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Accounting for the horizon 2001
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Title | Accounting for the horizon |
Creator |
Șabac, Florin M. |
Date Created | 2009-09-25 |
Date Issued | 2009-09-25 |
Date | 2001 |
Description | Firms have long-term relationships with economic agents such as managers, auditors, and suppliers that can be characterized as sequences of shorter lived agents interacting with a longer lived firm. In this framework, the agent's tenure with the firm becomes the object of investigation. Thus, when the firm starts a multi-period relationship with an agent, what is the role played by beliefs about the duration of the relationship? If the relationship is governed by long-term contracts and if the length of the contract can be chosen ex-ante, and commitment to a certain tenure is possible, is there an optimal ex-ante tenure from the firm's owners' point of view? This dissertation addresses several issues related to an agent's tenure in multi-period models, and is based on three essays. The first essay is on auditing and analyzes how beliefs about auditor tenure impact auditor independence and audit pricing. The second and third essays examine the question of optimal managerial tenure in a multi-period agency model, while at the same time investigating the value of commitment and how different assumptions about commitment impact the solution of the agency problem. One other common theme in the first and third essays is the lack of a "last period", so that, while the firm (the principal) has an infinite horizon and needs to consider a succession of auditors or managers, the agents (auditors or managers) have finite horizons. As a result, an agent's last period is not the last period of the model, the firm continues to exist and operate, and a new agent is hired. |
Extent | 9038468 bytes |
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Thesis/Dissertation |
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Text |
File Format | application/pdf |
Language | Eng |
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Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project |
Date Available | 2009-09-25 |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0099707 |
Degree |
Doctor of Philosophy - PhD |
Program |
Business Administration |
Affiliation |
Business, Sauder School of |
Degree Grantor | University of British Columbia |
Graduation Date | 2001-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/13160 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0099707/source |
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A C C O U N T I N G F O R T H E H O R I Z O N By Florin M . §abac 1987 B.Sc. Mathematics, University of Bucharest 1988 M.Sc. Mathematics, University of Bucharest 1995 Ph.D. Mathematics, University of South Carolina A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in T H E F A C U L T Y O F G R A D U A T E S T U D I E S A C C O U N T I N G D I V I S I O N , F A C U L T Y O F C O M M E R C E A N D B U S I N E S S A D M I N I S T R A T I O N We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A April 2001 © Florin M . §abac, 2001 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Accounting Division, Faculty of Commerce and Business Administration The University of British Columbia 2053 Main Mall Vancouver, Canada V6T 1Z2 Date: Abstract Firms have long-term relationships with economic agents such as managers, auditors, and suppliers that can be characterized as sequences of shorter lived agents interacting with a longer lived firm. In this framework, the agent's tenure with the firm becomes the object of investigation. Thus, when the firm starts a multi-period relationship with an agent, what is the role played by beliefs about the duration of the relationship? If the relationship is governed by long-term contracts and if the length of the contract can be chosen ex-ante, and commitment to a certain tenure is possible, is there an optimal ex-ante tenure from the firm's owners' point of view? This dissertation addresses several issues related to an agent's tenure in multi-period models, and is based on three essays. The first essay is on auditing and analyzes how beliefs about auditor tenure impact auditor independence and audit pricing. The second and third essays examine the question of optimal managerial tenure in a multi-period agency model, while at the same time investigating the value of commitment and how different assumptions about commitment impact the solution of the agency problem. One other common theme in the first and third essays is the lack of a "last period", so that, while the firm (the principal) has an infinite horizon and needs to consider a succession of auditors or managers, the agents (auditors or managers) have finite horizons. As a result, an agent's last period is not the last period of the model, the firm continues to exist and operate, and a new agent is hired. i i Table of Contents Abstract ii Acknowledgements V ii Table of Contents iii List of Tables v List of Figures vi 1 Introduction 1 1.1 The horizon problem 1 1.2 Agency costs, audit pricing, and auditor turnover 3 1.3 Correlated Noise, Commitment, and Ratcheting 8 1.4 Multiperiod Ratchet Effect and Managerial Tenure 12 2 Agency Costs, Audit Pricing, and Auditor Turnover 18 2.1 Introduction 18 2.2 Related literature 20 2.3 Audit pricing with exogenous costs and exogenous turnover 23 2.4 Auditor replacement in equilibrium 34 2.4.1 The game 35 2.5 Empirical implications and conclusion 51 3 Correlated Noise, Commitment, and Ratcheting 67 i i i 3.1 Introduction 67 3.2 The model 71 3.3 Full commitment, long-term contract 75 3.4 Commitment and renegotiation 77 3.5 Commitment to fairness 85 3.6 The agent cannot commit for two periods 96 3.6.1 Long-term contracts and renegotiation 96 3.6.2 Two agents 99 3.6.3 Random firing of the agent after one period 102 3.7 Correlation and the value of commitment 105 3.8 Appendix: proofs 107 4 Multiperiod Ratchet Effect and Managerial Tenure 123 4.1 Introduction 123 4.2 The principal-agent model 132 4.3 Full commitment, long-term contract 141 4.4 Commitment and renegotiation 144 4.5 Short-term contracts under commitment to fairness 152 4.6 Dynamic incentives and ratcheting 164 4.7 Managerial tenure with fixed switching cost 169 4.8 Conclusions 176 4.9 Appendix: proofs 180 Bibliography 191 List o f Tables 2.1 Type I history, first-period payoffs 58 2.2 Type I history, subsequent periods payoffs 60 2.3 Type II history, first period payoffs 60 2.4 Type II history, subsequent periods payoffs 61 2.5 Type III history, N = 1 payoffs 62 2.6 Type III history, N > 1, first period payoffs 63 2.7 Type III history, N > 1, subsequent periods payoffs 63 v Lis t of Figures 2.1 The game between manager and auditor 36 vi Acknowledgements The writing of a dissertation is a lengthy process that requires the effort and tests the patience of many other than the author. I thank: Jerry Feltham for taking a risk that many were unwilling to take, for the invaluable discussions, and the wonderful learning experience during my time at U B C ; Dan Simunic for his support throughout the program, and for those long conversations from which the idea for this thesis emerged; Peter Christensen for his insights and helpful discussions of the issues in Chapters 3 and 4, and particularly Chapter 3; Dorothee Feils for a first taste of business research, for patiently listening to half-baked research ideas at odd times, and for the constant support that simply made it all possible. Financial support from the Institute of Chartered Accountants of British Columbia Research Development Fund, the SSHRC through a doctoral fellowship, and the Win- spear Endowment is gratefully acknowledged. v i i Chapter 1 Introduction 1.1 The horizon problem Firms have long-term relationships with economic agents such as managers, auditors, and suppliers that can be characterized as sequences of shorter lived agents interacting with a longer lived firm. To stylize this to the extreme, the firm is infinitely lived, and the agents are finitely lived. In this framework, the agent's tenure with the firm becomes the object of investigation. For example, Hambrick and Fukutomi [15] propose a model of the dynamics of the CEO's tenure in office. Their analysis considers the CEO's performance in relation to the number of periods that have passed since the C E O started the current job. In this context, a natural question is to examine the role played by prior beliefs about the duration of tenure, where the duration of tenure is the total number of periods the agent will work for the firm. Thus, when the firm starts a multi-period relationship with an agent, what is the role played by beliefs about the duration of the relationship? If the relationship is governed by long-term contracts and if the length of the contract can be chosen ex-ante, and commitment to a certain tenure is possible, is there an optimal ex-ante tenure from the firm's owners' point of view? The same question arises when only short-term contracts are allowed. In this case, the commitment to a certain tenure may be implicit, as it appears to be in auditing. For example, if a manager is hired with a long-term contract, what is the importance of the length of the contract (as determined at contracting time, and assuming commitment 1 2 issues away)? In auditing, a similar question arises from the fact that auditors stay with the same firm for long periods of time. For auditors, beliefs about tenure are important for multi-period pricing under perfect competition which leads to lowballing. In principal-agent models, the agent's tenure is important because it impacts the amount and the characteristics of the information available for contracting. Specifically, if in each period the information system produces a piece of information that is used as a performance measure in contracting with the manager, the number of periods the manager is employed determines the informational environment for contracting. In all the cases discussed above, the agent's behavior and the principal's welfare are influenced by the (explicit or implicit) contracting horizon. A n auditor's pricing strategy depends on the horizon under consideration, resulting in more lowballing for longer horizons (expected tenure). The firm's auditing costs depend on the horizon, since keeping an auditor for a longer term spreads the switching costs over more periods. In a dynamic agency model with correlated performance measures, the agent's effort depends both on the horizon (tenure) and on how far the agent is from the first or the last period of his tenure. It follows that the principal's welfare also depends on the agent's horizon. This dissertation aims to address issues related to an agent's tenure in multi-period models, and is based on three essays. The first essay is on auditing and analyzes how beliefs about auditor tenure impact auditor independence and audit pricing. The second and third essays examine the question of optimal managerial tenure in a multi-period agency model, while at the same time investigating the value of commitment and how different assumptions about commitment impact the solution of the agency problem. One other common theme in the first and third essays is the lack of a "last period", so that, while the firm (the principal) has an infinite horizon and needs to consider a succession of auditors or managers, the agents (auditors or managers) have finite horizons. As a 3 result, an agent's last period is not the last period of the model, the firm continues to exist and operate, and a new agent is hired. The main contributions to the literature are: • A model of audit pricing with finite auditor tenure in which tenure beliefs determine auditor turnover in equilibrium. • A testable hypothesis regarding the existence of ex-ante tenure beliefs. • A better understanding of the multi-period L E N (Linear contracts, Exponential utility, Normal distributions) agency model with correlated periods. • A new commitment concept that extends the idea of renegotiation to a series of short-term contracts. • A new N-period model of ratcheting in a pure moral hazard context. • Results regarding the impact of inter-period correlation of (accounting-based) per- formance measures on the ex-ante optimality of a manager's tenure. The next three sections describe the main ideas and results of the three essays that are the main body of the thesis. Section 2 reviews Chapter 2 on audit pricing and auditor turnover in an infinite period horizon with finite auditor tenure. Section 3 presents Chapter 3 on commitment and ratcheting in a simple two-period world. Section 4 is devoted to Chapter 4, which deals mainly with the problem of optimal managerial tenure after generalizing some of the results from Chapter 3 to an N-period world. 1.2 Agency costs, audit pricing, and auditor turnover The impact of auditor tenure - the length of time a client retains a particular auditor - on the quality of audit services has long been the subject of both speculation and controversy. 4 For example, Mautz and Sharaf [28] suggest that the personal relationship that develops between an auditor and the client over time can be expected to adversely influence the auditor's vigilance. The same view was expressed by the U.S. Senate's Metcalf Committee [32] which concluded that a lengthy association with a client may lead an accounting firm to identify closely with the interests of the client management, thereby impairing auditor independence. These arguments have led critics of the profession to propose mandatory auditor rotation policies for publicly held companies, but such regulations have never been implemented in North America. The principal reason that auditor rotation has not been mandated is the presumption that such a policy would harm audit efficiency and be cost increasing to clients. Turning to research, to my knowledge, no formal analytical model of optimal auditor tenure has been developed and I do not develop such a model in Chapter 2. However, auditor tenure is an ingredient in multi-period pricing models where the traditional ap- proach, as in DeAngelo [5], is to assume an infinite time horizon. In the presence of auditor learning and/or client switching costs, the result is initial 'lowballing' of audit fees followed by an infinite number of periods in which incumbent auditors earn client- specific quasi-rents. Moreover, there is no auditor turnover because incumbent auditors price engagements at the amount necessary to deter auditor change by clients. However, infinite auditor tenure obviously does not exist in the real world. In the mid-1970's, average auditor tenure for audits of publicly held U.S. companies was about 18 years (Simunic [35]). More recently, average tenure appears to have decreased to about 10 years (O'Keefe, Simunic and Stein [31]). In order to introduce finite auditor tenure in analytical models, it is usual to assume that exogenous factors - such as the need for new financing, company growth, client financial distress, or auditor-client disagreement over a financial reporting issue (e.g. Dye [8]) - motivate voluntary auditor changes, whether initiated by the client or the auditor. 5 In Chapter 2, I develop a model of audit pricing and auditor change when investors, management, and the auditor all hold rational expectations that auditor tenure will be of finite length. I make no claim that this finite period is ex ante optimal in any sense, although - once specified - it is ex post optimal. Perfect competition among auditors bidding for an engagement is assumed. Lowballing and an implicit multi-period commitment are obtained, while the auditor is hired one period at a time. Consistent with intuition, the amount of lowballing is strictly increasing in the conjectured length of auditor tenure. Thus, the fee structure is a generalization of DeAngelo's infinite horizon model to a more realistic world with finite auditor tenure. I also examine the effects of the fee structure on incentives to replace the auditor, and auditor independence. I show that the quasi-rents earned by the auditor do not impair independence because management dismissal threats are not credible when auditors are identical and do not disagree on auditing issues as in Magee and Tseng [26]. Fees are such that management cannot compensate the auditor (except through an explicit bribe) for any additional risk incurred by the auditor from compromising independence during the auditor's expected tenure. The existence of the lowballing and the revision of beliefs about tenure by competing auditors ensure that management cannot threaten the auditor with replacement during the expected duration of the audit engagement. Thus, lowballing works to protect the auditor's independence through its impact on off the equilibrium path beliefs about tenure held by competing auditors. One can say that the other auditors punish the manager with higher fees if the lowball of the incumbent is not recovered. Conversely, the auditor is replaced at the end of the conjectured number of periods because retention would allow the auditor to earn economic rents from the engagement. While these rents do not result in a compromise of the auditor's independence, the total audit cost to the firm (audit fee) would incorporate the expected litigation costs faced 6 by the manager as if the auditor's independence were impaired. This is true because, once retained past the expected tenure, the auditor could only be replaced by one who would systematically compromise his independence. Thus, having rational expectations of finite auditor tenure can be sufficient to induce auditor change within the same class of auditors (e.g. the Big Five). No external event is necessary to trigger the auditor change. In deriving the pricing and auditor change model, I have assumed that tenure beliefs are exogenously given and common knowledge among the players. Maintaining the as- sumption that expected auditor tenure is an exogenous characteristic of the manager, it is sufficient to assume that the auditors' costs and fees are common knowledge. Indeed, once an auditor is hired, in the first period of his engagement, both competing auditors, and the investors can infer tenure beliefs from the observed amount of lowballing. In some sense, the manager can use lowballing to communicate his beliefs about tenure to the other players. Then, given a distribution of managers with different tenure beliefs, each manager can play an equilibrium based on his conjectured tenure N. Knowing that, auditors bidding for the engagement are more likely to be equally dispersed in their bids. If lowballing cannot be inferred because fees are not disclosed, it is more likely that tenure beliefs N are formed by competing auditors and by the investors independently of the manager (for example, conjectured tenure could be the mean observed auditor tenure for the industry). Then, if a manager tries to play an equilibrium based on a personal conjecture of tenure N' ^ N, he will at some point make an off-equilibrium move (from the point of view of the outsiders) which results in an increase of auditing costs. Therefore, given a distribution of managers with different tenure beliefs, and an outsiders' conjectured N, managers are more likely to try to be closer to N. Auditors that bid for the engagement would also base their bids on N. Empirically, one may expect to see more variance around the mean tenure in a cross-section of audit engagements 7 when audit fees are disclosed than when they are not disclosed, since with fee disclosure, managers have more discretion over TV. Another interesting question related to the formation of the initial tenure beliefs is that of the existence of ex-ante optimal tenure. My auditor change model assumes a flat residual agency cost, independent of the conjectured tenure N and, as a result, the total audit cost is decreasing in N. That would imply optimal auditor tenure N = oo. The empirical evidence of Johnson et al.[19] is consistent with the idea that very short and very long auditor tenure is suboptimal, leaving auditor tenure optimally undetermined in an interval of four to nine years in their paper. Their findings are consistent also with an U-shaped residual agency cost C(N) that depends only on N and is constant for each period of the auditor's tenure. Presumably, such an agency cost structure would result from a combination of increasing audit quality in the first periods and a significant decrease of audit quality (or perceived auditor independence) as the auditor's tenure increases. Assuming that C(N) increases sufficiently as N increases, it is possible to obtain an ex-ante optimal tenure that is neither 1 nor oo. However, in my model, once an equilibrium conjecture is agreed upon, the manager and the auditors are locked into it in all subsequent periods. Thus, there is no ex-ante optimal tenure, but common tenure beliefs, once reached, are ex-post optimal. More generally, all auditor changes are anticipated since they are based on the con- jectured N and convey no information to the market. This observation is consistent with the low explanatory power of empirical models of auditor change that relate auditor turnover to exogenous events. My model suggests that some auditor changes are endoge- nous, simply based on an expectation of auditor tenure, and are unrelated to exogenous events. Anticipated exogenous events that are usually associated with auditor changes are outside the scope of model in Chapter 2. For example, if a company anticipates a major 8 business transaction at a given point in time that requires a new auditor, any auditor engaged prior to that event will expect to be replaced exactly at that time. Such tenure expectations are consistent with the finite horizon model of Magee and Tseng [26] but not with my model in which tenure is an endogenous horizon, rather than an exogenous event date. As a result, the auditor change is no news to the market when triggered by an anticipated event. To conclude, the main empirical predictions of the model in Chapter 2 are that: 1. Lowballing is an increasing function of auditor tenure; this prediction provides a potential test between our endogenous horizon hypothesis and the exogenous event date hypothesis of Magee and Tseng. Recall that the model of Magee and Tseng predicts lowballing to be independent of tenure. 2. The variance of a cross-section of auditor tenure is higher in a market with fee disclosure than in a market without fee disclosure. The model is also consistent with the low explanatory power of empirical models of auditor change, and explains why some auditor changes are no news to the market. 1.3 Correlated Noise, Commitment, and Ratcheting The ratchet effect has been described in the economics literature in connection with centrally planned economies (see for example Litwack [25] and the references therein), and more generally in settings where the agent is privately informed. The book by Laffont and Tirole provides a detailed analysis and references [23]. Their description of the ratchet effect is as follows: " If [a regulated firm] produces at a low cost today, the regulator may infer that low costs are not hard to achieve and tomorrow offer a demanding incentive scheme. That is, the firm jeopardizes future rents by being efficient". The essence of 9 the ratchet effect with a privately informed agent is that the agent can obtain a rent in future periods by hiding his type in the current period. Weitzman [36] presents a multi-period model of the ratchet effect with moral hazard only, but in his model the ratcheting mechanism is exogenous. More recently, Milgrom and Roberts [29] and Indjejikian and Nanda [18] have shown that there is a ratchet effect in two period models with moral hazard but without adverse selection. In these models, the ratcheting is endogenous and driven by the lack of commitment by the principal regarding the use of available information. Milgrom and Roberts [29] define the ratchet effect as "the tendency for performance standards to increase after a period of good performance". Given that the principal will use today's outcome in writing tomorrow's contract creates for the manager a link between today's effort and tomorrow's standard of performance. The ratchet effect is always inefficient in the models with adverse selection since good types will mimic bad types and earn rents. In the pure moral hazard model, the ratchet effect is also inefficient with respect to the full commitment solution as shown by Indjejikian and Nanda [18]. Ratcheting results from the principal's ability to optimally adjust the agent's second- period incentive for the lower ex-post variance of the second performance measure by using the first-period performance measure when the principal cannot commit fully to a long-term contract. The assumption that the two periods are correlated is thus crucial, and differentiates this model of ratcheting in a pure moral hazard setting from other models of sequential action choice. Repeated moral hazard with independent periods is analyzed by Lambert [24], Rogerson [33], Holmstrom and Milgrom [17], and Fudenberg, Holmstrom and Milgrom [12]. Matsumura [27] presents an analysis of sequential action choice with correlated out- comes in a single period in which the agent observes a first outcome before selecting the second action. However, in Matsumura's model, there is no contracting after the 10 first outcome is observed, and this outcome is private agent information until the end of the period. As a consequence, second-period incentives are affected by the first-period performance, but there is no renegotiation. In Chapter 3, I extend the analysis of the Indjejikian and Nanda model [18] to include commitment issues and the possibility of agent turnover. Indjejikian and Nanda present two types of commitment: full commitment to a long-period contract (which they refer to as "commitment") and an intermediate form of commitment with a sequence of two short-term contracts (which they refer to as "lack of commitment"). I show that the two short-term contracts obtained by Indjejikian and Nanda correspond to a form of commit- ment which I call commitment to fairness. This form of commitment is an adaptation of the concept of fairness introduced by Baron and Besanko [1]. A contracting relationship is governed by fairness if the principal is restricted to fair wages and the agent must participate in all periods if he accepts the contract in the first period. Fair wages are paid when the agent gets his reservation wage as if he could leave in each period. That is, the agent's certainty equivalent of future compensation, conditional on available in- formation and on the principal's conjecture of the agent's first-period action is set to the reservation level at the start of each period.1 Thus, in addition to the usual contract acceptance constraint at the start of the first period, there is a second constraint that the second-period contract is acceptable to the agent as if the agent had other employment opportunities and had not committed to stay for both periods. The agent trades off his ability to leave in the second period for the guarantee of fair compensation in the second period. I also show that the two contracts under commitment to fairness and a long-term renegotiation-proof contract produce equivalent results. That is, the payoffs for the agent 1 Note that an equilibrium involves rational expectations regarding the agent's first-period action. Thus, second-period fair wages are based on the principal's conjecture of the agent's first-period action, which is correct in equilibrium. 11 and the principal, and the induced actions coincide for the renegotiation-proof contract and the sequence of contracts under commitment to fairness. The fairness constraint is not only sufficient for obtaining the solution of Indjejikian and Nanda, but also necessary. If the principal offers the agent the optimal sequence of contracts derived under commitment to fairness, and if the agent does not commit for both periods, the equilibrium breaks down. Restoring the agent's ability to leave in the second period allows the agent to act strategically in the first period, take an action other than that anticipated by the principal and then leave after the first-period compensation is paid. This situation parallels the "take-the-money-and-run" strategy that arises in ratcheting with adverse selection when the agent cannot commit to stay for both periods (see Laffont and Tirole [23]). Moreover, if the agent is able to leave in the second period, then there is no equilibrium with two short-term contracts in which the agent stays for both periods (see also Christensen and Feltham [2]). Thus, the fairness assumption helps to overcome the non-existence of an equilibrium problem. Removing the commitment to fairness assumption leads to a solution in which the principal optimally employs a different agent in each period. The only assumption neces- sary to obtain the two-agent solution is that the principal can commit to replace the first agent in the second period. At the time the principal replaces the agent in the second period, he is indifferent between retaining and replacing the agent. Thus, the principal's commitment to replace the agent is not a strong assumption. A n analysis of the principal's welfare under the different commitment assumptions reveals that for negatively correlated performance measures, full commitment is preferred to all other forms of commitment, or lack thereof. The situation is somewhat reversed with positively correlated performance measures in that, although full commitment is better than renegotiation and commitment to fairness, no commitment (two agents) is better than full commitment. The driving force behind this result is that with negative 12 correlation, having the same agent in both periods reduces the total risk to which the agent is exposed by the optimal incentive scheme. When the correlation is positive, using a different agent in each period eliminates the risk premium due to the correlation between the optimal compensation schemes for each action.2 1.4 Multiperiod Ratchet Effect and Managerial Tenure In Chapter 3, I extended the analysis of Indjejikian and Nanda [18] to include commit- ment issues and the possibility of agent turnover. The central theme of Chapter 3 is the analysis of different levels of commitment and a comparison of the principal's welfare un- der different commitment scenarios and different correlation of the performance measures. Regarding agent turnover, the main insight from Chapter 3 is that, in a two-period L E N model, the two agent solution (turnover or no commitment) is the least preferred when the performance measures are negatively correlated, with long-term full commitment and long-term commitment with renegotiation (or commitment to fairness) dominating. The result is the opposite for positive correlation of the performance measures, in that the two-agent solution (turnover or no commitment) dominates long-term full commitment. These results regarding turnover indicate that, in a two-period world, the principal prefers to replace the agent after one period when the performance measures are positively correlated. On the other hand, the principal prefers even the weakest form of long- term commitment with a single agent when the performance measures are negatively correlated. These results can be extended to more than two periods. In a setting with more than two periods, one can consider the choice of alternative tenures in a given time horizon. In a realistic setting, tenure is a choice variable in a world with more than two 2 T h e s i tuat ion is as if there is a compensat ion scheme for the first-period act ion c\ and a compen- sat ion scheme for the second-per iod act ion c V W i t h one agent, the r isk for wh ich the p r inc ipa l pays compensat ion is var (c i + £2) = var (c i ) + 2cov(ci,C2) + var (c2) . W i t h two agents,the r isk for wh ich the pr inc ipa l pays compensat ion is var (c i ) + var (c2) . 13 periods. In addition, when considering tenure (or replacement policies), the analysis can be framed in different ways. For example, regarding optimal tenure one may ask: 1. In a given finite number of periods, what is the optimal number of agents and their respective tenures? In other words, given a known finite life for the firm, how can one optimally partition that among several agents, and what are tenures of those agents? 2. In an infinite world, what is the optimal tenure for agents, if we assume a policy of replacing agents after the same number of periods (tenure)? The main difference between the two frameworks, and between the way questions are answered is that in one world there is a "last period", whereas in the other world, there is no "last period". The infinite period world permits the comparison of different tenure lengths in a natural way. A similar comparison of these two frameworks is undertaken in Chapter 2 regarding auditor tenure. In Chapter 4, I extend the two-period model of Chapter 3 in order to answer the second question posed above: what is the optimal (stationary) agent turnover policy in an infinite-period world? In Chapter 4 I develop an N-period model of the ratchet effect in a principal-agent problem with moral hazard but without adverse selection. Thus, while the agent's action is unobservable by the principal, in equilibrium the principal has rational beliefs regarding the agent's past actions and as a result, in equilibrium, information asymmetries do not develop over time between the principal and the agent. In addition, there is no learning of productivity or any agent characteristic that is unknown at the start. The only dynamic information effects are the adjustment of posterior beliefs about future performance measures, conditional on the sequential observation of past performance measures together with conjectures of the agent's past actions. The model generalizes the two-period model of Chapter 3, and most results derived therein remain valid in the 14 N-period case. Primarily, the conclusions regarding the role of commitment in obtaining different solutions remain the same. However, the N-period model gives insights into the importance of the contracting horizon, or tenure. In addition, the N-period model offers insights into the agent's long-term performance that cannot be inferred from the two-period model. While in Chapter 3 I have compared different commitment scenarios given a fixed two-period horizon, in Chapter 4 the emphasis is on tenure given a particular set of commitment assumptions. Since the very idea of tenure implies some form of implicit or explicit long-term commitment, the choice is between the three types of long-term com- mitment discussed in Chapter 3: full commitment to a long-term contract, commitment to a long-term contract with renegotiation, and commitment to fairness with short-term contracts. Full commitment to a long-term contract is too restrictive, in that, especially over longer horizons, renegotiation is more likely. Commitment to fairness is a mecha- nism that replicates the commitment to a long-term contract with renegotiation by using a sequence of short-term contracts. This makes the choice between the two forms of contracting almost a matter of taste. In Chapter 4, I choose to restrict the analysis to contracting under commitment to fairness. Besides capturing the idea of renegotiation, it only requires short-term contracts that are settled at the end of each period, allowing for a breakdown of performance and surplus period-by-period. In addition, the principal and the agent only need to commit to "fair contracts" and to a tenure duration, no other long-term commitments or contracts are necessary.3 Finally, the commitment to fairness solution to the dynamic agency problem provides consistency and ease of comparison to previous literature such as Milgrom and Roberts [29] and Indjejikian and Nanda [18]. 3 The reader who finds the concept of "commitment to fairness" unpalatable can interpret all results in the context of commitment to a long-term contract with renegotiation. 15 The driving force behind the "ratchet effect" with pure moral hazard described in Chapter 4 is the principal's inability to commit not to use past performance measures when setting a short-term contract with the agent in each period other than the agent's first period of tenure. The principal has an incentive to set the agent's compensation in a later period based on past performance measures in order to optimally adjust the risk in the agent's contract for the lower posterior variances. In doing so, the principal makes the fixed pay in future periods depend on past performance measures. For the agent, it means that from an ex ante (previous periods) perspective, future fixed pay depend on earlier effort choices. In other words, "fixed compensation" in one period is fixed only in that period, given the past actions and performance measures, but is variable when anticipated from earlier periods. It follows that the agent's incentives for effort in any one period are spread among the variable wage for that period and the fixed wages in all future periods. The nature of the solution to the dynamic agency problem is such that, for longer horizons, the manager's effort is close to some limit level for most periods. Thus, there are essentially two effort levels, the second best in the last period, and approximately a "third best" in most other periods. When correlation is positive, the "third best" is lower than the second best, generating inefficiencies relative to a repeated one-period problem (which is the multi-period problem when periods are independent). When the correlation is negative, the ratchet effect is efficient relative to the uncorrelated periods case since the "third best" effort level is closer to first best. To summarize, with positively correlated performance measures, incentives are stronger and the manager exerts less effort in most periods than in the last period. With negatively correlated performance measures, incentives are less strong and the manager exerts more effort in most periods than in the last period. The effort level in the last period serves as a benchmark because it coincides with the second-best solution, which is what one obtains with uncorrelated 16 performance measures. A commitment problem could arise in the case of negatively correlated performance measures due to the decreasing performance of the agent towards the end of his tenure. The principal needs to be able to commit not to fire the manager before term, otherwise the efficiency gains from getting effort levels close to first-best are lost. This would not be a problem in the positive correlation case since there the principal gets better performance towards the last period, and has no incentive to fire the manager before term. In either case, commitment to fairness assumes that the principal commits to retain the agent for all N periods. To answer the question of ex-ante choice of tenure by the principal, I examine op- timal stationary replacement policies, whereby the agents are hired for N periods at a time within an infinite horizon for the firm, and the noise in the performance mea- sures is firm-specific. The main result for positively correlated performance measures is that, in the presence of a switching cost, there exists a threshold switching cost such that optimal tenure is a single period whenever the switching cost is higher than the threshold value. On the other hand, optimal tenure is the maximum number of periods possible (the agent's maximum life) when the switching cost is lower than the threshold value. Thus, with positively correlated performance measures, the only optimal replace- ment (tenure) policies are the corner solutions of one period tenure or maximum possible tenure. The main result for negatively correlated performance measures is that the opti- mal replacement policy is always the maximum number of periods possible, irrespective of the switching cost. In the case when the noise in the performance measures is agent-specific, there is an additional "learning" effect in the first few periods of a manager's tenure due to the (rapid) reduction of the posterior variances of the performance measures towards their limit value. This effect results in an increase of managerial effort, since it becomes less 17 costly over time for the principal to motivate managerial effort due to lower risk premia. With negatively correlated performance measures and agent-specific noise, the manager exerts the second-best effort in the last period, a higher effort level in all other periods, and that effort has an inverted U shape. This finding is consistent with evidence from the management literature that firm performance increases at first, reaches a maximum, and then declines during a manager's tenure. For a discussion of these findings, see the papers by Eitzen and Yetman [9], Katz [21], and Hambrick and Fukutomi [15]. Chapter 2 Agency Costs, Audit Pricing, and Auditor Turnover 2.1 Introduction The impact of auditor tenure - the length of time a client retains a particular auditor - on the quality of audit services has long been the subject of both speculation and controversy. For example, Mautz and Sharaf [28] suggest that the personal relationship that develops between an auditor and the client over time can be expected to adversely influence the auditor's vigilance. The same view was expressed by the U.S. Senate's Metcalf Committee [32] which concluded that a lengthy association with a client may lead an accounting firm to identify closely with the interests of the client management, thereby impairing auditor independence. These arguments have led critics of the profession to propose mandatory auditor rotation policies for publicly held companies, but such regulations have never been implemented in North America. The principal reason that auditor rotation has not been mandated is the presumption that such a policy would harm audit efficiency and be cost increasing to clients. Turning to research, to our knowledge, no formal analytical model of optimal auditor tenure has been developed and we do not develop such a model in this paper. However, auditor tenure is an ingredient in multi-period pricing models where the traditional ap- proach, as in DeAngelo [5], is to assume an infinite time horizon. In the presence of auditor learning and/or client switching costs, the result is initial 'lowballing' of audit 18 19 fees followed by an infinite number of periods in which incumbent auditors earn client- specific quasi-rents. Moreover, there is no auditor turnover because incumbent auditors price engagements at the amount necessary to deter auditor change by clients. However, infinite auditor tenure obviously does not exist in the real world. In the mid-1970's, average auditor tenure for audits of publicly held U.S. companies was about 18 years (Simunic [35]). More recently, average tenure appears to have decreased to about 10 years (O'Keefe, Simunic and Stein [31]). In order to introduce finite auditor tenure in analytical models, it is usual to assume that exogenous factors - such as the need for new financing, company growth, client financial distress, or auditor-client disagreement over a financial reporting issue (e.g. Dye [8]) - motivate voluntary auditor changes, whether initiated by the client or the auditor. In this paper, we develop a model of audit pricing and auditor change when investors, management, and the auditor all hold rational expectations that auditor tenure will be of finite length. We make no claim that this finite period is ex ante optimal in any sense, although - once specified - it is ex post optimal. Perfect competition among auditors bid- ding for an engagement is assumed. Lowballing and an implicit multi-period commitment are obtained, while the auditor is hired one period at a time. Consistent with intuition, the amount of lowballing is strictly increasing in the conjectured length of auditor tenure. Thus, the fee structure is a generalization of DeAngelo's infinite horizon model to a more realistic world with finite auditor tenure. We also examine the effects of the fee structure on incentives to replace the auditor, and auditor independence. We show that the quasi- rents earned by the auditor do not impair independence because management dismissal threats are not credible when auditors are identical and do not disagree on auditing is- sues as in Magee and Tseng [26]. Fees are such that management cannot compensate the auditor (except through an explicit bribe) for any additional risk incurred by the auditor from compromising independence during the auditor's expected tenure. Conversely, the 20 auditor is replaced at the end of the conjectured number of periods because retention would allow the auditor to earn economic rents from the engagement. While these rents do not result in a compromise of the auditor's independence, the total audit cost to the firm (audit fee) would incorporate the expected litigation costs faced by the manager as if the auditor's independence were impaired. This is true because, once retained past the expected tenure, the auditor could only be replaced by one who would systematically compromise his independence. Thus, having rational expectations of finite auditor tenure can be sufficient to induce auditor change within the same class of auditors (e.g. the Big Five). No external event is necessary to trigger the auditor change. The remainder of the paper is organized as follows. In section 2 we examine relevant literature dealing with auditor tenure and auditor change. Section 3 presents the pricing model given exogenous agency costs and exogenous auditor turnover. Section 4 presents the auditor change model. Section 5 contains a discussion of empirical implications and concludes the paper. The appendices contain some of the proofs. 