lowest price.7 Lemma 1 below shows that deriving

the optimal pricing scheme of infinite periods is equivalent to deriving the optimal prices

for the first two periods.8

Lemma 1 For any pricing scheme {pt} under which the buyer does not act (choose either product

or leave the market) within t = 1 or t = 2, there exists another pricing scheme { pt} under which

the monopolist makes the same profits as under {pt}, and the buyer always acts within the first two

periods (t = 1 and t = 2).

Proof. Clearly, a monopolist will not set a pricing scheme under which the buyer never

investigates the seller’s product, because such pricing strategy is dominated by charging a

uniform price p ∈ (0, vA − sθA] at all periods. As the buyer can investigate first and wait in

the market for a lower price at no cost, she can always get search done within the first two

periods. For a pricing scheme {pt} under which the buyer investigates the seller’s product

at either the first period (ts = 1) or the second period (ts = 2), finds out that it is a good

match, and purchases the product at tp > 2 because ptp = min{pt}t≥ts , there exists a pricing

scheme { pt} which yields the same profit for the monopolist but induces the purchase at ts.

{ pt} is characterized by: pts = ptp and pt = pt (∀t 6= ts).

7 It is also because we allow for free recall and assume that there is no value discounted for future con-sumption, and therefore the consumer can investigate the product early on and buy it at a later period if sheis satisfied with the product. When the buyer discounts the future consumption, she simply purchases at theperiod which gives her highest discounted payoff.

8 We assume that the market operates for infinite periods for the purpose of endogenizing the length of timethat the low price lasts. Lemma 1 and our later analysis will show that even though a low price can potentiallylast more than one period, it is optimal for the seller to set it only for the first period.

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We will show that the two pricing schemes yield the same profit: suppose ts = 2, if the

seller sets p2 = ptp in the new pricing scheme { pt}, then the consumer will investigate the

outside option at t = 1, and be indifferent between buying the product at t = 2 and at t = tp;

If ts = 1, which means that the buyer finds out the monopolist’s product is a good match at

the first period, then the buyer will not investigate the outside option and she is indifferent

between buying the product at t = 1 and at t = tp if the seller set p1 = ptp . �

Lemma 1 shows that to obtain the optimal pricing strategy we only need to find the

optimal prices at t = 1 and t = 2: (p1, p2), and note that p1 can be different from p2 in our

dynamic setting. There are two possible search orders: the buyer investigates the product

first, or the buyer investigates the outside option first. The following lemma describes the

buyer’s search behavior after the first visit:

Lemma 2 Suppose that the buyer investigates option i first (i ∈ {A, B}), in equilibrium, the buyer

will not continue searching if option i turns out to be a good match, and thus there is no recall in

equilibrium.

The proof is provided in Appendix B. The result of no recall is due to our assumption

on the zero match value when it is a bad match. We deliberately consider a setting where a

consumer never revisits a seller so that we could isolate our “search invitation” effect from

“search deterrence” by Armstrong and Zhou (2015), as the latter one affects search behavior

through adding friction into the recall process.

Next, we compare the maximum profit of the monopolist under the two search processes.

If the monopolist wants the buyer to investigate the product at t = 2, the monopolist max-

imizes profit by setting the second-period price to vA − sθA

, which is the highest price that

induces the buyer to continue search after discovering that the outside option is a bad match

at t = 1. The buyer’s participation constraint is as follows:

θA(vA − p2)− s ≥ 0

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The optimal p2 makes the constraint binding, that is, p2 = vA − sθA≡ SVA. The price at

t = 1 can be any price such that the buyer prefers to investigate the outside option first, for

example, the same as p2. The monopolist in this case earns a profit of π = (1− θB)(θAvA −

s).

Alternatively, the monopolist can set a lower price at t = 1 to induce the buyer to in-

vestigate his product first. The monopolist aims to maximize π = θA p1 under the buyer’s

incentive compatibility constraint:

θA(vA − p1)− s + (1− θA)(θBvB − s) ≥ θBvB − s + max{0, (1− θB)[θA(vA − p2)− s]}

where the LHS is the expected surplus from searching in the order of A → B: the buyer

investigates option A first at a search cost s, and with probability θA, finds a good match

which yields a payoff (vA − p1); if option A turns out to be a bad match, the consumer

continues search at another cost s, and with probability θB, finds the outside option a good

match which yields a payoff vB. The RHS is the expected surplus from searching in the order

of B→ A.

The optimal price at t = 1 is then p1 = vA − sθA− (θBvB − s), and the second-period

price can be set at any level that deters the buyer from investigating the product at t = 2. A

sufficient condition is p2 ≥ vA − sθA

. The monopolist’s profit in equilibrium is π = θA p1. By

comparing π with π, we find that π > π if and only if SVA > SVB. The intuition is that, since

the monopolist can always extract all the incremental surplus relative to the outside option,

it is optimal for the monopolist to induce the socially efficient search order and extract the

highest surplus. The following proposition summarizes our results:

Proposition 1 When the product offers a higher expected search value than the outside option, the

optimal pricing strategy is that the seller sets a low buy-now price and a high buy-later price, such

that the buyer investigates the monopolist’s product first.

To compare our pricing strategy with the one in a standard static setting (Armstrong and

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Zhou, 2011; Ding and Zhang, 2018; Haan et al., 2018; Choi et al., 2018), we analyze a uniform

pricing scenario where the price does not change overtime. Under uniform pricing, if the

seller sets a price such that the buyer investigates his product at t = 2, he earns a profit of

π as derived before. To make the buyer investigates the monopolist’s product at t = 1, the

uniform price, denoted as pu, should satisfy the following incentive compatibility constraint

on the buyer:

θA(vA − pu)− s + (1− θA)(θBvB − s) ≥ θBvB − s + (1− θB)[θA(vA − pu)− s]

The optimal uniform price for the monopolist to induce being searched early is pu =

vA − vB + sθB− s

θAand the monopolist earns a profit of πu = θA pu.

It is easy to verify that p1 > pu and thus π > πu. It implies that the buyer purchases the

product at a higher price under dynamic pricing than uniform pricing, and for the monopo-

list, dynamic pricing strictly dominates uniform pricing. The intuition is that setting a high

buy-later price is equivalent to increasing future search cost, which makes it more effective

to induce consumer’s early visit and creates more surplus for the monopolist to extract.

We have derived that when SVA > SVB, it is optimal for the monopolist to use a limited

time offer. The condition is the same as the one under which A → B search order is socially

optimal, and besides, the buyer’s purchase decision in the monopoly market is also consis-

tent with the first-best scenario. Thus, we conclude that the monopolist’s optimal pricing

scheme achieves the first best allocation, as stated in the following corollary:

Corollary 1 The monopoly market equilibrium achieves the first-best market allocation.

4 Search Invitation in Duopoly

In the monopoly model we assume that the outside option is exogenous. Yet one natural

interpretation of the outside option is a product provided by another seller in the market.

