Centre De Referència En Economia Analítica Barcelona Economics Working Paper Series Codes of Best Practice in Competitive Markets for Managers Codes of Best Practice in Competitive Markets for Managers

We study …rms'corporate governance in environments where possibly heterogeneous shareholders compete for possibly heterogeneous managers. A …rm, formed by a shareholder and a manager, can sign either an incentive contract or a contract including a Code of Best Practice. A Code allows for a better manager's control but makes manager's decisions hard to react when market conditions change. It tends to be adopted in markets with low volatility and in low-competitive environments. The …rms with the best projects tend to adopt the Code when managers are not too heterogeneous while the best managers tend to be hired through incentive contracts when the projects are similar. Although the matching between shareholders and managers is often positively assortative, the shareholders with the best projects might be willing to renounce to hire the best managers, signing contracts including Codes with lower-ability managers.


Introduction
The last two decades have witnessed the creation and di¤usion of Codes of Best Practice (or Codes of Corporate Governance) all around the world. A Code of Best Practice is a list of rules promoted by a regulator suggesting how a …rm should supervise management. 1 Code's recommendations cover a wide range of Corporate Governance issues: for example, board structure, executive compensation, as well as the role played by institutional investors and capital structure (Bolton et al., 2002). Yet, the two most relevant features shared by any Code of Best Practice are, …rst, its voluntary nature 2 and, second, that they aim to improve manager's oversight. 3 In this paper, we analyze the implementation of Codes in markets where possibly heterogeneous shareholders compete for possibly heterogeneous managers. 4 We study which …rms use a Code and how this depends on the characteristics of the set of shareholders and managers, as well as each …rm's output market. We model each …rm as an agency relationship (Jensen and Meckling, 1976), where each shareholder hires one manager to conduct her project. A shareholder can hire a manager either through an incentive contract or through a contract that includes a Code of Best Practice. The success of each project depends on the decisions taken by the manager and also on product market conditions. Both shareholder and manager know the (ex-ante) distribution of the market conditions parameter; however, only the manager learns the actual realization of the market conditions once the contract has been signed.
The adoption of the Code is a mechanism that allows the shareholder to reduce man-ager's discretion (see, for instance, Dahya et al., 2002). However, this improvement in Board's control goes along with a decrease in ‡exibility at the manager's decision level.
We model this trade-o¤ in a simple way: managers'payment depends on the …nal outcome if he is hired through an incentive contract while a Code allows the shareholder to propose a contract specifying (ex-ante) the manager's decisions. 5 Thus, a shareholder chooses between a contract that easily adjusts to the environment but requires paying informational rents (an incentive contract) and a contract that allows to reduce manager's informational rents but imposes large costs if shareholder's predictions over the market conditions were erroneous since no adjustment is possible (a Code contract).
In the study of one isolated partnership, we show that a Code is more likely to be adopted when projects are highly pro…table, managers are e¢ cient, have low exogenous outside opportunities and there is low variance in market conditions.
The main purpose of our paper is the analysis of the adoption of Codes in environments where shareholders compete for managers, so that the identity of matched partners and their level of utility, in addition to the contract, is endogenous rather than exogenous.
With this purpose, we follow the approach adopted in Dam and Pérez-Castrillo (2006) and Serfes (2008) and we model a two-sided market where the transaction takes the form of a contract (instead of an object or a monetary transfer). 6 We take stability as the solution concept for this market. An outcome (i.e., a matching between shareholders and managers and a set of contracts) is stable if it can not be blocked by a shareholder-manager pair who would sign a more pro…table contract for both parties. 7 The analysis of the shareholder-manager competitive market gives rise to several inter- 5 Similar to Alonso-Paulí (2007), we consider that the adoption of the Code allows the shareholder to reduce manager's discretion. In Alonso-Paulí (2007), adopting the Code prevents the manager from taking certain (bad) actions. In this paper, adopting the Code lets the shareholder choose which actions the manager will take. 6 Dam and Pérez-Castrillo (2006) fully characterize a market with homogeneous principals and heterogeneous agents enjoying limited liability, whereas Serfes (2008) analyzes a market with heterogeneous principals and agents with CARA utility functions. See Roth and Sotomayor (1990) for a general presentation of matching markets with and without money. 7 Stability and competitive equilibrium are very close concepts (for matching models where the parties decide on money instead of contracts see, for instance, Shapley andShubik, 1972, Roth andSotomayor, 1990, and Pérez-Castrillo and Sotomayor, 2002). Any stable outcome is also a competitive equilibrium and viceversa.
esting results. First, in environments with shareholders own projects of di¤erent expected return while all managers have the same ability to conduct them, Codes are adopted by those shareholders owning the best projects. Second, also in this type of environment, identical managers end up with quite di¤erent ex-post utilities due to two di¤erent factors: a) managers signing contracts including Codes obtain lower utility than managers signing incentive contracts, and b) managers being hired through incentive contracts by shareholders with better projects also obtain higher utility. Third, when the market is composed by homogeneous shareholders and heterogeneous managers, the best managers are hired through incentive contracts in such a way that the Code of Best Practice is only implemented in the relationships involving lower-ability managers (contrary to the conclusion obtained in the analysis of an isolated relationship).
Fourth, when both sides are formed by heterogeneous agents, we provide conditions under which the matching is positively assortative, that is, shareholders with better projects hire better managers. This is always the case when either all the contracts (in a stable outcome) are based on incentives, or they all include Codes. We also show cases where the coexistence of incentive contracts and contracts including Codes makes that the matching is not positively assortative. In particular, a shareholder with a good project may opt for a contract including a Code of Best Practice, attracting a less e¢ cient manager when the more e¢ cient is "too expensive" because he is being hired in the market through an incentive contract. Finally, we discuss the welfare e¤ects of introducing Codes of Best Practice. Albeit its voluntary nature, the use of Codes is not always welfare enhancing.
We …nd that, in general, introducing Codes tends to be welfare enhancing if the environment faced by the …rms displays a low variance, while it decreases welfare in environments with intermediate variance. Regarding the disciplinary role of the market, the in ‡uential papers by Grossman and Hart (1980) and Scharfstein (1988) study the main e¤ects of the threat of takeovers and establish the takeover guidelines for this mechanism to be e¤ective. On the role played by large shareholders, Admati et al. (1994) and Huddart (1993)  Finally, the role played by an appropriately chosen executive compensation scheme has been extensively studied (see, for instance, the pioneer work by Jensen and Meckling, 1976, and the paper by Baker et al., 1988) and has been nicely summarized in Murphy (1999).
Finally and concerning the e¤ect of competition, it is worth mentioning the paper by Marin and Verdier (2008). 8 Although di¤erent from our approach, the authors analyze how the market structure, namely product market competition, may a¤ect the reorgani- 8 See also Marin and Verdier (2003).
zation of corporations. Product market competition leads managers to favor more ‡exible organizations by delegating choices to their subordinates. In our model, market structure has also a deep e¤ect in the choice of the governance structure, namely whether it is optimal to set a Code or a contract based on incentives. For instance, we show that it is more likely that contracts include strong incentives, delegating major decisions to the manager, when shareholders compete for the best managers, contrary to what we derive when an isolated partnership is analyzed.
The paper is organized as follows. In Section 2, we present the main features of the model and the corresponding solution concept. The properties that the contract displays in stable outcomes are stated in Section 3. Section 4 studies particular managershareholder markets and provides characteristics for the most general environment. The welfare e¤ect of introducing Codes is discussed in Section 5. Finally, Section 6 concludes and discuses some extensions of the model. All the proofs are in the Appendix.

