Relative Wealth Concerns, Executive Compensation, and Systemic Risk-Taking

K.7 Relative Wealth Concerns, Executive Compensation, and Systemic Risk-Taking Liu, Qi, and Bo Sun Please cite paper as: Liu, Qi, and Bo Sun (2016). Relative Wealth Concerns, Executive Compensation, and Systemic Risk-Taking. International Finance Discussion Papers 1164. http://dx.doi.org/10.17016/IFDP.2016.1164 International Finance Discussion Papers Board of Governors of the Federal Reserve System Number 1164 May 2016

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 1164 May 2016 Relative Wealth Concerns, Executive Compensation, and Systemic Risk-Taking Qi Liu and Bo Sun NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at www.ssrn.com.

Relative Wealth Concerns, Executive Compensation, and Systemic Risk-Taking1 Qi Liu Bo Sun Peking University Federal Reserve Board May, 2016 1 We thank seminar participants at Peking University, SAIF, University of Delaware, University of Hong Kong, and Wharton School of Business for their helpful comments. The views expressed herein are the authors’anddonotnecessarilyreflecttheopinionsoftheBoardofGovernorsoftheFederalReserveSystem.

Relative Wealth Concerns, Executive Compensation, and Systemic Risk Taking Abstract Given the recent empirical evidence on peer effects in CEO compensation, this paper theoretically examines how relative wealth concerns, in which a manager’s satisfaction with his own compensation depends on the compensation of other managers, affect the equilibrium contracting strategy and managerial risk-taking. We find that such externalities can generate pay-for-luck as an efficient compensation vehicle in equilibrium. In expectation of pay-for-luck in other firms, tying managerial pay to luck provides insurance to managers against a compensation shortfall relative to executive peers during market fluctuations. When all firms pay for luck, we show that an effort-inducing mechanism exists: managers have additional incentives to exert effort in utilizing investment opportunities, which helps them keep up with their peers during industry movements. In addition, we show that compensation arrangements involving pay-for-luck that are efficient from theshareholders’perspectivecannonetheless exacerbate aggregate fluctuationsintherealeconomy by incentivizing excessive systemic risk-taking, especially in periods of heightened risk. Keywords: Relative wealth concerns, Managerial compensation, Pay-for-luck, Excessive risk-taking JEL Classifications: D822, D86.

1 Introduction It has long been argued that relative wealth effects, in which a person’s satisfaction with their own wealth depends on the wealth of others, are a key component of utility (Veblen (1899), Frank (1985), andGal´ı(1994)). Relative wealth concernsareprevalentintheoverall population,andthey are especially common among successful experts and high-net-worth individuals.1 In particular, corproate executives enjoy lavish incomes and have frequent interactions within a social circle: most CEOs interact with peer CEOs by attending various social events and sharing membership in exclusive golf and country clubs, which inevitably invites comparisons and generates social ranks among CEOs based on various indicators of wealth. However, the academic literature that studies how to align the incentives of managers with those of shareholders largely ignores this trait. Since relative wealth concerns directly influence decision-making, it is logical to study the effects that relativeconsiderationshaveonmanagerialbehaviorandfirmvalue. Howdorelativewealthconcerns affect managers’ effort and investment decisions? How do compensation contracts optimally adjust for these effects? Can firms ever benefit from managerial relative wealth concerns? We develop a stylized contracting model that enables us to answer these questions. In our model,acontinuumofrisk-neutralfirmshirerisk-averse managerswithrelative wealthconcerns. In particular, we adopt a preferencespecification in which managers’ marginal utility of compensation depends not only on their own compensation, but also on how their compensation compares with that of others. We analyze this contracting problem first in a setting where managerial relative concerns are confined within the community of executives, and then in a generalized setting that allows for relative considerations to extend beyond the exclusive circle of managers. One key result of our model is that with managerial relative wealth concerns, tying CEO pay to observable industry events can emerge as an equilibrium compensation strategy. Empirical observations that managers are paid for exogenous and contractible shocks to performance — that is, the pay-for-luck phenomenon documented by Bertrand and Mullainathan (2001) — have received considerable academic attention and shareholder scrutiny. Standard contracting models 1Veblen (1899) argues that the possession of wealth is the basis of reputability and of social standing, and the amount of consumption necessary to maintain one’s social standing, as an important component of utility, increases over the course of wealth accumulation. Frank (1985) highlights the importance of relative wealth in determining social status. 2

suggest that rewarding executives for observable changes in firm performance that are beyond the executives’ control does not provide incentives and only makes the contract riskier (Holmstrom (1979)). Managerial power has been proposedas an alternative paradigm, interpreting pay-for-luck as suggestive of corporategovernance failures.2 We show that whenother firmsareexpected to pay their managers for luck, a manager with relative wealth concerns worries about falling behind his peers during market fluctuations if his contract does not give him as much exposure to the market component as others’. The insurance provision by pay-for-luck against a relative compensation shortfall delivers an equilibrium in which all firms pay for luck. Wealsoshowthatpay-for-luckcanhaveaneffort-inducingeffectinequilibrium. Whenmanagers are sensitive to the wealth of others, the market component in other managers’ pay provides effort incentives for an individual manager, because exerting effort in undertaking projects with payoffs that fluctuate with the market helps him keep up with peers. In the pay-for-luck equilibrium, this effort-inducing mechanism delivers higher shareholder payoff beyond that in the no-pay-for-luck equilibrium. Ourresultssuggestthatrelativewealthconcernsonthepartofmanagersmayenhance firms’ ability to discipline managers, generating the use of pay-for-luck that is value-maximizing in a setting with moral hazard. Shareholders can actually benefit from managers’ desire to catch up with executive peers and can improve firm value through efficient contracting. By the same token, our model indicates that the interaction of managerial relative concerns and pay structure generates additional incentives to take risk, uncovering an overlooked relation betweentheuseofpay-for-luck andcorporaterisk-taking. Becauseexposingfirmvaluetoaggregate fluctuations effectively helps managers maintain their relative income position in all states of the world, managers with pay-for-luck take risk more aggressively, especially in periods of increased aggregate risk. Standard models imply that firms take less risk to shy away from heightened aggregatefluctuations. Ourmodelthuspresentsacountervailingforcetothisconventional argumentand showsthatpay-for-luck, thoughlocally efficientwithinthefirmduetomanagerialrelative concerns, can nonetheless exacerbate fluctuations in the real economy in periods of intensified aggregate risk. Inordertostudytheimplicationsofrelativewealth concernsforsystemic risk-taking, weextend our model to allow managers to choose the firm’s exposure to market risk above and beyond what 2See, for example, Bertrand and Mullainathan (2001) and Bebchuk and Fried (2004). 3

is associated with their effort decision. In this context, as correlating one’s risk exposure with other firms’ helps prevent compensation downfalls relative to one’s peers, compensation contracts that optimally adjust for managerial relative concerns would actually allow for increased systemic risk-taking. Hence, managerial relative considerations lead to a trade-off between the build-up of systemic risk and the creation of productive effort from a social welfare perspective. To capture relative considerations in a global sense, we allow managerial relative concerns to extend beyond an exclusive circle of managers, and show that there exists a unique equilibrium in which all firms tie their managers’ compensation to market movements, as long as the degree of relative wealth concerns is not too small. We also generalize our specification of relative concerns in a variety of ways to examine the robustness of our theoretical results. For example, we allow managers to care about each and every other manager differently, possibly dependingon the peer’s proximity, similarity in background, or position in the pay distribution; care about only a selected subset of managers such as those better-paid ones; or care about some leading managers whose contracts affect others’. The results in those generalized versions confirm our results and also demonstratethataslightdegreeofrelative considerationscandeliverpay-for-luckasanequilibrium compensation strategy. Our paper is motivated by a number of recent empirical studies that suggest that peers are a crucial determinant of executive pay. Bouwman (2013) finds that CEO pay is strongly correlated with that of geographically close CEOs and presents evidence that the pattern is likely driven by managerial relative status concerns. Shue (2013) documents the phenomenon of “pay for friend’s luck” — pay responds to lucky industry-level shocks to the compensation of peers in distant industries. She also demonstrates the importance of contemporaneous social interactions by showing that peer similarities in compensation are more than twice as large in the year immediately following staggered alumni reunions. Bereskin and Cicero (2013) show that pay increases in a subset of firms in response to a governance shock affected compensation in other firms in the economy. Ang, Nagel, andYang(2013) findthatCEOcompensation contains acomponentthatispositively linked to social pressures due to interactions with other CEOs. Motivated by these empirical findings, we theoretically examine how managerial relative considerations affect the design of executive pay.3 3Previous studies on the effect of “social comparison” on executive pay are mainly concerned with the directors’ network; see, for example, Larcker et al. (2005), Kovacevic (2005), Barnea and Guedj (2006), and Hwang and Kim 4

It has long been puzzling that managers are rewarded for changes in firm performance that are beyond their control and that can be distinguished from their performance. To rationalize such a compensation practice, Hoffmann and Pfeil (2010) and Noe and Rebello (2012) show that pay-forluck can arise in a dynamic model if luck shocks are informative of future profitability. Gopalan et al. (2010) and Feriozzi (2011) propose alternative hypotheses based on strategic flexibility and implicitincentives createdbythelikelihoodofbankruptcy,respectively. ThemodelsofHimmelberg and Hubbard (2000), Oyer (2004), and Chaigneau and Sahuguet (2012) show that pay-for-luck can be driven by changes in CEOs’ reservation wages, as determined in a competitive labor market. In a calibration, Dittmann et al. (2013) find that pay-for-luck is not very costly to firms. None of these papers, however, explicitly examine the role of relative wealth concerns, which is at the heart of our analysis. The theoretical literature on relative wealth concerns is predominantly focused on the implications for financial markets (e.g., Abel (1990), Constantinide (1990), Gal´ı (1994), Campbell and Cochrane (1999), and DeMarzo et al. (2008)). We differ from this literature in that we examine the consequences relative wealth concerns have on corporate policies. A contemporaneous paper by DeMarzo and Kaniel (2015) shows that relative considerations can cause an inadequate use of relative performance evaluation. While they analyze the welfare efficiency in various compensation settings, we focuson theimplications for managerial risk-taking behavior, especially systemic risk.4 The closest paper to ours is Ozdenoren and Yuan (2015), which shows that with multiplicative effort, contractual externalities from relative performance evaluation can generate excessive systemic risk-taking. Their results are based on the assumptions that the principals are risk-averse, and the signal about the industry productivity shock (i.e., the aggregate shock) is noisy. In contrast, the principal in our model is risk-neutral, and the aggregate shock is assumed to be perfectly filtered out; incentives for excessive systemic risk-taking arise in our model are purely drive by managerial relative concerns.5 (2009). 4Also different from their use of average output of a discrete number of agents as the benchmark in relative evaluation,weallow theprincipaltoperfectly filteroutthecomponentoffirmvaluecausedbyaggregate shocksand tochoosewhethertoincludethatincompensation. Ourspecificationclearlydistinguishestheluckcomponentinpay and ensures that it does not contain any idiosyncratic movements. 5Ourpaperalsocontributestothevastliteratureontheroleofpaypackagesinfirms’risk-takingbydemonstrating aplausiblerelationbetweenpay-for-luckandmanagerialrisk-takingthathasnotbeenpreviouslyanalyzed. Although the role of compensation strategy in firms’ risk-taking has been extensively studied, a consensus on this subject has 5

