Waterfall versus Sprinkler Product LaunchStrategy: Influencing the Herd

Manaswini Bhalla

Department of EconomicsThe Pennsylvania State University

Abstract

The herd behavior literature has concentrated on the social learning phe-nomenon where players do not influence the rate and degree of learning. How-ever, under many economic scenarios agents can affect the speed of learningthrough their actions, e.g. prices. We analyze the problem of a monopolistthat can influence consumer learning about its product by manipulating itslaunching sequence in various regions and prices.Every period, the monopolist decides the number of launches and the price ofits product. The firm and consumers do not know the true underlying qual-ity of the product. However, consumers in each market inspect the good andreceive a private signal about its quality. They also learn about the qual-ity from observing previous sales and prices. We find that initially, the firmprefers prices and product launch strategies that allow greater transmission ofinformation from current to future buyers. The price sequence is found to besupermartingale, i.e. the price decreases with time. Initially, the monopolistprefers to launch the product at higher prices to influence learning. The num-ber of trials introduced by the firm is a monotonic function of the belief aboutthe quality of the product. As the belief of the product increases the likeli-hood of launching the product sequentially, decreases. The number of trials isfound to be a submartingale, i.e. the number of launches increases with time.However, the amount of learning encouraged by the monopolist is lower thanthat under a social planner.JEL Classification: C73, D62, D81, D82, D83Keywords: Product Launch, Social Learning, Endogenous

I would like to thank my advisor Professor Kalyan Chatterjee for his valuable suggestions.

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1 Introduction

Under uncertainty, agents usually rely on the decisions of others. For example, aconsumer deciding whether to purchase a particular good obtains valuable informa-tion by analyzing the consumption behavior of others. In cases when the qualityof the product is not verifiable, private information of other agents is more impor-tant to the decision maker. In this active learning literature, seminal papers likeBanerjee([3]) and Bikhchnadani etal ([4]) find that socially wasteful informationalcascade 1eventually arises when agents rationally learn from each others actions.This literature focuses on the social learning environment where agents do not in-fluence the rate and degree of learning. However, under many economic scenariosplayers can affect the speed of learning through their actions like the price charged bya firm. This paper analyzes how a monopolist can influence consumer learning andthe eventual rise of a herd by controlling the prices and the number and sequence ofmarkets to launch the product in each period. Unlike previous literature, the role ofthe monopolist is not passive and it is allowed to control learning through prices andnumber of trials. One of the major issues facing firms in their decision of a globallaunch of a new product is the pricing and sequencing of entry into the internationalmarkets. This paper studies how these strategies can be used to the advantage of amonopolist.

Waterfall or the hierarchical product launch strategy is a popular model for theglobal rollover. Pioneered by Ayal and Zif [1], in a waterfall launch, products arelaunched in a few regions, initially. The experience and success of the product inthese regions trickles down to the rest of the regions in a slow moving cascade. Thus,after the successful domestic launch of a new product in a few regions, firms areseen to launch it in other regions. Evidence of such a launching phenomenon hasbeen documented by Davidson and Harrigan [5]. They reported that during 1945-76, in planning for the global roll-over, U.S.-based multinationals initially focused onEnglish speaking markets (such as Canada or the United Kingdom), then on otherindustrial markets, and finally to the less developed countries. It is also documented2 that movies are released in a sequential order across the world. Consumers maydecide to observe the performance of the movie in other box offices before decidingwhether to see the movie when it is available. Production houses may wish to exploitthis consumer behavior in determining their launch and pricing strategies. Another

1Information Cascade is the event where consumers disregard their own private information andfollow the average actions of their predecessors

2www.imdb.com

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example of a sequential launch is that of Colgate Sensation. Colgate Sensation waslaunched sequentially first in Nordic countries then Mid European and then to theMediterranean countries. Another recent example of sequential product launch strat-egy is Apples i-phone. Apple launched its first iPhone only in U.S.A. on 29th June2007. Subsequently, it was released in Germany on 8th November,2007 for 628 eurosand on 28th November in France for 399 euros. It was later released on 14th March,2008 in Ireland, Austria for 399 euros(8GB). Recently, it was launched in 13 othercountries across the world. Though regulatory and preference differences across re-gions could be the stumbling blocks to a simultaneous product launch. Apple couldbe exploiting the social learning aspect of consumer behavior by introducing theproduct sequentially.

On the other hand, Omhae [6] and Riesenbeck and Freeling [7] advocate a sprin-kler diffusion strategy or a simultaneous world attack. They suggest that entryin all markets is the only viable choice in todays global marketplace as comparedto the waterfall diffusion strategy. It has been documented that Microsoft launchedWindows 95 in a similar fashion where 4-6 million customers worldwide bought theoperating system in the first three weeks after the launch. Thus, the marketing lit-erature is unclear about the optimality of these two launch strategies. This paperbridges the gap by shedding light on the optimal launch and pricing strategy in anenvironment where both the monopolist and the consumers learn about the under-lying quality of the product.

A few papers look at the role of prices in influencing the information flow acrossthe consumer base. For example, Welch [13] analyzes the optimal price to maxi-mize the success of a new security to a sequence of partially informed investors. Hehowever considers, fixed-price sales, since the Securities and Exchange Commissionhas banned variable-price sales in initial public offerings. Unlike Welch, this papercharacterizes the optimal flexible-price and launch strategy for a monopolist. Cem-inal and Vives([8]) studies the price dynamics of a strategic firm in the presence ofduopoly and consumer learning about the uncertain quality of the products. It isfound that the firms manipulate consumers beliefs through prices. Consumers inferthe quality of the good only through the past market shares as the prices are assumedto be unobservable. On the other hand, in our model, consumers take decisions basedon both the prices and the number of sales in the period. Bergemann and Valimaki[10, 11] in a sequence of papers analyse the role of resolving the uncertainty aboutthe products quality through prices. In their model, the number of consumers ineach period remains constant. In contrast, in our model, the number of consumers is

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endogenous. Also, they analyze the problem in a word of mouth envionment, whereconsumers share their ex-post experiences directly to each other. In our model, con-sumer purchase decisions reveals the ex-ante private information.

Sgroi[12] analyzes the problem of a social planner and a monopolist that may in-fluence the consumer learning by determining the number of offers to make in thefirst period. Our paper differs from this work in many ways. First, the monopolistin our problem can use both the number of offers and the price of the product asan instrument to influence the herd. Second, in Sgroi the monopolist knows thequality of the product whereas consumers are unaware of it. We on the other handanalyze both two sided and one sided problem of learning. Also, Sgroi analyzes astatic problem where the monopolist chooses the number of trials at the beginningof the game. We on the other hand solve a dynamic problem where the monopolistcan determine the number of offers in each period.

Bar-Issac[9] studies a dynamic learning model where the firm is privately informedabout the quality of the product and the consumers learn about the quality overtime. In contrast, in our paper the consumers are privately informed and both thefirm and the buyers learn the quality over time. Also, in Bar-Issac the seller only de-cides whether to trade or not. Once the trading decision is made, the product is soldto all the consumers in that period. In contrast, we focus on the sellers launchingand pricing strategy in each period where the firm not only decides whether to tradeor not but also how many offers to trade and at what price. This paper is thereforethe first to analyze the role of both the prices and number of launches in a model ofbilateral learning.