Shareholders and Managers
We consider the market for managers where n risk neutral shareholders S = fs 1 ; s 2 ; s 3 ; :::; s n g meet N risk neutral managers M = fm 1 ; m 2 ; m 3 ; :::; m N g. We denote shareholders by s, s i , s i 0 , etc. and, similarly, managers are represented by m, m j , m j 0 , etc. Each shareholder owns a project but she lacks the skills to develop it. Instead, each manager has the ability to conduct one project. Thus, shareholders and managers have to match in pairs to carry out projects and a contract is signed for each partnership with this objective. Managers enjoy limited liability over income, their wage can not be negative in any contingency.
Both shareholders and managers may be heterogeneous agents. Shareholders may differ in the pro…tability of the project they own while managers may diverge in their ability to conduct shareholders'projects. We allow for the possibility that both shareholders and managers can seek for alternative partners and sign new contracts. Hence, the matching between shareholders and managers will be endogenous.

Projects
Once a shareholder-manager pair is formed, a …rm is constituted and the manager is in charge of taking decisions concerning the project. We assume for simplicity that projects are independent in the sense that, once constituted, a …rm's pro…ts only depend on decisions taken in that …rm. The project yields a revenue R i > 0 for shareholder s i if it is successful, whereas the asset has value 0 in case of failure. The value of R i , for i = 1; :::n; is public information. Without loss of generality, we order projects as R 1 R 2 ::: R n > 0. The probability of success of the project depends on the manager's decision or e¤ort e and on some random shock h. In particular, we assume that the probability of success is eh.
Manager m j 's e¤ort is his own private information and it has a cost c j (e) = c j e 2 2 , with c j > 0. Managers'ability (the inverse of c j ) is public information, and we order managers depending on their ability: 0 < c 1 c 2 ::: c N ; that is, a lower index corresponds to a more e¢ cient manager.
The random variable h represents the uncertainty in the output market of a project.
This industry-speci…c component can re ‡ect di¤erences among sectors, countries, etc. It is ex-ante unknown to both parties. It is common knowledge that h is distributed according to F (h) on the interval h; h and it is revealed only to the manager after he accepts the contract and before he decides on the e¤ort. 9 We denote (h) =

Contracts and payo¤s
When shareholder s i and manager m j form a …rm, they sign the contract that will govern their relationship. The contract can take two forms. It can be based on an incentive 9 An alternative interpretation for the random variable h is that the value of success is industry-speci…c (hR) rather than the probability of success being industry-speci…c (he). This interpretation yields the same project's expected value. scheme (IS contract) or it can include a Code of Best Practices (CBP contract).
If the …rm (s i ; m j ) signs an IS contract, it has the form W IS s i ;m j = (w R ; w 0 ). The …rst component of the contract w R is the transfer to the manager in case revenue R i is obtained, the second part of the contract w 0 is the transfer in case of failure, when the result is 0. Under contract (w R ; w 0 ), the manager will select the e¤ort once he has observed the realization of the market conditions, h. He will select the contingent e¤ort e(w R ; w 0 ; h) that maximizes his utility, i.e., which implies that the level of e¤ort is: The previous equation represents the Incentive Compatibility Constraint (ICC). It states that the manager tends to exert a higher level of e¤ort if the bonus is large (w R w 0 ), if the market condition is particularly pro…table (high h), or if he has good skills (low c j ).
Alternatively, the …rm (s i ; m j ) can sign a CBP contract. A CBP is a monitoring technology that allows the shareholder to gather better information about the manager's decisions. We model the adoption of a Code in a very simple way. We assume that the board's control allows to make the manager's decisions ex-ante contractual, i.e., the shareholder can ask the manager for an speci…c level of e¤ort. Albeit manager's decisions are ex-ante contractible, shareholders do not know still the realization of the market conditions h. Therefore, the agency problem between the manager and the shareholder is not fully solved. A CBP contract for …rm (s i ; m j ) is then a vector W CBP s i ;m j = (w R ; w 0 ; e) that speci…es the payment to the manager in case of success and failure of the project, as well as the e¤ort he must exert. 10 Any contract must be acceptable for both the manager and the shareholder. Both agents must be better o¤ signing the contract than staying apart from the market. Contract W s i ;m j is acceptable for shareholder s i if it o¤ers her non-negative pro…ts. It is 10 We are aware that the shareholder would …nd pro…table to contract not only on e¤ort, an input measure, but also on the revenue obtained, an output measure, as shown by Baker (1992). We exclude this possibility assuming that it is prohibitively costly to do so. By doing that, we want to highlight the loss in ‡exibility incurred by adopting the Code, as compared to the IS contract.
acceptable for manager m j if his expected utility under W s i ;m j is not lower than the utility he would obtain by exiting the market. We call this the "outside utility" and denote it U . We write the previous Acceptability constraints as follows: where s i (m j ; W s i ;m j ) is shareholder s i 's expected pro…ts and V m j (s i ; W s i ;m j ) is manager m j 's expected utility when they sign the contract W s i ;m j .
Furthermore, contracts have to satisfy managers' limited liability which implies the following constraints: Contracts which are not acceptable for one of the parties or that do not satisfy limited liability constraints will be discarded. We say that they are not feasible contracts.

