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Essays on strategic and contractual relationships in oligopoly Zhang, Anming 1990

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ESSAYS ON STRATEGIC AND C O N T R A C T U A L RELATIONSHIPS IN OLIGOPOLY By Anmirig Zhang B. Sc. (Mathematics) Shanghai Jiao Tong University M . Sc. (Commerce) University of British Columbia A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY • in THE FACULTY OF GRADUATE STUDIES COMMERCE AND BUSINESS ADMINISTRATION We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June 1990 © Anming Zhang, 1990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Bri t ish Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Commerce and Business Administrat ion The University of Bri t ish Columbia 6224 Agricul tural Road Vancouver, Canada V 6 T 1W5 Date: Abstract The thesis consists of three essays. In the first essay, the strategic effects of bonus/penalty compensation contracts are analysed. This essay consists of the first three chapters of the thesis. In Chapter 1, viewing each firm as a "principal-agent" pair, we examine firms' rivalry in bonus/penalty contracts in oligopolistic markets when the agents are risk-neutral. Under standard assumptions concerning production, demand, and cost functions, we show that bonus/penalty contracts may be used for strategic purposes. We find that whether agents' actions would (in equilibrium) be encouraged through bonuses or discouraged through penalties would critically depend on the existing strategic relationships between agents' actions or firms' products. We further show that firms' capita] stocks can affect their strategic positions in the bonus/penalty rivalry. The social welfare implication of the bonus/penalty rivalry is also examined. In Chapter 2, using a general framework of rivalrous agency with risk-averse agents, we identify two distinctive effects of bonus/penalty contracting, namely, the strategic effect and the incentive effect. We find that the two effects may or may not work in the same direction for a principal-agent pair, depending on the nature of strategic relationships between agents' actions. In Chapter 3, we compare the strategic effects of bonus/penalty contracts with that of linear contracts. We find that, if only one principal is active in designing agency compensation contracts, then he/she would be indifferent between a bonus/penalty contract and a linear contract. If both principals are active in designing agency contracts, however, the choice between bonus/penalty and linear contracts would in general matter to the principals. In particular, we show in an example that both principals would non-cooperatively choose a bonus contract over a linear contract. n T h e second essay of the thesis , as in C h a p t e r 4, presents an analysis of c o m m o n sales agents based o n the i r p r e c o m m i t m e n t role when consumers are imper fec t ly in fo rmed abou t the p roduc t s on the marke t . W e show tha t an exclus ive channe l s tuc ture can cre-ate a cost due to exc lus ive channels ' i n a b i l i t y to c o m m i t themselves to sales i m p a r t i a l i t y . W e fur ther show tha t independent non-coopera t ive f irms m a y use c o m m o n agents as a p r e c o m m i t m e n t device to convince p o t e n t i a l consumers tha t the r i sk of be ing misrepre-sented has been reduced or e l im ina t ed . W e demons t ra te tha t a marke t i n v o l v i n g c o m m o n sales agents can arise as an e q u i l i b r i u m ou tcome . O u r analysis shows tha t c o m m o n sales agents can be welfare i m p r o v i n g for b o t h f irms and consumers . T h e t h i r d essay, as i n C h a p t e r 5, invest igates the d y n a m i c pa t t e rn of f i r m compe t i t i ve conduct, u s ing t ime-series and f i rm-specif ic d a t a for a set of d u o p o l y a i r l ine routes. W e es t imate the m e a n conduc t parameters for each f i r m and each p e r i o d , and infer whether the d a t a are consis tent w i t h the C o u r n o t , B e r t r a n d , F r i e d m a n , or G r e e n - P o r t e r models . W e find tha t a i r l ines ' c o m p e t i t i v e behav io r m a y swi t ch between the c o m p e t i t i v e and co l -lus ive regimes. M o r e o v e r , we find that a i r l ine profits in a co l lus ive p e r i o d appear less t h a n the ( s ing le-per iod) m o n o p o l y profi ts , and the degree of overa l l marke t compe t i t i ve -ness is between the C o u r n o t and m o n o p o l y so lu t ions bu t closer to the C o u r n o t so lu t ion . O u r d a t a suggest t ha t ma jo r carr iers m i g h t use quan t i t y vo lumes , ra ther t h a n pr ices , as the i r s t ra tegy var iables . W e also conduc t some B a y e s i a n analys is of seeing how our results w o u l d inf luence pr iors associated w i t h different models . W e i l lus t r a t e a m o d e l choice c r i t e r i on based o n B a y e s i a n analys is a n d use the c r i t e r ion to choose the "best" m o d e l a m o n g c o m p e t i n g models . in Table of Contents Abstract ii List of Tables vii List, of Figures ix Acknowledgement x 1 Rivalry in Bonus/Penalty Contracts in Oligopolistic Markets 1 1.1 Introduction 1 1.2 The Model 3 1.3 Agent Game and Effects of a Bonus on Efforts 6 1.4 The Strategic Bonus/Penalty Equilibria 13 1.5 Effort, Output, Payoff, and Welfare Comparisons 17 1.6 Effects of Firm Capital Stocks on Bonus/Penalty Rivalry 26 1.7 Concluding Remarks 28 2 Bonus/Penalty Contracts As Strategic and Incentive Devices 30 2.1 Introduction 30 2.2 The Model 30 2.3 Effects of a Bonus on an Agent's Effort 32 2.4 Strategic and Incentive Effects'of Bonus/Penalty Contracts 35 2.4.1 The incentive effect . 37 2.4.2 The strategic effect . 39 iv 2.4.3 Strategic and incentive effects 40 2.5 Concluding Remarks 43 3 Strategic Contracting: Bonus/Penalty and Linear Contracts 45 3.1 Introduction 45 3.2 The Strategic. Effect, of Linear Contracts 46 3.3 Choice between Bonus/Penalty Contracts and Linear Contracts 48 3.3.1 Only one principal is active in contract designing 49 3.3.2 Both principals are active in contract designing 50 3.4 Concluding Remarks 55 4 An Analysis of Common Sales Agents 57 4.1 Introduction 57 4.2 The Basic Model ' 60 4.3 Firm Sales Behavior and Consumer Buying Behavior 68 4.4 Information, Impartiality, and Common Sales Agents 73 4.5 The Channel Game 76 4.6 Common Sales Agents as an Equilibrium Outcome 80 4.7 Price, Profit, and Welfare Comparisons 85 4.8 Concluding Remarks 93 5 Strategic Stability of the Airl ine Industry: A n Empirical Study 95 5.1 Introduction 95 5.2 Theories of Firm Conduct in Oligopoly 97 5.3 Statistical Methodology - 102 5.4 Empirical Implementation 104 5.5 The Data 107 v 5.6 Results 1 1 6 5.7 Bayesian Analysis of Firm Conduct and Model Choice 125 .5.7.1 Firm Conduct 125 5.7.2 Model Choice 133 5.8 Sensitivity Analysis 136 5.9 Conclusion 1 4 3 Appendices 145 A Some Proofs 145 Bibliography 149 vi List of Tables 1.1 Pr incipals 'Payoff Mat r ix : Strategic Substitutes 20 3.1 Principals ' Payoff Mat r ix of the Example 54 3.2 Agents' Effort Ma t r ix of the Example 54 5.1 Duopoly Models and Conduct Parameters: the Case of Identical Costs . 101 5.2 Route, Distance, and Price and Volume of 4th Quarter, 1985 110 5.3 Mean Prices and Volumes, Quarterly 110 5.4 Costs per Passenger-Mile (Cents) 114 5.5 Average Flight Lengths (Miles) 115 5.6 Marginal Costs and Conduct. Parameters, 4th Quarter, 1985: Base Case . 117 5.7 Mean Marginal Costs (Dollars): Base Case 117 5.8 Estimated Mean Conduct Parameters, Quarterly: Base Case 117 5.9 Hotelling's T2 for Hypothesis H0it : vt = vtt0: Base Case 121 5.10 Conduct. Parameter Vectors, Quarterly 122 5.11 Estimated Overall Mean Conduct Parameters: Base Case 124 5.12 Bayesian Posterior Conduct Parameters, Cournot Prior: Base Case . . . 129 5.13 Bayesian Posterior Conduct Parameters, Bertrand Prior: Base Case . . . 129 5.14 Bayesian Posterior Conduct Parameters, Friedman Prior: Base Case . . . 129 5.15 Bayesian Posterior Conduct Parameters, GP-quant i ty Prior: Base Case . 131 5.16 Bayesian Posterior Conduct Parameters, GP-price Prior: Base Case . . . 131 5.17 Bayesian Posteriors of Overall Conduct Parameters: Base Case 132 5.18 Incompatibility of Prior and Sample (6^): Base Case 135 vi i 5.19 Sensitivity of Overall Conduct Parameters, American 136 5.20 Sensitivity of Overall Conduct Parameters, United 136 5.21 Sensitivity Analysis of Incompatibility of Prior and Sample, American . . 139 5.22 Sensitivity Analysis of Incompatibility of Prior and Sample, United . . . 139 5.23 Estimated Mean Conduct Parameters: Semi-Annually 141 5.24 Bayesian Posterior Conduct Parameters: Semi-Annually 142 A . l Payoff Matrices of the Games : 147 vm List of Figures 1.1 Effects of an Increase in 1^ on Equi l ibr ium Efforts 12 4.1 Gains from an Increase in a3 64 ix Acknowledgement I would like to thank my advisor, James Brander for all the time, effort, and help he has given me these years. Without his guidance and his many comments, this thesis would still be in progress. I am very grateful to the members of my thesis committee, Murray Frank, Hugh Neary, and Tae Oum for their time, effort, and their helpful comments. Thanks to Barbara Spencer for all her encouragement, and to Mike Tretheway for useful discussions. I would like to thank my parents and my wife, Y i who have shown much care, understanding, and support. This work is dedicated to them. I gratefully acknowledge the financial support from University of British Columbia Graduate Fellowship, Izaak Walton Killam Memorial Pre-doctoral Fellowship, and the Social Sciences and Humanities Research Council of Canada (Strategic Grant No. 494-85-0007). x Chapter 1 Rivalry in Bonus/Penalty Contracts in Oligopolistic Markets 1.1 Introduction Bonus and penalty contracts have been commonly observed in business settings. For instance, the senior managers of a firm may receive a bonus from owners if the firm reaches some pre-arranged profit level. Salesmen or distributors are sometimes paid a commission plus a bonus if sales exceed some target level. On the other hand, managers are often penalized if they fail to meet certain goals. In the case of bankrupcy, managers would normally be dismissed and would consequently suffer various personal costs. Viewing the managers as "agents" of the owners in a "principal-agent" relationship, it seems natural to use bonuses (or penalties) to alleviate the agent's incentive problem as suggested in Lewis (1980). If, however, the agent is risk-neutral, then the "first-best" outcome can be achieved by having the owner "sell-out" to the agent and bonus contracts will not be necessary. If the agent is risk-averse (or has a binding wealth constraint), then a bonus contract may be efficient. Another explanation for using bonus contracts by the principal is found in Brander and Poitevin (1988). They show that managerial compensation in the form of a bonus contract can reduce the agency costs of debt finance arising from the conflict of interest between shareholders and creditors. In this chapter we emphasize the strategic role of bonus/penalty contracts in a ri-valrous oligopolistic industry. The idea that the managerial contract may serve as pre-commitments in rivalrous agency1 has been emphasized by Fershtman and Judd (1987a) * A related idea, pointed out in Ross (1987), is that a profit maximiz ing firm may have an incentive 1 Chapter 1. Rivalry in Bonus/Penalty Contracts in Oligopolistic Markets 2 and Sklivas (1987). Using linear contracts they find that competition in managerial in-centives in an oligopolistic industry can result in non-profit maximizing behaviour even if owners maximize profits. Fershtman and Judd (1987b) introduce moral hazard and find that owners may alter internal incentives for strategic reasons, resulting in contracts that differ from those predicted by standard agency theory. Our contribution here is to demonstrate that bonus/penalty contracts can be used for strategic purposes, and to provide an analysis of firms' bonus/penalty rivalry in oligopolistic markets. Under standard assumptions concerning production, demand, and cost functions,2 we show that firms may use bonus/penalty contracts for strategic purposes. In the "normal" case where agents' actions or firms' products are strategic substitutes, each firm may use a bonus contract for strategic reasons. If one firm strategically uses bonuses alone, it would gain through more aggressive action in the output market relative to non-strategic behaviour, while the other firm would lose due to the rival's action. But if both use bonus contracts for strategic, purposes, then under reasonable symmetry both would suffer due to greater aggregate output than with non-strategic behaviour. The formal structure of the problem is similar to a "Prisoner's Delimma". On the other hand, the rivalry in bonus contracts is welfare-improving (for society as a whole) compared to non-strategic behaviour, because effort and output are closer to the competitive level. We find that whether an owner would in equilibrium give his managers a positive or negative incentive for aggressive action would critically depend on the existing strategic relationships between agents' actions or firms' products. When products are strategic substitutes, an owner would reward aggressive action by using bonuses. On the other hand, when products are strategic complements, an owner would discourage aggressive to commit, managers to a sales max imiz ing strategy. 2It is noted that in both Fershtman and J u d d (1987a) and Sklivas (1987), restrictive assumptions are made on demand and cost functions, while in Fershtman and J u d d (1987b) the output market is not explicit ly modeled. Chapter 1. Rivalry in Bonus/Penalty Contracts in Oligopolistic Markets 3 action. In the latter case, the rivalry in agency compensation would, in equilibrium, give rise to a penalty contract. Furthermore, under reasonable symmetry, both firms would produce less output and enjoy higher prices and (probably) higher payoffs at the strategic equilibria than at the non-strategic equilibrium. In this case the rivalry in agency compensation would be welfare-reducing for society as a whole. We further show that firms' capital stocks can affect their strategic positions in the bonus/penalty rivalry. For instance, owners of a larger firm might have a strategic ad-vantage over a rival in that they might use a larger magnitude of bonuses than the rival. W i t h sufficient asymmetry, the larger firm might gain from using strategic bonuses despite the use of strategic bonuses by its rival. Our work is also related to the theory of strategic commitment and the multi-stage oligopoly (see Shapiro (1989) and the references cited there). The chapter is organized in the following way. Section 1.2 sets up the model. Section 1.3 examines the action choice by the agents given compensation contracts. Section 1.4 then examines the bonus/penalty choice by the principals and characterizes the strategic bonus/penalty equilibria. Section 1.5 compares the strategic bonus/penalty equilibria with the non-strategic equil ibrium and does the welfare analysis. Section 1.6 examines the effects of firms' capital stocks on the bonus rivalry. The final section provides concluding remarks. 1.2 The Model Consider a two-firm industry. Each firm consists of a principal and an agent (shareholders and managers, or owner-managers and marketing divisions, for example). Each firm produces an output y ' , f = y\ei,s). (1.1) Chapter 1. Rivalry hi Bonus/Penalty Contracts in Oligopolistic Markets 4 In expression (1.1), e, is an action of agent, i (e, £ [0,e], e < oo), and s is the random state of nature. One may think of e, as the level of agent i's work effort, s is in the interval [6,0] with p.d.f. f(s) > 0. Using subscripts to denote partial derivatives, we assume Vi = > 0, y\ > 0. (1.2) That is, a firm's output rises as the agent increases his effort, and higher values of s are regarded as better states of nature. The two firms are labeled i = 1 and 2. Their inverse demand functions are pl(y1, y2) for i = 1,2. We assume downward-sloping demand curves and substitutable products, i.e., p\<0, p)<0 (1.3) for j ^ i. We also assume the following property for certain purposes. V\v\-V\v\>^- (1-4) Condit ion (1.4) means that own effects of output on price exceed cross effects. To produce output y1, the firm incurs a total cost of C^y1). F i r m i's profit, IP can be written as Ui(e1,e2,s) = f(y\y2)y> - W ) = R\y\y2) - C^). (1.5) Rt(y1,y2) is the total revenue of firm i . f?* = dRl/dyJ = ylp^ < 0 (due to substitutable products) and, hence, IT^ = dW/dej = R)y{ < 0- Thus, firm i's profit decreases with the effort of agent j for j ' ^ i . The information assumptions on principals and agents within a firm are those used in a standard principal-agent problem (Holmstrom (1979) and Shavell (1979), for exam-ple). Each agent chooses his action before knowing random states. The random state is unobservable to the principals. Further, a principal cannot directly observe the amount Chapter 1. Rivalry in Bonus/Penalty Contracts in Oligopolistic Markets 5 of effort his agent suppl ies . A p r i n c i p a l can nevertheless observe the ou tcome, namely , the f i rm 's o u t p u t level . T o focus on the s t ra tegic effects of b o n u s / p e n a l t y cont rac ts , we assume i n this chapter tha t b o t h p r i nc ipa l s and agents are r i s k - n e u t r a l . 3 W h e n the agent is r i sk -neu t r a l , it is we l l k n o w n tha t the c o m p e n s a t i o n contract TV' = IF — az is the first-best so lu t ion to the agency p r o b l e m i n the s t a n d a r d p r inc ipa l -agen t f ramework . al ( > 0), u sua l ly in terpre ted as the "rent" , is de t e rmined by the agent p a r t i c i p a t i o n cons t ra in t . It is no ted that i n a s t a n d a r d agency p r o b l e m , the p r i n c i p a l is concerned on ly w i t h the a c t i o n of his own agent and there is essent ia l ly no in t e rac t ion between one f i rm (cons i s t ing of a p r i n c i p a l and his agent) a n d other f i rms, i f any, i n externa] ou tpu t marke t s . If f i rms ' i n t e r ac t i on is e x p l i c i t l y recognized , the o p t i m a l paymen t cont rac ts may de-v i a t e f rom those a r i s ing f r o m the s t anda rd agency p r o b l e m . Suppose tha t each p r i n c i p a l rewards his agent acco rd ing to the fo l lowing b o n u s / p e n a l t y contrac ts : f n' - OL, i f yi < y\ BVuyl<*i)={ (1.6) I ( i r - a , ) + ^ i f y , : > y * / , is the amoun t of bonus ( if Z, > 0) or pena l ty ( if / , < 0). y\ is the target ou tpu t , 0 < y\ < yl{e,0). T h u s , agent i receives (LP — a,) plus a bonus (or pena l ty ) U if y% > yl; o therwise , he receives paymen t ( IP — at). W h e n / , = 0, Bl reduces to N\ the non-s t ra teg ic first-best con t rac t . N o t e tha t the above contrac t is p r o b a b l y the s implest c o n t r a c t u a l f o r m capable of c a p t u r i n g the essent ial character is t ics of a b o n u s / p e n a l t y con t rac t . R i v a l r y i n b o n u s / p e n a l t y contrac ts is mode led as a two-stage game. In the first stage, the p r i nc ipa l s s imu l t aneous ly choose the i r compensa t i on contracts (/;, y*, and Q , ) , subject to the agent p a r t i c i p a t i o n cons t ra in t s . In the second stage, after observ ing 3 W e shall in Chapte r 2 discuss the general case in which principals a n d / o r agents may be risk-averse. Chapter 1. Rivalry in Bonus/Penalty Contracts in Oligopolistic Markets 6 both compensation contracts, the agents simultaneously choose their actions or efforts. Final ly, uncertainty is resolved and payoffs are made. The equilibrium concept is that of Nash subgame perfect equilibrium. To solve the game, we start with the second-stage game and then proceed to the first-stage game. 1.3 Agent Game and Effects of a Bonus on Efforts Given BJ(I-i, y\, ) and B2(I2,yi2, a2), each agent tries to maximize his utility, denoted U\ U% = f[(Bl - Gi(ei))f(s)ds = E{Bl - G^a)), where G^e,) is the disutility of effort expressed in terms of money. (The expectation is in terms of 3, where 5 is suppressed for notational simplicity). Assume that G\ > 0, i.e., the agent disutili ty is increasing in the amount, of effort. Using F(s) to denote the cumulative distribution function of s, the i th agent's util i ty function can be written as follows. Ui(e1,e2) = EW - a , - G^e . ) + 7,(1 - F{s\)) (1.7) where s\ is derived from the following equation, yi(ei,sl) = yti. (1.8) s\ may be referred to as a critical state for agent i . If s > s\, then, other things equal, firm i's output would exceed the target and, consequently, agent i would receive the bonus. Equation (1.7) indicates that, given compensation contracts, the uti l i ty of each agent depends on his rival's action as well as his own action. The second-stage game is about competition between the agents. Each agent chooses his effort level to maximize his utility, given the effort level of the other agent. The first-order condition (foe) for agent i is JJ\ = dlll/det = 0, which can be written as follows: ds* EUl-G'l(er)-IJ(sl)-^=0, (1.9) Chapter 1. Rivalry in Bonus/Penalty Contracts in Oligopolistic Markets 7 where ds\j de% = — y*(e t, 5*)/y' s(ej, s-) < 0 (from (1.8) and (1.2)). Equations (1.9) contain two reaction functions, one for each agent. Interactions of the two functions determine the Nash equilibria of the agent subgame. The second-order conditions (soc's) are ^ = £;n|,. - G;'(eO + < o (1.10) for i = 1,2. In (1.10), can be calculated as follows. d2U'i d2st / a V x 2 where d^rm2 \-y*v" ~ y'y"dH + y-y"+y'y"d7j • The following condition is also considered, DB = U22 - U\2 U;,>0 (1.11) where U\3 = EVL\0 = E{R\0y\y\). Condit ion (1.11) means, given (1.10), that the agents' uti l ity functions are strictly concave. In this paper, both (1.10) and (1.11) are assumed to hold over the entire region of interest. This implies that the Nash equil ibrium of the agent, subgame exists and is unique (see Friedman (1977), among others). Before deriving comparative statics results with respect to a change in bonus levels, we examine conditions (1.10) and (1.11). The soc's of (1.10) hold if £ITJt < 0 a n d G ^ ' ( e t ) -IiUln i s either non-negative or sufficiently small (in absolute value) when negative. Since £1?., = E((Rl - C[)yle + (J& - Cl')(yl)2), we obtain that EU\T < 0 if, for 0 < s < 0, Rl- — C" < 0 and {R\ — C'i)ylee is either non-positive or sufficiently close to zero when positive. As for the term IJJ\U-, we have d2s%jd?e, > 0 if ylse = y\£ > 0 and ylss < 0. That yles > 0 means that the marginal return of effort is non-decreasing in s. Suppose (d2s\Id2ei) > 0. Then U\jz is negative if f'{s\) is either non-negative or sufficiently small in absolute value when negative. For the uniform distribution, we have /'(•) = 0 and, Chapter 1. Rivalry in Bonus/Penalty Contracts in Oligopolistic Markets 8 hence, U^Ii < 0. Thus for It > 0, the term is generally non-positive. Finally, (e^) > 0 if the marginal disutility is non-decreasing in effort. Next consider condition (1.11), which can be rewritten as DB = (EUl, - G'i{ei) + hU\n){EU\2 - G'2'{e2) + I2U2I2) - (EU\2)(EU221). It can be easily seen that DB > 0 if ^ ( n j j ^ n ^ ) > ^ ( n ^ ^ I i y and G'^e,) - I%XJ\U condition has just been discussed; while the former holds if (R\1 — C")(R22 — C2) > R\2R\ and R\2 = R2n, since (EUl^ETl2,) - (EU{2)(EU221) = £ ( ( 7 ^ - C[')(yl)2)E((R22 -C"){y2eY) - E(R\2yly2)E(R22iyly2e) > 0 ( t l i e l a s t inequality has involved the use of the Cauchy-Schwartz inequality). When linear demands come from uti l i ty maximization, use of Slutsky symmetry and condition (1.4) gives that Rltl — 2p\ < 0, R\2 = p\ = p\ = R\x, and R\^R\2 - R\2R\\ > v\v\ — v\v\ > 0- b i this case, therefore, conditions such as EW- < 0 and {EH\^){ED\2) > (f^n^X-ElT^) normally hold. For non-linear demand functions, these conditions would be satisfied at reasonably symmetric equilibria as long as products are sufficiently homo-geneous and demands are not too convex. Under conditions (1.10) and (1.11) the unique equilibrium of agents' efforts, denoted (e^, e2), can be obtained by jointly solving the two equations in (1.9). We can wri te 4 is either non-negative or sufficiently small (in absolute value) when negative. The latter ei = el(I1,I2,yl,yt2), e2 = e^ / j , ^ ,y* ,? /* ) . (1.12) We totally differentiate equations (1.9) with respect to e;, eJy and and then solve for de* jdli and de'jdl^, the effects of a bonus on the equilibrium levels of effort, obtaining del dli (1.13) DB 4 I t is obvious that changes in have no effect on equi l ibr ium levels of effort. Chapter 1. Rivalry in Bonus/Penalty Contracts in Oligopolistic Markets 9 dl, DB ' 1 V where Wu = d l ^ / d / , = -/(sj)(dst7de;) > 0. W i t h U]: < 0 and DB > 0, it follows immediately that de*'/dl, > 0 whereas the sign of de*/dl, is the same as the sign of U^. Following Bulow, Geanakoplos, and Klemperer (1985), we may say that the actions (or efforts) of the two agents are strategic substitutes, strategic independents, and strate-gic complements if U], < 0, = 0, and > 0, respectively. It can be easily seen that, if < 0, = 0, and > 0, then the two reaction functions (defined by equations (1.9)) are, respectively, downward-sloping, perpendicular with each other, and upward-sloping in the action space. Thus, an increase (decrease) in e, induces a decrease in e3 if actions are strategic substitutes (strategic complements), while changes in e t have no effect on e3 if actions are strategic independents. In our context, the above strategic relationships between agents' actions can be translated to the strategic relationships between firms' products. Since Ujx = ETV- — E(R),y\y{), it follows that U], < 0, = 0, and > 0 if, respectively, R].=p{+y:p], (1.15) < 0 ,= 0, and > 0 for 0 < s < 9. Similarly, the firms' products are said to be strategic, substitutes, strategic, independents, and strategic complements if, for 9 < s < 6, R1- < 0, = 0, > 0, respectively. Goods being strategic substitutes means that a higher own output would lower the rival firm's marginal revenue. It corresponds to downward-sloping reaction functions in the output space. This is regarded as the "normal" case in Cournot oligopoly (see, for example, Dixi t (1986)). From (1.15) and (1.3), R^ < 0 if p'ji is either non-positive or sufficiently small (in absolute value) when positive. Goods are strategic substitutes, for example, under linear demands. On the other hand, goods being strategic complements means that an increase in own output would increase the Chapter 1. Rivalry in Bonus/Penalty Contracts in Oligopolistic Markets 10 rival firm's marginal revenue. From (1.15) and (1.3), the strategic-complement property can hold for sufficiently homogeneous products and sufficiently convex demand functions. (A necessary condition for the property is that p1- > 0.) As was indicated earlier, too much convexity in demand functions may be inconsistent with the conditions (1.10) and (1.11). Thus, given (1.10) and (1.11), the products would be more likely to be strategic substitutes than to be strategic complements. Finally, the case of strategic independents is one in which the level of own output has no effect on the rival firm's marginal revenue. It is useful to note a relation between products being strategic complements (or sub-stitutes) and the nature of oligopolistic competition. Supposing that the output quantity is each firm's choice variable, then the case of strategic-complement products corresponds to the case where firms compete in price with upward-sloping reaction functions. 5 This may be seen as follows. Denote y% = 4>i{y3) as firm i's reaction function in the output space, with (f>\ > 0 (strategic-complement products). Recall that the inverse demand functions are p 1 = p1(y1,y2) and p2 = p2(y1 ,y2), which may implici t ly determine the demand functions as y1 = z1(p1,p2) and y2 = z2(p1,p2). Using the implici t function theorem one can show that z\ = plj/d and = —p\/d, where d = p\p\ — p\p\ > 0 (using (1.4)). Hence, z\ < 0 and zx- > 0 (using (1.3)). Substitute yl = zl(p%,p0) into the reaction function yl — ^(y3), and then regard it to be firm i's reaction function in price space (denoted pl = ^ (p3)). It follows that $ = (#zj - z))l[z\ - <\>\z\) > 0. The results of comparative statics with respect to a change in 7; are given in Propo-sition 1.1. Proposition 1.1: A n increase in the bonus set by a principal has the following effects on the equil ibrium levels of agent efforts: (i) his own agent wil l use more effort; (ii) his 5 U p w a r d - s l o p i n g reaction functions are generally regarded to be the normal case in a price game. T h i s holds, for example, for linear demand functions. Chapter 1. Rivalry in Bonus/Penalty Contracts in Oligopolistic Markets 11 rival's agent wi l l use less, the same, or more effort if agents' actions or firms' products are strategic substitutes, strategic independents, or strategic complements, respectively; and (iii) the total amount of effort wi l l rise under perfect symmetry. d(e; + e*2) I dli = de*/dli + de*/dlt = {Wu/DB){-U3n + ty) > 0. The last inequality arises because DA = (U3^ + TJ^U3- - U3-) > 0 under perfect symmetry 6 and U3- < U3t. Q.E.D. Proposition 1.1 is illustrated in Figure 1.1. For the case of strategic substitutes, depicted in Figure 1.1(a), it can be easily shown that the reaction functions of two agents are downward-sloping in the effort space and intersect at point A . A n increase in I\ by principal 1, given I2, wi l l increase U\, the marginal return to agent 1, and shift agent I's reaction function outward. The equilibrium point wi l l move along agent 2's reaction function from A to B . Consequently, agent 1 wil l supply more effort whereas agent 2 wi l l supply less effort. The situations for the other two cases can be similarly analyzed. The comparative static results of e\ and e 2 wi th respect to a change in target output y\ can be similarly found and are given as follows. Proof: The first two parts have been shown in the text. We now prove part (iii). del (1.16) rip* TP TP (1.17) dy\ DB where Thus, when there is no bonus or penalty (7 t = 0), the level of target output has no effect 6 T h e definit ion of perfect symmetry will be given in section 1.5. Chapter 1. Rivalry in Bonus/Penalty Contracts in Oligopolistic Markets 12 Figure 1.1: Effects of an Increase in I\ on Equilibrium Efforts (a) Strategic Substitutes e 2 Agent I's R . F . •ei (b) Strategic Independents — Agent 2's R . F . A B "— Agent I's R . F . -ei e 2 (c) Strategic Complements Agent 2's R . F . Agent I's R . F . Chapter 1. Rivalry in Bonus/Penalty Contracts in Oligopolistic Markets 13 o n the e q u i l i b r i u m levels of agent efforts: de\jdy\ = 0 and de* jdy\ = 0. T h e target level m a y affect the agent efforts o n l y when I, ^  0. 1.4 The Strategic Bonus/Penalty Equilibria P r i n c i p a l i 's payoff func t ion is V* = E(W - Br) = ES<S<{W - (IT - a;)) + Es>s<(IIl - ( IT - a , + /J) = a , - 1,(1 - F(s\)). In the first-stage, each p r i n c i p a l chooses Ii,y*,cti, g iven Ij,yj,ctj, to m a x i m i z e his payoff, subject, to his agent 's p a r t i c i p a t i o n cons t ra in t : UL > U0 where UQ is the reservat ion u t i l i t y , or o p p o r t u n i t y cost, of agent i . Suppose that a, is chosen by each p r i n c i p a l to make the p a r t i c i p a t i o n cons t ra in t b i n d i n g . (Hence , a l l the benefits f rom improvemen t s i n per formance go to the p r inc ipa l s . ) F r o m equa t ion (1.7), the b i n d i n g cons t ra in t gives EIT - G ,(e t) + 7,(1 - F(s\)) -lPo-ai = 0. P r i n c i p a l i 's payoff func t ion , V1 can then be r e w r i t t e n as VHluh^lyl) = EW - Cnie*) -17* = v\e\{h, /2; y\, y\), e*2{h, /2; y{, y\)). (1.19) V1 of (1.19) m a y be in te rpre ted as the value of f i r m i . In w r i t i n g out express ion (1.19), we have used the p r o p e r t y of N a s h subgame per fec t ion , tha t is , each p r i n c i p a l knows i n advance abou t the effects of a bonus i n in f luenc ing the e q u i l i b r i u m levels of efforts to be chosen by the agents. T h e foe's for the i t h p r i n c i p a l are ,• dVz ,de* ,de* v;srnl = <lt + M = 0' <120) Chapter 1. Rivalry in Bonus/Penalty Contracts in Oligopolistic Markets 14 Further, v) = E{fp)yl) < 0, and v\ = EU\ - G'^e*) = J 1 / ( ^ ) (^* /c / e t ) from (1.19) and (1.9). It follows immediately that Vj\ji=0 > 0 and < 0 if, respectively, products are strategic substitutes and complements (from Proposition 1.1) and that V \ | / . _ 0 = 0 (from equations (1.16)-( 1.18)). Consequently, if firms' products are strategically related, the non-strategic first-best contract Nl (= IT — a t ) cannot be optimal in a duopoly. Furthermore, the desirable deviation from Nl critically depends on the nature of strategic relationships between agents' actions (or firms' products.in the market). Bonuses would be a desirable policy if actions or products are strategic substitutes, while penalties would be desirable if they are strategic complements. Further using expressions (1.13),(1.14),( 1.16),( 1.17) reduces both (1.20) and (1.21) to the following: or (using v\ = — IiU-j and vl- = U1-) v\v~: =• o. (i .22) For i = 1,2, equations (1.22) determine two reaction functions, one for each princi-pal, in the (Ii,y\) — (72,2/2) s P a c e - Intersections of the two functions determine the equilibria of our two-stage bonus/penalty game. The equilibria w i l l be referred to as the strategic bonus/penalty equilibria, denoted as (7l512, y\, yl). F rom (1.22), we have Ii = —UjUjl/U*IUjj for i = 1,2 (where all the variables are evaluated at a strategic bonus/penalty equilibrium). A n examination of the equations gives the following result. Proposition 1.2: If the strategic bonus/penalty equilibria exist, they would involve a bonus or a penalty if agents' actions or firms' products are strategic substitutes or strategic complements, respectively. Strategic bonuses or penalties would not arise if actions or products are strategically unrelated. Chapter 1. Rivalry in Bonus/Penalty Contracts in Oligopolistic Markets 15 The motivation for a principal, say principal 1, to use bonuses when actions, or products, are strategic substitutes may be seen as follows. As was noted, V7|/1=o = (v}2(de2/dli))\j1=0 > 0. Compared with the non-strategic first-best contract, princi-pal 1 would find a bonus desirable at the margin, given B2. The gain from using the bonus is achieved purely through the strategic effect of bonuses represented by the term vl(de2/dli). A n introduction of a bonus into firm I's internal compensation contracts, given firm 2's, wi l l make agent 1 more aggressive. Agent I's commitment to working harder is credible because working harder is in his own interest. The best response of agent 2 is to act less aggressively, resulting in a lower firm 2's output. 