The rest of the paper is organized as follows. Section 2 describes the baseline model where managers are concerned about peer managers’ pay. Section 3 analyzes efficient contracting in equilibrium and the implied risk-taking behavior. Section 4 generalizes the specification of managerial relative concerns to allow managers to have relative considerations beyond the community of corporate executives.. Section 5 discusses the model’s empirical implications, and Section 6 concludes. The appendix contains the proofs and details on the model robustness to variants of relative concerns specifications. 2 Model There is a continuum of firms indexed by i [0,1], each owned by risk-neutral investors and oper- ∈ ated by a risk-averse manager. For a representative firm i, we consider a single-period contracting model with time t = 0,1,2. At t = 0, the manager is offered a contract. At t = 1, the manager exerts effort, denoted by a , which is unobservable to shareholders. At t = 2, firm value is realized, i and compensation is paid to the manager. We assume away heterogeneity in firms and managers; thus, we do not study efficient matching in managerial labor markets.6 2.1 Preferences We assume that the manager at firm i has preferences of the form E[u(w ,w˜,a )], wherew denotes i i i his compensation, w˜ characterizes executive peers’ pay, and a represents the manager’s choice of i effort. Specifically, we adopt a preference specification similar to that in Gal´ı (1994) and Garc´ıa and Strobl (2011), and assume that the utility of the manager at firm i is given by: u(w ,w˜,a ) = exp[ λ(w w˜ ψ(a ))], (1) i i i i − − − − 1 where ψ(a ) = a2 represents the cost of exerting effort to the manager. The one-step departure i 2 i from the standard utility is that in our specification, it is relative pay, instead of absolute pay, that not been reached. For example, Fahlenbrach and Stulz (2011) present evidence that bank CEOs lost a significant portionoftheirpayandarguethatpaypackageswerenotthelikelycauseofrisk-taking. Bebchuketal.(2010)show that prior to the crisis, executiveshad been granted compensation that was in great excess of what they lost during thecrisis. Cheng et al. (2014) also present evidencethat compensation payoutsare tied to risk-takingincentives. 6GabaixandLandier(2008)andEdmansandGabaix(2011)presentcompetitiveassignmentmodelsofmanagerial labormarkets. Archayaetal.(2013) studycompensation efficiencywhenfirmscompeteforscarcemanagerial talent. 6

determines the manager’s utility. A manager with relative considerations may care not only about the level of other managers’ pay butalsoaboutthestructureofothers’pay, whichdetermines howmuchthemanager’s paymay be behind others in each possible state. To capture both the level and the structure of executive peers’ pay, we assume that w˜ takes the following form: 1 1 w˜ = h E[w ]dk+h (w E[w ])dk, l k s k k − Z0 Z0 where the parameters h and h reflect the extent of the compensation externality, i.e., how much l s manager i cares about other managers’ pay: h 0 measures concerns regarding the absolute level l ≥ of peers’ pay, i.e., the average compensation of other managers across states; h 0 measures s ≥ concerns about the structure of peers’ pay, i.e., the average compensation of other managers in each state. E[w ] is the expected pay of manager k( [0,1]) across states. k ∈ This functional form in (1) captures the notion that managers care about the compensation of other managers in a parsimonious way. We note that this utility function satisfies the usual conditions with respect to a manager’s own compensation w : it is increasing and concave in w , i i and the coefficient of absolute risk aversion is u /u = λ. This utility specification also satisfies 11 1 − u /u = λ, which implies that an increase in the average managerial compensation in a state (or 12 1 acrossstates) raisesthemarginalutility ofcompensationwhenh (orh )ispositive, asthemanager s l tries to catch up his peers. Our specification is consistent with the findings in Miglietta (2014), which shows in a laboratory experiment that individuals’ utility increases in their wealth relative to their peers, and individuals are risk averse in their relative wealth. We want to emphasize that our contribution is to study the effects of relative considerations on compensation design, and the particular interpretation of the utility function introduced above is not crucial. We note that managers we consider here care about the compensation of others in the community of executives. We subsequently extend our analysis to study relative wealth concerns that are global, in the sense that managers care about their relative position with respect to the entire economy. We will also discuss a number of variations in utility specification in Appendix C. Lastly, we point out that the type of preferences we consider can be constructed from Maccheroni, 7

Marinacci, and Rustichini (2011). 2.2 Firm value The firm’s terminal value at time t = 2 is given by V = πa +(κ a +κ )(m˜ +η ), (2) i i 1 i 2 i whereκ 0, κ 0, π > 0 represents themanager’s productivity per unitof effort; m˜ N(0,σ2 ) 1 2 m ≥ ≥ ∼ is an aggregate shock that affects all firms; η N(0,σ2) is firm i’s idiosyncratic shock; and m˜ and i η ∼ η are independent of each other. i Similar to Feltham and Wu (2001) and Ozdenoren and Yuan (2015), we assume that the manager’s effort affects the firm’s exposure to productivity shocks, represented by κ a +κ . Tying 1 i 2 firm risk to managerial effort captures the idea that productive effort by managers can be crucial in implementing investment projects. That is, the manager may exert effort by undertaking a large number of investment opportunities, which consequently increases firm risk, as the success of these projects depends on the state realization. The case κ = 0 is the standard case in which the 1 manager’s effort influences only the expected firm value and has no effect on firm risk. By the same token, firm value is jointly determined by both managerial effort (a ) and firm i risk exposure, i.e., (κ a + κ )(m˜ + η ). It has been argued in the literature that one objective 1 i 2 i of managerial compensation is to induce managerial risk-taking actions that increase not only the variance but also the mean of firm value.7 Following Sung (1995) and Dittmann and Yu (2011), we assume that there is a first-best firm strategy, S∗(a ), that maximizes firm value given effort. i Let σ∗(a ) = σ(S∗(a )) denote the minimum firm risk that is associated with this strategy. Then i i (σ∗,a ) represents an efficiency frontier in the manager’s opportunity set. To increase value above i i (σ∗,a ), the manager will have to take more positive-NPV projects, which can be operationalized i i by allowing the manager to choose a single action (a ) that affects both the mean and variance i of firm value, as described in (2). In that sense, managerial effort in our model is interpreted as effort related to firm investments. The importance of this investment-related effort is particularly 7For example, risk-taking incentives are generally used to argue that options can be more efficient than stocks (Dittmann and Yu (2011) and Feltham and Wu (2001)). 8

pronounced in young, growth firms, such as those in the high-tech or knowledge-based industries, in which managers make significant investments in research and development activities. 2.3 Compensation contract Asiscustomaryintheliterature, werestrictourattention tolinearcontracts.8 Wealsoassumethat the component of firm value that depends on the aggregate shock is observable and contractible. That is, shareholders can gauge the part of firm value caused by the aggregate shock (m˜) and may pay the manager separately for this market-determined performance beyond managerial control if they so choose. We refer to the aggregate shock as the luck shock throughout the rest of the paper to be consistent with the related empirical literature, which uses industrial or economy-wide events to proxy for luck. Standardprincipal-agenttheorysuggeststhatamanagershouldbepaidrelativetoabenchmark that removes the effect of market or sector performance on the firm’s own performance. However, it has been argued that such indexation is not observed in the data; that is, executives can enjoy “pay-for-luck” as well as “pay-for-performance.” As we are interested in examining the rationale for pay-for-luck, we focus on the case in which the luck component of firm value can be filtered out.9 In particular, we decompose firm value into two components: V = (V V¯)+V¯, i i − where V¯ (κ a¯ + κ )m˜ represents the market-wide component of firm value that is caused by 1 2 ≡ luck; a¯ is the average effort choice by all other managers.10 The residual, (V V¯), represents the i − firm-specific component of firm value. We call V¯ the “luck component.” The compensation contract of the manager at firm i then takes the following form: w = α +β (V V¯)+γ V¯, i i i i i − 8Many papersspecify thelinear form of contracts for tractability (e.g., Holmstrom andTirole (1993), Jin (2002), Oyer(2004), and Bolton, Scheinkman and Xiong (2006)). 9Inamodelwheretheluckcomponentoffirmvaluecannotbedisentangledfromthetotalfirmvalue,pay-for-luck arises mechanically. 10Itisthesameastheaverageeffortofallmanagersinequilibrium,becauseeverymanagerisidenticalandinfinitely small. 9

that is, w = α +β [V (κ a¯+κ )m˜]+γ (κ a¯+κ )m˜, (3) i i i i 1 2 i 1 2 − where α 0 denotes the base salary, β 0 represents the pay-performance sensitivity, and γ 0 i i i ≥ ≥ ≥ measurestheload ofmanagerialpay onluckshocks. Expandingthespaceof γ toallow fornegative i values does not affect our main results (discussed in Appendix B). a¯ is the average effort choice by all other managers. Note that for each individual firm, the average effort by all other managers, a¯, is taken as given and is not influenced by the individual manager whose contract is under consideration. As the continuum of firms and managers are identical, in equilibrium compensation contracts are identical in all firms: α = α,β = β,γ = γ,a = a¯, i [0,1]. A positive loading on i i i i ∀ ∈ the luck component — a positive γ — implies the use of pay-for-luck in a firm, which is consistent i with the empirical literature.11 The exponential utility, normally distributed shocks, and linear payoffs yield a mean-variance equivalence in our model. That is, the certainty equivalent of the manager’s expected utility in our model is expressed as follows: 1 1 CE = E[w w˜] λVar[w w˜] a2. (4) i i − − 2 i − − 2 i 2.4 A partial-equilibrium regime As a prelude to studying the full-fledged general equilibrium of contracting, we first analyze the contractingproblemforonefirminapartialequilibriumsetting,takingalltheotherfirms’contracts as given. In the next section, we turn to studying a general Nash equilibrium in which all firms choose contracts simultaneously and act optimally given other firms’ strategies. Following Grossman and Hart (1983), we fix the target effort (a ) in the first stage when i solving for the optimal contract. Risk-neutral shareholders in a firm, indexed by i, choose a linear contract (that is, α ,β ,γ ) that minimizes the expected cost of implementing the target effort, i i i with the equilibrium feature that average effort equals target effort a¯ = a .12 The optimal effort is i 11In the model, we have assumed identical firms and managers for simplicity. When firms and managers are heterogeneous,wecanuseaweighted-averageeffortchoicebyallthemanagerstofilterouttheluckcomponentfrom firm value. 12AccordingtoGrossmanandHart(1983),wewillfindthecheapestcontracttoimplementanyefforta i inthefirst stage. Given any a i, a¯ is actually a parameter chosen by the shareholders to correctly filter out the luck component from themanager’s payoff. Thus,a i =a¯ always holds in solving for both thefirst and second stage problems. Later 10

subsequently solved in the second stage in Section 3.2. WeshowinAppendixAthatmanagerialrelative concerns,characterized byw˜,canbeexpressed as w˜ = h W +h Mm˜, l s where W = α+βπa¯ is the expected average pay of executive peers, and M = γ(κ a¯+κ ) is the 1 2 average sensitivityofothers’paytoluckshocks,where α,β,γ denotethecashcompensation,pay- { } for-performance, and pay-for-luck in the optimal contract all (identical) firms use in equilibrium. Here, a¯ is the average effort choice by all other managers, which also equals the effort choice of each manager in equilibrium. With this expression of w˜, we solve for the manager’s effort choice below, given his own contract. Lemma 1. Given the contract (α ,β ,γ ), the effort taken by the manager is i i i β π λ[β κ (γ (κ a¯+κ ) h M β κ a¯)σ2 +β2κ κ σ2] a = i − i 1 i 1 2 − s − i 1 m i 1 2 η . i 1+λβ2κ2(σ2 +σ2) i 1 m η Suppose the manager cares about his peers’ pay structure (h > 0): the manager is concerned s about the possible state-contingent pay differential, that is, how his pay relative to his executive peers varies depending on the realization of luck shocks. It is straightforward to see from Lemma 1 that other managers’ pay-for-luck (i.e., M) has an effort-inducing effect on an individual manager: When executive peers are paid for luck (i.e., M > 0), exerting effort increases the exposure of the manager’s pay to luck shocks and effectively helps the manager maintain his relative status during market fluctuations. We formally state this incentivizing effect below, which will be a key mechanismthatsupportsageneralequilibriumwithpay-for-luck tobeanalyzed inthenextsection. Corollary 1. The manager’s equilibrium effort is increasing in other managers’ pay-for-luck: ∂a i > 0 if κ > 0. 1 ∂M Shareholders face a trade-off when deciding whether to pay the manager for luck, taking all other firms’ pay as given. On the one hand, when executive peers are paid for luck (i.e., M > 0), tying managerial pay to luck can mitigate this manager’s net exposure to fluctuations in relative on,weshowthatonlysymmetricequilbriaexist. Therefore, a¯willequaltotheaverage(target)effortinequilibrium. 11

compensation,whichreducestheassociatedriskpremium. Thatis,ifothermanagers’compensation fluctuates with the market, the manager with relative wealth concerns worries about falling behind his peers during market fluctuations; giving the manager exposure to the market component thus provides insurance against a state-contingent compensation shortfall relative to other managers. On the other hand, this insurance effect lowers managerial incentives to exert effort as a means to catch up with peers (Lemma 1). When h M is not too small, the positive effect of pay-for-luck s dominates, rendering pay-for-luck optimal in this case. This argument establishes the following result. Proposition 1. Taking other firms’ contracts as given, there exists a threshold K 0 such that it ≥ is optimal to pay the manager for luck in the firm (i.e., γ > 0) if and only if h M > K. Moreover, i s K = 0 if and only if κ = 0. 1 When κ = 0, that is, the manager’s effort does not affect the firm’s exposure to luck shocks, 1 managerial effort incentives are not affected by their relative considerations and therefore are not mitigated by the insurance against a pay shortfall that is provided by pay-for-luck. The associated reduction in the required risk premium thus makes it optimal to pay the manager for luck as long as he cares about his peers’ pay structure (h > 0) and other managers are paid for luck (M > 0). s Note that even if executive peers are paid for luck, it is not optimal to include pay-for-luck in compensation if the manager cares only about the level of his pay relative to his peers. That is, managerial desire to keep up with his peers’ pay level does not make pay-for-luck optimal. We can see from Lemma 1 that when h = 0, pay-for-luck (γ ) reduces effort incentives — only the s i disincentivizing effect remains. In this case, it is efficient to avoid using pay-for-luck, because it increases the manager’s risk exposure. Therefore, when the manager does not care about his peers’ pay structure, an optimal contract does not pay the manager for luck even if other firms do choose to pay for luck. When other firms do not pay for luck or the manager does not care about other managers’ pay structure, managerial relative concerns are not contingent on the realization of luck shocks. Therefore, an optimal contract that avoids paying a risk premium to the manager never involves pay-for-luck. We summarize the results in Lemma 2. 12