We consider the situation of bilateral learning where neither the monopolist northe consumers are aware of the true value of the product. Though the monopolistwould know the characteristics of the product it could be unaware of the preferencesof the consumers. At the same time the consumers could be unsure of the truequality and characteristics of the product. A monopolist can influence the decisionsof buyers and the social information accumulation by the price and the number oftrials that it allows in every time period. Prices and the number of trials play tworoles- rent extraction and information revelation. On one hand, greater number oflaunches and higher prices increases current period rent. However, launching theproduct at less than the maximum number of segments and at a higher price, mayfavorably influence the belief about the product in the next period.

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All buyers in the same region have similar preferences, and decide whether or not topurchase a unit of the good offered by the monopolist. When the product is launchedin the market the consumers inspect the good and receive a private signal about thequality of the product. This signal may be generated by the local advertising cam-paign or by the media. Each consumer decides whether to buy after observing theprice, a signal about the quality of the good, the decisions of previous consumers andthe previous prices. Observation of the behavior of other players is crucial to learnabout the quality of the product. Future consumers learn from previous purchasesand update their belief about the uncertainty. Thus, launching performances in pre-vious markets influences sales in the remaining markets.

When the monopolist determines the number of regions to launch its product inthe current period, it may face a tradeoff between more informed purchase decisionsmade by future decisions and less informed purchase decisions made by current con-sumers. We find that initially, the firm prefers prices and product launch strategiesthat allow greater transmission of information from current to future buyers. That isthe monopolist sets the prices high and launches it at fewer than the maximum num-ber of segments. An the act of consumption at the high price positively influencesthe belief about the quality of the product in future. This favorable information toothers allows subsequent consumers to infer the private information of the buyer andvalue it highly. Price is a function of the last periods belief. The price sequenceis found to be supermartingale, i.e. the price decreases with time. The number oflaunches introduced by the firm is a monotonic function of the belief about the qual-ity of the product. As the belief of the product increases the likelihood of launchingthe product sequentially, decreases. The number of number of launches introducedby the firm is found to be a submartingale, i.e. the number of launches increaseswith time. We also observe that greater the horizon of the game and the numberof untapped consumers in the economy higher the probability that the seller wouldallow experimentation. However, the amount of learning encouraged by the monop-olist is lower than that under a social planner.

The paper is organized as follows: Section 2 explains the basic model of social learn-ing. We characterize the equilibrium for the finite and infinite period problem inSection 3. Section 4 compares the monopolists optimal policy with that of a socialplanner. In section 5 some extensions of the model are discussed. Section 6 concludes

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2 Model

This section describes the basic model of product launch and pricing decision of amonopolist.

2.1 The Environment

There is a countable set of dates, T = {1, 2, 3, . . . } and N number ofregions. The regions represent a set of consumers which could belong to differentgeographic or demographic groups. At each time period, t T , the monopolistdecides the number of regions, t and the price, pt < at which to launch itsproduct. The monopolists product is assumed not to be introduced in any regionat the beginning of the game. Each region i N has mi number of consumers.To begin with we assume mi = m i. Denote M t1 to be the set of all regionswhere the product can be launched at the beginning of t period. The product canbe launched in a region only once. Therefore, M t M t+1. For every t, i such thati M t1 denote lti = 1 if the product is available in region i at the end or duringthe time period t and 0 otherwise. We do not explore the possibility of the productbeing withdrawn from a region. Thus, if lti = 1 then l

hi = 1h t.

All consumers in region i such that lti = 1, decide whether to consume the productor not 3. Each consumer consumes the product only once. The consumers deriveutility from an unknown value of the good. The quality of the product of themonopolist, could either be superior to the outside option, i.e. = G or inferior, = B. At the beginning of the game all agents in all regions have the same commonprior about the state of the world, q0 = Pr(G)

4.

2.2 Signal Structure

The monopolists product is assumed to be an inspection good. Denote by ji theconsumers of region i. All consumers, jii M t1 s.t. lti = 1 inspect the productbefore their purchase and receive independent signals, sji {g, b} about the qualityof the product. For example, the consumers are allowed to inspect cars before apurchase or are able to preview a movie before viewing the entire film. Also, mostproducts have showrooms where the products are on display for inspection. There is

3The consumers are not strategic and assumed to be highly impatient. Thus, all consumersin the a region decide immediately. Currently, the game restricts information spill overs withincountries. By making the consumers more strategic we hope to introduce both intra and interregion diffusion.

4The common prior in all regions could be a result of uniform advertising across all regions.

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no cost of receiving such a signal. The signal sji is private information of each agentji. However, it is common knowledge that the signals are distributed independentlygiven the state of the world according to Table 1. 0.5. can be interpreted as

Table 1: Signal Structure

P (sji |) g bG 1 B 1

the precision of the signal received. This precision is assumed to be the same acrossall regions.Given the signal received sji and the common prior at the end of period t 1 orbeginning of period t, qt1, all consumers, jii M t1 such that lti = 1, update theirbelief about the quality of the product. Let fs(k)(q

t1) = Pr(G|s(k), qt1)s(k) {g, b} be the updated probability of the state 1 for the consumer that receives signals(k) {g, b} given qt1, the common prior at the end of period t 1.

fg(qt1) =

qt1

qt1 + (1 )(1 qt1)

fb(qt1) =

(1 )qt1(1 qt1) + (1 )qt1

Notice that fg(qt1) and fb(qt1) are increasing functions in q and concave and convex

respectively. Conditional on the updated prior, fk(qt1) each consumer, jii M t

such that lti = 1 decides whether to consume the product or not.

2.3 Strategy

The history at any time period t, ht H is a collection of all the actions taken bythe monopolist, k, pk and that of the consumers, kk < tMonopolistAt the beginning of every period, t the monopolist has two decisions to take. First,it has to decide the number of regions that it wishes to launch its product, t(ht).Second, it must also decide the price, pt(ht) at which to launch it. Since the con-sumers in every region are homogenous in preferences, the price would be same in allregions at any point in time t. The payoff relevant history for the monopolist is qt1,

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the common prior at the end of period t 1 and the number of regions where theproduct has not been introduced at the end of period t 1, M t1 Thus, the actionspace of the monopolist at every point in time t is

(qt1,M1t) = t(qt1,M t1) pt(qt1,M t1) M t1 RConsumersThe strategy of all consumers ji is given by

tji

(ht, sji) = {1, 0}ht H,ji, i. Where, tji = 1 implies that the consumer bought the monopolists product. For all ji suchthat i M t1, lti = 0, ji = 0. However, for all agents ji such that i M t1, lti = 1,the payoff relevant history is qt1. Thus the strategy can be written as

tji(qt1, sji) = {1, 0}ji, i M t1, qt1 [0, 1]tsji {g, b}

Once an act of consumption t is made and observed by all, the common prior isupdated. The updated common prior is denoted by, qt = Pr(G| t, t, qt1). Giventhe updated prior qt, the firm and the rest of the consumers take their decisions inthe next period.

2.4 Payoffs

ConsumersEach consumer derives utility, rji from the outside option. For now we assume thatall agents in all regions are homogenous and have the same reservation utility, rji = 05. The consumers utility is given by the following table.