7 Since the prod-ucts are substitutes, a decrease in firm 2's output wil l raise the price for firm I's product, increase firm I's value, and, hence, enlarge the pie to be divided between principal 1 and his agent. Since he takes all the gains himself, principal 1 would benefit at the margin from introducing a bonus into his agency compensation contract, given his rival's compensation contract. The use of penalties when actions or products are strategic complements is due also to the strategic effect of a penalty contract. A penalty policy set up by principal 1 would make his agent less aggressive. The best response of agent 2 now is to be less aggressive, bringing about a lower output for firm 2. Since the products are substitutes, a decrease in firm 2's output would raise the price of firm I's product and, consequently, increase firm I's value at the margin. As a principal uses a bonus (a penalty, respectively), v\ becomes negative (positive). Consequently, the agent uses more (less) than his principal's first-best level of effort, giving rise to an incentive distortion. The optimal level of bonus or penalty wi l l balance the strategic gain with the loss due to the incentive distortion. 7 T h e basic structure of agent level interaction is similar to the effects of f inancial structure on agent effort as examined in Brander and Spencer (1989). Chapter 1. Rivalry in Bonus/Penalty Contracts in Oligopolistic. Markets 16 It is interesting to compare our above results with those of Fershtman and Judd (1987) and Sklivas (1987). Using linear contracts, linear demands and constant marginal costs, they find that, in equilibrium, each owner will give his manager a positive incentive for sales in a Cournot-quantity game, but a negative incentive for sales (or equivalently, a positive incentive for keeping sales low) in a price game. As was noted, with linear demands, the reaction functions are downward-sloping in a quantity game while upward-sloping in a price game. In our model, contract instruments are bonuses and penalties. Each agent chooses his action input which, together with a random variable, determines the firm's output quantity. Hence, each manager indirectly chooses his firm's output, quantity. Our strategic substitutes case is thus basically equivalent to their Cournot-quantity game. On the other hand, our strategic-complement case, as was discussed earlier, might correspond to their price game. An Example. The following example illustrates that a "unique" bonus equilibrium can exist in our two-stage bonus/penalty game. (The example will be used again later.) Assume the production function yz(et,s) = re, + s. r is a positive parameter, and the random variable s is uniformly distributed over [ — 1/2,1/2]. Further assume linear demand, homogeneous products, and constant marginal cost c. We thus have p — a — bY, where p is price (a > c) and Y = y1 -f y2. Finally, assume the disutility function Gi{ez) = re2/2 with parameter r > 0. With linear demands products are strategic substitutes, giving rise to downward-sloping reaction functions in the effort space. Given the compensation contracts, the effort equilibrium is unique and is given by t (2bT2 + r)Il-bT2lJ + (bT2 + r)(q-c)T e, = e , ( / a , / 2 , y i , y 2 ) = (3&r> + r)(fcr'+ r) " Note that the level of target output has no effect on the equilibrium levels of agent efforts. Chapter 1. Rivalry in Bonus/Penalty Contracts in Oligopolistic Markets 17 (This is because, for this example, s is uniformly distributed, and production functions are separable in effort and s and are linear in 5.) Consequently, the strategic, bonus equilibria are characterized by the amount of bonuses. It can be shown that the two reaction functions of (1.22) are linear and downward-sloping in the Ix — I2 space. The unique amount of bonuses at equilibrium is given by l\ — T\ — (a—c)6 2 T 5 / (56 2 r 4+56T 2 r-)-r 2 ) > 0. It follows, using e, = e*{h, I2, y\, yX), that e\ = e\ = (a-c)(2bT2 +r)r/(5b2T4 + 5br2r+r2). 1.5 Effort, Output, Payoff, and Welfare Comparisons Suppose that there is strategic interaction between firms (that is, U3% 7^  0 or R3^ ^ 0). If a principal had ignored the firms' strategic interaction, the optimal contract would be Nl = IT — cti, which is the first-best solution for a principal in the standard agency problem. On the other hand, if the strategic interaction is recognized by principals, the optimal contract (within the class of our bonus/penalty contracts) would involve a bonus or a penalty. A t the strategic bonus/penalty equilibria, principals make a tradeoff between an incentive distortion and a strategic gain. Our concern in this section is to compare a strategic bonus/penalty equilibrium wi th the non-strategic first-best solution. The latter wi l l be simply referred to as the non-strategic equilibrium. Use overhead "tilde" and "hat" to denote variables evaluated at the non-strategic equilibrium and at the strategic equilibrium, respectively. A denotes any difference of valuables between the strategic and non-strategic equilibria. Since there are four unknown variables but only two equations (as in (1.22)), one may not be able to obtain a unique solution in the (Ii,y{) — (l2,y\) space. Such a uniqueness is nevertheless not required in obtaining the results of this section. Recall from equations (1.16)-( 1.18) that, when there is no bonus or penalty, as at the non-strategic equilibrium, the level of target output has no effect on the agents' efforts. Thus, given any strategic equil ibrium ( i i , l2,y\,yl), we Chapter 1. Rivalry in Bonus/Penalty Contracts in Oligopolistic Markets 18 can set y\ = y\ when doing the comparisons. In other words, the target outputs can be ignored in doing our comparisons and only the differences between the amount of bonuses or penalties matter. According to Proposition 1.2, if a strategic equilibrium exists, then / f e = ( /> , / ] )> (0,0) or P = ( / [ , / ? ) < (0,0). (1.23) P denotes the amount of bonuses at the strategic bonus equil ibrium whereas P denotes the amount of penalties at the strategic penalty equilibrium. For expositional purpose, we consider a single strategic equilibrium, either the strategic bonus equilibrium or the strategic penalty equilibrium, and use one of the two inequalities in (1.23) to characterize it. On the other hand, the non-strategic equilibrium is characterized by I — (I-i'l2) = (0,0). Most of the comparisons are carried out under perfect symmetry. B y perfect symme-try we mean that (i) firms have identical output functions y1(-) = y2{'), identical cost functions C\(-) = C2(-)1 face symmetric demands, and agents have identical disutility functions G'i(-) = G2(-)\ and (ii) at both the non-strategic equil ibrium and the strate-gic equilibrium under consideration, firms have identical levels of bonuses/penalties and target outputs, and agents have identical levels of efforts. The methodology used for the comparisons is that of mean-value theorem ( M V T ) methods. 8 Proposition 1.3: Assume perfect symmetry and strategic substitutes. Then, at the strategic bonus equilibrium (i) each agent supplies more effort (ii) each firm produces greater output (iii) prices are lower (iv) each principal has lower payoff (or each firm has lower value) than at the non-strategic equil ibrium. 8 A s has been demonstrated in Brander and Spencer (1983), the mean-value theorem method is very useful in compar ing the equi l ibr ia of two different models. Chapter 1. Rivalry in Bonus/Penalty Contracts in Oligopolistic Markets 19 Proof: (i) Using (1.12) and ignoring the target outputs give e\ — e*{i\,I2) a n d ei = e*i(I1J2). Apply ing the M V T to e*(7j, 7 2) yields dli dlj where de* / dl% and de*/dlj are evaluated at some point between Ib and I. Under perfect symmetry, A / , = AI3 = l\ > 0 and de*/dl: = de^/dli. Thus A e , = (d(e* + e))/dli)l^ > 0 (using Proposition 1.1). That is, e\ > e;. (ii) Similarly, Ay1 = y^Ae, > 0 (using (i)). (iii) Similarly, Apl = p\Ayl + p)Ay3 = (p\ + p))Ayl. Since pj + p) < 0 (using (1.4) and perfect symmetry) and A y ' > 0 (using (ii)), it follows that A p l < 0. (iv) Apply ing the M V T to V * ( J a , I2) yields A T / ' = VfAIi + VfAIj, where V / and V> are evaluated at I = (Iu I2) with 0 = 7, < 7,• < l\. Since A7, : = A7j• = l\ under perfect symmetry, AV1 = (V? + V?) / ) . Thus A V 1 < 0 if and only if V> + V] < 0 B y using expression (1.19) and perfect symmetry we have d(e* + e*2) v; + v; = (v} + v))-di, where v\ and are evaluated at ex = e^^,^) and e 2 = e*,(Ii,I2). Similarly to (i), one can show that ex < e\ < e\ for % — 1,2. Since = Iif(s\)(ds\/'dez) < 0 (using (1.19), (1.9), and h > 0), v) < 0, and d(e* + e*2)jdlx > 0, it follows that V> + V? < 0. Therefore, V''h < V\ Q.E.D. The situation with strategic, substitutes is depicted in Table 1.1. Each principal can make a choice between the non-strategic contract (Nl) and the strategic contract {B%). The payoffs are given in the table with e > 0 and n > V1 — Vl'b > 0. Thus, if principal 1 acts strategically while principal 2 does not, principal 1 wi l l gain relative to non-strategic Chapter 1. Rivalry in Bonus/Penalty Contracts in Oligopolistic Markets 20 Table 1.1: Principals ' Payoff Mat r ix : Strategic Substitutes ( e > 0, T] > V1 - Vl-b > 0 ) principal 2 principal 1 N2 B2 N' V\ V2 B1 V1 +£, V2 -77 Vl-b, V2'b behaviour, while principal 2 wi l l get hurt due to his rival's action. For each principal, strategically using internal payment contracts dominates non-strategically using con-tracts. But if both strategically manipulate, their compensation contracts, both would have lower payoffs than if both behaved non-strategically. Rivalry in agency compensa-tion leads to a lower payoff for both principals. The situation is similar to a Prisoners' Del imma. The results when actions or products are strategic complements can be quite different. Compared with non-strategic behavior, rivalry in penalty contracts would lead to less effort by agents, lower firms' outputs, higher prices, and would possibly lead to higher payoffs for both principals. Proposition 1.4: Assume perfect symmetry and strategic complements. Then, at the strategic penalty equilibrium (i) each agent supplies less effort (ii) each firm produces smaller output (iii) prices are higher (iv) if the sign of v\t -f- 2v\3 -f vl3J remains unchanged between (/;, Ij) = (0, 0) and (If, I?), Chapter 1. Rivalry in Bonus/Penalty Contracts in Oligopolistic Markets 21 each principal has higher payoff (or each firm has higher value) than at the non-strategic equilibrium. Proof: Parts (i)-(iii) can be shown by referring to the proof of Proposition 1.3 and by using (1.23). Now prove part (iv). Also referring to the proof of Proposit ion 1.3, we have -1> = (v\ + v ) ) d ( e ^ el)if, where v\ and vz are evaluated at ex = e\(Ii,I2) a n d e2 = e ^ / j , ^ ) and If < Ir < Ix = 0. W i t h d(e* + el)/dli > 0, it follows that Vl<v > V1 if (and only if) v\ -f v) < 0. v\ = hf{s\){ds\/dei) > 0 while vj = E(ylpljy3e) < 0. Thus, the sign of v\ + vx- is not clear. Nevertheless, one can show that d} + i}<0, v]'p + v)'p<0. (1.24) The former holds because v) = 0 and v\ < 0. The latter can be shown as follows. From equation (1.22), —v\'p/vjp = — Ujf/Ujf. Further using symmetry and DA = {Ujj + U]i)[U]J — U3-) > 0 gives (in particular) -Ujf/Ujf < 1. Thus, v]'p + vf < 0. Apply ing the M V T to v\ + vl- between points ( /? , / • ) and (I2,12) gives (using sym-metry ) A « + v)) = (v^ + 2v\3 + v } / { ^ £ l ) A I i , where v\% + 2v\ - + v1-- is evaluated at some point between (I}, Ij) and (If, Ij). If the sign of v\x + 2v\ - + vljj remains unchanged between (I,., If) = (0,0) and (If,Ip), then v\ + v* can be signed negative by applying the M V T to v) + v* between points (Jx, Ij) and either (Ii,Ij) (if v\% + 2v\j + vf is positive) or (If' ,IP) (if v\% -f 2v\0 -f v)3 is negative) and then using inequalities (1.24). Q .E .D . We now conduct a welfare comparison. Consider the following social welfare function: W = E(U(y\y2) - C^y1) - C2(y2)) - G^e,) - G2(e2) = w(eu e2). (1.25) Chapter 1. Rivalry in Bonus/Penalty Contracts in Oligopolistic Markets 22 U{yl,y2) is the consumer utility function in the usual industry (partial equilibrium) analysis. The inverse demand functions are p1 = dU/dy1. Letting u>i = dw/dez for i — 1,2, then can be calculated as follows. wl(eue2) = E((f - C;K) - Gfc) = v\ - E(yWe), (1-26) where the second equality follows from using (1.19). The following lemma suggests that from the social welfare point of view, the efforts supplied by oligopolists or monopolists are not enough. Lemma 1.1: (i) w\ > 0 if < pf and y\(e?ue*) = jr?(eP,e£); (ii) w% > 0 if p\ < p) and y\(eue2) = y2e{h,e2)\ and (iii) w\ > 0 if Ulf < fyf', pf < pf, and Proof: First show that v\ > i>* at these three equilibria. As noted before, v\ = IJisDidsl/de,) > 0 if I, < 0. Thus, v-p > 0 > vf and v\ = 0 > v). Further, from principal i's foe (1.22), v}'b/v}'b = Ujf/UJf. Thus if U]f < Uji\ then v^h > vf (recalling Uj-b < 0,{^'fe < 0,t)*'fc < 0). Next, at these three equilibria, W i = v\-E{tfp\y\)>v)-E{tfp\y\) = E(fp)yi) - E(fplyl) = E{yYe{p) - p\)) > 0, where the last second equality is due to the condition that y*(ei,e2) — y2(ei,e2) (at these equilibria) while the last inequality follows from p) < p*- < 0 (at these equilibria). Q.E.D. To have a unique solution to the social welfare function, we impose the following condition on the welfare function: wu < (1-27) Chapter 1. Rivalry in Bonus/Penalty Contracts in Oligopolistic Markets 23 for 0 < 5 < 1 (j / i , i,j = 1,2) over the entire region of interest. Since wz] = E(pljyly2) < 0, condition (1.27) has implied wu < 0, wiiw22 - w12w2i > 0. (1.28) (1.28) means that the welfare function is strictly concave, and implies a unique pair of socially efficient, level of effort. Note that conditions (1.27) and (1.28) are equivalent under perfect symmetry. Lemma 1.2: ^ ( e 2 , e2,) < w%(e\, e\) if e\ > e\ and e\ > e\; and w%(e\, e\) = w,(e\, e\) if and only if e\ = e\ and e\ — e\. Proof: Applying the M V T to wl(e1,e2) yields wt(el,e22) - Wi(e\,e\) = wtl(e2 - e\) + wz2{e22 - e2), where wu and wl2 are evaluated at some point between (e\,e\) and ( e ] ^ ) . Lemma 1.2 follows by noting that wrl < 0 and wl2 < 0. Q . E . D . Proposition 1.5: (i) Assume that pf < p-p and yl(ep,e2) = y2(e\,eV2)- Then, the welfare level is lower at the strategic penalty equilibrium than at the non-strategic equi-librium provided that the effort, level of each firm at the strategic penalty equilibrium does not exceed that at the non-strategic equilibrium, (ii) Assume that \J\f < U\f, p)'h < pjh, and yle(e\,eb2) = yli^nQ). Then, the welfare level is higher at the strategic bonus equilibrium than at the non-strategic equilibrium provided that the effort level of each firm at the non-strategic equilibrium does not exceed that at the strategic bonus equilibrium. Proof: Applying the M V T to W — to(ex, e2) yields W - W = u>i(ei - h) + w2(e2 - e2) (1.29) where wi and w2 are evaluated at some point between e and e. (i) Provided that e\ < e~\ and e2 < e2, it follows that < ea and e2 < e2. By using Lemmas 1.1 and 1.2 we Chapter 1. Rivalry in Bonus/Penalty Contracts in Oligopolistic Markets 24 know that w% > wx > 0. From (1.29), Wp < W. (ii) Provided that Since e\ > ex and e\ > e 2, it follows that e\ > eA and e2 > e 2. By using Lemmas 1.1 and 1.2 we know that Wi > w\ > 0. From (1.29), Wb > W. Q . E . D . The result that the pro-effort aspect of the bonus rivalry is welfare-improving arises because prices exceed marginal costs in oligopolistic industries. Nevertheless, duopolistic competition does not completely resolve the problem of too little effort from the social point of view, as seen in the following result. Proposition 1.6: Assume that U-{ < U-j: p\ < p)\ and y]^,^) = y 2 (e ! ,e 2 ) . Then, (i) the efforts supplied by the duopolists would be less than the socially efficient efforts; and (ii) the welfare level is lower at the strategic, equilibrium than at the socially efficient solution, provided that the effort level of each firm does not exceed the socially efficient level of effort. Proof: We prove the results only for the strategic substitutes case. (The strategic complements case can be easily seen.) (i) Denote e\ as the socially efficient level of effort for agent i , i = 1,2. From welfare function (1.25), Further, from Lemma 1.1, Awi = w\ — w\ = w\ > 0. Apply ing the M V T to wl(ei,e2) for i = 1,2 yields where Wij are evaluated at some point between eb and es. Solving the simultaneous equations for A e i and A e 2 and then adding the two gives (1.30) Awi =-- wjiAe-y + u ) 1 2 A e 2 , Aw2 = w2iAei -(- K ; 2 2 A e Ae, + Ae2 = — Awi(w22 - w21) + Aw2(wu — w12) (1.31) Chapter 1. Rivalry in Bonus/Penalty Contracts in Oligopolistic Markets 25 where A e t = e\ — e-, Ds = w-i\W22 — W\2W2\-. From (1.27), (1.28), and (1.31), we have A e i + A e 2 < 0 or equivalently, e\ + e\ < e^ + es2. (ii) Apply ing the M V T to W = u>(e1; e 2) yields AW = •?D1Aei + w2Ae2 where to,; are evaluated at some point between eb and es. Given A e a < 0 and A e 2 < 0, we know from part (i) that strict inequality holds for at least one of the two. Then using Lemma 1.2 yields u>, > w* = 0. Hence AW = Wb — Ws < 0. Q.E.D. The conditions specified in Propositions 1.5, 1.6, or Lemma 1.1 are satisfied under perfect symmetry. Corollary 1.1: Under perfect symmetry, (i) the welfare level is higher (lower, respec-tively) at the strategic bonus (penalty, respectively) equilibrium than at the non-strategic equilibrium, (ii) the efforts supplied by the duopolists are lower than the socially efficient levels of effort, and (iii) the welfare level is lower at the strategic equilibrium than at the socially efficient solution. We finally show that the socially efficient levels of effort may be achieved at the second-best optimum which arises from maximizing the welfare function subject to er = e*( i i , I2) for i = 1,2. Proposition 1.7: Assuming perfect symmetry, then the socially efficient levels of effort can be achieved at the second-best optimum which involves higher levels of bonus than those at the strategic bonus equilibrium. Proof: A t the second-best opt imum the welfare function is (ref (1.25)) W = w(el(IuI2),e;(IuI2)) = W(IUI2). (1.32) Denote Is = ( 7 j , / | ) to be the bonus level at the second-best optimum. Then W? = Wim,I2>) = 0. (1.33) Chapter 1. Rivalry in Bonus/Penalty Contracts in Oligopolistic Markets 26 From (1.32) and symmetry, W s Ae\ sde* sd(e* + e*2) dlr dl7 1 dli From (1.33) and Proposition 1.1, Wf = 0 = wl (1.34) A n examination of (1.30),(1.33),(1.34) shows that e\ = e*(I{Js2) for i = 1,2. App ly the M V T to e*(7i , / 2 ) : * - < = %0\ + -n). B y symmetry and Proposition 1.6, e\ - e\ = [d(e\ + e ^ / d l ^ l f - I?) < 0. Thus I* > Ib for i = 1,2. Q.E.D. Proposition 1.7 suggests that to encourage firms to use more bonuses might serve as a second-best solution to the problem of too little effort in oligopoly. 1.6 Effects of Firm Capital Stocks on Bonus/Penalty Rivalry The output of each firm may also depend on its capital stock, denoted k^. kt and k2 are here considered as being historically given when firms engage in our two-stage bonus/penalty game. To examine the effects of firm capital stocks on the bonus/penalty rivalry, we conduct a comparative statics analysis of effects of a change in the stock of one firm, say firm 1, on firms' equil ibrium payoffs. From (1.19), one obtains dVl - , dh - • dl2 - , dy] - , dyX ^ = v ' > t + V ! ^ + v « t + v i i : - ( 1 3 5 ) Using foe's (1.20) and (1.21), (1.35) becomes (for i = 1,2) ^ = V / ^ + y i M (1.36) dkX h dk, y2 dkX* V J Chapter 1. Rivalry in Bonus/Penalty Contracts in Oligopolistic Markets 27 dV2 « , d / a %7dy\ , ^ * r = v ^ + v3sit ( 1 ' 3 7 ) We now show that Vf < 0 and V2 < 0, i.e., a higher level of bonus by one principal wi l l decrease the equil ibrium payoff to the other. From expression (1.19), T>1 ~ i ^ e i , « l ^ E 2 v'>=VldT2+v>dT3-v\ can be determined by (1.20). Substituting the expression for v\ gives h V 2de*/dljdl2 2dl2 del/dh 1 1 2 1 where use of (1.13) and (1.14) yields the second equality. One can similarly show that V2 < 0. Thus, for any given level of target outputs, dV1 /'dkx > 0 and dV2/dki < 0 if dli/dki > 0 and dl2jdkx < 0. That is, for any given level of target outputs, if a higher capital stock increases the equilibrium level of its own bonus and decreases the equil ibrium level of its rival's bonus, then a higher capital stock would in equilibrium increase own payoff while decrease the rival's payoff. A firm's capital stock, kt can refer to factors such as size, human capital, union-management relationship, etc. For concreteness we may regard ki as the size of firm i . Presumably, the larger the firm, the greater its output, that is, dy1 /dki > 0. Consider the example given in Section 4. Now the production function is modified to incorporate the effect of firm size: yl(ei, s, ki) = TelJrsJryo(ki) with y\ = y'0 > 0. For this example one can show that there exists a bonus equil ibrium which is characterized by the unique amount of bonuses. Further, one can show that dlx/dki > 0, d^/d^ < 0, and V 1 , = V2t = 0. Consequently, from (1.36) and (1.37), we have dV1/dki > 0 and dV2/dk1 < 0. For this example, therefore, a larger firm would Use a larger amount of bonus and have a higher payoff at the strategic bonus equilibrium than a smaller firm. The advantage with a larger firm in this example may be seen from the fact that V}ki > 0. Note that the foe of principal 1 is (ref (1.20)) V / = v\(de\ldh) + vl(de*2/dh). Chapter 1. Rivalry in Bonus/Penalty Contracts in Oligopolistic. Markets 28 As mentioned, the first term in Vj is a loss to principal 1 due to the incentive distor-tion, while the second term is a gain to the principal due to the strategic effect of a bonus. The optimal amount of bonus is set to balance the gain with the loss. Since d(v\(de*2ldlx))Idki = (de*2l dh)E(y\p\yl) > 0, the strategic effect would be stronger with a large firm than with a small firm. Further, since d(y\(de\l'<£/i))/'dkx = 0, the incentive distortion would remain unchanged as a firm gets larger. As a result, a larger firm enables its owner to. use a larger bonus at equilibrium, resulting in a greater effort of its own agent and a smaller effort by the rival agent. Since an increase in the bonus level raises the price for its own product due to the lower effort, and hence lower output, of the rival, own profit would increase more for a large firm than for a small firm. The above analysis might suggest a three-stage game. In the first stage firms choose i capital investments; in the second stage principals decide bonus/penalty contracts; and finally agents choose action levels. The added feature of this three-stage game is that levels of capital stocks may be chosen strategically by firms who understand the rivalry in compensation contracts in later stages. 1.7 Concluding Remarks This chapter has presented an analysis of rivalry in bonus/penalty contracts in an oligopolistic industry when both firm owners and managers are risk-neutral. Under standard assumptions on production, demand, and cost functions, we have shown that bonus/penalty contracts may be used for strategic purposes. If principal 1 uses bonuses or penalties strategically while principal 2 does not, 1 would gain relative to non-strategic behaviour, while 2 would get hurt due to I's action. In the normal case of strategic sub-stitutes, if both use bonuses for strategic purposes, then under reasonable symmetry both Chapter 1. Rivalry in Bonus/Penalty Contracts in Oligopolistic Markets 29 would expect lower profits than if both act non-strategically. The rivalry in bonus con-tracts can hurt both firms, and the formal structure of the problem is similar to that of a Prisoners' Delimma game. The players would thus have an incentive to collude tacitly in their choice of payment, contracts. If they interact with each other from time to time, we might see the non-strategic first-best solution emerge as the equilibrium contract. The reason for the rise of non-strategic compensations is, however, different from the one offered by the standard agency theory. The former recognizes strategic interaction while the latter does not. We find that the welfare implication of firms' strategic rivalry in agency compensation critically depends on the nature of strategic relationship of firms. When agents' actions or firms' products are strategic substitues, the rivalry in bonus contracts would be welfare-improving, compared with non-strategic behaviour, due to increased amount of efforts and thereby increased outputs. But if they are strategic complements, the firms' rivalry would lead to a more collusive outcome than firms' non-strategic behaviour. Nevertheless, the amount of effort, supplied in an oligopolistic industry is less than the socially efficient level of effort. One way to extend the analysis of this paper is to relax the risk-neutrality assumption on principals and agents. The next chapter will provide an approach of this kind. Chapter 2 Bonus/Penalty Contracts As Strategic and Incentive Devices 2.1 Introduction The purpose of this chapter is to discuss effects of bonus/penalty contracts in rivalrous agency in a more general framework than Chapter 1 where both principals and agents are assumed risk-neutral. In this chapter, principals and agents may be risk-averse. The model structure is laid out in the next section. The effects of a bonus on agents' rivalry are discussed in section 2.3. We then in section 2.4 identify two distinctive effects associated with bonus/penalty contracting: the strategic effect and the incentive effect. Section 2.5 summarizes and interprets the main results of the chapter. 2.2 The Model The basic model structure remains the same as before. There are two principal-agent pairs. Given the compensation contracts by both principals, each agent chooses an unobservable action, such as his work effort, e, (e, G [0,e], e < oo). The payoff or profit, of each pair, denoted II1, depends on both agents' actions (ez and ej). Given e; and ej, the realization of profits depends on the realization of a random state of nature, 5 : I T ( e i , e2, 5 ) . 5 is normalized to be in [0,1] interval with p.d.f. f(s). Higher values of s are regarded as better states of nature, that is, ITS > 0. Each agent chooses his action before observing the random state, s. Finally, s is revealed to the agents (but not to the principals), and payoffs are made. 30 Chapter 2. Bonus/Penalty Contracts As Strategic and Incentive Devices 31 Principals' decisions on agency compensations are treated as strictly prior to agents' actions. Assume that each principal observes only the profit level of his firm, IP, and rewards his agent according to the following class of bonus/penalty contracts: Similarly to the interpretation of Chapter 1, I, is the amount of bonus (if It > 0) or penalty (if Ix < 0); whereas II* is the target profit (0 < LT* < TP(e,e, 1)). Thus, agent i receives payment W;(IP) plus a bonus (or penalty) 7, if IP > II'; otherwise, he receives payment W?(rP). When Iz = 0, Bl reduces to the initial compensation contract W%. In this chapter we consider the situation where only one principal is actively seeking the use of bonuses and penalties. Specifically, given that I2 = 0, i.e., principal 2 does not use bonuses (or penalties), principal 1 contemplates whether or not to use a bonus (or penalty). If principal 1 decides not to use a bonus, then 7a = 0 and B1 reduces to Wi, a non-bonus contract. Our analysis focuses on the marginal gain to the first principal-agent pair from using bonuses and penalties under ^ = 0 or equivalently, under W^II 1 ) and Assume that under W^IT 1) and ^ ( I I 2 ) , the payoff to the first principal-agent pair is increasing with the agent's effort ei, while decreasing with the effort of the rival agent ej (i.e., agents' actions are substitutes), for the relevant range of s. We shall also consider a situation in which III = 0. In such a situation, the pair consisting of principal 1 and agent 1 is said to be a monopoly pair. (2-1) W2(U2). u\ >0, < 0 (2.2) Chapter 2. Bonus/Penalty Contracts As Strategic arid Incentive Devices 32 2.3 Effects of a Bonus on an Agent's Effort Given compensation contracts B1 and B2 (= W2), the objective function, or the utility, of agent i is Ul = E(Ul(Bl - Gi(ei))) where G',;(e,) is the disutility of effort expressed in terms of money. A n agent is either risk-neutral or risk-averse with u\ > 0 and u" < 0. Denote Zl(e1,e2, s) = Wl(Ul(e1, e2, s)) — Gj(e;). Then Ul can be written as C/ i(e1,e2;7i,nJ) = + Es>s<{Ul\Zl + / , )) , (2.3) where s\ is derived from the following equation n i(e1,e2,SJ) = nl. (2.4) Thus, given compensation contracts, the uti l i ty of each agent depends not only on his action but on his rival's action. This stage game is about competition between agents. Each agent chooses his action to maximize his utility, given the rival's action. The foe's for agent 1 and agent 2 are, respectively, Ul = ^f-=0: E^u^Z^Z]) + E^u'^Z1 + h)Z\)--f(s\)^(u1(Z1(e1,e2,s\)-rIi)-u1(Z\e1,ei,s\))) = 0 , (2.5) 77-7-2 U2 = —— = 0 : EAu'(Z2)Z2) = 0. (2.6) de2 Equations (2.5) and (2.6) determine the two reaction functions, one for each agent. The Nash equilibria of the agent game, denoted (e^e^), are the intersections of the two reaction functions. To examine the effects of a bonus introduced by principal 1 on the equil ibrium levels of agents' efforts, we carry out comparative statics exercises. Totally Chapter 2. Bonus/Penalty Contracts As Strategic and Incentive Devices 33 de\ -ui:u22 dh ~ D deX U1IU2J D differentiating equations (2.5) and (2.6) w.r.t. ej, e 2, and Ix and then solving for de'/dl^ and de*2jdl\ yields (2.7) (2.8) where D = U^U22 - U^U^ and U\j = dU'/dh. We now try to sign de*/dli and deX/dli at 7] = 0. We assume that under VK^II 1 ) and ^ ( I T 2 ) in question, the following conditions are satisfied over the range of interest: (i) Ui < 0 for i = 1,2, and (ii) D > 0. Thus under W ^ I I 1 ) and iy 2(n 2), the Nash equil ibrium of the agent game, to be referred to as an init ia l equil ibrium, exists and is unique. Note that when = 0, Ulir = E{u'l(Z\)2 -f u[Zl7). Hence, C7>. < 0 if Z\% = {I\\)2W>> + W\^[ - G" < 0. We now sign at I\ = 0. L e m m a 2.1: ^ i /Ui=o > 0 f ° r a sufficiently high target profit. P r o o f : It can be verified that tfi/U=o = Ea>s{(v^{Zx)Z\) - f i s l f f i u i i Z 1 ) , (2.9) where all the variables on the right-hand side of equation (2.9) are evaluated at i i = 0. Using condition (2.2), one obtains ds\jde^ = — Ul/U] < 0. If the agent is risk-neutral, i.e., u'[ = 0, then the first term on the right-hand side of (2.9) vanishes. The second term and, hence, U\J\J1=Q are positive as long as f(s\) ^  0. If the agent is risk-averse, then the first term tends to zero as s\ —> 1" (meaning that s\ tends to 1 from below). Suppose that l i m ^ i ^ j - f(s\) > 0. Then the second term and, hence, /Ui=o a r e positive as s\ —> 1~ or as the target profit is sufficiently high. Suppose now that l im s t _tl- f(s\) = 0. In this case, we impose two assumptions on p . d i . f(s): Chapter 2. Bonus/Penalty Contracts As Strategic and Incentive Devices 34 (i) the left-hand derivative / ' ( l ) exists and f'(s) is left-hand continuous at 5 = 1, and (ii) F(t) = Jof{s)ds < 1 for any t, 0 < t < 1. That UlI\]l=o > 0 can be shown as follows. W i t h the existence of / ' (1~) , l i m 5 « ^ i - f(s\) = / ( l ) = 0. Condit ion (ii) is equivalent to F(t) — 1 t = 1, essentially saying that 5 = 1 is an "effective" point. It would natually arise when one normalizes random variable 5. Moreover, it wi l l guarantee f'(l~) < 0. (Otherwise, if / ' (1~) > 0, there exists some t, 0 < t < 1, such that / ( l — t) < / ( l ) = 0 because of the second part of condition (i). But f(s) > 0, we must have / ( l — t) = 0 or equivalently F(l — t) = 1 with 0 < 1 — t < 1 which contradicts condition (ii).) From (2.9), {Uu\ii=o)st = : = 0 and (,d{U^)Il=0/ds\)3ti=1 = -f'(l-)(dst1/de1)u'1(Z1) < 0 (due to / ' (1~) < 0). The result follows immediately. Q.E.D. c i^/|/i=o > 0 s a y s that the use of a bonus by principal 1 increases Uj, the (expected) marginal return to agent 1 from taking an action (such as effort) at the margin. This is essentially due to the fact that, in the presence of a bonus, effort would have an additional effect of lowering the critical state of nature and, hence, increasing the probability of obtaining bonuses. W i t h U\j\h=0 > 0, we have (del/dl^j^ > 0 from (2.7) (recalling Ul2 < 0 and D > 0 at Ii = 0). O n the other hand, at 7a = 0, sign (de^/dli) = sign [U21) (from (2.8)). Note that, at Ii = 0, Ul = E(u'2'(Z2)Z2Z22 + u'2(Z2)Z221). (2.10) Thus, when agent 2 is risk-neutral and W ^ L T 2 ) is a linear contract, U21 — W^n 2 )^]! 2 , !) . Supposing W^n 2 ) > 0, then sign (U21) = sign ( E T I ^ ) . This , together with the analysis of Chapter 1, suggests the following definition. Definition 2.1: Actions (such as efforts) taken by the two agents are said to be strate-gic substitutes, strategic, independents, and strategic complements under compensation Chapter 2. Bonus/Penalty Contracts As Strategic and Incentive Devices 35 contracts W ^ f l l 1 ) and W^IT") if U2\ < 0, = 0, and > 0, respectively, where JJ\\ is defined by (2.10). The above discussions lead to our comparative statics results as follows. Proposition 2.1: A bonus with a sufficiently high target profit used by principal 1 would have the following effects on the levels of agent efforts at an ini t ia l equilibrium: (i) his own agent wi l l use more effort, and (ii) his rival's agent wi l l use less, the same, and more effort if agents' actions are strategic substitutes, strategic independents, and strategic complements, respectively. 1 2.4 Strategic and Incentive Effects of Bonus/Penalty Contracts The objective function of principal 1 is V1 = Eiv^W — B1)), whereas the objective function of principal 2 is V2 = E(v2(U2 — W2)) (noting B2 = W2). Each principal is either risk-neutral or risk-averse with v[ > 0 and v" < 0. Pr incipal I 's objective function can be further written as E ^ v ^ - W ^ U 1 ) - ^ ) ) . (2.11) The above equation has implied that, principal 1 anticipates correctly the effects of his bonuses on the equilibrium levels of efforts chosen by both agents. In other words, principal I ' s expectation of e\ and e.2 wi l l be confirmed in equilibrium. As said, we are interested in the question concerning whether a bonus or a penalty is worth using for principal 1 or, more generally, for principal 1 and agent 1 as a whole, given the original contracts W ^ I I 1 ) and i y 2 ( I I 2 ) . We first give the following definition. 