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In this section, we extend the benchmark to a duopoly model with two strategic sellers,

offering product A and B respectively. Product A’s match value to the buyer is a random

variable that takes the value vA > 0 (a good match) with probability θA ∈ (0, 1) and the

value 0 (a bad match) with probability 1− θA. Product B’s match value to the buyer also

follows a two-point distribution and takes the value vB with probability θB ∈ (0, 1) and the

value 0 with probability 1− θB. Both sellers announce and commit to their pricing schemes

simultaneously at t = 0: {pAt } and {pB

t }. Without loss of generality, we assume that SVA ≥

SVB > 0. Other model settings remain the same as in the benchmark. Similar to Lemma 1,

the following lemma states that the equilibrium prices can simply be characterized by two

periods:

Lemma 3 For any equilibrium where the consumer purchases a product at t > 2, there exists another

equilibrium where the consumer purchases within the first two periods and all agents’ payoffs remain

the same.

A formal proof is provided in Appendix B. {(pA1∗, pA

2∗), (pB

1∗, pB

2∗)} denotes the equilib-

rium prices hereafter. Before deriving the equilibrium, we list all the five possible consumer

search paths for any given prices announced by seller A and seller B. The possible paths are:

Path I (A→ B): at t = 1, the buyer investigates product A. If product A is of match value

vA, the buyer purchases it at price pA1 ; if product A is of match value 0, the buyer continues

search to find out the match value of product B at t = 2. The buyer only purchases product

B at price pB2 if its match value turns out to be vB.9

Path II (B→ A): at t = 1, the buyer investigates product B. If product B is of match value

vB, the buyer purchases it at price pB1 ; if product B is of match value 0, the buyer continues

search to find out the match value of product A at t = 2. The buyer only purchases product

A at price pA2 if its match value turns out to be vA.

9 For any given pricing scheme, it can never be optimal that the buyer investigates product B after A re-gardless of product A’s match value. It is strictly dominated by a search strategy of investigating B first andonly investigating A if B is a bad match. More details can be found in the proof of Lemma 2 in Appendix B.

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Path III (∅ → A or ∅ → B): at t = 1, the buyer does not search. She investigates either

product A or product B at t = 2.

Path IV (A → ∅ or B → ∅): at t = 1, the buyer investigates either product A or product

B. Independent on the search result, the buyer does not search at t = 2.

Path V (∅ → ∅): the buyer never searches. It is easy to see that this path does not

constitute any equilibrium nor deviation, as this path yields zero profit for both sellers.

Equilibrium Search Path

Path III could happen if first-period prices are too high and Path IV could happen if second-

period prices are too high. However, neither of them can be the equilibrium path because the

seller of the non-searched product at the non-searched period will deviate to a lower price

to induce search. Moreover, we will prove that Path II cannot constitute an equilibrium as

described in the following proposition:

Proposition 2 (Equilibrium Search Path) When SVA > SVB, there exists no pure-strategy Nash

Equilibrium in which the buyer first investigates product B and continues investigating product A if

product B is a bad match (Path II).

A formal proof is provided in Appendix B. To sustain an equilibrium in which the con-

sumer searches a la B→ A, two incentive compatibility constraints have to hold simultane-

ously. First, there is no profitable deviation for seller B to induce the consumer to search a la

A→ B or ∅→ B. Second, it is also not profitable for seller A to undercut seller B and direct

the consumer search order to A → B or A → ∅. When SVA > SVB, the incentive for seller

A to compete for prominence is very high, and it is not that profitable for seller B to lower

price for maintaining the prominent position. The two constraints cannot hold together, and

therefore an equilibrium with B→ A does not exist.

Thus far, we have proved that when SVA > SVB, all of Path II - V cannot constitute an

equilibrium. Therefore, the only possible search path in the equilibrium is (A → B): the

16

buyer investigates product A first at t = 1, and if it turns out to be of zero match value, the

buyer investigates product B at t = 2. There are many potential equilibria that feature such

search path. Next, we show that there indeed exist such equilibria in pure (pricing) strategies

and we analytically characterize the profit maximizing equilibria among all possible equilibria.

A profit maximizing equilibrium is denoted by {(pA1∗, pA

2∗), (pB

1∗, pB

2∗), (A → B)}, in which

both sellers obtain their highest profits among all pure strategy Nash equilibria and the

buyer investigates product A first.10

Profit Maximizing Equilibria

If the price set {( pA1 , pA

2 ), ( pB1 , pB

2 )} constitutes a Nash equilibrium, then pB2 = vB − s

θBhas

to hold. The reason is that product B will only be investigated in the second period in

equilibrium, therefore for any pB2 < vB − s

θB, the seller B could deviate to pB

2 = vB − sθB

to

extract all the surplus from the buyer at t = 2. The equilibrium price pA1 needs to satisfy two

conditions: incentive compatibility constraint on seller B (ICB), and incentive compatibility

constraint on the consumer (ICC). The consumer’s ICC condition is such that the consumer

will indeed search from A to B given the prices:

θA(vA − pA1 ) + (1− θA)[max{0, θB(vB − pB

2 )− s}] ≥

θB(vB − pB1 ) + (1− θB)[max{0, θA(vA − pA

2 )− s}](ICC)

If seller B deviates, the only possible profitable deviation is to induce the consumer to

investigate B at t = 1. For the same intuition as in the benchmark, the most profitable

deviation for seller B is to set a high pB2 such that investigating product B at t = 2 is not an

option, and pB1 such that the buyer is just indifferent from investigating B and investigating

A at t = 1. As such, given ( pA1 , pA

2 ), the most profitable after-deviation price pB1 satisfies the

10 When SVA = SVB, the profit maximizing equilibria can also display B→ A search order.

17

following equation:

θB(vB − pB1 ) + (1− θB)[max{0, θA(vA − pA

2 )− s}] = θA(vA − pA1 )

From the previous equation, we can solve for the most profitable deviation price pB1 . To

ensure that seller B will not deviate, we have the ICB condition as follows after substituting

pB1 into the right-hand side:

(1− θA) pB2 ≥ vB −

θA

θB(vA − pA

1 ) +(1− θB)

θB[max{0, θA(vA − pA

2 )− s}] (ICB)

In the profit maximizing equilibria, seller A’s prices (pA∗1 , pA∗

2 ) have to satisfy both the

ICC and ICB constraints. Note that a high enough pA2 , such that pA

2 ≥ vA − sθA

, (weakly)

minimizes the RHSs of both (ICC) and (ICB) and yields the highest pA1 . Therefore, in the

profit maximizing equilibria, we have pA∗2 ≥ vA − s

θA, and pA∗

1 should be:

pA1∗ = vA −

sθA− (θBvB − s) (1)

which makes the constraint (ICB) binding. Interestingly, this is also the first-period equilib-

rium price set by the monopolist in the benchmark model (see Proposition 1).

After pinning down(

pA1∗, pB

2∗, pA∗

2), we need to find all pB

1∗ that constitute the profit

maximizing equilibria. Two constraints have to be satisfied here. The first is (ICC): it is

optimal for the consumer to search from A to B. The second is the incentive compatibility

condition on seller A (ICA): seller A does not want to induce the consumer to investigate

product A at t = 2.

Firstly, pB∗1 cannot be too low otherwise (ICC) fails: the consumer would rather follow

the search path B → ∅ rather than A → B. This condition determines the lower bound of

pB∗1 as follows:

θA(vA − pA1∗)− s ≥ θB(vB − pB

1∗)− s

18

which yields a lower bound for pB∗1 : pB

1∗ ≥ (1− θA)(vB − s

θB).