2.2 Related literature Following DeAngelo [5], the issues surrounding multi-period audit pricing - particularly the existence of 'lowballing' and its effects on auditor independence - have continued to interest researchers. Dye [8] introduced the notion of relative 'bargaining power' of the auditor vs. the client into the analysis and argued that DeAngelo's pricing model implicitly assumes that the auditor possesses all (or most) of the bargaining power in setting audit fees. This allows the auditor to earn quasi-rents. Dye argues that if the client possessed all of the bargaining power, then an auditor would always be constrained to perform audits for a fee equal to the auditor's costs, thereby eliminating quasi-rents and consequently lowballing. To resurrect the possibility of lowballing, Dye goes on to model 21 a situation where there may be information asymmetries between the auditor and the client about the client's financial statements being audited. In this setting, auditors earn quasi-rents (hence, lowballing occurs) only when quasi-rents are not disclosed to investors. This is so because the absence of disclosure prevents the (assumed) negative implications of quasi-rents on the auditor's independence from being impounded by investors in the firm's market value1. Moreover, an incumbent auditor may be replaced if management's information reflected in proposed financial statements is more favorable than the auditor's information about the client, and the auditor is unwilling to attest to management's more favorable information. Note that in this more complex (relative to DeAngelo's) model, an auditor will still be used for an indefinite number of periods unless an information asymmetry concerning the client's financial statements arises. Kanodia and Mukherji [20] develop a model of audit pricing, lowballing and auditor turnover under the assumption that the client has all of the bargaining power, but the auditor has better information about true audit costs. In their setting with information asymmetry about costs and a competitive market where clients make take-it-or-leave-it offers to purchase audit services, there will be some auditor turnover. However, in this model, auditor replacement also becomes less likely over time. More recently, Morgan and Stocken [30] take an intermediate view of bargaining power in which an incumbent auditor sets fees subject to competition from outside auditors. Their model provides for some probabilistic auditor replacement in an equilibrium with mixed strategies. Finally, Magee and Tseng [26] analyze the complex and controversial issue of the effect of quasi-rents on auditor independence. That is, under what conditions will the existence of quasi-rents - independent of the causes that allow quasi-rents to arise - impair an auditor's independence? They show that in a rational market, these conditions are 1 One common feature of bo th the DeAnge lo and Dye arguments is the imp l ic i t assumpt ion that quasi rents impa i r audi tor 's independence because managers can make credible threats to replace the audi tor . 22 quite limited, requiring - among other things - disagreement among auditors about the requirements of G A A P , a lack of auditor knowledge about his or her own position on a G A A P matter before it arises, and a lack of client knowledge about the incumbent auditor's position on a matter. In the absence of these conditions, the client simply employs the net benefit maximizing auditor and - as in DeAngelo - there is no motivation for auditor changes. It is also worth noting that while Magee and Tseng use a finite horizon pricing model, the horizon applies both to the life of the client firm as well as to the auditor's tenure. None of the existing models predict an auditor change simply because an auditor has been utilized a certain number of years by a client that will continue to exist in the future. Empirical work, on the other hand, has identified a variety of circumstances under which auditor changes occur. Williams [37] finds that the most significant variables explaining auditor changes are the auditor's industry specialization, the length of auditor tenure (long tenure increases the probability of replacement), and the client's desire to signal favorable information following adverse publicity. Haskins and Williams [16] find a series of variables associated with auditor changes within the Big 8 group: client financial distress, client size and growth rate, and the industry dominance of audit firms. DeFond [6] finds that auditor changes occur in anticipation of, and in response to changes in client agency costs as measured by decreases in management ownership of shares and increases in leverage. These agency cost factors are significant after controlling for changes in client firm growth and the issuance of new securities. In general, while existing empirical work provides insights into factors motivating auditor changes, the studies are scattered (there is little replication) and the statistical models are characterized by low explanatory power. However, the evidence (weakly) suggests that an auditor change is more likely as the length of tenure increases. Finally, there is limited empirical evidence linking auditor tenure with decreased audit 23 quality. Besides the negative effect of length of tenure on earnings response coefficients reported by Johnson et al [19], Krishnan and Krishnan [22] find that long auditor tenure is less likely to be associated with auditor resignation than with auditor dismissal by the client. Note that auditor dismissal could be motivated by weaker capital market reaction to positive unexpected earnings as auditor tenure increases. More directly, Deis and Giroux [7] and Copley and Doucet [4], using public sector data, find that the probability of receiving a substandard audit increases with the length of the auditor-client relationship. The model of audit pricing, auditor tenure, and auditor change developed in this paper is consistent with the empirical findings of a negative relation between length of tenure and audit quality, and that auditor change is more likely as length of tenure increases. Moreover, the model is also consistent with the hypothesis that changes in agency costs can lead to auditor changes. However, the changes in agency costs arise endogenously, rather than being associated with exogenous changes in management ownership or other factors. 2.3 Audit pricing with exogenous costs and exogenous turnover In this section we develop a model of audit pricing that extends DeAngelo [5] by allowing for finite tenure and a more general cost structure. We recover DeAngelo's results as a particular case (infinite tenure) and we show how lowballing depends on tenure when the auditor's tenure is finite. Our model differs from Magee and Tseng [26] in that we do not have a last period and the firm continues to exist after an auditor is replaced. Accordingly, the results on lowballing are different: in the Magee and Tseng model, with finite tenure, lowballing is independent of tenure, while in our model, lowballing is increasing with expected tenure. 24 Assumpt ions The time variable is discrete and the unit of measure is one time period, typically one year. The firms exist for infinitely many periods and engage auditors one period at a time. Auditing is mandatory. A firm can switch to a new auditor each period. An incumbent auditor can choose not to continue providing audit services to the firm. The value of a firm is determined by outside investors through trading on the securities market. The quality of audit for any firm impacts the value of that firm through the impact of auditing on agency costs. For the rest of the paper, we analyze audit pricing and auditor switches for a single firm, referred to as the firm. The manager of this firm will be referred to as the manager. The decision to hire an auditor is made by the manager. The manager has an ex- ogenously specified compensation package and is risk neutral. The auditor is also risk neutral. The manager's compensation is increasing in firm value, and the manager incurs no personal cost in replacing the auditor. The manager's choice of auditor affects his wealth only through the effect of an audit on the value of the firm. Consequently, the manager will minimize the total cost of auditing when choosing an auditor. The firm hires an auditor at the start of period 1. The manager conjectures what the auditor's tenure with the firm is going to be. The auditor and the investors also have a conjecture of what the auditor's tenure will be. In a rational expectations equilibrium, the man- ager's conjecture, the auditor's conjecture, and the investors' conjecture of auditor tenure coincide. We denote this equilibrium conjecture of auditor tenure by N. In addition to finite tenure, we also allow for N to be infinite. The impact of auditing on the value of the firm in any given period is represented by a residual agency cost function denoted by C. The cost C is defined as the difference between agency costs of the firm given the auditor it employs and agency costs in a world 25 with perfect auditing. The cost C can be equivalently characterized as the difference between the value of the firm in a world with perfect auditing and the value of the firm given the auditor it employs (and assuming the firm pays the same price for auditing in both worlds). The cost function C is an implied cost and characterizes the impact on the value of the firm of investors' perceptions of audit quality and of other agency cost variables such as leverage or management ownership of the firm. As a result, C will depend on determinants of audit quality such as brand name, reputation, size, and extent of consulting services. In addition to these variables, we assume that C also depends on two time variables: • t, the number of periods the auditor has been employed by the firm; • N, the conjectured auditor tenure. In what follows we assume that all auditors are identical and that the firm character- istics that determine agency costs do not change in time. As a result, we can concentrate only on the dependence of C on N and t. Thus, C is a function of N and t only, and we denote it by C(N,i). In general, the residual agency cost is endogenously determined. To simplify the analysis, we begin by assuming that the residual agency cost function is exogenously given, and we first solve the pricing problem. Once we know the structure of audit fees for any given cost function, we will endogenize the residual agency cost, given tenure beliefs and auditor independence considerations. In auditing the firm's financial statements, an auditor incurs costs A(t) in period t. The function A{t) is nonincreasing and is the same for all auditors. The function A(t) is also exogenously specified, and incorporates both production costs and expected litigation costs. Later, we will endogenize the expected litigation costs when we introduce auditor independence issues. As a result of perfect competition among auditors, the ex 26 ante net present value of the audit engagement to the auditor is zero.2 The audit fees are such that in each period the net present value of future audit fees less audit costs is nonnegative. The firm incurs a cost S whenever it switches to a new auditor. This cost is incurred in the first period of the audit engagement. The manager and the investors estimate that the impact of audit costs on firm value is the net present value of all future audit costs as incurred by the firm. The audit costs incurred by the firm are the sum of audit fees and agency costs. The expectations for future audit costs are based on repeating identical auditor engagements to infinity. The audit pr ic ing model The firm hires an auditor at the start of period 1. The firm pays the auditor a fee F(N, t) at the end of period t. For uniformity, all other costs such as A(t), 5, and C(N,t) are also considered as end of period costs. To abstract from all considerations other than audit pricing, we assume, in this section only, that the manager can commit to replace the auditor at the end of N periods. The switching cost S, the auditor's costs A(t), and the residual agency cost function C(N,t) are all exogenously specified. In the next section, we remove the assumption on commitment to turnover on the principal's part and we endogenize A(t),C(N,t) given auditor independence considerations. The auditing market is assumed to be perfectly competitive in the following sense. The incumbent auditor sets fees at the start of each period, other than the first period of the engagement, such that the audit cost to the firm, conditional on retaining the in- cumbent are less than or equal to the audit cost conditional on replacing the incumbent. The audit cost to the firm is the present value of all future audit fees plus the residual 2The zero NPV condition applies only at the beginning of the first period of the audit engagement and is based on the conjectured N and the implied fee schedule derived from the conjectured N. 27 agency cost. Retaining the incumbent means retaining for the remainder of his conjec- tured tenure and continuing to replace auditors every N periods thereafter. Replacing the incumbent is also assumed to be followed by continuing to replace auditors every N periods. In the first period, the fee is set such that the net present value of the entire audit engagement (present value of future audit fees less audit costs) is zero. For simplicity, we treat competitive pricing as an exogenous assumption in this sec- tion. Later, we show how competitive pricing arises in conjunction with the replacement of auditors and its impact on tenure beliefs. For now, setting aside the fact that an auditor switch may impact tenure beliefs, competitive pricing can be derived as follows. An incumbent auditor sets the highest fee such that he is retained. The firm will switch to a new auditor if the present value of all future auditing costs conditional on retaining the incumbent exceeds the present value of all future auditing costs conditional on hiring a new auditor. When these costs are equal, the firm retains the incumbent. An auditor switch does not change either the tenure beliefs under which a replacement is hired or the manager's commitment to switching auditors at the end of their conjectured tenure. In the first period, all auditors compete for the engagement, which drives the N P V to zero. We shall refer to this set of assumptions as commitment to tenure and competitive pricing. The following proposition gives the equilibrium audit fee structure for any exoge- nously specified tenure N and costs S, A(t), C(N, t) under the assumption of commitment to tenure and competitive pricing. Proposition 2.3.1 Let N be an exogenously given auditor tenure belief, and let A(t), C(N,t), S represent exogenously given auditing costs, residual agency costs, and switch- ing cost, respectively. Assume further commitment to tenure N and competitive pricing. Then, the following audit fee schedule obtains in equilibrium for the N periods of the 28 auditor's tenure: Q{N)\pS + Y£1P(A(t) + C(N,m-C(N,l)-Sift = l F(N,t) = { (2.1) Q(N)[PS + EL + C(N, t))] - C(N, t)ift>2 N where Q(N) = (J^/?*)"1 is an annuity factor. t=i Proof. In each period the total auditing costs to the firm are denoted by D(N, t), D(N,1) := F(N,l) + C(N,l) + S (2.2) D(N,t) := F(N,t) + C{N,t)ior2<t<N. (2.3) Let K(N) be the present value of all costs of hiring and retaining an auditor for N periods, N KW-^^pDWt), (2.4) t=i where 8 represents the discount factor.3 Let K^N) be the present value of all future auditing costs conditional on the firm hiring an auditor every N periods in perpetuity, and based on the assumption that the firm's agency costs determinants remain unchanged,4 oo K^N) := Y,?NK(N) = (1 - B^KiN). (2.5) In particular, if N = oo, oo K^oo) = K(oo) = fftD(oo, t). t=i The audit fees for the N periods are determined by perfect competition among au- ditors. Let R(N, t) denote the present value of all future auditing costs to the firm 3 A s usual, (3 = (1 + r ) - 1 . 4 Tha t is, the cost function C(N,t) remains the same in the future. 29 conditional on retaining the incumbent, discounted to time t, TV R(N, t): = Y / 9 i _ t + 1 - D ( / Y , i) + P^^K^N) i=t (2.6) = 8[D(N, t) + R(N, t + 1)] if t < N . For the firm, the present value of all future auditing costs conditional on hiring a new auditor is K^N). Thus, the incumbent sets the fees F(N, t) such that R(N, t) < Koo(N) for all 2 < t < N. It follows that R(N, t) = K^N) for t = N,..., 2 . (2.7) Equation (2.7) is equivalent to an entry-preventing condition, since it represents the highest fees for which the manager of the firm is indifferent between retaining and re- placing the incumbent. It follows that the present value of future audit costs for the firm is constant through periods 2 , . . . , N of the audit engagement. Furthermore, using equation (2.7) and backwards induction, D(N, t) = rK^N) for t — 2 , . . . , N. For t = 1, equations (2.4) and (2.5) determine D(N, 1) = rKoa(N). For a proof, see Appendix A . Thus, D(N,t) depends only on and not on t. Let D(N) denote the common value of D{N, t) for 1 < t < N. The next task is to determine the equilibrium value of F(N, t) as a function of N, and the variables S, A(t), and C(N,t). For the first period, perfect competition implies the zero N P V condition TV YPt(F(N,t)-A(t)) = 0. (2.8) t-i Once F(N, t) is known for 2 < t < N, the fee F(N, 1) is determined by solving equation (2.8). For clarity of exposition, we will express the auditor's fees and the objective function of the firm in terms of D(N). First, the equations that define D(N,t), (2.2) and (2.3), 30 imply that F{N,1) = D(N)-C(N,1)-S (2.9) F(N, t) = D(N) - C(N, i) for 2 < i < TV . (2.10) Substituting (2.9) and (2.10) into (2.8) gives an equation for D(N), 0(D(N) - A(l) -S- C{N, l)) + . . . + 0N(D{N) - A(N) - C(N, *)) = 0. (2.11) Rearranging terms on both sides of (2.11) gives TV TV TV D(N) = + £ pA(t) + Yl r*C{N, t), which implies t=i t=i t=\ TV TV D(N) = Q(N) /35 + ^ / 5 ^ ( t ) + ^ / 3 t C ( i V , i ) , (2.12) \ t=\ 4=1 TV where Q(N) = ^^/Pt j I s a n annuity factor.5 Once the values of S and N, and the function C(N,i) are known, D(N) is uniquely determined by (2.12), and the audit fees in each period are uniquely determined by (2.9) and (2.10).D The fee structure is such that, in each period, the auditor's fee and the residual agency costs add up to the same amount, which is the sum of the annual equivalent values of the switching cost, the auditor's costs, and the residual agency costs over the auditor's tenure.6 The perfect competition among auditors ensures that, in present value terms, the auditor's fees add up to the same amount as the auditor's costs over the duration of 5Q(N) specifies the amount per year for TV years that has a present value of one. 6 T h e annual equivalent value of a stream of cash flows c ( l ) , . . . , c(N) with discount factor (3 is the TV amount Q(N) /3*c(i) that, if substituted for c(t) in each period, would have the same N P V as the t=i original cash flows. For a zero net discount rate (/3 = 1), the annual equivalent value is the average cash flow over TV periods. See also Copeland and Weston [3], pp. 49-55. 31 the engagement. The firm's total cost is then the sum of the switching cost, the auditor's costs, and the residual agency costs. Let B(N) denote the sum of the annual equivalent value of the switching cost and the annual equivalent value of the auditing costs, B(N) := Q(N) (pS + . (2.13) To simplify the analysis that follows, we assume that C(N, t) does not depend on t for t = 1,..., N, that is the residual agency cost is the same in each period of the audit engagement. Let C(N) = C(N, t) for t = 1,..., iV denote this common value. Then, the annual equivalent value of the residual agency costs is C(N), and the firm's total cost of audit in each period becomes D(N) = B(N) + C(N). (2.14) The first term, B(N) is a decreasing function of N, since we assumed A(t) to be a nonincreasing function of t. For a proof, see Appendix B . The equilibrium fee structure is then deduced from the equilibrium value of D(N) according to (2.9) and (2.10), f D(N)-C(N)-S if * = 1 F(N,t)={ (2.15) [ D(N)-C{N) if t > 2 . From (2.14) it follows that f B(N)-S if * = 1 F{N,t)={ (2.16) [ B{N) if t > 2 . Lowballing occurs if F(N, 1) < A(l). The net amount of lowballing is determined by (2.9), LB = A(l) - F(N, 1) = A(l) + S- B(N) (2.17) 32 and is increasing in N since B(N) is decreasing in N. Furthermore, from equation (2.13) it follows that TV \ / T V LB = Q(N) Y,pt)w) + s)-hs+Y,p*Mt) yt=\ J \ 4=1 TV (2.18) = Q(N)Y/Pt(Ml)-A(t) + S) • 4=2 Thus, there is lowballing whenever ^4(1) > A(t) for at least one period or 5 > 0. Low- balling is driven by perfect competition and either by the auditor's learning (if we in- terpret lower costs of auditing in later periods as a result of learning) or by the firm's switching costs (or by both). In order to put the above results into perspective by comparing them to the pricing in the models of DeAngelo [5] and Magee and Tseng [26], we assume that all the auditor's learning costs are incurred in the first period: ( A + L if t = 1 A(t) = { (A if * > 1 . Given the simplifying assumptions for the audit costs, we can rewrite (2.13) as B(N) := A + Q{N)8(S + L). (2.19) The fee structure then becomes f A-r 0Q(N)(L + S)- S if t = 1 F(N,t)={ { A + 6Q{N){L + S) if t > 2 . In particular, for N = oo, 8Q(N) = 1 — 8 and the corresponding fee structure is the same as DeAngelo's, ' A + (1 - 8)L - 8S if t = 1 F(N, t) ' A-r (1 -0)(L + S) if t > 2 . 33 Lowballing occurs if F(N, 1) < A(l). The net amount of lowballing is determined by (2.9): LB = A(l) - F(N, 1)=A + L + S- B(N) . (2.20) Using (2.19), we can rewrite (2.20) as LB = (1 - Q{N)8) (S + L) . (2.21) It follows that there will be lowballing (that is positive LB) if either L > 0 or S > 0. Thus, lowballing is driven either by the auditor's learning costs or by the switching cost (or by both). In particular, in the absence of switching costs (5 = 0), the audit fee structure is flat (F(N,t) = B(N)) and there will be lowballing if the auditor has learning costs. In this case, there is lowballing without first period discounts. This shows that the implications of the model regarding lowballing generalize DeAngelo's [5] results to the case of finite tenure. Since LB is increasing as a function of N, there will be a one-to-one correspondence between the amount of lowballing and the conjectured N. We also note for future refer- ence that there is a one-to-one correspondence between the first period fee F(N, 1) and N. This result differs from the finite horizon pricing and lowballing of Magee and Tseng [26] in that lowballing depends on the time horizon. The difference in pricing between our model and theirs is due to the fact that we assume tenure beliefs of outside auditors to be independent of the number of periods remaining for the incumbent. Thus, our model is based on the assumption that tenure beliefs are a moving horizon, while Magee and Tseng's model is based on the assumption that the end of the auditor engagement is exogenously fixed and even if an incumbent auditor is replaced, the replacement auditor's 34 tenure does not extend beyond that fixed end date. With our notation, the fee structure of Magee and Tseng is F(N,t)={ A+(l- 3)L- pS if * = 1 A + (l-p)(L + S) tfl<t<N A + L + S if i = JV . (2.22) The outside auditors bid F — A + (1 — P)L — PS in every period except when there is only one period left until an auditor change occurs, in which case they bid F = A + L. The lowball is then LB — p(L + S) which corresponds to lowballing for infinite tenure in our model. 2.4 Auditor replacement in equilibrium In the preceding section, we have derived the audit fee structure given the manager's commitment to replacing the auditor after N periods, competitive pricing, and exoge- nously specified auditing and residual agency costs. We remove now the assumption on the manager's ability to commit to replace the auditor after a given number of periods and we endogenize the residual agency costs and part of the auditor's costs. We maintain all the other assumptions we have made and introduce accounting issues over which the manager and the auditor may disagree, giving rise to the possibility of compromise and impaired auditor independence. We also make detailed assumptions on how tenure beliefs are affected by the way in which the auditor is replaced/retained relative to the existing tenure beliefs. Tenure beliefs have to be consistent in equilibrium. However, we need to specify tenure be- liefs off the equilibrium path, that is how tenure beliefs are updated by an observed off-equilibrium retention/replacement decision. The rules by which tenure beliefs are 35 updated are exogenously specified.7 We now proceed to describe the game between manager and auditor in which the manager decides whether to retain or to replace the auditor, and the auditor submits a fee bid and may have to decide on compromising/not compromising on an accounting issue in each period. The manager and the incumbent auditor maximize utility, that is the manager min- imizes the audit cost for the firm, net of any possible personal benefit that may result from the auditor compromising on an accounting issue. The auditor maximizes his audit fees less audit costs, where the audit costs include any increase in expected litigation costs that may arise from the auditor compromising on accounting issues. The compet- ing auditors and the investors are not, properly speaking, players in the game since they do not maximize an objective function and their "strategies" are exogenously specified. However, the outside auditors submit competitive bids based on tenure beliefs, such that they earn a normal return on the audit engagement. Also, the investors are price pro- tected, in that they increase the residual agency cost whenever they believe that the auditor's independence is impaired. 2.4.1 The game In each period, the game between manager and auditor unfolds as follows (for a graphical description, see Figure 1). 1. At the beginning of the period, all auditors submit bids to the manager. We assume bidding to be costless. 2. The manager chooses an auditor. Given the bids and the beliefs regarding tenure, 7 Note that the use of the word "beliefs" is not in the usual sense. Tenure is not random, beliefs are not about a random variable, and there is no Bayesian updating involved. 36 | auditor] submits bid | manager"] accounting issue | auditor] compromise no com promise | manager] | manager"] retain replace retain replace Figure 2.1: The game between manager and auditor 37 the manager retains the incumbent if indifferent between future audit costs con- ditional on retaining the incumbent and future audit costs conditional on hiring a new auditor. The manager anticipates that the agency costs may increase if the incumbent is retained. In the first period, there is no incumbent and an auditor whose fees are consistent with the tenure beliefs is chosen.8 The fee that was bid by the auditor who was ultimately engaged, or the fee bid by the incumbent in case an incumbent auditor was retained, is not subject to renegotiation for the remainder of the period. If the auditor is not replaced later in the period and provides the audit service, the fee bid at the start of the period is the final fee for that period. 3. The investors anticipate whether there is a positive probability of a compromise by the incumbent auditor (see 5 and 7 below). The investors cannot observe at any point in the game whether the auditor has compromised his independence or not. If they believe that there is a positive probability that the auditor's independence is impaired, agency costs increase by A as a result of a drop in share price. The amount A is assumed to be independent of the probability of impaired independence and the nature of the accounting issue. We assume the investors believe that even the smallest compromise in auditor independence has very large consequences. When the investors believe that the auditor maintains his independence with probability one, they trust the auditor to be independent. When the investors believe that there is a positive probability of impaired independence, they do not trust the auditor anymore and lower the firm value by A , which is a penalty borne by the manager.9 As shown in 8 below, the 8 We do not model how a common conjecture of auditor tenure is reached, or how the manager chooses one. In the first period we simply assume that the manager has chosen one auditor consistent with some tenure beliefs and that all other auditors and the investors have the same auditor tenure beliefs. 9 The investors trust the auditor if, and only if, they believe that the auditor is independent with probability one. Equivalently, the investors trust the auditor if they believe the auditor's strategy in 38 penalty A represents the expected decrease in firm value, given that the investors know the distribution of accounting issues (see 5 and 7 below), but cannot observe the accounting issues themselves. As it turns out, whether the investors trust the auditor to be independent or not depends on tenure beliefs at that time and on whether the auditor has been retained beyond the conjectured tenure.10 4. If the auditor is replaced at this stage, he is replaced either for pricing reasons (that is bidding too high) or because of an anticipated increase in agency costs in case he is retained. In either case, the competing auditors do not revise their beliefs regarding tenure. 5. Nature randomly picks an accounting issue over which the manager and the auditor may or may not disagree. This move by Nature is unobservable by investors and by the other auditors and is represented by the realization of two random variables (e,5) (see 7, 8 below). The investors and the competing auditors know that each period there is a positive probability of a disagreement between the incumbent and the manager. In the first period of the engagement, the auditor incurs the learning cost and the firm incurs the switching cost before a disagreement can arise. Neither cost can be recovered if the auditor is replaced. Thus, at the next stage, when the manager and the auditor choose their actions, these cost are sunk. equ i l ib r ium is to reject any compromise. 1 0 T h e assumpt ions above amount to the investors par t i t ion ing their beliefs about the audi tor 's com- p rom ise /no compromise strategy into two groups: the audi tor does not compromise w i th p robab i l i t y one (audi tor bel ieved to be independent) and the audi tor compromises w i th posi t ive probabi l i ty (audi - tor bel ieved not to be independent) . Thus , the quest ion of investors hav ing ra t iona l beliefs about the audi tor 's strategy, and their response to those beliefs, is great ly s impl i f ied. For a more detai led s t ructure and i ts impl icat ions to the mode l , see A p p e n d i x D . 39 6. The manager chooses between asking for compromise and threatening dismissal versus doing nothing. 1 1 7. The incumbent accepts or rejects the proposed compromise. The auditor cannot report the accounting issue over which there is disagreement.12 If the auditor rejects the proposed compromise, there is no change in the expected litigation costs for the auditor. If the auditor agrees to compromise, he believes that there is an increase in litigation risk resulting in an expected litigation loss amount 8 > 0. Thus, the auditor's cost in a period in which he compromises on an accounting issue is increased by 8, the expected litigation cost given the particular accounting issue over which he compromised. We assume that 8 is the realization of a random variable 8 with support in {0}U[5m;„, 8max] and with a mass point at zero. The mass point at zero corresponds to the cases when no disagreement arises. An example of such a distribution would be a uniform distribution on an interval [8min,8max] with a mass point at zero. The auditor incurs all the learning costs in the first period, and these are denoted by L. Otherwise, the auditor's cost of producing the audit is the same in each period and denoted by A. Let jt denote a period t variable that is equal to one whenever the auditor compromises, conditional on an accounting issue disagree- ment being present, and equal to zero otherwise. Thus, jt describes the auditor's compromise/no compromise strategy. Then, the auditor's cost in each period is, 1 1 T h e manager does noth ing i f there is no disagreement. T h e manager cannot commi t to d ismissal if the audi tor does not compromise. 1 2 T h e idea here is that , if the audi tor accepts the proposed compromise, he issues an unqual i f ied op in ion . O n the other hand , if the audi tor rejects the proposed compromise, he does not issue a qual i f ied op in ion ; ins tead, he makes a take-i t-or- leave-i t offer to the manager consist ing of two opt ions: replace the audi tor or reta in the audi tor and agree to modi fy the f inanc ia l statements accord ing to the audi tor 's judgement. T h i s mechan ism also replaces the possib i l i ty of the audi tor resigning instead of issuing a qual i f ied report i n case the manager refuses to modi fy the f inancia l statements as required by the audi tor . 40 conditional on the auditor being retained until the end of the period, { A + L + ^iS if t = 1 A + 7tc5 if t > 2 . 8. The manager either replaces the incumbent or retains him. The manager benefits if the auditor is retained after having accepted the proposed compromise because the manager's compensation is higher under the accounting policy over which there is disagreement with the auditor. Specifically, let £ represent the expected litigation cost against the manager in case the auditor compromises, and let OJ represent the decrease in firm value if the auditor does not compromise and the manager changes the accounting policy accordingly. Note that, for consistency, we measure the litigation cost £ by the equivalent decrease in firm value that would have the same effect on the manager's compensation. We assume that, for each accounting issue, the decrease in firm value due to the auditor not compromising the issue is always larger than the expected litigation cost to the manager in case the auditor compromises.13 Thus, the manager's incremental cost between the compromise/no compromise cases is defined as e = Co — ( > 0. With this notation, A = E[u] represents the expected decline in firm value when the investors believe that the auditor has compromised, but cannot observe the accounting issue over which the auditor compromised. In terms of the residual agency cost borne by the manager, C(N, t, 7 t, 7 t ) = C + %A + jtC + (1 - it)u , where jt represents the investors' beliefs (see also point 3 above and Appendix D). Note that, when the auditor compromises, the manager bears the cost of litigation 1 3 Note that the auditor does not report the issue. No compromise by the auditor always results in a change of accounting policy by the manager, and thus in a modified financial report. 41 (,. On the other hand, if the auditor does not compromise, the manager bears the cost of changing the accounting policy Co. In both cases, the penalty A is imposed by the investors if they believe the auditor's independence has been compromised. Given that the investors do not observe the disagreement, the compromise, or the auditor's refusal to compromise, the investor's beliefs are in general independent of the auditors compromise/no compromise decision. The differential cost to the manager between the compromise/no compromise cases is —7*£ and can be thought as the manager's benefit from an auditor compromising. Like the auditor's expected litigation cost 5, e is the realization of a random variable e with the property that it has support in {0} U [emin,emax\. The distribution has a mass point at zero corresponding to the cases where there is no disagreement. A n example of such a distribution would be a uniform distribution on [£min,£max] with a mass point at zero. The manager does not replace the incumbent if the cost of audit increases as a result. If the manager is indifferent between replacing and retaining the incumbent, and he threatened to replace the auditor, he will carry through the dismissal threats. Note also that the two variables (e, 6) completely characterize the accounting issue over which the manager and the auditor may disagree through its impact on the payoffs of both players. In addition, we assume that S < E[e] < emin and that Smax < E[S] + L. 9. If the incumbent auditor is replaced at this point, the competing auditors revise their tenure beliefs in a way that is detailed below. If the incumbent is retained, the beliefs about tenure remain unchanged. 