'Moreover , d(e{ + e'2)/dh = (de\ldh) +(de*2/dh) = Ul, {-U222 + U^)/D. Hence, for both strategic-independent and strategic-complement actions, the total amount of efforts will rise; while for strategic-substitute actions, the total amount of efforts will rise if U22 < U2l-Chapter 2. Bonus/Penalty Contracts As Strategic and Incentive Devices 36 Definition 2.2: The bonus (and penalty, respectively) contracts B1(Ii,Ht1;W1(H1)) are said to dominate the non-bonus/penalty contract W1(n1) if and only if {d(Ul + V1 )/dIi)} Q > 0 (and < 0, respectively). A n alternative expression for Definition 2.2 is that B1 is said to dominate Wt if and only if a bonus/penalty contract is more desirable at the margin than the non-bonus contract W\. It is noted that a bonus (or a penalty) policy which is good for principal 1 and agent 1 as a whole may not be good for both of them. But. if principal 1 designs agency compensation, then by using a payment transfer as part of the bonus/penaty contract, principal 1 can make himself (strictly) better off while his agent is not made worse off. We now examine {d(Ul -f V1)/dIi)I Q. Using expression (2.3) and et = e*(Ii,H\), we have d.Ul - r / i ^ f l i 771 ^ f l , 7-/1 - 1 AT ^ A T + 7' 7i=o dli ah dh where all the variables on the right-hand side of the equation are evaluated at h = 0, which is not explicitly indicated for notational simplicity. From the foe of agent 1 (2.5), U{ = 0. Hence, dU1 , deX , •irl=v^v] (2'12) where U\ = ^ ( u ' ^ Z ^ W^n 1)^) (2.13) U) =£i>a«K(Z1)). (2.14) Further using (2.11) we obtain dV1 , r1 de*. . de% . . J 1 = 0 ah dh dh where v? =E(v'1(u1- w1(n1))(i-iy1'(n1))nl), . (2.16) Chapter 2. Bonus/Penalty Contracts As Strategic and Incentive Devices 37 vl = E(v[(W - wv(n'))(i - wftn 1))^), (2.17) V} = -E.^WIL1 -W^TL1))). (2.18) Using (2.12) and (2.15) we obtain diU1 + V1) = V?d4 + (Ul + V ? ) ^ > + U) + Vl\ (2.19) dh where V? is given by (2.16), U] by (2.13), V2 by (2.17), Uj by (2.14), V? by (2.18), and de\jdli and de*2j dl2 by, respectively, (2.7) and (2.8) evaluated at = 0. The third and fourth terms on the right-hand side of (2.19) tend to zero as s\ —» 0 or as the target output IT j becomes sufficiently high. Basically, they represent the (expected) payment transfer between principal 1 and agent 1 as a direct result of the use of bonuses and penalties. In what follows, we give an interpretation to the first and second terms on the right-hand side of (2.19). 2.4.1 The incentive effect Proposition 2.2: Consider a monopoly pair of principal 1 and agent 1. If V* > 0, i.e., principal 1 would like to see his agent increase effort under W ^ I l 1 ) , then the bonus contract B1(I1,Il\;W1(Il1)) dominates W^U1). Proof: As said, a monopoly pair of principal 1 and agent 1 implies that U\ = 0 (under WX and W2). Thus, U2 — V2l = 0, and for sufficiently high target profit, de\/dli = -UIJ/UI-L > 0. Using (2.19) and the condition > 0, we know that (d(£ / 1 + Vl)/dli)h=0 > 0 for sufficiently high target profit. Q.E.D. The proposition says that bonus contract can be benefitial if, under W ^ f l 1 ) , the agent supplies less effort than the principal wishes. This result is similar to Lewis (1980). Chapter 2. Bonus/Penalty Contracts As Strategic and Incentive Devices 38 For a single principal-agent pair, the gain from using bonuses is achieved purely through the incentive effect of bonus contracts, which is reflected by the first term on the right-hand side of (2.19), namely, Vj(de* / dli). This may be seen by examining the condition that Vj > 0 given ^ ( I I 1 ) . Note that in the proposition, W ^ L l 1 ) can be any payment schedule. The principal and the agent have different objective functions. The agent has a disutility for his effort, and only the agent supplies effort. However, both principal and agent wi l l share the benefits from the agent's effort. Thus, there may be an incentive problem in the sense that the agent would supply less than the optimal (from the principal 's point of view) amount of effort. In the traditional principal-agent problem, one is interested in the Pareto optimal payment schedule. Let W ^ n 1 ) denote the optimal payment schedule within a certain class of payment schedules (such as non-bonus continuous contracts). Holmstrom (1979) shows that, if the agent's disutility rises with effort and effort increases profit (stochastically), then Vj > 0, i.e., the principal would like to see the agent increase effort given M^n1). Thus, M ^ n 1 ) generally does not solve the incentive problem. In such a situation, bonuses would have the agent supply more effort and would alleviate the incentive problem. If the agent is risk-neutral, it is well known that, the optimal contract, namely, VK^n 1 ) , is the first-best contract. In effect, VK^n 1 ) = U1 — c\i where constant a a is determined by the agent's participation constraint. In other words, the principal simply charges the agent a fixed amount of rent which is independent of the outcome, and the agent consequently becomes the residual claiment. The agent therefore would have no dilution of incentives: Vj = 0 given W ^ I I 1 ) . W i t h Vj = 0, we would have d(Uv+ V^/dh = 0 if the principal is also risk-neutral (here 'ui(-) = v'i(-) is implic i t ly assumed). Bonus contracting in this case adds no value to the principal-agent pair, and it merely transfers (expected) bonus money from the principal to the agent. Wi thout the incentive problem, the value of bonus contracting generated by its incentive effect would be lost. Chapter 2. Bonus/Penalty Contracts As Strategic and Incentive Devices 39 2.4.2 The strategic effect When one moves from a single principal-agent pair to rivalrous agency, bonus/penalty contracting may be used as as a strategic device. Moreover, the strategic, effect of bonus contracts can be independent of their incentive effect. Both points, which have been illustrated in Chapter 1, are seen from the following proposition. Proposition 2.3: Assume that agent 1 is risk-neutral and W ^ L T 1 ) = II 1 — Q j . Then, the bonus (and penalty) contracts !![; ^ ( L T 1 ) ) dominate ^ ( f l 1 ) if agents' actions are strategic substitutes (and strategic complements, respectively). Proof: ^ ( n 1 ) = I I 1 - a , => W^U1) = 1. Consequently, V? = V} = 0, but U\ = E(u'1(Z1)W;{U1)Il12) < 0 (because IT^ < 0). Thus, for sufficiently high target profit, U2(de%/ dli) > 0 and < 0 if agents' actions are strategic substitutes and strategic complements, respectively. Using (2.19) we know that {d(Ul + V1)/dI-l)I^_Q > 0 and < 0 for sufficiently high target profit if agents' actions are strategic substitutes and strategic complements, respectively. Q .E .D. We now identify the source of gain. From (2.19), (4t/1 + V 1 ) / d / 1 ) / i = 0 = U2\de*2l'<ZJi)+ U] -+- Vj . The incentive problem just discussed is not presented here and there is no role for incentive effects of a bonus contract. Further, as has been indicated, Uj and V}1 tend to zero as the target becomes sufficiently high. Hence the source of gain is from the term U\{de*2j'dli) which captures the strategic effect of bonus/penalty contracts. In the case of strategic substitutes, the term is positive. W i t h a possibility of obtaining a bonus for high profit, it is in agent I's interest to work harder. Since agent I's commitment to more effort is credible, the best agent 2 can do, given his compensation contract, is to supply less effort. In the case of strategic complements, on the other hand, the term is negative. W i t h a possibility of getting a penalty, it is in agent I's interest to work less. The best response from agent 2 is also to work less. In both cases, therefore, use of bonus/penalty Chapter 2. Bonus/Penalty Contracts As Strategic and Incentive Devices 40 contracting by principal 1 can make the rival's agent supply less effort, which would at the margin benefit his own agent. Proposit ion 2.3 suggests that, when an agent is risk-neutral, the desired strategic contracting critically depends on the nature of strategic relationships between agents' actions. If the actions are strategic substitutes, then bonus contracting is desirable. On the other hand, if the actions are strategic complements, then penalty contracting is desirable. 2.4.3 Strategic and incentive effects We have illustrated in the special cases two distinctive effects of bonus contracting, namely, the incentive effect and the strategic effect. In the general case of rivalrous agency with risk-averse agents, it is conceivable that both effects may be actually functioning. To illustrate this, we consider the following setting. The original compensation contract, ^ (n 1 ) is in the form of linear payment schedules (linear in profit): ^ ( i l 1 ) ~ B-[Ill — ct-y with /3j > 0. The parameter B\ measures the degree of continuous dependence of agent I's payment on the pair's final payoff. We assume that, under and W2, Expression (2.20) says that as agent I's compensation becomes more related to the outcome, the marginal return to the agent from supplying effort wi l l increase. This would be the case when either the agent is risk-neutral, or the payoff function is sepa-rable in action and state of nature: n 1 ( e 1 , e 2 , 3 ) = 7r 1 ( e 1 , e 2 ) + h(s). The justification is as follows. Under linear contracts, agent I's objective function can be written as dU\ > 0. (2.20) dfc U1 (ei,e2,s) ctt — G r 1 (e 1 ))) . The foe for agent 1 is U} = E(u1(-)(f31n{-G'1)) = 0. (2.21) Chapter 2. Bonus/Penalty Contracts As Strategic and Incentive Devices 41 Thus ule = £K(-)nj) + Eiu'li^ifaUl - G\)). (2.22) The first term on the right-hand side of (2.22) is positive. The second term is zero if the agent is risk-neutral. Alternatively, if f l 1 is separable, EU\ — ^7 r J ( e 1 , e 2 ) = ^l(ei,e2) = f l ] . Thus the second term on the right-hand side of (2.22) is E^I!1^!!] - G[)) = ( E « n 1 ) ) ( / 3 1 n 1 - G[) = 0 (where the last equality follows from the foe (2.21) and the fact that EU\ = U\. Therefore, U\p > 0. Proposition 2.4: Assume that principal 1 is risk-neutral and agent 1 is risk-averse. As-sume further that W ^ I l 1 ) is the optimal contract within the class of linear contracts sub-ject to the constraint that agents' actions are taken by principal 1 as if they were strategic independents. Then, the bonus contract B1(IuIl\]W1(U1)) dominates W M I l 1 ) if agents' actions are strategic substitutes; while the bonus/penalty contract B1(]-1,Ut1;Wi(U1)) may not dominate W ^ I T 1 ) if agents' actions are strategic complements. Proof: Let ^ ( I I 1 ) = A l l 1 - a a . We first show that 0 < & < 1. Under Wy and W2, the foe's for agents 1 and 2 are, respectively, [/^(ej, e 2;/3i) = 0 and t / 2 (e 1 , e 2 ; •) = 0. Total differentiation of the foe's w.r.t. ei, e2, and 8\ yields U^de! + U\2de2 + Ulpdfa = 0, (2.23) U^dej + U\2de2 = 0. (2.24) When agents' actions are taken as if they were strategic independents, we have U\2 = U21 = 0. Consequently, de\jdd\ = —U\^jU\x > 0 and de2jdd\ = 0 (using (2.23) and (2.24)). Pr incipal I's objective function is V1 — (1 — /3 1 )£ , ( I I 1 ) -f (assuming for simplicity that v\ = 1 and Ui(0) = 0). Hence, a a = V1 - (1 -B^EiU1). (2.25) Chapter 2. Bonus/Penalty Contracts As Strategic and Incentive Devices 42 The participation constraint for agent 1 is U1 = £ , (w 1(/9 1 i l 1 — a x — G-i)) > [/J, where UQ is the reservation uti l i ty for agent 1. We consider the situation in which principal 1 chooses Q j so that the constraint is binding, i.e., E ( u 1 ( - y 1 + ( i - /3 1 ) J Bn 1 +/3 1 n 1 - G i ) ) = U1 (2.26) where has been substituted using (2.25). B y totally differentiating (2.26) w.r.t. 8\ and then by setting dV1 /d8\ = 0, we obtain the foe for principal I's optimization problem: (Eu',(.))(EUl)(l - 8,)^- + ( E ^ - X A I I , + (1 - p\)EH\)))^-+ + Eiu'^-XU1 - EU1)) = 0. (2.27) Since de2/dli = 0 (as was indicated), equation (2.27) reduces to (£ U ; ( - ) ) (£?n i ) ( i - A ) ^ 1 + tfKC-xn1 - EU1)) = o. (2.2s) dpi Suppose that 81 < 0. Then the first term on the left-hand side of equation (2.28) is positive. The second term is the covariance between u'J-) and (fl 1 - EU1). Note that u;(.) = u'^-V1 + Ell1 - G a + ^ {U1 - E n 1 ) ) and <(•) < 0 (risk-averse). Hence, (n 1 -EU1) t=4- S^U1 - Ell1) \ (non-increasing) for ^  < 0, u[(-) / (non-decreasing); similarly, ( n 1 - Ell1) j = ^ - ^ {U1 - EU1) u[(-) \ . Thus, the covariance is non-negative, and equation (2.28) cannot hold. The reasoning for 8\ > 1 is similar, but here the first term is non-positive and the second term is negative. W i t h 0 < ^ ' ( n 1 ) < 1, we have U1 < 0 from (2.13), V1 > 0 from (2.16), and V2X < 0 from (2.17). Thus, for sufficiently high target profit, ( U 1 + V1)^*/dh) > 0 and V\{de\l dli) > 0 if agents' actions are strategic, substitutes; while ( U 2 + V21)(de*,/dli) < 0 and V1(de\/dlx) > 0 if agents' actions are strategic complements. The results then follow from using (2.19). Q.E.D. Chapter 2. Bonus/Penalty Contracts As Strategic and Incentive Devices 43 Proposition 2.4 shows that both incentive and strategic effects of bonus/penalty con-tracts can arise at the same time. When agents' actions are strategic substitutes, the two effects would work in the same direction: both would enhance the payoff for the principal-agent under consideration. In this case, a bonus would be desirable at the margin. When agents' actions are strategic complements, however, the two effects would work in opposite directions: the incentive effect of bonus contracting would alleviate the agent's incentive problem but its strategic effect would be against the interest of the principal-agent pair. Here, a penalty, which is desirable in inducing a favourable action from the rival agent, would worsen the incentive problem of the own agent and, hence, might not be desirable at the margin. 2.5 Concluding Remarks In a general framework of rivalrous agency with risk-averse agents, we have identified two distinctive effects of bonus/penalty contracting, namely, the strategic, effect and the incentive effect. The two effects can arise simultaneously in a bonus (or penalty) contract. They may or may not work in the same direction for a principal-agent pair, depending on the nature of strategic, relationships between agents' actions. If agents' actions are strategic substitutes, both effects would make bonus contracting more desirable. In effect, a bonus policy aimed at alleviating the incentive problem of own agent can have a secondary, and desirable, strategic effect, and vice versa. In such a situation, the basic results of Chapter 1 would remain in the general case of risk-averse principals and agents. But if agents' actions are strategic complements, a principal would have to trade-off between the incentive effect of bonus contracting and the strategic effect of penalty contracting, both of which are desirable at the margin. Thus, if the incentive problem at hand is more important than the strategic problem, then bonus contracting would be Chapter 2. Bonus/Penalty Contracts As Strategic and Incentive Devices 44 pursued; otherwise, penalty contracting would be used. As has been seen in Chapter 1, strategic rivalry among oligopolists in the case of strategic complements would generate a more collusive outcome than if players behaved non-strategically. Our analysis here shows that the significance of that result is reduced once we moves from risk-neutral to risk-averse agents. Essentially, policies that discourage effort supply may worsen the incentive problem of own agents under the condition of moral hazard. The strategic, effects of bonus/penalty contracts have not been formally discussed in the literature. One contribution of this chapter is to show that in a rivalrous agency with risk-averse agents, principals may use bonus/penalty contracts for strategic purposes. Our analysis shows that strategic interaction between different principal-agent pairs, principals at one level and agents at another, can have non-trivial impacts on the design of optimal compensation contracts. In other words, the optimal contracts can be different in a systematic way from those obtained in the standard principal-agent framework where such a strategic interaction is essentially assumed away. Chapter 3 Strategic Contracting: Bonus/Penalty and Linear Contracts 3.1 Introduction In Chapter 1, we have shown that in rivalrous agency, bonus/penalty contracts may be used by principals for strategic purposes. The idea that managerial compensation can serve as a strategic device in oligopoly is described in Fershtman and Judd (1987a) and Sklivas (1987). Using linear contracts they find that competition in managerial incentives can result in non-profit maximizing behaviour even if owners maximize profits. Their work suggests that linear contracts may be used by principals for strategic purposes. In this chapter, we first show that principals may explore strategic interaction by using linear contracts. The result is obtained under standard assumptions on demand and cost, functions. (Both Fershtman and Judd (1987a) and Sklivas (1987) use linear demands, homogeneous products, and constant marginal costs.) We then examine the principals ' choice between bonus/penalty contracts and linear contracts. We find that, if only one principal is active in designing agency compensation contracts, then he would be indifferent between a bonus/penalty contract and a linear contract. If both principals are active in designing agency contracts, however, the choice between the two contract methods would in general matter to the principals. In particular, we show in an exam-ple that both principals would non-cooperatively choose a bonus contract over a linear contract. The welfare implication of principals ' choice is also discussed. Section 3.2 discusses the rivalry in linear contracts. Section 3.3 examines principals' 45 Chapter 3. Strategic Contracting: • Bonus /Penalty and Linear Contracts 46 choice between bonus/penalty contracts and linear contracts. Finally, section 3.4 provides concluding remarks. 3.2 The Strategic Effect of Linear Contracts Consider the following class of compensation contracts: . 8XIY - oti if yl < y\ B^Ii,yloi^i)={ H ~ y z (3.1) (An ' -a , ) + /, if > 2/| where 8X (> 0) is a contract parameter. Note that the class of our bonus/penalty contracts in Chapter 1 is a special case of (3.1) when 8X is set to be unity, i.e., Bl = Bp(Ii, y\, ctx, 1). Also note that by letting / , = 0 in (3.1), one obtains the following linear contract (linear in outcome), L\ax,8i) = B^,y\,ax,8x) = 8XW - ax. (3.2) We now give an examination of rivalry in linear contracts. The elements of model structure remain the same as Chapter 1 except that we are now considering the linear contact, L\ rather than the bonus/penalty contract, Bl. In the second-stage of the game, given compensation contracts i 1 ( a 1 , / 3 1 ) and L2(a2,82), the agents simultaneously choose their actions to maximize their utilities. The agents' foe's are u\ = MEni) - G ; = o (3.3) for i = 1, 2. The comparative statics of a change in 8t on equil ibrium effort levels, (e^, e 2), can be calculated as follows. (Changes in a, have no effect on equilibrium effort levels.) d^ = _uyuji d^^ui^ dBx DL ' d8i DL K ' ' In expressions (3.4), U\p = EIL\ > 0 (using (3.3)), U]x = 8fEW)x) = 8,( E( H::,:y;,yi)), U]3 = 8r(E%) - G't, and DL = ( A ^ I L ^ ) - G'{)(82(EU222) - G'2') - 8MEIi\2)(EIl2n). Chapter 3. Strategic Contracting: Bonus/Penalty and Linear Contracts 47 We assume, as we did in Chapter 1, that f/j- < 0 and DL > 0. It can be easily seen that U]3 < 0 and DL > 0 if G'j > 0, EWn < 0, and {EUl^iEUl,) - (EUl2)(EU221) > 0. The comparative statics exercise gives the following result. Proposition 3.1: A n increase in by principal i has the following effects on the equilibrium levels of agent efforts: (i) his own agent wi l l use more effort; (ii) his rival's agent wi l l use less, the same, and more effort if agents' actions or firms' products are strategic substitutes, strategic independents, and strategic complements, respectively; and (iii) the total amount of effort wi l l rise under perfect symmetry. Proposition 3.1 is similar to Proposition 1.1 of Chapter 1. The first two parts of the proposition have just been shown in the text, whereas the last part can be shown by referring to the proof of Proposit ion 1.1. Turn to the first-stage game. Anticipat ing the agents' responses, each principal chooses 3i to maximize his payoff (or the value of his firm), given the rival's decision. (Assume that each principal chooses az to make his agent just will ing to participate.) Using (3.3), the foe for the i th principal can be written as ( i - A ) ( ^ n : ) ^ | + (Jen})^| = o. (3.5) Letting i = 1,2, then equations (3.5) determine two reaction functions, one for each principal, in the 3\ — 32 space. Intersections of the two functions determine equilibria of our two-stage game in linear contract, rivalry. The equilibria wi l l be referred to as the strategic "linear" equilibria, denoted as (Bi,32). Using (3.5) and (3.4) we have Ell) W, 3, = I m K (where all the variables are evaluated at a strategic linear equilibrium) for i — 1,2. Examining the above equations leads to Proposit ion 3.2. Chapter 3. Strategic Contracting: Bonus/Penalty and Linear Contracts 48 Proposition 3.2: If a strategic linear equilibrium exists, then Bz > 1, = 1, and < 1 if agents' actions or firms' products are strategic substitutes, strategic independents, and strategic complements, respectively. Proposition 3.2 is similar to Proposition 1.2 (of Chapter 1). In the absence of strategic interaction between firms, N1 (= IT — a t ) is the first-best compensation contract. If there is strategic interaction between firms, Proposition 1.2 has indicated that a bonus or a penalty added on Nl would be desirable. In the above proposition, however, we see a different contracting deviation from i V 1 . Rather than a bonus (or penalty), the strategic contracting instrument now is parameter f3l: which measures the degree of continuous dependence of agent i's reward upon the firm's final profit. When B{ = 1, L1 reduces to Nl. In equil ibrium, however, principals would like to over-relate payment to outcome (i.e., Bi > 1) if firms' products are strategic substitutes, while they would under-relate payment to outcome (i.e., B, < 1) if firms' products are strategic complements. In the case of strategic substitutes, for instance, a principal would be better off by having payment over-relate to outcome since more effort would be induced from his own agent and less effort from his rival's agent. Consider the example given in the fourth section of Chapter 1. One can show that, for this example, fa = f32 = J3 > 1 and (3 satisfies the equation 0 — l)(26r 2 /3 + r) = 6 2 r 4 /3 2 . The equil ibrium levels of agent efforts are given by e\ — e2 = eL = (a — c)(2bT2B -f-r ) r / (56 2 r 4 / 3 + (2/3 + 3)6r 2 r + r 2 ) . 3.3 Choice between Bonus/Penalty Contracts and Linear Contracts We have seen that both a bonus/penalty contract, Bl, and a linear contract, Ll, can be used for strategic purpose in rivalrous agency. A n interesting question is which one a principal would choose between the two alternative contract methods. We examine Chapter 3. Strategic Contracting: Bonus /Penalty and Linear Contracts 49 the question in this section. As has been specified in Chapters 1 and 2, principals' decisions on agency compensations are treated as strictly prior to agents' actions and are observable to agents; and the equilibium concept is that of Nash subgame perfect equilibrium. Also assume that both principals and agents are risk-neutral so that we can focus on the strategic effect of the two contract methods. 3.3.1 Only one principal is active in contract designing Proposition 3.3: Assume that only one principal is active in pursuing optimal agency compensation. Then in equilibrium, use of a bonus/penalty contract, B\ and use of a linear contract, L1, by the principal would yield identical levels of agent efforts, of outputs, of prices, of principal payoffs, and of welfare. Proof: Consider the class of compensation contracts (3.1), which contains B1 and Lz as two special cases. For given Bp and Bp, the equilibrium levels of agent efforts are jointly determined by the agents' foe's as follows. U11(e1,e2;I1,yt1,/31) = 31(EU\) - G[ - IJ(s\)^ = 0 (3.6) ds1 Ul(eue2] I2,yl62) = / ^ M I 2 ) - G'2 - hf^-r1 = 0 (3.7) de2 where s\ is derived from equation yl(el,sti) = y\ for i = 1,2. F i x Bp and assume that only principal 1 is active in pursuing optimal agency compensation. For any given Bp, principal 1 chooses (I\,y\,3\) to maximize his payoff V" 1 , which is the same as the value of his firm, V1 = EU1 - G a - Ula (3.8) where C/J is the reservation uti l i ty of his agent. It can be shown that the three foe's (w.r.t. I\,y\,B-\) for principal 1 wi l l reduce to a single equation as EU\ - G[ - (EUl)1^ = 0. (3.9) Chapter 3. Strategic Contracting: Bonus/Penalty and Linear Contracts 50 Using (3.6) one can rewrite (3.9) as ft(JBnl) - hf(4)^ = EU\ - (EUl)^-. (3.10) In equilibrium, (3.6) becomes (using (3.10)) En\-(EIll)^-G[=0. (3.11) U22 Note that the left-hand side of equation (3.11) is a function of (e a , e2,12, y\, B2)- In equi-l ibr ium, therefore, the levels of agent efforts, which are jointly determined by equations (3.11) and (3.7), are independent of which particular contract variable(s) are used by principal 1 and in particular, are independent of whether a bonus contract, Bl, or a linear contract, L\ is used. The rest of the proposition follows immediately from the fact that the equilibrium levels of outputs, prices, principal payoffs, and welfare are all functions of the equilibrium levels of agent effort. Q .E .D. Thus, if one principal is a first mover, he would be indifferent between a bonus/penalty contract and a linear contract. Moreover, the two contract methods would result in identical welfare outcomes. 3.3.2 Both principals are active in contract designing What happens if both principals are active in seeking optimal agency compensations? We would examine a situation in which given the rival's decision, each principal first chooses which contract, method to use between B% and Wl and then, given the chosen contract method, he chooses contract parameters. The objective function of each principal is his payoff or equivalently, the value of his firm (given by (3.8)). When two principals are in a symmetric position, one can derive, similarly to (3.11), the following equation for the second principal-agent pair: £ n 2 - ( M I 2 ) ^ - G 2 = 0. (3.12) Chapter 3. Strategic Contracting: Bonus/Penalty and Linear Contracts 51 The equil ibrium levels of agent efforts, (e x, e 2 ), now are jointly determined either by (3.6) and (3.7) or by (3.11) and (3.12). Clearly, e, = e,( 7 l 5 y\, Qx; 7 2 , y 2 , / 3 2 ) . Hence, aprincipal 's payoff would in equil ibrium depend on both his own and his rival's contract methods. The choice between bonus/penalty and linear contracts would matter to principals in general. To show that bonus/penalty contracts can play a role different from linear contracts, we shall in the remaining part of this section examine the example given in Section 1.4 of Chapter 1. Recall the example as follows. The production function is yl(el,s) = r e t + s with r > 0. The random state s is uniformly distributed over [—1/2,1/2]. We have also assumed linear demand, homogeneous products, and constant marginal cost c. Using p to denote price, the demand function can be written as p = a — b(yljry2) wi th a > c. Finally, the disutili ty function Gj(e,) = re2/2 with r > 0. Note that in the above specification, the two principal-agent pairs are perfectly symmetric. If both principals use Bl, we would obtain a "unique" strategic bonus equilibrium (see Section 1.4). Further, the equilibrium levels of agent efforts are eB = e\ — eb = (a — c)(26r 2 + r)r/(5b2T4 + 5br2r -f r2). On the other hand, if both principals use L\ we would have strategic linear equilibria, and the equil ibrium levels of agent efforts are given by eL = ef = ef = (a - C)(26T2/3 + r ) r / (56 2 r 4 / 3 + (2/3 + 3)br2r + r2) where /? > 1 (see Section 3.2). A principal's payoff, Vl(ei, e 2) is given by (3.8) (the expression for principal 2 can be similarly given). Under symmetry, e-y = e 2 = e in equilibrium and the equilibrium payoff to each principal can be written as V(e) = -(2bT2 + T-)e2 + (a - c)re - g - U0, (3.13) where U0 is the agent's reservation utility. Thus, in equilibrium, a principal's payoff is quadratic in agent effort. Principals would receive the highest payoffs if they could Chapter 3. Strategic. Contracting: Bonus/Penalty and Linear Contracts 52 observe their agents' actions and they could cooperate with each other in agency com-pensation. In other words, the effort levels would be chosen to maximize V(e) . The maximum effort levels, denoted e M , are eM = = e^f = (a — C)T/(46T 2 + r). Another interesting case is where principals cannot observe agents' actions, as assumed in the standard agency problem, but principals could cooperate with each other in agency com-pensation. In this case, principals would act as if they were non-strategically-minded. In other words, they would use Nl (= IP — a t ) , the non-strategic first-best contracts. The equilibrium levels of agent efforts, denoted eN, are eN = = e2 = (a — c ) r / (36 r 2 + r . It can be easily shown that eL > eB > eN > eM (3.14) and that VL <VB <VN < VM. (3.15) (It can be shown that VL > 0 as long as (a — c) is sufficiently large.) The ordering in (3.15) suggests that: 1) cooperative principals would be better off if they could observe their agents' actions; and 2) rivalrous principals would be better off if both could agree to use bonus contracts. The welfare implicat ion of these different equilibria or solutions is now discussed. Our linear demand structure with homogeneous products can arise from util i ty U(y\y2) = a{y1 + y2)--2{y1+y2)2 + m , (3.16) where m is consumption of numeraire good. Using (3.16), the welfare function, given by (1.25) of Chapter 1, can be written as br2 r b ^ ( e 1 , e 2 ) = - ^ - ( e 1 + e 2 ) 2 - ^ ( e 2 + e 2 ) - - + m . (3.17) z 2 o Under symmetry, t t» (e 1 , e 2 ) can be written as w(e) = - ( 2 6 r 2 + r)e2 + 2(a - c)re - - + m. 6 Chapter 3. Strategic Contracting: Bonus/Penalty and Linear Contracts 53 w(e) achieves a unique maximum at e 5 = (a, — C)T/(26T 2 + r ) , which are the second-best opt imum levels of agent efforts. It can be easily shown that (ref (3.14), (3.15)) (It can be shown that wM > 0 as long as (a — c) is sufficiently large.) The welfare ordering and the principal-payoff ordering are just opposite. In particular, for cooperative principals, welfare would be higher if the agency relation exists (such that principals could not observe their agents' actions); and for rivalrous principals, welfare would be higher at the strategic linear equilibrium than at strategic bonus equilibrium. Using welfare function (3.17) and orderings (3.18) and (3.19), one is able to draw the second-best, welfare contours in the effort space. The interesting question now is which contract form independent and non-cooperative principals would use. To show that the use of a bonus contract can be a dominant strategy, we present a numerical example using a — c = 4, b = T = r = 1. It can be shown, after tedious calculations, that there is a unique equilibrium of our two-stage game for each of the strategy combinations, namely, ( L , L ) , ( B , B ) , and ( L , B ) or ( B , L ) . The (equilibrium) principal-payoff matrix is shown in Table 3.1. It can be easily seen from the table that using a bonus contract strictly dominates using a linear contract for both principals. Also note that the highest payoff to a principal in the table arises in a situation where he uses a bonus contract while the other player uses a linear contract. It may also be interesting to look at Table 3.2. Table 3.2 gives the equilibrium effort levels for the two agents under all the possible strategy combinations. If one principal uses a bonus contract given the other's strategy, then the agent of the first principal es > eL > eB > eN > e M and that ws > wL > wB > wN > w M (3.19) Chapter 3. Strategic Contracting: Bonus/Penalty and Linear Contracts 54 Table 3.1: Principals ' Payoff Mat r ix of the Example ( VL = 1.1999, VB = 1.2218, VBL = 1.2222, VLB = 1.2026 ) principal 2 principal 1 L B L VL, VL VLB, VBL B vBL, VLB VB, VB Table 3.2: Agents' Effort Ma t r ix of the Example ( e L =.1.1056, eB = 1.0909, e B L = 1.1111, eLB = 1.0833 ) ^ ^ ^ ^ p n n c i p a l 2 principal 1 \ ^ L B L e\eL eLB, iBL B eBL, eLB eB,eB Chapter 3. Strategic Contracting: Bonus/Penalty and Linear Contracts 55 would supply more effort while the rival's agent would supply less effort than if the first principal uses a linear contract. But if both principals use a bonus contract, both agents would supply less effort than if both use a linear contract. The lower effort levels from both agents would nevertheless benefit, both principals due to a more collusive output outcome. 3.4 Concluding Remarks The main purpose of this chapter is to compare the strategic effect of bonus contracts with that of linear contracts. It is found that if only one principal is active in designing compensation contracts, he/she would be indifferent between the two contract methods. In this case, the two methods would in equilibrium yield identical welfare levels. If both principals are active in designing agency contracts, then which contract method to be used would matter in general. We present an example in which use of bonus contracts is a dominant strategy for both principals. Nevertheless, welfare is lower in this example at the strategic bonus equilibrium than at the strategic linear equilibrium. The chapter also shows that linear contracts can be used for strategic purposes under standard assumptions on demand and cost functions. It is noted that the comparisons of effort, price, output, payoff, and welfare between strategic and non-strategic equilibria, as done for bonus/penalty contracts in Chapter 1, can also be carried out for linear contracts and similar results can be derived. In Chapters 1, 2, and 3, we have assumed that the compensation contracts of a principal-agent pair are observable to the agents of the other pairs. Suppose that agency contracts are unobservable to the competitors. In such a situation, if agents are risk-neutral, the optimal agency compensation would be the non-strategic first-best contract. 