Lemma 2. If h = 0 or M = 0, then γ = 0. s i Here, we take executive peers’ pay as given and find that pay-for-luck may arise if executive peers are paid for luck. However, why are some managers paid for luck in the first place? To gain insight into the economic trade-off that gives rise to pay-for-luck in practice, we next turn to a general-equilibrium setting in which all firms decide their pay structure simultaneously. 3 Efficient contracting in a general equilibrium What happens when all firms set their managerial pay simultaneously? In this section, we study theinteraction of managerial statusconcerns andrisk-takinginthecontext ofageneral equilibrium of optimal contracting. In the proof of Proposition 2 and Proposition 4, we show that given other firms’ contracts, each firm has a unique optimal target effort and a unique optimal contract to implement the effort. Therefore, the model only has symmetric equilibria. We show that there exists an equilibrium that is characterized by pay-for-luck with increased risk-taking. 3.1 Pay-for-luck Here we solve for the Nash equilibrium in which each firm chooses the optimal pay contract, taking into account other firms’ compensation schedule. Recall that the value of each firm is given by V =πa +(κ a +κ )(m˜ +η ), i [0,1]. i i 1 i 2 i ∀ ∈ If κ > 0, productive effort exerted by the manager also increases the firm’s exposure to risk. That 1 is, diligent managers undertake more positive-NPV projects, which also leads the firm value to be more exposed to fluctuations in the underlying state. κ = 0 corresponds to the case in which 1 managerial effortdoes notaffect firmrisk. In orderto disentangle the effects on pay structurewhen managerial effort influences firm risk, we consider two cases: κ = 0,κ > 0 and κ > 0,κ = 0.13 1 2 1 2 13Wealsofocusonthecaseinwhichh s 6=1. h s=1correspondstoatrivialcasewheretheexposureofamanager’s own compensation to theluck shock and his disutility from his peers’ pay-for-luck (dueto relative wealth concerns) are exactly canceled out. Therefore, the manager’s expected utility is independent of aggregate risk σ2 . When m κ1 =0,κ2 >0, and h s =1, any γ can be an equilibrium. When κ1 >0,κ2 =0, and h s =1, γ will be infinity in an equilibrium. 13

We show that when managerial relative wealth concerns are sufficiently strong and managerial effort affects firm risk, there is a general-equilibrium outcome in which optimal contracts exhibit properties of pay-for-luck: when other firms are expected to pay their managers for luck, paying a manager for luck provides insurance against a compensation shortfall relative to executive peers and reduces the required risk premium in compensation. These results are summarized below. Proposition 2. Suppose that κ > 0,κ = 0, and h = 1. 1 2 s 6 1) If h < 1, then there exists only one equilibrium, in which γ = 0; s 2) If h > 1, then there exist two equilibria. In one equilibrium, γ = 0; in the other equilibrium, s γ > 0. Shareholders’ payoff is increasing in σ2 in the pay-for-luck equilibrium, and it is greater m than that in the no-pay-for-luck equilibrium as long as σ2 > 0. m Interestingly, pay-for-luck has an effort-inducing effect in equilibrium: When managerial effort is necessary in implementing projects (i.e., κ > 0), a higher effort increases the exposure of 1 managerial pay to fluctuations in luck shocks. If other managers are paid for luck, higher effort can help the manager catch up with his peers. Thus, managerial relative wealth concerns provide additional incentives for the manager to exert effort in this case (Corollary 1). This mechanism supports the existence of an equilibrium in which shareholders in all firms tie managerial pay to luck. Relative wealth concerns on the part of managers thus create a potential source of value for shareholders by committing managers to exert effort. We find that the associated benefits of this effort-inducing mechanism increase shareholder payoff in the pay-for-luck equilibrium beyond that intheno-pay-for-luckequilibrium(part2ofProposition2). Thereisactuallyalsoasimpleargument for this result: the shareholders can always set γ κ a¯ = h M (i.e., exactly net out the manager’s i 1 s exposure to luck shocks) and use the same level of pay-for-performance (β ) as in the no-pay-fori luck equilibrium to induce desired effort. In so doing, shareholders can obtain a payoff identical to that in the no-pay-for-luck equilibrium, which implies that shareholder payoff in the pay-for-luck equilibriumisatleastasgoodasthatintheno-pay-for-luck equilibrium. Thisargumentisbasedon the assumption that the pay-for-luck equilibrium exists, and we provide the proof for the existence in Appendix D. 14

Up to this point, we have shown that relative wealth concerns on the part of managers may enhance firms’ ability to discipline managers, generating the use of pay-for-luck that is valuemaximizing in a setting with moral hazard. Shareholders can actually benefit from managers’ desire to catch up with executive peers and can improve firm value through efficient contracting. Since the effort-inducing effect of pay-for-luck is stronger when aggregate risk is more volatile, i.e., a higher σ2 (by Lemma 1), greater benefits of managerial relative concerns accrue to shareholders m during periods of high market fluctuations. Here, our model has two equilibria: if shareholders of a firm believe that some firms are paying theirmanagersformarketmovements, theyhaveanincentivetopayforluck aswellinordertohelp their manager keep up with his peers; if a firm expects others not to pay for luck, the shareholders do not find it worthwhile to tie managerial pay to market fluctuations. In equilibrium, these beliefs are self-fulfilling. In Section 4, we show that a unique equilibrium containing pay-for-luck exists when relative wealth concerns are global, in the sense that managers care about their relative positions with respect to the entire economy. We also generalize the specification of relative wealth concerns inavariety of ways inAppendixC andillustratetherobustnessofourresults. Theresults in those generalized versions in Section 4 and Appendix C also demonstrate that a slight degree of relative considerations can deliver pay-for-luck as an equilibrium compensation strategy. Whenmanagerialeffortdoesnotaffectthefirm’sriskexposure(κ = 0),relativewealthconcerns 1 do not render additional incentives for effort. Pay-for-luck by all firms cannot be supported as an equilibrium in this case. In equilibrium no manager is paid for luck, as summarized below. Proposition 3. If κ = 0 and h = 1, there is only one equilibrium in which γ = 0. 1 s 6 Whenmanagerial relative concernsareconfinedwithinthecommunity ofexecutives, thebundle of a manager’s choice of risk with his productive effort is crucial in generating the complementarity in managers’ effort, which underpins the use of pay-for-luck in equilibrium. We show in Section 4 that when managerial relative concerns extend beyond an exclusive circle of managers, insurance effect itself is sufficient to generate pay-for-luck. Therefore, a (unique) pay-for-luck equilibrium exists when risk selection is completely independent of effort, that is, the case with κ = 0. 1 15

3.2 Risk-taking In the discussion thus far, we have fixed the target effort and shown that in the pay-for-luck equilibrium, managerial relative wealth concerns provide additional incentives to exert effort. We endogenizetheoptimaleffortinthissubsection,andfindthattheequilibriumeffortisindeedhigher in the pay-for-luck equilibrium. As productive effort is instrumental in implementing investment projects, firm risk in the pay-for-luck equilibrium also exceeds that in the no-pay-for-luck equilibrium (in which relative wealth concerns do not play a role). We summarize the results for the case in which κ > 0 and κ = 0 as follows. 1 2 Proposition 4. Suppose that κ = 0 and h > 1. The following holds in the pay-for-luck equilib- 2 s rium: 1) Given other firms’ contracts, each firm has a unique target risk, which is increasing in σ2 m and is decreasing in σ2. η 2) Firm risk is greater than that in the no-pay-for-luck equilibrium as long as σ2 > 0. m Taking other firms’ compensation as given, the greater the aggregate risk, the stronger the incentivizing effects for effort. That is, in response to heightened market fluctuations, the manager is more keen to keep up with peers’ contingent pay on luck shocks by also increasing his own exposure to luck shocks through putting forth effort. As a result, both firm risk and equilibrium effort increase with aggregate risk. As in our no-pay-for-luck equilibrium, standard models argue that as market risk increases, firms take less risk in order to shy away from increased fluctuations in the underlying state. The corporate response to market risk is more nuanced when managers who care about their relative compensation are paid for luck. Managers would then also have incentives to take more investment projects when aggregate risk is pronounced in an effort to catch up with their peers upon lucky market events. Our model thus presents a countervailing force to the impact of aggregate risk on corporate risk-taking and managerial effort. In a general-equilibrium setting of contracting, the additional effort incentives provided by relative wealth concerns — managers want to catch up with their peers’ pay for luck by exerting effort—increasetheequilibriumeffortandthelevelofrisktoleratedinthepay-for-luckequilibrium. 16

That is, relative wealth concerns on the part of managers render pay-for-luck efficient, which, in turn, leads to an increased level of corporate risk-taking. 3.3 Excessive pay We now turn to analyzing the level of managerial pay. We show that the greater managerial concerns are about their pay level relative to their executive peers (h ), the higher managerial pay l is in equilibrium: A small initial increase in compensation in one firm can lead to a magnified pay raise across firms in equilibrium due to the compensation externality. Our model implies that the considerably highlevel of managerial compensation may beattributed in partto managerial desires to catch up with peers and their frequent social interactions within the executive circle. Lemma 3. Given a target effort, the expected managerial pay in each firm in equilibrium is represented by 1 E[w] = (risk premium+cost of effort+u¯), 1 h l − where u¯ is the certainty equivalent of the reservation utility. As excessive risk-taking by financial institutions and overly generous executive pay are widely regarded as key factors in the run-up to the 2007-09 crisis, there have been advocates of pay reductions in the financial services industry. How would these reductions change the equilibrium compensation in our model of relative wealth concerns? In the model, reductions of managerial compensation in a subset of firms would have spillover effects on other firms’ pay. We interpret a pay cut in some firms (indexed by j = i) as causing the outside option of the manager, indexed by 6 i, to drop, that is, a decrease in u¯ . Suppose that some firms (indexed by j = i) impose a pay cut i 6 that results in a reduction of u¯ by an amount of δ. As the manager i cares about his pay relative i to peers, firm i can further reduce the expected pay of the manager i by h δ without compromising l managerial effort. Given firm i’s move, other firms can also further reduce their managers’ pay by h (δ +h δ), which leads to further reductions in compensation in firm i, and so on and so forth. l l In equilibrium, the spillover effects caused by relative wealth concerns will eventually lead to a δ reduction of in managerial pay in all firms in equilibrium. (1 h ) l − 17

3.4 Correlated risk-taking Alimitationofthemodelconsideredsofaristhatitprecludesmanagersfrommakingariskselection that is independent of effort decisions. In this section, we consider a more general version of the model that allows managers to choose effort and risk separately, specifically by letting managers make an additional choice of risk above and beyond the risk-taking associated with effort. This extension enables us to explicitly study the role of managerial relative considerations in a firm’s choice regarding systemic risk-taking, which proved critical in the wake of the recent financial crisis that highlighted corporate herding behavior, especially corporations’ collective exposure to real estate bubbles, as a main source of systemic risk. The literature suggests that correlation of risk across banks is a major prudential concern, as joint failures are socially costly (Acharya (2009); Acharya et al. (2012)). In particular, we modify the firm value to be V = πa +κ a (m˜ +η )+θ m˜, i i 1 i i i where θ is an additional choice of firm exposure to luck shocks — the manager can take on extra i risk above and beyond the risk-taking associated with effort choice. θ can take values in a closed i interval [θ ,θ ] with θ > θ 0. l h h l ≥ To examine the optimality of pay-for-luck, we allow the luck component of firm value to be filteredout. Forillustrationpurposes,weassumeκ = 0withoutlossofgenerality. Thecompensation 2 contract of the manager at firm i then takes the following form: w = α +β [V (κ a¯+θ¯)m˜]+γ (κ a¯+θ¯)m˜, i i i i 1 i 1 − where a¯ denotes the average effort choice by all other managers, and θ¯ denotes the average risk choice by all other managers. Note that for an individual firm, a¯ and θ¯are taken as given and are not affected by the individual manager whose contract is under consideration. As in the baseline model,inequilibriumcompensationcontracts areidenticalacrossfirms,thatis,α =α,β = β,γ = i i i γ,a = a¯,θ =θ¯, i [0,1], because all firms and managers are identical. i i ∀ ∈ 18