Matching
In our model, the identity of the partners in any …rm is endogenous. A manager will not only choose between signing a contract with a particular shareholder or staying out of the market, but he also has the option to form a …rm with any other shareholder. We will represent the identity of the partners forming …rms through a matching function that associates shareholders with managers. We now describe a matching in this economy. In addition, we need to describe which contract governs any relationship. The only requisite is that contracts within a …rm must be feasible.
De…nition 3 A menu of contracts W compatible with a matching for the market fS; M; R; cg is a vector of feasible contracts, one for each …rm formed under .
A matching and a set of contracts determine a possible organization of our market that we will refer to as an outcome. The objective of our paper is to characterize the equilibrium outcomes in the shareholders/managers market. That is, we identify the characteristics of both, the contracts emerging and the …rms formed.
To be an equilibrium, the outcome ( ; W) shall be immune to potential blocking from any shareholder-manager pair. This idea corresponds to the concept of stability. It states that it is not sensible to expect ( ; W) to be an (stable or equilibrium) outcome of the market if there exists any shareholder-manager pair that can form a …rm by signing a feasible contract such that both the shareholder and the manager are better-o¤ under the new deal compared to the initial situation ( ; W).
Stability requires that there does not exist any shareholder-manager pair that can block the current outcome, signing a feasible contract W 0 for them. Furthermore, since all contracts in a stable outcome are feasible, a stable outcome is also individually rational.

Contracts in a stable outcome
In this section, we have a …rst look at the characteristics of contracts signed in stable outcomes of market fS; M; R; cg. The …rst characteristic implied by stability is that it is not possible for the partners of any existing …rm to sign an alternative contract that both …nd better than the current contract since, otherwise, it would exist a pro…table deviation for the partners. That is, contracts in a stable outcome are Pareto optimal among those feasible contracts that satisfy (in case of incentive contracts) incentive constraints. We refer to this notion as (constrained) Pareto Optimality; it is formalized in the following de…nition.
; with at least one strict inequality.

Proposition 1 states the optimality property.
Proposition 1 All the contracts in a stable outcome for the market fS; M; R; cg are constrained Pareto optimal.
The property of constrained Pareto optimality allows to identify any contract in a stable outcome once we know the identity of the partners (s i ; m j ) and the utility obtained by manager m j . Indeed, the contract is the one that maximizes shareholder s i 's expected pro…ts under the constraint that manager m j gets this utility level.
In the rest of the section, we will characterize the best contract from shareholder s i 's point of view as a function of any possible "reservation utility" level U j which has to be achieved by the manager. We denote such a contract as W s i ;m j (U j ). 11 Note that the level U j will be an equilibrium reservation utility level, determined by the possibility that manager s i forms a partnership with other shareholders, when we will perform the complete analysis of stable outcomes in next section.
Remember that the contract W s i ;m j (U j ) either takes the form of an Incentive Scheme We …rst identify the contract if (s i ; m j ) sign W IS s i ;m j (U j ) then, we calculate the best contract including a CBP W CBP s i ;m j (U j ) and …nally, as a function of the reservation utility U j , we state which type of contract is chosen.
Under an IS contract, limited liability constraint (LL 0 ) makes the incentive condition (ICC) costly for shareholder s i . However, the impact of limited liability on contracts and on payo¤s di¤ers depending on the level of manager's reservation utility U j . For low values of U j , the optimal payment scheme depends only on the value of the project (R i ).
The shareholder shares half of the value in case of success and the manager ends up with a utility larger than U j . For large values of U j , the optimal payment scheme also depends on U j as the participation constraint binds. The threshold, denoted by b U ij , that divides both regions depends on the value of the project and on the distribution of the market speci…c component as well as on the e¢ ciency of the manager. Formally, Finally, note that the shareholder s i will not …nd acceptable a contract with the manager m j if she obtains negative earnings. This situation arises for We summarize these …ndings and identify the contract W IS s i ;m j (U j ) in Proposition 2.
Proposition 2 If (s i ; m j ) sign an IS contract W IS s i ;m j (U j ) in a stable outcome for the market fS; M; R; cg, then: (a) U j e U ij and the manager's expected utility is U j = max  Under a CBP contract W CBP s i ;m j (U j ), shareholder and manager not only agree on the payment scheme but also on a pre-speci…ed level of e¤ort. Since the e¤ort is contractual, the only objective of the payment scheme is that the manager accepts to enter into the relationship. Furthermore, given that both shareholder and manager are risk neutral agents, only the expected wage matters. For the sake of simplicity, and without loss of generality, we stick to …xed wages. Proposition 3 summarizes the characteristics of the contract when the Code is adopted: in a stable outcome for the market fS; M; R; cg, then: and the manager obtains a utility of U j ; The optimal ex-ante level of e¤ort that maximizes shareholder's pro…ts depends on the value of the project R i , the manager's e¢ ciency (the inverse of c j ) and also on (h): the average value of the market conditions. Since manager's e¤ort must be ex-ante selected and the shareholder does not know the true market conditions, the optimal level of e¤ort is taken as if the true realization was in fact the mean of the distribution (because this is the choice that minimizes the potential losses from an ex-post deviation/mistake). Due to the adoption of the Code, the shareholder does not need to pay informational rents.
Once we have studied the characteristics of W IS s i ;m j (U j ) and W CBP s i ;m j (U j ), we proceed to analyze whether shareholder s i prefers to propose an IS contract or a CBP contract as a function of U j . The main advantage of implementing a Code versus providing incentives is that the former allows a better control of the manager's actions. Yet, since the market-speci…c component h is not veri…able, adopting a Code causes a loss in ‡exibility (ex-post) in the manager's decision taking. Therefore, the optimal contract will depend on market conditions. Market conditions can be easily captured by the Sharpe ratio. This ratio, also known as reward-to-variability index, has been extensively used to study the excess of return per unit of risk in an investment strategy or a trading strategy. 12 In what follows, we de…ne the Sharpe ratio of a distribution F (h) as This ratio states how good ex-ante market conditions are, once market volatility is taken into account. That is, a larger ratio signals better (ex-ante) market conditions. Environments with high variance are less appealing than markets with low variance, keeping the mean of the distribution …xed.
Proposition 4 characterizes the optimal contract. 13 Proposition 4 Shareholder s i obtains higher pro…ts with the contract W CBP Proposition 4 states the conditions for the optimal contract to include a Code of Best Practice. Note that a CBP is never adopted if Sr(h) 1. In a nutshell, adopting a CBP requires, keeping …xed the mean of the market conditions, an environment with low enough 12 For a more detailed analysis of the Sharpe ratio, see Sharpe (1966 and1994). 13 We take the convention that the IS contract will be selected in case of indi¤erence between the two contracts.
variance (Sr(h) > 1) and a low manager's utility level (U j < U ij ). Intuitively, a CBP becomes very e¤ective in situations where the environment is not too volatile since the pre-speci…ed level of e¤ort (the optimal corresponding to a situation around the mean of the distribution) does not di¤er too much from the optimal ex-post e¤ort. 14 On the other hand, the IS contract allows the manager to adapt better to current circumstances, which also bene…ts the shareholder (although the manager's decisions will not be optimal). 15 Besides, the Code allows the shareholder to control better the manager which, in turn, implies a lower wage expected cost. This latter e¤ect is less relevant when the manager's utility is large as, in this case, it is cheaper to give the proper incentives to the manager also expost. In fact, under an IS contract, manager's e¤ort increases with the reservation utility and towards the …rst best level of e¤ort (Proposition 2), while the e¤ort is independent of U j when a CBP is adopted (Proposition 3). Hence, a CBP is more useful when manager's reservation utility is not too large. Figure 2 illustrates the e¤ect of manager's utility on the adoption decision when the environment is not too volatile. be dependent on input (e¤ort) rather than on output (i.e., revenues). 15 Similarly, the threshold U ij depends on market conditions, as well. In markets with lower variance or larger variance, the threshold shifts to the right making the shareholder more willing to adopt CBPs. The adoption of a CBP also depends on the ratio R i =c j , that is, on the value of the project and on manager's e¢ ciency. The higher this ratio, the higher the e¤ort that will be asked from the manager; given the more acute agency problem, the shareholder s i …nds the CBP more appealing as it allows a better manager's control. This result is true unless the partnership copes with a very volatile environment.

