THREE ESSAYS ON COMPETITION UNDER UNCERTAINTY By Jongwoo Park A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY ECONOMICS 2011 ABSTRACT THREE ESSAYS ON COMPETITION AND REGULATION By Jongwoo Park Chapter 1: Designing Contests with Heterogeneous Agents This paper studies how to design a contest between agents with heterogeneous abilities under the uncertainty of their performances. We find that the level effort crucially depends on marginal winning probabilities of effort, more rigorously, on the probability density of the expected output gap. In particular, we emphasize that the contest mechanism with heterogeneous agents should be qualitatively different from that with homogeneous ones. The principal often chooses to adopt a worse monitoring technology, to assign less positively correlated tasks, and to announce a winner only if the agent outdoes the rival conspicuously. These schemes are beneficial to the principal only when the agent's abilities are sufficiently different. Chapter 2: Entry Decision with Tying under Quality Uncertainty and Switching Costs This paper studies primary market monopolist's entry decision into the competitive subsidiary market through tying strategy, in which both primary and subsidiary goods are non-depreciating durables with periodically upgraded. Under the quality uncertainty and switching costs in the subsidiary market, we show that, as switching cost goes up, a primary market monopolist would be more likely to make an early-entry into the subsidiary market to capture future profits from periodic upgrades. From a policy perspective, this result implies that, when antitrust authorities decide whether or not to prohibit primary market monopolist's tying behavior, they have to consider a technical aspect of the good as well that determines the size of switching cost. If switching cost is high, they need to scrutinize tying behavior more strictly at the early stage of market evolution. On the contrary, if the switching cost is low they need to pay more attention as the market progresses more. Chapter 3: Customer Return Policy as a Signal of Quality This paper presents a signaling model in which the length of return period is used as a signaling device for product quality. Without consumer's interim benefit, we show that there exist multiple separating equilibria, where a seller with a high-quality good offers a longer return period than a specific minimum level while a seller with a low-quality good does not offer return service. All the separating equilibria satisfy the Cho-Kreps intuitive criterion. We find, however, no pooling equilibrium exists since any pooling strategy would be dominated by high-quality seller's deviation to the strategy offering a perfectinformation price and a maximum refund period. With interim benefits there could be multiple separating equilibria, but the smallest return period among them satisfies the intuitive criterion. Multiple pooling equilibria could also exist and not all of them would necessarily be eliminated by the intuitive criterion. Dedicated to my parents, sisters and beloved Jin Young iv ACKNOWLEDGMENTS First of all, I would like to express my deepest appreciation to my advisor, Jay Pil Choi, for his invaluable guidance and support during my graduate studies and the writing of the dissertation. He is such a great mentor who always inspires me in every step. This dissertation would not have been completed without his advice and encouragement. I would also like to extend my special thanks to all my committee members, Anthony Creane, Thomas Jeitschko and Suzanne Thornsbury, for their valuable comments and suggestions. They are willing to provide me with their profound knowledge and thoughtful insights, which helped me to improve my dissertation in many fruitful ways. I would like to extend my heartfelt thanks to my beloved Jin Young for her infinite love. To meet her is the best thing ever happened in my life. She has stimulated me intellectually as a colleague and inspired me spiritually as a life companion. I also owe my deepest gratitude to my parents and sisters for their endless affection. They have always been with me. I would like to extend my appreciation to my friend and colleagues in the Department of Economics at Michigan State University. They have shared their knowledge and friendship with me, although I cannot list all of them here. Finally, I thank the Bank of Korea for supporting me financially support. v TABLE OF CONTENTS LIST OF FIGURES ........................................................................................... viii CHAPTER 1: DESIGNING CONTESTS WITH HETEROGENEOUS AGENTS 1.1 Introduction ....................................................................................... 1.2 Model ................................................................................................ 1.3 Moral Hazard and Monitoring ........................................................... 1.4 Job Assignment ................................................................................ 1.5 Margin Rule ...................................................................................... 1.6 Concluding Remarks ........................................................................ Bibliography ............................................................................................ 1 1 4 8 13 14 18 20 CHAPTER 2: ENTRY DECISION WITH TYING UNDER QUALITY UNCERTAINTY AND SWITCHING COST .................................................................................. 23 2.1 Introduction ...................................................................................... 23 2.2 Model ................................................................................................ 27 2.3 Analysis ............................................................................................ 31 2.4 Policy Implication .............................................................................. 42 2.5 Concluding Remarks ........................................................................ 44 Bibliography ............................................................................................ 46 CHAPTER 3: CUSTOMER RETURN POLICY AS A SIGNAL OF QUALITY .... 48 3.1 Introduction ....................................................................................... 48 3.2 Model ............................................................................................... 52 3.3 Basic Analysis .................................................................................. 54 3.4 Equilibrium in Signaling Game.......................................................... 59 3.5 Extension: Equilibrium with Consumer's Interim Benefits ................. 63 3.6 Concluding Remarks .......................................................................... 68 Appendix .................................................................................................. 70 Bibliography .............................................................................................. 80 vi LIST OF FIGURES 1.1 Equilibrium Effort ......................................................................................... 7 1.2 Equilibrium Effort under Different Monitoring Technologies ........................ 10 2.1 Game Tree .................................................................................................. 30 2.2 Switching Cost and the Probability of Firm i's Entry at t=1 .......................... 42 3.1 Seller’s Profits ............................................................................................. 59 3.2 Separating Equilibria ................................................................................... 61 A.1 Example ...................................................................................................... 74 vii Chapter 1 Designing Contests with Heterogeneous Agents 1.1 Introduction A rank-order payment scheme has been widely used when a principal interacts with multiple agents. Compared with piece-rate payments, it has several virtues as a relative evaluation scheme. When there are common shocks, a rank-order payment scheme dominates a piece-rate scheme by …ltering them out (Green and Stokey; 1983). It also reduces principal’ s incentive to betray agents by undervaluing their performances or reneging on the contract ex post (Malcomson; 1984). Moreover, intrinsically, some competitions cannot be speci…ed without a tournament or contest mechanism, such as promotion, R&D races, or sport events. When a principal adopts a rank-order reward scheme, she endeavors to design a contest in a way to enhance agents’e¤orts, thereby to increase total outputs. A main purpose of this paper is to study how the principal can manage competition to induce more favorable outcomes under this reward scheme. We set up a model, in which two agents with heterogeneous abilities compete to win the prizes with some uncertainties of their performances and the principal has imperfect observability on their e¤orts. The important …nding from this basic 1 model is that the level of agents’e¤ort is determined by their marginal winning probability of e¤ort, or, more speci…cally, it depends crucially on the probability density of the output gap on their competitive front. Moreover we …nd that total outputs are proportional to the level of output gap at the equilibrium. Therefore the principal can induce more outputs by adjusting factors that a¤ect the probability density of the output gap. Keeping those results in mind, we made a further analysis of three representative ways that could a¤ect contest mechanism, and found some interesting and empirically testable results. The most striking result is related to the choice of monitoring technology. We argue that principal’ attempt to improve monitoring technology, which reduces uncertainty of the s contest, does not necessarily increase total outputs. In other words, asymmetric agents may make more e¤orts when they are monitored less intensively rather than more intensively, especially when they have a large enough gap in their abilities. Actually the improvement of monitoring technology a¤ects the contest mechanism in two ways. On the one hand it increases principal’ valuation of agents’e¤orts. This means that, with a little more e¤orts, s agents can persuade the principal more convincingly than before that he deserves to receive a winner’ prize. Therefore both high-ability and low-ability agents would have an incentive s to make more e¤orts. On the other hand, it makes the outcome of the contest more obvious. This implies that the high-ability agent does not need to dominate his contestant anymore with a large output margin while the low-ability agent gives up too quickly. Therefore both agents would have an incentive to make less e¤ort as well. We call the former a substitution e¤ect and the latter an income e¤ect. When the two agents are su¢ ciently heterogeneous, a negative income e¤ect dominates a positive substitution e¤ect. Therefore a better monitoring technology may reduce agents’e¤orts and bring less total outputs to the principal. This result is sharply contrasting with the idea of traditional moral hazard literature, in which a moral hazard problem stems basically from the unobservability on agent’ action. In this paper, s however, we argue that reducing uncertainty by better observing agents’action can lead to a more severe moral hazard problem, especially when there is large heterogeneity in agents’ 2 abilities. Another interesting result is on how a principal assigns jobs to competing agents. When the principal has various tasks, she may choose to assign positively, negatively or independently correlated tasks to them. We show that the principal always prefers to assign more positively correlated tasks to symmetric agents and less positively correlated tasks to asymmetric agents. This is because assigning highly positively correlated tasks leads to increasing observability on the di¤erence between two agents. Technically, assigning more positively correlated tasks to them is equivalent to choosing the better monitoring technology. Thus, by the same reason above, it would be better for the principal to assign less positively correlated tasks if agents are more asymmetric in their abilities. The last result we found is on what we call ’ margin rule.’ Sometimes a principal the requires agents to outdo her competitors su¢ ciently to be the winner. In other words, to win the contest, an agent must do better than their rivals by a large enough margin. Unscheduled promotion or a huge bonus for outstanding job performance among agents can be exempli…ed in this context. We show that this margin rule may increase the e¤orts of heterogeneous agents while it is not optimal for symmetric agents. The reason is that it stimulates a low-ability agent by giving him a relative advantage while spurring a high-ability agent by penalizing his shirking. All these three applications, that is, choosing a monitoring technology, assigning jobs and using the margin rule, are at times observed in the real world and cannot be explained without considering the essential characteristics of contest between heterogeneous agents. We explain more in detail in remaining sections. Our model is developed based on Lazear and Rosen (1981). This seminal paper addresses the contest mechanism in the presence of costly monitoring for worker’ e¤orts. They show s that a rank-order payment scheme can be optimal if agents are homogeneous while it is no longer true if the agents have heterogeneous abilities.1 Meanwhile they pay little attention 1 Yet Bhattacharya and Guasch (1988) show that if wages can be made contingent on performance the tournament can again attain the …rst-best outcome. A crucial premise in their model is that a principal is assumed to know the heterogeneity of abilities across agents. That is, the principal knows who is a high-ability agent or a low-ability agent, and can o¤er 3 to the problem of designing a better contest mechanism, which is the issue we focus on in this paper. In the rent-seeking literature, there are several papers that discuss implications of player asymmetry; Katz et al. (1990), Baik (1994), and Nti (1999). In most of them the source of asymmetry is players’di¤erent valuations, not their abilities themselves. Moreover they study the properties of the Nash equilibrium e¤orts and payo¤s in di¤erent settings rather than an issue of designing a contest mechanism. Regarding Section 3 to 5, there are also a lot of papers in the literature of contest design. For example, Che and Gale (1998) analyze the e¤ect of caps on bidding and Moldovanu and Sela (2001) study the way of allocating prizes. Compared to them, the issues addressed in this paper are the change of monitoring technology, job assignment, and the margin rule. The brief review of each issue and our contribution to the literature will be discussed below. The remaining parts of this paper are as follows. In section 2, we analyze basic model, focusing on the di¤erence between symmetric and asymmetric cases. In section 3, we study the choice of monitoring technologies linked with moral hazard issues and provide a striking result di¤erent from traditional moral hazard literature. In section 4 and 5, we explore job assignment and the margin rule in turn. In section 6, we present concluding remarks. 1.2 Model There are two agents, i = A; B, who contest …xed prizes. A principal awards v w to the winner and v L to the loser. The output of agent i is qi = i xi + i , where xi indicates the level of e¤ort. Each agent’ ability is parameterized by s i , the marginal products of an e¤ort.2 Each i is known to both agents and a principal, but the principal does not know which agent has the higher ability than the other. A random shock, i 2 ( 1; 1), is di¤erent prizes to di¤erent agents. On the other hand, we still assume that a principal can o¤er only a uniform and …xed prize to them irrespective of their types. This assumption is more reasonable in the example of promotion and R&D race. 2 Alternatively, we can also represent the heterogeneity of agents’abilities with di¤erent marginal costs of exerting their e¤orts. All the results below are equivalent in both ways. 4 drawn from a known symmetric distribution with mean 0 and variance 2 , and distributed independently and identically across the agents. What i implies here is that the principal can observe the level of each agent’ e¤ort with some uncertainty. It may come from agent’ s s production luck or principal’ measurement error. s The winner of the contest is the agent who produces more outputs than the other. Then the probability for the agent A to win can be written as Pr(qA > qB ) = Pr( A xA = G( A xA where B xB > B B xB ) A ) = Pr( A xA B xB > ) G( ) is de…ned as the di¤erence in outputs between agent A and B, or simply an output gap, and G( ) is the symmetric distribution of we obtain E( ) = 0 and V ar( ) = 2 A to lose is [1 A ). Since A and B are i.i.d., (B 2 2 . Correspondingly, the probability for the agent B xB )]. Both agents have the same cost function C(x) with G( A xA C 0 > 0 and C 00 > 0. Without loss of generality, we consider the following cost function, s C(xi ) = 2 x2 , whose marginal cost is C 0 (xi ) = xi . Then agent A’ maximization problem i at a given degree of uncertainty can be written as M ax xA G( A xA B xB )v w + [1 G( A xA B xB )]v l 2 x2 . A The …rst-order condition is A (v w v l )g( A xA B xB ) xA = 0:3 From this condition, we can …nd four factors that determine agent’ optimal level of e¤orts. s Obviously, the e¤ort of agent A is increasing in (v w v l ), the winner’ gain, and decreasing in s 3 The second-order condition is 2 (v w v l )g 0 ( A xA A 0 ( ) is positive for always satis…ed if we further assume g holds actually in most of well-known symmetric densities. 