The certainty-equivalent expected utility of the manager in a firm is then expressed as 1 1 CE = α +β πa h W λ[(β (κ (a a¯)+θ θ¯+γ (κ a¯+θ¯) h M)2σ2 +β2κ2a2σ2] a2. i i i i − l − 2 i 1 i − i − i 1 − s m i 1 i η − 2 i Recall that in equilibrium M = γ (κ a¯+θ¯) holds. We can see that for sufficiently strong relative i 1 wealth concerns (i.e., h > 1), the manager will choose θ as large as possible given M, because s from the manager’s standpoint, a project of aggregate risk constitutes a source of insurance against falling behind his executive peers. If γ = 0 (i.e., compensation packages do not pay for luck), it i β π i can be verified that any value of θ , together with a = , can be an equilibrium. In i i 1+λκ2σ2β2 1 η i this case we assume that the manager will choose θ = θ . i l Weareprimarilyinterestedinhowmanagerschoosecorrelatedriskandhowtheefficientcontract optimally adjusts. Formally, we state the following results: Proposition 5. Suppose that h > 1,κ > 0,κ = 0. There is an equilibrium that involves pays 1 2 for-luck, i.e., γ > 0. 1) Managers in the pay-for-luck equilibrium optimally choose θ = θ . h 2) Shareholders’ payoff is greater than that in the no-pay-for-luck equilibrium. This proposition implies that when managers are concerned about their wealth relative to executive peers, they are inclined to increase their aggregate risk exposure (θ) and take correlated risks. In addition, an efficient contract that optimally adjusts for managerial relative concerns would allow for increased systemic risk-taking. The rationale for pay-for-luck is the same as in the baseline model: in expectation of pay-for-luck adopted by other firms, tying an individual manager’s compensation to market movements insures the manager against fluctuations in relative compensation. In addition, pay-for-luck incentivizes productive effort in equilibrium, as effort is instrumentalinselectinginvestmentprojectsthathelpmanagerskeepupwiththeirpeerswhosepay fluctuates with the market. Hence, the key implication that executives’ relative wealth concerns lead to the use of pay-for-luck, which enhances shareholder payoff, is also present in this more general version of the model. More importantly, managerial relative considerations make the correlation of risks — such as companiesinvestinginrealestatesecuritiesandbankslendingtothesamesector—moreattractive. 19

Whenmanagerialstatusconcernsarestrong,firmsherdintheirchoiceofassetsandprojects,which, in turn, leads to greater systemic risk being built up in the system. 4 Relative concerns with respect to the economy Ouranalysis thusfar hasbeenconductedundertheassumptionthatmanagers areconcernedabout their relative compensation in the community of corporate executives. The theoretical literature has also focused on settings in which relative wealth concerns are global, in the sense that agents care about their relative position with respect to the overall wealth in the economy. To capture this broader notion of relative wealth concerns, in this section we modify w˜ to take the following form: 1 1 w˜ = h E[V ]dk+h (V E[V ])dk, e l k s k k − Z0 Z0 where V is the value of firm k in the economy. Thus, the first term represents the extent of k externality fromthe absolutewealth level in theoverall economy, that is, managers’ concerns about their relative position with respect to (the average wealth of) the economy across states. The second term represents the extent of externality from the state-contingent realization of wealth in the economy, that is, managers’ concerns about their relative position with respect to (the average wealth of) the economy in each state. It can be shown that w˜ can be rewritten as w˜ = e e h W +h M m˜, where W = πa¯ represents the unconditional average wealth in the economy, and l e s e e M = κ a¯+κ represents the average sensitivity of others’ wealth to the state of the economy (i.e., e 1 2 luck shocks, m˜). In this analysis, we again consider two cases: κ > 0,κ = 0 and κ = 0,κ > 0. 1 2 1 2 When relative concerns are global, the insurance provision of pay-for-luck makes it an equilibrium compensation strategy as long as the degree of managerial relative concerns (h ) is not too s small. Pay-for-luck also incentivizes effort and improves shareholders payoff in equilibrium. We summarize the results below. Proposition 6. Suppose that κ > 0,κ = 0. Fixing the target effort a, there exists a threshold hˆ 1 2 such that 1) if h < hˆ, there exists a unique equilibrium with γ = 0; s 2) if h > hˆ, there exists a unique equilibrium with γ > 0. s 20

Specifically, hˆ goes to 0 as κ approaches 0. As long as κ > 0, the shareholders’ payoff is 1 1 higher in the pay-for-luck equilibrium (when h > hˆ) than in the no-pay-for-luck equilibrium (when s h < hˆ). s Now we turn to examining the equilibrium risk-taking behavior. Similar to the baseline model, in response to heightened market fluctuation, each manager is more keen to keep up with peers’ contingent pay on luck shocks by also increasing his own exposure to luck shocks through putting forth effort. As a result, both firm risk and equilibrium effort increase with aggregate risk. Proposition 7. Suppose that κ > 0 and h > hˆ. The following holds in the pay-for-luck equilib- 1 s rium: 1) Given other firms’ contracts, each firm has a unique target risk, which is increasing in σ2 m and decreasing in σ2. η 2) Firm risk is greater than that in the no-pay-for-luck equilibrium (when h < hˆ). s As in Section 3.4, now we let managers make a risk choice independent of their effort, that is, managers choose θ [θ ,θ ],θ > θ 0, as the firm’s additional exposure to luck shocks, an extra l h h l ∈ ≥ risk above and beyond that associated with effort. We show the robustness of model results to this generalized specification of relative concerns below. Proposition 8. Suppose that κ > 0 and h > hˆ. There exists only one equilibrium with pay-for- 1 s luck, i.e., γ > 0. 1) Managers in the equilibrium optimally choose θ = θ . h 2) Shareholders’ payoff is greater than that in the no-pay-for-luck equilibrium (when h < hˆ). s Asinourbaselinemodel,tyingmanagerialpaytomarketmovements providesinsuranceagainst shortfalls in relative wealth and therefore remains an equilibrium strategy. Proposition 8 also confirms the key trade-off illustrated in Section 4: Pay-for-luck as a value-enhancing compensation arrangement from the shareholders’ perspective actually leads to excessive systemic risk built up in the economy. Proposition 9. Suppose that κ = 0,κ > 0. Fixing the target effort a, if h > 0, then there exists 1 2 s a unique equilibrium with γ > 0. Moreover, if the manager is allowed to make the additional risk 21

choice θ, there exists an infinite number of equilibria with γ > 0, in which any value in [θ ,θ ] can l h be an equilibrium. When relative concerns are global, the insurance provision of pay-for-luck always takes effect, as the wealth in the economy is always positively correlated with the aggregate (luck) shock. As a result, even if κ = 0 (i.e., the incentivizing mechanism goes away), the insurance mechanism itself 1 is sufficient to generate a pay-for-luck equilibrium, which is the unique equilibrium. The purpose of this section is to demonstrate that if managerial relative considerations extend beyond the community of corporate executives, the insurance provision of pay-for-luck can deliver a unique equilibrium in which all firms tie their managers’ compensation to market movements, as long as the degree of relative wealth concerns is not too small. 5 Empirical implications 5.1 Pay-for-luck: an efficient contracting view Standard contracting models that optimally design incentive pay to maximize firm value imply that shareholders will not reward executives for observable luck, that is, observable changes in firm performance that are beyond the executives’ control. However, Bertrand and Mullainathan (2001) find that executives at oil companies receive pay raises when their company performance improves as a result of changes in global oil prices beyond their control. Similar pay-for-luck is also observed at multinational businesses when currency exchange rates fluctuate. In response to the empirical findings, which are inconsistent with a standard principal-agent model, managerial power has been proposed as an alternative paradigm, most notably by Bertrand and Mullainathan (2001) and Bebchuk and Fried (2004). Pay-for-luck has been widely interpreted as suggestive of corporate governance failures. Motivated by the empirical studies on executive peer effects,14 our model shows that when managers are sensitive to the wealth of their peers, pay-for-luck can be consistent with optimal contracting and need not reflect inefficiency. Tying managerial pay to observable industry events provides managers with insurance against compensation shortfalls relative to their peers. In ad- 14See Bouwman (2013), Shue(2013), Bereskin and Cicero (2013), and Ang,Nagel, and Yang(2013). 22

dition, pay-for-luck can provide effort incentives in equilibrium and therefore raises shareholder payoff. Should managers be rewarded for luck? Our answer is that the envious ones should. Ourmodelsuggeststhatpay-for-luckcanbeespeciallyvalue-enhancinginindustrieswithstrong growthopportunitiesandinregionswithahighconcentrationofgrowthfirms,suchasSiliconValley, home to various high-tech companies. Managerial effort in growth industries is more likely to be associated with launching new products and undertaking investment opportunities (i.e., a larger κ ). Recall that paying for luck motivates effort in equilibrium, because productive effort allows 1 executives to catch up with their peers by increasing firm exposure to market fluctuations. Thus, whenmanagers areconcerned aboutexecutive peerswhoareinthesameindustryor geographically close, pay-for-luck as part of equilibrium contracting would have a stronger effort-inducing effect in industries and regions with growth companies. 5.2 Excessive risk-taking under pay-for-luck Compensationpractices thatincentivize excessive risk-taking, especially of correlated risk, atfinancial institutions have often been mentioned as one key factor contributing to the recent financial crisis.15 The financial crisis renewed interest in the potential for compensation to affect managerial risk-taking, though empirical evidence on the role of compensation in the crisis is mixed (e.g., FahlenbrachandStulz(2011); DeYoung,PengandYan(2013); ShueandTownsend(2013)). Extant empirical studies on compensation and its implications for risk-taking, however, do not examine pay-for-luck. Our model uncovers a relation between pay-for-luck and risk-taking. In the presence of relative wealth concerns,tyingpaytoluckmay beanoptimalway toefficiently incentivize managerialeffort and therefore emerge in compensation practices. In this case, exposing firm value to aggregate fluctuations serves as a source of insurance against falling behind one’s executive peers. This mechanism suggests an overlooked link between pay-for-luck and corporate risk-taking, especially of systemic risk, in the cross section of firms. Interestingly, ourmodelalsoshowsthatmanagershaveincentivestotakerisksmoreaggressively in periods of heightened aggregate risk. Without relative wealth concerns, a standard model that 15See, for example, Rajan (2005), Kashyap et al. (2008) and Clementi et al. (2009). 23

separates the systematic component of firm performance would imply that managerial risk-taking isunrelated toaggregate fluctuations, and, inamodelinwhichtheluck componentis notseparable from the total firm performance, managers would take less risk when aggregate risk is pronounced in order to shield themselves from intensified fluctuations. In contrast to these predictions from existing models, however, managers with relative considerations tend to take more risk, especially correlated risk, precisely when industry prospects are volatile, in an effort to avoid falling behind their executive peers upon industry movements. Our model generates a clear implication for how volatilities in industry fundamentals affect corporate risk-taking that highlights a contrast with existing theories and warrants further empirical examination. More importantly, this mechanism, builtonmanagerialrelative wealth concerns,exacerbates thebuild-upofsystemicriskinbadtimes. 6 Conclusion Economists have long believed that relative wealth concerns are important. Not only are relative considerations prevalentin thepopulation, butthey are also likely to beprevalent amongcorporate executives — those who care about relative wealth that determines social status are more likely to pursue careers as managers. Indeed, empirical studies suggest that peers are a crucial determinant of managerial pay. To date, the theoretical literature has primarily focused on the asset pricing implications relative concerns have andignored thepossibility thatsuch compensation externalities affect corporate policies. That is, relative wealth concerns have been studied independently from incentives. In this paper, we study the interaction of managerial relative concerns and compensation in the context of managerial effort and investment policy. We show that with managerial relative wealth concerns, tying CEO pay to observable industryevents can emerge as an equilibrium compensation strategy and need not necessarily reflect contracting inefficiency. In expectation of pay-for-luck in other firms, tying managerial pay to luck provides insurance to managers against compensation shortfall relative to executive peers during market fluctuations. In addition, when all firms pay for luck, managers may have additional incentives to exert effort in utilizing investment opportunities, which helps them keep up with their peers during industry movements. Our model suggests that 24

relative wealth concerns can create one potential source of firm value by committing managers to exert effort, raising shareholder payoff in equilibrium. It is important to interpret our results with caution. They should not be seen as advocating for paying managers generously for luck. Rather, we argue that criticism of such practices should be balanced by the insurance mechanism that takes effect when managers are concerned about their wealth relative to others. We also show that managerial relative concerns generate incentives to invest aggressively, especially inprojects whosepayoffs arecorrelated with peerfirms’performance. From a social welfare perspective, the equilibrium compensation strategy that exhibits pay-for-luck represents a trade-off between the build-up of systemic risk and the creation of productive effort. Our study does not aim to find complete explanations for each of the compensation and risktakingphenomenaconsidered. Wepaintthesetofphenomenawithanintentionally broadbrush,as our objective is to examine the contracting implications when managers are concerned about their social standing within a closely interacting social circle. Given the mounting empirical evidence for theimportanceofpeereffectsoncompensation,ourgoalistoinitiateafirstattempttotheoretically examine how managerial relative considerations influence compensation structure and risk-taking in a general-equilibrium contracting framework. Certainly more work lies ahead to develop a richer understanding of how managerial relative wealth concerns play a role in structuring executive pay. 25