Next, we look for the upper bound of pB1∗. It is obvious that if pB

1∗ > vB− s

θB, seller A will

deviate and charge pA1 = vA − s

θA. Therefore, pB

1∗ should be smaller than vB − s

θB. Moreover,

seller A will not deviate to induce search path (B → A) where she can optimally charge

pA2 = vA − s

θA. The reason is that this deviation yields profit (1− θB)θA(vA − s

θA), which

is smaller than θA pA1∗ as long as the assumption holds: vA − s

θA≥ vB − s

θB. Therefore, the

only possible profitable deviation for seller A is to price at (pA1 > vA − s

θA, pA

2 ) to induce a

search path (∅→ A), which can be profitable if pB1∗ is high. The buyer prefers to follow the

(∅→ A) path if:

θA(vA − pA2 )− s ≥ θB(vB − pB

1∗)− s + (1− θB)(θA(vA − pA

2 )− s)

The highest pA2 to make the buyer skip the first period and only investigate product A

at t = 2 is when pA2′ = 1

θA(θAvA − vB + pB

1∗ − s + s

θB). To ensure that this deviation is not

profitable, we must have:

θA pA1∗ ≥ θA pA

2′

We then obtain that pB1∗ ≤ (1− θAθB)(vB − s

θB) is required to satisfy the incentive com-

patibility constraint on seller A. Thus far, we have found the conditions on pB1∗ and pA

2∗ to

sustain the profit maximizing equilibria in which pA1∗ and pB

2∗ yield the highest profits for

seller A and seller B. The results are summarized in the following proposition:

Proposition 3 (Profit Maximizing Equilibria) There exist profit maximizing pure strategy equi-

libria: {(pA1∗, pA

2∗), (pB

1∗, pB

2∗), (A → B)}, where pA

1∗ = vA − s

θA− (θBvB − s), pB

2∗ = vB − s

θB,

pA2∗ ∈ [vA − s

θA,+∞), pB

1∗ ∈ [(1 − θA)(vB − s

θB), (1 − θAθB)(vB − s

θB)]. In equilibrium, the

buyer first investigates product A at t = 1, and only if product A is not a good match, the buyer

investigates product B at t = 2. Both seller A and seller B obtain the highest profits in the profit

maximizing equilibria among all pure strategy Nash equilibria: πA∗ = θA(vA − sθA− θBvB + s)

19

and πB∗ = (1− θA)θB(vB − sθB). When SVA = SVB, in all profit maximizing equilibria the search

order can be either A→ B or B→ A, and both sellers make the same profit.

By comparing the transaction prices of product A and product B (pA1∗ and pB

2∗ accord-

ingly), we find a counter-intuitive result that when SVB < SVA < (1+ θB) · SVB, an advanta-

geous product is sold at a cheaper price than its competitor. The reason is that in a duopoly

price-directed search market, seller A competes hard in price to earn search prominence,

because being investigated first in equilibrium yields a higher profit than being investigate

later. Meanwhile, seller B is able to charge a monopoly price because he is essentially a

monopolist for the consumer who fails to find a good match in product A. Therefore, even

though product A enjoys a higher search value, its final price might be lower due to the

competition for search prominence. Moreover, in terms of consumer purchase likelihood

and seller’s profit, seller A might also do worse than seller B when θA < (1− θA) · θB and

θA · SVA < θB · SVB respectively. These counter-intuitive results occur especially when the

search values of product A and product B do not differ much, and/or the matching proba-

bility of product B is higher than product A.

We are also interested in analyzing the effect of search cost on the market price. It is easy

to see that the equilibrium prices for both sellers fall when the search cost rises, which is

in line with the past literature on price-directed search (Haan et al., 2018; Ding and Zhang,

2018; Choi et al., 2018). The non-prominent seller (seller B) extracts all the surplus from

the consumer’s second visit conditional on the first product being a bad match, and thus a

higher search cost reduces the surplus that seller B could extract. For the prominent seller

(seller A), the search cost affects pA∗1 directly and indirectly. The direct effect is shown in

the ”− sθA

” part. A higher search cost reduces the consumer’s total surplus from conducting

search and therefore seller A needs to reduce the pre-announced price. The indirect effect

is reflected in the ”−θB(vB − sθB)” part. The direct effect causes a lower first-period price of

product A, and thus the profit in seller B’s most profitable deviation is smaller. Therefore,

seller B is less likely to undercut seller A (ICB is less binding), which allows the equilibrium

20

price of A to be higher. Overall, the direct effect dominates the indirect effect for seller A

and the equilibrium price of product A also falls with a higher search cost.

Last but not least, we could compare the equilibrium discounts that seller A and seller

B offer, which are defined by ∆A = pA2∗ − pA

1∗ and ∆B = pB

2∗ − pB

1∗ respectively. It is

straightforward to get that ∆A ∈ [θB(vB − sθB),+∞) and ∆B ∈ [θAθB(vB − s

θB), θA(vB − s

θB)].

The discount offered by seller A is most likely to be bigger than seller B, especially if the

matching probability of product B is higher (θB > θA). This finding describes a possible real-

life situation: when facing limited time offers that only contain discount information instead

of price information (the sale price and the future price), a consumer first investigates the

more heavily discounted product.

4.1 Welfare Analysis: Limited Time Offer vs Uniform Pricing

Thus far, we have discussed the market equilibrium where strategic sellers manipulate

prices over time to earn the consumer’s early visit. In the profit-maximizing equilibrium,

both sellers offer the buyer a lower buy-early price and a higher buy-later price. One way to

interpret the results is that sellers end up competing in limited time offers. In the monopoly

model, we find that limited time offer decreases consumer surplus because the discounted

buy-early price is higher than the equilibrium price in a uniform-pricing regime. For policy

implications, we would like to address the same question here: how do limited time offer

affects total welfare and consumer surplus in a competitive market?

To answer the question, we need to solve for the market equilibrium when both sellers

adopt uniform pricing in the first two periods. First, we find that there is no pure strategy

equilibrium under uniform pricing as predicted in the previous literature.

Lemma 4 No Nash equilibrium exists in pure uniform pricing strategies.

Proof. If the pure strategy equilibrium exists, the product that the buyer investigates in the

second period must be sold at a price that extracts all the buyer’s surplus, otherwise there

21

exists a profitable deviation. Suppose that in equilibrium the buyer investigates product

A at t = 1, and thus pB = vB − sθB

. A potential deviation of seller B is to reduce price to

attract buyer’s investigation at t = 1. To do so, seller B could price lower but not below (1−

θA)(vB− sθB), otherwise seller B makes strictly lower profit compared with before deviation.

When product B is priced at (1 − θA)(vB − sθB), we solve for product A’s price which

makes the buyer indifferent between investigating A and B at t = 1. Denoted as pA, the

price must satisfy the following equation:

θA(vA − pA)− s + (1− θA){θB[vB − (1− θA)(vB −s

θB)]− s} =

θB[vB − (1− θA)(vB −s

θB)]− s + (1− θB)[θA(vA − pA)− s]

After solving the equation, we get pA = vA − sθA− θA(vB − s

θB). To ensure that seller

B charges pB = vB − sθB

and would not undercut seller A to change the buyer’s search

order, the price for product A cannot be higher than p. Otherwise, seller B has a profitable

deviation in which he undercuts A to a price higher than (1− θA)(vB − sθB) and makes a

higher expected profit.