When the true state of the world is Good(G) (or Bad(B)) the utility from con-

Table 2: Realized utility of agent ji

= G = B

aji = 1 z zaji = 0 z z

suming(or not consuming) the product is greater than that from not consuming (orconsuming) the product. If instead of z we assume that the payoff from takingan incorrect action is z z, the explicit value of the cutoffs would change but the

5Section 6 discusses the extension of the model where the reservation utilities are different acrossregions.

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nature of the solution would remain the same. Hence, for mathematical conveniencewe assume z = z.

MonopolistThe monopolist maximizes the discounted expected payoff. The realized utility ofthe monopolist at time period t,

t(qt1,M t1) = pt(qt1,M t1)ct(qt1,M t1)t Twhere, ct(qt1,M t) is the number of adoptions of the product at time period t. Sincewe assume that the consumers are impatient and hence have no incentive to wait, thenumber of consumers that would try the product at time period t would be equal tothe sum of the population of the regions where the product was launched in periodt. That is ct1(qt1,M t) =

imii s.t. lti = 1.

The dynamic problem solved by the monopolist is the following.

V t(qt1,M t1) = Max,pEV [p(qt1,M t1)c((qt1,M t1)] + EV (q,M )

3 Equilibrium Analysis

3.1 Example, T=2

In this section the discounted dynamic optimization problem of the monopolist whichintends to launch the product in N regions over two time periods is solved. In anyperiod the monopolist can decide not to sell, which gives a payoff of zero. However,if the firm wishes to launch the product then it must do so at either of these twoprices: Separating price, PS(q) = 2fg(q) 1, the maximum price that the agentwhich receives a good signal is willing to pay to for the good; and the Pooling price,PP (q) = 2fb(q) 1, the maximum price at which the consumer that receives a badsignal wishes to purchase the product, where q is the common public belief aboutthe state of the world at the beginning of the period. PS(q) sells with probabilityPr(s1|q), and yields an immediate payoff of q (1 ), while PP (q) yields theimmediate payoff of 2fb(q) 1 with certainty. The pooling price PP (q) stops sociallearning. This is because both type of current consumers purchase the product at thisprice. The future consumer can not infer the signal of the immediate predecessor whobuys the good regardless of the signal received. This implies that after a pooling pricethe public belief in the next period will remain the same. However, the separatingprice, PS(q) sells only to part of the market or only to the high type consumer. Apurchase at this price would reveal the signal received by the current consumer to

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the future consumers. It allows for revelation of signals and hence for social learning.The three prices Separating Price, Pooling Price and Exit are the strictly dominantstrategies of the firm. Therefore, from now on only these three dominant prices ofthe monopolist would only be consideredThe pricing decision of the firm that wishes to launch the product simultaneously isgiven by the following lemma. This result also applies for the pricing decision of thefirm in the last period of a finite horizon sequential launch.

Lemma 1The pricing decision of the firm in a simultaneous launch is the following:

1 0.50 q 1x Exit yx Screening yx Pooling y

Proof. In any time period the monopolist can decide not to sell. If instead themonopolist wishes to sell she has two prices to choose from. PS(q) = 2fg(q) 1, themaximum amount that the high type consumer or the consumer that receives a highsignal is willing to spend; and the Pooling Price, PP (q) = 2fb(q) 1, the maximumprice that the low type is willing to spend on the good. All other prices beingstrictly dominated by one of these. Notice that PS(q) > PP (Q). The separatingprice,PS(q), 2fg(q) 1 sells with probability Pr(s1|q) and leads a profit of ES =Pr(s1|q)(2fg(q) 1) = q (1 ). The pooling price, 2fb(q) 1 sells with certaintyand leads a profit of EP = 2fb(q) 1.q 1 , PP (q) < PS(q) < 0. Hence the optimal strategy of the firm in the

last period is to not launch the product q 1 .1 q , PP (q) < 0, PS(q) > 0. Hence the optimal strategy of the firm wouldbe to sell only to the high type at the separating price.q , the separating price will be charged if the associated expected payoff q (1 ) is higher than 2fb(q) 1. As can be observed from the diagram, thatthe pricing strategy of the firm is to charge a separating price for all q q =(1)2+

(1)4+2(21)(21)

For a low perception about the quality of the product the firm would not bother

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q1

[1 ]

1

q

2f0(q) 1

q (1 )

0

Figure 1: Last period pricing strategy.

to enter the market. The monopolist can choose one of the three above mentionedprices. An act of consumption at the separating price would imply that the agentreceived a good signal and hence would favorably update the belief. This wouldenable the seller to sell at a higher price, though with a lower probability of makinga sale. If on the other hand the monopolist chooses the pooling price it would ensurea sale, though at a lower price. The difference in the expected utility of the two typesand hence the two prices decreases with the increase in the quality of the product.Thus, the benefit of launching the product at a separating price would decrease asq increases. Hence, there exists a threshold precision level such that the expectedprofit from selling to all equals the profit from selling only to the high type. Thus,the firm would sell only to the high type for intermediary beliefs and sell to all forthe high level beliefs.

Lemma 2If the monopolist launches its product sequentially at time period t, i.e. t < N thenpt would be the separating price. i.e. pt = 2fg(q

t1) 1Proof. Let us assume that if t(qt1,M t) = n < N then pt(qt1) 6= PS(qt1) =2fg(q

t1) 1. Or else, if t(qt1,M t) = n < M t then pt(qt1) = PP (qt1) =

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2fb(qt1) 1. The expected utility of the firm would then be

V (qt1,M t; t = n, pt = 2fb(qt1)1) =in

mi[2fb(qt1)1]+V (qt1,M t

in

mi)

However, if pt(qt1) = PP (qt1)

V (qt1,M t) = Max{0,M t(q (1 )),M t[2fb(qt1) 1]}

Thus V (qt1,M t;M t) V (qt1,M t; t = n, pt = (2fb(qt1) 1)) = t[2fb(qt1) 1] + V (qt1,M t n). This however, contradicts the assumption that t(qt1) = n.Hence, if t(qt1,M t) = n < N then pt(qt1) = PS(qt1) = 2fg(qt1) 1.

The lemma above shows that if the monopolist launches the product sequentiallyit must do so at a separating price. Launching the product at a pooling price wouldimply that the prior about the quality of the product would not change in the nextperiod. Launching the product sequentially comes at a cost of lower current expectedprofit. Sequential launch is profitable only if it favorably changes the prior in thenext period. A sequential launch at a pooling price would not change the prior in thenext period. Thus, if the firm finds it profitable to launch the product at a poolingprice in less than M t regions, it would do better by launching the product simulta-neously in all regions. Hence, the firm would always launch the product sequentiallyat a separating price.

Before we proceed let us introduce some notation. The updated common prior atthe end of t periods, qt can be written as a function of the number of regions thatthe product was launched, t (or the number of trials induced in period t, mt), thenumber of adoptions in period t, kt and the last periods common prior, qt1

q(t, kt, qt1) = Pr(1|t, kt, qt1)

Proposition 1The launch strategy of the firm for N 2 when quality is private information is thefollowing:

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0 q21(M) 1 q22(M) q 1x Exit y x Seq,n=1 y x Seq,n 1 y x Sim. PS y x Sim. PP y

Proof. Let q0 be the common prior about the quality of the product at the beginningof period 1. Also, let M be the number of regions left to launch at the beginning oftime period 1. Since, this is the first period M1 = N .