References [1] Abel, Andrew, 1990, “Asset Prices under Habit Formation and Catching Up with the Jones,” American Economic Review 80, 38-42. [2] Ang, James, Gregory Nagel, and Jun Yang, 2013, “The Effect of Social Pressures on CEO Compensation,” Working Paper. [3] Barnea, Amir, and Ilan Guedj, 2006, “But Mom, All the Other Kids Have One! - CEO Compensation and Director Networks,” Working Paper. [4] Bebchuk, Lucian, Alma Cohen, and Holger Spamann, 2010, “The Wages of Failure: Executive Compensation at Bear Stearns and Lehman 2000-2008,” Yale Journal on Regulation 27, 257- 282. [5] Bebchuk, Lucian, and Jesse Fried, 2004, “Pay Without Performance: The Unfulfilled Promise of Executive Compensation,” Cambridge, MA: Harvard University Press. [6] Bereskin, Frederick, and David C. Cicero, 2013, “CEO Compensation Contagion: Evidence from an Exogenous Shock,” Journal of Financial Economics 107, 477-493. [7] Bertrand, Marianne, and Sendhil Mullainathan, 2001, “Are CEOs Rewarded for Luck? The Ones Without Principals Are,” Quarterly Journal of Economics 116, 901-932. [8] Bizjak, John M., Michael Lemmon, and Lalitha Naveen, 2008, “Has the Use of Peer Groups ContributedtoHigher Pay andLess EfficientCompensation,”Journal of Financial Economics 90, 152-168. [9] Bolton, Patrick, Jos´e Scheinkman and Wei Xiong, 2006, “Executive Compensation and Short- Termist Behaviour in Speculative Markets,” Review of Economic Studies 73, 577-610. [10] Bouwman, Christa H.S., 2013, “The Geography of Executive Compensation,” Working Paper. [11] Chaigneau, Pierre, and Nicolas Sahuguet, 2012, “The Effect of Monitoring on CEO Pay Practices in a Matching Equilibrium,” Working Paper. [12] Chan, Yeung Lewis, and Leonid Kogan, 2002, “Catching Up with the Joneses: Heterogeneous Preferences and the Dynamics of Asset Prices,” Journal of Political Economy 110, 1255-1285. [13] Cheng, Ing-Haw, Harrison Hong, and Jose A. Scheinkman, 2015, “Yesterday’s Heroes: Compensation and Risk at Financial Firms,” Journal of Finance, 70, 839-879. 26

[14] Clementi, G. L., T. Cooley, M. Richardson, and I.Walter, 2009, “Rethinking Compensation in Financial Firms,” Restoring Financial Stability, ed. by V. Acharya, and M. Richardson. John Wiley and Sons. [15] DeMarzo, Peter M., Ron Kaniel, and Ilan Kremer, 2008, “Relative Wealth Concerns and Financial Bubbles,” Review of Financial Studies 21, 19-50. [16] DeMarzo, Peter M.andRonKaniel, 2015, “Relative Pay forNon-Relative Performance: Keeping up with the Joneses with Optimal Contracts,” Working Paper. [17] Dittmann, Ingolf, Ernst Maug, and Oliver Spalt 2013, “Indexing Executive Compensation Contracts, ” Review of Financial Studies 26, 3182-3224. [18] Dittmann, Ingolf, and Ko-Chia Yu, 2011, “How Important are Risk-Taking Incentives in Executive Compensation?” Working Paper. [19] Edmans, Alex, and Xavier Gabaix, 2011, “The Effect of Risk on the CEO Market,” Review of Financial Studies 24, 2822-2863. [20] Fahlenbrach, Ru¨diger, and Ren´e Stulz, 2011, “Bank CEO Incentives and the Credit Crisis,” Journal of Financial Economics 99, 11-26. [21] Feltham, Gerald A., and Martin GH Wu. 2001, “Incentive Efficiency of Stock versus Options,” Review of Accounting Studies 6, 7-28. [22] Feriozzi, Fabio,2011, “PayingforObservableLuck,”RANDJournal of Economics 42,387-415. [23] Frank, Robert H., 1985, “Choosing the Right Pond: Human Behavior and the Quest for Status,” Oxford University Press. [24] Gabaix, Xavier, and Augustin Landier, 2008, “Why has CEO Pay Increased So Much?” Quarterly Journal of Economics 123, 49-100. [25] Gal´ı, Jordi,1994, “KeepingUpwiththeJoneses: ConsumptionExternalities, PortfolioChoice, and Asset Prices,” Journal of Money, Credit and Banking 26, 1-8. [26] Garc´ıa, Diego, and Gu¨nter Strobl, 2011, “Relative Wealth Concerns and Complementarities in Information Acquisition,” Review of Financial Studies 24, 169-207. [27] Gopalan, Radhakrishnan, Todd T. Milbourn, and Fenghua Song, 2010, “Strategic Flexibility andtheOptimality ofPayforSectorPerformance,”Review of Financial Studies 23, 2060-2098. [28] Grossman, Sanford, and Oliver Hart, 1983, “An Analysis of the Principal-Agent Problem,” Econometrica 51, 7-45. 27

[29] Himmelberg, Charles, and Glenn Hubbard, 2000, “Incentive Pay and the Market for CEOs: An Analysis of Pay-for-Performance Sensitivity,” Working Paper. [30] Hoffmann,Florian, andSebastianPfeil, 2010, “Reward forLuckinaDynamicAgency Model,” Review of Financial Studies 23, 3329-3345. [31] Holmstrom, Bengt, 1979, “Moral Hazard and Observability,” The Bell Journal of Economics 10, 74-91. [32] Holmstrom, Bengt and Jean Tirole, 1993, “Market Liquidity and Performance Monitoring,” Journal of Political Economy 101, 678-709. [33] Hwang, Byoung-Hyoun, and Seoyoung Kim, 2009, “It Pays to Have Friends,” Journal of Financial Economics 93, 138-158. [34] Jin, Li, 2002, “CEO Compensation, Diversification, and Incentives,” Journal of Financial Economics 66, 29-63. [35] Kashyap, Anil, Raghuram Rajan, and Jeremy Stein, 2008, “Rethinking Capital Regulation,” Proceedings of the 2008 Jackson Hole Symposium organized by the Kansas City Fed. [36] Kovacevic, Savo. 2005, “A Social Comparison Perspective of Executive Remuneration Committees in Australian Companies,” Working Paper. [37] Larcker, David, Scott Richardson, Andrew Seary, and Irem Tuna, 2005, “Back Door Links Between Directors and Executive Compensation,” Working Paper. [38] Maccheroni, Fabio, Massimo Marinacci, and Aldo Rustichini. 2012, “Social decision theory: Choosing within and between groups” The Review of Economic Studies, 79, 1591-1636. [39] Miglietta, Salvatore, 2014, “Incentives and Relative Wealth Concerns,” Quarterly Journal of Finance, 4(04), 1450013. [40] Ozdenoren, Emre and Kathy Yuan, 2015, “Contractual Externalities and Systemic Risk, ” Working Paper. [41] Noe, Thomas H., and Michael J. Rebello, 2012, “To Each According to Her Luck and Power: Optimal Corporate Governance and Compensation Policy in a Dynamic World,” Working Paper. [42] Oyer, Paul, 2004, “Why Do Firms Use Incentives That Have No Incentive Effects?” Journal of Finance 59, 1619-1650. 28

[43] Rajan, Raghuram, 2005, “Has Financial Development Made the World Riskier?,” Proceedings, Federal Reserve Bank of Kansas City, August, 313-369. [44] Steven C. Salop, 1979. “Monopolistic Competition with Outside Goods,”The Bell Journal of Economics 10(1) 141-156. [45] Shue, Kelly, 2013, “Executive Networks and Firm Policies: Evidence from the Random Assignment of MBA Peers,” Review of Financial Studies 26, 1401-1442. [46] Shue, Kelly, and Richard Townsend, 2013, “Swinging for the Fences: Executive Reactions to Quasi-Random Option Grants,” Working Paper. [47] Veblen, Thorstein, 1899, “The Theory of the Leisure Class: An Economic Study of Institutions,” New York: Penguin. 29

Appendix A. Derivation of w˜ 1 1 1 1 w˜ = h E[w ]dk+h (w E[w ])dk = h (α+βπa )dk +h [ (β(κ a κ a¯)+γ(κ a¯+ l 0 k s 0 k − k l 0 k s 0 1 k − 1 1 1 κ 2 ))m˜dkR + 0 β(κ 1 a k +Rκ 2 )η k dk]. Note that inR equilibrium, a k = a¯Rfor all k ∈ [0,1]. So w˜ = 1 h l W+h s Mm˜R+h s β(κ 1 a¯+κ 2 ) 0 η k dk. Since(η k ) k∈[0,1] areidenticallydistributedandindependentof 1 eachother,bythelawoflargeRnumbers, 0 η k dk converges to0almostsurely. Sow˜ = h l W+h s Mm˜. R Appendix B. Allowing for negative γ In the main paper, we restrict γ (loading on luck) to be bounded below by zero, due to empirical relevance;empiricalstudieshavedocumentedcompensationpracticethatpaysmanagersforpositive changes in firm performance beyond managerial control, and we do not observe firms penalizing managers for market upswings after all. For theoretical generality, in this section we allow for negative γ, that is, a negative exposure to luck shocks, and re-derive our main results. When managerial relative considerations are confined within the community of executives, we modify Proposition 2 to Proposition 10, stated below, that incorporates a choice of negative γ. Proposition 10. Suppose that managers are only concerned about their peer executives’ pay and κ > 0,κ = 0, and h = 1. 1 2 s 6 1) If h < 1, then there exists two equilibria. In one equilibrium, γ = 0; in the other equilibrium, s γ < 0. 2) If h > 1, then there exist two equilibria. In one equilibrium, γ = 0; in the other equilibrium, s γ > 0. Shareholders’ payoff is increasing in σ2 in the pay-for-luck equilibrium, and it is greater m than that in the no-pay-for-luck equilibrium as long as σ2 > 0. m As we have argued in the main text, when κ > 0, relative concerns incentivize the manager to 1 exerteffortaslongashisnetexposuretotheluckshockisnegative. Inthemodelwherethemanager is only concerned about his peer executives’ pay, his net exposure is (1 h )γκ a¯. Therefore, our s 1 − main result (an equilibrium with γ > 0 exists) stays unchanged, and the shareholders would set a negative γ to generate the effort incentivizing mechanism in the case when h < 1. s By the same token, when the manager is concerned about the entire economy, the shareholders would set a negative γ to generate the effort-incentivizing mechanism whenh < hˆ. In all the other s cases, the results remain the same as in the main text. We summarize the results below. Proposition 11. Suppose that κ > 0,κ = 0. Fixing the target effort a, there exists a threshold 1 2 hˆ such that 30