Stable outcomes in the shareholder-manager market
In this Section, we analyze stable outcomes in the market formed by shareholders and managers. The main objectives are to highlight the e¤ect of competition for managers on the contracts signed by …rms and to state the composition of the …rms: who is matched with whom. We …rst prove, in Proposition 5, that stable outcomes always exist.
Proposition 5 The set of stable outcomes in the market fS; M; R; cg is always nonempty.
We prove this existence result in the Appendix by adapting to our environment the proof developed by Crawford and Knoer (1981) for assignment games. 16 In this proof we …rst show, as they did, that a stable outcome always exists in any "discrete economy" where the possible levels of reservation utility of the managers are discrete, di¤ering by a (possible very small) amount. We then prove that, if the market fS; M; R; cg has no stable outcome, then also a discrete economy with levels of minimum utility close enough can not have a stable outcome.
An alternative way to prove existence is to propose an outcome for the market and show that it is indeed stable. Since this approach may help to understand how stable outcomes look like, we now develop such a constructive method when there are either two or three partners in each side.
Consider an economy with two shareholders and two managers where forming partnerships is always pro…table, that is s 2 (m 2 ; W s 2 ;m 2 (U )) 0. Imagine a …ctitious auction between the two shareholders to hire the best manager. We denote by U s i m 1 , for i = 1; 2, the maximum utility level for manager m 1 that makes shareholder s i indi¤erent between hiring m 1 with a reservation utility of U s i m 1 and hiring m 2 with the outside utility U : We can interpret U s i m 1 as the "bid" of s i for m 1 . It is easy to check that if manager s 1 is ready to pay more than s 2 to hire the better manager, i.e., U s 1 m 1 U s 2 m 1 , then the matching ((s 1 ; m 1 ) ; (s 2 ; m 2 )) together with the contracts W s 1 ;m 1 (U s 1 m 1 ); W s 2 ;m 2 (U ) , form a stable outcome. The other case is similar. Remark also that we can easily extend the argument to markets with more than two shareholders or more than two managers.
When there are three shareholders and three managers, we need to proceed in several steps. In the …rst step, we denote by U s i m j 1 , for i = 1; 2; 3 and j = 1; 2, where the subindex refer to step 1,"the bid of s i for m j against m 3 ", that is, the maximum utility level such that: and we de…ne U for all k (note that, by our 16 See also Gale (1984), Demange and Gale (1985) and Sotomayor (2002) for existence results in assignment games. The main di¤erence between our model and the assignment game is that in ours, matched agents agree on contingent payments and, possibly, on e¤ort, and not only on a …xed monetary transfer.
convention, S(m 3 ) 1 = fs 1 ; s 2 ; s 3 g): If there exists a matching such that (m j ) 2 S(m j ) 1 for all j, we take . The matching with the contracts W (m j );m j (U (m j )m j 1 ) for j = 1; 2; 3; form a stable outcome. If we can not associate a shareholder to each manager, we need a second step. We take the unique shareholder s whose bid is the highest for both m 1 and m 2 : s = S(m 1 ) 1 = S(m 2 ) 1 . We decrease simultaneously U

Homogeneous shareholders and heterogeneous managers
Consider the case where the project in hands of all the shareholders o¤ers identical returns, i.e., R i = R for all i = 1; :::; n while managers di¤er in their ability, c 1 < c 2 < ::: < c N .
Also for simplicity, we are denoting s N the less e¢ cient manager with whom a shareholder makes non-negative pro…ts (if there are other managers in the market, we can discard them as they will never be matched in a stable outcome). Proposition 6 characterizes the stable outcomes in such a market. In the proposition, we denote byñ = min fn; N g the number of …rms that will be formed. Also, to identify the threshold levels that will separate regions in the proposition, we use the following notation: if Sr(h) > 1; we denote by j = s W s;m j (U j ) the level of pro…ts that the shareholder obtains when manager m j 's utility is such that she is indi¤erent between a CBP contract and an IS contract. 17 It is easy to check that j increases with the manager's e¢ ciency, hence it is decreasing in j. (c) W s;m j is the optimal contract for the manager that gives pro…ts to the shareholder. (d3) W s;m j is a CBP contract if j > J and W s;m j is an IS contract if j J, when Since all shareholders hold projects with the same return, the …rst characteristic highlighted in Proposition 6 is that their pro…ts must be equal. If this was not the case, shareholder s i getting lower pro…ts than s i 0 could attract manager (s i 0 ) by proposing a contract that slightly increases his utility. Second, the level of pro…ts depends on the strength of the competition for managers. Shareholders achieve positive pro…ts when competition for managers is smooth, that is, they are in the short side of the market.
Furthermore, their pro…ts are higher when they could hire better managers with lower outside option who are not hired in this market. Third, since CBPs allow better manager's control, they arise if competition between shareholders is not very strong (so that managers' level of utility is low), while shareholders are more prone to o¤er incentive contracts in situations where competition is tough. Also, similar to the conclusion that we obtained in the analysis for an isolated …rm (Proposition 4), CBPs are not adopted unless ex-ante markets conditions are good enough (Sr(h) > 1). In other words, keeping the mean …xed, CBPs are adopted in environments with low variance, while IS contracts are signed under volatile environments.
Finally, Proposition 6 (d3) states who adopts a CBP when a market sustains simultaneously IS and CBP contracts: e¢ cient managers end up being hired through IS contracts while a CBP is used to attract ine¢ cient ones. This contrasts with the conclusion obtained after Proposition 4 suggesting that a CBP would be adopted for e¢ cient managers.
Proposition 6 (d3) shows that, when shareholders compete for the best managers, the conclusion is reversed. To attract e¢ cient managers, shareholders o¤er them a high utility level and this now makes IS contracts more appealing than CBP contracts. This result stresses the relevance of the study of manager-shareholder relationships in a framework where not only contracts but also the matching is endogenous. 18