However, pA cannot constitute an equilibrium. Seller A will deviate to a higher price

approaching vA − sθA

, given that seller B charges pB = vB − sθB

. As such, a pure-strategy

equilibrium with a search path starting from A does not exist. The same logic applies to the

proof on the nonexistence of a pure-strategy equilibrium with a search path starting from B.

To summarize, no Nash equilibrium exists in pure uniform pricing strategies. �

Having shown that there is no equilibrium in pure strategies, we proceed to show the

existence of an equilibrium in mixed strategies. Each seller’s strategy is a probability distri-

bution: FA and FB with support [pA, pA] and [pB, pB] respectively.

Suppose that the upper boundaries of both supports are the monopoly solution (vA −s

θA, vB − s

θB) and neither has a mass point at the upper boundary. Seller B can guarantee an

expected profit of θB(1− θA)(vB − sθB) by setting price to vB − s

θBand selling at t = 2. There-

22

fore, seller B would never price below (1− θA)(vB− sθB), which can attract the buyer at t = 1

and yield the same level of expected profit as what is guaranteed. The same logic applies for

seller A. Seller A would not price below (1− θB)(vA − sθA) which yields an expected profit

equal to pricing at vA − sθA

.

If pA = (1− θB)(vA − sθA) and pB = (1− θA)(vB − s

θB), the buyer investigates A first if

θA(vB − sθB) < θB(vA − s

θA), which comes from the following equation:

θA(vA − pA)− s + (1− θA)[θB(vB − pB)− s] >

θB(vB − pB)− s + (1− θB)[θA(vA − pA)− s]

In this scenario, seller A could raise price to pA = vA − sθA− θA(vB − s

θB) which is higher

than pA = (1− θB)(vA − sθA), and also gain a higher expected profit: θA pA.

Conversely, if θA(vB − sθB) > θB(vA − s

θA), the buyer strictly prefer investigating product

B first when pA = (1− θB)(vA − sθA) and pB = (1− θA)(vB − s

θB). Therefore, seller B could

raise price to pB = vB − sθB− θB(vA − s

θA) and earn a higher expected profit: θB pB. The

following proposition summarizes the properties of mixed-strategy equilibrium:

Proposition 4 When both firms have to set an uniform price over time, in the mixed-strategy equi-

librium:

1. when θA(vB − sθB) < θB(vA − s

θA), the price distribution of A has support over the interval

[vA − sθA− θA(vB − s

θB), vA − s

θA), and the equilibrium profit that seller A obtains must

equal its best alternative, namely, θA(vA− sθA− θA(vB− s

θB)). The price distribution of B has

support over the interval [(1− θA)(vB − sθB), vB − s

θB], and the equilibrium profit that seller

B obtains is θB(1− θA)(vB − sθB).

2. when θA(vB − sθB) > θB(vA − s

θA), the price distribution of A has support over the inter-

val [(1− θB)(vA − sθA), vA − s

θA], and the equilibrium profit that seller A obtains is θA(1−

θB)(vA − sθA). The price distribution of B has support over the interval [vB − s

θB− θB(vA −

23

sθA), vB − s

θB), and the equilibrium profit that seller B obtains is θB(vB − s

θB− θB(vA − s

θA)).

Thus far, we have obtained the equilibrium results under dynamic pricing and under

uniform pricing. We next discuss the welfare implications for each agent.

Total Welfare: the dynamic pricing competition ensures that the consumer search starts

from a high search-value product (product A) to a low search-value product (product B),

which is socially optimal as proved in the monopoly model. The purchase decision in equi-

librium also leads to the first best allocation. Whereas under the uniform pricing scheme,

the mixed-strategy equilibrium can sometimes be with an inefficient search order (B → A).

Therefore, allowing for competition in limited time offers (in expectation) strictly increases

total welfare in our setting. Our results are opposite to Armstrong and Zhou (2015) as they

find that exploding offers and buy-now discounts lead to welfare loss. In their model, the

welfare loss is plausibly caused by a decrease in both buy-later demand and the quality of

the match between product and consumer due to search deterrence.

Seller A: when θA(vB − sθB) > θB(vA − s

θA), or when θA(vB − s

θB) < θB(vA − s

θA) but

θA > θB, seller A’s expected profit is higher under dynamic pricing; Otherwise, seller A is

better off if the price competition is static.

Seller B: seller B is (weakly) worse off under dynamic pricing competition with all pa-

rameters. This is because, when sellers are competing in limited time offers, the product

with a lower search value will be investigated later with probability one; while in a uniform-

pricing regime, sometimes the search order reverses.

The Buyer: when θA < θB, the buyer is strictly better off under dynamic pricing. How-

ever, when θA > θB, the effect of banning limited time offers on consumer surplus is am-

biguous.

Moreover, when θA = θB, both seller A and B earn the same profit under dynamic and

static pricing, and thus the buyer is strictly better off under dynamic pricing because the

total welfare is higher. Starting from here, keep other parameters constant, (1) if θA increases

a bit and becomes slightly higher than θB, seller B’s profit remains the same, seller A earns

24

more under dynamic pricing, and the consumer surplus might decrease but would still be

higher under dynamic pricing. In this scenario, every agent becomes (weakly) better off

under dynamic pricing; (2) if θA gets higher and higher, seller B becomes worse off, and

seller A’s profit increases but perhaps at the cost of consumer surplus; (3) if θA is smaller

than θB, the total welfare and consumer surplus strictly increase but both sellers earn less

under dynamic pricing. The follow proposition summarizes our welfare analysis:

Proposition 5 (Welfare) When sellers compete in limited time offer, compared with a market under

uniform pricing, limited-time-offer equilibrium achieves the socially optimal search order and a higher

total welfare. Depending on parameters, it might be the case that all agents become better off, or only

the consumer becomes better off, or only the seller with a higher search value becomes better off.

5 Extensions

In this section, we consider two extensions of the monopoly model to examine the robust-

ness of “search invitation” mechanism. First, we investigate the situation where outside

option is private information. A consumer knows her own outside option’s value, however,

the monopolist is only informed about the value distribution. We will show that Proposi-

tion 1 still holds in this case when outside option is not observable. Second, we extend the

two-point distribution of match values in our benchmark model to a general distribution

function. We find sufficient conditions under which limited time offer is the optimal pricing

strategy.

5.1 Private Outside Options

A consumer’s outside option could be private information and not observable by firms11.

For example, in the context of the travel agency story in the introduction, consumers might11 Similar to the benchmark, the following two interpretations are equivalent: one is to assume that there is

a representative consumer with a private outside option. The outside option’s match value to the consumeris randomly drawn from a publicly known distribution. Another one is to assume that there is a unit mass of

25

be very different in their travel planning skills, but the travel agency cannot observe the

information to customize their limited time offers. To generalize our results, we consider a

case where the representative consumer’s outside option (vB, θB) (also denoted as the con-

sumer’s type) is drawn from a joint distribution with p.d.f f (v, θ), where v > 0 and θ ∈ (0, 1)

for all (v, θ) ∈ supp( f ). (vB, θB) is private information and unknown to the seller. We as-

sume that even for the worst possible draw, the outside option is worth investigating, that

is, v− sθ > 0, ∀(v, θ) ∈ supp( f ). We summarize our findings in the following proposition.

