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Noncooperative games of information sharing and investment : theories and applications Kao, Jennifer L. 1991

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NONCOOPERATIVE GAMES OF INFORMATION SHARING AND INVESTMENT: THEORIES AND APPLICATIONS B y J E N N I F E R L . K A O B . C O M M . , The University of Alberta, 1981 C A . The Institute of Chartered Accountants of Alberta, 1983 A THESIS S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y i n T H E F A C U L T Y O F G R A D U A T E S T U D I E S Faculty of Commerce and Business Administration We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A June 1991 ® Jennifer L . Kao, 1991 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British 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. Department of Accounting The University of British Columbia Vancouver, Canada D a t e August 1 9 . 1991 DE-6 (2/88) ABSTRACT ii The thesis considers the effects of public policy with respect to disclosure on rivalrous competition in duopolies. Principal contributions to the stochastic oligopoly theories of information sharing include imbedding an investment stage in the standard noncooperative two-stage information sharing/output (pricing) games under a wide range of assumptions about effects of investment or distributions of returns, and investigating the extent to which altering the usual preference assumption to allow risk aversion may affect information choice. At an information sharing level, it is established that, besides being sensitive to the type of competition and the nature of information asymmetries, as previously reported in the literature, private incentives to disclose also depend importantly on both the impact of investment and risk preferences. In this regard, equilibrium levels of investment are shown to be affected nontrivially by variance effects from investment, risk attitude of decision makers, as well as the type of industry rivalry. Also characterized are the welfare orderirtgs obtained as a consequence of these varying disclosure effects on the decisions and hence profits of competing firms. If alternative accounting practices are distinguishable along the informativeness dimension, then, at an abstract level, the analysis can be seen as approaching the study of many accounting issues from a distinctly different perspective than the familiar tax or wealth transfer viewpoints. The principal contributions to the accounting literature include iii identifying the impact of disclosure rules or rule changes on industry, consumers, and society as a whole, and offering insights to the accounting policy makers concerning the results of disclosure regulation. T A B L E OF CONTENTS iv Page ABSTRACT ii LIST OF FIGURES vii ACKNOWLEDGEMENTS viii CHAPTER 1. INTRODUCTION 1 CHAPTER 2. INFORMATION SHARING AND INVESTMENT IN A RISK NEUTRAL WORLD 2.1 Opening Remarks 5 2.2 Literature Review 7 2.3 Description Of Three-Stage Information Sharing/Investment/Output (Pricing) Games Given Two-Sided Information Asymmetry 9 2.4 Model For Three-Stage Information Sharing/Investment/Output (Pricing) Games 15 2.5 Equilibrium Outcomes From The Third-Stage Output (Pricing) Game 19 2.6 Equilibrium Outcomes From The Second-Stage Investment Game 23 2.7 Equilibrium Outcomes From The First-Stage Information Game 25 2.8 Welfare Analysis 31 2.9 . Summary And Discussion 36 2.10 Model For Two-Stage Information Sharing/Output (Pricing) Games Given One-Sided Information Asymmetry 38 CHAPTER 3. INFORMATION SHARING AND INVESTMENT IN A RISK AVERSE WORLD 3.1 Opening Remarks And Literature Review 45 3.2 Model For Two-Stage Information Sharing/Output Games 46 3.3 Analysis 49 3.4 Model For Two-Stage Investment/Output Games 56 3.5 Summary And Discussion 63 CHAPTER 4. ACCOUNTING APPLICATIONS 4.1 Opening Remarks 66 4.2 Accounting Policy-Induced Changes To The Level Of Investment 67 4.3 Self-Selection Of Accounting Methods And Information Sharing 71 4.4 Lobbying Efforts And Information Sharing 75 4.5 International Accounting Standards And Industry Differences 76 4.6 Alternative Channel For Information Exchange 78 4.7 Caveats In Applying The Theory To The Study Of Accounting Method Choice 79 4.8 Caveats In Applying The Theory To The Empirical Domain 82 CHAPTER 5. CONCLUSION 5.1 Concluding Remarks 85 5.2 Directions For Future Research 87 BIBLIOGRAPHY 90 FIGURE 1. THE SEQUENCE OF EVENTS AND ACTIONS FOR THREE-STAGE INFORMATION SHARING/ INVESTMENT/OUTPUT (PRICING) GAMES 94 FIGURE 2. THE SEQUENCE OF EVENTS AND ACTIONS FOR TWO-STAGE INFORMATION SHARING/OUTPUT (PRICING) GAMES GIVEN ONE-SIDED INFORMATION ASYMMETRY 95 APPENDIX 1.CHAPTER 2 PROOFS 96 APPENDIX 2.CHAPTER 3 PROOFS 124 vii LIST OF FIGURES Page The Sequence Of Events And Actions For The Three-Stage Information Sharing/Investment/Output (Pricing) Games 94 2. The Sequence Of Events And Actions For The Two-Stage Information Sharing/Output (Pricing) Games Given One-Sided Information Asymmetry 95 viii ACKNOWLEDGEMENTS I would like to express my sincere gratitude to my thesis chairman Dr. Gerald A. Feltham, for his invaluable insights and suggestions on this and earlier drafts. I also gratefully acknowledge the helpful comments of Dr. Ronald Giammarino as regards the modelling of risk averse case. I am especially indebted to my thesis co-chairman and supervisor, Dr. John S. Hughes, for his generous guidance throughout the duration of this project. Without his timely advice and unfailing support, it would have been that much more difficult to deal with some of the seemingly insurmountable problems that I have encountered. I would also like to thank the Social Sciences and Humanities Research Council of Canada and the University of British Columbia Accounting Development Fund for their financial support. A special note of thanks is extended to my sister Dr. Glenda Wenchi Kao who, by sharing with me her own personal experience, has helped me put into a proper perspective the ups and downs of a Ph.D. student life. Finally, I would like to express my heartfelt appreciation to my parents, Mr. & Mrs. Chi-Kun Kao, for being there to lend me their encouragements and moral support when I needed them most. 1-C H A P T E R 1 I N T R O D U C T I O N Over the last decade or so, there has been considerable interest in public disclosure policies. Some studies have appealed to the notions of costly contracting and political processes to explain accounting practice in the absence of an uniform measurement rule or disclosure policy.1 More recently, other studies have turned to signalling arguments for reasons behind voluntary earnings forecasts and other proprietary information releases by firms or firm managers. Typically modelled as noncooperative games among oligopolists, this second branch of research speaks to choices pertaining to a particular signal rather than a specific disclosure rule.2 In making an information choice, ex post, individual firms seek to ( balance the counterveiling incentives created by their different constituencies, while bearing in mind the tension with competitors in the imperfectly competitive product market. On the other hand, traditional information sharing models consider situations in which all firms anticipate receiving private information with respect to their own cost or an industry-wide demand realization. In these games, firms decide prior to observing the actual signal whether the information contained therein will be exchanged with rival firms. ! A comprehensive review of this so-called positive accounting or economic consequences research appears in Watts and Zimmerman [1986]. 2Darrough and Stoughton [1990], Feltham and Xie [1991], Wagenhofer [1990]. 2 Commitments to disclose are assumed to be credible and binding. These commitments determine the kind of information available to each firm at the output or pricing stage, and hence the type of competitive environment. Notwithstanding the interest it has generated in the industrial organization literature, the notion that public policy with respect to disclosure may affect or be affected by rivalrous competition in the product market has received little attention from accounting researchers.3 The purpose of the thesis is two-fold. First, it extends the information sharing literature by relaxing, either individually or collectively, standard assumptions to more closely approximate the real world. Second, it demonstrates with specific examples how insights from economics may help one gain a deeper understanding of the extant empirical as well as theoretical research relating to accounting policy issues, as regards both voluntary accounting method choice and mandated rule changes. The thesis begins with a three-stage duopoly game of incomplete information whereby investment decisions are preceded by a selection of disclosure policy and succeeded by output (pricing) choices. Adding an investment stage to the standard two-stage information sharing games does not affect earlier results, even when such investment affects both the mean and variance of returns. Bilateral (no) information exchange of firm-specific cost information continues to dominate other forms of information arrangements for all firms engaged in Cournot quantity (Bertrand price) competition. 3Studies addressing this issue include Feltham, Gigler, and Hughes [1991], and Hughes and Kao [1991]. 3 The robustness of information choice, however, does not carry over to situations wherein risk aversion is present. Given an aversion to risk, firms whose choice variable is output (price) may, in some cases, favor a policy of no (bilateral) information exchange whether both firms or only one of the firms are endowed with private information about cost. Examining the policy-induced changes to investment, it is found that equilibrium levels of investment spending are invariant to changes in reporting requirements provided duopolists are risk neutral, and only expected mean effects are present. Varying one or more of the assumptions can lead to major qualitative changes to the results. Introducing an increasing variance effect from investment causes an expansion of investment when industry-wide information sharing is replaced by no information sharing. The converse is true if, instead, a decreasing variance effect or risk aversion is assumed. With the exception of Bertrand industries producing substitute goods, the information decisions made by risk neutral firms are in general consistent with social welfare. Nevertheless, just as the information choice is sensitive to risk, these welfare results are also dependent importantly on the assumption about risk preferences. If accounting reports are a credible communication vehicle, then, to the extent that the maintained structural"and distributional assumptions represent a first-order approximation for external corporate financial reporting settings, these findings form an useful basis on which many familiar accounting policy issues may be reexamined. The importance of disclosure on competition in the product market may have significant implications for empirical accounting research. It calls to the empiricists' attention a need to redefine tests 4 designed not only to detect market reaction and discern altered behavior in relation to an impending rule change, but to explain prevailing accounting practices prior to official policy pronouncements. From the accounting policy makers' viewpoint, results from present investigation provide guidance as to when a laisser-faire policy by the SEC or FASB may be warranted, and when an intervention by those authorities may be called for. The remainder of the thesis is organized as follows. Chapter 2 presents a formulation of noncooperative games of information sharing and investment given risk neutrality, along with the main results under a wide range of assumptions about investment and competition. A parallel analysis in a risk averse setting follows in Chapter 3. Chapter 4 discusses potential accounting applications of the above results, including a reference to the extant studies of accounting method choice. Closing remarks and a brief layout of directions for future research appear in Chapter 5. 5 CHAPTER 2 INFORMATION SHARING AND INVESTMENT IN A RISK NEUTRAL WORLD 2.1 Opening Remarks A hallmark of oligopolistic industries is the presence of strategic interaction among rival firms. Using game theory to study these strategic interactions has become a topic of considerable interest in the field of industrial organizations. These games are in general multistage. In the initial stage, firms make commitments to undertake some irreversible strategic actions which influence the competitive environment for a subsequent stage. Included in this literature4 are two-stage investment/output games and, of particular interest here, two-stage information sharing/output (pricing) games. Applications in the first category are typically modelled as games of complete information in which one or more firms have the opportunity in the first stage to undertake an investment that brings about a predictable reduction in their unit production costs one stage hence. In the second category, elements of uncertainty regarding industry-wide or firm-specific conditions are considered, making these games of incomplete information. In the usual model, rival firms reveal or withhold private information obtained as a consequence of the first stage. Firms then compete in the next stage by choosing outputs "See Shapiro [1987] for a survey of such applications. 6 or prices conditional on the information, if any, pursuant to commitments made available in the previous stage. The extant stochastic oligopoly theories of information sharing can be extended along several dimensions. This chapter begins the investigation by combining both lines of research cited above in a fashion depicted in Figure 1. The economic motivation for adding an investment stage to the standard two-stage information sharing/output (pricing) game comes from the effect that levels of investment are likely to have an impact on privately observed marginal costs of production. By shifting marginal cost distributions, investment decisions have the potential to alter the welfare implications of information sharing in a fashion similar to a change from linear to convex cost functions. Relating the characterization of equilibrium investment decisions to accounting context, one may view such an addition as an attempt to address some of the issues raised by Wolfson [1980] in his discussion of a study by Horwitz and Kolodny [1980] regarding the impact of SFAS No. 2 on levels of investment in research and development (R&D). In that pronouncement, the FASB took away the option by firms to capitalize R&D costs, and required that such costs be written off immediately. One of the findings reported by Horwitz and Kolodny was that, pursuant to SFAS No. 2, R&D spending did go down for some smaller high technology companies.5 However, Wolfson [1980] observes that empirical research similar to this one, designed to discern altered investment behavior 5This result was consistent with that found in Elliott, Richardson, Dyckman and Dukes [1984]. 7 surrounding a new policy pronouncement, fails to consider the consequences of such alterations from an equilibrium perspective and as a result generally does not explain why investment levels might change. 2.2 Literature Review Since Spence's 1977 paper " Entry, Capacity, Investment and Oligopolistic Pricing ", strategic investment models in oligopoly theory have focused on the role of irreversible investment as a means to deter entry or expand market share within the context of complete information. The focal point of these studies is to contrast the levels of investment derived from strategic versus non-strategic models. When entry deterrence is the object of interest, there is a strategic incentive to over (under) invest given Coumot quantity (Bertrand price) competition (Dixit [1979,1980], Schmalensee [1983], Shapiro [1987], Ware [1984]). A similar conclusion is also reached in a simultaneously played two-stage investment/output game where a firm's primary concern is to enhance its market power (Brander and Spencer [1983]). These models differ importantly from the one employed in the thesis in that they assume marginal costs to be known deterministic functions of investment, thereby implying no informational issues. On the other hand, the oligopoly literature on incomplete information with respect to uncertain industry-wide or firm-specific parameters has been principally concerned with the incentives among oligopolists to share privately observed signals. When the stochastic element relates to firm-specific cost, it is common practice to assume that private signals 8 are independently or non-negatively correlated across firms, and that every signal perfectly reveals a parameter of the model distinct for each firm. Most studies have characterized the information game as a choice between releasing either all or none of the information contained in the signal (Li [1985], Shapiro [1986]). Others, however, have allowed firms to endogenously determine a garbling strategy whereby their imperfectly observed private signals may be partially transmitted to rival firms (Darrough [1990], and Gal-Or [1986]). Notwithstanding these differing assumptions, all four studies have concluded that bilateral information exchange is the dominant information choice of all firms engaged in Cournot competition in the production of substitute goods. However, with Bertrand competition, firms are shown to reverse their previous decision and, instead, favor a policy of complete privacy (Darrough [1990], and Gal-Or [1986]). By comparison, when the underlying uncertainty originates from the demand side of the market, firms are generally assumed to receive a noisy signal with respect to a parameter of the model common to all firms. Once again, studies vary in how they model the transmission of information about demand,6 and whether these private signals can differ in the degree of precision7 across firms. Despite the divergent assumptions, no information exchange has prevailed as an industry-wide dominant information choice so long as firms behave in a Cournot fashion, 6Clarke [1983a,1983b,1985], Kirby [1988], for instance, require either complete or no information exchange. Conversely, partial revelation of private demand signals is permitted in Darrough [1990], Gal-Or [1985], Li [1985], and Vives [1984]. divergence in the degree of precision is modelled in Clarke [1983a,1983b,1985], and Vives [1984]. 9 produce substitutes, and have production technologies displaying constant returns to scale. However, as Kirby shows, relaxing the linearity assumption may result in a reversal in preference over alternative disclosure regimes. As before, changing the form of competition from Cournot quantity to Bertrand price competition induces firms to revise their information decisions, in this case, to favor bilateral exchange (Vives [1984]). 2.3 Three-Stage Information Sharing/Investment/Output (Pricing) Games Given Two-Sided Information Asymmetry 2.3.1 The Sequence Of Events And Actions Consider a three-stage information sharing/investment/output (pricing) game which unfolds in the manner described below. In the first stage, firms decide whether they will commit themselves to subsequent information exchange with rival firms. This is followed by an independent selection of investment by both firms in the second stage. It is assumed throughout the thesis that the pair of actual investment outlays are public knowledge even though, given Cournot (Bertrand) behavior, results will continue to hold in the absence of such an assumption. Next, each firm observes a signal which discloses with certainty the consequences of its investment on the firm's own marginal cost. This information is then released or withheld in accordance with the decision reached at the beginning of the game. On the basis of information they have obtained, firms proceed to make an output or pricing choice in the final stage. Consistent with the convention in the information sharing literature, the thesis relies 10 importantly upon the following assumptions. First, in contrast to the ex post voluntary disclosure literature, the ex ante commitment to disclose following the receipt of private signal is assumed to be enforceable. Second, the reporting of firm-specific cost information must be truthful, i.e., costlessly verifiable. Third, the credibility of ex post voluntary disclosure can only be established at a prohibitively high cost to the disclosing firm. Fourth, firms behave non-cooperatively at every stage of the three-stage game. 2.3.2 Information Structure And Equilibrium Concept For The Third-Stage Output (Pricing) Game  This stochastic duopoly is modelled as a Bayesian game. The game's information structure describes the extent of each firm's private knowledge about a firm-specific stochastic cost component which characterizes the environment, and about its rival's investment. The game's exogenous random variables are Zj e Z, i = 1,2, where Z represents the set of all possible realizations. Prior probability distributions over Z are common knowledge and are assumed to be normal,8 zs - N(0,a2), i = 1,2. This specification implies that the Normal distributions have been assumed in most of the literature (Clarke [1983a, 1983b,1985], Darrough [1990], Gal-Or [1985,1986], Kirby [1988], Vives [1984]). However, normality is not necessary for the results in this chapter to hold. As pointed out by previous studies, an affine information structure in which conditional expectations obey a linear property will suffice. This stronger assumption is maintained to simplify the mathematics once risk aversion is introduced in Chapter 3. 11 stochastic component of each firm's cost is drawn from an identical distribution,9 and that the cost distribution for firm i is normal.10 According to the game structure, each firm receives a perfect signal regarding its uncertain marginal cost of production. These private signals are assumed to be independently distributed across firms. Upon learning z^  firm i's best estimate of its opponent's private signal Zj is given by E(Zj | z() = 0,11 i,j = 1,2, j ^ i. Cournot-Nash (Bertrand-Nash) equilibria12 are considered under three information structures: (a) Both firms publicly report their private cost. (b) Both firms do not publicly report their private cost. (c) Only one firm chooses to publicly report its private cost. These cases are referred to as bilateral information exchange, no information exchange, and 9By focusing on the symmetric information case and by further requiring prior means to equal zero, subsequent notation is greatly reduced without affecting results qualitatively. c C 10One implication of the normality assumption is that marginal costs may become negative. This can be mitigated if marginal costs are assumed to center around a large positive mean relative to their variance. "In general, E(Zj | z{) = E(Zj) + (p)(aj/ai)[zi - E(zj)]. But, by assumption E(Zi) = E(Zj) = 0, 0"j = 0"j = a , and p = 0, where p is the correlation coefficient between signals Z; and Zj, i,j = 1,2, j iM. Hence, firm i's best estimate of firm j's signal is equal to 0. 