Heterogeneous shareholders and homogeneous managers
We now consider an economy formed by heterogeneous shareholders (R i > R i 0 for all i < i 0 ) and equally e¢ cient managers (c j = c for all m j 2 M). We assume, for simplicity, that there are more managers than shareholders, i.e., N > n and that the partnership between s n and a manager is feasible. A direct implication of the above is that the reservation utility shareholders will need to provide to managers is the outside utility U . 19 Proposition 7 characterizes the unique stable outcome for this market (for notational simplicity, we denote U i instead of U ij ). and U I+1 U < U I . 18 Similar to Barros and Macho-Stadler (1998) and Dam and Pérez-Castrillo (2006), the use of incentive contracts has also a positive e¤ect on …rm's e¢ ciency. The better the manager, the closest the e¤ort to its e¢ cient (…rst-best) level. 19 If N n, managers will tipically end up with a higher utility than U but the qualitative results will hold.

Heterogeneous shareholders and heterogeneous managers
We now analyze stable outcomes in markets where both shareholders and managers are heterogeneous. We …rst state a proposition that provides interesting information about the level of utility obtained by di¤erent managers. We have seen that homogeneous managers can end up with di¤erent utility levels. However, a general property allows to rank the level of managers' utility as a function of their ability: when shareholders compete for managers, better managers always obtain larger utility. 20 The main question about the shape of matchings in stable outcomes is whether shareholders with good projects end up hiring e¢ cient managers. If this is a characteristic of 20 Gabaix and Landier (2006)develop a simple equilibrium model to analyze how CEO's level of remuneration depends upon their ability. The authors show that competition for managers leads to a situation where CEO's pay increases with ability but it is mainly determined by …rm size, which in our model, is captured by the value R i : the matchings, we say they are positively assortative. If, for instance, all participants were hiring managers through a CBP contract, then a negatively assortative contract could not be stable. The rationale is based on the fact that it is optimal (in terms of total surplus) that the best managers run the best projects, hence any negative assortative matching will be blocked by at least one alternative shareholdermanager partnership creating more value for them.
However, the matching is not necessarily positively assortative when both types of contracts coexist in a stable outcome. In that case, how does a non-positively assortative matching in a stable outcome look like? Proposition 10 provides useful information in that respect. Therefore, in a non-positively assortative matching in a stable outcome, shareholders with pro…table projects hire low-ability managers through CBPs while high-ability managers sign incentive contracts in less pro…table …rms. This is due to the following trade-o¤. On the one hand, maximizing total surplus requires a positively assortative matching. Hence, stable outcomes tend to be possitively assortative. On the other hand, the better the shareholder, the more likely that she prefers proposing a CBP contract.
Therefore, it can be the case that, say, shareholder s 2 's best contract to attract manager m 1 is an IS contract giving him a rent not lower than b U 21 while a shareholder with a more pro…table project, s 1 , would prefer to hire this manager through a CBP. If s 1 is forced to pay m 1 the rent that s 2 is ready to o¤er, then she would rather attract a less e¢ cient manager through a CBP, as long as the di¤erence in e¢ ciency between the two managers and in the value of the two projects is not too large.
To gain further intuition about why a negatively assortative matching can be stable, consider a situation similar to the one displayed in Figure 3, except that now m 1 is slightly more e¢ cient than the rest of the managers, c 1 < c. The white dots in Figure 4 represent the maximum utility U i1 shareholder s i for i I, is ready to pay to match with m 1 (we know that she will do it through an IS contract). If shareholder s 1 wants to hire m 1 , she has to o¤er him at least U I1 . Given the preference of s 1 to o¤er a CBP, she will prefer keeping a "normal"manager at the prize U instead of attracting the slightly better manager at the prize U I1 .