5 < 0. This condition is < 0 and negative for > 0, which B xB ) , the parameter of marginal costs. Moreover, it depends crucially on the marginal winning probability of e¤ort, A g( A xA B xB ), which can be decomposed into A , the parameter for agent’ ability, and g( ), the probability density of output gap. Actually g( ) stands for s the marginal winning probability of output gap, that is, the measurement of the change in agent A’ winning probability caused by a marginal change of output gap at a given level. s The condition above shows that, as the density increases, the agent exerts more e¤orts, in other words, competition becomes more aggressive. Similarly, the …rst-order condition for agent B is B (v w v l )g( A xA B xB ) xB = 0: Comparing two conditions, we can see that the high-ability agent always makes more e¤orts than the low-ability agent. This result comes from the fact that the marginal gain of a unit increase in e¤ort is greater to the high-ability agent than the low-ability agent. Combining two …rst-order conditions, we can get the following condition that holds at the equilibrium. g( ) = ( 2 A 2 )(v w B where vl ) = A xA B xB : (1.1) indicates the equilibrium level of output gap, which also represents the location of competitive front on the equilibrium. We can immediately observe that is increasing in the degree of asymmetry in agents’abilities. Also, it is increasing in the winner’ gain and des creasing in the marginal cost parameter . Figure 1.1 illustrates these features well. Thus we have the following proposition. Proposition 1 is increasing in ( 2 A 2 ) and (v w B v l ), but decreasing in . Now we …nd the total output in equilibrium. Using the …rst-order conditions, we can 6 Figure 1.1: Equilibrium E¤ort represent the expected total outputs as follows. E qA + qB = A x A + B x B = 2 + A 2 B (v w v l )g( ) (1.2) The total expected output is increasing in agents’ abilities and the winner’ gain, and is s decreasing in the marginal cost. In particular, in equilibrium the total expected output depends crucially on g( ), a marginal winning probability of output gap. Substituting the equilibrium condition (1.1) into (1.2), the expected total output can be rewritten as A xA + B xB = This shows that the total output is increasing in 2 + A 2 A 2 B 2 B . . The next proposition summarizes the results above. Proposition 2 In equilibrium, the total output is increasing in the equilibrium output gap between agents and the marginal winning probability of the gap, given everything else remains. 7 These two propositions are very useful for the future analysis.4 In what follows, we will consider three representative ways of managing a contest to increase the level of agents’ e¤orts and thereby total outputs. 1.3 Moral Hazard and Monitoring The issue what we address in this section is whether and when the better monitoring technology on agents’e¤orts would increase or decrease their e¤orts. In traditional principalagent literature, agent’ moral hazard comes from imperfect observability on his behavior. s So intensi…ed monitoring over agents’e¤orts is considered as a very good way to prevent it. However what we argue is that it may have an adverse e¤ect when the degree of asymmetry in agent’ ability is large enough. To show this, we focus on how a change in the uncertainty s of monitoring a¤ects each agent’ e¤orts. s Suppose that there are two monitoring technologies that are represented by two possible distribution functions with the same mean, but with di¤erent variances. The better monitoring technology allows to have the lower variance, l , while the worse monitoring technology to have the higher variance, h , where h > l . This means that we consider two distribution functions, G( ; h ) and G( ; l ), that can be ordered by the mean-preserving spread (MPS) R R R R such that g( ; l )d = g( ; h )d and G( ; l )d G( ; h )d . The MPS in this model implies that the principal has more uncertainty in monitoring the agents’e¤ort levels, indicating the less accurate monitoring technology. We further assume the following Single Crossing Property. g 0 ( ; l ) > g 0 ( ; h ) for 2 ( 1; 0) and g 0 ( ; l ) < g 0 ( ; h ) for 2 (0; 1): 4 Sometimes, the principal may want to maximize the e¤ort of high-ability agent, for example, in R&D race. Both Proposition 1 and 2 in our model still hold in this case. 8 This assumption ensures that the MPS moves the probability mass from the center toward both tails smoothly, so two distribution functions must be crossing only once at 0: In other words, G( ; l ) has …rst-order stochastic dominance over G( ; h ) for 2 ( 1; 0], while G( ; h ) does over G( ; l ) for 2 [0; 1). This assumption also guarantees that two density functions cross once in each positive and negative region.5 We de…ne these unique crossing point as and , where > 0, such that g( ; l ) = g( ; h ) = g( ; l ) = g( ; h ): To economize on notation, we often denote a representative distribution and density function by G( ) and g( ) respectively. For simplicity, we ignore any cost the principal might incur in changing her monitoring technology. As a benchmark, let us begin with the case where agents have symmetric abilities, i.e., A= . From (1.1), the level of symmetric equilibrium e¤orts is B = xA = xB = x = (v w v l )g(0)= : In this case, an output gap is zero regardless of the degree of uncertainty. Since we have g(0; l ) > g(0; h ), competition becomes more intense under less uncertainty, in other words, under the better monitoring technology. This result is well consistent with common sense in that the agents are forced to work harder since the principal can monitor them better. Put it di¤erently, the improvement in monitoring technology reduces the moral hazard problem of the agents. However, this result can dramatically change if agents are asymmetric in their abilities. Without loss of generality, we now assume that agent A has better ability than agent B, i.e., 5 Again, this assumption is fairly general in that most known distribution functions satisfy this property. For example, fors Normal distribution with the density function f ( ) = a . 2 2 2 h l . p 1 exp( ), we obtain = ln h 2 2 2 2 2 2 l 9 h l Figure 1.2: Equilibrium E¤ort under Di¤erent Monitoring Technologies A> B . Then k is positive, which represents the competitive front under the distribution function with the variance k , where k = h; l. Now we compare two di¤erent competitive fronts and their intensities of competition under two di¤erent levels of uncertainty. As illustrated in Figure 1.2, we can easily identify that g( l ; l ) R g( h ; h ) and, correspondingly, l R h according to k Q . Then, by Proposition 2, the total output is greater under less uncertainty if the output gap is relatively small, and so it is under more uncertainty if the output gap is large enough. Then the following proposition summarizes the previous two results. Proposition 3 For symmetric agents, total outputs always increase under less uncertainty. However, for asymmetric agents, total outputs increase under less uncertainty if and under more uncertainty if > l > h < l < h. Interestingly, the better monitoring does not always result in the better outcome. Intuition behind this surprising result is as follows. When a principal improves her monitoring 10 technology, it a¤ects agents’e¤orts in two ways. On the one hand, from the relationship between a principal and agents, the improvement of monitoring technology makes both agents’ e¤ort more valuable than before since it brings a more honest result to the agents. Putting it di¤erently, with a little more e¤orts, agents could persuade the principal more convincingly that he deserves to receive a winner’ prize. This brings a positive e¤ect on agent’ e¤orts. s s We call this a substitution e¤ect in that the value of their e¤orts appreciated by the principal is heightened by the improvement of monitoring technology. Actually this is the very principal-agent problem that traditional moral hazard literature addresses. A thing worthy of note here is that this substitution e¤ect widens the output margin between two agents due to the gap in their abilities. On the other hand, from the relationship between agents, the improvement of monitoring technology makes the outcome of the contest more predictable, or even obvious. Putting it di¤erently, the principal would be more likely to declare the high-ability agent as a winner since she could be more convinced that the output gap results from the di¤erence in abilities between the agents. If the high-ability agent knows that his winning probability is su¢ ciently high, he does not need to dominate his competitor by a large output margin anymore to convince the principal that he deserves to win the contest. Similarly, if the low-ability agent knows his winning chances are su¢ ciently low, he has no more reason to exert an e¤ort to catch up with his opponent, thus would give up the contest very quickly. This brings a negative e¤ect on both agents’e¤orts. We call this an income e¤ect in that the change of total winning probability causes both agents to shirk. This income e¤ect is the unique characteristic, which is present only in contest framework. Again we need to note that the income e¤ect always narrows the output gap. One more thing to mention is that the size of income e¤ects de…nitely depends on the degree of asymmetry in agents’abilities, thereby on the level of their output gap. Consequently, we can expect that the total e¤ect of adopting better monitoring technology is determined by the relative size of the substitution e¤ect and the income e¤ect on 11 the given competitive front. When the output gap is small enough, the substitution e¤ect is greater than the income e¤ect because the latter is relatively small. In this case better monitoring always brings a favorable result to the principal. However, when the output gap is large enough, the income e¤ect dominates the substitution e¤ect. In this case the better monitoring causes the agents to reduce their e¤orts while the worse monitoring induces them to exert more e¤orts. According to the Proposition 1, the output gap in equilibrium is determined by the di¤erence of agents’ abilities, given the prize. Moreover, by Proposition 3, the principal chooses di¤erent monitoring technology depending on the size of di¤erence of the abilities. In particular, if agents’ abilities are su¢ ciently di¤erent, the principal prefers the worse monitoring. The next proposition summarizes it. Proposition 4 As ( 2 A 2)R , B (v w v l )g( ) l S h . If the di¤erence of abilities is large (small) enough, the principal prefers worse (better) monitoring. The most striking point from this proposition is that when the di¤erence of agents’abilities is su¢ ciently large, principal’ endeavor to intensify the monitoring on agent’ e¤ort with a s s view to reaping more outputs from them can bring an adverse e¤ect. This result is sharply contrasting with the idea of traditional moral hazard literature. Many papers, followed by Holmstrom (1979) and Grossman and Hart (1983), argue that a moral hazard problem basically stems from unobservability on agent’ actions, thus a s worse monitoring system leads to a severe moral hazard problem. It is obvious that they pay attention only to the substitution e¤ect in arriving at this conclusion. In this paper, however, we argue that less uncertainty in observing agent’ action can lead to a more severe s moral hazard problem, especially when there is large heterogeneity in agents’abilities, due to the income e¤ect. One of few exceptions is Cowen and Glazer (1996). They show that better monitoring can induce less e¤ort. However, there are several di¤erences between 12 their paper and our work. They consider one agent case, and the paper is mainly based on the graphical analysis without analytical treatment. Since their model is based on the dynamic environment, time inconsistency is an important problem, which is not an issue in our paper. By contrast, we have studied the contest model. Dubey and Wu (2001) and Dubey and Haimanko (2003) also drive a similar result to this paper that worse scrutiny could result in a better performance, but they use very speci…c multi-period performance evaluation scheme. Meanwhile we use more general framework, which is a lot applicable. 1.4 Job Assignment In this section, we address how the principal would assign tasks between agents. The principal can make competing agents work on various sets of tasks. The outputs of the tasks may be positively correlated, negatively correlated, or independent. For example, if the principal makes two salesmen work in the same area, their performance would be more likely to have positive correlation at given levels of e¤ort. By contrast, if they work in di¤erent areas, the outcomes would be less likely to be correlated. Interestingly this job correlation plays a crucial role in con…guring a winner. This implies that the possible correlation between tasks assigned to asymmetric contestants can be considered as a very important strategic tool to increase outputs. To see this, let us change agent’ output function slightly into qi = i xi + (1 + ) i . Here s captures a potential correlation between both agents’ outputs facing a common shock.6 Suppose that the principal can choose the tasks with various correlations.7 We denote such di¤erent situations with h and l respectively where h > l . Then we can obtain V ar( ) = 2 = 2(1 2 2 , k = h or l. Given the same monitoring technology, the variance k) 6 We should be very careful in understanding the nature of this correlation. It comes from the possible correlation from the shock, not from the di¤erences between agents abilities. 7 Actually tasks are very highly and positively correlated in most of the contests, such as sport competitions, arts concourses, or essay contests. Nevertheless in some contests like the promotion within the company, we can often witness negative correlation or independence between tasks. 13 can di¤er only in the correlation of tasks that the agents perform. Putting it di¤erently, considering these di¤erent correlations between tasks is equivalent to treating two possible distribution functions with di¤erent variances. For example, if the principal assigns more positively correlated tasks to the agents, she can observe more correctly that the output gap truly comes from the agents’heterogeneity, not from output shocks. Now, without loss of generality, we can set g ( ; l ) = g( ; h ) and g ( ; h ) = g( ; l ). ^ ^ Then all the previous analysis can be applied immediately. We …nd g ( h ) R g ( l ) according ^ ^ ^ to k Q , where k is the equilibrium competitive front under g ( ; k ). Correspondingly, we obtain h R l according to k Q . The next proposition summarizes it. Proposition 5 If the di¤erence of abilities are large (small) enough, the principal assigns less (more) correlated tasks. Waldman(1984) and Meyer (1994) study the way of assigning tasks in di¤erent settings. The common theme of the papers is that the principal attempts to gather information about agent’ types through designing a di¤erent way of task assignments. Compared to these s papers, we study the way of task assignments to induce most e¤orts, based on the degree of heterogeneity in agents’abilities. 1.5 Margin Rule In this section we address an issue on what we call ’ margin rule.’ In many cases, the the principal awards the prize only when the winning is conspicuous. For example, a CEO picks up an employee as an o¢ cer only when the employee shows an extraordinary performance in a given project compared with the others. This kind of unscheduled promotion happens often in many organizations. If the employee has performed relatively-well, the CEO would not make such a big decision. We study why the contest designer requires such a huge margin for the agent to be awarded the prize. 14 Suppose that the principal announces the winner only if the agent’ output is greater s than her rival by a positive margin, will be a winner if qA > qB + . Then there are three possible outcomes. Agent A , and a loser if qA < qB . For jqA qB j < , the game will end up with a draw and both agents divide the prizes evenly. Setting v L = 0 for simplicity, we can write the agent’ problem as follow. s M ax G( A xA xA )v w B xB +[G( A xA B xB + = [G( A xA B xB + ) G( A xA ) + G( A xA B xB B xB )] )] vw 2 vw 2 x2 2 A 2 x2 A One interpretation for the second equation is that the agent competes with both an advantage of and a disadvantage of [g( A xA at the same time. The …rst-order condition is B xB + ) + g( A xA B xB )] A v w 2 xA = 0: This shows that the agent competes on two competitive fronts, at g( A xA at g( A xA B xB + B xB + ) and ). This means the marginal winning probability of e¤ort depends on the average of probability densities at these two points. Now the equilibrium condition is written as g(e + ) + g(e )= ( 2 A 2 e 2 )v w B where e = A xA B xB : (1.3) Let us study the symmetric case …rst. Both agents choose the same e¤ort levels, and thereby ~ = 0. Since g( ) = g( ) by the symmetry of the density function, the Nash equilibrium e¤ort level of each agent is x = xA = xB = v w g( )= : 15 Although the agents are symmetric, they compete virtually at maximized when . However, g( ) is = = 0. Thus the contest designer will never use the margin rule for symmetric agents. On the other hand, when the agents are asymmetric, the expected total outputs are A xA + h Therefore, if 1 g(e + 2 B xB = ) + g(e 2 + A h 2 g(e + B vw i ) ) + g(e 2 : i ) > g(e ), the total outputs increase with the margin rule. The following proposition summarizes the results above and provides the condition under which there exists a positive that increases the total outputs. Proposition 6 For symmetric agents, the principal prefers no margin rule. For asymmetric agents, there exists an optimal margin that maximizes the total output as long as g 0 (e + ) > g 0 (e ). Under the Normal distribution, the principal prefers the margin rule if the di¤erence of agents’abilities is large enough. Proof. By Proposition 2, the principal wants to maximize e . Applying implicit di¤erenti- ation to equation (1.3), we obtain de = d g 0 (e + g 0 (e + ) ) + g 0 (e g 0 (e ) ) 2 e 2 ( A 2 )v w B . The denominator is always negative. Thus, as long as g 0 (e + ) > g 0 (e can increase the total e¤orts.9 With the Normal distribution, e the sign of d is the sign of d p 1 2 3 exp (e + )2 2 2 h (e + ),8 the principal N (0; 2 ), it is shown that ) exp (e )2 2 2 i (e 8 Note that this is exactly the same condition as 1 g(e + ) + g(e ) > g(e ). 2 9 Note that cannot be greater than e . Otherwise, g 0 (e ) becomes positive, the total output must be reduced. 16 ) . e Then d R 0 corresponds to d exp e + e !2 Q e e + (1.4) : While the left-hand side is decreasing in e , the right-hand side is increasing in e . This implies that when e is greater than a threshold value, the presence of e¤orts. In other words, the optimal choice of induces more to maximize e¤orts must satisfy equation (1.4). Implementing the margin rule has the same e¤ect as giving an advantage to a low-ability agent and a disadvantage to a high-ability agent at the same time. In other words, this margin rule makes the result of the contest more uncertain. Therefore, by the similar logic in the previous sections, if agent’ abilities are more asymmetric the principal can make the s contest more productive by implementing the margin rule. In this context, the margin rule is closely related to the handicapping theory, in which a principal confers a relative advantage or disadvantage on agents to make an uneven contest more competitive. For example, Baye et al (1993) show that a politician can maximize political rents by excluding the lobbyist who values the prize most. Che and Gale (2003) show that, when contestants are asymmetric, it is optimal to handicap the strongest contestant through imposing a maximum on an allowable prize.10 Fu (2006) analyzes the e¤ect of a¢ rmative action in admission policy, showing that favoring the weaker group increases 10 In our setting, it is easy to show that the principal designs a biased contest toward the low-ability agent. For example, let represent the relative advantage on the agent 2 B. Then agent A maximizes G( A xA )v w B xB 2 xA , and agent B maximizes w 2 G( B xB A xA + )v 2 xB . The equilibrium with the asymmetric agents is characterized by g(b )= ( 2 A b 2 )v w where B 2+ 2 A B v w g(b as A xA + B xB = The principal sets to satisfy g(b winning probability. b = Ax A B xB :The total output is again written ): There is the unique optimal that maximizes b . ) = g(0), which attains the maximum of the marginal 17 competition and, thereby, investments in education. However, most of the papers assume that the principal knows which agent is a high-ability agent or a low-ability agent. Therefore the principal could induce the …rst-best outcome through o¤ering di¤erent prizes or conferring a relative advantage or disadvantage according to agent’ type. Meanwhile the premise s of this paper is that the principal does not know which agent has higher ability than the other although she recognizes the degree of heterogeneity. A notable exception is the study by Meyer (1991), in which she shows that, even though a principal could not di¤erentiate agent’ types, handicapping the contest favoring the …rst-period winner in the second period s competition provides better information on agent’ types, thus results in better outcome. s However, the margin rule can induce better performance from agents even without a multiperiod framework or any other devices. In this sense, the margin rule can be used more ‡ exibly in a variety of settings. 1.6 Concluding Remarks This paper has examined how agent’ optimal level of e¤orts and the total output are s determined under the regime of contest when there is uncertainty in their performance. We …nd that the marginal winning probability of e¤ort, more speci…cally, the probability density of output gaps, plays a crucial role in determining them. Thus the principal can design the contest better by adjusting the factors that a¤ect on the density. Based on this result, we also provided three ways of improving the mechanism of the contest: choosing monitoring technology, assigning tasks, and using the margin rule. One major theme throughout the paper is that the contest mechanism with asymmetric agents should be qualitatively di¤erent from that with symmetric ones. At …rst glance, it appears that more monitoring, assigning highly correlated tasks, and not using the margin rule can induce more e¤orts. This is true when the agents are symmetric. However, we challenge this intuition in the case of heterogeneous agents. The principal prefers to adopt the counter- 18 intuitive alternatives such as less monitoring, assigning less correlated tasks, and using the margin rule, as the agents are more heterogeneous in their abilities. 19 BIBLIOGRAPHY 20 BIBLIOGRAPHY Baik, K. H. "E¤ort Levels in Contests with Two Asymmetric Players." Southern Economic Journal, 1994, 61, pp. 367-378. Bhattacharya, S. and Guasch, J. L. "Heterogeneity, Tournament, and Hierarchies." Journal of Political Economy, 1988, 96, pp. 867-881. Baye, M. R., Kovenock, D. and de Vries, C. "Rigging the Lobbying Process: An Application of the All-Pay Auction." American Economic Review, 1993, 83(1), pp. 289-294. Che, Y.-K. and Gale, I. L. "Caps on Political Lobbying." American Economic Review, 1998, 88, pp. 643-651. Che, Y.-K. and Gale, I. L. "Optimal Design of Research Contests." American Economic Review, 2003, 93, pp. 646-671. Cowen, T. and Glazer, A. "More Monitoring Can Induce Less E¤ort." Journal of Economic Behavior & Organization, 1996, 30, pp. 113-123. Dubey, P. and Wu, C-W. "Competitive Prizes: When Less Scrutiny Induces More E¤ort." Journal of Mathematical Economics, 2001, 36(4), pp. 311-336. Dubey, P. and Haimanko, O. "Optimal Scrutiny in Multiperiod Promotion Tournaments." Games and Economic Behavior, 2003, 42(1), pp. 1-24. Fu, Q. "A Theory of A¢ rmative Action in College Admissions." Economic Inquiry, 2006, 44(3), pp. 420-428. Green, J. R. and Stokey, N. L. "A Comparison of Tournaments and Contracts." Journal of Political Economy, 1983, 91, pp. 349-364. Grossman, S. J. and Hart, O. D. "An Analysis of the Principal-Agent Problem." Econometrica, 1983, 51, pp. 7-45. Holmström, B. "Moral Hazard and Observability." Bell Journal of Economics, 1979, 10, pp. 74-91. Katz, E., Nitzan, S. and Rosenberg, J. "Rent-seeking for Pure Public Goods." Public Choice, 1990, 65, pp. 49-60. Lazear, E. P. and Rosen, S. "Rank-Order Tournaments as Optimum Labor Contracts." Journal of Political Economy, 1981, 89, pp. 841-864. Malcomson, J. "Work Incentives, Hierarchy, and Internal Labour Markets." Journal of Political Economy, 1984, 92, pp. 486-507. 21 Meyer, M. A. "The Dynamics of Learning with Team Production: Implications for Task Assignment." Quarterly Journal of Economics, 1994, pp. 1157-1184. Moldovanu, B. and Sela, A. "The Optimal Allocation of Prizes in Contests." American Economic Review, 2001, 91, pp. 542-558. Nti, K. O. "Asymmetric Rent-seeking with Asymmetric Valuations." Public Choice, 1999, 98, pp. 415-430. Waldman, M. "Job Assignments, Signalling, and E¢ ciency." Rand Journal of Economics, 1984, pp. 255-267.. 22 Chapter 2 Entry Decision with Tying under Quality Uncertainty and Switching Costs 2.1 Introduction Tying strategy has been widely used when a primary good monopolist tries to extend his monopolistic power to a relevant subsidiary market. By tying his subsidiary good to the primary good, the primary good monopolist can force consumers to buy his subsidiary good and exclude the rival e¤ectively out of the competitive subsidiary market. Especially in software market, this strategy has been very successful. For example, Microsoft could exclude its rivals in application software markets by tying its applications, such as Internet Explorer or Windows Media Player, to its operating system, Windows.1 Microsoft cases have triggered …erce debates among economists and antitrust authorities. However, most of 1 The Internet Explorer case was …led in 1994 by the Department of Justice of the United States, and settled in 2001 by Microsoft’ agreement on not tying Internet Explorer to s Windows. In Windows Media Player case, the European Union ruled that Microsoft had abused its dominant position in the operating system to exclude its competitors out of the multimedia player market, and …ned 479 million euro in 2004. 23 debates have focused on traditional market structures without considering important features that are unique in software industry. Software has a few characteristics that make it di¤erentiated from other goods. First, although an operating system and an application program are closely related as complementary goods, the relative status between them is not symmetric. An operating system has intrinsic stand-alone value even without an application program, but the latter is of no use without the former. Second, software is a durable good without physical depreciation, but with periodic upgrades. This implies that consumers can make a choice between an old version and a new version. Putting it di¤erently, the new generation of software has to compete with the old generation. Third, software-users incur costs when they switch between two alternative softwares, and the size of switching cost is di¤erent across the kinds of softwares. For instance, a user of one word-processing program has to incur a relatively high switching cost in moving to another word-processing program since they have to learn a lot of new function keys until getting used to it. Meanwhile, a user of one music player program can easily move to another music player program with incurring a low switching cost since those programs can be played well with several common function keys. Due to these features, tying behavior in software market shows a couple of interesting patterns. First, most of tying cases that have been issues among antitrust authorities occurred when a market matures to some extent, not in the early stage of the market.2 When a …rm launches a new software, it faces a substantial risk since a new software turns out quickly to be one of two extremes, a big hit or a big failure, due to the network externalities. As long as the late-entry into the subsidiary market after observing consumer’ response is s possible, a primary good monopolist would prefer it to avoid this risk. In this sense primary good monopolist’ tying strategy for early-entry could be less criticized than that for s late-entry in view of risk-taking, even though both tying behaviors have negative e¤ect on social welfare as shown in section 4. Second, most of tying cases, especially for late-entry, 2 For example, Microsoft began to tie Internet Explorer to its operating system from Windows 95 onwards. 24 have been related with the softwares of which switching costs are relatively small.3 At the beginning of the market evolution, a primary good monopolist can surely take up the relevant subsidiary market by simply tying the new software with his operating system. At the later stage, however, he has to compensate consumers for switching costs to succeed in making a new entry into the subsidiary market even though he ties them. In this sense …rm’ s intertemporal entry decision crucially depends on the probability of succeeding in late-entry and the probability, in turn, would be determined by the size of switching costs. If switching cost is su¢ ciently high compared to the upgrade bene…t of primary good, consumers would not switch to the newly-introduced subsidiary good of the primary good monopolist. Thus the monopolist is less likely to succeed in late-entry and fails to exclude his competitor out of the market, and vice versa otherwise. Therefore the smaller the switching costs are, the more likely a primary good monopolist is to make a late-entry. In this paper we present the economic explanation for these featured patterns; that is, what determines …rm’ decision s on whether or not to tie, and how to choose the optimal timing of tying? Here we focus on the role of switching costs in determining them. To identify monopolist’ optimal entry-decision through tying, we set up a three-period s entry model where a primary good monopolist has to decide whether or not to enter into the subsidiary market in each period, with or without tying respectively. The main results are as follows. In the presence of uncertainty in consumer’ valuation on a subsidiary good, s a primary good monopolist decides when to enter into the subsidiary market with tying by comparing the of probability of the good being valued high and the chances of succeeding in late-entry. If switching costs are su¢ ciently low a primary good monopolist would always prefer to enter late into the subsidiary good market rather than to enter early. If switching costs fall on intermediate range the …rm would prefer a late-entry rather than an early-entry 3 For example, Microsoft tied Internet Explorer and Windows Media Player to Windows and it was very successful. Actually, both application softwares incur very little or no switching cost. Meanwhile, in more complicated software like word-processing software, Microsoft does not tie MS-Word to its operation system. Especially in Korea, Microsoft didn’ try to tie MS-Word to Windows even though it has been dominated by the Hangul t of Haansoft in word-processing software market. 