12The relevant equilibrium concept for case (a) is Nash in that both firms have perfect and complete information about every aspect of the game. The final two information structures, however, call for a stronger equilibrium concept, i.e. Bayesian-Nash, because in those cases the output (pricing) stage is characterized by simultaneously-played games of incomplete information. 12 unilateral information exchange, respectively. The equilibrium solutions for regimes (a) and (b) are labelled by superscripts "dd" and "nn", respectively. In the case of unilateral information exchange, equilibrium solutions are superscripted by "dn" ("nd") when firm 1 (2) chooses to disclose without a reciprocal action by firm 2(1). All the prior information with respect to the stochastic cost components of the model is represented by a vector 0. In the symmetric case, this vector consists of the prior mean and variance, i.e., 0 = [0, o2]. Firm i's information set, C O ; e Q.^ where Q, is its information space, consists of this commonly known vector 0, the publicly reported equilibrium levels of investment (x^Xj), firm i's own private cost signal zi5 and any information transmitted by its rival. Since, irrespective of the sharing arrangements, both © and (xj,x2) are available to all firms at the beginning of final stage, they will henceforth be suppressed in future reference to information sets. Under bilateral information exchange, the information sets are homogeneous and contain Zj and z2. However, when firm I's decision to share is not met with a similar exchange by firm 2, their information sets become CO, = [z,] and c o 2 = [Zj.zJ, respectively. Finally, if both firms choose a policy of no information exchange, then they face C0i = [z,] and a>2 = [zj, respectively. From the earlier description of the model, it is clear that the third-stage output (pricing) decisions must be conditioned on an individual firm's own information and the game's information structure. Denote firm i's output (pricing) strategy by yt = y^cOj) (pj = YJ(COJ)), where y{ e Y; (pj e Pj), the set of all possible outputs (prices), and Y e r;, the set of all decision rules which map the information space Qj into the decision space Y ( (Ps). 13 Under the assumption of risk neutrality, the decision problem facing firm i, i = 1,2, at the output (pricing) stage can be formulated as: Max Ji(Yi,Y 2,x„(Oi) (1) where J^Yi^.*! . 0 ^) is firm i's expected payoff conditioned on its information set 0} and investment level x;. If firm j discloses, then Ji(Yi>Y2.xi,coi) = n1(Yi(o>i),Y2(o)2).Xi,zi), i = 1,2, (2) where 7ti(Y1(co1),Y2(c02)>xi,zi) is firm i's end-of-period payoff13 given actual output production (or prices), yt and y2 (or p, and p2), investment level xi5 and cost component z;. On the other hand, if firm j does not disclose, then Ji(Yi.Y2.Xi,a>i) = 17ti(Yi(coi),Yj(cOj),xi,zi)f(zj)dzj, (3) i,j = 1,2, j * i. A set of output (pricing) strategies [*Yi»*Yj defines a Bayesian-Nash equilibrium if Ji(*YI.*Y2.XI.G>I) ^ Ji(Yii*Y2.Xi,a),) V y, e T, , and (4) J2OY1.Y2.X2.GJ2) ^ J 2(*Yi .Y2. 2) v Y2 e r 2 . 'Full specifications of profit functions nif i = 1,2, are presented in Equation (14). 14 2.3.3 Information Structure And Equilibrium Concept For The Second-Stage Investment Game  Prior to making investment decisions, neither firm is aware of either its own or its rival's eventual cost realizations. However, having committed itself to an information regime at an earlier stage, it does know what kind of information will become available to each firm at the conclusion of the present stage. Based on this understanding, both players choose a level of investment so as to maximize their expected profits at the beginning of second stage: Max Jj(x) (5) where J^x) = E(0i[Ji(*Y1(x,co1),*Y2(x,co2),xi,coi) | x] and x = (x1;x2). A pair of equilibrium investment outlays ("x^ 'xj) is Cournot-Nash (Bertrand-Nash) if JjCX/xa) > J^x^'xj), V x, € X! (6) J2(*Xj,*x2) > J2(*Xj,X2), V x2 e X 2 where X;, i = 1,2, is the set of all possible levels of investment for firm i. Once the equilibrium output (pricing) strategies and optimal investment choices have been determined, pairwise comparisons of ex ante profits, tJi(tx),14 across three information structures reveal the type of disclosure favored by both duopolists at the start Superscript "t", t = dd,dn,nd,nn, is used as a generic representation of equilibrium strategies (outcomes) obtained. 15 of information stage. Implicit in this discussion is, of course, the notion that the appropriate solution technique for the overall three-stage information sharing/investment/ output (pricing) game is backward induction.15 2.4 The Model For Three-Stage Information Sharing/Investment/Output (Pricing) Games Given Two-Sided Information Asymmetry 2.4.1 Cournot Duopoly Market Setting Firms in this Cournot duopolistic industry produce a homogeneous good. The inverse demand function for this product is assumed to be linear of the form:16 p(Y0) = a - b(y, + y2), (7) where lower case y; denotes firm i's output production, upper case Y 0 represents the aggregate industry output, and "a" is the demand intercept known by all firms. Following the convention of several earlier studies (Fried [1984], Shapiro [1986]), the deterministic slope "b" for the inverse demand function is set equal to 1 . The total cost functions facing the firms are:17'18 15This approach ensures subgame perfection because for any given pair of investment and marginal cost realizations, equilibrium strategies for the three-stage game are also Bayseian-Nash at the output (pricing) stage. 16Linear inverse demand functions imply linear reaction functions. 17This specification encompasses cases wherein total cost functions are linear, convex, or concave, and scale returns are constant, decreasing, or increasing in own level of output production. 16 Q = Ky 2 + + xh i = 1,2, (8) where K is some constant, Xj is firm i's level of investment, and C i = u(Xi) - vCxX i = 1,2. (9) It is further assumed that the function u.(Xj) has the following properties: p.'(Xi)<0, u"(Xi) > 0, i = 1,2. (10) Three distinct assumptions about the variance effect from investment, as captured by the function, v(x;), i = 1,2, are made. (a) The Benchmark Case v(Xj) = 1, v'(Xi) = 0, [BVarCqVaxJ =0. dD Investment does not affect the variance of cost distributions. As in the studies of information sharing, Q, i = 1,2, is assumed to contain all the relevant information concerning input costs and production functions through cost minimization. 17 (b) The Variance Increasing Case v(Xi) > 0, v'(Xi) > 0, v"(Xi) < 0, (12) [aVar(Ci)/8xJ = 2v(xi)v'(xi) > 0. Investment raises the uncertainty regarding cost. (c) The Variance Decreasing Case' .20 v(: x;) = [1/C(xi)] > 0, C'(x;) > 0, V'(x) < 0, (13) [8Var(c;)/a: »xj = [^CXx^/C^xJ] < 0. Investment lowers uncertainty regarding cost. The scope of investigation defined above is much broader than that followed in the literature in that total cost functions, in addition to -being linear,21 can be convex or concave in output production, and cost variances can be increasing or decreasing, as well as constant in the level of investment. This broader scope enables one to enquire into the robustness of previous results on information choice. 1 9An example of stochastic processes described by this cost specification can be found in Bernoulli experiments. When n identical experiments are independently and successively carried out, it can be easily verified that the variance of the resultant Binomial distribution, given by nx(l - x)82, is decreasing in n. Note that x is the probability of success in each trial, and 8 is the unit cost savings associated with each success which, for illustrative purposes, is assumed to be (l/n)(1/4). 20If experiments are, instead, undertaken simultaneously, then the variance of these n independent but identical Bernoulli trials becomes xn(l - xn)82, which is decreasing in n provided x is less than (l/2)(1/n). Both x and 8 are as defined in footnote 18. Only one other study by Kirby.[1988] departs from this structural assumption to examine the consequences of making Cournot duopolists face convex total cost functions. 18 Finally, the end-of-peripd profit functions are given by: = [p(Yo)]y, - C, - Xj, i = 1,2. (14) 2.4.2 Bertrand Duopoly Market Setting An investment stage also can be added to the standard two-stage information sharing/pricing game (Gal-Or [1986], Vives [1984]). Investment is relevant in industries where firms compete in price because investment, by lowering cost of production, may allow firms to take a more aggressive pricing stance relative to their opponents. Consider a Bertrand duopoly wherein each firm produces a differentiated good.22 Both the inverse and the direct demand functions for these products are linear, and they take the form in (15) and (16), respectively: Pi(Y 0) = a - btfj - b 2y j ; b, > b 2 > 0, (15) y, = [a/0), + b2)] - [b.Ab,2 - b22)]Pi + [b2l(bx2 - b22; :)]Pj (16) where a and bj, i = 1,2, are common knowledge. Alternatively, demand can be expressed as: ft s a - P i P i + P 2 P j j ' (17) 22Contrary to the earlier Cournot duopoly model, product differentiation is required here because, otherwise, optimal pricing strategies will be trivially given by individual firms' marginal cost of production. 19 where a = [a/ib, + b2)], p, = [b./Cb,2 - bj)], and (32 = [^/(b,2 - b22)], i,j = 1,2, j * i with a > 0, pj > P2 > 0. It follows from the last two inequalities, describing the sign and relative size of Pt and P2, that the model assumes the products to be imperfect substitutes. For analytical tractability, this part of study stays within the framework of previous literature by considering only the case of constant returns to scale and linear production functions.23 The definitions of cost components, along with the means and variances of their distributions follow directly from (9)-(13). 2.5 Equilibrium Outcomes From The Third-Stage Output (Pricing) Game 2.5.1 Cournot Duopoly Market Setting The set of first-order conditions associated with decision problem (1) is given by: [9Ji(tY1.tY2.xi)coi)/ayi] (18) = a - 2(1 + K/YM) - E(fYj I coj) - c; = 0, i,j = 1,2, j ^ i, t = dd,dn,nd,nn. In their Propositions #1 and #2, Basar and Ho [1974] show that Nash equilibrium solutions to this output game exist and are uniquely defined by strategies that are linear in Though complicated, it can be shown that results from the subsequent analysis also hold in cases where total cost functions are either concave or convex in own level of output production. 2 0 a firm's own information.24'25 In terms of the present study, these linear output strategies26 take the following form: V i = T u a + T 2 iu(X l) + T3iu(x2) + T4iv(x,)2, + T&ixfa, (19) i,j = 1,2, j + i, t = dd,dn,nd,nn. , To compute coefficients T h i, h = 1,2,3,4, i = 1,2, the conditional expectations E(tyj | C0j), i,j = 1,2, j =M, t = dd,dn,nd,nn, must be determined and they, of course, depend on the information structure. Solving for T h i under each of the three information structures leads to: Bilateral Information Exchange d dy j = [(1 + 2K)a - 2(1 + K)|i(X i) + u(Xj) + 2(1 +K)v(xi)zi - v(Xj)Zj] (20) x[l/(l + 2 K ) ( 3 + 2 K ) ] , i,j = 1,2, j ^ i . ^Profit functions along with the linearity of inverse demand functions defined earlier ensure that an unique Cournot equilibrium to the third-stage output game exists. This is because, given these two assumptions, the second-order conditions will be satisfied and, in addition, own output effect on own profit will exceed the cross effect. ^The fact that equilibrium output strategies are linear in a firm's own information is intuitive in the linear case with additive separability. 26The linearity of equilibrium outputs in the stochastic components implies that output distributions are normal and, moreover, they can be completely characterized by their means and variances. 21 No Information Exchange -y, = {(1 + 2K)a - 2(1 + K)U(X|) + u(Xj) + [(1 •+ 2K)(3 + 2K)/2(1 + K)]v(xi)zi} x [1/(1 + 2K)(3 + 2K)] , i,j = 1,2, j * i. (21) Unilateral Information Exchange ^y, = [(1 + 2K)a - 2(1 + K)U-(X , ) + u(x2) + 2(1 + K ) V ( X i ) Z i ] (22) x [1/(1 +'2K)(3 + 2K)] **y2 = {(1 + 2K)a - 2(1 + K ) U ( X 2 ) + (i(Xl) + [(1 + 2K)(3 + 2K)/2(1 + K)]v(x2)z2 - v(Xl)z,} x [1/(1 + 2K)(3 + 2K)] (23) Whereas the model reduces to the usual linearity case when the quadratic coefficient, K , equals zero, it corresponds to a technology that depicts a strictly decreasing (increasing) scale returns in output if K is strictly positive (negative). Intuitively, firm i's actual level of output production reflects, in addition, to its own cost, its opponent's marginal cost to the extent that such- information is publicly available to firm i . 2 7 Absent that disclosure however, the latter can only appear in firm i's output in an expectation sense.28 27For instance, according to Equations (20) and (23), y2 depends on the realized z, given that firm 1 is the one that discloses. 28Recall that both the prior and conditional expectations of zj( j = 1,2, j ^ i, are assumed to be zero. 22 2.5.2 Bertrand Duopoly Market Setting An important consequence of the product differentiation assumption is the propriety of a first-order approach to determining optimal prices as described below:29 = a - 28/7,(0),) + fWYj I co,) + p\c, = 0, i,j = 1,2, j ^ i, f = dd,dn,nd,nn. The attention will again be restricted to equilibrium pricing strategies linear in each firm's own information hypothesized to be: fp, = T „ a + T2,u(x,) + T3iu(x2) + T4,v(x,)Zi + T5,v(Xj)Zj, i,j = 1,2, j + i, t = dd,dn,nd,nn. The equilibrium strategies associated with each of the three information structures considered earlier are: Bilateral Information Exchange [3Ji(VY*: .Xi.cOiVdp,] (24) (25) dd 'Pi = [IW, 2 - P22)]{(2p, + p2)oc + p1p2[^i(xj) - v(Xj)Zj] + 2p12[u(xi) - v(x,)z,]}, (26) i,j = 1,2, j + i. It can be easily verified that the set of second-order conditions for the pricing game is satisfied. 23 No Information Exchange "P, = [l/(4p\2 - P22)][(2P1 + p 2)a + P^.UCXJ) + 2pi2u(xi)] - ( l ^ M X i t e , (27) i,j = 1,2, j Unilateral information Exchange ""P, = [1A4P,2 - p22)][(2p1 + p 2)a + p,P2|i(x2) + 2pi2u.(x1)] - [2f312/(4f312 - p 2 2)]v( X l) Z l ^P, = [l/(4p,2 - p22)][(2p i + p 2)a + p1p2p:(x1) + 2p12u(x2)] - (l/2)v(x2)z2 - [WWi2 - P22)]v(x1)z1 (28) (29) As in the Cournot model, firm i's pricing strategies differ across the information regimes of interest here in the way its rival firm's cost information is captured. Moreover, it should be noted that, in both markets, a similar unit improvement in own outcome realization may have differential effects on the firm's level of output production or market price depending on the information regimes in question.30 2.6 Equilibrium Outcomes From The Second-Stage Investment Game Substituting from Equations (20)-(23) ((26)-(29)) into (14) allows one to express the expected profit functions at the beginning of the second-stage as a function of the first two This observation is formalized in footnote 33. 24 parameters of the output (unit profit) distributions. Specifically, for a given pair of investment levels x = (xj,x2), they are given by: J,(x) = (1 + KMtEjVi)] 2 + [VarOYi)]} '- x, (30) Ji(x) = R1{[Ecoi(tY i-c i)]2 + [Var( tY i-c i)]}-x i 5 (31) i = 1,2, f = dd,dn,nd,nn, when the choice variable is quantity and price, respectively. Faced with the decision problem given by (5), each Cournot or Bertrand duopolist determines its own optimal investment level fXj, i = 1,2, from the following set of first-order conditions: [aJiOO/BxJ = (1 + K){2[E<oi(tYi)][3ECDi(tYi)/axJ + [3Var(tYi)/axi]} - 1 (32) = 0. [aiiW/axj = p1{2[E(0i(tYi - C . M P E ^ Y , - c,)/9xj + [3Var(tYi - c^/3xj) - 1 (33) = 0, i = 1,2, t = dd,dn,nd,nn, respectively. It follows from the expressions for the optimal third-stage strategies, (20)-(23) ((26)-(29)), that the comparative statics of expected outputs (prices31) with respect to investment, evaluated at the same levels of investment, are identical across all regimes of And hence expected unit profits, Cp{ - c), i = 1,2. 25 interest here. Thus, pairwise Pareto-orderings of the equilibrium investment levels are reduced to pairwise comparisons of output (unit profit) variances at relevant spending levels. This is now formalized.32 Proposition 1 Assume that private cost signals of both firms are independent of one another. (a) In Cournot duopolies, when the variance of marginal costs is decreasing (increasing) with investment, the equilibrium levels of investment under bilateral information exchange are less (greater) than those under no information exchange. 6 (b) In Bertrand duopolies, when the variance of marginal costs is decreasing (increasing) with investment, the equilibrium levels of investment under bilateral information exchange are greater (less) than those under no information exchange. (c) Irresepective of the nature of competition, changes in disclosure do not lead to different investment decisions when the variance of marginal cost distributions is independent of the level of investment. 2.7 Equilibrium Outcomes From The First-Stage Information Game 2.7.1 Analysis And Results From an industry standpoint, complete sharing of private cost information contributes to better coordination of output strategies, irrespective of the nature of underlying competition. If the choice variable is quantity, the resultant improvement in coordination leads to a reduced output correlation, and greater profits at an individual firm level. In the absence of information exchange, firms are forced to react to each other's expected rather See Appendix for proof. 26 than actual quantity decisions. In this case, the output correlation is prescribed by prior beliefs. At the other extreme, given access to a rival firm's private cost information, it is to each firm's own advantage to contract (expand) output production whenever its opponents report a favorable (unfavorable) cost realization, and hence produce a higher (lower) level of output. These mutually beneficial adjustments to planned output strategies, made possible by bilateral information exchange, eliminate the likelihood of over or under production by either firm compared to what is in the best interest of both parties ex post. From Equation (30), it is clear that expected profit functions are convex in an individual firm's own optimal output strategies. All else held equal, these equilibrium decisions are more sensitive to an unit change in the firm's own cost when private information is bilaterally exchanged than when it is not.33 Consequently, better (worse) than average cost savings in the former regime can be expected to be accompanied by a comparatively larger (smaller) level of output production. Relatively speaking, this translates into a greater increase (decline) in profit in the equilibrium. If both favorable and unfavorable draws occur with equal probability, then, according to Jensen's inequality, the gain from revealing a favorable cost signal outweighs any loss sustained due to sharing an unfavorable cost signal with other firms, so much so that, ex ante, bilateral information exchange is strictly preferred to complete privacy. Summarizing this discussion yields the Simple comparative statics of Equations (20) and (21) with respect to z, yield: (o^y/dZi) - 0-y^z,) = [1/(1 + 2K)(3 +• 2K)]{[4(1 + K ) 2 - (1 + 2K)(3 + 2K)]/[4(1 + K)2]}v(Xj) > 0, i = 1,2. 27 following proposition 34 Proposition 2 Assume that the quadratic coefficient K is greater than or equal to zero, and let fx be a vector of optimally chosen levels of investment Cx1*x2) for one of the information structures under consideration. In addition, define c, according to (9), and v(Xj) by reference to either (11), (12), or (13). In Cournot duopolies, so long as the demand intercept "a" is sufficiently large such that expected profit functions, % i = 1,2, t = dd,dn,nd,nn, are concave in the level of investment,35 bilateral information exchange is the unique dominant information choice for both duopolists. In other words, ""J^x) > ""J^Cx) , ^ (^x) > ^ r x ) and ddJ2(ddx) > ""J^x) , ndJ2(ndx) > ""JjCx) If firms, instead, engage in Bertrand competition, then the improved coordination given information exchange can raise (lower) correlation in pricing strategies36 (unit profit37) 34See Appendix for proof. 35This ensures that the set of second-order conditions to the second-stage investment game is satisfied. 36dd--Kplp2 " Kplp2 = {[Covrp^^p l^/tVarrpOVarrp,)]0^} - 0 = [4B1(32/(P22 + 4p\2)] > 0, where Cov(d d P l,d dp2) = [4p13P2rj2/(4p12 - p22)2], Var(ddPi) = [(P,2p22 + 4pi4)o2)/(4p12 - P22)2], C o v r p , , ^ ) = 0, and VarCPi) = (a2/4), i = 1,2. 37ddn _ nn-P(pl-cl)(p2-c2) F(pl-cl)(p2-c2) = {[Cov((ddp1-c1),(ddp2-c2))]/[Var(ddp1-c1)Var(ddp2-c2)](1/2)} -0 28 across firms. Although products are by assumption differentiated, the distinction is not substantial enough to discourage substitution from taking place. In this market, if one firm initiates a price reduction and conveys such information to its opponents, it will be met with a similar cut of a somewhat lesser amount from those firms. Conversely, if unexpected cost savings are passed on to the consumers by one firm, the rival will counter with a price hike of a lesser magnitude.38 Therefore, pricing decisions reflect individual firms' anticipation of a potential erosion to their existing market share. In anticipation of these aggressive actions from opponent firms when private information about cost is publicly reported, both duopolists find themselves better off, ex ante, not sharing information. The next proposition states this formally.39 = {(-2p\p2o-2)(2(y - pY)/[p\2ft22 + (2fr2 - pV)2]} < 0, where Cov[(ddprCl),(ddp2-c2)] = [-2R1R2o2(2p12 - P22)/(4P,2 - p2 2)2], Var(ddprCi) = {[p/p, 2 + (2P,2 - p 2 2 ) 2 ]o 2 / (4p 1 2 - p 2 2) 2], Cov((>rCl),rp2-c2)) = 0, VarCprCi) = (o2/4), i = 1,2. 38Oddp/aZi) - OddPj/3z,) = [l/(4p7 - p^Jt^pWxi)] - [1/C4P,2 - P22)][-p1p2v(xi)] = [1/(4P,2 - p22)](-2p,2 + p,p2)v(Xi) < 0, i,j = 1,2, j + i. 39See Appendix for proof. 