The value function faced by the monopolist at the first time period is given bythe following expression

V 2(q0,M) = Max{0;n[q0 (1 )] + EV 1[q,M n]n < M ;M [q0 (1 )];M [2fb(q0) 1]} (1)

Step 1 q0 1 The monopolists value function can be reduced to

V 2(q0, n,M) = Max{0;n[q0 (1 )] + EV 1[q,M n]}We know, V 1(q, n,M) = Max{0,M [q (1 )];M [2f0(q) 1]},q. V 1(q,M) andhence EV 1(q,M) are increasing in q. Thus, L(q0) = n[q0(1)]+EV 1[q,Mn] isincreasing in q0. L(q0 = 0) 0, L(q+0 = 1) > 0. Thus q21(n,M) [0, 1] suchthat L(q21(n,M)) = 0. There exists q

21(M) = min[q

21(M,n)]n such that q q21(M)

the monopolist would exit the market.Step 2 q [1 , q]The monopolists value function is

V 2(q0, n,M) = Max{n[q0 (1 )] + EV 1[q,M n],M [q0 (1 )]}Define R(q0, n,M) = n[q0 (1)] +EV 1[q,M n]M [q0 (1)]. R(q0, n,M)is decreasing in q0 as

f0(q)q 0.5q0 q. Also, R(q0 = 1 , n,M) 0, R(q0 =

q, n,M) 0. Thus, there q22(M,n) [1, q] such that R(q0 = q22(M,n), n,M) = 0and q0 q22(M,n) the monopolist wishes to launch the product simultaneously inall M regions than to launch it in n < M regions. Thus there exists, q22(M) =

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Max[q22(M,n)]n such that q q22(M) the monopolist would launch the productsimultaneously.

Step 3 q qThe monopolists value function is

V 2(q0, n,M) = Max{n[q0 (1 )] + EV 1[q,M n],M [2f0(q0) 1]}

Define, T (q0, n,M) = n[q0 (1)] +EV 1[q,M n]M [2f0(q0) 1]. T (q0, n,M)is decreasing in q0 as

f0(q)q 0.5q0 q. Also, T (q0 = q,M, n) = R(q0 = q, n,M)

0, T (q0 = 1,M, n) < 0. Thus, for all q q the firm would launch the productsimultaneously.Step 4 q [q21(M), q22(M)] The proof for this step is relegated to Proposition 2 wherethe optimal launching sequence for the general finite horizon game is solved.

In the first period, for q < 1 the monopolist can either exit the market orlaunch the product sequentially at a separating price. If the monopolist exits themarket it receives a payoff of 0. However, if it launches the product sequentially themonopolist would bear a current loss by selling at price, PS(q) < 0. However, sellingat a separating price sequentially would lead to a possibility of an improved q in thenext period and hence a higher revenue. This is because an event of a sale at theseparating price would indicate that the current consumer received a high signal. Asuccessful experiment in the first period may ensure sales at a higher belief and pricein the next period. The benefit of an improved belief about the quality of the prod-uct in the next period and hence a higher future expected revenue increases with q.However, the loss of making a sale in the current period at a negative price of a dis-count decreases with q. Thus, there exits a threshold qN1 (M) such that q qN1 (M)the firm is willing to bear loss in the current period to improve the belief favorablyin the next period. Hence, the firm launches the product sequentially only to oneregion q qN1 (M).For a belief greater than 1 the monopolist can either launch the product sequen-tially or simultaneously at a separating price. Launching the product sequentiallyimproves the belief about the quality of the product in the next period. However,this advantage comes at a cost of a lower immediate discounted payoff in the firstperiod. Launching the product simultaneously would give an advantage of a greaterimmediate discounted payoff. However, the firm would lose out on the possibility ofmaking a sale at a higher price to future consumers. This advantage of making asale at a higher price decreases with an increase in q. Thus, there exists a threshold,qN2 (M) such that q qN2 (M) the firm would launch the product simultaneously.

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Also, q [qN1 (M), 1 ] the firm launches the product in one region sequentially.q [1, qN2 (M)] the firm launches the product in more than or equal to 1 region.

3.2 Finite Time Horizon

The monopolist solves a discounted problem with bounded returns per stage. Beforeproceeding with the explicit solution of the problem for T

Also, the value function of the problem with N periods left is larger than thevalue function of the problem with N 1 periods left to go.Lemma 4 (Monotonicity of the finite horizon problem)The value function V N(.) of the N period to go problem is non decreasing in thenumber of periods to go N . i.e.

V N(q) V N1(q)

Proof. For the N periods left to go, it is always possible to adopt the strategy optimalfor the problem with N1 periods left and not to sell and get zero in the last period.Thus, the value function of the N period problem is non-decreasing in the numberof periods left.

It is now shown that the value function of the finite-horizon problem is concavein M .

Lemma 5 (Concavity of the Value function in M ,V N(q,M) MV N(q, 1))

Proof. Proof by induction. Let N=1

V 1(q,M) = Max{0;M [q (1 )];M [2f0(q) 1]}V N(q, 1) = Max{0, [q (1 )]; [2f0(q) 1]}N

V 1(q,M) is linear in M as it is a max of two linear functions in M and 0. Therefore,V 1(q,M) = MV 1(q,M) and (EV 1(q,M) MEV 1(q, 1)) Now suppose, V N1(q,M) MV N1(q, 1)

V N(q,M) = Max{0, n[q(1)]+EV N1(q,Mn)n M,M [q(1)],M [2f0(q)1]}V N(q, 1) = Max{0, [q (1 )]; [2f0(q) 1]}N

It is obvious to see that V N(q,M) MV N(q, 1)q,N,MIn the rest of the sub-section the Bellman equation is analyzed in order to estab-

lish that the optimal solution is a simple cutoff policy.

For q 1 the comparison is between the first and the second argument.V N(q,M) = Max{0 + V N1(q,M);n[q (1 )] + EV N1[q,M n]]}

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This is because q 1 ,M [q (1 )],M [2f0(q) 1] < 0.q [1 , q] the maximum is achieved by either the second or the third argu-

ment.

V N(q,M) = Max{n[q (1 )] + EV N1[q,M n];M [q (1 )]}

This is due to the fact that q 1 ,M [q (1 )] 0 and M [2f0(q) 1] < 0.On the other hand, q [q, 1] the value function can be re-written as

V N(q,M) = Max{n[q (1 )] + EV N1[q,M n];M [2f0(q) 1]}

This can be inferred from Lemma 1.Consider the optimal policy q 1 . The optimal policy of the monopolist withN periods left to go is such that the monopolist does not sell at all to the currentconsumer whenever the belief is lower than the cutoff level qN1 (M), and sells forhigher beliefs to only one region q [qN1 (M), 1 ]. The firm once exists neverreverses the decision to stop selling. This is because {qN1 (M)} is decreasing in thenumber of periods, N left to go. In the two period problem, for q < q21(M) it isoptimal for the firm not to sell in both the current and the next period rather thanquoting the separating price, PS(q) that sells at loss to the consumer in the currentperiod. The general result on the optimality and monotonicity of the cutoff policyfor exit is stated in the next result.