25 with a positive probability and the probability is decreasing in the size of switching costs. If switching costs are su¢ ciently large the …rm would prefer a late-entry with a positive probability but the probability is …xed at such a low level. These results have important policy implications. When antitrust authorities determine whether a …rm’ tying behavior is anti-competitive or not, they consider several factors s together. Typically, indices on market structure are used; such as concentration ratio, market shares, etc. This paper suggests that they should also consider technical aspects of the goods and the status of market evolution as well. That is, if switching costs are high enough antitrust authorities should pay more attention at the early stage of market maturity because after that stage exclusive power of tying decreases gradually. On the contrary, if switching costs are relatively small they should pay more attention as the market develops more. There is voluminous literature on tying as an exclusion strategy. Traditional leverage theory sees tying as a strategic device for a monopolist in one market to extend his monopoly power to the other market. However, well-known Chicago School argument refutes it stating there is no incentive for the monopolist to tie them since he could extract entire rents without tying by charging a su¢ ciently high price on the primary good. This argument made the previous leverage theory vulnerable. In response to Chicago School argument, Winston (1990) shows that, in a di¤erentiated independent good case, tying can be a pro…table strategy since a monopolist could deter potential entry by committing to tie in the future. For a complementary good case, Choi and Stefanadis (2001) shows that, under risky R&D, tying can also deter entry by reducing the expected return of entry in each market since entry is pro…table only when entries into both markets are all successful. Nalebu¤ (2004) also analyzes an integrated …rm facing di¤erent competitors in each market and shows the …rm can be bene…ted by tying through larger market shares. Meanwhile Carlton and Waldman (2002) studies tying incentives on the presence of economies of scope. However these studies do not consider periodic upgrades and switching costs, which play a crucial role in software market. Carlton and Waldman (2005) showed that, assuming continuous demand space, switching costs and upgrades, the monopolist’ incentive to tie is s 26 intensi…ed because it guarantees future pro…ts resulting from upgrades. Kim (2007) also draws a similar conclusion assuming unit demands for heterogeneous consumers. Nevertheless, both papers are less realistic in that they assume both goods can be useful only when they are consumed together. In software market a primary product generally has its own intrinsic value without joint use of subsidiary goods, while not vice versa. Moreover in both papers entry decision always takes place at the …rst period in two-period model, which is a very restrictive assumption. The analysis presented here considers stand-alone value of the primary good and the possibility of late-entry. The remaining parts of this paper are as follows. In section 2, we set up a basic model, and, in section 3, analyze it comparing monopolist’ pro…ts between under an early-entry s and under a late-entry. Section 4 provides welfare analysis and policy implications. Section 5 gives concluding remarks. 2.2 Model There are two …rms, i and j, and two non-depreciating durable goods, A and B. Firm i produces both A and B, but …rm j produces B only. A is of use by itself, but B is of use only if used together with A. For example, in software industry, we may think A as an operating system and B as an application program. In this sense, we call A a primary good and B a subsidiary good. We consider three-period competition, t = 1; 2; 3. Both A and B are newly introduced at t = 1 and upgraded periodically at t = 2 and 3 respectively. Here we assume any upgrade of B is compatible to the previous version of A. This means that any upgrade of A have to compete with its older versions to capture potential buyers for the upgrade of B. Both …rms have to incur research and development (R&D) costs in developing and upgrading the products. The costs are RA and RB in initially developing A and B respectively. Upgrades of both goods also require the same R&D costs as those of initial development. Here we introduce " as R&D cost savings in developing B and upgrading it at the same time, 27 where " < RB . This means the costs of developing and upgrading B simultaneously at t = 2 are smaller than those of developing it at t = 1 and upgrading it at t = 2 subsequently. The existence of " also implies that there is no case in which a …rm would invest R&D costs in a given period without being able to sell the product in that period. Marginal costs are all normalized to zero. Consumers, whose size is normalized to one, are homogeneous in their preference for the goods and have a unit demand for each good. Good A gives consumers an intrinsic standalone value of V A per-period and each upgrade gives them an additional per-period bene…t of V A . Thus entire per-period bene…t of the second-period version would be 2A and that of the third-period version would be 3A. Good Bi and Bj give consumers per-period bene…ts of ViB and VjB respectively only when used together with A. To study a primary good monopolist’ tying incentive to extend s its monopoly power to the relevant competitive subsidiary market where his product has inferior quality, we endow …rm j with the quality superiority as follows: ViB V B and VjB ViB = > 0: Meanwhile, the upgrades of B give consumers the same additional per-period bene…t, V B , across the …rms in subsequent periods. This means that the quality di¤erence would be maintained throughout the periods. Again, this upgrade bene…t could be obtained only when consumed with good A, but it does not matter which version of A is combined.4 Furthermore we introduce the following uncertainty on consumer’ valuation of V B before s …rm i’ investing R&D costs in B in the …rst period: s o n B V B 2 V B; V where Pr(V B ) = 1 B and Pr(V ) = . 4 In reality, a new version of application software often functions better under new oper- ating system than under the old one. However, this simpli…cation doesn’ a¤ect main results t at all. 28 The true value of V B is realized just before being released in the market in the …rst period. Consumers choose either Bi or Bj at t = 1 and can switch to the other at t = 2 with incurring switching costs, s 2 [0; 1). We further assume that there is no switching at t = 3. This indicates that if one …rm prevails in the second period it will do so in the third period as well. This assumption re‡ ects the reality that the longer consumers use one good the more they are accustomed to it so that they would be unlikely to switch to the other. Moreover this keeps the focus on …rm’ entry decision only between period 1 and 2. s In addition, to simplify the model and prevent trivial cases, we will use the following assumption throughout the analysis. Assumption 1. 2V A > (2 B Assumption 2. R3 s). B < V B < R3 and V B > (2 + s + 3RB ) ". Assumption 3. Tie-breaking rule: tying strategy will be used if it gives at least the same pro…t as separate-selling does. Assumption 4. If neither …rm enters into the subsidiary market in a given period, the market will collapse and no …rm enters onwards. By Assumption 1, as shown later, we can exclude the case in which the quality superiority of Bj is extremely high so that …rm j can make a new entry into the market B at t = 2 in spite of …rm i’ tying. By Assumption 2, as shown later as well, the supports of V B s and V B are de…ned su¢ ciently low and high respectively such that if V B = V B investment in market B is unpro…table in any case while if V B = V B even late-entry into market B might be pro…table. Assumption 3 can be justi…ed by the fact that, for …rm i to sell its subsidiary good, the less stringent incentive compatibility constraints are required with tying than without tying. Assumption 4 means, if neither …rm enters, no future pro…ts would be expected at all. In this situation, in the …rst period, …rm j invests R&D costs and enters into the market B. At the same time, …rm i decides whether or not to enter into the market B. If …rm i decides to enter, it can choose whether or not to tie its subsidiary good to the primary good. 29 t=1 t=2 E T E T N T N E N T E Firm i NE (B) T NT T E NT NE (B) E E Firm i NE N T Firm j Firm i N E Firm j N E N T Firm i E Firm i Firm j t=3 (B) (B) (B) (B) (A) (A) (A) E: Enter. NE: Not enter. u : Firm j’ nods for entry-decision. l : Firm i’ nods for tying-decision. s s T: Tie. ¡ : Firm i’ nods for entry-decision. n : Final payoff nods. s NT: Not tie. Figure 2.1: Game Tree Depending on the …rst period outcome, both …rms decide whether or not to enter into the market in the subsequent periods and …rm i chooses whether or not to tie as well. To make a new entry in the second or the third period, each …rm has to release the corresponding upgraded products. Finally they compete in a Bertrand way under the restriction of nonnegative price. An extensive-form representation of the entire game is shown in Figure 2.1. 30 2.3 Analysis In this section, we solve the model and …nd a subgame perfect Nash equilibrium that speci…es both …rms’entry decision and …rm i’ tying decision. If V B turns out to be V B in s the …rst period, neither …rm invests in market B anymore. So, for the analysis in the second and third periods, we restrict our attention to the case that V B was revealed as V B in the …rst period. Then, for the analysis in the …rst period, we will consider both cases together. 2.3.1 In the third period In market A, …rm i will invest RA and release the …nal upgrade version of his primary good with the quality of 3V A at the price of V A (= 3V A 2V A ), gaining a pro…t of (V A RA ). In market B, meanwhile, the equilibrium con…guration depends on which …rm prevailed in the previous period. Since there is no switching in this period, the …rm who prevailed in the previous period will invest RB and release the …nal upgrade version of B at the price of V B , with appropriating the whole monopoly pro…t of (V B RB ).5 The following lemma summarizes …rm’ equilibrium strategies and pro…ts at t = 3. s Lemma 1 Suppose that V B = V B and no switching happens in the third period. Then a …rm that prevailed at t = 2 prevails at t = 3 as well. Pro…ts at t = 3 are e3 = (V A i e3 = (V A i RA ) + (V B RB ) and e3 = 0 j B RA ) and e3 = (V j RB ) if Bi sold at t = 2, if Bj sold at t = 2. 5 Alternatively, if we assume the possibility that consumers switch in the last period as well, the equilibrium characterization in the last period would be changed in a way to reduce …rm i’ incentive to prevail in the second period because it would lower the expected future s pro…t from the last period’ upgrade. So it lowers the critical level of switching cost that s allows …rm i to enter late in the second period. However this modi…cation does not change the entire analysis qualitatively as long as a positive third-period pro…t is expected. If we assume in…nite periods instead of three periods, no switching assumption in the last period could be relaxed. 31 We have a couple of things to notice here. First, contrary to the Chicago School argument, the release of primary good upgrade does not allow …rm i to appropriate the surplus in market B by charging a high price on the primary good because the upgrade of the primary good has to compete with its old version to attract potential buyers. Second, at t = 3, tying strategy is of no use in …rm i’ entering into the market B because no switching happens in s this period. 2.3.2 In the second period The second period equilibrium also depends on the …rst period equilibrium con…guration. We will …rst solve two cases separately, depending on which …rm prevailed at t = 1, and then provide overall equilibrium con…guration at t = 2. (Case 1: when Bj sold at t = 1.) In this sub-section we analyze a case that …rm j prevailed in the …rst period. We have four sub-cases to consider depending on both …rms’entry decision into the market B at t = 2; both …rms enter, either i or j enters, or neither does. In the last case, both …rms would get zero pro…t in market B, so we will analyze the remaining three sub-cases one by one. First, suppose that only …rm i enters into the market B at t = 2. Then …rm i would newly B introduce a good B with an upgraded per-period quality of 2V , investing (2RB "). Since all consumers were using Bj in the previous period, they can either keep using it without upgrade with getting a per-period bene…t of (V with getting a per-period bene…t of 2V B B + ), or switch to a newly-introduced Bi minus switching cost, s. So incentive compatibility for consumers to switch to Bi is 2 Note that 2V B 2V B and (V s B + pB i 2 (V B + ) () pB i 2V B (2 + s). ) are multiplied by two because consumers have two remain- ing periods onwards. Since 2V B > (2 + s) by Assumption 2, …rm i would sell Bi at 32 h 2V B (2 i h B + s) in market B and make a pro…t of 2V (2 + s) (2RB i ") . In market A, …rm i would sell an upgrade of A with a quality of 2V A at the price of 2V A , which comes from 2 (2V A V A ), and make a pro…t of 2V A RA . Moreover, from Lemma 1, he will take the whole pro…ts from upgrade at t = 3 in both markets as well. Thus the overall pro…ts for t = 2 through 3 would be 2 + e3 i i = (3V A 2RA ) + (3V B 2 s 3RB ) + " and 2 + e3 j j (2.1) = 0. Note that …rm i makes the same pro…t whether he would tie or not because he is the only …rm in market B. By tie-breaking rule, however, …rm i sells both goods in a bundle. Second, suppose that only …rm j enters at t = 2. In this case there is no switching in market B. Therefore …rm j would sell its upgrade of B at the price of 2V B and …rm i would sell its upgrade of A at 2V A respectively. Then overall pro…ts for t = 2 through 3 are 2 + e3 i i = 3V A 2RA and 2 + e3 j j = 3V B 2RB . (2.2) Finally, suppose that both …rms enter into the market B at t = 2. Now consumers can either keep using Bj and upgrade it, or move to a newly-introduced Bi . In this case the market outcome hinges crucially on …rm i’ tying decision. With …rm i’ tying, consumer’ s s s incentive compatibility to switch to Bi turns out 2 (V A + V B ) s pAB i 2 V B pB () pAB j i pB + 2V A j 2 s, where pAB is the price of …rm i’ bundle of an upgrade of A and a newly-released Bi . Note s i that, under tying regime, if consumers choose to keep using Bi , they cannot upgrade A. If 2V A > 2 + s, …rm i would sell its bundled products, but …rm j would sell nothing. h i A A B + " for …rm The resulting second period pro…ts would be 2V (2 + s) R 2R i and RB for …rm j. Meanwhile, if 2V A < 2 + s, …rm j could sell the upgrade of its product in market B, while …rm i could sell nothing in both markets. The pro…ts would be 33 h i (RA + 2RB ) + " for …rm i and 2 2V A +s RB for …rm j. Then overall pro…ts for t = 2 through 3 are 2 + e3 i i 2 + e3 j j 8 > < (3V A 2RA ) + (V B 2 = > : (V A 2RA ) 2RB + " = 8 > < 3RB ) + " if 2V A > 2 +s if 2V A < 2 s +s (2.3) RB if 2V A > 2 > V B + (2 : + s) 2V A if 2V A < 2 2RB +s + s: Without …rm i’ tying, …rm i could not sell his subsidiary good in market B due to the s quality inferiority and non-negative price constraints. So, consumer’ incentive compatibility s to switch to Bi turns out 2 (V B ) s pB i 2 V B pB () pB j j pB + (2 i In market B, therefore, …rm j would sell the upgrade of Bj at (2 2 +s RB , while …rm i make = (3V A 2RA ) + s) and make a pro…t of 2RB + " . In market A, …rm i would sell the upgrade of A at 2V A , resulting in a pro…t of 2V A 2 + e3 i i + s) . RA . Then overall pro…ts for t = 2 and 3 are 2RB + " and 2 + e3 j j =V B +2 +s 2RB . (2.4) Comparing …rm i’ pro…ts in (2.3) and (2.4) under Assumption 2, we can verify that …rm i s would tie the goods if 2V A > 2 + s, otherwise he would sell them separately. Therefore 34 the pro…ts for t = 2 through 3 are as follows: 2 + e3 i i 2 + e3 j j 8 > < (3V A = > (3V A : = 8 > < 2RA ) + (V 2RA ) B 3RB ) + " if 2V A > 2 s +s if 2V A < 2 2 +s 2RB + " (2.5) RB > VB +2 : if 2V A > 2 +s if 2V A < 2 2RB +s + s: Now we can characterize an overall equilibrium strategy and expected pro…ts at t = 2 when …rm j prevailed in the previous period as the following lemma. Lemma 2 Suppose that V B = V B and Bj has been sold at t = 1. (i) If 2V A > 2 + s, …rm i would enter into both market at t = 2 and sell its tied products while …rm j would not enter into the market B. Pro…ts for t = 2 through 3 are i h 2 + e3 = (3V A B ) + " and e2 + e3 = 0. A ) + 3V B (2 + s + 3R ei 2R j j i (ii) If 2V A < 2 + s, …rm i would enter into only the market A while …rm j would sell its upgraded product in market B. Pro…ts for t = 2 through 3 are e2 + e3 = 3V A i i 2RA and e2 + e3 = 3V j j B 2RB . Proof. From (2.1), (2.2), and (2.5), we can construct the following payo¤ matrix. Firm j enter f(3V A Firm i enter f(3V A not enter B 2 2RA ) +s 2RB + "; V 2RB g if 2V A < 2 n 3V A 2RA ; 3V B 3RB ) s RB g if 2V A > 2 +"; +s Firm i 2RA ) + (V Firm j not enter B +2 f(3V A 2 2RA ) + 3V + s + 3RB + "; 0g +s 2RB o Finding Nash equilibria in each case directly proves the lemma. 