1) if h < hˆ, there exists a unique equilibrium with γ < 0; s 2) if h = hˆ, there exists a unique equilibrium with γ = 0; s 3) if h > hˆ, there exists a unique equilibrium with γ > 0. s Specifically, hˆ goes to 0 as κ approaches 0. As long as κ > 0, the shareholders’ payoff is 1 1 higher in the pay-for-luck equilibrium (when h > hˆ) than in the no-pay-for-luck equilibrium (when s h = hˆ). s When κ = 0, there is no effort-incentivizing mechanism, so all the results in this case are 1 unchanged. As can be seen in the main text, we are primarily interested in the cases when h is s not too small. So our main results are not affected if γ is allowed to be negative. Appendix C. Generalizing relative concerns with respect to peers In the baseline model in Section 3, the relative wealth concerns consist of two components: the average level of others’ wealth across states and the average level of all others’ wealth in each state. In a closely interacting circle of executives, a manager with relative wealth concerns might care about his wealth relative to not only an average level, but also each individual manager’s. The wealth of each and every other manager in each state can be an essential element in managerial relativeconcerns. Tocapturethisadditionalcomponent,weconsidersomegeneralizedspecifications ofrelativewealthconcernsinthissection. Weshowthatourresultsinthebaselinemodelarerobust to generalizations of managerial relative concerns with respect to their peers.16 Each and every other manager To capture managers’ concerns about each and every other individual manager, we generalize the specification of relative wealth concerns using the following form:17 1 1 w˜ = h E[w ]dk+h f(k)(w E[w ])dk, l k s k k − Z0 Z0 where f(k) 0 denotes the differential weighting an individual manager, indexed by i, attaches ≥ to his peer indexed by k = i. That is, managers are concerned about each individual peer’s pay 6 and exhibit a varying degree of “envy” toward each and every other manager. The insurance provision and effort-inducing effects of pay-for-luck are robust to this generalization of relative 16Ourresults hold as long as thenormality of w˜ is preserved,which is required for tractability. 17An alternative form, w˜ = h lR 0 1 E[w k ]dk +h sR 0 1 (w k −E[w k ])dk +h sR 0 1 f(k)(w k −E[w k ])dk, whose first two components coincide with w˜ in the baseline model, collapses to the current form. 31

wealth concerns; therefore, shareholders across firms consequently find it optimal to pay managers for luck, which raises shareholder value in equilibrium. We formalize this claim in Proposition 12. Proposition 12. Suppose that κ > 0,κ = 0, f(k) is bounded for k [0,1], and h 1 f(k)dk = 1 1 2 ∈ s 0 6 holds. R 1 1) If h f(k)dk < 1, there exists only one equilibrium, in which γ = 0; s 0 1 2) If h sR 0 f(k)dk > 1, there exist two equilibria. In one equilibrium, γ = 0; in the other equilibrium,Rγ > 0. Shareholders’ payoff in the pay-for-luck equilibrium is greater than that in the no-pay-for-luck equilibrium. Managers can weigh their peers using a variety of criteria, that is, f(k) can take various forms depending on managerial preferences. To provide some economic interpretation of managerial relative considerations, we discuss a few plausible forms of f(k) below. Example 1: Proximity It is natural to think that managers benchmark themselves more against their peers who are geographically close or in similar industries than against those in distant locations and businesses. Weighing executive peers based on proximity implies that the f(k) used by a manager is inversely related to his distance from any other manager, denoted by d(i,k) for any k = i. This distance can 6 represent the extent of separation in the nature of industries, geographic locations, and network relations. For example, following Salop’s circular city (1979) model, we can assume that a continuum of managers are uniformly distributed on a circle with a unit diameter. Then one possible form of ψπ f(k) would be f(k) = 1 d(i,k), and d(i,k) = denotes the arc length between the manager − 360 under consideration, indexed by i, and any other manager, indexed by k = i, where ψ is the angle 6 in degrees. Example 2: Similarity Managers may have a stronger tendency to compare themselves with peers who have a similar background in education, ethnicity, alumni association, age and tenure year, among others. Let us use a vector to denote the personal characteristics managers identify themselves by and use X to select their reference peers. The weighting function f(k) used by a manager, indexed by i, in this case can be any monotonically decreasing function of a pairwise similarity score, proxied by , for any k = i. k i |X −X | 6 Example 3: Quantiles Managers may also care differently about their peers in varying quantiles of the pay distribution. For example, managers may be more concerned about the top percentile compared to the bottom percentile and exhibit relatively stronger envy toward certain subgroups depending on the 32

rankingin compensation. Thissuggests thatthereferencepoints inmanagerial relative wealth concerns include not only the average level of their peers’ pay, but also other distributional moments. Specifically, we can write f(k) as a step function: f(k) = f for the compensation of managers in k the kth percentile, k 1,2, ,100 . ∀ ∈{ ··· } Position oneself In the previous subsection, we analyze the optimality of pay-for-luck when managers care about each and every other manager in the executive circle. Psychology theory suggests that individuals identify their relative income position and may only envy up.18 This tendency suggests that managers may exclusively exhibit relative considerations toward a selected subset of their peers — for example, thebetter-paidpeers. Inthiscase, theweightingfunctionf(k)usedbyamanagerindexed by i is represented by an indicator function I that equals 1 if w > w and 0 otherwise. {w k >wi} k i Distinct from the specifications of f(k) in Section 6, which are deterministic functions, f(k) in this case becomes a random variable that dependson realized pay. We show that w˜ can be simplified to w˜ = h W + hsMm˜ +constant in this case. Because of the same mechanisms in the baseline model, l 2 we obtain similar contracting results and formalize them below. Proposition 13. Suppose that κ > 0,κ = 0, and h = 2. 1 2 s 6 1) If h < 2, then there exists only one equilibrium, in which γ = 0; s 2) If h > 2, then there exist two equilibria. In one equilibrium, γ = 0; in the other equilibrium, s γ > 0. Shareholders’ payoff inthe pay-for-luck equilibriumisgreater thanthat inthe no-pay-for-luck equilibrium. Leading managers So far in this section, we have studied various cases in which each individual firm and manager are infinitely small such that each individual’s compensation contract does not affect other firms and managers, as in the baseline model. In reality, there may be some leading managers, whose compensation contracts can affect others. To capture this possibility, we adopt an alternative specification of relative wealth concerns as in the following form: 1 1 w˜ = h E[w ]dk+h (w E[w ])dk+ h (w E[w ]). l k s k k j j j − − Z0 Z0 j=i X1,···,in 18See, for example, Fiske, “Envy Up,Scorn Down: How Comparison Divides Us,” Am Psychol. 2010, 65(8). 33

Thefirsttwo components areidentical to thespecification of w˜ in thebaseline model, and the third component consists of the pay of some individual managers, called leading managers hereafter, where h represents a manager’s concern about his pay relative to leading managers indexed by j j for j i , ,i , n. 1 n ∈ { ··· } ∀ Nowleadingmanagers’compensationcontracts wouldactually haveaneffectonothermanagers in the executive circle. The leading firms take this into account when designing their compensation contracts, whichaddscomplications inderivingaclosed-formsolution. For tractability, weconsider the case with one leading manager, indexed by 0. This leading manager’s relative wealth concerns, denoted by w˜ , remain unchanged from those in the baseline model: 0 1 1 w˜ = h E[w ]dk+h (w E[w ])dk. 0 l k s k k − Z0 Z0 For the rest of the managers, indexed by i (0,1], their relative wealth concerns are expressed as ∈ 1 1 w˜ = h E[w ]dk+h (w E[w ])dk+h (w E[w ]), l k s k k 0 0 0 − − Z0 Z0 whereh denotes other managers’ relative considerations toward theleading manager. Pay-for-luck 0 can remain part of equilibrium contracting and increase shareholder payoff in this case. We state these results in Proposition 14 below. Proposition 14. Suppose that κ > 0,κ = 0, h is small, and h h < 1. 1 2 l s 0 1 1) If h < , then there exists only one equilibrium in which all managers are not paid for s 1+h 0 luck. 1 2) If h > , then there exist two equilibria. In one equilibrium, γ = 0; in the other s 1+h 0 equilibrium, γ > 0. Shareholders’ payoff in the pay-for-luck equilibrium is greater than that in the no-pay-for-luck equilibrium. Appendix D. Proofs Proof of Proposition 1, Lemma 1, and Lemma 2: The CEO’s compensation is w = α +β [πa +(κ a +κ )η]+[β κ (a a¯)+γ (κ a¯+κ )]m˜. i i i i 1 i 2 i 1 i i 1 2 − So we can calculate that the certainty-equivalent of the expected utility is CE = α + β πa i i i − h W 1λ[(β κ (a a¯)+γ (κ a¯+κ ) h M)2σ2 +β2(κ a +κ )2σ2] 1a2. Taking the first-order l − 2 i 1 i − i 1 2 − s m i 1 i 2 η −2 i condition yields β π λ[β κ (β κ (a a¯)+γ (κ a¯+κ ) h M)σ2 +β2κ (κ a +κ )σ2] a = 0, i − i 1 i 1 i − i 1 2 − s m i 1 1 i 2 η − i 34

which implies that β π λ[β κ (γ (κ a¯+κ ) h M β κ a¯)σ2 +β2κ κ σ2] a = i − i 1 i 1 2 − s − i 1 m i 1 2 η . (5) i 1+λβ2κ2(σ2 +σ2) i 1 m η To minimize the cost of the contract, the base salary α must be set such that the participation i constraint is binding. Thus, the shareholders’ objective is 1 min λ[(β κ (a a¯)+γ (κ a¯+κ ) h M)2σ2 +β2(κ a +κ )2σ2] βi,γi 2 i 1 i − i 1 2 − s m i 1 i 2 η subject to (5). Since a = a¯ in equilibrium, the objective function can be simplified to i 1 min λ[(γ (κ a¯+κ ) h M)2σ2 +β2(κ a +κ )2σ2] βi,γi 2 i 1 2 − s m i 1 i 2 η subject to (5). When h M = 0, increasing γ from zero to positive will always increase the risk premium and s i thus increase the cost of the contract. From (5), we can see that increasing γ also has a negative i effect on the manager’s effort. So β has to rise to induce the target effort as γ increases, which i i further boosts the cost of the contract. Therefore, when h M = 0, the optimal γ equals zero, i.e., s i no pay-for-luck. When h M > 0, plugging a = a¯ into (5), we can obtain that s i λκ (κ a +κ )σ2β2 [π λκ (γ (κ a¯+κ ) h M)σ2 ]β +a = 0. 1 1 i 2 η i 1 i 1 2 s m i i − − − Thus, we can solve that the optimal β is i 2a i β = . i [π λκ (γ (κ a¯+κ ) h M)σ2 ]+ [π λκ (γ (κ a¯+κ ) h M)σ2 ]2 4λκ a (κ a +κ )σ2 1 i 1 2 s m 1 i 1 2 s m 1 i 1 i 2 η − − − − − q We can calculate that ∂βi = λκ1(κ1a¯+κ2)σ m 2 βi > 0. Then it is easy to ∂γi √[π−λκ1(γi(κ1a¯+κ2)−hsM)σ m 2 ]2−4λκ1ai(κ1ai+κ2)σ η 2 check that the objective function 1λ[(γ (κ a¯+κ ) h M)2σ2 +β (γ )2(κ a +κ )2σ2] is a convex 2 i 1 2 − s m i i 1 i 2 η functioninγ . Hence, theoptimalγ ispositiveifandonlyifthederivativeoftheobjective function i i w.r.t. γ is negative at γ = 0, which is equivalent to i i λκ (κ a¯+κ )(κ a +κ )2σ2(β )2 (κ a¯+κ )h M + 1 1 2 1 i 2 η i | γi=0 < 0. 1 2 s − (π+λκ h Mσ2 )2 4λκ a (κ a +κ )σ2 1 s m 1 i 1 i 2 η − q 35

