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Vicky ACKNOWLEDGMENTS I thank David Sappington for his invaluable help and advice. I also thank Sanford Berg, Steven Slutsky, Jon Hamilton and Chunrong Ai for their comments. TABLE OF CONTENTS page A C K N O W L E D G M E N T S ..............................................................................................................4 L IS T O F T A B L E S ................................................................................. 7 LIST OF FIGURES .................................. .. ..... ..... ................. .8 ABSTRAC T ........................................................................................... CHAPTER 1 MULTIAGENT CONTRACTS WITH UNKNOWN COST CORRELATION ..................11 Introdu action ................... .......................................................... ................. 11 T h e M o d el ................. .......... .......... ............................. ................ 14 Uninformed Principal: Limits on Communication.............................................................. 21 If B oth A gents Observe the Correlation................................... .................................... 22 If Only One Agent Observes the Correlation ............... ..........................................32 Uninformed Principal: No Limits on Communication.........................................................33 If B oth A gents Observe the Correlation................................... .................................... 34 If Only One Agent Observes the Correlation............................................................... 34 Lim its on Com munication and Exclusion ........................................ ......................... 40 C onclu sions.......... ............................... ................................................43 2 EFFICIENCY IN THE BRAZILIAN SANITATION SECTOR................ ..................53 Introdu action .............................................................................. ................. 53 Overview of Brazil's Water and Sewerage Industry ...........................................................56 M e th o d o lo g y ..................................................................................................................... 5 9 R e su lts ................... ...................6...................4.......... S en sitiv ity C h eck s ................................................................66 C onclu sions.......... ..........................................................67 APPENDIX DERIVATION OF THE SOLUTIONS TO CHAPTER 1 ..............................79 L im its on C om m unication ................................................................................................ 79 W hen Correlations are Relatively Sim ilar............................................... 79 W hen Correlations are R relatively D different .................................................................. 81 N o Lim its on C om m unication ............................................................83 W hen Correlations are Relatively Sim ilar............................................... 83 W hen Correlations are Relatively Different ................................................................ 87 Limits on Communication and Exclusion ................................. ...............90 L IST O F R E F E R E N C E S ....................................................................................................92 B IO G R A PH IC A L SK E T C H ............................................................................... .....................97 LIST OF TABLES Table page 21 Average statistics by operatortype for 2004.................... .......... ............ .............. 70 22 Summary statistics for firststage regressions....................................... 71 23 Firststage LSD V regression results ............................................................................ 72 24 Secondstage regression results .............................................................. .....................73 25 Ranking of firmspecific costs across firmtypes. .................. ...........................74 26 Firststage LSDV regression results using a balanced panel..........................................75 27 Secondstage regression results using a balanced panel ....................................... 76 28 Firststage LSDV regression results excluding the Regional type ..................................77 29 Secondstage regression results excluding the Regional type .......................................78 LIST OF FIGURES Figure page 11 Timing at [PNOC], when both agents observe but do not report the correlation.............46 12 A solution to [PCM] that can also be a solution to [PNOC] ............... ............. .....47 13 Welfare under the two alternative mechanisms at [PNOC]. ..........................................48 14 Timing at [PCO], when only agent A observes and reports the correlation...................49 15 Welfare at [PEXCL] below welfare at [PNOC] .......................................................50 16 Welfare at [PEXCL] sometimes larger than welfare at [PNOC] .................................51 17 Welfare at [PEXCL] almost always larger than welfare at [PNOC].............................52 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THEORETICAL AND EMPIRICAL ANALYSES OF INCENTIVES AND PUBLIC OWNERSHIP By Guillermo Sebastian Sabbioni Perez August 2007 Chair: David Sappington Major: Economics This dissertation includes both theoretical and empirical research in economic efficiency. The first chapter theoretically evaluates how to design procurement models when two agents may have better information than the principal about their production costs and about the similarity (i.e., correlation) of those costs between agents. The results indicate that if the uncertainty about the correlation is not severe, the principal should design a contract as if she knew that correlation is the lowest possible. The drawback with such mechanism is that the high cost agent earns rent if correlation is high. In contrast, if the uncertainty about the correlation is more severe, the principal should not tailor the solution to any of the potential correlations. This contract reduces the rent of the highcost agent if correlation is high by giving rent to the low cost agent if correlation is low. Although the welfare level may be different, the characterization of the results holds when both agents observe the correlation and do not report it, as well as in the case where only one agent observes and reports the correlation to the principal. Finally, when the probability of the lowcorrelation setting is very small, the principal may find optimal to exclude the highcost agent from the contract if correlation is low, allowing her to achieve the socially optimal situation if correlation is high. The second chapter empirically evaluates the relative efficiency of different type of water and sewerage operators in Brazil. The analysis consists of a cost comparison between public and private operators, and between statelevel and municipallevel operators. In a first stage, a cost function is estimated utilizing a fixedeffects panel data model. In a secondstage, the firm specific costs from the first stage are explained by means of firmtype indicator variables. The results illustrate that water and sewerage provision in Brazil is characterized by substantial economies of scale, indicating that statelevel provision is economically more efficient. The results also show that there is not an economically significant difference between the costs of private and public firms. CHAPTER 1 MULTIAGENT CONTRACTS WITH UNKNOWN COST CORRELATION Introduction This chapter analyzes a multiagent principalagent model where the agents are privately informed not only about their production costs but also about the extent to which those costs are correlated. The analysis is performed with and without limits on communication, and considers the possibility that either one or both agents may be perfectly informed about the correlation. The results show that full rent extraction is impossible if the informed agents cannot communicate the realization of the correlation to the principal. Additionally, the results also illustrate that even with communication on the correlation the principal may not be able to fully extract rent from all agents if only one of them is informed about the exact correlation. The most important finding of the analysis under limited communication is that the high cost agent earns positive rent if correlation is high. When limits on communication on the correlation prevail, the principal cannot tailor the payments to the exact correlation coefficient. As a result, the extreme lottery of payments for the highcost agent should ensure that non negative expected rent is obtained under both potential correlations. Extreme payments for the highcost agent are necessary to prevent the lowcost agent from exaggerating cost. Since expected rent is calculated using two different sets of conditional probabilities (one for each correlation), and since the high payment is obtained with higher probability (and the low payment with lower probability) under the highcorrelation environment, the principal can eliminate rents for the highcost agent only if correlation is low. Similarly, the most important finding of the analysis without limits on communication is that if only one of the agents is perfectly informed about the extent of cost correlation, again the highcost agent enjoys positive rent when correlation is high. Although the rent for the highcost agent if correlation is high parallels the results with blocked communication, the source of this rent is slightly different. Instead of being explained by the inability of the principal to tailor payments to each correlation, when communication on the correlation is available the mechanism designer needs to prevent the highcost agent from untruthfully underreporting the correlation. If correlation is high, the highcost agent would claim that correlation is low because he would then be assigned a high payment more frequently (and a low payment less frequently) than what he should if the principal designed two independent mechanisms, one for each correlation report. When limits on communication prevail, the rent for the highcost agent if correlation is high is explained by his required participation if correlation is low combined with the fact that his rent is always higher under the highcorrelation scenario than under the lowcorrelation setting. A natural question is then whether the principal should always require nonnegative rent for the highcost agent if correlation is low. Numerical examples illustrate that if the probability of the low correlation scenario is low enough, it could be optimal to exclude the highcost agent from the contract if correlation is low. This work is related to two streams of research in the mechanism design literature. First, since the agents) have private information on more than one dimension, the results contribute to the literature on multidimensional mechanism design. Dasgupta and Maskin (2000) show that if bidders' types are multidimensional and independently distributed there may be no efficient auction (i.e., one that assigns the object to the bidder that values it the most). Jehiel and Moldovanu (2001) illustrate that efficient mechanism design may be impossible in a social choice setting, where each agent can receive multidimensional signals from all other agents and where the signals are independently drawn for each agent. The difference between the former two analyses and the present study is that here the two elements of the twodimensional type are not independently distributed. Trying to rank the revenue between a First Price and a Second Price auction, Fang and Morris (2006) consider an auction where the bidders, besides knowing their own type, also receive a signal about the other bidders' types. In Fang and Morris' model there is a separation between each player's valuation and his signal about his opponent's valuation, since both realizations are drawn from two independent probability distributions. In their model, the beliefs of the first bidder about the second bidder depend on the type of the second bidder rather than on the type of the first bidder. In this study, on the other hand, the agent's privately observed cost at the same time provides information about the other agent's cost. Miller et al. (2007) consider a problem where agents' types are multidimensional and continuous, but they relax Jehiel and Moldovanu's assumption that the agents' private information is independently distributed. Miller et al. show that incentivecompatible implementation of any decision rule is possible provided that the agents' types satisfy one of their correlation conditions. The present work is different in that the principal is not required to utilize budgetbalanced transfer payments and in the discrete support of the agents' types. Second, when addressing the case with limited communication on the correlation, the present chapter also makes a contribution to the literature on robust mechanism design. This stream of research attempts to relax the assumption of too much common knowledge at the time of contract design. Too much common knowledge refers to perfect information about all the relevant parameters of the problem at hand, as explained by Bergemann and Morris (2005), Bergemann and Valimaki (2006) and Heifetz and Neeman (2006). The influential statement by Wilson (1987) has been the starting point of this emerging literature: "Game Theory is deficient to the extent it assumes features to be common knowledge, such as one player's probability assessment about another's preferences or information." One way of relaxing the assumption of too much common knowledge is by focusing on contracts that are not finetuned to the exact environment faced (e.g., to the exact correlation between the agents' costs). The origins of both streams of research can be traced back to the seminal work by Demski and Sappington (1984) and Cremer and McLean (1985, 1988), who showed that if agents are sufficiently riskneutral and if they face no limited liability constraints, mechanisms can be designed to ensure costless truthful revelation of private information provided the agents' types are correlated. McAfee and Reny (1992) extended the analysis to the case where agents may have infinitely many possible types. This chapter starts with a description of the model employed. First, the case with limited communication on the correlation is analyzed. Second, the situation with no limits on communication is presented. A following section evaluates the possibility of excluding some agents from the game if their participation is too costly in terms of welfare. Conclusions are presented in the final section. The Model A principal contracts with two agents, A and B, each producing one unit of output. For each agent, the cost of producing one unit of output is given by C = e. The parameter p e {/f, JH )} is the agent's initial cost, with A /,H / L > 0. Meanwhile, e denotes the cost reducing effort the agent exerts. Effort e generates disutility yr(e) > 0 for the agent, where /y(e) is increasing and convex. The final cost C is perfectly observable by the principal, but she cannot distinguish between the cost parameter / and the effort level e. Each agent knows his own initial cost, but not the cost of his counterpart.1 The principal reimburses each agent for his 1 The principal is subsequently referred as "she", while the agents are referred as "he." observed final cost C and may provide additional payment (t). The utility of each (riskneutral) agent is given by the difference between the transfer payment t and the disutility yr(e) of the costreducing effort exerted: U = t y/(e). The agents' initial costs are assumed to be positively and imperfectly correlated. The prevailing probability structure consists of p, = pHH = a / 2 and pLH = p = (1 a)/ 2, where p, denotes the joint probability that agent A (B) has cost t, and agent B (A) has cost /j, with i,j e {L,H}. This formulation presumes that the probability that either agent faces an identical counterpart is invariant to the agent's cost realization. For example, the probability that agent A (B) has low cost is a and the probability that he has high cost is 1 a when agent B (A) has low cost.2 The positive and imperfect correlation between the agents' costs is captured by a e (2, 1) .3, 4 This ensures that the probability that an agent has low cost is higher when the other agent also has low cost than when the other agent has high cost: a > 1 a if a e (Y, 1). The parameter a takes on the value a, with probability ,l and the value a, with probability u0, where a, > a, and u + u0 = 1. Thus, when alpha is a, the agents' costs are highly correlated. This setting is referred to as the highcorrelation scenario. When alpha is a0, the agents' costs are less correlated (although still positively). This setting is referred to as the lowcorrelation scenario. This information is common knowledge at the start of the game. 2 From Bayes' rule. 3 The correlation coefficient is given by 2ca 1 > 0. See Es6 (2005). 4 The values a = 0.5 and a = 1 are excluded from the analysis. These values would constitute no correlation and perfect (positive) correlation, respectively. 5 The prior presented satisfies the BeliefsDeterminePreferences (BDP) property from Neeman 211114). In short, a prior has the BDP property if each agent has different beliefs about the other agents' types depending on his own type. Neeman shows that full rent extraction results hinge on the BDP assumption. The principal never observes the exact realization of the correlation. In contrast, at least one of the agents acquires precise information about the correlation before contracting. This means that besides her information disadvantage about the agents' initial costs, the principal is also less informed than (at least one of) the agents about the exact correlation. In addition, two alternatives are considered regarding limits on communication. In the first alternative setting, the agents) who are informed about the correlation are unable to communicate its realization. The second case consists of a framework with no limits on communication, in which the informed agents) report the actual correlation observed. The principal designs a mechanism where simultaneously and independently both agents report (some or all of) their private information.6 Each agent always reports his privately observed cost ,/, as it is standard in the literature. Additionally, the principal can also request the more informed agents) to report the actual correlation. The decision of whether to ask for a correlation report depends on the particular circumstances faced by the principal. As explained later in more detail, contracts that do not depend on the exact correlation between the agents' types constitute one important example of the issues addressed by the emerging literature on robust mechanism design. Agent A submits the report rA and agent B submits the report rB Based on the joint report r = (rA, rB), the principal assigns a pair [t(r), C(r)] to each agent. A pair entails a transfer t(r) from the principal to the agent and a final cost C(r) that the agent must meet.7 Both the transfer (t) and the final cost (C) depend on all information (r) reported to the principal. Therefore, each agent knows that the [t, C] pair received can vary with the report of his 6 Collusion possibilities are not considered here. 7 From the cost function C = e, a finalcost target is equivalent to an effort target. counterpart. In the cases with asymmetric agents (i.e., when only one agent observes the correlation), the analysis assumes that the principal offers the same menu of options to both agents. The reason behind this assumption is that the agents may refuse to accept different menus of contracts, on the grounds that the principal would be treating them unfairly. The agents will only contract with the principal if they anticipate nonnegative expected rent from doing so. However, the computation of the expected rent depends on the agent's knowledge about the correlation. For example, if a lowcost agent observes a low correlation, he knows that the other agent has low cost with probability a, and high cost with probability 1 a,. On the other hand, if an agent is uninformed about the correlation, he only knows the expected probability of facing each type of counterpart.9 An agent who is uninformed about the correlation only knows that the other agent has similar cost with probability a and different cost with probability 1 &, where &c /o0ao + ala . The principal maximizes the sum of the value derived from both units produced, plus the agents' profits, minus the cost of social funds. 10 The principal values each unit produced at S, where S is assumed constant and sufficiently large. The social funds are the transfers and the reimbursed costs received by the agents, which the principal delivers at a cost of 1+ A per unit.11 If the principal observed the agents' costs:12 W = 2S + Y[t' (' C')] (1+ A) [t + C']. IG{A,B} IG{A,B} 8 Opportunity rents are normalized at zero and no expost limited liability constraints are imposed. 9 Every time an agent is described as "informed" or "uninformed", it is meant from the point of view of whether the agent has perfect information on the correlation. As a remainder, all agents are always perfectly informed about their own production costs. 10 See Laffont and Tirole (1986). 1 The parameter > 0 measures the distortion introduced by the excisetax system utilized to raise social funds. 12 qf(PC) is used when specific reference to the cost parameter / needs to be made. Otherwise, y(e) constitutes more compact notation. Standard manipulation yields:13 W = (+ A) [e' + y(e')] A [t' (e')] (11) It{A,B} It{A,B} Equation 11 shows that if the principal observed the agents' costs, the solution would be characterized by t = Iy(e) and e = e for both agents, where y' (e*) = 1.14 The socially efficient level of effort e would be delivered without any rent being afforded to any agent. Demski and Sappington (1984) and Cremer and McLean (1985) (CM) illustrated how this result can be replicated if the principal does not observe the agents' costs but she knows exactly how these costs are correlated.15 Let t,, y(B, C,) denote the expost rent for an agent (A or B) that reports cost 8, when he has cost /, and when the other agent reports cost 8/k, for i, j, k e {L,H}. This problem [PCM] constitutes the benchmark situation of this chapter: Maximize W= (1+A) I2p, [ej +,(ej)] A 2p,[tj , (ej)] (12) i,j~{L,H} ij~ L,H} subject to: u, c[t r(P, C,, )] ( a[ 8 i, je {L,H},i j (13)(14) + (1 a)[t,j V(/, C, )] > 0 u, > a[t, r,(P, C ,)] i, je L,H},i j (15)(16) + (1 a)[t, V(A, C, )] 13 The constants 2S and the 's are dropped for the sake of exposition. 14 From C = e, an extra unit of effort reduces the final cost also by one unit. Therefore, q '(e) = 1 denotes the equality of the marginal cost and the marginal benefit of exerting costreducing effort. 15 It is well documented that risk aversion and/or limited liability prevent the principal from achieving the first best. See Sappington (1983) for the singleagent case. See also Cremer and McLean (1988), Demski, Sappington and Spiller (1988) and Robert (1991) Equation 12 incorporates the four possible cost combinations that can arise and their respective probabilities. Equations 13 and 14 ensure nonnegative expected rent for the agents. Equations 15 and 16 ensure that the agents report their costs truthfully in equilibrium. 16 The solution to [PCM] permits payments that fully extract (expected) rent while ensuring e, = e *, for i,j e {L,H} To accomplish this, the principal sets relatively high payments for cost reports that match and relatively low payments when cost reports do not match. These two degrees of freedom allow for an infinite number of solutions to [PCM]. After selecting tHH > tH and tLL> tm the principal finds tH and tLH such that no rent is obtained in expectation by either agent, and so UH = UL = 0. In particular, consider the bound tH > tH , where:17 la t" r((eHH) + [ [ HL + (1 a HH > V(eH), (17) 2a 1 with H, V(e) (eH, A) > 0, for i e {L,H} .18 The lower bound tHH > tH prevents the lowcost agent from exaggerating his cost while ensuring that a highcost agent earns nonnegative rent in expectation. This is achieved with a high payment tHH > V(eHH) if both agents report high cost and a low payment tHL < (eHL) if a highcost report is not matched with the same report from the other agent. If the lowcost agent lies, he is relatively likely to receive the low tHL payment and relatively unlikely to receive the 16 This is known as BayesNash implementation. Stronger solutions would be obtained if ExPost or Dominant Strategy implementation was required. ExPost implementation consists on the requirement that the incentive constraints are satisfied even after the other agents make their (truthful) report. Dominant Strategy implementation consists on the requirement that an agent is always better off by reporting the truth, even when the other agents may lie. 17 The lower bound tLLM" guarantees that a highcost agent does not claim low cost, which is usually not constraining for the principal. 18 (H is the profit earned by a lowcost agent in the singleagent case. This profit is explained by the savings in disutility of effort that the lowcost agent would enjoy if he exaggerates his cost. high tHH payment because, due to the positive correlation among costs, the other agent is relatively likely to report low cost. It is important to notice that tm" is increasing in H, which means that t"n is increasing in effort eH, for i e {L,H} .19 The effort exerted by the high cost agent has a positive impact on the "reward" (i.e., payment above disutility of effort) that an agent receives if a highcost report is matched by his counterpart. Similarly, the effort exerted by the high cost agent positively affects (the absolute value of) the "penalty" (i.e., payment below disutility of effort) that an agent receives if a highcost report is not matched by the other agent. In other words, the larger is e,, the higher tHH has to be (and the lower tHL has to be) to become unattractive for a lowcost agent. This feature of the payments has important implications for the solution to the principal's problem when she is imperfectly informed about the correlation. The subsequent analysis focuses on settings where the principal, unlike the agentss, does not know exactly the extent of the correlation between the agents' costs. In such settings, one or both agents have better information than the principal about both their costs and the exact correlation. As it will be seen, the principal cannot achieve the firstbest outcome in those settings. The ensuing analysis is performed with and without limits on communication about the correlation. The situation with full communication of all private information follows the standard optimal contracting approach. The case with blocked communication on the correlation deserves additional explanation. It is important to understand why the principal may not utilize all available communication channels to gather as much information as possible. 19 It also means that tHH.n is increasing in the cost advantage of the lowcost agent, A. Uninformed Principal: Limits on Communication Consider, first, the setting where the informed agents) are unable to report the correlation to the uninformed principal. In this context, it is impossible for the principal to finetune the contract to the exact correlation observed by the agentss. As explained below, specific circumstances prevent the principal from making a contract conditional on the actual correlation. Consequently, the principal designs a mechanism where both agents report only their cost: rA =/ and rB =/ with i,j{L,H}. For example, this could be the situation faced by a regulator when publiclyowned firms are privatized quickly.20 In the preprivatization stage, the regulator may not know the identity of the operators that will take control of the soontobe privatized firms. Alternatively, the regulator may not know exactly which production technologies the new firms will utilize. As a result of this uncertainty, the regulator may ignore the exact correlation between the firms' costs when she designs the regulatory contract.21 The inability of the agents) to subsequently report the realized correlation can be a stylized means for capturing prohibitive costs of acquiring more precise information about the correlation at a later stage. For example, it may be too costly or it may take a long time for an expert auditor to provide accurate information about the correlation.22 Limited communication on the correlation is a key element of the new literature on robust mechanism design. This strand of research attempts to relax the implicit assumption of too much common knowledge at the time of contract design.23 As such, this literature focuses on contracts 20 When a government makes a tender for the construction of highways may serve as another example. 21 In the auction setting presented by Arya et al. (2005), for example, the authors suggest that the contract designed can be utilized with different pairs of bidders (i.e., with pairs of bidders that may have different correlations among their valuations for the object auctioned). 22 In the context of a fast privatization process, political pressure sometimes requires that a regulatory framework be set up early enough. 23 See Bergemann and Morris (2005) and Heifetz and Neeman (2006). that are not finetuned to the exact environment faced (e.g., on contracts that do not rely on the exact correlation between the agents' types). According to researchers, the main motivation is that realworld mechanisms seem to be simpler than what it would be required by a theoretically optimal contracting approach. For example, Arya et al. (2005, p. 15) suggest that the robustness problem might "help us better understand observed institutions." Additionally, Bergemann and Valimaki (2006, p. 3) say that "practitioners have often been led to argue in favor of using simpler but apparently supoptimal mechanisms." Also, Bergemann and Morris (2005, p. 1) emphasize that "the optimal mechanisms solving the welldefined planner's problem seem unreasonably complicated." Two cases are examined below: one in which both agents are informed about the correlation and one in which only a single agent is so informed. In the context of the privatization process introduced before, the situation with only one agent informed about the correlation could arise when only one of the firms has been operating in the industry for a sufficiently long time (perhaps in other geographical areas). Therefore, only one of the firms would be endowed with sufficient expertise to permit an accurate assessment of the extent of cost correlation in the environment. This experience would allow the firm to be well informed about the correlation between its own cost and the cost of the (less experienced) counterpart.24 On the other hand, if both firms have been operating in the industry for a similar length of time, they could both know exactly the extent to which their costs are correlated. If Both Agents Observe the Correlation The principal computes expected welfare: 24 Although plausible, this chapter does not consider the possibility that a more experienced agent could presumably be more likely to have low cost. W=(1+2) 2a,[eJ+w(e,)]A 2pr[t, W(e,)], (18) I,jG{L,H} i,j]{L,H} with pJ = lP,\ +oPj, and where p,' denotes the probability of joint costs /, and fLJ when a = a,, for i,j e {L,H} and s e {0,1}. The difference between Eq. 18 and Eq. 12 is explained by the fact that p, is perfectly known by the principal at [PCM], whereas it can only be estimated (PJ ) when correlation is uncertain and its communication is blocked. To ensure that both agents participate and report their costs truthfully for all cost and correlation realizations, the next eight constraints have to be satisfied:25 t 'i, e {L,H},i j se {0,1} (19)(112) + ([t )[ VC, ) > 0 S i, jJ {L,H},i j se {0,1} (113)(116) +(1 a)[t, v(, C )] With no communication the principal solves problem [PNOC]: Maximize Eq. 18 subject to Eq. 19 to Eq. 116. The only difference between the constraints at [PNOC] and the constraints at [PCM] is that the constraints at [PNOC] consider the possibility of two potential correlations (alpha could be either a1 or ao). The timing is depicted in Figure 11. At the solution to [PNOC], the two degrees of freedom that the principal had available at [PCM] are utilized to satisfy the four additional constraints. There are two alternative ways in which the principal can perform this task. Which of the two mechanisms is optimal depends on the specific values of the parameters, as is later illustrated in detail.26 25 Before concluding, the last section discusses whether it is always optimal to require participation and truthful revelation by all agents in all scenarios. 