35 B n B 3V o 2RB ; 0 This lemma implies that, for a primary good monopolist to succeed in making a late-entry into the subsidiary market, the value of primary good upgrade should be su¢ ciently high compared to the sum of quality superiority of his competitor’ good and switching cost. s (Case 2: when Bi sold at t = 1:) In this sub-section we analyze a case that …rm i prevailed in the …rst period. We will consider the same sub-cases as in Case 1. First, supposing …rm i enters only, he would sell an upgrade of B at the price of 2V B and an upgrade of A at the price of 2V A . By the similar calculation to that in Case 1, …rm i’ pro…t at t = 2 is (2V A s RA ) + (2V B RB ). Then, from Lemma 1, overall pro…ts for t = 2 through 3 are 2 + e3 i i = (3V A 2RA ) + (3V B 2RB ) and 2 + e3 j j (2.6) = 0. It does not matter whether he would tie or not since …rm i is the only seller in both market. By tie-breaking rule, however, …rm i will use tying strategy. Second, supposing …rm j only enters at t = 2, …rm j would introduce a product B with the quality of (2V B + ) in market B, with investing RB . Now consumers can either B keep using Bi without upgrade and get a per-period bene…t of V , or switch to a newlyintroduced Bj and get a per-period bene…t of (2V B + ) with incurring a switching cost, s. So consumer’ incentive compatibility to switch to Bj turns out s 2 (2V B + ) s pB j 2 V B () pB j 2V B +2 s. That is, …rm j would sell his product with an upgraded quality at the price of 2V B +2 s . Thus overall pro…ts for t = 2 through 3 are 2 + e3 i i = (3V A 2RA ) and 2 + e3 j j 36 = (3V B 2RB ) + (2 s) . (2.7) Finally, suppose that both …rms enter into the market B at t = 2. With …rm i’ tying, s consumer’ incentive compatibility to switch to Bj is, s B 2(V A + V ) pAB i 2(V B + ) pB () pAB j i s pB + 2V A j (2 where pAB is the price of …rm i’ bundle of upgrades. Since 2V A > (2 s i s), s) by Assumption 1, …rm i could sell its bundled products at the price of 2V A (2 s). So pro…ts at t = 2 h i are 2V A (2 s) (RA + RB ) for …rm i and RB for …rm j. Thus total pro…ts for t = 2 through 3 are = (3V A 2 + e3 i i 2RA ) + V B (2 s) 2RB and 2 + e3 j j RB . = (2.8) Without tying, consumer’ incentive compatibility to switch to Bj turns out s 2V B pB i 2(V B + ) s pB () pB j j pB + (2 i s). > s, …rm j would sell a good in market B. So pro…ts at t = 2 are (2V A RA RB ) i h for …rm i and (2 s) RB for …rm j. If 2 < s, …rm i would sell a good in market If 2 B. So pro…ts are (2V A RA ) + (s RB for …rm i and 2 ) RB for …rm j. Thus total pro…ts for t = 2 through 3 are 2 + e3 i i 2 + e3 j j 8 > < (3V A = > (3V A : 2RA ) RB 2RA ) + V 8 > B < V + (2 = > RB : B if 2 (2 s) 2RB >s if 2 s if 2 s) < s. Comparing …rm i’ pro…ts in (2.8) and (2.9), we can easily verify that, when both …rms s enter the market B, …rm i would always make a higher pro…t under tying strategy because V B (2 s) 2RB > 0 by Assumptions 2 and 3. The equilibrium pro…ts are those in (2.8). 37 Now the following lemma describes an equilibrium at t = 2 when …rm i prevailed in the …rst period. Lemma 3 Suppose that V B = V B and Bi has been sold at t = 1. Then …rm i would sell its bundled upgrades of A and B at t = 2 and 3 while …rm j would not enter into the market B at t = 2. Pro…ts for t = 2 through 3 are e2 + e3 = (3V A i i 2RA ) + (3V B 2RB ) and e2 + e3 = 0. j j Proof. From (2.6), (2.7), and (2.8), we can construct the following payo¤ matrix. Firm j enter Firm i f(3V A 2RA ) + V not enter (2 s) 2RB ; f(3V A n (3V A 2RA ); (3V B B 2RB ); 0g o 2RA ); 0 +(3V 2RB ) + (2 2RA ) n (3V A RB g enter Firm i B Firm j not enter o s) Finding Nash equilibria in each case directly proves the lemma. This lemma is simply saying that, if the primary good monopolist prevailed in the subsidiary market in the …rst period, he could take the whole market for the rest of the period as well. From Lemma 2 and 3, we can get the following proposition. Proposition 7 Under the given Assumptions 1 to 4, if …rm Bi prevailed at t = 1 it will take the whole market for the rest of the periods. If …rm Bj prevailed at t = 1 …rm i can take the market B away from …rm j only if s < 2(V A ). This proposition indicates that the only case where …rm i could not sell his subsidiary good at t = 2 onwards is that …rm i did not enter into the market B at t = 1 and the switching cost at t = 2 is su¢ ciently high. 38 2.3.3 In the …rst period We analyze a subgame perfect Nash equilibrium for the entire game in this subsection. In the …rst period, …rm i has two alternative strategies on entering into the subsidiary market; to enter in the …rst period, or to enter in the second period when it is possible and pro…table. First, suppose that …rm i does not enter into market B at t = 1 and makes a new entry at t = 2. Since, at t = 1, good B is of no use without good A, …rm i can exploit the entire h i A + V B + ) RA surplus from both markets. So …rm i’ …rst period pro…t would be 3(V s h i B if V B turns out V with a probability of or 3(V A + V B + ) RA if V B turns out V B with a probability of (1 ). Meanwhile …rm j’ pro…ts would be s RB . In this case the second period problem corresponds to the Case 1. Thus, from Lemma 2, total pro…ts for the entire 8 > > > > < i= > > > > : j = 8 > > > > < > > > > : game are (6V A B 3RA ) + (6V (6V A 3RA ) + 3(V + ) (6V A 3RA ) + 3(V B + 1 ) RB 3V B 3RB ) + B C A with if s > 2 V A with (1 1 if s < 2 V A 3RB s + " if s < 2 V A C A if s > 2 V A RB ), with with (1 ). Then expected pro…ts are E( e i ) = E( e j ) = 8 > (6V A > > > > > > < > (6V A > > > > > > : 8 > < > : RB h B 3V 3RA ) + h (6V B 3RB ) + i s + " + (1 h ) 3(V B + if s < 2 V A h i h B A) + ) 3(V B + 3R 3(V + ) + (1 if s > 2 V A i ) i ) . (2.10) 3RB i if s < 2 V A (1 )RB 39 if s > 2 V A . Second, suppose that …rm i enters into market B in the …rst period. In this case …rm i’ s entering at t = 1 without tying is always dominated by entering at t = 2, due to the existence of ", R&D cost savings. So, if …rm i enters in the …rst period, it always accompanies tying strategy. With tying …rm i’ pro…t at t = 1 would be (3V A s probability of and (3V A RA ) + (3V B RA ) + (3V B RB ) with a probability of (1 RB ) with a ) while …rm j’ s RB . In this case the second period problem corresponds to the Case 2. pro…t would be Thus, from Lemma 3, total pro…ts for the entire periods are 8 > < (6V A 3RA ) + (6V B 3RB ) with i= > (6V A 3RA ) + (3V B RB ) with (1 : ), and RB . j = Then …rm’ expected pro…ts are s E( e i ) = (6V A E( e j ) = 3RA ) + (6V B 3RB ) + (1 )(3V B RB ) (2.11) RB : Now comparing …rm i’ expected pro…ts between (2.10) and (2.11), we can …nd …rm i’ s s optimal entry decision. When s < 2 V A (6V B 3RB ) + (1 )(3V B h RB ) 7 () 7 , (6V B i 3RB ) + RB + 3 RB + s + 2 s + " + (1 (6V ) 3(V B + 1. " That is, …rm i makes an entry at t = 1 rather than at t = 2 if s>2 VA h > 1 . Meanwhile when , B 3RB ) + (1 )(3V B () That is, …rm i makes an early-entry if RB ) 7 7 h 3(V RB + 3 3V B > 2. 40 2R B + 2. i ) + (1 h ) 3(V B + i ) i ) Then the following proposition describes the subgame perfect equilibrium of the entire game. Proposition 8 Subgame perfect equilibrium of the game is as follows. (i) When s < 2 VA , …rm i enters into the market B at t = 1 if otherwise. (ii) When s > 2 V A > 1 but enters at t = 2, , …rm i enters into the market B at t = 1 if > 2 but enters at t = 2, otherwise. (iii) Equilibrium expected pro…ts, E( e i ) and E( e j ), are those de…ned in (2.10) if …rm i enters at t = 2 and those de…ned in (2.11) if …rm i enters at t = 1. By entering in the …rst period, …rm i have to take the risk of B not being successful. On the contrary, if B is valued high by consumers it can secure the future pro…ts coming from periodic upgrades. So this proposition states that the higher the probability of B’ being s successful, the more incentive …rm i has to enter in the …rst period. Moreover, from the relation between 1 , 2 and s, we get the following proposition. Proposition 9 The probability for the …rm i to enter into the market B at t = 1 is 8 > 0 > > > < (1 > > > > : (1 1) 2) if s 2 [0; ) h if s 2 ;2 V A h if s 2 2 V A ;1 , where 1 is increasing in s but 2 is independent of s, such that (1 1 ) < (1 2) Note that 1 > 2 by Assumption 2. If switching costs are su¢ ciently low, …rm i has no incentive to enter into the market B in the …rst period. This is because, in spite of lateentry, …rm i can always take the market away from …rm j while reducing the risk of B’ s being unsuccessful. For the intermediate switching costs, the probability is monotonously increasing in s because high switching costs reduce …rm i’ second period pro…t even when s …rm i succeeds in taking the market away from its rival. Finally, if switching costs are 41 Figure 2.2: Switching Cost and the Probability of Firm i’ Entry at t = 1. s su¢ ciently high, the probability is …xed at (1 2 ), which is always higher than (1 1 ). The reason is that …rm i0 s pro…t does not depend on switching costs anymore, since in this case …rm i’ choice would be constrained between two alternatives, entering in the …rst period s or not entering at all. Figure 2.2 plots this relationship. 2.4 Policy Implication In this model, all consumers demand both one primary good and one subsidiary good in each period, whether they are initial products or upgrades. Moreover the entire value of the product is distributed between consumers and producers in the form of consumer’ surplus s and …rm’ revenue. Then the total welfare is the sum of the product values sold in both s markets net of switching costs and R&D costs. First, suppose that tying is prohibited. Then the subsidiary good with high quality would always prevail in the market and no switching 42 happens. So the total welfare is would be (6V A 3RA ) + (6V B 3RB ) + (1 +3 )(3V B + 3 RB ). (2.12) Second, suppose that tying is allowed and …rm i enters in the …rst period. Then …rm i would prevail in market B for the entire period and …rm j would only incur R&D cost in the …rst period. Thus total welfare would be (6V A 3RA ) + (6V B 3RB ) + (1 )(3V B RB ) RB . (2.13) Third, suppose that tying is allowed and …rm i does not enter in the …rst period. Then …rm j would prevail in the …rst period. From the second period, …rm j would prevail for the rest of the period as well if switching cost is su¢ ciently high, but …rm i would prevail otherwise. Thus total welfare would be, if s > 2 V A (6V A 3RA ) + (6V and, if s < 2 V A (6V A 3RA ) + (6V B +3 , 3RB ) + (1 )(3V B + 3 RB ) and (2.14) , B + 3RB ) (s + RB ") + (1 )(3V B + 3 RB ). (2.15) From (2.12) and (2.13), it is apparent that if …rm i’ entry into the market B through s tying from the …rst period it brings negative e¤ect on social welfare by (3 + RB ). This re‡ ects the fact that consumers have to bear utility loss due to the use of low-quality good and there is waste of resources resulting from overlapping investments between two …rms. From (2.12) and (2.14), we can see that there is no loss in social welfare if …rm i did not enter in the …rst period and switching cost is su¢ ciently high. The reason is that in this case there is no switching and no overlapping investments since high switching cost prevent …rm i from entering late. However, from (2.12) and (2.15), if …rm i didn’ enter in the …rst period t 43 and switching cost is not su¢ ciently high, tying has again negative e¤ect on social welfare by (2 + s + RB "). In this case the welfare loss comes from the quality inferiority of the good, consumer’ switching costs, and …rm’ overlapping investment. s s The above welfare analysis veri…es again the fact that primary good monopolist’ tying s strategy may have negative e¤ect on social welfare. This negative e¤ect asks antitrust authorities to pay careful attention in scrutinizing this type of tying behavior. In this respect, the results in the previous section provide important policy implication. It suggests another criterion that should be considered in judging whether monopolist’ tying behavior is antis competitive or not. So far most of the criteria have focused on indices related with market structure, such as concentration ratio, market shares, etc. This paper suggests that they have to consider both technical aspects of the goods that determine the size of switching cost and the status of market evolution as well. If a subsidiary product incurs high enough switching costs so that late-entry into a subsidiary market through tying is di¢ cult, a primary good monopolist is more likely to enter early with tying than to enter late. In this case antitrust authorities should pay more attention in the early stage of market maturity since after that the monopolist incentive to enter into the subsidiary market decreases gradually. Meanwhile if a subsidiary product incurs relatively low switching costs, the monopolist is more likely to enter late through tying than to enter early. In this case the authorities should pay more attention as the market becomes more matured. 2.5 Concluding Remarks We study an issue on primary good monopolist’ entry decision to expand its monopoly s power to the relevant subsidiary market. A focus is taken on the timing of entry with tying strategy. When the quality of the subsidiary good is uncertain, the …rm’ entry decision s depends on the size of switching cost. If switching costs are su¢ ciently low, the …rm always makes a late-entry. As the switching costs go up, the probability for the …rm to make an 44 early-entry also increases. If the switching cost is su¢ ciently high, this probability is …xed at certain high level. Once the monopolist succeeds in entering into the subsidiary market, the …rm can take up the market forever and exploit entire expected future pro…ts from upgrades. Therefore antitrust authorities need to scrutinize …rm’ tying behavior more strictly in the s early stage of market evolution if switching costs are high, vice versa if those are low. In this paper, I analyze a case of selling, while Carlton and Waldman (2005) considers cases of both selling and renting. If both …rms rent the product instead of selling, the result is quite simple. In this case the new version does not need to compete with its old version anymore. Then the monopolist will not tie the products. Instead he exploits whole market pro…t by charging a high price on the primary good, as addressed by Chicago School argument. 45 BIBLIOGRAPHY 46 BIBLIOGRAPHY Bulow, J. "Durable Goods Monopolists." Journal of Political Economy, 1982, 90, pp. 314-332. Carlton, D.W. and Waldman, M. "The Strategic Use of Tying to Preserve and Create Market Power in Evolving Industries." Rand Journal of Economics, 2002, 33, pp. 194-220. Carlton, D.W. and Waldman, M. "Tying, Upgrades, and Switching Costs in DurableGoods Markets." NBER Working Paper, 2005, June No. 11407. Choi, J.P. and Stefanadis, C. "Tying, Investment, and the Dynamic Leverage Theory." Rand Journal of Economics, 2001, 32, pp. 52-71. Farrell, J. and Shapiro, C. "Dynamic Competition with Switching Costs." Rand Journal of Economics, 1988, 19, pp. 123-137. Fudenberg, D. and Tirole, J. "Upgrades, Tradeins, and Buybacks." Rand Journal of Economics, 1988, 29, pp. 235-258. Gilbert, R.J. and Katz, M.L. "An Economist’ Guide to U.S. v. Microsoft." Journal of s Economic Perspectives, 2001, 15, pp. 25-44. Kim, B.