29 Proposition 3 Let the quadratic coefficient K be equal to zero and conditions, apart from type of competition, otherwise the same as those described in Proposition 2. In Bertrand duopolies, so long as the demand intercept "a" is sufficiently large such that expected profit functions, % i = 1,2, f = dd,dn,nd,nn, are concave in the level of investment,40 no information exchange is the unique industry-wide dominant information choice. In other words, ^("x) < ""J^x) .^(""x) < M J,rx) and d dJ2(d dx) < Mj^x) , n dJ2(n dx) < ""JjC-x). 2.7.2 Discussion To recapitulate, when the uncertainty pertains to a stochastic element in firm-specific marginal cost, the information choice of firms is clear. Ex ante, bilateral (no) information exchange is strictly preferred by all Cournot (Bertrand) competitors with or without the addition of an investment stage to the usual two-stage information sharing models. In addition, varying assumptions regarding the variance effect from investment does not result in disclosure policies qualitatively distinct from those concluded earlier. Within the framework of the study, these results continue to hold for Cournot industries whose members face total cost functions linear or convex in output. Nevertheless, information choice is not always robust with respect to scale returns assumption. For example, Kirby [1988] shows that, given industry-wide demand uncertainty, convex total cost functions and decreasing returns to scale are sufficient to reverse the commonly-reported finding in favor 'See footnote 35. 30 of bilateral information exchange. However, this same set of conditions fails to effect an analogous switch in information choice in the present setting. Relative to the linearity case, ceteris paribus, the convexity of total cost functions makes it more costly to over or under produce due to lack of output coordination. If total cost functions are sufficiently convex, or equivalently stated, if the mistakes from poorly coordinated outputs result in sufficiently large penalties, then, regardless of the sources of uncertainty with which they are confronted, firms have a strict preference for committing to a policy of complete disclosure. In the case of common demand uncertainty, this points to a preference reversal, whereas with firm-specific cost uncertainty, this new cost structure simply makes bilateral information exchange an even more attractive option. Not surprisingly, results contrary to the above can be obtained if total cost functions are concave as opposed to convex. More specifically, when production technologies exhibit increasing returns to scale, there exist conditions under which no exchange of private cost information dominates.41 Despite a rather limited range over which the information reversal occurs, the qualitative importance of cost structure, first brought to notice by Kirby, is generalizable to situations beyond those envisioned in that study. For instance, reversal occurs when the quadratic cost coefficient K is bounded from below by -1, and from above by -0.64645. Within this range, the set of second-order conditions for both output and investment games is satisfied. 31 2.8 Welfare Analysis 2.8.1 Welfare Definitions For The Three-Stage Games The discussion has hitherto centered on the effects of disclosure policies on firms. In this regard, sharing (withholding) information was shown to be the Pareto preferred information choice of all firms competing in quantity (price), and investment was shown to depend nontrivially on the extent of disclosure. Attention is now given to the question of how the welfare of consumers and society as a whole may be affected when firms are permitted to choose their own disclosure policies. In keeping with the industrial organization literature,42 this issue is approached from a producer-oriented perspective whereby a Marshallian definition of consumer surplus43 is employed. Given the separability of aggregate utility function and constant marginal utility of money,44 the representative consumer is assumed to maximize: CS(y1,y2) = U(y„y 2) - (TI, + 7t 2) (34) where U(y1;y2) = a(y, + y2) - (b1/2)(y,2 + y22) - b2y,y2 is the total surplus (TS), and is, by assumption, quadratic and strictly concave. The inverse demand functions Pi(Y0) in (15), 42Brander and Spencer [1983], Cheng [1988], Clarke [1983], Kirby [1988], Shapiro [1986], Shubik [1984], Singh and Vives [1984], Vives [1984]. 4 3By definition, consumer surplus is the difference between what the consumers actually paid and what they would be willing to pay rather than forgo the purchase. ^In other words, there are no income effects on the consumer side. 32 i = 1,2, facing individual firms which produce differentiated products can be obtained by partially differentiating U(yl5y2) with respect to y^45 In a homogeneous market such as the Cournot duopoly examined in the present study, (34) can be straightforwardly simplified to:46,47 CS(y„y 2) = (1/2)Y02 + (u, - z,)y, + OA - zjy2 + (x, + x2) (35) If information is incomplete and imperfect, then one can derive the ex ante consumer surplus for each of the information structures discussed earlier by taking expectations over Zj, i = 1,2, after substituting optimal strategies into (35).** To illustrate, when the products 45Strictly speaking, the inverse demand functions are given by: P i = OU/9y,)/[OU/9y1)yi + OU/ay2)y2], i = 1,2. This simplifies to Equation (15) when the level of income is, for convenience, fixed at 1, and the proportional factor X, the Lagrange multiplier for the budget constraint in the utility maximization problem, is ignored. 46This, of course, assumes the slope of inverse demand function is equal to 1. 47This is because p = a - (y, + y2) and n{, i = 1,2, are as defined in (14), hence (34) becomes: a(yi + y2) - (V2)(yi2 + y22) - y,y2 - [a - (y, + y2)](y, + y2) + (U-, - zjy, + OA - z2)y2 + (xj + x2) = - (l/2)(y,2 + y22) - y,y2 + (y, + y2)2 + (u, - z,)y, + (u^ - z2)y2 + (x, + x2) = (l/2)(yt + y2)2 + (u1 - z,)y, + (u^ - z2)y2 + (x, + x2) = (1/2)Y02 + (u, - z,) y i + (|i2 - Z2)y2 + (x, + x2) 48Because, according to Rogerson [1980], the stochastic price variation lies entirely on the supply side of the market. 33 are perfect substitutes, the expected consumer surplus becomes: E r C S C V y a ) ] = (l/^UECYo)] 2 + Var/Yo)} (36) + E[(Hi - z jVi + (A - z2)fy2 + (\ + fx2)] t = dd,dn,nd,nn. From the viewpoint of consumers, clearly the most desirable disclosure regime is the one that achieves the largest expected consumer surplus defined in (36). It follows that, ex ante, consumers are better off when expected firm outputs are high than when they are low. Similarly, greater output variation at the industry level, while detrimental to the producers, is beneficial to the consumers. On the other hand,, ex ante, the total surplus, E[U(y1,y2j], is simply an equally weighted sum of E[ tCS( tY0)] and ^ Cx), t = dd,dn,nd,nn. It should be noted however that, since the above analysis has not explicitly modelled consumers as strategic players, any ensuing welfare statements made on the basis of either welfare measure should in no way be construed as general equilibrium results. 2.8.2 Analysis And Results To simplify the analysis, the attention henceforth will be limited to comparing expected consumer surplus under two extreme information regimes where either both firms disclose or they do not disclose. In addition, total cost functions are linear in the level of output production. 34 Case 1: Constant Variance Effect From Investment In this case, both investment and expected output (pricing) strategies are independent of information availability. It follows from (36) that the expected consumer surplus can be ordered in accordance with aggregate output variances. Simple algebra leads to the following proposition.49 Proposition 4 In Cournot (Bertrand) duopolies, given that investment affects the mean but not the variance of cost distributions, consumers are strictly worse off (strictly better off) with bilateral information exchange than without. When producer surplus is given equal consideration in the welfare measure, then, ex ante, equilibrium information exchange in Cournot (Bertrand) duopolies is consistent with (contrary to) the maximization of expected total surplus. In much of the information sharing literature, the interests of consumers and firms are in conflict regarding information decisions (Clarke [1983a], Fried [1984], Shapiro [1986], Vives [1984]). Vives [1984], however, shows that these interests may be aligned in at least some Bertrand industries whose products are close substitutes. Since Vives models the demand side uncertainty, his result is neither directly comparable with nor contradictory to that described in Proposition 4. What is new, but not surprising50 from the present analysis of the benchmark case, is the persistence of disagreement between the consumers and Cournot duopolists over the 'See Appendix for proof. 'Since equilibrium levels of investment are independent of information. 35 latter's information choice, notwithstanding the addition of an investment phase. On the other hand, as will become evident below, investment is not as innocuous once one departs from the benchmark case. Case 2: Nonconstant Variance Effect From Investment In A Cournot Industry If, in addition to the mean, the variance of costs is also affected by investment, then, expected output at an individual firm and hence overall industry level is higher (lower) given bilateral information exchange so long as the variance effect from investment is increasing (decreasing).51 This implies the effect on consumer surplus must be jointly determined by both mean and variance of industry output distributions. To the extent that the aggregate output variance is in general smaller with more information,52 ex ante, consumers are worse off in Cournot industries where the variance of marginal costs decreases with investment when regulatory authorities adopt a laisser-faire policy with respect to disclosure. Alternatively, with an increasing variance effect, a tradeoff between two opposing effects must be made in order to assess how consumers 51For the variance increasing case, this is given by: E[ddY0(ddx)] - ErY0Cx)] = (1/3) ([2a - u( d d X l) - u(ddx2)] - [2a - uCx,) - urx 2)]} = (l/3){[urx1) - u(d d X l)] + [uTx2) - uTx2)]} >0. The variance decreasing case can be shown by analogy. "Recall this is because output strategies are less correlated in that reporting regime. 36 fare. More specifically, in that latter scenario, it is difficult to unambiguously determine which effect will dominate, a priori. With this in mind, several numerical examples53 are given in Appendix 1 to illustrate conditions under which bilateral information exchange is preferred by consumers as well as firms. This welfare result, a marked contrast to the constant variance case, is made possible because the mean effect is sufficient to compensate for any decline in the consumer welfare brought about by the unfavorable variance effect. Under Cournot competition and industry-wide uncertainty, Kirby [1988] has identified conditions for an analogous interest alignment. Despite the similarity in conclusion, the approaches taken are distinctly different. As indicated earlier, Kirby draws upon sufficiently convex total cost functions to establish her result. In contrast, the present study makes use of the structure provided by an increasing variance effect from investment, while remaining with the more common assumption of linear total cost functions. 2.9 Summary And Discussion In the context of three-stage information sharing/ investment/output (pricing) games, public policy on disclosure has been shown to impact competition in duopoly industries. For a wide range of scale returns and variance assumptions, ex ante, complete sharing (withholding) of private cost signals dominates when the rivalry is described by Cournot These are by no means knife-edged cases as is evident from the relatively broad regions within which the result holds. 37 quantity (Bertrand price) competition. Similar to the previous literature, which has demonstrated the importance of cost structure in altering the incentives not to exchange privately observed industry-wide demand information, this study shows that incentives may also be sensitive to the impact of investment on cost variances given firm-specific cost uncertainty. Although within the structure of the thesis information choice is invariant to the addition of an investment stage, endogenizing investment decisions adds another dimension to the potential impact of disclosure on production. That is, equilibrium levels of investment may depend on the disclosure regime and differences in those levels entail nontrivial strategic effects on rival firms' output or pricing decisions in the final stage product market competition. The above analysis has assumed that investment is chosen after the decision to exchange private information has been made. If the sequence of play is reversed, it seems likely that the strategic consequences will change as well in cases where disclosure affects levels of investment. Without formally analyzing this modified three-stage game, it is difficult to predict what the consequences of reordering might be. In the benchmark case, however, it would appear the ordering of first two stages is innocuous. Irrespective of the assumption about the sequence of play, an useful way to think about sequential play in multistage games is to view them as imbedding a Stackelberg relationship between one firm's earlier stage decision and its rival's later stage choice. Accordingly, the ordering becomes important whenever the decisions at one stage have strategic implications for those at another stage. 38 From the society's standpoint, allowing firms the freedom to choose their own disclosure policy in general does not maximize consumer surplus, however, society as a whole may not necessarily be hurt by such a laisser-faire policy due to compensating effects on industry profits. Under the maintained assumption that commitments are enforceable, duopolists competing in the same choice variable have been found to reach unanimous decisions with respect to disclosure decisions, as well as production and investment decisions. The next section speaks to the robustness of present findings should information asymmetry become one-sided. 2.10 Two-Stage Information Sharing/Output (Pricing) Games Given One-Sided Information Asymmetry 2.10.1 Opening Remarks And Literature Review It seems likely that in some markets information acquisition can be asymmetric owing to the experience gained over time by the relatively more established firms, and the comparatively higher cost of learning, at least at the beginning, faced by new entrants. At issue is whether some firms may find it cost-ineffective to resolve all the uncertainty in -own marginal cost prior to engaging in output or price competition with their more informed opponents. In Feltham, Gigler, and Hughes [1991], a two-period output/output (pricing/pricing) game is modelled wherein a private signal revealing an industry-wide demand or a firm-specific cost parameter is available to an incumbent firm at the conclusion of the first period. The entrant must extract what information it can from the reports that the 3 9 incumbent provides. A t times these reports may yield less than full disclosure. Although their discussion refers to a specific mandatory accounting rule, namely, the line-of-business accounting, its application goes beyond that particular policy. Based on the rule in force, they show that the incumbent wi l l strategically choose its first-period output (price) so as to influence the entrant's belief about its private information, and thereby alter the complexion of the post-entry competitive environment. When present, these strategic effects result in under or over production by the incumbent in comparison to first-period optimum under full disclosure. However, depending on a combination of assumptions about both the nature of competition and the sources of uncertainty, the welfare of both the incumbent and the consumers may or may not be advanced by such distortions. Apart from strategic effects, the information asymmetry also affects equilibrium play in the second period. The purpose of this section is to characterize similar effects in a setting where disclosure rules are chosen by the competition rather than being preset by policy makers as in Feltham et al. 2.10.2 The Model Eliminating the investment stage from the earlier three-stage game, the game is modelled as a two-stage information sharing/output (pricing) duopoly. The events and actions, described in Figure 2, are identically defined as before with the following major exception. Pursuant to a decision process outside the realm of present analysis, one of the duopolists (labelled firm 2) elects not to become informed of its own unit cost. 40 The identity of both the informed and the uninformed players, as well as the prior distributions on Z j is common knowledge. On the other hand, will be contained in the information sets for both firms only if a policy of sharing is chosen by the informed. Otherwise, z, is only available to firm 1. -~ At the beginning of second stage, firm i chooses a decision rule, % to maximize the following profit function: Max JiCYi.Ya.ODj) (37) where J, = E [7ti(Yi(co1),Y2(co2),coi)]) i = 1,2. Marginal costs of production c{ = zi5 i = 1,2, are independently and normally distributed across firms with mean c and constant variance a2. Finally, y{ e r,, i = 1,2, are the set of all possible decision rules defined earlier. A set of output (pricing) strategies ['YI.'YJ constitutes a Bayesian-Nash equilibrium if Ji(*Yi.*Y2.Q>i) ^ JI(YI»*Y2.G>I) V Yi e r „ and (38) J2(*Yi, Y 2,« 2) ^ J2CY1.Y2.tO2) V Y 2 e T 2. 2.10.3 Analysis And Results For The First-Stage Information Sharing Game Disclosure rule preference can be determined by comparing each firm's expected profit in two disclosure regimes, i.e., information sharing denoted by "du" and no information sharing denoted by "nu". Given Cournot quantity competition, the industry as a whole attains a higher expected 41 profit when firm 1 reveals its marginal cost than when it does not.54 The greater share of the expanded expected profits, however, goes to firm 1. Nonetheless, in spite of its relatively smaller share, firm 2 is still better off compared to the alternative choice by firm 1. With Bertrand competition, as before, the industry's interest will be advanced if all the firm-specific cost information remains private. Once again, the distribution of the expected gains from no information exchange is disproportionally in favor of the informed player, so much so that firm 2 is made worse off. That is, not only does one firm (i.e., firm 1) capture the entire increase in industry profit resulting from no disclosure, it also takes away some of the profit previously accorded to firm 2. 5 5 When the choice variable is quantity (price), from firm I's point of view, sharing (withholding) is strictly preferred because the improvement in (deterioration to) output 'Note in this case, expected firm profits are given by: 'J, = E C X ) = (l/9)[(a - c)2 + 4a 2 ] ; % = E(du7t2) = d/9)[(a - c)2 + a 2] 'J, = EC"*,) = (l/9)[(a - c)2 + (9/4)o2]; nu 'J2 = E ( n u TC 2 ) = (l/9)[(a - c)2] where J, = E M i UiCjii^),^^),^], i =1,2. 55 Expected firm profits are as follows: duf 'J, = E(du7t,) = [8,7(28, - p2)2][cx - (p, - p 2)c] 2 + [(2p,2 - p 2 2) 2p,/(4p, 2 - p 2 2) 2]o 2; % = E(du7t2) = [P,/(2P, - p 2) 2][a - (p, - p 2)c] 2 + [p, 2p 2 2p,/(4p, 2 - p ^ a 2 ; nu 'J, = E(»%) = [Pi/(2p, - p 2) 2][a - (p, - p 2)c] 2 + (P,/4)a2; nu % = E( n u K 2 ) = [p1/(2pi - p 2) 2][a - (p, - P2)c]2. 42 (pricing) coordination has a positive effect on its expected profit.56 By comparison, firm 2 is consistently better off with the elimination of the information asymmetry in all circumstances. The rationale for Coumot duopolies runs parallel to what has been discussed before. On the other hand, in Bertrand industries, this somewhat surprising result for firm 2 is driven by the greater gains it receives due to more accurate pricing strategies than the losses caused by more correlated pricing strategies. Summarizing this discussion leads to the following proposition.57 Proposition 5 In duopolies characterized by two-stage information sharing/output (pricing) games with one-sided information asymmetry, the firm with superior information is strictly better (worse) off when it reveals its cost information, than when it does not. The uninformed firm strictly prefers a policy of full disclosure by its rival. 2.10.4 Welfare Analysis Continuing to suppress the investment stage, the following statements on consumer and social welfare can be made.58 56This is because outputs are in general less correlated, while prices more correlated when private cost information is shared. du Pyly2 nu, 'P y i y 2 = (-D - 0 = (-1) < 0, and du Pplp2 nu Pplp2 = (+D " 0 = (+1) > 0. 57 See Appendix for proof. 58, ;See Appendix for proof. 43 Proposition 6 Let welfare measures be as defined in Section 2.8. In Cournot or Bertrand duopolies with one-sided information asymmetry, consumers are strictly worse off under the information choice of the informed than under the alternative. On the other hand, whether firms compete in price or quantity, society as a whole is strictly better off when there is more rather than less information. These results mirror those reported in Proposition 4 given that the variance effect59 from investment is constant. Suppose expected total surplus is an appropriate welfare measure for policy making purposes. Then, these two propositions suggest that if private disclosure is to be regulated, policy makers need only direct their efforts toward Bertrand industries. 2.10.5 Summary And Discussion Before society invests resources to regulate disclosure policy, it might first examine the incentives of the informed firms to disclose, and how the welfare of the uninformed may be impacted as a result. In this section, it was shown that varying assumptions about the nature of the information asymmetry can result in differential welfare effects on the players. Not only does the interest of consumers continue to be controverted by the disclosure choice of the informed in some markets, but more interestingly, the wellbeing of the less informed firms may also be adversely affected. These results suggest that conflicting disclosure rule preferences may be present across 'Recall in this case, the orderings are determined by aggregate output variances only. 44 industries as well as among firms within the same industry. However, as will become evident in the next chapter, the condition that information advantage is one-sided in some industries, yet two-sided in others, is not necessary for either inter-industry or intra-industry variation in disclosure policies to occur. 