26 See the Appendix for a formal derivation of all the results. In the first potential solution to [PNOC], one degree of freedom is utilized to set t, = L y(e, ), while the other degree of freedom is used to set tHH equal to the lower bound that [PCM] would require if correlation is known to be low (i.e., to set tHH equal to t"n from Eq. 1 7 if a = a ). This payment structure is subsequently referred to as the CMLow mechanism, because it is identical to one of the solutions to [PCM] if the principal knows that correlation is low. The fact that t, = Vy(eL) implies that tLH = I(eLH), regardless of which correlation occurs. Since t, = yI(e, ), the principal can do no better than to set tLH = I(eLH). Setting tLH < I(eLH) would imply a violation of the participation constraints of the lowcost agent, while setting tLH > I(eH ) would imply unnecessary positive rent for him. This is no different than what the principal can accomplish at [PCM], when fully extracting expected (and expost) rent from the lowcost agent by giving him nonstochastic transfers. These payments automatically guarantee that the participation constraints for the lowcost agent are satisfied under both potential correlations because they are also satisfied expost (i.e., for any cost report of the other agent). At the same time, tHH = tH (a0) ensures that the incentive compatibility constraints for the lowcost agent are satisfied for both potential correlation coefficients. This happens because the lottery of payments for the highcost agent (tHH, tHL ) tailored to the lowcorrelation setting is sufficiently "extreme" if correlation is high.27 In other words, if the payments for the highcost agent are not attractive for the lowcost agent if correlation is low, they are even less attractive under the highcorrelation setting. The reason for this result is simple. If the lowcost agent lies 27 A lottery of payments for the highcost agent is subsequently called more "extreme" when the difference between tHH and tL gets larger. under the highcorrelation setting, he is "extremely" likely to receive the low tL payment (i.e., even more likely than in the lowcorrelation scenario). This happens because the highcorrelation framework imposes on the lowcost agent a very high chance of facing a similar agent, from al > ao. Figure 12 depicts this solution, showing the CM wellknown result of increasingly extreme payments for the highcost agent when correlation becomes arbitrarily small.28 The problem with the CMLow mechanism comes from the fact that t"m > yr(eHH), as shown by Eq. 17. This is a key feature of the solution to [PCM]: in order to prevent untruthful reporting from a lowcost agent, a highcost agent obtains positive (expost) rent if he faces an identical agent. This rent is not a problem at [PCM], because the principal offsets the high payment tHH > I(eH ) with a low payment tL < r(eH ), such that the highcost agent earns no expected rent. However, the positive and negative (expost) rents for the highcost agent come at a cost at [PNOC]: they prevent the principal from fully extracting rent from the highcost agent under both correlations, as is now illustrated. At [PNOC], the principal does not know which correlation is realized. Therefore, she has two sets of potential weights to use when calculating the expected rent of the highcost agent. As she does at [PCM], the principal would like to average positive and negative (expost) rents such that (exante) rent UH is reduced to zero. However, achieving this goal under two sets of conditional probabilities is impossible. This is apparent after a reexamination of the participation constraints for the highcost agent: U aU HH+ ( )UHL > 0, s E{0,1 with UHH tHH (eHH ) > 0 and UHL tHL (eHL ) < 0, from Eq. 17. 28 This result can also be obtained analytically by taking the derivative of Eq. 17 with respect to alpha. If the principal extracts all rent for the highcost agent at the highcorrelation scenario, she would not provide him with at least zero rent in the lowcorrelation setting. That is, U 0 = 0 is impossible without violating Uo > 0. This result is explained by the necessary positive and negative expost rents for the highcost agent (UHH > 0 and UH < 0), combined with a, > ao. Therefore, the expected rent for the highcost agent is always larger if correlation is high: aUHH + (1 a )UHL > OUHH + (1 0)UHL, As a result, the principal can only extract rent from the highcost agent under the low correlation setting, leaving him with positive rent if correlation is high:29 = a o [aD + (1 0 )HH ] >0 S2a 1 H Recalling that H, '(eH,) (e A) > 0, it can be seen that rent U, increases with effort level e,, for i e {L,H}. As it happens at [PCM], the payments for the highcost agent include rewards and penalties which discourage untruthful reporting by the lowcost agent. The problem is that these rewards and penalties increase (in absolute value) with effort eH,. That is, the higher the effort required from a highcost agent, the more extreme his lottery of payments needs to be (i.e., higher tHH and lower tL) to become unattractive for a lowcost agent. That is not a problem at [PCM], where no rent is provided to either agent. At [PNOC], however, there is rent for the highcost agent if correlation is high. This rent depends on the effort he delivers, because his effort has a positive impact on the rewards and penalties included in his payments. 29 The principal provides rent to the highcost agent if correlation is high because she requires his participation if correlation is low. This assumption is relaxed in the last section of this chapter, where the principal no longer requires that the highcost agent earns nonnegative rent if correlation is low. Consequently, the principal asks for suboptimal effort levels eH < e and eHH < e in order to reduce the rent U, of the highcost agent if correlation is high:30 V' (e=H) HH < 1 (1 + A) (2ao 1) ia (1 + 2) (2a0 1) (1a) The CMLow mechanism works well if correlation is low, where no rent is commanded by either agent. The problem arises if correlation is high, in which case rent is obtained by the high cost agent. The welfare loss generated by rent UV is severe if the potential correlations are very different. The reason is that under the CMLow mechanism, the principal tailors one of the CM solutions to the lowcorrelation setting (recall tHH = t (a0)). If this scenario does not occur, however, the welfare loss depends on how "wrong" the principal is with respect to the exact degree of correlation (i.e., how different the actual correlation is high with respect to what the principal thought low). Depending on this difference, therefore, the welfare loss under the CM Low mechanism calls for an alternative solution to [PNOC]. Intuitively, this second contract should work better than the CMLow contract when the potential correlations are substantially distinct with respect to the correlation at the lowcorrelation setting. In the second potential solution to [PNOC], the two degrees of freedom available at [P CM] are utilized in a different way than under the CMLow mechanism. This alternative mechanism does not consist of a CM solution tailored to any of the potential correlation coefficients. In this alternative contract, the two degrees of freedom available at [PCM] are employed to guarantee expost truthful reporting by the lowcost agent. That is, one degree of 30 The efficient level of effort is delivered after a low cost report: eL = eH = e*. freedom is used to set tLL y(eLL) = tHL (eHL A) while the other degree of freedom is used to set tLH I(eH ) tHH y(e A). As a result, both incentive compatibility constraints for the lowcost agent are satisfied because they are satisfied expost (i.e., for any cost report of the other agent): a, [tLL y(eLL)] [tHL (eHL A)] + (1 a, ){[tLH (e )] [tHH (eHH A)]} = 0 This mechanism is subsequently referred to as the ExPost mechanism. The disadvantage of the ExPost contract is that tLL # y(eL) which means that the principal cannot fully extract rent from the lowcost agent under both correlations. The reason for this result is, again, the impossibility of setting equal to zero the average of positive and negative (expost) rents under two possible sets of weights. At most, expected rent can be zero under only one of the potential correlations. As a consequence, under the ExPost mechanism the lowcost agent obtains rent when correlation is low: U o0 a'a0 [a HH +(1 )OHL]>0 a, + a, 1 If correlation is high, however, the rent for the highcost agent is lower than under the CM Low mechanism:31 U, = l a [aL HL +(IaLHH]>O a, +a, 1 The principal cannot completely eliminate rent U' because positive and negative (ex post) rents for the highcost agent are still required. Recall that tHH > I(eH ) and tHL < I(eHL) 31 The fact that rent UH' is lower under the ExPost mechanism than under the CMLow mechanism is easily verified from the effect of al > ao in the denominator of both expressions: a, + ao 1 > 2ao 1. The term in brackets [ ] has only a secondorder effect on rent. prevent untruthful reporting from the advantaged lowcost agent: he would probably receive a low payment if he claims high cost. It can be seen that rents under the ExPost mechanism depend on the difference between the potential correlations (a1 a ) measured with respect to the "middle" correlation coefficient (a, + ao 1). Recalling that 2a 1 denotes the correlation coefficient between costs, the "middle" correlation coefficient al + ao 1 is defined as the correlation at the middle point of the distance between al and ao. For example, if a, =0.8 and ao =0.6, the middle correlation would be the correlation coefficient if alpha was 0.7. It is important to notice that the middle correlation coefficient is not the expected correlation 2d 1, which would incorporate the probabilities of each scenario in the calculation. This confirms that the ExPost mechanism does not consist of a CM mechanism tailored to any of the potential correlations. Like under the CMLow mechanism, rents under the ExPost mechanism also increase with the effort e., delivered by the highcost agent. The explanation resides again on the positive impact of his effort on the rewards and penalties that his payments include to discourage untruthful reporting by the lowcost agent. Therefore, the principal reduces rents under the Ex Post mechanism through the suboptimal effort levels eHL < e and e < e* :32 '(e = (a, a) pu, (1a,) + pO < (1+ A) (ao + a, 1) a '(eHL) 1 A (a1 ao) pj1a, +p o (1 ao) =' (eHL L<1 (1+ A) (ao +a 1) 1 a 32 Again, the efficient level of effort is delivered after a low cost report: eL = eLH = e As it was mentioned before, which of the two alternative mechanisms yields a higher welfare level depends on the parameters of the problem. The CMLow mechanism is optimal for [PNOC] if: a < (117) 2ao 1 P,i Equation 117 can be evaluated in terms of how different the two potential correlations are, measured with respect to the correlation at the lowcorrelation setting.33 For similar correlations (small (a, a ) /(2ao 1)), the CMLow mechanism yields a higher welfare level. This happens because the welfare loss that the CMLow mechanism entails is directly proportional to how different the potential correlations are. As explained above, the rent that the CMLow mechanism provides to the highcost agent under the highcorrelation scenario is too large if the potential correlations are very distinct. Remember that this rent if correlation is high depends on how "wrong" the principal is with respect to the exact degree of correlation (i.e., how different the actual correlation is high with respect to what the principal thought low). Then, the principal should utilize the CMLow mechanism only if the potential correlation coefficients are relatively similar. In contrast, if the potential correlations become too distinct, the principal should employ the ExPost mechanism. This contract does not consist of a CM contract tailored to any of the potential correlation coefficients. Figure 13 plots the behavior of welfare under both alternative contracts as a function of how different the potential correlations are.34 Equivalently, Eq. 117 can also be analyzed in terms of the relative likelihood of each correlation coefficient. When /u is so large that Eq. 117 holds, the principal ensures that no rent is afforded under the (relatively likely) lowcorrelation scenario. She does so by implementing 33 Remember that 2a 1 denotes the correlation coefficient. 34 The graph assumes a constant value of a2 while al is shown in the horizontal axis. the CMLow mechanism, which provides no rent to either agent if correlation is low. Although this payment structure provides rent UH1 to the highcost agent if correlation is high, the principal still applies the CMLow mechanism if the highcorrelation setting is relatively unlikely (i.e., if u, is so low that Eq. 117 holds). On the other hand, if /u is so large that Eq. 117 does not hold, the principal reduces the (relatively likely) rent U. as much as possible by means of the ExPost mechanism. She cannot completely eliminate this rent because tHH > I(eHH) and tL < VI(eHL) still prevent untruthful reporting from the lowcost agent. The drawback to the ExPost mechanism is that the lowcost agent obtains rent UL if correlation is low. Yet, the principal is willing to accept rent U if the lowcorrelation scenario is relatively unlikely (i.e., if /u is so low that Eq. 117 does not hold). The results obtained are similar to those of Arya et al. (2005). In an auction environment, they also find a condition that determines which of two mechanisms the principal should apply. Furthermore, the proposed mechanisms and the circumstances under which they should be employed are similar to the ones presented in this study. If the uncertainty about the correlation is not severe, the principal should utilize a CMLow type of mechanism.35 On the other hand, the principal should employ an ExPost type of mechanism if the uncertainty about the correlation is 36 more severe. Additionally, after considering the parameter values assumed by Arya et al., the condition that determines optimality in their paper is the same as Eq. 117 in this study. Instead of a discrete support for the parameter alpha, they assume a uniform distribution between a lower bound a > 0.5 and 1. Given such a probability structure, they find that the principal should 35 BayesianNash is the name given by Arya et al. to such a contract. 36 DominantStrategy is the name given by Arya et al. to such a contract. employ the CMLow type of mechanism if a > 2/3. Now notice the result of substituting U0 = ,1 = 0.5 and a, = 1 into Eq. 117: 1 ao 0.5 2ao 1 0.5 Manipulation yields a0 > 2/3, which means that their condition can be considered a special case ofEq. 117 in this chapter.37 If Only One Agent Observes the Correlation Consider now a situation where communication of the correlation is still blocked, but where only agent A observes the exact correlation before contracting. This is a framework where only agent A is endowed with a strong knowledge and expertise about the industry. Meanwhile, agent B and the principal remain ignorant about the correlation. They never acquire such privileged information, knowing only the probabilities of facing either a, or a Returning to the previous privatization example, this asymmetry of agents could arise when firm A has been operating in the industry for a long time (perhaps in other markets), while firm B is relatively new in the industry. In such a context, it is plausible that not all the agents are perfectly informed about the correlation between their costs. Since it is assumed that the principal does not design a different menu of options for each agent even when she distinguishes the experienced agent A from the inexperienced agent B, the solution to this problem is identical to the solution to [PNOC]. The explanation for this result is simple. To ensure A's participation, the principal must guarantee that participation and incentive 37 There is only one difference between their results and the ones in the present study. Unlike here, the ExPost type of mechanism from Arya et al. does not provide any rent to the lowvaluation bidder (which is the equivalent of the highcost agent). (See their Corollary 2, on page 11). The linearity of the agents' utility in their model is the conjectured explanation for such a difference (i.e. there is no convexity like the one coming from the disutility of effort in the present model). A formal proof is beyond the scope of this study. compatibility constraints are satisfied under both potential correlations. Therefore, the constraints are automatically satisfied for the inexperienced agent B, who computes expected rent not only across both potential types of counterpart, but also across both potential correlations. Formally, if the constraints from [PNOC] are satisfied, then the equivalent constraints for an agent that only observes a are also satisfied: i, j.E {L,H},i j + (1 ()[tj y( C, )] > 0 i, j f{L,H},i j + (1 a)[t, 6(#, C, )] As a consequence, the solution to [PNOC] also applies if only one of the agents observes the correlation but the principal offers the same menu of options to both of them. Uninformed Principal: No Limits on Communication In the second case of interest, nothing prevents the principal from asking for a correlation report from the informed agentss. Unlike in the case with limits on communication, the Revelation Principle applies in this context. Therefore, the principal can restrict attention to truthful and direct mechanisms where the agents report all their private information. This situation is interesting because an agent who is informed about the correlation becomes a two dimensional type of agent. The (simple) situation where both agents observe the correlation is presented first. Next, the (more interesting) setting with only one agent informed about the correlation is investigated. If Both Agents Observe the Correlation It is well known that if some element of private information is common knowledge for both agents, the principal can elicit that information at no cost.38 She can do so by designing a mechanism where both agents report their individual cost and the realized correlation: rA = (/,, a,) and rB = (/6 ah), with i,j {L,H} and s,h e {1,0}. Since it is impossible to have different (and truthful) correlation reports, the principal can design a "forcing" contract that threatens the agents to very large negative rents if the correlation reports differ.39 Therefore, even when the principal has imperfect information about the correlation, the firstbest situation can still be replicated if both agents can communicate the actual correlation observed. Hence, the solution to this problem is identical to the solution to [PCM].40 If Only One Agent Observes the Correlation If the correlation is known only by agent A, the principal designs a mechanism where the inexperienced agent B only reports his cost, rB = (/), while the experienced agent A reports his cost and the realized correlation, rA = (/j as), with i, j e L, H) and s e {0,1}. The timing of the game is depicted in Figure 14. As explained above, both agents are offered the same menu of contracts even when the principal knows about their asymmetry in terms of their information on the exact correlation. Let ts y(/8 C ) denote the expost rent for an agent (A or B) that reports cost /, when he has 38 See Fudenberg and Tirole (1991). 39 In the work of Tangeras (2002), for example, the element of private information that is common knowledge for both agents is the "industry" portion of their costs, which has to be added to each agent's idiosyncratic cost. Their cost structure follows the model proposed by Auriol and Laffont (1992). 40 There also exists a Nash equilibrium in which both agents lie about the correlation. As pointed out by Fudenberg and Tirole, the possibility of multiple equilibria in "shootthemall" mechanisms "gave rise to a large literature on unique Nash implementation" (1991, p. 293). cost /,, when the other agent reports cost /k and when the experienced agent A reports that alphais as, for i, j,k e {L,H} and s {0,1}. The principal computes expected welfare: W= /((1+A) I2pi[e + The difference between Eq. 118 and Eq. 18 is that the principal now assigns different contracts for different correlations. To ensure that the experienced agent participates and reports his cost truthfully for all cost and correlation realizations, conditional on a truthful correlation report, the next constraints have to be satisfied: u, : n,[tl ',( CC )] i, je {L,H},i # j s e {0,1} (119)(122) + (1 a,)[ti y(, C;)] > 0 > a,[t (?,c )] i, jeL,H},i j s {0,1} (123)(126) + (1 a)[t; (A C )] Additionally, conditional on a truthful cost report, the experienced agent A should not lie about the correlation: Us as[th Vf'8 C,,)] Sath i, j{e L,H},i # j s, h {0,1} (127)(130) + (1 )[t, (# C,)] Finally, agent A should not simultaneously misreport his cost and the correlation: U'" 2a [th Vf(,8 _Ch)] i, je {L,H},i j s,h {0,1} (13)(134) + (1_a,)[th _( _Ch)] '\ U/II ii~\r With only one agent informed and with communication of the correlation, the principal solves problem [PCO]:41 Maximize Eq. 118 subject to Eq. 119 to Eq. 134. Intuition could suggest that once a correlation report becomes available, the principal should again be able to achieve the firstbest by means of two different CM mechanisms, one for each correlation. That is not the case, however. The problem is that at [PCO], the agent who is informed about the correlation may not have the incentive of truthfully reporting its realization. In particular, a highcost agent would gain by claiming that correlation is low when it is actually high. An intuitive explanation follows. Consider a payment structure that consists of two independent CM mechanisms (i.e., one for each correlation). Based on the correlation report, the principal could offer nonstochastic payments to the lowcost agent and an extreme enough lottery (i.e., low tL and high t' ) to the highcost agent, for s e {0,1}. Under each correlation, the highcost agent's lottery would extract all his rent and it would also prevent a lowcost agent from exaggerating his cost. The problem of these two independent CM mechanisms is that if the experienced agent has high cost, he would enjoy rent under the highcorrelation setting if he claims that correlation is low. The explanation behind this incentive follows. After the highcost agent underreports the correlation, he would be assigned the high payment toH with probability a, (instead of with probability ao) and he would be assigned the low payment to with probability 1 a (instead of with probability 1 a ). Since a, > a, this average of positive and negative expost rents yields rent for the highcost agent if correlation is 41 As in the case with no communication and only one agent informed, it is not necessary to impose participation and incentive compatibility constraints for the inexperienced agent B. Those constraints are automatically satisfied when the principal offers the same menu of contracts to both agents. high. Therefore, a highcost agent would want to report that correlation is low when the high correlation environment arises. Although they prevent untruthful cost reporting by the lowcost agent, extreme payments to the highcost agent generate a new incentive if there is asymmetric information about the correlation. A highcost agent would earn positive rent if correlation is high by reporting that correlation is low. As a result of this incentive, full rent extraction is not possible at [PCO]. The twodimensional type of the informed agent does not allow the principal to consider the problem as if they were two independent CM problems. The rent for the highcost agent if correlation is high is unavoidable.42 This result is similar to what happens at [PNOC], although the source of the rent at [PCO] is different. At [PNOC], the rent for the highcost agent at the highcorrelation setting occurs because the principal cannot tailor his payments to the actual correlation. This comes as a result of designing only one set of payments that has to work well regardless of the correlation. At [PCO], however, the principal can design different payments for different correlations. Yet, the rent for the highcost agent if correlation is high still prevails. At [PCO], this rent exists to discourage him from claiming that correlation is low when it is actually high.43 The solution to [PCO] is similar to the solution to [PNOC] in that Eq. 117 again determines the optimal mechanism. If the lowcorrelation scenario is relatively likely (i.e., if u0 is so large that 117 holds), the principal only affords rent to the highcost agent in the (relatively unlikely) event that correlation is high: 42 The incentive of the highcost agent to claim that correlation is low when it is actually high is always binding at [PCO]. 43 If [PCO] could be considered as two independent CM problems, there would be four degrees of freedom (recall that a standard CM problem has two degrees of freedom). Since one degree of freedom is utilized to prevent untruthful correlation reporting from the highcost agent if correlation is high, problem [PCO] has three degrees of freedom instead of four. 1 = a [ao0o ]+(Iao)o j>0 S 2a0 1 The principal provides rent to the highcost agent if correlation is high by increasing the payment t' (without reducing t' accordingly) above the t, that an independent CM mechanism would require (i.e., one designed specifically for the highcorrelation setting).44 As a result, the lottery (t ,, ti ) under the lowcorrelation scenario is relatively less attractive than the lottery (t' tG ) obtained if truthfully revealing that correlation is high. Consequently, the rent provided by the lottery (t ,, tE ) prevents the highcost agent from underreporting the correlation.45 To reduce the rent of the highcost agent if correlation is high, the principal also requests less than the efficient effort from the highcost agent if correlation is low. This suboptimal effort e < e reduces how extreme his payments need to be if correlation is low, because it reduces the rewards and penalties included in those payments as a means to avoid untruthful cost reporting by the lowcost agent:46 V'(eHH) = 1 '< ( )HH 1; and (1 + A) (2a 1) /oua, v"(eZL) =1 A (a a\ ma \L 1. (1 + A) (2a 1) ( ao) 44 Equation 17 would determine this value, for a = al. 45 Since the lottery for the highcost agent if correlation is high is even more extreme than what an independent CM mechanism would require, a lowcost agent does not have the incentive of claiming highcost under the high correlation environment. He also does not have the incentive of claiming high cost and low correlation. The reason is as follows. If the highcost/lowcorrelation lottery is unattractive for a lowcost agent if correlation is low, it is even more unattractive for a lowcost agent if correlation is high (the low payment tm0 would arise even more frequently, from al > ao). 46 Since the effort exerted by the highcost agent in the highcorrelation scenario does not have any impact on rents, its socially efficient level is required: eHH' em e*. Also, eLL eLH eLL eLHO e* As a consequence of this effort distortion, the less extreme payments if correlation is low become less attractive for a highcost agent who would untruthfully underreport the correlation. Making the lottery (tH ti ) under the lowcorrelation scenario less extreme contributes to the rent reduction for the high cost agent if correlation is high, because the required increase in tH is smaller than what an independent CM mechanism would dictate. The contract just described is similar to the CMLow mechanism at [PNOC].47 As such, welfare decreases substantially if the principal applies this mechanism when the correlation coefficients are very different (with respect to the correlation at the lowcorrelation setting).48 The reason comes again from the rent for the highcost agent, which increases with the difference between both potential correlations. Therefore, if Eq. 117 does not hold, the highcost agent commands smaller rent in the (relatively likely) event that correlation is high: S1 al ao (1 a + L]> a, +a, 1 Like at [PNOC], the principal reduces rent U1 at the cost of affording rent to the low cost agent in the (relatively unlikely) event that correlation is low: U,=o [a 0 +] >0 20 k'o 1D [aHOH + ((1 aO0)D]HL a + aO 1 The suboptimal effort levels in this case are as follows ,,o z (l ao) [1oo + (1 ) a o0 ,1 y'(eHH) =1 HH < 1; and (1+ A)(ao +a 1) / oa /', A (a, ao) [PO(1 ao)+/]i] O' (1 + A) (ao + a 1) /0( (1 ao) 47 Although it is not required due to the degrees of freedom available, the principal could offer nonstochastic payments to the lowcost agent under both correlations: t1 = tLH t = tLHO v(e*) 48 Again, recall that 2a 1 denotes the correlation coefficient. This contract resembles the ExPost mechanism at [PNOC]. Rents again depend on the difference between the potential correlations (a, a ) with respect to the middle correlation coefficient (a, + a, 1), which refers to the correlation at the middle point of the distance between al and a Another similarity with the ExPost mechanism from [PNOC] resides on the fact that the incentive constraints that prevent the lowcost agent from exaggerating cost are binding under both correlations instead of only at the lowcorrelation setting.49 Limits on Communication and Exclusion The analysis so far has assumed that the principal never excludes any agent from the game. That is, participation and truthful revelation of private information has been required from all agents under both potential correlations. The results from the optimization problems performed have illustrated the welfare loss imposed by this assumption. In particular, problem [PNOC] showed that rent for the highcost agent is always higher under the highcorrelation scenario than under the lowcorrelation setting (U1 > U ). Recall that the explanation resides on the implications of the link between UHH > 0 and UHL < 0 (which prevent untruthful reporting by the lowcost agent) and a, > ao. As a consequence, the best the principal could do is reduce rent U to zero and ad allow positive rent U A natural question then arises: should the principal always require nonnegative rent UO > 0 for the highcost agent if correlation is low? The answer depends on the welfare achieved with and without the participation of the highcost agent if correlation is low. Intuitively, if the probability of the lowcorrelation scenario is low enough, it may be optimal to exclude the highcost agent from the contract. Then, consider 49 Recall that this is the reason why this mechanism was given the ExPost name. the following problem, where the principal designs a mechanism that would provide negative rent for the highcost agent if correlation is low. The principal computes expected welfare: W = (1+ 2) 2T,, [eL, + Y(eL,)] + L 2pH [eH + ze{L,H} JE{L,H} S(135) A, 2 L, [t, I(e, )] + ,L2p' [tH  )] j z{L,H} {jL,H} Equation 135 arbitrarily excludes the highcost agent from the game if correlation is low (i.e., if alpha is ao). To ensure that the highcost agent does not participate (whether truthfully or lying) if correlation is low, the next two constraints have to be satisfied: U a [tHH /H CHH ) (136) +(1 ao)[tHL V( CHL)] <0 ao [tLH (H CLH)] (137) + (1 ao)[t, y(,H CLL)] <0 Like at [PNOC], to ensure that the lowcost agent participates and reports his cost truthfully for both correlations, the next four constraints have to be satisfied: Us a,[t Vf(68 C, )A U [C sE {0,1} (138)(139) + (1 a)[tLH (fL CL)] > 0 Us > a,[t Vf(68 CH )A U[ HL ( CL)] e {0,1} (140)(141) + (1 as )[tHH (8L CHH)] Like at [PNOC], to ensure that the highcost agent participates and reports his cost truthfully if correlation is high, the next two constraints should be verified: UH = at [HH 6(H CHH) (142) + (1 a, )[tHL 68H CHL )] >0 U', > a [tL H LH H I ILH H CLH)] (143) + (1 a )[t, y(H CLL)] With no communication on the correlation and excluding the highcost agent from the game if correlation is low, the principal solves problem [PEXCL]: Maximize Eq. 135 subject to Eq. 136 to Eq. 143. At the solution to [PEXCL], the socially efficient effort level is restored: e = e *, for i, j e {L, H}. No rent is provided to the lowcost agent under any correlation (U1 = U = 0) and no rent is earned by the highcost agent if correlation is high (U1 = 0). As expected, the high cost agent would obtain negative rent if correlation is low (U < 0), so he does not participate in that scenario. The principal need only ensure that tL is low enough: t, Like in the previous problems, by setting tL small enough the principal guarantees that a lowcost agent does not exaggerate his cost. Additionally, tL small enough ensures that the highcost agent earns negative rent if correlation is low (U < 0) because all rent is extracted if correlation is high (U' = 0). Recall from problem [PNOC] that U1 > U implies that U < 0 if U, = 0. The welfare loss at the solution to [PEXCL] is attributed to the zero effort delivered by the highcost agent if correlation is low. Therefore, if the probability of the lowcorrelation setting is sufficiently low, the solution to [PEXCL] may be an alternative to the ExPost mechanism obtained at [PNOC]. Recall that the ExPost contract is optimal at [PNOC] when the lowcorrelation setting is relatively unlikely (i.e., when po is so low that Eq. 117 does not hold). Some conclusions can be drawn from numerical examples. Figure 15 shows that welfare at the solution to [PEXCL] may be still below the welfare achieved under the