Proposition 6 (Private Outside Option) Let U = min θv, ∀(v, θ) ∈ supp( f ). Denote the set of

(v, θ) ∈ supp( f ) such that θv = U as B. We have:

1. If ∀(v, θ) ∈ supp( f ), vA − sθA≤ v− s

θ , limited time offers is not a profitable tactic compare

to using simple uniform prices.

2. If ∀(v, θ) ∈ B, vA − sθA

> v− sθ , limited time offer is strictly more profitable than uniform

pricing.

In the optimal limited time offer, the monopolist sets a high buyer-later price that extracts all search

value (p2 = vA − sθA

) and a low buy-early price p1.

A formal proof is provided in Appendix B. Proposition 6 shows that the results of our

benchmark model remain robust: it is optimal for the monopolist who has relatively better

product to use limited time offer to induce some consumer types to visit him first. The

intuition is that when the monopolist’s product is inferior to all types’ outside option in

terms of expected search value, it is too costly to compete for prominence. Therefore, the

monopolist’s optimal strategy is to set a high flat price to extract all search value p = vA− sθA

and only sell at t = 2 when the consumer finds out the outside option is a bad match at t = 1.

When the monopolist’s product is superior to some types’ outside options, it is strictly more

consumers in the market. Consumers are heterogeneous in their outside option values, which are i.i.d. drawnfrom the same distribution.

26

profitable for the monopolist to set a low price at t = 1 to induce early search, which is

similar to the argument in the Proposition 1.

5.2 General Match Value

In this section, we extend our benchmark model to incorporate general match values rather

than a two-point distribution. F(u) and G(v) denote the distributions of the consumer’s

match value to the monopolist’s product and her outside option respectively. We assume

that both f (u) and g(v) are log-concave12 and on the support [u, u] and [v, v] respectively.

Both u and v are non-negative. Similar to the benchmark, we assume∫ u

u udF(u) ≥ s and∫ vv vdG(v) ≥ s, that is, both options are worth investigating for the consumer.

Under the general match value distribution, the search deterrence mechanism in Arm-

strong and Zhou (2015) naturally arises and gives the monopolist the incentive to provide a

limited time offer. To better illustrate our search invitation mechanism, we assume that the

consumer cannot recall: once she continues search, the past option becomes unavailable to

her in the future.13 When recall is not allowed, the “search deterrence” incentive is absent

for setting limited time offers.

Similar to the benchmark, there are three possible pricing strategies. The monopolist

uses a flat price in strategy 1 and 2. In strategy 1, the monopolist sets a relatively low price

to induce the consumer to investigate him first. Whereas in strategy 2, the monopolist sets a

relatively high price to induce the consumer to investigate her outside option first. Stategy 3

is limited time offer: the monopolist sets different prices at t = 1 and t = 2 to induce search

prominence. We will show that limited time offer pricing yields strictly higher profit than

uniform pricing (strategy 1 and 2) under some parameters.

12 Log-concaveness of distribution function implies a increasing hazard rate f (·)1−F(·) and g(·)

1−G(·) , which guar-antees the uniqueness of equilibrium in oligopoly price competition games.

13 This is a more plausible assumption for off-line search market.

27

Case 1: flat price to induce A→ B

Obviously price should be such that∫ u

p (u − p)dF(u) ≥ s, otherwise search yields strictly

negative expected utility for the consumer. At price p, if the consumer investigates A first

and finally purchases A, she obtains utility u− p− s. If she abandons A and investigates B,

she gets utility∫ v

v vdG(v)− 2s. Therefore, we can find a threshold u such that:

u = p +∫ v

vvdG(v)− s (2)

and 1− F(u) is the demand for product A. The consumer’s expected utility if she chooses

the A→ B search path is:

WA→B = −s +∫ u

u(u− p)dF(u) + F(u)(u− p)

Similarly, if the consumer investigates her outside option B first, the threshold value that

makes the consumer stop searching and consume B is denoted by v, which solves:

v =∫ u

p(u− p)dF(u)− s (3)

The consumer’s expected utility if she chooses B→ A search path is:

WB→A = −s +∫ v

vvdG(v) + G(v)v

Therefore, if the monopolist aims to set a flat price p to induce the consumer to investigate

his product first, the optimization problem becomes:

maxp

π1 = p(1− F(u)) (4)

28

such that: ∫ u

u(u− p)dF(u) + F(u)(u− p)−

∫ v

vvdG(v)− G(v)v ≥ 0 (5)

where (u, v) are determined by Equation (2) and (3). To interpret the optimization problem,

the objective function Equation (4) is equivalent to a classic monopolist’s profit maximiza-

tion problem, where the consumer has an outside option of value∫ v

v vdG(v)− s. As F(v) is

assumed to be log-concave, the unconstrained solution is unique. Equation (5) is the con-

straint requiring that it is optimal for the consumer to search a la path A → B rather than

B→ A under price p.

It is easy to see that the RHS of Equation (5) is decreasing in p. Let p defined by p =

1−F(u)f (u) , and p1 denotes the price that makes Inequality (5) binding. The optimal price for the

monopolist in case 1 is: p∗1 = min{ p, p1}.

Case 2: flat price to induce B→ A

If the seller sets a high enough price, it is optimal for the consumer to investigate the outside

option first. Given the monopolist’s price p, the threshold that the consumer stops searching

and consuming B is again denoted by v in Equation (3): v =∫ u

p (u− p)dF(u)− s. Conditional

on that the consumer visits the monopolist, with probability 1− F(p) the consumer finds a

match value greater than p and makes the purchase. Therefore, the optimization problem

for the monopolist becomes:

maxp

π2 = G(v)(1− F(p))p (6)

such that: ∫ v

vvdG(v) + G(v)v−

∫ u

u(u− p)dF(u)− F(u)(u− p) ≥ 0 (7)

and the constraint guarantees that the consumer would prefer to investigate the outside

option first.

29

Case 3: limited time offer to induce A→ B

Similar to the argument in the benchmark model, to induce the consumer to search a la

A → B, it is optimal for the monopolist to commit to a limited time offer in which the price

at t = 2 is too high to investigate the product.

Both threshold u and the consumer’s expected utility if she chooses A → B path remain

the same as in case 1. If the consumer chooses to investigate B first, she loses the opportunity

of investigating A when the monopolist sets a limited time offer. The consumer’s expected

utility from investigating B first is:

WB = −s +∫ v

vvdG(v)

It is immediate that WB < WB→A. For the monopolist’s optimization problem, the

objective function π3 is the same as π1 described in Equation (4), with a new constraint:

WA→B −WB ≥ 0, that is,∫ u

u (u − p)dF(u) + F(u)(u − p) −∫ v

v vdG(v) ≥ 0. Let p3 be the

solution of WA→B −WB = 0. As WB < WB→A, we have p3 > p1. The optimal price for the

monopolist to set in case 3 is: p∗3 = min{ p, p3}.

Proposition 7 (General Match Value) Under general match value F(u), G(v) with no recall:

1. when the search cost is high enough (s >∫ u

p uF(u)du), providing a limited time offer (case 3)

is strictly more profitable than a flat price offer (case 1) to induce prominence.