10. If the incumbent is retained, the audit service is performed, costs are incurred, and 42 fees are paid at the end of the period. Note that the game between manager and auditor is a game of symmetric information with observed actions. The realizations of the random variables associated with the ac- counting issue are known both by the manager and by the auditor. The investors and the outside auditors do not observe (e, S), but they do know the distribution of 5 and e. The key issue in the information asymmetry between investors on one hand, and the manager and the auditor on the other hand, is that there is a positive probability of an accounting issue arising in each period. The investors do not know whether a disagreement over an accounting issue has occurred or not. The distributions of accounting issues in different periods are mutually independent. Tenure beliefs The competing auditors' tenure beliefs determine their bidding strategy at the bidding stage in each period and whenever an incumbent auditor is replaced. The beliefs about tenure are specified as a function of the history of the game and are consistent on the equilibrium path. Tenure beliefs are central to the model, since they determine the cost of audit in the event an incumbent auditor is replaced, and thus determine whether the manager's threats of replacing the auditor are credible or not. The next task is to describe how tenure beliefs are revised when an auditor is replaced or retained in a way that is not consistent with existing tenure beliefs at the time. There are two possibilities: either the incumbent auditor is replaced before the expected last period of his engagement, or the incumbent auditor is retained beyond the expected last period of the engagement. The manager and an incumbent auditor have beliefs about the auditor's tenure that in order to be rational must be consistent on the equilibrium path. In addition, the manager and the auditor have beliefs about play in the event of a deviation from the 43 equilibrium path. Since all auditors are identical, a replacement auditor is assumed to follow the same equilibrium strategies given the tenure beliefs at that point in the game. Thus, the credibility of dismissal threats depends on current tenure beliefs and tenure beliefs conditional on the (off-equilibrium) replacement. When tenure beliefs are finite, TV < oo, the off-equilibrium moves are a replacement before TV periods or retaining an incumbent beyond TV periods. When an incumbent auditor is replaced before the expected tenure TV, the new tenure beliefs should be TV' < TV. For simplicity, we assume that TV' = 1, that is replacing an auditor before term makes multi-period engagements not credible, and replacement auditors are only willing to believe single-period tenure possible. When an auditor is replaced before the end of his tenure, there is a loss of quasi-rents, which means that the auditor could not recover his lowball. Potential replacement auditors recognize the possibility that they will be faced with a similar loss, and so they cease to lowball, which is equivalent to bidding based on tenure TV = 1. When an incumbent auditor is retained beyond the conjectured tenure TV, we assume that replacement auditors believe finite tenure is not credible anymore, and their tenure beliefs change to TV = oo. The key is that there are no equilibria consistent with a finite tenure TV' > TV after an incumbent is retained beyond the expected tenure. The intuition is that if auditors believe it is possible to be retained longer, then an incumbent would be able to deter entry in perpetuity at the end of their tenure (as long as that initial tenure belief is TV' < oo). When tenure beliefs are TV = oo, the only off-equilibrium move is replacing the incumbent auditor. There are two possibilities: contingent on an auditor replacement, tenure beliefs become finite, TV' < oo, or tenure beliefs remain unchanged, TV = oo. In the first case, we say that finite tenure is credible, and we assume that TV' = 1 as in the case of replacement before term with finite tenure. The second case corresponds to a situation 44 in which finite tenure beliefs are no longer possible as a result of an incumbent being retained beyond term. In this case we assume that any subsequent auditor replacement will leave tenure beliefs unchanged at N = co (since the alternative is finite tenure, which we assume no longer credible). This assumption is implicitly present in the models of DeAngelo [5] and Zhang [38], where tenure is assumed infinite, and that a replacement auditor also bids based on infinite tenure. To summarize, when we say that tenure beliefs are 1 < N < oo, that means both that replacement is believed to take place at the end of N periods (finite tenure credible) and that replacement does not take place before the N periods (multi-period tenure credible). Beliefs of N = 1 require credible finite tenure, and replacement before the end of the period does not affect beliefs. Beliefs of N = oo can occur, however, both when finite tenure is credible and when it is not. In the former case, replacement of the incumbent changes tenure beliefs to = 1, while in the latter case, tenure beliefs remain unchanged. The above discussion of tenure beliefs is formalized by the following definitions and assumptions. Tenure beliefs are a pair (N,A/B), where N is the number of periods an auditor is conjectured to be retained by the firm, and A/B are belief types as defined below. Tenure beliefs are said to be of type A if tenure is believed to be infinite and this belief is not affected by a replacement of the incumbent. Type A beliefs are also referred to as "finite tenure not credible". Tenure beliefs are said to be of type B if tenure is believed to be finite, or if tenure is believed to be infinite but this belief is affected by a replacement ofthe incumbent. Type B tenure beliefs are also referred to as "finite tenure credible". Tenure beliefs (oo, A) remain unchanged whenever an incumbent is replaced. Tenure beliefs (N,B) change to (l,B) whenever an incumbent is replaced before the end of periods, and this holds for both finite and infinite N. Tenure beliefs (N, B) with finite N change to (oo, A) whenever an incumbent is retained beyond the end of period N. Tenure 45 beliefs (N,B) remain unchanged as long as retain/replace decisions are consistent with N (auditor never replaced if TV = oo, or auditor replaced every N periods otherwise). Based on the assumptions we made regarding tenure beliefs and their revision, all possible histories in the game can be described as one of three types. A history here simply means all publicly available information at any given stage in the game. Thus a history consists of audit fees, and retain/replace decisions by the manager at all stages preceding the current one. Tenure beliefs can be inferred from past history and the above assumptions regarding their revision. Type I. At some point in the history of the game, an incumbent auditor has been retained beyond N periods when beliefs were (N, B) with finite N. As a result, beliefs are (oo,^4), and either a new auditor is being hired, or there is an incumbent auditor such that beliefs were (oo, A) at the time he was hired. Type I I . There is an incumbent auditor that was hired when beliefs were (N, B) with finite N, and who was retained beyond period N. As a result tenure beliefs are (oo,i4). Type I I I . No auditor hired under tenure beliefs (AT, B) has been retained beyond term. As a result, tenure beliefs are (N,B). We assume that, at the start of the game, beliefs can only be of Type B. It follows that the game history starts out as Type III and can either remain Type III, or it can change to Type II and remain Type II, or it can go through Type II to Type I. Once the game history becomes Type I, it remains as such for ever. On the equilibrium path, only Type III histories can occur, Type II and Type I are results of off-equilibrium behavior. A strategy profile specifies actions at each possible history. We shall define separately strategy profiles for each history type. If the game were to be started at either type of history, we show that the given strategies are subgame perfect. An equilibrium consists of a set of subgame-perfect strategies that are consistent with 46 tenure beliefs at each stage of the game. The manager and an incumbent auditor believe that a replacement auditor will play the same equilibrium strategy (given tenure beliefs in that event).14 Since off-equilibrium path moves can only change the history types from III—>II—>I, and these changes are irreversible, we characterize equilibrium strategies starting with Type I histories and then moving backwards to Types II and III. The following proposi- tion characterizes equilibrium strategies when the game has reached a Type I history. Proposition 2.4.1 Assume that the players are at a Type I history in the game. Then the following strategy profile is subgame perfect. • auditor bids { Fi — S in the first period Fj in subsequent periods Fj = A + E[5] + (l-0)(L + S) • auditor compromises on all accounting issues that arise • manager retains the auditor in the event of a disagreement if, and only if, the auditor compromises • the total cost of audit to the manager is Kj = Y^(A + m + A + £[CD + 0(L + S) . Proof. See Appendix C. • Note that the auditor's cost in each period is { A+S+LHt=l A + S\it>2 1 4 Some of the off-equilibrium actions are specified for a replacement auditor, and it is important that the manager and the incumbent auditor have common beliefs about the strategy of a new auditor in the event of auditor turnover. 47 because the auditor's strategy is to compromise on all accounting issues that the manager asks him to. Since the pricing stage precedes the disagreement stage, the realization of 5 for the current period is not known to the auditor at the bidding stage. The fact that the auditor is risk neutral and 5 is independent from period to period implies that the auditor perceives his costs when setting fees as { A + E[c51 + L if t = 1 A + E[5] if t > 2 . The principal's residual agency cost is C(N,t,£\jt = 1) = C + A + C in each period, since the auditor compromises on each accounting issue that arises, and an accounting issue arises with positive probability in each period, resulting in the investors imposing the agency penalty A and the manager bearing the litigation cost £. This cost to the manager is from an ex-ante perspective C(N,t\jt = 1) = C + A + E[(] (the Cs are also independent through time). Without loss of generality, we have normalized the residual agency cost such that C = 0, so C(N, t) = A + E[£]. Given A(t) and C(N, t) as described above, and tenure beliefs (co, A) , the audit fee structure follows from Proposition 2.3.1. The strategies presented in Proposition 2.4.1 are a subgame perfect equilibrium if the game were to reach a Type I history and will be referred to as Type I equilibrium strategies. We do not claim this to be a unique equilibrium in this case. However, there is no pure strategy equilibrium in which the auditor maintains his independence in each period. As a result, it is likely that if other equilibria exist, there will be a positive probability of the auditor compromising his independence in each period. 1 5 In that case, the investors will impose the penalty A , and the expected litigation cost E[£] 1 5 T h e penalty A is imposed in every period. This would not occur if the equilibrium is such that the auditor never compromises, but such an equilibrium does not exist. Suppose, by contradiction, that it does exist. Then, the cost of a replacement auditor that does not compromise is Kj = (5A(\ — /?) + P(L + S). In the first period, the dismissal threat is not credible since the switching cost is sunk when dismissal threats are made, and this increases the manager's end of period payoff by S if the incumbent is retained (in fact the incumbent's fee is F — S, while a replacement bids F to which the switching 48 will make such an equilibrium a very costly one. Our result is consistent with Zhang's modelling [38] of the fact that quasi-rents impair auditor independence under Type A tenure beliefs. The key in our model is that dismissal threats become credible once the auditor has lowballed and the competing auditors do not revise their tenure beliefs when they see an incumbent replaced (before term). We now turn to a Type II history, that is we assume that there is an incumbent auditor who has been retained longer than expected (and was engaged under tenure beliefs (N, B), and tenure beliefs are now (oo, A). The following proposition characterizes subgame perfect strategies in this case. Proposition 2.4.2 Assume that the players are at a Type II history in the game and that the incumbent auditor's fee bid in the first period in which he was retained beyond his expected tenure is F ' . 1 6 Then the following strategy profile is subgame perfect. • auditor bids F = Fu in each period other than the first in which he is retained beyond his expected tenure where F u = Fj + E[(] • auditor makes no compromise on any accounting issues that arise • manager retains the auditor in each period whether the auditor compromises or not on accounting issues cost must be added). However, once the switching cost is not subtracted from the incumbent's fee, the manager becomes indifferent between retaining the incumbent and replacing him after making the dismissal threats, in which case we have assumed that threats are credible. It follows that the penalty A is imposed in each period, since investors rationally believe that there is a positive probability of compromise in each period. 1 6There is no unique F' determined in equilibrium. Moreover, a Type II history is only reached after an incumbent submitted the bid F' and has been retained beyond the conjectured last period. Thus, for a Type II history, the fee F' is given as part of the information that characterizes the history of the game. On the equilibrium path, the auditor is never retained beyond his expected tenure. The strategies described in Proposition 2.4.2 characterize the off-equilibrium play for any possible bid F' off the equilibrium path. 49 The cost of audit to the manager is KJJ = Kj — 0 in all periods other than the first period in which the incumbent is retained beyond his expected tenure, in which case the cost of audit is B{F' + KN). Proof . See Appendix C. • Note that the auditor's cost is A(t) = A in each period and that the residual agency cost is C(N,t) = C + A , since the auditor is an incumbent and does not compromise on any accounting issue. The pricing is derived along the lines of Proposition 2.3.1, although the proposition itself is not applicable here for determining audit fees due to the fact that perfect competition does not reduce the N P V of the engagement to zero for the incumbent in the first period. If the manager has deviated from the equilibrium path by retaining the incumbent longer than expected, he will then optimally retain that incumbent in perpetuity. The incumbent in turn maintains his independence, but can extract very high rents because the replacement auditors would be playing a Type I equilibrium where they always com- promise, the agency penalty A is always imposed, and the manager bears the litigation cost E[Q. As a result, the equilibria that are played at Type I and Type II histories are both highly undesirable to the manager, since in both cases the total audit cost incor- porates E[C], either as an expected litigation cost, or through the auditor's fees. In both cases the cost is as if the auditor's independence was impaired. The following proposition characterizes the equilibrium strategies at game histories of Type III given tenure beliefs (N,B). Propos i t ion 2.4.3 Assume that the players are at a Type III history in the game, and that tenure beliefs are N. Then the following strategy profile is subgame perfect. 50 • auditor bids { FJJI — S in the first period Fin in subsequent periods Fm = A + PQ(N)(L + S) • auditor makes no compromise on any accounting issues that arise • manager retains the auditor in each period whether the auditor compromises or not on accounting issues The cost of audit to the manager is Kin = K^N) = YTpFm = + A + 8Q{N)(L + S)) . Proof . See Appendix C. • Note that the auditor's cost is A + L in the first period of the engagement, and A thereafter, while the residual agency cost is constant C(N, t) = C + A'm all periods. The equilibrium strategies presented in the proposition allow for infinite tenure if beliefs are (00, B). Since Q(N) is increasing in N, the total audit cost Km is decreasing in N, and the lowest value is attained when tenure beliefs are (00, B). The highest auditing costs are attained for tenure beliefs (1 ,5) . 1 7 The three propositions together present a complete strategy profile that is subgame perfect in the game starting at any history. We are interested in the game starting at a Type III history with finite tenure beliefs and finite tenure credible. Then, the auditor is not replaced before term because that would increase the audit cost (tenure beliefs (1,B) produce the costliest audit under (N,B) beliefs). The auditor is not retained beyond term because the game would switch to a Type II equilibrium and that has a higher cost 1 7 T h i s shows, in particular, that the tenure beliefs revision rule can be modified without altering the results significantly as follows: if an auditor hired under tenure beliefs (N, B) is replaced after N' < N periods, the new tenure beliefs are N'. Such a rule would be consistent in equilibrium, since it would keep the manager's threats of dismissal from being credible, thus preserving auditor independence. 51 to the manager than the highest cost for a Type III equilibrium (beliefs (1,73)). As a result, on the equilibrium path, if an auditor is engaged under tenure beliefs (N, B) with finite N, the firm will maintain a stationary cycle of replacing auditors every N periods. The auditors maintain their independence and lowballing depends on the conjectured tenure N. 2.5 Empirical implications and conclusion In this paper we develop a model of auditor change driven by agency costs and rational expectations of auditor tenure. When a new auditor is engaged, investors, management, and the auditors reach a rational expectations equilibrium belief of auditor tenure. Perfect competition among identical auditors ensures that the auditors earn only normal returns on the audit engagements (zero N P V over the entire expected duration of the audit engagement). Lowballing and a multi-period commitment are obtained while the auditor is hired one period at a time. The amount of lowballing is strictly increasing in the conjectured length of auditor tenure. The existence of the lowballing and the revision of beliefs about tenure by competing auditors ensure that management cannot threaten the auditor with replacement during the expected duration of the audit engagement. Thus, lowballing works to protect the auditor's independence through its impact on off the equilibrium path beliefs about tenure held by competing auditors. We can say that the other auditors punish the manager with higher fees if the lowball of the incumbent is not recovered. The auditor is replaced at the end of the conjectured number of periods (tenure) because of increased agency costs and litigation costs to the manager. Retaining an incumbent auditor beyond the expected tenure would allow the auditor to earn economic rents from the engagement. While these rents do not result in a compromise of the auditor's independence, the total audit cost 52 to the firm (audit fee) would incorporate the litigation costs that would be faced by the manager if the auditor's independence were compromised. This is true because, once retained past the expected tenure, the auditor could only be replaced by one who would systematically compromise his independence. In deriving the pricing and auditor change model, we have assumed that tenure be- liefs are exogenously given and common knowledge among the players. Maintaining the assumption that expected auditor tenure is an exogenous characteristic of the manager, it is sufficient to assume that the auditors' costs and fees are common knowledge. Indeed, once an auditor is hired, in the first period of his engagement, both competing auditors, and the investors can infer tenure beliefs from the observed amount of lowballing. In some sense, the manager can use lowballing to communicate his beliefs about tenure to the other players. Then, given a distribution of managers with different tenure beliefs, each manager can play an equilibrium based on his conjectured N. Knowing that, auditors bidding for the engagement are more likely to be equally dispersed in their bids. If lowballing cannot be inferred because fees are not disclosed, it is more likely that tenure beliefs N are formed by competing auditors and by the investors independently of the manager (for example, conjectured tenure could be the mean observed auditor tenure for the industry). Then, if a manager tries to play an equilibrium based on a personal conjecture of tenure N' ^ N, he will at some point make an off-equilibrium move (from the point of view of the outsiders) which results in an increase of auditing costs. Therefore, given a distribution of managers with different tenure beliefs, and an outsiders' conjectured N, managers are more likely to try to be closer to N. Auditors that bid for the engagement would also base their bids on N. Empirically, we may expect to see more variance around the mean tenure in a cross-section of audit engagements when audit fees are disclosed than when they are not disclosed, since with fee disclosure, managers have more discretion over N. 53 Another interesting question related to the formation of the initial tenure beliefs is that of the existence of ex-ante optimal tenure. Our auditor change model assumes a flat residual agency cost, independent of the conjectured tenure N and, as a result, the total audit cost is decreasing in N. That would imply optimal auditor tenure N = oo. The empirical evidence of Johnson et al.[19] is consistent with the idea that very short and very long auditor tenure is suboptimal, leaving auditor tenure optimally undetermined in an interval of four to nine years in their paper. Their findings are consistent also with an U-shaped residual agency cost C(N) that depends only on N and is constant for each period of the auditor's tenure. Presumably, such an agency cost structure would result from a combination of increasing audit quality in the first periods and a significant decrease of audit quality (or perceived auditor independence) as the auditor's tenure increases. Assuming that C(N) increases sufficiently as N increases, it is possible to obtain an ex-ante optimal tenure that is neither 1 nor oo. However, in our model, once an equilibrium conjecture is agreed upon, the manager and the auditors are locked into it in all subsequent periods. Thus, we have no ex-ante optimal tenure, but common tenure beliefs, once reached, are ex-post optimal. More generally, in our model, all auditor changes are anticipated since they are based on the conjectured N and convey no information to the market. The low explanatory power of empirical models of auditor change that relate auditor turnover to exogenous events leaves room for alternate theories of auditor turnover. Our model offers the al- ternative that some auditor changes are endogenous, simply based on an expectation of auditor tenure, and are unrelated to exogenous events. Anticipated exogenous events that are usually associated with auditor changes are outside the scope of our model. For example, if a company anticipates a major business transaction at a given point in time that requires a new auditor, any auditor engaged prior to that event will expect to be replaced exactly at that time. Such tenure expectations 54 are consistent with the finite horizon model of Magee and Tseng [26] but not with our model in which tenure is an endogenous horizon, rather than an exogenous event date. As a result, the auditor change is no news to the market when triggered by an anticipated event. To conclude, the main empirical predictions of our model are that: 1. Lowballing is an increasing function of auditor tenure; this prediction provides a potential test between our endogenous horizon hypothesis and the exogenous event date hypothesis of Magee and Tseng. Recall that the model of Magee and Tseng predicts lowballing to be independent of tenure. 2. The variance of a cross-section of auditor tenure is higher in a market with fee disclosure than in a market without fee disclosure. The endogenous auditor changes described in our model might explain some of the nu- merous changes which empirical research reveals are unrelated to exogenous events. Fur- thermore, these endogenous changes might explain why, empirically, auditor changes generally convey little news to the market. A p p e n d i x A The entry-preventing condition (2.7) determines D(N,t) as follows. For t = N, 6D{N,N) + dK^N) = K^N) PD{N,N) ^-K^N) = rK^N). D{N, N) 55 For t = N - 1, 0D(N,N-1) + 02D{N,N) + 02KOO(N) = KOO(N)^ 0D(N,N-1) = ( l - l-^-02 - /3 2 ) Koo( iV) = ( l - / 3 + / 3 2 - ^ 2 ) / v o o ( / V ) = ( l - / ? ) / , 0 O ( / V ) ^ For t = N - 2, 0D(N,N-2) + / 3 2 D ( ^ , ^ - l ) + / 3 3 / J ( / V , / V ) + / 3 3 A ' 0 0 ( / V ) = K^N) 0D(N,N-2) = (^-l^02-l-^-0z-0^jKoo{N) = (l - 0 + 02 - 02 + 0* - $*) K^N) = (1-/3) # « , ( # ) = * • D(N,N-2) = ^-K00(N) = rK00(N). D(N,N-1) Similar calculations for i = AT — 3 , . . . , 2 give D{N, t) = K^Ny for 2 < i < / V . Let D(N) denote this common value, D(N) := Koo(N)r. (2.23) From (2.4) it follows that K(N) 0D(N,1) + ^oo(VV) 0 ' 1 - /5 / 3 D ( / V , l ) + ( ^ - / 3 i V ) / x 0 0 ( / V ) . (2.24) 56 Using (2.5) into (2.24) yields (1 - d^K^N) = 3D{N, 1) + (0 - 0N)Koo(N). (2.25) Equation (2.25) can be now solved for D(N, 1), D(N, 1) = ^-J-K^N) = rK^N) = D(N). A p p e n d i x B Given two continuously differentiable functions / , g, defined on [0, co) let fXf(t)g(t) dt Tgf(x) := ^ - p . (2.26) / g(t)dt Jo Given any discrete stream of cash flows c(t), t > 1, there exist / , g as above such that f(t) = c(t), g(t) = 0l for all integers t > 1, and y ( i v ) = | ^ ' ) ( | ^ c ( i ) ) ' ( 2 ' 2 7 ) The operator Tg corresponds to what is known in capital budgeting as annual equiva- lent value of a stream of cash flows discounted over a finite period of time. Tg is a linear positive operator if the function g is positive. In addition, if / is decreasing over the interval [0, XQ], the function Tgf is also decreasing over the same interval. Conditions on / that guarantee the convexity of Tgf are trickier. f(x)g(x) f g{t)dt-g{x) f f(t)g(t)dt MM = - A ~ 7 i r t dt 'o / ait) g(x) (Jjf(x) - f(t)]g(t) dt 2 (2.28) 57 Thus, given a positive function g(x), Tgf is an increasing function of x for all x such that Appendix C The proofs for the three propositions are based on the one-stage-deviation principle. A strategy profile is subgame perfect if neither player has an incentive to deviate at the current stage and then return to playing the strategy (Fudenberg and Tirole [14], Theorem 4.1, Theorem 4.2). Proof of Proposition 2.4.1 The proof shows that for the given strategies, no player has an incentive to deviate at the current stage and then return to playing his strategy. The table below indicates the continuation payoffs to the manager and the auditor in the first period of an audit engagement when at the start of the engagement tenure beliefs are (oo, .4), conditional on disagreement over an accounting issue (e, <5).18 The fee structure is given by Proposi- tion 2.3.1 with 1 8Recall that in the absence of a disagreement, i.e. e = S = 0, the manager cannot replace the auditor after the pricing stage. and decreasing for all x such that A + E[5] if t > 2 and C(N,t) = A + E[C]. 58 Type I, first period Retain Replace Compromise Manager 0{e - E[e] + S) - Kj -Kj Auditor 0{E{5] -S + L) 0 No Compromise Manager -0E[e] + 8S-K1 Auditor 0E[5] + 0L 0 Table 2.1: Type I history, first-period payoffs The continuation payoffs in the table are derived as follows. If the auditor is replaced, his continuation payoff is zero, given the perfect competition and zero N P V assumption on a new engagement. If the manager replaces the auditor, his cost is the total cost of all future engagements with an auditor that compromises in every period, that is residual agency cost C(oo, r) = E[Q + A . This cost is given by Kj below based on Proposition 2.3.1. If the auditor is retained, his continuation payoff in case he compromised consists of the present value of all future audit fees less audit costs. The audit fees are A + E[6] + (1 — 8)L — 0S in the first period, and A + E[5] + (1 — 0)(L + S) in subsequent periods, while the costs are A + S in the first period (since at this time the issue is known and the learning cost is sunk), and A + E[5] in subsequent periods. If the auditor is retained and did not compromise, his payoff is greater by exactly 05, the avoided increase in expected litigation costs. Similarly, for the manager, if the auditor is retained, the same argument works with S instead of L and e instead of 5. The payoffs to the manager are the present value of all future audit costs. These are 0(A + C) + A + E[5] + (1 — 8)L — 8S in the first period, and 0(A. + E[C] + A + E[5]•+ (1 — 0)(L + S) in subsequent periods if the auditor compromises, 59 and lower by Be if the auditor does not compromise. In all these cases, compromise/no compromise refers only to the current period, in all subsequent periods the auditor is assumed to return to the equilibrium strategy, which is to compromise in each period. Intuitively, the key to this payoffs is in the differences between the retain/replace columns. On the first line, by retaining the incumbent, the manager gains by avoiding an additional switching cost, the benefit from the current auditor compromising on a specific issue less the expected benefit of the auditor compromising which works like an opportunity cost. On the second line, by being retained, the auditor gains from the fee compensating for the learning cost (which is sunk and not counted at this stage), from passing the expected litigation cost to the manager, less the litigation cost associated to the particular accounting issue. Assuming that S < E[e], the threat of replacement when the auditor does not com- promise is credible.1 9 As a result, if E[5] — S + L > 0 for all 6, that is Smax < E{5] + L, the auditor always prefers to compromise. The manager retains the auditor that com- promises since we assumed emin > E[e]. The term Kj represents the cost of audit to the manager on the equilibrium path, and is given by Kj = YZTp(A + E f i + EK~] + A ) + P(L + S) • Next we examine the continuation payoffs to the manager and the auditor in sub- sequent periods of an audit engagement when at the start of the engagement tenure beliefs are (oo, A), conditional on disagreement over an accounting issue (e, 8). These are derived in a similar manner to those for the first period (see the discussion above). The manager's replacement threat is still credible if the auditor does not compromise, the manager always prefers to retain the auditor if the auditor compromises, and the 1 9 T h i s assumption is not critical, since if S < E[e], the auditor's strategy will be no compromise if the disagreement arises in the first period of the engagement, and compromise on all disagreements in subsequent periods. 60 Type I, subsequent periods Retain Replace Compromise Manager 0{e - E[e]) - Kj -Kj Auditor 0(E[5] - 5) + 0(L + S) 0 No Compromise Manager -0E[e] - Kj ~Kj Auditor 0E[5] + 0(L + S) 0 Table 2.2: Type I history, subsequent periods payoffs auditor prefers to compromise and be retained instead of losing the engagement since SmaX < E[5] + L + S.D P r o o f of P ropos i t ion 2.4.2 The proof shows that for the given strategies, no player has an incentive to deviate at the current stage and then return to playing his strategy. The table below indicates the continuation payoffs to the manager and the auditor in the first period in which the incumbent has been retained longer than expected, given that the incumbent's fee bid was F ' . The tenure beliefs are (co, A) as a result of the incumbent being retained longer than expected. The payoffs are derived as in the proof of Proposition 2.4.1. Type II, first period Retain Replace Compromise Manager 0e - 0(F' + Kn) -Kj Auditor -05 + 0(F' + Kn)-0A/(l-0) 0 No Compromise Manager -0(F' + Kn) -Kj Auditor 0(F> + Kn) - BA/(I - 0) 0 Table 2.3: Type II history, first period payoffs The cost KJJ represents the total audit cost on the equilibrium path in subsequent periods. If 0(F' + Kn) < Kj, the manager's replacement threat is not credible in case 61 the auditor does not compromise. We note that there is a whole range of F' such that the auditor earns rents and this upper bound is satisfied. Otherwise, the fee F' is not uniquely determined. The payoffs are derived as in the proof of Proposition 2.4.1. Type II, subsequent periods Retain Replace Compromise 2 0 Manager SE - KJJ -Kj Auditor -dS + Kjj-PA/il-B) 0 No Compromise Manager -KJJ ~Kj Auditor KJJ - 0A/(1 - 0) 0 Table 2.4: Type II history, subsequent periods payoffs The same discussion as in the first period applies here, the manager's threats are not credible as long as Kn < Kj. Thus, in equilibrium, the incumbent auditor can set audit fees such that the total audit cost is just under what it would be after a replacement. A replacement at this stage would move the game to a type / equilibrium, for which the cost is Kj. We write then Kn = Kj — 0. The incumbent tries to get the highest fee and earns rents for a range of audit fees. In addition, the total audit cost on the equilibrium path is either 0(F' + Kn) or Kn, both of which are strictly less than the total audit cost on the Type I equilibrium path. If the manager has deviated from the equilibrium path by retaining the incumbent longer than expected (under a Type III equilibrium), he will then optimally retain that incumbent in perpetuity. The incumbent in turn maintains his independence, but can extract very high rents because the replacement auditors would be playing a Type I equilibrium where they always compromise, the agency penalty A is always imposed, and the manager faces the litigation cost E[Q. • Proof of Proposition 2.4.3 62 The proof shows that for the given strategies, no player has an incentive to deviate at the current stage and then return to playing his strategy. The table below indicates the continuation payoffs to the manager and the auditor in each period under the assumption that tenure beliefs are (1, B). The payoffs are derived as in the proof of Proposition 2.4.1. Type III, N = 1 Retain Replace Compromise Manager 0{e + S) - K^l) - # o o ( l ) Auditor -06 + BL 0 No Compromise Manager PS-K^I) --fiToo(l) Auditor PL 0 Table 2.5: Type III history, N = 1 payoffs The cost 1̂ 00(1) represents the total cost of replacing the auditor each period. The manager strictly prefers to retain the auditor regardless of whether the auditor compro- mises or not due to the presence of switching costs, and given that dismissal threats are not credible the auditor will never compromise. In case the manager retains an incum- bent for a second period or longer, a Type II equilibrium is played with a total audit cost of at least PKn (corresponding to a fee F' = 0). We assume E[C] is large enough so that pKH > i^oo(l). In other words, the costs of retaining the auditor beyond the expected term are higher than the costs of replacing the auditor every period. The table below indicates the payoffs to the manager and the auditor in the first period under the assumption that the tenure expectation is N > 1 and tenure beliefs are of type B. The payoffs are derived as in the proof of Proposition 2.4.1. 