ExPost mechanism (which constitutes the solution to [PNOC]), even for a relatively small probability of the lowcorrelation setting (5% in the example). Figure 16 shows that a slight decrease of that probability (to 3%) can raise welfare at the solution to [PEXCL] above the welfare achieved at the solution to [PNOC]. Finally, Figure 17 illustrates that when the probability of the lowcorrelation scenario drops substantially (to 1%), the welfare under the ExPost mechanism is almost always below the welfare attained at the solution to [PEXCL]. Therefore, depending on the parameters of the problem, the figures show that it may not be optimal to require participation and truthful revelation of private information from all agents under all scenarios.50 Conclusions This chapter contributes to the recent and growing literature on both multidimensional and robust mechanism design. The results from [PNOC] show that full rent extraction is impossible if two informed agents cannot communicate the realization of the correlation to the principal. This finding coincides with the results obtained by other researchers on robust mechanism design, like Arya et al. (2005), Bergemann and Morris (2005), Heifetz and Neeman (2006) and Neeman (2004). Additionally, the solution to problem [PCO] illustrates that even with communication on the correlation the principal may not be able to fully extract rents from the more informed agents. These results coincide with the findings by authors investigating multidimensional mechanism design, like Miller et al. (2007). When limits on communication on the correlation prevail, the principal cannot tailor the payments to the exact correlation coefficient. As a result, the highcost agent earns positive rent 50 Although the analysis was not performed, it is conjectured that the ExPost type of mechanism that constitutes a solution to [PCO] if Eq. 17 does not hold could also be replaced by a contract that excludes the highcost agent from the game if correlation is low. if correlation is high. This result is the most important finding of the analysis under limited communication. The reason behind the rent for the highcost agent under the highcorrelation setting is that his extreme lottery of payments should ensure that nonnegative expected rent is obtained under both potential correlations. Since expected rent is calculated using two different sets of weights (one for each correlation), and since the high payment is obtained with higher probability (and the low payment with lower probability) under the highcorrelation environment, the principal can reduce rents for the highcost agent to zero only if correlation is low. As a remainder, extreme payments for the highcost agent prevent the lowcost agent from exaggerating cost. When full communication is restored, and if only one agent is perfectly informed about the extent of cost correlation, the principal is still unable to fully extract rent from both agents under both correlations. Even when the mechanism designer can tailor payments to each correlation coefficient, the highcost agent still obtains positive rent if correlation is high. This is the most important result of the analysis without limits on communication. Although the rent for the high cost agent if correlation is high parallels the results with blocked communication, the source of this rent is slightly different. Instead of being explained by the inability of the principal to tailor payments to each correlation, when communication on the correlation is available the principal has to worry about a highcost agent untruthfully underreporting the correlation. He would do so because he would obtain a high payment more frequently (and a low payment less frequently) than what he should if the principal designed two independent CM mechanisms, one for each correlation report. The rent for the highcost agent if correlation is high is driven by the fact that nonnegative rent is required for him under the lowcorrelation setting. If the principal could exclude the high cost agent from the game if correlation is low, the socially efficient effort level can be restored without affording any rents to the (participating) agents. The cost of such a contract is that no effort is delivered by the highcost agent when correlation is low, because he does not participate due to the negative rent he would obtain. If the probability of the lowcorrelation scenario is small enough, numerical examples show that the principal can achieve higher welfare by designing a contract that excludes the highcost agent from the game when correlation is low. Finally, the usual criticism of the risk neutrality assumption also applies to this study. The solutions obtained make use of that assumption, since agents here voluntarily participate in the mechanism proposed by the principal even when they could potentially obtain large negative ex post rent. Therefore, the impossibility of the fullrentextraction result would probably be more pronounced if risk aversion or limited liability constraints were imposed. All Each agent The principal Both agents Each A specific The agents parties learns his offers two observe the agent (t,C) pair is produce, learn al, own cost contracts correlation reports assigned to transfers are ao and parameter (3 but do not his cost each agent made and p1, po report it depending costs are on both reimbursed reports Figure 11. Timing at [PNOC], when both agents observe but do not report the correlation. Payments under the CMLow contract  tLL(CMLow)  tHH(CMLow)  tHL(CMLow)  tLH(CMLow)    0.56 0.60 0.63 0.67 0.71 0.74 0.78 0.82 0.85 0.89 0.93 0.96 1.( ..  0.20 0.40 0.60 0.80 Parameters: ao= 0.55 Po= 0.01 1t = 0.99 X= 0.60 A= 0.20 1.00 0.80 0.60 0.40 0.20 0.00 Figure 12. A solution to [PCM] that can also be a solution to [PNOC]. l(e)= 1 e 4 '' '' '' '' Welfare at PNOC \   .... " ... 2.42 2.40 2.38 2.36 2.34 2.32 2.30 2.28 2.26 2.24 0 Figure 13. Welfare under the two alternative mechanisms at [PNOC].   CMLow PNOC  ExPost PNOC (X1 Parameters: ao= 0.55 o = 0.70 pi = 0.30 S= 0.60 A= 0.20 l(e)= 1 e 4 .56 0.60 0.63 0.67 0.71 0.74 0.78 0.82 0.85 0.89 0.93 0.96 1.( All Each agent Agent A The principal Each agent A specific The agents parties learns his observes offers four reports his (t,C) pair is produce, learn a,, own cost the contracts cost and assigned to transfers are ao and parameter (3 correlation agent A each agent made and p1, po reports the depending costs are correlation on both reimbursed reports Figure 14. Timing at [PCO], when only agent A observes and reports the correlation. Welfare at PNOC vs PEXCL 2.40 2.39  \ 2.38 2.37 2.36 2.35 2.34   ExPost PNOC Altern PEXCL 0.56 0.60 0.63 0.67 0.71 0.74 0.78 0.82 0.85 0.89 0.93 0.96 1. 0.56 0.60 0.63 0.67 0.71 0.74 0.78 0.82 0.85 0.89 0.93 0.96 1.( Figure 15. Welfare at [PEXCL] below welfare at [PNOC] Parameters: ao= 0.55 lo = 0.05 pi = 0.95 S= 0.60 A= 0.20 2.33 2.32 2.31 l(e)= 1 e 4 Welfare at PNOC vs PEXCL 2.40 2.39 2.38 \ 2.37 / / 2.36  2.35  ExPost PNOC 2.34 Altern PEXCL 2.33 ao 2.32 . 0.56 0.60 0.63 0.67 0.71 0.74 0.78 0.82 0.85 0.89 0.93 0.96 1.CI Figure 16. Welfare at [PEXCL] sometimes larger than welfare at [PNOC] Parameters: ao= 0.55 to = 0.03 t1 = 0.97 S= 0.60 A= 0.20 Tl(e) = 1 e 4 Parameters: I S ao= 0.55 to = 0.01 t1 = 0.99 S= 0.60 A= 0.20 l(e) = 1 e 4 4 2.40 2.39 2.38 2.37 2.36 2.35 2.34 2.33 2.32 0.56 0.60 0.63 0.67 0.71 0.74 0.78 0.82 0.85 0.89 0.93 0.96 1.AC Figure 17. Welfare at [PEXCL] almost always larger than welfare at [PNOC] Welfare at PNOC vs PEXCL  ExPost PNOC  Altern PEXCL L CHAPTER 2 EFFICIENCY IN THE BRAZILIAN SANITATION SECTOR Introduction Early work on the relative performance of water and sewerage (WS) utilities by Crain and Zardkoohi (1978) tried to determine whether private U.S. water utilities attained a more efficient level of operation than public ones. Since then, a number of papers have been published on the efficiency of WS utilities. Some authors have also focused on the private vs. public issue, while others have tested other hypotheses, like the existence of economies of scale, economies of scope, or the possible homogeneity or homotheticity in the production technology. Data availability limited the types of studies: early papers focused mostly on utilities in the U.K. and U.S. because these countries pioneered the collection and publication of data on WS firms. Until a decade ago, little research was conducted into the efficiency of WS utilities in developing countries. Then, studies began to address the performance of water systems using quantitative techniques. A number of papers focused on Asian and African water utilities, most of them supported by the World Bank.1 These studies provided insights for countries implementing World Bank policies to increase coverage and quality of WS services in their regions. Politically potent and economically important, WS utilities generate public concern over efficiency. Using data from Brazil, this study extends the standard approach by comparing not only the performance of public and private WS firms, but also the performance of different types of public WS operators. There are four different types of WS providers in Brazil. The first type consists of regional public operators, which provide services at the state level. The other three types of WS operators 1 For example, Estache and Rossi (1999), Estache and Rossi (2002), and Estache and Kouassi (2002). provide services at the local (municipal) level. The first local type consists of private firms that have signed concession contracts with the municipalities where they operate. The other two local types both consist of publiclyowned operators, but they differ in their legal status. One type consists of local public providers that are organized similarly to a corporate business. They are called publiccorporative operators throughout this chapter. The other type consists of local public providers that are run like notforprofit organizations. They are called publicnon corporative providers throughout this study. Besides the public vs. private discussion, controversy exists in Brazil about whether municipalities or states should be responsible for WS provision. The Association of StateOwned Sanitation Firms (AESBE), for example, argues that WS services should be provided at the state level for two reasons: one, the larger scale of operation permits scale economies; two, there is a possibility of crosssubsidization between poorer and richer municipalities.2 In contrast, the National Association of Municipal Sanitation Services (ASSEMAE) favors municipal provision on the grounds that WS services are an essential necessity for the population.3 ASSEMAE does not explain in detail why the essential nature ofWS services calls for municipal and not state provision, but standard arguments are based on local control, responsiveness to citizen concerns and awareness of local conditions. This study finds that WS provision in Brazil is characterized by economies of scale. Therefore, since an increase in output generates a lessthanproportional increase in (operating) costs, WS provision at the state level should be preferred. The potential efficiency gains are not trivial when one recalls that Brazil has a population of 187million people. 2 Associaqio das Empresas de Saneamento Basico Estaduais. See Ihip \ \ .aesbe.org.br (last visit: March 26, 2007). 3 Associaqao Nacional dos Serviqos Municipais de Saneamento. See http://www.assemae.org.br (last visit: March 26, 2007). In a first stage, a fixedeffects panel data model with data for 20002004 is employed. A cost function is proposed, identifying firmspecific (operating) costs which account for inefficiency and other unobserved heterogeneity. In a second stage, those firmspecific costs are explained by means of firmtype and other timeinvariant indicator variables. The results show that regional operators have lower firmspecific costs than localprivate and localpublic providers. This finding indicates that the efficiency gains of the state providers from their scale of operation are augmented by their lower firmspecific costs. This work also shows that the localpublic vs. localprivate comparison depends on how the local public provider is organized. Private and publiccorporative providers have lower firm specific costs than publicnoncorporative providers. This finding indicates that the WS operators organized as notforprofit organizations have the highest firmspecific costs in Brazil. In spite of the firmspecific cost differences found, it is worth mentioning that these differences represent a small portion of operating costs. The firststage regressions illustrate that the output produced, input prices and other technological factors explain most of the variation of operating cost, regardless of the firmtype. As a result, the firmspecific cost differences found are not substantial from an economic point of view. Quantifying the relative efficiency of the Brazilian regional operators, Tupper and Resende (2004) use Data Envelopment Analysis (DEA) with data for 19962000. The efficiency scores obtained are considered in the construction of a proposed linear reimbursement rule that constitutes a yardstick mechanism. However, the authors acknowledge that its implementation is constrained by the weak current regulatory framework. Utilizing DEA with data for 19982002, Seroa da Motta and Moreira (2006) argue that the government level at which conceding authority resides is not a crucial barrier to the Brazilian sanitation sector's development when looking at the operators' performance. Unlike this study, they find that ownership does not matter for productivity gains for municipal services. Like this chapter, they find that regional operators benefit from larger scale economies. Evidence on the beneficial effects of private sector participation (PSP) in the Latin American sanitation sector is not conclusive. In Argentina, Bolivia and Brazil, for example, Clarke et al. (2004) find that even when connection rates to piped water improved following the introduction of PSP, connection rates similarly improved in the control regions that never privatized. In contrast, Galiani et al. (2005) find that child mortality in Argentina fell 8% in the areas that privatized their water services. They conclude that while privatization is associated with reductions in deaths from infectious diseases, it is uncorrelated with deaths from causes unrelated to water conditions. The ambiguity on the beneficial effects of PSP in the Latin American sanitation sector coincides with the results from other regions.4 This chapter first presents an overview of Brazil's water and sewerage industry. The study then illustrates the twostage methodology utilized. A following section present the results obtained. After performing some sensitivity checks, conclusions are provided in a final section. Overview of Brazil's Water and Sewerage Industry In 1971, Brazil created a national plan for WS provision (PLANASA).5 This plan delegated authority for the provision of WS services to twenty seven newly born stateowned companies. According to PLANASA, these public companies were the only sanitation entities authorized to obtain financing from the National Housing Bank (Banco Nacional de Habitacao  4 For the US, Bhattacharyya, Parker and Raffiee (1994) found evidence of greater efficiency in public utilities, Crain and Zardkoohi (1978) found evidence in favor of private operators, and Byrnes et al. (1986), Feigenbaum and Teeples (1984) and Fox and Hofler (1986) found no difference between public and private operators. For Asia, Estache and Rossi (1999) found evidence in favor of private operators while in a later study Estache and Rossi (2002) found no difference. For Africa, Estache and Kouassi (2002) found evidence in favor of private firms. 5 Plano Nacional de Saneamento. See Soares (2001) for a detailed description. BNH). This feature made PLANASA attractive for the municipalities that were interested in expanding their sanitation systems.6 About 3,200 municipalities joined the new plan, awarding concessions to the stateowned companies for 20 to 30 years.7 On the other hand, about 1,800 municipalities never adhered to PLANASA, providing WS services on their own ever since. The three types of local operators (private, publiccorporative and publicnoncorporative) provide WS services within the municipalities that never joined the system. The PLANASA model started to decline in the 80's. After 1986, the BNH was unable to finance the required expansion of the WS sector due to a weak fiscal situation of the federal government.8 Antiinflationary policies may have also played a role, since the government at that time pressed for low water tariffs to keep inflation under control. The 80's decade was also characterized by an emphasis on decentralization, best illustrated in the constitutional reform of 1988. The centralizing concept of PLANASA, on the other hand, was more in accordance to the military regimes of earlier decades. There have recently been some attempts to define a new framework for the WS sector. Bill 4147/2001, for example, intended to allow for more private participation. This bill defined the states to be the conceding authority in metropolitan areas. However, the constitutional reform of 1988 granted to municipalities the right to make concessions for public services of "local" 6 See Faria (2005). SIn many cases, however, there was never any formal contract between the municipality and the stateowned companies. SPLANASA formally extinguished in 1992. See Parlatore (1999). interest.9 Due to that controversy about the interpretation of the Constitution, Bill 4147 never became law. 10 11 A new Bill 5296/2005 also attempted to redefine the rules for the WS sector. A Parliamentary Commission approved it on July 2006 after many modifications and it recently became Law 11445 in January 2007. This new bill specifies that municipalities have the conceding authority over services of local interest. Nevertheless, the implications of the new Law 11445 are yet to be seen. Due to the heated debate about where the conceding authority resides, only some municipalities that never adhered to PLANASA have made concessions to private operators.12 These private companies provide WS services to less than 4% of the population. 13 The Brazilian Association of Private Water and Sewage Operators (ABCON) suggests that only through a more active private participation will the WS sector meet the high investment levels required. 14 According to 2004 data, approximately 112,000 people are directly employed in the WS sector, almost 90% by the stateowned companies. The national coverage for water services is roughly 85%, although the sewage coverage is below 60%. On average, almost 30% of the treated water produced is unaccounted for, due to leaking through broken pipes and illegal 9 Water distribution and sewerage collection are defined as services of "local" interest. On the other hand, water catchment and water and sewerage treatment are defined to be of local interest only in case of exclusive use by the municipality. See Minist6rio das Cidades, http://www.cidades.gov.br (last visit: March 26, 2007). 10 There was also a strong opposition from the public and representative institutions, like the Brazilian Association of Sanitary and Environmental Engineering (Associaqio Brasileira de Engenharia Sanitaria e Ambiental  ABES). See http://www.abesdn.org.br (last visit: March 26, 2007). 11 A Nacional Water Agency (Agencia Nacional de Aguas ANA) was also created in 2000. However, the main function of ANA is to monitor the utilization of water resources. Its role as a regulator is yet to be defined. 12 See Vargas and De Lima (i 2' 4). 13 This figure contrasts with the situation in other infrastructure sectors like telecommunications, railroads and electricity, where private participation is much more active. See Oliveira and Fujiwara (2005) and Pinheiro (2003). 14 Associaq~o Brasileira de Concessiondrias de Serviqos Puiblicos de Agua e Esgoto. See http://www.abcon.com.br (last visit: March 26, 2007). connections. Furthermore, only 50% of the sewage volume collected receives some type of treatment. Table 21 shows these statistics by operatortype. Methodology Duality theory implies that the production technology of a firm can be modeled with a cost function, where the firm's cost depends on its output level and the prices of the inputs employed in production. Other factors may also affect the firm's technology and hence the firm's costs. Specifically: c =c(q,w,z) (21) where c denotes cost, q denotes output level, w denotes input prices and z includes other control variables.15 To empirically estimate Eq. 21, a panel data framework is adopted: Y, = Xtfl + u, + E, (22) where Y,, denotes the dependent variable for individual i at time t, X,, denotes the vector of explanatory variables, u, accounts for timeinvariant heterogeneity at the individual level and E,, denotes random statistical noise. Heterogeneity is the denomination of the observed and unobserved unique individual characteristics. Fixed or random effects models can be adopted for panel data. The fixed effects model allows unobserved heterogeneity to be correlated with the explanatory variables. In contrast, the random effects model assumes that any unobserved heterogeneity is distributed independently of the covariates. In the context of this study, correlation between unobserved heterogeneity and the explanatory variables is hard to rule out. Such a correlation would exist, for example, if the firm 15 Control variables have been sometimes denominated hedonic measures, referring to the approach introduced by Spady and Friedlaender (1978) for the trucking industry. They emphasize that the service dimensions provided by the firm should enter the cost function as explanatory variables. Feigenbaum and Teeples (1983) first introduced the hedonic approach to the water sector. can modify its output level based on private information about its unobserved inefficiency. Therefore, a fixed effects model for panel data is employed. The fixed effects formulation allows unobserved inefficiency to be captured by the firmspecific coefficients. 16 In a panel data framework, the cost function from Eq. 21 takes the following form: c,, = /o +, fq,, +8 zI, + Pz,, + E, +u,, (23) where the f 's are parameters to be estimated and u, denotes cost inefficiency and any additional unobserved heterogeneity. It is assumed that the ,, are i.i.d and uncorrelated with the covariates. In contrast, the u are allowed to be potentially correlated with the explanatory variables. 17 The least squares dummy variables (LSDV) estimator is utilized, including also a yearspecific effect:18 c,, = a, + P q,t + Pf. i",, + l z,, + Yt + ,, (24) where the firmspecific intercepts a, f, + u, account for inefficiency and any other unobserved heterogeneity. Utilizing this formulation, Schmidt and Sickles (1984) proposed the measure a,* = a, min(a) to construct a ranking of relative inefficiency. Their approach permits the computation of individual inefficiency terms relative to the most efficient firm in the sample. 19 That calculation might be appropriate when one is concerned about efficiency at the individualfirm level. For example, a regulator could set the tariff of firm "A" based partly on its 16 According to Greene (2005), assuming that inefficiency is timeinvariant is not a problem in short panels. This is especially true in the water industry, which is characterized by low technological change. 17 It is unnecessary to make any distributional assumption on the inefficiency term p If one is willing to make distributional assumptions on the [i, Maximum Likelihood would theoretically allow for more efficient estimates than both fixed and random effects models. Nevertheless, Kumbhakar and Lovell (2000) and MurilloZamorano (21 114) mention several papers that after performing empirical comparisons of the three approaches generate similar efficiency rankings, especially at the top and bottom of the distribution. 18 The LSDV estimator is equivalent to the withingroups estimator. 19 Ashton (2000) constructs an efficiency ranking of British water firms utilizing that measure. relative efficiency with respect to firm "B", as suggested by Shleifer (1985). However, the focus of this study is not yardstick comparison between individual operators but rather between different types of firms. The goal of this chapter is to identify whether regional, localpublic or localprivate firms are relatively more efficient in providing WS services in Brazil. Thus, an alternative analysis is pursued. After estimating Eq. 24, the predicted firmspecific costs a, are computed. A high &, indicates an inherently high cost for firm i, even when controlling for output, input prices and other exogenous variables. Once the predicted firmspecific costs are obtained, an additional regression is performed. In this secondstage regression, the dependent variable is the predicted firmspecific cost (per unit of output), while firmtype indicators and regional dummies are the explanatory variables:20 = +A 3*Type, + *Region, + (25) q, The vectors 3 and contain the coefficients for each of the Type and Region indicator variables. Meanwhile, q, represents the average output of firm i for the period under analysis. If the coefficients 3 for the firmtype dummy variables are statistically significant in Eq. 25, there will be evidence of relatively distinct firmspecific costs between the different operatortypes. Following the extant literature, Operating Cost is utilized to represent the dependent variable c on Eq. 24.21 Wage is employed to represent input prices w, since they account for 20 Although output is present in the first stage regression, the firmspecific costs are going to be correlated with output by the nature of the fixedeffects model. That is the explanation behind the utilization of firmspecific costs per unit of output. 21 Although it is also conceivable to use total cost as the dependent variable, that would require data on the price of capital, since depreciation charges constitute a large share of total costs. Since reliable data on the price of capital are unavailable, this study focuses only on operating cost, which excludes depreciation. more than 40% of the operating cost. 22 Wages were calculated as the ratio of total labor expenses divided by the number of employees, as it is standard in the literature.23 Although the volume of water produced seems like the most appealing output variable, the number of connections has also been widely used by researchers.24 Thus, both Volume and Connections are employed as two alternative measures of output q. As control variables, this study includes Network Length,25 the Percentage of Urban Population,26 a Metering Index, 27 a Fluorination Index28 and a Sewerage Dummy that equals 1 if the firm also provides sewerage collection (not all operators provide both services).29 22 Data on other input prices are limited. Some data on energy consumption suggests that energy is the second most important input, representing around 20% of operating cost. 23 More detailed data on input prices would theoretically allow for more efficient estimates utilizing the Seemingly Unrelated Regressions (SUR) model proposed by Zellner (1962). This model consists of a multivariate regression system. Besides the cost function, the inputdemand shareequations are utilized, enhancing the efficiency of the estimation because the same coefficients participate not only in the cost function but also in the inputdemand shareequations. 24 The number of connections is employed by Ashton (2000), Estache and Rossi (1999), Estache and Rossi (2002) and Teeples and Glyer (1987). The volume of water produced is utilized by Antonioli and Filippini (2001), Aubert and Reynaud (2005), Bhattacharyya, Harris et al. (1995), Bhattacharyya, Parker and Raffiee (1994), Bottaso and Conti (2003), Corton (2003), Crain and Zardkoohi (1978), Cubbin and Tzanidakis (1998), Estache and Rossi (1999), Estache and Rossi (2002), Fabbri and Fraquelli (2000), Fox and Hofler (1986), Kim (1987), Stewart (1993) and Teeples and Glyer (1987). The number of customers is used by Antonioli and Filippini (2001), Aubert and Reynaud (2005), Fabbri and Fraquelli (2000) and Saal and Parker (2000). 25 The length of pipes is utilized by Antonioli and Filippini (2001), Bottaso and Conti (2003), Corton (2003), Cubbin and Tzanidakis (1998), Fox and Hofler (1986), Kim (1987) and Stewart (1993). 26 A proxy of density is used by Bottaso and Conti (2003), Fabbri and Fraquelli (2000) (ratio between population served and the length of pipelines) and Teeples and Glyer (1987) (connections per mile of line). The percentage of nondomestic consumers is employed by Bottaso and Conti (2003), Estache and Rossi (1999), Estache and Rossi (2002), Fox and Hofler (1986), Kim (1987) and Stewart (1993). 27 The percentage of metered connections is used by Cubbin and Tzanidakis (1998), Estache and Rossi (2002), Feigenbaum and Teeples (1983) and Teeples and Glyer (1987). 28 A proxy for quality is used by Antonioli and Filippini (2001) (dummy indicating if water has to be chemically treated before distribution), Estache and Rossi (1999) (continuity), Estache and Rossi (2002) (continuity), Feigenbaum, and Teeples (1983) (water treatment index), Fox and Hofler (1986) (tests of water quality and tests of organic contamination), Saal and Parker (2000) (percentage of water that is compliant with key parameters relative to the compliance percentage for England and Wales) and Teeples and Glyer (1987) (water treatment index). 29 Other control variables have been also employed in previous literature. The percentage of water losses is used by Antonioli and Filippini (2001) and Bhattacharyya, Harris et al. (1995). The storage capacity is used by Feigenbaum and Teeples (1983), Fox and Hofler (1986) and Teeples and Glyer (1987). A dummy indicating if the utility has to purchase water from other utility is employed by Aubert and Reynaud (2005), Feigenbaum and Earlier literature suggests that a longer network should be associated with higher costs due to its maintenance (fixing leaks, for example). Previous research also suggests that a higher metering index should be associated with higher costs due to the reading and maintenance of the meters. The fluorination index should also be associated with higher costs due to a more intense chemical treatment of water before delivery. The sewerage dummy is also expected to show a positive sign, capturing the higher operating cost of providing both water and sewerage services. Finally, the effect of a higher proportion of urban population in the area served is difficult to predict. On the one hand, many researchers argue that having customers densely located in a small area reduces costs. On the other hand, Feigenbaum and Teeples (1983) argue that "we should expect that it is more costly to supply more densely developed service areas, which requires more hydrants, higher water pressure and greater peak capabilities for fire protection."30 The main source of data is the National System of Sanitation Information (SNIS) of Brazil.31 Operators voluntarily join the SNIS, which started collecting data in 1995. The number of firms providing data has increased each year ever since. This study utilizes an unbalanced panel for 20002004. There are approximately 180 observations for 2000 and 340 observations for 2004, with almost 1200 observations in total.32 The SNIS is part of the Modernization Teeples (1983), Fox and Hofler (1986) and Teeples and Glyer (1987) (they use the proportion of water that is purchased). The number of districts served is used by Corton (2003). A dummy indicating if the utility obtains water from surface sources is used by Aubert and Reynaud (2005), Bhattacharyya, Harris et al. (1995), Bottaso and Conti (2003) (they use river sources), Estache and Rossi (1999) and Fox and Hofler (1986). I ignore the reasons that support the inclusion of a proxy for capital stock, as done by Antonioli and Filippini (2001) (number of water wells), Aubert and Reynaud (2005) (average net base rate divided by the estimated price of capital), Bhattacharyya, Harris et al. (1995) (residual of the revenue less variable costs) and Bottaso and Conti (2003) (replacement costs of net tangible assets). Such a variable would be an appropriate covariate in a production function, but not in a cost function 30 Feigenbaum and Teeples (1983, p.674). They confirm that result in their paper. 31 Sistema Nacional de Informacoes sobre Saneamento. See www.snis.gov.br (last visit: May 7, 2007). 