Note that the left-hand side in the above inequality is decreasing in h M and nonnegative at s h M = 0. Thus, there must exist a cut-off K 0 such that the left-hand side in the above s ≥ inequality is less than zero if and only if h M > K. Moreover, since the left-hand side in the above s inequality is positive at h M = 0 unless κ = 0, the cut-off K equals to zero if and only if κ =0. s 1 1 Proof of Proposition 2: When κ = 0, the shareholders’ objective is 2 1 min λ[(β κ (a a¯)+γ κ a¯ h M)2σ2 +β2κ2a2σ2] βi,γi 2 i 1 i − i 1 − s m i 1 i η subject to a = βiπ−λβiκ1(γiκ1a¯−hsM−βiκ1a¯)σ m 2 , which can be rewritten as i 1+λβ i 2κ2 1(σ m 2 +σ η 2) λκ2[(a a¯)σ2 +a σ2]β2 [π λκ (γ κ a¯ h M)σ2 ]β +a = 0. 1 i − m i η i − − 1 i 1 − s m i i Similarly,wecancalculatethatβ = 2ai , i [π−λκ1(γiκ1a¯−hsM)σ m 2 ]+√[π−λκ1(γiκ1a¯−hsM)σ m 2 ]2−4λκ2 1ai[(ai−a¯)σ m 2 +aiσ η 2] and ∂βi = λκ2 1a¯σ m 2 βi > 0. Thus, it is easy to check that the ob- ∂γi √[π−λκ1(γiκ1a¯−hsM)σ m 2 ]2−4λκ2 1 ai[(ai−a¯)σ m 2 +aiσ η 2] jective function 1λ[(β (γ )κ (a a¯)+γ κ a¯ h M)2σ2 +β (γ )2κ2a2σ2] is convex in γ . So if there 2 i i 1 i − i 1 − s m i i 1 i η i exists an optimal solution with γ > 0, we must have that at the optimal γ , i i ∂β ∂β i κ (a a¯)+κ a¯ (β (γ )κ (a a¯)+γ κ a¯ h M)σ2 + i β κ2a2σ2 = 0. ∂γ 1 i − 1 i i 1 i − i 1 − s m ∂γ i 1 i η i i (cid:18) (cid:19) Since in equilibrium, a = a¯, γ = γ, and M = γκ a¯, the left-hand side simplifies to κ2a¯2γ(1 h )+ i i 1 1 − s ∂βiβ κ2a2σ2, which is positive if h < 1. So when h < 1, there is only one equilibrium, in which ∂γi i 1 i η s s γ = 0. i When h > 1, there could be two equilibria. In one equilibrium, we still have γ = 0 for each s i firm. In the other equilibrium, γ > 0. The Lagrangian function (with the Lagrangian multiplier i ρ) is 1 L = λ[(β κ (a a¯)+γ κ a¯ h M)2σ2 +β2κ2a2σ2] 2 i 1 i − i 1 − s m i 1 i η ρ[β π λβ κ (γ κ a¯ h M β κ a¯)σ2 (1+λβ2κ2(σ2 +σ2))a ]. − i − i 1 i 1 − s − i 1 m − i 1 m η i 36

Taking the FOC w.r.t. β and γ yields that i i λ[κ (a a¯)(β κ (a a¯)+γ κ a¯ h M)σ2 +β κ2a2σ2] 1 i − i 1 i − i 1 − s m i 1 i η = ρ[π λκ ( 2κ a¯β +γ κ a¯ h M)σ2 2λβ κ2(σ2 +σ2)a ], − 1 − 1 i i 1 − s m − i 1 m η i λκ a¯(β κ (a a¯)+γ κ a¯ h M)σ2 = ρλβ κ2a¯σ2 . 1 i 1 i − i 1 − s m − i 1 m Taking the ratio of the two equalities can simplify to (β κ (a a¯)+γ κ a¯ h M)[π λκ σ2 (β κ (a a¯)+γ κ a¯ h M) 2λβ κ2σ2a ]= λκ3β2σ2a2. i 1 i − i 1 − s − 1 m i 1 i − i 1 − s − i 1 η i − 1 i η i Note that a = βiπ−λβiκ1(γiκ1a¯−hsM−βiκ1a¯)σ m 2 implies that π λκ σ2 (β κ (a a¯)+γ κ a¯ h M)= i 1+λβ i 2κ2 1(σ m 2 +σ η 2) − 1 m i 1 i − i 1 − s ai +λβ κ2σ2a . So (β κ (a a¯)+γ κ a¯ h M) ai λβ κ2σ2a = λκ3β2σ2a2. Thus, β κ (a βi i 1 η i i 1 i − i 1 − s βi − i 1 η i − 1 i η i i 1 i − a¯)+γ κ a¯ h M = λκ3 1β i 3σ η 2ai . Plugging it int (cid:16) o a = βiπ−λβiκ1(cid:17)(γiκ1a¯−hsM−βiκ1a¯)σ m 2 yields that i 1 − s −1−λβ i 2κ2 1σ η 2 i 1+λβ i 2κ2 1(σ m 2 +σ η 2) λ2κ4σ2 σ2a β4 λκ2σ2a β2 πβ +a 1 m η i i = 0. (6) 1 η i i − i i − 1 λκ2σ2β2 − 1 η i Since the risk premium in this case is 1 1 λ2κ4σ2 σ2β4 λ[(β κ (a a¯)+γ κ a¯ h M)2σ2 +β2κ2a2σ2] = λκ2a2σ2β2 1 m η i +1 , 2 i 1 i − i 1 − s m i 1 i η 2 1 i η i (1 λβ2κ2σ2)2 " − i 1 η # which is increasing in β . The optimal β must be the minimum (positive) solution to (6). Let i i F(β ) denote the left-hand side of (6). Since F(0) > 0 and F 1 <0, 0< β < 1 . In i λκ2 1σ η 2 i λκ2 1σ η 2 (cid:18) (cid:19) equilibrium, a = a¯ and M = γ κ a¯, so we can confirm that inqthis second equilibrium,q i i 1 λκ2β3σ2 1 i η γ = > 0. (7) i (h 1)(1 λβ2κ2σ2) s − − i 1 η We use β and β to denote the optimal pay-performance sensitivity in the no-pay-for-luck equii0 iγ librium and pay-for-luck equilibrium, respectively. Then β is the minimum (positive) solution to i0 2 the equation λκ2σ2a β2 πβ +a = 0,19 which is a π i − r (cid:16) a π i(cid:17) −4λκ2 1 σ η 2 . β is the minimum (positive) 1 η i i − i i 2λκ2 1σ η 2 iγ solution to (6). Note that F(0) > 0 (F(β ) is the left-hand side of (6)), so ∂F < 0. Since i ∂βi| βi=βiγ 19More rigorously, β i0 should be the minimum (positive) solution to the equation λκ2 1[(σ m 2(a i −a¯)+σ η 2a i]β i 2 − πβ i+a i=0. Butsinceatthefirststagethetargetefforta i isfixedandequalsa¯ inequilibrium,theequationcanbe simplified to λκ2 1σ η 2a i β i 2−πβ i+a i =0. 37

∂F < 0, by the Implicit Function Theorem, ∂βi < 0. Thus, β β with equality holds ∂σ m 2 ∂σ m 2 | βi=βiγ iγ ≤ i0 only when σ2 = 0. m Recall that the risk premium in the pay-for-luck equilibrium is 1λκ2a2σ2β2 λ2κ4 1σ m 2 σ η 2β i 4 +1 2 1 i η i (1−λβ i 2κ2 1σ η 2)2 2− πβi h i (with β = β ), by (6), it can be simplified to 1λκ2a2σ2β2 ai . We can calculate that i iγ 2 1 i η i 1−λκ2 1σ η 2β i 2 β2 2 πβi β 4 3πβ + πλκ2σ2β3 ∂ i − ai = i − ai i ai 1 η i . (8) ∂β 1 (cid:16)λκ2σ2β(cid:17)2 (cid:16) (1 λκ2σ2β2)2 (cid:17) i − 1 η i − 1 η i   Since β < 1 , ∂ 4 3πβ + πλκ2σ2β3 = 3π(1 λκ2σ2β2) < 0 for any β between β iγ λκ2 1σ η 2 ∂βi − ai i ai 1 η i −ai − 1 η i i iγ and 1 . q Note that β (cid:16) = 2 (cid:17) implies that π 2 λκ2σ2, and thus, β 2ai q λκ2 1σ η 2 i0 a π i + r (cid:16) a π i(cid:17) 2 −4λκ2 1σ η 2 ai ≥ q 1 η i0 ≤ π ≤ 1 . It is easy to check that 4 3πβ + πλκ2σ2β3 > 0, and β < β 1 , so λκ2 1σ η 2 − ai i ai 1 η i | βi=βi0 iγ i0 ≤ λκ2 1σ η 2 q (cid:16) (cid:17) 2− πβi q 4 3πβ + πλκ2σ2β3 > 0. Since the risk premium 1λκ2a2σ2β2 ai is affected by − ai i ai 1 η i | βi=βiγ 2 1 i η i 1−λκ2 1σ η 2β i 2 (cid:16)σ2 only through its effe(cid:17)ct on β , we have m i ∂ 1 2 πβi ∂ 1 2 πβi ∂β λκ2a2σ2β2 − ai = λκ2a2σ2β2 − ai i < 0 at β = β . ∂σ2 2 1 i η i 1 λκ2σ2β2 ∂β 2 1 i η i 1 λκ2σ2β2 ∂σ2 i iγ m " − 1 η i # i " − 1 η i # m So the risk premium is decreasing in σ2 . Note that when σ2 = 0, we will have the same risk m m premium in both the no-pay-for-luck equilibrium and the pay-for-luck equilibrium. Therefore, the shareholders’ payoff is better in the pay-for-luck equilibrium than that in the no-pay-for-luck equilibrium as long as σ2 > 0. m Proof of Proposition 3: When κ = 0, the shareholders’ objective is 1 1 min λ [(γ κ h M)2σ2 +β2κ2σ2] βi,γi 2 i i 2 − s m i 2 η subject to a = β π. Obviously, γ must be set such that γ κ h M = 0. But since in equilibrium, i i i i 2 s − γ = γ and M = γκ , this is impossible to get γ κ = h M when h = 1. i 2 i 2 s s 6 Proof of Proposition 4: We have shown that if h > 1, there are two equilibria. In one s equilibrium, γ = 0. In this case, a = βiπ+λβ i 2κ2 1a¯σ m 2 with a¯ = a . Thus, the problem can be i i 1+λβ i 2κ2 1(σ m 2 +σ η 2) i simplified to a = πβi , and the shareholders’ objective is to maximize i 1+λκ2 1σ η 2β i 2 1 1 πa λκ2a2σ2β2 a2. i − 2 1 i η i − 2 i 38

2 Plugging β = a π i − r (cid:16) a π i(cid:17) −4λκ2 1σ η 2 into the above objective function, it is easy to derive that the i 2λκ2 1 σ η 2 optimal target effort is a∗ = π . √1+4λκ2 1 σ η 2 Intheotherequilibrium,γ > 0. Recallthattheriskpremiuminthiscaseis 1λκ2a2σ2β2 λ2κ4 1σ m 2 σ η 2β i 4 γ +1 , i 2 1 i η iγ (1−λβ i 2 γ κ2 1σ η 2)2 (cid:20) (cid:21) where β is the minimum (positive) solution to (6). We use RP(a ,β (a )) to denote this risk iγ i iγ i premium. Then the optimal target effort is to maximize πa RP(a ,β (a )) 1a2. Taking the i − i iγ i − 2 i FOC w.r.t. a yields that i λ2κ4σ2 σ2β4 ∂RP ∂β π λκ2a σ2β2 1 m η iγ +1 iγ a = 0. (9) − 1 i η iγ (1 λβ2 κ2σ2)2 − ∂β ∂a − i " − iγ 1 η # iγ i By (6), λ2κ4 1σ m 2 σ η 2β i 4 γ +1= 2− π a β i iγ . Together with (8), we can rewrite (9) as (1−λβ i 2 γ κ2 1σ η 2)2 1−λκ2 1σ η 2β i 2 γ 2 πβiγ ∂ 1 2 πβiγ ∂β π λκ2a σ2β2 − ai λκ2a2σ2β2 − ai iγ a = 0. − 1 i η iγ1 λκ2σ2β2 − ∂β 2 1 i η iγ1 λκ2σ2β2 ∂a − i − 1 η iγ iγ − 1 η iγ! i Let G(a ,β ) denote the left-hand side of the above equation. We first rewrite (6) as follows: i iγ π λ2κ4σ2 σ2β4 λκ2σ2β2 β +1 1 m η i = 0. (10) 1 η i − a i − 1 λκ2σ2β2 i − 1 η i β is the minimum (positive) solution to (10). Let J(β ,a ,σ2 ) denote the left-hand side of (10). iγ i i m By the Implicit Function Theorem, ∂βiγ = ∂J /∂J . We can calculate that ∂J = πβi, and ∂ai −∂ai ∂βi| βi=βiγ ∂ai a2 i ∂ ∂ β J i = 2λκ2 1 σ η 2β i − a π i − 2λ2κ4 1σ ( m 2 1− σ λ η 2 κ β 2 1 i 3 σ (2 η 2 − β i λ 2) κ 2 2 1σ η 2β i 2) = 2λκ2 1 σ η 2β i − a π i − 2 (cid:16) λκ2 1σ η 2β β i 2 i − (1 a π − i β λ i κ + 2 1 1 σ (cid:17) η 2 ( β 2 i 2 − ) λκ2 1σ η 2β i 2) = 4− 3 a π i βi+ a π i λκ2 1σ η 2β i 3 . Together with (8), we can obtain that − βi(1−λκ2 1σ η 2β i 2) ∂ 1 2 πβiγ ∂β λκ2πσ2β3 λκ2a2σ2β2 − ai iγ = 1 η iγ . ∂β 2 1 i η iγ1 λκ2σ2β2 ∂a 2(1 λκ2σ2β2 ) iγ − 1 η iγ! i − 1 η iγ Thus, 2 πβiγ λκ2πσ2β3 G(a ,β )= π λκ2a σ2β2 − ai 1 η iγ a . i iγ − 1 i η iγ 1 λκ2σ2β2 − 2(1 λκ2σ2β2 ) − i − 1 η iγ − 1 η iγ Note that ∂G < 0, ∂G < 0, and ∂βiγ > 0, so dG = ∂G + ∂G ∂βiγ < 0. Therefore, there is a ∂ai ∂βiγ ∂ai dai ∂ai ∂βiγ ∂ai unique solution to (9), i.e., the optimal target effort is unique. Moreover, by the Implicit Function Theorem, ∂ai has the same sign as ∂G , which equals ∂G ∂βiγ > 0. So the optimal target effort ∂σ m 2 ∂σ m 2 ∂βiγ ∂σ m 2 39