1  1 see K a t z (1988) for a discussion of unobservable contracts as commitment strategies. Chapter 3. Strategic Contracting: Bonus/Penalty and Linear Contracts 56 When agents are risk-averse, bonus contracts, rather than linear contracts, might arise at equilibrium. Here, the incentive effect of bonuses, which serves as the primary motivation for using a bonus contract, might generate a secondary effect, namely, the strategic effect of bonus contracts. In other words, a principal might sti l l benefit from the strategic effect of bonus contracts even in the case of unobservable agency compensation. The rigorous analysis of this problem would be a natural subject for further study. Chapter 4 An Analysis of Common Sales Agents 4.1 Introduction It has been observed that many sales transactions occur through the use of a common agent who conducts sales on behalf of several independent clients. In fact, common sales agents have been an important sales channel in a number of markets. Examples include travel agents in the air travel market and real estate agents in the residential housing market. In these markets, firms (or individual sellers) are usually free to choose the exclusive sales channel structure in which each firm conducts the sales itself or uses an exclusive retailer. On the other hand, potential consumers could also buy the products directly through firms or sellers. Then, why would common sales agents arise in these markets? In the airline market, for example, it was expected that the role of travel agents in the distribution system would decline after deregulation. It appears that the opposite has occured (Levine (1987)). Deregulation has enhanced the role of travel agents. About 80% of all airline tickets are now issued by travel agents. It has been widely recognized that imperfect information is prominent in the deregulated air travel market. But the explanation of how imperfect information may affect marketing channels has not been formally presented. In this chapter, we present an analysis of common sales agents based on their precom-mitment role under imperfect consumer information. Specifically, we consider a world in which potential consumers lack full information about products on the market. As 57 Chapter 4. An Analysis of Common Sales Agents 58 a result, a consumer may choose a product which is not his (or her) best-available buy. We show that the exclusive channel structure can create a cost due to lack of coordi-nation between exclusive channels. The idea is this. A n exclusive sales agent would not normally have an incentive to turn over a mismatching customer to the other sell-ers. The self-interested sales action of exclusive sellers would reduce the average product quality as perceived by consumers, and would be rationally taken into account by poten-tial consumers in their consumption decisions. The marginal consumers of the industry would consequently decide to stop consuming. The market under the exclusive channel structure therefore functions poorly. Moreover, the reduction in the overall market size can hurt all the sellers if consumer mismatching is reasonably symmetric with respect to firms' products. Thus, the cost is borne by firms as well as consumers. We then introduce a common sales agent, and illustrate how a common agent can coordinate sales for the client firms. Essentially, firms' competition in sales is inter-nalized within a common agent, enabling the agent to act in an unbiased way and, in effect, transmit information to consumers. We show that independent non-cooperative firms may use common agents as a precommitment device to convince potential buyers that their risk of being misrepresented has been eliminated. The sales impartiali ty of a common agent is credible to consumers because it is consistent with a common agent's incentives. Valuing information and impartiality, the marginal consumers would buy the products through common agents rather than through firms or sellers. B y having sales actions more in line with consumers' best interest through common agents, firms have effectively alleviated the conflict of interest between consumers and individual sellers. It is noted that under perfect information, consumers are able to choose their best-available products, i.e., no mismatching would occur. Our reasoning for the use of com-mon agents does not apply here and the existence of exclusive channels could be implied by efficiency. Our results thus show the importance of imperfect consumer information in Chapter 4. An Analysis of Common Sales Agents 59 designing marketing channels, which has received little analytical attention, and suggest possible ways to handle problems of information. This work is related to Bernheim and Whinston (1985). In a model different from our's, they show that a common marketing agency can serve as a device for firms to achieve collusion. The essence of their analysis is that common agency provides an indirect, mechanism through which competing firms may "sell out" to a single party, thereby creating incentives which generate a collusive outcome. Their representation of common agency suggests a negative effect of common agency on the competitiveness of the client industry. In contrast, our analysis shows that common sales agents can be welfare improving for both firms and consumers. Essentially, the use of common agents reduces the cost, which is borne by both firms and consumers, due to a failure in coordination between exclusive channels when consumers are imperfectly informed. The idea that third party contracting may be used as a precommitment device to alleviate interest conflicts between the original parties has been recognized years ago (see Schelling (1956)). In our example, conflicts of interest occur between individual firms and potential consumers under vertical integration. A common agent functions as a "neutral" third party who has no bias towards any particular firm. The sales impartiali ty of common agents would be rationally perceived by potential consumers if the latter could observe the incentive schemes of common agents. References on strategic agency delegation may also be useful. In the marketing liter-ature, two early studies here are McGui re and Staelin (1983) and Jeuland and Shugan (1983). The former shows that in oligopoly a manufacturer may decentralize via an ex-clusive independent retailer for strategic purposes, whereas the latter argues that proper channel contracting may help a single manufacturer achieve a desirable coordination within his channel members. It is noted that these studies focus on the question of vertical integration, or the exclusive channel structure. The general studies on the role Chapter 4. An Analysis of Common Sales Agents 60 of agency contracting as precommitment devices include Fershtman and Judd (1987), Sklivas (1987), and Ka tz (1988). The chapter is organized as follows. Section 4.2 presents the basic, model. Section 4.3 examines firm sales behavior and consumer buying behavior when each firm conducts its own sales activity. The market performance under the exclusive channel is evaluated. We then introduce common sales agents in Section 4.4 and illustrate how a common agent can coordinate sales for the client, firms. B y developing a channel game in Section 4.5, we demonstrate in Sections 4.5 and 4.6 that a market involving common sales agents can arise as an equilibrium outcome. We further examine the results in Section 4.7 when product prices may vary with channel structures. Section 4.8 provides concluding remarks. 4.2 The Basic Model Consider a market consisting of two variants (products) of a differentiated product class and a population of potential consumers. The two variants are produced by two inde-pendent firms and are sold through the firms and, perhaps, through other marketing channels. In other words, we assume that consumers can always buy products directly through the firms under the channel structures to be considered. Each buyer consumes at most one product or none. The consumption decision of a potential buyer might be viewed as the result of a choice hierarchy. In the first step of the hierarchy, the buyer de-cides whether to consume the product class. In the second step, if he decides to consume, he must decide which particular product to buy. Given the product attributes (prices, quality aspects, etc.), one product suits h im better, i.e., a good match; otherwise, a bad match. The potential buyer is assumed rational in the sense that when making the deci-sion of "buy" or "do not buy", he would compare the util i ty from his most desired good Chapter 4. An Analysis of Common Sales Agents 61 with that from "do not buy" and then choose the alternative yielding a higher utility. A n important element yet to be addressed is the buyers' ability to make a good match. A buyer is said to be perfectly informed if he is able to choose with certainty a product which gives h im a higher uti l i ty than the other product. Under perfect information, therefore, no mismatching occurs. In this paper we consider a world where consumers are imperfectly informed in the sense that they may suffer mismatches. Let q be the probability of a good match perceived by consumers. This probability may be affected by the consumer's effort to search for information, denoted el for the i th consumer, and by certain marketing actions of the sales channels. Suppose that channels have identical levels of marketing actions towards the two products, denoted a, at equilibrium, a may be a vector of several action components, a = (a 1 , . . . , a J , . . . , a n ) . Let the utilities to consumer i of a good match be denoted a{i) and a bad match B{i). These utilities wi l l depend on product attributes, namely, prices (as a cost to a consumer) and non-price attributes. We have 5(i) = a(i) — B(i) > 0. Using / to denote the consumer income, the expected utilities the i th consumer derives from the alternatives can be written as follows: f q(e\a)a(i) + (l-q(e\a))B(i) + I-g(ez) if buy U(t,e\a)=i _ (4.1) [ I if do not buy where g(el) is the search cost or the disutility of effort. Consumer i's expected net benefit function from buying is S(i, e\ a) = q(e\ a)a(i) + ( l - q(e\ a))8{i) - g(el). (4.2) We assume that potential buyers are risk-neutral and wi l l choose their effort levels opti-mally. This implies that the optimal level of effort taken by consumer i , denoted e7, is (implicit ly) determined by the following first-order condition. (Subscripts wi th a letter Chapter 4. An Analysis of Common Sales Agents 62 denote partial derivatives). Se = qe8(i) — g = 0 (assuming that the second-order condition is satisfied). As can be seen from the above equation, a necessary condition for the existence of optimal efforts is that qe and g' have the same sign. One such condition is that qe > 0 and g' > 0, i.e., a consumer is less likely to make an error as his search effort increases, but search is costly. Suppose that the optimal effort exists, e1 — e(i,a) with 0 < e1 < e for the domain of (i,a), where e is some positive number. Taking into account the optimal choice of effort by consumer i , i's expected net benefit function (4.2) becomes s(i,a) = MaxeS(i,ez,a) = 5(z, e(i, a), a), (4-3) and i's expected uti l i ty function (4.1) becomes { s(i, a) + I if buy (4.4) I if do not buy The potential consumers are heterogeneous in tastes: the utilities from buying varies from person to person. Let the consumers be ordered in such a way that those who derive higher expected net benefits from buying are indexed with higher numbers, that is, dS/di = qa'(i) + (l — q)B'(t) > 0. It follows immediately that Si = ds/di = dS/di > 0 (where the use of the envelope theorem gives the second equality). In other words, the expected net benefit from buying increases as i increases. We can now define the marginal consumer of the sector or the industry. Let i* be such that s(i*,a) = 0. Then those indexed by i > t* w i l l buy some product of the industry because of positive expected net benefit from buying, while those with i < i* wi l l not. Consumer i* is just indifferent between "buy" and "do not buy" and is thus referred to as the marginal consumer, i* = i*(a). Chapter 4. An Analysis of Common Sales Agents 63 Suppose that the ordered consumers are distributed according to density function f(t) over interval [0,1], with cumulative distribution function F(i). The industry demand, xd can be written as xd(a) = 1 - F(i*(a)). (4.5) Aggregate consumer surplus , us can be written as us(a) = / u(i,a)f(i)di = [ s(i, a)f(i)di + I. (4.6) Proposition 4.1: Suppose that a3 is an action component and that qaJ > 0 for 0 < e < e, i.e., the higher the level of marketing action a3, other things equal, the higher the probability of a good match perceived by consumers. Then, given product attributes, higher a3 increases i) the expected net benefit from buying, ii) the industry demand or the market size, and i i i) aggregate consumer surplus. Proof: i) Using the envelope theorem, sa, = Sa, = qa,8(i) > 0. ii) xd} = — f(i*)i*i — f(i*)Sa,/Si > 0. i i i) ul = £ Sa,f(i)di + s(i\a)xdaJ > 0. Q.E.D. Proposition 4.1 can be seen from Figure 4.1. The welfare gain has two components: an increase in the benefit from buying (the vertically shaded area) and an increase in the market size (the horizontally shaded area). (Further note that, in Figure 4.1, if i*(0) < 1, then consumers within the interval [i*(0), 1] would consume some product of the industry regardless of the level of marketing actions. To focus on the problem of our interest, we may assume that 0 < i* ( l ) < i*(0) < 1, i.e., there are consumers in the population whose buying decision is affected by the marketing actions.) We have not yet specified channels' marketing actions, a. The specification may depend on the circumstances under investigation. The circumstance considered in this Chapter 4. An Analysis of Common Sales Agents 64 Figure 4.1: Gains from an Increase in a° (1 > c^1 > a,i<* > 0) s(i,a) Chapter 4. An Analysis of Common Sales Agents 65 paper is now described. Let q0 denote the probability that a consumer wi l l make the cor-rect matching decision. We impose the following condition on consumer choice behavior: 0 < g 0 < 1 (4.7) (for simplicity we assume that consumers' search effort has no effect on q0). Condition (4.7) says that consumers of our population may suffer mismatches. This would be the case if consumers lack information about the products on the market and there are limits on time and abili ty to obtain and process information. Condit ion (4.7) is an expression for what we mean by imperfect consumer information. Given a channel structure, a potential consumer decides whether or not to consume the product class and, if deciding to consume, he must choose a particular product and approach an agent who sells the product. The agent may conduct sales for the other product as well. The agent wi l l either sell the requested product to the customer or not sell the requested product. In the latter case, the agent wi l l either sell the other product to the customer if he carries out sales for both products, or turn over the customer to an agent selling the other product if he only carries the product the consumer has requested. Suppose that each sales agent is perfectly informed about the products on the market such that he can identify a bad (or good) match made by his customer. We shall consider the following class of agents' sales strategies: an agent wi l l sell the requested product to a customer with probability o} if the customer chooses correctly; and with probability 1 — a? if the customer makes a choice error (0 < aJ < 1, j = 1,2). Lett ing a = (a 1 , a 2 ) , then by choosing a value of a, each agent sets up his sales strategy or action. Note that in above specification of sales strategies, we have implici t ly assumed that a sales agent woidd treat all his customers in the same manner. In particular, if there is a positive probability that some consumers have their transactions done before others, then the specification has ruled out the possibility that an agent might use a sales strategy which Chapter 4. An Analysis of Common Sales Agents 66 is contingent on the transaction order. Also note that we have implici t ly assumed that the marginal profit from sales is positive to agents. If a sale yields a negative profit, an agent wi l l not complete the sale regardless the customer's choice outcome. We now provide an interpretation for a 2 . 1 A sales agent may or may not correct a customer's choice error. If the agent corrects the error, it is said that the agent adopts an impartial sales attitude (denoted / ) towards the customer; otherwise, the agent adopts an (anti-consumer) biased sales attitude (denoted B). W i t h a 2 , the agent plays I with probability a? and plays B with probability 1 — a 2 , a 2 = 0 and a? — 1 are the two pure strategies corresponding, respectively, to playing B and playing I towards any customer. As a? increases, agents' sales action becomes more impartial or less biased. Given a channel structure, the probabilities that a consumer wi l l actually have a good match and a bad match are, respectively, given by (4.8) and (4.9). q(a) = q0al + (1 - q0)a2, (4.8) 1 - q(a) = q0(l - a1) + (1 - q0)(l - a2). (4.9) Using condition (4.7) we obtain that q' = (qai,qa2) = (g 0,1 — q0) > 0 and (1 — q)' < 0, i.e., as a sales agent's action becomes more impartial , the probability that a consumer wil l be represented (misrepresented) by the agent rises (falls). Given imperfect consumer information, it might be reasonable to assume that before a sale is completed, a buyer does not know which sales action, a is taken by a sales agent. A post-purchase buyer may discover whether or not he has been misrepresented by his sales agent, but we assume that there is no post-purchase contact between a buyer and an agent. Each potential consumer builds up his expectation about agents' sales action based on what he has observed. Suppose a positive probability that there are consumers 1 A l t h o u g h interpretation for a 1 may be similarly given, it is less interesting in this paper than that for a2. A s we shall see, a 1 remains the same ( = 1) at equi l ibr ium under the channel structures considered in the paper. Chapter 4. An Analysis of Common Sales Agents 67 who make consumption decision after some other consumers have completed their trades, and a positive probability that an earlier consumer's information about his representing outcome wil l be received by a later consumer. Then a buyer may be able to update, using Bayes rule, his expectation based on information from earlier trades. This means that, since a consumer's representing outcome is only affected by an agent's (actual) sales action o (ref equations (4.8) and (4.9)), consumers' expectation on a, denoted as a, may be affected by d. We shall consider situations in which the effect of a on a, referred to as the "image" effect of a, exists but is arbitrarily small. W i t h an expectation level of a = (a1, a2) about a = (a 1 , a 2 ) , a potential consumer assigns the following probability to a good match: q(a) = q0al + (1 - q0)a2. (4.10) Note that condition (4.7) has implied qa = (g a i , g 0 2) — (q0, 1 — go) > 0, suggesting that higher expectation about the sales impartiali ty increases the probability of a good match perceived by a consumer. Also note that, given positive marginal profit of sales, imperfect consumer information has created a role for firms and other marketing channels to play in the consumer-product matching. It is clear that in the l imit ing case of perfect consumer information, go = 1 and consumers can get their desired products directly through the firms. We shall consider the "rational" expectation by consumers in the sense that the expec-tation has the property that it wi l l be confirmed at equilibrium. W i t h our specification of channels' sales strategies, this implies that consumers have identical levels of expectation about channels' sales action at equilibrium. Corollary 4.1 then follows Proposition 4.1 immediately. Corollary 4.1: Given product attributes, higher expectation about the sales im-partiality of channels increases i) the expected net benefit from buying, i i) the industry Chapter 4. An Analysis of Common Sales Agents 68 demand or the market size, and iii) aggregate consumer surplus. 4.3 Firm Sales Behavior and Consumer Buying Behavior In this section we consider a channel structure in which each firm conducts its own sales. The channel structure wi l l be referred to as the exclusive channel. Under the exclusive channel, firms themselves are the only sales outlets. Suppose that the marginal profit from a sale is positive for the firms. The positive marginal profit is known to potential consumers even though they may not have full information about prices and/or costs. Consumers' decision of whether or not to buy depends (partly) on their expectations of firms' sales actions. Suppose for the moment that consumers' expectation takes value a, 0 < a < 1. Such an expectation wil l yield industry demand xd(a) (given by equation (4.5)). Given xd, consumers' product choice behavior may be analyzed as follows. First introduce the matching demand xk to be the set of consumers who should match with product k, k = 1,2. Those in xk w i l l be referred to as product k's consumers. Since X\ + x2 = xd, Xk may be alternatively expressed as xk = 0kxd, where 0k (0 < 0k < 1) is the proportion of xk in xd, which is assumed to be independent of consumers' expectation value a. The consumer preliminary demand for product k, denoted xpk, contains consumers who choose product k. Using x\ and xk, the probability, g0,fc, that a consumer makes the correct choice in choosing product k, may be further specified as: (ref condition (4.7)) 0 < q0ik = P(i <E xpk | z <E xk) < 1 (4.11) k = 1,2. x\' depends on consumers' product choice behavior which has been assumed to be random. It is useful to decompose xpk into two components, (xp fl xk) (= dk) and (xpkC\Xj) (= dk), j ^ k. The former are consumers who have chosen correctly, whereas the latter have by error chosen product k when it is not their best buy. The expected numbers Chapter 4. An Analysis of Common Sales Agents 69 of them are, respectively, E(dk) = fj, P(t G dk)f{i)di = // , Okq0:kf(i)di = 9kq0<kxd > 0 and E{dk) = 8j(l — qo,j)xd > 0. E(dk) > 0 means that some mismatching is expected for consumers of each product under imperfect information. In effect, xvk = xk (i.e., the two sets contain the same consumers) with probability 1 if and only if information is perfect. The final demand for each product (or each firm), denoted xf., depends on both the consumer preliminary demand and the sales actions taken by the firms, xf, = a\dk -f (1 — Q-Vldk + (1 — 5.})^' + a]dj, where j ^ k and ak = (a^a 2 . ) is firm k's sales action. Taking the expectation of xf, w.r.t. consumers' product choice behavior yields E{x{) = ak{~a)xd{a) (4.12) where ak(a) = a\0kqQtk + (1 - a\)9j(l - q0<j) + (1 - a))03qOj + ~a26k{l - q0>k), and k = 1,2. Assume that firms are risk-neutral. Then the equilibria of sales actions are reached when each firm chooses its sales action, ak to maximize its expected demand, E(xf), given the other firm's action. It is useful to examine the effects of ak on E(xf). From (4.12), dE(xf) •, , dxd{a) dE(xd) - , dxd(a) The first terms on the right hand side of equations (4.13) and (4.14) are positive and negative respectively, capturing the direct effect of sales action on keeping or losing a sale. The second terms capture the (indirect) effect of sales action on total industry demand, via the effect of sales action on consumers' expectation. As indicated earlier, we consider situations where the effect of o on a is arbitrarily small. Consequently, the second-term effect is negligible. This implies a dominant strategy for each firm, namely, a\ = 1 and a?k = 0 for any expectation level consumers entertain. In other words, each firm wil l sell its product to al l its customers no matter whether the product is a customer's best-available buy. Chapter 4. An Analysis of Common Sales Agents 70 W i t h a = (1, 0), the final demand is the same as the preliminary demand, i.e., xf = xTk for k = 1,2. The expected demand for each product is E(xk) = E(xpk) = pkxd where pk denotes the proportion of E(xpk) in xd. It can be easily shown that pk > , = , < 6k if E(dk) >,=,< E(dj), respectively. Thus, given the total demand, a firm wil l expect a higher demand than its matching demand if (and only if) the expected number of mismatching consumers of the rival product is greater than that of its own product. Now consider the formation of consumers' expectations about firms' sales actions. The actual sales actions taken by firms are determined by channels' incentives which are known to consumers. Given that each firm conducts the sales itself and that the image effect of its sales action is arbitrarily small, consumers wil l rationally expect a biased action from both firms, i.e., a = (1,0). The expectation has the property that it wi l l be confirmed at equilibrium, i.e., a = a. = (1,0). If instead the expectation were a2 > 0 (with a1 = 1), then consumers in interval [ i* ( l , a 2 ) , i * ( l , 0)] who decided to buy would obtain a negative net benefit from buying. Our analysis of firm sales behavior and consumer buying behavior shows that when each firm conducts its own sales, in equilibrium, consumers' expected net benefit from buying wil l be s(i, 1,0), the industry demand wil l be xd(l,0), and the expected demand of each firm wi l l be pkxd(l, 0). It is useful to compare this with the case of perfect consumer information. Under perfect information, consumers' net benefit from buying wil l be s(i, 1,1), the industry demand wil l be xd(l, 1), and the demand of each firm wil l be 6kxd(l, 1). According to Corollary 4.1, both the expected net benefit from buying and the industry demand are lower under imperfect consumer information than under perfect information. So aggregate consumer surplus falls as information becomes imperfect. The result is summerized as follows. Proposition 4.2: Assume the exclusive channel. Then under imperfect consumer information, i) a consumer expects a lower ut i l i ty from buying, ii) the market has a smaller Chapter 4. An Analysis of Common Sales Agents 71 size, and iii) aggregate consumer surplus is smaller than under perfect informtion. Although imperfect information would hurt consumers under the exclusive channel, it might not be the case for every firm. In effect, one firm (say, firm k) would expect a higher demand as an exclusive seller under imperfect consumer information than under perfect informtion if (and only if) pk > (xd(l, l)/xd(l,0))9k. Consequently, firm k would expect a profit gain at the expense of the other firm (recalling positive marginal profit). In such a situation, firm k has no incentive to participate in common agency and the common-agent channel might not be viable. When would this happen? A necessary condition for pk > (xd(l, l)/xd(l, 0))9k is, since 3^(1,1) > xd(l,0), that pk > 6k or equivalently, E(dk) > E(dj). That E(dk) > E(dj) is a condition for asymmetric mismatching: the expected number of mismatching consumers of product j is greater than that of product k. We shall consider symmetric mismatching of the following kind, E(dx) = E(d2). (4.15) Condit ion (4.15), referred to as symmetric consumer mismatching, says that the number of product. 2's consumers making a choice error is on average equal to the number of product I's consumers making an error. The condition holds if and only if 92(1 — 90,2) = 0 i ( l - ? o , i ) -Lemma 4.1: Under symmetric consumer mismatching, pk — 9k for k = l , 2 , i.e., the expected proportion of the preliminary demand for each firm is the same as the proportion of the matching demand for each firm. Lemma 4.1 can be easily shown. A n immediate implication of the lemma is that each firm, as an exclusive seller, expects a lower demand under imperfect consumer information than under perfect information. Chapter 4. An Analysis of Common Sales Agents 72 We assume that firms earn strictly positive profits under perfect information. That is, the market exists under perfect information (by existence of the market we mean both firms are producing). The following result is a corollary of Proposition 4.2. Corollary 4.2: Assume the exclusive channel and symmetric consumer mismatching. Then under imperfect consumer information, the market may not exist; if the market exists, it is smaller, consumers expect lower utilities from buying, and each firm expects a lower profit than under perfect information. Proof: Under imperfect consumer information, the (expected) profit of a firm is lik(0kxd(l, 0)) (using Lemma 4.1) while under perfect information, Iik('dicxd(l,l)). Tlk(0kxd(l,O)) < Uk(0kxd(l,l)) since xd(l,0) < xd(l,l) and U'k > 0 (positive marginal profit). If Uk(9kxd(l,0)) < 0 (< Uk(0kxd(l,l))), then the market does not exist under imperfect information. If llk(0kxd(l, 0)) > 0 (k = 1,2), then the market exists, and the rest of the results follows Proposition 4.2 immediately. Q .E .D. If the sellers are not better informed than the buyers, they may not be able to identify buyers' mismatching. Each firm would naturally complete a sale for all its customers. The loss identified in Proposition 4.2 and Corollary 4.2 may be regarded as the value of information. 2 The interesting point emphasized in Proposition 4.2 and Corollary 4.2, however, is that even if the sellers are perfectly informed, they would not utilize their superior information at all . The waste of information is attributed to the inability for exclusive sellers to coordinate under imperfect consumer information. The lack of coordination may be seen by noting that if one seller turns over a mismatching buyer to the other seller, the second seller wi l l not do the same thing in return. The first seller wil l 2 E v e n condit ional on imperfect information on both consumers and sellers, the exclusive sales channel, if privately opt imal , might not be socially opt imal . Consider the situation in which, given product attr ibutes, llk(pkxd(l,0)) < 0, but Ylk ^k(PkXd{l,0))+ f.1,^  0 ) s(i,l,0)f(i)di+I > I. In such a situation, t rading is socially opt imal , but t rading would not happen because the sellers would not part icipate. Chapter 4. An Analysis of Common Sales Agents 73 thus lose a sale while the second gain one. In effect, an exclusive seller can credibly do nothing other than take a biased action. The natural bias of vertical integration would become highly visible. As a result, a feasible market may simply not exist. If the market exists, many feasible trades (between sellers and buyers in interval [ i*( l , 1) ,z*(l , 0)]) are not made and consumers engaging in trade get lower utilities. These results suggest that the market under consideration functions very poorly. Recapturing the loss identified here may be regarded as the.potential value of coordination. Moreover, Corollary 4.2 shows that, when consumer mismatching is reasonably sym-metric with respect to firms' products, the information waste by firms would hurt both of them. Firms would therefore have an incentive to find ways to explore their information. This is equivalent to seeking institutions which not only possess superior information but can be used to convince the potential buyers that they would not be misrepresented. Recall that i) the sales action of a firm is (almost) unknown to the other parties, and ii) there is no post-purchase contact between a seller and a buyer. In such a circumstance, a seller's claims to be truly representing the customers are unverifiable and consequently meaningless. Both the other seller and the potential buyers wi l l rationally ignore them. We have also ruled out possible use of warranties or penalties for misrepresentation by the firms. Final ly, there is virtually no role for reputation effects. 4.4 Information, Impartiality, and Common Sales Agents We define a common sales agent to be an independent agent who i) conducts sales on behalf of both firms and by doing so, is able to obtain information about the products on the market, and ii) receives from both firms identical net compensations which are proportional to total industry sales. B y "independent", we mean that a common agent can decide whether or not to participate in delegation and, if deciding to participate, Chapter 4. An Analysis of Common Sales Agents 74 he can choose his sales action which is unobservable to the client firms. We also note that our common agent is paid the same level of "net" compensations by different clients. Thus, firms with a higher selling cost, other things equal, wi l l offer a higher compensation than those with .a lower selling cost. Consider now that a common sales agent is available in the market. Since consumers can always buy products directly through firms themselves, the possible sales outlets include both firms and the common agent. This channel structure is referred to as the common-agent channel. As discussed, each firm, as a seller, wi l l take a biased action. Now examine the sales action of the common agent. A common agent is perfectly informed about the products of both firms and, hence, is able to identify any choice error made by his customers. Further, since our common agent has been paid identical compensations by the clients, he has no reason to work, or not to work, for any particular client. The only incentive a common agent has is to increase total industry demand, xd(a), which depends on consumers' expectation of channels' sales action. The higher the expectation level of a, the higher the industry demand (Corollary 4.1). The highest expectation value, a = (1,1) is realized when all the consumers expect that the common agent wil l (with probability 1) use a strictly impartial sales action, denoted a 0 = (1,1). The reason is this. If consumers expect an impartial action from a common agent while a biased action from firms' own selling offices, consumers wil l then get products only through the common agent because this way they wil l obtain a higher expected net benefit from buying (Corollary 4.1). As a result, consumers' expectation of the sales action under the common-agent channel is the same as their expectation of the common agent, namely, a = a0 = (1,1). Thus, the common agent wi l l have his profit maximized when a0 = (1,1). In our situation, a necessary condition for ao = (1,1) is that do = (1,1). Suppose that do = (dj, dp) ^ (1,1), i-e., at least one of a0 and d2, is less than unity. Using equation (4.9), Chapter 4. An Analysis of Common Sales Agents 75 the probability that a consumer is misrepresented by the common agent is positive. B y applying the image effect indicated in section 4.2, one wi l l obtain a positive probability that there is at least one consumer with an expectation level ao 7^  (1)1); which contradicts the definition of a 0 = (1,1). The above discussions show that strict impartiality is a dominant strategy for a com-mon agent. Consumers wil l rationally expect this, indicating that a = d = (1,1) at equilibrium. Consequently, consumers' expected net benefit from buying wi l l be s(i, 1,1) and the market size wil l be xd(l, 1). Proposition 4.3: Assume the common-agent channel. Then, given product at-tributes, the common agent, wi l l , in equilibrium, emerge as the only actual sales outlet. Moreover, under imperfect consumer information, i) a consumer expects the same utility from buying, ii) the market, has the same size, and iii) aggregate consumer surplus is the same as under perfect information. The first part of Proposition 4.3 is an interesting result because consumers could buy products direcly through firms if they wished to. Our analysis shows that the emergence of an outside sales agent, such as a common agent of our model, can be attributed to factors other than geographical or locational advantages of an outside agent perceived by consumers. In our model, the advantage of a common agent lies in his sales impartiality perceived by consumers, and in his superior information (or his willingness to search for information for consumers). Under imperfect information, consumers value both information and impartial i ty of sales channels. Thus, if pleasure travellers could deal with travel agents located on or near airports, perhaps they would stop dealing directly with airlines (since directly dealing with airlines risks bias). 