63 Type III, N > 1, first period Retain Replace Compromise Manager P(e + S) - K^N) - t f o o ( l ) Auditor -06 + pL 0 No Compromise Manager - K n W + pS - t f o o ( l ) Auditor PL 0 Table 2.6: Type III history, N > 1, first period payoff's The cost /CjofT) represents the total cost of replacing the auditor every N periods. The manager strictly prefers to retain the auditor regardless of whether the auditor compromises or not, and given that dismissal threats are not credible the auditor will never compromise. If the manager replaces the incumbent at the disagreement stage, tenure beliefs change to N = 1. As a result, the manager's cost of replacement is —pS — A'oo(l). In subsequent periods, given tenure beliefs N > 1, the payoffs are given by the table below. The payoffs are derived as in the proof of Proposition 2.4.1. Type III, N > 1, subsequent periods Retain Replace Compromise Manager Pe-K^N) - t f o o ( l ) Auditor -ps + vt 0 No Compromise Manager Auditor vt 0 Table 2.7: Type III history, N > 1, subsequent periods payoffs The auditor's payoff in case he is retained depends on how many periods are left in his engagement and is denoted by Vt. The value of incumbency Vt represents the present 64 value of future expected quasi-rents and is determined by the fee structure (2.16), (2.19). As before, the manager cannot make credible threats, and the auditor never compromises. We note that the same arguments are valid for N = oo, that is infinite tenure could be an equilibrium as long as finite tenure is credible and dismissing the auditor leads to a revision in beliefs about tenure. • Appendix D In equilibrium, the investors have rational beliefs about the auditor's strategy on com- promising. Let r = (7((e,5))t denote the auditor's strategy, where jt — 1 if the auditor compromises and 7< = 0 otherwise. The auditor's strategy is a function of the game history and of the particular accounting issue over which there is disagreement. Let T denote the investors' beliefs about T. Then, in equilibrium T = T. For an accounting issue (e,8), let Vi denote the firm value given financial statements and the investors' belief that the auditor has not compromised and let V2 denote the firm value given the same financial statements, the investors' belief that the auditor has compromised and assuming the investors could observe the accounting issue. Thus, if the investors know what the accounting issue is and they believe the auditor has compromised, the same financial statement information leads to different beliefs about the firm's future cash flows, and thus V2 is in general less than V\ (we assume that, whatever the accounting issue is, the manager tries to "fool" the investors into believing the firm value is higher than the auditor is willing to attest to). Now let co = Vi — V2 denote the difference in firm value under the two scenarios. Note that u is associated with the accounting issue the same way as e and 8, and therefore is the realization of a random variable Co. If there is no accounting issue, to — 0. Since investors cannot observe either the accounting issue or whether it arises or not, 65 their beliefs are summarized by knowledge of the distribution of Co. Combined with the investors' beliefs about the auditor's strategy T, the residual agency cost borne by the manager is C(N, t) = C + E[TCo + f C + (1 - t)u>] (2.29) = C + E[ 7 t (e, ~6)Co + 7 i (e , ~S)C + (1 - ^(i, 5))Cu]. In particular, if T = "always compromise", that is % = 1 at all histories, then A = E[^t(e,5)to} = E[to], and it follows that C(N,t) = C + A + E[C]. If, on the other hand, f = "never compromise", that is 7 t = 0 at all histories, then C(N,t) = C + A . The assumptions made in point 3) in the description of the game correspond to a restriction of the auditor's strategies to strategies that are independent of e and 5. In that case, given the other assumptions, the auditor is never indifferent between compromising and not compromising, so mixed strategies are ruled out. It follows that the auditor is restricted to pure strategies independent of the accounting issue and that, in this case, C(N, t) = C + jt(E[Q] + E[C]) + (1 - 7t)E[u>] (2.30) = C + 7 t ( A + E[C]) + (1 - it)A - C + A + 7 tE[C] • The above description of the agency cost describes exactly the situation in which the investors impose the agency cost A if, and only if , the anticipate the auditor to compro- mise, 7t = 1. On the equilibrium path, the investors' beliefs and the auditor's strategies coincide, T = T. It follows that, in equilibrium, the differential cost to the manager between an auditor that compromises and an auditor that maintains his independence is, all else equal, the expected litigation cost E[£]. Under the assumptions made in this Appendix, none of the results presented in the paper change, since the equilibria presented involve pure strategies on the auditor's part of either "never compromise" or "always compromise". Of course, this still leaves the possibility that there may be strategies that depend on the accounting issue, and are 66 thus ex-ante, or from the point of view of the uninformed investors, random. We do not explore these possibilities, as they are intractable and do not add any new insights to the auditor replacement question. At the time the manager decides to replace/retain the auditor after an accounting disagreement, the residual agency cost borne by the manager conditional on retaining the auditor is C(N,t,e,%jt) = C + 7tA-"fte, where jt represents the investors' beliefs and jt the auditor's decision on the compromise. The same cost borne by the manager, conditional on replacing the auditor, is given by C(N,t,e,jt) = C + %A - jtE[e], where jt represents the manager's and the investors' common beliefs regarding the strat- egy of a replacement auditor. Chapter 3 Correlated Noise, Commitment, and Ratcheting 3.1 Introduction The ratchet effect has been described in the economics literature in connection with centrally planned economies (see for example Litwack [25] and the references therein), and more generally in settings where the agent is privately informed. The book by Laffont and Tirole provides a detailed analysis and references [23]. Their description of the ratchet effect is as follows: " If [a regulated firm] produces at a low cost today, the regulator may infer that low costs are not hard to achieve and tomorrow offer a demanding incentive scheme. That is, the firm jeopardizes future rents by being efficient". The essence of the ratchet effect with a privately informed agent is that the agent can obtain a rent in future periods by hiding his type in the current period. Weitzman [36] presents a multi-period model of the ratchet effect with moral hazard only, but in his model the ratcheting mechanism is exogenous. More recently, Milgrom and Roberts [29] and Indjejikian and Nanda [18] have shown that there is a ratchet effect in two period models with moral hazard but without adverse selection. In these models, the ratcheting is endogenous and driven by the lack of commitment by the principal regarding the use of available information. Milgrom and Roberts [29] define the ratchet effect as " the tendency for performance standards to increase after a period of good performance". Given that the principal will use today's outcome in writing tomorrow's contract creates for the manager a link between today's effort and tomorrow's standard of 67 68 performance. The ratchet effect is always inefficient in the models with adverse selection since good types will mimic bad types and earn rents. In the pure moral hazard model, the ratchet effect is also inefficient with respect to the full commitment solution as shown by Indjejikian and Nanda [18]. Ratcheting results from the principal's ability to optimally adjust the agent's second- period incentive for the lower ex-post variance of the second performance measure by using the first-period performance measure when the principal cannot commit fully to a long-term contract. The assumption that the two periods are correlated is thus crucial, and differentiates this model of ratcheting in a pure moral hazard setting from other models of sequential action choice. Repeated moral hazard with independent periods is analyzed by Lambert [24], Rogerson [33], Holmstrom and Milgrom [17], and Fudenberg, Holmstrom and Milgrom [12]. Matsumura [27] presents an analysis of sequential action choice with correlated out- comes in a single period in which the agent observes a first outcome before selecting the second action. However, in Matsumura's model, there is no contracting after the first outcome is observed, and this outcome is private agent information until the end of the period. As a consequence, second-period incentives are not affected by the first-period performance. In this paper, I extend the analysis of the Indjejikian and Nanda model [18] to include commitment issues and the possibility of agent turnover. Indjejikian and Nanda present two types of commitment: full commitment to a long-period contract (which they refer to as "commitment") and an intermediate form of commitment with a sequence of two short-term contracts (which they refer to as "lack of commitment"). I show that the two short-term contracts obtained by Indjejikian and Nanda correspond to a form of commit- ment which I call commitment to fairness. This form of commitment is an adaptation of the concept of fairness introduced by Baron and Besanko [1]. A contracting relationship 69 is governed by fairness if the principal is restricted to fair wages and the agent must participate in all periods if he accepts the contract in the first period. Fair wages are paid when the agent gets his reservation wage as if he could leave in each period. That is, the agent's certainty equivalent of future compensation, conditional on available in- formation and on the principal's conjecture of the agent's first-period action is set to the reservation level at the start of each period.1 Thus, in addition to the usual contract acceptance constraint at the start of the first period, there is a second constraint that the second-period contract is acceptable to the agent as if the agent had other employment opportunities and had not committed to stay for both periods. The agent trades off his ability to leave in the second period for the guarantee of fair compensation in the second period. I also show that the two contracts under commitment to fairness are equivalent to the long-term renegotiation-proof contract: the payoffs for the agent and the principal, and the induced actions coincide for the renegotiation-proof contract and the sequence of contracts under commitment to fairness. The fairness constraint is not only sufficient for obtaining the solution of Indjejikian and Nanda, but also necessary. Allowing the agent to leave in the second period gives the opportunity to the agent to take another action in the first period than that antici- pated by the principal and then leave after the first-period compensation is paid. This situation parallels the "take-the-money-and-run" strategy that arises in ratcheting with adverse selection when the agent cannot commit to stay for both periods (see Laffont and Tirole [23]). Moreover, if the agent is able to leave in the second period, then there is no equilibrium with two short-term contracts in which the agent stays for both periods (see also Christensen and Feltham [2]). Thus, the fairness assumption helps to overcome 1 Note that an equilibrium involves rational expectations regarding the agent's first-period action. Thus, second-period fair wages are based on the principal's conjecture of the agent's first-period action, which is correct in equilibrium. 70 the non-existence of an equilibrium problem. Removing the commitment to fairness assumption leads to a solution in which the principal optimally employs a different agent in each period. The only assumption neces- sary to obtain the two-agent solution is that the principal can commit to replace the first agent in the second period. At the time the principal replaces the agent in the second period, he is indifferent between retaining and replacing the agent. Thus, the principal's commitment to replace the agent is not a strong assumption. An analysis of the principal's welfare under the different commitment assumptions reveals that for negatively correlated performance measures, full commitment is preferred to all other forms of commitment, or lack thereof. The situation is somewhat reversed with positively correlated performance measures in that, although full commitment is better than renegotiation and commitment to fairness, no commitment (two agents) is better than full commitment. The driving force behind this result is that with negative correlation, having the same agent in both periods reduces the total risk to which the agent is exposed by the optimal incentive scheme. When the correlation is positive, using a different agent in each period eliminates the risk premium due to the correlation between the optimal compensation schemes for each action. 2 The remainder of the paper is organized as follows. Section 2 presents the basic assumptions of the model. Sections 3, 4, 5, and 6 derive the full commitment solution, the commitment and renegotiation solution, the commitment and fairness solution, and the no commitment solution, respectively. Section 7 concludes with an analysis of the principal's welfare under the different commitment assumptions. The proofs are collected in an Appendix. 2The situation is as if there is a compensation scheme for the first-period action C\ and a compen- sation scheme for the second-period action ci- With one agent, the risk for which the principal pays compensation is var(ci + 62) = var(ci) + 2cov(ci,C2) + var(c2). With two agents,the risk for which the principal pays compensation is var(ci) + var(c2). 71 3.2 The model In this section I develop a simple two-period agency model that will serve as the basis for the analysis of commitment and ratcheting in a pure moral hazard context. A risk- neutral principal owns a production technology that requires productive effort from an agent in two periods 2 = 1,2. The agent is risk- and effort-averse with exponential utility and quadratic effort cost of the form u(w, a) = — exp[—r(w — ^-(a2 + a2,)], where w is the agent's terminal wealth and a = (ai, a 2) is the agent's effort at the start of periods 1 and 2. The agent's certainty equivalent of terminal wealth w and effort a is, assuming w to be normally distributed, ACE(w, ai, a2) = E[w] - var(w) - ^(aj + a22) . (3.31) The output from agent's effort at EM. is, for 2 = 1,2, xt = btat + A t , (3.32) where Xt is an arbitrary mean zero noise term which does not depend on at in any way. Both outcomes, x\ and x2, are not observed until after the termination of the contract at the end of period 2. Hence, the output xt only determines the principal's expected surplus, and since the principal is risk-neutral, no further distributional assumptions are needed regarding A t . The agent's actions are unobservable. Hence, neither the output nor the agent's actions are contractible. A contractible performance measure yt is observed at the end of each period. The agent's effort in period 2 affects only the mean of the performance measure in that period, yt = mtat + et , (3.33) where et are mean zero noise terms. The noise terms in the performance measures are 72 joint normally distributed with variance-covariance matrix 1 P (3.34) The variances of the noise terms et are normalized to one in order to focus on the inter- period correlation of the two performance measures. An alternative description of the performance measures can be given as follows. When the correlation is positive, the noise terms can be decomposed as ex = ui + 8 (3.35) e2 = i>2 + 8 , (3.36) where ux, v2 are period-specific, and 8 is a common component in both periods. A l l three terms are independent of each other and var(<5) = p. The common component 8 can be thought of as a shock that persists from the first period to the second period. When the correlation of the performance measures is negative, the noise terms can be decomposed as £i = vx + 8 (3.37) e2 = i>2 - 8 , (3.38) where i>\, v2 are period-specific, and 8 is a common component in both periods. A l l three terms are independent of each other and var(<5) = \p\. The common component 8 can be thought of as first-period accruals that have to be reversed in the second period. Thus, negatively correlated noise in accounting-based performance measures reflects the nature of the accrual process. In this model, however, the accruals, as with all the other components of the noise in the performance measure, are outside the manager's control. The principal owns the production technology for both periods and needs an agent to supply productive effort. There is more than one agent that the principal can employ 73 in each period. A l l agents are identical (the agents have the same ability and the same utility functions) and have alternative employment opportunities. Each agent's reserva- tion certainty equivalent is normalized to zero in each period. Both the principal and the agents are assumed to have discount rates of zero. Utility functions, discount rates, reservation wages, the nature ofthe production technology, and the information structure are common knowledge. Throughout the paper, I maintain the above assumptions and I analyze the principal- agent problem under different commitment assumptions. Specifically, I consider the following types of commitment: 1. full commitment to a long-term contract; 2. commitment with renegotiation (with a long-term contract); 3. commitment to fairness (with a sequence of two short-term contracts); 4. no commitment (with a sequence of two short-term contracts). For each of these cases, there is a slightly different time line of events that will be detailed when that case is analyzed. In all four cases, the following basic six-date time line is present. At the start of the first period, the principal and the agent sign a contract. After contracting, in the first period, the agent provides effort ax, after which the first performance measure is reported. At the end of the first period, the agent may receive some compensation depending on the particular contracting setting. At the start of the second period, recontracting, or contracting for the second period may take place, depending on the particular setting. After contracting, the agent provides effort a 2 , and then the second performance measure y2 is reported. At the end of the second period all remaining contracts are settled. After the end of the second period (i.e., after all contracts are settled), the outcomes xx, x2 are revealed to the principal. Thus, in each period, all or 74 some of the following events occur in this order: contracting (at the start of the period), productive effort supplied by agent, performance measure reported, contract(s) settled (at the end of the period). The first two types of commitment (full commitment, commitment with renegotiation) assume that the agent is compensated by a long-term contract c(yi,y2) to be settled at the end of the second period. The other two types of commitment (commitment to fairness, no commitment) assume that the agent is compensated by a series of short-term contracts (ci(yi), c 2 (yi , 2/2)) to be settled at the end of each period. Contracts are always assumed to be linear, and the only contractible information is given by the two performance measures. The resulting contracts are thus the optimal linear contracts in each case, although linear contracts are not optimal. 3 The main issue here is the role of commitment in a two-period L E N 4 framework and how lack of commitment gives rise to ratcheting. Given linear contracts and normally distributed performance measures, the agent's wealth is normally distributed as well, which implies that the agent's certainty equivalent of wealth and effort is given by equation (3.31). More precisely, given a linear contract in both performance measures, the certainty equivalent of compensation at the start of the first period is given by ACE(u), 01,02) = E[w] - ^rvar(w) - \ {a\ + a\) . (3.39) Similarly, for any contract that is linear in the second performance measure, the agent's 3 T h e H o l m s t r o m and M i l g r o m [17] f ramework is not appl icab le here p r imar i l y because the per iods are not independent (the fact that the performance measures are correlated is a key assumpt ion i n my mode l , whi le independence is a key assumpt ion in the Ho lms t rom and M i l g r o m model ) . H o l m s t r o m and M i l g r o m [17] present sufficient condi t ions under wh ich l inear contracts are op t ima l in a mu l t i -per iod agency. In their mode l , the agent's act ions generate a sequence of independent b inary signals (one in each per iod) and the agent's u t i l i ty is exponent ia l . 4 L i n e a r contracts, Exponen t i a l ut i l i ty , N o r m a l d is t r ibut ions. 75 certainty equivalent at the start of the second period is given by ACE(w,a2\yi,a1) = E[w\yi,a1]-^rYa.T(w\y1,ai) - ^a2, . (3.40) Note that, while the conditional expectation E[u>|yi,ai] depends both on the observed value of yi and the agent's first-period action, the posterior variance var(w\yi,a,i) does not depend on either the realized value of y 1 ; or on the agent's action a\. The agent's wealth w represents the total compensation to be received by the agent, and the agent is indifferent as to the timing of consumption. Thus, the agent's utility ensures that there are no intertemporal consumption smoothing issues. In addition, the agent's exponential utility eliminates wealth effects, in that compensation paid (earned) does not impact the agent's risk preferences. The quadratic cost of effort, additive across tasks, means that the agent is not indifferent to the allocation of effort among tasks in the two periods.5 3.3 Full commitment, long-term contract In this section I assume that the principal can commit at the start of the first period to a two-period contract. The terms of this contract are not subject to renegotiation. Furthermore, if the agent accepts the contract at the start of the first period, he commits for both periods, and cannot leave after the first period. These assumptions about the parties' ability to commit make the model equivalent (within the L E N framework) to a two task, two correlated performance measures, as analyzed by Feltham and Xie [11] and Feltham and Wu [10]. The time line of events is as follows. 1) At the start of the first period, the principal offers the agent a linear contract based 5 T h e agent's exponent ia l u t i l i t y u(w,a) = — exp[—r(iu — ^ ( a f + a 2 ) ] impl ies that , over two per iods, his cer ta inty equivalent is ACE(u>,a) = E[w] — | r v a r ( w ) — \{a\ + a 2,), wh ich is not a funct ion only of to ta l effort ax + a-i. 76 on the two performance measures that are contractible information: c = a0 + 0xyx + 02V2 • (3.41) The coefficient «o represents the fixed wage to be paid the agent, and the coefficients 0X, 02 represent the variable wages to be paid the agent for each unit of the performance measures. 2) If the agent accepts the contract, he provides productive effort ax in the first period and a2 in the second period. 3) After the agent has provided effort at, which is not observed by the principal, the performance measure yt is publicly reported. 4) At the end of the second period, the contract is settled. After the end of the second period, the outcomes xx,x2 are revealed to the principal. Note that in this case, the timing of information and the timing of payments to the agent is irrelevant, the only constraint is that the performance measures are available before the contract is settled. The assumption that the principal can commit to a long-term contract guarantees that after yx is observed, the principal will not be able to modify the contract. The L E N assumptions guarantee that the agent's choice of action in the second period will be independent of the observation of the first-period performance measure and of the agent's first-period action ax. The reason is that, at the start of the second period, the part of the agent's certainty equivalent of compensation conditional on yx,ax that is variable in a,2, and thus provides incentives for a2, does not depend on either yx or ax. Here, each of the assumptions of the L E N model is essential. The exponential utility guarantees that wealth does not affect the agent's risk preferences. The linearity of the contract and the fact that the performance measures are joint normally distributed with additive noise ensure that yx and ax impact on the agent's compensation separately from a 2 . In the L E N 77 model with full commitment, the timing of y\ is not important, only its contractibility matters. However, in other settings, the timing of information is important, and if the agent observes the first-period performance measure y\ before selecting the second-period action a2, then a2 may depend on yx. See for example the model of sequential action choice with correlated periods of Matsumura [27]. With full commitment to a long-term contract, the only role played by the correlation between the two performance measures is through the impact on the total risk to which the agent is exposed for incentive purposes as measured by var(c) = 8\ + 2p0\02 + /3f • The principal's problem is to maximize, at the start of the first period, the expected total outcome net of the agent's compensation, subject to the agent's participation con- straint and the agent's incentive compatibility constraint. Proposition 3.3.1 The optimal linear contract c = ao+0iyi+02y2 with full commitment for both periods and the optimal actions are characterized by ai = 0\mi, a2 = 02m2, «o = - ^ ( a 2 + a 2) + i r ( 4 + 2 p ^ + 4 ) > (3-42) 2 2 m\ mxm2 mi, _ mt(r + m$)bi - rpm2b2 (r + m{){r + m2) — rlpA m2(r + m\)b2 -rpm^ 02 — 7—; 2T7—I 2T 2~2 • (A 4 4 ) (r + mf)[r + vnif) — rzpl The principal's surplus is 1 c? da, 2 1 yt = M i + b2a2 - -r(^\ + 2 p ^ - + - - (a 2 + a 2) . (3.45) 2 m\ mim2 m2 2 Proof. See Feltham and Xie [11] or the Appendix. • 3.4 Commitment and renegotiation In this section, I assume that the principal and the agent commit to a two-period contract subject to renegotiation in the second period. Specifically, the principal commits to 78 a two-period contract and the agent commits to stay for both periods if he finds the initial contract acceptable. However, the principal and the agent cannot commit not to renegotiate after the first performance measure is observed. The terms ofthe contract are subject to renegotiation in the usual sense: the existing contract can only be replaced by a new contract if both parties agree to it. The renegotiation takes the form of a take-it-or-leave-it offer by the principal. The time line of events is as follows. 1) At the start of the first period, the principal offers the agent a linear contract based on the two contractible performance measures: c1 = a1, + 0[yx + 0{y2 . (3.46) The coefficient ao represents the fixed wage to be paid the agent, and the coefficients 02 represent the variable wages to be paid the agent for each unit of the performance measures. 2) If the agent accepts the contract, he commits to the terms of the contract, unless both parties agree later to replace it by a new contract. 3) In the first period, after the agent has provided effort ax unobservable by the principal, the performance measure yx is publicly reported. 4) At the start of the second period, the principal can make the renegotiation offer. The principal cannot fire the agent without the agent agreeing to leave at the start of the second period, and the agent cannot leave in the second period without the principal agreeing to it. In addition, a new contract cR = a§(yi) + 02(yi)y2 can replace the old one only if both the principal and the agent weakly prefer it. The renegotiation offer is exogenously assumed to be linear in keeping with the restriction to linear contracts in the model. The coefficients of the renegotiation offer may depend on the first performance 79 measure which is known at the time of renegotiation.6 5) After the renegotiation stage, the agent provides effort a2 and then y2 is reported. 6 ) After y2 has been reported, the contract is settled at the end of the second period. After the end of the second period, the outcomes xi,x2 are revealed to the principal. As in the full-commitment case, the contract is settled at the end ofthe second period. The main difference is that the contract can be renegotiated after the agent has taken his first-period action and the first-period performance measure has been reported. In this context, a renegotiation-proof contract is a contract c = ao + Bijji + B2y2 such that, once agreed upon at the start of the first period, there does not exist a contract at the renegotiation stage which is weakly preferred by both parties and at least one party strictly prefers. The fact that a linear contract can be renegotiation-proof is due to the particulars.of the L E N framework. It turns out that, for every initial contract that is linear in yi, y2, the optimal renegotiation offer (which is restricted to be linear in y2) has a second-period incentive independent of y\ and a fixed wage that is linear in y\. In other words, the renegotiation offer is also linear in the two performance measures from an ex-ante (start of the first period) perspective. A n equilibrium in the principal-agent renegotiation game7 consists of a pair of con- tracts (c1,cR), the agent's actions a i , a 2 , and principal's beliefs about the agent's actions ai,a2 such that: (i) the agent accepts both the initial contract and the renegotiation offer, and rationally anticipates the renegotiation offer and its acceptance when selecting action a\\ (ii) the principal's beliefs are correct di = o,\,a2 = a2; (iii) the pair (c1,cR) is ex-ante (start of the first period) optimal and cR is ex-post (start of the second period, 6 A t the time of renegotiation, the initial contract is linear in the second performance measure, which is still uncertain. I assume that a renegotiation offer is restricted to having the same linear form. 7 Note that, in general, an equilibrium in the renegotiation game is not given by a renegotiation-proof contract. 80 conditional on y\,a\) optimal from the principal's point of view. The following proposition shows that the analysis of the equilibrium can be restricted without loss of generality to renegotiation-proof contracts. Propos i t ion 3.4.1 If (c7, cR, ax, a2) is an equilibrium in the principal-agent renegotia- tion game and cR is linear in y2 at the time of renegotiation, then cR is also ex ante (at the start of the first period) linear in y~i, y2 and (cR, cR, a\, a2) is an equivalent equilibrium with a single renegotiation-proof contract. Proof . First, I show that any renegotiation offer that gives the optimal second-period incentive and is acceptable to the agent is linear in the two performance measures from an ex-ante perspective. At renegotiation time, the principal is restricted to offer a linear contract cR = a§{yi) + 0R(yi)y2, whose coefficients may depend on the first-period performance measure. The second-period incentive 6R does not depend on y\ since it is determined only by the conditional variance var(y 2 |yi) which is independent of the actual value of y\. Furthermore, from the participation constraint at renegotiation time it follows that Since the initial contract is linear in yi, y2 and since the conditional expectation E[j/2|yi] is linear in yi (due to the joint normality of the distributions), it follows, solving the equation implied by assuming the participation constraint(3.47) to be binding for aR(yi), that aR is linear in yi, ,oi,a2] - -rvar(c7|yi) - -a 2, < kCE(cR\yuax,a2) a?(yi) +/?2RE[2/2|2/i,ai,a2] - Jrvar(cR|2/!) - \a\ . (3.47) a. R — 81 It then follows that, conditional on being accepted by the agent, the renegotiation offer is of the following form cR = E[cI\y1,a1,a2} + pR(y2 - E [ j / 2 | j / i , « i , a2}) - ^r[var(c%i) - var(c i J | |/i)] x (3-49) = c 7 + (0* - 0[)(y2 - E[y2\yi,ai, a2)) - -r[var(c / |y 1) - var(c i ?|y 1)] . From the above equation it follows that ACE(cR) = ACE(c 7 ) at the start of the first period, and that cR is acceptable at the start of the first period if, and only if, c 7 is acceptable. Thus, cR is accepted by the agent if offered as a first-period contract. Since cR is an optimal renegotiation offer given c 7, and since the efficient second-period incentive is independent of y i , a i , the principal has no incentive to renegotiate cR if it is offered as an initial contract. This proves that cR is renegotiation-proof. It remains to show that cR induces the same actions as (c 7 ,c R ) , since the payments to the agents and the principal's surplus are determined by ai,a2. The second-period action is determined by cR in both cases since that is the contract in effect at the time the agent provides effort a 2 . Since (c7, cR, a i , a 2 ) is an equilibrium, the first period action is chosen in anticipation of the second-period renegotiation, and so is determined by the (linear) incentive contained in aR(y\). The reason is that it is suboptimal for the agent to select an action different from a i and then reject the renegotiation offer. Thus, the agent's first-period action is determined only by the incentives in cR, and so is the same when cR is offered as the first-period contract. • The main idea in the proposition is that if the contract is renegotiated in equilibrium, the agent's actions are completely determined by the renegotiated contract. The principal cannot gain by offering the agent a contract that will later be renegotiated as long as the agent anticipates the renegotiation. The principal's problem is to maximize at the start of the first period the expected 82 total outcome net of the agent's compensation, subject to the agent's participation con- straint, and the agent's incentive compatibility constraints. The contract must also be renegotiation-proof at the start of the second period. Since at the time the contract can be renegotiated, the first-period action has already been taken, and the first-period performance measure has been reported, the only part of the agent's compensation that remains uncertain is the second-period variable wage /32y2. The requirement that the contract is renegotiation-proof means that at the start of the second period, the contract maximizes the principal's expected total outcome net of the agent's compensation condi- tional on first-period information and subject to the agent's participation constraint and the agent's second-period incentive compatibility constraint. Propos i t ion 3.4.2 The optimal linear two-period renegotiation-proof contract c = «o + PiVi + 02V2 o,nd the optimal actions are characterized by di = miB\, d 2 = m282 a0 = -\{a\ + a\) + J r ( 4 + 2p-™- + % > (3-50) 2 2 m{ m i m 2 [m2, + r ( l - p 2)]bimi - rpfr2m2 , . P l (mj + r)(ml + r(l-p*)) ' 1 j ^ = 2 + 27* 2, • (3-52) mj + r ( l — p2) The principal's surplus is given by TT = biai + b2a2 - Jr(-4 + 2 p ^ - + -4) - k & i + a ' ) • ( 3 - 5 3 ) 2 m\ m\m,2 m\ 0 Proof . See the Appendix. • The renegotiation concept I use here is the same as that of Fudenberg and Tirole [13] in that both parties must agree to the renegotiated contract, but the timing is different. In my model, renegotiation takes place after the first performance measure is observed, while Fudenberg and Tirole have the renegotiation take place between the time the agent. 83 takes the action and the time the performance measure is observed in a single period model. Having the renegotiation take place after y\ is observed and before a2 is taken avoids the insurance/adverse selection problem of Fudenberg and Tirole. 8 The driving force behind renegotiation in my model is the principal's desire to op- timally adjust the agent's induced effort in the second period to match the (reduced) ex-post variance of the second performance measure. In this context, the fact that y\ is observed is important, and not the actual value of y\. As a result, the second-period ac- tion and the second-period incentive are independent of y:, the agent first-period action a\, and the principal's conjecture of the agent's first-period action a\. The inability of the principal to commit not to renegotiate is ex-ante inefficient relative to the full commitment case. The reason is that the renegotiation-proof contract imposes an additional binding constraint on the optimal contract. In both cases, the principal chooses the agent's actions to be induced by the contract in order to maximize his surplus as given by (3.53). In the full commitment case, the principal maximizes 7r unconstrained, while with renegotiation in the second period, the second-period action is constrained to a2 = 62^2 [ m i + r ( l ~ P 2 ) ] - 1 - Since the unconstrained optimum is at least as high as the constrained one, the full commitment contract is at least as efficient as the renegotiation- proof contract. If p = 0, that is if the performance measures in the two periods are independent, there is no difference between the two contracts, which take the form of repeatedly inducing the optimal action from the one period problem. In general, there are additional conditions under which the full commitment contract is the same as the renegotiation-proof contract. The idea is that whenever the full commitment contract is renegotiation-proof, the two coincide. The precise conditions 8 I f renegotiation takes place before yi is observed, the agent is offered insurance by the principal for the risk in y\ when the agent has private information about the action ai. As a result, the principal would like to perfectly insure the agent in order to avoid paying compensation for risk, and this destroys the agent's incentive to provide effort because the agent anticipates the renegotiation. Fudenberg and Tirole show that the only equilibria involve randomization by the agent in choosing his action. 84 under which the full commitment contract is renegotiation-proof are summarized in the following proposition (see Christensen and Feltham [2]). Propos i t ion 3.4.3 The principal's surplus is the same under full commitment and under commitment and renegotiation if, and only if, at least one of the following conditions is satisfied: 1. the two periods are independent, p = 0; 2. the first-period performance measure is uninformative with respect to the first-period action, m\ = 0; b\ b2m2 3. pmx 7̂ 0 and = — ^ — r r . pm\ 777-2 + r ( 1 ~~ P ) Proof . The principal's surplus is given both in the full commitment case and in the renegotiation case by the same function of the action induced 71(01,62) (see equations (3.53) and (3.45)). The principal's surplus is strictly concave in the two induced actions, which implies that the global maximum is unique. Let of, of represent the actions induced by the full commitment contract as given in (3.43), (3.44) and let of, a2 represent the actions induced by the commitment and renegotiation contract as given in (3.51), (3.52). Since of maximizes 71(61,62) given 6 f and of maximizes 77(61,62) given dR, it follows that of = of if, and only if, o f = dR. Since the unique global maximum of 7T(6I ,6 2) is attained only at (of, of), it follows that the principal is strictly better off with full commitment relative to commitment and renegotiation if and only if o f ^ dR. Note that the above argument also shows that of = 6 f if of = a2. Thus, the two contracts coincide if and only if 6 f = d2. The remainder of the proof is straightforward algebra and shows that (l)-(3) are equivalent to o f = d2.9 • 9 S e e also Chr is tensen and Fe l tham [2] for more detai ls. 85 Another interesting special case is when the two periods are identical: rri\ = ra2 = m and b\ = b2 — b. Then, if p = —1, the contracts again coincide, and in addition induce the first-best action in both periods a\ = a\ = b2; the principal's surplus is the first-best TT* = b2. When the two periods are identical, the full commitment contract induces the same action in both periods. When the noise terms in the two performance measures are perfectly negatively correlated, the total variance in the contract is zero since var(c) — df + 2p/3i02 + 8\. Thus, the principal can motivate the agent to work without imposing any risk, and thus need not pay the agent any risk premium. The principal can then motivate the agent by paying only the cost of effort whenever 8\ = 82, and thus achieve first-best. In the renegotiation case, the second-period action is the first-best one because the conditional variance of the second-period performance measure is zero. From the previous argument comparing the two contracts, it follows that the first-period actions are also the same, and the renegotiation contract also achieves first-best. 3.5 Commi tmen t to fairness In the previous two sections, I presented the two types of commitment that relate to two- period contracts: full commitment and renegotiation-proofness. In this section, I begin the analysis of short-term contracts in a two-period relationship. Short-term contracts are used whenever the parties cannot write long-term contracts. The inability to write a long- term contract does not necessarily imply that there is no long-term commitment (that is commitment beyond the first-period) in the principal-agent relationship, only that the principal cannot commit to a specific contract in a period other than the current one, or that the contracting environment is such that it prohibits the use of long-term contracts. The main idea in this section is an intermediate form of commitment, whereby the agent commits to stay for both periods, while the principal commits to offer contracts that 86 provide the agent with his reservation certainty equivalent for each period based on the public information at the contracting date and the principal's conjectures of the agent's actions. This form of commitment is an adaptation of the concept of fairness introduced by Baron and Besanko [1]. A contracting relationship is governed by fairness if the principal is restricted to fair wages and the agent must participate in all periods if he accepts the contract in the first period. Fair wages are paid when the agent gets his reservation wage as if he could leave in each period. That is, the agent's certainty equivalent of future compensation, conditional on available information and conjectured actions is set to zero at the start of each period. Thus, in addition to the usual contract acceptance constraint at the start of the first period, there is a second constraint that the second-period contract is acceptable to the agent as if the agent had other employment opportunities, had not committed to stay for both periods, and had taken the conjectured first-period action. The key fact here is the agent's ability to commit to stay for both periods; removing the agent's ability to commit for both periods leads to a situation where there is no equilibrium in which the agent stays in the second period. The agent gives up his ability to leave in the second period for the guarantee of fair compensation in the second period. The time line of events is as follows. 1) At the start of the first period, the principal offers the agent a linear contract based on the first-period performance measure y\. ci = «i + phyi • (3.54) The coefficient a.\ represents the fixed wage to be paid the agent, and the coefficient 0i represents the variable wages to be paid the agent for each unit of the performance measure. 2) If the agent accepts the contract, he commits to stay for both periods. The principal 87 commits to offer the reservation certainty equivalent of wages in each period, given all the information available at the time of contracting and given that the agent's actions correspond to the principal's conjectures. 3) In the first period, after the agent has provided effort ax unobservable by the principal, the performance measure y\ is publicly reported. 4) After the performance measure yi is publicly reported, the first-period contract is settled at the end of the first period. 5) At the start of the second period, a new contract is offered by the principal c 2 = a2 + fom . (3.55) This contract is subject to the agent reservation wage restriction and is accepted by the agent (since I assumed commitment for the second period on the agent's part). The terms of the second-period contract may depend on the performance measure yi and on the principal's conjecture about the agent's first period action, since they are informative about y2, and therefore can be used to reduce the noise of the second-period performance measure. 