45 CHAPTER 3 INFORMATION SHARING AND INVESTMENT IN A RISK AVERSE WORLD 3.1 Opening Remarks And Literature Review Throughout both the foregoing analysis and much of the literature on information sharing, it has been concluded that, for a wide range of conditions, Cournot oligopolists endowed with firm-specific cost information will independently commit to a policy of full disclosure ex ante, provided commitment is enforceable. However, according to Hvid [1989], the apparent robustness of information choice is due, in a large measure, to the standard assumption of risk neutrality. Within a context of common demand uncertainty rather than firm-specific cost uncertainty, Hvid illustrates with examples that, for some parameter values, risk averse Cournot duopolists may express strict preference for industry-wide information exchange over complete withholding, a result contrary to earlier studies. What is captured in his risk averse model, but absent from earlier risk neutral models, is a further sensitivity to the reduced output variation that accompanies an exchange of private information. The aim of this chapter is two-fold. At a more transparent level, the analysis addresses the question of whether a reversal similar to that reported by Hvid can be found given that the source of uncertanity is firm-specific. More subtly, the results establish that risk 46 aversion can modify the investment and output strategies taken in response to mandated changes in disclosure requirements. 3.2 Model For Two-Stage Information Sharing/Output (Pricing) Games The first part of the analysis models a standard two-stage information sharing/output game in which the game structure, notation, and distributional assumptions are defined similarly to the three-stage game. As before, the analysis focuses only on linear decision rules of the form: Vi = THa + T 2 ic + T 3 i Z i + T 4 iz j ; (39) i,j = 1,2, j ± i, t = dd,dn,nd,nn. These conjectures are fulfilled in equilibrium. For analytical tractability, firms60 are assumed to have negative exponential utility for profits: U(TC ;) = -expHpjcJ, i = 1,2, (40) where (p is a positive constant risk aversion parameter, identical for all firms.61 In (40), management compensation is assumed to be proportional to the end-of-period profits. 60They can be either firms or firm managers. The thesis has assumed away agency problems, hence both definitions will be used interchangeably. 61(p can be interpreted as the manager's personal risk aversion parameter times his share of Ttj. 47 Moral hazard due to unobservable effort may be one of the reasons why managers are compensated in this fashion when they do not own the firm. Although moral hazard is not modelled in the thesis, one can envision the duopoly game being imbedded in agency contracting problems wherein managers are agents and absentee owners are principals. In such a setting, a principal faces a tension between imposing risk on firm manager through profit-dependent compensation, and inducing less aggressive behavior with respect to his or her choice of outputs.62 When there is no residual uncertainty regarding own as well as rivals' unit costs at the time firms select their output levels, the equilibrium strategies for the second-stage output game follow in a straightforward manner from those presented in (20). This is because utility is monotone in profits. On the other hand, under no or unilateral information exchange, there is still uncertainty at the output stage. Nonetheless, the linearity of the profit functions, 7 X i ( 6 3 and the normality of Z j simplify the objective function to:64 62In the absence of moral hazard due to hidden effort, the owner has incentive to induce more aggressive behavior by the manager in the output game. Fershtman and Judd [1987] demonstrate that optimal linear compensation to a risk neutral manager would depend on both profit and sales. However, relaxing the linearity restriction, optimal contracts based on profit alone can be shown to induce a reduction in risk aversion precisely to obtain the advantage described later in this chapter, when the risk attitudes of rival managers are asymmetric. 63The fact that 71, is linear in zi follows directly from the above conjecture about the rival firm's output. "Maximizing (41) results in precisely the same decisions as maximizing the expected exponential utility function in (40) even though (41) does not equal the expected utility. 48 EUCTti | oOj.Yj) = E(7CS | <o„Yj) - (tp^^arCKj | (41) i,j = l ,2 , j^ i . The set of first-order conditions associated with maximizing (41) is given by: a - 2tYi(coi) - E ( f Y j I co,) - c, - (ptYi(u)i)Var(tyj I = 0 (42) i,j = 1,2, j ^ i, t = dd,dn,nd,nn. As before, expected firm profits and the variance of firm profits can be expressed as follows:65 Bilateral Information Exchange "J, = [E(ddyi)]2 + Var(ddyi) (43) VarCX) = 4[E(ddyi)]2Var(ddyj) + 2[Var(ddyi)]2 (44) E( d d y i) = [(a - c)/3] ; Var(ddyi) = (5a2/9) .. (45) No Information Exchange. ml = [(1 - T3)/T3]{[Eryi)]2 + V a r r V i ) } (46) VarrTti) = {[4(1 - T3)2/T32] + 1}[Eryi)]2Varryi) (47) + {[2(1 -T 3) 2/T 3 2] + l}[Varr y i)] 2 Ery,) = [T3(a - c)/(l + T3)] ; VarC^) = T3W (48) 'Expressions for the unilateral information exchange regimes can be similarly defined. 49 where i = 1,2, T 3 = T 3 1 = T 3 2 in equilibrium, and 0 < T 3 < (1/2) for all (p > 0.66,67 In principle, T 3 can be determined endogenously from the following equilibrium relation: T/cprj2 + 2T3 = 1. (49) Nevertheless, the solution is too complex to use in the subsequent analysis; therefore, a closed-form solution for ""y^  i = 1,2, will not be derived. Finally, T 3 and hence EC"^) are both decreasing functions of risk parameter, (p.68,69 3.3 Analysis 3.3.1 From First-Stage Information Sharing Game Rather than reacting to each other's expected output, complete information sharing allows firms to set their own production level in response to rivals' actual output. When 66Note when (p = 0 (i.e., risk neutral), T 3 = (1/2). 6 7 A detailed derivation of E(™y), i = 1,2, can be found in the proof for Observation 1 in the Appendix. 68Partially differentiate T33cpa2 + 2T3 = 1 with respect to T 3 and cp yields Oiyacp) = [(- T33c2)/(2 + ST/cpa2)] < 0. 69From Equation (48). It is straightforward to show that [3Ery,)/99] = [(a - c)/(l + T3)2]&TJd<f>) < 0, i = 1,2. The above follows because (a - c) > 0 by assumption, and (dTJdy) < 0. 50 the source of uncertainty lies with firm-specific cost, this implies greater output variation, and hence greater profit variance because the latter, according to Equations (44) and (47), is a convex function of the former. But, in addition to being associated with the largest dispersion in end-of-period outcomes, this disclosure regime also has, as established previously, the highest expected firm profits. These two counterbalancing forces, along with the inability to endogenously determine coefficient T 3, underscore potential problems facing researchers upon moving into a world of risk aversion. Not only is the Pareto dominance of one of the information regimes situation-specific, it could also become analytically intractable to draw a general conclusion on the pairwise orderings of the expected utility for profits. Notwithstanding these difficulties, as in Hvid, weaker statements, supported by specific numerical examples, can be made to demonstrate the nonrobustness of information choice with respect to risk preferences. Without loss of generality, the variance of marginal costs, o2, will be set equal to l . 7 0 Scenarios contradicting previous findings on the relative Pareto preference of bilateral versus no information exchange are presented next.71,72 70This assumption is innocuous since it is the magnitude of o 2 relative to cp and c that is germane. 71The examples described in Observation 1 are in no way exhaustive. 72See Appendix for derivation. 51 Observation 1 Under any one of the following combinations of risk parameter cp and unit profit margin (a - c), risk averse Cournot duopolists enjoy higher expected utility for profits when private cost information is withheld than when it is publicly reported: a cp = 0.01 ; (a - c)2 > 488.575. b cp = 0.10 ; (a - c)2> 51.314. c cp = 1.00 ; (a - c)2> 7.320. d <P = 5.00 ; (a - c)2 > 2.522. e <P = 10.00 ; (a - c)2 > 1.603. f <P = 100.00 ; (a - c)2> 0.329. g <P = 500.00 ; (a - c)2> 0.111. h cp = 2,000.00 ; (a • -c) 2> 0.045. The above observation suggests that as firms become more risk averse, i.e., as cp increases, then, even marginally profitable firms tend to favor keeping information private. Intuitively, this is because as dislike for the rising profit variation grows stronger, firms find the corresponding increase in expected profits inadequate to justify information sharing. Hvid also makes a similar remark given industry-wide demand uncertainty. Observation 1 however does not imply the dominance of no information exchange, for that requires additionally, pairwise orderings of EUC^Ttj) versus EUrTcJ and EU( d d7T,j) versus EUC\), i =1,2. Although these computations are considerably more complicated, examples contrary to the risk neutrality case can still be found. For instance, if the risk parameter cp is very small (say, cp = 0.1), and operations are relatively profitable (say, unit profit margin (a - c) is no less than 5.437), a strategy of completely withholding information can dominate.73 When the degree of risk aversion See Appendix for an example. 52 is intensified to between 1.31 and 4.35, the same conclusion will continue to hold, this time, for all the marginally profitable Cournot industries (i.e. (a - c) > 0).74 In general, it appears most profitable Cournot industries favor no information exchange so long as cp is less than 10.5. At the other end, no one form of information arrangement seems to clearly dominate for firms that are at least moderately risk averse (say, constant absolute risk aversion parameter is at or above 10.5). In other words, even though according to Observation 1, many profitable industries may strictly prefer no infomiation exchange to bilateral information exchange, the former cannot be supported as a noncooperative solution.75 A similar observation reported by Hvid prompts him to suggest that there may therefore be a need for a coordination device in these industries. 3.3.2 Welfare Analysis Recall that in the risk neutral Cournot industries, consumers strictly prefer a regime where no information is exchanged. However, the sensitivity to risk displayed by firms and the resultant changes to output strategies can also affect expected consumer surplus, (36), such that risk neutral consumers may end up better off with more rather than less public See Appendix for an example. 75Analytically speaking, firm i would strictly prefer not to disclose given that firm j chooses to disclose, whereas the opposite is true if firm j keeps its information private. This of course is a classic prisoners' dilemma problem. 53 information. The rationale is clear. Although output strategies under full disclosure are independent of risk preferences, the same statement does not extend to no information exchange.76 In that regime, on average, aggregate outputs are declining in (p. As cp increases, so does the difference between equilibrium outputs expected to be produced in these two extreme reporting regimes, reflecting a need to modify outputs by both firms absent information exchange. At some.point, the gap may be so wide as to compensate for any corresponding decline in industry-wide output variation caused by the exchange of information. 3.3.3 Extensions The sensitivity of public policy with respect to disclosure to the risk attitude of decision makers is by no means confined to symmetric Cournot industries. Take a Cournot duopoly with one-sided information asymmetry for instance. The above analysis points to a preference for no information exchange by the informed firm, provided it is at least marginally profitable and displays some degree of aversion toward risk (say, (p > 1.713). 7 7 The uninformed firm, however, continues to be worse off in an expected utility sense, if its more informed opponent's private cost information is suppressed because the ensuing decrease in expected profit exceeds the reduction in profit 76Refer to expressions in (45) and (48). 77The derivation, though complex, follows directly from two-sided information asymmetry case, thus will not be repeated here. 54 variance. While Cournot behavior has been assumed so far, the impact of risk attitude on information choice may be no less pronounced in Bertrand duopolies. Intuitively, changing the preference assumption, in this case, results, in a positive weight being assigned to the reduced unit profit variance due to information exchange. Gains are higher, the more risk averse firms are. Difficulties arise, however, because these gains must be balanced against an escalating decline in expected profits. Since the equilibrium pricing strategies cannot be completely characterized, numerical examples must once again be used78 to establish the existence of a noncooperative solution for information choice distinct from that reported in the literature, whether information asymmetry is one-sided or two-sided.79 3.3.4 Summary And Discussion Apart from Hvid, the extant information sharing literature has assumed that firms are motivated solely by their desire to have as high an expected profit as possible. Although this branch of research sheds light on cross-industry variation in disclosure, with the exception of a study by Gal-Or [1987], it has by and large failed to explain why variation 78In particular, examples assume that p, and Pj defined in (17) can be related as follows: Pi = riPj, where r| is some positive constant, i,j = 1,2, j ^ i. More general examples without imposing conditions on pi5 i = 1,2, are analytically intractable. 79 See Appendix for an example. 55 persists even within an industry characterized by the same assumption regarding the nature of competition. In her study, Gal-Or departs from the usual simultaneity assumption to model the oligopoly rivalry as a sequentially-played game. When approached this way, firms which take on a Stackelberg leadership role in either output or pricing decision have been found to favor a disclosure policy contrary to what would have been the choice had the game been played simultaneously. However, it is clear from the above analysis, intra-industry disparity in information choice can be brought about without either altering the order of play, or limiting private information acquisition to one firm. All else held equal, a multitude of disclosure policies may prevail in a risk averse industry to the extent that its constituent members display varying degrees of aversion toward risk and/or enjoy different levels of expected profits from operation.80 It remains an empirical issue whether predictions from a risk averse model, a one-sided information asymmetry model, or a Stackelberg leader-follower model are more descriptive of disclosure choice in real life, and hence what the consequences of public policy choice might be. We have not yet characterized equilibrium with differing attitudes toward risk. Such asymmetries are considered in the next section. 56 3.4 Model For Two-Stage Investment/Output Games Under Alternative Disclosure Rules 3.4.1 Opening Remarks Besides discretionary disclosure, many private decisions may be affected by the decision makers' perception about risk. Of particular interest here are investment and output decisions, which, for analytical convenience, will be examined using a two-stage investment/output game model, holding aside the issue of information choice. This line of enquiry is motivated, in part, by Palfrey [1982], who demonstrates that a less risk averse Cournot duopolist enjoys an advantage over its more risk averse opponent when both firms are identically uninformed of a stochastic realization in demand. By comparison, if the less risk averse firm alone is awarded exclusive access to information resolving some of the demand uncertainty, then, the risk advantage it previously enjoyed can substantially dissipate. In the end, the decline in risk advantage can be so great that the informed yet less risk averse firm is worse off, whereas at the same time its more risk averse uninformed rival is better off. In addition to characterizing the sensitivity of decisions to reporting requirements81 in a risk averse setting, much of the analysis that follows will be devoted to extending Palfrey's findings on risk advantage to situations in which either all the Cournot duopolists 81The question of how risk advantage may be traded off against information advantage however will not be dealt with. In other words, both duopolists in the present model are either equally uninformed or equally informed of each other's private cost. 57 are symmetrically endowed with private information82 regarding an uncertainty attaching to their own supply, or they are asymmetrically endowed of risk. Within the framework of the present model, the consequences can be established by contrasting the equilibrium output and investment strategies in a risk neutral industry with those obtained in a risk averse industry under otherwise identical conditions. 3.4.2 The Model Consider two exogenously imposed reporting regimes: full disclosure, corresponding to earlier bilateral information exchange, and no disclosure, analogous to no information exchange. It is common knowledge which one of the two rules is in force. With the exception of this difference, events and actions follow directly from the three-stage Cournot output game. Recall, given risk neutrality and the simplified assumption of constant variance effect, firms are expected to take exactly the same actions in equilibrium for all information regimes. This observation may no longer hold once risk aversion is introduced, because, firms or firm managers are now sensitive to the dispersion in end-of-period outcomes. Instead of maximizing their expected firm profits, firms select both actions with a view to maximizing the expected utility for profits (41). Suppose production technologies exhibit constant returns to scale, then, for any given 82This is to be contrasted with the scenario in Palfrey described above, where either none of the firms receive any private signal, or only one of the firms possesses private information (i.e. information asymmetry is one-sided). 58 level of investment, the equilibrium output strategies under a full disclosure regime are given by (20), with K = 0 and v(x;) = v (Xj) = 1, i,j = 1,2. By backward induction, the first-stage investment in turn is determined from maximizing: Wi(x) = II U(7ti(ddY1,ddY2>xi))f(z1,z2)dz1dz2, (50) i = 1,2. Integrating (50) and differentiating the resultant expression with respect to X ; , i = 1,2, lead to the following set of first-order conditions: [aWiOO/axJ = (-4/9)[a - 2u(x.) + iK^u. '^ ) - [(9 + lOqxj2)^] (51) = 0, i,j = 1,2, j + i. On the other hand, with partial disclosure, equilibrium output strategies become: ""v, = [1/(1 - T3 1T3 2)][T3 i(l - T3j)a - T^x.) + T3iT3ju(Xj)] + T 3 iz i ; (52) i,j = 1,2; j ± i, where T 3 i , i = 1,2, have the properties described in Section 3.2. The decision problems for the investment game are: Max W,(x) (53) x i where W,(x) = JJ U(7ti(nnY1,nnY2,xi))f(z1,z2)dZldz2, i = 1,2. The first-order conditions associated with (53) can be characterized by: [aw,(x)/9xj = - {[(i - T3yr3]a - (I/T^U.^) + u^)}^) - [(1 + T 3) 2(l - T 3) 4/T 3 4] = 0, i,j = 1,2, j $ i. 59 (54) 3.4.3 Results And Analysis Compared to risk neutral firms, risk averse firms behave more conservatively in every respect. To mitigate their exposure to the increased profit variation brought about by disclosure regulation stipulating a free exchange of privately observed cost information, firms, in principle, will contract their output production from a level considered optimal previously. But, for a given pair of investment levels, output strategies are invariant to changes in risk attitude under full disclosure. As a consequence, risk created due to the variation in profit can only be ameliorated through the remaining decision variable, i.e., investment.83 More specifically, both firms will cut back their investment spending to lower expected output production, and hence the profit variance.84 Output strategies are however available as a risk modifying device85 when access to the rival firm's private signals is denied. Relative to the preceding reporting regime, in this case, the conservative nature of output, along with investment decisions by risk averse "Alternative means of risk shedding are assumed away. ^From Equation (44), it is clear that profit variance is increasing in own optimal output strategy, which in turn is directly related to the level of investment. 'See footnote 69. 60 firms, magnifies the extent of reduction in both output and the variation in firm profit,86 ex ante. Inasmuch as risk can be lessened through investment regardless of disclosure requirements, and output can only be used for a similar purpose in the absence of information exchange, then, ceteris paribus, the following results are obtained.87 Proposit ion 7 (a) For a given pair of investment levels, on average, risk neutral Cournot duopolists produce as much as (more than) their risk averse counterparts when both firms are (are not) required to disclose their privately observed cost information, i.e., E[d d y R N(x)] = E[d d y R A(x)]; E[nnyRN(x)] > E[nnyRA(x)]. (b) For a given disclosure rule, i. the equilibrium levels of investment are higher when both Cournot duopolists are risk neutral than when they are risk averse, i.e., dd ^ d d v . nn„ ^ nn„ A R N ^ ARA> A R N A R A -i i . the equilibrium levels of expected output production are higher when both Cournot duopolists are risk neutral than when they are risk averse, i.e., E[dV,(dVN)] > Ery^x^)] ; . E [ " n y R N r x R N ) ] > E r y R A r x R A ) ] . . 86This is because, according to (47), the dispersion in profit is an increasing function of the expected output. 87See Appendix for proof. 