0 qN1 (M) 1 qN2 (M) q 1x Exit y x Seq,n=1 y x Seq,n 1 y x Sim. PS y x Sim. PP y

Proposition 2(a) (Existence and Monotonicity of cutoff levels for exitin N)In the problem with N periods to go and M regions left to launch the product in,it is optimal to exit the market when the belief is below the cutoff level qN1 (M). Inaddition, the cutoff levels qN1 (M) is non-increasing in the number of periods to go N ,i.e. qN1 (M) qN11 (M) and non-increasing in M , i.e. qN1 (M) qN1 (M 1)

17

Proof. By induction.Step 1We know from Proposition 1 that q21(M,n)n,M such that q q21(M,n) themonopolist prefers to exit the market in the second period when there are M regionsleft to launch the product than to launch the product in n < M regions. Weshow that if qN1 (M,n)n such that q qN1 (M,n) the monopolist prefers to exitthe market than sell sequentially in n markets then qN+11 (M,n)n,M such thatq qN+11 (M,n) the monopolist prefers to exit the market than sell in n < Mmarkets.Step 2Suppose, it is optimal to follow a cutoff policy of exit for N period q qN1 (M,n).i.e. qN1 (M,n) satisfies the following equation

n[qN1 (M,n) (1 )] + EV N1[qN1 (M,n),M n] = 0We know from Lemma 4, EV N [q,M ] EV N1[q,M ]. Therefore,

n[qN1 (M,n) (1 )] + EV N [qN1 (M,n),M n] 0

Step 4 Existence and Monotonicity of qN1 (M) in NDefine, LN+1(q, n) = n[q(1)]+EV N [q,Mn]. Since, qN+11 (Mn, n) > 0n,LN+1(q = 0, n) < 0 and LN+1(q = qN1 (M,n), n) 0. Since, LN+1(q, n) is contin-uous and strictly increasing in q. qN+11 (M,n) [0, qN1 (M,n)] such that LN+1(q =qN+11 (M,n), n) = 0. Also, q

N+11 (M,n) qN1 (M,n)M,n. Define, qN+11 (M) =

min[qN+11 (M,n)]n. Then, qN+11 (M) min[qN1 (M,n)]n = qN1 (M)M M .

Step 5 Monotonicity in MSuppose, it is optimal to follow a cutoff policy of exit for N period q qN1 (M,n).

i.e. qN1 (M,n) satisfies the following equation

n[qN1 (M,n) (1 )] + EV N1[qN1 (M,n),M n] = 0We know from Lemma 3, EV N [q,M ] EV N [q,M ]M M . Therefore,

n[qN1 (M,n) (1 )] + EV N1[qN1 (M,n),M n] 0Define, LN(q, n) = n[q(1)]+EV N1[q,M n]. Since, qN11 (M n, n) > 0n,LN(q = 0, n) < 0 and LN(q = qN1 (M,n), n) 0. Since, LN(q, n) is continuous andstrictly increasing qN1 (M , n) [0, qN1 (M,n)] such that LN(q = qN1 (M , n), n) = 0.Also, qN1 (M

, n) qN1 (M,n)n. qN1 (M ) = min[qN1 (M , n)]n min[qN1 (M,n)]n =qN1 (M).

18

The intuition for the existence of qN1 (M) is similar to that given before. We alsofind that qN1 (M) increases as N increases. Thus, as the horizon of the game increasesthe firm is more likely to exit the market.Now consider the optimal product launch strategy for q [1 , q]. The decision tocontinue learning involves a current cost and a future benefit. To keep the learningprocess going, it is necessary to price high, even though it would be myopically opti-mal to sell to all the consumers at the pooling price. The cost of learning is the loss incurrent expected profit from inducing a high price and is equal to (Mn)(q(1)).However, the benefit in the future from higher profits is EV N1[q,M n]. In par-ticular, it is optimal to stop the learning process q [qN2 (M), 1] for the problemwith N periods to go and M regions left to launch the product. The sequence of thecutoff levels {qN2 (M)} are increasing in the number of periods to go N . This derivesfrom the fact that experimentation has a larger value with a larger horizon N ahead.

Proposition 2(b)(Optimality and Monotonicity of cutoff levels for the re-gion to launch simultaneously)In the problem with N periods to go and M regions left to launch the product in, it isoptimal to launch the product simultaneously q [qN2 (M), 1]. In addition, the cutofflevels qN2 (M) are increasing in the number of periods to go N , i.e. q

N2 (M) qN12 (M)

and qN2 (M) qN2 (M )M M . This implies that the region where the firm launchesits product simultaneously is decreasing in the number of periods to go N .

Proof. Appendix A

The firm gets a lower current expected payoff by launching the product sequen-tially. However, it gets a chance of selling at a higher price in future in the event ofa sale in the current period. The advantage of launching the product sequentiallydecreases as q increases. If q is low, it is beneficial to give up some current expectedpayoff in return for a possibility of a sale at a higher q in the future. But as qincreases the advantage of a higher future expected returns, decreases. Thus, thereexists a cutoff level qn2 (M) such that q qN2 (M) the firm would launch the productsimultaneously. The range where the firm launches the product simultaneously de-creases with the increase in the horizon of the game. Also, this range increases withthe number of regions where the product has not been launched. This is because itpays to encourage learning when there is a larger time horizon and greater numberof regions left.

19

Corollary : (Optimality and Monotonicity of cutoff levels for the regionto launch sequentially)

In the problem with N periods to go and M regions left to launch the product in,it is optimal to launch the product sequentially, q [qN1 (M), qN2 (M)]. In addi-tion, the cutoff levels qN2 (M) are increasing in the number of periods to go N , i.e.qN2 (M) qN12 (M). This implies that the region where the firm launches its productsequentially is increasing in the number of periods to go N .

Proof. From Proposition 2 we know that the monopolist would launch the productsequentially q [qN1 (M), qN2 (M)]. In this proof we show that the firm would launchin only one region q [qN1 (M), 1 ].q [qN1 (M), 1 ] launching the product in n regions gives an expected return ofnm(q(1))+EV N1[q,Mn]. The difference between the expected utility fromlaunching the product in n+ 1 and n regions is the following. Since, q, Ln+1Ln =k(q (1 )) + EV N1(q,M n 1) EV N1(q,M n)q [qN1 (M), 1 ],q (1 ) 0 the monopolist is willing to introduce the product at a discount inthe current period so as to encourage learning in the next period. Also, a launchin an additional period would imply greater losses in the current period and fewerregions in the future to launch the product at a positive price.