6) In the second period, the agent provides effort a 2 , then the performance measure y2 is reported. The second-period contract is settled at the end of the second period. After the end of the second period, the outputs Xi,x2 are revealed to the principal. Short-term contracting with fair wages relies on the principal's conjectures regarding the (unobservable) agent actions. At the start of the second period, when c 2 is set, the terms of the contract depend on E [^ 2 |2 / i , where &i is the principal's conjecture. The concept of fair wages in the second period assumes that the principal's conjecture is correct, that is the agent actually has provided effort a\ in the first period. At the start of the first period, fair wages refer to the total future compensation paid to the agent 88 over the two periods C\ + c2, and involve both the principal's and the agent's conjectures of future actions d i , d 2 . The principal and the agent are assumed to be in agreement over the conjectured future actions. For example, the principal states his conjecture d i as part of the first-period contract C\. The agent then agrees to that conjecture when accepting the initial contract, and that will be the basis for setting the second-period contract.1 0 Thus, an optimal sequence of contracts under commitment to fairness is also charac- terized by a rational expectations equilibrium regarding the agent's actions. The agent's commitment to stay for both periods is essential in sustaining this rational expectations equilibrium, as will be shown later in this section. The principal's problem is to maximize at the start of each period the remaining expected total outcome net of the agent's compensation, subject to the agent's reservation wage constraint and the agent's rationality constraint. Since the agent commits to stay for both periods at the start of the first period, the second-period contract c 2 is anticipated at the time the first-period contract is set. P ropos i t ion 3.5.1 The optimal sequence of short-term linear contracts (c\ = O i + PiVi, c 2 = 0:2(2/1)+ $22/2) and the optimal actions induced under commitment and fairness are characterized by d\ = mi(0i — pj32), d 2 = m202, 1 + P (3.56) (3.57) (m\ + r)(ml + r(l - p2)) m\ + r ( l — p2) ' a2{yi) = -p—(yi - miai) + x«2 —2il - P2) - 1 (3.58) b2m2 (3.59) 777.2 + r ( l — P2) 1 0 A n alternative would be that both parties solve for the unique equilibrium conjecture a,\. In this case, the principal can use an expert witness (professor of accounting) to prove that the second-period wage is fair. 89 The principal's surplus is given by Ti = M i + M2 - \r{^\ + 2 p ^ 2 - + ^ ) - \{a\ + a22) . (3.60) 2 m\ mim2 m2 0 Proof . See the Appendix. • Note also for future reference that n d\ do 0i = — + P— , (3.61) m\ m2 61m? mi d2 ai = —5- rp—^ mf + r mf + r m 2 _ (rai + r ( l - p 2))6i?n 2 - rpb2mxm2 ^ (mf + r){m\ + r ( l - p2)) The optimal actions induced by the sequence of contracts in Proposition 3.5.1 are the same as the actions induced by the renegotiation-proof contract . From (3.53) and (3.60), the principal's surplus is the same in both cases. Moreover, from an ex-ante (start of the first period) perspective, the contract c = 5i + c2 is the same as the renegotiation-proof contract described in Proposition 3.4.2. Under commitment to fairness, as with renegotiation, the second-period action and the second-period incentive do not depend on the first-period action, or on the specific value of first-period performance measure. The reason, as before, is that when contracting at the start of the second period, the role played by the available information is to reduce the variance in the second-period performance measure, and that variance does not depend on either the specific value of the first-period performance measure or the first-period action. It follows that the principal's optimal choice of risk for the agent when setting the second-period incentive does not depend on either the specific value of the first-period performance measure or the first-period action. Thus, the principal's problem in setting the incentive for the second period is the same in both cases, and as a result the induced second-period action is the same. 90 Once the second-period incentive rate is set, the fixed wage is given by the fairness re- striction. Then, the second-period fixed wage depends linearly on y\ because E[y 2|?/i,ai] is linear in y\. From an ex-ante perspective, the second-period contract contains a (ran- dom) linear term in yi, which contributes in addition to the first-period incentives from the first-period contract to the agent's choice of action in the first period. At the start of the first period, the agent's cumulative future compensation C\ + c 2 is linear in y\ and y 2 , while the second-period incentive rate has the same value as in the renegotiation case. Both the principal and the agent anticipate the second-period contract c 2 and have a common conjecture regarding its terms. It follows that the principal's problem is the same as in the renegotiation case and that the total incentive for the first-period action is the same. The main difference is that, with commitment to fairness, the first-period incentives are split between the first-period variable wage and the second-period fixed wage. The agent's actions are the same and the principal's surplus is the same in both cases. Thus, the two short-term contracts under commitment to fairness replicate the payoffs of the long-term renegotiation-proof contract. This particular sequence of contracts is the one derived by Indjejikian and Nanda [18] to illustrate how lack of commitment in a dynamic agency relationship leads to ratcheting. The principal's inability to commit to a long term-contract without renegotiation in the second period creates the same inefficiency with short-term contracts under commitment and fairness as the inefficiency of the renegotiation-proof contract. Here, as before, the inefficiency is relative to the full commitment long-term contract. The results in this section show that commitment to fairness is a sufficient assumption for obtaining the ratcheting with short-term contracts described by Indjejikian and Nanda [18]. However, their description of the commitment assumptions that are necessary for the solution they derive is incomplete and referred to as "the absence of commitment". 91 If "absence of commitment" means that the agent cannot commit to stay for both periods, then, as is shown in the following, there is no equilibrium with a sequence of short-term contracts in which the agent stays for both periods. As a result, the commitment to fairness assumption is not only sufficient, but also necessary to obtain the solution described in Proposition 3.5.1, which is to say Indjejikian and Nanda's solution. The fairness assumption is necessary since, once the agent has committed to stay for both periods, there must be a restriction in the second period to prevent an "infinite" transfer from the agent to the principal through the second-period contract. The anticipation of such a contract renders any first-period contract in which the agent commits for both periods unacceptable to the agent. Thus, the simplest solution to this problem is a lower bound on the agent's reservation certainty equivalent, which is precisely what the fairness constraint provides. The following proposition shows that the sequence of optimal contracts derived under commitment to fairness always induces the agent to leave in the second period if the agent's ability to leave is restored. P ropos i t ion 3.5.2 If pm\Qi 7̂ 0> and the principal offers the sequence of contracts (ci,C2) from Proposition 3.5.1, the agent works for the principal only in the first period, leaves in the second period, and earns a positive surplus. Proof . Suppose that the agent is unable to commit to the firm for both periods. Then, the agent's ability to leave in the second period guarantees the same participation con- straint for the second period as the assumption of fair wages. If the principal assumes that the agent will stay with the firm for both periods, he will offer the agent the contract Ci in the first period, anticipating c~2 for a second-period contract (where c*i, c 2 are the same as those in Proposition 3.5.1). At the time the agent chooses his first-period action, he can plan to stay for the 92 second period, in which case he will maximize 1 1 ACE(c i + c2,ai,a2\ai) = E[cx + c2\aua2] - - rvar(c i + c2) - -{a\ + a\) , which is the agent's certainty equivalent of terminal wealth given that the agent stays in both periods and the principal's conjectures of the first- and second-period actions. The agent's certainty equivalent of compensation over the two periods depends on the agent's action choice in the first period ai , on the agent's conjectured second-period action a 2 , and on the contract terms, which depend on the principal's conjectures regarding the agent's actions fii, a2. In this case, the agent and the principal have a common conjecture for the agent's equilibrium action choice in the second period. The main point is that the agent's action choice in the second period is independent of the action chosen in the first period and of any other first-period information, which implies that the second-period action can be conjectured to be the same by both the principal and the agent at the start of the first period. If the agent plans on leaving in the second period, he will maximize ACE(c i , a i | a i ) = E[ci|ai] - ^rvar(ci) - \^a\ , which is the agent's certainty equivalent of terminal wealth given the agent collects the payment C\ at the end of the first period and then leaves. The agent's certainty equivalent depends on his first-period action choice a\ and on the contract terms which depend on the principal's conjecture of the agent's first-period action h\. As in the previous case, the principal's conjecture a\ is based on the assumption that the agent stays for both periods. The principal's conjectures of the agent actions are the equilibrium actions described in Proposition 3.5.1, and the contracts offered to the agent are determined by these conjectures as in equations (3.56)-(3.59). The key here is that the agent's certainty equivalent of compensation depends on the principal's conjectured actions (which are based on the assumption that the agent stays 93 in the second period) and on the agent's decision to leave/stay in the second period. The agent chooses his action strategically in conjunction with his decision to leave/stay. The contracts offered to the agent are based on the principal's conjectures 0,1,0,2, and are designed to be accepted by the agent conditional on the agent taking the conjectured actions. Also, the agent gets only his reservation certainty equivalent when he takes the conjectured actions. Thus, the agent may benefit by deviating from the actions conjectured by the principal. If the agent plans to stay in the second period , the action he chooses is the same as the one the principal expects Si , and in equilibrium his certainty equivalent of compensation is zero. On the other hand, if the agent plans to leave in the second period, Ci = ai+Ayi, and the certainty equivalent of first-period compensation is ACE(c i | a i , Si) = a\ + B\m,\a,\ — -rd\ — -a\ , (3.63) where CL\, Q\ are given by (3.56), (3.57). The first-order condition for the agent's action choice implies ai = mi@i = a,i+ p—-0,2 • (3.64) Substituting in the agent's certainty equivalent of compensation gives, for all p ^ 0, ACE(c i | a i , a 1 ) = ax + a\ - ^r(32 - 1-2 m i - - l a ? aia2 2 aj " o a i _ P—Gifl2 + o r ( ~ 2 + 2P +P — 2 m 2 2 mf mi?n 2 (3.65) + - 0 ( 4 + 2 ^ - ^ + ^ 4 ) 2 mf m,\m,2 = \P2<~2 > 0 • 2 mi The agent can thus earn a positive surplus if the principal offers a first-period contract in anticipation of a second-period contract, expecting the same agent to stay in the 94 second period, while the agent plans on leaving in the second period after the first-period compensation has been paid. The agent has a strict incentive to act strategically in the first period and then "take-the money-and-run" whenever he is offered the first contract of the sequence (ci, c2) and the correlation between the performance measures is non-zero. Now I will show that it is also optimal ex-post for the agent to leave in the second period, given that he acted strategically (i.e., he took the action (3.64)) in the first period. At the start of the second period, the principal offers the contract c 2 = 0:2(2/1)+ P2J/2 given by (3.58) and (3.59), and the agent's certainty equivalent of second-period compensation is A C E ( c 2 | y i , a\,a2) = a2 + 32{m2a2 + p(yx - m,ia\)) - 7^/3 2(l - p2) - ^ a 2 , (3.66) where a\ = a\ + pmia2/m2 is the action that the agent took in the first period. If the agent were to accept the second-period contract, his action choice in the second period would be the same as that anticipated by the principal a2. The key here is that the second-period action a2 and the second-period incentive pay 32 are independent of the actual action the agent took in the first period and of the specific value of the first- period performance measure. The second-period fixed wage is the one determined under commitment to fairness since I assumed that the principal designs the contracts based on having the same agent in both periods, 02 ffll. „ 1 d o , 2\ 1-2 / o „r,\ a2 = -p—yx + p—axa2 + - r —(1 - p ) - -a2 . (3.67) m2 m2 2 mi, 2 It follows that the agent's certainty equivalent of compensation at the start of the second 95 period is A C E ( c 2 7/i, a{, a 2) = -p—V\ + p—aia2 + ^r—x (1 - P ) m 2 m 2 2 ra\ d 2 / / t \ \ 1 ^ 2 / - I H (m2a2 + - mia|)) - - r — ( 1 777.2 2 7772 m l - / - t \ 2 m l - = p—a2(ai - a!) = -p — a 2 < 0 , 777,2 7772 which means that the agent will always reject the second-period contract, independent Of 7/i. • The conclusion of the proposition holds for both positive and negative correlation between the performance measures in the two periods. However, the source of the agent's surplus is different in the two cases. If the performance measures are positively correlated, the agent works harder than the principal expects him to and the surplus comes from "too much" incentive pay. If the performance measures are negatively correlated, the agent works less than the principal expects him to, and the surplus comes from the fixed pay. As shown by Christensen and Feltham [2], the fact that there is no equilibrium in which the agent stays for both periods when the agent is able to leave holds in greater generality, without having to specify that the principal offers the particular contract sequence from Proposition 3.5.1. The restriction to short-term contracts in addition to the agent's ability to leave is also essential for the equilibrium to break down. For example, it can be shown that if long-term contracting is possible, there are renegotiation equilibria such that the agent never leaves in the second period if he accepts the contract in the first period. - p2) - \a\ (3.68) 96 3.6 The agent cannot commit for two periods Up to this point, I have assumed that the agent can commit to stay for both periods, either by signing a long-term contract (that may be subject to renegotiation) or by entering into a contractual arrangement based on commitment to fairness. In this section, I remove the agent's ability to commit to stay for more than one period and analyze two settings: one in which long-term contracts are possible, but are subject to renegotiation, and a second setting in which only short-term contracts are possible and the only commitment the principal can make is commitment to firing the agent hired in the first period. 3.6.1 Long-term contracts and renegotiation In order to analyze the agent's ability to leave under commitment and renegotiation, a slight modification of the long-term contract is necessary. In the section on renegotiation, I assumed that the agent commits to stay for both periods and that the (renegotiated) contract is settled at the end of the second period. I now remove both assumptions as follows: 1. The long-term contract is settled over both periods as follows. Instead of a single payment at the end of the second period c = a0 + 0oiVi + Po2V2, the principal and the agent agree at the start of the first period on two payments to the agent, one at the end of each period, c.\ = a.\ + Q\y\, c2 = a2(yi) + p2y2- The total payments to the agent are set to be the same in both cases, C\-\-c2 — c. 2. The agent can leave at the start of the second period. If the agent leaves, he takes payment C\ and gives up c2. Thus, the agent cannot renege on c\ after observing yi and then leave. 97 Note that this modification of the long-term contracting with renegotiation is different from the setting with short-term contracts. The key difference is that, with short-term contracts, the first-period contract c.\ has to be acceptable to the agent on its own at the start of the first period. With long-term contracts, the agent has to accept the total contract c = c\ + c2 at the start of the first period. The renegotiation mechanism is modified in a straightforward way: given an initial contract c1 = [c[,c2), a renegotiation offer from the principal replaces c\ by cR. At the start of the second period, the agent can respond to a renegotiation offer from the principal in three ways: accept the renegotiation offer and stay, reject the renegotiation offer and stay, and reject the renegotiation offer and leave. An equilibrium in the renegotiation game in this modified form consists of an ini- tial contract, a renegotiation offer, and the agent's actions (c[,C2,cR, ax,a2) with the same properties of an equilibrium as described in Section 4. Note again that an equi- librium assumes that the agent accepts the renegotiation offer and stays in the second period. This introduces an additional constraint on the renegotiation offer. When the agent is committed to stay, the renegotiation offer has to be weakly preferred to the initial contract, ACE(c2\yi,ai,a2) < ACE(cR\yi, ax, a2). When the agent can leave, the additional constraint that staying is preferred to the reservation certainty equivalent, ACE(c2 \yi, ai, a2) > 0, is needed to ensure the agent does not leave. As before, the analysis can be restricted without loss of generality to renegotiation-proof contracts (see Proposition 3.4.1). The following proposition shows that a linear renegotiation-proof contract can be always rewritten so that the agent stays for both periods even when he can leave in the second period. Thus, having access to long-term linear contracts, makes the distinction whether the agent can leave or not unimportant. 98 Propos i t ion 3.6.1 For any renegotiation-proof equilibrium (0,01,02), where the long- term contract c is linear in y\, y2 and the agent is committed to stay in the second period, there exist c.\,ci linear in y\ and y2 respectively such that (c*i, 62, a\, o 2) is an equivalent equilibrium, c = C\ + C2, and the agent always stays in the second period without having committed to do so. Proof . The main idea of the proof is to rearrange the agent's payments across periods so that the acceptance decision in the second period, which in general may depend on the specific realization of yi when the agent is not committed to stay, does not depend on yi. To understand this idea, I start with a long-term renegotiation-proof contract for which the payments are all at the end of the second period. The restriction to a renegotiation-proof contract does not restrict the generality of the problem and eliminates the renegotiation constraint at the start ofthe second period, leaving only the acceptance constraint relative to the reservation certainty equivalent. Then, given the contract c = o;o + piyi + 02y2, the agent's participation constraint at the start of the second period is ACE(c I yi,al,a2) = a0 + 8m + B2E[y2 \ yi] - ^rvar(c| yi) - ^(a2 + a\) = a0 + + 02(m2a2 + p(yi - m^)) - \r0\(\ - p2) - ^(a2 + a2,) (3.69) = c*o + (0! + p02)yi + p2{m2a2 - pmiai) - ^0^(1 - p2) - ^(a2 + a2.) > 0 . It is easy to see that, starting with a long-term renegotiation proof contract and allowing the agent to leave at the start of the second period makes the agent's decision to stay depend on the particular realization of y\. If, instead of the whole payment, only one part is allocated to the second period c 2 = a02 + 02iVi + 022^2, the agent's decision to 99 stay is given by A C E ( c 2 | y i , a u a 2 ) = a 0 2 + p2iVi + P22EW2 I Vi] - ^rvar(c 2 | yi) - ^ ( a 2 + al) 1 1 = «02 + (#21 + P022)yi + p22(m2a2 - pmiai) - - r / ? | 2 ( l - p2) - - ( a 2 + a 2) > 0 . (3.70) Then the agent will always stay if P21+PP22 = 0, and a02 is set such that the participation constraint (3.6.1) is satisfied with a strict inequality. The second-period incentive rate is set such that Q22 = 02. The payment for the first period C\ = a 0 i + PuVi + P12V2 is set such that c = C\ + c 2. Thus /? 1 2 = 0, 0n + /?2i = Pi, and a 0 i + «02 = «o- It follows that the sequence of payments (ci,c 2) is acceptable to the agent at the start of the first period since it is the same as the payment from the old contract c, while at the same being also ex-post (start of the second period) acceptable. • 3.6.2 T w o agents I now remove all abilities to commit beyond a single period for both the principal and the agent. Thus, the principal cannot commit in the first period to anything that may happen in the second period. That means not only the terms of the second-period contract, but also terms such as fair wages as described in the previous section. The agent in turn, cannot commit to stay with the firm for more than one period and can leave at the end of the first period. In particular, long-term contracts are not possible in this setting. Under the no commitment assumption, the sequence of contracts derived under the commitment to fairness assumption is no longer an equilibrium, as shown at the end of the previous section. Thus, the commitment to fairness assumption is also necessary for the contracts derived in that section to arise in equilibrium. The key problem is that if the principal offers a first-period contract based on the assumption that the same agent will be employed in the second period, the agent has an incentive to act strategically and 100 "take-the -money-and-run". On the other hand, the principal may design the contracts based on the assumption that he can commit to firing the first agent at the end of the first period, and will employ a different agent in each period. The time line of events is as follows. 1) At the start of the first period, the principal offers the first agent a linear contract based on the first-period performance measure y\. c1 = ai+ Pm . (3.71) The coefficient a.\ represents the fixed wage to be paid the agent, and the coefficient 0i represents the variable wages to be paid the agent for each unit of the performance measure. The principal commits to firing the agent at the end of the first period. 2) If the agent accepts the contract, he provides effort a\ in the first period. 3) After the agent has provided effort a\ unobservable by the principal, the performance measure y% is publicly reported. The contract is settled at the end of the period. 4) After the contract is settled, the principal fires the first agent. 5) At the start of the second period, a new contract is offered by the principal to a second agent c2 = a2 + 02y2 . (3.72) This contract is subject to the agent participation constraint and is accepted by the agent. The terms of the second-period contract may depend on the performance mea- sure yi since it is informative about y2j and affects the posterior beliefs (at the time of contracting) about the variance of the second-period performance measure. 6) In the second period, after the agent has provided effort a2, the second performance measure y2 is reported. The second-period contract is settled at the end of the second period. After the end of the second period, the outputs x\,x2 are revealed to the principal. 101 The principal's problem is to maximize, at the start of each period, the remaining expected total outcome net of compensation for the agent(s), subject to the participation constraints and the agents' rationality constraints. In addition, the principal is committed to employing a different agent in each period. Propos i t ion 3.6.2 The optimal sequence of short-term linear contracts (c\ = a.\ + PiVii c2 = ai2 + 82112) and the optimal actions induced with two different agents are char- acterized by di = m\(3i, a2 = m282, A = 4 ^ • (3-74) "2 = -\a22 + ]-r(l - p2)^\ - p{yi - m i a i ) — , (3.75) 2 2 1712 m2 02 = 2 , n 2T • (3.76) The principal's surplus is given by 1 1 b\m\ ^ = 2 M l = 2 rAVr ' (3"77) "2 = 2ha* = 2 ml + r(l-?) ' ( 3 " 7 8 ) Proof . See the Appendix. • Since I have proved in the previous section that the sequence of contracts based on a two-period tenure for the agent hired in the first period is not an equilibrium when the agent can leave in the second period, it follows that the two-agent solution is the only one possible in the absence of commitment over more than one period and under the assumption that the first agent's tenure is certain (either one period or two periods). That is, I assume that the principal and the agent are restricted to pure strategies regarding 102 firing/leaving decisions in the second period. Note, in addition, that the principal is assumed to be able to commit to firing the first agent before contracting in the second period takes place. This is a mild assumption on commitment since, in the absence of switching costs, the principal has no incentive to renege on the commitment to fire the first agent. 3.6.3 R a n d o m firing of the agent after one per iod I now turn to the analysis of random firing of the agent at the end of the first period. The time line of events is the same as described at the start of the section on two agents. The only difference is that the principal commits to a random mechanism of firing for the first agent instead of committing to fire the first agent with certainty. To keep the analysis tractable and within the L E N framework, I assume that the principal can commit to firing the first agent at the end of the first period with probability 1 — p, independent of ?/i,2/2- The firing decision is described by a random variable {1 if the agent is fired at the end of the period (3.79) 0 if the agent is offered a contract in the second period , which is independent of the performance measures. Thus, the agent's wealth from an ex-ante (start of the first period) perspective is w = ci + (1 — / )c 2 . The agent's expected utility of wealth is then given by EU(w, Si , d 2) = pEU(ci + c 2, d x , a 2) + (1 - p)EU(c x , d x) (3.80) = —pexp(—rACE(cx + c 2 ,a x ,a 2 )) — (1 — p)exp(—rACE(c x,d x)) . In the second period, the contract offered by the principal is the same, whether the first agent has been retained or not. The key is that the principal sets the second period fixed wage as a function of the observed value of the first-period performance measure 103 and his conjecture regarding the agent's first-period action. The principal's second-period surplus is then independent of his conjecture di . Conditional on the first agent not being fired, the agent's acceptance of the second-period contract depends on the agent's first- period action and on the principal's conjecture: A C E ( c 2 | y i , a 1 ; di , d2) = p/? 2 ^i(di - ax) . (3.81) Since the agent's acceptance of the second-period contract is independent of y i , and the principal's firing decision is also independent of y i , it follows that the agent's beliefs about his continuing employment with the principal in the second period are not affected by the observation of y 2 . Thus, at the time the agent selects the first-period action, the only thing that matters is whether the agent believes that the principal's conjecture di will be correct, since that will determine whether the agent will accept the second- period contract conditional on not being fired. In a rational expectations equilibrium, the principal's conjecture is correct, hi = a\. It follows that, in a rational expectations equilibrium, the agent will always accept the second-period contract conditional on not being fired. I now turn to proving that a rational expectations equilibrium is not possible when the agent can leave at the end of the first period. Assume, by contradiction, that there exists a rational expectations equilibrium in which di = ax. Then, the principal offers a first-period contract based on the assumption that the agent accepts the second-period contract conditional on not being fired at the end of the first period. I now show that for the contract based on this assumption, the agent has an incentive to deviate from the equilibrium value of di and that he will, as a result, always turn down the second-period contract when not fired. Thus, an equilibrium in which di = a\ is not possible when the agent is not fired with certainty. As before, from the agent's participation constraint it follows that c 2 = a 2 + /32y2 with 104 d 2 = m202 and the fixed wage given by o 2 = —/?2p(yi —di) + -/3|[r(l —p2)—m2]. Then, for any conjectured first-period action d 1 ; the two contracts are independent ex ante (start of the first period). 1 1 It follows that ACE(c i + c 2, d x , d2)) = ACE(c"i,di) + ACE(c 2 , d i , d 2), and equation (3.80) becomes E\](w, au a2) = exp( - rACE(c 1 , ai))[-p exp( - rACE(c 2 , aua2)) - (1 - p)] . (3.82) At the time the agent selects his first-period action, let Ai = ACE(c"i, ax, di) and A2 = A C E ( c 2 , a i , di , d 2). Then the first-order condition for the agent's action is — E U ( w , d 1 , d 2 ) = pr oai &4j dA2 H - dai dai exp(-r ( i4 i + A2)) + (1 - p ) r - — e x p ( - r A i ) = 0 u(l\ (3.83) where dAi dai dA2 = mipi - ai (3.84) = -pmxfc (3.85) da i Setting = di in the above equations simplifies the first-order condition (3.83) to p(mid\ - di - mip02) + (1 - p)(miPi - dx) = mx0x - ax - pmxp02 = 0 . (3.86) It follows that the equilibrium first-period action under random firing and rational ex- pectations is di = mx{0x -ppd2) . . (3.87) Comparing the equilibrium action under rational expectations with random firing (3.87) to the equilibrium action under rational expectations and commitment to fairness as described in Proposition 3.5.1, it follows that the contracts offered with random firing i i This is a straightforward calculation: cov(ci ,C2) = cov(/3i2/i,p2V2 — 02P{yi — « i ) ) — 0. 105 are equivalent to the contracts under commitment to fairness if the correlation p is replaced by pp. Then, following the same argument as in Proposition 3.5.2, if the agent is offered such a pair of contracts, he has the incentive to act strategically in the first period, deviate from the equilibrium conjecture a\, and then leave in the second period taking a positive surplus. 3.7 Correlation and the value of commitment In this section, I conclude by comparing the principal's surplus given the four different assumptions on commitment when the correlation between the two performance mea- sures varies. Let irc denote the principal's surplus under long-term contracting with full commitment (see Proposition 3.3.1), and let TTR denote the principal's surplus under long-term contracting with second-period renegotiation (see Proposition 3.4.2). Let nf, t = 1,2 denote the first- and second-period surplus under commitment to fairness (see Proposition 3.4.2), and let TTJ , t = 1,2 denote the first- and second-period surplus with two agents (see Proposition 3.6.2). Let TTf = irF 4- irF and let TTT = irf + irT denote the total surplus under short-term contracting. Proposition 3.7.1 If the two performance measures are uncorrelated, p = 0, the prin- cipal's surplus is the same in all cases and there is no difference between contracts: T T C = TTR = 7TF = TTT . (3.88) Full commitment is weakly preferred to commitment with renegotiation or commitment to fairness: N C > T T * = 7TF . (3.89) The inequality is strict whenever none of the conditions from Proposition 3-4-3 is satisfied. 106 Furthermore, assume that the correlation between the performance measures is pos- itive, p > 0. Then, full commitment is preferred to renegotiation, but no commitment (two agents) dominates both Proof. The only thing that is left to prove is that for positive correlation, the two-agent solution dominates the full commitment solution. Suppose that the incentive rates from the full commitment contract are offered to two separate agents in the two periods. Then, the same actions are induced for the two agents as for the single agent. However, the risk premium paid to the two agents is lower since Thus, two agents provide the same effort at a lower risk premium, which means that the principal's surplus is higher with two agents. The advantage of using two agents comes both from not having to pay compensation for the part of the total variance of the single-agent contract that is due to the performance measures being positively correlated (the term 2pQ\Q2 in the above equation), and from the ability to use the lower ex-post variance with the second agent. • The above result can be strengthened under the additional assumption that the two periods are identical, in that for negatively correlated performance measures, the two agent solution is the least preferred. Proposition 3.7.2 Assume that the two periods are identical, b\ = b2 and mi = m2. If the two performance measures are positively correlated, p > 0, full commitment is better than renegotiation, but no commitment (two agents) dominates both: IT > 7T > 7T = 7T . (3.90) \r[3l + 2p0i02 + Pi] > \r\Pl + P22(l - p2)] . (3.91) 7T > 7T > 7T = 7T (3.92) 107 If the two performance measures are negatively correlated, p < 0, full commitment is better than renegotiation, and better than no commitment (two agents) : 7TC > TTR = TJF > 7TT . (3.93) Finally, if p = — 1, TT* = 7TC = TTR = TTF > 7TT , (3.94) where ir* = b2 is the first-best surplus. Proof. See the Appendix. • With negatively correlated performance measures, the risk premium paid to a single agent through a long-term full commitment contract is reduced through the aggregation of the two performance measures. The key is that the incentive weights are identical on the two performance measures when the periods are identical. When the periods are not identical, the result regarding negatively correlated performance measures does not hold in general, but depends on the particular values of the parameters. 3.8 Appendix: proofs Proof of Proposition 3.3.1 The optimal contract is a solution of the following optimization problem: max E [xi + x2 — c] (3.95) subject to the participation constraint ACE(c | a 1 , a 2 ) > 0 (3.96) and the incentive compatibility constraint di , d 2 G argmax ACE(c |a i , a 2) , (3.97) a i , ( i 2 108 where di,ci2 are the actions to be induced by the contract. Given the basic structure of the model (4.274), (3.31), (3.33), and the linear contract (4.281), the principal's surplus is given by 7T = E [X\ + X2 — c] = M l + M2 — CXQ — d\m\CL\ — 02m2a2 • (3.98) The agent's certainty equivalent of compensation is given by (3.99) ACE(c |a i , a2) = a0 + dim-^ai + 02m2a2 - ^rven(01yi + 02y2) - ^ (a 2 + a\) = a0 + / M i + 02a2 - K{0\ + 2p0x02 + 0\) - l-[a\ + a2) Since the agent's participation constraint is binding at the optimum, a0 + 0 i m i a i + 02m2a2 = l-r{0\ + 2p0x02 + 0\) + l-(a\ + a\) , (3.100) which in turn gives for the principal's expected surplus TT = M i + b2a2 - l-r{0\ + 2p0x02 + 0\) -\{d\-r a\) . (3.101) The first-order conditions for the incentive compatibility constraint are —ACE(c\a1,a2)=m101-a1=O (3.102) oa-y ^ - A C E ( c | a i , a2) = m202 - a2 = 0 , (3.103) 0(l2 which imply that the actions induced by the contract are di = mi 0i, d 2 = m202 • (3.104) It follows that the principal's problem can be rewritten as an unconstrained opti- mization problem (since the incentives 0i, 02 uniquely determine the agent's actions, the principal chooses the optimal actions to be induced by the contract) max M i + b2a2 - J r ( - ^ + 2p-^- + %) - ^(a 2 + d2.) . (3.105) a i , a 2 2 m\ mim 2 m2 2 109 The first-order conditions are Solving for di, d2 gives 61 - ai - r% - rp-^— = 0 (3.106) m\ m,\m2 b 2 - a 2 - r% - rp-^— = 0 . (3.107) mi mim2 „ _ m\{r + m%)bi - rpmxm2b2 Q>\ — —, TyT} 0 0 lO.lUO) (r + m f ) ( r + rrij) — r^/tr m\{r+ m\)b2-rpmlm2bl a2 — ——• — • 5-5- . (d.iuy) (r + mf)(r + m^) — rlpl Substituting di,d2 in the agent's participation constraint (3.100) gives ao + d2 + d2, - \r(^\ + 2 p ^ 2 - + ^%) - J(d2 + al) = 0 , (3.110) which can be used to determine a0 as 1 1 d2 a a, 2 « o = -x(a? + d2) + - r ( ^ + 2 p ^ * - + ^ | ) . (3.111) 2 2 m f m,im2 m\ The principal's surplus is TT = M i + b2a2 - + 2p-^- + ^)-\{a\ + a\) . (3.112) • Proof of Proposition 3.4.2 At the start of the second period, the renegotiation-proof contract is a solution of the following optimization problem: max E [xi + x2 — c | yx, dx, d2] (3.113) ao,/3i,/32 subject to the participation constraint ACE(c I yi,a\,a2) > w0 (3.114) 110 and the incentive compatibility constraint d 2 € argmaxACE(c | yi,a\,a2) , (3.115) where a\ is the actual first-period action taken by the agent, yx is the reported value of the first-period performance measure, and d 2 is the action to be induced by the contract in the second period. The participation constraint requires that the agent's certainty equivalent of compensation conditional on the first-period information is no less than the certainty equivalent of compensation under the existing contract w0. In addition, di represents the principal's conjecture of what the agent's first-period action has been. In equilibrium, the principal's conjecture is correct, ax = a\.12 From now on, I will use only di for both the principal's conjecture and for the agent's actual first-period action. Moreover, before either action has been taken, the conjectured actions from the point of view of both the principal and the agent are also di and d 2 . Both the principal and the agent have rational expectations regarding their own behavior and each other's behavior. The information structure (3.33), (4.276) implies that E[y 2 | yi] = rn2a2 + p{yx - mxax) (3.116) var(y2 | yx) = 1 - p2 . (3.117) The principal's expected surplus at the start of the second period can be written as TT2 = E [xx + x 2 - c\yu di , d2] = E[£i|j/i] + E[x 2 |j/i] - a0 - 0xyx - foEfalVi] (3.118) = bi&i + b2a2 + E[Ai + A 2 |yi] - a 0 - dxyx - /5 2(m 2d 2 + p(yx - ax)) . 1 2 I n fact this is a subtle point: there is an equilibrium in which the principal's conjecture is correct. Thus, when renegotiation takes place, or at the start of the first period, there is no uncertainty about the agent's past or future actions; that is, although the agent's actions are unobservable by the principal, there is no contracting with private agent information. I l l The agent's certainty equivalent of compensation at the start of the second period is ACE(c | yx,ai,a2) = a0 + 0xyx + 02E[y2 | yx] - var(/31?/1 + 02y2 | yx) - ^ (d 2 + a2) = a0 + 0xyx + 02E[y2 \ yx] - ^r/3fvar(?/2 | yi) - ^ (d 2 + a%) = a0 + 0xyx + 02{m2a2 + p(yx - mxax)) - ^r /? | ( l - p2) - ^(d 2 + d2.) . (3.119) Since the agent's participation constraint is binding at the optimum, «o + Pm + 02(m2a2 + p(yx - mxax)) = w0 + ^ r /? | ( l - p2) + ^ (d 2 + a2,) , (3.120) which in turn gives for the principal's expected surplus TT2 = bxax + b2a2 + E[Ai + A 2 | yx) - w0 - l-r02(\ - p2) ~\{o?x + o2) . (3.121) The first-order condition for the incentive compatibility constraint is d — A C E ( c | yx,ax,a2) = m202 - a 2 = 0 . (3.122) oa2 It follows that the action induced by the contract is a2 = m202 , (3.123) and the principal's problem can be rewritten as an unconstrained optimization problem max M i + b2a2 + E[XX + A 2 | yx] - -r-\(l - p2) - -(a2 + a2) - wQ . (3.124) a.2 l m2 The first-order condition is 6 2 - r ( l - p 2 ) - ^ - d 2 = 0 . (3.125) mi. Solving for d 2 gives a2 = 02m2 = — . 3.126) m^ + r ( l — p2) 112 In summary, the restriction that the contract is renegotiation-proof at the start of the second period is equivalent to restricting the second-period incentive pay to the value above (4.300). At the start of the first period, the optimal contract is a solution of the following optimization problem: max E [xi + x2 - c] (3.127) subject to the participation constraint ACE(c |o i , a 2 ) > 0 , (3.128) the incentive compatibility constraint ai,a2 G argmax ACE(c |a i , a2) , (3.129) and the renegotiation-proofness constraint A = 2 , , (3-130) m2 + r ( l — p2) where Si , a2 are the actions to be induced by the contract. It follows that the principal's problem is the same as the long-term contract problem (3.95), (3.96), (3.97) with the additional constraint (3.130). As a result, the optimiza- tion problem can be rewritten as (4.292) subject to (3.130), which gives the first-order condition (4.293). Solving for a : gives i a 2 Oi - rp ai = p\m\ = —- 1 + raJ (3-131) _ [ml + r ( l — p2)]bimf — rpb2m\m2 ~ (m\ + r)(ml + r ( l - p2)) 113 The fixed wage is determined in the same way as in (4.290), (4.291), ao = ~\{a\ + a\) + \r{^ + 2 p ^ + 4 ) • (3-132) 2 2 mf mxm2 Similarly, the principal's surplus is (see (4.292)) vr = b i a x + & 2a 2 - J r ( 4 + 2 p - ^ - + 4 ) " \ft + d2) . (3.133) 2 mf mxm2 2 • P r o o f of P ropos i t ion 3.5.1 The second-period contract specifies the second-period incentive for the action a 2 based on all information available at the time of contracting: maxE [xi + .x2 - c 2 | yx, ax, a2] (3.134) subject to the participation constraint A C E ( c 2 | y i , d i , a 2 ) > 0 (3.135) and the incentive compatibility constraint a2 G argmaxACE(c 2 | yi,a\,a2) , (3.136) 0.2 where yx is the reported value of the first-period performance measure, and a2 is the action to be induced by the contract in the second period. The first-period action a\ taken by the agent is the same as the principal's conjecture d i . The principal's expected surplus at the start of the second period can be written as 7T2 = E [xx + x2 - c2 | yu di , a2] = E[xx \ yx] + E[x 2 | yx] - a2 - 02E[y2 \ yx] (3.137) = bi&i + b2a2 + E[Ai + A 2 | yx] - a 2 - /3 2(m 2d 2 + p{yx - mxax)) . 114 The agent's certainty equivalent of compensation at the start of the second period is 1 1 A C E ( c 2 | a\, a2) = a2 + 32E[y2 I y\] - - r var(a 2 + (hy* \ V\) - ^ = a2 + 02E[y2 I yi] - ^r/?