61 The above proposition is equally applicable to industries in which some firms are risk neutral, while others are risk averse. In that context, the present analysis can be viewed as extending Palfrey's findings from a null information regime to regimes where firm-specific private cost information either is present yet not exchanged, or is not only present but also fully disclosed. Drawing an analogy with Palfrey, so long as there remains some residual uncertainty at the time output decisions are made, risk neutral firms can be expected to exploit those averse to risk not only by investing, but also by producing more output for a given level of investment. This aggressive stance enables them to derive a greater share of overall industry outputs. On the other hand, under full disclosure, such a risk advantage does not completely dissipate88 in that the risk neutral firms' pursuit of a relatively more aggressive investment policy89 once again enables them to achieve a higher level of output and hence a larger market share, ex ante, than their risk averse counterparts. Evident from the above discussion is that both investment and output decisions are sensitive to risk attitude. However, the extent of modification made by risk averse firms to one or more of these decisions may be distinctively different depending on what disclosure requirements have been set. Intuitively, the first-stage investment can be expected to be more conservative when This observation is in contrast to the one-stage output game model examined by Palfrey, where risk advantange, as one may recall, completely dissipate in the presence of information advantage. 89Investment, as one may recall, is not available to the less risk averse firms in Palfrey. 62 risk averse firms exchange private information than when they do not because that decision variable is the only available means whereby risk may be shed in the former regime. By the similar arguments, this same reporting regime can be expected to be associated with comparatively less conservative second-stage output strategies. These two contrasting actions pose problems when attempting to order the expected output levels across reporting regimes.90 This is now formalized.91 Proposition 8 (a) In equilibrium, risk averse Cournot duopolists undertake less investment under bilateral than under no information exchange, i.e., ddx < ""x i = 1 1 (b) For a given pair of investment levels, in equilibrium, risk averse Cournot duopolists have higher output production under bilateral than under no information exchange, i.e., E[ddy iRA(x)] > EnyiRA(x)], i = 1,2. Recall in a setting where decision makers are risk neutral, it was established in Proposition 1 that disclosure policy does not affect equilibrium levels of investment so long as variance effect from investment is constant. But, it is evident from the above proposition that investment is far from innocuous once one moves to a risk averse world. 9 The ambiguity arises, to a large extent, due to the inability to completely characterize equilibrium output, ""y^  i = 1,2, as alluded to earlier. 9 1 See Appendix for proof. 63 This suggests a game that models investment helps one capture and gain a deeper understanding of the strategic behavior in a stochastic oligopoly. 3.4.4 Welfare Analysis For a given disclosure rule, consumer surplus is consistently less if Cournot duopolists are sensitive rather than indifferent to risk. The rationale is clear. Less aggressive actions in this market, on average, lead to lower aggregate outputs and smaller variation in industry outputs, both of which are contrary to the best interest of consumers. Hence, the next • . 92 proposition. Proposition 9 For a given disclosure rule, consumers are better off in risk neutral Cournot duopolies than in risk averse Cournot duopolies, i.e., (a) E ^ C S ^ x ™ ) ] > ErcS^CV)]. (b) E r c s ^ r x ^ ) ] > Ercs^rxRA)] 3.5 Summary And Discussion To the extent that firms are sensitive to profit variation, previous work which approaches the study of information sharing based on the premise that oligopolists are risk neutral does not completely characterize the consequences of different disclosure rules. In See Appendix for proof. 64 particular, the preferences of risk averse duopolists to share firm-specific cost information with rival firms are often in direct contrast with the preferences of their risk neutral counterparts. As a result of differences in risk attitudes among firms, reporting methods may vary not only across-industries, but also within the same industries. Similarly, models that consider risk aversion were shown to have the potential to explain both inter-industries and intra-industry variation in adjustments made to the production and investment decisions following changes to exogenously imposed disclosure rules. The above observations are however subject to a caveat. Neither two-stage game employed to examine the sensitivity of information choice and investment to risk preferences formally models the agency problems which may give rise to compensation schemes of the type assumed. For instance, one might envision a situation in which firms are owned by risk neutral individuals who can observe manager decisions regarding disclosure and output or pricing. Management compensation, in this case, would involve a fixed payment to the risk averse manager. Such a flat wage contract would tend to induce less aggressive behavior from the risk averse manager than the alternative contingent compensation scheme assumed in the thesis. Nonetheless, the issue of why firms choose to base management compensation on the end-of-period profits is not addressed. Recently, several studies have begun to incorporate product market considerations into traditional agency research in a risk neutral world.93 With the added dimension of risk Fershtman and Judd [1987], Gal-Or [1991]. 65 aversion, a priori, one might expect to see further complexities in that line of enquiry. Specifically, firms would have to structure compensation contract to deal with the less aggressiveness of managerial behavior induced by not only moral hazard but also risk aversion. These research questions are interesting, but they are clearly beyond the scope of present investigation. Even though the analysis has been conducted under a set of relatively strong assumptions throughout the thesis, the predictions obtained from this precommitment-based disclosure research can nevertheless be shown to have potential to either explain or offer new insights into a wide range of policy-related issues that are of interest to accountants. It is to this end, the discussion now turns. 66 CHAPTER 4 ACCOUNTING APPLICATIONS 4.1 Opening Remarks Both the present analysis and its principal antecedents from the information sharing literature are distinct from the studies of voluntary disclosure in their assumptions about the timing and object of information choice. Information sharing models typically require the decision concerning a particular disclosure rule to be reached prior to the resolution of uncertainty. In contrast, the ex post voluntary disclosure studies defer disclosure decisions until after signals are privately observed. Using this latter approach, several theoretical studies have modelled the incentives for voluntary management earnings forecasts and other discretionary releases in oligopolistic markets (Darrough and Stoughton [1988], Dontoh [1989], Feltham and Xie [1991], Wagenhofer [1990]). At the same time, others have applied the strategic incentives characterized in the former research paradigm to study specific accounting issues (Feltham, Gigler, and Hughes [1991], Hughes and Kao [1991]). They argue that alternative accounting rules disclose different private information about firms in oligopolistic industries, hence there is potential for these rules to affect the equilibria that emerge. The economic consequences identified are quite apart from the more widely understood effects, such as those related to taxes, agency conflicts, or capital structure. 67 As in the last two studies cited above, the purpose of this chapter is to offer a discussion relating the predictions set forth in the preceding theoretical analysis of noncooperative games of information sharing, investment, and production to accounting contexts. Specific issues considered in this discussion include changes in investment activities surrounding new policy pronouncements, self-selection of accounting methods in the absence of an uniform disclosure policy, and variations in lobbying efforts in response to the impending rule changes. 4.2 Accounting Policy-Induced Changes To The Level Of Investment Propositions 1 and 8 distinguish cases in which the level of investment would be either increased, decreased, or unaffected as one moves from more to less disclosure. The crucial aspects are the effect of investment on the variance of marginal costs, the type of industry, and the risk attitudes of managers. Holding, risk attitudes fixed, a constant variance results in no change in investment activities, whereas an increasing variance leads to a contraction (expansion) in Cournot (Bertrand) duopolies. Investment may also be expanded in Cournot duopolies if managers are risk averse and the variance effect remains fixed. These results have implications for many unanticipated changes in accounting policies that affect the extent of disclosure concerning the impact of investment on production costs. One application is found in Hughes and Kao [1991] who model a two-stage R&D/output game to analyze the impact of Statement of Financial Standard (SFAS) No. 68 2 on the equilibrium levels of investment in research and development (R&D) by Cournot duopolists. In their study, R&D is defined as an activity which affects marginal costs of production. By eliminating the option to selectively capitalize, SFAS No. 2 is said to, in effect, alter the disclosure regime, and thus the subsequent competitive environment under which rival firms make such decisions. However, before the disclosure effect of mandated accounting rule changes on investment can be characterized, one needs to consider the relative informativeness of alternative policies sanctioned by the Financial Accounting Standards Board (FASB) in the absence of an uniform disclosure requirement. Hughes and Kao [1991] argue that, compared to the expense-only rule, selective capitalization conveys more firm-specific information. This is because the estimates of the timing as well as amount of future benefits from R&D activities are disclosed and independently verified by an external auditor only under the capitalization method. Given this assumption, they go on to show that the policy-induced changes to R&D investment depend importantly on the characteristics of firms and the makeup of industries subject to those rules. To make the empirical connection, Hughes and Kao [1991] liken the effect of R&D spending on the variance of unit cost savings to the degree of diversification in R&D activities. The latter, in turn, is proxied by size. If small firm size and high technology are equated with less diversified activities, then the predictions from their risk neutral model can be seen as consistent with empirical evidence. In particular, Horwitz and Kolodny [1980] (and to a lesser extent Elliot, Richardson, Dyckman and Dukes [1984]) report that R&D spending did go down for some firms pursuant to SFAS No. 2, most 69 noticeably the smaller high technology companies. Hughes and Kao [1991] also use firm size to proxy for differences in risk attitudes. Since smaller firms' ownership is in general less diversified than larger firms', managers for the former may be more sensitive to the risk reflected in profit distributions. However, if this is so, then their prediction of greater R&D spending under less disclosure runs contrary to the evidence in the aforementioned studies.94 Shifting to the issue of accounting for oil and gas exploration costs, the same prediction from the risk averse model may explain why many smaller oil producers voiced strong opposition during public hearings held to consider the proposed SFAS No. 19.9S More specifically, these firms claimed that their exploration and drilling activities would be substantially reduced if only the successful effort (SE) method, one of the two accounting principles sanctioned by FASB prior to SFAS No. 19, was allowed. The empirical linkage hinges of course on the assumption that the full costing (FC) method is less informative than the successful effort method (SE). But this' is reasonable since under FC all the costs of exploration and drilling must be capitalized, whether or not they are identifiable with reserves that have commercial potential. '"Note the converse is true if one, instead, holds the view that smaller firm managers are not as risk averse as larger firm managers. Such an argument is founded on the belief that moral hazard problems may be less severe in smaller firms, and thus making it unnecessary for the owners of such firms to impose, by way of incentive scheme, as much risk on their managers. 95The idea that the ownership of smaller firms is less well diversified is consistent with the testimony these firms presented to the Security Exchange Commission. In particular, the full cost advocates alleged that they tended to be smaller than the nonmajor successful effort companies and, therefore, unable to diversify the risks of exploration as effectively (Deakin [1979]). 70 The idea that mandated accounting policy choices may bring about changes in the production, investment, and/or financing decisions in the private sector has been considered in the accounting literature. (Collins, Rozeff, and Dhaliwal [1981], Watts and Zimmerman [1986], Wolfson [1980]). What is more difficult to predict, according to Wolfson [1980],96 is how such behavior will be altered. In the past, some researchers have appealed to contracting theory to explain the reported changes in R&D spending pursuant to the implementation of SFAS. No. 2 (Dukes, Dyckman, and Elliott [1980], Horwitz and Kolodny [1980], Vigeland [1981]). That approach, however, has produced conflicting empirical results. More importantly, its theoretical underpinning was called into question by Ball [1980] in his review of an article by Dukes et al [1980]. This thesis offers an alternative perspective which is based on the economic consequences that arise from the strategic effects of rivalry on investment. The arguments presented above are intuitive, and provide a deeper understanding of why investment activities may be affected by mandated accounting policy pronouncements than what has been achieved in the extant literature.97 ^his remark was made in his discussion of a study by Horwitz and Kolodny [1980] regarding the impact of SFAS No. 2 on levels of investment in R&D. ^However, it should be pointed out that at a macro level accounting policy change itself may, according to Ball [1980], be a reaction to shifts in the underlying environment. But, due to its complexity, such dynamics are assumed away in the thesis. 71 4.3 Self-Selection Of Accounting Methods And Information Sharing The crucial assumption in the above discussion is that audited financial statements prepared in accordance with generally accepted accounting principles (GAAP) constitute a vehicle through which firms may share their firm-specific information with competing firms in the same industry. The previous section dealt with changes in mandated rules. However, one can step back and consider information sharing as a matter of firm choice. GAAP often permits the same underlying economic event to be captured by more than one equally acceptable accounting principles in the financial statements. Each of these accounting principles can, by assumption, convey a varying degree of information about the reporting firms. Accordingly, Observation 1, and Propositions 2, 3, and 5 may offer insights into self-selection of accounting methods in the absence of an uniform measurement rule or disclosure policy.98 Returning to the R&D example,99 it follows from Proposition 2 that firms in Cournot industries were likely to have favored selective capitalization in order to better coordinate their production decisions. The incremental benefit of capitalization over the expense-only rule may have been the greatest for firms whose variance in marginal cost is increasing in 98In a study of the relationship between competition and the duopolists' incentive to disclose private information, Darrough [1990] also raises the possibility that ex ante information choice may be viewed as firms agreeing or subscribing to mandated accounting disclosure policies. "Recall selective capitalization was previously assumed to reveal more information concerning the effects of R&D on marginal costs than the alternative expense-only rule. 72 R&D spending.100 The fact that not all industries chose a policy of selective capitalization in the pre-SFAS No. 2 era suggests that assumptions such as Cournot competition, firm-specific cost uncertainty, and risk neutrality might not have been met. Specifically, the ordering of expected profits across disclosure regimes reverses if the nature of competition is changed from Cournot (quantity) to Bertrand (price), or the source of uncertainty from firm-specific to industry-wide. In either case, results from Proposition 3 and the information sharing literature predict a strict preference for less rather than more disclosure. That is, it is possible the expense-only rule would be favored in some industries. A similar switch in information choice and hence accounting practice may likewise be obtained when firm managers display at least some aversion toward risk. Under the condition that firm size is an appropriate proxy for risk attitudes, one may draw on the findings reported in Observation 1 and Proposition 2 to conclude that industries which self-selected the expense-only rule prior to its adoption by FASB in SFAS No. 2 were, on average, smaller in size than those that did not. Accounting method choice can vary not only across industries but also within the same industry because firms may not be symmetrically endowed with proprietary information First note that the following comparative static result: [9(ddJ - ""jyao2] > 0 i.e., the relative gains of bilateral information exchange (selective capitalization) over no information exchange (expense-only rule) are an increasing function of the uncertainty about the benefits of investment. 73 about their own marginal cost. If the informed firms are generally more established and thus larger than the uninformed and less established firms, then, based on Proposition 5, larger firms would (not) tend to favor selective capitalization in Cournot (Bertrand) industries. On the other hand, smaller firms, uninformed of their own marginal cost due to the cost of acquiring such private information, would stay with the less informative expense-only rule in all cases. In other words, before SFAS No. 2, there could exist intra-industry variation with respect to the choice of accounting treatments for R&D expenditures as well as inter-industry variation. In a similar fashion, one can reconsider the choice between the SE and FC methods in oil and gas accounting. To make the point, assume that the Coumot risk averse model does capture the competitive environment in a petroleum industry whose member firms are relatively small in size. Then, the fact that many smaller oil and gas producers chose to account for their drilling and exploration costs by the less informative FC method prior to SFAS No. 19 could be seen as in support of Observation 1. In SFAS No. 14, "Financial Reporting for Segments of a Business Enterprise"* the FASB required industrial companies engaged in more than one line of businesses to report profits by major product lines. When information asymmetry is two-sided, one would expect the line-of-business (LOB) approach to prevail in cases where firms behave as Coumot competitors, and the alternative aggregate reporting method to prevail in Bertrand industries. This is because the former, clearly the more informative of the two previously generally accepted reporting methods, corresponds to bilateral information exchange, while 74 the latter corresponds to no information exchange.101 Exceptions may be found if firms are risk averse, or if they are motivated to use their information advantage to influence rival firms' production or entry decisions. For example, according to Feltham, Gigler and Hughes [1991], one-sided information asymmetry can develop in the second period of a two-period entry game if the incumbent firm uses aggregate reporting in the pre-entry market. However, such an information advantage may not arise if LOB reporting is employed because in this case entrants can deduce the unknown demand or cost parameter102 by inverting the incumbent's first period reported accounting profits. The above discussion hinges on the researchers' ability to empirically partition industries on the type of competition, the source of uncertainty, the endowment of proprietary information, and/or the decision makers' attitudes toward risk.103 If this is possible, then viewing accounting method choice from an intrinsic value maximization perspective can expand the scope of plausible motives beyond the well-documented tax, agency, or political arguments which empiricists have so often employed. An analogy between these predictions and empirical evidence cannot be drawn however because, unlike the R&D or oil and gas accounting, no profile of the underlying characteristics peculiar to those that voluntarily chose LOB is available from previous research. 102The unknown unit cost can only be imperfectly revealed if total reported cost of production includes an allocation of common facility costs. 103Section 4.8 discusses empirical difficulties one might encounter in selecting sample to reflect the type of industry competition. 