2. Define a k-shifted distribution of F(u) as Fk(u + k) = F(u). For any (F(u), G(v), s), there

exists a threshold k ∈ R. For any k ≥ k, under (Fk(u), G(v), s), it is optimal for the monopolist

to induce prominence (case 3) rather than not (case 2).

The first statement compares π∗1 and π∗3 . Note that π∗3 ≥ π∗1 is trivially true as there is

less restriction on pricing for case 3. And π∗3 > π∗1 ⇐⇒ p > p1. Or, if the monopolist’s

problem in the case 1 has a corner solution where the constraint is binding, a limited time

offer is strictly better than a flat price to induce search prominence.

30

A sufficient condition for π∗3 > π∗1 is s >∫ u

p uF(u)du. Note that when the search cost

is high enough, the unconstrained optimal price p cannot induce search even if the mo-

nopolist’s product is the only option for the consumer. Therefore, such price cannot make

the consumer investigate product A before her outside option B. It implies that under p,

Equation (5) cannot hold if s >∫ u

p uF(u)du.

The second statement compares π∗2 and π∗3 , which is consistent with our benchmark

result: when the monopolist’s product is good enough relative to the outside option, using

limited time offer to induce prominence is the globally optimal strategy. We provide a formal

proof for the second statement in Appendix B.

We also provide an example to illustrate Proposition 7, where the match value distribu-

tion of both options are uniformly distributed. We set the distribution of outside option to be

g(v) ∼ U[0, 2]. The match value distribution of the monopolist’s product is f (u) ∼ U[0, u],

where we vary u from 1.5 to 3.5. We also vary the search cost from 0.01 to 0.75. Figure I

below shows the simulation result of the monopolist’s optimal strategy:

Figure I: Uniform distribution example

31

In the blue region, the monopolist is indifferent in using a flat price or limited time offer

to induce prominence. In the yellow region, using limited time offer to induce prominence is

the strictly dominant strategy, whereas in the green region it is optimal to be non-prominent

in search. This example confirms Proposition 7 and our benchmark model: a seller with a

good product is more likely to use limited time offer to induce prominence; and when search

cost is not very low, a limited time offer is strictly more profitable than a uniform low price

in inducing prominence.

6 Conclusion

This paper provides a novel perspective to understand a commonly seen sales tactic, lim-

ited time offer. Our idea is inspired from what happens in the real world: it has become

very easy for consumers to obtain price information, and therefore sellers could use price

to direct a consumer’s order of inspecting options. We underline that a seller’s motive for

actively advertising limited time offers is to be visited first. We present a model in which

a representative consumer is informed about a product’s current and future prices, but the

product’s match value to the consumer is unknown and it costs time to research into it. In

a monopoly framework, as long as the product offers a higher expected search value than

the outside option, it is the most profitable for the monopolist to offer a limited-time dis-

count, which attracts the consumer to investigate the product before the outside option in

equilibrium. In the duopoly framework, we find that the product with a higher search value

is investigated first in equilibrium, and both sellers provide limited time offers with a low

buy-now price and a high buy-later price.

The analysis on the duopoly model also delivers policy implications for policymakers

and platform designers. Market competition using limited time offers (for example, Black

Fridays) increases total welfare, and in most cases, also increases consumer surplus com-

pared with the market equilibrium under uniform pricing. This is different from the result

32

of Armstrong and Zhou (2015) due to a novel channel illustrated in our model, through

which limited time offers improve total welfare. That is, limited time offers direct the con-

sumer search process to the most efficient order. As such, we argue that in most cases there

is no reason for policymakers to ban limited time offers. However, the limited time offer

competition may make all sellers worse off as it facilitates competition for being prominent.

If a platform is financially dependent on retailers’ revenues, the platform designer may want

to regulate limited time offers to reduce competition among retailers.

For the future research, it would be interesting to expand the scope of “limited time

offer” beyond price discounts. For example, some advertisements explicitly state that the

quantity is limited. Intuitively, such quantity limit works in a similar way to a limited-

time price reduction. By creating search friction at a later period, i.e., the risk of stock-out,

it induces consumers to inspect the deal first before other options. In some other cases,

consumers can observe the status of the inventory. The lightening deals at Amazon and

Taobao usually include a predetermined quantity of stock, and consumers can see a status

bar indicating the percentage of deals that have already been claimed. The less stock left,

the higher search friction it creates. On top of that, the number of products bought by other

consumers also signals information about the product’s quality. The faster the status bar

progresses, the more optimistic consumers get about the product. The quality signaling goes

hand in hand with search friction, which together make the invitation-to-search mechanism

more effective.

33

References

Simon P Anderson and Regis Renault. Pricing, product diversity, and search costs: A

bertrand-chamberlin-diamond model. The RAND Journal of Economics, pages 719–735,

1999.

Maria Arbatskaya. Ordered search. The RAND Journal of Economics, 38(1):119–126, 2007.

Mark Armstrong. Ordered consumer search. Journal of the European Economic Association, 15

(5):989–1024, 2017.

Mark Armstrong and Jidong Zhou. Paying for prominence. The Economic Journal, 121(556):

F368–F395, 2011.

Mark Armstrong and Jidong Zhou. Search deterrence. The Review of Economic Studies, 83(1):

26–57, 2015.

Mark Armstrong, John Vickers, and Jidong Zhou. Prominence and consumer search. The

RAND Journal of Economics, 40(2):209–233, 2009.

Susan Athey and Glenn Ellison. Position auctions with consumer search. The Quarterly

Journal of Economics, 126(3):1213–1270, 2011.

Yongmin Chen and Chuan He. Paid placement: Advertising and search on the internet. The

Economic Journal, 121(556):F309–F328, 2011.

Michael Choi, Anovia Yifan Dai, and Kyungmin Kim. Consumer search and price competi-

tion. Econometrica, 86(4):1257–1281, 2018.

Yucheng Ding and Tianle Zhang. Price-directed consumer search. International Journal of

Industrial Organization, 58:106–135, 2018.

Laura Doval. Whether or not to open pandora’s box. Journal of Economic Theory, 175:127–158,

2018.

34

Daniel Garcia and Sandro Shelegia. Consumer search with observational learning. The

RAND Journal of Economics, 49(1):224–253, 2018.

Marco A Haan and Jose L Moraga-Gonzalez. Advertising for attention in a consumer search

model. The Economic Journal, 121(552):552–579, 2011.

Marco A Haan, Jose L Moraga-Gonzalez, and Vaiva Petrikaite. A model of directed con-

sumer search. International Journal of Industrial Organization, 61:223–255, 2018.

Mandy Hu, Ivy Dang, and Pradeep K Chintagunta. Search and learning at a daily deals

website. Forthcoming in Marketing Science, 2019.

J Jeffrey Inman, Anil C Peter, and Priya Raghubir. Framing the deal: The role of restrictions

in accentuating deal value. Journal of Consumer Research, 24(1):68–79, 1997.

Itamar Simonson. The influence of anticipating regret and responsibility on purchase deci-

sions. Journal of consumer research, 19(1):105–118, 1992.

Martin L Weitzman. Optimal search for the best alternative. Econometrica: Journal of the

Econometric Society, pages 641–654, 1979.