|var(y 2 | Vi) - ^ a 2 (3.138) = a2 + 02(m2a2 + p{yi - mxa\)) - 7,r02(l - p2) - ^«2 • Since the agent's participation constraint is binding at the optimum, and in equilibrium &i = a\, a2 + 32{m2a2 + p(yx - m^)) = \r0l(l - p2) + l-a\ , (3:139) which in turn gives for the principal's expected surplus TT2 = M i + M2 + E[Ai + A 2 I yx\ - \r&\{\ - p2) - \a\ • (3.140) The first-order condition for the incentive compatibility constraint is — A C E ( c I y i , a\,a2) = m202 - a2 = 0 . (3.141) oa2 It follows that the action induced by the contract is a2 = m202 , (3.142) and the principal's problem can be rewritten as an unconstrained optimization problem I d 2 1 max M i + M 2 + E[Ai + A 2 | Vl] - - r - | ( l - p2) - -a2 . (3.143) a,2 Z 777-9 ^ The first-order condition is Solving for a2 gives b2-r(l- p2)^-a2 = 0 . (3.144) 6 2 = m202 = 2 ^ } 2 . • (3.145) m2 + r ( l — pz) 115 Thus, the second-period action induced by this contract is the same action induced by the renegotiation-proof contract. In addition, the second-period fixed wage is determined by (3.139), nn 1 - r r 1 , (3.146) « 2 = -p—{yi - mioi) + \a\ m2 2 mi, At the start of the first period, the principal and the agent anticipate the terms of the second-period contract c 2 as determined in (3.146) and (3.145). The optimal first-period contract is a solution of the following optimization problem: max E [xx + x2 — cx — c2] (3.147) ai,/3i subject to the participation constraint A C E ( c i + c 2 | a i , a 2 ) > 0 , (3.148) and the incentive compatibility constraint ax € argmaxACE(c x + c 2 |a x ,a 2 ) , (3.149) 0,1 where a x is the action to be induced by the contract in the first period, and the second period contract is anticipated to be the one given by (3.146) and (3.145), with a second- period action a2 = m202. The principal's expected cumulative surplus is given by 7T = E [xx + x2 - Ci - c2] = bxdx + b2a2 - ax - 0xmxdx - E[a2] - 02m2d2 . (3.150) The agent's certainty equivalent of cumulative compensation is given by ACE(c i + c2\dx,d2) = ax + 0xmxdx + E[a2) + B2m2a2 - ^rvar(c x + c 2) - ^(a 2 4- a\) = ax + 0xmxax + E[a2] + 32m2d2 - (var(cx) + var(c2)) - ^(a 2 + a 2) = ax + 0xmxax + E[a2] + 02m2a2 - [0\ + /32(1 - p2)] - ^(a 2 + a 2) , (3.151) 116 where I have used that cx and c2 are independent random variables, a fact which follows directly from the expression (3.146) for 0:2- At the time the agent chooses his first-period action, E[a2\au ax] = -p\p(E[yx] - mxax) + \Q\ [r(l - p2) - m22] (3.152) = -mY32p(ax - ax) + -/?f [r(l - p2) - m22] , which implies that the agent's certainty equivalent is ACE(c i + c 2 | a i , ax,a2) = ax + p\mxax - 32mxp(ax - ax) + ^/3 2 [r(l - p2) - m22] + 02'm2a2 - \r [32 + 32{l - p2)] -\(% + a22) . (3.153) The first-order condition for the agent's incentive compatibility constraint is mi8X — pmx32 — ax = 0 . (3.154) Thus, the first-period action to be induced by the contract is given by &i = mx(0x - pfo) , (3.155) and the first-period incentive is 3, = ^ + p ^ . (3.156) mi m2 The agent chooses his first-period action in anticipation of the second-period adjustment to his fixed wage based on the first-period performance, and this is the essence of the ratchet effect. The second-period fixed wage, to be determined at the start of the second period has a component that depends on yx, which makes it contribute to the incentives in the first period when yx is still uncertain. 117 In equilibrium, the agent's participation constraint is binding, and the principal's conjecture of the agent's action is correct. As a result, using (3.151) and ACE(c i + c2\ai,a2) — 0 in (3.150) gives for the principal's expected surplus TT = M i + b2a2 - l-r [3\ + /?2(1 - p2)] + %) • (3.157) It follows that the principal's problem can be rewritten as max&ia! + b2a2 - J r ( - % + 2 p - ^ - + %) - \{a\ + d2,) . (3.158) ai 2 m\ m\m2 m2 2 The first-order condition is b\ — r—^ — rp—— di = 0 , (3.159) m\ mxm2 which gives the first-period action as 1 (, _ &2 \ m °i = f~ ( 6 i - rPZTZT j • (3.160) 1 + — V m1m2 Substituting d 2 from (3.145) gives . _ hmKml + r ( l - p2)) - rpb2mxm2 C\\K\\ G l (m 2 + r ) ( m 2 - f - r ( l - p 2 ) ) " K ' The first-period incentive is then a di d 2 0i = — + P — mi m2 bimi(m\ + r ( l - p2)) - rpb2m2 + 6 2 m 2 (3.162) (mf + r)(m 2 + r ( l — p2)) m 2 + r ( l — p 2) The first-period fixed wage is determined by the agent's participation constraint (see (3.151)), on = -p\miai - -0\ [r(l - p2) - m 2] - 62m2a2 + -r [B\ + 02(l - p2)] + - (a? + a 2) = -pimiai + ^r(32 + ^d? 1-9 m l - - 1 / a l ^ a l ° 2 9 d o , - a x - p — a i a 2 + - r ( — + 2p + p — , z m2 2 mf mxm2 m2 (3.163) 118 • Proof of Proposition 3.6.2 In the first period, the principal offers a contract C\ = OL\ + 6\y\ anticipating that a new agent will be hired in the second period and offered a contract c 2 = a 2 4- Q2y2. The second-period contract can make use of all available information from the first period, so that the fixed and the variable wage in the second period may depend on yx. The second-period contract is a solution of the following problem: maxE [xi + x2 — c 2 | yx, a,i] (3.164) subject to the participation constraint A C E ( c 2 | y i , a i ) > 0 (3.165) and the incentive compatibility constraint a2 € argmaxACE(c 2 | y i , a i , a 2 ) , (3.166) 0-2 where y\ is the reported value of the first-period performance measure, a2 is the action to be induced by the contract in the second period, and a x is the second agent's and the principal's common conjecture of what the first agent's action has been. Since E[£i + x2\ yi,di] = bxai + b2a2 + E[Ai + A 2 | yi ,ai] and a2 has no impact on E[Ai + A 2 | 2/1, fii], I can substitute ir2 = x2 — c 2 for the principal's objective function in (3.164). Since the second agent's certainty equivalent of compensation is ACE(c 2 | i / i , a i , a2) = a2 + /32E[y2|2/i, aua2} - ^r/3|var(y2|2/i) - ]-a22 1 1 ( 3 - 1 6 7 ) = a2 + 02{m2a2 + p(yx - m i O i ) ) - - r ^ | ( l - p2) - -a22 , 119 the first-order condition for the agent's incentive compatibility constraint gives a 2 = m202 as the action induced by the contract. The participation constraint is binding at the optimum and determines the fixed wage as «2 = -p2(m2a2 + p(yi - mxax)) + \rPl(l ~ P2) + = - o a 2 + ^ ( i -P)—2- —pfa - m i a i ) • Z Z / / i 2 if 1-2 Substituting in the principal's second-period expected profit gives TT2 = b2a2 + E[A 2 | yu ax] - a2 - 62E[y2\yx, ax] (3.168) 1 a 2 1 = b2a2 + E[A 2 | yx, ax] - - r ( l - p2)-\ - -a. (3.169) 2 • Solving for the action chosen by the principal from the unconstrained problem gives a2= 2 ^ } 2, , (3.170) m 2 + r ( l — p2) and for the principal's second-period surplus 1 . a2 7T2 = 6 2a 2 + E[A 2 | yi,ax] - - [ m 2 + r ( l - p 2)]—| = ^ 2 a 2 + E [ A 2 | y 1 , a 1 ] (3.171) 1 & 2 m 2 + E[A 2 | y ^ a ^ 2 m 2 + r ( l - p2) From (3.171) it follows that E[7r2] = \b2a2 which does not depend on ax or yx. As a result, in setting the first-period incentives, the principal can ignore 7r2. The principal's first-period problem is then maxE[5i-C!] (3.172) <*i,/8i subject to the participation constraint ACE(ci) > 0 (3.173) 120 and the incentive compatibility constraint ai € argmax ACE(ci |a i ) . (3.174) a i (3.175) Since the agent's certainty equivalent of compensation is 1 1 ACE(ci |a i ) = ai + Pimiai - -rp2var(yi) - -a\ = ai + Pimiai - - ^ a 2 , it follows that the agent's induced action is ai = m-idi. Since the participation constraint is binding at the optimum, the fixed wage is * = - 5 & ? + 5 r 5 - ( 3 ' 1 7 6 ) and the principal's expected first-period profit is 0 a l 1 2 2 m\ vTi = M i + ^ 2 ( l - - a 2 (3.177) Thus, the principal's optimal choice of action for the first agent is = h™L (3.178) and the expected first-period surplus is 1 1 1 6 2 m 2 TTI = M i - r M i = « M l = o o • * • (3.179) 2 2 2 m{ + r The principal's total expected surplus for both periods is n = TTI + E ^ ] . • P r o o f of P ropos i t ion 3.7.1 The proof is a straightforward computation. First, I calculate the principal's surplus separately for the two periods under commitment and fair wages. The expected second- period surplus is (see (3.143), (3.145)) Z m 2 1 (3.180) b2 m 2 m 2 + r ( l - p2) ' 121 For the first period, a similar calculation gives (see equations (3.163), (3.162), and (3.161)) 7 r f = 6di — OJI — mdiCbi Since bai ~ ~—- (ai + pa.2)2 ~ x d 2 2 777/ 2 o a i _ ^~^( a i + 2 P a i ° 2 + P2al) - \a\ 2 777/ 2 1 2 a 2 aia2 1 2 a 6ai - - ( m 4- r ) — - rp^^y- - - r p (3.181) 2 m 2 m 2 2 m : 777/ 4- r it follows that 2 F b2m2 — rpba2 1 (6m2 — rpd 2 ) 2 „ 6m 2 — rpd2 1 2 d 2 = o ^ x rpa 2 , , 0 r - -rp — mz + r 2 mz(mz + r) mz(mz + r) 2 mz = ^ 9/ I ,—\ [26m2(6m2 - rpa2) - (6m2 - rpa2)2 2 ml\ml 4- r) L -2rpa2(bm2 - rpa2) - (m 2 + r)rp2a22] (3.183) [(bm2 — rpd 2 ) 2 — rp 2 d 2 (m 2 4- r)] 2 m 2 (m 2 + r) 1 6 2 m 2 1 1 2 m 2 + r 2 m 2 + r From Proposition 3.6.2, I have that rpd2(2b + pa2) . T T T 1 6 2 m 2 1 b2m2 , „ „ „ , s 7 r T = T r f 4- TTJ = — — 4- „ , (3.184) 1 2 2 m 2 4- r 2 m 2 4- r ( l - p2) ' V 1 which implies that 1 b2m2 TTF - 7 T T = - - — rpa2(2b 4- pa2) . (3.185) 2 ml 4- r The last calculation is for the difference in principal's surplus under full commitment and 122 no commitment b2m2 1 b2m2 1 b2m2 K C - H T • m2 + r 4- rp 2 m2 + r 2 m 2 + r(l — p2) b2m 2 2 2(m2 4- r)(m2 + r — rp2) — (2(m2 4- r) — rp2)(m2 + r + rp) 2(m2 4- r)(m2 4- r — rp2)(m2 4 r 4 rp) 2(m2 4- r) 2 - 2(m2 4- r)rp2 — 2(m2 4- r)2 — 2rp(m2 + r) 4- (m2 4- r)rp2 4- r 2 p 3 2(m2 4- r)(m2 4- r — rp2)(m2 4 r 4 rp) r n Irn 2 — 9 .2 2 r P [rP2 ~ 2(m2 4- r) — p(m2 4- r)] 2(m2 4- r)(m2 + r — rp2)(m2 + r + rp) _ 2 2 -rp [m2 4- r(l - p2) 4- (m2 4- r)(l 4- p)] 2(m2 4- r)(m2 4- r — rp2)(m2 4- r + rp) (3.186) Chapter 4 M u l t i p e r i o d Ratchet Effect and Manager ia l Tenure 4.1 In t roduct ion Firms have long-term relationships with economic agents such as managers, auditors, and suppliers that can be characterized as sequences of shorter lived agents interacting with a longer lived firm. To stylize this to the extreme, the firm is infinitely lived, and the agents are finitely lived. In this framework, the manager's tenure with the firm becomes the object of investigation. For example, Hambrick and Fukutomi [15] propose a model of the dynamics of the CEO's tenure in office. Their analysis considers the CEO's performance in relation to the number of periods that have passed since the C E O started the current job. In this context, a natural question is to examine the role played by beliefs about the duration of tenure, where the duration of tenure is the total number of periods the agent will work for the firm. Thus, when the firm starts a multi-period relationship with an agent, what is the role played by beliefs about the duration of the relationship? If the length of the contract can be chosen ex-ante, and commitment to a certain tenure is possible, is there an optimal ex-ante tenure? For example, if a manager is hired with a long-term contract, what is the importance of the length of the contract (as determined at contracting time, and assuming away commitment issues)? In auditing, a similar question arises from the fact that auditors stay with the same firm for long periods of time. For auditors, beliefs about tenure are important for multi-period pricing under perfect competition which leads to lowballing. 123 124 These issues are examined by §abac and Simunic [34], who show that there is no ex-ante optimal tenure, but that tenure beliefs can lead to auditor replacements in equilibrium. In principal-agent models, the agent's tenure is important because it impacts the amount and the characteristics of the information available for contracting. Specifically, if in each period the accounting system produces a piece of information that is used as a performance measure in contracting with the manager, the number of periods the manager is employed determines the informational environment for contracting. As Feltham and Xie [11] show, multiple correlated performance measures are valuable because insurance effects allow the principal to reduce the risk imposed on the agent. Their setting corresponds to a multi-period world where the principal can commit to not renegotiate contract terms at later dates as subsequent information is revealed. If such commitment is not possible, and the principal chooses the sequentially optimal contract instead, then the agent's effort is influenced by the ratchet effect. The ratchet effect has been described in the economics literature in connection with centrally planned economies (see for example Litwack [25] and the references therein), and more generally in settings where the agent is privately informed. The book by Laffont and Tirole provides a detailed analysis and references [23]. Their description of the ratchet effect is as follows: " If [a regulated firm] produces at a low cost today, the regulator may infer that low costs are not hard to achieve and tomorrow offer a demanding incentive scheme. That is, the firm jeopardizes future rents by being efficient". The essence of the ratchet effect with a privately informed agent is that the agent can obtain a rent in future periods by hiding his type in the current period. Weitzman [36] presents a multi-period model ofthe ratchet effect with moral hazard only, but in his model the ratcheting mechanism is exogenous. More recently, Milgrom and Roberts [29] and Indjejikian and Nanda [18] have shown that there is a ratchet effect in two period models with moral hazard but without adverse selection. In these models, 125 the ratcheting is endogenous and driven by the lack of commitment by the principal regarding the use of available information. Milgrom and Roberts [29] define the ratchet effect as " the tendency for performance standards to increase after a period of good performance". Given that the principal will use today's outcome in writing tomorrow's contract creates for the manager a link between today's effort and tomorrow's standard of performance. The ratchet effect is always inefficient in the models with adverse selection since good types will mimic bad types and earn rents. In the pure moral hazard model, the ratchet effect is also inefficient with respect to the full commitment solution as shown by Indjejikian and Nanda [18]. Ratcheting results from the principal's ability to optimally adjust the agent's second- period incentive for the lower ex-post variance of the second performance measure by using the first-period performance measure when the principal cannot commit fully to a long-term contract. The assumption that the two periods are correlated is thus crucial, and differentiates this model of ratcheting in a pure moral hazard setting from other models of sequential action choice. Repeated moral hazard with independent periods is analyzed by Lambert [24], Rogerson [33], Holmstrom and Milgrom [17], and Fudenberg, Holmstrom and Milgrom [12]. Matsumura [27] presents an analysis of sequential action choice with correlated out- comes in a single period in which the agent observes a first outcome before selecting the second action. However, in Matsumura's model, there is no contracting after the first outcome is observed, and this outcome is private agent information until the end of the period. As a consequence, there is no renegotiation in the second period based on the first-period performance. In Chapter 3, I extended the analysis of Indjejikian and Nanda [18] to include com- mitment issues and the possibility of agent turnover. The central theme of Chapter 3 is 126 the analysis of different levels of commitment and a comparison of the principal's wel- fare under different commitment scenarios and different correlation of the performance measures. Regarding agent turnover, the main insight from Chapter 3 is that, in a two- period L E N model, the two agent solution (turnover or no commitment) is the least preferred when the performance measures are negatively correlated, with long-term full commitment and long-term commitment with renegotiation (or commitment to fairness) dominating. The result is the opposite for positive correlation of the performance mea- sures, in that the two-agent solution (turnover or no commitment) dominates long-term full commitment. Both results are driven by the way in which the correlation of the performance mea- sures impacts the risk premia that have to be paid to the agent(s). Due to the particular nature of the L E N framework, which essentially puts the agent in a mean-variance world, these risk premia are completely determined by the variances of the performance mea- sures. The driving force behind this result is that, with negative correlation, having the same agent in both periods reduces the total risk to which the agent is exposed by the optimal incentive scheme. When the correlation is positive, using a different agent in each period eliminates the risk premium due to the correlation between the optimal compensation schemes for each action. The situation is as if there is a compen- sation scheme for the first-period action c"i and a compensation scheme for the second- period action c2. With one agent, the risk for which the principal pays compensation is var(cx + c2) = var(ci) + 2cov(c"i, c2) + var(c2). With two agents,the risk for which the principal pays compensation is var(ci) + var(c2). These results regarding turnover indicate that, in a two-period world, the principal prefers to replace the agent after one period when the performance measures are positively correlated. On the other hand, the principal prefers even the weakest form of long- term commitment with a single agent when the performance measures are negatively 127 correlated. These results can be extended to more than two periods. In a setting with more than two periods, one can consider the choice of alternative tenures in a given time horizon. In a realistic setting, tenure is a choice variable in a world with more than two periods. In addition, when considering tenure (or replacement policies), the analysis can be framed in different ways. For example, regarding optimal tenure one may ask: 1. In a given finite number of periods, what is the optimal number of agents and their respective tenures? In other words, given a known finite life for the firm, how can one optimally partition that among several agents, and what are tenures of those agents? 2. In an infinite world, what is the optimal tenure for agents, if we assume a policy of replacing agents after the same number of periods (tenure)? The main difference between the two frameworks, and between the way questions are answered is that in one world there is a "last period", whereas in the other world, there is no "last period". The infinite period world permits the comparison of different tenure lengths in a natural way. A similar comparison of these two frameworks is undertaken in Chapter 3 regarding auditor tenure. In what follows, I extend the two-period model of Chapter 3 in order to answer the second question posed above: what is the optimal (stationary) agent turnover policy in an infinite-period world? In this paper I develop an N-period model of the ratchet effect in a principal-agent problem with moral hazard but without adverse selection. Thus, while the agent's action is unobservable by the principal, in equilibrium the principal has rational beliefs regarding the agent's past actions and as a result, in equilibrium, information asymmetries do not develop over time between the principal and the agent. In addition, there is no learning of productivity or any agent characteristic that is unknown at the start. The only dynamic information effects are the adjustment of posterior beliefs about future 128 performance measures, conditional on the sequential observation of past performance measures together with conjectures of the agent's past actions. The model generalizes the two-period model of Chapter 3, and most results derived therein remain valid in the N-period case. Primarily, the conclusions regarding the role of commitment in obtaining different solutions remain the same. However, the N-period model gives insights into the importance of the contracting horizon, or tenure. In addition, the N-period model offers insights into the agent's long-term performance that cannot be inferred from the two-period model. While in Chapter 3 I have compared different commitment scenarios given a fixed two-period horizon, in this paper the emphasis is on tenure given a certain choice of commitment assumptions. Since the very idea of tenure implies some form of implicit or explicit long-term commitment, the choice is between the three types of long-term com- mitment discussed in Chapter 3: full commitment to a long-term contract, commitment to a long-term contract with renegotiation, and commitment to fairness with short-term contracts. Full commitment to a long-term contract is too restrictive, in that, especially over longer horizons, renegotiation is more likely. Commitment to fairness is a mecha- nism that replicates the commitment to a long-term contract with renegotiation by using a sequence of short-term contracts. This makes the choice between the two forms of contracting almost a matter of taste. In this paper, I choose to restrict the analysis to contracting under commitment to fairness. Besides capturing the idea of renegotiation, it only requires short-term contracts that are settled at the end of each period, allowing for a breakdown of performance and surplus period-by-period. In addition, the principal and the agent only need to commit to "fair contracts" and to a tenure duration, no other long-term commitments or contracts are necessary.1 Finally, the commitment to fairness solution to the dynamic J T h e reader who finds the concept of "commitment to fairness" unpalatable can interpret all results 129 agency problem provides consistency and ease of comparison to previous literature such as Milgrom and Roberts [29] and Indjejikian and Nanda [18]. The driving force behind the "ratchet effect" with pure moral hazard described in this paper is the principal's inability to commit not to use past performance measures when setting a short-term contract with the agent in each period other than the agent's first period of tenure. The principal has an incentive to set the agent's compensation in a later period based on past performance measures in order to optimally adjust the risk in the agent's contract for the lower posterior variances. In doing so, the principal makes the fixed pay in future periods depend on past performance measures. For the agent, it means that from an ex ante (previous periods) perspective, future fixed pay depend on earlier effort choices. In other words, "fixed compensation" in one period is fixed only in that period, given the past actions and performance measures, but is variable when anticipated from earlier periods. It follows that the agent's incentives for effort in any one period are spread among the variable wage for that period and the fixed wages in all future periods. The nature of the solution to the dynamic agency problem is such that, for longer horizons, the manager's effort is close to some limit level for most periods. Thus, there are essentially two effort levels, the second best in the last period, and approximately a "third best" in most other periods. When correlation is positive, the "third best" is lower than the second best, generating inefficiencies relative to a repeated one-period problem (which is the multi-period problem when periods are independent). When the correlation is negative, the ratchet effect is efficient relative to the uncorrelated periods case since the "third best" effort level is closer to first best. To summarize, with positively correlated performance measures, incentives are stronger and the manager exerts less effort in most periods than in the last period. With negatively correlated performance in the context of commitment to a long-term contract with renegotiation. 130 measures, incentives are less strong and the manager exerts more effort in most periods than in the last period. The effort level in the last period serves as a benchmark because it coincides with the second-best solution, which is what one obtains with uncorrelated performance measures. A commitment problem could arise in the case of negatively correlated performance measures due to the decreasing performance of the agent towards the end of his tenure. The principal needs to be able to commit not to fire the manager before term, otherwise the efficiency gains from getting effort levels close to first-best are lost. This would not be a problem in the positive correlation case since there the principal gets better performance towards the last period, and has no incentive to fire the manager before term. In either case, commitment to fairness assumes that the principal commits to retain the agent for all N periods. To answer the question of ex-ante choice of tenure by the principal, I examine op- timal stationary replacement policies, whereby the agents are hired for periods at a time within an infinite horizon for the firm, and the noise in the performance mea- sures is firm-specific. The main result for positively correlated performance measures is that, in the presence of a switching cost, there exists a threshold switching cost such that optimal tenure is a single period whenever the switching cost is higher than the threshold value. On the other hand, optimal tenure is the maximum number of periods possible (the agent's maximum life) when the switching cost is lower than the threshold value. Thus, with positively correlated performance measures, the only optimal replace- ment (tenure) policies are the corner solutions of one period tenure or maximum possible tenure. The main result for negatively correlated performance measures is that the opti- mal replacement policy is always the maximum number of periods possible, irrespective of the switching cost. In the case when the noise in the performance measures is agent-specific, there is 131 an additional "learning" effect in the first few periods of a manager's tenure due to the (rapid) reduction of the posterior variances of the performance measures towards their limit value. This effect results in an increase of managerial effort, since it becomes less costly over time for the principal to motivate managerial effort due to lower risk premia. With negatively correlated performance measures and agent-specific noise, the manager exerts the second-best effort in the last period, a higher effort level in all other periods, and that effort has an inverted U shape. This finding is consistent with evidence from the management literature that firm performance increases at first, reaches a maximum, and then declines during a manager's tenure. For a discussion of these findings, see the papers by Eitzen and Yetman [9], Katz [21], and Hambrick and Fukutomi [15]. The remainder of the paper is organized as follows. In Section 2, I present the principal-agent model and the informational environment for contracting. In Section 3, I present the full-commitment long-term contract solution to the agency problem. In section 4,1 discuss long-term contracts with renegotiation, and the optimal renegotiation- proof contract. In section 5,1 turn to short-term contracts under commitment to fairness and I derive the first key result, the optimal sequence of "fair" short-term contracts and induced actions. I also show that the commitment to fairness mechanism replicates the payoffs of the renegotiation-proof contract. Section 6 is devoted to the discussion of the dynamics of effort and incentives given the "fair contracts" solution derived in Section 5. In Section 7 I examine the question of agent replacement policies with a fixed switching cost and derive the second key result of the paper, the solution to the optimal tenure problem. Section 8 concludes the paper and the Appendix contains some of the more technical proofs. 132 4.2 T h e principal-agent model In this section I present an N-period generalization of the principal-agent model from Chapter 3, in which the agent's performance measures are correlated. The model is a pure moral hazard model similar to the one found in Indjejikian and Nanda [18]; the agent does not have any private information other than his own effort level. A risk-neutral principal owns a production technology that requires productive effort from an agent in N periods t = 1 , . . . , ^ . The agent is risk- and effort-averse with exponential utility and quadratic effort cost of the form u(w, a) = — exp[—r(w — — (a\ + ..., +a2N)], where w is the agent's terminal wealth and a = ( a 1 ; . . . , a^) is the agent's effort at the start of periods 1 through N. The agent's certainty equivalent of terminal wealth w and effort a is, assuming w to be normally distributed, ACE(t(),a 1, ...,aN) = E[w] - irvar(ty) - ^(a 2 + V a2N) . (4.187) The output from agent's effort at G R is, for t = 1,..., N, zt = btat + A t , (4.188) where Xt is an arbitrary mean zero noise term which does not depend on at in any way. The outcomes zt are not observed until after the termination of the contract at the end of period N. Hence, the output zt only determines the principal's expected surplus, and since the principal is risk-neutral, no further distributional assumptions are needed regarding Xt. The agent's actions are unobservable. Hence, neither the output nor the agent's actions are contractible. A contractible performance measure xt is observed at the end of each period. The agent's effort in period t affects only the mean of the performance measure in that period, xt = mtat+it , (4.189) 133 where et are mean zero noise terms. The noise terms in the performance measures are joint normally distributed with variance-covariance matrix E/v- In addition,for each rj, denote by E 4 the variance-covariance matrix of xx,..., xt. The variance-covariance matrix E ^ is defined recursively as follows. S i cr a pe[(t-l) pex{t-l) E t _ i (4.190) (4.191) where e[(t) = (1 ,0 , . . . , 0) is a row vector of length rj. For each 1 < rj < N, let a t = (ax,..., at), and x t = (xx,..., xt) denote the histories of means and outcomes from the first t variables in the sequence. Important notat ion convention. Throughout the paper the variance-covariance matrices are written such that the lower right corner corresponds to the lowest value of the time index and the upper left corner corresponds to the highest value of the time index. For example, the matrix E t is written as follows: cov(xt,xt) cov(xuxt_i) . . . cov(xuXi) cov(xt-i,xt) cov(xt-i,Xt-i) ••• cov(xt-i,Xi) cov(xi,xt) cov(£i ,£ t _i) cov(xi, Xi) In terms of the interpretation of the signals xt as managerial action (effort) plus noise, the noise terms are joint normally distributed and et ~ N(0, a), cov(£t,et±i) = p and cov(et,et±k) = 0 for all k > 2. The particular form of E/v gives the simplest variance-covariance matrix in which there is correlation among periods. The structure is such that, in the sequential model, the entire history is relevant in conditioning the current variable, although variables more 134 than one period in the past are independent ofthe current one. When p = 0, T,N = aIN, where 1^ is the N x N identity matrix and in this case the noise terms it are i.i.d. The variances and covariances of the noise terms need to be restricted for the matrix E/v to be positive definite. The key requirement is that the absolute value of the covari- ance between two adjacent periods' noise does not exceed half the variance of each noise term. The following Lemma formally proves this result. L e m m a 4.2.1 Let Dt — det(T,t) for all t > 1 and set D0 = 1. Then, for all t > 1, A + i = oDt - p 2 A - i • (4.192) In addition, a > 0 and 0 < \p\ < a/2 are sufficient conditions for T,t to be positive definite for all t < N. Proof . See the Appendix. • An alternative description of the performance measures can be given as follows. When the correlation is positive, the noise terms can be decomposed as et-i = i>t-i + St-2 + 8t-i (4.193) it = h + St-i + 6t (4.194) i t + i = vt+i + 5t + 5t+i , . (4.195) where ut, are period-specific, and 8t is a common component in adjacent periods. A l l terms are mutually independent and var(^ t) = p. The common (between adjacent pe- riods) component 8t can be thought of as a shock that persists from one period to the next period. The noise component 8t first appears in period t, persists for one period and then disappears. In period t + 1, a new noise term 8t+i appears, and so on. 135 When the correlation of the performance measures is negative, the noise terms can be decomposed as et-i = ut-i ~ St-2 + St-i (4.196) it = ut - 6^! + 6t (4-197) et+l = ut+i -6t + St+i , (4.198) where i>t, are period-specific, and St is a common component in adjacent periods. A l l terms are mutually independent and v a r ^ ) = p. The common (between adjacent peri- ods) component St can be thought of as period t accruals that have to be reversed in the next period, t + 1. Thus, negatively correlated noise in accounting-based performance measures reflects the nature of the accrual process. In this model, however, the accruals, as with all the other components of the noise in the performance measure, are outside the manager's control. In view of the above discussion, the result of Lemma 4.2.1 can be easily interpreted. First note that, given the mutual independence of the terms in the decompositions of noise et, a = var(e t) = var(z>t) + var(5 t_i) + var(J t_i) = var(P t) + 2p . (4.199) It then follows, that unless there is no period-specific noise ut, the covariance p is always less than half the variance a. The limit case is when there is no period-specific noise, and the noise terms consist only of the components that carry over from period to period. This restriction is specific to the A^-period model, since there are two terms that carry over from period to period in each of the "middle" noise terms it with t =fi 1,N. By contrast, in the two-period case, there is only one term that carries over from period to period, and the correlation is unrestricted between -1 and 1. 136 The principal owns the production technology for N periods and needs an agent to supply productive effort. There is more than one agent that the principal can employ in each period. A l l agents are identical (the agents have the same ability and the same utility functions) and have alternative employment opportunities. Each agent's reserva- tion certainty equivalent is normalized to zero in each period. Both the principal and the agents are assumed to have discount rates of zero. Utility functions, discount rates, reservation wages, the nature ofthe production technology, and the information structure are common knowledge. Throughout the paper, the following basic time line is present. At the start of the first period, the principal and the agent sign a contract. After contracting, in the first period, the agent provides effort ax, after which the first performance measure is reported. At the end of the first period, the agent may receive some compensation depending on the particular contracting setting. At the start of period t > 1, recontracting, or contracting for period t may take place, depending on the particular setting. After contracting, the agent provides effort at, and then the performance measure xt is reported. At the end of period N, all remaining contracts are settled. After the end of period N (i.e., after all contracts are settled), the outcomes zx,..., Zjv are revealed to the principal. Thus, in each period, all or some of the following events occur in this order: contracting (at the start of the period), productive effort supplied by agent, performance measure reported, contract(s) settled (at the end of the period). As in the two-period case, four types of commitment can be considered: full commit- ment to a long-term contract, commitment with renegotiation, commitment to fairness, and no commitment. The first two types of commitment (full commitment, commit- ment with renegotiation) assume that the agent is compensated by a long-term contract c ( x i , . . . , x/v) to be settled at the end of period N. The other two types of commitment (commitment to fairness, no commitment) assume that the agent is compensated by a 137 series of short-term contracts (ci(xi),. . . , C/v(xi,.. . , x^)) to be settled at the end of each period. Contracts are always assumed to be linear, and the only contractible information is given by the N performance measures. The resulting contracts are thus the optimal linear contracts in each case, although linear contracts are not optimal. 2 The main issue here is agent tenure in an N-period L E N 3 framework and how lack of commitment gives rise to ratcheting. The above description of the information environment gives the ex-ante (time zero) structure. In developing the principal-agent model in which contracting and action choice take place sequentially, I need to describe the information structure at each point in time given a realization of history up to that point. Since xi,...,x^ are joint normally distributed, it follows (see for example Greene(1997), Theorem 3.6) that the conditional distribution of xt+i,..., SAT given x\,..., xt is also normal. L e m m a 4.2.2 The conditional distribution of future information given the history from the first t periods, specified past actions at, and specified future actions at+i,... ,a^ is normal and given by (xt+i, xN)\{xt,at, dt+i, ...,aN)~ N((Et[xt+i],Et[xN}), Et+i,N) where the conditional means are determined by Et[xt+i] = mt+lat+i + pei(£)E7/ J(£ - mtat)' (4.200) 2 T h e Ho lms t rom and M i l g r o m [17] f ramework is not appl icable here p r imar i l y because the per iods are not independent (the fact that the per formance measures are correlated is a key assumpt ion i n my mode l , whi le independence is a key assumpt ion i n the Ho lms t rom and M i l g r o m model ) . H o l m s t r o m and M i l g r o m [17] present sufficient condi t ions under wh ich l inear contracts are op t ima l i n a mul t i -per iod agency. In their mode l , the agent's act ions generate a sequence of independent b inary signals (one in each per iod) and the agent's u t i l i ty is exponent ia l . 3 L i n e a r contracts, Exponen t i a l ut i l i ty , N o r m a l d is t r ibut ions. 138 Et[xt+2] = mt+2at+2, • • •, Et[xN] = mNaN (4.201) and the conditional variances are determined by E; 't+i,t+i — cr t + i where o~t+i = o — pe\(t)Y,t 1 pe\(t) (4.202) E; •t+l,k a pe[{t-l) p e i ( t - l ) E s + i i f c _ i for k > t + 2 (4.203) In other words, the information structure is the same at time t as at time zero except that the conditional expectation and the conditional variance of xt+i are affected. I will use the notation Et[-] for conditional expectation given history x t and cov t(-,-) for conditional covariance given history xt. Throughout the paper, the notation m t a 4 refers to multiplication component by component, that is m t a t = {m\a\,... ,mtat). Since the means of the performance measures given past outcomes are influenced by managerial effort, conditional expectations will depend on either observed or conjectured past managerial effort. Specifically, the manager knows his past actions so from his perspective, Et[-] = E j - jx^aJ . The principal, on the other hand, does not observe the manager's actions, but has conjectures about the agent's past actions ht = ( d i , . . . ,d f). Thus, from the principal's perspective, Et[-] = E t[- |x 4 ,a t]. In addition, the conditional expectations at time t for any future performance measures Et[rrt+fe] implicitly assume conjectured future actions at+k both from the principal's and the agent's perspective. Note that at = var(x t |x t_ 1) is the conditional variance of xt given history x t - 1 . The conditional variances do not depend on the agent's actions and represent the common posterior beliefs of the principal and the agent about the variance of future performance measures given past observations of the performance measures. In addition, note that while the conditional expectations depend on past observed values of the performance 139 measures, conditional variances do not depend on these values, but only on time and the observability of these performance measures. I now turn to the derivation of explicit formulas for at and E t [ x t + i ] . L e m m a 4.2.3 l?t[xt+i] = m i + 1 a 4 + 1 + Rt • (xt — mtat)', where Rt = ^-(Dt.1,-pDt_2,...,(-p)t-1) = J l j ~P ^ ( -P ) 2 \Ot OtOt-\ OtO-t-\Ot-2 ' o t . . . O x and at+i = vart(xt+i) = —=—. Ut In addition, (o~t)t>\ is a decreasing sequence and at > Pi, whei Proof . See the Appendix. • Lemma 4.2.3 shows how past history influences conditional expectations for the cur- rent period's variable. Thus, to obtain the current conditional expectation, the uncon- ditional mean is corrected by a sum of weighted deviations of past results from their means. The signs of the weights of these corrections are alternating if the correlation between periods is positive; otherwise, all weights are negative. In absolute value, the weights attached to events from the more distant past are smaller relative to more recent events. This follows from the fact that p < a/2 < p,\ < at and the fact that each weight is a product of the form ±(p/ok-i)... (p/ak-p). The intuition is that the influence of distant events is weaker than that of closer ones because the correlation is not direct but through a chain of pairwise correlated variables. Also note that Rt can be rewritten as Rt = ^-{l,Rt-i) , (4.205) (4.204) epi a+d'o2- 4p2 140 which implies Et[xt+1] = mt+lat+l + ̂ -{xt- Et_i[xt}) . (4.206) When the correlation between periods is positive, the current conditional expectation is adjusted upwards from the mean by a term proportional to the amount that the last outcome exceeded previous expectations. Thus, an increase in past expectations E t _i[x t ] will decrease the current expectation E t [ .x t + i ] . When the correlation between periods is negative, the opposite is true; an increase in the last period's outcome will decrease the current expectation, while an increase in last period's expectation will increase the current expectation. The proportionality factor in equation (4.206), p/at is essentially a conditional correlation since p/at = covt-i(xt,xt+i)/vari(xt). Given linear contracts and normally distributed performance measures, the agent's wealth is normally distributed as well, which implies that the agent's certainty equivalent of wealth and effort is given by equation (4.187). More precisely, given a linear contract in all N performance measures, the certainty equivalent of compensation at the start of the first period is given by ACE(w,au...,aN) = E[w] - ^rvar(w) - ^(a 2 + h a2N) , (4.207) LJ LJ where ax,..., aN are the actions that the agent expects to take in the N periods. Sim- ilarly, for any contract that is linear in performance measures xt,...,x^, the agent's certainty equivalent at the start of period t is given by A C E (w, a t , a N |x 4_!, a t_x) = Et[w\x*-i>&-i] - \r™xt(w\xt-i>**-i) ~ \(a2t + • • • + a/v) > (4.208) where at_j is the history of actions already taken, and at,..., are the actions that the agent expects to take in periods t,... ,N. Note that, while the conditional expectation 141 Ej[iy|x 4_ 1, a^-J depends both on the observed values of x\,... ,xt~i and the agent's pre- vious actions, the posterior (conditional) variance var4 (w|x(_1, a t_ a) does not depend on either the realized value of xx,..., xt-\, or on the agent's actions ax,..., at-\. Through- out the paper I consider the actions at_1 a sunk cost at the start of period t, and therefore not directly included in the agent's certainty equivalent (4.208). Past actions impact the agent's welfare at the start of period t only through their impact on the means of past performance measures in the conditional expectation E t[ ,tD|x t_ 1, a t_i]- The agent's wealth w represents the total compensation to be received by the agent, and the agent is indifferent as to the timing of consumption. Thus, the agent's utility ensures that there are no intertemporal consumption smoothing issues. In addition, the agent's exponential utility eliminates wealth effects, in that compensation paid (earned) does not impact the agent's risk preferences. The quadratic cost of effort, additive across tasks, means that the agent is not indifferent to the allocation of effort among tasks in the TV periods.4 4.3 Full commitment, long-term contract In this section, I assume that the principal can commit at the start of the first period to a TV-period contract. The terms of this contract are not subject to renegotiation. Furthermore, if the agent accepts the contract at the start of the first period, he commits for TV periods, and cannot leave in a later period. These assumptions about the parties' ability to commit make the model equivalent (within the L E N framework) to a TV task, TV correlated performance measures, as analyzed by Feltham and Xie [11] and Feltham and Wu [10]. 