75 4.4 Lobbying Efforts And Information Sharing Focusing on the strategic interactions that characterize the behavior in oligopoly settings also allows one to examine why firms might take opposing lobbying stances when accounting policy makers propose changes to disclosure rules.104 In the case of oil and gas accounting, the intense lobbying against SFAS No. 19 undertaken by smaller oil and gas producers might have reflected a concern for an erosion of their ability to compete effectively rather than, or in addition to, a fear of violating debt covenants or lowering management compensation were the SE method adopted. By the same token, this concern for the potential impairment to their own competitive position might have also played a role in the decision by those, which had previously used the consolidated method in the pre-SFAS No. 14 period, to voice discontent with the FASB's Discussion Memorandum on LOB reporting. A similar observation is also made by Feltham, Gigler, and Hughes [1991]. Notwithstanding differences between their analysis and that presented in this thesis, they also make the point that the impact of such rules depends on the underlying assumptions regarding the nature of competition and the source of uncertainty. This approach is to be contrasted with the positive accounting theory-based studies of lobbying behavior. In that literature, firms are hypothesized to be more vocal in expressing their support for (or displeasure with) an impending rule change if, as a result of that change, their political exposure is substantially lowered (or heightened). 76 4.5 International Accounting Standards And Industry Differences In addition to explaining the inter-industry as well as intra-industry variation in accounting method choices among firms in the same country, the sensitivity of predictions to industry characteristics may also contribute to disparities in international accounting standards. For example, the Canadian Institute of Chartered Accountants (CICA) requires firms to capitalize costs incurred in developing a new product or process if a set of conditions105 are met.106 In order to comply with these accounting recommendations, Canadian firms must provide external auditors with sufficient and appropriate evidence to establish the propriety of any R&D costs capitalized. In future years, these deferred R&D expenditures will be subject to continuous monitoring by the firms' auditors, in connection with annual audit, tb ensure that none of the criteria for deferral are violated.107 One reason why these rules differ from those in the U.S. may be that Canadian firms are, on average, smaller and hence less diversified than their American counterparts. This implies a decreasing variance model may be more descriptive of the former group of firms, while a constant variance model of the latter. In other words, according to Proposition 4 5These conditions are described in Section 3450.21 of the CICA Handbook. 106This is in sharp contrast to the corresponding expense-only rule in SFAS No. 2 prescribed by its counterpart, the FASB, in U.S. 107The fact that, under selective capitalization, the expected benefits from deferred R&D costs must be independently verified every year appears to give some credence the earlier conjecture that such an accounting method may be more informative than the alternative expense-only rule. 77 and the numerical examples in the Appendix, the relatively more informative R&D disclosure requirements stipulated by the CICA may very well be justified by the dissimilar consumer welfare effects which result from varying variance assumptions about investment. R&D accounting is, of course, not the only, accounting principles that can differ across international boundaries. In fact, in an exposure draft on the comparability of financial statements in January, 1989, the International Accounting Standards Committee (IASC) identified no fewer than thirteen international accounting standards which, in its opinion, need to be changed to severely reduce the wide range of free choices currently available within its member states. However, the IASC's efforts to reconcile these diverse international accounting practices has so far been greeted with some reservation from both the CICA and the FASB. One explanation for such lukewarm response is that how a piece of accounting regulation on disclosure may affect domestic consumers' wellbeing depends, in a large measure, on the competitive and reporting environment in that country. As a consequence, a case may be made against harmonization of international accounting standards. This observation is, however, subject to several caveats. First, the welfare of consumers is only one of many factors considered by policy makers in setting their respective country's accounting standards. Second, the model employed in the thesis does not capture the dynamics of standards setting process because neither the policy makers nor the consumers have been given a strategic role.108 108That is, the disclosure regulation is implicitly assumed to be a singleplay game as opposed to a multiplay game suggested by Amershi, Demski, and Wolfson [1982], and Wolfson [1980]. 78 4.6 An Alternative Channel For Information Exchange Throughout this chapter, the focus has been on corporate financial reporting as the device through which private cost information can be exchanged. A no less appealing information exchange mechanism is provided by industry trade associations. Information transmitted through trade association publications is, arguably, not as restricted by accounting conventions as is the case with audited financial statements. Moreover, the membership in a trade association is usually discretionary. Kirby [1988] examines trade associations in this role.109 One of the several industry associations mentioned in that study is the Semiconductor Industry Association (SIA). SIA was founded in 1977 and represents over 30 firms which collectively provide 90% of US semiconductor production for electronics, the largest industry110 in the country since 1986. One of SIA's principal activities is to gather and disseminate industry data through its Semiconductor Trade Statistics Program. More recently (1987), the semiconductor industry formed SEMATECH, a consortium of 14 firms whose purpose is to facilitate cooperation in R&D in part through the transfers of information on new technology. Even though the information shared in the semiconductor industry would appear to be 109However, as Kirby pointed out, her study was intended to give a theoretical interpretation of trade associations as bona fide information exchange mechanisms. In fact, out of 64 trade associations surveyed, Kirby [1985] was only able to find 24 operated an information exchange program. This suggests, in many trade associations, other channels existed through which member firms could exchange information. n oThe trade association in that industry, the American Electronics Association, according to Kirby [1988], also plays a role in exchanging private information among its members. 79 industry-wide rather than firm-specific, the presence of these institutions in that industry points to alternative channels to audited financial statements via which information may be disseminated. And more importantly, they can be seen as lending substance to the notion that firms heavily involved in investment activities, such as R&D exploration, might better coordinate their production if outcomes of those activities are shared with rival firms. 4.7 Caveats In Applying The Theory To The Study Of Accounting Method Choice 4.7.1 Ex Post Voluntary Disclosure Earlier it was suggested that, when given the option, firms within the same industry may self-select different accounting methods if they are asymmetrically informed about their own costs. The reason offered was that firms were concerned with how the strategic interaction with rival firms may affect their ability to maintain or enhance their competitive position in the product market. Implicit in this discussion is the assumption that restrictions on ex post voluntary disclosure are self-sustaining. . However, when there are no costs or sanctions to prevent voluntary disclosure, it is in the best interest of firms receiving private signals of low marginal costs to disclose that information voluntarily, notwithstanding their ex ante decision not to disclose. Since rival firms may infer high costs if no information to the contrary is provided, no information exchange is not sustainable as an equilibrium unless opposing incentives, such as positive 80 proprietary costs of disclosure or financial market valuation of firm value are modelled.111 Some reflection makes it clear that, within the framework of the thesis, the only equilibrium would be one involving bilateral information exchange. If, instead of making a nonsignal-contingent commitment on disclosure as is assumed in the thesis, firms commit themselves to disclose either good news or bad news, ex ante, then the conjecture might be that in equilibrium bilateral information exchange will once again emerge. This is because the absence of disclosure would imply bad news for firms that have previously committed to disclose good news in an earlier stage. Limiting the possibility that firms may renege on commitments not to disclose is the observation that the credibility of ex post disclosure requires that lying be detectable and subject to penalties. In the accounting context, since disclosures made under generally accepted accounting principles are independently verified by auditors, it seems reasonable to assume that disclosures outside and subsequent to the release of audited financial statements would be incrementally costly to verify. In fact, the results of previous sections will still go through if such costs exceed the incremental benefits of bilateral over no information exchange. mSee Verrecchia [1983], Darrough and Stoughton [1990], and Feltham and Xie [1991] for illustrations. 81 4.7.2 The Consequence of Nontrathtelling On Ex Post Disclosure From modelling perspective, truthtelling has generally been maintained in the information sharing as well as ex post voluntary disclosure literature. Relaxing that assumption could have as yet undetermined effects on the analysis. Within the accounting context, truthtelling comes into play when information is disclosed in an medium outside the set of audited annual financial statements prepared in accordance with GAAP. Further costs have to be incurred by the disclosing firms to establish the credibility of such disclosure. If the costs of verification are high or if they exceed the benefits from ex post disclosure, then the model predictions regarding the equilibrium information choice of no disclosure would continue to apply.112 Holding aside the issue of truthtelling, the requirement that GAAP be consistently applied from one year to another limits a firm's ability to renege on prior commitment regarding accounting method choice. Turning to lobbying behavior vis-d-vis a proposed accounting rule change, firms may have incentives to misrepresent their true views in order to strategically manipulate policy makers. For example, they may publicly announce a position which they have no intention of implementing. The announcement would advance their own welfare if rival firms could be led to take a similar stance and follow the rule that results. However, the institutional constraints provided by accounting and auditing standards identified above could once again Even without costly verification, equilibria could exist for no or partial disclosure if additional tension is introduced as shown in Feltham and Xie [1991]. 82 mitigate such behavior by eliminating the option to defect. 4.8 Caveats In Applying The Theory To The Empirical Domain The three-stage information sharing model can provide a number of empirical predictions that vary nontrivially with changes in assumptions about the type of competition, the effect of investment on variance, the existence of information asymmetry, and the risk aversion bf managers. The present study has modelled the choice of investment and disclosure in situations where only one of the last three factors is considered. However, any depiction of a real life disclosure environment is likely to require considerations of elements beyond those just mentioned. Holding aside the dynamics of disclosure, at the sample design stage, researchers may be faced with difficulties such as how to define a market,113 and what criteria to use in determining if an industry is monopolistic, oligopolistic, or imperfectly competitive. In addition, large firms often undertake a broad range of unrelated business activities. It is not uncommon to see these firms behave quite competitively in some markets, yet, at the same time, command some monopoly power in others. Another relevant question is whether an investment/output or an investment/pricing model better reflects economic n3There is no simple recipe for defining a market, as is demonstrated by the many debates among economists and antitrust practitioners about the degree of monopoly power in specific industries. 83 reality.114 Under certain conditions,115 Kreps and Scheinkman [1983] show the outcome of a two-stage capacity constrained Bertrand competition game is the same as that of a one-stage Cournot game. This suggests the task of identifying industries along the dimension of pricing or output competition may prove problematic. Rather than making a choice between these two decision variables on a priori grounds, researchers might consider matching the model of oligopolistic behavior to technologies of production and exchange in that industry.116 Further complicating the attempt to go beyond a descriptive extension of the theories to empirical domain is the fact that strategic rivalry is generally mutidimensional. It involves not just investment, risk attitudes, output, and/or price as modelled in the thesis, but also factors such as product quality, post-sale services, geographic location, reputation, advertising, and entry deterrance. Thus, unless empirical tests of predictions from the model control for such factors, they are unlikely to have much power. Finally, the discussion in this chapter suggested firm size as an empirical proxy for the variance effect, proprietary information, and/or risk attitudes. In the past, this same size variable has been drawn upon quite liberally to explain many unexplainable phenomena, as well as to 1 1 4 A common view shared by many economists is that pricing competition more accurately reflects actual behavior, but predictions of Coumot theory are closer to matching the evidence. 115That is, firms produce homogeneous product, demand function is concave, and, given capacity constraint, the rationing rule used is efficient. 116For instance, Shapiro [1987] notes that competition via sealed bids between firms without capacity constraint fits the Bertrand model well, whereas competition to install sunk productive capacity corresponds to Cournot. In general, firms face a more elastic demand and correspondingly lower equilibrium price in the case of pricing competition than in quantity competition. 84 operationalize a host of economic variables, such as political costs. Results on the size variable in this type of empirical research are often conflicting and less than satisfactory.117 It remains an open issue whether using firm size in the present context would fare any better. In conclusion, there are a number of technical difficulties to overcome in applying the usual capital-market based research techniques to test the theory presented in the thesis. Without getting into the specifics, it appears some of these problems may be better understood and solutions identified if predictions are first subject to a rigorous examination in an experimental design setting.118 Of course, that research design by no means represents a panacea.119 However, at a minimum, insights gained from experiments may help point to ways whereby the design of market research may be strenghtened. In any event, it is well beyond the scope of present study to undertake a thorough investigation of either type. Daley and Vigeland [1983] for instance comment that the role of size variables in the study of accounting method choice is not yet fully understood. This observation is also shared by Watts and Zimmerman [1978] in their examination of managers' lobbying positions concerning the proposed general price level accounting. 1 1 8For instance, subjects taking part in the experiment may be instructed to behave or compete in a certain manner. Their risk preferences can be induced through a proper payoff structure. I19The first and foremost problem with experimental studies is how well does the controlled environment approximate real life decision making process. 85 CHAPTER 5 CONCLUSION 5.1 Concluding Remarks Since early 1980's, a large number of analytical research on public policies with respect to disclosure has appeared in the accounting literature. Traditionally, studies have focused on the firms' private incentive to exchange information following signal realization. The thesis departs from this ex post voluntary disclosure-based research to approach the issue of public disclosure in a setting where decision is committed to prior to the receipt of firms' own private signals. At an analytical level, the innovations to previous precommitment-based studies of information sharing in the industrial organization literature include imbedding an investment stage into usual two-stage information sharing games, and allowing risk aversion on the part of decision makers. Irrespective of the type of competition, earlier results on information choice are found to be quite robust to the addition of investment and the introduction of different variance effects from investment, provided the standard assumptions of risk neutrality and two-sided information asymmetry are maintained. Reversal does occur however if, as in a study by Kirby [1988], total cost functions are allowed to be nonlinear in the level of output production. 86 Using numerical examples, it is demonstrated that disclosure preferences are sensitive the assumption about risk preferences. More specifically, for a given form of competition, the aversion to variation in profit distributions often leads to an information choice contrary to that obtained in a risk neutral setting. This offers a theoretical explanation for inter-industry as well as intra-industry variation in disclosure choice. Disparity in reporting methods within the same industry can also arise if only one of the competing firms acquires private information about its own marginal cost. The three-stage game structure allows a rigorous analysis of disclosure effects on investment. In a risk neutral world, investment is innocuous so long as only expected marginal costs are affected by such investment. However, similar statements cannot be made when the variance effects from investment are nonconstant. In general, the equilibrium levels of investment are a function of information as well as competition. With the introduction of risk aversion, however, disclosure policy is shown to entail a nontrivial effect on investment even in the benchmark case. Previous studies have generally reported that consumer surplus is not maximized by the firms' information choice. Nonetheless, the thesis identifies a broad range of demand parameter values and some increasing cost variance functions under which both the consumers' and the producers' interest is aligned. The alignment is a direct consequence of earlier results on the disclosure effects on investment and the passive role consumers assumed to play in the model. Moving to applications, predictions from the thesis may have a number of implications vis-d-vis accounting disclosure policies. At a minimum, the intrinsic value maximization 87 view of public policies with respect to disclosure brings new insights into many widely-researched accounting issues by drawing upon economic consequences quite apart from the familiar tax or wealth transfer arguments often-cited in the accounting literature. More importantly, results from the thesis suggest that empiricists need to have a better understanding of firms in the industries affected by the policy pronouncements when they construct hypotheses and design samples for testing purposes. Factors omitted before but are potentially important ^ include the nature of product market competition, the variance effects from investment, the risk preferences of decision makers, and the asymmetry in the cost of acquiring private information. However, it is beyond the scope of the present investigation to address the issues of how one might discriminate empirically between theories set forth in the thesis and those previously advanced in the literature. Despite its relevancy, it is likewise not the focus here to pursue the extent to which the present results may enhance the power of tests intended to explain both cross-sectional variation in voluntary accounting method choice and other policy-induced reactions to mandated accounting rule changes. 5.2 Directions For Future Research There are several directions along which the thesis may be extended. Throughout most of the analysis, duopolists are assumed to be symmetric in every respect. However, divergence in intra-industry firm-specific features, such as firm size, was relied upon in applying the results to reexamine several empirical issues. A reasonable avenue of future 88 research would be to explicitly model differences in firm size. Another variant to the present study would be to allow an increasing variance effect from investment for one of the duopolists, and a decreasing or constant variance effect for the other. On the other hand, models may be restructured such that either one firm is risk neutral while its opponent is risk averse, or both firms display varying degrees of risk aversion. Sources of asymmetries that set firms apart in real life, of course, are not limited to those considered in this study. Size differences, for instance, may be captured by the presence of fixed costs or capacity constraints facing some but not all of the firms. It remains to be shown if similar predictions and empirical implications might emerge notwithstanding these changes. It is well known that there is no general theory of oligopolies. The decision to focus on Coumot and Bertrand duopolies in the thesis reflects largely the popularity accorded both forms of competition in the industrial organization literature, rather than a conviction that such behavior holds in the real world. As an alternative, the noncooperative games of information sharing and investment may be modelled as Stackelberg leader-follower games. It follows from studies by Gal-Or [1985,1987], all else held equal, the firm which takes the lead in setting output level or price will reverse its information choice and/or investment decisions from what has been predicted in the thesis. That is, changing the order of play can yield a similar effect as that brought about by the introduction of either risk aversion or one-sided information asymmetry assumption in a simultaneously-played setting. 