Asher Wolinsky. True monopolistic competition as a result of imperfect information. The

Quarterly Journal of Economics, 101(3):493–511, 1986.

Jidong Zhou. Ordered search in differentiated markets. International Journal of Industrial

Organization, 29(2):253–262, 2011.

35

Appendix A

Examples of Limited Time Offers

Figure II: Black Friday Advertisement of Kmart in 2013

36

Figure III: Black Friday Advertisement of Forever 21 in 2014

37

Appendix B

Proof of Lemma 2

We extend the proof to a more general setting (footnote 9), where option B is priced by a

strategic seller. Lemma 2 is a special case when pB1 = pB

2 = 0.

Without loss of generality, suppose that for a given set of prices {(pA1 , pA

2 ), (pB1 , pB

2 )}, the

buyer first investigates product A, and no matter whether A is a good match or not, she

still investigates product B; if B is a bad match, the buyer revisits and buys A at pA2 ; if B

is a good match, the buyer purchases B at pB2 . For such search path to exist, we must have

vB − p2B > vA − p2

A, otherwise the buyer will not investigate B when product A is a good

match. The buyer’s utility from this search path is:

−2s + θB(vB − p2B) + (1− θB)θA(vA − p2

A)

However, there exists a weakly dominant strategy for the buyer: to investigate product

B at t = 1, if it is a good match, the buyer buys B at t = 2; if product B turns out to be a bad

match, the buyer continues investigating product A. This strategy yields an expected utility

of

−s + θB(vB − p2B) + (1− θB)[θA(vA − p2

A)− s]

Therefore, if the buyer is satisfied with the first product she investigates, she will pur-

chase it without further search, and thus recall will never happen.

�

Proof of Lemma 3

Assume that there exists a equilibrium with prices of two firms ({pAt }, {pB

t }), and the con-

sumer may purchase a product at t > 2. There are two possible cases: case 1, only one

38

product might be purchased at t > 2 and the other can only be purchased at t ≤ 2. Case 2,

both products in equilibrium might be purchased at t > 2.

In case 1, w.l,o,g assume that it is product A that can be purchased at t > 2. We show that

given {pBt }, there exists another price plan { pA

t } for firm A, such that ( pAt , pB

t ) is an Nash

equilibrium and the payoff for all parties remains the same. Since waiting is free, the buyer

can always investigate both options within 2 periods of time and purchase later. Let tAs ≤ 2

be the time that the consumer investigates product A and tAp > 2 is the purchase time. Let

a new pricing scheme { pAt } be such that pA

tAs= pA

tAp

and pAt = pA

t , ∀t 6= tAs . Since there is no

discounting, it has to be pAtA

p= min{pA

t }t≥tAs

.

If tAs = 2, the consumer investigates the value of B at t = 1 and the consumer would only

investigate A at t = 2 when she finds a bad match with B. Therefore, at the subgame of bad

match with B, firm A is de facto a monopolist, and the consumer is indifferent between buy

at t = 2 or tAp when facing the price plan { pA

t } When at t = 1 there is a good match with

B, under new price schemes the consumer does not gain more from searching A, so she will

purchase at the same time and price from B as under ({pAt }, {pB

t }). Thus the payoff for all

parties remains unchanged. To see this is an equilibrium, simply notice that the price scheme

for B and search rule of consumer is unchanged, so if ({pAt }, {pB

t }) is in an equilibrium,

under ( pAt , pB

t ) firm A still has no profitable deviation. For firm B, as pA1 and min{ pA

t }t≥2

remains unchanged, he makes the same highest profit if deviates the prices and induces

A→ B search path as under ({pAt }, {pB

t }). Therefore ( pAt , pB

t ) is in fact an equilibrium price

scheme.

If tAs = 1, replacing pA

1 by pAtA

ponly makes the consumer can purchase at the same price

but an earlier time. Therefore under the new price schemes the search strategy of buyer

remains unchanged: search order is A → B and purchases A at t = 1 if finds a good match

with A. Otherwise investigates B at t = 2 and purchases B at min{pBt }t≥2. Therefore the

payoff of all parties remains unchanged. For seller A, as the strategy of seller B and the con-

sumer remains unchanged, he has no profitable deviation as ({pAt }, {pB

t }) is an equilibrium.

39

For seller B, since she is visited secondly, it has to be min{pBt }t≥2 = vB − s

θBand she has no

profitable deviation if remains A → B search order. If she deviates to a price scheme that

induces B → A search order, since under { pAt }, we have min{ pA

t }t≥2 = min{pAt }t≥2, the

seller B makes the same deviation profit as if she is under price scheme ({pAt }, {pB

t }). Thus

( pAt , pB

t ) is an equilibrium price scheme.

In case 2, by applying the argument of case 1 twice, there exists a new pricing scheme

({ pAt }, { pB

t }) such that pitis= pi

tip

and pit = pi

t, ∀t 6= tis and i = A, B. The same argument as

above guarantees that the search order and transaction prices under ({ pAt }, { pB

t }) remains

unchanged comparing to ({pAt }, {pB

t }), and ({ pAt }, { pB

t }) is an equilibrium. �

Proof of Proposition 2

Proof. We use proof by contradiction to show that pricing schemes resulting in Search Path

II cannot constitute an equilibrium.

Assume that there exists an equilibrium {( pA1 , pA

2 ), ( pB1 , pB

2 ), (B → A)}. The equilibrium

pricing scheme must satisfy the following constraints: incentive compatibility constraint on

A (ICA), incentive compatibility constraint on B (ICB), incentive compatibility constraint on

the buyer/consumer (ICC) and finally the participation constraint on the buyer/consumer

(PCC). Next, we will compare the most non-binding constraints of ICA and ICB, and prove

that they contradict with each other.

For seller A, the incentive compatibility constraint requires that there is no profitable de-

viation in prices which makes the buyer investigate product A first. Given a pair of equi-

librium prices of seller B: ( pB1 , pB

2 ), we first find the most profitable deviation for seller A:

arg maxpA πA. There are two cases:

(1) The first case is when pB2 is so high that the buyer essentially loses the option to

purchase product B at t = 2. Similar to the intuition provided in the benchmark, the most

profitable deviation for seller A is to set a very high price at t = 2 and force the buyer to

40

search A at t = 1. The highest first-period price, denoted as p, that seller A can charge is the

price making a consumer indifferent between investigating A and B at t = 1:

θB(vB − pB1 )− s ≤ θA(vA − pA

1 )− s (8)

We obtain pA1 = vA − θB

θA(vB − pB

1 ) which is the choice on pA1 to make the equation bind-

ing. Seller A’s highest profit from deviation is then θA p and thus the incentive compatibility

constraint on seller A is:

(1− θB)θA pA2 ≥ θA p (9)

(2) The second case is when pB2 is moderate and the buyer could investigate product B at

t = 2. Similar to the previous case, the most profitable deviation for seller A is to set a very

high second-period price in order to induce first-period search. The highest first-period price

in this case, denoted as pA1 , that seller A can charge is the pA

1 making a consumer indifferent

between investigating A and B at t = 1:

θB(vB − pB1 )− s ≤ θA(vA − pA

1 )− s + (1− θA)(θB(vB − pB2 )− s) (10)

Comparing Inequality (8) and (10), we obtain pA1 > pA

1 and seller A profits more than in

the previous case. The incentive compatibility constraint on seller A in this case is:

(1− θB)θA pA2 ≥ θA p (11)

Comparing Inequality (9) and (11), since pA1 > pA

1 , among all the possible incentive

compatibility constraints on seller A, the most non-binding one is Inequality (9).