Introducing Codes: Welfare Considerations
Does the introduction of CBPs yield a welfare improvement? As we have stressed, CBPs are voluntary mechanisms; shareholders only adopt them if they earn higher pro…ts, given the managers'reservation utilities. Therefore, a welfare improvement would be expected as a reasonable outcome if shareholders choose to implement CBPs. However, this shall not always be the case due to three e¤ects. First, the CBP allows a shareholder to avoid paying informational rents in those cases where an IS contract would ensure the manager a level of utility higher than his reservation utility. Therefore, even if welfare (shareholder's expected pro…ts plus manager's expected utility) decreases, the shareholder may be willing to adopt the CBP. Second, the introduction of CBP contracts may have an e¤ect on the managers' reservation utility level. Finally, the introduction of CBPs may have a deep e¤ect on the market structure; a negative assortative matching may arise.
We should be aware that the set of stable outcomes in this economy needs not to be unique and henceforth a comparative statics analysis for welfare becomes hard to derive. Nonetheless, we are still able to draw appealing welfare implications from analysing particular market structures. We claim that the introduction of CBPs is likely to enhance welfare in those environments where market conditions forecasts are high (Sr(h) p 3) and to worsen it in environments with moderate ex-ante market conditions (Sr(h) 2 (1; p 3)). We do not need to discuss those environments forecasting low market conditions (Sr(h) 1) since the …rms that operate in them will never adopt CBPs, hence its introduction will be without consequence in these industries. We brie ‡y make a case for our claim.
The welfare obtained by each pro…table partnership depending on the governance mechanism is: Under IS contracts, manager's m j utility, when he forms a …rm with shareholder s i , is at least b U ij . If the introduction of CBPs would not change managers'utility level, then the constrained pareto optimal contract is a CBP contract only if variance is low (and If this is the case, the decision to adopt a CBP contract is always welfare improving. However one would expect that the introduction of CBPs a¤ect managers'utility; not only to the managers hired by …rms implementing CBPs, but also to those who signing IS contracts coexist with …rms that adopt CBP contracts. To disentangle these e¤ects, consider …rst a market with many managers of similar ability whose outside utility U is low. If only IS contracts are allowed, managers'level of utility is given by Second, we discuss situations where the introduction of voluntary CBPs may a¤ect the managers' utility of the rest of the managers, as well. We argue that the previous claim often holds, although some new e¤ects make CBPs less appealing from a welfare point of view. To discuss this case let us focus on a market with one "good"manager and many identical "standard" managers ( shareholders being identical). The e¤ect of the CBPs in the …rms involving standard managers is the same as before: good for welfare if low variance and bad if intermediate variance. This is also the e¤ect in the …rm where the good manager is hired if it adopts a CBP, as the welfare is in this case independent on the manager's utility. However, this is not necessarily true if this …rm uses an IS contract. The introduction of CBP's, in this case, relaxes competition for managers between shareholders. Given that the other …rms can now adopt CBPs, they may make higher pro…ts with their standard managers. Therefore, hiring the good manager through an IS contract might become less appealing. The (equilibrium) reservation utility of the good manager may decrease if CBPs can be adopted, which would imply a lower welfare in the …rm where he is working.
A third e¤ect is due to the fact that the matching may not longer be possitively assortative if IS and CBP contracts coexist. This e¤ect is also typically detrimental for welfare, as a possitively assortative matching is optimal from a total surplus point of view. Therefore, we can conclude that, in our model, the introduction of voluntary CBPs decreases welfare in environments with intermediate volatility. On the other hand, it certainly increases welfare in environments with low volatility as long as it does not induce too many changes in the structure of the market, either through drastic decreases in managers'level of utility or through changes in the equilibrium allocation of managers to …rms.

Conclusion and Extensions
This paper has explored the way market conditions and competition a¤ects shareholders' willingness to adopt Codes of Best Practice. We have modeled a Code as a monitoring mechanism that allows shareholders to control ex-ante managers'decisions. Adopting the Code, though, impedes a ‡exible managers'reaction to changing market conditions. Due to ex-post in ‡exibility, CBPs tend to be adopted in environments where predicting market conditions is not a complex task for shareholders. More mature industries, such as utilities, banking, food and drink sector, should be, according to our predictions, examples of industries where CBPs are more likely to be adopted. On the contrary, when the environment faces high volatility, the best a shareholder can opt for is to leave manager's hands free (i.e., to o¤er him an incentive contract). High-tech sectors such as dot-com industries, or pharmaceutical companies should tend to use incentive contracts, letting the manager take the major decisions.
Our analysis may have implications regarding when …rms will be more willing to adopt CBPs. Indeed, market conditions vary depending, for instance, on macroeconomic conditions or business cycles. As we have mentioned, the choice of the governance structure takes into account expected market conditions (measured through the Sharpe ratio Sr(h)).
Hence, CBPs are more likely to be adopted after recessions, when expected market conditions tend to improve, whereas contracts rather based on incentives should be expected during (or ending) booms where forecasts over market conditions tend to worsen.
Our …ndings suggest that the characteristics of the set of shareholders and the set of managers in the market have a deep e¤ect on the decision of adopting a CBP. When shareholders have similar projects, i.e., …rms'technologies are similar, the CBPs do not seem to be the right mechanisms to attract the best managers. Indeed, partnerships will agree on this governance structure only in environments with low level of competition for managers. In addition, when both types of governance structures coexist, the lower the manager's ability, the more likely that a Code is adopted. Instead, when managers are of similar ability, the shareholders with best projects prefer to adopt the Code since this governance pattern allows a shareholder to reduce her manager's rents. In fact, our analysis suggests that the best shareholders might be willing to renounce to hire the best managers, who would be o¤ered incentive contracts, and hire instead lower ability managers through CBPs. Hence, although the matching between shareholders and managers is always positively assortative when only one type of governance structure exists in the market, the property may fail to hold due to the coexistence of both governance structures.
Since the characteristics of the market are linked to the type of industry a …rm operates in and to the type of project, it seems natural to ask how the conclusions of our analysis would be a¤ected if shareholders'heterogeneity was not due to the value of their project but to the distribution function of the market speci…c characteristics (that is, we would have a distribution function F i (h) for each shareholder i). Similar to the conclusions obtained when heterogeneity is due to the value of the project, also here the matching is "positively assortative" when all the …rms end up under the same type of governance structure. However, the meaning of "positively assortative" depends now on the type of governance. If all contracts include a Code, then the best managers are hired by those shareholders whose market-speci…c component has a higher mean. However, if they are all incentive contracts, the "best" shareholders are those in markets where the combination of mean and variance (expressed in the variable V ar(h) + (h) 2 ) is higher; these are the shareholders who end up forming …rms with the higher-ability managers. Also, we have some information about how a non-positively assortative matching in a stable outcome looks like. For example, if all shareholders own projects in markets whose shocks have the same average then, in a non-positively assortative matching, shareholders in markets with higher volatility hire low-ability managers through incentive contracts while highability managers sign contracts including a Code in …rms producing in markets with lower variance.
Finally, we have not considered the possibility that …rms, once created, could compete against each other in the product market. In our model, there was competition among shareholders to catch the best managers and among managers to work for the shareholders that o¤er the best contracts, but there was no competition among …rms. A …rm's pro…ts were independent of the composition of the other …rms. Extending our model to explicitly allow …rms'market competition seems computationally demanding. However, the analysis developed so far provides enough elements to be able to anticipate the e¤ects of …rms' competition on the use of Codes of Best Practices.
For our purposes, …rms'market competition should have two main implications. First, it makes the best managers even more appealing than before as shareholders will be ready to o¤er good salaries to these managers not only because of their value for the …rm but also to avoid that they are hired by the market competitors. According to our results, such an increase in managers'utility should favour the use of incentive contracts. Second, market competition typically allows improving incentive contracts by making use of yardstick contracts, where a manager is paid according not only to his absolute performance, but also as a function of his relative performance with respect to others. Both e¤ects go in the direction of making incentive contracts more appealing. Therefore, we should expect to observe less use of Codes in those markets characterized by tough competition.