-C. "The strategic Tying with Consumer Switching costs." mimeo, 2007 Klemperer, P. "Markets with Consumer Switching Costs." Quarterly Journal of Economics, 1987, 102, pp. 375-394. Nalebu¤, B. "Bundling as an Entry Barrier." Quarterly Journal of Economics, 2004, 119, pp. 159-187. Whinston, M.D. "Tying, Foreclosure, and Exclusion." American Economic Review, 1990, 80, pp. 837-859. Whinston, M.D. "Exclusivity and Tying in U.S. v. Microsoft: What We Know, and Don’ Know." Journal of Economic Perspectives, 2001, 15, pp. 63-80. t 47 Chapter 3 Customer Return Policy as a Signal of Quality 3.1 Introduction When you place an order for a good, especially online, you may not be completely sure of product quality. When you receive the item, its true quality might be higher or lower than anticipated. If the expected value of the good is greater than your willingness to pay, you will buy it in spite of the quality uncertainty. Otherwise, the uncertain quality of the product may prevent purchase, even though true quality is high. In the latter case customer return policy, which allows customers to return the item within a speci…c period, may attract potential buyers by reducing customer risk. After receiving the item, product quality becomes known. If you are satis…ed with the quality, you would keep and use it; otherwise you could simply ship it back to the seller and get a refund of it. In this sense a seller’ return policy enables customers to defer their purchase decision until more is known s about product quality. We can observe a variety of return periods across sellers and types of goods. One feature in common with them is that most of the sellers with high reputation on product quality o¤er a relatively long return period while smaller retailers or private sellers o¤er a relatively 48 short one. For example, on Ebay where various qualities of goods are sold, private or usedgood sellers o¤er shorter return period than commercial or new-good sellers.1 In this paper, we address economic rationale for this variation of return periods, focusing on its role as a signaling device for product quality. Consumers always bene…t by a longer return period since they can collect more information on product quality. Meanwhile seller pro…ts are a¤ected by a longer return period in two di¤erent ways. On the one hand, a longer return period increases the depreciation loss from the returned items. On the other hand, it increases the chances that consumers get information about true product quality. The former unilaterally lowers seller’ pro…ts. s The latter, however, brings di¤erent e¤ects on seller’ pro…ts depending on the qualities of s the products they sell. Since the precision of information is positively correlated with the length of return period, a longer return period leads to a high return rate for a low-quality seller and a low return rate for a high-quality seller. Therefore the high-quality seller would bene…t whereas the low-quality seller would be harmed. In this sense a length of return period can be used as an e¢ cient signaling device for product quality. That is, by o¤ering a longer return period that cannot be imitated by a low-quality seller, a high-quality seller can di¤erentiate himself from a low-quality seller. To address this issue, we set up a model where a seller o¤ers both a price and a length of return period. We adopt the speci…c information structure, in which only true information on product quality is revealed during the return period and precision of information is increasing function of the length of return period.2 In other words, if customers receive a signal on the product quality during return period, they can perfectly …gure out the true quality of the product, but a probability to get the signal is in proportion with the length of the period. Taking this information structure into consideration, a seller quotes a price and a return 1 On Ebay, conditions of an item are labeled as ’ New, New others, Used, and For parts or not working’according to the quality of the product. See more details on Ebay website. (http://pages.ebay.com/help/sell/contextual/condition_11.html) 2 The loss of depreciation comes not only from actual damage of being used, but also from the devaluation by being re-categorized as an open box item after the return is accepted. In this case the loss of depreciation may not be proportional to the length of return period. Meanwhile throughout the paper we assume time-proportional depreciation loss. 49 period. After observing the o¤er and updating their belief on product quality, customers decide whether or not to accept the o¤er. The main results are as follows. First, without consumer’ interim bene…ts3 during the s return period, there exist multiple separating equilibria, where a high-quality seller o¤ers a positive length of return period that is longer than a speci…c critical level, but a low-quality seller does not o¤er return service. By doing so, a high-quality seller can fully convince customers of the quality of the product he sells. Interestingly, at the separating equilibria, both types of sellers charge the same price as that of perfect information. Moreover, all the separating equilibria satisfy the Cho-Kreps intuitive criterion.4 However, we …nd that there is no pooling equilibrium in which both types choose the same level of price and return period. This is because any pooling strategy is dominated by high-quality seller’ deviation s to the strategy in which he o¤ers a perfect information price and a maximum refund period. Second, with consumer’ interim bene…ts, both separating and pooling equilibria could exist s depending on the speci…c forms of information and depreciation function. Even if there exist multiple separating equilibria, the unique separating equilibrium that satis…es the intuitive criterion is the one with the smallest return period. Some of pooling equilibria could also survive the intuitive criterion. This paper builds on two veins of literature; the one is return policy and the other is signaling quality. There is a lot of economic literature on customer return policy. Che (1996) studies customer return policy with experience goods and shows that a seller will adopt return policy if customers are highly risk-averse or retail costs are high. Risk-averseness plays a crucial role in seller’ adopting return policy. In this paper, however, we assume s risk-neutral customers and implement a potential depreciation loss from the returned items as a restriction on seller’ behavior. Ben-Shaha and Posner (2010) assume depreciation loss s 3 Consumer’ interim bene…ts are the utilities from the trial use of the items during the s return period. 4 Cho and Kreps (1987) provide a way to re…ne equilibrium concept by restricting beliefs o¤-the-equilibrium path. That is, if a deviation is observed, the receiver believes that the deviation is not made by a type for whom the deviation is equilibrium-dominated. 50 and information structure similar to our paper5 and address the e¤ect of implementing a mandatory return policy. They propose that the mandatory return policy should be neither too strict nor too generous to balance consumer protection and seller’ depreciation loss due s to customer’ abuse of the policy. Meanwhile we address the return policy in view of seller’ s s signaling device. There is a vast amount of literature on signaling quality. Among them, the closest signaling device to customer return policy is seller’ warranties for durability of the product. s For example, Spence (1977) shows that there is a unique separating equilibrium in a competitive market where a high-quality seller o¤ers better warranty than a low-quality seller because the low-quality seller has to incur higher costs when an actual break-down happens. Grossman (1981) also draws a similar result assuming a single seller, in which warranty service is provided according to ex post veri…able events that depend on product quality. Meanwhile Gal-Or (1989) shows that, in a duopoly model, the signaling e¤ect of warranty could be limited if product durabilities net of warranty are either too close or too di¤erent. However, as pointed out by Moorthy and Srinivasan (1995), warranty is somewhat di¤erent from return service or money-back guarantee; it takes the focus on performance-based warranties, it induce consumer’ moral hazard due to relatively long warranty periods, and s it compensates only partial value of the good. Moreover, the ultimate purchasing-decision is made at the initial purchase stage under warranties, while the decision would be made at the end of return period under customer return policy. In this sense, customer return policy gives better protection for consumers, thereby stronger signaling e¤ect than warranty. Beside them, there are so many papers studying a variety of signaling devices; uninformative advertising in Milgrom and Roberts (1986), high and declining prices in Bagwell and Riordan (1991), product compatibility in Kim (2002), and bundling in Choi (2003). Recently Bourreau and Lethiais (2007) evaluated the use of free contents as a signaling device, using the same information structure as ours, and show that at separating equilibria a high-quality 5 The information structure in Ben-Shaha and Posner (2010) is a little bit di¤erent from that in ours in that a low-quality seller can send either a true or a false signal. In this paper, however, we assume signals are always true. 51 seller provides positive amount of free contents. In this paper, we study the length of return period instead. The remaining parts of the paper are as follows. In section 2, we present a signaling model, focusing on the information structure. In section 3, we solve basic maximization problems for a high-quality seller and a low-quality seller respectively. In section 4, we explain separating and pooling equilibria without consumer’ interim bene…ts. In section 5, s we address separating and pooling equilibria with consumer’ interim bene…ts. In section 6, s we give concluding remarks. 3.2 Model There is one …rm who sells an experience good. The quality of the good, chosen by nature, can be either vH or vL , where 0 < vL < vH and vH 2vL , meaning the quality di¤erence is not extremely large. A seller with a type of i 2 fH; Lg quotes a price, pi , to a buyer and at the same time o¤ers a return period, ti , as well. So seller’ strategy pro…le s is si = (ti ; pi ). If the item is returned at the end of the return period, the seller incurs loss caused by depreciation damage, (t)vi , which is continuous function of depreciation ratio. We assume that 0 (t) > 0 and 00 (t) < 0, which means the depreciation ratio is increasing in t at a diminishing rate There is a continuum of consumers whose preferences for the good are heterogeneous. Preference for quality, , is assumed to be uniformly distributed between 0 and 1, i.e., u[0; 1]. Before the seller makes an o¤er, consumers have common prior belief on the quality of the good, denoted by = Pr(vi = vH ) and (1 ex ante expected quality is vH + (1 ) vL ) = Pr(vi = vL ), thus consumer’ s v. Consumers have a unit demand for the good, so they will buy it if the net utility is greater than 0, that is, U =( v 52 p) > 0: So consumers ex ante expected net utility is ( v p). After purchasing the good, consumers can get information about product quality by inspecting or using it during the o¤ered return period. If they are not satis…ed with the realized quality or their ex post expected net utility is negative, they can return it to the seller at no cost. We assume no buyer’ interim bene…t s during the return period and no discount factor in consumer utility. We further assume that during the return period consumers do not incur any opportunity cost in keeping the item. This assumption enables them to make a purchase decision at the end of the return period, not during the period, so simpli…es subsequent analysis. Information structure is as follows. During the return period, consumers receive a signal, H or L depending on seller types, with a probability of , which contains true information on the product quality. That is, if they receive a signal they can …gure out the quality of the good for sure. Meanwhile if they do not receive a signal, which denoted by ?, with a probability of (1 ), they will maintain their prior belief on product quality. We assume that the precision of the information, which is equal to the probability to receive a signal, is (t) is continuous in t such that 0 (t) > 0 and increasing in the length of return period; 00 (t) < 0, which means the probability is increasing in t at a diminishing rate. We further assume (0) = 0 and (t) = 1, where t is su¢ ciently large so that consumers always receive a signal with a probability of 1. This information structure can be formalized as follows: Pr( i jvj ) = (t) if i = j and Pr(?jvj ) = 1 0 if i 6= j (t), where i 2 fH; Lg and j 2 fH; Lg, and Pr(vH j H ) = 1, Pr(vH j L ) = 0, and Pr(vH j?) = . The timing of the game is as follows. After the quality of the good determined by nature, a seller with vi o¤ers si = (ti ; pi ) depending on his type. After observing seller’ o¤er, s consumers update their beliefs on the quality of the product and choose whether or not to 53 accept the o¤er. After purchasing the good, consumers receive a signal i during the return period and decide whether or not to return it at the end of the return period. If return is claimed by consumers, the seller will accept it and incur depreciation loss. 3.3 Basic Analysis Under perfect information, a consumer with such that (vi 0, where i = fH; Lg, p) will buy the good. The location of the critical consumer who is indi¤erent between buying v v p and not buying is i = v . So a seller will charge pi = 2i and obtain a pro…t of i = 4i i depending on his type. Under the quality uncertainty, if there is no customer return policy, a p critical consumer locates at v = v , and a seller will charge v and get a pro…t of v regardless 2 4 of his type. So a seller with a low-quality good will bene…t from quality uncertainty, while a seller with a high-quality good will be harmed since vL < v < vH . Now suppose that a seller adopts a customer return policy. Let denote the location of the critical consumer who is indi¤erent between buying and not buying. We need to consider two cases separately according to the location of . First, suppose that 2 [ v ; L ).6 Under a given length of return period, if consumers do not receive a signal, with probability of (1 (t)), consumers who purchased the good will expect a net utility of (v receive a signal, with a probability of (t), their net utility would be (vH p). If they p) if the signal is H , or 0 if L . Therefore the total expected utility will be U = (1 (t))(v p) + (t) (vH p); and the critical consumer locates at = So 1 1 (1 (1 ) (t) vL ) (t) v p p < = v. v v cannot be in between v and L . Second, suppose that 2 [ H ; v ). If consumers 6 Note that a critical consumer should always be between H and L . 54 do not receive a signal, those who purchased a good will return it and get a utility of 0. If they receive a signal, their net utility would be (vH p) if the signal is H , but 0 if L . Therefore the total expected utility will be U = (t) (vH locates at p); and the critical consumer = H. consumer location j— — — — — j— — — — — -j— — — — – — — — – j— j 0 = H v Therefore, under customer return policy, consumers with L 1 2 [ H ; 1] will buy the good. The intuition is quite obvious. Since there is no return fee and no cost in keeping the item, consumers who can get at least positive net utility from the best possible quality will purchase the good. Then we can characterize consumer responses pro…le as follows: 8 > 2 [0; ): Not buy at all. > > H > > > > Consumer < 2 [ H ; v ): Buy, and return if not receiving H , otherwise retain. > responses > 2 [ v ; L ): Buy, and return if receiving L , otherwise retain. > > > > > : 2 [ ; 1]: Buy, and always retain. L Here we have four segments which have di¤erent response pro…les respectively. Consumers in the …rst segment will not buy the good at purchase stage since their preferences for the good is su¢ ciently low while consumers in the last segment will buy and never return since their preference is su¢ ciently high. Consumers in the second segment have relatively-low preference, so they will retain the good only if they are sure of high quality. Meanwhile, consumers in the third segment have relatively-high preference, so they will retain the good as long as they are not sure that the quality is low. Based on these consumer responses, the seller faces di¤erent demand and pro…t functions depending on the quality of the product he sells. In the remaining parts of this section, we solve each seller’ pro…t maximization problem when a return period is given and the belief s is …xed at , and …nd an equilibrium price and pro…t as a function of return period. 