increases in σ2 . It is easy to check that when σ2 = 0, the two equilibria have the same optimal m m target effort. So the optimal target effort in the pay-for-luck equilibrium is higher than that in the no-pay-for-luck equilibrium as long as σ2 > 0. m To derive the negative relation between the optimal a and σ , we need to show that, fixing a , i η i G is decreasing in σ . First, we define y = σ β . By (10), y is the minimum (positive) solution to η η iγ λκ2y2 π y+1 λ2κ4 1σ m 2 y4 = 0. Then by the Implicit Function Theorem, it is easy to derive 1 − aiση − σ η 2(1−λκ2 1σ η 2β i 2) that ∂y > 0. Second, we show that X = σ η 2β i 3 γ is increasing in σ . The proof is as follows: ∂ση 1−λκ2 1σ η 2β i 2 γ η if β increases in σ , then note that ∂X and ∂X are both positive, so dX > 0. If β decreases iγ η ∂βiγ ∂ση dση iγ in σ , then by (10), X = 1 λκ2σ2β π + 1 . Since β < 1 , λκ2σ2β π + 1 η λ2κ4 1σ m 2 1 η iγ − ai βiγ iγ λκ2 1σ η 2 1 η iγ − ai βiγ increases in σ if β decreases in h σ . Thus, we prove t i hat X = σ η 2βq i 3 γ is increasing in σ . The η iγ η 1−λκ2 1 σ η 2β i 2 γ η left-hand side of (9) (i.e., G) can be rewritten as π λ3κ6σ2 a X2+λκ2ay2 λκ2 1πX a , which − 1 m i 1 − 2 − i is obviously decreasing in σ η for fixing a i . Thus, by th(cid:2)e Implicit Function Th(cid:3)eorem, the optimal a i decreases in σ2. η Proof of Lemma 3: The CEO pay must be binding in the participation constraint, so we have 1 1 α +β πa h W λ[(β κ (a a¯)+γ (κ a¯+κ ) h M)2σ2 +β2(κ a +κ )2σ2] a2 = u¯, i i i − l − 2 i 1 i − i 1 2 − s m i 1 i 2 η − 2 i where u¯ is the certainty-equivalent of the reservation utility. Note that W = α +β πa¯ and a = a¯ i i i in equilibrium; thus, 1 E[w] = α +β πa¯ = (risk premium+cost of effort+u¯). i i 1 h l − Proof of Proposition 5: As in the baseline model, we can calculate that the certaintyequivalent of the expected utility is 1 1 α +β πa h W λ[(β (κ (a a¯)+θ θ¯)+γ (κ a¯+θ¯) h M)2σ2 +β2κ2a2σ2] a2. i i i − l − 2 i 1 i − i − i 1 − s m i 1 i η − 2 i Since in equilibrium γ = γ, θ = θ¯, M = γ(κ a¯+θ¯), a = a¯, and h > 1, β (κ (a a¯)+θ θ¯)+ i i 1 i s i 1 i i − − γ (κ a¯+θ¯) h M < 0. Therefore, given M, the manager will choose θ as large as possibleto catch i 1 s i − up with his peers’ exposure to the aggregate shock. So the manager will choose θ = θ optimally. i h Following the similar procedure as in the baseline model, we can show that κ (a a¯)+θ 1 i i − − θ¯+γ (κ a¯+θ¯) h M = λκ3 1β i 3σ η 2ai . Then it is easy to check that β is the minimum (positive) i 1 − s −1−λβ i 2κ2 1 σ η 2 i 40

solution to (6), and the optimal target effort is the solution to (9). Then using the same procedure as in the baseline model, we can show that as long as σ2 > 0, the shareholders’ payoff is better in m the pay-for-luck equilibrium than in the no-pay-for-luck equilibrium, and the optimal target effort in the pay-for-luck equilibrium is higher than that in the no-pay-for-luck equilibrium. Proof of Proposition 6, 7, and 8: Given that the relative concerns with respect to the economy can take the form: w˜ = h W +h M m˜, the proofs of these three propositions are just e l e s e similar to the proofs of Proposition 2, 4, and 5. Moreover, similar to the proof of Proposition 1, the threshold hˆ goes to 0 as κ approaches 0. 1 Proof of Proposition 9: Since when the relative concerns are global, h M = h κ , the s e s 2 shareholders will set γ = h to avoid paying the risk premium associated with luck shocks. Thus, s γ > 0 as long as h > 0. When the manager is allowed to make the additional risk choice θ, the s manager’s net exposure to the luck shocks is β (θ θ¯)+ γ (κ + θ¯) h (κ +θ¯), which always i i i 2 s 2 − − equals to zero in equilibrium. Thus, it is easy to verify that any value in [θ ,θ ] for θ can be an l h equilibrium. Proofs for Appendix C ProofofProposition12: Itiseasytoshowthatw˜canbesimplifiedtow˜ = h W+h 1 f(k)dkMm˜. l s 0 Thus, the rest of the proof is similar to the proof of the baseline model. R Proof of Proposition 13: I = I is a random variable that equals 1 or 0 with {w k >w} {η k >η} equal probability, and I ,k [0,1] are independent of each other. Then w˜ = h W + {w k >w} ∈ l 1 1 1 h Mm˜ I dk+h β κ a¯ I η dk. By the law of large numbers, I dk converges s 0 {η k >η} s i 1 0 {η k >η} k 0 {η k >η} to E[I { Rη k >η} ] = 1 2 , and 0 1 I {η k R>η} η k dk converges to E[I {η k >η} η k ], which is aR constant. Therefore, w˜ can be rewritten as w˜R= h l W + 1 2 h s Mm˜ +constant. Then the rest of the proof is similar to the proof of the baseline model. Proof of Proposition 14: We use α ,β ,γ ,a to denote the compensation and effort choices i i i i for the firms and managers indexed by i (0,1]. For the leading manager, we use α ,β ,γ ,a 0 0 0 0 ∈ to denote the corresponding compensation and effort choices. Similar to the specification in the baseline model, let a¯ be the average effort choice that equals a in equilibrium, and let a¯ be i 0 a weighted-average effort choice that equals a in equilibrium. Then the compensation for the 0 managers indexed by i (0,1] has the form: w = α +β (πa +κ a η)+[β κ (a a¯)+γ κ a¯]m˜. i i i i 1 i i 1 i i 1 ∈ − Thecompensationfortheleadingmanagerisw = α +β (πa +κ a η )+[β κ (a a¯ )+γ κ a¯ ]m˜. 0 0 0 0 1 0 0 0 1 0 0 0 1 0 − 41

The certainty-equivalent utility for the managers indexed by i (0,1] is ∈ 1 CE = α +β πa h W λ(h β κ a )2σ2 i i i i − l − 2 0 0 1 0 η 1 1 λ[(β κ (a a¯)+γ κ a¯ h M h (β κ (a a¯ )+γ κ a¯ ))2σ2 +(β κ a )2σ2] a2. − 2 i 1 i − i 1 − s − 0 0 1 0 − 0 0 1 0 m i 1 i η − 2 i Taking the first-order condition w.r.t a yields that i β π λβ κ σ2 [ β κ a¯+γ κ a¯ h M h (β κ (a a¯ )+γ κ a¯ )] a = i − i 1 m − i 1 i 1 − s − 0 0 1 0 − 0 0 1 0 . i 1+λβ2κ2(σ2 +σ2) i 1 m η Note that the managers’ effort is affected by the leading manager’s compensation contract. In particular, ∂ai = h κ (a a¯ )X, ∂ai = h κ a¯ X, where X = λβiκ1σ m 2 . ∂β0 0 1 0 − 0 ∂γ0 0 1 0 1+λβ i 2κ2 1(σ m 2 +σ η 2) Then following the same procedure as in the proof of the baseline model, we can show that β κ (a a¯)+γ κ a¯ h M h (β κ (a a¯ )+γ κ a¯ ) = λκ3 1β i 3σ η 2ai , and β is the minimum i 1 i − i 1 − s − 0 0 1 0 − 0 0 1 0 −1−λβ i 2κ2 1σ η 2 i positive solution to (6). The solution for the leading manager is complicated because his compensation contract will affect other managers’ effort choices. The certainty-equivalent utility for the leading manager is CE = α +β πa h (α +β κ a ) 0 0 0 0 l i i 1 i − 1 1 λ[(β κ (a a¯ )+γ κ a¯ h (β κ (a a¯)+γ κ a¯))2σ2 +β2κ2a2σ2] a2. − 2 0 1 0 − 0 0 1 0 − s i 1 i − i 1 m 0 1 0 η − 2 0 Taking first-order condition w.r.t a yields that 0 β π λσ2 β κ ( β κ a¯ +γ κ a¯ h (β κ (a a¯)+γ κ a¯)) a = 0 − m 0 1 − 0 1 0 0 1 0 − s i 1 − i 1 . 0 1+λβ2κ2(σ2 +σ2) 0 1 m η Following the same procedure as in the proof of the baseline model, we can show that λκ3 1 β 0 3σ η 2a 0 h l β i κ 1 h 0 X γ κ a¯ h γ κ a¯ = . 0 1 0 − s i 1 −1 λβ2κ2σ2 − λσ2 (1 h β κ h X) − 0 1 η m − s i 1 0 β is a solution to λκ2σ2a β2 πβ +a λκ σ2 β λκ3 1σ η 2a0β0 3 + h l βiκ1h0X = 0. Recall 0 1 η 0 0 − 0 0 − 1 m 0 1−λβ0 2κ2 1σ η 2 λσ m 2 (1−hsβiκ1h0X) that for other managers, we have that γ κ a¯ h γ κ a¯ (cid:16) h γ κ a¯ = λκ3 1β i 3σ η 2ai . (cid:17) i 1 − s i 1 − 0 0 1 0 −1−λβ i 2κ2 1σ η 2 Ifh h < 1,itiseasytocheckthat1 h β κ h X > 0. DenoteU = λκ3 1β0 3σ η 2a0 + h l βiκ1h0X > s 0 − s i 1 0 1−λβ0 2κ2 1σ η 2 λσ m 2 (1−hsβiκ1h0X) 0, and V = λκ3 1β i 3σ η 2ai > 0. If h = 0, then following the same procedure as in the proof of the 1−λβ i 2κ2 1σ η 2 l baseline model, we can check that the optimal a and β equal the optimal a and β , respectively. 0 0 i 42

Thus, U = V when h = 0. Moreover, we can also show that if h < 1 , there is only one equilibl s 1+h0 rium with no pay-for-luck; if h > 1 , there exist two equilibria. In one equilibrium, γ = γ = 0; s 1+h0 0 i in the other equilibrium, γ ,γ > 0. The shareholders’ payoff in the pay-for-luck equilibrium is 0 i greater than that in the no-pay-for-luck equilibrium. By the continuity, when h is small enough, l the results still hold. 43