Appendix
Proof. of Proposition 1. Assume ( ; W) is stable, but the contract W s i ;m j 2 W signed by (s i ; m j ), where (s i ) = m j , is not constrained Pareto optimal. (a) First, suppose there exists a feasible contract W 0 for the partnership (s i ; m j ) such that s i (m j ; W 0 ) > s i ( (s i ); W s i ; (s i ) ) and V m j (s i ; W 0 ) > V ( (m j ); W m j ; (m j ) ). In that case, (s i ; m j ) will block ( ; W) with W 0 . This contradicts the initial fact that ( ; W) is stable.
(b) Second, suppose there exists a feasible contract W 0 for the partnership (s i ; m j ) such ). Consider the contract W " that includes both salaries higher that W 0 by > 0. Manager's e¤ort will be the same under W " and W 0 (if these contract include a CBP this will happen by contract; if they are incentive contracts, the ICC does not change). If is small enough, W " satis…es s i (m j ; W ") > s i ( (s i ); W s i ; (s i ) ) and V m j (s i ; W ") > V ( (m j ); W m j ; (m j ) ); which we already know contradicts the stability of ( ; W).
The analysis of the third possibility requires a more detailed understanding of the (optimal) contracts between shareholder and manager. We will develop such an analysis in the rest of Section 3 using (b), i.e., there can not exist a contract that leaves the manager indi¤erent by improving shareholder's pro…ts. We can check afterwards that, among those contracts, it is not possible to improve manager's expected utility without lowering strictly shareholder's pro…ts.
Proof. of Proposition 2. The contract W IS s i ;m j (U j ) is the solution to the following programme: We can rewrite the programme by plugging (ICC) into the objective function and the last constraint and let us denote, for notational convenience, (h) V ar(h) + (h) 2 . After some calculations we obtain: where we have omitted (LL R ) since it is implied by (ICC) and (2).
Let and be the Lagrange multipliers corresponding to (1) and (2), respectively.
The Kuhn-Tucker (…rst-order) conditions of the above maximization problem are (1), (2), 0, 0, and: w 0 = 0: First, simplifying (3) we get: and plugging (7) into (4) we obtain: We study the di¤erent regions where Kuhn-Tucker conditions may be satis…ed: (1) and (2) are binding). Payment in case of failure is w 0 = 0 (2). Finally, from (7) and (8) Also, it is easily checked that the solution at the borders of cases 3 and 4 coincide with the solution at the borders of Case 1 (the solution is continuous). Therefore, the optimal contract has the shape found in Case 1, which proves part (b). Finally, parts (c) and (d) follow from the contracts in Case 1.
Proof. of Proposition 3. First, since there is no need to give the manager incentives, a …xed wage is optimal, we denote it by w (= w R = w 0 ). The contract W CBP s i ;m j (U j ) is the solution to the following problem: w 0: From (9), w c j e 2 2 +U j which implies that (10) is not binding. Also, since w a¤ects negatively shareholder s i 's expected pro…ts, (9) is binding. This proves part (c) of Proposition 3. Therefore, shareholder's problem is: hence, e CBP = R i c j (h). This shows part (a). Part (b) is obtained by plugging e CBP into (9) binding. Finally, to prove part (d), we substitute e CBP and the optimal wage into shareholder s i 's expected pro…ts.
Proof. of Proposition 4. We compare the pro…t functions of propositions 2 and 3.
First, we compare both pro…t functions at the extreme values U j = 0 and e U ij : where the inequality holds since V ar(h) > 0. Moreover, ) is a decreasing and concave function of U j for Therefore, the functions s i (W CBP s i ;m j (U j )) and s i (W IS s i ;m j (U j )) cross at most once in The previous properties imply …rst, that is strictly decreasing w.r.t. U j . We evaluate pro…ts at b U ij and we Therefore, if V ar(h) 2 1 3 (h) 2 ; (h) 2 the Code is adopted if and only if U j < U j , where U j 2 ): By the properties of the derivatives of the pro…t functions, there exists a unique U ij 2 ( b U ij ; e U ij ) such that the Code is adopted if U j < U ij .
Level U ij is the smallest of the two values for which U ij . It corresponds to the expression stated in part (a) of the Proposition.
Proof. of Proposition 5. We are going to adapt to our environment the proof of existence developed by Crawford and Knoer (1981) for assignment games. We …rst consider a discrete economy, where the possible levels of reservation utility of the managers are U , U + 1, U + 2, and so on. The numbers can be as small as wished, we denote U + r instead of U + r for notational simplicity.
Given that the participation constraint in a contract W s i ;m j (U j ) might not be binding when the optimal contract is an IS contract, we denote as W + s i ;m j (U j ) the optimal contract when s i is imposed only to select contracts that provide m j a …nal utility of U + r, where r must be some non-negative natural number.
We apply the following algorithm that de…nes the possible o¤ers by shareholders to managers and the way they should act at any time t: R2. Each shareholder s i makes an o¤er W + s i ;m k (U k (0)) to the manager m k with whom she obtains the largest non-negative pro…ts, given the level of reservation utility U j (0) that she must guaranty to any manager m j . That is, k 2 arg max j2N R3. Each manager who receives one or more o¤ers, rejects all but his favorite, which he tentatively accepts. Ties are broken at any time in any manner.
R4. O¤ers not rejected in previous periods remain in force. If manager m j rejected an o¤er from some shareholder in period t 1, then U j (t) = U j (t 1)+1; if not, U j (t) = U j (t 1).
Rejected shareholders continue to make o¤ers to their favorite managers, taking into account the current permitted reservation utility levels, as long as they make non-negative pro…ts.
R5. The process stops when no rejections are issued in some period. Managers then accept the o¤ers that remain in force from the shareholders they have not rejected.
Claim 1. After a …nite number of periods, no rejections are issued, every manager gets at most one o¤er, and the process stops.
This claim follows the fact that the increments in the minimum utility are discrete and that a shareholder's pro…ts are negative if the reservation utility she needs to o¤er to the manager is high enough.
Claim 2. The process converges to a discrete stable allocation in the discrete market previously de…ned.