Let Gl(qN+1) be the cumulative density distribution of the next period common

prior, qN+1 when l number of trials are allowed in the Nth period. It can be shownthat Gl(q

N+1) Gl+2(qN+1)qN+1 qN . Thus, gl(qN+1) gl+2(qN+1)qN+1 q.Thus, EV N1(q,Mn1)EV N1(q,Mn) =qN+1q[gn+1(qN+1)V N1(qt+1,M(n 1)k)] [gn(qN+1)V N1(qN+1,M (n)k)] 0. Hence, Ln+1 Ln 0n. Thusthe firm would launch the product in only one region for q 1 However, q [1, qN2 (M)] there exists a unique maximum number,H(q,M) of launches that thefirm would initiate. H(q,M) is the n maximizes kn(q(1))+EV (q,Mn).q [q1, 1 ] the firm launches the product only in one region. This is because

launching the product sequentially q [qN1 (M), 1 ] implies selling the productat a discounted price. Increasing the number of launches increases this loss. Also,launching the product in greater number of regions, now implies that the productwould be launched in fewer number of regions in the future. However, the number oflaunches q [1 , qN2 (M)] is n(M) 1. Thus, for some regions of q the numberof launches could be greater than 1. In the next proposition we show that the pricesdecrease and number of launches increase on an average with time.

Proposition 3 (Prices Supermartingale, Launches Submartingale)

20

Prices, a function of the past beliefs are supermartingale. i.e. They decrease onan average in time. Also, number of launches, a function of last period belief aresubmartingale. i.e. They increase on an average in time.

Proof. The game moves to the next period,t+ 1 only when the product is launchedsequentially in period t. Thus, the relevant price at time period t, P t = 2f1(q) 1.There are three possible prices in the next period. P t+11 =

j Pr(s = g)

jPr(s =

b)nj(2fg(q) 1)C(n, j) where, n is the number of launches in period t and q isthe next period belief; P t+12 =

j KPr(s = g)jPr(s = b)nj(2fg(q) 1)C(n, j)

and P t+13 =j HPr(s = g)jPr(s = b)nj(2fg(q) 1)C(n, j) +

j HPr(s =

g)jPr(s = b)nj(2fb(q)1)C(n, j). It can be seen that P t+12 , P t+13 P t+11 . Consider,P t+11 =

j Pr(s = g)

jPr(s = b)nj(2fg(q) 1)C(n, j). Since, fg is concave in q,P t+11 2fg[

j Pr(s = g)

jPr(s0)nj q]1 2fg[

j

j(1)njq]1 2fg(q)1 =P t. Since, P t+12 , P

t+13 P t+11 P t, prices are supermartingale.

3.3 Infinite Horizon

As it is clear from the analysis of the previous section, this model of monopolypredicts that the monopolist would follow a cutoff policy for its product and pricingstrategy. For low values of the belief about the quality of the product the firm wouldexit the market. However, it would launch the product sequentially for intermediarylevels of q and launch simultaneously for large values of q. In this section the infinitehorizon problem is solved. We solve for the stationary solution of the problem for Nvery large. The value function of the infinite horizon problem of the monopolist canbe written as

V (q,M) = max{0, n(q (1 )) + EV (q,M),M(2f0(q) 1),M(q (1 ))}The value function of the infinite-horizon problem is continuous and, when strictlypositive, strictly increasing in q. The first argument corresponds to exiting themarket, the second argument to the separating price strategy that keeps the learningprocess on, and the third to simultaneous launch strategy that allows the monopolistto stop the learning process of the consumers.Note that for q small enough it is optimal to post the separating price(even thoughit is negative), and sell with positive probability at a current loss.

Proposition 3(a) (Exit threshold)For belief lower than q1(M) (0, 1) the monopolist exists the market. q1(M)

is non-increasing in M .

21

Proof. Let q1(M,n) be such that n(q1(M,n) (1 )) + EV (q1(M,n),M) = 0.Such a value uniquely exists and is in the interval (0, 1 ), because Ln(q) = n(q (1))+EV (q,M) is a continuous, strictly increasing function of q, Ln(q = 0) 0and Ln(q = 1) > 0. Therefore, q q1(M,n) the firm would prefer to launch theproduct sequentially in n regions than to exit. Let q1(M) = min[q1(M,n)]. q1(M)exists because there are a finite number of q1(M,n).q1(M,n) be such that n(q1(M,n) (1)) +EV (q1(M,n),M) = 0. By Lemma 3,EV (q,M) EV (q,M )M M . Hence, n(q1(M,n)(1))+EV (q1(M,n),M ) 0. Since, n(q (1 )) + EV (q,M ) is a continuous, strictly increasing functionof q, Ln(q = 0) 0 and Ln(q = 1 ) > 0, there exists q1(M , n) [0, 1 ]such that it is optimal to launch the product in n regions q q1(M , n). Moreover,q1(M

n) q1(M,n)n. Since, q1(M,n) is increasing in n, q1(M ) = min[q1(M , n)] min[q1(M,n)] = q1(M).

Learning will be eventually stopped if the belief q is between (q2(M), 1)

Proposition 3(b)(Simultaneous Launch)For belief between (q2(M), 1) the monopolist would launch its product simultane-

ously. q2(M) are non-increasing in M .

Proof. Appendix B

The intuition of the threshold policy of the monopolist for the infinite horizonproblem is similar to that of the finite horizon problem.

4 Welfare Comparisons

In this section we discuss how the monopolists optimal product launch strategy isdifferent from that of a social welfare maximizer. Consider the problem of a socialplanner that has to use the price system and the number of experiments to achieveoptimal information disclosure and maximize welfare. The value function faced bythe social planner is the following

WN(q,M) = Max{0, n[q (1)] +EWN1[q,M n],M [q (1)],M [2q 1]}Lemma 6The value function of the social planner is greater than that of the monopolist,

WN(q,M) V N(q,M)

22

Proof. Consider, W 1(q,M) = Max{0,M [q (1 )],M [2q 1]}. V 1(q,M) =Max{0,M [q(1)],M [2f0(q)1]}. It is obvious to see that W 1(q,M) V 1(q,M).Therefore, EW 1(q,M) EV 1(q,M).Suppose, WN(q,M) V N(q,M)M, q. WN+1(q,M) = Max{0,M [q(1)],M [2q1], n[q (1 )] + EWN [q,M n]} and V N+1(q,M) = Max{0,M [q (1 )],M [2f0(q)1], n[q(1)]+EV N [q,Mn]}. Since, WN(q,M) V N(q,M)M, qand q f0(q) we can conclude that WN+1(q,M) V N+1(q,M)M, q

The social planner maximizes the problem of both the firm and that of the con-sumer. Therefore, the value function of the social planner is atleast as much as thatof the monopolist. The optimal policy of the social planner for the last period is

=

Exit if q 1 PS(q) = (2f1(q) 1) q [1 , ]PP (q) = (2f0(q) 1) q

Lemma 7The social planner does not sell q qN1 (M) [0, 1 ], sells simultaneously in allregions q [qN2 , 1]. Also, the social planner sells sequentially q [qN1 (M), qN2 (M)].

0 qN1 (M) 1 qN2 (M) 1x Exit y x Seq,n=1 y x Seq,n 1 y x Sim. PS y x Sim. PP y

Proof. The proof of Lemma 7 follows from that of Proposition 3 with the W (q,M)defined above as the value function.

The optimal product launch and pricing strategy of the social planner is similarto that of a monopolist. Only the thresholds of the two policies vary. The followingproposition explains how the two policies are different. It also reflects that the mo-nopolist discourages learning more than the social planner would optimally like to.