89 Another type of extension would be to bring in capital structure. Using a two-stage capital structure/output game model, Brander and Lewis [1986] show that debt-financed firms generally act more aggressively than pure-equity firms in the product market. In their model, there is no resolution of demand uncertainty before financing decisions are made. One might extend that line of enquiry to a stochastic duopoly by imbedding the financing decisions in place of investment decisions in a three-stage information sharing game. One may then speak to the consequences of disclosure on a firm's financing choice. A conjecture would be that there exists a nontrivial relationship between information choice and capital structure. To the extent this is true, then there may also be scope for empirical research. However, characterizing such relationship is likely to be as difficult as it would be interesting. Beyond these, the precommitment based disclosure research also has the potential to capture why there are differences in the international accounting standards, or how the international trade policies and national tax policies are determined. Although results from this type of research are likely to be context-dependant, it is not atypical of studies of oligopolistic behavior. At a minimum, this new research paradigm offers an alternative view of the world. In the process, it contributes to the goal of eventually coming up with an unifying theory of oligopoly. 90 BIBLIOGRAPHY Amershi, A., J. Demski, and M. Wolfson. "Strategic behavior and regulation research in accounting." Journal of Accounting and Public Policy, [1982], pp. 19-32. Ball, R. "Discussion of accounting for research and development costs: The impact on research and development expenditures." Journal of Accounting Research, Vol. 18, [supplement 1980], pp. 27-37. Basar, T. and Y. Ho. "Informational properties of the Nash solutions of two stochastic nonzero-sum games." Journal of Economic Theory, 7, [1974], pp. 370-387. Brander, J. and B. Spencer." Strategic commitment with R & D : The symmetric case." The Bell Journal of Economics, Vol. 14, No. 1, [Spring 1983], pp. 225-235. Cheng, L. "Assisting domestic industries under international oligopoly: The relevance of the nature of competition to optimal policies." American Economic Review, Vol. 78, No. 4, [September 1988], pp. 746-758. Clarke, R. " Collusion and the incentives for information sharing." The Bell Journal of Economics, Vol. 14, [1983a], pp. 383-394. Clarke, R. " Duopolists don't wish to share information." Economics Letters Vol. 11, [1983b], pp. 33-36. Clarke, R. " Unilateral announcement of information in stochastic duopoly." Economic Policy Office Discussion Paper, U.S. Department of Justice Antitrust Division, [1985]. Collins, D., M. Rozeff, and D. Dhaliwal. "The economic determinants of the market reaction to proposed manatory accounting changes in the oil and gas industry: A cross sectional analysis." Journal of Accounting and Economics 3, [March 1981], pp. 37-71. Daley, L. and R. Vigeland. "The effects of debt covenants and political costs on the choice of accounting methods: The case of accounting for R&D costs." Journal of Accounting and Economics 5, [December 1983], pp. 195-211. Darrough, M. "Disclosure policy and competition: Cournot and Bertrand." Working Paper, The Columbia University, [September 1990]. Darrough, M. and N. Stoughton. "Financial disclosure policy in an entry game." Journal of Accounting and Economics, Vol. 12, [1990], pp. 219-243. 91 Deakin, E. "An analysis of differences between non-major oil firms using successful efforts and full cost methods." The Accounting Review, [October 1979], pp. 722-734. Dixit, A. " A model of duopoly suggesting a theory of entry barriers." The Bell Journal of Economics, Vol. 10, No. 1, [Spring 1979], pp. 20-32. Dixit, A. " The role of investment in entry-deterrence." Economic Journal, Vol. 90, [March 1980], pp. 95-106. Dontoh, A. "Voluntary disclosure." Journal of Accounting Auditing and Finance, Vol. 4, No. 4, [Fall 1989], pp. 480-511. Dukes, R., T. Dyckman and J. Elliott. "Accounting for research.and development costs: The impact on research and ddevelopment expenditures." Journal of Accounting Research, Vol. 18, [supplement 1980], pp. 1-26. Elliott, J., G. Richardson, T. Dyckman, and R. Dukes. "The impact of SFAS No. 2 on firm expenditures on research and development: Replications and extensions." Journal of Accounting Research, Vol 22, No. 1, [Spring 1984], pp. 85-102. Feltham, G.A., F.B. Gigler, and J.S. Hughes. "The effects of line of business reporting on competition in oligopoly settings." Working Paper, The University of Minnesota, [July 1991]. Feltham, G.A., and J. Xie. "Voluntary financial disclosure in an entry game with continua of types" Working Paper, The University of British Columbia, [May 1991]. Fershtman, C , and K. Judd. "Equilibrium incentives in oligopoly." The American Economic Review, Vol. 77, No. 5, [December 1987], pp.925-940. Fried, D. " Incentives for information production and disclosure in a duopolistic environment." Quarterly Journal of Economics, 99, [May 1984], pp. 367-381. Gal-Or, E. " Information sharing in oligopoly." Econometrica, Vol. 53, No. 2, [March 1985], pp. 329-343. Gal-Or, E. " First mover and second mover advantages." International Economic Review, Vol. 26, No. 3, [October 1985], pp. 649-653. Gal-Or, E. " Information transmission - Cournot and Bertrand equilibria." Review of Economic Studies, LIII [1986], pp. 85-92. Gal-Or, E. " First mover disadvantages with private information." Review of Economic Studies, LIV [1987], pp. 279-292. 92 Gal-Or, E. "Managerial Compensation in Oligopoly." Working Paper, The University of Pittsburgh, [January 1991]. Horwitz B. and R. Kolodny. "The economic effects of involuntary uniformity in the financial reporting of R&D expenditures." Journal of Accounting Research, Vol. 18, [supplement 1980], pp. 38-74. Hughes, J. and J. Kao." Economic implications of alternative accounting rules for research and development costs." Contemporary Accounting Research, Vol. 8, No. 1, [Fall 1991]. Hvid, M. "Risk-averse duopolists and voluntary information transmission." The Journal of Industrial Economics, Vol. XXXVIII, No. 1, [September 1989], pp. 49-64. Kirby, A. "Trade associations as information exchange mechanisms." Ph.D. Dissertation, Stanford University, [December 1985]. Kirby, A. " Trade associations as information exchange mechanisms." RAND Journal of Economics, Vol. 19, No. 1, [Spring 1988], pp. 138-146. Kreps, D. and J. Scheinkman "Quantity pre-commitment and Bertrand competition yield Coumot outcome" Bell Journal of Economics, Vol. 14 [1983], pp. 326-337. Li, L. " Cournot oligopoly with information sharing." RAND Journal of Economics, Vol. 16, No. 4, [Winter 1985], pp. 521-536. Palfrey, T. "Risk advantages and information acquisition." The Bell Journal of Economics, Vol. 13, [1982], pp. 219-224. Rogerson, W. " Aggregate expected consumer surplus as a welfare index with an application to price stabilization." Econometrica, Vol. 48, No. 2, [March 1980], pp. 423-436. Schmalensee, R. " Advertising and entry deterrence." Journal of Political Economy, Vol. 90, pp. 636-653. Shapiro, C. " Exchange of cost information in oligopoly." Review of Economic Studies, LEI, [1986], pp. 433-446. Shapiro, C. " Theories of oligopoly behaviour." in Handbook of Industrial Organization, North Holland, [1987], pp. 1-100. Shubik, M. A game-theoretic approach to political economy, The MIT Press, [1984]. 93 Singh, N and X. Vives "Price and quantity competition in a differentiated duopoly" Rand Journal of Economics, Vol. 15, No. 4, [Winter 1984], pp. 546-554. Verrecchia, R. "Discretionary Disclosure." Journal of Accounting and Economics, 5, pp. 179-194. Vigeland, R. "The market reaction to Statement of Financial Accounting Standards No. 2." The Accounting Review, Vol. LVI, No. 2, [April 1981], pp. 309-325. Vives, X ." Duopoly information equilibrium: Cournot and Bertrand." Journal of Economic Theory, 34, [1984], pp. 71-94. Wagenhofer, A. "Voluntary disclosure: Effects of political costs and entry." Journal of Accounting and Economics, Vol. 12, No. 4, [March 1990], pp. 341-364.. Ware, R." Sunk costs and strategic commitment: A proposed three-stage equilibrium." The Economic Journal, 94, [June 1984], pp. 370-378. Watts, R. and J. Zimmerman. "Towards a positive theory of the determination of accounting standards." The Accounting Review 53, [January 1978], pp. 112-134. Watts, R. and J. Zimmerman. Positive Accounting Theory, Prentice Hall, [1986]. Wolfson, M. "Discussion of the economic effects of involuntary uniformity in the financial reporting of R&D expenditures." Journal of Accounting Research, Vol. 18, [supplement 1980], pp. 75-83. FIGURE 1 94 T H E SEQUENCE OF EVENTS AND ACTIONS FOR T H E THREE-STAGE INFORMATION SHARING/ INVESTMENT/OUTPUT (PRICING) GAMES 1st stage 2nd stage Information Investment sharing decision decision 3rd stage Output or pricing Information decision sharing Nature draws random component of marginal cost functions for each duopolist End-of-period Profits are realized by both duopo-lists 95 FIGURE 2 T H E SEQUENCE OF EVENTS AND ACTIONS FOR THE TWO-STAGE INFORMATION SHARING/OUTPUT (PRICING) GAMES GIVEN ONE-SIDED INFORMATION ASYMMETRY 1st stage Information sharing decision Information sharing 2nd stage Output or pricing decision Nature draws random component of marginal cost functions for each duopolist End-of period Profits are realized by both duopo-lists F i r m 1 observes that private cost signal; f irm 2 does not. 96 APPENDIX 1 CHAPTER 2 PROOFS Proof of Proposition 1 Cournot Duopoly First note that in reaching their investment decisions, firms consider the following set of first-order conditions: = 2[Eari(tYi)] [3E0)i(tYi)/9xi] + [3 Varivax J - 1 = 0, i = 1,2, t = dd,nn. But, from Equations (20)-(21), it is clear that for a pair of arbitrarily fixed levels of investment x = (x,,x2), E(ddYj) = EC 1^), and thus [3E(Vi)/3xj] are identical for all f = dd,nn. Constant a2 By assumption, output variance is independent of investment. The set of first-order conditions (Al) therefore becomes: [9tJi(x)/9xi] ( A l ) [^(xyaxj (A2) = 2[Emi(Vi)][aEa)i(tYi)/dxi] - 1 = 0, i = 1,2, t = dd,nn. 97 This implies d \ = ""x., i = 1,2. Increasing g 2 Case In this case, comparing pairwise output variances at relevant spending levels is equivalent to ordering the equilibrium levels of investment across alternative information structures. {[aVarryiVaxJ-^Varryp/axJ} (A3) ( X j , X 2 ) = ( " " X j , x2) = {[8v(nnxi)v'(nnxi)o2]/[16(l + 2K)2(3 + 2K)2(1 + K ) 2 ] } x [16(1 + K ) 4 - (1 + 2K)2(3 + 2K) 2 ] >0 The inequality follows from the fact that v'(x) > 0 (by assumption), and [16(1 + K ) 4 - (1 + 2K)2(3 + 2K) 2 ] > 0. Decreasing a 2 Case In this case, v'(x) < 0 by assumption. Hence, (A3) < 0, in other words, ("x,,^ < Bertrand Duopoly It follows from (A17)/(A18) that the sets of first-order conditions characterizing optimal investment under bilateral and no information exchange are given by (A4) and (A5), respectively. 2B.(p7 - 2(312)[l/(4f312 - p22)2] (A4) x [(2p\ + p2)cc + k k u T x p + (p2 2 - 2p1 2)u(d dx i)]p.'(d dx i) + [2p1(2p i 2 - p22)2cy2/(4f312 - p 2 2) 2]v( d\)v'( d dx l) - 1 = 0, i,j = 1,2, j * i. x [(2p, + p 2)a + p ^ r x j ) + (p 2 2 - 2p i 2 )p : rx i ) ]p . ' r x i ) + (^o2/2)vrxxr^ -1 = 0, i,j = 1,2, j j= i. Replacing ( d d X i , d d x 2 ) with O^,1"^) in (A4) and equating to (A5) result in + [2p1(2p12 - P ^ V ^ P , 2 - p 2 2 ) 2 ]vrx i )vTx i ) - 1 = P p . a 2 / ^ 2 - P22)2]{(2P12 - p 2 2) 2 - [(4p,2 - PaWlJvrxPv'C-x,) But, {•} < 0. Thus, If v ' f x , ) = 0 (constant a 2), then, (A6) = 0, i.e., d d Xj = ""x^ If v ' C X j ) > 0 (increasing o 2), then, (A6) < 0, i.e., < mx-If vTxD < 0 (decreasing a2), then, (A6) > 0, i.e., d d Xj > 2P4(P22 - 2p12)[l/(4p12 - p22)2] (A5) 1 - (P 1a 2/2)vrx i)vTx i) (A6) QED. 99 Proof of Proposition 2 First note that V*) ' (A7) = (1 + K J K E ^ X ) ] } 2 + VartV*)]} - % i = 1,2, t = dd,dn,nd,nn. Constant G 2 From Proposition 1, it is easy to see that, for a pair of optimally selected levels of investment (tx1,tx2), the expected output levels are identical for all information structures. Therefore, the preference orderings of expected profits are completely determined by those of output variances as described in Lemma 1: Lemma 1 Var(d d y i) > VarCy,.), Var^y.) > V a r ^ ) , Var(ddy2) > Var^y,), VarC^) > V a r ^ ) -Proof of (A8) in Lemma 1 [Var(ddyi) - Var(ndy,)] = [cr7(l + 2K)2(3 + 2K)2] {[1 + 4(1 + K)2] - [(1 + 2K)2(3 + 2K)2/4(1 + K)2]} = [cr74 (1 + K)2(1 + 2K)2(3 + 2K)2] [l + 16(1 + K)4 - (1 + 2K)2(3 + 2K)2] > 0 . (A8) (A9) 100 rVarTyi) - Var ryO] = [oVd + 2K)2(3 + 2K)2]{4(1 + K) 2 - [(1 + 2K)2(3 + 2K)2/4(1 + K)2]} = [cr74(l + K)2(1 + 2K)2(3 + 2K)2][16(1 + K)4 - (1 + 2K)2(3 + 2K)2] >0. • . (A9) can be analogously proved. Therefore, d d J , ( d d x ) > n d J , ( n d x ) , ^ J ^ x ) > ""Jj.Cx) (A10) d d J 2 ( d d x ) >- d n J 2 ( d D x) , n d J 2 ( n d x ) > ""JjCx) Increasing cr2 Case It follows directly from the constant variance case that, for a pair of arbitrarily fixed levels of investment x = (xj,x2), (a) E(ddyi) = ECY) = ECY) = ECVi) , i = 1,2 (b) VarCVO > VarCVj, VarCy,) > VarC?,.); Var(ddy2) > V a r ^ ) , VarCdy2) > VarCy,). (c) ^ ( x ) > "^(x) , ^ ( x ) > ^ ( x ) ; d d J 2 (x ) > ^ ( x ) , n d J 2 (x ) > mJ2(x). But, conditions can be imposed on the inverse demand intercept "a" such that expected profit functions, ^Cx), i = 1,2, are concave in the levels of investment, and that reaction functions are stable. For instance, under bilateral information exchange, this is given by: 101 a > [1/4(1 + K)(1 + 2K)u"(d dx,)]{8(l + K) 2 [v'( d d x,)] 2 + 8(1 + K O M ^ V C ^ •i) (All) + 8(1 + K^tn'C^Xi)] 2 + 8(1 + K)V(ddx,)li"(ddx,) - 4(1 + K)u(xddx2)u"(' :ddx1)} The concavity condition, along with (a)-(c) and Proposition 1, ensures that bilateral information exchange is the dominant information choice for both duopolists. Decreasing c 2 Case Proof is similar to the above case, thus will not be repeated. Proof of Proposition 3 Constant G 2 The levels of output production for each of the four information regimes can be derived by substituting equilibrium pricing strategies (26)-(29) to (17). Expressing the end-of-period profit functions (14) in terms of the equilibrium pricing strategies and the resultant output level, and taking expectation yield the following expected profits at the beginning of first stage: = - PV)2] x {[(2(3, + p2)oc + p \ M d d X j ) + (P? - 2p 1 2)u( d dx i)] 2 + [p,2R22 + (2p,2 - p,2)2]^} - d d X i QED E ( d d 7 t i ) (A12) 1 0 2 E("X) (A13) = [R1/(4R12 - p22)2] x {[(2p, + p2)rx + P ^ r x j ) + (p 2 2 - 2p 1 2 )urx i ) ] 2 + [(4P,2 - P ^ o 2 } - " x , E(dn7ci) (A14) = [Pi/(4pt2 - P22)2] x {[(2pt + p 2)a + - P ^ r x j ) + (P22 - 2p,2)u(d nx i)]2 + (2P,2 - B^o 2 } - ""x, E C \ ) (A15) = [PI/(4PI2 - P22)2] x {[(2P, + p 2)a + P^UTXJ ) + (p 2 2 - 2p 1 2)uf dx i)] 2 + + (4ft 2 - p ^ W } - N D X I where i,j = 1,2, j ^ i. However, according to Proposition 1, d d Xj = ""xj = "% = D\, thus, E P V ^ x ) ] - E[ d d7C i( d dX)] = [p1p22/4(4p12 - p22)2](-3p22 + SP^cr2 >0. E p ^ x ) ] - E t ^ ^ x ) ] = [p1P22/4(4p12 - p22)2](-3p22 + g p ^ o 2 > 0. In other words, 103 •"J^x) > ndJ!(ndx), ^C-x) > "L.Cx) (A16) ddJ2(ddx) > ""JaC'x), ndJ2(ndx) > ""Jarx) Increasing g 2 For an arbitrarily fixed levels of investment, the expressions for the expected profit functions in each of the information regimes at the beginning of first stage are as follows: EO J (A17) = [p,/(4p,2 - p22)2] x {[(2^ + p2)cc + p\|32|i(Xj) + (\V - 2P12)|i(xi)]2 •'+ [pY(32V(x2) + (2P,2 - P 2 2)V(x,)]a2} - xs ECX> (Ai8) = [Pi/(4p!2 - P22)2] x .{[(2B, + p 2)a + p1p2u(x j) + (p 2 2 - 2p1 2)u(x i)]2 + v2(x i)[(4p12 - p ^ W } - x ; 'ECiTi) (A19) = [f31/(4f312 - f322)2] x {[(2p, + p 2)a + p ip2u(x j) + (p2 2 - 2p i 2)u(x i)] 2 + v 2(x i)(2p i 2 - p ^ V } - x, ECX) (A20) = [P,/(4p,2 - p22)2] x {[(2p, + P2)a + p,p2u(X j) + (P22 - 2p i 2)M.(x i)]2 + {^(xpp^p,2 + v2(x i)[(4p12 - P22)2/4] Ja2} - X i 104 where i,j = 1,2, j ^ i. It follows directly from the above case that no information exchange is a dominant information choice for both firms. Once again, (A 16) holds provided the intercept for the direct demand functions "oc" are sufficiently large so that both the second-order conditions to the investment game and the stability conditions are satisfied. In the case of bilateral information exchange, this is given by: a > [1/(2(3, + (32)]{-P1(32u(x2) + (2pt2 - P22)[u.(x,) + [u.'(Xl)]2 + v(x1)v"(x1)o2 (A21) + [v'(x1)]2a2]} Decreasing a2 By analogy. QED. Proof of Proposition 4 Cournot Duopoly Recall from Equation (35) that: CS(ty1,ty2) (A22) = (l/^Y,,2 + [uTXl) - ZJVI + M\) - zJVz + (\ + %) where t = dd,nn. When K = 1 and v(Xj) = 1, optimal output strategies, according to Equations (20)-(21) become: dd. (A23) = (l/3)[a - 2urXi) + UTXJ) + 2z; - Zj] nn. 105 °yi (A24) = (l/3)[a - 2u(nnxi) + UOCJ) + (3/2)zJ i,j = 1,2, j ^  i, giving rise to the following aggregate outputs at the industry level: d d Y 0 (A25) = (l/3)[2a - u.(ddx,) - u.(ddx2) + z, + zj ""Yo (A26) = (l/3)[2a - uCx,) - urx 2 ) + (3/2)(z, + z2)] Var(ddY0) = (2o2/9) ; VarrY 0 ) = (o2/^. (A27) Hence, expected consumer surplus, defined in Equation (36), is given by: E(ddCS) (A28) = (l/2){(l/9)[2a - |i(ddXl) - p.(ddx2)]2 + (2cr79)} •+ uC'VEryi) + u(ddx2)E(ddy2) - (402/3) + (dd Xl + ddx2) ErCS) (A29) = (l/2){(l/9)[2a - uTx,) - urx 2)] 2 + (a2/2)} + nrx^EC-y,) + urx2)Ery2) - a 2 + r x , + "%) The total surplus, ,on the other hand, is: +TS = U( ty1, ty2) (A30) = a^o - (1/2XV + Va 2) " V i V z / t = dd,nn. 106 Thus, the following expected total surplus: E(ddTS) (A31) = (a/3)[2a - nC^) - u(ddx2)] - (l/2)(l/9){[a - 2M.(ddXl) + u(ddx2)]2 + [a - 2u<ddx2) + uTx,)]2} - (l/2)(10o2/9) + (4o2/9) - (l/9)[a - 2u(ddXl) + u(ddx2)][a -2u(ddx2) + ECTS) (A32) = (a/3)[2a - M-rxJ - urx 2)] - (l/2)(l/9){[a - 2 u C x , ) + p.rx2)]2 + [a - 2 u C x 2 ) + uPx,)]2} - (U2)(c?l2) - (l/9)[a - 2p.rxt) + p.rx2)][a - 2urx 2) + urx,)] From (A28)/(A29), expected consumer surplus given bilateral information exchange is strictly lower because: E(ddCS) - ErCS) = (cr79) - (4o2/3) - (o74) + a 2 = -(17a736) <0. From (A31)/(A32), expected total surplus given bilateral information exchange is strictly higher because: E(ddTS) - ECTS) = -(5cr79) + (4cr79) + (a2^) = (5a736) > 0. 107 Bertrand Duopoly With constant variance effect, v(Xj), i = 1,2, is equal to 1. Evaluating direct demand functions, Vi = a - PifPi + P2tpj, ij = 1,2, j ^ i, at optimal pricing strategies given in Equations (26) and (27) to yield firm i's output level at the end of 3rd stage for each of the two extreme information regimes: "Vi (A33) = [py(4p7 - P22)][(2p\ + B2)cx + PiP2u(ddxj) + (P22 - 2p12)u(ddxi) - p.p^ + (2^2 - p 2 2)Z i] n ny i (A34) = [ p ^ p , 2 - P22)][(2p, + p2)a + p^^rxj) + (P22'- 2p12)p.rxi)] + ( p ^ ) ^ where i,j = 1,2. j + i. For ease of exposition in the subsequent analysis, the following shorthand notations will be used: , D, = [(2p, + p2)a + P.p.uCx,) + (p 2 2 - 2p12)u( tx1)] (A35) D 2 = [(2P, + P2)a + p.P^Cx,) + (P22 - 2p12)u( tx2)] (A36) where t = dd,nn. The aggregate levels of output production for both regimes are as follows: D D Y 0 (A37) = [P1/(2P1 - p2)][2cx - (P, - P2)p.(ddXl) - (P, - p2)u.(dx2) + (Pt - P2)z, + (P, - P2)zJ " " YQ (A38) = ['P,/(2P1 - P2)][2a - (P, - P2)M.rx,) - (P, - p2)uTx2)] + [p!(Zl + zJ/2] According to Equation (34), consumer surplus is given by: l dCS (A39) a d dY 0 - (b1/2)[B1/(4B12 - P22)]2{(D,2 + D2 2) + ( Z l 2 + z22)[(2p12 - B22)2 + Bj2B22] - 4B1B2(2B12- - p 2 2)z l Z 2 + 2D,[(2Pi2 - P22)z, - P,p2zJ + 2D2[(2p 1 2-P 2 2)z 2-P IP 2z 1]} - b. tp^P, 2 - P22)]2{D1D2 + D^tf - P^z, - P^z,] + D2[(2p1 2 - P22)z, - p.fczj + Z1Z2[(2P12 - p 2 2) 2 + p,2p22] - (ZL2 + Z22)(2P12 - B/Jfck} - [p1/(4p12 - P22)2]{D1[(2p1 + p 2)a + p,p2u(ddx2) + 2p12u(ddx1) - p,P2z2 - 2p,2z1] + D2[(2P1 + P2)a + p^u^x,) + 2pi2u(ddx2) - p ip 2 z 1 - 2p,2z2] + [(2P, + P2)a + pIP#(ddx2) + 2p12u(ddx1)][(2p12 - p 2 2 ) Z l -p,P2zJ • ' . + [(2pt + p 2)a + P.p^OO + 2p,V(ddx2)][(2p12 - p^z j - P,P2ZL] - 2P1p2[(2p12 - P22) - 2p1 2]z1z2 + PT2[P22 - 2(2P,2 - p 2 2)]( Z l 2 + z,2)} + [u(ddXl) - Z l ] d d y i + [|i(ddx2) - z j * ^ + (ddXl.+ ddx2) m CS (A40) a™Y0 - (b1/2)[P1/(4f312 - p22)]2(D,2 + D2 2) - (b1/2)[p1/(4p12 - p 2 2)](D l Z l •+ D2z2) - (b1/2)(z12 + z22)(p12/4) - b2[P1/(4p12 - P22)]2D,D2 - b2(P12/4)z1z2 - b2[P12/2(4P12 - P2 2)](D l Z 2 + D2z,) - [p1/(4p i2 - p22)2][(2p i + p2)cc + p!p 2urx 2) + 2p 1V( n nx1)]D 1 - [P./C4P,2 - p22)2][(2p1 + P2)tx + P ^ r x . ) + 2p12u(nnx2)]D2 - [P1/2(4p12 - p22)][(2p, + p 2)a + 2p 1 2|irx 1) + p.p.urx^z, - [P1/2(4p12 - P22)][(2P1 + p 2)a + 2p12u(nnx2) + P^uCx,)^ + [p1/2(4p i2 - p 2 2)](D l Z ) + D 2 Z 2) + (p^Kz , 2 + z,2) + [iiCx,) - z j - y , + [urx 2) - zTyi + Tx, + mx2) 1 0 9 Note that the terms involving a, b u b2, in (A39) and (A40) above will all appear in the expressions for the total surplus in the bilateral and no information exchange regimes, respectively. -Expected consumer surplus, E(ddCS) and E^CS), can be easily derived by taking expectations of (A39) and (A40) over zx, respectively. Expected consumer surplus given bilateral information exchange is strictly higher because: E(ddCS) - EPCS) = - 2(b1/2)[P1/(4pY - B22)]2[(2p7 - p7)2 + JVPVJO2 - 2(b1/2)(-l)(B12/4)o2 + 2b2[B1/(4B12 - p22)]2(2p,2 - p22)p1B2o2 - [p^p, 2 - p22)2]2o2p12[p22 - 2(2P,2 - p22)] - 2[p1/(4p12 - p22)](2pi2 - p 2 V - 2(p1/4)o2 + 2(p1/2)o2 . = - 2(b1/2)[p1/4(4p12 - P22)]2o2[(16P14 - 12p,2p22 + 4p24) - (16p,4 - Sp^p,2 + p24)] + 2b2[p1/(4pi2 - p22)]2(2p,2 - P 2 2)P1p2o2 - 2[p1/4(4p12 - M?W{(\2$ftz2 - 16P,4) + (32P,4 - 24p12p22 + 4p24) - (16P,4 - 8p t 2P 2 2 + p24)] = -2(b1/2)[P1/4(4p12 - p , 2 ) ] 2 ^ ^ 2 + 3p24)Q2 t 2b2[P1/(4p12 - p22)]2(2P,2 - ^ P^ce - 2[p1/4(4p12 - p V f l H P f o 2 + 3 p 2 V >0. On the other hand, expected total surplus is also strictly higher under bilateral information exchange because the first two terms after the last equality above are both positive. QED. Numerical Examples Assume. K = 0, marginal cost functions take the following functional form: q = U-(Xi) - v ( X i ) Z i ; p = 0 ; • -• : • . ' •' - x t ' u(xi).= e ; u ' ( X i ) = - e <0; (A41) " X i ^"(Xi) = e >0 ; ; v(x,) = (X i) ( 1 / 2 ); v'(Xl) = (l/2)(Xi)(-1/2) > 0 ; v"(x,) •'= (-l/4)(xi)<-"3/2> < 0 , i"= 1,2. [dECXVdx,] (A42) =.' (-4/9)"[a - 2u.(ddxj) + |i(ddxj)]u'(ddxi) + (8/9)v(ddxi)v'(d\)o2 - 1 = (4/9)(a - ex)ex + (8/9)( 1/2)0^ - 1 : (4/9)(a - ex)e-x + (4/9)a2 - 1 = 0. '. a = {[1 - (4/9)0* + (4/9)e2x]/[(4/9)e2x]} (A43) [a2E(d\)/a2Xi] (A44) : (-4/9)[a - 2u(ddXi) + u(ddxp]^ ;+(8/9)v(ddx1)v,,(ddXi)o2+.(8/9)[y'(ddxi)]2o2 • (8/9)[M.'