By construction, the algorithm always provides individually-rational outcomes. Hence, we prove that the …nal allocation of the process is indeed discrete stable if we show that it can not be blocked by a shareholder-manager pair. By Claim 1, the process converges to an outcome, denote it by ( ; W). Suppose that ( ; W) is not discrete stable. Then, there exists a couple (s i ; m j ) such that where U j and i are, respectively, the utility and pro…ts currently obtained by s i and m j under ( ; W) (remember that U j = U + r, for some natural number r). We note that, denoting by T the period where the process has stopped, U j U j (T ); in fact, if the …nal contract signed by m j includes a CBP, but U j can be strictly larger than U j (T ) if m j signs an IS contract. However, we know that shareholder s i has preferred (s i ) to m j when the minimum utility level to o¤er to m j was, at most, U j (T )+1 (the utility level can have been smaller at the time s i made her decision, as she might have been provisionally matched with (s i ) for some periods before T ). That is, s i (W + s i ;m j (U j (T ) + 1)) i . Hence, (s i ; m j ) can not block the outcome. Therefore, we have proven that there exists H > 0 such that, for any individually rational outcome ( ; W) we can …nd a pair (s i ; m j ) whose bene…ts by deviating are larger than H. Note that we can always split the extra pro…ts B obtained by the shareholder between her and her manager by increasing both manager's salaries by B=2; which does not alter his incentives. Hence, if we choose the unit of measurement smaller than H=2; this would imply that any individually rational allocation can be improved upon by a least one partnership (s i ; m j ) in the corresponding discrete market as well, which would contradict Claim 2.
Proof of Proposition 6. We prove part (a) is necessary. Suppose, without loss of generality, that shareholder s 1 obtains lower pro…ts than s 2 . Then, s 1 could hire the manager who is currently with s 2 , o¤er him a slightly better utility level than before (through, say, the same type of contract than s 2 was o¤ering) and make strictly higher pro…ts. This is not possible in a stable outcome. Part (b) easily follows from the maximum and minimum pro…ts that the shareholder hiring the worst manager (or not hiring at all) can make. Part (c) follows after Proposition 1. Finally, it is immediate that if the contracts satisfy (a) -(c), then the outcome is stable.
To prove part (d1), we note that if V ar(h) (h) 2 , the CBP is never adopted according to Proposition 4. If V ar(h) < (h) 2 and n , then j for all managers, since j is decreasing in j. This implies that U j U j since the constrained Pareto optimal contract between the shareholder and the manager m j that provides the shareholder a pro…t level smaller or equal than j shall give m j higher utility level than U j (which he would obtain in the case the shareholder would get j ). Therefore (Proposition 4), An incentive contract is optimal for all j. A very similar argument allows to prove part (d2 ).
When V ar(h) < (h) 2 and J is such that J > J+1 then J if and only if j J. As we have argued above, j implies that the constrained Pareto optimal contract between a shareholder and m j is an IS contract. In fact, W s ; m j is a IS contract if and only if j > J, as stated in part (e) of the Proposition.
Proof. of Proposition 7. Given N > n and that managers are homogeneous, any contract in any stable outcome should be the best contract for the shareholder when she only needs to o¤er U to the manager. It is then immediate that (a) and (b) characterize the (unique) stable outcome.
Proof of Corollary 1. Under the conditions set in part (c) of Proposition 1, a manager matched with a "bad"shareholder is hired through an IS contract, which implies that he achieves b U i , whereas a manager matched with a "good" shareholder achieves Proof. of Proposition 8. Let m j and m j 0 , with c j < c j 0 , be two managers that are matched under the stable outcome ( ; W) and let U j and U j 0 be the level of utility they obtain. Suppose, by contradiction, that U j U j 0 . By inspection of a shareholder's pro…ts in Propositions 2 and 3, it is easily checked that they are increasing in manager's e¢ ciency, for a given level of manager's utility. Therefore, if s i 0 = (m j 0 ) o¤ers W s i 0 ;m j (U j 0 ) to manager m j , she will obtain higher pro…ts than in the outcome ( ; W) while m j achieves the utility level U j 0 U j . It is always possible to modify that contract to make sure that both s i 0 and m j obtain higher pro…ts than under ( ; W); that is, they can block the outcome, which contradicts the fact that it is stable.
Proof of Propositions 9 and 10. Similar to the Proof of Proposition 2, i.e., for the sake of clearness, let us denote (h) V ar(h) + (h) 2 . We do the proof by contradiction.
Take two matched shareholders s i and s i 0 , with m j = (s i ) and m j 0 = (s i 0 ), such that R i > R i 0 while c j > c j 0 : Denote by U j and U j 0 , with U j < U j 0 according to Lemma 8, the level of utility obtained by managers m j and m j 0 in the stable outcome ( ; W). The contracts signed by shareholders s i and s i 0 are, respectively, W s i ;m j (U j ) and W s i 0 ;m j 0 (U j 0 ).
We are going to prove that unless W s i ;m j (U j ) = W CBP s i ;m j (U j ) and W s i 0 ;m j 0 (U j 0 ) = W IS s i 0 ;m j 0 (U j 0 ); the following inequality hods: s i (W s i ;m j 0 (U j 0 )) + s i 0 (W s i 0 ;m j (U j )) > s i (W s i ;m j (U j )) + s i 0 (W s i 0 ;m j 0 (U j 0 )): Therefore, either s i (W s i ;m j 0 (U j 0 )) > s i (W s i ;m j (U j )) or s i 0 (W s i 0 ;m j (U j )) > s i 0 (W s i 0 ;m j 0 (U j 0 )).
However, this cannot happen in a stable outcome. Indeed, suppose for instance that the …rst inequality was true. Shareholder s i could o¤er to manager m j 0 a contract that would guaranty this manager an expected utility slightly larger than U j 0 while keeping for herself expected pro…ts larger than s i (W s i ;m j (U j )). That is, the partnership (s i ; m j 0 ) could block the outcome ( ; W).
(a) Consider …rst that both are IS contracts, i.e., W s i ;m j (U j ) = W IS s i ;m j (U j ) and W s i 0 ;m j 0 (U j 0 ) = W IS s i 0 ;m j 0 (U j 0 ): From Proposition (2), the optimal payment has a di¤erent shape depending on the level of utility. We know that U j b U ij and U j 0 b U i 0 j 0 . This also implies that U j b U i 0 j : We consider now two cases: (a1) U j 0 b U ij 0 . In this case, equation (11) is equivalent to: which holds given that: and (11) is implied by: which is equivalent to: We see that @f (R i ;R i 0 ) @R i > 0 if and only if U j < b U ij 0 c j c j 0 , which always holds in this region.