3 3 I n this result, product attributes such as prices and services are assumed to be the same for a common agent and firms' own selling offices. In the airline market, for instance, it doesn't cost a consumer anything directly addit ional to buy a ticket f rom a travel agent. T h e agent's commission is Chapter 4. An Analysis of Common Sales Agents 76 A n immediate corollary of Propositions 4.3 and 4.2 is as follows. Corollary 4.3: W i t h the common-agent channel, i) a consumer expects a higher utility from buying, ii) the market has a larger size, and iii) aggregate consumer surplus is greater than with the exclusive channel. The essence of our analysis of common agents lies in the ability for common agents to coordinate sales for independent sellers. B y obtaining product information from all sellers, a common agent is informed at least as well as each seller. More importantly, the incentive structure of a common agent makes h im less "hungry" in selling a particular product than the exclusive sellers and more "hungry" in transmitting information to consumers. Essentially, firms' competition in sales is internalized within a common agent. On the other hand, sales impartiali ty of a common agent is credible to the consumers since being impartial is consistent with a common agent's incentives. Since common agents would bring about an increase in consumer surplus, there may be a public, policy rationale for requiring common agents, even when common agency is not in the interest of every firm. 4.5 The Channel Game As has been seen in Corollary 4.2, when consumer mismatching is reasonably symmetric, both firms would be worse off under the exclusive channel due to firms' inabili ty to commit themselves to a sales action other than misrepresentation. Firms would thus have an incentive to look for devices to signal their sales impartiality. We shall argue that independent non-cooperative sellers may use common agents as such a precommitment device, and demonstrate that a market involving common sales agents can arise as an paid by the airline. If a common agent charges a higher product price than firms' own selling offices, then an equi l ibr ium channel structure may involve both firms and common agents as sales outlets. W e discuss the issue further in the concluding remarks. Chapter 4. An Analysis of Common Sales Agents 77 equil ibrium outcome. The general setting is as follows. There is an independent risk-neutral sales agent in the market. The agent is assumed to be willing to accept any contract that offers h im non-negative expected profit, which would be the case if potential agents are competitive. If a firm decides to delegate its sales activity to the agent, the firm wi l l supply its product information to the agent and wi l l offer the agent a compensation contract. If the agent accepts the contract(s), he wil l conduct sales on behalf of the client(s), incur costs, and get paid according to the compensation contracts) . The sales action taken by the agent is unobservable to the firms and is almost unobservable to consumers. A client firm can nevertheless observe the number of sales completed by the agent. The costs borne by the agent are assumed to be the same as the corresponding costs borne by a firm. Thus, we rule out the possibility that the use of an outside agent by the client firms is because of cost-saving (due perhaps to economies of scale or scope, transaction economies, or other factors). 4 Finally, we assume symmetric consumer mismatching. The general model is a two-stage game. In the first stage, each firm decides whether or not to delegate and, if it does delegate, it chooses the agent's compensation contract subject to the agent's participation constraint. In the second stage, observing firms' decisions concerning the marketing channel and the compensation contracts, a potential buyer decides whether or not to consume any product, of the industry and, if deciding to consume, he (or she) must choose which particular product to buy and which sales channel to use. Finally, sales are concluded and payoffs are made. The model wi l l be referred to as the channel game. To demonstrate the basic insights, we consider in this section a simple version of the general model. In this simple channel game, it is assumed that i) common agency is 4 W e have also assumed that agents have no cost disadvantage. If using the agent is more costly than having the firm sell its own output , then there will be a pr ice /qual i ty tradeoff in f irms' channel decision. Chapter 4. An Analysis of Common Safes Agents 78 intrinsic in the sense that the agent serves either both firms or neither, and ii) the net compensation proportional to total industry sales is standardized such that the agent receives identical compensation from two firms and the agent's participation constraint is satisfied. We now solve for subgame perfect Nash equilibrium of the simple channel game. The analysis starts with the second-stage game. There are only two possible channel structures in the stage: the exclusive channel and the common-agent channel. Channel sales behavior and consumer buying behavior under the exclusive channel and under the common-agent channel have been examined in sections 4.3 and 4.4, respectively. The expected demand for each firm is 0kxd(O) under the exclusive channel and is 0kxd(l) under the common-agent channel. (Since the first argument of a, a 1 , remains unchanged ( = 1) under the two sales structures, it is suppressed hereafter for notational simplicity). Next consider firms' decisions in the first stage. In this simple channel game, each firm makes only one decision, either to delegate or not to delegate. If a firm chooses not to delegate, it obtains Ilf = Uk(E(xk)) where E(xd) is its expected demand under the exclusive channel. Taking the second-stage game into account, E(xf) = 0kxd(O). If a firm chooses to delegate, its payoff wi l l depend on the other firm's decision. It expects I l f if the other chooses not to delegate, and IiA otherwise. Ilf is the kth firm's payoff under the common-agent channel and may be written as follows, HA = RA — Ckk — §A where, under the common-agent channel, RA is the revenue, QA is firm k's payment to the agent, and CAk are other costs the firm incurs (e.g., production costs). Let CkQ be the cost the agent incurs in conducting sales for firm k. Then $A — Ck0 = $A, where $ A is the standardized net compensation to the agent. The common agent earns 2§A (> 0), whereas each firm expects to earn UA = (RA — Ckk — CkQ) — $A = IIk(0kxd(l)) — $A (the second equality follows from the assumption that the cost borne by the agent is the same as the cost borne by a firm if the firm conducts its sales). Since xd(l) > xd(0) and Chapter 4. An Analysis of Common Sales Agents 79 the marginal profit is positive, I l j 4 > IIJT. Thus, both firms would choose to delegate as long as 0 < $ A < 11^  - nf. The analysis leads to Proposition 4.4: Common sales agency can be an equilibrium outcome of the simple channel game. Moreover, under symmetric consumer mismatching, each firm expects a higher profit with the common-agent channel than with the exclusive channel. The intuit ion is as follows. W i t h imperfect information consumers may mismatch themselves, lowering their utilities from buying. Therefore, potential consumers value both information and impartial i ty of sales channels. Since an individual firm cannot credibly commit itself to a sales action other than misrepresentation, the marginal con-sumers of the industry would decide to stop consuming if firms were the only sales outlets available. On the other hand, the marginal consumers would buy the products if they could get them through a common agent. This is because sales impartial i ty of common agents is credible to consumers. Since the increase in market size can benefit all sellers, it is in each seller's own interest to participate in common agency. As a result, institutions such as common sales agents arise. As has been seen, common sales agents can be used as a precommitment device to convince potential consumers that they would be more likely to get their "best-buys". B y having sales action more in line with consumers' best interests through common agents, the sellers have effectively alleviated the conflict between potential consumers and exclu-sive sellers. Our analysis reveals that the coordination achieved by common sales agents can improve the welfare for both consumers and firms. Essentially, the coordination re-duces the cost, which is borne by both consumers and firms, due to a failure in sales coordination between exclusive sellers under imperfect consumer information. It is also noted that, under reasonably symmetric consumer mismatching, common sales agency is a first-best contract in the sense that, given product attributes, it yields Chapter 4. An Analysis of Common Sales Agents 80 the highest net surplus (the sum of expected profit and consumer surplus). In this case existing information is fully utilized. 4.6 Common Sales Agents as an Equilibrium Outcome In this section we examine the more general channel game specified in the last section. We assume for simplicity that the production cost is zero 5 and that the only cost incurred to each sales channel is the selling cost. Denote the selling cost to channel j as 1^  for j = 1,2,0 (using channel 1 to denote firm 1, channel 2 firm 2, and channel 0 the outside agent). Using T to denote the cost of the industry, then Ylj Tj = F . Further, it is assumed that the proportion of Vj in T is equal to the proportion of xd in xd, where xd is the number of consumers served by channel j . F xd for j — 1,2,0. Thus, a relatively high demand for a channel would result in a relatively high selling cost to the channel. It is noted that under the above cost specification, the cost borne by the outside independent agent is the same as the cost borne by a firm. The class of compensation contracts considered here is a constant per-unit fee: a commission of Xk for each unit sold by the agent. W i t h positive selling costs, a necessary condition for an agent to participate in a delegation is that the (net) commission rate is positive. We solve for subgame perfect Nash equilibria of the two-stage channel game. The analysis starts with the second-stage game. When the game enters this stage, the possi-ble channel structures are i) the exclusive channel, ii) the common-agent channel, i i i) the exclusive-agent channel, and iv) the unequal-commission channel. W i t h positive com-mission rates, the common-agent channel arises when both firms have delegated with 5 T h e results of this section would stil l be valid when different firms face different, non-zero, product ion costs. Chapter 4. An Analysis of Common Sales Agents 81 identical commission rates and the agent has accepted both delegations. The exclusive-agent channel is the channel structure where only one firm has delegated and the agent has accepted the delegation. Finally, the unequal-commission channel is the channel structure in which both firms have delegated with unequal commission rates and the agent has accepted both delegations. Recall that under all the channel structures, con-sumers can always buy products directly through the firms. Channel sales behavior and consumer buying behavior under the first two channel structures have been examined in sections 4.3 and 4.4. Now consider channel behavior and consumer behavior under the exclusive-agent channel. Since the commission rate is positive and the sales action is (almost) the agent's private information, an exclusive outside agent, like a firm's own selling office, wil l take a biased action. (Note that the reason for the use of a biased action by an exclusive agent can be independent of whether this outside agent is perfectly informed or not). If the agent works for, say, firm 1, then, given product attributes, consumers who preliminarily demand for product 1 are indifferent between approaching the agent and approaching firm 1 itself. Consumers entertain an expectation of a = 0 about the sales action of the channels and, hence, the industry demand is xd(0). Finally, under the unequal-commission channel, the agent would allocate all his customers to the firm offering h im a higher commission (recalling that the sales action taken by the agent is his private information, that there is no post-purchase contact between the agent and a buyer, and that the image effect is arbitrarily small). Since consumers can observe differentials in firms' commission rates, they would rationally treat the agent as an exclusive agent of the firm offering h im a higher commission, even if both firms have offered compensation contracts which satisfy agent participation constraint. The industry demand would consequently be xd(0). Next examine the first-stage game when firms make their channel and compensation decisions and then, given firms' decisions, the independent agent makes his participation Chapter 4. An Analysis of Common Sales Agents 82 decision. Examine first the agent's participation decision. Defining A = T/xd(l), we obtain the following result. (The proof of this result is in Appendix A ) . Lemma 4.2: Suppose a delegation offer with commission A/,. Then, i) if the other firm does not delegate or delegates with Xj < Xk, the agent wil l participate iff (if and only if) Xk > T/xd(0); ii) if the other firm delegates with the same commission, Xj = Xk, the agent wi l l participate iff A*. > A; and iii) if the other firm delegates with Xj > Xk, the agent wil l be indifferent between participation and non-participation if Xj > T/xd(0), and wi l l not participate if Xj < T/xd(0). Turn to firms' channel and compensation decision. B y defining A = min{Xi, A2} and Xk = X + (xd(l) - xd(0))pk/xd(l) for k = 1,2, we first state the following result. (The proof of Lemma 4.3 is given in Appendix A ) . Lemma 4.3: Assuming equal commission rates, Xx — X2 = X, then common sales agency can arise as an equilibrium outcome if and only if A < A < A. Each firm is free to choose its commission rate. If a firm decides to delegate, then its commission must be in the range [A, A] regardless of the other firm's decision. (This can be seen by noting that firm k wi l l not set Xk > A, while the equilibrium effect of Afc < A is equivalent to non-delegation.) Suppose that firm 2 has chosen to delegate with commission A < A 2 < A. The strategies available to firm 1 are i) not to delegate, ii) delegation with commission Ai = A 2 , and iii) delegation with Ai / A 2 . From Lemma 4.3, not to delegate is dominated by delegation with Aj = A 2 . Further, it can be easily shown (see the proof of Proposit ion 4.6) that by choosing delegation wi th Aa / A 2 , firm 1 expects a payoff no more than one by choosing not to delegate. Therefore, firm I's best response to the choice of firm 2 is delegation with Aj = A 2 . In other words, delegation wi th A 2 = A 2 is a Nash equil ibrium of the channel game. Chapter 4. An Analysis of Common Sales Agents 83 In effect, the channel game has a continumn of equilibria at which common sales agency arises with (common) commission rates varying from A to A. Thus, the equilibria differ only in the division of profits between the firms and the common agent. One notable equilibrium is delegation with A f c = A (k = 1,2) at which, with the agent just earning his opportunity cost, each firm has the highest expected profit among all the equilibria. Serving as a focal point, the equilibrium might actually emerge. The discussions lead to the following result. Proposition 4.5: A common sales agency can arise as an equilibrium outcome of the channel game. The basic message of Proposit ion 4.5 is that common sales agents can arise as an equilibrium outcome in the absence of regulatory intervention. The existence of com-mon sales agents is the result of non-cooperatively seeking own profit maximization by independent firms. The result may provide a formal explanation for the existence, and an even enhanced role, of travel agents after airline deregulation. The complexity of fare structures, flight schedules, and traffic, connections in the post-deregulation era made it less likely for a consumer to find best-available flights himself. Deregulation increased the value to consumers of having an expert to search for them. A travel agent, who is less biased than individual airlines, serves the role of the expert. In the circumstance of this paper, firms have virtually no conflicting views about which action should be chosen by a common agent. Firms independently, and non-cooperatively, would choose agency compensations which induce the agent's impartiality. In a sense, firms "jointly" use the marketing channel structure as a signal to convince potential consumers that their risk of being misrepresented has been eliminated. 6 6 B e r n h e i m and W h i n s t o n (1986) consider situations in which a number of r isk-neutral pr incipals, who have confl icting views about a c o m m o n agent's decision, independently attempt to influence the decision of a common agent by using agency compensat ion. T h e y find that some degree of cooperat ion Chapter 4. An Analysis of Common Sales Agents 84 One point deserves comment. The coordination effect of our common agents may also be achieved by a two-product monopoly if the two products yield the same marginal profit to the monopolist. If, however, the products yield different marginal profits, the mo-nopolist would secretly allocate all his customers to the more profitable good and would virtually stop producing and selling the less profitable good. Consumers would rationally expect the monopolist's biased behavior and marginal consumers would stop consuming. On the other hand, the possibility that duopolists may have different marginal profits on their products is allowed in Proposition 4.5. As Proposition 4.5 shows, duopoly ri-valry can give rise to a common sales agent through which both products are available to consumers. In this sense we may say that duopoly is superior to monopoly. Finally, it may be interesting to consider a regulation game in which the (net) com-missions to the independent agent are standardized to be at the same level for different firms. In the pre-deregulation airline market, for example, travel agent commissions were standardized by law to prevent a travel agent from working for certain airline at the expense of consumers' best interest. The channel game in this case may be modified as a three-stage game, referred to as the regulation game. In the first stage, a regulatory agent, whose objective is to maximize aggregate consumer surplus, sets a common com-mission rate, A. In the second stage, each firm decides whether or not to delegate and then the agent decides whether or not to participate. The final stage game remains the same as before. Proposition 4.6: A common sales agency is the unique equilibrium outcome of the regulation game. The proof of this result is in Appendix A. From that proof we can see that the change in channel contracting, from exclusive channels to common agents, would bring about can still be achieved with common agency. Chapter 4. An Analysis of Common Sales Agents 85 a gain to firms and common agents. The division of the gain between firms and the independent agent would depend on A, the level of standardized commissions. When A = A, the agent earns his opportunity cost (i.e., zero profit here) while the firms capture all the gain. As A increases, the agent's share rises while the firms' share falls. The actual division is likely to be affected by existing channel power. In the example of the pre-deregulation airline industry, although commission rates were set after public hearings, these hearings tended to be airline dominated. 4.7 Price, Profit, and Welfare Comparisons In the previous sections, product attributes (prices, quality aspects, etc.) are held con-stant for different channel structures. This would be the case if, for example, product attributes are fixed before firms make the channel decision. We have seen that given product attributes, less impartial sales action reduces consumers' expected net benefit from buying and consequently the industry demand. The reduction in market size can result in lower demands and lower profits for both firms. The welfare implication is clear. A channel structure such as common sales agents is welfare improving if it is capable of inducing impart ia l action from sales channels. Since channel structures would determine channels' sales actions which would in turn affect consumer demand functions, different product attributes may be chosen by firms under different, channel structures. Specifically, when prices are observable to consumers and pricing decision comes after firms' decision on marketing channels, equil ibrium prices may vary with channel structures. As has been seen in the previous sections, a channel structure may imply certain sales action. Thus equil ibrium prices may be written as a function of sales action. Write equil ibrium prices as a function of sales actions: p = p(a) (0 < a < 1). If p' > 0, then as sales action becomes less impart ial , both lower demands Chapter 4. An Analysis of Common Sales Agents 86 and lower prices wi l l reduce firms' profits. But lower prices provides a welfare gain for consumers. The interesting question is whether the gain wi l l outweigh the consumers' welfare loss due to the risk of a bad match. Suppose now that p' < 0. As sales action becomes less impartial , consumers' net benefit from buying wi l l decrease due both to rising prices and to the risk of bad match, while firms wil l benefit from higher prices. The question again is whether the profit gain due to higher prices wi l l outweigh the profit loss due to lower demands. We examine these issues in this section. Recall that the uti l i ty to consumer i of a good match is ct(i) and a bad match is 3(i). Suppose that the uti l i ty functions are separable in price and non-price attributes and that the price paid adds a direct cost to a consumer. ct(i) = — p -f a0(i) and 8{i) = —p + 3Q(I). The marginal consumer of the industry, i* is then defined by the following (using equations (4.2) and (4.3)): q(a)ct0(i*) + (1 - q(a))(30(i*) - p = 0. It can be shown that when S(i) = 6, i.e., the disutili ty of a bad match is assumed to be identical among consumers, i* = a - 1 ( p + (1 — q(a))8) where C K _ 1 ( - ) is the inverse of function a(-). The industry demand, xd can be written as follows. xd = 1 — F(i*) = xd(p-\- (1 — q(a))8) = xd(a) where a is the sum of product price and consumers' expected uti l i ty loss from a bad match, a may be interpreted as the "full price" or "risk-adjusted price" of consuming a unit of the industry's output. Note that dxd/da = — f(i*)/a' < 0. W i t h two products in the industry, we have a — (o^, 172) and ak = Pk + (1 — qj(ak))8 for j 7^  fc, k, j = 1, 2. Since product k's consumers may by mistake choose product j , the probability of a good match for product k's consumers, qk, w i l l depend on firm j ' s sales action, ar In effect, from equations (4.7) and (4.10), qk — qo,k + (1 — qo,k)a] = Qfc(aj)-The industry demand may be written as follows: xd = xd(aua2) = xd(Pl + (1 - q2(a1))6,p2 + (1 - 9 l ( a 2 ) ) 5 ) . (4.17) Chapter 4. An Analysis of Common Sales Agents 87 Note that symmetric consumer mismatching combined with symmetric sales action, &\ = a 2 ( = a), yields ^ ( 1 — q2[a1)) = 82(1 — qi(a>2)), indicating that a product 2 consumer has the same probability of having a bad match as a product 1 consumer. Further, E(xdk) = E(xpk) = 9kxd. Assuming zero costs, the expected profit for each firm, given channel structures or a, can be written as follows. U1(PuP2-l - 9 i ( a ) ) = Pl8xd(Pl + j—-(1 -qi(a))6,p2 + {l r-qi(a))S), . (4.18) n 2 ( p i , p 2 ; l - 9 i ( a ) ) = p 2 ( l -8)xd(pi + —~e{l - g 1 ( a ) ) 6 , p 2 + ( l -qi(a))S), (4.19) where 9i = 0(p2—pi) wi th 8(0) = 1/2. It should be noted that under perfect information, the two products have identical values of non-price attributes. The product differentia-tion is then due to heterogeneous tastes of individual consumers. We suppose that the products are imperfectly substitutable, 8' > 0. Thus, a decrease in price has two effects on demands. One is to take away consumers from the other product (the market share effect): d8jdp\ = —8' < 0 and 88jdp2 > 0. The other is to attract potential consumers (the market size effect): dxd/dPk = dxd/dak < 0, k = 1, 2. The market share effect hurts the rival while the market size effect benefits both. Let r = 1 — qi(a). Then a one-to-one correspondence exists between a and r: a — 1 iff r — 0, and 0 < a < 1 iff 1 — g 0 , i > r > 0. r may be interpreted to be the probability of a bad match since r — 1 — q\{a) = 1 — q2(a) at equilibrium. As the probability of making a choice error goes to zero (g° —*• 1), we have r —> 0. In what follows we shall for convienence work with r , although working with r is qualitatively equivalent to working with a. Given 0 < o . < l o r 0 < r < l — g 0 , i , the equil ibrium prices would be the solution of the following simultanious equations: m a x l l i ( p i , p 2 ; r ) + ! / IJ 2 (pi ,p 2 ; r), (4.20) Chapter 4. An Analysis of Common Sales Agents 88 maxIJ2(pi,p2\r) + uU1(p1,p2;r), (4.21) P2>0 where v takes value either 0 or 1. If the two firms act like a cartel in pricing, then v = 1. On the other hand, if each firm chooses its product price, given the rival's price, to maximize own profit, then v = 0. In the latter case firms engage in Bertrand competition with product differentiation. We shall carry out our analysis under a small deviation from impartiality which refers to either an impartial sales action (a = 1 or r = 0) or a channel structure inducing channels' impartial action. We assume that there exists a solution to the system and that firms have identical level of prices at equilibrium. Let pm and pn denote, respectively, equilibrium prices under price collusion and price rivalry. Let A 7 " and A n be A evaluated at pm and r = 0, and pn and r = 0, respectively, where A is defined as follows. ,d2Xd d2Xd \ , ,dxds2 dxd A = (——+ - — — )xd-2[—) -4—xd9'. 4.22 Proposition 4.7: Assume cartel pricing behavior. Then, a small deviation from impartiality would result in i) lower, unchanged, and higher equilibrium prices if A m < 0, = 0, and > 0, respectively; ii) higher equilibrium full-prices and lower expected net benefit from buying; iii) smaller market size; and iv) smaller profit for each firm. Proof: (i) We want to show that (dpm(r)/'dr)r=Q <,—,> 0 if A m <,=,> 0 respec-tively. The first-order condition (foe) for firm 1 is dn = d<n, + n,) = 0 dpi dpi The foe for firm 2 can be similarly derived. The equilibrium prices {p™TP™) a r e deter-mined by the intersection of the two foe's. Under our symmetric conditions p™ — p2l = pm, and pm will satisfy equation (4.24): Qxd,m 0(O)xd-m + pm — (1 + Zm{r)) = 0, (4.24) Chapter 4. An Analysis of Common Sales Agents 89 where dxd/dp1 = (8xd/da1)(1 + rS{d(9/l - 6)/dp1)) and Zm(r) = -A0'(O)r8 (note: 0'(O) = (d')p=pm). The suppressed arguments in xd'm and dxd'm/dPl are (p m + r£ , jDm + r £ ) . Also note dxd/dpk = dxd/dak (k = 1,2) at r = 0. On the other hand, the foe for a multi-product monopolist is dn an dPl an dP2 an 1 an 1 = — H = 1 = 0. dxd dpi dxd dp2 dxd dPidxd/dp1 dp2dxd/dp2 It can be easily seen that (oTI/ 'dx d ) x d = x d m = 0 where xd,m — xd(pm-\-r8,pm +rS). Suppose that the second-order condition (soc) is satisfied at xd'm, i.e., (d2U/d2xd)xd=xdi7Tl < 0. (d 2n/ 'd 2x d) x d = x d, m will be a function of r. At ?- = 0, the soc is equivalent to ,d2xm | 82xm x m ifdxm\2 < Q ( 4 2 5 ) ^ a 2 ^ a^ dp2 * ^ ' The proof for this is given in Appendix A. The equilibrium price pm, given r, is implicitly determined by equation (4.24). Let pm = pm(r). Totally differentiating (4.24) w.r.t. r gives pm'(r) _ pm(l + Zm)(d2xd'm/d2ai + d2xd'mlda1da2) + dx^/da^ + Z?  -8 ~ pm(l + Zm)(d2xd'm/d2ai + d2xd-mlduxdc2) + dxd-m/da1 + (dxd-m/6VX)(1 + Zm) where Z™ = -4pm{dxd'm/da1)0'{<d) > 0. Further using equation (4.24) gives pm'(r) _ (d^-x^/d2^ + d2xd'm/darda2)xd'm - 2(dxd'm/da^2 + Z™xd-m/(pm(l + Zm)) -8 ~ (d2xd'm/d2ai + d2xd-m/da1da2)xd'm - 2{dxd>m / do^)2 - 2(dxd-m/da1)2(l + Zm) Thus F X ' (4.26) r=o A m + 4(dxm/dp1)6'(0)xm - 2(dxm/dp1)2' The first two cases can be seen by noting (dxm/dPi)0'(0)xra > 0, while the last by noting inequality (4.25). ii) and iii) We only need to show that (dam(r)/'dr)r=0 > 0 and (dxd'm(r)/'dr)r=0 < 0 when (dpm(r)/dr)r_0 < 0 (the results are obvious when (dpm(r)/dr)r=Q > 0). Since Chapter 4. An Analysis of Common Sales Agents 90 A m < 0 from i), <Jm'(r)\r=0 = pm'(0) + 8 > 0 (using equation (4.26)). (dxd'm(r)/dr)r=Q = 2(dxm/dPl)am'(0) < 0. Finally, [dsm/da)a=1 = (1 - q0)(rm'(r)\r=0 > 0. iv) (^n^(r) /dr) r = 0 = -xm8/2 < 0 (using equation (4.24)). Q.E.D. Next consider the case of price rivalry. We introduce two conditions here: i) non-increasing marginal industry profit, and ii) grossly substitutable products. By non-increasing marginal industry profit, we mean (d2U/d2xd)xd_=xd,„ < 0 at r — 0, which is equivalent to ,d2xn d2xn \ n , 6 V \ 2 f ^ The proof for this is almost identical to the proof for inequality (4.25). By grossly substitutable products, we mean (dx^'d/dPj)(pn/x^) > 0 at r — 0, which is equivalent to dxlF _dx-pn 39,(0) pn > Q dPj xi dPk xn dPk 0(0) -for k ^  j, k,j — 1, 2. The first term of the gross cross-price elasticity (dxdkn/dp3)(pn/xdk'n) is negative, saying that the higher the price of a product, the lower the industry demand. The second term is positive: an increase in price of one product will turn away some of its consumers to the rival product. Condition (4.28) might be interpreted as saying that the products within an industry are more substitutable than the products across industries. The gross substitutes property has been used by, for example, Anderson, et al. (1989). Proposition 4.8: Assume Bertrand pricing behavior, non-increasing marginal indus-try profit and grossly substitutable products. Then, a small deviation from impartiality would result in i) lower, unchanged, and higher equilibrium prices if A n < 0, = 0, and > 0, respectively; ii) higher equilibrium full-prices and lower expected net benefit from buying; iii) smaller market size; and iv) smaller profit for each firm. Chapter 4. An Analysis of Common Sales Agents 91 Proof: We shall prove the following: a) if A n < 0, then (dpn(r)/dr)r=0 < 0, (dan{r)/dr)r=0 > 0, and (dxd'n(r)/dr)r=0 < 0; b) if A n > 0, the marginal industry profit is non-decreasing, and the products are grossly substitutable, then (dpn(r)/dr)r_0 > 0 and (d.lll{r)/dr)r=0 < 0 for k = 1,2. Note that (dsm/da)a=1 = (1 - q0)(dan(r)/dr)r=0 and that the results in the other cases are obvious. a) Each firm chooses its product price, given the product price of its rival, to maximize its own profit. The Nash equilibrium (w™,^) i s determined by the intersection of the two foe's. Under our symmetric conditions p™ = pV, = pn, and pn satisfies the following equation. dxd'n 0(Q)xd'n + 6(0)pn——(1 + Zn{r)) - pnxd-ne'(0) = 0, (4.29) where Zn{r) = —48'(0)r8, and the suppressed arguments in xd,n and dxd,n/dpi are (p" + rS,pn + r6). pn, given r , is implici t ly determined by equation (4.29). Let pn = pn(r). Totally differentiating (4.29) w.r.t. r and further using equation (4.29) gives pn'(r)l A n £- x-L\ = (4 30) -S lr=o A n + 4{dxn/dp1)8'(0)xn + (xn/Pn)(dxn/dPl) - 2(xn/pn)6'(0)xn' v ' Part a) can be seen by inspecting equation (4.30). b) First note that condition (4.28) combined with foe (4.29) implies xn dx11 xndx?pn h 2 — - = — — ^ — > 0. (4.31) pn dpi pn op3 x\ B y applying conditions (4.27) and (4.31), one can show that the denominator on the right hand side of expression (4.30) is negative. Since the numerator is positive, it follows that (dpn(r)/dr)r_0 > 0. Further, using D to denote the denominator and applying equa-tions (4.30) and (4.27), we can show that ( d n £ ( r ) / d r ) r = 0 < 46pn(dxn/dpi)2e\0)xn/D < 0. Q.E.D. Chapter 4. An Analysis of Common Sales Agents 92 Propositions 4.7 and 4.8 show that a small deviation from impartiality wi l l result in lower, higher, or unchanged equilibrium prices, depending on demand functions under consideration. As an example, consider the logit model as follows. 0(p2 -Pl) = , (4.32) g -MPl _j_ e ~ M P 2 x(pi,p2) = , (4.33) where p is a positive scale parameter and vo (< 0) is the util i ty loss by choosing not to consume. The demand system (4.32) and (4.33) can arise under our demand specification at r = 0 (see, for example, Domencich and McFadden (1975)). Note that the demand functions for individual options (buy either product 1 or product 2 or none) can be derived from (4.32) and (4.33). Under the logit demand function, it can be shown that A m > 0, A " > 0, and both conditions (4.27) and (4.28) are satisfied. Thus, under the logit model, a small deviation from impartiali ty would result in higher equilibrium prices. Nevertheless, each firm would expect less profit and, hence, has no incentive to deviate, at least locally, from the sales impartiality. The welfare implication under Propositions 4.7 and 4.8 is clear. The social welfare is the sum of the aggregate consumer surplus, u s and the industry profit, II = pxd. Taking pricing behavior into account, the welfare function can be written as follows: w(a,p(a)) = u s(a,p(a)) + fl(a,p(a)). Thus dw yi ds(i,a,p{a)) S , . S J . , dxd(a,p(a)) dU(ajP(a)) Ta = I da  f ( l ) d t + S(t '  a'P(a)) da.  + "  (4' 34) Under the conditions of Propositions 4.7 or 4.8, all three components on the right hand side of equation (4.34) are positive and, therefore, (dw/da)a=i > 0. In other words, when the marginal industry profit is non-increasing and products are grossly substitutable, Chapter 4. An Analysis of Common Sales Agents 93 a small deviation from impartiality wil l reduce both consumers' and sellers' welfare no matter whether products are later priced cooperatively or non-cooperatively. In partic-ular, the welfare loss to consumers has two components: a reduction in average product quality and a reduction in the overall market size. Propositions 4.7 and 4.8 suggest that our insights developed in the earlier sections may still prevail in situations where product prices vary with channel structures. 