Proposition 4

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The monopolist exists the market for a belief larger (q1(M)) than the socially optimal(q1(M)). Also, q2(M) q2. Thus, the social planner encourages learning more thanthe monopolist.

Proof. q 1 , WN(q,M) = Max{0, n[q (1 )] + EWN1[q,M n]},V N(q,M) = Max{0, n[q (1 )] + EV N1[q,M n]}. From Lemma 7 we knowq1(M), q1(M) exist. W (q,M), V (q,M) are increasing in q and from Lemma 6 weknow thatWN1(q,M) V N1(q,M)q,M (hence, EWN1(q,M) EV N1(q,M)).Thus, q1(M) q1(M). Hence, the monopolist exists the market for a belief largerthan the socially optimal.q2(M,n) is defined such that

n[q2(M,n) (1 )] + EV (q2(M,n),M n) = M [q2(M,n) (1 )]

From Lemma 6 we know that EW (q,M n) EV (q,M n). Thus,

n[q2(M,n) (1 )] + EW (q2(M,n),M n) M [q2(M,n) (1 )]

q2(M,n) is defined such that

n[q2(M,n) (1 )] + EW (q2(M,n),M n) = M [q2(M,n) (1 )]

For 0.5, n[q(1)]+EW (q,Mn)M [q(1)] is decreasing in q. Thus,q2(M,n) q2(M,n). Thus, q2(M) = Max(q2(M,n)) q2(M) = Max(q2(M,n)).

The social planner encourages learning more than the monopolist. The monop-olist exits the market earlier than the social planner. Also, the range for which thesocial planner encourages a sequential launch is larger for a social planner than thatof a monopolist.

5 Extensions

5.1 Heterogeneity in population

In this section we solve the model where the regions have different population sizes.Each region i has mi 0 number of consumers. L denotes an ordering of theregions in the decreasing order of their population. Therefore, L(1) is a region wheremL(1) = Max{mi}iN and mL(N) = Min{mi}iN . At the beginning of each period, tthe monopolist decides on the number of regions, t N t1t T , the identity

24

of these regions, rt and the price at which to sell them, pt Rt T .For every t, i such that i M t denotes lti = 1 if the product is available in regioni at time period t and 0 otherwise. The history at any time period t, ht H is acollection of all the actions taken by the monopolist, k, pk and that of the consumers, kk t.MonopolistAt any point in time t the monopolist has two decisions to take. First, it has todecide the number of regions that it wishes to launch its product and the identityof these regions. The firms utility does not depend upon the the number of regionsthat the product was launched but only upon the total number of trials introducedin the previous period. Thus, reinterpret M t as the number of consumers that theproduct has not been introduced to, till period t. Since, product can be introducedto each consumer only once M t M t+1. The firm only decides on ,(ht), the totalnumber of consumers that the product was introduced to. Also, it must also decidethe price, p(ht) at which to launch it. The payoff relevant history for the monopolistis qt1, the last period belief about the quality of the product and the number ofconsumers to which the product has not been introduced, M t. The strategy of themonopolist is

(qt1,M t) = (qt1,M t) p(qt1,M t) {g(qg1,M g) pg(qg1,M g)}g=t+1 (3)qg1 [0, 1],M t M

The monopolist maximizes the discounted expected payoff. The realized utility ofthe monopolist at time period t is (qt1,M t) = p(qt1,M t)c(qt1,M t)t T ,where, c(qt1,M t) is the number of people that adopt the product. c(qt1,M t) =

imiis.t.lti = 1After the reinterpretation of M t as the number of consumers to which the producthas not been introduced, the results for finite horizon and infinite horizon productlaunch and pricing problem of the firm remain the same.

5.2 Heterogeneity in the reservation utilities

We only consider the case where the monopolist decides to launch its product in tworegions which differ in terms of their reservation utility. Let the reservation utilitiesof the regions be r1, r2. Let without loss of generality r1 r2. Both the regionshave the same population, m. At the beginning of each period, t the monopolistdecides on the number of regions, t N t1t T and the price at which to sellthem, pt Rt T. For every t, i such that i M t denotes lti = 1 if the productis available in region i at time period t and 0 otherwise. The history at any time

25

period t, ht H is a collection of all the actions taken by the monopolist, k, pk andthat of the consumers, kk t.The monopolist can charge one of the four prices, separating price for region 1,PS1 = 2f1(q)1r1 ; separating price for region 2, PS2 = 2f1(q)1r2; Pooling pricefor region 1, PP1 = 2f0(q) 1 r1; Pooling price for region 2, PP2 = 2f0(q) 1 r2.The respective profits derived from these prices are, S1 = PS1Pr(s1|q)(2m); S2 =PS2Pr(s1|q)(m); P1 = PP1(2m); P2 = PP2(m). The following lemma describesthe pricing strategy of the firm in the last period or when it launches the productsimultaneously.

Lemma 8The pricing decision of a firm in a simultaneous launch is the following:

qs2 qs1s20 qp1p2 1x Exit y x Screening in Reg 1 y x Pool in Reg 2 y x Pool in Reg 1 y

Proof. The monopolist has a choice to launch the product at either of the prices,PS1, PS2, PP1, PP2.

Note that PS2 PS1, PS1 PP1, PS2 PP2. Denote, qS1 by the q such thatS1 = 0. Similarly, qs2. It can be noted that qS1 qS2

Lemma 9If the firm launches the product sequentially then it does so only with a separatingprice

Proof. First we will prove that the monopolist would not launch its product at anyof the pooling prices. That is the optimal price is not PP1, PP2.

Let us suppose that the firm launches the product sequentially in only region 1at price, PP1. Then the profits received would be

m[2f0(q) 1 r1] + V 1(q)where, V 1(q) = Max{S1,S2,P1,P2}. The firm can do better by launching theproduct together in both the regions at the price which maximizes V 1(q). Thus, it

26

is never optimal for the firm to launch the product at the price, PP1. Similarly, forthe price PP2.

5.3 Private Information

This section analyzes the one sided problem where the monopolist is aware about thequality of its product. Let us assume that the monopolists product can either beof Good(G) or Bad(B) type. We only focus on the pure strategy of the monopolist.If the utility from consuming the bad product is less than the reservation utility ofall agents, then all pure strategies equilibrium would be pooling equilibrium. Thatis the bad type monopolist would never find it in its advantage to distinguish itselffrom the good type and hence always adopt the same strategy as that of the hightype.

The launch strategy of the high type monopolist is also a threshold policy. Theproposition below characterizes the optimal launch strategy for the high type mo-nopolist and shows that the firms allow for greater experimentation when it is awareof its product quality than if it is notProposition 4:The amount of experimentation allowed by the firm is greater when it is aware of theproduct quality than when it is not.

Proof. Appendix C

The firm that knows that its product is of high quality is willing to take a greaterrisk by encouraging greater learning. The firm that is unaware of the quality of itsproduct assigns a higher probability to a bad herd than the firm that is aware thatits product is good. This knowledge of the quality encourages the firm to take boldersteps of introducing the product sequentially so as to obtain a higher expected futurepayoff.