55 3.3.1 Seller with a low-quality good Suppose that a seller provides a low-quality good, that is v = vL . When he o¤ers (pL ; tL ), consumers with 2 [ H ; 1] would buy the good at purchase stage. At the end of the o¤ered return period, consumers with 2 [ H ; v ] would return it for sure and those with 2 [ v ; L ] would return it with a probability of (t). Thus a low-quality seller’ expected s pro…t after the return period ends is 8 > > p [(1 > > < = p [(1 L > > > > : (t)) (1 L ) + (1 v )] (t)vL [ (t) (1 (t)) ( L (t)vL [ (t) ( L v) + ( v H )] if L > 1, (3.1) v )] v) + ( v H )] if L 1, where L denotes the pro…t for a seller with a low-quality goods.7 Solving pro…t maximizing problem, we can …nd seller’ equilibrium price and pro…t, given the length of return period, s and characterize consumer’ equilibrium response pro…le. The following lemma summarizes s it. Lemma 4 Suppose that consumers are uncertain on the quality of the product and a seller o¤ers both a price and a return policy. Then, when consumer’ belief on the quality is s …xed at with given the length of return periods, a low-quality seller o¤ers pL and consumers 2 [ H ; 1] will purchase the good. At the end of the return period consumers with 2 [ H ; v (pL )) would return it for sure, those with 2 [ v (pL ); L (pL )) would do so if they received a signal, and the rest of them would keep it. An equilibrium price and seller’ pro…t s 7 The case that v > 1 is of no interest since no one would buy and keep the good. 56 are pL (t) = v 2 v L (t) = 4 vL (t)v + (1 vL (t)v + (1 (t))vL 2 (t))vL 2 (t)v + (1 41 (t) v (t)v + (1 41 (t)) vL (t) (t)) vL v 3 v vL v L H 5, v 32 (3.2) vL v L H 5 . Proof. See Appendix. From this lemma, we can verify that an equilibrium price and a pro…t are#decreasing in t, " v h i (t)v+(1 (t))vL vL v L vL @ 1 @ H (t) have negative since both @t (t)v+(1 (t))v ) and @t v L signs. The intuitions are as follows. As for the decreasing-price in t, a low-quality seller would face higher returns if he o¤ers a longer return period while the amount of sale is …xed at (1 H ). This, in turn, would force a low-quality seller to lower the price in order to reduce the returns. As for the decreasing-pro…t in t, the better information harms a lowquality seller in two ways; it lowers …nal demand for the good and increases depreciation loss. Actually the pro…t of a low-quality seller decreases from vL 4 v (v vL )+vL (vH vL ) (t) H vv 1 H 2 L (0) = v to 4 L (t) = as t increases from 0 to t. If t is su¢ ciently large, a low-quality seller would make a pro…t lower than that of perfect information case, which provides a basis on which a high-quality seller can di¤erentiate himself from a low-quality sell by using return period as a signaling device for product quality. 3.3.2 Seller with a high-quality good Suppose that a seller provides a high-quality good, that is v = vH . When he o¤ers (pH ; tH ), consumers with 2 [ H ; 1] will buy the good at purchase stage. At the end of the o¤ered return period, consumers with (1 2 [ H ; v ] will return it with a probability of (t)) and the rest of them will keep it. Thus high-quality seller’ expected pro…t after s 57 the return period ends is 8 > < p [ (t) (1 (t)vH (1 H )] = H > p [(1 : v ) + (t) ( v H )] where (t)) (1 if v > 1, H) (t)vH (1 (t)) ( v (3.3) if v < 1, H) 8 H denotes the pro…t for a seller with high-quality good. Solving pro…t maximizing problem again, we can …nd seller’ equilibrium price and pro…t, given the length of return s period, and characterize consumer’ equilibrium response pro…le. The following lemma sums marizes it. Lemma 5 Suppose that consumers are uncertain on the quality of the product and a seller o¤ers both a price and a return policy. Then, when consumer’ belief on the quality is …xed s at given the length of return periods, a high-quality seller o¤ers pH and consumers with 2 [ H (pH ); 1] will purchase the good. At the end of the return period consumers with 2 [ H (pH ); v (pH )) will return it if they did not receive a signal, and the rest of them will keep it. An equilibrium price and the seller’ pro…t are s pH (t) = vH (t)v + (1 v 2 (t))vH 1 (t) (1 (t)) (vH v v) , (3.4) H (t) = v 4 vH (t)v + (1 (t))vH 1 (t) (1 (t)) (vH v v) 2 . Proof. See Appendix. On the contrary to the low-quality seller’ case, we cannot unilaterally determine the s h i vH @ movement of equilibrium price and pro…t as t changes. While @t v is al(t)(v v) H @ ways positive, @t 1 0 (t) 1 (t) (t) (1 (t))(vH v ) v 0 (t) H can be positive or negative depending on whether is greater or smaller than (t) . This re‡ ects the fact that a high-quality seller faces 8 The case that H > 1 is of no interest since no one would buy the good. 58 Figure 3.1: Seller’ Pro…ts s trade-o¤ between revealing quality and incurring depreciation loss when he increases t. So the entire sign of @PH (t) @ H (t) and will be determined by the speci…c functional forms @t @t of both (t) and (t). Intuitively, when t is relatively small, the e¤ect of depreciation loss dominates the e¤ect of revealing quality. Speci…cally if the good incurs a huge depreciation loss in earlier period, the pro…ts might decrease from v H (0) = 4 . However, as t increases, the quality revealing e¤ect would dominates depreciation loss e¤ect, so the pro…t would eventually increase and converge to vH H (t) = 4 , which provides an incentive for a high-quality seller o¤er longer return period. In Figure 3.1, three possible pro…t curves of high-quality seller are shown, coupled with that of low-quality seller. 3.4 Equilibrium in Signaling Game In this section we analyze equilibrium in signaling game, in which a return period is used as a signaling device. We will look into the existence of both separating and pooling 59 equilibrium, and re…ne them by Cho-Kreps ’ intuitive criterion.’ 3.4.1 Separating Equilibrium In this sub-section we …nd separating equilibria. Suppose that there exists a separating equilibrium, where a high- and a low-quality seller o¤er sH = (tH ; pH (tH )) and sL = (tL ; pL (tL )) respectively. At any separating equilibria consumers can perfectly differentiate seller’ types, that is consumer’ posterior beliefs after observing seller’ o¤er s s s are Pr(vH ; sH ) = 1 and Pr(vH ; sL ) = 0. Substituting = 0 into pL (t) and L (t) in h i v v vL (tL ) H L and Lemma 4, low-quality seller’ price and pro…t are pL (tL ) = 2 1 s vH i2 h vL v v (tL ) H L on the separating equilibria. This implies that the lowvH L (sL ) = 4 1 v L quality seller would set tL = 0 and pL (tL ) = 2 , resulting a pro…t of Meanwhile, substituting L (sL ) = vL 4 . = 1 into pH (t) and s H (t) in Lemma 5, high-quality seller’ price vH vH and pro…t at the separating equilibria are …xed at pH (tH ) = 2 and L (sH ) = 4 . Then the next proposition characterizes the separating equilibria. Proposition 10 (i) There exist multiple separating equilibria such that sL = (tL ; pL (tL )) = vL vH (o; 2 ) and sH = (tH ; pH (tH )) = (tH 2 ftse ; tg ; 2 ) where tse satis…es (tse ) v (tse ) + L vH The resulting pro…ts are 1 H (sH ) = vL vH + vH 4 and r (tse ) + (1 L (sL ) = v (tse )) L = 1: vH vL 4 . At any separating equilibria consumer’ posterior belief is Pr(vH ; sH ) = 1 and Pr(vL ; sL ) = 0. (ii) All the separating s equilibria satisfy the Cho-Kreps ’ intuitive criterion.’ Proof. See Appendix. Proposition 10 shows that we have multiple separating equilibria, in which a seller with a high-quality good o¤ers a positive return period beyond a certain speci…c level while a seller 60 Figure 3.2: Separating Equilibria with a low-quality good provides no return service. As we’ seen in the previous section, ve low-quality seller’ pro…t is decreasing in t and smaller than that of perfect information if s t > tse . Therefore, by o¤ering return period longer than tse , a high-quality seller can prevent low-quality seller’ imitation and convince consumers that he is selling a high-quality good. s Interestingly, the prices and the pro…ts are the same as those under perfect information. The reason is that at the separating equilibria seller’ type is perfectly inferred by customers, so no s return happens. This also guarantees that at any separating equilibria the intuitive criterion is always satis…ed since all of the pro…ts on the equilibrium path are the highest ever pro…t a high-quality seller could make. From the welfare perspective, at separating equilibria, the social welfare is optimal since the outcome is the same as that of perfect information and, moreover, no return happens. 61 3.4.2 Pooling Equilibrium In this sub-section we look into the pooling equilibria. Suppose that there exists a pooling ~~ equilibrium, where sH = sL = s = (t; p). Consumers do not update their beliefs, so their ~ ex post beliefs are still Pr(vH ; s) = ~ and Pr(vL ; s) = 1 ~ . As for the belief on o¤-the- equilibrium path, we assume that Pr(vH ; s 6= s) = 0. Then the best o¤-the-equilibrium ~ pro…t for the low-quality seller is v M ax L (s 6= s) = M ax L 1 ~ t t 4 v vL 2 (t) H , vH vL and the maximized pro…t is 4 at t = 0. Similarly, the best o¤-the-equilibrium pro…t for the high-quality seller is v M ax H (s 6= s) = M ax L ~ t t 4 vH (t)vL + (1 (t))vH 1 (t) (1 (t)) (vH vL vL ) 2 , vH vH and the maximized pro…t is 4 at t = t. Note that 4 is the best pro…t that the high-quality seller could get under perfect information. This means that, at any pooling equilibrium, the pro…t for the high-quality seller cannot exceed the best o¤-the-equilibrium pro…ts, where the v H seller charges 2 and o¤ers the return period of t. Thus we have the following proposition. ~~ Proposition 11 There exists no pooling equilibria, such that sH = sL = (t; p). That is, at any pooling strategy, the high-quality seller has an incentive to deviate to the strategy o¤ering a price of perfect information and a maximum refund period. Thus any pooling strategy cannot be supported as equilibrium. The result of the analysis in this section can be summarized that, without consumer’ s interim bene…ts, there exist only multiple separating equilibria in signaling game and all of them satisfy intuitive criterion. This may not seem to be consistent with the reality in that it implies that sellers would be indi¤erent between tse and t. In reality, however, the 62 seller seems to prefer shorter return period if other things are equal, especially when the return actually happens. This di¤erence basically comes from the assumption that there is no interim bene…t for consumers.9 On the contrary, if we assume consumer’ interim s bene…ts, the smallest t among separating equilibria satis…es the intuitive criterion, as will be discussed in next section. 3.5 Extension: Equilibrium with Consumer’ Interim s Bene…ts In the previous section, under the assumption of no consumer’ interim bene…t, we s found that there exist multiple separating equilibria that satisfy intuitive criterion. Now we relax that assumption and show that, with consumer’ interim bene…ts, the separating s equilibria that satisfy the intuitive criterion boils down to the unique one with the smallest return period among them. When there exist consumer’ interim bene…ts, a seller would s not o¤er excessively long return period because it results in a surge of interim bene…tpoaching consumers, who have no intension of retaining the good. Actually the maximum ^ return period, t, is limited to the level at which the pro…t maximizing price is non-negative, ^ ^ speci…cally (t) = 1 as shown later. Thus we restrict our attention to t 2 [0; t] in this section. 2 Suppose that there are consumer’ interim bene…ts and, for simplicity, the size of interim s bene…t is the same as that of depreciation loss. At purchase stage, all consumers would buy the product, that is = 0, whether the quality of the product is high or low. This is because consumers can get at least positive interim bene…ts without incurring return fee or any other opportunity cost in keeping it. At the end of return period, consumers would compare the remaining expected value of the good with the price he paid, and keep the good if the former is greater than the latter, or return it otherwise. Then we can identify new 9 Meanwhile consumer’ return fee doesn’ a¤ect the previous two propositions qualitas t tively. It only shifts the tse to the right as discussed in next section. 63 locations of critical consumers, similar in section 3, as follows: _H (1 Then consumers with p (t))vH p _v (1 (t))v _L (1 p . (t))vL 2 [ _ L ; 1] would not return it in any case, those with 2 [ _ v ; _ L] would return it if receive L , those with 2 [ _ H ; _ v ] would return it if not receive H , and those with 2 [0; _ H ] would return it for sure. In this case seller’ pro…ts would be s h _ L = p (1 _ L ) + (1 (t)) _ L _v i (t)vL h (t) _ L _v + _v i (3.5) i (3.6) for a low-quality seller10 , and h _ H = p (1 _ v ) + (t) _ v _H i h (t)vH (1 (t)) _ v _H + _H for a high-quality seller11 . Solving maximization problem respectively, we have the following 10 In this section we pay attention only to the case that _ L h1 since, otherwise, ilo_ L > 1, _ L = p (1 cal maximum doesn’ exist. Formally, supposing t (t)) 1 _ v h i (t)vL (t) 1 _ v + _ v . Solving the maximization problem, and substituting pL into _ L , _ h i vL _L = v 1 (t)(1 + v ) 1 since vH 2vL . This contradicts to the given assumption. 2v L 11 Similarly we pay attention only to the case that _ v 1 since, otherwise, i loh cal maximum doesn’ exist. Formally, supposing _ v > 1, _ H = p (t) 1 _ H t h i (t)) 1 _ H + _ H . Solving the maximization problem, and substituting (t)vH (1 h i vL _ v , _ v = vH 1 pL into _ (t) 1 + v 1 since vH 2vL . This contradicts to the given 2v H assumption. 64 equilibrium prices and pro…ts, without updating belief, similar to those in section 3, pL = _ v (1 (t))vL 1 2 (t)v + (1 (t)) vL (t) (t)v + (1 (t)) vL , (1 (t))v (3.7) 2 = v (1 (t))vL 1 4 (t)v + (1 (t)) vL (t) (t)v + (1 (t)) vL (1 (t))v pH = _ (1 (t))vH v 1 2 (t)v + (1 (t)) vH (t) (t)v + (1 (t)) vH , (1 (t))v _ L , (3.8) _ v (1 (t))vH 1 H = 4 (t)v + (1 (t)) vH (t) (t)v + (1 (t)) vH 2 . (1 (t))v Note that _ L (pL ) < 1 for a high-quality seller and _ v (pH ) < 1 for a low-quality seller12 , thus _ _ the local maximums, which are global maximum as well, exist respectively. Comparing (3.2) and (3.7), and (3.4) and (3.8), we can easily verify that both sellers’pro…ts decrease when there exist consumer’ interim bene…ts. This is because the number of consumers who end s up with buying and retaining the good eventually is smaller than that of without interim bene…ts. In other words, more sales but more returns reduce seller’ pro…ts. s = 0 into pL (t) and _ L (t) in _ vL (3.7), low-quality seller’ price and pro…t are pL (tL ) = 2 (1 2 (tL )) and _ L (sL ) = s _ vL (1 2 (tL ))2 vL on the separating equilibria. Note that, from pL (tL ) = 2 (1 2 (tL )) 0, _ 4 1 (tL ) h i 1 (tL ) should be smaller than 1 and, for (tL ) 2 0; 2 , L (sL ) is decreasing in t. Thus the 2 Now let us …nd separating equilibria. Substituting vL vL low-quality seller would set tL = 0 and pL (tL ) = 2 , resulting a pro…t of _ L (sL ) = 4 , _ which is the same as that without interim bene…ts. Meanwhile substituting = 1 into pH (t) _ vH and _ H (t) in (3.8), high-quality seller’ price and pro…t are pH (tH ) = 2 (1 s _ 12 With a similar logic in the proofs of Lemma 4 and 5, io n h vL 1 v 1 (t)( 1 + (t) + (1 (t)) v 2 (t)v+(1 (t))vL n h vH _ v (p ) = 1 _H 1 (t) 1 + (t) + (1 (t)) 2 (t)v+(1 (t))vH 65 2 (tH )) for a low-type seller, _ L (pL ) = _ 1, and, for a high-type seller, io vH 1. v h i vH (1 2 (t ))2 and _ H (sH ) = 4 1 (tH ) at separating equilibria. For (tL ) 2 0; 1 , _ H (t) is also 2 H decreasing in t. Then next proposition characterizes the separating equilibria. _se Proposition 12 Suppose there are interim bene…ts for consumers. Let us de…ne tL such r h i h i _se _se _se v _se _se _se v that (tL ) 1 + (tL ) + 1 (tL ) v L + (1 (tL )) (tL ) + (1 (tL )) v L = 1, H H 2 _se 1 2 (tH ) v _se _se _se and tH such that = v L . (i) Then, if tL < tH , there exist multiple sep_H ) H 1 (tse h i vL vH _se _se arating equilibria, in which sL = o; 2 and sH = tH 2 tL ; tH ; 2 (1 2 (tH )) , vL vH (1 2 (t ))2 _se _se resulting in _ L (sL ) = 4 and _ H (sH ) = 4 1 (tH ) . If tL = tH , there exists a unique H vL _se vH _se separating equilibrium, in which sL = o; 2 and sH = tL ; 2 1 2 (tL ) , resulting 2 _se 1 2 (tL ) _ (sL ) = vL and _ (sH ) = vH in L . Otherwise, there exists no separating equi4 4 H _se 1 (tL ) librium. (ii) Even if there exist multiple separating equilibria, the unique one that satis…es v L the Cho-Kreps ’ intuitive criterion’is sL = o; 2 _se vH and sH = tL ; 2 1 _se 2 (tL ) . Proof. See Appendix. The …rst part of the proposition shows that the existence of consumer’ interim bene…t s puts an upper bound on the return period of separating equilibria. Without consumer’ s interim bene…ts, a high-quality seller has no incentive to deviate. So we have an incentive compatibility constraint only for a low-quality seller. However, with the interim bene…ts, a high-quality seller also has an incentive to deviate when the return period is high enough. Compared to the case without interim bene…t, more consumers purchase initially but less consumers end up with …nal purchase under a given return period and the size of …nal purchasers is decreasing in the return period.13 This brings negative e¤ect on the seller providing longer return period in two ways. First it encourages strategic consumers who poach only interim bene…ts but have no intention to purchase …nally. Consumers with 13 Note that _ , _ , and _ , which are increasing in t, are larger than v H L respectively. 