4 The agent's exponential utility u(w, a\,..., ayv) = — exp[—r(w — ̂ -{a\ H 1-Q/v)] implies that, over N periods, his certainty equivalent is ACE(u), a\,..., a^) = E[w] — | r var(w) — \{a\ H h a?N), which is not a function only of total effort ai + • • • + a^. 142 The time line of events is as follows. 1) At the start of the first period, the principal offers the agent a linear contract based on the Af performance measures that are contractible information: c = a0 + p1x1 + --- + /32xN . (4.209) The coefficient a0 represents the fixed wage to be paid the agent, and the coefficients Pi,..., PN represent the variable wages to be paid the agent for each unit of the perfor- mance measures. 2) If the agent accepts the contract, he provides productive effort a 1 ; . . . ,a^ in each of the A^ periods. 3) After the agent has provided effort at, which is not observed by the principal, the performance measure xt is publicly reported. 4) At the end of period N, the contract is settled. After the end of the period N, the outcomes Z\,..., are revealed to the principal. Note that in this case, the timing of information and the timing of payments to the agent is irrelevant, the only constraint is that the performance measures are available before the contract is settled. The assumption that the principal can commit to a long-term contract guarantees that after x\,... ,xt are observed, the principal will not be able to modify the contract. The L E N assumptions guarantee that the agent's choice of action in period t will be independent of the observation of performance measures x t - 1 and of the agent's actions &t_i- The reason is that, at the start of period t, the part of the agent's certainty equiv- alent of compensation conditional on xt_i,&t_i that is variable in at, and thus provides incentives for at, does not depend on either x t - 1 or a t_ x . Here, each of the assumptions of the L E N model is essential. The exponential utility guarantees that wealth does not affect the agent's risk preferences. The linearity of the contract and the fact that the 143 performance measures are joint normally distributed with additive noise ensure that xt_x and a t_! impact on the agent's compensation separately from at. In the L E N model with full commitment, the timing of xx,... ,xt-i is not important, only their contractibility matters. However, in other settings, the timing of information is important, and if the agent observes the first-period performance measure yx before selecting the second-period action a 2 , then a2 may depend on yx. See for example the two-period model of sequential action choice with correlated periods of Matsumura [27]. With full commitment to a long-term contract, the only role played by the correlation between the two performance measures is through the impact on the total risk to which the agent is exposed for incentive purposes as measured by var(c) = var(/3i5;i + • • • + 0N%N)- The principal's problem is to maximize, at the start of the first period, the expected total outcome net of the agent's compensation, subject to the agent's participation con- straint and the agent's incentive compatibility constraint. Before presenting the solution to the agency problem under full commitment, I need some notation. Let M be a di- agonal N x N matrix with m- i , . . . , m^r on the main diagonal. Then, the performance measures can be written in vector form as x = Ma! + e, where a! is a column vector with components a\,..., a^. Let b = (61,..., b^)' be the column vector with compo- nents bi,..., b?{ such that the principal's gross payoff is z = b'a + A\ + • • • + XN- With this notation, the N-period full commitment contract is characterized by the following Proposition. Proposition 4.3.1 The optimal linear contract c = ao + 0 i X i + - • - + 0NXN with full com- mitment for N periods and the optimal actions are characterized by d\ = 0\m\,..., ajv = 144 (3NmN, 0 = QMb , (4.210) a = M0 = M QMb (4.211) where Q = (M2 + rE/v) 1 - The principal's expected surplus is TT = b • a - -r var{0iXi H h 0N%N) - H r- d2N) • (4.212) Proof . See Feltham and Xie [11]. • 4.4 Commi tmen t and renegotiation In this section, I assume that the principal and the agent commit at the start of the first period to a iV-period contract subject to renegotiation in subsequent periods. Specifically, the principal commits to a TV-period contract and the agent commits to stay for N periods if he finds the initial contract acceptable. However, the principal and the agent cannot commit not to renegotiate after the performance measures are observed. The terms of the contract are subject to renegotiation in the usual sense: the existing contract can only be replaced by a new contract if both parties agree to it. The renegotiation takes the form of a take-it-or-leave-it offer by the principal. The time line of events is as follows. 1) At the start of the first period, the principal offers the agent a linear contract based on the N contractible performance measures: The coefficient Ofo represents the fixed wage to be paid the agent, and the coefficients 0i,..., 0N represent the variable wages to be paid the agent for each unit of the perfor- mance measures. + 0{1x1 + --- + PNXN . (4.213) 145 2) If the agent accepts the contract, he commits to the terms of the contract, unless both parties agree later to replace it by a new contract. 3) In the first period, after the agent has provided effort a\ unobservable by the principal, the performance measure x\ is publicly reported. 4) At the start of the second period, the principal can make a renegotiation offer. The principal cannot fire the agent without the agent agreeing to leave at the start of the second period, and the agent cannot leave in the second period without the principal agreeing to it. In addition, a new contract cR1 = aR1(xi) + 0Rl(xi)x2 H h (3§l(xi)xN can replace the old one only if both the principal and the agent weakly prefer it. The renegotiation offer is exogenously assumed to be linear in keeping with the restriction to linear contracts in the model. The coefficients of the renegotiation offer may depend on the first performance measure which is known at the time of renegotiation.5 5) After the renegotiation stage, the agent provides effort a2 and then x2 is reported. At the end of the second period, the contract in effect after renegotiation is denoted by CIZ since this is also the contract in effect at the start of the next period. 6) At the start of period t, the principal can make a renegotiation offer cRt = a 0 I J i(x t_ 1) + /^ f (x<_i)£t + • • • + /3^(xr-i)^/v- At this time, the contract under renegotiation is cn, which is the outcome of period t — 1 renegotiation. The contract that is renegotiated consists of a fixed payment determined by the history of past performance measures and variable payments for the remaining performance measures that are still uncertain at the time of renegotiation. The renegotiation offer is then restricted to a contract of the same form, that is the principal may offer a different fixed payment and different variable pay- ments only for the performance measures that are reported in the future. A l l payments, fixed and variable, may in general depend on past history. 5 A t the time of renegotiation, the initial contract is linear in the performance measures x^,... ,XN, which are still uncertain. I assume that a renegotiation offer is restricted to having the same linear form. 146 7) After xN has been reported, the contract is settled at the end of period 7Y. After the end of period N, the outcomes Z \ , . . . , Z N are revealed to the principal. As in the full-commitment case, the contract is settled at the end of period N. The main difference is that the contract can be renegotiated each time after the agent has taken period rj action and the period rj performance measure has been reported. In this context, a renegotiation-proof contract is a contract c = «o + + • • • + 0N%N such that, once agreed upon at the start of the first period, there does not exist a contract at any later renegotiation stage which is weakly preferred by both parties and at least one party strictly prefers. The fact that a linear contract can be renegotiation-proof is due to the particulars of the L E N framework. It turns out that, for every initial contract that is linear in x\,..., XN, the optimal renegotiation offers (which are restricted to be linear in xt,...,XN) have period t,... ,N incentives independent of x t _ x and a fixed wage that is linear in xt_1. In other words, the renegotiation offers are also linear in the N performance measures from an ex-ante (start of the first period) perspective. An equilibrium in the principal-agent renegotiation game6 consists of a sequence of contracts ( c n , cR1,..., cIN, cRN), the agent's actions a\,..., aN, and principal's beliefs about the agent's actions ai,...,art such that: (i) the agent accepts both the initial contract and all the renegotiation offers, and rationally anticipates all renegotiation offers and their acceptance when selecting his actions; (ii) the principal's beliefs are correct at = at for all rj; (iii) the sequence ( c n , cRl,..., cIN', cRN) is ex-ante (start of the first period) optimal and (c7*, cRt,..., cIN, cRN), is ex-post (start of period rj, for all rj, conditional on x.t-n&.t-i) optimal from the principal's point of view. The following proposition shows that the analysis of the equilibrium can be restricted 6 Note that, in general, an equilibrium in the renegotiation game is not given by a renegotiation-proof contract. 147 without loss of generality to renegotiation-proof contracts. Propos i t ion 4.4.1 / / the sequence (c11, cR1,..., cIN, cRN, a i , . . . , a/v) of contracts and actions is an equilibrium in the principal-agent renegotiation game and cRt is linear in xt,..., XN at the time of renegotiation, then cRi is also ex ante (at the start of pe- riod l,...,t — 1) linear in X\,...,XN and offering the contract cRN in every period (cRN,..., cRN, a i , . . . , a/v) is an equivalent equilibrium with a single renegotiation-proof contract. Proof . First, I show that any renegotiation offer that gives the optimal last period incentive and is acceptable to the agent is linear in the performance measures from an ex-ante perspective. At renegotiation time, the principal is restricted to offer a linear contract cRN = aRN(xN_1) + /3^A r(xw_ 1)x/v, whose coefficients may depend on the first N — l performance measures. The period N incentive 0RN does not depend on x J V _ 1 since it is determined only by the conditional variance var(xAr|xjv-i) which is independent of the actual values of X J V - I • Furthermore, from the participation constraint at renegotiation time it follows that A C E ( c / A r , a J v | x i V _ l J a i V _ 1 ) = E ^ l x ^ , a i V _ 1 , aN] - var(c / |xA,_ 1) - ^a2N < ACE(c R A r , a jv | x A f _ 1 , a J V _ 1 ) = a; 0 , i V(x i V_ 1, a ^ ) + /^E^/vIx^, a,N_u aN] - ^ v a r ^ l x ^ ) - ^a2N . (4.214) Since the initial contract is linear in x\,...,xx and since the conditional mean of the last period performance measure E [ £ / V | X J V - I ] is linear in x\,..., xjy-i (due to the joint normality ofthe distributions), it follows, solving the equation implied by assuming the 148 participation constraint(4.214) to be binding for aRN(xN_x), that aRN is linear in X / V - D aRN = E[c / A r |x A r _ 1 ,a A r _ 1 ,a A f ] - /3A^ArE[x iV|xAr_1, a A r_ 1 , aN] ! (4-215) - -r[var(cIN\xN_1) - var(c f l J V |x J V_ 1)] . It then follows that, conditional on being accepted by the agent, the renegotiation offer is of the following form cRN = E[c / A r |x A r_ 1 ,a A r_ 1 ,a Ar] + BRN{xN - E[xN\xN_x, §.N_X, aN}) - ]-r {V&T{CIN\XN_1) - var(cR 7 V|x i V_1)] Z (4.216) = C I N + ( / ^ - /&")(% - E M x ^ , ^ , ^ ] ) - [var(cIN\xN_1) - var(c i i i V|xA r_1)] . From the above equation it follows that A C E ( c R J V ) = A C E ( c / A f ) at the start of period N — 1, and that cRN is acceptable at the start of period N — 1 if, and only if, cIN is acceptable. Thus, cRN is accepted by the agent if offered as a period N — 1 renegotiation offer. (Note that, in equilibrium, cIN is the outcome of renegotiation from period N — 1.) Since cRN is an optimal renegotiation offer given cIN, and since the efficient period N incentive is independent of x A r _ 1 , a A f_ 1 , the principal has no incentive to renegotiate cRN if it is the initial contract in period N — 1. This proves that cR is renegotiation-proof in period N and it can replace c R i V _ 1 i n period N — 1. It remains to show that cRN induces the same actions as (cIN, cRN), since the payments to the agent and the principal's surplus are uniquely determined by the agent's actions. The period N action is determined by cRN in both cases, since that is the contract in effect at the time the agent provides effort aN. Since ( c / A r _ 1 , c R J V , ajv-i, ajv) is an equilibrium in the last two periods, the period N — 1 action is chosen in anticipation of the period N renegotiation, and so is determined by the (linear) incentive contained in aRN(XIM-I). The reason is that it is suboptimal for the agent to select an action different 149 from a^v-i and then reject the renegotiation offer. Thus, the agent's period N — 1 action is determined only by the incentives in cRN, and so is the same when cRN is the period N — 1 renegotiation offer. In the same way, it is shown by backwards induction that cRN can be offered as a renegotiation offer in periods N — 2,... ,2, and as an initial contract c11 at the start of the first period. • The main idea in the proposition is that if the contract is renegotiated in equilibrium, the agent's actions are completely determined by the renegotiated contract. The principal cannot gain by offering the agent a contract that will later be renegotiated as long as the agent anticipates the renegotiation. The principal's problem is to maximize at the start of the first period the expected total outcome net of the agent's compensation, subject to the agent's participation con- straint, and the agent's incentive compatibility constraints. The contract must also be renegotiation-proof at the start of subsequent periods. Since at the time the contract can be renegotiated, the actions a i , . . . , at-\ have already been taken, and the performance measures x\,..., xt-i have been reported, the only part of the agent's compensation that remains uncertain is the period t,... ,N variable wage Btxt 4- • • • + PN%N- The requirement that the contract is renegotiation-proof means that at the start of periods t = 2 , . . . , N, the contract maximizes the principal's expected total outcome net of the agent's compensation conditional on information from periods 1,... ,t — 1 and subject to the agent's participation constraint and the agent's period t,... ,N incentive compat- ibility constraints. Propos i t ion 4.4.2 Given the optimal linear N-period renegotiation-proof contract c = ag + 3\X\ -\— • + QN%N) the optimal actions are characterized by at = mt0t, for all t and "o = - + • • • + a2N) + \r vai{pxxx + ••• + BNxN) . (4.217) 150 The principal's surplus is given by 7r = M i H h bNaN - var(0iXi -\ h 0N£N) - ^(a? H r- a^) . (4.218) Proof . At the time the agent selects action at at the start of period t, his certainty equivalent of compensation given conjectured future actions S 4 + 1 , . . . , aN is A C E ( c , a t + i , . . . ,aN I X t - u a ^ ! , ^ ) = o 0 + H h /3t-ixt-i + BtE[xt I x t - i , at] Note that, at the time the agent chooses action at, effort in the previous periods is sunk, and does not enter the agent's calculations. Furthermore, the fixed portion of the agent's compensation ao + 0\X\ + • • • + 0t-ixt-i, the variable payments in future periods 3t+1E[xt+i I x t _i,a t _i,a t ,a 4 +i] -\ \- 0NE[xN \ x 4 _ l 5 a ^ , at, aN], the risk premium for the variance of the contract, and the cost of future actions do not depend on the choice of period t action. It then follows that the only part of the agent's certainty equivalent (4.219) that depends on at is (4.219) + 0t+iE[xt+1 \ xt_l,&t_1,at,at+1] + h 3NE[xN | x t _ i , a 4 _ 1 ; at, aN] - ^rvar(c | x ^ ) - ^(a 2 + a 2 + 1 + • • • + a?N) . 0tE[xt I a*] - - a 2 (4.220) = 0t{mtat + pe[(t - 1) mt_ia t _i ) ) - -at . The first-order condition for the incentive compatibility constraint is _d_ dat ACE(c, at+i, ...,aN \ xt_x, a ^ , at) = mt0t - at = 0 (4.221) It follows that the action induced by the contract is at = mt0t • (4.222) 151 As in the full commitment case, the principal's expected surplus, 7T is the gross benefit b i d + • • • + frjv&/v less compensation for the agent's effort \{a\ + • • • + a2N) and a risk premium for the variance the performance measures |rvar(/3ix x + • • • + B^x^). d The renegotiation concept I use here is the same as that of Fudenberg and Tirole [13] in that both parties must agree to the renegotiated contract, but the timing is different. In my model, renegotiation takes place after the performance measure xt is observed, while Fudenberg and Tirole have the renegotiation take place between the time the agent takes the action and the time the performance measure is observed in a single period model. Having the renegotiation take place after xt is observed and before at+\ is taken avoids the insurance/adverse selection problem of Fudenberg and Tirole. 7 The driving force behind renegotiation in my model is the principal's desire to op- timally adjust the agent's induced effort in subsequent periods to match the (reduced) ex-post variance of the performance measures. In this context, the fact that xt is ob- served is important, and not the actual value of xt. As a result, period t,... ,N actions and period t , . . . , N incentives are independent of X f _ i , the agent's actions a 4 _j, and the principal's conjectures of the agent's actions a^ , . The inability of the principal to commit not to renegotiate is ex-ante inefficient rel- ative to the full commitment case. The reason is that the renegotiation-proof contract imposes additional binding constraints on the optimal contract. In both cases, the prin- cipal chooses the agent's actions to be induced by the contract in order to maximize his surplus as given by (4.218). In the full commitment case, the principal maximizes 7T unconstrained, while with renegotiation in periods 2 , . . . , N, the contracts are con- strained. Since the unconstrained optimum is at least as high as the constrained one, 7 I f renegotiation takes place before xtl is observed, the agent is offered insurance by the principal for the risk in xt when the agent has private information about the action at. As a result, the principal • would like to perfectly insure the agent in order to avoid paying compensation for risk, and this destroys the agent's incentive to provide effort because the agent anticipates the renegotiation. Fudenberg and Tirole show that the only equilibria involve randomization by the agent in choosing his action. 152 the full commitment contract is at least as efficient as the renegotiation-proof contract. If p = 0, that is if the performance measures in the two periods are independent, there is no difference between the two contracts, which take the form of repeatedly inducing the optimal action from the one period problem. Explicit formulas for the optimal actions will be derived in the next section, where a mechanism for replicating the renegotiation-proof solution by short-term contracts is presented. 4.5 Shor t - term contracts under commitment to fairness In the previous two sections, I presented the two types of commitment that relate to N-period contracts: full commitment and renegotiation-proofness. In this section, I turn to the analysis of short-term contracts in an N-period relationship. Short-term contracts are used whenever the parties cannot write long-term contracts. The inability to write a long-term contract does not necessarily imply that there is no long-term commitment (that is commitment beyond the first-period) in the principal-agent relationship, only that the principal cannot commit to a specific contract in a period other than the current one, or that the contracting environment is such that it prohibits the use of long-term contracts. The main idea in this section is an intermediate form of commitment, whereby the agent commits to stay for N periods, while the principal commits to offer contracts that provide the agent with his reservation certainty equivalent for each period based on the public information at the contracting date and the principal's conjectures of the agent's actions. This form of commitment is an adaptation of the concept of fairness introduced by Baron and Besanko [1]. A contracting relationship is governed by fairness if the principal is restricted to fair wages and the agent must participate in all periods if he accepts the 153 contract in the first period. Fair wages are paid when the agent gets his reservation wage as if he could leave in each period. That is, the agent's certainty equivalent of future compensation, conditional on available information and conjectured actions is set to zero at the start of each period. Thus, in addition to the usual contract acceptance constraint at the start of the first period, there are additional constraints that the period t contract is acceptable to the agent as if the agent had other employment opportunities, had not committed to stay for all periods, and had taken the conjectured actions in periods 1,... ,t — 1. The key fact here is the agent's ability to commit to stay for N periods; removing the agent's ability to commit for all N periods leads to a situation where there is no equilibrium in which the agent stays in all N periods (see also the detailed analysis in Chapter 3 of the agent's commitment and its impact on the equilibrium). The agent gives up his ability to leave in subsequent periods for the guarantee of fair compensation in periods 2 , . . . , N. The time line of events is as follows. 1) At the start of the first period, the principal offers the agent a linear contract based on the first-period performance measure x\\ c1 = ai + p\xx . (4.223) The coefficient ct\ represents the fixed wage to be paid the agent, and the coefficient Pi represents the variable wages to be paid the agent for each unit of the performance measure. 2) If the agent accepts the contract, he commits to stay for all N periods. The principal commits to offer the reservation certainty equivalent of wages in each period, given all the information available at the time of contracting and given that the agent's actions correspond to the principal's conjectures. 3) In the first period, after the agent has provided effort ax unobservable by the principal, 154 the performance measure X\ is publicly reported. 4) After the performance measure X\ is publicly reported, the first-period contract is settled at the end of the first period. 5) At the start of the second period rj, t > 2, a new contract is offered by the principal ct = at(*t-i>Qt-i) + Pt(xt-i,Qt-i)xt • (4.224) This contract is subject to the agent reservation wage restriction and is accepted by the agent (since I assumed commitment for all N periods on the agent's part). The terms of the period rj contract may depend on the past observed values of the performance measures and on the principal's conjectures about the agent's past actions ( x ^ ^ a ^ ) , since they are informative about xt, and therefore can be used to reduce the noise of the period rj performance measure. 6) In period rj, the agent provides effort at, then the performance measure xt is reported. The period rj contract is settled at the end of period rj. 7) After the end of period N, the outputs z\,..., are revealed to the principal. Short-term contracting with fair wages relies on the principal's conjectures regarding the (unobservable) agent actions. At the start of period rj, when ct is set, the terms of the contract depend on E ^ x ^ , a t - 1 ] , where a t - 1 are the principal's conjectures of the agent's actions in periods 1,..., rj — 1. The concept of fair wages in period rj assumes that the principal's conjectures are correct, that is the agent actually has provided effort in the first rj — 1 periods. At the start of the first period, and any subsequent period rj, fair wages refer to the total future compensation paid to the agent over the remaining N—t+1 periods c tH \-cN, and involve both the principal's and the agent's conjectures of future actions at,...,aN. The principal and the agent are assumed to be in agreement over the conjectured future actions. For example, the principal states his conjecture &i as part of 155 the first-period contract C\. The agent then agrees to that conjecture when accepting the initial contract, and that will be the basis for setting the second-period contract.8 While the agreement on conjectured actions restricts the principal's contract choices (given fairness and past performance measures), the agent's actions are unconstrained, even though, in equilibrium, the agent will choose to implement the conjectured actions. Thus, an optimal sequence of contracts under commitment to fairness is also charac- terized by a rational expectations equilibrium regarding the agent's actions. The agent's commitment to stay for all N periods is essential in sustaining this rational expectations equilibrium, as will be shown later in this section. The principal's problem at the start of each period is to maximize the remaining expected total outcome net of the agent's compensation, subject to the agent's reservation wage constraint and the agent's rationality constraint. Since the agent commits to stay for all N periods at the start of the first period, subsequent contracts c t + i , . . . ,CN are anticipated at the time the period t contract is set. Let wt denote the agent's cumulative compensation net of personal effort cost from period t to period N wt = ct-^at + h cN - ^a2N . (4.225) At the start of period t (at contracting time), the manager's objective is the certainty equivalent of wt conditional on all available information x t _ l 5 a t_ x , and on conjectures about future actions at, . . . , aAT , denoted ACEt_i(?/;t|x f_ 1, &t-ii at, • • •, a/v)- A C E t - i ^ t l x t . ! , ^ ) = Et-i[ct\lk-i>*t-i] ~ \hl i l x (4.226) + r -E t _i[c / V |x t _ 1 ) a ( _ 1 ] - -a2N - -rvaxt-i(ct + h cN) 8 A n alternative would be that both parties solve for the unique equilibrium conjecture h\. In this case, the principal can use an expert witness (professor of accounting) to prove that the second-period wage is fair. 156 The values at,..., a/v represent at this time the agent's conjectures of his future actions. The agent correctly anticipates that the terms of future contracts will depend on all history available at contracting time. At the start of period t, the principal chooses at and 8t, taking into account all available information xt_1 and his conjectures regarding the manager's actions, past and future, ai,...,Ojv. In a rational expectations equilibrium, at each point in time, the * principal's conjectures about past managerial actions are correct, and the principal and the manager have the same conjectures regarding the manager's future actions. In other words, at = dt for all values of t. The following is a key technical result needed for the solution to the agency problem. The Lemma proves that, from the perspective of the start of period t + 1, future perfor- mance measures in periods k > t + 2, adjusted for conditional expectations given start of period k information, are mutually independent. L e m m a 4.5.1 Given information history xt, the following relations hold: Proof . See the Appendix. • The intuition for this result is that adjusting for expectations based on available information removes the common components in the decompositions (4.193)-(4.195) or (4.196)-(4.198), so that (4.193)-(4.195) become covt(xk - Ek_i[xk],xi - Ei^[xi)) = 0fork^l>t + 2 (4.227) covt(xt+1,xk - Ek_x[xk)) = 0 for k>t + 2 . (4.228) et-i - Et-2[er*-i] = i>t-\ •+ 8\ (4.229) et - Et-i[it] = ut + 6t (4.230) (4.231) 157 The Lemma implies that, if the agent's certainty equivalent of period k compensation, conditional on information available at the start of period k, is zero for k > t + 1, then the contracts c t , . . . , c /v are mutually independent as seen from the start of period t. The key is that 4CE f c _i ( c j b , a f c | x A _ 1 , a 4 ._J = 0 implies E j f c_ 1[c f c|a f c, x f c _ 1 , a ^ J - \a\ - \r varfc_i(cfc) = 0 and, as a result the fixed wage in period k is determined by ak = —Pk^k-i[xk\ak,^k-i^k-i] + ̂ al + ^r vaTk-i{^k)• It follows that the period A; contract can be rewritten as 4 = 0k(xk - E f c_i[f f c]) + \^a\ + ^rvar f c_i(c f c) , (4.232) and that the variable compensation for the agent is based on the performance measure adjusted for expectation. These adjusted performance measures are mutually indepen- dent as shown in Lemma 4.5.1, and from an ex-ante perspective (relative to period k), the agent's compensation in period k consists of the risk premium, the cost of effort, and a zero-mean variable compensation term. Thus, although the performance measures are correlated, it turns out that the fair contracts mechanism decouples the contracts by adjusting the fixed wage based on the history available at contracting time. Note also that the fixed wage in period k provides incentives for the agent's actions in periods 1,..., k — 1, since the period k fixed wage depends on the realized values of past perfor- mance measures X f c - i - As a result, the fixed wage ak depends on the random variables x ; , . . . , ik-i when anticipated at the start of period I < k. The fact that future contracts are independent of each other implies that the variance of the agent's future compensation in the agent's certainty equivalent (4.226) can be easily calculated as var t_i(c t H h cN) = var t_i(c 4) H 1- vart_i(c/v) . The following lemma shows that, in a rational expectations equilibrium, the fairness constraints determine the fixed wage for the agent in the period t contract as a function 158 only of the incentives in ct and conditional expectations of period t variables given history x t - 1 and conjectures at_1 (note that the fixed wage and all other terms of the contract are set by the principal based on his conjectures of the agent's actions). Moreover, as in a single-period problem, the fixed wage compensates the agent for his cost of effort plus a risk premium for the posterior variance of the period t compensation contract (performance measure). L e m m a 4.5.2 / / the fairness constraint is satisfied at the start of each period, that is if ACEt-^Wt , S j , . . . , a/v|x4_1, a i_ 1) — 0 for all 1 < t < N, then the agent's certainty equivalent of future compensation at the start of period t is the same as the certainty equivalent of period t compensation ACEt-i{wt,at,... ,01^x^,0,^) = Et-i[ct\xt-\i a^, at] - ^ a 2 - vart-i{ct) , (4.233) and the fixed wage set by the principal is at = -BtEt^Xtlx^a^, at] + ^ a 2 + vart„i(ct) . (4.234) In addition, at the time the agent selects the period t action, ACEt-i{wt,auat+i,... ,aN\xt_x, 0,^,0^,0^) - Et-\[ct\xt_l,at_x,at] - \a2t - \r vart_x{ct) 1 1 (4.235) + Pt+iEt-i [xt+i - [xt+i , Ot] H h QnEt-x [XN ~ ^/v-i[5/v|aAr]Ut_i, at_i,a«] Proof . See the Appendix. • At contracting time, the agent and the principal have common conjectures regarding the agent's future actions at,..., a^. Based on these common conjectures, and assuming 159 that the agent's past actions correspond to the principal's conjectures, = a t_ x , the agent's certainty equivalent at the start of period t is zero ACEt-i(wt) = 0. Furthermore, each line in equation (4.235) is equal to zero. However, when the agent's action in period t becomes a choice variable, each of the lines in equation (4.235) captures the impact of action at on compensation in periods t,... ,N. The compensation in future periods is affected by at since the fixed part of the agent's compensation in future periods is adjusted to extract all surplus from the agent based on past information. After the agent has accepted the contract ct, at the time of choosing the action at, let at denote the principal's conjecture of what action the agent will choose. Then, equation (4.294) in the proof of Lemma 4.5.2 implies E t _ i [xk -Ek-i[xk\at]\at] = Et-i [-Rk-i • ( x ^ - mk-i&k-i)\at,at] (E t _i [xt - mtat\at] - Rt_x • ( x ^ - mt-i^t-i)) (4.236) -mt{at - at) . o~k-i • • - o-t {-pf-1 ak-i •••ot Substituting in the agent's certainty equivalent (4.235) gives the agent's objective func- tion at the time of selecting action at, ACEt-^wt, at\at) = Et-i[ct\at,at] - ^a2t - irvar^^Ct) + mt -P ( -p ) W _ * Pt+i — {at ~ at) H h BN {at - at) at aN-i...at (4.237) From equation (4.234), given the principal's conjecture at, I have that at = -0tEt-i[xt\at] + \a2t + l-ratQ2t and (4.238) E t_i[ct|o t, dt] = mt0t{at - dt) + \ratd2t . (4.239) 160 As a consequence, the agent's objective function (4.237) becomes ACEt-i(wt,at\at) + mt8t+1 — (at -dt) + h mtBN <7jV_i • • - Ot N-t [at ~ dt) (4.240) Equation (4.240) captures the essence of the ratchet effect, in that it shows the agent's expectation of the impact that the period t action will have on setting performance standards in future periods. Note that an increased effort in period t has a mixed impact in future periods if p is positive; expected compensation will be reduced in some periods and increased in others. The magnitude of the effect decreases as the distance in time increases. In addition, future incentives j3t+i, • • •, PN directly influence these future impacts of at. If the correlation between periods is negative, increased effort in the current period increases compensation in all future periods. In addition, note that in each period, the agent's action choice is independent of past history and only depends on the impact of that action on future compensation as described by equation (4.240). The principal's problem is to maximize at the start of each period the remaining expected total outcome net of the agent's compensation, subject to the agent's reservation wage constraint and the agent's rationality constraint. maxE t _i [(zt - ct) H h (zN - CN)^^, a t _ l 5 dt,..., dN] (4.241) subject to the reservation wage constraint ACEt_i(wt) = 0 (4.242) and the agent's rationality constraint at e argmax ACE t _i(?j ; t |aud t) (4.243) 161 Theorem 4.5.1 The actions induced by the sequence of optimal contracts and the se- quence of optimal incentive intensities are given by mNbM aN = mN0N = 9 N N (4.244) mN + raN mtbt + — -^j—mt+lbl+l U w m* a* cr, mf , , „ 0t = pr ' m A + 1 4.245 p m* . a* = mtft - - — - a t + 1 . 4.246 °t mt+i The fixed part of the agent's compensation is at = - A ^ - i [ ^ t l a - i . a t - i ] + \at + \r°tPt • (4.247) Proof . See the Appendix. • The next task is to show that the short-term contracts with commitment to fairness replicate the payoffs of the renegotiation-proof contract as in the two-period case. This is done by showing that the optimal actions induced by the sequence of contracts in Theorem 4.5.1 are the same as the actions induced by the renegotiation-proof contract described in Proposition 4.4.2. From (4.218) and (4.247), the principal's surplus is the same in both cases, in that the agent is only compensated for his effort and a risk premium for the posterior variance of each performance measure. Since the payoffs are uniquely determined by the induced actions it follows that, from an ex-ante (start of the first period) perspective, the contract c = C\ A V cN is the same as the renegotiation-proof contract described in Proposition 4.4.2. The proof that both types of contracts induce the same actions is by backwards induc- tion, starting with the last period. Under commitment to fairness, as with renegotiation, the period 7Y action and the period N incentive do not depend on the previous actions, or on the specific value of previous performance measures. The reason, as before, is that 162 when contracting at the start of period N, the role played by the available information is to reduce the variance in the period N performance measure, and that variance does not depend on either the specific values of the previous performance measures or the previous actions. It follows that the principal's optimal choice of risk for the agent when setting the period TV incentive does not depend on either the specific value of the previous performance measures or the previous actions. Thus, the principal's problem in setting the incentive for period N is the same in both cases, and as a result the induced period iV action is the same. Once the period t incentive rate is set, the fixed wage is given by the fairness restric- tion. Then, the period t fixed wage depends linearly on x t - 1 because E[ f t | x t _ 1 , a ^ ] is linear in x t - 1 . From an ex-ante perspective, the period t contract contains a (random) linear term in which contributes in addition to the incentives from the period t — 1 contract to the agent's choice of action in period t — 1. At the start of period t — 1, the agent's cumulative future compensation c t _i + • • • + c/v is linear in xt-i, • • • ,5jv, while the induced actions at,..., are the same as in the renegotiation case. Both the prin- cipal and the agent anticipate future contracts c t , . . . , C J V and have a common conjecture regarding their terms. It follows that the principal's problem is the same as in the rene- gotiation case and that the total incentive for the period t — 1 action is the same. The main difference is that, with commitment to fairness, the period t — 1 incentives are split between the period t variable wage and the period t,...,N fixed wages. The agent's actions are the same and the principal's surplus is the same in both cases. Thus, the sequence of short-term contracts under commitment to fairness replicates the payoffs of the long-term renegotiation-proof contract. This particular sequence of contracts generalizes the one derived by Indjejikian and Nanda [18] for the case of two periods to illustrate how lack of commitment in a dynamic agency relationship leads to 163 ratcheting. The principal's inability to commit to a long term-contract without renegoti- ation in the second period creates the same inefficiency with short-term contracts under commitment and fairness as the inefficiency of the renegotiation-proof contract. Here, as before, the inefficiency is relative to the full commitment long-term contract. The results in this section show that commitment to fairness is a sufficient assumption for obtaining the ratcheting with short-term contracts described by Indjejikian and Nanda [18]. However, their description of the commitment assumptions that are necessary for the solution they derive is incomplete and referred to as "the absence of commitment". If "absence of commitment" means that the agent cannot commit to stay for all N periods, then, as is shown in the following, there is no equilibrium with a sequence of short-term contracts in which the agent stays for all N periods. As a result, the commitment to fairness assumption is not only sufficient, but also necessary to obtain the solution described in Theorem 4.5.1. The fairness assumption is necessary since, once the agent has committed to stay for all N periods, there must be a restriction in subsequent periods to prevent an "infinite" transfer from the agent to the principal through the period t > 2 contracts. The anticipation of such a contract renders any first- period contract in which the agent commits for all N periods unacceptable to the agent. Thus, the simplest solution to this problem is a lower bound on the agent's reservation certainty equivalent, which is precisely what the fairness constraint provides. The following proposition shows that the sequence of optimal contracts derived under commitment to fairness always induces the agent to leave in a later period (rj > 2) if the agent's ability to leave is restored. P ropos i t ion 4.5.1 If pm\8\... m/v/3jv ^ 0, and the principal offers the sequence of contracts (ci,..., C/v) from Theorem 4-5.1, the agent can always earn a positive surplus by leaving before the end of the last period. 