6 Conclusion

Prices and the number of previous sales of a product at any point in time revealinformation about the unknown quality of the product. Social learning literature sofar, has focused on the scenario where economic agents do not influence learning.This paper sheds light on how a monopolist can influence the degree of learningamong consumers by strategically choosing the prices and the number of sales of itsproducts at every point in time.

27

We study a model where consumers and the monopolist are unaware of the qualityof the product and learn about it through the prices and number of sales. Whenthe belief of the product is extreme the firm either exits the market or launches theproduct sequentially. However, for intermediate levels of the belief of the product thefirm launches the product sequentially in different regions. We find that for gameswith greater time horizon the firm is more likely to introduce the product sequen-tially. Also, the price sequence is found to be supermartingale, i.e. the prices on anaverage decrease with time. However, the number of sales or trials are found to besubmartingale, i.e. the number of launches increases on an average with time. Wealso find that the amount of learning introduced by the monopolist is not efficient.

Appendix

A Proposition 2(b)

Proof. We will first prove the existence of qN2 (M)

(a)Existence of qN2 (M)

Step 1: Proof by induction. We know from Proposition 1 that there existsq22(M,n)n < M such that

n[q22(M,n) (1 )] + EV 1[q22(M,n),M n] = M [q22(M,n) (1 )]n < M

Step 2:q [1 , q],N,M

V N(q,M) = Max{n[q (1 )] + EV N1[q,M n],M [q (1 )]}

Suppose, qN2 (M,n) such that n[qN2 (M,n) (1)] + EV N1[qN2 (M,n),M n] =M [qN2 (M,n)(1)]. We know from Lemma 3 that EV N [q,Mn] EV N1[q,Mn]q. Therefore,

n[qN2 (M,n) (1 )] + EV N [qN2 (M,n),M n] M [qN2 (M,n) (1 )]

Step 3: RN+1(q, n) is strictly decreasing in q

In this step we define RN+1(q, n) = n[q (1)] +EV N [q,M n]M [q (1)]

28

and show that it is continuous and strictly decreasing in q.Step 3(a):RN+1(q, n) is a composite function of continuous function and hence iscontinuous.Step 3(b): To prove that RN+1(q, n) is strictly decreasing,

RN+1

q= (M n) + EV

N [q,M n]q

We will show by induction that RN+1(q,n)q

0.We know, V 1(q,M n) = Max{0, (M n)[q (1 )], (M n)[2f0(q) 1]}q [1 , q]. Since, 2f0(q)

q 1q q ,EV 1[q,Mn]

qM n and hence, R2(q,n)

q< 0.

Now suppose, EVN1[q,Mn]q

M nM,n < M .V N(q,M n) = Max{0, n[q (1 )] + EV N1(q,M 2n), (M n)(q (1 )), (M n)(2f0(q) 1)}.If V N(q,M n) = n[q (1 )] + EV N1(q,M 2n) then EV N [q,Mn]

q=

[n + EVN1(q,M2n)

q] n + M 2n. The last inequality follows from 1

and EVN1[q,Mn]q

M n,M,n < M .Since, 2f0(q)

q 1,q q, EV N [q,Mn]

qM n

Step 5: RN+1(q = q, n) 0, RN+1(q = qN2 (M,n), n) 0. Therefore from Step 2and 3, qN+12 (M,n) [qN2 (M,n), q]n,M such that RN+1(q = qN+12 (M,n), n) = 0 orn[qN+12 (M,n)(1)]+EV N [qN+12 (M,n),Mn] = M [qN+12 (M,n)(1)]n < MLet, qN+12 (M) = Max[q

N+12 (M,n)]n. Hence, q

N+12 (M) qN2 (M)M,N .

Monotonicity in M

B Proposition 3(b)(Simultaneous Launch)

Proof. We will begin by first proving the existence and the monotonicity of q2(M)then that of q3(M).(a) q2(M)q [1 , q] the value function can be written as

V (q,M) = Max{n[q (1 ) + (M n)EV (q, 1),M(q (1 ))}

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Now, Rn(q) = n[q (1 )] + EV [q,M n] M [q (1 )] is continuous andstrictly decreasing in q, Rn(q = q) 0 and Rn(q = 1 ) > 0. Therefore, thereexists q2(M,n) such that

n[q2(M,n) (1 )] + EV [q2(M,n),M n]M [q2(M,n) (1 )] = 0Therefore, q q2(M,n) the firm would prefer to launch the product simultaneouslyin all regions than to launch it sequentially in n regions. Let q2(M) = max[q2(M,n)].q2(M) exists because there are a finite number of q2(M,n).Now we would prove the monotonicity of q2(M). Let q2(M,n) be such that

n[q2(M,n) (1 )] + EV [q2(M,n),M n]M [q2(M,n) (1 )] = 0It can be shown that (MM )[q(1)] EV 1[q,Mn]EV 1[q,M n]M M . Thus, 0 = n[q22(M,n)(1)]+EV [q2(M,n),Mn]M [q2(M,n)(1)] n[q2(M,n) (1 )] + EV [q2(M,n),M n] M [q2(M,n) (1 )]. Since,n[q (1 )] + EV [q,M n]M [q (1 )] is decreasing in q. q2

C Proposition 4:

Proof. The product launch strategy of the monopolist when it is aware of the qualityof the product is the following:The pricing decision of the firm in a simultaneous launch is the following or the laststage for a finite horizon problem:

1 0.50 qP 1x Exit yx Screening yx Pooling y

In the problem with N periods to go and M regions left to launch the product in, itis optimal to exit the market when the belief is below the cutoff level qN1P (M). It isoptimal to launch the product simultaneously q [qN2P (M), 1]. In addition, the cutofflevels qN1P (M), q

N2P (M) is non-increasing and increasing, respectively in the number of

periods to go N , i.e. qN1P (M) qN11P (M), qN2P (M) qN12P (M) and non-increasing inM , i.e. qN1P (M) qN1P (M 1). This implies that the region where the firm launchesits product simultaneously is decreasing in the number of periods to go N .

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The proof of the statements above are very similar to proposition 2(a)(b). In thisproof we would show that the thresholds qN1PM qN1 (M) and qN2PM qN2 (M).First, qN1PM qN1 (M). qN1P (M,n) is defined such that

n[2fg(qN1P (M,n)) 1] + EV N1[qN1P (M,n),M n] = 0

Similarly, qN1 (M,n) is defined such that

n(qN1 (M,n)+(1)(1qN1 (M,n)))[2fg(qN1 (M,n))1]+EV N1[qN1 (M,n),Mn] = 0

Since, qN1 (M,n) + (1 )(1 qN1 (M,n)) and the function n(q (1 )) +EV N1(q,M n) is increasing in q. We have that qN1 (M,n) qN1P (M,n). Hence,qN1 (M) qN1P (M).Similarly, qN2P (M,n) is defined such that

(nM)[2fg(qN2P (M,n)) 1] + EV N1[qN2P (M,n),M n] = 0

And, qN2 (M,n) is defined such that

(nM)[2fg(qN2 (M,n)) 1] + EV N1[qN2 (M,n),M n] = 0

Since, qN2 (M,n) + (1 )(1 qN2 (M,n)) and the function n(q (1 )) +EV N1(q,M n) is increasing in q. We have that qN2 (M,n) qN2P (M,n). Hence,qN2 (M) qN2P (M).

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