4.8 Concluding Remarks One major objective of the paper is to explain the existence of common sales agents in a number of markets including travel and real estate. Our basic point is that the exclusive channel structure may be not optimal if consumers are imperfectly informed. Since firms' own selling offices would have been perceived by consumers to sell only their own products, these sales outlets would have had limited appeal. The conflicts of interest between consumers and exclusive channels creates a cost which may be borne by both consumers and firms. The cost can be reduced by the use of a common sales agent as the third contracting party. Essentially, by delegating their sales activities to a common agent, all the firms make a clear commitment to fair dealing. Anticipat ing a higher net benefit from buying, consumers would get products through common agents rather than by dealing directly with the original sellers. The marginal consumers would consequently buy the products, increasing the demand for each firm. We thus suggest that business institutions such as common sales agents in these markets may arise as response to the difficulties that the exclusive channel structure has in dealing with imperfect consumer information. Under imperfect and asymmetric information, institutions such as common sales agents can be justified as public policy attempts to provide consumer protection. Our Chapter 4. An Analysis of Common Sales Agents 94 ana lys i s fur ther suggests tha t the marke t i tself m a y spontaneous ly give rise to c o m m o n sales agents i f agency compensa t ions are k n o w n to consumers and p roduc t marke t s are suff icient ly c o m p e t i t i v e (such tha t the size of the marke t under cons ide ra t ion is sensi-t ive to the f i rms ' sales ac t ions) . T h e p o l i c y i m p l i c a t i o n is tha t enhanc ing the marke t m e c h a n i s m might be an a l te rna t ive to r egu la t ion i n ce r t a in c i rcumstances . It has been assumed i n the paper tha t different channels p r o v i d e the same p roduc t w i t h i den t i c a l a t t r ibu tes . Fo r c o m m o n agent and sales m a d e d i r ec t l y by the firm, for ins tance , p r i ce is the same th rough ei ther channel . If a c o m m o n agent is a l lowed to set his re ta i l p r i ce for a p r o d u c t , he m a y be able to charge a h igher p r i ce t han the firm i tse l f s ince the former is perceived by consumers as a more accura te i n f o r m a t i o n source t han the la t te r . If a c o m m o n agent is more cost ly t h a n a firm's o w n sel l ing office, then there w i l l be a p r i c e / a c c u r a c y tradeoff i n consumers ' dec is ion of w h i c h channel to use. In e q u i l i b r i u m , we m a y see b o t h forms of m a r k e t i n g channels a c t u a l l y func t ion ing . F o r future studies we note two th ings . F i r s t , i t is in te res t ing to e x a m i n e the re t a i l level in a more de ta i l ed fashion and in t roduce the in te rac t ion between different agents. Second, consider s i tua t ions where agency compensa t ions are more c o m p l e x a n d are more diff icult for consumers to observe and unde r s t and . In such s i tua t ions , a firm m a y have incent ives to i nduce b iased act ions by a c o m m o n agent by m a n i p u l a t i n g agency compensa t i on . T o the extent tha t the image of c o m m o n agents ' i m p a r t i a l i t y is s t i l l m a i n t a i n e d , at least p a r t i a l l y ( re la t ive to exc lus ive channels ) , by consumers , the firm m a y benefit . These cons idera t ions are b e y o n d the scope of the current paper . Chapter 5 Strategic Stability of the Airline Industry: An Empirical Study 5.1 Introduction A n a l y s i s of the d y n a m i c pa t t e rn of f i r m c o m p e t i t i v e conduc t i n o l i gopo ly is i m p o r t a n t to i n d u s t r y po l icy . T o the extent tha t firm conduc t is s table over t ime , the knowledge of conduc t i n some p e r i o d can be used to predic t and evaluate marke t per formance in the future per iods . Since the passage of the A i r l i n e D e r e g u l a t i o n A c t i n 1978, the U.S. a i r l ine i n d u s t r y has seen a significant increase i n in t e r - f i rm r iva l ry . A l m o s t a l l the major carr iers have adop ted the hub-spoke ne twork strategy. A signif icant p r o p o r t i o n of a i r l ine c i t y -pa i r routes have been p r i m a r i l y served by two or three a i r l ines . Observa t ions of the deregula ted a i r l ine i n d u s t r y w o u l d suggest tha t a i r l ine c o m p e t i t i v e behav iour is non-s table over t ime . T h e rise i n air fares d u r i n g the most recent cycle raises concerns about a i r l ine c o m p e t i t i v e conduc t on o l igopo l i s t i c c i t y -pa i r routes , especia l ly those connec t ing to hub a i r p o r t s . 1 In th is paper we examine the dynamic , pa t t e rn of firm conduc t us ing t ime-series a n d firm-specific d a t a for a set of d u o p o l y a i r l ine routes . O u r m a i n purpose is to t r y to infer f r o m the d a t a whether firm conduc t is s table over t ime , and i f not , wha t the d y n a m i c pa t t e rn looks l ike , a n d wha t the degree of c o m p e t i t i o n is i n ( N a s h ) c o m p e t i t i v e per iods and wha t the degree of co l lus ion is i n col lus ive per iods . B y so do ing we may o b t a i n an idea of overa l l compet i t iveness of the a i r l ine marke t s under cons ide ra t ion . T h e development of the economic theory of o l i gopo ly has seen a great n u m b e r of a l t e rna t ive models . However , few a t tempts have been made i n e m p i r i c a l inves t iga t ion 1 see Morr ison and W i n s t o n (1990) for a discussion on this. 95 Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 96 of the relative usefulness of different theoretical models for a particular industry. A principal objective of the paper is to conduct such an exercise. Specifically, we estimate "conduct parameters" for each duopolist over several periods and examine, within a small set of oligopoly models, which model is most consistent with the obtained results. We (explicitly) conduct some Bayesian analysis of seeing how our results would influence priors associated with different models under consideration. We also illustrate a model choice criterion based on Bayesian analysis and use the criterion to choose the "best" model among competing models. Exist ing empirical studies of firm conduct in oligopoly have generally focused on single-period firm interaction. In particular, these papers (including Iwata (1974), Gol -lop and Roberts (1979), Appelbaum (1982), Dix i t (1988), Brander and Zhang (1989)) basically estimate or calibrate static "conjectural variation" models. Two very interest-ing studies which explicitly consider time-varying market conduct are Porter (1983b) and Bresnahan (1987). The former investigates cartel stability for the Joint Executive Com-mittee of the U.S . railroad industry during the periods from 1880 to 1886, whereas the latter investigates changes of firm conduct in mid-1950s for the U.S . automobile indus-try. Taking firms' choice of strategy variables as exogenous and using industry aggregate data, both studies find evidence of switches of firm conduct between a (Nash) competi-tive regime and a. collusive regime. 2 B y using carrier-specific quarterly airline data, we find that airlines' competitive behavior may switch between the competitive and collu-sive regimes. Moreover, we find that airline profits in a collusive period appear less than the (single-period) monopoly profits, and the degree of overall market competitiveness is between the Cournot and monopoly solutions but closer to the Cournot solution. Our data also suggest that major carriers might use quantity volumes, rather than prices, as 2 R o b e r t s and Samuelson (1988) analyze advertising compet i t ion in the U .S . cigarette industry by using a dynamic model of nonprice compet i t ion in oligopoly. Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 97 the i r s t ra tegy var iables . T h e past 20 years or so have ev idenced a considerable number of e m p i r i c a l studies of the a i r l i ne indus t ry . M o s t recent studies have i n general e x p l i c i t l y recognized the i n d u s t r y as one of o l i gopo l i s t i c marke t s . M o r r i s o n and W i n s t o n (1987), W h i n s t o n and C o l l i n s (1988), B e r r y (1989), a n d H u r d l e , et a l . (1989) examine e m p i r i c a l l y the con tes t ab i l i ty of the a i r l ine marke t s . B o r e n s t e i n (1989, 1990) and B e r r y (1990) inves t igate re la t ionships be tween hub-spoke route s t ruc ture and marke t power. B r a n d e r and Z h a n g (1989) look at a i r l ine conduc t parameters for a set of d u o p o l y routes i n the t h i r d quar ter of 1985. L e v i n e (1987) offers a comprehens ive d iscuss ion about var ious issues of ( imper fec t ly ) c o m p e t i t i o n i n deregula ted a i r l ine marke t s . T h e pape r is o rgan i zed as fol lows. W e star t w i t h i n Sec t ion 5.2 a set of ex i s t i ng o l i g o p o l y theories tha t offer c o m p e t i n g exp lana t ions about firm in t e rac t ion over t ime . W e then in Sect ions 5.3 and 5.4 discuss, respect ively , our s t a t i s t i ca l me thodo logy a n d several issues encoun ted in our e m p i r i c a l i m p l e m e n t a t i o n . Sec t ion 5.5 describes the d a t a and d a t a c o n s t r u c t i o n . Sect ions 5.6 and 5.7 present and analyse the "base case" results . Bayes i an analys is of firm conduc t a n d m o d e l choice is conduc ted in Sec t ion 5.7. Sens i t i v i t y of the results to var ia t ions i n some parameters as w e l l as the uni t l eng th of a s t ra tegy p e r i o d is e x a m i n e d i n Sec t ion 5.8. Sec t ion 5.9 provides conc lus ion . 5.2 Theories of Firm Conduct in Oligopoly C o n s i d e r a d u o p o l i s t i c i n d u s t r y p r o d u c i n g a homogeneous p r o d u c t . Le t pt be the marke t p r ice of the p r o d u c t i n t i m e p e r i o d t, and x\ be the co r re spond ing o u t p u t of firm i , 2 = 1,2. T h e inverse d e m a n d m a y be w r i t t e n as pt = p{Xt), where Xt = x) -f x2. Cos t s for f i r m i i n p e r i o d t are denoted CI = Cl(x\), and profits at t can be w r i t t e n *i=ptx\-C\x\). (5.1) Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 98 Fi rms ' conduct under a set of oligopoly theories may be captured by the following equations'. P t = c\-x\p't(l+vi) (5.2) where c\ are firm i's marginal costs, i — 1,2. The Cournot model of oligopoly assumes that equil ibrium arises when each firm optimally chooses its output level, given the output decision of the other. If the firms make one choice of strategy variable (here, quantity) per period, this translates into v\ = 0 for all i and t in equation (5.2). From (5.2) the Cournot model predicts a positive price-cost margin. The Bertrand model considers similar situations as the Cournot model with one differ-ence: the strategy variable is price in the Bertrand model. This difference would, however, result in quite different assessments of firms' competitive conduct. The Bertrand model predicts, in the homogeneous product case, that the oligopolists price at marginal costs. As a result, if the firms of the Bertrand kind have the same costs, then v\ = —1 for all i and t. In such a situation, the Bertrand model of oligopoly yields the same outcome as that under the perfectly competitive strategy which has the firm to believe that it can sell as much output as it likes at the current market price. 3 Both the Cournot and Bertrand models are examples of one-shot, Nash non-cooperative game in which the Nash equil ibrium is inefficient (to the firms). The firms would have obtained higher payoffs than both Cournot. and Bertrand behavior if they could manage to collude and maximize the single-period joint profits. Since pt -f- Xtp't is the (single-period) monopoly marginal revenue, from (5.2) the (static) cartel solution arises if (1 -f v\)x\ — Xt — x\ -f x\, or v\ = x\jx\. Under identical costs (and hence identical market shares), therefore, the static cartel solution to the duopoly requires that v\ = 1 for all i and t. Note that the static cartel solution is independent of which strategy variable, 3 I f the two firms have different costs, then the f i rm with the higher marginal cost at the solution will have v) = —1, while the other f i rm has v\ > —1. Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 99 quantity or price, one takes the firms to be choosing. One l imitat ion of the Cournot and Bertrand models is their static character. If games firms play are not one-time affairs but repeated over and over, it would appear more appropriate to use the repeated game version of oligopoly. Using such an approach, Friedman (1971) has shown that the static cartel solution can be supported as a non-cooperative equilibrium of the (infinitely) repeated game. The basic idea is that firms might use a "trigger" strategy in which they act according to the cartel solution in a period if no deviation from the solution is detected in the previous periods, and respond to any deviation by reverting to Nash behavior for the one-shot game thereafter. If future profits are sufficiently important, then a firm would be better off overall by not deviating at any time. The trigger strategy outlined by Friedman has the property that "punishment" pe-riods, or price wars, are never actually triggered. This is due to the fact that in the Friedman model, any deviation from the cartel allocation can be detected with certainty (and without incurring costs). B y introducing uncertainty about changes in market con-ditions, Green and Porter (1984) and Porter (1983a) have proposed a theory of oligopoly that can explain price wars. In their theory, firms cannot tell wi th certainty which one is responsible for a fall in own profits in the short-run: deviation from cartel solutions or worsened market condition. Unexpected, bad enough conditions wi l l trigger price wars even if there is no secret deviation. The theory predicts that there wi l l in general be alternating periods of price war and of collusion under uncertainty.. This takes the form of time-varying v\ in equation (5.2). In the Green-Porter model, the degree of competition in price wars is taken as exoge-nous. Thus, if quantity is the choice variable (as assumed in Green and Porter (1984) and Porter (1983a)), then vl = 0 in a price war; if price is the choice variable (as assumed in Porter (1983b)) and costs are identical, then v% = —1 in a price war. Turn now to Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 100 the degree of collusion in collusive periods. Porter (1983a) has shown, using quantity as the choice variable, that, in general, the optimal quantity in cooperative periods wil l exceed that which would maximize joint profits in any single period. This essentially is due to the "noisiness" effect: imperfect information about market conditions makes tacit collusion more difficult. We may introduce a parameter p and interpret it as a measure of the degree of noisiness or uncertainty, associating a higher degree of uncertainty with a higher value of p. p is normalized in [0,1] interval. If information is too noisy (p — 1), then no equilibrium output less than the Cournot output can be supported. On the other extreme, when the noisiness goes to zero (p = 0), we essentially have the certainty world of Friedman in which the static cartel output can be supported as an equilibrium. Thus, both the Cournot and Friedman outcomes might be viewed as special cases of the Green-Porter model: Cournot arises if environment is too noisy while Friedman arises under certainty. In both cases parameters v\ remain constant over time. In general, when environmental uncertainty has a "normal" level (or 0 < p < 1), we would, under identical costs, have vl = 0p+ 1(1 — p) = 1 — p in collusive periods if output is the choice variable. Similarly, if price is the choice variable, then too much uncertainty and no uncertainty would give rise to the Bertrand and Friedman outcomes respectively, while normal levels of uncertainty would, under identical costs, give rise to vl = ( — l)p + 1(1 — p) — \ — 2p in collusive periods. Thus degrees of collusion in the collusive regime may be reflected through values of p. The foregoing discussions suggest that the parameter v\ may be used to indicate the degree of collusiveness of firms' conduct for any given period. As v\ rises (from -1), the conduct of firm i becomes more collusive. Following Brander and Zhang (1989) we shall refer to v\ as a "conduct parameter". Note that in our current context a conduct parameter may be time-dependent. Each of the oligopoly theories we have reviewed so far imposes different values on Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 101 Table 5.1: Duopoly Models and Conduct Parameters: the Case of Identical Costs Models M i Cournot M> Bertrand M 3 Friedman M 4 GP-quanti ty M 5 GP-price Conduct Parameters v\ = 0 all t vl = - 1 all t vt = i all t v\ = 0,l-p 0 < p < 1 periodic switches ^ = - 1 , 1 - 2 ^ 0 < p < 1 periodic switches conduct parameters. The situation is summerized in Table 5.1. The first three models of oligopoly, namely, the Cournot, Bertrand, and Friedman (which gives rise to the cartel outcome in each period) models, require that vlt be constant over time and equal 0, -1, and 1 respectively. To the extent that firm conduct is stable over time (as implied by these three models), the knowledge of conduct in some period t can be used to predict and evaluate market performance in the future periods. As indicated by Geroski, Phlips, and Ulph (1985), this observation makes it important to examine and test the stability of firm conduct over time, since these three models imply quite different degrees of collusion, or competition, in industries. The stability issue is addressed in the Green-Porter model. The fourth and fifth models in Table 5.1 are the Green-Porter model wi th , respectively, quantity and price as the choice variables. As discussed above, they predict that in general there wi l l be periodic switches of firm conduct between the (Nash) competitive and collusive regimes, and the degree of firm collusion in the collusive regime wil l be lower than that the static cartel solution would imply. Our principal objective in this paper is to estimate conduct parameters for each firm over several time periods for a duopoly airline market and consider whether the results support the Cournot, Bertrand, Friedman, GP-quantity, or GP-pr ice models. Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 102 5.3 Statistical Methodology Our basic statistical methodolgy follows Brander and Zhang (1989). We regard vlt, the conduct parameter for firm i and period t, as an unknown variable. We then estimate it by trying to use informative "observations". Given period t, the "observed" conduct parameters are calculated for a set of duopoly airline routes. Consider one route, route k, in the set. We rewrite equation (5.2) as follows: ^ = ^ ~ C k 7 ? ( X t f c ) - l (5.3) t k Ptk s\k K ) where 77 ( A ^ ) = — (dXtkldpt).)(ptkjXtk) is the (positive) elasticity of market demand, and s\k is the market share of firm i . The conduct parameter for the route, v\k, can thus be calculated if one knows the variables on the right hand side (rhs) of equation (5.3) at the observed point. These calculated route-specific conduct parameters are informative in estimating the underlying conduct parameter if each airline route used in our analysis represents a similar strategic situation in the period, which we regard as one of our main statistical assumptions. The above procedure is repeated for each period in our data set. Our general statistical specification may be written as <4 = < + 4 (5.4) where i = 1, 2 refers to a firm, k = 1, 2 , n refers to an airline route, t = 1, 2 , T refers to a period, and e\k is an error term with mean 0. The error term is due to corrupted observability to the econometrician. We estimate the means and standard deviations of the conduct parameters for each firm and each period, and infer whether the data are consistent with the Cournot, Bertrand, Friedman, GP-quanti ty, or GP-pr ice models. For Chapter 5. Strategic, Stability of the Airline Industry: An Empirical Study 103 two firms and T periods, there are T by 2 conduct parameters to be estimated V (5.5) As we have seen in the previous section, different models may impose different values on the matrix V. In the case of homogeneous product and identical costs, the Cournot, Bertrand, and Friedman models impose, respectively, 0, -1, and 1 on V: while the two versions of the Green-Porter model can explain periodic switches of firm conduct between the competitive and collusive regimes: for the GP-quant i ty model, v\ switch between 0 and 1 — p, whereas for the GP-price model, between -1 and 1 — 2p, where 0 < p < 1. It is rather difficult, however, to identify the expected pattern of the switches according to the Green-Porter theory. Moreover, there are various theories (including the Green-Porter theory) which all predict that oligopoly might have periods of competition and periods of collusion under uncertainty but differ somewhat in the (expected) switch pattern (see Bresnahan (1989) for a review). In our empirical testing of the five models, rather than attempt to specify the switch patterns theoretically, we shall use the data to choose "sensible" switch patterns for each firm. The data wil l also be used to help determine p, the possible level of uncertainty. Our main purpose in this study is to try to infer from the data whether market conduct is stable over time, and if not, when the switches occur, and what the degrees of competition and collusion, respectively, are in competitive and collusive periods. Thus, the Green-Porter theory is exploited to the extent it predicts that in the face of uncertainty, such switches wi l l occur over time and the firms' profit in a collusive period wi l l be less than the (single-period) monopoly profit. It may be worth noting that, with a finite number of time periods, we might fail to observe "Green-Porter" behavior even if it exists. It is possible, for example, that in Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 104 the periods under study the firms are engaging in price wars. If this happens to be the case, then the Cournot and GP-quanti ty models (or the Bertrand and GP-pr ice models) are indistinguishable given the data. For this reason and since empirical studies of this kind must necessarily contain a finite number of time periods, a study which favors the Cournot, or Bertrand, model cannot be used as the only evidence in predicting market performance in the future. Such a study would nevertheless improve one's confidence in the Cournot, or Bertrand, model as a useful model for a particular industry, and might be incorporated as one's prior knowledge in the face of new evidence. The above example may also be useful in seeing the rationale for using Bayesian analysis. In this paper we shall conduct Bayesian exercises of seeing how our results would affect priors associated with the Cournot, Bertrand, Friedman, GP-quantity, or GP-price models. (This approach is very similar to Brander and Zhang (1989), and is very similar in spirit to Learner (1986).) We shall also consider a model choice criterion based on Bayesian analysis and use the criterion to choose the "best" model among competing models. 5.4 Empirical Implementation In our empirical implementation, we use a cross-sectional and time series data set of Chicago-based routes involving American Airlines and United Airlines as duopolists. Dealing with the same two airlines on every route corrects for many firm-specific effects. Restricting the routes to Chicago-based routes reduces the importance of variations in route-specific idiosyncratic factors such as airport delays and climate. It also intends to alleviate the airline "network" effect associated with whether or not a route involves a hub city for a particular airline. From equation (5.3), the higher is the market share of a carrier, the smaller is its own route-specific conduct parameter but the greater is its Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 105 rival's. Since a higher market share of an airline on a route may be caused exogenously by its hubbing on the route, an asymmetric network would result in biased estimates of conduct parameters for both carriers. Chicago is a major hub for both American Airl ines and United Airl ines, while virtually all the connecting cities in our route set duopolists) over time increases comparability between different periods and enhances our confidence in addressing the stability issue of firm conduct. (A disadvantage of the same-route requirement is that it would limit the size of the data set, as is seen in the data construction of the following section.) In section 5.2 a duopoly version of oligopoly theories is presented. As is noted in the data section, it proved almost impossible to have a reasonably large number of "strict" duopoly routes for our data set. Airlines other than American and United were operating on some of the routes in our sample periods. In such a situation we assume that the other airlines' output or, if they behave as perfect competitors, their supply functions, were exogenous to the two major firms. We now discuss what implications this assumption might have to data construction and to inferences. Consider a route market and suppose that the other airlines were a "competitive fringe". Let the aggregate supply function of the fringe be X[k = X^(ptk). If the industry demand was Xtk = X(ptk), then the (residual) demand facing the "duopolists" would be output of the duopolists. It can be shown that in such a situation, the route-specific conduct parameter v\k equals. are hubs for neither airline. Final ly, working with the same set of routes (for the same Xd = X(ptk)-Xj(ptk). From the residual demand, price may be written as a function of X = x 1 + x 2 , the total (5.6) Chapter 5. Stra,tegic Stability of the Airline Industry: An Empirical Study 106 where v\k is given by equation (5.3), and <f)tk = (dXtk/dptk){ptkl Xtk) is the positive price elasticity of the fringe firm supply function. From equation (5.6) v\k is less than or equal to v\k, and tends to v\k if X[kjx\k is sufficiently small (v\k = v\k if and only if X(k = 0, assuming ptk — c\k 7^  0) Thus v\k of equation (5.3) might under-estimate the degree of collusion, but nevertheless might still be used provided that the operation scale of carriers other than the two majors is sufficiently smaller than that of each major. The implication to route construction is to try to select "duopoly routes" on which other carriers, if any, together had sufficiently small operations compared with each of the two major carriers under consideration. 4 The theoretical models of section 5.2 consider the (simple) case where the firms pro-duce a homogeneous product. In effect, airline production is the joint production of seats in various fare classes, which in general are categorized into first-class, standard economy, and discount. First-class and standard economy seats are produced mainly for business travelers who are will ing to pay for service quality (flexibility and convenience), while discount seats are targeted at leisure travelers who are price-sensitive but not particularly service-sensitive. The discount category has by far the largest volume share among the three. Our analysis focuses just on the discount category. More specifically, we consider the discount category with "local traffic". Local traffic for a city-pair includes passengers if the end points of that city-pair are the origin and destination of those passengers' one-way trip with no stop in between. In such a situation one may regard the products of American Airlines and United Airlines as homogeneous, noting that the two airlines are reasonably symmetric in factors (most notably, network) which might affect non-price product attributes of the two airlines. Since the discount category usually consists of sev-eral component fareclasses (multiple levels of fares with different conditions), an airline's 4 A n o t h e r impl icat ion of equation (5.6) is that variations in conduct parameters v\k over time may be due partly to the variations of the second term on the rhs of (5.6). Us ing the "duopoly routes", this second term would be sufficiently small and , hence, the specif ication error would be smal l . Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 107 average discount fare is used as the product price for the airline. The theoretical models of section 5.2 also assume that the firms make one choice of strategy variable (quantity or price) per period. In empirical implementation, an important, and difficult, issue arises concerning the length of each "period". Further, the period length might depend on which strategy variable is being chosen by the firms. One observation of the airline industry may suggest that large carriers normally set capacity on a quarterly basis. (Capacity is jointly determined by flight frequency and aircraft type.) Such an observation would imply, if load factor is (stochastically) constant, that quantity (volume) is being chosen by American Airlines and United Airlines in each quarter. If price (fare) is the choice variable, our prior knowledge about the length of carriers' planning horizon is vague. In any case, the shortest length for each period our currently available data permit, is a quarter (our principal data source for quantity and price contains only quarterly information). We consider the quarterly planning horizon as our "base" case and use the semi-annually data for sensitivity analysis. 5 1 3 5.5 The Data The data requirements, as implied by our discussions in the previous sections and equa-tion (5.3) in particular, are price, market, share, demand elasticity, and marginal cost, all of which are route-, firm-, and quarterly-specific. Also, the product is a seat in the discount category, and quantity of the product is measured by local trafic. We obtained 5 I t is not entirely clear (to us), a prion, which strategy variable(s) are being chosen by airlines. In analysis of the regulated airline industry, most authors assume that airlines (non-cooperatively) choose capacity (or (somewhat) equivalently, frequency) while fares are fixed by regulation (see Schmalensee (1977) and studies cited there). Panzar (1979) models the unregulated airline industry by assuming that firms (non-cooperat ively) choose prices under product differentiation. Us ing a data set of Chicago-based duopoly routes in the third quarter of 1985, Brander and Zhang (1989) f ind that the Cournot model (Nash in quantity) seems much more consistent with the data than the Ber t rand model . 6 Empi r ic .a l studies of the airline industry normal ly use quarterly or annually data , most of which are based on static framework. Berry (1989) estimates a two-stage entry game using quarterly data. Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 108 quantity and price data from L P . Sharp Associates. The L P . Sharp data derives from Databank I A of the U.S . Department of Transportation ( D O T ) Origin and Destination Survey. The data set, or O D 1 A , is a 10% sample of all tickets that originate in the U.S. on significant domestic carriers. We have used the following procedure to obtain the airline routes in our data set. 20 routes were ini t ial ly selected by taking al l Chicago-based city-pair routes for the third quarter of 1985 on which American Airlines and United Airlines together had a market share exceeding 95%, and on which each carrier had at least 100 passengers in the 10% sample. The quantity data for the 20 routes were then collected for each quarter from 1981 to 1986. ( O D 1 A data is quarterly, beginning on January 1, 1981, and is updated quarterly with a lag of six to nine months.) The six-year quantity data for the 20 routes revealed quite a bit of structural changes on several routes over time. About half of the routes were not characterized as duopoly of the two airlines before 1983. In the first quarter of 1984, American Airlines had no flights operating on five city-pairs (United had a monopoly on these five routes). American entered four of the five routes in the second quarter of the year, and entered the remaining route in the third quarter. Thus, the fourth quarter of 1984 marked the first period when both American Airlines and United Airlines had a full quarter operation on the 20 routes. We next examined the 20 routes for the 9 quarters from the fourth quarter of 1984 to 1986. Over the time span, the combined market share of American Airlines and United Airlines remained high and were reasonably stable for most of the 20 routes. One notable exception here was the Chicago-Las Vegas route on which the joint market share had experienced a dramatic decline: it averaged 97% over the first five quarters while only 66% in 1986 (it was 51% in the third quarter). We eliminated the Chicago-Las Vegas route for airlines other than American and United had sizable production in 1986. For each remaining route, American Airlines and United Airlines together had a market Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 109 share exceeding 70% (and each had at least 100 passengers in the 10% sample) for every quarter. Our final concern is about a duopolist's market, share relative to the other, i.e., s\k in equation (5.3) (s\k > 0, s]k + s2k — 1). From (5.3) the calculated conduct, parameter v\k is quite sensitive to s\k for small values of sltlc, and v\k tends to (positive) infinity when s\k tends to 0 (or s\k tends to 1). Thus, exceptionally small s\k could have a major impact on the estimates of underlying conduct parameters v\. If the duopolists incur similar costs and charge similar prices, then small enough s\k over time would likely be attributed to exogenous factors such as asymmetric airline network on the route. We calculated, for each route, the average (relative) market share over the 9 quarters, and we then dropped the Chicago-Des Moines and Chicago-Omaha routes on which the average market share of American Airlines were 24% and 27% respectively (American Airlines started service ori the two routes in the second quarter of 1984). For each remaining route, the average market share of each duopolist over the 9 quarters was at least one third. Our data set therefore consists of the same 17 Chicago-based city-pair routes for each quarter from 1984 to 1986. As discussed above, these routes were characterized by duopoly of American Airlines and United Airlines over time. Table 5.2 lists the name of each connecting city and distance of each route (according to the ascending order of distance). It also gives the directional fare, which is taken to be half of the excursion (or round-trip) fare, and the number of directional (one-way) passengers in the 10% sample for each duopolist and each route for the fourth quarter of 1985, which is the median quarter of the 9-quarter sample period. Table 5.3 gives the mean values of price and volume over the 17 routes for each carrier. The rows contain information for each quarter wi th the bottom row containing overall averages for the 9 quarters. The second and third columns show that average fares were highest in the fourth quarter of 1984 (4Q84) for both carriers. Average fares were Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 110 Table 5.2: Route, Distance, and Price and Volume of 4th Quarter, 1985 Route Distance A A Fare U A Fare A A Pass U A Pass Grand Rapids 134 94 96 601 565 Buffalo 467 129 128 776 1055 Rochester 522 154 157 783 854 Tulsa 587 155 150 692 366 W i c h i t a 591 148 169 394 297 Syracuse 601 149 149 635 661 Oklahoma 692 155 156 635 514 Albany 717 148 156 683 469 Hartford 778 153 171 997 2597 Providence 842 143 164 575 779 Aus t in 972 129 140 558 291 Phoenix 1440 162 144 2599 2427 Tucson 1441 146 150 886 494 Reno 1680 153 146 243 398 Ontario, C A 1707 177 179 768 845 Sacramento 1790 182 205 271 410 San Jose 1837 202 238 579 672 Table 5.3: Mean Prices and Volumes, Quarterly A A U A A A U A Industry A A & U A Quarter Fare Fare Pass Pass Pass Share 4Q84 166 172 857 781 1725 .96 1Q85 163 155 714 846 1631 .97 2Q85 150 132 1038 832 1927 .97 3Q85 149 154 865 885 1784 .98 4Q85 152 159 746 806 1581 .98 1Q86 149 160 1204 741 1983 .99 2Q86 151 152 788 850 1739 .96 3Q86 157 153 662 849 1641 .95 4Q86 153 157 637 798 1575 .95 Average 154 155 836 821 1732 .97 Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 111 relatively low from 2Q85 to 2Q86 for American Airlines whereas, for United Airlines, from 1Q85 to 3Q85 and from 2Q86 to 3Q86. The overall average fares were about the same for the two carriers. The next three columns contain average passenger volumes for the two airlines and the industry (all carriers). The overall traffic volumes for the two carriers were roughly equal, wi th American slightly ahead of United. (They were almost equal if 2Q85 is excluded. As is noted below, the industry might not be in equilibrium for 2Q85 as a result of a strike against United Airlines.) The average combined market shares of American Airlines and United Airlines are given in the last column of Table 5.3. They were ranging from 95% to 99% for the sample quarters, suggesting that our sample routes were on average highly concentrated by American Airlines and United Airlines in the 9-quarter period. From Table 5.3 one may also notice that American Airlines had "unusually" high traffic volumes in 2Q85 and in 1Q86. American's high volume in 2Q85 was largely attributed to a United pilots' strike. The strike, starting May 17, prevented United from operating its full schedule for 29 days. 7 At the beginning of the strike, United's daily operations, both departures and arrivals, at O'Hare International Airpor t , Chicago were down by 70% from normal. Systemwidely, American and other major U.S . airlines picked up extra passenger traffic left by United. (American Airlines had recorded its 10 busiest days in history (by then) during the strike.) Immediately after the strike ended June 14, Uni ted began rebuilding its strike-reduced operations by offering a series of fare promotions. A t the beginning of July, United was operating 90% of its pre-strike capacity. For our 17 routes we see that United Airlines had its lowest average fare in 2Q85 ($132) accompanied by a normal production level. For the same quarter, American's average fare was at about the same level as the succeeding four quarters (including the same 7 T h e strike and its effects have been reported in various issues of A v i a t i o n Week and Space Technology ( A W & S T ) between M a y and J u l y in 1985. Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 112 quarter of the following year, 2Q86). The 1Q86 situation appears different from the 2Q85 one. In 1Q86, American Airlines had one of the lowest average fares and the highest traffic volume among the 9 quarters, while United Airlines had the second highest average fare accompanied by the lowest volume among the quarters. Compared with the same quarter of the previous year (1Q85), American's average fare fell by 8.6% and its traffic increased by 69%, while United's average fare increased by 3.2% and its traffic fell by 12.4%. Reading industry publications such as A W & ST does not seem to reveal any major exogenous events which might to a certain degree explain the 1Q86 phenomenon. (In the first quarter of 1986 United Airlines was starting service on the 10-nation transpacific route system it was acquiring from Pan American. Some of its domestic equipment were diverted to Pacific routes.) Systemwidely, there were widespread and deep discounting of fares in the U.S . domestic, market in 1Q86, reflecting heated competition (see, for example, A W & ST May 12). Finally, it is also interesting to note that the 1Q86 production expansion of American Airlines as well as of the industry has not taken place in a high demand period: the first quarter is a traditionally low traffic period. As indicated earlier, our estimation of conduct parameters requires marginal cost information for each route, each carrier, and each quarter. Costs raise difficult issues of workable marginal cost definition and of measurement. Consider the linear specification of the total cost for an airline on a city-pair route market (this specification has been used by Douglas and Mil le r (1974), Schmalensee (1977), and Panzar (1979)): C\x\k) = a\kx\k + b\kfi(x\k) + Fntk (5.7) where / t is the number of the airline's flights, a\k is (direct) cost of serving a passenger, b\k is cost of a flight, and Ffk is (firm-specific) fixed cost. The first two components of Cx form the variable cost. Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 113 The marginal cost of carrying a passenger is J(x\k) = a\k + b\J<(x\k), (5.8) For carrying a standby passenger we would expect f[ = 0 and hence c\k = a\k. If an additional passenger would lead the airline to consider adding one more flight (f- > 0), items such as flight operating cost (eg. crew, fuel, landing fee), aircraft maintenance and depreciation are going to be part of marginal cost. Moreover, expansion of ground facilities resulted from the added flight may also be considered as part of marginal cost. Generally, maginal cost of serving an "average" passenger is the direct cost of passenger service plus a share of cost incurred to make flights and seats available. (If rewriting equation (5.8) into cl(x\k) = f-(x\k){a\k + b\k) + (1 - f-(x\k))a\k, we might interpret / / as the probability of adding a flight resulted from an additional passenger.) Suppose that g\k, the average number of seats per airline i's flight, is exogenously given. The average load factor over all the flights is by definition equal to = x\kl(g\kfi) = l2(xltk). B y using this identity we rewrite equation (5.8) as ii i \ i i ^tkl 9tk ii h(Xtk) r / i \ / r n \ c (xtk) = <*tk + —j htkJTT\fr\xtk)- (5-9) Thus, if a carrier's average load factor is (stochastically) independent of its passenger levels, then the marginal cost can be approximated using the first two terms on the rhs of equation (5.9). That is, c\x\k) = a\k + b\kl(g\kk) = (a\kx\k + b\kf,(x\k))jx\k. In this case the marginal cost is taken to be the average variable cost (i.e., the variable cost devided by the number of passengers). We use the average variable cost to approximate the marginal cost. 8 One proxy for variable costs is the use of operating expenses. U.S . D O T reports quarterly, based on D O T Form 41, operating expenses for each carrier (see A i r Carrier Finant ial Statistics). 8 O n e useful exercise here is to see whether a carrier's average load factor and its traffic output are statistically unrelated. Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 114 Table 5.4: Costs per Passenger-Mile (Cents) American United 1984 1985 1986 13.14 12.33 11.27 11.89 12.61 11.00 Operating expenses are costs incurred in the performance of air transportation and air transportation related services, including direct aircraft operating expenses and ground and indirect operating expenses. The D O T also provides monthly (revenue) passenger miles, which equal (available) seat miles devided by load factor (see A i r Carrier Traffic Statistics). We divide operating costs by passenger miles to obtain a measure of variable cost per passenger mile. Table 5.4 provides the yearly operating costs per passenger mile (cpm\) for each carrier from 1984 to 1986. It can be seen from the table that American Airlines had a higher cost than United in 1984, and the two carriers had roughly the same costs in 1985 and 1986. American's cost declined during the three-year period, and both carriers enjoyed the lowest cost in 1986. A n examination of quarterly data suggests that United's relatively high cost in 1985 was due partly to its relatively high cost in the second quarter, a likely consequence of its pilots' strike in May and June. It is noted that price (and volume) data discussed earlier are those of the discount category while the costs are those for all fareclasses. Marginal cost for the discount seats is likely to be lower than that for the first-class and standard economy seats. As a result, the cost figures we used may, other things equal, be biased upwards. The bias caused by this factor is nevertheless not likely to be large since the discount seats are normally much more than the first-class and standard economy seats combined. The cost per passenger-mile may be directly used to calculate a carrier's marginal cost (or cost per passenger) for a route whose distance is equal to its average flight length for the U . S . market as a whole, denoted AFL\. We calculate AFL\ through dividing aircraft Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 115 Table 5.5: Average Flight Lengths (Miles) American United 1984 831 773 1985 802 751 1986 769 745 revenue miles by aircraft revenue departures performed for each carrier and each year (data are from A i r Carrier Traffic Statistics). The results are given in Table 5.5. From the table we see that American Airlines had higher average stage lengths than United Airlines in each sample year. Calculat ion of marginal costs for other routes requires knowledge on the elasticity of cpmr wi th respect to flight distance. Our first prior knowledge on this is that cost per passenger increases but is concave in distance (or cost per passenger mile falls with distance). 9 Using 0 (0 > 0) to denote the (constant) elasticity of cost per passenger mile with respect to distance (£>), a carrier's marginal cost for a route may be calculated by itk = C V < i j t i k \ ) (5-10) Equation (5.10) can be rewritten as c\k = cpml(AFL\)eD\~6. Thus, if 0 = 0, then a carrier's marginal cost for any particular route is simply equal to multiplication of cpm1 by the route distance. On the other hand, if 0 — 1, then for given carrier and given period, cost per passenger is the same for all the routes of different distances, c\k = cpm\AFL\. Our prior knowledge concerning which particular value 0 is taking, however, is much more vague than our first prior knowledge, namely, ^ / 0,1. Several studies in the literature seem to suggest a value of about .5 for 9 (see Brander and Zhang (1989) for a discussion). We use 9 = .5 in our base-case analysis and discuss sensitivity using other values of 0. To complete our data collection we need to know elasticity of market demand for each 9 A i r l i n e costs can be separated into l ine-haul costs and terminal costs. Te rmina l costs are costs related to the amount of traffic carried but independent of the mileage it travels. Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 116 carrier in each period, ntk- In this paper we plan to use demand elasticity estimates from outside studies. Oum, Gi l len , and Noble (1986) estimate a two-stage neoclassical con-sumer demand system for U.S . domestic air travel routes using a cross-sectional sample of 200 intra-U.S. routes in 1978. They find that the own price elasticity of the discount fareclass varies from 1.5 to 2.0 with most of the "vacational routes" falling in the upper half of the range. It is not clear in their paper, however, how other route characteristics, such as route distance and route traffic density, affect the route-specific demand elasticity. Nevertheless, O u m , et al. (1986) seems to be the most carefully done demand study of airline markets (where fareclasses are subdivided into first-class, standard economy, and discount, and a translog demand system is employed). We thus use the average estimate of 1.7 from Oum, et al. for our sample routes in each period, that is, ntk = V — 1-7. We use rj = 1.7 as the base-case value and conduct sensitivity analysis for changes in rj. 5.6 R e s u l t s This section and the next section report the results from the base case where 77 = 1.7, 9 = .50, and the unit of a "period" is a quarter. Section 5.8 provides a sensitivity analysis. The in i t ia l results are the calculated route-specific marginal costs for each airline and each quarter, c\k from equation (5.10), and the calculated route-specific conduct parameters for each airline and each quarter, v\k from equation (5.3). The results are reported in Table 5.6 for the fourth quarter of 1985, the median quarter in our 9-quarter sample period. The routes are listed from the short- to long-haul routes. The first two columns in Table 5.6 show that marginal costs in the fourth quarter of 1985 were very similar between the two airlines, with American marginal costs slightly higher than United's. Since we are using yearly data on both costs per passenger-mile (given in Table 5.4) and average flight length (given in Table 5.5), the route-specific Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 117 Table 5.6: Marginal Costs and Conduct Parameters, 4th Quarter, 1985: Base Case Route A A Cost U A Cost A A C P U A C P Grand Rapids 40.42 40.00 .88 1.05 Buffalo 75.46 74.68 .67 .23 Rochester 79.78 78.95 .71 .62 Tulsa 84.60 83.73 .18 1.18 Wich i t a 84.89 84.01 .27 .98 Syracuse 85.60 84.72 .48 .44 Oklahoma 91.86 90.91 .26 .58 Albany 93.50 92.53 .06 .69 Hartford 97.40 96.39 1.21 .03 Providence 101.32 100.28 .18 .15 Aust in 108.86 107.74 -.60 .15 Phoenix 132.51 131.13 -.40 -.69 Tucson 132.55 131.18 -.75 -.42 Reno 143.12 141.64 -.71 -.92 Ontario, C A 144.27 142.78 -.34 -.34 Sacramento 147.73 146.21 -.20 -.19 San Jose 149.66 148.11 -.05 .20 Table 5.7: Mean Marginal Costs (Dollars): Base Case American United 1984 1985 1986 114.45 105.50 94.43 99.88 104.41 90.71 Table 5.8: Estimated Mean Conduct Parameters, Quarterly: Base Case American United Quarter mean st. error 95% conf. hit mean st. error 95% conf. int 4Q84 .15 .16 (-.20, .50) .58 .16 (.24, .93) 1Q85 .39 .12 (.14, .65) .15 .16 (-.19, .49) 2Q85 -.08 .07 (-.23, .07) -.03 .22 (-.50, .44) 3Q85 .02 .13 (-.26, .31) .14 .18 (-.23, .52) 4Q85 .11 .14 (-.19, .40) .22 .15 (-.09, .53) 1Q86 .03 .11 (-.21, .27) 1.04 .27 (.47, 1.61) 2Q86 .39 .15 (.08, .71) .42 .17 (.06, .78) 3Q86 .62 .18 (.25, .99) .31 .16 (-.03, .64) 4Q86 .47 .18 (.08, .86) .33 .17 (-.02, .68) Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 118 marginal costs were the same for each quarter in a given year. Table 5.7 gives the mean costs per passenger (over the 17 routes) from the base case for the three years. It can be seen from the table that the two airlines had rather similar costs in 1985 and 1986 but slightly different costs in 1984. (This is true not only for the base case but for all cases.) As indicated earlier, different marginal costs for duopolists could affect the values of firm-specific conduct parameters under the Bertrand and cartel solutions. Since marginal costs were sufficiently close for 8 out of 9 quarters, we would nevertheless continue to regard conduct parameters -1 and 1 as representing the Bertrand and cartel solutions respectively. The Cournot conduct parameter (which equals 0) is not affected by the cost difference. 1 0 The ini t ia l results of calculated route-specific conduct parameters are then used to estimate the underlying conduct parameters v\. Table 5.8 reports the estimated (base-case) mean conduct parameters for each airline and each quarter, along with the standard errors of the means and 95% confidence interval for each individual mean. (The indi-vidual 95% confidence interval is obtained by mult iplying the critical t-value of 2.120 by the standard error and adding and subtracting from the mean.) Table 5.8 shows that estimated mean conduct parameters of the two airlines are somewhat similar for the first two quarters (4Q84 and 1Q85), and are similar for the succeeding quarters except 1Q86. In 1Q86, the estimated mean conduct parameter is remarkably close to the Cournot value of 0 for American Airlines, whereas it is close to the (static) cartel solution of 1 for United Airlines. Table 5.8 also shows that the Bertand model is strongly inconsistent with the base-case data. The Bertrand value of -1 is well outside all the 18 individual 95% confidence intervals. As a result, the GP-price model is unlikely to be supported by 1 0 A n o t h e r interesting observation f rom Table 5.6 is a negative correlation between conduct parameters and distances which is quite significant in the fourth quarter of 1985. T a k i n g the 9 quarters as a whole, the correlat ion, though becomes less significant, is still negative. Several possible explanations for this negative correlation f rom the base-case parameters are given in Brander and Zhang (1989). Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 119 the base-case data. On the other hand, the Friedman value of 1 is outside 17 (out of 18) 95% confidence intervals, suggesting inconsistency between the Friedman model and the base-case data. The Cournot model also appears to be inconsistent with the data. The Cournot conduct parameter of 0 is outside American's 95% confidence intervals in 1Q85, 2Q86, 3Q86, and 4Q86, and is outside United's 95% confidence intervals in 4Q84, 1Q86, and 2Q86. The base-case data of Table 5.8 appear to favor the GP-quant i ty model with the degree of firm collusion in the collusive regime of about .50. In effect, American Airlines appear to behave in the Cournot fashion in 4Q84 and in the 2Q85-4Q85 period while collusively in 1Q85 and in 2Q86-4Q86. United's conduct is collusive in 4Q84, switches to (Nash) competitive in 1985, and then switches back to collusive in 1986. The base-case data thus suggest a p value of about .50. Recall from the discussions in section 5.2 that parameter p was ini t ial ly introduced to reflect the degree of environmental uncertainty. A value of .50 is halfway between the "sufficient" noisiness (p = 0) and certainty (p = 1). In what follows, we shall use p = .50 as representing the degree of environmental uncertainty. (The use of p = .50 is for concreteness. Strictly speaking, a continum of p values in the neighbourhood of .50 could be used for our purpose.) Consequently, from Table 5.1 the Green-Porter theory predicts periodic switches of v: between 0 in the competitive regime and .5 in the collusive regime if quantity is the choice variable, whereas between -1 and 0 if price is the choice variable. A n implicit assumption in using the 95% confidence interval is that observations are a random sample from a normal population. We assess the assumption of normality by examining the normal plots (or Q-Q plots) as well as histograms for the calculated route-specific conduct parameters for each airline. The statistical package SAS is used. Based on the normal plots using one-quarter data (17 observations), we would not, for the majority of the cases, reject the notion that these observations are normally distributed. Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 120 The normal plots (as well as histograms) using 9-quarter observations (153) show that the normality assumption appears satisfied for each airline. To obtain a joint assessment of conduct parameters for the two airlines in any partic-ular period, we need to analyse the two variables jointly. (The individual 95% confidence intervals-are not sufficient in this regard since they ignore the covariance structure of the two variables.) Let y'th. = (v\k,v2k). We assume Assumption 5 .1. For each time period t, ytl, yt2, ytn are a random sample from a bivariate normal population N2(vt,T,t), where v't = (v],v2) is the unknown conduct parameter vector and E t is the unknown 2 x 2 variance-covariance matrix. Assumption 5.1 leads to a test of the hypothesis Hot : vt = vtp versus Hiit : vt ^ vt0. A t the a level of significance, reject Hot in favor of Hit if 2(n — 1) T2 = n(yt - v^yS-1^ - vt>0) > - i - ^ F 2 , n _ 2 ( a ) (5.11) n — 2 where yt — Yf!k=\Vtkln is fhe estimated mean vector of conduct parameters, St is the sample variance-covariance matrix, and F 2 ,n-2(«) is the upper (100a)th percentile of the E-dis t r ibut ion with 2 and n — 2 degrees of freedom. The statistic T2 is called Hotelling's T2. Table 5.9 reports Hotelling's T2 for the null hypotheses H0>t : (vl,v2) = (-1,-1), (0,0), (.5,.5), or (1,1) for each quarter. Useful inferences may be drawn from Table 5.9. 1 1 The critical value of the test, given by the rhs of equation (5.11), is 7.85 at level of significance a = .05 and is 5.76 at a = .10. We see from Table 5.9 that both hypotheses Vt = — 1 and vt — 1 are rejected for every quarter of our sample periods, and the significance of rejection against the Bertrand hypothesis is to a greater degree than that against the cartel one. This implication of Table 5.9 confirms the similar result from Table 5.8. For the first two sample quarters, the (static) Cournot hypothesis (i.e., vt = 0 1 1 I n mult ivariate analysis, it is rare to be content with a small number of tests of Hott '• i't — vt,o-Norma l ly it is preferable to f ind confidence regions for values of vt. Chapter 5. Strategic. Stability of the Airline Industry: An Empirical Study 121 Table 5.9: Hotelling's T2 for Hypothesis Hot : vt — vtfl- Base Case Ko> u t 2 .o) (-1.-1) (0, 0) (.5, .5) (1,1) 4Q84 ' 156.98 14.38 4.53f 35.65 1Q85 164.58 11.02 5.03t 46.23 2Q85 188.71 1.68f 77.02 264.93 3Q85 62.08 .89f 12.92 53.38 4Q85 83.22 2.30t 8.15 44.87 1Q86 94.17 19.58 42.44 105.40 2Q86 119.42 9.91 .59f 21.56 3Q86 116.35 13.15 2.68f 19.65 4Q86 78.36 6.73| 1.50f 16.66 f: Hot is not rejected at both the 5% and 10% levels of significance. {: Hot is n ° t rejected at the 5% level of significance but is rejected at the 10% level. given t) is rejected at a — .05 or .10 while the (dynamically) collusive hypothesis (i.e., vt = .5) is not rejected. Then for the next three quarters starting with 2Q85, the collusive hypothesis is rejected while the Cournot hypothesis is not rejected. In 1Q86, both the Cournot and collusive hypotheses are rejected. In effect, as seen earlier, the two airlines have rather different conduct parameters for the quarter. Using 1 and 2 to denote American Airlines and United Airlines respectively, the hypotheses that (vl,v2) — (0,1) or (v],v2) = (0, .5) for t = 1Q86 are not rejected at a = .05 or .10 (the Hotelling's T2 are .09 and 4.98 respectively). Following 1Q86, the positions of the two hypotheses are reversed again, with the Cournot hypothesis being rejected at a — .10 while the collusive hypothesis not. Thus, Table 5.9 seems to suggest that there might be a "price war" during the period beginning at 2Q85 and lasting about a year. The second transition (from price war to collusion) appears to occur in the first quarter of 1986. Our general statistical specification, namely, equation (5.4), may be written in the Chapter 5. Strategic. Stability of the Airline Industry: An Empirical Study 122 Table 5.10: Conduct Parameter Vectors, Quarterly 1Q 2Q 3Q 4Q 1984 1985 V2 ^3 v4 vs 1986 v& V7 V8 v9 matrix form as Y = ZV + e (5.12) where the N x 2 matrix Y has the N (= 9n) "observed" route-specific conduct parameter vectors for rows, Z (N X 9) is the appropriate design matrix (containing either 1 or 0 in each column and only one 1 in each row), V, given by (5.5), is the unknown conduct parameter matrix, and the N x 2 matrix e contains the zero-mean error terms. The conduct parameter vectors u t 's may be written in the way of Table 5.10. It may be de-sirable to statistically compare conduct parameter vectors among different time periods. For instance, the Bertrand, Cournot, and Friedman models all predict time-invariant conduct parameters. Thus, a statistically testable hypothesis of these models is that Vi = v-2 = • • • = v9, i.e., there is no overall time-period effect. To carry out such an analysis, we introduce the following assumption. Assumption 5.2. For each time period t, yti,yt2,- • • ,Vtn a r e a random sample from a bivariate normal population N2(vt, S ) . Moreover, the random samples of different periods are independent. Compared with Assumption 5.1, two further assumptions have been made in Assump-tion 5.2. First , different time periods are statistically treated as independent groups. Second, all populations have a common covariance matrix. Under Assumption 5.2, one may test the following class of multivariate general linear hypotheses: LVM = 0. Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 123 The matrix L refers to hypotheses on the elements within given columns of the parameter matr ix V, while the matrix M permits the generation of hypotheses between parameters among different time periods) so we take M as the identity matrix. The hypothesis of vy = v2, for instance, is corresponding to L = (1, —1,0, • • • , 0 ) l x 9 , while the hypothesis of v3 = v4 = v5 can be represented by The test statistic used here is that of Roy's maximum root . 1 2 As expected, the hypothesis of no overall period effect is strongly rejected at a ~ .05. Further analysis suggests that two factors are mainly responsible for the rejection. The first is the 1Q86 factor. For instance, the hypothesis of v2 = v& (the first quarters in 1985 and 1986 respectively) is rejected at the 5% level. The second factor is that the market conduct appears more competitive in the third quarter of 1985 than in the third quarter of 1986. The test criterion rejects that u 4 = v8 at a = .05, but it does not reject the following individual hypotheses at a = .05: vx = v2, or v3 = v4 = v5, or v7 = v8 = v9, or v3 — v7: or vi = v5 = v9. In effect, firm conduct in the base case resembles the Cournot behavior in the third quarter of 1985 while it resembles the (dynamically) collusive behavior in the same quarter a year later. This is an interesting result in that Brander and Zhang (1989), using a data set consisting of similar, but more, routes in the third quarter of 1985, find that (base-case) firm conduct in that quarter is consistent wi th the Cournot description. B y looking at time series data, we see that airline conduct appears more collusive in the third quarter of 1986 than that of the third quarter of 1985, and, hence, firm conduct may change over time even within the same industry. 1 2 F o u r test statistics, W i l k s ' l a m b d a , Pi l la i 's trace, Hotel l ing-Lawley trace, and Roy 's m a x i m u m root, are reported in SAS with F approximat ions. T h e y usually give the same test results. of the two airlines. We are concerned here only with the first kind of hypothesis (i.e. / 0 0 1 - 1 0 0 0 0 0 L \ 0 0 1 0 - 1 0 0 0 Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 124 Table 5.11: Estimated Overall Mean Conduct Parameters: Base Case American United 4Q84 -4Q86 mean st. error 95% conf. int .23 .05 (.14, .33) mean st. error 95% conf. int .35 .06 (.22 .48) We also test a "seasonal effect" on firm conduct. The Roy's maximum root test does not reject the hypothesis that (v3 + v7)/2 — (v4 + v8)/2 = (v5 + v9)/2 at the 5% level. This suggests that the seasonal effect is not likely to be significant. Finally, to obtain an idea of overall competitiveness of the airline markets under consideration, we estimate mean conduct parameters over both the 17 routes and the 9 quarters. The estimated overall mean for each airline, the standard errors of the means, and 95% confidence intervals for each mean are reported in Table 5.11. (The critical t-value of TV — 1 = 152 degrees of freedom is taken to be 1.975.) Table 5.11 shows that the overall base-case market conduct of each airline is closest to a level predicted by the Cournot model among the three "static" models (Bertrand, Cournot, and Fr iedman) . 1 3 In effect, the overall market conduct in the base case is between the Cournot level v = 0 and the (dynamically) collusive level of v = .5. The hypothesis that (vl,v2) = (.25, .25), for example, would not be rejected based on individual 95% confidence intervals or the Hotelling's T2 test at the 5% (or 10%) level of significance. This is not an unexpected result. We have indicated earlier that the base-case data yields periodic switches of firm conduct between the collusive regime with p = .5 and the competitive (Nash in quantity) regime, and each regime has about half of the time. Table 5.11 also shows that the 95% confidence intervals are very tight. 1 3 C l a s s i c a l hypothesis tests (under A s s u m p t i o n 5.2) would very strongly reject the Ber t rand model (v = —1), and would strongly reject the Fr iedman model (v = 1). T h e tests would also reject the Cournot model (v = 0). B u t significance of the rejection to the Cournot model is to a much lesser degree than that to either the Ber t rand or Fr iedman models. Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 125 5.7 Bayesian Analysis of Firm Conduct and Model Choice Analysis in the previous section shows that the base-case results most favor the G P -quantity model among our five models (the Cournot, Bertrand, Friedman, GP-quantity, and GP-pr ice models). In this section we conduct a Bayesian analysis of seeing how our base-case results would affect, or update, priors associated with each of these models. We also illustrate a model choice criterion based on the Bayesian approach and use the criterion to choose the "best" model among the five competing models. 5.7.1 Firm Conduct Our analysis begins with Assumption 5.1. Under the assumption, the general statistical model of equation (5.4), or (5.12), is taken to be Vtk = vt + etk, ytk ~ N2(vt,T,t) (5.13) (recalling y'tk = (vlk,v2k), etc.) where vt is the (unknown) conduct parameter vector and E t is the (unknown) 2 x 2 covariance matrix, k — 1, 2, • • • , n, and t = 1, 2, • • • , T. Consider things for a given period. The subscript t may be suppressed, and the likelihood function may be written as L(yu- • • , yn | v, S ) cx | S | - ^ 2 e x p | - ^ "£(yk - i / ) ' ! ] " 1 ^ - v ) } , (5.14) where oc means "proportional to". We now specify prior distributions used in our analysis. Again , the subscript t is suppressed. Assumption 5.3: For each time period, the prior distributions for ( v , S _ 1 ) take the following Normal-Wishart conjugate form: Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 126 / „ ) 2 - I ( T ; , E 1 | ttl0) oc | E 1 | 1 / 2ecc_p| — ^ ^ ( ^ - vl0)'T, \v - u,- 0) | | S - i | ( ^ - m - i , / 2 E A ; P | _ 1 ^ ( 2 - 1 ^ ) ^ ( 5 . 1 5 ) . where m = 2 (the bivariate case), fi;0 = (^0,^0,^0,^0) is the set of prior parameters associated with the i th model or hypothesis, and tr(-) is the trace operation. Thus, the prior conditional distribution of the conduct parameter vector v, given the inverse of E , is Normal with expected conduct vector vl0 and precision measure of the prior assessment n ; 0 , and the prior marginal distribution of E - 1 is Wishart with parameters Sl0 and T,0.14 The Bayesian posterior distribution can be obtained through mult iplying the prior (5.15) by the likelihood function (5.14). After a certain amount of matrix manipulation, the posterior distribution of (v, E - 1 ) is seen to remain in the Normal-Wishart family with the posterior parameter set Sli — (v^fii, St,fi) as 1 5 riioVio + ny v% — (5.lb) n t 0 + n n% = nl0 + n • (5-17) Si — Sl0 + S H ——(y - vi0)(y - viQy (5.18) nl0 + n rt = r,o + n (5.19) where y = J2kVk/n and S = Y^kiVk — y){yk ~ y)' contain the sample information. Thus the posterior distribution is determined by both the prior distribution and the current 1 4 T h e Wishar t distr ibution with an integer r;o can be viewed as the distr ibut ion of X^ J=i where r j 0 is called the degree-of-freedom parameter and the £j are each independently distr ibuted as Nm(0, S^1). W i t h m = 1 and Sio = 1, this Wishar t d istr ibut ion reduces to the univariate chi-square distr ibut ion with Tio degrees of freedom. T h e more general univariate correspondent of the Wishar t distr ibut ion is the G a m m a distr ibut ion with parameters s;o and r^o. 1 5 L e a m e r (1978, pp . 85) considers the same problem as this, but contains an error in the expression of Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 127 data set, and in particular, the posterior mean is a weighted average of the prior mean estimate and the sample mean. The posterior marginal distribution of the conduct parameter vector, v = (v1 ,v2), is seen to be a (generalized) Student with parameters v%, S t / n t , and T1 — m - f 1. This result may be used to make joint posterior inferences about v1 and v2. Rather than attempt to obtain such a joint assessment, we shall here focus just on individual assessments of conduct for each airline. The posterior marginal distribution of the j th airline's conduct, v3, is a (generalized) univariate Student's t distribution as follows: v3 — x>\ . yjsi/hi(fi - m + 1) where m — 2 and s\ is the j th diagonal element of the 2 x 2 matrix Si, j = 1,2. To use (5.20), we need to specify (vl0, nl0, Sl0, riQ), the prior parameters for each model in question. For the Cournot, Bertrand, and Friedman models, the prior assessments of v3 are 0, -1, and 1 respectively. For the GP-quant i ty model (the GP-price model, respectively) with p — .50, the v3 are either 0 or .50 (either -1 or 0, resp.), depending on which regime, (Nash) competitive or collusive, the period is in . The results given in section 5.6 are to be used to classify regime for each period. The rest of the prior parameters, namely, (nl0, Sl0, TI0), may be interpreted as "confidence weights" on their priors, relative to the sample, assigned by various interested people. Two extreme cases are conceivable. One extreme case is total prior ignorance in which all the individuals assign no weights on their priors. From equations (5.16) - (5.19) the prior ignorance case is corresponding to setting nl0, Sl0, and r , 0 equal to zero. The results in this case have been reported in section 5.6. The other extreme involves interested people who are completely confident in their respective models (and, hence, who assign no weights on the data part). In such a situation, the current data (or any data) would not affect their prior beliefs. We consider here the "equal weights" situation in which all the individuals Chapter 5. Strategic. Stability of the Airline Industry: An Empirical Study 128 would assign equal weights between their prior beliefs and the current data. Assumption 5.4: The equal-weight case is specified by setting nl0 = TI0 = n and slo — sl f ° r j — 1,2 and all i . Under these assumptions, the posterior base-case mean conduct parameters for each airline and each quarter, and 95% confidence interval for the mean are calculated using (5.20) as well as (5.16) - (5.19). The results are reported in Tables 5.12, 5.13, 5.14, 5.15 and 5.16, which, respectively, correspond to the five oligopoly models of Table 5.1. For each model, there are a total of 18 posterior mean conduct parameters (2 airlines x 9 quarters). Tables 5.12, 5.13, and 5.14 report the Bayesian posterior mean conduct parameters based on the Cournot, Bertrand, and Friedman priors, or M i , M 2 , and M3 in Table 5.1, respectively. Among the three models, the Bertrand prior is most significantly affected by the base-case data. Its posterior conduct parameters are well away from the Bertrand value of -1 for all the 18 instances. The Friedman prior is also significantly affected by the sample. The majority of its 95% posterior confidence intervals (14 out of 18) do not include the Friedman value of 1. Of the .three models, the Cournot model is least affected by the base-case evidence. The Cournot prior yields posterior conduct parameters that are far closer to the Cournot value than both the Bertrand and Friedman values for almost, all the instances. The majority of its 95% posterior confidence intervals (14 out of 18) include the Cournot value of 0. Although the Cournot model performs reasonably well against the current data, one might s t i l l feel uneasy about it since there are at least four instances (1Q85 and 3Q86 for American Airl ines, 4Q84 and 1Q86 for United Airlines) in which the Cournot belief could be seriously challenged by the data. As a result, a Cournot follower might give up his or her prior belief after seeing this quarterly base-case data. Recall that the Cournot model Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 129 Table 5.12: Bayesian Posterior Conduct Parameters, Cournot Prior: Base Case American United Quarter prior mean 95% conf. int prior mean 95% conf. int 4Q84 0 .08 (-.16, .31) 0 .29 (.04, .55) 1Q85 0 .20 (.01, .38) 0 .08 (-.15, .30) 2Q85 0 -.04 (-.14, .06) 0 -.01 (-.33, .30) 3Q85 0 .01 (-.18, .20) 0 .07 (-.18, .32) 4Q85 0 .05 (-.14, .25) 0 .11 (-.10, .32). 