32 Missing data on some variables explain the variation in the number of observations across the two models presented in the next section. The online data are split into several files, each containing only a certain group of variables (financial, descriptive, operational, etc) and a certain type of firm (regional, local, etc). These spreadsheets were pooled together for this work. Program of the Sanitation Sector (PMSS), which Brazil started in 1992 with financial support from the World Bank.33 To adjust monetary figures for inflation, data on the Brazilian Consumer Price Index (IPCA) are utilized.34 Results Summary statistics of the variables used in the firststage LSDV regressions are presented in Table 22, discriminated by operator type. The size difference between the regional and local operators is evident when observing the average output, network length and operating cost of each type of firm. For example, from the point of view of the number of connections, the average regional operator is 14 times bigger than the average publiccorporative provider. Table 23 presents the results from both LSDV regressions, according to Eq. 24. Alternatively, both Connections and Volume are positive and statistically significant. The hypotheses that their coefficients are equal to 1 is rejected, which provides evidence of increasing returns to scale regardless of the output variable chosen. For example, a 10% increase in the volume of water produced generates only a 0.98% increase in operating cost.35 This confirms the argument made by AESBE, which favors statelevel provision due to economies of scale. As expected, Wage also has a positive and statistically significant effect on cost. From the set of control variables, the Metering Index shows the expected positive sign and statistical significance. This means that when the fraction of metered connections is higher, 33 Program de Modemizacao do Setor de Saneamento. The PMSS resides on the sphere of the National Secretariat of Environmental Sanitation (Secretaria Nacional de Saneamento Ambiental), which depends on the Ministry of Cities (Minist6rio das Cidades). See www.cidades.gov.br (last visit: May 7, 2007). 34 The Indice Nacional de Precos ao Consumidor Amplo (IPCA) is constructed by the Brazilian Institute of Geography and Statistics (Instituto Brasileiro de Geografia e Estatistica IBGE). See www.ibge.gov.br (last visit: May 7, 2007). 35 To test for economies of scale that may vary with output, specifications including the square of the output variable were run. The results were not satisfactory. The Volume2 variable was not statistically significant, while Volume remained statistically significant with little change in the value of its coefficient. Meanwhile, both Connections and Connections2 became statistically insignificant. operating cost increases. This is in line with earlier research suggesting that reading and maintaining the meters has a positive impact on cost. The Sewerage Dummy coefficient is positive and statistically significant. This means that collecting and treating sewage increases the operating cost of a water provider. Finally, Network Length, the Percentage of Urban Population and the Fluorination Index are statistically insignificant. The firststage LSDV regressions in Table 23 also illustrate that output, input prices and other technological factors explain most of the variation of operating cost.36 As a result, the firm specific costs provided by the firststage regressions represent a small portion of the operating cost (less than 1%). After running the firststage LSDV regressions, the predicted firmspecific costs &, are obtained. Following Eq. 25, a secondstage regression is then performed, where the firm specific cost (per unit of output) plays the role of the dependent variable. Firmtype and five regionindicators are the timeinvariant explanatory variables in the secondstage regression. Dummies for the five different regions in which Brazil is divided are included because geographic heterogeneity may differently affect the cost of access to raw water.37 Even when the specific effect of regional heterogeneity is not the focus of this study, it is important to control for that timeinvariant characteristic. If the firmtype dummy variables are statistically significant in the secondstage regression, there will be evidence of relatively distinct firmspecific costs between the different types of operators. Table 24 presents the results of the secondstage regression. The negative and statistically significant coefficient of the Regional dummy indicates that regional providers have lower firm 36 The high explanatory power of both models remains even if the firm fixedeffects (not reported) are excluded. In that case, the R2 is still above 0.95. 7 These regional dummies could partially capture different energy prices as well. specific costs than all other operators. At the local level, meanwhile, the firmspecific cost comparison between private and public operators depends on how the public provider is organized. The positive and statistically significant coefficient of the PublicNonCorp dummy variable indicates that the localpublicnoncorporative operators have higher firmspecific costs than both the localprivate and the localpubliccorporative providers. Table 24 also shows that there is not a statistically significant difference between the firmspecific cost of localprivate and localpubliccorporative operators. Table 25 illustrates the value of the firmspecific costs, obtained from the results of the secondstage regression. The table shows that the regional public operators are the lowestcost WS providers, while the localpublicnoncorporative operators are the highestcost WS providers in Brazil. Although the differences are substantial and statistically significant, it is worth recalling that the firmspecific costs are not a significant portion of the operating costs. Sensitivity Checks As a first sensitivity check, a balanced panel was used. Since the sample size increases over the years, it is important to check that incorporating new firms does not affect the analysis. The results are presented in Table 26. The positive and significant effect of both output variables (Connections and Volume) remains. The same is true for the Wage variable across both specifications. Table 27 contains the results of the second stage regressions. It can be verified that the lower firmspecific cost for the Regional type is confirmed when using a balanced panel, regardless of the output variable chosen. However, the higher firmspecific cost for the public noncorporative type is not obtained when a balanced panel is employed. The reason behind this statistically imprecise result could be the loss of observations when utilizing a balanced panel. That is, most of the observations dropped when using a balanced panel correspond to local firms (which include the publicnoncorporative type) rather than Regional ones. As a second sensitivity check, and utilizing again the larger unbalanced panel, the Regional type was excluded from the sample. The reason for this sensitivity check is that the size differential between the Regional and Local operators could affect the conclusions. Table 28 shows the results of the first stage LSDV regression and Table 29 contains the results of the second stage regression. The first stage regressions show that the positive and statistically significant effect of output, wage and the metering variable are again verified, along with the sewerage dummy. Furthermore, the coefficients obtained are similar to those in Table 23 and Table 24. For example, the measure of economies of scale indicates that a 10% increase in the volume of water produced generates a 0.94% increase in operating cost. The second stage regressions show that the higher firmspecific cost for the publicnoncorporative type is again confirmed, providing confidence about the conclusions drawn earlier. The statistically insignificant difference between the private and publiccorporative operators is also verified. Conclusions Brazil is a country that lacks perfect access to WS services. Efficiency improvements could free up funds for network expansion, which would constitute a step towards a desired full service situation. Therefore, greater attention for costcontainment is needed, regardless of the jurisdictional and ownership/organizational status of the WS operators. Improving our understanding of relative performance can help policy makers focus on the sources of differential cost patterns. The results of this study suggest that, at least for Brazil, evidence of economies of scale is enough to claim that WS provision at the state level is more efficient than WS provision at the municipal level. Economies of scale generate substantial cost savings, which far outweigh any potential differential in firmspecific costs. As such, the argument made by AESBE seems more compelling than the argument made by ASSEMAE. In addition, this chapter finds evidence of inherently lower firmspecific costs (per unit of output) for regional WS firms than for all other types of WS operators in Brazil. These lower costs reinforce the efficiency gains the regional firms achieve through actual economies of scale. Finally, this study shows evidence of higher firmspecific costs (per unit of output) for local publicnoncorporative providers than for localprivate and localpubliccorporative providers. Future research could examine what features generate the intrinsic cost differences among operator types. In particular, the higher firmspecific cost for the publicnoncorporative type deserves further attention. It may be important to check whether the notforprofit motive of those organizations actually drives their higher firmspecific costs. Even when cost differences between the different types of WS operators were found, it is worth noting that these differences represent a small portion of operating costs. The firststage regressions illustrated that output, input prices and other technological factors explain most of the variation of operating cost, regardless of the firmtype. As a result, the firmspecific cost differences presented are significant from a statistical point of view, but less significant from an economic perspective. Clearly, much work remains. For the purpose of rewarding good performance and penalizing weak performance, scholars and practitioners need to develop efficiencymeasuring procedures that can pass legal challenges. The process must continue to build on the pioneering research of those whose work is cited in the references. In particular, the publication of league tables is one way to put pressure on the weakest performing WS utilities. Similarly, the managers of WS utilities in the top 20 percent might be awarded some share of the cost savings that can be attributed to their efforts. Those promoting improvements in WS sector performance can take steps to reduce production costs and free up cash flows for network rehabilitation and expansion. Identifying, implementing, and evaluating good incentive systems represent a challenge for regulators. A final issue that also deserves future research follows. The analysis in this chapter only considered relative measures of efficiency. The goal was to identify sources of cost differences between the different types of operators. However, cost savings for the entire industry could also be estimated utilizing the results obtained. For example, reducing water losses would also free up funds for network expansion. Table 21 shows that water losses in 2004 stood at almost 30% on average, while they were 49% for the state operators. Thus, a reduction of water losses by 10% should not be hard to achieve. Yet, it could represent an almost 1% lower operating cost. Some researchers suggest that the only explanation behind the lack of incentives for reducing water losses is that it may be cheaper to produce more water instead. Although geography might validate this statement in some cases, the issue deserves further exploration. Table 21. Average statistics by operatortype for 200438 Type of firm # Connect. Empl. Water Sewer. Water Sewer. cover, cover, losses Treatm. Private 31 30470 88.9 80.1% 50.4% 29.1% 54.4% Public NonCorp 296 18851 99.2 86.3% 63.0% 26.6% 46.5% Public Corp 11 75180 400.3 98.8% 68.5% 41.3% 36.6% Regional 25 1104748 2978.6 71.3% 33.8% 48.7% 74.5% Total 369 91660 300.7 85.1% 58.5% 28.9% 51.3% 38 The sample also includes six Microregional operators, which are not the focus of the analysis. These are public operators that are neither regional nor local, since they provide services to just a few municipalities. Table 22. Summary statistics for firststage regressions39 Variable Private Public Non Corp. Public Corp. Regional N=62 N=913 N=42 N=121 Operating Cost Connections Volume Wage Network Length Dummy sewerage Urban % Metering % Fluorination % 9,102,144 (12,500,360) 50,539 (67,902) 22,239 (41,240) 15,959 (6,953) 642 (840) 0.82 (0.39) 0.88 (0.09) 0.88 (0.19) 0.33 (0.46) 5,088,097 (13,124,270) 24,251 (39,282) 8,552 (17,315) 11,512 (6,256) 294 (457) 0.56 (0.50) 0.77 (0.21) 0.77 (0.32) 0.30 (0.44) 25,661,470 (34,681,050) 74,573 (67,960) 34,899 (33,649) 18,082 (9,349) 1,005 (1,044) 0.98 (0.15) 0.95 (0.05) 0.86 (0.21) 0.35 (0.47) 239,346,200 (338,851,600) 1,021,909 (1,234,135) 396,122 (583,563) 31,147 (10,613) 11,398 (12,930) 0.94 (0.23) 0.77 (0.12) 0.74 (0.28) 0.20 (0.36) N=1163 (Standard deviations in parenthesis) 39 Volume is in 1000m3/year and Network length is in Km. Operating cost and Wage are in Reais/year deflated using 2000 as the base year. For the Volume variable, summary statistics are for 1172 observations. The 1163 observations include 25 observations for the Microregional category, which are not reported. Table 23. Firststage LSDV regression results40 Dependent Variable: Operating Cost C Connections Volume Wage Network Length Dummy sewerage Urban % Metering % Fluorination % Constant Observations Rsquared Year and firm fixedeffect not reported Standard errors clustered at the statelevel * significant at 10% ** significant at 5% *** significant at 1% 40 Operating Cost, Volume, Connections, Wage and Network length are in In form. The statistical significance of all coefficients is very similar when the standard errors are clustered at the Region level. connections Volume 0.427 (0.112)*** 0.150 (0.043)*** 0.024 (0.073) 0.122 (0.058)** 0.070 (0.246) 0.382 (0.177)** 0.001 (0.035) 8.496 (1.133)*** 1163 0.99730 0.098 (0.040)** 0.157 (0.040)*** 0.089 (0.088) 0.138 (0.067)** 0.082 (0.258) 0.409 (0.165)** 0.005 (0.035) 11.274 (0.713)*** 1172 0.99729 Table 24. Secondstage regression results Dependent Variable: (In) FirmSpecific Cost per Unit of Output (from LSDV regressions) Public Non Corp Public Corp Regional Constant Observations Rsquared Omitted type: Private Region fixedeffect not reported Robust standard errors in parenthesis * significant at 10% ** significant at 5% *** significant at 1% Connections 0.331 (0.127)*** 0.009 (0.217) 1.154 (0.190)*** 0.960 (0.124)*** 380 0.31726 Volume 0.294 (0.124)** 0.073 (0.176) 0.270 (0.158)* 2.640 (0.116)*** 380 0.04428 Table 25. Ranking of firmspecific costs across firmtypes. Firm type FirmSpecific cost per unit of output ($/Connection) ($/1000m3) Index (Regional= 100) Regional 0.12 10.70 100 100 Public Corp 0.38 13.03 314 122 Private 0.38 14.01 317 131 Public Non Corp 0.53 18.80 441 176 Table 26. Firststage LSDV regression results using a balanced panel Dependent Variable: Operating Cost Connections Volume Connections 0.631 (0.152)*** Volume Wage Network Length Dummy sewerage Urban % Metering % Fluorination % Constant Observations Rsquared 0.185 (0.022)*** 0.014 (0.074) 0.034 (0.048) 0.158 (0.135) 0.002 (0.203) 0.023 (0.028) 0.127 (0.043)*** 0.186 (0.026)*** 0.082 (0.104) 0.046 (0.045) 0.150 (0.139) 0.009 (0.188) 0.023 (0.030) 13.287 (1.053)*** 766 0.99711 7.894 (1.155)*** 758 0.99713 Year and firm fixedeffect not reported Standard errors clustered at the statelevel * significant at 10% ** significant at 5% *** significant at 1% Table 27. Secondstage regression results using a balanced panel Dependent Variable: (In) FirmSpecific Cost Connections Volume per Unit of Output (from LSDV regressions) Public Non Corp 0.176 (0.219) 0.005 (0.301) Public Corp Regional Constant Observations Rsquared 1.966 (0.259)*** 2.702 (0.328)*** 170 0.54194 0.265 (0.179) 0.056 (0.166) 0.339 (0.182)* 2.795 (0.249)*** 170 0.11137 Omitted type: Private Region fixedeffect not reported Robust standard errors in parenthesis * significant at 10% ** significant at 5% *** significant at 1% Table 28. Firststage LSDV regression results excluding the Regional type Dependent Variable: Operating Cost Connections Volume Connections 0.439 (0.130)*** Volume Wage Network Length Dummy sewerage Urban % Metering % Fluorination % Constant Observations Rsquared 0.145 (0.045)*** 0.022 (0.072) 0.119 (0.058)* 0.062 (0.252) 0.424 (0.194)** 0.094 (0.040)** 0.152 (0.041)*** 0.083 (0.087) 0.135 (0.066)* 0.077 (0.264) 0.460 (0.179)** 0.005 (0.039) 11.37 (0.721)*** 1047 0.99575 0.004 (0.039) 8.433 (1.326)*** 1042 0.99579 Year and firm fixedeffect not reported Standard errors clustered at the statelevel * significant at 10% ** significant at 5% *** significant at 1% Table 29. Secondstage regression results excluding the Regional type Dependent Variable: (In) FirmSpecific Cost Connections Volume per Unit of Output (from LSDV regressions) Public Non Corp 0.331 0.286 (0.127)*** (0.120)** Public Corp 0.037 0.077 (0.220) (0.170) Constant 1.005 2.711 (0.124)*** (0.113)*** Observations 354 354 Rsquared 0.1322 0.01567 Omitted type: Private Region fixedeffect not reported Robust standard errors in parenthesis * significant at 10% ** significant at 5% *** significant at 1% APPENDIX DERIVATION OF THE SOLUTIONS TO CHAPTER 1 Limits on Communication When Correlations are Relatively Similar IfEq. 117 holds, solve [PNOC] imposing only Eq. 19 (multiplier denoted with PL), Eq. 110 (multiplier denoted with PLO), Eq. 112 (multiplier denoted with PH0) and Eq. 114 (multiplier denoted with IL ). Solving from the first order conditions with respect to payments yields: PL1 =u1 >0, PLO =A O a >a0, S2ao I) Io = aa >0 and P, =A L +) a+ao >0. 2ao 1 2aoI ) Setting equal to zero the first order conditions with respect to the effort vector: aL = (1+ A)[p/la + p/oao][1+ y'(eLL)]+ /2[pla + uoao ]0'(eLL) BeLL PL llj (eLL) PLOaO (eLL) LO aO (eLL) = 0 Substituting the value of the multipliers and simplifying: (eL) = 1 DL = (1 + ,)[p,(1 a) +o(1 ao)][1 +'(eL)] aeLH + A[p/ (1 a) + / (1 ao)]y' (eLH) PL(1 a )y'(eLH) PLo(1 ao)y'(eLH) ILO(1 ao)'(eLH) =0 Substituting the value of the multipliers and simplifying: W' (eH) = 1 aL = (1 + A)[1ua, + Oao[ ][1+ V' (eH)] + A[/ a, + ,0ao ]W' (eHH) SeHH  PHOaOV' (e,) + LO (1 ao )' (eH, A) = 0 Substituting the value of the multipliers and simplifying: A al aO 1(1ao) < y' (eHH 1 A a HH<1 (1+ 2) (2a, 1) ( u a, + 0ao) QL = (1 + A)[p~(1 a) + o (1 a)][1 + (eHL)] + [p1 (1 ) + o(l o)]I' (eL) PHo (1 ao )y(eHL)+ IOa (eHL A) = 0 Substituting the value of the multipliers and simplifying: A a1 a0 11a0o D (1 + A) (2ao 1) (1 p/1a /o0) Solve from the binding constraints to obtain the payments. From Eq. 19 and Eq. 110: tLL = y(e*) and tLH = V(e) Substituting in Eq. 114 and simplifying: tHH = I(eHL A) + (eHH A) HL 1 a 1 a Rearranging terms in Eq. 112: tHH = (eHH )+ HL ) L ao ao Equating the righthandsides of the last two equations: tHL = y(eHL) a [(1 l )HH + aoDHL] 2ao 1 Substituting back in Eq. 112: 1 a tHH = (eHH )+ a [(1 a 0)HH +a oHL 2a, 1 The excluded constraints (Eq. 111, Eq. 113, Eq. 115 and Eq. 116) are satisfied (not binding) by the payments above. In particular, substituting the payments in Eq. 111 yields the rent for the highcost agent if correlation is high: UH [(1_ a HH + aHL] > 0 2a, 1 When Correlations are Relatively Different IfEq. 117 does not hold, solve [PNOC] imposing only Eq. 19 (multiplier denoted with PL1), Eq. 112 (multiplier denoted with PH ), Eq. 113 (multiplier denoted with ILI) and Eq. 1 14 (multiplier denoted with IL0 ). Solving from the first order conditions with respect to payments: ILI 'U (a, ao) 0 (2a 1) S 1> 0, > ILo /o >0, a, +a o 1 2a' 1 2a 1 PLi =A 2a o >0 and PH,,A 2l1 >0 a0' + a 1 a1 + a( 1 Setting equal to zero the first order conditions with respect to the effort vector: dL = (1+ A)[/a, + oao + ][1 + (eLL)] + A[p/a, + pOao ] '(eLL) OeLL PLj/af' (eLL) IL,,ay' (eLL) ILoao (eLL) = 0 Substituting the value of the multipliers and simplifying: Vy'(eLL) = 1  = (1 + A)[/p/ (1 a) +p /o(1 ao)][1+ (eLH)] + A[//i (1 1) + //o (1 ao )] (eLH) SLH PLI1 al '(eLH) L (1 al)'(eLH) I (1 c o)i(eLH) = 0 Substituting the value of the multipliers and simplifying: y' (eLH) = 1 aL = (1 + +)[pla, + p0ao ][1+ q' (eHH)]+ A[p/a1 + /pOa ]y' (eHH) BeHH  PHOo' (eHH ) + IL1(1 a l)' (e A) + ILo (1 Ca )'(eHH A) = 0 Substituting the value of the multipliers and simplifying: (eH) =1 a ao u, (1 a1 )+ Oo H < w' (eHH HH (1+2A) (a, +ao 1) (P, + aoo) aL = (1 + )[/i (1 a) + a o(1 ao)[1 + '(eHL)] ,eHL + Z[/l (1 a) + /0 (1 ao )]t' (eHL) PHO (1 a0 )' (eHL) + Ila1' (eHL A) + ILOa (eHL A) = 0 Substituting the value of the multipliers and simplifying: Ay a1 ao0 ja,1+ + ( o (1 o),H < '( )= ( ^1 <1 (1+ A) (a1 + ao 1) (1 p, poao) Solve from the binding constraints to obtain the payments. From Eq. 19: 1a, tLL = ry(eLL ) [tLH ry(eLH )] From Eq. 112: tHH = y(eH ) [tHL (eHL ) ao 50 Substituting the last two expressions into Eq. 113 and Eq. 114 yields: tHL = y(eHL) [(1 _1)aHH +1 a HL] a, +ao 1 tLH = y (eLH )+ a [aoHH +( 0 o)HL] a, +a, 1 Substituting tL in Eq. 19 yields: tLL = I(eLL) 1 [a0 HH +(1 0 )HL]. a1 +ao 1 Substituting t, in Eq. 112 yields: tHH = y( )+ a [(1 a1)(D + a HL al +aO 1 The excluded constraints (Eq. 110, Eq. 111, Eq. 115 and Eq. 116) are satisfied (not binding) by the payments above. In particular, substituting the payments in Eq. 110 and Eq. 1 11 yields the rent for the highcost agent if correlation is high and the rent for the lowcost agent if correlation is low: UH1 = 0 [ 1 HH 1 H) > 0 a, +oao 1 ULo = 1 [a HH++ (1 ) HL> 0 a + ao 1 No Limits on Communication When Correlations are Relatively Similar IfEq. 117 holds, solve [PCO] imposing only Eq. 119 (multiplier denoted with y19 ), Eq. 120 (multiplier denoted with 720), Eq. 121 (multiplier denoted with 21 ), Eq. 123 (multiplier denoted with 23 ) and Eq. 130 (multiplier denoted with 30 ). Solving from the first order conditions with respect to payments yields: a23 A1 ao > 0(2a 1) + /1 (a, + ao 1)> Y23 = jl/d1 > 0, y20 = 2 > 0, 2a 1 2a 1 719 A Uo U1  > 0, 21 = A > 0 and 30 = Ai > 0. 2ao 1 Setting equal to zero the first order conditions with respect to the effort vector: 8L o = .',,"i''(eo) (1 + A)oao + y19ao~"'(e) + y c',,i''(eL) =0 LL Substituting the value of the multipliers and simplifying: y'(e ) = 1 8L o = poaoV'(eHH) (1 + ))/ 0 + y.,_, ,HH) eHH  723(1 ao) (eH A) y,30ol '(eH) =0 Substituting the value of the multipliers and simplifying: (l+ A) po ao (2ao 1)H aL  = o0(1 ao)'(eH )(1+A)o0(1 ao) + 79 (1 a, )' (e0 ) + 23 (1 o )I'(eH) =0 Substituting the value of the multipliers and simplifying: y' (e ) = 1 aL oL= 0 0(1 o)yI'(eL) (1 + A)Po(1 ao) + 720 (1 Lo) '(eL) SeHL 7. ',,'(eo A) 30 (1 )' (eL)=0 Substituting the value of the multipliers and simplifying: Vy'(eo A 1 A2 ( a, (' (o o , aL = p, a, '(e1) (1+ A),la, + y2aly'(eV )=o ae (la,)'(eLL 11 =0211 L)=0 eLL Substituting the value of the multipliers and simplifying: ey'(e) = 1 aL  =/la, y'(e, )(1+ A)j)ia + y3oal (e,) =0 elHH Substituting the value of the multipliers and simplifying: y'(e) = 1 aL e (LH =1,( l)'( )(11a)yl+)(1 a) + Y2(1 1) L'(e) =0 Substituting the value of the multipliers and simplifying: (e Substituting the value of the multipliers and simplifying: V'(eLH ) = aL L 1 (e)V)=o ae=H ,(1 ct )y'(eL) (1 +A)M1 1 ) +/3o (1 a L) =0 OeHL Substituting the value of the multipliers and simplifying: /' (e) 1 Solve from the binding constraints to obtain the payments. The five equations are as follows: a [t (e )]+ (1 )[t e)] 0 (119) a0 [to H (eO)] + (1 a )[toL (eL)] 0 (120) a1 [tL (eL)] +(1 )[tH r(el )] 0 (121) a [t y(eL,) t + (eL A)] (1 0[ o o(123) + (1 a)[t y(eH )t + y(eo, A)]= a, [t y(e, ) t H + y(e H, )](1 (130) + (1 a, )[tL y(e, ) t + y(e)] 0 The system has an infinite number of solutions. Selecting arbitrary values for tH tHL and tL a solution can be characterized as follows: S(0 o) o tLL (eLL) [LH LH+(eL) 2ao  2a0 1 1i = Vy(el) (I [LH V(el tLL L 1 H H a, t, =V((e,) [t ( (el +)] o +1(I o)LO a1 a, (2a0 1) Some of the excluded constraints are automatically satisfied by the payments above. For the rest of the excluded constraints to be satisfied, the following bounds on the arbitrary payments should hold: I(eH )< tH y(e1 ) a o ( 0) o )2a+ 21 2a 1 1+ [ + (1 a )LL i Y/(el H)+ 01 [a H + (1 a aO)WL a, + a, 1 where YL r(e, + A) (e' )> 0, for i {L,H} and se {1,0}. 1 ao Ho L 1 V(eHL ) 0~ t L < min HL 2ao 1 +(1 )0 HH I )ao IH Vf(elHL 2al 11+(1a )HH1 (1a )(ag ao) oaoHL (2 1)(2o 1) +(1 1 1 L I0 IOHL HLao +a +1ao) HH (1 ao)(a ao) oaoHL (ao + a, 1)(2ao 1) + (1 a0 )0H I(eH ) 2 1(0, H L 2ao 1 ,(eH )<+H y(eH)+ 0 2ao 1 al +ao 1 L + ) + + [alo +(1_ 1 0 a l ) In particular, substituting the payments into Eq. 122 yields the rent for the highcost agent if correlation is high: U1 = a( a [( +(1ao)(o> 0 H(2a 1) When Correlations are Relatively Different IfEq. 117 does not hold, solve [PCO] imposing only Eq. 120 (multiplier denoted with y20), Eq. 121 (multiplier denoted with y21), Eq. 123 (multiplier denoted with Y23), Eq. 130 (multiplier denoted with y30) and Eq. 133 (multiplier denoted with y33). Solving from the first order conditions with respect to payments yields: 2a, 1 2ac 1 720 = 2a'1 >0, y21 = 2a 1>0, O +aI 1 ao +a 1 723 0= 'UO>0, 730 = 1 >0 and 7/ (,33 = ) (2a 1)>0 aO + a 1 Setting equal to zero the first order conditions with respect to the effort vector: aL oL = oaoVy'(eO ) (1+ A)p/ao +y. y, ei''( )=0 eLL Substituting the value of the multipliers and simplifying: y'(e = 1 8L oI = ,,,,i '(eo ) (1 + A)p/oa + y ,,.,,i '' (e H) SeHH 23(1 ao0)Vy(eH A) y30aV y'(eHH) y33(1 ,e A) Substituting the value of the multipliers and simplifying: /' (eH) [/0a + /1 (1 a )] (a, a) ( ,< w '(eHH HH (1 + ) /ao0a ( + a1 1) aL oL = 0(1 ao)W'(eH)(1+A)/o (1 a) +7 23(1 o)'(eH )=0 eLH Substituting the value of the multipliers and simplifying: "y'(eH ) 1 8L a = e (1 ao)y'(eL) (1 + A)/o(1 ao) + 7o(1 o)'/(eL) SeHL y c( '.'(e'L A) 3o0( l )y'(eoL) 33c/' (eL A) =0 Substituting the value of the multipliers and simplifying: /,(eL )=I A [/uo(1 ao)+/,u ] (a, o ) , w' (e /1 'i HL (1+A) ,o(1 o) (ao + a1) L I a = la, '(eL ) (1 + A),Ia + y/21 a '(e ) YL 33l(eLL) =0 eLL Substituting the value of the multipliers and simplifying: y'(el) 1 aL / = ula '(eH1) (1 + A) a, + y,30 a (e,) =0 eHH Substituting the value of the multipliers and simplifying: y'(e) = 1 8L  /1(1 aI)y'(e (H (1 +)/I(1 a) + 21(1 a)y'(e ) OeLH + 33(1 a )'(el )=0 Substituting the value of the multipliers and simplifying: y' (e ) = 1 aL L /(1 )'(eL) (1 (1 a + /30(1 a)'(e})=0 OeHL Substituting the value of the multipliers and simplifying: y'(e) =1 Solve from the binding constraints to obtain the payments. The five equations are as follows: a, [to (eO )]+ (1 ao )[to (eL )] 0 (120) a [t~L L(e )] +(1 )[tH y(e H)] 0 (121) a [to y(e) tL +(e A)] (123) + (1 )[t (123) +(1a )[to (eH ) t, + V(e, A)]=  a1 [t'H (eH)toHH + y(eH )] (130) + (1 a )[tL (eL) t + y (e,)]= 0 1[ [tLL L L [L (eLL) HL +(eL (133) +(1 a)[tH I(e ) tHH + H(e A)]= The system has an infinite number of solutions. Selecting arbitrary values for tH, tHL and tLH, a solution can be characterized as follows: tLL  r(el) [tLH V/(el)] a1 to =V(eo o (1 ) + tHL = y(eHL) a0 1 1 al)HH + aDHL] a, + 1 o = o 1ao 1[(1 ) + 0]H a1 + a0 1 o /(a1 ) [0 o H ]o 1 a0 [to H /(eHo ] tLL =o )+ HH + (1 o0 )IL LH a, (a, + a, 1) a, tHH =V (e1H)+ al ao 1[( al)( + a00 (HL1 ) (e1 ) a, (a1 + a 1) a Some of the excluded constraints are automatically satisfied by the payments above. For the rest of the excluded constraints to be satisfied, the following bounds on the arbitrary payments should hold: a 0 a0 OH 0 HH Vf(e )+ ao 0 L (1ao)(a ao) a HH y(e0L)(Do 2ao 1 + (1 ao)YTL (a +a 1)(2o 1) _+(1 L tLH < min< (eH) + a HL + [(1 i a, ) L + a, LH a1 +ao 1 al + ao 1 )+(e) + a [(I)oH +(1 O)DL] (a, + a, 1) L (ef1 ) HH1 ( 1 a, H 0 H1L (L 2D1_+ 0 (a,+ 01)(2a,1) + a 0 t+' 1 (a' + ao 1) YV(el ) + a (I i )T' + al(a a) (HH 2a, 1 L+a (a +ao1)(2a 1) + a, I t < min< (e ) + + ( a) ao + ao 1 a) )7(el)+( +(H o)0L (a, + ao 1) lowcost agent if correlation is low and the rent for the highcost agent if correlation is high: U (a,+ a,  L (a= + ao >0, [, ( yz9 + (/ I > and 4 D 0 > 0. (a1 + 1) Limits on Communication and Exclusion Solve [PEXCL] imposing only Eq. 138 (multiplier denoted with 73,), Eq. 139 (multiplier denoted with y3g) and Eq. 142 (multiplier denoted with 742). Solving from the first order conditions with respect to payments yields: /73 = A0 >0, /39 A1 >0 and 42 Ai1 >0. Setting equal to zero the first order conditions with respect to the effort vector and substituting the value of the multipliers yields '(e, ) = 1, for i,j e {L,H}. Solving from the binding constraints yields the payments. The solution is given by: tH =(eLH), t =(eLL) and tHH = y(eH ) 1 HL y(eHL)], with tiL arbitrary. a1 Some of the excluded constraints are automatically satisfied by the payments above. For the rest of the excluded constraints to be satisfied, the following upper bound on tHL should hold: y,(e*); tHL < min y(e*) l [y(e*) y(e A)]; 2a, 1 y(e*) a [y(e*) y(e A)] ao +a, 1 LIST OF REFERENCES Antonioli, B. and Filipini, M. (2001) 'The use of a variable cost function in the regulation of the Italian water industry', Utilities Policy, 10, 181187. Arya, A., Demski, J., Glover, J. and Liang, P. (2005) 'Quasirobust multiagent contracts', Carnegie Mellon Working Paper. Ashton, J. (2000) 'Cost efficiency in the UK's water and sewerage industry', AppliedEconomics Letters, 7, 455458. 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Neeman, Z. (2004) 'The relevance of private information on mechanism design', Journal of Economic Theory, 117, 5577. Oliveira, G. and Fujiwara, T. (2005) 'Brazil's regulatory framework: predictability or uncertainty?', Texto para Discussdo No. 147, Escola de Economia de Sao Paulo. Parlatore, A. (1999) 'Privatization of the water utility sector in Brazil', Essay No. 8 in 'Privatization in Brazil: the case ofpublic utilities', BNDES. Pinheiro, A. (2003) 'Regulatory reform in Brazilian infrastructure: where do we stand?', Texto para Discussdo No. 964, IPEA. Robert, J. (1991) 'Continuity in auction design', Journal of Economic Theory, 55, 169179. Saal, D. and Parker, D. (2000) 'The impact of privatization and regulation on the water and sewerage industry in England and Wales: a translog cost function model', Managerial and Decision Economics, 21, 253268. Sappington, D. (1983) 'Limited liability contracts between principal and agent', Journal of Economic Theory, 29, 121. Schmidt, P. and Sickles, R. (1984) 'Production frontiers and panel data', Journal of Business and Economic Statistics, 2(4), 367374. Seroa da Motta, R. and Moreira, A. (2006) 'Efficiency and regulation in the sanitation sector in Brazil', Utilities Policy, 14, 185195. Shleifer, A. (1985) 'A theory of yardstick competition', Rand Journal ofEconomics, 16(3), 314 327. Soares, D. (2001) 'Privatization of sanitation and water distribution in Brazil: a general overview of the current market and outlook for private investors', Journal of Project Finance, 7(1), 3551. Spady, R. and Friedlaender, A. (1978) 'Hedonic cost functions for the regulated trucking industry', Bell Journal, 9, 159179. Stewart, M. (1993) 'Modeling water cost 199293', OFWAT Research Paper No. 4. Tangeras, T. (2002) 'Collusionproof yardstick competition', Journal of Public Economics, 83, 231254. Teeples, R. and Glyer, D. (1987) 'Cost of water delivery systems: specification and ownership effects', The Review of Economics and Statistics, 69(3), 399408. Tupper, H. and Resende, M. (2004) 'Efficiency and regulatory issues in the Brazilian water and sewage sector: an empirical study', Utilities Policy, 12, 2940. Vargas, M. and De Lima, R. (2004) 'Concessoes privadas de saneamento no Brasil: bom negocio para quem?, Ambiente & Sociedade, VII(2). Wilson, R. (1987) 'GameTheoretic Approaches to Trading Processes', in T. Bewley (ed.) Advances in Economic Theory: Fifth World Congress, Ch. 2, Cambridge University Press, 3377. Zellner, A. (1962) 'An efficient method of estimating seemingly unrelated regression equations and tests for aggregation bias', Journal of the American Statistical Association, 57, 348 368. BIOGRAPHICAL SKETCH Guillermo Sebastian Sabbioni Perez was born in 1975 in Argentina. In March 2000, he graduated with honors from the Universidad Cat6lica Argentina with a Licenciatura en Economia (Bachelor of Arts in economics). After graduation, he worked for one year as an Analyst in the Corporate Finance Department of the largest media group in Argentina. He then worked for two and a half years as a Category Manager in the Commercial Department of a supermarket chain store, also in Argentina. While getting experience in the private sector, he also engaged in parttime teaching at the Universidad Cat6lica Argentina. He was awarded a Fulbright Scholarship at the end of 2002; this allowed him to return to school to pursue graduate education, this time in the United States. He started a doctorate in economics at the University of Florida in the fall of 2003. While pursuing his doctorate, he worked parttime as a Research Assistant for PURC, Public Utility Research Center. His work at PURC provided him with the opportunity to present his research at international meetings. He was the instructor of undergraduate Game Theory in the summer semester of 2006. He also presented one of his research papers at the Second Summer School on "Economic Analysis of Heterogeneity in Social Organizations" in CORE, LouvainlaNeuve, Belgium, in June 2006. As recognition of his work, he was awarded the Madelyn M. Lockhart International Travel Award and the Walter Lanzillotti Research Grant by the Department of Economics, both in 2006. He graduated in August 2007, his dissertation titled Theoretical and Empirical Analyses of Incentives and Public Ownership. PAGE 1 1 THEORETICAL AND EM PIRICAL ANALYSES OF INCENTIVES AND PUBLIC OWNERSHIP By GUILLERMO SEBASTIAN SABBIONI PEREZ A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007 PAGE 2 2 2007 Guillermo Sebastian Sabbioni Perez PAGE 3 3 To my wife Carolina, to my daughter Francisca, to my parents Jorge and Nelyta and to Inda, Maru and Vicky PAGE 4 4 ACKNOWLEDGMENTS I thank David Sappington for his invaluable he lp and advice. I also thank Sanford Berg, Steven Slutsky, Jon Hamilton and Chunrong Ai for their comments. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........7 LIST OF FIGURES................................................................................................................ .........8 ABSTRACT....................................................................................................................... ..............9 CHAPTER 1 MULTIAGENT CONTRACTS WITH UNKNOWN COST CORRELATION..................11 Introduction................................................................................................................... ..........11 The Model...................................................................................................................... .........14 Uninformed Principal: Limits on Communication.................................................................21 If Both Agents Observe the Correlation..........................................................................22 If Only One Agent Observes the Correlation..................................................................32 Uninformed Principal: No Limits on Communication...........................................................33 If Both Agents Observe the Correlation..........................................................................34 If Only One Agent Observes the Correlation..................................................................34 Limits on Communication and Exclusion..............................................................................40 Conclusions.................................................................................................................... .........43 2 EFFICIENCY IN THE BRAZI LIAN SANITATION SECTOR...........................................53 Introduction................................................................................................................... ..........53 Overview of Brazils Wate r and Sewerage Industry..............................................................56 Methodology.................................................................................................................... .......59 Results........................................................................................................................ .............64 Sensitivity Checks............................................................................................................. .....66 Conclusions.................................................................................................................... .........67 APPENDIX DERIVATION OF THE SOLU TIONS TO CHAPTER 1................................79 Limits on Communication......................................................................................................79 When Correlations are Relatively Similar.......................................................................79 When Correlations are Relatively Different....................................................................81 No Limits on Communication................................................................................................83 When Correlations are Relatively Similar.......................................................................83 When Correlations are Relatively Different....................................................................87 Limits on Communication and Exclusion..............................................................................90 LIST OF REFERENCES............................................................................................................. ..92 PAGE 6 6 BIOGRAPHICAL SKETCH.........................................................................................................97 PAGE 7 7 LIST OF TABLES Table page 21 Average statistics by operatortype for 2004.....................................................................70 22 Summary statistics for firststage regressions....................................................................71 23 Firststage LSDV regression results..................................................................................72 24 Secondstage regression results.........................................................................................73 25 Ranking of firmspecifi c costs across firmtypes..............................................................74 26 Firststage LSDV regression re sults using a balanced panel.............................................75 27 Secondstage regression resu lts using a balanced panel....................................................76 28 Firststage LSDV regression re sults excluding the Regional type....................................77 29 Secondstage regression result s excluding the Regional type...........................................78 PAGE 8 8 LIST OF FIGURES Figure page 11 Timing at [PNOC], when both agents obs erve but do not report the correlation.............46 12 A solution to [PCM] that can also be a solution to [PNOC]...........................................47 13 Welfare under the two alterna tive mechanisms at [PNOC].............................................48 14 Timing at [PCO], when only agent A observes and reports the correlation.....................49 15 Welfare at [PEXCL] be low welfare at [PNOC]..............................................................50 16 Welfare at [PEXCL] sometimes larger than welfare at [PNOC]....................................51 17 Welfare at [PEXCL] almost always larger than welfare at [PNOC]...............................52 PAGE 9 9 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THEORETICAL AND EM PIRICAL ANALYSES OF INCENTIVES AND PUBLIC OWNERSHIP By Guillermo Sebastian Sabbioni Perez August 2007 Chair: David Sappington Major: Economics This dissertation includes both theoretical and empirical res earch in economic efficiency. The first chapter theoretically evaluates how to design procurement models when two agents may have better information than the principa l about their producti on costs and about the similarity (i.e., correlation) of those costs between agents. Th e results indicate that if the uncertainty about the correlation is not severe, th e principal should design a contract as if she knew that correlation is the lowest possible. The drawback with su ch mechanism is that the highcost agent earns rent if correlation is high. In cont rast, if the uncertainty about the correlation is more severe, the principal should no t tailor the solution to any of the potential correlations. This contract reduces the rent of the highcost agen t if correlation is high by giving rent to the lowcost agent if correlation is low. Although the we lfare level may be different, the characterization of the results holds when both agents observe th e correlation and do not re port it, as well as in the case where only one agent observes and reports th e correlation to the pr incipal. Finally, when the probability of the lowcorrelation setting is very small, the principal may find optimal to exclude the highcost agent from the contract if correlation is low, allowing her to achieve the socially optimal situatio n if correlation is high. PAGE 10 10 The second chapter empirically ev aluates the relative efficiency of different type of water and sewerage operators in Brazil. The analysis consists of a cost comp arison between public and private operators, and between statelevel and muni cipallevel operators. In a first stage, a cost function is estimated utilizing a fixedeffects pa nel data model. In a secondstage, the firmspecific costs from the first stage are explained by means of firmtype indicator variables. The results illustrate that water and sewerage provi sion in Brazil is characterized by substantial economies of scale, indicating that statelevel provision is economically more efficient. The results also show that there is not an economi cally significant difference between the costs of private and public firms. PAGE 11 11 CHAPTER 1 MULTIAGENT CONTRACTS WITH UNKNOWN COST CORRELATION Introduction This chapter analyzes a multiagent principalagent model where the agents are privately informed not only about their produc tion costs but also about the ex tent to which those costs are correlated. The analysis is performed with a nd without limits on communication, and considers the possibility that either one or both agents ma y be perfectly informed about the correlation. The results show that full rent extraction is impossi ble if the informed agents cannot communicate the realization of the correlation to the principal. A dditionally, the results also illustrate that even with communication on the correlatio n the principal may not be able to fully extract rent from all agents if only one of them is in formed about the exact correlation. The most important finding of the analysis under limited communication is that the highcost agent earns positive rent if correlation is high. When limits on communication on the correlation prevail, the principal cannot tailor the payments to the exact correlation coefficient. As a result, the extreme lottery of payments for the highcost agent should ensure that nonnegative expected rent is obtai ned under both potential correla tions. Extreme payments for the highcost agent are necessary to prevent the lowcost agent from exaggerating cost. Since expected rent is calculated using two different sets of conditional probabilities (one for each correlation), and since the high payment is obtained with hi gher probability (and the low payment with lower probability) under the hi ghcorrelation environment, the principal can eliminate rents for the highcost agent only if correlation is low. Similarly, the most important finding of the analysis wit hout limits on communication is that if only one of the agents is perfectly inform ed about the extent of co st correlation, again the highcost agent enjoys positive rent when correla tion is high. Although the rent for the highcost PAGE 12 12 agent if correlation is high parall els the results with blocked communication, the source of this rent is slightly different. Instead of being explained by the inabil ity of the principal to tailor payments to each correlation, when communication on the correlation is available the mechanism designer needs to prevent the highco st agent from untruthfully underreporting the correlation. If correlation is high, the highcost agent would claim that correlation is low because he would then be assigned a high payment more frequently (and a low payment less frequently) than what he should if the principal designe d two independent mechanisms, one for each correlation report. When limits on communication prevail, the rent for the highcost agent if correlation is high is explained by his required pa rticipation if correlation is low combined with the fact that his rent is always higher under the highcorrel ation scenario than under the lowcorrelation setting. A natural question is th en whether the principal should always require nonnegative rent for the highcost agent if correlation is low. Numeri cal examples illustrate that if the probability of the low correlation scenario is low enough, it c ould be optimal to excl ude the highcost agent from the contract if correlation is low. This work is related to two streams of resear ch in the mechanism design literature. First, since the agent(s) have private information on more than one dime nsion, the results contribute to the literature on multidimensiona l mechanism design. Dasgupta and Maskin (2000) show that if bidders types are multidimensional and independ ently distributed there may be no efficient auction (i.e., one that assigns the object to the bidder that values it the most). Jehiel and Moldovanu (2001) illustrate that efficient mechanism design ma y be impossible in a social choice setting, where each agent can receive multidimensional signals from all other agents and where the signals are independently drawn for each agent. The difference between the former PAGE 13 13 two analyses and the present study is that here the two elements of the twodimensional type are not independently distributed. Trying to rank the revenue between a First Price and a Second Price auction, Fang and Morris (2006) consider an auction where the bidders, besides knowing their own type, also receive a signal about the other bidders t ypes. In Fang and Morris model there is a separation between each players valuation and his signal about his opponents valuation, since both realizations are drawn from two independen t probability distributions. In their model, the beliefs of the first bidder a bout the second bidder depend on the type of the second bidder rather than on the type of the fi rst bidder. In this st udy, on the other hand, the agents privately observed cost at the same time provides information about the other agents cost. Miller et al. (2007) cons ider a problem where agents types are multidimensional and continuous, but they relax Jehiel and Mol dovanus assumption that the agents private information is independently distributed. Mille r et al. show that incentivecompatible implementation of any decision rule is possible provided that the agents types satisfy one of their correlation conditions. The present work is di fferent in that the principal is not required to utilize budgetbalanced transfer payments and in the discrete support of the agents types. Second, when addressing the case with limited communication on the correlation, the present chapter also makes a contribution to the literature on robust mechanism design. This stream of research attempts to relax the a ssumption of too much common knowledge at the time of contract design. Too much common knowledge refers to perf ect information about all the relevant parameters of the problem at hand, as explained by Bergemann and Morris (2005), Bergemann and Vlimki (2006) and Heifetz and Neeman (2006). The influential statement by Wilson (1987) has been the starting point of this emerging literature: Game Theory is deficient to the extent it assumes features to be comm on knowledge, such as one players probability PAGE 14 14 assessment about anothers preferences or inform ation. One way of relaxing the assumption of too much common knowledge is by focusing on cont racts that are not finetuned to the exact environment faced (e.g., to the exact co rrelation between the agents costs). The origins of both streams of research can be traced back to the seminal work by Demski and Sappington (1984) and Cremer and McLean (1985, 1988), who showed that if agents are sufficiently riskneutral and if they face no limited liability c onstraints, mechanisms can be designed to ensure costless truthf ul revelation of private inform ation provided the agents types are correlated. McAfee and Reny ( 1992) extended the analysis to the case where agents may have infinitely many possible types. This chapter starts with a description of th e model employed. First, the case with limited communication on the correlation is analyze d. Second, the situation with no limits on communication is presented. A following section ev aluates the possibility of excluding some agents from the game if their participation is too costly in terms of welfare. Conclusions are presented in the final section. The Model A principal contracts with two agents, A and B, each producing one unit of output. For each agent, the cost of producing one unit of output is given by e C The parameter } {H L is the agents initial cost, with 0 L H Meanwhile, e denotes the costreducing effort the agent exerts. Effort e generates disutility 0 ) ( e for the agent, where ) ( e is increasing and convex. The final cost C is perfectly observable by the principal, but she cannot distinguish between the cost parameter and the effort level e. Each agent knows his own initial cost, but not the cost of his counterpart.1 The principal reimburses each agent for his 1 The principal is subsequently referred as s he, while the agents are referred as he. PAGE 15 15 observed final cost C and may provide additional payment ( t ). The utility of each (riskneutral) agent is given by the difference between the transfer payment t and the disutility ) ( e of the costreducing effort exerted: ) ( e t U The agents initial costs are assumed to be positively and imperfectly correlated. The prevailing probability structure consists of HH LLp p = 2 / and HL LHp p = 2 / ) 1 ( where ijp denotes the joint probability that agent A (B) has cost i and agent B (A) has cost j with } { ,H L j i This formulation presumes that the probabi lity that either agen t faces an identical counterpart is invariant to the ag ents cost realization. For exampl e, the probability that agent A (B) has low cost is and the probability that he has high cost is 1 when agent B (A) has low cost.2 The positive and imperfect correlation between the agents costs is captured by ) 1 (2 1.3, 4 This ensures that the probability that an agent has low cost is higher when the other agent also has low cost than when the other agent has high cost: 1 if ) 1 (2 1 The parameter takes on the value 1 with probability 1 and the value 0 with probability 0 where 0 1 and 10 1 Thus, when alpha is 1 the agents costs are highly correlated. This setting is referred to as the highcorrelation scenario. When alpha is 0 the agents costs are less correla ted (although still positively). This se tting is referred to as the lowcorrelation scenario. This information is common knowledge at th e start of the game.5 2 From Bayes rule. 3 The correlation coefficient is given by 2 1 > 0 See Es (2005). 4 The values = 0.5 and = 1 are excluded from the analysis. These values would constitute no correlation and perfect (positive) correlation, respectively. 5 The prior presented satisfies the BeliefsDeterminePrefe rences (BDP) property from Neeman (2004). In short, a prior has the BDP property if each ag ent has different beliefs about the ot her agents types depending on his own type. Neeman shows that full rent extraction results hinge on the BDP assumption. PAGE 16 16 The principal never observes the exact realization of the correl ation. In contrast, at least one of the agents acquires precise information a bout the correlation befo re contracting. This means that besides her informati on disadvantage about th e agents initial cost s, the principal is also less informed than (at least one of) the agen ts about the exact correlation. In addition, two alternatives are considered rega rding limits on communication. In th e first alternative setting, the agent(s) who are informed about the correlation are unable to communicate its realization. The second case consists of a framework with no lim its on communication, in which the informed agent(s) report the actual correlation observed. The principal designs a mechanism where si multaneously and indepe ndently both agents report (some or all of) th eir private information.6 Each agent always reports his privately observed cost as it is standard in the literature. Additionally, the pr incipal can also request the more informed agent(s) to repor t the actual correlation. The d ecision of whether to ask for a correlation report depends on the particular circum stances faced by the principal. As explained later in more detail, contract s that do not depend on the exact correlation between the agents types constitute one important example of th e issues addressed by the emerging literature on robust mechanism design. Agent A submits the report A r and agent B submits the report B r Based on the joint report ) (B Ar r r, the principal assigns a pair )] ( ), ( [ r C r t to each agent. A pair entails a transfer ) ( r t from the principal to the agent and a final cost ) ( r C that the agent must meet.7 Both the transfer (t) and the final cost (C) depend on all information (r) reported to the principal. Therefore, each agent knows that the ] [ C t pair received can vary with the report of his 6 Collusion possibilities are not considered here. 7 From the cost function C = e a finalcost target is equivalent to an effort target. PAGE 17 17 counterpart. In the cases with asymmetric ag ents (i.e., when only one agent observes the correlation), the analysis assumes that the prin cipal offers the same menu of options to both agents. The reason behind this assumption is that the agents may refuse to accept different menus of contracts, on the grounds that the principal would be treating them unfairly. The agents will only contract with the princi pal if they anticipate nonnegative expected rent from doing so.8 However, the computation of the e xpected rent depends on the agents knowledge about the correlation. Fo r example, if a lowcost agent observes a low correlation, he knows that the other agent has low cost with probability 0 and high cost with probability 01 On the other hand, if an agent is uninfo rmed about the correlation, he only knows the expected probability of facing each type of counterpart.9 An agent who is uninformed about the correlation only knows that the other agen t has similar cost with probability and different cost with probability 1, where 1 1 0 0 The principal maximizes the sum of the value derived from both units produced, plus the agents profits, minus the cost of social funds.10 The principal values each unit produced at S, where S is assumed constant and suff iciently large. The social f unds are the transfers and the reimbursed costs received by the agents, whic h the principal delivers at a cost of 1+ per unit.11 If the principal observed the agents costs:12 } { } {] [ ) 1 ( )] ( [ 2B A i i i B A i i i iC t C t S W 8 Opportunity rents are normalized at zero and no expost limited liability constraints are imposed. 9 Every time an agent is described as informed or uni nformed, it is meant from the point of view of whether the agent has perfect information on th e correlation. As a remainder, all ag ents are always perfectly informed about their own production costs. 10 See Laffont and Tirole (1986). 11 The parameter > 0 measures the distortion introduced by the excisetax system utilized to raise social funds. 12 ( C) is used when specific reference to the cost parameter needs to be made. Otherwise, (e) constitutes more compact notation. PAGE 18 18 Standard manipulation yields:13 } { } {)] ( [ )] ( [ ) 1 (B A i i i B A i i ie t e e W (11) Equation 11 shows that if the principal observed the agents costs, the solution would be characterized by ) ( e t and *e e for both agents, where 1 *) ( e .14 The socially efficient level of effort *e would be delivered without any re nt being afforded to any agent. Demski and Sappington (1984) and Cremer and McLean (1985) (CM) illustrated how this result can be replicated if th e principal does not observe the agents costs but she knows exactly how these costs are correlated.15 Let ) (jk i jkC t denote the expost rent for an agent (A or B) that reports cost j when he has cost i and when the other agent reports cost k for k j i } { H L This problem [PCM] constitutes the benchmark situation of this chapter: Maximize } { } { ,)] ( [ 2 )] ( [ 2 ) 1 (H L j i ij ij ij H L j i ij ij ije t p e e p W (12) subject to: 0 )] ( )[ 1 ( )] ( [ ij i ij ii i ii iC t C t U j i H L j i }, { (13)(14) )] ( )[ 1 ( )] ( [jj i jj ji i ji iC t C t U j i H L j i }, { (15)(16) 13 The constants 2S and the s are dropped for the sake of exposition. 14 From C = e an extra unit of effort reduces the fi nal cost also by one unit. Therefore, (e) = 1 denotes the equality of the marginal cost and the margin al benefit of exerting costreducing effort. 15 It is well documented that risk aversion and/or limited liability prevent the principal from achieving the firstbest. See Sappington (1983) for the singleagent case. See also Cremer and McLean (1988), Demski, Sappington and Spiller (1988) and Robert (1991) PAGE 19 19 Equation 12 incorporates the four possible co st combinations that can arise and their respective probabilities. Equations 13 and 14 en sure nonnegative expected rent for the agents. Equations 15 and 16 ensure that the agents report their costs trut hfully in equilibrium.16 The solution to [PCM] permits payments that fully extract (expected) rent while ensuring *e eij for j i,} {H L. To accomplish this, the principal sets relatively high payments for cost reports that match and relatively low paymen ts when cost reports do not match. These two degrees of freedom allow for an infinite num ber of solutions to [PCM]. After selecting min HH HHt t and min LL LLt t the principal finds HLt and LHt such that no rent is obtained in expectation by either agent, and so 0 L HU U. In particular, consider the bound min HH HHt t where:17 ) (min HH HHe t ] ) 1 ( [ 1 2 1HH HL ) (HHe (17) with ) (Hi Hie 0 ) ( Hie for } {H L i .18 The lower bound minHH HHt t prevents the lowcost agent from exaggerating his cost while ensuring that a highcost agent ea rns nonnegative rent in expecta tion. This is achieved with a high payment ) (HH HHe t if both agents report hi gh cost and a low payment ) (HL HLe t if a highcost report is not matched with the same repor t from the other agent. If the lowcost agent lies, he is relatively likely to receive the low HLt payment and relatively unlikely to receive the 16 This is known as BayesNash implementation. Stronger solutions would be obtained if ExPost or Dominant Strategy implementation was required. ExPost implementation consists on the requirement that the incentive constraints are satisfied even afte r the other agents make their (truthful) report. Dominant Strategy implementation consists on the requirement that an agent is always better off by reporting the truth, even when the other agents may lie. 17 The lower bound tLL min guarantees that a highcost agent does not claim low cost, which is usually not constraining for the principal. 18 H is the profit earned by a lowcost agent in the singleagent case. This profit is explained by the savings in disutility of effort that the lowcost agent would enjoy if he exaggerates his cost. PAGE 20 20 high HHt payment because, due to the positive corr elation among costs, the other agent is relatively likely to report low cost. It is important to notice that minHHt is increasing in Hi which means that minHHt is increasing in effort Hie, for } { H L i .19 The effort exerted by the high co st agent has a positive impact on the reward (i.e., payment above di sutility of effort) that an agen t receives if a highcost report is matched by his counterpart. Similarly, the e ffort exerted by the high cost agent positively affects (the absolute value of) the penalty (i.e., payment below disutility of effort) that an agent receives if a highcost report is not matched by the othe r agent. In other words, the larger is Hie, the higher HHt has to be (and the lower HLt has to be) to become unattractive for a lowcost agent. This feature of the payments has important implications for the solution to the principals problem when she is imperfectly informed about the correlation. The subsequent analysis focuse s on settings where the principal, unlike the agent(s), does not know exactly the extent of th e correlation between the agents co sts. In such settings, one or both agents have better inform ation than the principal about both their costs and the exact correlation. As it will be seen, the principal ca nnot achieve the firstbest outcome in those settings. The ensuing analysis is performed with a nd without limits on communication about the correlation. The situation with full communication of all private information follows the standard optimal contracting approach. The case with bl ocked communication on the correlation deserves additional explanation. It is important to understand why the principa l may not utilize all available communication channels to gather as much information as possible. 19 It also means that tHH min is increasing in the cost advantage of the lowcost agent, PAGE 21 21 Uninformed Principal: Limits on Communication Consider, first, the setting where the informed agent(s) are unable to report the correlation to the uninformed principal. In this context, it is impossible for the pr incipal to finetune the contract to the exact correl ation observed by the agent(s). As explained below, specific circumstances prevent the principal from making a contract conditional on the actual correlation. Consequently, the principal designs a mechanis m where both agents report only their cost: i Ar and j Br, with } { H L j i For example, this could be the situation f aced by a regulator when publiclyowned firms are privatized quickly.20 In the preprivatization stage, the regulator may not know the identity of the operators that will take contro l of the soontobe privatized firm s. Alternatively, the regulator may not know exactly which production technologies the new firms will utilize. As a result of this uncertainty, the regu lator may ignore the exact correlation between the firms costs when she designs the regulatory contract.21 The inability of the agent(s) to subsequently re port the realized correlation can be a stylized means for capturi ng prohibitive costs of acquiring more precise information about the correlation at a later stage. For example, it may be too costly or it may take a long time for an expert auditor to provi de accurate information about the correlation.22 Limited communication on the correlation is a key element of the new literature on robust mechanism design. This strand of research attemp ts to relax the implicit assumption of too much common knowledge at the time of contract design.23 As such, this literature focuses on contracts 20 When a government makes a tender for the construction of highways may serve as another example. 21 In the auction setting presented by Arya et al. (2005), for example, the authors suggest that the contract designed can be utilized with different pairs of bidders (i.e., with pairs of bidders that may have different correlations among their valuations for the object auctioned). 22 In the context of a fast privatization process, political pressure sometimes requires that a regulatory framework be set up early enough. 23 See Bergemann and Morris (2005) and Heifetz and Neeman (2006). PAGE 22 22 that are not finetuned to the exact environment faced (e.g., on contracts that do not rely on the exact correlation between the agents types). Acco rding to researchers, the main motivation is that realworld mechanisms seem to be simpler than what it w ould be required by a theoretically optimal contracting approach. For example, Arya et al. (2005, p. 15) sugg est that the robustness problem might help us better understand observed institutions. Additionally, Bergemann and Vlimki (2006, p. 3) say that practitioners have often been led to argue in favor of using simpler but apparently supoptimal mechanisms Also, Bergemann and Morris (2005, p. 1) emphasize that the optimal mechanisms solvi ng the welldefined planners problem seem unreasonably complicated. Two cases are examined below: one in wh ich both agents are informed about the correlation and one in which only a single agen t is so informed. In the context of the privatization process introduced before, the situ ation with only one agen t informed about the correlation could arise when only one of the fi rms has been operating in the industry for a sufficiently long time (perhaps in other geographi cal areas). Therefore, only one of the firms would be endowed with sufficient expertise to permit an accurate assessment of the extent of cost correlation in the environment. This expe rience would allow the firm to be well informed about the correlation between its own cost and th e cost of the (less e xperienced) counterpart.24 On the other hand, if both firms have been operati ng in the industry for a similar length of time, they could both know exactly the extent to which their costs are correlated. If Both Agents Observe the Correlation The principal computes expected welfare: 24 Although plausible, this chapter does not consider the possibility that a more experienced agent could presumably be more likely to have low cost. PAGE 23 23 } { } { ,)] ( [ 2 )] ( [ 2 ) 1 (H L j i ij ij ij H L j i ij ij ije t p e e p W (18) with 0 0 1 1ij ij ijp p p and where s ijp denotes the probability of joint costs i and j when s for } { ,H L j i and } 1 0 { s. The difference between Eq. 18 and Eq. 12 is explained by the fact that ijp is perfectly known by the principa l at [PCM], whereas it can only be estimated (ijp ) when correlation is uncertain an d its communication is blocked. To ensure that both agents participate and re port their costs truthfully for all cost and correlation realizations, the next eigh t constraints have to be satisfied:25 0 )] ( )[ 1 ( )] ( [ ij i ij s ii i ii s s iC t C t U j i H L j i }, { } 1 0 { s (19)(112) )] ( )[ 1 ( )] ( [jj i jj s ji i ji s s iC t C t U j i H L j i }, { } 1 0 { s (113)(116) With no communication the principal solves problem [PNOC]: Maximize Eq. 18 subject to Eq. 19 to Eq. 116. The only difference between the constraints at [PNOC] and the constraints at [PCM] is that the constraints at [PNOC] c onsider the possibility of two pot ential correlations (alpha could be either 1 or 0 ). The timing is depicted in Figure 11. At the solution to [PNOC], the two degrees of freedom that the principal had available at [PCM] are utilized to satisfy the four additiona l constraints. There are two alternative ways in which the principal can perform this task. Which of the two mechanisms is optimal depends on the specific values of the parameters, as is later illustrated in detail.26 25 Before concluding, the last section discusses whether it is always optimal to require participation and truthful revelation by all agents in all scenarios. 