164 Proof . The proof is by contradiction. Assume that the agent does not leave until the end of period N. Then the agent is faced with the pair of contracts (CN-\,CN) at the start of period N — 1. If the agent has not left until the start of period N — 1, it is assumed that his actions are the ones conjectured by the principal up to this point, a iV_2 = QLN-2- I n that case, a direct application of Proposition 5.2 from Chapter 2 shows that the agent acts strategically in period N — 1, chooses an action aN_x / 6/v-i different from the principal's conjecture, and leaves in period N. This contradicts the assumption that the agent stays for all N periods and concludes the proof. • 4.6 D y n a m i c incentives and ratcheting In this section, I examine the impact of noise correlation on the dynamics of incentives and induced managerial effort based on the commitment to fairness solution developed in the previous section. As opposed to the two-period case, to make different periods readily comparable, I assume that all periods are identical, in that the manager's effort level impacts the performance measures and the outcome in the same way in each period. That means m\ = • • • = = m and bi = • • • = bpf = b. With this simplification, equations (4.244), (4.245), and (4.246) become rr?b aN = m0N = - 5 - (4.248) mz + row p\ = mb G t - p r - ^ 3t+l (4.249) 77?/ + rot 777/ + rat at = mdt - - a t + 1 . (4.250) I now turn to the dynamic behavior of at and 3t. First, I use equation (4.299) in the expression for at (4.250) to derive m?b - pr at = — 5 - =- a w . (4.251) 777/ 4- rat TT?/ + rot 165 Using equations (4.248), (4.249), and (4.251), it can be shown that, for positively corre- lated performance measures p > 0, a^-i < o/v and 0N-I > PN- These results are the same as those from the two-period model (see Chapter 2 or Indjejikian and Nanda [18]). The next task is to see what happens in periods N — 2, N — 3, etc. First, I note that the posterior variances at converge rapidly to a limit value, denoted by <7oo. I will then use the limit variance as an approximation for posterior variances. a + v/c 2 — Ap2 L e m m a 4.6.1 Let = px = (see also Lemma 4-2.1), and let e\ = ot - C T O Q . Then, lim at = (4.252) t—>oo .2 e°t+l = -^—el . (4.253) Proof . From Lemma 4.2.3 it follows that the sequence at is decreasing and bounded from below by p\, so it converges and l im t _ > 0 O cr t > px. On the other hand, l i m ^ o o at is p2 a solution to the equation a = x (rec; x largest of the two solutions, l i m ^ o o at = p\. For the imply that p2 all that at+i = a ), and since ax is the at p2 p2 For the second part of the lemma, eat+x = < 7 t + 1 — C o o and at+\ = o~ , = a — — 0~t O"oo 2 2 2 2 P , P P ( N P a • In particular, if \p\ < a/2, < + i < 4 < < ( 4 ) ^ 1 , (4-254) uoo \uoo/ and |p| < a^, which imply that at converges exponentially fast to a^. Note that when |p| is increasing from 0 to a/2, the convergence factor p2/a2^ increases from 0 to 1. The smaller the convergence factor, the faster the convergence, which implies that the smaller 166 the correlation across periods, the faster the convergence of at to a^. In the limit case when |p| = a/2, a^ = a/2, and ef+ 1 = — ef, which implies e° = a°° ef , (4.255) and the convergence is slower. The above result means that for a long enough history, the posterior variances are approximately constant for the later periods of the manager's tenure. In particular, if we assume that at the time the manager is hired, all previous history is common knowledge, and if we assume that this previous history is long enough, the assumption that posterior variances are approximately constant holds for all periods of the manager's tenure. It is also implicitly assumed that the noise terms in the performance measures are firm- specific and not agent-specific. Thus, when a new agent is hired, past history is relevant to determining conditional expectations and conditional variances of the performance measures. The other possibility is that noise is agent-specific, and history that precedes the current agent's tenure is irrelevant. In this case, the assumption that posterior variances are constant throughout the agent's tenure is not reasonable. The problem is particularly significant during the first few periods of the agent' tenure, when most of the change in posterior beliefs about variance takes place. In what follows, I will assume that noise is firm-specific, and that beliefs about the distribution of the performance measures are conditioned on a long history that precedes the hiring of the current agent.9 Thus at = C o o is a constant for 1 < t < N. Given this 9 As a limiting case, if the firm has been in existence for infinitely many periods, the posterior variances are actually constant and equal to the limit value. 167 assumption, the recursive relations (4.249) and (4.251) become 0t = rab CT~ ^ A + 1 (4.256) Tfi2b pr ,, at = - j - r i <h+i • (4-257) 777/ + moo ?rr 4- rffno Then, pr at - Ot-i = 5— (a t + i - at) (4.258) m 2 4- ra^ Pt ~ Pt-i = r j 1 — ( A + i - A) , (4-259) TTT/ 4- ra^ which implies that if p is positive the two sequences are alternating, while if p is negative, the two sequences are monotonic. Specifically, if p is positive, then ajv > ajv-i, a/v-i < aN-2, a/v-2 > a/v-3, etc, and 8N < 8N-i, PN-I > PN-2, PN-2 < PN-Z, etc. In this case, the ratchet effect looks like a "sawtooth effect". If p is negative, actions are increasing a/v < ajv-i < a/v-2 < • • •, and incentives are decreasing 8^ > PN-I > PN-2 > In both cases at and pt converge to limit values1 0 which I denote a and J3. Taking limits in both sides of equations (4.256) and (4.257) gives mz 4- ruoo rrr 4- raoo a = —9 a . 4.261) mz 4- rUoo rrr 4- raoo solving for the limit values J3 and a gives m2b a = lim aN-t = 9 , , ;—r , (4.262) t->oo 77?/ 4- r (aoo + P) 1 4 - ^ - P = lim /3„_ t - mb . (4.263) t-400 77?/ 4- r(aoo 4- p) 1 0 A l l the analysis is backward in time, so these limits are reached towards the first periods of the agent's tenure. 168 The convergence is very fast for both sequences since i - i \p\r I - i \at - a\ = — r — \at+i - a\ , mz + rcToo \Pt-P\= ^ l f t f i - f l , m2 + raoo and \p\r/(m2 -rrooo) < 1 for all admissible values of p including the extremes |p| = <7oo = a /2 . 1 1 To gain some insight into the nature of the limit effort level a, let p = ka so that A; represents the ex-ante correlation between adjacent period performance measures (out- comes). Then, a^ = | a ( l + \ / l - 4A;2), and <7oo + P = |cr(l + 2A;-l-Vl - 4fc2). A separate analysis shows that for A; G [—1/2,0], as A; increases from —1/2 to 0, + p increases from 0 to <J. On the other hand, for k G [0,1/2], as k increases from 0 to 1/2, + p has a maximum of (1 + y/2)a/2 at k = 1/2V2 and has its minimum equal to a at both endpoints. For a the implications are that for negative correlation A;, a decreases from b to bm2/(m2 + ro) when k increases from —1/2 to 0; for positive correlation k, a decreases from bm2/(m2 + ra) to a minimum value of bm2/(m2 + ra(l + V2) /2) , and then in- creases back to bm2/(m2 + ra) when k increases from 0 to 1/2. To put these results into perspective, note that the first-best effort level in the static one-period problem is a* = b, and that in the dynamic problem, if effort is contractible, the first-best solution is also a*t = b in every period with a fixed wage that compensates the manager for his effort. Thus, when the performance measures are negatively correlated, the manager's effort is moved towards first-best. In the extreme case in which k = —1/2, the first-best effort level is approximated arbitrarily close (limited only by the total number of periods available). On the other hand, when the performance measures are positively correlated, 1 1 Since, in general \p\ < a/2 < a^, it follows that \p\r < m2 + raa 169 the effort level is always less than or equal to the second-best from the one-period prob- lem a* = bm?/(rn? + ra). In the next section I wi l l analyze the impact of managerial effort on the principal's surplus which wi l l shed some light on the inefficiencies or the efficiencies (relative to the single-period static solution) associated with the multiperiod ratchet effect. 4.7 Manager ia l tenure w i t h fixed switching cost In this section, I analyze the principal's ex-ante preferences for the manager's tenure assuming that the principal can commit to retain the manager a given number of periods. The principal and the agent sign one-period contracts subject to commitment to fairness. Alternately, the results in this section can be interpreted as relevant to the optimal tenure under long-term contracting with renegotiation. Al though a similar question can be asked regarding the full commitment long-term contract, it is a question of optimal inter-period insurance effects for the agent. In that case, the principal's problem is to determine the optimal length of contracts so that a particular correlation structure for the performance measures gives the optimal amount of risk for the agent's compensation. Here I do not address the question of optimal tenure with full commitment to long-term contracts and focus instead on the relation between the impact of renegotiation on managerial effort ("ratcheting") and managerial tenure. As in the two-period case, the principal's welfare is lower wi th either a renegotiation- proof contract or its equivalent, a sequence of contracts under commitment to fairness, than wi th the full commitment long-term contract for the same duration. The other results derived in Chapter 3 regarding the principal's welfare with or without agent turnover can be generalized only in a l imited way to the Af-period case. For example, the natural generalization of the fact that, with positive correlation, turnover dominates 170 full commitment is as follows. Proposition 4.7.1 Given a fixed finite horizon of N periods, and any 1 < T < N, assume that the inter-period covariance is positive p > 0 and that all periods are identical. Then, the principal is always better off employing two agents with full commitment long- term contracts of length T and N — T, respectively than employing a single agent with a full commitment long-term contract for all N periods. Proof. The proof is a natural extension of the proof of Proposition 7.1 in Chapter 2. Suppose that the incentive rates from the full commitment contract are offered to two separate agents in the two periods. Then, the same actions are induced for the two agents as for the single agent. However, the risk premium paid to the two agents is lower since i rva r ( /3 1 x 1 + h 0NxN) > i r var(/3 1x 1 + • • • + BTxT) + ^r v a r ( / 3 T + 1 x T + 1 + • • • + 0NxN) (4.264) > i r v a r ^ x i + h p\xT) + va,r(0T+1xT+1 H h 0NxN\xT) Thus, two agents provide the same effort at a lower risk premium, which means that the principal's surplus is higher with two agents. The advantage of using two agents comes both from not having to pay compensation for the part of the total variance of the single-agent contract that is due to the performance measures being positively correlated and from the ability to use the lower ex-post variance with the second agent. • The above proposition shows that, within a given TV-period horizon, and without switching costs, using more than one agent is always preferred. The optimum in this case is to use a different agent in each period. However, this type of result is not relevant in the case of infinite horizons with stationary replacement policies and non-zero switching costs. 171 For the remainder of the paper, I consider only the case of short-term contracts under commitment to fairness. From equation (4.297) it follows that ct = 0t(xt - E t _i[x t ]) + \a\ + l-rat(52 , (4.265) which implies E0[~zt - ct] = btat - \a\ - ^ratfi , (4.266) since E 0 [xt — Et-i[xt]] = 0 for all 1 < t < N. Using at = bt — ratrallBt to substitute for at in equation (4.266) gives the principal's ex-ante expected surplus for period t Eofa] = E0[zt - ct] The principal's total ex-ante expected surplus is then E o M + • • • + E o M = \ J2 b\ - \r J2 °t ( l + fit • (4-268) A similar calculation gives E[7T f] as a function of at only, 1 1 m 2 Eofa] = btat - -a2 - ̂ rat-^ik - atf Z Z V cx^ = btat-U-1-^(b2-2btat + al) (4.269) 2 2 rat -(m2t + rat)a2t + 2(m 2 + ro^a* - m2tb2] . 2rat Since dEJTTt] (l + (bt -at)> 0 (4.270) 172 for all at < bt, it follows that the expected period t profit is strictly increasing in man- agerial effort and attains a maximum at at = bt. This simple observation, together with the discussion of the impact of the correlation k (see the discussion at the end of the previous section) on managerial effort imply the following. When the performance measures are positively correlated, all else equal, the principal is best off with the extreme values of the correlation, 0 or 1/2, and better off the farther that correlation is from l/2y/2. In all cases, the principal is worse off than in the one-period setting with the second-best solution. In the case of negatively correlated performance measures, the worst performance is in the last period, and the best is at most a. The higher the absolute value of the correlation, the higher a, and the better off is the principal. In the extreme case k = 1/2, the first- best solution (first-best effort and first-best surplus) can be approximated arbitrarily close given sufficiently long tenure. A commitment problem could arise in this case due to the decreasing performance of the agent towards the end of his tenure. The principal needs to be able to commit to not fire the manager before term, otherwise the efficiency gains from getting effort levels close to a are lost. This would not be a problem in the positive correlation case since there the principal gets better performance towards the last period, and has no incentive to fire the manager before term. In either case, commitment to fairness assumes that the principal commits to retain the agent for all N periods. I turn now to addressing the issue of ex-ante optimal tenure. The efficiencies or inefficiencies generated by the ratchet effect depend on knowing when the last period is. Since there is managerial turnover, I will assume that there is a long-lived principal and a shorter-lived agent, such that the firm continues to operate after replacing a manager. First, I will assume away any problems that the principal may have in committing to a certain managerial tenure. The principal can offer a contract that guarantees employment for a number of periods, with compensation determined at the start of each period. The 173 manager can commit not to leave as long as the contract terms are acceptable at the start of each period. The setting is commitment to fairness for N periods at a time, for an infinite sequence of identical agents. In order to simplify the analysis, I will assume again that all periods are identical, mt = m and bt = b for all rj. In addition, I will assume that the entire history of the firm prior to hiring the current agent is common knowledge, and that the history is long enough so the approximation of posterior variances by a constant holds: o~t ~ o~oo- The history (x 0 ,a 0 ) represents all history available at the time a manager is hired, and I will use the notation Eo[-] to denote expectations conditional on available history at the start of the manager's tenure, Eo[- |xo,a 0]. Since I need to compare different lengths of managerial tenure from the point of view of a long-lived firm, I assume that the firm is infinitely lived and decides on a stationary tenure policy, that is managers are replaced every N periods in perpetuity. Then, the available history at the start of a manager's tenure is always the same (x 0 ,a 0 ) . If past history is at all relevant to the tenure decision, it only matters at the start of a manager's tenure, since that is the time at which the principal decides on tenure, and both the principal and the agent commit to it. Note in addition that, in this context, given the infinite horizon, although the output zt may be observed by the principal, it is never publicly reported and is not contractible.1 2 I now turn to the principal's objective function when choosing the agent's tenure. Given an infinite horizon and identical histories (x 0 ,a 0 ) e a c n time a new manager is hired, the principal's expected surplus for each future period is E o N = Eo[A4] + \b2 - ^rcioo ( l + ) /32 . (4.271) The sequence E 0[A f] which reflects the principal's posterior beliefs about noise in the output does not depend on the agent's actions, and therefore does not depend on the 1 2 The results that follow do not change by assuming instead that either the output is never revealed to the principal, or that the output is uncorrelated to any of the performance measures. 174 (4.272) tenure variable N. It follows that the principal will ignore Eo[A4] when choosing tenure for the agents.13 Since the principal only chooses among stationary tenure policies, and the discount rate is assumed to be zero, the principal's objective function is the average surplus per period for the entire tenure of one agent, and ignoring the noise terms Eo[At] in equation (4.271), t=i L where N n(iv) = - i t - * ) . t=i In other words, II(N) is the total surplus for the N periods of a manager's tenure ignoring the noise terms in the output. The average surplus per period for the entire tenure of one manager is the natural measure with infinite periods and no discounting, since it corresponds to the annual equivalent value of the infinite stream of surpluses (see also the discussion of the total auditing cost in Chapter 2). In addition, I assume that there is an upper limit on the agents' tenure (mandatory retirement) Nmax. Assuming a fixed switching cost C that is incurred only when a new manager is hired and that the principal's objective function is expected surplus per period net of switching costs, I have the following theorem. Theorem 4.7.1 Assume that the correlation between adjacent periods is positive, p > 0. There exists Co such that ifC< Co, ex-ante optimal tenure is one period, and ifC> Co, and there is an upper bound on tenure Nmax, then the optimal tenure is Nmax. Proof . See the Appendix. • 1 3 T h i s term is completely eliminated if we assume that the noise in the performance measures is manager-specific and thus Eo[-] = E[-]. The same is true if we assume that the performance measures are uninformative about the output, in other words if the noise terms At and it are mutually independent. 175 This theorem is a mainly negative result, in that it shows that there is no ex-ante interior optimal tenure when the variance-reducing "learning" at the start of the man- ager's tenure is absent. The manager's effort is higher, but if there is no benefit from retaining the manager for the first periods, either the switching cost is low enough to allow turnover in each period, or the switching cost is so high that, once the manager is retained for more than one period, he will be retained forever. The intuition behind this result is that, with positive correlation switching agents dominates any longer tenure. It is only when the switching cost is high enough that longer tenure is preferred. Specifically, E0[II(./V)] grows (almost) linearly in N, and as a result, the average benefit per period is essentially of the form 7 l + (TTi — C)/N, which is either strictly increasing or strictly decreasing over the entire domain depending on whether Hi > C oi H\ < C. The first case leads to maximum tenure, while the second case leads to turnover every period. For negative correlation, the optimum is always Nmax regardless of the magnitude of the switching cost as the following result shows. Theorem 4.7.2 Assume that the correlation between adjacent periods is negative, p < 0. Then, for any value of the switching cost C, if there is an upper bound on tenure Nmax, the optimal tenure is Nmax. Proof . The main idea is to show that H(N)/N is increasing in N. This is proved in a straightforward way by induction. Let a f + 1 , . . . , ajv+i a n d ai > • • • > aN denote the agent's actions in each period for the case of tenure N + 1 and N, respectively. Then, the following inequalities hold: a N and N+l > a. ,2Y+1 > a? for all t . 176 It then follows that „/V+l _ JV N+1 > N N+1 N and 7T f + 1 > > for all t . The above imply that and ^ + 1 > ^ + --- + *NN)- This last equation implies that which concludes the proof by showing that (TT^ + • • • + TV^)/N is increasing in N. So, regardless of the switching cost, the principal's gross benefit per period increases with tenure; the presence of the switching cost only adds to this conclusion since the average switching cost per period decreases with tenure. • 4.8 Conclusions In this paper I develop an N-period model of the ratchet effect in a principal-agent problem with moral hazard but without adverse selection. Thus, while the agent's action is unobservable by the principal, in equilibrium the principal has rational beliefs regarding the agent's past actions and as a result, in equilibrium, information asymmetries do not develop over time between the principal and the agent. In addition, there is no learning of productivity or any agent characteristic that is unknown at the start. The 177 only dynamic information effects are the adjustment of posterior beliefs about future performance measures, conditional on the sequential observation of past performance measures together with conjectures of the agent's past actions. The model generalizes the two-period model of Chapter 3, and most results derived therein remain valid in the N-period case. Primarily, the conclusions regarding the role of commitment in obtaining different solutions remain the same. However, the N-period model gives insights into the importance of the contracting horizon, or tenure. In addition, the N-period model offers insights into the agent's long-term performance that cannot be inferred from the two-period model. While in Chapter 3 I have compared different commitment scenarios given a fixed two-period horizon, in this paper the emphasis is on tenure given a certain choice of commitment assumptions. Since the very idea of tenure implies some form of implicit or explicit long-term commitment, the choice is between the three types of long-term com- mitment discussed in Chapter 3: full commitment to a long-term contract, commitment to a long-term contract with renegotiation, and commitment to fairness with short-term contracts. Full commitment to a long-term contract is too restrictive, in that, especially over longer horizons, renegotiation is more likely. Commitment to fairness is a mecha- nism that replicates the commitment to a long-term contract with renegotiation by using a sequence of short-term contracts. This makes the choice between the two forms of contracting almost a matter of taste. In this paper, I choose to restrict the analysis to contracting under commitment to fairness. Besides capturing the idea of renegotiation, it only requires short-term contracts that are settled at the end of each period, allowing for a breakdown of performance and surplus period-by-period. In addition, the principal and the agent only need to commit to "fair contracts" and to a tenure duration, no other long-term commitments or 178 contracts are necessary.14 Finally, the commitment to fairness solution to the dynamic agency problem provides consistency and ease of comparison to previous literature such as Milgrom and Roberts [29] and Indjejikian and Nanda [18]. The driving force behind the "ratchet effect" with pure moral hazard described in this paper is the principal's inability to commit not to use past performance measures when setting a short-term contract with the agent in each period other than the agent's first period of tenure. The principal has an incentive to set the agent's compensation in a later period based on past performance measures in order to optimally adjust the risk in the agent's contract for the lower posterior variances. In doing so, the principal makes the fixed pay in future periods depend on past performance measures. For the agent, it means that from an ex ante (previous periods) perspective, future fixed pay depend on earlier effort choices. In other words, "fixed compensation" in one period is fixed only in that period, given the past actions and performance measures, but is variable when anticipated from earlier periods. It follows that the agent's incentives for effort in any one period are spread among the variable wage for that period and the fixed wages in all future periods. The nature of the solution to the dynamic agency problem is such that, for longer horizons, the manager's effort is close to some limit level for most periods. Thus, there are essentially two effort levels, the second best in the last period, and approximately a "third best" in most other periods. When correlation is positive, the "third best" is lower than the second best, generating inefficiencies relative to a repeated one-period problem (which is the multi-period problem when periods are independent). When the correlation is negative, the ratchet effect is efficient relative to the uncorrelated periods case since the "third best" effort level is closer to first best. To summarize, with positively 1 4 T h e reader who finds the concept of "commitment to fairness" unpalatable can interpret all results in the context of commitment to a long-term contract with renegotiation. 179 correlated performance measures, incentives are stronger and the manager exerts less effort in most periods than in the last period. With negatively correlated performance measures, incentives are less strong and the manager exerts more effort in most periods than in the last period. The effort level in the last period serves as a benchmark because it coincides with the second-best solution, which is what one obtains with uncorrelated performance measures. A commitment problem could arise in the case of negatively correlated performance measures due to the decreasing performance of the agent towards the end of his tenure. The principal needs to be able to commit not to fire the manager before term, otherwise the efficiency gains from getting effort levels close to first-best are lost. This would not be a problem in the positive correlation case since there the principal gets better performance towards the last period, and has no incentive to fire the manager before term. In either case, commitment to fairness assumes that the principal commits to retain the agent for all N periods. To answer the question of ex-ante choice of tenure by the principal, I examine op- timal stationary replacement policies, whereby the agents are hired for AT periods at a time within an infinite horizon for the firm, and the noise in the performance mea- sures is firm-specific. The main result for positively correlated performance measures is that, in the presence of a switching cost, there exists a threshold switching cost such that optimal tenure is a single period whenever the switching cost is higher than the threshold value. On the other hand, optimal tenure is the maximum number of periods possible (the agent's maximum life) when the switching cost is lower than the threshold value. Thus, with positively correlated performance measures, the only optimal replace- ment (tenure) policies are the corner solutions of one period tenure or maximum possible tenure. The main result for negatively correlated performance measures is that the opti- mal replacement policy is always the maximum number of periods possible, irrespective 180 of the switching cost. In the case when the noise in the performance measures is agent-specific, there is an additional "learning" effect in the first few periods of a manager's tenure due to the (rapid) reduction of the posterior variances of the performance measures towards their limit value. This effect results in an increase of managerial effort, since it becomes less costly over time for the principal to motivate managerial effort due to lower risk premia. With negatively correlated performance measures and agent-specific noise, the manager exerts the second-best effort in the last period, a higher effort level in all other periods, and that effort has an inverted U shape. This finding is consistent with evidence from the management literature that firm performance increases at first, reaches a maximum, and then declines during a manager's tenure. For a discussion of these findings, see the papers by Eitzen and Yetman [9], Katz [21], and Hambrick and Fukutomi [15]. 4.9 A p p e n d i x : proofs P r o o f of L e m m a 4.2.1. For the first part, simply write a p 0 . . . 0 p a 0 . . . 0 = 0 p .... 0 0 For the second part of the proof, let pi be the largest solution 1 5 of the equation a x x, or equivalently, x2 — ax + p2 = 0. Since A A A - i A A - i A-2 "D0 ' 15 This is where I need a2 > 4p2. 181 it is sufficient to show that > 0 for all t. I prove by induction the stronger inequality A - i Dt A - i > Pi a + Jo2 - 4p2 P , DX First, since pi = — , it follows that —— = a > Hi- Then, I assume that 2 DQ 1 > fj,i, and I show that * + 1 > From Dt+i = aDt — p2DT-\, I have that Dt-i Dt Dt+i 2Dt-i D ,. Dt . 2 A - i ^ P2 . ,,. ——— = a — p ———. By assumption, —— > pi, which implies — p ——— > —. Adding Dt Dt A - i Dt HI a on both sides gives the desired result since • A+i 2 A - i . P Dt A Hi P r o o f of L e m m a 4.2.3. For the first part of the proof, I need to calculate the first row of the matrix E t l . a p 0 0 .0 0 . . . 0 p a p 0 0 0 . . . 0 0 p cr p 0 0 . . . 0 0 0 p a p 0 . . . 0 0 0 0 p a p . . . 0 0 0 0 0 p 0 0 0 0 0 A straightforward calculation then gives the first row of E t 1 as eiWEi" 1 = T T t A - ! , - P A - 2 , ( - P ) 2 A - 3 , • • •, (-P)'- 1 A ) (4.273) 182 As for the variance, c 4 + i = a — pe\(t)T,t 1 pe\(t) implies 2A-1 ' (4.274) = oDt - p 2 A - i = A + i Dt A ' To conclude the proof of equation (4.204), I use equation (4.274) to rewrite (4.273). For the second part of the proof, I note that P2 (jt+i = <J ~ — • (4-275) P2 Thus, o\ = a > a = o2- Assuming at < at-i, it follows that a 1 ^ 1 P2 ^ P2 P2 ^ P2 ^ < — =>• < => a < a =>• et+i < (Jt (Tt-l (Tt (Tt-l (Tt (Tt-\ (Tt which proves the inequality by induction. • P r o o f of L e m m a 4.5.1. From Lemma 4.2.3 it follows that xk - E f c_i[x f c] = x k - (mkak + Rk-i • (x f c_i - nifc-iajfe_i)) (4.276) P - -pUk-3, \~P) * 1 1 - p ( -P ) 2 ( -p) f c - 2 Rk-i = -ry- (A-2, - P A - 3 ( - ) 2 A - 4 , • • •, ( - P ) f c - 3 A , {-P)k-2D0) 0~k-\ ' (Tk-\(Tk-2 ' (Tk-\(Tk-2(Tk-Z ' ' • • • 0"i (4.277) Denote by RV(C | xt) the random part of ( given x^, and denote (,k = EV(xk — Ek-i[xk]\xt). With this notation I have _p f_p)2 t_p\k-t+i Ck = ik + —-xk-X + x f c _ 2 + • • • + - i - ^ xt+i (4.278) 0~k-l (Tk-lO~k-2 &k-i . . . (Tt+i Similarly, -p (_p)2 ( - p ) ' ~ m 0 = ^ + — + + • • • + x t + 1 (4.279) (Tl-\(Tl-2 a (_ i . . . (7 t + i 183 It follows that covt(xk - Ek-i[xk],xi - Ej_i[£/]) = covt(Cfc,C/) (4.280) To calculate the above covariance, I analyze separately the cases k = l + l, k = l + 2, etc. For k = I + 1, C* = — 0 + x m . (4.281) It follows that covt(C*,Ct) = c o v t ( — 0 + 0 ) = c o v t ( 0 , 5 / + 1 ) vart(Ci) • (4.282) 0"l cr/ Since cov t (^ ,x / + i ) = p, it remains to show that var t(£*) = o~\. If I = t + 2, ~ P - p _ .T; H X;_i and / \ 2 v a r T ( 0 ) = var t(x/) + 2 cov 4(x (, x/_i) H = — v a r ^ - i ) 2 2 = a - 2 - ^ - + ^ - a , _ ! (4.283) P 2 = cr - = 0X Oi-\ Now I prove by induction that var t(£j) = o\. It is sufficient to show that var t(£;) = o~i =>• var T (0+i) = cr/ + i . From 0 + i = — 0 + /̂+i> and assuming that v a r T ( 0 ) = ot, I have / \2 v a r 4 ( 0 + i ) = v a r t ( 5 m ) + 2 — c o v t ( x m , Q) + — y - vart(Cj) Ol Oi 2 2 = a - 2^- + ^oi (4.284) ffj erf P 2 = a = a ( + i Ol Thus, I proved that vart(Ci) = o~i for all I and that cov t(£;, 0+i) = 0 for all /. Next, I show that covT(0,Cfc) - 0 implies cov T (0 , Cfc+i) = 0 f ° r all k > 1 + 2. Since (k+i = — (k + xk+1, Ok cov T (0 , C*+i) = - — c o v t ( 0 , Ct) + cov T (0 , £fc+i) = 0 • (4.285) Ok 184 This concludes the proof of the first part of the lemma by induction. For the second part of the lemma, I need to show that covt(xt+i,Ck) = 0 for all k > t + 2. This is straightforward since covt(xt+1,Ck) = ~—covt(xt+i,xt+2) + —covt{xw,xt+1) ak-\ • • .crt+2 (-P)k-1 P + _ ^ _ (Tt+1 = 0 . (4.286) &k-l- • -0~t+2 CTfc-1 • • • • Proof, of L e m m a 4.5.2 The proof is by backwards induction. For t = N, there is nothing to prove since by definition ACE/v_i(u;t) = EAT-I[CAT] — \aN — \rvar/v_i(c/v), and aAT is set by the reservation wage constraint. Now assume that ACEfc_i(iWfc) = E^-ifc^] — \a\ — \r varfc-i(cfc) = 0 for a l H +1 < k < N, and that ak = — 0kEk-\[xk] + \&\ + varjt-i(cfc). I need to show that ACE i _i (u ; i ) = E t_i[c t] — \a\ — | rvar t _i(c t ) = 0. By definition, A C E ^ K ) = Et_![ct] - l-a\ + • • • + Et-itcjv] - ^a2N - irvart_1(ci + h cN) . (4.287) Substituting a f c in equation (4.287) for t + 1 < k < N, I obtain A C E t _ i ( w t ) = E t_i[ct] - Ja2 + E t _ i + . . . + E f _ ! 1 - A + i E t [ x 4 + i ] + ^a2 + 1 + ^rvart(cm) + 0t+iXt+i 2 (4.288) -3NEN-i[xN] + ̂ a2N + varjv-i(cjv) + pV^jv - 2 « / v rvar t _i(c t H h CAT) . To simplify equation (4.288) further, I write Ck = -/?fcEfc_![xfc] + ^a2. + ^rvar f c_i(c f c) + , (4.289) 185 and I note that the random part of ck conditional on x<_i, &t-i x s simply 0k(£k — E f c - i ^ ] ) . It follows from Lemma 4.5.1 that C\,...,CN are independent at the start of period t, and as a consequence var4_!(c t H h cN) = var4_!(c t) H h var t_i(c/v) . (4.290) Substituting back in equation (4.288) gives the following ACE t _i(wt) = E t_i[ct] - ^a2 - ^ rva r ( _i (ct) + A + i E t _ i [ x t + i - E 4 [x t + 1 ] ] + . . . + BNEt_x [5jv - E J V . ^ X J V ] ] (4.291) To conclude the proof of the lemma, I show that Et_i [xk — Efc_i [a ]̂] = 0 for all t + 1 < k < N. Since Et_i [xk] = 77^% and Efe_i[xjfc] = m,kak + Rk-\ • (xjt_i — in f c_ 1a f c_ 1) it suffices to show that E t _ i [i?fc_i • (x f c_ 1 — m ^ a ^ ) ] = 0. Expanding the term Rk-i • (Xfc-i ~ -Rk-i • (xfc-i -m-i^k-i) = ——( xk-i - mk-iak-i) O'k-l (-p)2 (-p)^1'1 + — —(xk-2 - mk-2ak-2) H 1 (Zt+i ~ fnt+1at+i) Ok-,Ok-2 • (4.292) (—p)k~t + (xt - mtat) O'k-l •••O't (_p)*-*+i ( - p ) f c - 1 + — (xt-i - mt-idi-i) .+ 1 (xi ~ rraicti) . (Jk-l • • • 0~t-l Ok-\---<J\ 186 Taking expectations of both sides in equation (4.292), it follows that {-P)k-1 E4_x [-Rk-i • (x f c_! - ma-la*.!)] = E t _ i {-pf~t+1 + 0~k-l • • • Vt-l •(xt-i - mt-idt-i) H K 0~k-l • • - ox (xt - mtat) (x\ — mjai) (4.293) + C f c - l •••O't (-Pf-1 E f _ ! [xt - mtat] - P <7fc-l . . . <7t \ C T t _ i Finally, I have ( - P ) t _ 1 (xt-i - mt_!at_i) H 1 (xi - miax) at-i ...ox -p (-py-1 •(xt-i - mt-iat-i) + h (%i - m^ai) = -Rt-i • (x 4_i - m ^ a ^ ) , o-t-i which gives o~t-i • • • <7i E 4 _ ! [-i2 f c_i • (xfc_x - in f c _ 1 a f c _ 1 )] (-P) fc-i (E t _! [xt - mtat) - Rt_i • ( x ^ - m ^ a ^ ) ) = 0 . (4.294) • Proof, of Theorem 4.5.1 The first-order condition for the agent's rationality con- straint determines at given at and 3t (see (4.240)). d dat A C E t _ i(wt | at,at) \N-t It follows that = mt3t -at + mt8t+\ — -\ h mt3N ^ ^ = 0 at aN-i...at \N-t (4.295) at = mt [ 8t + 0t+i— + • • • + 3 N - ^ - at aN_i ...ot \N-t-l = mt8t - m t - (Pt+i + A + 2 — + • • • + 0N-t-^ o~t \ o-t+i CT/V-I . . . at+i o P mt+i = mt8t at+i . ot mt (4.296) 187 In particular, for t = N, = TTINPN- The agent's reservation wage constraint is used to determine at. From Lemma 4.5.2, I have at = -0tEt-i[xt] + + ^rvari_i(c 4) = - & E t _ i [ z t ] + l-a\ + l-r 02t var t_i(x t) (4.297) I can use now equations (4.296) and (4.297) to substitute in the principal's problem for the action induced by Qt. The first-order condition is d d0i •E t _ i \{zt -ct) + h {zN - cN) = [zt - ct] d zt + 0tEt-i[xt. -r0fvaxt-i(xt) - 0txt (4.298) Z t ~ 2°* ~ 2 r ^ Q t , dat dat n O 0 t O 0 t dat Since ——- = mt, it follows that dpt rptOt = mt(bt - at) (4.299) Using equation (4.299) to substitute for at and at+\ in equation (4.296), I obtain rptcrt at = k mt and bt = mtPt (&H-1 ) mt at mt+1 mt+1 Solving for pt gives m t + r — /3t = bt + ot+i - rp o~Pw , mt) at m + i <?t mf+i (4.300) 188 dt = {m2t + rat)-1 n . P rn\ at+1 m\ mtPt H h+i - rp o—Pt+i at mt+1 ot mf+1 which simplifies to Pt P mt i mtbt H h— mt+1bt+i °~t mj+1 m\ + ro~t °~t rnt+1 pr—2———0t+i mf + rot (4.301) • Proof, of Theorem 4.7.1 The proof is based on describing the asymptotic behavior of U(N)/N. The main idea is that there exist coefficients 71 > 0, 72 > 0, 73 < 0, and 8 < 0, with \5\ < 1 such that U(N) = 7 liV + 7 2 (1 -6N)+ 73(1 - 52N) . (4.302) Then, I will show that taking C 0 = 72(1 — 8) + 73(1 — 82) > 0 gives the desired result. I now turn to the details of the proof. Let ef = Bt — B. To simplify notation, let 8 = —pr m2 + rac Note that \8\ < 1 determines how fast the actions and the incentive rates converge to their respective limits. It follows that ef = 5ef+ 1 and that m2fi eN-k-8 eN-8 r { p + ( J o o ) Since eN_k = Bx-k — P, it follows that BN-k = l3 + ePN_k = Q\\ + m ^ ( P + C T O O ) 8k+i 189 Then (4.272) gives 1 1 N ~ l E0[U{N)] = -Nb2 - - r o - o o ( l + ra^m-2) £ 2 2 fc=0 = - AT*2 - ^ ( 1 + r c r o o m - 2 ) ^ V (1 + 2 . f / + 1 + „ ™' , 2 < ^ + 2 ) 2 2 °°V £ ^ V r( /9 4-CT00) ^ ( p + f J o o ) 2 / fc=0 = n N b ~ o m ° o ( l + ra^mr2)^ N + -j—• + 2 ' " 2 ' " o o v * ' °° V r ^ + aoo) 1 - 5 r^p + a^)2 1 - 5 2 = 7 i N + 72(1 -SN) + 7 3 ( 1 - O . (4.303) The three coefficients on the last line of equation (4.303), 71, 72, and 73 are given by = 1 2 y n 2 m2a +r(al-p2) n 2 a 0 O ( m 2 + r ( p + a 0 O ) ) 2 V ' 7 2 = m V ^ ± ^ # ^ # > 0 (4.305) 1 4,2 rp2(m2 + ra00) 73 = - 7 : ^ 0 7—̂ 7 U „ , 7 < 0 (4.306) 2 a 0 O ( m 2 + r ( p + c r 0 O ) ) 3 ( m 2 + r ( a 0 O - p ) ) Let C 0 = 7 2 ( 1 - 6) 4- 73(1 - 5 2). Also note that 72(1 - 5) + 73(1 - 52) > 0. Given the switching cost C, the principal's problem is the maximization of (E 0[Il(A r)] — C)/N. The principal's objective function can be rewritten as follows: E0[n(iV)] - c l 2 { 1 - S N ) + l 3 ( l - 5 2 N ) - C N = 7i + — N • Now rewrite C = C 0 + £, where £ > 0 if C > C0, and e < 0 if C < C0. Then Eo[n(AQ] - C 72(1 - SN) + 7 3 ( 1 - 0 - Co - e , , AT = + AT ( 4 - 3 0 7 ) " C = 71 + 72(1 - 5) 4- 73(1 - <52) - Co - £ (4.308) = 71 - £ . (4.309) 190 If C > C0, and thus e > 0, for TV > 2, Eo[n(AQ] - C l2(S-6N)+^(62-S2N)-e N = 7 1 + N ( 4 3 1 0 ) < 7i " ^ , (4-311) since j2(5 - 8N) + 7 3 (52 - S2N) < 0. It follows that E0[n(/V)] - C e N < 7 1 " N for all N. On the other hand, E0[U(N)]-C lim = 7i . N->oo N Thus, for any given N, there exists N' > N such that E0[n(iv)] - c E0[U(N')]-C N N' In addition, there exists N0 such that the function (E0[n(iV)] — C)/N becomes increasing for N > N0. Then, for all N < N0, {E0[U{N)} - C)/N < 71 - e/N0. The maximum cannot be reached between N = 1 and N = No, because for large enough N', (Eo[Il(JV')] — C)/N' > 71 — e/N0. The maximum is attained at Nmax since the function (E0[n(A/')] — C)/N is increasing for N > N0 and we assumed Nmax to be large enough. If Co > C, and thus e < 0, Eo[n(AQ] - C j2(5-5") + K(62 - 52N)-e N = 7 1 + N ( 4 - 3 1 2 ) < 7 i " ^ < 7 i - e , (4-313) for all N > 2. 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