1Q86 0 .02 (-.14, .18) 0 .52 (.10, .95) 2Q86 0 .20 (-.03, .42) 0 .21 (-.04, .46) 3Q86 0 .31 (.04, .58) 0 .15 (-.08, .39) 4Q86 0 .23 (-.04, .51) 0 .16 (-.08, .40) Table 5.13 : Bayes ian Posterior Conduct Parameters, Bertrand P rior: Base Case American United Quarter prior mean 95% conf. int prior mean 95% conf. int 4Q84 -1 -.42 (-.73,-.11) -1 -.21 (-.57, .16) 1Q85 -1 -.30 (-.60, .00) -1 -.43 (-.73, -.12) 2Q85 -1 -.54 (-.73, -.35) -1 -.51 (-.87, -.16) 3Q85 -1 -.49 (-.75, -.23) -1 -.43 (-.75,-.11) 4Q85 -1 -.45 (-.72,-.17) -1 -.39 (-.69, -.09) 1Q86 -1 -.48 (-.73, -.24) -1 .02 (-.50, .55) 2Q86 -1 -.30 (-.63, .02) -1 -.29 (-.64, .06) 3Q86 -1 -.19 (-.57, .19) -1 -.35 (-.67, -.02) 4Q86 -1 -.27 (-.63, .10) -1 -.34 (-.67, .00) Table 5.14 : Bayesian Posterior Conduct Parameters, Friedman P rior: Base Case American United Quarter prior mean 95% conf. int prior mean 95% conf. int 4Q84 1 .58 (.30, .85) 1 .79 (.55, 1.03) 1Q85 1 .70 (.50, .90) 1 .58 (.30, .85) 2Q85 1 .46 (.25, .68) 1 .49 (.12, -.85) 3Q85 1 .51 (.25, .77) 1 .57 (.28, .87) 4Q85 1 .55 (.30, .81) 1 .61 (.36, .86) 1Q86 1 .52 (.28, .75) 1 1.02 (.64, 1.40) 2Q86 1 .70 (.46, .93) 1 .71 (.45, .97) 3Q86 1 .81 (.55, 1.07) 1 .65 (.40, .91) 4Q86 1 .73 (.46, 1.01) 1 .66 (.40, .93) Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 130 (as well as the Bertrand and Friedman models) requires that conduct parameters be constant over time. To the extent that firm conduct is actually non-stable, the Cournot model might, no longer be the best in the face of new models capable of accommodating the non-stability. The Green-Porter theory provides such a model. It can explain periodic switches of firm conduct between a Nash competitive regime (denoted N) and a collusive regime (denoted C ) . B y using the results given in section 5.6, we classify the quarterly regime switch patterns for each airline as follows: ( N C N N N N C C C ) for American Airl ines, and (C N N N N C C C C) for United Airlines. Note that the patterns of the two airlines are similar but not identical. There are five quarters in the (Nash) competitive regime and four in the collusive regime for American, and four quarters in the competitive regime and five in the collusive regime for United. Thus, if taking the two airlines together, we would have equal number of quarters in each regime. Also using the results of section 5.6, we set p — .50. From Table 5.1 these give rise to the prior estimates of conduct parameters for the two versions of the Green-Porter model, which are presented in Tables 5.15 and 5.16, along with the Baysian posterior means and corresponding 95% confidence intervals. Table 5.15 shows that the GP-quant i ty model fits the base-case data remarkably well. The posterior estimates are remarkably close to the GP-quant i ty priors, with relatively tight 95% posterior confidence intervals which contain the prior values for all the 18 instances. On the other hand, the prior associated with the GP-pr ice model would be significantly changed in the face of the data (Table 5.16). Only 5 (out of 18) 95% confidence intervals include its prior values. The poor performance of the GP-pr ice model lies in its over-competitive prediction of firm conduct in both regimes. From these tables, however, we are unable to see a consensus emerging among all the models under consideration. The 95% confidence intervals for about 7 Bertrand posterior mean conduct parameters would instead include the Cournot value of 0, indicating that Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 131 Table 5.15: Bayesian Posterior Conduct Parameters, GP-quanti ty Prior: Base Case American United Quarter prior mean 95% conf. mt prior mean 95% conf. int 4Q84 0 .08 (-.16, .31) .5 .54 (.31, .77) 1Q85 .5 .45 (.28, .62) 0 .08 (-.15, .30) 2Q85 0 -.04 (-.14, .06) 0 -.01 (-.33, .30) 3Q85 0 .01 (-.18, .20) 0 .07 (-.18, .32) 4Q85 0 .05 (-.14, .25) 0 .11 (-.10, .32) 1Q86 0 .02 (-.14, .18) .5 .77 (.38, 1.17) 2Q86 .5 .45 (.23, .66) .5 .46 (.22, .70) 3Q86 .5 .56 (.31, .81) .5 .40 (.18, .63) 4Q86 .5 .48 (.22, .74) .5 .41 (.18, .65) Table 5.16: Bayesian Posterior Conduct Parameters, GP-price Prior: Base Case American United Quarter prior mean 95% conf. int prior mean 95% conf. int 4Q84 -1 -.42 (- .73,-11) 0 .29 (.04, .55) 1Q85 0 .20 (.01, .38) -1 -.43 (-.73,-.12) 2Q85 -1 -.54 (-.73, -.35) -1 -.51 (-.87,-.16) 3Q85 -1 -.49 (-.75, -.23) -1 -.43 (-.75,-.11) 4Q85 -1 -.45 (-.72, -.17) -1 -.39 (-.69, -.09) 1Q86 -1 -.48 (-.73, -.24) 0 .52 ( .10, .95) 2Q86 0 .20 (-.03, .42) 0 .21 (-.04, .46) 3Q86 0 .31 (.04, .58) 0 .15 (-.08, .39) 4Q86 0 .23 (-.04, .51) 0 .16 (-.08, .40) Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 132 Table 5.17: Bayesian Posteriors of Overall Conduct Parameters: Base Case American United prior mean 95% conf. int mean 95% conf. int Cournot 0 Bertrand -1 Friedman 1 GP-quant i ty .25 GP-pr ice -.50 .12 (.05, .19) -.38 (-.48, -.28) .62 (.54, .70) .24 (.17, .31) -.13 (-.21,-.05) .18 (.08, .27) -.32 (-.44, -.21) .68 (.58, .77) .30 (.21, .39) -.07 (-.18, .03) a Bertrand follower might switch to the Cournot model in light of the data. Those following the Friedman model, however, would remain (totally) unconvinced about the Cournot model: no Friedman posterior confidence intervals include the Cournot value of 0. A Friedman follower might switch to the GP-quant i ty model from the posterior point of view, but conversion vto the GP-quant i ty model would (almost) not happen to those holding the Bertrand prior. As for the relative accuracy of the Cournot and GP-quant i ty models, we note that a Cournot follower is more likely to update his prior towards the GP-quant i ty model after seeing this set of data than a GP-quant i ty follower updating towards the Cournot model. The above Bayesian analysis of time-series firm conduct shows that, of the five oligopoly models reviewed, the GP-quant i ty model seems most favored by our base-case data. This result is supported by looking at posteriors of overall mean conduct param-eters over both the routes and the quarters. The Bayesian posteriors based on the five models are given in Table 5.17. Since each airline has about the same number of quarters in each of the two regimes during our sample period, the prior values of firm conduct for the two Green-Porter models are taken to be the average of competitive degrees of the two regimes. It can be seen from the table that only the GP-quant i ty model yields 95% posterior confidence intervals which contain its own prior assessments. (The critical t value of 27V — 1 = 305 degrees of freedom is taken to be 1.965.) Pr ior values of the Chapter 5. Strategic. Stability of the Airline Industry: An Empirical Study 133 other four candidates are not within almost all the posterior confidence intervals (the only exception is that the posterior confidence interval would include the Cournot value if one starts with the GP-price prior). 5.7.2 Model Choice We now consider a model choice criterion based on what is called "Bayes factors" (Learner (1978)). We illustrate that the higher is the degree of compatibility of a model's prior and sample information about firm conduct, other things equal, the more preferable would be the model. We write our general statistical specification of (5.12) for each airline as Y3 = ZV3 + e3 (5.21) where j represents the airline in question, j = 1,2, Y3 (N x 1) contain N (= Tn) "observed" route-specific conduct parameters for the airline, Z (N x T) is the design matrix, V3 ( T x l ) contain the unknown conduct parameters of the airline, and e3 (N x 1) is the error vector. The Bayes factor in favor of the i th model relative to the kth model, given the sample data D3 = (Yj,Z), is defined as BFik = P(D3 \ Mi)/P(D3 \ Mk) (where P(-) denotes probabili ty). Bayesian analysis of model selection based on a loss function suggests that, under some symmetric conditions, M t is preferred to Mk, denoted Mi > Mk, if BFlk > l . 1 6 1 6 ( T h e discussion in this note is very similar to one given in Zellner (1971).) Consider a situation in which one (and only one) model must be chosen for making decisions from a set of compet ing models. These models can be ordered in terms of their (performance) "superiority" which is regarded as the state of nature and , hence, unknown to the decision makers. Consider now the choice between Mi and Mk- A decision of choosing Mi, denoted Mi, may suffer a loss if Mu is superior to Mi according to the nature. T h e expected loss associated with Mi, given D°, may be writ ten as (the one with Mk is similar) E(L | Mi,D3) = P(Mi \ EP)L{Mi,Mi) + P(Mk \ D>)L(Mk,Mi) where L — L(-, •) denotes the loss funct ion with the first argument of L referring to the superior model of the state of nature. Mt is said to be preferred to Mk if (and only if) E(L \ Mi, D1') < E(L \ Mk,Dj). Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 134 To calculate the Bayes factor for each airline, we use the univariate version of As-sumption 5.2. That is, e3 ~ N ( Q N I N ) where h3 is the unknown precision (the reciprocal of the variance). The prior distribution of (V- 7 , h3) takes the Normal -Gamma conjugate form for each prior model. To be specific, for the ith model, the prior condi-tional distribution of V3, given h3, is Normal with mean vector V/0 = (vli0, • • • ,vJTl0)' and covariance matrix (h3)"1 Nx0, and the prior marginal pdf for h3 is Gamma with param-eters S j 0 and r / 0 . Under these assumptions the Bayes factors can be computed (Zellner (1971) and Learner (1978) give the expression of P(D3 \ Mx) for regression equation (5.21) when Z is any N x T observable matrix) . In our context, Z is the design ma-trix, and we further assume that people of different models in question all assign equal confidence weights on their respective priors and the sample. The equal weights are characterized by setting N?0 — Z'Z, a T X T diagonal matrix with elements equal to n, sio = s3 = (Y3 - ZY3)'(Y3 - ZY3) where Y3 = (yl, • • • ,fT)', and T/0 = N. For the equal-weight case, the Bayes factor can be calculated as 23 j + ( n / 2 ) E L ( y ? " - < i o ) 2 N ~ " 2 ^ + ( n / 2 ) £ L i ( ^ - < f c o ) 2 . Let 6j denote the sum of squared deviations of sample means from the i th prior over Suppose that no loss would occur if a choice matches the nature while a positive loss would occur if not. Further , apply ing the Bayes ' theorem to P(Mi \ D]) gives P(Di) where P(M{) is prior assessment that JW, is superior (to Mj . ) , P(D3 \ Mi) is the marginal probabil i ty density funct ion (pdf) for the sample observations given that M z is used, and P{D3) = £ , P(Mi)P(DJ | Mi). T h u s , L(Mk,Mi)P(Mk) Mi > Mk if BFik > L(Mi,Mk) P(Mi If the loss funct ion is symmetr ic , i.e., L(Mk,Mi) = L(Mi, Mk), and the prior assessment of M% equals that of Mk, then Mt > Mk if BFik > 1. Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 135 Table 5.18: Incompatibility of Pr ior and Sample (8f): Base Case prior American United M j Cournot .95 1.90 M2 Bertrand 14.17 17.25 M3 Friedman 5.73 4.56 M4 GP-quanti ty .08 .47 M 5 GP-price 6.43 6.88 time periods, that is, % = 't(yi-<io)2- (5-23) 8j may be interpreted as measuring the degree of incompatibility of prior and sample mean on firm conduct. From equation (5.23), BFlk < 1 if 8\ > 83k. That is, the higher the degree of incompatibility of sample mean and prior of the i th model is relative to prior of the k th model, the smaller wi l l be the Bayes factor, BFlk, in favor of M%. Note that. 8\ is a very simple measurement which only uses information of prior assessments and sample means over time. Table 5.18 reports the 8\ for each airline and each of our five models, whose priors are specified in Tables 5.12 - 5.16. Table 5.18 shows that the results from the two airlines are consistent with each other. The model choice criterion based on Bayes factors would suggest M 4 > M ] > M3 > M5 > M2, given our base-case sample information. Thus, the models assuming price as airlines' choice variable (M2 and M 5 ) appear least supported by the base-case data, while the models assuming quantity as airlines' choice variable ( M 4 and Mi) appear most preferred. As expected, the GP-quant i ty model, which is capable of explaining periodic switches of firm competitive conduct, is the most preferred model among the five oligopoly models. Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 136 Table 5.19: Sensitivity of Overall Conduct Parameters, American 0 = .35 0 = .50 0 = .75 77 = 1.4 -.05 (.05) .02 (.04) .09 (.04) rj = 1.7 .16 (.06) .23 (.05) .32 (.04) 7] = 2.0 .36 (.07) .45 (.06) .55 (.05) Table 5.20: Sensitivity of Overall Conduct Parameters, United 0 = .35 0 = .50 0 = .75 7? = 1.4 .05 (.06) .11 (.05) .19 (.04) 77 = 1.7 .27 (.08) .35 (.06) .44 (.05) 77 = 2.0 .50 (.09) .59 (.08) .70 (.06) 5.8 Sensitivity Analysis Sections 5.6 and 5.7 report the results obtained from the base-case situation where 77 = 1.7, 9 = .50, and the unit of a strategy period is taken to be a quarter. (Recall that 77 is the elasticity of market demand for the discount fareclass, 0 is the elasticity of cost per passenger mile with respect to flight distance, and a strategy period is a time horizon when firms make one choice of strategy variable.) In this section we conduct sensitivity analysis to see how the results would respond to alternative values of 77 and 0. We are also interested in seeing what results would emerge if the planning horizon is taken to be semi-annually. Consider first the effects of changing elasticity estimates on the overall competitiveness of the airline market under consideration. We would take 1.4 < 77 < 2.0 and .35 < 0 < .75 to be the ranges of elasticity estimates of serious interest (see Brander and Zhang (1989) for a discussion). Table 5.19 reports the estimated mean conduct parameters of American Airlines when the values of 77 are 1.4, 1.7, and 2.0, and the values of 0 are .35, .50, and .75. Table 5.20 reports the corresponding results for United Airl ines. There are 9 different combinations of cases, including the base case, for each airline. Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 137 From equation (5.3) the higher is the demand elasticity estimate, other things un-changed, the higher is the calculated route-specific conduct parameter and hence the higher are the estimated mean conduct parameters. Tables 5.19 and 5.20 show that an increase in 77 of .30 would induce an increase in the overall conduct parameters of .21 to .26. (Thus, an increase in 77 of .10 would result in an increase in the conduct parameters of about .07 or .08.) Such a degree of sensitivity does not seem significant. For variations in the conduct parameters increase with 0 but the effect appears small. (The tables also show that the standard errors of the means increase with 77 but fall as 8 becomes large.) When both the 77 and 6 effects are combined, we see that the highest degree of compet-itiveness (or the smallest conduct parameter) is reached at the lowest elasticity estimates of the ranges (77 = 1.4 and 6 = .35), while the lowest degree of competitiveness (or the largest conduct parameter) is reached at the highest elasticity estimates of the ranges (77 = 2.0 and 0 = .75). The highest competitiveness resembles the Cournot conduct, whereas the lowest competitiveness yields estimated conduct parameters that are be-tween the Cournot conduct of 0 and the (static) cartel conduct of 1 and are somewhat closer to 1 than 0. It can be shown that, based on Bayesian posteriors of conduct param-eters, both the Bertrand and cartel hypotheses appear unlikely to hold even for the above extreme cases. Thus, the obtained result concerning the overall market competitiveness of the sample period appears not very sensitive to variations in elasticity estimates. We next examine the sensitivity of our model choice to variations in 77 and 9. As the values of 77 and 0 remain the same for all the periods, changes in these values would not affect the basic patterns of conduct parameters over time. It would still be the case, for example, that firm conduct is more collusive in the 2Q85-4Q85 period than in the 2Q86-4Q86 period. Changes in i] and 0 do, however, affect the levels of competitiveness in each period, as seen above. To the extent that the levels of competitiveness are not Chapter 5. Strategic. Stability of the Airline Industry: An Empirical Study 138 significantly affected by the ?/ and 6 variations, our basic results concerning choices of the five models would not be seriously altered. One useful exercise here is to examine compatibility, or incompatibility, of prior and sample information for each of the 9 combinations of 77 and 8 values. As discussed earlier, one simple measurement for this purpose is Sf, the sum of squared deviations of sample means from the i th prior values over time for airline j . Table 5.21 reports American 8t for each of our five models and each of the 9 cases, using the quarterly data. It also lists the model preference ranking for each case based on the Bayes factor analysis. As illustrated earlier, the smaller the b\ is relative to the 83k (k ^ 7), the more preferred the i th model (or prior) is relative to the kth model (or prior). Table 5.22 does the same thing for United. Perhaps the clearest implication of Tables 5.21 and 5.22 is that the Bertrand model ( M 2 ) is the least preferred model among the five candidates. It stays at bottom for 17 out of the total 18 instances, and is ranked the fourth in the remaining instance (the lowest elasticity values with American Airl ines). The GP-price model ( M 5 ) , which shares the same feature as the Bertrand model in assuming price as the choice variable, also performs poorly. The GP-pr ice model is ranked the third for small elasticity values and is ranked the fourth for large elasticity values. Thus competition in model performance is among the Cournot, Friedman, and G P -quantity models. For the low end of elasticity values (and therefore the high end of degrees of competitiveness), the Cournot model (Mi) performs very well and is three times ranked the first in the extreme instances. On the other extreme, the Friedman model (M3) gets competitive as one moves to the very high end of elasticity values, and it tops the rest in the highest elasticity case with United Airl ines. Nevertheless, performance of the GP-quant i ty model ( M 4 ) appears the strongest and most consistent. The GP-quanti ty model is the most preferred model among the five candidates in 14 instances where Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 139 Table 5.21: Sensitivity Analysis of Incompatibility of Prior and Sample, American M a M2 M3 M 4 M 5 Cournot Bertrand Friedman G P - q G P - p model preference (1.4, .35) .30 8.43 10.18 .78 3.38 Mx > M 4 > M5 > M 2 > M3 (1.4, .50) .31 9.61 9.02 .48 3.94 M i > M4 > M5 > M3 > M2 (1.4, .75) .42 10.97 7.88 .24 4.59 M 4 > Mx > M5 > M3 > M2 (1.7, .35) .64 12.44 6.83 .14 5.45 M 4 > Mx > M5 > M 3 > M2 (1.7, .50) .95 14.17 5.73 .08 6.43 M4 > Mx > M3 > M 5 > M2 (1.7, .75) L44 16.17 4.71 .14 7.58 M 4 > Mi > M3 > M5 > M2 (2.0, .35) 1.74 17.21 4.27 .28 8.28 M 4 > Mi > M3 > M5 > M2 (2.0, .50) 2.47 19.61 3.33 .57 9.80 MA > Mi > M3 > M 5 > M2 (2.0, .75) 3.46 22.39 2.54 1.05 11.57 M 4 > M 3 > Mi > M5 > M2 Table 5.22: Sensitivity Analysis of Incompatibility of Prior and Sample, United M i M 2 M 3 M 4 M 5 Cournot Bertrand Friedman G P - q G P - p model preference (1.4, 35) .55 10.41 8.69 .79 3.38 M a > M 4 > M5 > M3 > M2 (1.4, 50) .65 11.70 7.60 .57 4.04 M 4 > Mi > M5 > M3 > M2 (1.4, 75) .87 13.26 6.47 .42 4.88 M 4 > Mi > M 5 > M3 > M2 (1.7, 35) 1.45 15.35 5.55 .40 5.74 M 4 > Mi > M3 > Ms > M2 (1.7, 50) 1.90 17.25 4.56 .47 6.88 M 4 > Mi > M3> M5 > M2 (1.7, 75) 2.57 19.56 3.59 .70 8.31 M 4 > Mi > M3 > M5 > M2 (2.0, 35) 3.30 21.25 3.36 .96 9.06 M 4 > Mi > M3 > M 5 > M2 (2.0, 50) 4.23 23.87 2.60 1.44 10.79 M 4 > M3> Mi > M5 > M2 (2.0, 75) 5.50 27.07 1.93 2.19 12.95 M3 > MA > Mi > Ms > M2 Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 140 elasticity estimates take non-extreme values, whereas it comes second in the remaining 4 extreme instances. These may suggest that the (relative) performance superiority of the GP-quant i ty model for this sample appears reasonably robust to variations in elasticity estimates. Analysis in the remaining section involves changing the unit lengths of a strategy period. Observation of the airline industry suggests that carriers might choose capacity on a semi-annually basis with a "high demand season" and a "low demand season" in each model year. The high demand season consists of the second and third quarters of a year, whereas the low demand season consists of the first quarter of a year and the last quarter of the previous year. For our sample data, 2Q85-3Q85 and 2Q86-3Q86, denoted 85-85 and 86-86 respectively, are the two high demand seasons, whereas 4Q84-1Q85 and 4Q85-1Q86, denoted 84-85 and 85-86 respectively, are the two low demand seasons. The only single quarter of the sample, 4Q86, is not included in the following analysis. The quantity volumes of a season are obtained by adding the volumes of its two component quarters, while the prices (and marginal costs, respectively) of a season are calculated as a weighted average of the two quarterly prices (marginal costs, respectively), using the volumes of the quarters as the weights. Table 5.23 reports the estimated mean conduct parameters for each airline and each season using the base-case elasticity estimates, along with the standard errors of the means and 95% confidence interval for each mean. Table 5.23 is similar to Table 5.8 (the latter uses the quarterly data). Table 5.23 shows that among the sample seasons, American mean conduct resembles the Cournot conduct in the two middle seasons, and appears more collusive than Cournot in the first and last seasons (especially, the last season). United mean conduct is close to the Cournot value in the 85-85 season, while more collusive in the other seasons. Based on this information, we may classify the semi-annually regime switch patterns for each airline as (recalling that N and C represent the Nash competitive regime and the collusive Chapter 5. Strategic. Stability of the Airline Industry: An Empirical Study 141 Table 5.23: Estimated Mean Conduct Parameters: Semi-Annually American United season mean st. error 95% conf. int mean st. error 95% conf. int low 84-85 high 85-85 low 85-86 high 86-86 .26 .14 (-.03, .54) -.04 .10 (-.24, .16) .07 .12 (-.19, .33) .49 .16 (.15, .83) .33 .15 (.01, .64) .08 .20 (-.35, .50) .57 .19 (.17, .98) .37 .16 (.03, .71) regime respectively): (C N N C) for American, and (C N C C) for United. The degree of collusion in the collusive regime appears slightly lower than p = .50, the degree of collusion when the quarterly data are used. For the sake of illustration and comparison we would continue to use p — .50. We now report the semi-annually results based on the Bayesian analysis of firm con-duct discussed earlier. The posterior mean conduct parameters for each airline and each season, along with 95% confidence interval for each (posterior) mean, are reported in Ta-ble 5.24 for each of our five models. Table 5.24 is similar to Tables 5.12 - 5.16 (the latter use the quarterly data). Table 5.24 shows that the Bertrand conduct appears unlikely to be a good description of the semi-annual data from the posterior point of view. The 95% posterior confidence intervals do not include the Bertand conduct of -1, no matter which priors one starts with . In the similar sense but to a lesser degree, the Friedman and GP-pr ice models also appear not favored by the data. One important, implication of Table 5.24 is that inferences concerning the relative accuracy of the Cournot model and the GP-quant i ty model is more fragile, or less robust, based on the semi-annual data than based on the quarterly data. We see from Table 5.24 that, not only the GP-quant i ty model, but would the Cournot model yield 95% posterior confidence intervals which contain their respective prior assessments. In other words, a person who holds the Cournot prior could also find "convincing" evidence that firms behave in the Cournot fashion, just as a GP-quant i ty follower could find similar evidence Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 142 Table 5.24: Bayesian Posterior Conduct Parameters: Semi-Annually American United Cournot Cournot prior mean 95% conf. mt prior mean 95% conf. int 84-85 0 .13 (-.07, .33) 0 .16 (-.05, .38) 85-85 0 -.02 (-.15, .11) 0 .04 (-.24, .32) 85-86 0 .04 (-.14, .21) 0 .29 (-.00, .58) 86-86 0 .24 (.00, .49) 0 .18 (-.05, .42) Bertrand Bertrand prior mean 95% conf. int prior mean 95% conf. int 84-85 -1 -.37 (-.67, -.08) -1 -.34 . (-.65, -.02) 85-85 -1 -.52 (-.74, -.30) -1 -.46 (-.80, -.12) 85-86 -1 -.46 (-.72,-21) -1 -.21 (-.60, .17) 86-86 -1 -.26 (-.60, .09) -1 -.32 (-.65, .02) Friedman Friedman prior mean 95% conf. int prior mean 95% conf. int 84-85 1 .63 (.39, .86) 1 .66 (.42, .90) 85-85 1 .48 (.25, .71) 1 .54 (.21, .86) 85-86 1 .54 (.30, .78) 1 .79 (.51, 1.07) 86-86 1 .74 (.50, .99) 1 .68 (.43, .94) GP-quant GP-quant prior mean 95% conf. int prior mean 95% conf. int 84-85 .5 .38 (.18, .58) .5 .41 (.20, .62) 85-85 0 -.02 (-.15, .11) 0 .04 (-.24, .32) 85-86 0 .04 (-.14, .21) .5 .54 (.27, .81) 86-86 .5 .49 (.27, .72) .5 .43 (.20, .66) GP-pr ice GP-price prior mean 95% conf. int prior mean 95% conf. int 84-85 0 .13 (-.07, .33) 0 .16 (-.05, .38) 85-85 -1 -.52 (-.74, -.30) -1 -.46 (-.80, -.12) 85-86 -1 -.46 (-.72, -.21) 0 .29 (-.00, .58) 86-86 0 .24 (.00, .49) 0 .18 (-.05, .42) Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 143 that firms behave according to the GP-quanti ty model. Consequently, neither Cournot followers nor GP-quant i ty followers would find their prior beliefs significantly affected by this semi-annual data. A n important reason for the relative fragility of the semi-annual data is the "average effect". Consider United Airlines in the 85-86 season, which consists of the fourth quarter of 1985 and the first quarter of 1986. From Table 5.8 United conduct is fairly competitive in 4Q85 (the estimated mean conduct parameter of .22 is fairly close to the Cournot value), while it is very collusive in 1Q86 (the estimated mean conduct parameter of 1.04 is remarkably close to the cartel or Friedman value). Use of the semi-annual data has, however, "averaged" the two "opposite" degrees of firm conduct, yielding the mean conduct parameter of .57 for the season (Table 5.23) which is somewhat between the Cournot and cartel values. As a result, the Cournot prior would induce a posterior which is close to the Cournot value while the cartel prior would induce a posterior which favors the cartel value. Indeed, Table 5.24 shows that both priors yield 95% posterior confidence intervals which include their respective prior values. This average effect thus makes it less likely to use data to distinguish between competing, and close, hypotheses from the posterior point of view. For our case, this average effect is also evident in the 84-85 season for both airlines. 5.9 Conclusion B y using a data set of Chicago-based duopoly airline routes from the fourth quarter of 1984 to the fourth quarter of 1986, we find some evidence that airlines' competitive behavior may switch between the competitive and collusive regimes. "Price wars" appear to occur between the second quarter of 1985 and the first quarter of 1986. Air l ine profits in the collusive regime appear less than the (single-period) monopoly profits. The degree Chapter 5. Strategic Stability of the Airline Industry: An Empirical Study 144 of overall market competitiveness for this set of data is beteen the Cournot and monopoly solutions but closer to the Cournot solution. The data also suggest that major carriers might use quantity volumes, rather than prices, as their choice variables. We have seen that, of the five oligopoly models considered, the quantity version of the Green-Porter model seems most favored by our base-case results. Only those following the GP-quant i ty model could find convincing evidence that firms behave in the G P -quantity fashion. Followers of the other models might change their prior beliefs after seeing the base-case data. Nevertheless, the Cournot model also performs reasonably well and the base-case results are not robust enough to produce a consensus among all the five models. The sensitivity analysis shows that our basic results appear not very sensitive over the ranges of plausible elasticity estimates. On the other hand, we find that inferences concerning the relative accuracy of the five models (in particular, of the GP-quanti ty model and the Cournot model) is more fragile, or less robust, based on the semi-annually data than based on the quarterly data. Appendix A Some Proofs The proofs of Lemmas 4.2, 4.3, Proposition 4.6, and Inequality (4.25) follow. Proof of Lemma 4.2: Suppose first that both.firms delegate with Xk = = A. If the agent decides to participate, he will become a common agent and consequently attract all the consumers in the industry (Proposition 4.3). W i t h an industry demand of xd(l), his profit is UQ = Xxd(l) — V, which is non-negative iff X > X. Next suppose that only firm k ( k = l , say) delegates. If the agent decides to participate, he wi l l become an exclusive agent of firm 1. As discussed, consumers whose preliminary demand is product 1 are indifferent between approaching firm 1 and approaching the agent. Let # n and #io (> 0) be, respectively, the proportions of x\ approaching firm 1 and approaching the agent: 8U +O10 = 1- W i t h an industry demand of xd(0), the agent's expected profit, UQF can be written as follows. I L ^ = XE{xd) - E(T0) = E(xd)(X - T/xd(0)), where condition (4.16) and Lemma 4.1 have been used and E(x%) = E(xpQ) = E{6l0x\) = 0w6iXd(O) > 0. It is clear that IT5 > 0 iff X > T/xd(0). Finally, suppose that the other firm delegates with different commission rates. Then, after taking the second stage game, the agent wi l l actually be an exclusive agent of the firm offering h im a higher rate. Thus, only the higher rate is relevant to the agent's participation decision. Apply ing the foregoing results completes the proof. Q .E .D. Proof of Lemma 4.3: From Lemma 4.2 common agency is viable iff A > A-Suppose A > A and suppose that firm j has decided to delegate. If firm k (k ^ j) chooses 145 Appendix A. Some Proofs 146 delegation, then the channel structure wil l be the common-agent channel. The expected payoff to firm k wil l be Ilf = (Pk — X)9kXd(l), where pk > 0 denotes the price of product k, A: = 1, 2. If firm k instead chooses not to delegate, the channel structures wi l l be either the exclusive channel or the exclusive-agent channel. Under either channel structure, firm k wil l expect the same payoff, namely, I l f = pk9kXd(0) — 9kT. F i r m k wil l choose delegation if and only if the benefit from delegation is at least equal to the benefit from non-delegation, i.e., LT^ > I l f or A < A f c . Thus, given A < A < A, delegation is a Nash equilibrium. Q .E .D. Proof of Proposition 4.6: Given A, common agency can be an equilibrium iff A 5: A < A (Lemma 4.3). Now we prove that, given A, common agency is the unique equilibrium iff A < A < T/xd{0). The game with T/xd(0) < A < A is depicted in Table A.1(a). As was known, delegation, denoted A, is a Nash equilibrium of the game. Suppose that one firm (firm 2, say) has chosen not to delegate, denoted F. Then if firm 1 chooses F, it wi l l obtain I l f . If instead firm 1 chooses A, then the agent wi l l accept the delegation (Lemma 4.2) and the sales outlets wi l l be firm 1, firm 2, and the outside agent as an exclusive sales agent of firm 1. As in the proof of Lemma 4.2, we let 0 n and 0 1 O (> 0) be, respectively, the proportions of x\ approaching firm 1 and approaching the agent: 9\\ + #10 = 1- W i t h industry demand xd(0), the expected profit to firm 1, UAF can be computed as follows. UAF = (piE(xd) - £ ( r i ) ) + ( P l -X)E(x d) = I l f - 0 l o0i(Xx d (0) - T), where ^ ( T j ) = E(xd)T/x(0) (using condition (14)), E(xd) - 91091xd(0), and E(xd) = E(x\) - E(xd) = 91161xd(0). Since 61Q > 0 and Xxd(0) - T > 0, it follows that TlAF < I l f . Thus, firm I's best response to firm 2's choice of F is F. In other words, (F, F) is another (and the only other) Nash equil ibrium of the game. Turn to the game with A < A < T./xd(0), depicted in Table A.1(b). Suppose that firm 2 has chosen F. If firm 1 chooses to delegate, the agent w i l l not accept the delegation Appendix A. Some Proofs 147 Table A . l : Payoff Matrices of the Games (a) T/xd(0) < A < A firm 2 firm 1 F A F nf ,n? n f , n £ F A n f ^ i r f (b) A < A < T/xd(0) firm 2 firm 1 F A F n f , n f n f , n f A n f , n f (Lemma 4.2) and the sales structure uses exclusive channels. Therefore, whether firm 1 delegates or not, the sales structure remains the same and firm 1 expects the same payoff. Consequently, delegation is a dominant strategy for both firms. Final ly , consider the choice of A by the regulatory authority. From the above dicus-sions, the equilibrium channel structure is a function of A: ( F , F ) if A < A or A > A; ( A , A ) if A < A < T/xd(0); and ( A , A ) or ( F , F ) if T/xd(0) < A < A. Aggregate consumer surplus is us(0) under (F ,F ) and us(l) under ( A , A ) . Since us(l) > us(0) (Corollary 4.1), the regulatory authority wi l l set A < A < T/xd(0). Q.E.D. Proof of Inequality (4.25): The industry profit function is (from equations (4.18) and (4.19)) 2 2 Q U = ^ Uk = YiPk0kxd(p, + -^rS,P2 + rS), k=i k=i \ V2 / where 9\ = 9(p2 — P i ) and 92 = 1 — 9(p2 — pi). It follows that dU i an + I an W l i th dxd (dxd/dPl)dPl (dxd/dp2) dp an n . dxd ,89 n dxd. . 39, — = 9kxd + P k 9 k — + pkxd— + Pj03~ + P]xd—L oPk opk dpk dpk dpk Appendix A. 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