26 See the Appendix for a formal derivation of all the results. PAGE 24 24 In the first potential solution to [PNOC], one degree of freedom is utilized to set ) (LL LLe t while the other degree of freedom is used to set HHt equal to the lower bound that [PCM] would require if correlation is known to be low (i.e., to set HHt equal to minHHt from Eq. 17 if 0 ). This payment structure is subsequently referred to as the CMLow mechanism, because it is identical to one of the solutions to [PCM] if the pr incipal knows that correlation is low. The fact that ) (LL LLe t implies that ) (LH LHe t regardless of which correlation occurs. Since ) (LL LLe t the principal can do no better than to set ) (LH LHe t Setting ) (LH LHe t would imply a violation of the participati on constraints of the lowcost agent, while setting ) (LH LHe t would imply unnecessary positive rent for him. This is no different than what the principal can accomplish at [PCM], when fully extracti ng expected (and expost) rent from the lowcost agent by giving him nonstochast ic transfers. These pa yments automatically guarantee that the participation constraints fo r the lowcost agent are satisfied under both potential correlations because they are also satisfied ex post (i.e., for any cost report of the other agent). At the same time, ) (0 minHH HHt t ensures that the incentive compatibility constraints for the lowcost agent are satisfied for both potentia l correlation coefficients. This happens because the lottery of payments for the highcost agent (HHt, HLt) tailored to the lowcorrelation setting is sufficiently extreme if correlation is high.27 In other words, if the payments for the highcost agent are not attractive for the lowcost agent if correlation is low, they are even less attractive under the highcorrelation setting. The reason for this result is simple. If th e lowcost agent lies 27 A lottery of payments for the highcost agent is subsequently called more extreme when the difference between tHH and tHL gets larger. PAGE 25 25 under the highcorrelation setting, he is extremely likely to receive the low HLt payment (i.e., even more likely than in the lo wcorrelation scenario). This happens because the highcorrelation framework imposes on the lowcost agent a very high chance of facing a similar agent, from 0 1 Figure 12 depicts this solu tion, showing the CM wellknown result of increasingly extreme payments for the highcost agent when correlation becomes arbitrarily small.28 The problem with the CMLow mech anism comes from the fact that minHHt ) (HHe as shown by Eq. 17. This is a key feature of the so lution to [PCM]: in order to prevent untruthful reporting from a lowcost agent, a highcost agen t obtains positive (expost) rent if he faces an identical agent. This rent is not a problem at [PCM], because the pr incipal offsets the high payment ) (HH HHe t with a low payment ) (HL HLe t such that the highcost agent earns no expected rent. However, the positive and negative (expost) rents for the highcost agent come at a cost at [PNOC]: they prevent the principal from fully extracting rent from the highcost agent under both correlations, as is now illustrated. At [PNOC], the principal does not know which correlation is realized. Therefore, she has two sets of potential weights to use when calculati ng the expected rent of the highcost agent. As she does at [PCM], the principal would like to average positive and ne gative (expost) rents such that (exante) rent HU is reduced to zero. However, achie ving this goal under two sets of conditional probabilities is impossibl e. This is apparent after a r eexamination of the participation constraints for the highcost agent: 0 ) 1 ( HL s HH s s HU U U } 1 0 { s with 0 ) ( HH HH HHe t U and 0 ) ( HL HL HLe t U from Eq. 17. 28 This result can also be obtained analytically by taki ng the derivative of Eq. 17 with respect to alpha. PAGE 26 26 If the principal extracts all rent for the highcost agent at the highcorrelation scenario, she would not provide him with at least zero rent in the lowcorrelation setting. That is, 01HU is impossible without violating 00HU This result is explained by the necessary positive and negative expost rents for the highcost agent (0 HHU and 0 HLU), combined with 0 1 Therefore, the expected rent for the highcost agent is always larger if correlation is high: HL HH HL HHU U U U ) 1 ( ) 1 (0 0 1 1 As a result, the principal can only extract re nt from the highcost agent under the lowcorrelation setting, leaving him with positive rent if correlation is high:29 1 HU HH HL ) 1 ( 1 20 0 0 0 1 >0 Recalling that ) (Hi Hie 0 ) ( Hie it can be seen that rent 1 HU increases with effort level Hie, for } { H L i As it happens at [PCM], the pa yments for the highcost agent include rewards and penalties wh ich discourage untruthful repor ting by the lowcost agent. The problem is that these rewards and penalties increase (in absolute value) with effort Hie. That is, the higher the effort required from a highcost ag ent, the more extreme his lottery of payments needs to be (i.e., higher HHt and lower HLt) to become unattractive for a lowcost agent. That is not a problem at [PCM], where no rent is provided to either agen t. At [PNOC], however, there is rent for the highcost agent if correlation is high. This rent depends on the effort he delivers, because his effort has a positive impact on the re wards and penalties included in his payments. 29 The principal provides rent to the highcost agent if correlation is high because she requires his participation if correlation is low. This assumption is relaxed in the last section of this chapter, wh ere the principal no longer requires that the highcost agent earns no nnegative rent if correlation is low. PAGE 27 27 Consequently, the principal asks for suboptimal effort levels *e eHL and *e eHH in order to reduce the rent 1 HU of the highcost agent if correlation is high:30 1 ) 1 ( ) 1 2 ( ) ( ) 1 ( 1 ) ( '0 1 0 0 1 HH HHe 1 ) 1 ( ) 1 2 ( ) ( ) 1 ( 1 ) ( '0 1 0 0 1 HL HLe The CMLow mechanism works we ll if correlation is low, where no rent is commanded by either agent. The problem arises if correlation is high, in which case rent is obtained by the highcost agent. The welfare loss generated by rent 1 HU is severe if the potenti al correlations are very different. The reason is that unde r the CMLow mechanism, the pr incipal tailors one of the CM solutions to the lowcorrelation setting (recall ) (0 minHH HHt t ). If this scenario does not occur, however, the welfare loss depends on how wrong the principal is with respect to the exact degree of correlation (i.e., how diffe rent the actual correla tion is high with respect to what the principal thought low). Depending on this difference, therefore, the welfare loss under the CMLow mechanism calls for an alternative solution to [PNOC]. Intuitively, this second contract should work better than the CML ow contract when th e potential correlations are substantially distinct with respect to the correla tion at the lowcorrelation setting. In the second potential solution to [PNOC], the two degrees of freedom available at [PCM] are utilized in a different way than unde r the CMLow mechanis m. This alternative mechanism does not consist of a CM solution ta ilored to any of the potential correlation coefficients. In this alternative contract, the two degrees of freedom available at [PCM] are employed to guarantee expost truthful reporting by the lowcost agent. That is, one degree of 30 The efficient level of effort is delivered after a low cost report: eLL = eLH = e* PAGE 28 28 freedom is used to set ) (LL LLe t = ) ( HL HLe t while the other degree of freedom is used to set ) (LH LHe t = ) ( HH HHe t As a result, both incentive compatibility constraints for the lowcost agent are satisfied because they are sa tisfied expost (i.e., for any cost report of the other agent): 0 )] ( [ )] ( [ ) 1 ( )] ( [ )] ( [ HH HH LH LH s HL HL LL LL se t e t e t e t } 1 0 { s This mechanism is subsequently referred to as the ExPost mechanism. The disadvantage of the ExPost contract is that ) (LL LLe t which means that the prin cipal cannot fully extract rent from the lowcost agent under both correlat ions. The reason for this result is, again, the impossibility of setting equal to zero the average of positive and negative (expost) rents under two possible sets of weights. At most, expected rent can be zero under only one of the potential correlations. As a consequence, under the ExPos t mechanism the lowcost agent obtains rent when correlation is low: 0 LU HL HH ) 1 ( 10 0 0 1 0 1 >0 If correlation is high, however, the rent for the highcost agent is lower than under the CMLow mechanism:31 1 HU HH HL ) 1 ( 11 1 0 1 0 1 >0 The principal cannot comp letely eliminate rent 1 HU because positive and negative (expost) rents for the highcost agent are still required. Recall that ) (HH HHe t and ) (HL HLe t 31 The fact that rent UH 1 is lower under the ExPost mechanism than under the CMLow mechanism is easily verified from the effect of 1 > 0 in the denominator of both expressions: 1 + 0 1 > 2 0 1 The term in brackets [ ] has only a seco ndorder effect on rent. PAGE 29 29 prevent untruthful reporting from the advantaged lowcost agent: he w ould probably receive a low payment if he claims high cost. It can be seen that rents under the ExPos t mechanism depend on the difference between the potential correlations (0 1 ) measured with respect to the middle correlation coefficient (10 1 ). Recalling that 1 2 denotes the correlation coefficient between costs, the middle correlation coefficient 10 1 is defined as the correlation at the middle point of the distance between 1 and 0 For example, if 1 =0.8 and 0 =0.6, the middle correlation would be the correlation coefficient if alpha was 0.7. It is important to notice that the middle correlation coefficient is not the expected correlation 1 2 which would incorporate the probabilities of each scenario in the calculation. This confirms that the ExPost mechanism does not consist of a CM mechanism tailored to any of the potential correlations. Like under the CMLow mechanism, rents unde r the ExPost mechan ism also increase with the effort Hie delivered by the highcost agent. The explanation resides again on the positive impact of his effort on the rewards and pe nalties that his payments include to discourage untruthful reporting by the lowcost agent. Theref ore, the principal reduc es rents under the ExPost mechanism through the suboptimal effort levels *e eHL and *e eHH:32 1 ) 1 ( ) 1 ( ) ( ) 1 ( 1 ) ( '0 0 1 1 1 0 0 1 HH HHe 1 1 ) 1 ( ) 1 ( ) ( ) 1 ( 1 ) ( '0 0 1 1 1 0 0 1 HL HLe 32 Again, the efficient level of effort is delivered after a low cost report: eLL = eLH = e* PAGE 30 30 As it was mentioned before, which of the two alternative mechanisms yields a higher welfare level depends on the parameters of th e problem. The CMLow m echanism is optimal for [PNOC] if: 1 0 0 0 11 2 (117) Equation 117 can be evaluated in terms of how different the tw o potential correlations are, measured with respect to the correl ation at the lowcorrelation setting.33 For similar correlations (small ) 1 2 /( ) (0 0 1 ), the CMLow mechanism yields a higher welfare level. This happens because the welfare loss that the CMLow mech anism entails is direct ly proportional to how different the potential correlations are. As explained above, the rent that the CMLow mechanism provides to the highcost agent under the highcorrelation scenario is t oo large if the potential correlations are very di stinct. Remember that this rent if correlation is high depends on how wrong the principal is with respect to the exact degree of correlation (i.e., how different the actual correlation is high with respect to what the pr incipal thought low). Then, the principal should utilize the CMLow mechanism only if the potentia l correlation coefficients are relatively similar. In contrast, if the potential correlations become too distinct, the principal should employ the ExPost mechanism. This contra ct does not consist of a CM contract tailored to any of the potential correlation coefficients. Figure 13 plots the be havior of welfare under both alternative contracts as a function of how different the potential correlations are.34 Equivalently, Eq. 117 can also be analyzed in terms of the re lative likelihood of each correlation coefficient. When 0 is so large that Eq. 117 holds, the principal ensures that no rent is afforded under the (relatively likely) lowco rrelation scenario. She does so by implementing 33 Remember that 2 1 denotes the correl ation coefficient. 34 The graph assumes a constant value of 2 while 1 is shown in the horizontal axis. PAGE 31 31 the CMLow mechanism, which provides no rent to either agent if correlation is low. Although this payment structure provides rent 1 HU to the highcost agent if correlation is high, the principal still applies the CMLow mechanism if the highcorrelation setting is relatively unlikely (i.e., if 1 is so low that Eq. 117 holds). On the other hand, if 1 is so large that Eq. 117 does not hold, the principal reduces the (relatively likely) rent 1 HU as much as possible by means of the ExPost mechanism. She cannot completely eliminate this rent because ) (HH HHe t and ) (HL HLe t still preven t untruthful reporting from the lowcost agent. The drawback to the ExPost mechanism is that the lowcost agent obtains rent 0 LU if correlation is low. Yet, the principal is willing to accept rent 0 LU if the lowcorrelation scenario is relatively unlikely (i.e., if 0 is so low that Eq. 117 does not hold). The results obtained are similar to those of Arya et al. (2005). In an auction environment, they also find a condition that determines whic h of two mechanisms the principal should apply. Furthermore, the proposed mechanisms and th e circumstances under which they should be employed are similar to the ones presented in this study. If the uncertain ty about the correlation is not severe, the principal should ut ilize a CMLow type of mechanism.35 On the other hand, the principal should employ an ExPost type of mechan ism if the uncertainty about the correlation is more severe.36 Additionally, after considering the parameter values assumed by Arya et al., the condition that determines optimality in th eir paper is the same as Eq. 117 in this study. Instead of a discrete support for the parame ter alpha, they assume a uniform distribution between a lower bound 5 0 and 1. Given such a probability structur e, they find that the principal should 35 BayesianNash is the name given by Arya et al. to such a contract. 36 DominantStrategy is the name given by Arya et al. to such a contract. PAGE 32 32 employ the CMLow type of mechanism if 3 / 2 Now notice the result of substituting 5 01 0 and 11 into Eq. 117: 5 0 5 0 1 2 10 0 Manipulation yields 3 / 20 which means that their condition can be considered a special case of Eq. 117 in this chapter.37 If Only One Agent Observes the Correlation Consider now a situation where communication of the correlation is still blocked, but where only agent A observes the exact correlation be fore contracting. This is a framework where only agent A is endowed with a strong knowledge and expertise about the industry. Meanwhile, agent B and the principal remain ignorant abou t the correlation. They never acquire such privileged information, knowing only the probabilities of facing either 1 or 0 Returning to the previous privatization example, this asymmetr y of agents could arise when firm A has been operating in the industry for a long time (perhaps in other market s), while firm B is relatively new in the industry. In such a cont ext, it is plausible th at not all the agents are perfectly informed about the correlation between their costs. Since it is assumed that the principal does not design a different menu of options for each agent even when she distinguishes the experience d agent A from the inexperienced agent B, the solution to this problem is identical to the solutio n to [PNOC]. The explanation for this result is simple. To ensure As participation, the principa l must guarantee that participation and incentive 37 There is only one difference between their results and th e ones in the present study. Unlike here, the ExPost type of mechanism from Arya et al. does not provide any rent to the lowvaluation bidder (which is the equivalent of the highcost agent). (See their Corollary 2, on page 11). The linearity of the agents utility in their model is the conjectured explanation for such a difference (i.e. there is no convexity like the one coming from the disutility of effort in the present model). A formal proof is beyond the scope of this study. PAGE 33 33 compatibility constraints are satis fied under both potential correlations. Theref ore, the constraints are automatically satisfied for th e inexperienced agent B, who co mputes expected rent not only across both potential types of c ounterpart, but also across both pot ential correlations. Formally, if the constraints from [PNOC] are satisfied, then the equivalent c onstraints for an agent that only observes are also satisfied: 0 )] ( )[ 1 ( )] ( [ ij i ij ii i ii iC t C t U j i H L j i }, { )] ( )[ 1 ( )] ( [ jj i jj ji i ji iC t C t U j i H L j i }, { As a consequence, the solution to [PNOC] also applies if only one of the agents observes the correlation but the principal offers th e same menu of options to both of them. Uninformed Principal: No Limits on Communication In the second case of interest, nothing preven ts the principal from asking for a correlation report from the informed agent(s). Unlike in the case with limits on communication, the Revelation Principle applies in this context. Ther efore, the principal can restrict attention to truthful and direct mechanisms where the agen ts report all their priv ate information. This situation is interesting because an agent who is informed a bout the correlation becomes a twodimensional type of agent. The (simple) situa tion where both agents observe the correlation is presented first. Next, the (more interesting) setting with only one ag ent informed about the correlation is investigated. PAGE 34 34 If Both Agents Observe the Correlation It is well known that if some element of private information is common knowledge for both agents, the principal can elic it that information at no cost.38 She can do so by designing a mechanism where both agents report their indi vidual cost and the realized correlation: A r ) (s i and B r ) (h j with } { ,H L j i and } 0 1 { h s. Since it is impossible to have different (and truthful) correlation reports, the principal can design a f orcing contract that threatens the agents to very large negativ e rents if the correlation reports differ.39 Therefore, even when the principal has imperfect information about the correlation, the fi rstbest situation can still be replicated if both ag ents can communicate the actual correlation observed. Hence, the solution to this problem is iden tical to the solution to [PCM].40 If Only One Agent Observes the Correlation If the correlation is known only by agent A, the principal designs a mechanism where the inexperienced agent B only reports his cost, ) (i Br while the experienced agent A reports his cost and the realized correlation, ) (s j Ar with } { ,H L j i and } 1 0 { s. The timing of the game is depicted in Figure 14. As explained above, both agents are offered the same menu of contracts even when the principal knows about their asymme try in terms of their information on the exact correlation. Let ) (s jk i s jkC t denote the expost rent for an ag ent (A or B) that reports cost j when he has 38 See Fudenberg and Tirole (1991). 39 In the work of Tangeras (2002), for example, the el ement of private information that is common knowledge for both agents is the industry portion of their costs, which has to be added to each agents idiosyncratic cost. Their cost structure follows the model proposed by Auriol and Laffont (1992). 40 There also exists a Nash equilibrium in which both agen ts lie about the correlation. As pointed out by Fudenberg and Tirole, the possibility of multiple equilibria in shootthemall mechanisms gave rise to a large literature on unique Nash implementation (1991, p. 293). PAGE 35 35 cost i when the other agent reports cost k and when the experienced agent A reports that alpha is s for } { H L k j i and } 1 0 { s The principal computes expected welfare: } 1 0 { } { } { ,)] ( [ 2 )] ( [ 2 ) 1 (s H L j i s ij s ij s ij H L j i s ij s ij s ij se t p e e p W (118) The difference between Eq. 118 and Eq. 18 is that the principal now assigns different contracts for different correlations. To ensure that the experienced agent participates and reports his cost truthfully for all cost and correlation realizations, conditional on a truthful correlation report, the next constraint s have to be satisfied: 0 )] ( )[ 1 ( )] ( [ s ij i s ij s s ii i s ii s s iC t C t U j i H L j i }, { } 1 0 { s (119)(122) )] ( )[ 1 ( )] ( [s jj i s jj s s ji i s ji s s iC t C t U j i H L j i }, { } 1 0 { s (123)(126) Additionally, conditional on a trut hful cost report, the experi enced agent A should not lie about the correlation: )] ( )[ 1 ( )] ( [h ij i h ij s h ii i h ii s s iC t C t U j i H L j i }, { } 1 0 { h s (127)(130) Finally, agent A should not simultaneously mi sreport his cost a nd the correlation: )] ( )[ 1 ( )] ( [h jj i h jj s h ji i h ji s s iC t C t U j i H L j i }, { } 1 0 { h s (131)(134) PAGE 36 36 With only one agent informed and with commun ication of the correl ation, the principal solves problem [PCO]:41 Maximize Eq. 118 subject to Eq. 119 to Eq. 134. Intuition could suggest that once a correlati on report becomes available, the principal should again be able to achieve the firstbest by means of two different CM mechanisms, one for each correlation. That is not th e case, however. The problem is th at at [PCO], the agent who is informed about the correlation may not have the in centive of truthfully reporting its realization. In particular, a highcost agent would gain by clai ming that correlation is low when it is actually high. An intuitive explanation follows. Consider a payment structure that consists of two independent CM mechanisms (i.e., one for each correlation). Based on the correlation repo rt, the principal could offer nonstochastic payments to the lowcost agent and an extreme enough lottery (i.e., low s HLt and high s HHt) to the highcost agent, for } 1 0 { s. Under each correlation, the highco st agents lottery would extract all his rent and it would also pr event a lowcost agent from exa ggerating his cost. The problem of these two independent CM mechanisms is that if the experienced agent has high cost, he would enjoy rent under the highcorrelation setting if he claims that correlation is low. The explanation behind this incentive follows. After the highcost agent unde rreports the correlation, he would be assigned the high payment 0 HHt with probability 1 (instead of with probability 0 ) and he would be assigned the low payment 0 HLt with probability 11 (instead of with probability 01 ). Since 1 >0 this average of positive and negative expost rents yields rent for the highcost agent if correlation is 41 As in the case with no communication and only one agent informed, it is not necessary to impose participation and incentive compatibility constraints for the inexperi enced agent B. Those constraints are automatically satisfied when the principal offers the same menu of contracts to both agents. PAGE 37 37 high. Therefore, a highcost agent would want to report that correlati on is low when the highcorrelation environment arises. Although they prev ent untruthful cost repo rting by the lowcost agent, extreme payments to the highcost agent generate a new incentive if there is asymmetric information about the correlation. A highcost agent would earn pos itive rent if correlation is high by reporting that correlation is low. As a result of this incen tive, full rent extraction is not possible at [PCO]. The twodimens ional type of the informed ag ent does not allow the principal to consider the problem as if they were two independent CM problems. The rent for the highc ost agent if correlati on is high is unavoidable.42 This result is similar to what happens at [PNOC], although the source of the rent at [PCO] is different. At [PNOC], the rent for the highcost agent at the highcorrel ation setting occurs beca use the principal cannot tailor his payments to the actual correlation. This comes as a result of designing only one set of payments that has to work well regardless of th e correlation. At [PCO] however, the principal can design different payments for different correla tions. Yet, the rent for the highcost agent if correlation is high still prevails. At [PCO], this rent exists to discourag e him from claiming that correlation is low when it is actually high.43 The solution to [PCO] is similar to the solution to [PNOC] in that Eq. 117 again determines the optimal mechanism. If the lowco rrelation scenario is rela tively likely (i.e., if 0 is so large that 117 holds), the principal only affo rds rent to the highcost agent in the (relatively unlikely) event that correlation is high: 42 The incentive of the highcost agent to claim that correlation is low when it is actually high is always binding at [PCO]. 43 If [PCO] could be considered as two independent CM problems, there would be four degrees of freedom (recall that a standard CM problem has two degrees of freedom). Since one degree of freedom is utilized to prevent untruthful correlation reporting from the highcost agent if correlation is high, problem [PCO] has three degrees of freedom instead of four. PAGE 38 38 1HU 0 0 0 0 0 0 1) 1 ( 1 2HH HL >0 The principal provides rent to the highcost agent if correlation is high by increasing the payment 1 HHt (without reducing 1 HLt accordingly) above the 1 HHt that an independent CM mechanism would require (i.e., one designed specifically for the highcorrelation setting).44 As a result, the lottery (0HHt, 0HLt) under the lowcorrelation scenario is relatively less attractive than the lottery (1HHt, 1HLt) obtained if truthfully revealing that correlation is high. Consequently, the rent provided by the lottery (1HHt, 1HLt) prevents the highcost agent from underreporting the correlation.45 To reduce the rent of the highcost agent if co rrelation is high, the pr incipal also requests less than the efficient effort from the highcost ag ent if correlation is low. This suboptimal effort 0 Hie<* e reduces how extreme his payments need to be if correlation is low, because it reduces the rewards and penalties includ ed in those payments as a means to avoid untruthful cost reporting by the lowcost agent:46 1 ) 1 ( ) 1 2 ( ) ( ) 1 ( 1 ) ( '0 0 0 0 1 0 0 1 0 HH HHe ; and 1 ) 1 ( ) 1 2 ( ) ( ) 1 ( 1 ) ( '0 0 0 0 1 0 0 1 0 HL HLe 44 Equation 17 would determine this value, for = 1. 45 Since the lottery for the highcost agent if correlation is high is even more extreme than what an independent CM mechanism would require, a lowcost agent does not have the incentive of claiming highcost under the highcorrelation environment. He also does not have the incentive of claiming high cost and low correlation. The reason is as follows. If the highcost/lowcorrelation lotte ry is unattractive for a lowcost agent if correlation is low, it is even more unattractive for a lowcost agent if correlation is high (the low payment tHL 0 would arise even more frequently, from 1 > 0). 46 Since the effort exerted by the highco st agent in the highcorrelation scenario does not have any impact on rents, its socially efficient level is required: eHH 1 = eHL 1 = e* Also, eLL 1 = eLH 1 = eLL 0 = eLH 0 = e* PAGE 39 39 As a consequence of this effort distortion, the less extreme payments if correlation is low become less attractive for a highcost agent w ho would untruthfully unde rreport the correlation. Making the lottery (0 HHt, 0 HLt) under the lowcorrelation scenario less extrem e contributes to the rent reduction for the high cost agent if correla tion is high, because the required increase in 1 HHt is smaller than what an independent CM mechanism would dictate. The contract just described is simila r to the CMLow mechanism at [PNOC].47 As such, welfare decreases substantially if the principal applies this mechanism when the correlation coefficients are very different (with respect to the correlation at the lowcorrelation setting).48 The reason comes again from the rent for the highcost agent, which increases with the difference between both potential co rrelations. Therefore, if Eq. 117 does not hold, the highcost agent commands smaller rent in the (relativel y likely) event that correlation is high: 1 HU 0 1 0 1 0 1 0 1) 1 ( 1HL HH >0 Like at [PNOC], the principal reduces rent 1 HU at the cost of affording rent to the lowcost agent in the (relatively unlikel y) event that correlation is low: 0 LU ) 1 ( 10 0 0 0 0 1 0 1HL HH >0 The suboptimal effort levels in this case are as follows 1 )] 1 ( [ ) 1 ( ) ( ) 1 ( 1 ) ( '0 0 0 1 1 0 0 1 0 0 1 0 HH HHe ; and 1 ) 1 ( ] ) 1 ( [ ) 1 ( ) ( ) 1 ( 1 ) ( '0 0 0 1 1 0 0 1 0 0 1 0 HL HLe 47 Although it is not required due to the degrees of freedom available, the principal could offer nonstochastic payments to the lowcost agent under both correlations: tLL 1 = tLH 1 = tLL 0 = tLH 0 = (e*) 48 Again, recall that 2 1 denotes the correlation coefficient. PAGE 40 40 This contract resembles the ExPost mechan ism at [PNOC]. Rent s again depend on the difference between the potential correlations (0 1 ) with respect to the middle correlation coefficient (10 1 ), which refers to the correlation at the middle point of the distance between 1 and 0 Another similarity with the ExPost mechanism from [PNOC] resides on the fact that the incentive constraints that prev ent the lowcost agent from exaggerating cost are binding under both correlations instead of only at the lowcorrelation setting.49 Limits on Communication and Exclusion The analysis so far has assumed that the prin cipal never excludes any agent from the game. That is, participation and truthf ul revelation of private informa tion has been required from all agents under both potential correl ations. The results from the optimization problems performed have illustrated the welfare loss imposed by th is assumption. In part icular, problem [PNOC] showed that rent for the highcost agent is alwa ys higher under the highco rrelation scenario than under the lowcorrelation setting (0 1H HU U ). Recall that the expl anation resides on the implications of the link between 0 HHU and 0 HLU (which prevent unt ruthful reporting by the lowcost agent) and 0 1 As a consequence, the best the principal could do is reduce rent 0HU to zero and allow positive rent 1HU A natural question then arises: should the principal always require nonnegative rent 00HU for the highcost agent if correlation is low? The answer depends on the welfare achieved with and without the participation of the highcost agent if correlation is low. Intuitively, if the probability of the lowcorrelation scenario is low enough, it may be optimal to exclude the hi ghcost agent from the contract. Then, consider 49 Recall that this is the reason why this mechanism was given the ExPost name. PAGE 41 41 the following problem, where the principal desi gns a mechanism that would provide negative rent for the highcost agent if correlation is low. The principal computes expected welfare: } { 1 } { } { 1 } {)] ( [ 2 )] ( [ 2 )] ( [ 2 )] ( [ 2 ) 1 (H L j Hj Hj Hj H L i Li Li Li H L j Hj Hj Hj H L i Li Li Lie t p e t p e e p e e p W (135) Equation 135 arbitrarily exclude s the highcost agent from th e game if correlation is low (i.e., if alpha is 0 ). To ensure that the highcost agent doe s not participate (whe ther truthfully or lying) if correlation is low, the next two constraints have to be satisfied: 0 )] ( )[ 1 ( )] ( [0 0 0 HL H HL HH H HH HC t C t U (136) 0 )] ( )[ 1 ( )] ( [0 0 LL H LL LH H LHC t C t (137) Like at [PNOC], to ensure that the lowco st agent participates and reports his cost truthfully for both correlations, the next four constraints have to be satisfied: 0 )] ( )[ 1 ( )] ( [ LH L LH s LL L LL s s LC t C t U } 1 0 { s (138)(139) )] ( )[ 1 ( )] ( [HH L HH s HL L HL s s LC t C t U } 1 0 { s (140)(141) Like at [PNOC], to ensure that the highco st agent participates and reports his cost truthfully if correlation is high, the ne xt two constraints should be verified: 0 )] ( )[ 1 ( )] ( [1 1 1 HL H HL HH H HH HC t C t U (142) )] ( )[ 1 ( )] ( [1 1 1LL H LL LH H LH HC t C t U (143) PAGE 