66 H, v , and L 2 [0; _ H ) correspond to these strategic consumers. Second it deters some of potential buyers from making …nal purchase due to lowered residual values. This restricts high-quality seller’ s incentive to provide excessively long return period. Therefore, in this case, we have incentive compatibility constraints for both types of sellers. The second part of the proposition is saying that, since consumers are su¢ ciently sophisticated to make use of interim bene…ts during the return period, the high-quality seller would o¤er the shortest return period among separating equilibria to minimize depreciation loss and, thereby, maximize pro…ts as long as he can di¤erentiate himself from a low-quality seller. Now let us look into pooling equilibria. At the pooling equilibrium, both sellers o¤er s = ~ ~~ ~ (1 (t))v . Substituting this into (3.5) and (3.6), the pro…ts on-the-equilibrium t; p = t; 2 path are _ L (~) = p 1 s ~ _ H (~) = p 1 s ~ p ~ 1 p ~ 1 (t) 1 (t) + vL v (t) (t) 1 + vH v (t)vL (t) (t)vH (t) , . v L Again, the best o¤-the-equilibrium pro…ts are for the low-quality seller is 4 at t = 0. For the high-quality seller, the best o¤-the-equilibrium pro…t is v M ax _ H (s 6= s) = M ax H ~ t 4 (1 (t) + (1 (t)) v (t)) vH L 2 41 (t) + (1 (t) (1 v 32 (t)) vH L 5 . (t)) Then the next proposition characterizes the pooling equilibria under consumer’ interim s bene…ts. Proposition 13 Suppose there are interim bene…ts for consumers. Then, there exist multi~~ ~~ ple pooling equilibria, s = (t; p), if (t; p) satis…es ~ _ L (~) s vL 4 and _ H (~) s 67 M ax _ H (s 6= s). ~ In case of pooling equilibria, we cannot specify the critical values of return period without further assuming speci…c functional forms on (t) and (t). Moreover the intuitive criteria cannot eliminate all the pooling equilibria; that is, the degree of elimination depends on the range of return period that supports pooling or separating equilibria respectively and the size of each equilibrium pro…ts. In summary, when there is interim bene…t during the return period, there may exist multiple separating equilibria, but the minimum return period among them could survive the intuitive criterion. There could also exist multiple pooling equilibria. The intuitive criterion could eliminate some of them, but not all necessarily. Throughout the paper, we assume that there is no return fee for consumers. Relaxing the assumption, however, does not change the previous results qualitatively. It only changes the critical level of return period that can support the separating equilibria. Suppose there is a return fee. Then consumer’ incentive to buy and return would decrease and the location s of the critical consumer moves to the right, meaning fewer consumers would purchase. The speci…c e¤ects of this shift on both seller’ pro…ts depend on the location of s . Intuitively, we can conjecture that a low-quality seller bene…ts more from it than a high-quality seller. This is because most of withdrawn buyers belong to for-sure returners, who might harm the low-quality seller more if they purchased. This narrows the pro…t margin between two sellers, and, in turn, shifts the minimum return period supporting separating equilibria to the right. 3.6 Concluding Remarks In this paper we show that the length of return period can be used as an e¤ective signaling device. Without interim bene…ts, we show that there exist multiple separating equilibria, where a high-quality seller o¤ers a positive length of return period above a speci…c level, while a low-quality seller does not provide customer return policy. All the separating equilibria satisfy the intuitive criterion. We also …nd that there is no pooling equilibrium since the 68 high-quality seller always has an incentive to deviate to perfect information price and the maximum return period. With interim bene…ts, in the meanwhile, there could be multiple separating equilibria, but the smallest return period survives the intuitive criterion. Multiple pooling equilibria could also exist and not all of them would be necessarily eliminated by the intuitive criterion. The result drawn in this paper re‡ ects the reality well. As shown in the example of Ebay, the seller with high-quality good o¤ers a longer return period than the seller with low-quality good. Moreover, as a signaling device, customer return policy is more e¤ective than warranty in that warranty generally does not guarantee a full refund. 69 APPENDIX 70 Appendix Proofs omitted in the text We …rst …nd a local equilibrium when L < 1 and L > 1 respec- Proof of Lemma 4. tively. Then, comparing both pro…t functions, we show that the global equilibrium always corresponds to the case in which L < 1. (Case 1: L < 1) From (3.1), the pro…t function is L1 = p [(1 L ) + (1 (t)) ( L (t) 1 (t) + vL v =p 1 v )] p (t)vL [ (t) ( L v) + ( v (t) 1 (t) + vL v (t)vL 1 vH H )] p, and the …rst-order condition is @ L1 =1 @p 2 (t) 1 (t) + vL v The second-order condition is satis…ed; p 2 (t)vL (t) 1 (t) + vL v 1 (t) (t) vL + v 1 vH = 0. < 0. Then the pro…t maximizing price and the corresponding pro…t are pL1 (t) = L1 (t) = v 2 v 4 vL (t)v + (1 vL (t)v + (1 (t))vL 2 (t))vL 2 (t)v + (1 41 (t) 41 (t) (t)) vL v (t)v + (1 (t)) vL v 3 v vL v L H 5, 32 v vL v L H 5 . p For this to be a local maximum, it must be the case L (pL (t)) < 1. Because L (pL (t)) = vL , L L (pL1 (t)) = 1 2 v (t)v + (1 (t))vL 2 41 71 (t)v + (1 (t) (t)) vL v 3 v vL v L H 5, Note that 2 41 0 h since 0 (t)v + (1 v= [ (t)v + (1 vL =v of vH [1 (t)v + (1 (t) v 2 (t)] = [2(1 i v vL v L H (t)) vL (t))vL ] (t)) vL 3 v vL v L H 5 1, 1. Then we need to show =v 2. Rearranging it, the condition is simpli…ed as 1 (t))]. The minimum of the left-hand side is 2 by the assumption 2vL , and the maximum of the right-hand side is 1 at (t) = 0. So the condition 2 always holds. (Case 2: L > 1) From (3.1), we have a pro…t function as follows; L2 = p [(1 = (1 (t)) (1 v )] p v (t))p 1 (t)vL [ (t) (1 (t)vL v) + ( v p + v (t) 1 H )] p v p vH 1 vH = 0. , and the …rst-order condition is @ L2 = (1 @p (t)) 1 1 (t) The second-order condition is satis…ed; p v (t)vL 2(1 2 2 (t)) v < 0. Then the pro…t maximizing price v and the corresponding pro…t are pL2 (t) = v 1 + (t) 2 v L2 (t) = 4 (1 1 1 vL (t) vH (t)) 1 + (t) vL v 1 1 , vL (t) vH vL v 2 (t) (t)vL . For this to be a local maximum, it must be the case that L (pL (t)) > 1. Because L (pL (t)) = pL vL , L (pL (t)) = v 1 + (t) 2vL 72 1 1 vL (t) vH vL v . So the local equilibrium exists if and only if v 1 + (t) 2vL () (t) 1 1 1 1 vL (t) vH vL (t) vH vL v vL v > 2vL v >1 1. v vL Let’ denote (t) 1 1 (t) v L s as L (t). We will show that there exist a unique t such v H 2v 2vL @ (t) that L (t) = vL 1. Note that 1 2 [0; 1], and L (0) = 0 and L (t) = 1. @t v v would be negative when t is very small since v L H for t such that (t) > 1 vL v ; but should be unilaterally positive v vH , which is equivalent to L (t) > 0. This means that L (t) B should be increasing in t for L (t) 0. So there exists a unique critical value, tL , such that h i 2v B B (tL ) = vL 1. Therefore there exists a local maximum if t 2 tL ; 1 . L (Overall equilibrium) B For t 2 [0; tL ), L (t) = B L1 (t) since there is no local maximum in Case 2. For t 2 [tL ; 1], since there exists a local maximum in Case 2, we need to compare both pro…ts for the given interval. However we cannot explicitly compare two pro…ts, due to the complexity of the model, without assuming speci…c functional forms on (t) and (t). Nevertheless we can intuitively explain that the seller would always choose the strategy in Case 1. Choosing a strategy in Case 2 means that the seller wants to extract surplus from the small size of high-preference consumer group, by charging high price to the state-contingent buyers rather than by charging a low price and increasing for-sure buyers. Thus to be bene…ted from this strategy, the seller needs to charge a su¢ ciently high price. pL2 (t) decreases in t when t is relatively small and then starts to increase as t goes up. However, the price he could charge is restricted by the value of the product. The maximum price he can charge is v, since the seller is a low type. From v [1 + L (t)] v, we can …nd tB 2 L h i B such that L (tB ) = 1. So only t 2 tL ; tB can be supported as an equilibrium in Case L L h i B 2. However note that L2 (tB ) < 0. This implies that for t 2 tL ; tB , L1 (t) > L2 (t). L L Therefore the global equilibrium is pL = pL1 (t) and 73 L (t) = L1 (t). The examples are based on the assumption that vH = 2, vL = 1, and solid lines are L1 (t), L2 (t), and 1 = 2 . Dashed, dotted L (t) respectively. 1 0.75 0.5 0.25 0 0 0.25 0.5 0.75 1 ρ(t) -0.25 -0.5 (a) (t) = q t , (t) = t q t t 1 0.75 0.5 0.25 0 0 0.25 0.5 0.75 1 ρ(t) -0.25 -0.5 (b) (t) = q Figure A1: Example 74 q t , (t) = 4 t t t We …rst …nd a local equilibrium when v < 1 and v > 1 respec- Proof of Lemma 5. tively. Then, comparing both pro…t functions, we show that the global equilibrium always corresponds to the case in which v < 1. (Case 1: v < 1) From (3.3), the pro…t function is H = p [(1 v ) + (t) ( v 1 = p 1 (t) v + H )] (t) vH (t)vH (1 p (t)vH (1 (t)) ( v 1 (t)) v H) 1 vH p, and the …rst-order condition is @ H =1 @p 2 1 (t) v + (t) vH The second-order condition is satis…ed; 2 p (t)vH (1 1 (t) v (t) (t)) 1 v 1 vH = 0. < 0. Then, the pro…t maximizing + v H price and the corresponding pro…t are as follows, v 2 vH (t)v + (1 v H1 = 4 vH (t)v + (1 pH1 = (t))vH (t))vH 1 1 (t) (t) (1 (t)) (vH v v) (1 (t)) (vH v v) 2 , . pH For this to be a local maximum, it must be the case that v (pH ) < 1. From v (pH ) = v , v (pH1 ) = Note that 0 v 1 is (t)v+(1H (t))v H (t) 1 2 vH (t)v + (1 (1 (t))(vH v ) v (t))vH 1 1 since 0 (t) (1 vH v v (t)) (vH v v) 1. Then all we have to show 2. Rearranging it, the condition is simpli…ed as vv H The minimum of the left-hand side is 1 by the assumption of vH 2 1 of the left-hand side is 2 at (t) = 1. So the condition always holds. 75 . 1 1 . 2 (t) 2vL , but the maximum (Case 2: v > 1) From (3.3), we have a pro…t function as follows: = p [ (t) (1 H )] = p H p vH (t) 1 (t)vH (1 (t)) (1 H) (t)vH (1 (t)) 1 + (t)(1 p vH , (t)) = 0. and the …rst-order condition is @ H = (t) 1 @p 2 The second-order condition is satis…ed; p vH 2 (t) vH < 0. Then, the pro…t maximizing price and the corresponding pro…t are pH2 = H2 = vH (1 (t)) 1 + (t) , 2 (t) vH (t) 1 4 (t) (1 (t)) 2 . (t) pH Now we need to check if v (pH ) > 1. From v (pH ) = v , v (pH2 ) = vH (1 (t)) 1 + (t) . 2v (t) So the local equilibrium exist if and only if vH 1 (t) (1 (t)) 2v 1 + (t) > 1 () (t) > 2v (t) (t) vH Let (t) (1 (t)) denote as (t) 1. H (t). On the contrary to the case of a low-quality seller, the local maximum does not always exist. It depends on the relative size of growth rates in (t) 0 (0) 2v 1 2 [0; 1], and H (0) = H (t) = 0 and lim H (t) = 0 by vH (0) t !0 0 (t)(1 (t)) 0 (t) 0 (t) @ H (t) (t) l’ Hospital’ rule. s = (t) 2 ? 0 depending on 0 ? (t)(1 (t)) , @t (t) (t) (t) and (t). Note that 76 but @ 2 H (t) < 0 by the assumption that @t2 00 (t) < 0. This means 00 (t) < 0 and H (t) could increase or decrease in t when t is relatively small, but should eventually decrease and 0 (0) converge to 0 as t increases. If 0 > 1, that is, the depreciation rate increases faster than (0) 2v the information precision rate, there always exists a unique tB such that H (tB ) = v 1. H H H B ], in which the local maximum exists. If So we could always …nd an interval, t 2 [0; tH 0 (0) 2v 1 , and, 0 (0) < 1, however, the existence of local maximum depends on the size of vH B if exists, the local maximum exist for t 2 [max(0; tH ); tB ]. H (Overall equilibrium) Even though there exists a local maximum in Case 2, it cannot be the global equilibrium since, for the interval that satis…es price constraint, H1 (t) is always greater than H2 (t). i h vH (t)(1 (t)) Note that in Case 2 the price cannot exceed vH . That is pH2 = 2 1 + < vH , (t) which is equivalent to 1. Now let us compare H (t) H1 (t) = vH 4 H2 (t) = vH (t) 1 4 v (t)v + (1 (t) 1 (t))vH (1 H1 (t) and (t) (1 H2 (t); (t)) (vH v v) 2 (t)) 2 . (t) (1 (t))(vH v ) First compare the terms in the squared brackets. We can easily verify 1 (t) v h i2 v v (1 (t)) 1 1 (t) (t) since Hv 1, (t) > 1 and both terms in the brackets are posi- tive due to the restriction that that (t)v+(1v (t))v H 1 H (t) 1. Next, compare the rest terms. The condition 1 (t) is equivalent to (vH =v) 1 2 [1; 1). This, in turn, means that H1 (t) (vH =v) 1 equilibrium price and pro…t are pH (t) = pH1 (t) and (t) > 0, which always hold since H2 (t) for all t. Therefore, global H (t) = H1 (t). Proof of Proposition 10. For a high-quality seller, he has no incentive to deviate in any 77 2 case since (ICL ) vH vL H (sL ) = 4 < 4 . For a low-quality seller, incentive compatibility condition is vH vL L (sH ) = 4 (t)v + (1 H vL = L (sL ), < 4 () (t) v (t) + L vH 1 vL vH + r (t) v (t)) L vH (t) + (1 Note that L (t) is increasing in (t) and L (0) = se se 1. So there exists a unique tse such that 1 (t))vL ) v (t) + L vH qv 1 2 vL vH L (t) > 1. se v v 1 and L (t) = (t) H L + 1 se v L vH H L (t ) = 1. Thus, for t 2 [t :t], the separating se se se equilibria can be supported. Finally, since on-the-equilibrium path the pro…ts of both types of seller are …xed with the levels of vL L (sL ) = 4 and H (sH ) = vH 4 respectively, all the separating equilibria survive the intuitive criterion. Proof of Proposition 12. In this case, to support separating equilibria, we have to consider incentive compatibilities of both sellers as follows: vH (1 (t)) (t)vH + (1 (t)) vL 2 _ (sH ) = vL (ICL ) L 1 (t) 4 (t)vH + (1 (t)) vL (1 (t))vH vL = _ L (sL ), 4 v vH (1 2 (t))2 (ICH ) _ H (sL ) = L = _ H (sH ), 4 4 1 (t) which are simpli…ed as (ICL )0 (ICH )0 (t) 1 + (t) + (1 (1 2 (t))2 1 (t) _ H (t) se v (t)) L + vH s (1 (t)) (t) + (1 vL . vH Note that _ L (t) is increasing in t but _ H (t) is decreasing in t; se se 78 v (t)) L vH _ L (t) se 1, h 2 (1 + | 0 (t) (t)) (t) + (1 {z (t) + (1 v (t)) L + (t) vH {z " | 1 r 2 (1 0 (t) (1 (t)) h (t) + (1 {z h i 2 (t)) [3 [1 (t)]2 {z h i 7 7 i5 > 0 vL H 2 (t)] } 3 (t)) v 1 (+) for (t)2 0; 2 # i 0 (t) + 1 (+) 6 61 4 @ _ H (t) se = @t v (t)) v L H (+) | 2 | 0 (t)) vL vH 0 (t) + (1 @ _ L (t) se = 0 (t) + r @t 0 (t) vL vH } } <0 } 1 (+) for (t)2 0; 2 _se _se So, similarly in Proposition 10, we can verify that there exist a unique tL and a tH such v _ _ that _ L (tL ) = 1 and _ H (tH ) = v L respectively. Then the existence of the separating se se se se H equilibria counts on the size of two critical values, which in turn depend on the speci…c functional forms of (t) and (t). Therefore there would be multiple separating equilibria if _se _se _se _se tL < tH , a unique separating if tL = tH , and no separating equilibrium otherwise. Finally, if separating equilibria exist, the unique separating equilibria that satis…es the intuitive _se _ _se criterion is (tL ; pH (tL )) since _ H (sH ) is decreasing in t. 79 BIBLIOGRAPHY 80 BIBLIOGRAPHY Bagwell, K. and Riordan, M. H. "High and Declining Prices Signal Product Quality." American Economic Review, 1991, 81, pp. 224-239. Ben-Shahar, O. and Posner, E. A. "The Right to Withdraw in Contract Law." U of Chicago Law & Economics, Olin Working Paper, 2010, No. 514. Bourreau, M. and Lethiais, V. "Pricing Information Goods: Free vs. Pay Content." Internet and Digital Economics, ed. by E. Brousseau, and N. Curien, 2007, pp. 345-367. Che, Y.-K. "Customer Return Policies for Experience Goods." Journal of Industrial Economics, 1996, 44, pp. 17-24. Cho, I.-K. and Kreps, D.M. "Signaling Games and Stable Equilibria." Quarterly Journal of Economics, 1987, 102, pp. 179-221. Choi, J.P. "Bundling New Products with Old to Signal Quality, with Application to the Sequencing of New Products." International Journal of Industrial Organization, 2003, 21, pp. 1179-1200. Gal-Or, E. "Warranty as a Signal of Quality." The Canadian Journal of Economics, 1989, 22(1), pp. 50– 61. Grossman, S.J. "The Role of Warranties and Private Disclosure about Product Quality." Journal of Law and Economics, 1981, 24, pp. 461– 483. Kim, J.-Y. "Product Compatibility as a Signal of Quality in a Market with Network Externalities." International Journal of Industrial Organization, 2002, 20, pp. 949-964. Milgrom, P. R. and Roberts, J. "Price and Advertising Signals of Product Quality." Journal of Political Economy, 1986, 94, pp. 796-821. Moorthy, S. and Srinivasan, K. "Signaling Quality with a Money-Back Guarantee: The Role of Transaction Costs." Marketing Science, 1995, Vol. 14, No. 4, pp. 442-466. Okuno-Fujiwara, M. Postlewaite, A. and Suzumura, K. "Strategic Information Revelation." Review of Economic Studies, 1990, 57, pp. 25-47. Spence, M. "Consumer Misperceptions, Product Failure, and Product Liability." Review of Economic Studies, 1977, 44 (3), pp. 561– 572. 81