For seller B, the incentive compatibility constraint also requires that there is no profitable

deviation in prices which makes the buyer investigate product A first. Given a pair of equi-

librium prices of seller A: ( pA1 , pA

2 ), we first find the most profitable deviation for seller B:

41

arg maxpB πB. There are two cases:

(1) if pA1 is so high that the buyer will not investigate product A at t = 1. The deviation

from the equilibrium is to induce a search path (∅ → B). The most profitable deviation

for seller B is to set the highest pB2 , denoted as pB

2, which satisfies the following incentive

constraint on the buyer (ICC) at t = 2:

θB(vB − pB2 ) ≥ θA(vA − pA

2 )

We obtain pB2= vB− θA

θB(vA− pA

2 ) and the most profitable deviation in this case generates

θB pB2

profits. Hence, the incentive compatibility constraint on seller B to not deviate is:

θB pB1 ≥ θB(vB −

θA

θB(vA − pA

2 ))

The most non-binding ICA, Inequality (9), could be rewritten as:

θB pB1 ≤ θBvB − θAvA + (1− θB)θA pA

2 < θBvB − θAvA + θA pA2

Comparing the above two inequalities, we find that ICA and ICB of the equilibrium are

contradictory to each other. We then check the second case of seller B’s possible deviation.

(2) If pA1 is moderate and the buyer could search and investigate product A at t = 1.

Then the deviation from the equilibrium (B → A) is to induce a search path (A → B). As

product B is only investigated at the second period, the most profitable deviation of seller

B is to charge pB2 = vB − s

θBwhich makes the buyer’s search participation constraint (PCC)

binding. The incentive compatibility constraint on seller B is:

θB pB1 ≥ (1− θA)θB(vB −

sθB

)

Once again, we compare the ICB with Inequality (9). Since it must be satisfied that pA2 ≤

vA − sθA

, based on Inequality (9) we have:

42

θB pB1 ≤ θBvB − θAvA + (1− θB)θA pA

2 ≤ θBvB − θBθAvA − (1− θB)s

Comparing the above two inequalities, we also find a contradiction if and only if vA −s

θA> vB − s

θB.

To summarize, using proof by contradiction, we find that when vA − sθA

> vB − sθB

, there

does not exist any equilibrium in which the buyer searches and investigates product B first

at t = 1 and investigates product A at t = 2 if match value of product B is zero. �

Proof of Proposition 6

This Proposition contains two statements. Part 1 is if ∀(v, θ) ∈ supp( f ), vA − sθA≤ v− s

θ ,

limited time offers cannot make strictly higher payoff than uniform pricing. Part 2 is if

∀(v, θ) ∈ B, vA − sθA

> v− sθ , limited time offers are strictly better than uniform pricing. We

will show the two parts separately.

To begin with, note that conditional on the price plan makes the consumer searches her

outside option first, it is always optimal to set a price at t = 2 equals to p2 = vA − sθA

.

Thus under uniform pricing, the monopolist optimally set price p = vA − sθA

so to induce

all types of consumers investigate A at t = 2, and makes an expected profit equals to π =

(vA − sθA)∫ ∫

(1− θ) f (v, θ)dvdθ. For limited time offers price scheme, it is obvious that all

price schemes are at least weakly dominated by setting p2 = vA − sθA

: on the one hand,

this gives the highest incentive for the consumer to investigate A first as if the consumer

chooses B → A search path, she gets 0 expected payoff if finds a bad match with B. On the

other hand, for the consumer with relatively good outside option, p2 = vA − sθA

extracts the

most surplus from consumers that follows B→ A path. Thus the optimal limited time offer,

( p1, p2), must have p2 = vA − sθA

.

Part 1: we prove by contradiction. Assume that there exists a limited time offer (p1, p2 =

vA − sθA) that gives strictly higher payoff than that under optimal uniform pricing, π. Note

43

that p1 must be low enough such that some types of the consumer will search A first, other-

wise profit would be exactly the same. For a consumer with type (vB, θB), she investigates

product A first if and only if:

θBvB − s ≤ θA(vA − p1)− s + (1− θA)(θBvB − s)⇐⇒ θBvB ≤ vA −s

θA+ s− p1 (12)

By selling to a consumer of type (vB, θB) with price p1 at t = 1, the profit is strictly higher

than the selling at p2 = vA − sθA

at t = 2 if and only if:

p1 > (1− θB)(vA −s

θA)⇐⇒ vAθB −

θBsθA

+ s > vA −s

θA+ s− p1 (13)

Combine (12) and (13), we have:

vAθB −θBsθA

+ s > θBvB ⇐⇒ vA −s

θA> vB −

sθB

(14)

and (14) contradicts with the assumption that ∀(v, θ) ∈ supp( f ), vA − sθA≤ v− s

θ .

Part 2: we construct a limited time offer (unnecessarily optimal) that gives strictly higher

profit to the monopolist than the optimal uniform price profit π. Let U be the smallest θv

among low types consumers. Then an limited time offer (p1 = vA − sθA− (U − s), p2 =

vA − sθA) is strictly more profitable than a uniform price at vA − s

θA.

To see this, note that for a consumer with type (vB, θB), she investigates A first if and

only if θBvB ≤ vA − sθA

+ s− p1. Thus under (p1 = vA − sθA− (U − s), p2 = vA − s

θA), all

consumers with type θv = U will search product A in the first period. It is strictly more

profitable for the monopolist to sell to these consumers, as at p1 = vA − sθA− (U − s), it is

easy to verify that Equation (13) always holds.

Note that this price scheme is unnecessarily the optimal for the monopolist. This is the

highest price that makes the consumers who have the worst outside option to visit the mo-

nopolist first. By further reducing p1, the monopolist can attract more types of consumer to

44

investigate product A at t = 1, but at cost of losing profits from lower types and making

some unfavorable consumers (those with types (vB, θB) such that vA − sθA≥ vB − s

θB) visit

him first. �

Proof of Proposition 7, statement 2

For a k-shift of F(u), Define the foreclosure price p f (k) as:

p f (k) = u + k−∫ v

vvdG(v) + s (15)

Intuitively, under p f (k) the consumer visits the monopolist first and will purchase from

the monopolist even if she has the lowest realization, u + k. Thus it is the lower bound of all

possible prices a monopolist might charge in case 3. Denote the highest profit under strategy

of case 3 as π∗3 . We have π∗3 ≥ p f (k).

On the other hand, in case 2, the lowest price such to make the constraint Inequality (7)

satisfy is when it is binding. Denote the price as p2(k). It is easy to see that p

2(k) = p

2(0)+ k,

thus for all k, v(p2(k)) (as defined in Equation (3)) remains the same and is in the interior of

(v, v). We denote it as v. Then we can find an upper bound for π∗2 : π∗2 < G(v)(u + k). Since

both u and u are finite, as long as k ≥ G(v)u−u+∫ v

v vdG(v)−s1−G(v) , we have π∗3 > π∗2 . �

45