42 42 With no communication on the correlation and excluding the highcost agent from the game if correlation is low, the principal solves problem [PEXCL]: Maximize Eq. 135 subject to Eq. 136 to Eq. 143. At the solution to [PEXCL], the socially efficient effort level is restored: e eij for j i ,} { H L No rent is provided to the lowcost agent under any correlation (00 1 L LU U ) and no rent is earned by the highcost agent if correlation is high (01HU ). As expected, the highcost agent would obtain negative rent if correlation is low (00HU ), so he does not participate in that scenario. The principal need only ensure that HLt is low enough: 1 *) (1 0 1 e tHL Like in the previous problems, by setting HLt small enough the principal guarantees that a lowcost agent does not exagge rate his cost. Additionally, HLt small enough ensures that the highcost agent earns negative rent if correlation is low (00HU ) because all rent is extracted if correlation is high (01HU ). Recall from problem [PNOC] that 0 1H HU U implies that 00HU if 01HU The welfare loss at the solution to [PEXCL] is attributed to the zero effort delivered by the highcost agent if correlation is low. Theref ore, if the probability of the lowcorrelation setting is sufficiently low, the solution to [P EXCL] may be an alternative to the ExPost mechanism obtained at [PNOC]. Recall that the ExPost contract is optimal at [PNOC] when the lowcorrelation setting is relatively unlikely (i.e., when 0 is so low that Eq. 117 does not hold). Some conclusions can be drawn from numerical examples. PAGE 43 43 Figure 15 shows that welfare at the solution to [PEXCL] may be still below the welfare achieved under the ExPost mechanism (which c onstitutes the solution to [PNOC]), even for a relatively small probability of the lowcorrelation setting (5% in the example). Figure 16 shows that a slight decrease of that probability (to 3%) can raise welf are at the solution to [PEXCL] above the welfare achieved at the solution to [P NOC]. Finally, Figure 17 illustrates that when the probability of the lowcorrelation scenario dr ops substantially (to 1% ), the welfare under the ExPost mechanism is almost always below the welfare attained at th e solution to [PEXCL]. Therefore, depending on the parameters of the problem, the figures show that it may not be optimal to require participation and truthful revelation of privat e information from all agents under all scenarios.50 Conclusions This chapter contributes to the recent and growing literature on both multidimensional and robust mechanism design. The results from [PNOC ] show that full rent extraction is impossible if two informed agents cannot communicate the re alization of the correla tion to the principal. This finding coincides with the results obtai ned by other researchers on robust mechanism design, like Arya et al. (2005) Bergemann and Morris (2005), Heifetz and Neeman (2006) and Neeman (2004). Additionally, the solution to pr oblem [PCO] illustrates that even with communication on the correlation the principal may not be able to fully extract rents from the more informed agents. These results coincide with the findings by authors investigating multidimensional mechanism design, like Miller et al. (2007). When limits on communication on the correlatio n prevail, the principal cannot tailor the payments to the exact correlation coefficient. As a result, the highcost agent earns positive rent 50 Although the analysis was not performed, it is conjectured that the ExPost type of mechanism that constitutes a solution to [PCO] if Eq. 17 does not hold could also be replaced by a contract that excludes the highcost agent from the game if correlation is low. PAGE 44 44 if correlation is high. This re sult is the most impo rtant finding of the analysis under limited communication. The reason behind the rent for the highcost agent under the highcorrelation setting is that his extreme lotter y of payments should ensure that nonnegative exp ected rent is obtained under both potential correlations. Since e xpected rent is calculat ed using two different sets of weights (one for each correlation), and since the high payment is obtained with higher probability (and the low payment with lowe r probability) under the highcorrelation environment, the principal can re duce rents for the highcost agent to zero only if correlation is low. As a remainder, extreme payments for the highcost agent prevent the lowcost agent from exaggerating cost. When full communication is restored, and if onl y one agent is perfectly informed about the extent of cost correlation, the principal is still unable to fully extract rent from both agents under both correlations. Even when the mechanism desi gner can tailor payments to each correlation coefficient, the highcost agent still obtains positiv e rent if correlation is high. This is the most important result of the analysis without limits on communication. Although the rent for the highcost agent if correlation is high parallels the re sults with blocked comm unication, the source of this rent is slightly different. Instead of being explained by the in ability of the principal to tailor payments to each correlation, when communicati on on the correlation is available the principal has to worry about a highcost agent untruthfull y underreporting the correlation. He would do so because he would obtain a high payment more frequently (and a low payment less frequently) than what he should if the principal designe d two independent CM mechanisms, one for each correlation report. The rent for the highcost agent if correlation is high is driv en by the fact that nonnegative rent is required for him under th e lowcorrelation setting. If the principal could ex clude the high PAGE 45 45 cost agent from the game if correlation is low, the socially efficient effort level can be restored without affording any rents to the (participating) agents. The cost of such a contract is that no effort is delivered by the highcost agent when co rrelation is low, because he does not participate due to the negative rent he would obtain. If th e probability of the lowcorrelation scenario is small enough, numerical examples show that the principal can achieve higher welfare by designing a contract that excludes the highcost agent from the game when correlation is low. Finally, the usual criticis m of the risk neutrality assumpti on also applies to this study. The solutions obtained make use of that assumption, since agents here voluntarily participate in the mechanism proposed by the principa l even when they could potent ially obtain large negative expost rent. Therefore, the impossibi lity of the fullrentextraction result would probably be more pronounced if risk aversion or limited liability constraints were imposed. PAGE 46 46 All parties learn 1, 0and 1, 0Each agent learns his own cost parameter The principal offers two contracts Each agent reports his cost A specific (t,C) pair is assigned to each agent depending on both reports The agents produce, transfers are made and costs are reimbursed Both agents observe the correlation but do not report it Figure 11. Timing at [PNOC], when both agents observe but do not report the correlation. PAGE 47 47 Parameters: 0.55 0.01 0.99 0.60 0.20 (e) 1e44 Payments under the CMLow contract0.80 0.60 0.40 0.20 0.00 0.20 0.40 0.60 0.80 1.00 0.560.600.630.670.710.740.780.820.850.890.930.961.0 0 tLL(CMLow) tHH(CMLow) tHL(CMLow) tLH(CMLow) Figure 12. A solution to [PCM] that can also be a solution to [PNOC]. PAGE 48 48 Parameters: 0.55 0.70 0.30 0.60 0.20 (e) 1e44 Welfare at PNOC2.24 2.26 2.28 2.30 2.32 2.34 2.36 2.38 2.40 2.42 0.560.600.630.670.710.740. 780.820.850.890.930.961.0 0 CMLow PNOC ExPost PNOC Figure 13. Welfare under the two al ternative mechanisms at [PNOC]. PAGE 49 49 All parties learn 1, 0and 1, 0Each agent learns his own cost parameter The principal offers four contracts Each agent reports his cost and agent A reports the correlation A specific (t,C) pair is assigned to each agent depending on both reports The agents produce, transfers are made and costs are reimbursed Agent A observes the correlation Figure 14. Timing at [PCO] when only agent A observes a nd reports the correlation. PAGE 50 50 Parameters: 0.55 0.05 0.95 0.60 0.20 (e) 1e44 Welfare at PNOC vs PEXCL2.31 2.32 2.33 2.34 2.35 2.36 2.37 2.38 2.39 2.40 0.560.600.630.670.710.740.780.820.850.890.930.961.0 0 ExPost PNOC Altern PEXCL Figure 15. Welfare at [PEXC L] below welfare at [PNOC] PAGE 51 51 Parameters: 0.55 0.03 0.97 0.60 0.20 (e) 1e44 Welfare at PNOC vs PEXCL2.32 2.33 2.34 2.35 2.36 2.37 2.38 2.39 2.40 0.560.600.630.670.710.740. 780.820.850.890.930.961.0 0 ExPost PNOC Altern PEXCL Figure 16. Welfare at [PEXCL] some times larger than welfare at [PNOC] PAGE 52 52 Parameters: 0.55 0.01 0.99 0.60 0.20 (e) 1e44 Welfare at PNOC vs PEXCL2.32 2.33 2.34 2.35 2.36 2.37 2.38 2.39 2.40 0.560.600.630.670.710.740. 780.820.850.890.930.961.0 0 ExPost PNOC Altern PEXCL Figure 17. Welfare at [PEXCL] almost always larger than welfare at [PNOC] PAGE 53 53 CHAPTER 2 EFFICIENCY IN THE BRAZI LIAN SANITATION SECTOR Introduction Early work on the relative performance of water and sewerage (WS) utilities by Crain and Zardkoohi (1978) tried to determine whether private U.S. water utilities attained a more efficient level of operation than public ones. Since then, a number of pa pers have been published on the efficiency of WS utilities. Some authors have al so focused on the private vs. public issue, while others have tested other hypot heses, like the existence of economies of scale, economies of scope, or the possible homogeneity or homot heticity in the production technology. Data availability limited the types of studies: early pa pers focused mostly on utilities in the U.K. and U.S. because these countries pioneered the co llection and publication of data on WS firms. Until a decade ago, little rese arch was conducted into the effi ciency of WS utilities in developing countries. Then, studies began to address the performance of water systems using quantitative techniques. A numb er of papers focused on Asia n and African water utilities, most of them supported by the World Bank.1 These studies provided insights for countries implementing World Bank policies to increase cove rage and quality of WS services in their regions. Politically potent and economi cally important, WS utilities generate public concern over efficiency. Using data from Br azil, this study extends the sta ndard approach by comparing not only the performance of public and private WS firm s, but also the performance of different types of public WS operators. There are four different types of WS providers in Brazil. The first type consists of regional public operators, which provide serv ices at the state level. The ot her three types of WS operators 1 For example, Estache and Ro ssi (1999), Estache and Rossi (2002), and Estache and Kouassi (2002). PAGE 54 54 provide services at the lo cal (municipal) level. The first local type consists of private firms that have signed concession contracts wi th the municipalities where they operate. The other two local types both consist of pub liclyowned operators, but they differ in their legal status. One type consists of local public providers that are organized similarly to a corporate business. They are called publiccorporative operators throughout this chapter. The ot her type consists of local public providers that are run like notforpro fit organizations. They are called publicnoncorporative providers throughout this study. Besides the public vs. private discussion, c ontroversy exists in Brazil about whether municipalities or states should be responsible for WS provisi on. The Association of StateOwned Sanitation Firms (AESBE), for example, argues that WS services should be provided at the state level for two reasons: one, the larger scale of operation permits scale economies; two, there is a possibility of crosssubsidization be tween poorer and rich er municipalities.2 In contrast, the National Association of Municipal Sanitation Services (ASSEMAE) favors municipal provision on the grounds that WS services are an essential necessity for the population.3 ASSEMAE does not explain in detail why the essential nature of WS services calls for municipal and not state provision, but standard arguments are based on local control, responsivene ss to citizen concerns and awareness of local conditions. This study finds that WS provision in Braz il is characterized by economies of scale. Therefore, since an increase in output genera tes a lessthanproportional increase in (operating) costs, WS provision at the state level should be preferred. The pot ential efficiency gains are not trivial when one recalls that Braz il has a population of 187million people. 2 Associao das Empresas de Sa neamento Bsico Estaduais. See http://www.aesbe.org.br (last visit: March 26, 2007). 3 Associao Nacional dos Servios Municipais de Saneamento. See http://www.assemae.org.br (last visit: March 26, 2007). PAGE 55 55 In a first stage, a fixedeffects panel data model with data for 20002004 is employed. A cost function is proposed, identifying firms pecific (operating) costs which account for inefficiency and other unobserved heterogeneity. In a second stage, those firmspecific costs are explained by means of firmtype and other timei nvariant indicator variables. The results show that regional operators have lower firmspeci fic costs than localprivate and localpublic providers. This finding indicates th at the efficiency gains of the state providers from their scale of operation are augmented by their lower firmspecific costs. This work also shows that the localpublic vs. localprivate comparison depends on how the local public provider is organized. Private and publiccorporative providers have lower firmspecific costs than publicnoncorporative providers. This finding indicates that the WS operators organized as notforprofit organizations have the highest firmspecific costs in Brazil. In spite of the firmspecific cost differen ces found, it is worth mentioning that these differences represent a small portion of operating cost s. The firststage regressions illustrate that the output produced, input prices and other technol ogical factors explain most of the variation of operating cost, regardless of the firmtype. As a result, the firmspecifi c cost differences found are not substantial from an economic point of view. Quantifying the relative efficiency of the Br azilian regional operators, Tupper and Resende (2004) use Data Envelopment An alysis (DEA) with data for 19962000. The efficiency scores obtained are considered in the construction of a proposed linear reimbursement rule that constitutes a yardstick mechanism. However, the authors acknowledge that its implementation is constrained by the weak current regulatory fr amework. Utilizing DEA with data for 19982002, Seroa da Motta and Moreira (2006 ) argue that the government leve l at which conceding authority resides is not a crucial barrier to the Brazilian san itation sectors development when looking at PAGE 56 56 the operators performance. Unlike this study, they find that ownership does not matter for productivity gains for municipal services. Like th is chapter, they find that regional operators benefit from larger scale economies. Evidence on the beneficial effects of privat e sector participati on (PSP) in the Latin American sanitation sector is not conclusive. In Argentina, Bolivia and Brazil, for example, Clarke et al. (2004) fi nd that even when connection rates to piped water improved following the introduction of PSP, connection rates similarly improved in the control regions that never privatized. In contrast, Galiani et al. (2005) find that child morta lity in Argentina fell 8% in the areas that privatized their water services. They conclude that while privatization is associated with reductions in deaths from infectious dise ases, it is uncorrelated with deaths from causes unrelated to water conditions. The ambiguity on the beneficial effects of PSP in the Latin American sanitation sector coincides with the results from other regions.4 This chapter first presents an overview of Brazils water and sewerage industry. The study then illustrates the twostage methodology uti lized. A following section present the results obtained. After performing some sensitivity checks, conclusions are provided in a final section. Overview of Brazils Water and Sewerage Industry In 1971, Brazil created a national plan for WS provision (PLANASA).5 This plan delegated authority for the provi sion of WS services to twenty seven newly born stateowned companies. According to PLANASA, these public companies were the only sanitation entities authorized to obtain financing from the Nationa l Housing Bank (Banco N acional de Habitaao 4 For the US, Bhattacharyya, Parker and Raffiee (1994) found evidence of greater efficiency in public utilities, Crain and Zardkoohi (1978) found evidence in favor of private operators, and Byrnes et al. (1986), Feigenbaum and Teeples (1984) and Fox and Hofler (1986) found no difference between public and private operators. For Asia, Estache and Rossi (1999) found evidence in favor of private operators while in a later study Estache and Rossi (2002) found no difference. For Africa, Estache and Kouassi (2002) found evidence in favor of private firms. 5 Plano Nacional de Saneamento. See So ares (2001) for a detailed description. PAGE 57 57 BNH). This feature made PLANASA attractive for the municipalities that were interested in expanding their sanitation systems.6 About 3,200 municipalities jo ined the new plan, awarding concessions to the stateowned companies for 20 to 30 years.7 On the other hand, about 1,800 municipalities never adhered to PLANASA, providi ng WS services on their own ever since. The three types of local operators (private, publiccorporative and publicnoncorporative) provide WS services within the municipalities that never joined the system. The PLANASA model started to decline in th e 80s. After 1986, the BNH was unable to finance the required expansion of the WS sector due to a weak fiscal situation of the federal government.8 Antiinflationary policies may have also pl ayed a role, since the government at that time pressed for low water tariffs to keep in flation under control. The 80s decade was also characterized by an emphasis on decentralization, best illustrated in the constitutional reform of 1988. The centralizing concept of PLANASA, on the other hand, wa s more in accordance to the military regimes of earlier decades. There have recently been some attempts to de fine a new framework for the WS sector. Bill 4147/2001, for example, intended to allow for more private participation. This bill defined the states to be the conceding authority in metropol itan areas. However, the constitutional reform of 1988 granted to municipalities the right to make concessions for public services of local 6 See Faria (2005). 7 In many cases, however, there was never any formal contract between the municipality and the stateowned companies. 8 PLANASA formally extinguished in 1992. See Parlatore (1999). PAGE 58 58 interest.9 Due to that controversy about the interpretation of the Constitution, Bill 4147 never became law.10, 11 A new Bill 5296/2005 also attempted to redefine the rules for the WS sector. A Parliamentary Commission approved it on July 2 006 after many modifications and it recently became Law 11445 in January 2007. This new bill specifies that municipalities have the conceding authority over services of local interest. Nevertheless, the implications of the new Law 11445 are yet to be seen. Due to the heated debate about where the conceding authority re sides, only some municipalities that never adhere d to PLANASA have made con cessions to private operators.12 These private companies provide WS serv ices to less than 4% of the population.13 The Brazilian Association of Private Water and Sewage Operators (ABCON) suggests that only through a more active private participation will the WS s ector meet the high inve stment levels required.14 According to 2004 data, approximately 112,000 people are directly employed in the WS sector, almost 90% by the stateowned companies. The national coverage for water services is roughly 85%, although the sewage coverage is below 60%. On average, almost 30% of the treated water produced is unaccounted for, due to leaking through broken pipes and illegal 9 Water distribution and sewerage collection are defined as services of local interest. On the other hand, water catchment and water and sewerage treatment are defined to be of local interest only in case of exclusive use by the municipality. See Ministrio das Cidades, http://www.cidades.gov.br (last visit: March 26, 2007). 10 There was also a strong opposition from the public and representative institutions, like the Brazilian Association of Sanitary and Environmental Engi neering (Associao Brasileira de Engenharia Sanitria e Ambiental ABES). See http://www.abesdn.org.br (last visit: March 26, 2007). 11 A Nacional Water Agency (Agencia Nacional de Agua s ANA) was also created in 2000. However, the main function of ANA is to monitor the utilization of water resources. Its role as a regulator is yet to be defined. 12 See Vargas and De Lima (2004). 13 This figure contrasts with the situation in other infrastructure sectors like tel ecommunications, railroads and electricity, where private participati on is much more active. See Oliveira and Fujiwara (2005) and Pinheiro (2003). 14 Associao Brasileira de Concessionrias de Servios Pblicos de gua e Esgoto. See http://www.abcon.com.br (last visit: March 26, 2007). PAGE 59 59 connections. Furthermore, only 50% of the sewa ge volume collected receives some type of treatment. Table 21 shows these statistics by operatortype. Methodology Duality theory implies that the production tec hnology of a firm can be modeled with a cost function, where the firms cost depends on its ou tput level and the prices of the inputs employed in production. Other factors may also affect th e firms technology and hence the firms costs. Specifically: ) ( z w q c c (21) where c denotes cost, q denotes output level, w denotes input prices and z includes other control variables.15 To empirically estimate Eq. 21, a panel data framework is adopted: it i it itu X Y ', (22) where itY denotes the dependent variable for individual i at time t itX denotes the vector of explanatory variables, iu accounts for timeinvariant heteroge neity at the individual level and it denotes random statistical noise. Heterogene ity is the denomination of the observed and unobserved unique individua l characteristics. Fixed or random effects models can be adopt ed for panel data. The fixed effects model allows unobserved heterogeneity to be correlated with the explanat ory variables. In contrast, the random effects model assumes that any unobserved he terogeneity is distribut ed independently of the covariates. In the context of this study, correlation betwee n unobserved heterogeneity and the explanatory variables is hard to rule out. Such a correla tion would exist, for example, if the firm 15 Control variables have been sometimes denominated hedonic measures, referring to th e approach introduced by Spady and Friedlaender (1978) for the trucking industry. They emphasize that the service dimensions provided by the firm should enter the cost function as explanatory variables. Feigenbaum and Teeples (1983) first introduced the hedonic approach to the water sector. PAGE 60 60 can modify its output level based on private in formation about its unobserved inefficiency. Therefore, a fixed effects model for panel data is employed. The fixed effects formulation allows unobserved inefficiency to be capture d by the firmspecific coefficients.16 In a panel data framework, the cost function from Eq. 21 takes the following form: i it it z it w it q itu z w q c 0, (23) where the s are parameters to be estimated and iu denotes cost inefficiency and any additional unobserved heterogeneit y. It is assumed that the it are i.i.d and uncorrelated with the covariates. In contrast, the iu are allowed to be potentially correlated with the explanatory variables.17 The least squares dummy vari ables (LSDV) estimator is utilized, including also a yearspecific effect:18 it t it z it w it q i itz w q c (24) where the firmspecific intercepts i iu 0 account for inefficiency and any other unobserved heterogeneity. Utilizing this formulati on, Schmidt and Sickles (1984) proposed the measure ) min(* i i i to construct a ranking of relativ e inefficiency. Their approach permits the computation of individual inefficiency te rms relative to the most efficient firm in the sample.19 That calculation might be appropriate when one is concerned about efficiency at the individualfirm level. For example, a regulator co uld set the tariff of firm A based partly on its 16 According to Greene (2005), assuming that inefficiency is timeinvariant is not a problem in short panels. This is especially true in the water industry, which is characterized by low technological change. 17 It is unnecessary to make any distribu tional assumption on the inefficiency term i. If one is willing to make distributional assumptions on the i, Maximum Likelihood would theoretically allow for more efficient estimates than both fixed and random effects models. Neverthele ss, Kumbhakar and Lovell (2000) and MurilloZamorano (2004) mention several papers that after performing em pirical comparisons of the three approaches generate similar efficiency rankings, especially at the top and bottom of the distribution. 18 The LSDV estimator is equivalent to the withingroups estimator. 19 Ashton (2000) constructs an efficiency rank ing of British water firms utilizing that measure. PAGE 61 61 relative efficiency with respect to firm B, as suggested by Shle ifer (1985). However, the focus of this study is not yardstick comparison betw een individual operator s but rather between different types of firms. The goal of this chapter is to identify whether regional, localpublic or localprivate firms are relatively more efficien t in providing WS services in Brazil. Thus, an alternative analysis is pursued. After estimating Eq. 24, the predicted firmspecific costs i are computed. A high i indicates an inherently high cost for firm i even when controlling for output, input prices and other exogenous variables. Once the predicted fi rmspecific costs are obtained, an additional regression is performed. In this secondstage regression, the depe ndent variable is the predicted firmspecific cost (per unit of output), while fi rmtype indicators and regional dummies are the explanatory variables:20 i i i i iq Region Type (25) The vectors and contain the coefficients for each of the Type and Region indicator variables. Meanwhile, iq represents the average output of firm i for the period unde r analysis. If the coefficients for the firmtype dummy variables are sta tistically significant in Eq. 25, there will be evidence of relatively distinct firmspecific costs between the different operatortypes. Following the extant literature, Operating Cost is utilized to re present the dependent variable c on Eq. 24.21 Wage is employed to represent input prices w since they account for 20 Although output is present in the first stage regression the firmspecific costs are going to be correlated with output by the nature of the fixedeffects model. That is the explanation behind the utilization of firmspecific costs per unit of output. 21 Although it is also conceivable to use total cost as the dependent variable, that would require data on the price of capital, since depreciation charges constitute a large share of total costs. Since reliable data on the price of capital are unavailable, this study focuses only on operating cost, which excludes depreciation. PAGE 62 62 more than 40% of the operating cost. 22 Wages were calculated as the ratio of total labor expenses divided by the number of employees as it is standard in the literature.23 Although the volume of water produ ced seems like the most appealing output variable, the number of connections has also been widely used by researchers.24 Thus, both Volume and Connections are employed as two alternative measures of output q As control variables, this study includes Network Length ,25 the Percentage of Urban Population ,26 a Metering Index,27 a Fluorination Index28 and a Sewerage Dummy that equals 1 if the firm also provides sewerage collection (not all operato rs provide both services).29 22 Data on other input prices are limited. Some data on energy consumption suggests that energy is the second most important input, representing around 20% of operating cost. 23 More detailed data on input prices would theoretically allow for more efficient estimates utilizing the Seemingly Unrelated Regressions (SUR) model proposed by Zellner (1962). This model cons ists of a multivariate regression system. Besides the cost function, the inputdemand shareequations are utilized, enhancing the efficiency of the estimation because the same coefficients participate not only in the cost function but also in the inputdemand shareequations. 24 The number of connections is employed by Ashton (2 000), Estache and Rossi (1999) Estache and Rossi (2002) and Teeples and Glyer (1987). The volume of water produced is utilized by Antonioli and Filippini (2001), Aubert and Reynaud (2005), Bhattacharyya, Harris et al (1995), Bhattacharyya, Parker and Raffiee (1994), Bottaso and Conti (2003), Corton (2003), Crain and Zardkoohi (1978), Cubbin and Tzanidakis (1998), Estache and Rossi (1999), Estache and Rossi (2002), Fabbri an d Fraquelli (2000), Fox and Hofler (1986), Kim (1987), Stewart (1993) and Teeples and Glyer (1987). The number of customers is used by Antonioli and Filippini (2001), Aubert and Reynaud (2005), Fabbri and Fraquelli (2000) and Saal and Parker (2000). 25 The length of pipes is utilized by Antonioli and Filippini (2001), Bottaso and Conti (2003), Corton (2003), Cubbin and Tzanidakis (1998), Fox and Hofler (1986), Kim (1987) and Stewart (1993). 26 A proxy of density is used by Bottaso and Conti ( 2003), Fabbri and Fraquelli (2000) (ratio between population served and the length of pipelines) and Teeples and Glyer (1987) (connections per mile of line). The percentage of nondomestic consumers is employed by Bottaso and Conti (2003), Estache and Rossi (1999), Estache and Rossi (2002), Fox and Hofler (1986), Kim (1987) and Stewart (1993). 27 The percentage of metered connections is used by Cubbin and Tzanidakis (1998), Estache and Rossi (2002), Feigenbaum and Teeples (1983) and Teeples and Glyer (1987). 28 A proxy for quality is used by Antonioli and Filippini (2001) (dummy indicating if water has to be chemically treated before distribution), Estache and Rossi (1999) (continuity), Estache and Rossi (2002) (continuity), Feigenbaum, and Teeples (1983) (water treatment index), Fox and Hofler (1986) (tests of water quality and tests of organic contamination), Saal and Parker (2000) (percentage of water that is compliant with key parameters relative to the compliance percentage for England and Wa les) and Teeples and Glyer (1987) (water treatment index). 29 Other control variables have been also employed in previous literature. The percentage of water losses is used by Antonioli and Filippini (2001) and Bhattacharyya, Harris et al. (1995). The storage capacity is used by Feigenbaum and Teeples (1983), Fox and Hofler (1986) and Teeples and Glyer (1987). A dummy indicating if the utility has to purchase water from other utility is employed by Aubert and Reynaud (2005), Feigenbaum and PAGE 63 63 Earlier literature suggests that a longer network should be as sociated with higher costs due to its maintenance (fixing leaks, for example). Previous research also suggests that a higher metering index should be associated with higher costs due to the reading and maintenance of the meters. The fluorination index shoul d also be associated with higher costs due to a more intense chemical treatment of water before delivery. Th e sewerage dummy is also expected to show a positive sign, capturing the higher operating cost of providing both water and sewerage services. Finally, the effect of a higher proportion of urban population in the area served is difficult to predict. On the one hand, many researchers argue that having customers densely located in a small area reduces costs. On the other hand, Fe igenbaum and Teeples (1 983) argue that we should expect that it is more costly to supply more densely developed service areas, which requires more hydrants, higher wate r pressure and greater peak cap abilities for fire protection.30 The main source of data is the National System of Sanitation Information (SNIS) of Brazil.31 Operators voluntarily join the SNIS, which started collecting data in 1995. The number of firms providing data has increas ed each year ever since. This study utilizes an unbalanced panel for 20002004. There are approximately 180 observations for 2000 and 340 observations for 2004, with almost 1200 observations in total.32 The SNIS is part of the Modernization Teeples (1983), Fox and Hofler (1986) and Teeples and Glyer (1987) (they use the proportion of water that is purchased). The number of districts served is used by Corton (2003). A dummy indicating if the utility obtains water from surface sources is used by Aubert and Reynau d (2005), Bhattacharyya, Harr is et al. (1995), Bottaso and Conti (2003) (they use river sources), Estache and Rossi (1999) and Fox and Hofler (1986). I ignore the reasons that support the inclusion of a proxy for capital stock, as done by Antonioli and Filippini (2001) (number of water wells), Aubert and Reynaud (2005) (average net base rate divided by the estimated price of capital), Bhattacharyya, Harris et al. (1995) (r esidual of the revenue less variable costs) and Bottaso and Conti (2003) (replacement costs of net tangible assets). Such a variab le would be an appropriat e covariate in a production function, but not in a cost function 30 Feigenbaum and Teeples (1983, p.674). They confirm that result in their paper. 