(ddxJ2 - (4/9)[a - 2u(ddXi) + ii(d\)]\i"(d\) • (8/9)e'2x - (4/9)(a - e-.x)ex = (4/9)ex(3ex - a) I l l < 0, ij = 1,2, j ± i. where the inequality follows from the second-order conditions. It must be true that a > 3 (because 3e"x < 3, V x > 0). By fixing a2, one can derive a range of values for "a" within which E(ddCS) > ECCS). Three sets of exogenously specified parameters (a.o2) are identified below for illustrative purposes: (a) a 2 = (1/200), a e (3.245, 9.6939343) (b) a 2 = (1/20) , a e (3.200, 9.6535678) (A45) (c) o 2 = (1/4) , a e (3.000, 9.4763686). Remarks: (1) Lower bound for "a" is determined from (A43), first by setting o2 at one of the three values indicated above, then letting x —> 0. (2) Since "a" cannot be lower than 3, this implies that o2 must be bounded from above by (1/4) in these examples. A step-by-step derivation leading to the conclusion that E(ddCS) > E(""CS) will next be presented for one of numerical examples, a = 6.3480984, and a 2 = (1/20). Bilateral Information Exchange From [3E(dd7i)/8x] = (-4/9X6.3480984 - ex)(-ex) + (8/9)x1/2(l/2)(l/20)x1/2 - 1 = 0 we get ddx = 1, (a global maximum). 112 [a2E(dd7i)/3x2] ddx = 1 = (8/9)(-e1)2 - (4/9)(6.3480984 - e'V 1 + (S^Xl^Wa^O) + (8/9)(l)1/2(-l/4)(l)"3/2(l/20) = -0.8574799 <0. ddJ(ddx) = (l/9)[(6.3480984 - e1)2 + (1/20)(5)(1)] -1 = 3.0014462. E[ddy(ddx)] = (1/3X6.3480984 - e1) = 1.9934062. Var[ddy(ddx)] = (1/9X1/20X5X1) = 0.027777. Var[ddY0(ddx)] = (l/9)(l/20)(2) = 0.011111. E[ddY0(ddx)] = (l/3)[(6.3480984)(2) - e1 - e'1] = 3.9868124. E[ d dCS( d dx)] = (1/2)[(3.9868124)2 + 0.011111] = 7.952892. No Information Exchange From [dEOO/dx ] = (-4/9X6.3480984 - ex)(-e-x) + (l/2)(x)1/2(l/2)(x)-,/2(l/20) - 1 = 0. we get ""x = 0.98945, (a global maximum). [a2EC7t)/ax2] ""x = 0.98945 = (8/9)(-e-°- 9 8 9 4 5) 2 - (4/9)(6.3480984 - e-°- 9 8 9 4 5)(e- 0- 9 8 9 4 5) = - 0.8646403 <0. • " J r x ) = (l/9)[(6.3480984 - e0'98945)2 + (9/4)(l/20)(0.98945)] - 0.98945 = 2.9808528. E r y r x ) ] = (l/3)(6.3480984 - e 0 9 8 9 4 5) = 1.9921056. Varryrx)] = (1/20)(1/16)(4)(0.98945) = 0.0123681. VarrY o rx)] = (1/20)(1/16)[(4)(0.98945) + (4)(0.98945)] = 0.0247362. Ery 0rx)] = (l/3)[(2)(6.3480984) - e 0 9 8 9 4 5 - e098945] = 3.9842113. ErCSrx) ] = (1/2)[(3.9842113)2 + (0.0247362)] = 7.9493375. Remarks: Given that ddJ(ddx) - ""JCx) = 0.0205934 > 0, and E[ddCS(ddx)] - EpCSCx)] = 0.0035545 > 0, bilateral information exchange is clearly consistent with the interests of both the producers and the consumers. QED. Proof of Proposition 5 115 Cournot Duopoly Sharing of z, With Firm 2 At the beginning of second-stage output game, decision problems facing firms are: Max TtjCzj) . (A46) where TI^Z,) = (a - yt - y2 - c + z,)y,; 7i 2(zi) = (a - y, - y2 - c)y2. Solving first-order conditions to yield the following optimal output strategies and actual market price for every Z j € Z: d uy i = (l/3)[a - c + 2z,]; d uy 2 = (l/3)[a - c - z,]; (A47) P = (a - d u y i - duy2) = (l/3)(a + 2c - z,). (A48) Not Sharing z, with Firm #2 The decision problems at the beginning of second-stage output game become: Max 7^ (z,) and Max 7t 2 (A49) Yi y2 where T t ^ Z j ) = (a - y, - y2 - c + zjy^ 7ti(zj) = (a - y, - y2 - c + zjyj. Solving first-order conditions to yield the following optimal output strategies and actual market price for every zl e Z.: n uy i = (l/3)[a - c + (3/2)Zl]; n uy 2 = (l/3)(a - c); (A50) P = (a - n u y i - nuy2) = (l/3)[a + 2c - (3/2)Zl]. (A51) Ordering Expected Profits Across Information Regimes First note that the expected firm profits for both information regimes are as described in footnote 53. E(du7X,) - E<r%) = (cr79)[4 - (9/4)] = (7o*/36) >0. E(du7t2) - Ecx) = ( l^Xr /Kl - 0) > 0. Therefore, both the informed and uninformed firms prefer a policy of full disclosure by the informed. Bertrand Duopoly Sharing of z, with Firm #2 At the beginning of second-stage pricing game, decision problems facing firms are: Max 7Ci(z,) (A52) Pi where ^ (z,) = (p2 - c + z^y^ T C 2 ( Z 1 ) = (p2 - c)y2. Solving first-order conditions to yield the following optimal pricing strategies, for every Z l e Z: d u p l = [l/(4p,2 - p22)][(2p, + P2)cc•+ (p,p2 + 2p12)c - 2p12z1] 117 (A53) d up 2 (A54) = [1/C4P,2 - P22)][(2P1 + p2)ct + (P,P2 + 2P12)c - PJBJZ,] The associated outputs are: duy, (A55) = a - P^P, + p2duP2 = [p1/(4pi2 - P22)][(2p, + p2)a + (P,p2 + p22 - 2p12)c + (2p!2 - p22)Zl] d uy 2 (A56) = a - p,duP2 + p^P, = [P/C4P,2 - p22)][(2p1 + p2)a + (P,p2 + p22 - 2p12)c - PJP^] Not Sharing z, with Firm #2 The decision problems at the beginning of second-stage pricing game become: Max 71,^) and Max 7t2 (A57) Pi P 2 where ^(zi) = (a - yt - y2 - c + zjy^ 7t2 = [a - E(y,) - y2 - c]y2. Solving first-order conditions to yield the following optimal pricing strategies, for every z, e Z: n uPi (A58) = [l/(4p,2 - p22)][(2p1 + p2)oc + (p,p2 + 2p,2)c] - (Zl/2) = [l/(4p7 - B22)][(2B1 + B2)a + (PJP2 + 2pi2)c] The associated outputs are: nu y i = a - p^Px + P2nuP2 = [P^p!2 - p22)][(2p1 + p2)a + (p!p2 + p22 - 2P,2)c] + (p1/2)z1 n u y 2 = a - prP 2 + P2DUP! = [p^p, 2 - p22)][(2p1 + p2)a + (ftp, + p22 - 2p12)c] - (P2/2)z, Ordering Expected Profits Across Information Regimes First note that the expected firm profits for both information regimes are as described in footnote 54. ECX) - E O O = [p.Op,4 - 8p12p22)/4(4p12 - p 2 2) 2]a 2 <0; E( d u7C 2) - ECih) = [PxP.WW, 2 -> 0. 118 (A59) (A60) (A61) Therefore, the informed prefers not to release its private cost information, while the uninformed is strictly better off if such information is disclosed. QED. Proof of Proposition 6 Cournot Duopoly From (A47)/(A50), aggregate industry outputs are given by: d u Y 0 = (l/3)(2a - 2c + z,); Var(duY0) = (o2/9). (A62) n u Y 0 = (l/3)[2a - 2c + (3z,/2)]; Var(duY0) = (o2/4). (A63) Hence, expected consumer surplus, defined in Equation (36), is given by: E(duCS) (A64) = (l/2)[(l/9)(2a - 2c)2 + (o79)] + [E(d u y i) + E(duy2)]c - (2o2/3) EfCS) (A65) = (l/2)[(l/9)(2a - 2c)2 + (o74)] + [E(n u y i) + ECy2)]c - (o72) The total surplus, on the other hand, is: fTS = a^o - (l/2)Cyi2 + V22) - V1V2. t = du,nu. (A66) 120 Thus, the following expected total surplus: E(duTS) (A67) = (a/3)(2a - 2c) - (l/2)(l/9)[2(a - c)2 + 5a2] - (l/9)[(a - c)2 - 2a2] ECTS) (A68) = (a/3)(2a - 2c) - (l/2)(l/9)[2(a - c)2 + (9/4)(o2)] - (l/9)(a - c)2 From (A64)/(A65), expected consumer surplus under bilateral information exchange is strictly lower because: E(duCS) - ECCS) = V / l 8) - (2o2/3) - (o 2^) + (a 2^) = (-17 /^72) <0. From (A67)/(A68), expected total surplus under bilateral information exchange is strictly higher because: E(duTS) - E(nuTS) = -(5a2/! 8) + (2a2/9) + (a2/*) = (5^/72) > 0. 121 Ber t rand Duopoly From (A55)/(A56)/(A60)/(A61), aggregate industry outputs are: d u Y 0 (A69) = [8/(28, - p2)][2a - 2(8, - 82)c + (8, - p2)z,] n u Y 0 (A70) = [P1/(2p1 - 82)][2a - 2(8, - p2)c] + [(p, -p2)/2]z, For ease of exposition in the subsequent analysis, the following shorthand notation will be used: D 3 - [(28, + p2)a +. (p,p2 + p2 2 - 2p,2)c] (A71). According to Equation (34), consumer surplus is given by: d uCS ' , • (A72) = a d uY 0 - (b,/2)[P,/(4p,2 - p22)]2{2D32 + [(2p,2 - p 2 2) 2 + P,2p22]z,2 + 2D3[(2p,2 - p22) - B,pjz,} - b2[p,/(4p,2 - P22)]2{D32 + D3Zl[(2p,2 - p22) - P,pJ - (2p,2 - pVmz,2} - [ p ^ P , 2 - P22)2]{2D3[(2P, + P2)a + (P,p2 + 2P,2)c] - (P,P2 + 2p,2)z,D3 + [(2P, + p2)a + (p,p2 + 2p12)c][(2p,2 - p22) - ftpjz, + p,2[p22 - 2(2p,2 - p 2 2)] Z l 2} + (c - z , ) d u y i + (c - z2)duy2 n uCS (A73) = a n uY 0 - (b1/2){2[B1/(4S12 - p22)]2D32 + [(P,2 + p22)/4]Zl2 + (p, - MMW? - p22)]D3Zl} - b2{[p,/(4p,2 - p22)]2D32 + [P,/(4p!2 - p22)][(p, - P2)/2]D3Zl - (P,P2Zl2/4)} - 2[P1/(4p12 - P22)2][(2p1 + P2)a + (P,p2 + 2p,2)c]D3 - (l/2)[p1/(4pi2 - P22)]D3Zl - [(2P, + p2)a + (P,p2 + 2p12)c][(p1 + P2)/2(4P,2 -p22)]Zl + (PlZl2/4) + (c - Z l ) n u y i + (c - zjry2 Note that the terms involving a, bu b2, in (A72) and (A73) above will all appear in the expressions for the total surplus in the bilateral and no information exchange regimes, respectively. Expected consumer surplus, E(duCS) and E^CS), can be easily derived by taking expectations of (A72) and (A73) over zv Zj, respectively. Hence, the ranking of expected consumer surplus is as follows: E(duCS) - ECCS) = - O W P ^ P , 2 - p22)]2[(2pi2 - p22)2 + p^p^o2 - (b1/2)(l/4)(-l)(P12 + p^o 2 + b J P^f t 2 - p22)]2(2p42 - fcWAo2 - b^PaWo 2 - [PiA4Pi2 - P22)2]p!2[P22 - 2(2p,2 - P^Jo2 . - [Pi/(4p!2 - P22)](2p12 - p 2 V - (P,/4)CT2 + (p1/2)c52 = - ( b ^ l W p , 2 - B,2)2]^ x [(16P,6 - 12pt4p22 + 4pt2p24) - (16P,6 - 8p t 4p22 + p,2P24 + 16pt4p22 - SP^p,4 + p26)] + b2[l/4(4p12 - p/^c^tCSP^P, - 4Pt3p23) - (\6&% - 8P,3p23 + p,p25)] + [p1/4(4p12 - p V J V x [(16ft4 - 12&%2) - (32P,4 - 16P,2P22 - 8p,2P22 + 4P24) + (16p,4 - 8p,2p22 + P24)] = - (b1/2)[l/4(4p12 - fcfii-WW + 11P.2P24 - P 2 V + b2[l/4(4p12 - p22)2](-8p,5p2 + 4p,3p23 - p ip2 5)o2 + [p1/4(4p12 - p22)2](4p12p22 - 3p24)o2 123 = [l/4(4p7 - B22)2][(b1/2)(20B14S22 - lip^p, 4 + p^o 2 + b2(-8p,5p2 + 4p 1 3P 2 3 - p,p25)] + [f31/4(4p12 - P22)2](4P,2P22 - 3 p 2 V = [1/4(48,* - P, 2) 2]^ 2 - b . V l p J l O p ^ 2 - (ll/^p^p, 4 + (1/2)P26] + p2(-8pi5p2 + 4p 1 3p 2 3-p ip 2 5)} + [p1/4(4p12 - p22)2](4p12P22 - 3 p 2 V = [1/4C4P,2 - P, 2) 2]^ 2 - b . V p p ^ 2 - (3/2)P13P24 - (l/2)pip26] + [P .Wfr 2 - p22)2](4pi2p22 - -3PJV > 0. On the other hand, expected total surplus is also strictly higher under information exchange because the term involving bj and b2 above is negative. QED. 124 APPENDIX 2 CHAPTER 3 PROOFS Proof of Observation 1 Bilateral Information Exchange Suppressing any reference to investment, optimal output strategies are as described in (20). This gives rise to the following profits for every pair of realized ( z^ ) e (Z,Z): d d 7 X i (A74) = (l/9)[(a - c) + (2zt - Z j)]2 where i j = 1,2, j i. The associated expected utility for profits at the beginning of first-stage information game is: EU( d < X) = E(d\) - ((p/2)Var(dd7ti), i = 1,2, and (A75) E(d\) (A76) = JI d d7t if(z1,z2)dz1dz2 : (l/9)[(a - c)2 + 5a2] Var^Tti) (A77) = II [d<X - E r T t O T z ^ d z ^ : (50o781) + [20(a - c)2a2/81] 125 No Information Exchange In this case, the set of linear decision rules becomes: y~i = T n a .+ T 2 i C + T 3 1 z, (A78) Y2 = 1" I2a + f 22C + T32Z2 At the beginning of second-stage output game, firm i maximizes (41) for every realized Zj € Z, where E(7ti(yi)yj(zj),zi)) (A79) = [a - y; - E(Vj I Z;) - c + zjy, Var(jt1(yi,yJ(zJ),zi)) (A80) = y i 2 Var( y j |z i) , • = y i 2 T , 2 2 a 2 i,j = 1,2, j ^ i. Solving the set of first-order conditions (A81): a - ys[2 + cpVarty | z,)] - Ety | z{) - c + z, = 0 (A81) results in: ^ (A82) = [a - Efy I Zj) - c + zJ/[2 + cpVar(yj | Zi)] = (a - T l j a - T 2 jc - c + Zj)/(2 + cpT3j2a2) = [(1 - T ^ a - (1 + T2 j)c + zJ/(2 + cpo 2^ 2) 126 =Tna + T2 ic + T 3 i Z i where i,j = 1,2, j + i. In principle, T's can be solved from the Equations (A83)-(A85): T 3 i = 1/(2 + cpo%2) (A83) => T 3 i = (2 + c p t T 2 ^ 2 ) 2 / ^ + (po2); T 2 i = -(1 + T2j)/(2 + cpa2T3j2) = -(1 + T2 j)T3 i; (A84) T u = (1 - TJ/Q. + (po2T3j2) = (1 - TXi)Tr„ (A85) where i,j = 1,2, j ^ i. From T n = (1 - T,j)T3i, both T u and T i 2 can be expressed as a function of T 3 1 and T 3 2 : T 1 4 (A86) = T 3 i(T 3 j - l)/(T3 iT3 j - l),i,j = 1,2, j ^ i . From T 2 i = -(1 + T 2 j)T 3 i, both T 2 1 and T 2 2 can be expressed as a function of T 3 1 and T 3 2 : T 2 i (A87) = "T3i(T3j - l)/(T3 jT3 i - 1), i,j = 1,2, j * i. Thus, for every zi e Z, optimal output strategies are given by: "Vi (A88) = T 3 i(T 3 j - l)/(T3.T3j - l)(a - c) + T 3 i Z j , i,j = 1,2, j * i. In equilibrium, T 3 i(T 3 j - 1) = T 3 j(T 3 i -1) implies T 3 1 = T 3 2 = T 3. Therefore, 127 n Yi (A89) = [TJ/CTJ + l)](a - c) + T 3 Zi, for every zx e Z, where T 3 and cpo2 can be related as follows: iVqjo2 + 2T3 = 1. (A90) For every realized zt e Z, firm i's profit is: nnTCj (A91) = T y i ) [ a - " " y , - E C y j I Zi) - c + zj = {[Ta/fTa.+ Dl(a - c) + T3Zi}{a - [Ty(T3 + l)](a - c) - T3z, - [T^T, + l)](a - c) - c + z,} = {[T3/CT3 + l)](a - c) + T3Zi}{{l - [2Ty(T3 + l)]}a - {1 - [2T,/(T3 + l)]}c + (1 - T3)Zj} = { [ W , + l)](a - c) + T3Zi}{[(l - T3)/(T3 + l)](a - c) + (1 - T3)Zj} Firm i's expected utility for profit at the beginning of first-stage is determined from: EUC%) = EOtj) - ((p/2)VarrTti), i = 1,2, and (A92) E O t i ) (A93) = J -TiifCzJdz, = [T3(l - T3)/(T3 + l)2](a - c)2 + T 3(l - T 3)a 2 VarPTti) (A94) = J J P K J - E(^ i)]2f(zI,z2)dz1d.z2 = CJ^ a - c)2[l/(l + T3)2][T34 + 4T32(1 - T3)2] + o W + 2T32(1 - T3)2] 128 EUfrt . ) versus EU(nnn:) First note that, after some simplification, EUCX) - E U ( d \ ) = [1/162(1 + T3)2] x {(a - c)2[(-18 + 126T3 - I8OT3 2 ) + y<?(20 + 40T3 - 304T32 + 648T33 - 405T34)] + c*[(-90 - I8T3 + 72T32 - 162T33 - 162T34) - (pa2T2(162 + 243T32 + 162T33 + 243T34)]} Let a 2 = 1. By fixing (p, and making use of the equilibrium relationship (A90), one can solve for the corresponding T 3. cp and T 3 can then be substituted into above to determine the conditions (i.e., in terms of restrictions on (a - c)) under which ElIC"?^) may be greater or less than E U C X ) . Take (p = 0.01, for instance, T 3 in this case is equal to 0.4994. Hence, > EUCX) - E U ( d \ ) =0 < > <=> (a - c)2(0.229093415) - (111.9292429) =0 < > « , ( a - c ) 2 = 488.5746735 < The remaining cases can be likewise verified. QED. Proof of Example from Footnote 73 Unilateral Information Exchange -- dn Note the following set of linear decision rules: y, = T„a + T 2 ,c + T 3 1 z, (A95) y 2 = T 1 2 a + T 2 2 c + T 3 2 z 1 + T 4 2 Z 2 Firm 1 maximizes (41) at the beginning of second-stage for every z, e Z, where E[7t1(y1,y2(z2),z1)] (A96) = [a - y, - E(y21 zt) - c + z,](y,) Var[7i1(y1,y2(z2),z1)] (A97) = y,2Var(y21 z,) = y X V Solving the set of first-order conditions (A98)-(A99): a - y,[2 + <pVar(y21 z,)] - E(y2 | 2 l ) - c + zl = 0 (A98) a - 2y2 - E(y, | z,^) - c + z, = 0 (A99) Expressing y, (y2) as a function of the remaining terms in (A98) ((A99)), and equating it with (A95) to get the following optimal output strategies: • " . 130 "toy, (AlOO) = [2/(6 + (per2)] [(a - c) + 2 z J , for every zx e Z ^ 2 (A101) = [(4 + 90^/2(6 + 9a2)] {(a - c) + [(6 + cpo2)/^ + (pcj2)]z2 - [4/(4 + 9C2)]z1} for every pair of realized z l t Zj €. Z. Profits for both duopolists are given by (A 102) and (A 103), respectively: (A102) = [(4 + 90^/(6 + (pa2)2] [(a - c) + 2zJ2 ^ (A 103) = [(4 + (po2)/(6 + (po^ )]2{(a - c) + [(6 + 9C2)/(4 + y<?)]z2 - [4/(4 + (pa2)]^}2 Firm I's expected utility for profits at the beginning of first-stage is determined from: EUpTt,) = £(*%) - (cp/2)Var(dn7t1), where (A104) ECX) (A 105) = J "Xf^dz , ) = [(4 + 90^/(6 + 9a2)2][(a - c)2 + 4a2] VarpTi,) (A106) = j[[dDK1-E(danl)]2f(zi,z2)dz1dz2 = [o^a - c)7(6 + 9a2)4] [(6 + 9a 2) 2 + 16(4 + 9a2)2] + [a4/(6 + 9a2)4] [4(6 + 9a 2) 2 + 32(4 + 9a2)2] Unilateral Information Exchange — nd By symmetry, one can easily show that at the beginning of first-stage: EUC\) = EC\) - ((p^VarCX), where (A107) E C X ) (A108) = 11 "VCzi.z^dZjdza = [(4 + cpo 2)^ + (pa2)]{(a - c)2 + a2![(6 + 90^/(4 + (pa2)]2 + [4/(4 + (pa2)]2}} VarCX) (A 109) = 11 - E(dnK1)]2f(z1,z2)dz1dz2 = a2(a - c)2{[(4 + (po2)/2(6 + (po2)]2 + [2(4 + 90^/(6 + (pa2)2]2} + a 4 {[32/(6 + (pa2)4] + [2/(6 + (pa2)]2 + (1/8)} EU(dd7r,) ygrst/s EU(nd7c,): EUrjt,) versus EU(dn7t,) First note that after some simplification, EUC%) - EUCX) = [1/6(6 + (po2)]4{(a - c)>a2(-103680 - 565929a2 - l l O ^ c 4 - 780(p3a6 + 2(p4a8) + a2[326592 + 2799369a2 + 80352qrV + 950493a6 + 39 69 4a 8 - 902(299376 + 2445129a2 + 663129V + 76569V + 3199V)]} 132 EUCX) - EUC^,) = [1/2(1 + T3)2(6•+ <po2)4]{(a - c)2{[2(6 + 902)4T3(1 - T3) - 2(1 + T3)2(4 + 9C2)(6 + (pa2)2] + (pa2[(l + T3)2(292 + 1409O2 + HqrV) - (6 + cpa2)4(5T34 - 8T33 + 4T32)]} + a2{2(l + T3)2(6 + 9C2)4T3(1 - T3) - 8(1 + T3)2(4 + (pa2)(6 + (pa2)2 + (po2[(l + T3)2(656 + 304CPO 2 + 3 6 9 V -(6 + 90 2 ) 4 (3T 3 4 -4T 3 3 + 2T32)]}} Once again, a 2 is set equal to 1. By fixing 9 at 0.1, T 3 , as determined from (A90), is 0.494. Hence, < EU( d d7tj) - EUCX) = 0 > < <=> (a - c)2(-109450.1398) + (322948.82341) = 0 > > <=» (a - c)2 = 2.950647883 < and, in addition, EUPTli) - EUC^ii) < = 0 > 133 > <=> (a - c)2(36.63699089) - (1082.911661) = 0 < > <=> (a - c)2 = 29.55787674 < Therefore, no information exchange dominates for all (a - c)2 > 29.55787674, or equivalently, (a - c) > 5.437. QED. Proof of Example From Footnote 74 Note that from (A90), T 3 = 0.4431 when 9 = 1.31. Substituting 9, a2, and T 3 to [EU(d d7t,) - EUCX)] and [EUCX) -EU(dnJt1)] above, it can be easily verified that both utility differences are strictly positive V (a - c) > 0. In other words, no information exchange dominates for all marginally profitable firms. QED. Proof of Example From Footnote 79 To simplify the analysis, assume that Bj = 2 and B2 = 1. Technical details are analogous to those presented in Observation 1, and thus will not be repeated here. The following only summarizes optimal pricing strategies and expected utility Tor profits for each of the information structures examined. 134 Bilateral Information Exchange ""Pi (A110) = (l/15)(5cc + 10c - 2Zj - 8Zi), ij = 1,2, j * i. EUCX) (All l) = (50/225)(a - c)2 + (1067225)02 - (11236/50625)(po4 - (10600/50625)cpcj2(a - c)2 where i = 1,2. Note that, in this case, the first two parameters of profit distributions are given by: E(d\) (A112) = (2/225)[(5cc - 5c)2 + 53c2] Var( dX) (A 113) = (2/225)2[212o2(5a - 5c)2 + 5618a4] No Information Exchange T j (A114) = [1/(2 + t3)][-t3oc + 2(t3 + l)c] + t3Zj where i = 1,2. And, as before, the above expression invokes the equilibrium condition that t31 = t32 = t3. Even though t3 cannot be explicitly determined, t3 and cpo2 are related through: t/cpo2 + 4t3 = -2 (A115) where -0.5 < t3 < 0 V cp > 0. Hence, 135 = [-2t3(l + t,)/(2 + t3)2](a - c)2 - 2t3(t3 + lja 2 - (9/2)t32(t3 + l^cpa4 - (a - c)V(p{4[(t3 + l) 2 + t32]2 + t,4}[l/(2 + t3)22] (A116) reflecting the following first two parameters of profit distributions: ECX) : [-21,(1 + t3)/(2 + t3)2](a - c)2 - 2t3(t3 +' lja 2 . (A117) Var("X) : 9t32(t3 + 1)V + (a - c)2a2{4[(t3 + l) 2 + t32]2 + t34}[l/(2 + t3)2] (A118) Unilateral Information Exchange — dn dn P i = [1/(15 +'<po2)][5a + (10 + (pa2)c - 8zJ dnr = [1/4(15 + (pa 2 )]^ + (po^a + (40 + 3^)c - 8z,] - (1/2)^ (A119) (A120) EUCX) = Et^Tt,) - ((p/2)Var(dn7t1), where (A121) = [25(8 + ^)/4(l5 + 9a2)2](a - c)2 + [14(7 + (po2)/^ + (pa2)2]^2 (A 122) 136 V a r C X ) (A123) = (a - c)V{{[5(7 + (pa2)^ + (po2) + 280]2[16(15 + (po2)4]} + [25/4(15 + (po2)2]} + a 4 {[(7 + ( poWClS + (po2)2] + [392(7 + (pa2)2/(15 + (pa2)4]} Unilateral Information Exchange -- nd n d P , = [1/4(15 + (pa2)][(20 + (pa2)©: + (40 + Scpcr^c - 8zJ - (l/2)z, (A124) E U C T C O = E(" d7C,) - ((p/2)Var(nd7X,), where E("V) = (a - c)2[(20 + (pa2)2/^^ + (pa2)2] + ^ [8/(15 + (pa2)2 + (1/2)] Var(ndjc1) = (a - c)2a2[(20 + (pa 2) 2^^ + cpa2)2]{l + [16/(15 + a2)2]} + ^tiHS/ilS + (pa2)4] + [16/(15 + (pa2)2] + (1/2)} (A125) (A126) (A127) EU(d d7t ;) versus EU(nn7t,) Once again, a 2 is set equal to 1. By fixing (p, t3 can be determined from (A115). In particular, it can be verified that: (1) .When 0 < (p < 0.485, "Eli , > ""EUj <=> (a - c)2 > cv, where cv stands for critical value, and it declines as (3 increases. (2) .When cp > 0.485, "EU, > ""EU,, V (a - c)2 > 0. 1 3 7 EUC"^.) versus EU(B<4c.) and EU(dn7t,) versus EUC'TC.) Examples whereby "dd" dominates other information regimes: (1) .When 0.01 < (p < 0.998, and V (a - c) 2 > cv. Once again, cv declines as (p increases. (2) .When 1.08 < <p < 3.93, and V (a - c) 2 > 0. (3) .When (p > 4.83134, and V (a - c) 2 < cv. cv declines as (p increases. eg. when (p = 5, cv = 3.470009624, and when (p = 50, cv = 0.302084207. Proof of Proposition 7 First note that the equilibrium investment t x R N , t = dd,nn, can be characterized by: Recall from (51)/(54) that the equilibrium investment levels for the risk averse case in each of two reporting regimes examined can be determined from the following first-order conditions: QED. (-4/9)[a - u( t x R N )]u ' ( t x R N ) - 1 = 0 (A128) (-l)[a u r x ^ u r s j [(9 + lOcpo2)^] = 0 (A 129) (-l)[a [(1 + T 3 ) 2 ( l - T 3) 3]/T 3 3 = 0 (A 130) 138 (a) . Replacing (ddx8N,ddxRN) in (A 128) with (ddxRA,ddxRA) from (A 129) to get: (-4/9)[a - u C ^ l u ' r x R A ) - 1 = [(9 + lOcpo2)^] - 1 = (10902/9) / >0. • • ( D X R N ' ^ R N ) ^ ( X R A < X RA) Replacing Cx^."*™) in (A128) with C x ^ x ^ ) from (A130) to get: (-4/9)[a - P-rx^luTxRA) - 1 = (-4/9)[-(l + T3)2(l - T3)Yr33] - 1 = [4(1 + T 3) 2(l - T 3) 3 - 9T33](1/9T33) = (4 - 4T3 - 8T 3 2 - T 3 3 + 4T 3 4 - 4T35)(1/9T33) = (1 - 2T3)[2 + 2(1 + 2T3) + 2T3 4 - T33](1/9T33) > 0, V 0 < T 3 < (1/2), and (p > 0. C X j ^ , ™ ^ ^ ) > ( XRA, X ^ ) . (b) . E[ddyRN(x)] - E[ddyRA(x)] = (l/3)[a - |i(x)] - (l/3)[a - u(x)] = 0. E r w x ) ] - EryRAW] = (l/3)[a - u(x)] - [ T^l + T3)][a - u(x)] = [(1 - 2T3)/3(1 + T3)][a - U(x)] > 0, V 0 < T 3 < (1/2), and (p > 0. (c). E[ddyRN(ddXjuv])] - E[ d dy R A( d dx R A)] = (l/3)[u(ddxRA) - uTx™)] > 0 , v "x^ > d d X R A, u'(x)< 0, /. u( d d X R N) < u( d d X R A). E r y ^ r x ^ ) ] - Ery^rx^)] = (l/3)[a - UPXRN)] - [17(1 + T3)][a - uCXJ] = [(1 - 2T3)/3(1 + T3)]a + [ T 3iir X R A)/(l + T3)] - [uTx^yS] > [(1 - 2T3)/3(1 + T3)]a + [ T 3 u r X R N)/(l + T3)] - [ur X R N )/3] = [(1 - 2T3)/3(1 + T3)]a - [urx^Jtd - 2T3)/3(1 + T3)] = [a-urx R N)][(l -2T 3 ) /3(l+T 3 ) ] > 0, v a > ""XRN by assumption. 140 Proof of Proposition 8 (a) . Replace (ddXRA, ""x^) in (A129) by r x ^ P x ^ ) from .(A 130) to get: -[a - uTx R A)]u'rx R A) - [(9 + lOcpo2)/^ = [(1 + T 3) 2(l - T3)Yr33] - [(9 + lOcpo2)/^ = [(1 + T3)2(l - T3)3/T3 3] - (9/4) - (10/4)[(1 - 2T3)/T33] (follows from (A90)) = (1/4T33)(1 - 2T3)[(-2)(1 - 2T3) - 2 (1 - T3 4) - 2 - T3 3] < 0, V 0 < T 3 < (1/2), and (p > 0. • • ( X R A ' X RA) < O ^ R A ' X R A ) . (b) . E[ddyiRA(x)] - EryiRA(x)] = (l/3)[a - n(x)] - [(T3)/(l + T3)][a - u(x)] = [(1 - 2T3)/3(1 + T3)][a - uXx)] >0. QED. 1 4 1 Proof of Proposition 9 First recall that ddxm > d d X R A , and hence HC^RN) < nCVO; moreover, > " V , , and hence urx^) < uCx (A131) (A 132) (A133) (A134) Hence, E(ddCSm) - EpCS^) = (l/9)[u( d d X R A) - u( d d X R N)] >0. (b). ""Y^ = (2/3)[a - UCXKN)] + d/2)(Zl + zj; (A135) E r Y ^ ) = (2/3)[a - UTXRN)]; VarCY^) = (1/4)0 ;^ (A136) ""YRA = [2Ty(l + T3)][a - H-OCRA)] + T3(z, + zj; (A137) ECYRA) = [277(1 + T3)][a - HC-XRA)]; (A138) VarrY^) = 2 T 3 V . (A139) (a). "Ym = (2/3)[a - u( d d X R N) + z, + zj; EC^YRN) = (2/3)[a - UTXRN)]; Var^Y^) = (4/9)(o2); % = (2/3)[a - u( d d X R A) + z, + zj; E ^ Y ^ ) = (2/3)[a - u( d d X R A)]; Var(d dYR A) = (4/9)(o2). Hence, 142 E r c s ^ ) - Ercs^) = 2{(l/9) - [T32/(l + T3)2]}(a - c)2 + (1/4)(1 - 4T32)a2 = 2[(1 + 4T3)(1 - 2T3)/9(1 + T3)2](a - c)2 + (1/4)(1 - 4T32)a2 >0. QED. 


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