31 Sistema Nacional de Informaoes sobre Saneamento. See www.snis.gov.br (last visit: May 7, 2007). 32 Missing data on some variables explain the variation in the number of observations across the two models presented in the next section. The online data are split into several files, each containing only a certain group of variables (financial, descriptive, op erational, etc) and a certain type of firm (regional, local, etc). These spreadsheets were pooled together for this work. PAGE 64 64 Program of the Sanitation Sector (PMSS), whic h Brazil started in 1992 with financial support from the World Bank.33 To adjust monetary figures for infl ation, data on the Brazilian Consumer Price Index (IPCA) are utilized.34 Results Summary statistics of the variables used in the firststage LSDV regressions are presented in Table 22, discriminated by operator type. Th e size difference between the regional and local operators is evident when observing the average output, network length a nd operating cost of each type of firm. For example, from the point of view of the number of connections, the average regional operator is 14 time s bigger than the average publiccorporative provider. Table 23 presents the results from both LSDV regressions, according to Eq. 24. Alternatively, both Connections and Volume are positive and statistically significant. The hypotheses that their coefficients are equal to 1 is rejected which provides evidence of increasing returns to scale regardless of the out put variable chosen. For example, a 10% increase in the volume of water produced generates only a 0.98% increase in operating cost.35 This confirms the argument made by AESBE, which favor s statelevel provision due to economies of scale. As expected, Wage also has a positive and statisti cally significant effect on cost. From the set of control variables, the Metering Index shows the expected positive sign and statistical significance. This means that when the fraction of metered connections is higher, 33 Programa de Modernizaao do Setor de Saneamento. The PMSS resides on the sphere of the National Secretariat of Environmental Sanitation (Secretaria Nacional de Saneamento Ambiental), which depends on the Ministry of Cities (Ministrio das Cidades). See www.cidades.gov.br (last visit: May 7, 2007). 34 The ndice Nacional de Preos ao Consumidor Amplo (IPCA) is constructed by the Brazilian Institute of Geography and Statistics (Instituto Brasileir o de Geografia e Estatistica IBGE). See www.ibge.gov.br (last visit: May 7, 2007). 35 To test for economies of scale that may vary with output, specifications including the square of the output variable were run. The results were not satisfactory. The Volume2 variable was not statistically significant, while Volume remained statistically significant with little change in the value of its coefficient. Meanwhile, both Connections and Connections2 became statistically insignificant. PAGE 65 65 operating cost increases. This is in line with earlier research suggesting that reading and maintaining the meters has a positive impact on cost. The Sewerage Dummy coefficient is positive and statistically significant. This means th at collecting and treating sewage increases the operating cost of a water provider. Finally, Network Length the Percentage of Urban Population and the Fluorination Index are statistically insignificant. The firststage LSDV regressions in Table 23 also illustrate th at output, input prices and other technological factor s explain most of the variation of operating cost.36 As a result, the firmspecific costs provided by the firststage regressions represent a small portion of the operating cost (less than 1%). After running the firststage LSDV regres sions, the predicted firmspecific costs i are obtained. Following Eq. 25, a secondstage re gression is then performed, where the firmspecific cost (per unit of output) plays the role of the dependent variable. Firmtype and five regionindicators are the timeinvariant explanat ory variables in the secondstage regression. Dummies for the five different regions in which Brazil is divided are included because geographic heterogeneity may differently a ffect the cost of access to raw water.37 Even when the specific effect of regional heteroge neity is not the focus of this st udy, it is important to control for that timeinvariant characteristic. If the firm type dummy variables are statistically significant in the secondstage regression, there will be evidence of relatively distinct firmspecific costs between the different types of operators. Table 24 presents the results of the secondst age regression. The nega tive and statistically significant coefficient of the Regional dummy indicates that regiona l providers have lower firm36 The high explanatory power of both models remains even if the firm fixedeffects (not reported) are excluded. In that case, the R2 is still above 0.95. 37 These regional dummies could partially capture different energy prices as well. PAGE 66 66 specific costs than all other operators. At the local level, meanwhile, the firmspecific cost comparison between private and public operato rs depends on how the public provider is organized. The positive and statisti cally significant coefficient of the PublicNonCorp dummy variable indicates that the loca lpublicnoncorporative operators ha ve higher firmspecific costs than both the localprivate and the localpublicc orporative providers. Tabl e 24 also shows that there is not a statistically signi ficant difference between the firms pecific cost of localprivate and localpubliccor porative operators. Table 25 illustrates the value of the firmspeci fic costs, obtained from the results of the secondstage regression. The table shows that the region al public operators are the lowestcost WS providers, while the localpublicnoncorp orative operators are the highestcost WS providers in Brazil. Although the differences are substantial and statisti cally significant, it is worth recalling that the firmspecific costs are not a significant portion of the operating costs. Sensitivity Checks As a first sensitivity check, a balanced panel was used. Since the sample size increases over the years, it is important to check that inco rporating new firms does not affect the analysis. The results are presented in Table 26. The positive and significant effect of both output variables ( Connections and Volume ) remains. The same is true for the Wage variable across both specifications. Table 27 contains the results of the second stage re gressions. It can be verified that the lower firmspecific cost for the Regional type is confirmed when using a balanced panel, regardless of the output variable chosen. Howeve r, the higher firmspecific cost for the publicnoncorporative type is not obtai ned when a balanced panel is employed. The reason behind this statistically imprecise result could be the loss of observations when utilizing a balanced panel. That is, most of the observations dropped when using a balanced panel correspond to local firms (which include the publicnoncorporativ e type) rather than Regional ones. PAGE 67 67 As a second sensitivity check, and utilizing agai n the larger unbalanced panel, the Regional type was excluded from the sample. The reason fo r this sensitivity check is that the size differential between the Regional and Local operators could affect the conclusions. Table 28 shows the results of the first stage LSDV regre ssion and Table 29 contains the results of the second stage regression. The firs t stage regressions show that the positive and statistically significant effect of output, wage and the meteri ng variable are again verified, along with the sewerage dummy. Furthermore, the coefficients obtained are similar to those in Table 23 and Table 24. For example, the measure of economies of scale indicates that a 10% increase in the volume of water produced generates a 0.94% in crease in operating cost. The second stage regressions show that the higher firmspecific cost for the public noncorporative type is again confirmed, providing confidence about the co nclusions drawn earlier. The statistically insignificant difference between the private and p ubliccorporative operators is also verified. Conclusions Brazil is a country that lacks perfect access to WS services. Efficiency improvements could free up funds for network expansion, which would c onstitute a step toward s a desired full service situation. Therefore, greater attention for costcontainment is needed, regardless of the jurisdictional and owne rship/organizational st atus of the WS operators. Improving our understanding of relative performa nce can help policy makers focus on the sources of differential cost patterns. The results of this study suggest that, at least for Brazil, evidence of economies of scale is enough to claim that WS provision at the state level is more effi cient than WS provision at the municipal level. Economies of s cale generate substantial cost savings, which far outweigh any potential differential in firmspecific costs. As such, the argument ma de by AESBE seems more compelling than the argument made by ASSEMAE. PAGE 68 68 In addition, this chapter finds evidence of inherently lower fi rmspecific costs (per unit of output) for regional WS firms than for all other types of WS operators in Brazil. These lower costs reinforce the efficiency gains the regional firms achieve through actual economies of scale. Finally, this study shows evidence of higher firm specific costs (per un it of output) for localpublicnoncorporative providers than for localp rivate and localpublic corporative providers. Future research could examine what features generate the in trinsic cost differences among operator types. In particular, the higher firms pecific cost for the publicnoncorporative type deserves further attention. It may be important to check whether the notforprofit motive of those organizations actually drives their higher firmspecific costs. Even when cost differences between the differe nt types of WS operators were found, it is worth noting that these differences represent a small portion of operating costs. The firststage regressions illustrated th at output, input prices and other technological f actors explain most of the variation of operating cost, regardless of the firmtype. As a result, the firmspecific cost differences presented are significant from a statistical point of vi ew, but less significant from an economic perspective. Clearly, much work remains. For the purpose of rewarding good performance and penalizing weak performance, scholars and practit ioners need to develop efficiencymeasuring procedures that can pass legal ch allenges. The process must con tinue to build on the pioneering research of those whose work is cited in the references. In part icular, the publication of league tables is one way to put pressure on the weakest performing WS utilities. Similarly, the managers of WS utilities in the top 20 percent might be awar ded some share of the cost savings that can be attributed to their efforts. T hose promoting improvements in WS sector performance can take steps to reduce production costs and free up cash flows for netw ork rehabilitation and expansion. PAGE 69 69 Identifying, implementing, and evaluating good incentive systems represent a challenge for regulators. A final issue that also deserves future research follows. The analysis in this chapter only considered relative measures of efficiency. The goal was to identify sources of cost differences between the different types of operators. However, cost savings for the en tire industry could also be estimated utilizing the results obtained. For example, reducing water losses would also free up funds for network expansion. Table 21 shows th at water losses in 2004 stood at almost 30% on average, while they were 49% for the state ope rators. Thus, a reduction of water losses by 10% should not be hard to achieve. Yet, it could represent an almo st 1% lower operating cost. Some researchers suggest that the onl y explanation behind the lack of incentives for reducing water losses is that it may be cheaper to produce more water instead. Although geography might validate this statement in some cases, the issue deserves further exploration. PAGE 70 70 Table 21. Average statistics by operatortype for 200438 Type of firm # Connect. Empl. Water cover. Sewer. cover. Water losses Sewer. Treatm. Private 31 30470 88.9 80.1% 50.4% 29.1% 54.4% Public NonCorp 296 18851 99.2 86.3% 63.0% 26.6% 46.5% Public Corp 11 75180 400.3 98.8% 68.5% 41.3% 36.6% Regional 25 1104748 2978.6 71.3% 33.8% 48.7% 74.5% Total 369 91660 300.7 85.1% 58.5% 28.9% 51.3% 38 The sample also includes six Microregional operators, wh ich are not the focus of the analysis. These are public operators that are neither regional nor local, since th ey provide services to just a few municipalities. PAGE 71 71 Table 22. Summary statistic s for firststage regressions39 Variable Private N=62 Public Non Corp. N=913 Public Corp. N=42 Regional N=121 Operating Cost 9,102,144 5,088,097 25,661,470 239,346,200 (12,500,360) (13,124,270) (34,681,050) (338,851,600) Connections 50,539 24,251 74,573 1,021,909 (67,902) (39,282) (67,960) (1,234,135) Volume 22,239 8,552 34,899 396,122 (41,240) (17,315) (33,649) (583,563) Wage 15,959 11,512 18,082 31,147 (6,953) (6,256) (9,349) (10,613) Network Length 642 294 1,005 11,398 (840) (457) (1,044) (12,930) Dummy sewerage 0.82 0.56 0.98 0.94 (0.39) (0.50) (0.15) (0.23) Urban % 0.88 0.77 0.95 0.77 (0.09) (0.21) (0.05) (0.12) Metering % 0.88 0.77 0.86 0.74 (0.19) (0.32) (0.21) (0.28) Fluorination % 0.33 0.30 0.35 0.20 (0.46) (0.44) (0.47) (0.36) N=1163 (Standard deviations in parenthesis) 39 Volume is in 1000m3/year and Network length is in Km. Operating cost and Wage are in Reais/year deflated using 2000 as the base year. For the Volume variable, summary statistics are for 1172 observations. The 1163 observations include 25 observations for the Microregional category, which are not reported. PAGE 72 72 Table 23. Firststag e LSDV regression results40 Dependent Variable: Operating Cost Connections Volume Connections 0.427 (0.112)*** Volume 0.098 (0.040)** Wage 0.150 0.157 (0.043)*** (0.040)*** Network Length 0.024 0.089 (0.073) (0.088) Dummy sewerage 0.122 0.138 (0.058)** (0.067)** Urban % 0.070 0.082 (0.246) (0.258) Metering % 0.382 0.409 (0.177)** (0.165)** Fluorination % 0.001 0.005 (0.035) (0.035) Constant 8.496 11.274 (1.133)*** (0.713)*** Observations 1163 1172 Rsquared 0.99730 0.99729 Year and firm fixedeffect not reported Standard errors clustered at the statelevel significant at 10% ** significan t at 5% *** significant at 1% 40 Operating Cost, Volume, Connections, Wage and Network length are in ln form. The statistical significance of all coefficients is very similar when the standa rd errors are clustered at the Region level. PAGE 73 73 Table 24. Secondst age regression results Dependent Variable: (ln) FirmSpecific Cost per Unit of Output (from LSDV regressions) Connections Volume Public Non Corp 0.331 0.294 (0.127)*** (0.124)** Public Corp 0.009 0.073 (0.217) (0.176) Regional 1.154 0.270 (0.190)*** (0.158)* Constant 0.960 2.640 (0.124)*** (0.116)*** Observations 380 380 Rsquared 0.31726 0.04428 Omitted type: Private Region fixedeff ect not reported Robust standard errors in parenthesis significant at 10% ** significan t at 5% *** significant at 1% PAGE 74 74 Table 25. Ranking of firmsp ecific costs across firmtypes. Firm type FirmSpecific cost per unit of output Index (Regional=100) ($/Connection) ($/1000m3) Regional 0.12 10.70 100 100 Public Corp 0.38 13.03 314 122 Private 0.38 14.01 317 131 Public Non Corp 0.53 18.80 441 176 PAGE 75 75 Table 26. Firststage LSDV regres sion results using a balanced panel Dependent Variable: Operating Cost Connections Volume Connections 0.631 (0.152)*** Volume 0.127 (0.043)*** Wage 0.185 0.186 (0.022)*** (0.026)*** Network Length 0.014 0.082 (0.074) (0.104) Dummy sewerage 0.034 0.046 (0.048) (0.045) Urban % 0.158 0.150 (0.135) (0.139) Metering % 0.002 0.009 (0.203) (0.188) Fluorination % 0.023 0.023 (0.028) (0.030) Constant 7.894 13.287 (1.155)*** (1.053)*** Observations 758 766 Rsquared 0.99713 0.99711 Year and firm fixedeffect not reported Standard errors clustered at the statelevel significant at 10% ** significan t at 5% *** significant at 1% PAGE 76 76 Table 27. Secondstage regressi on results using a balanced panel Dependent Variable: (ln) FirmSpecific Cost per Unit of Output (from LSDV regressions) Connections Volume Public Non Corp 0.176 0.265 (0.219) (0.179) Public Corp 0.005 0.056 (0.301) (0.166) Regional 1.966 0.339 (0.259)*** (0.182)* Constant 2.702 2.795 (0.328)*** (0.249)*** Observations 170 170 Rsquared 0.54194 0.11137 Omitted type: Private Region fixedeff ect not reported Robust standard errors in parenthesis significant at 10% ** significan t at 5% *** significant at 1% PAGE 77 77 Table 28. Firststage LSDV regression results ex cluding the Regional type Dependent Variable: Operating Cost Connections Volume Connections 0.439 (0.130)*** Volume 0.094 (0.040)** Wage 0.145 0.152 (0.045)*** (0.041)*** Network Length 0.022 0.083 (0.072) (0.087) Dummy sewerage 0.119 0.135 (0.058)* (0.066)* Urban % 0.062 0.077 (0.252) (0.264) Metering % 0.424 0.460 (0.194)** (0.179)** Fluorination % 0.004 0.005 (0.039) (0.039) Constant 8.433 11.37 (1.326)*** (0.721)*** Observations 1042 1047 Rsquared 0.99579 0.99575 Year and firm fixedeffect not reported Standard errors clustered at the statelevel significant at 10% ** significan t at 5% *** significant at 1% PAGE 78 78 Table 29. Secondstage regression results excluding the Regional type Dependent Variable: (ln) FirmSpecific Cost per Unit of Output (from LSDV regressions) Connections Volume Public Non Corp 0.331 0.286 (0.127)*** (0.120)** Public Corp 0.037 0.077 (0.220) (0.170) Constant 1.005 2.711 (0.124)*** (0.113)*** Observations 354 354 Rsquared 0.1322 0.01567 Omitted type: Private Region fixedeff ect not reported Robust standard errors in parenthesis significant at 10% ** significan t at 5% *** significant at 1% PAGE 79 79 APPENDIX DERIVATION OF THE SOLUTIONS TO CHAPTER 1 Limits on Communication When Correlations are Relatively Similar If Eq. 117 holds, solve [PNOC] imposing only Eq. 19 (multiplier denoted with 1LP), Eq. 110 (multiplier denoted with 0LP ), Eq. 112 (multiplier denoted with 0HP ) and Eq. 114 (multiplier denoted with 0LI ). Solving from the first order conditions with respect to payments yields: 01 1 LP, 0 1 20 0 1 1 0 0 LP, 0 1 20 0 1 1 0 LI and 0 1 2 10 0 1 1 0 0 HP. Setting equal to zero the first order conditi ons with respect to the effort vector: 0 ) ( ) ( ) ( ) ( ] [ )] ( 1 ][ )[ 1 (0 0 0 0 1 1 0 0 1 1 0 0 1 1 LL L LL L LL L LL LL LLe I e P e P e e e L Substituting the value of the multipliers and simplifying: 1 ) ( LLe 0 ) ( ) 1 ( ) ( ) 1 ( ) ( ) 1 ( ) ( )] 1 ( ) 1 ( [ )] ( 1 )][ 1 ( ) 1 ( )[ 1 (0 0 0 0 1 1 0 0 1 1 0 0 1 1 LH L LH L LH L LH LH LHe I e P e P e e e L Substituting the value of the multipliers and simplifying: 1 ) ( LHe 0 ) ( ) 1 ( ) ( ) ( ] [ )] ( 1 ][ )[ 1 (0 0 0 0 0 0 1 1 0 0 1 1 HH L HH H HH HH HHe I e P e e e L Substituting the value of the multipliers and simplifying: PAGE 80 80 1 ) ( ) 1 ( ) 1 2 ( ) 1 ( 1 ) ( '' 0 0 1 1 0 1 0 0 1 HH HHe 0 ) ( ) ( ) 1 ( ) ( )] 1 ( ) 1 ( [ )] ( 1 )][ 1 ( ) 1 ( )[ 1 (0 0 0 0 0 0 1 1 0 0 1 1 HL L HL H HL HL HLe I e P e e e L Substituting the value of the multipliers and simplifying: 1 ) 1 ( ) 1 2 ( ) 1 ( 1 ) ( '' 0 0 1 1 0 1 0 0 1 HL HLe Solve from the binding constrai nts to obtain the payments. From Eq. 19 and Eq. 110: ) (*e tLL and ) (*e tLH Substituting in Eq. 114 and simplifying: HL HH HL HHt e e t0 0 0 01 ) ( ) ( 1 Rearranging terms in Eq. 112: HL HL HH HHt e e t0 0 0 01 ) ( 1 ) ( Equating the righthandsides of the last two equations: HL HH HL HLe t 0 0 0 0) 1 ( 1 2 ) ( Substituting back in Eq. 112: HL HH HH HHe t 0 0 0 0) 1 ( 1 2 1 ) ( The excluded constraints (Eq. 111, Eq. 113, Eq. 115 and Eq. 116) are satisfied (not binding) by the payments above. In particular, su bstituting the payments in Eq. 111 yields the rent for the highcost agen t if correlation is high: PAGE 81 81 0 ) 1 ( 1 20 0 0 0 1 1 HL HH HU When Correlations are Relatively Different If Eq. 117 does not hold, solve [PNOC] imposing only Eq. 19 (multiplier denoted with 1LP), Eq. 112 (multiplier denoted with 0HP ), Eq. 113 (multiplier denoted with 1LI) and Eq. 114 (multiplier denoted with 0LI ). Solving from the first order conditions with respect to payments: 0 1 ) 1 2 ( ) (0 1 0 0 0 1 1 1 LI, 00 0 LI 0 1 1 20 1 0 1 LP and 0 1 1 20 1 1 0 HP Setting equal to zero the first order conditi ons with respect to the effort vector: 0 ) ( ) ( ) ( ) ( ] [ )] ( 1 ][ )[ 1 (0 0 1 1 1 1 0 0 1 1 0 0 1 1 LL L LL L LL L LL LL LLe I e I e P e e e L Substituting the value of the multipliers and simplifying: 1 ) ( LLe 0 ) ( ) 1 ( ) ( ) 1 ( ) ( ) 1 ( ) ( )] 1 ( ) 1 ( [ )] ( 1 )][ 1 ( ) 1 ( )[ 1 (0 0 1 1 1 1 0 0 1 1 0 0 1 1 LH L LH L LH L LH LH LHe I e I e P e e e L Substituting the value of the multipliers and simplifying: 1 ) ( LHe 0 ) ( ) 1 ( ) ( ) 1 ( ) ( ) ( ] [ )] ( 1 ][ )[ 1 (0 0 1 1 0 0 0 0 1 1 0 0 1 1 HH L HH L HH H HH HH HHe I e I e P e e e L Substituting the value of the multipliers and simplifying: PAGE 82 82 1 ) ( ) 1 ( ) 1 ( ) 1 ( 1 ) ( '' 0 0 1 1 0 0 1 1 0 1 0 1 HH HHe 0 ) ( ) ( ) ( ) 1 ( ) ( )] 1 ( ) 1 ( [ )] ( 1 )][ 1 ( ) 1 ( )[ 1 (0 0 1 1 0 0 0 0 1 1 0 0 1 1 HL L HL L HL H HL HL HLe I e I e P e e e L Substituting the value of the multipliers and simplifying: 1 ) 1 ( ) 1 ( ) 1 ( ) 1 ( 1 ) ( '' 0 0 1 1 0 0 1 1 0 1 0 1 HL HLe Solve from the binding constraints to obtain the payments. From Eq. 19: )] ( [ 1 ) (1 1 LH LH LL LLe t e t From Eq. 112: )] ( [ 1 ) (0 0 HL HL HH HHe t e t Substituting the last two expressions into Eq. 113 and Eq. 114 yields: HL HH HL HLe t 1 1 0 1 0) 1 ( 1 ) ( HL HH LH LHe t ) 1 ( 1 ) (0 0 0 1 1 Substituting LHt in Eq. 19 yields: HL HH LL LLe t ) 1 ( 1 1 ) (0 0 0 1 1 Substituting HLt in Eq. 112 yields: HL HH HH HHe t 1 1 0 1 0) 1 ( 1 1 ) ( PAGE 83 83 The excluded constraints (Eq. 110, Eq. 111, Eq. 115 and Eq. 116) are satisfied (not binding) by the payments above. In particular, substituting the payments in Eq. 110 and Eq. 111 yields the rent for the highcost agent if correlation is high and the rent for the lowcost agent if correlation is low: 0 ) 1 ( 11 1 0 1 0 1 1 HL HH HU 0 ) 1 ( 10 0 0 1 0 1 0 HL HH LU No Limits on Communication When Correlations are Relatively Similar If Eq. 117 holds, solve [PCO] imposing only Eq. 119 (multiplier denoted with 19 ), Eq. 120 (multiplier denoted with 20 ), Eq. 121 (multiplier denoted with 21 ), Eq. 123 (multiplier denoted with 23 ) and Eq. 130 (multiplier denoted with 30 ). Solving from the first order conditions with respect to payments yields: 0 1 20 0 1 1 23 0 1 2 ) 1 ( ) 1 2 (0 0 1 1 0 0 20 0 1 20 0 1 1 0 19 01 21 and 01 30 Setting equal to zero the first order conditi ons with respect to the effort vector: 0 0 0 0 0 0) 1 ( ) ( LL LLe e L) ( '0 0 19 LLe ) ( '0 0 23 LLe =0 Substituting the value of the multipliers and simplifying: 1 ) ( '0LLe 0 0 0 0 0 0) 1 ( ) ( HH HHe e L ) ( '0 0 20 HHe PAGE 84 84 ) ( ) 1 (0 0 23 HHe ) ( '0 1 30 HHe =0 Substituting the value of the multipliers and simplifying: 1 ) 1 2 ( ) ( ) 1 ( ) 1 ( 1 ) ( '0 0 0 1 0 0 0 1 0 HH HHe ) 1 ( ) 1 ( ) ( ) 1 (0 0 0 0 0 0 LH LHe e L ) ( ) 1 (0 0 19 LHe ) ( ) 1 (0 0 23 LHe =0 Substituting the value of the multipliers and simplifying: 1 ) ( '0LHe ) 1 ( ) 1 ( ) ( ) 1 (0 0 0 0 0 0 HL HLe e L ) ( ) 1 (0 0 20 HLe ) ( '0 0 23 HLe ) ( ) 1 (0 1 30 HLe =0 Substituting the value of the multipliers and simplifying: 1 ) 1 2 ( ) ( ) 1 ( ) 1 ( 1 ) ( '0 0 0 1 0 0 0 1 0 HL HLe 1 1 1 1 1 1) 1 ( ) ( LL LLe e L ) ( '1 1 21 LLe =0 Substituting the value of the multipliers and simplifying: 1 ) ( '1LLe 1 1 1 1 1 1) 1 ( ) ( HH HHe e L ) ( '1 1 30 HHe =0 Substituting the value of the multipliers and simplifying: 1 ) ( '1HHe ) 1 ( ) 1 ( ) ( ) 1 (1 1 1 1 1 1 LH LHe e L) ( ) 1 (1 1 21 LHe =0 Substituting the value of the multipliers and simplifying: 1 ) ( '1LHe PAGE 85 85 ) 1 ( ) 1 ( ) ( ) 1 (1 1 1 1 1 1 HL HLe e L ) ( ) 1 (1 1 30 HLe =0 Substituting the value of the multipliers and simplifying: 1 ) ( '1HLe Solve from the binding constraints to obtain the payments. The five equations are as follows: 0 )] ( )[ 1 ( )] ( [0 0 0 0 0 0 LH LH LL LLe t e t (119) 0 )] ( )[ 1 ( )] ( [0 0 0 0 0 0 HL HL HH HHe t e t (120) 0 )] ( )[ 1 ( )] ( [1 1 1 1 1 1 LH LH LL LLe t e t (121) 0 )] ( ) ( )[ 1 ( )] ( ) ( [0 0 0 0 0 0 0 0 0 0 HH HH LH LH HL HL LL LLe t e t e t e t (123) 0 )] ( ) ( )[ 1 ( )] ( ) ( [0 0 1 1 1 0 0 1 1 1 HL HL HL HL HH HH HH HHe t e t e t e t (130) The system has an infinite number of so lutions. Selecting arbitrary values for 1LHt 1HLt and 0LHt a solution can be characterized as follows: )] ( [ ) 1 ( ) (0 0 0 0 0 0LH LH LL LLe t e t 0 0 0 0 0 0 0 0) 1 ( 1 2 ) (HH HL HL HLe t 0 0 0 0 0 0 0 0) 1 ( 1 2 1 ) (HH HL HH HHe t )] ( [ ) 1 ( ) (1 1 1 1 1 1LH LH LL LLe t e t 0 0 0 0 0 1 0 1 1 1 1 1 1 1) 1 ( ) 1 2 ( )] ( [ ) 1 ( ) (HH HL HL HL HH HHe t e t PAGE 86 86 Some of the excluded constraints are automati cally satisfied by the payments above. For the rest of the excluded constr aints to be satisfied, the following bounds on the arbitrary payments should hold: 1 0 1 0 1 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1) 1 ( 1 ) ( ; ) 1 ( ) 1 ( 1 2 1 2 ) ( min ) (LL LH LH LL LH HH HL LH LH LHe e t e where 0 ) ( ) ( s Li s Li s Lie e for } { H L i and } 0 1 { s 0 0 0 0 0 1 0 0 1 0 1 0 1 0 1 0 1 1 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 1) 1 ( ) 1 2 )( 1 ( ) )( 1 ( ) 1 ( 1 ) ( ; ) 1 ( ) 1 2 )( 1 2 ( ) )( 1 ( ) 1 ( 1 2 ) ( min ) 1 ( 1 2 ) (HH HL HH HL HL HH HL HH HL HL HL HH HL HLe e t e 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0) 1 ( ) 1 ( 1 2 1 ) ( ; ) 1 ( 1 2 ) ( min ) (LL LH HH HL LH LL LH LH LH LHe e t e In particular, substituting the payments into Eq. 122 yields the rent for the highcost agent if correlation is high: 0 ) 1 ( ) 1 2 (0 0 0 0 0 0 1 1 HH HL HU PAGE 87 87 When Correlations are Relatively Different If Eq. 117 does not hold, solve [PCO] impos ing only Eq. 120 (multiplier denoted with 20 ), Eq. 121 (multiplier denoted with 21 ), Eq. 123 (multiplier denoted with 23 ), Eq. 130 (multiplier denoted with 30 ) and Eq. 133 (multiplier denoted with 33 ). Solving from the first order conditions with respect to payments yields: 1 1 21 0 1 20 >0, 1 1 21 0 0 21 >0, 0 23 >0, 1 30 >0 and 1 ) 1 2 ( ) (1 0 0 0 0 1 1 33 >0 Setting equal to zero the first order conditions with respect to the effort vector: 0 0 0 0 0 0) 1 ( ) ( LL LLe e L ) ( '0 0 23 LLe =0 Substituting the value of the multipliers and simplifying: 1 ) ( '0LLe 0 0 0 0 0 0) 1 ( ) ( HH HHe e L ) ( '0 0 20 HHe ) ( ) 1 (0 0 23 HHe ) ( '0 1 30 HHe ) ( ) 1 (0 1 33 HHe =0 Substituting the value of the multipliers and simplifying: 1 ) 1 ( ) ( )] 1 ( [ ) 1 ( 1 ) ( '0 1 0 0 1 0 0 1 1 0 0 0 HH HHe ) 1 ( ) 1 ( ) ( ) 1 (0 0 0 0 0 0 LH LHe e L ) ( ) 1 (0 0 23 LHe =0 Substituting the value of the multipliers and simplifying: 1 ) ( '0LHe ) 1 ( ) 1 ( ) ( ) 1 (0 0 0 0 0 0 HL HLe e L ) ( ) 1 (0 0 20 HLe PAGE 88 88 ) ( '0 0 23 HLe ) ( ) 1 (0 1 30 HLe ) ( '0 1 33 HLe =0 Substituting the value of the multipliers and simplifying: 1 ) 1 ( ) ( ) 1 ( ] ) 1 ( [ ) 1 ( 1 ) ( '0 1 0 0 1 0 0 1 1 0 0 0 HL HLe 1 1 1 1 1 1) 1 ( ) ( LL LLe e L ) ( '1 1 21 LLe ) ( '1 1 33 LLe =0 Substituting the value of the multipliers and simplifying: 1 ) ( '1LLe 1 1 1 1 1 1) 1 ( ) ( HH HHe e L ) ( '1 1 30 HHe =0 Substituting the value of the multipliers and simplifying: 1 ) ( '1HHe ) 1 ( ) 1 ( ) ( ) 1 (1 1 1 1 1 1 LH LHe e L) ( ) 1 (1 1 21 LHe ) ( ) 1 (1 1 33 LHe =0 Substituting the value of the multipliers and simplifying: 1 ) ( '1LHe ) 1 ( ) 1 ( ) ( ) 1 (1 1 1 1 1 1 HL HLe e L ) ( ) 1 (1 1 30 HLe =0 Substituting the value of the multipliers and simplifying: 1 ) ( '1HLe Solve from the binding constraints to obtain the payments. The five equations are as follows: 0 )] ( )[ 1 ( )] ( [0 0 0 0 0 0 HL HL HH HHe t e t (120) 0 )] ( )[ 1 ( )] ( [1 1 1 1 1 1 LH LH LL LLe t e t (121) 0 )] ( ) ( )[ 1 ( )] ( ) ( [0 0 0 0 0 0 0 0 0 0 HH HH LH LH HL HL LL LLe t e t e t e t (123) PAGE 89 89 0 )] ( ) ( )[ 1 ( )] ( ) ( [0 0 1 1 1 0 0 1 1 1 HL HL HL HL HH HH HH HHe t e t e t e t (130) 0 )] ( ) ( )[ 1 ( )] ( ) ( [0 0 1 1 1 0 0 1 1 1 HH HH LH LH HL HL LL LLe t e t e t e t (133) The system has an infinite number of so lutions. Selecting arbitrary values for 1 LHt 1 HLt and 0 LHt a solution can be characterized as follows: )] ( [ ) 1 ( ) (1 1 1 1 1 1 LH LH LL LLe t e t 0 1 0 1 0 1 0 0 0) 1 ( 1 ) (HL HH HL HLe t 0 1 0 1 0 1 0 0 0) 1 ( 1 1 ) (HL HH HH HHe t ) ( 1 ) 1 ( ) 1 ( ) ( ) (0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 LH LH HL HH LL LLe t e t ) ( ) 1 ( ) 1 ( ) 1 ( ) (1 1 1 1 0 1 0 1 0 1 1 0 1 1 1 HL HL HL HH HH HHe t e t Some of the excluded constraints are automati cally satisfied by the payments above. For the rest of the excluded constr aints to be satisfied, the following bounds on the arbitrary payments should hold: 0 0 0 0 0 1 1 0 0 1 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0) 1 ( ) 1 ( ) ( ; ) 1 ( 1 1 ) ( ) ( ; ) 1 ( ) 1 2 )( 1 ( ) )( 1 ( ) 1 ( 1 2 ) ( minHL HH LH LH LL HL LH HL HH LL LH LH LHe e e t PAGE 90 90 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 1 0 0 1 1 0 1 0 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1) 1 ( ) 1 ( ) ( ; ) 1 ( 1 1 ) ( ; ) 1 ( ) 1 2 )( 1 ( ) )( 1 ( ) 1 ( 1 2 ) ( minHL HH HL HL HH HH HL HL HH HL HH HL HLe e e t ) 1 ( ) 1 ( ) ( ; ) 1 ( 1 ) ( ; ) 1 ( ) 1 2 )( 1 ( ) ( ) 1 ( 1 2 ) ( min0 0 0 0 0 1 1 1 1 0 1 0 1 0 1 1 0 1 0 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 HL HH LH LL LH LH HL HH LH LL LH LHe e e t In particular, substituting the payments into Eq. 119 and Eq. 122 yields the rent for the lowcost agent if correlation is low and the rent for the highcost agent if correlation is high: 0 ) 1 ( ) 1 ( ) (0 0 0 0 0 1 0 1 0 HL HH LU 0 ) 1 ( ) 1 (0 1 0 1 0 1 0 1 1 HL HH HU Limits on Communication and Exclusion Solve [PEXCL] imposing only Eq. 138 (multiplier denoted with 38 ), Eq. 139 (multiplier denoted with 39 ) and Eq. 142 (multiplier denoted with 42 ). Solving from the first order conditions with respect to payments yields: 0 38 >0, 1 39 >0 and 1 42 >0. Setting equal to zero the firs t order conditions with respec t to the effort vector and substituting the value of the multipliers yields 1 ) ( ije for }. { H L j i Solving from the binding constraints yields the payments. The solution is given by: PAGE 91 91 ) (LH LHe t ) (LL LLe t and )] ( [ ) 1 ( ) (1 1HL HL HH HHe t e t with HLt arbitrary. Some of the excluded constraints are automati cally satisfied by the payments above. For the rest of the excluded c onstraints to be satisfied, the following upper bound on HLt should hold: )] ( *) ( [ 1 *) ( )]; ( *) ( [ 1 2 *) ( *); ( min1 0 1 1 1e e e e e e e tHL PAGE 92 92 LIST OF REFERENCES Antonioli, B. and Filipini, M. (2001) The use of a variable cost function in the regulation of the Italian water industry, Utilities Policy 10 181187. Arya, A., Demski, J., Glover, J. and Liang, P. 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In March 2000, he graduated with honors from the Univer sidad Catlica Argentina with a Licenciatura en Economa (Bachelor of Arts in economics). After gr aduation, he worked for one year as an Analyst in the Corporate Finance Department of the largest media group in Argentina. He then worked for two and a half years as a Category Manager in the Commerci al Department of a supermarket chain store, also in Argentina. While getting experience in the private sector, he also engaged in parttime teaching at the Universidad Catlica Argentina. He was awarded a Fulbright Scholarship at the end of 2002; this allowed him to return to school to pursue graduate education, this time in the United States. He started a doctorate in economics at the University of Florida in the fall of 2003. While pursuing his doctorate, he worked parttime as a Research Assistant for PURC, Public Utility Research Center. His work at PURC provided him with the opportunity to presen t his research at in ternational meetings. He was the instructor of undergraduate Game Th eory in the summer semester of 2006. He also presented one of his research papers at the Second Summer School on "Economic Analysis of Heterogeneity in Social Or ganizations" in CORE, LouvainlaNeuve, Belgium, in June 2006. As recognition of his work, he was awarded th e Madelyn M. Lockhart International Travel Award and the Walter Lanzillotti Research Gran t by the Department of Economics, both in 2006. He graduated in August 2007, his dissertation titled Theoretical and Empirical Analyses of Incentives and Public Ownership 