EXECUTIVE COMPENSATION IN A DYNAMIC MODEL

orfov, Stanimir Georgiev

International College of Economics and Finance

State University – Higher School of Economics

Moscow

During the last years, there has been a revived interest in the theory of dynamic contracting1. However, although most of the research incorporates some form of limited commitment/enforcement, little has been done in terms of extending the notion of commitment per se. In particular, there is no reason to believe that (the value of) the outside option is constant across the history of observables. For example, it is unrealistic to treat the reservation utility of a CEO as fixed regardless of the situation in his/her firm, industry, or the economy as a whole. The dependence could come through many channels- externalities, different types of agents, a certain structure of beliefs, but more importantly, it can significantly influence the nature of the relationship and the form of the optimal contract. It would be interesting to see how the agent is actually compensated for variability in the value of his/her outside options. When would his/her participation constraint bind? How is the agent's wealth affected in the short and the long run? In fact, would there be a limiting distribution and how would it depend on initial conditions? Such questions can only be analyzed in a generalized framework allowing for history-dependent reservation utilities. Moreover, extending the notion of commitment can bring some important insights into various contractual problems. For example, in order to address the wide use of broad-based stock option plans, Oyer (2004) builds a simple two-period model where adjusting compensation is costly and employee's outside opportunities are correlated with the firm's performance.

The current paper generalizes the notion of commitment by defining the outside options on the history observed in a dynamic contractual setting. I prove existence and obtain the first in the literature characterization of such an environment. The characterization is very general in terms of assumptions and, more importantly, is fully recursive. Its convergence properties make it perfect for computing the optimal contract for a general class of dynamic hidden action models.

I consider a moral hazard problem in an infinitely repeated principal-agent interaction while allowing the reservation utilities of both parties to vary across the history of observables. More precisely, to keep the model tractable, the reservation utilities are assumed to depend on some finite truncation of the publicly observed history. The rest of the model is standard in the sense that the principal wants to implement some sequence of actions which stochastically affect a variable of his/her interest, but suffers from the fact that the actions are unobservable. For this purpose, the optimal contract needs to provide the proper incentives for the agent to exercise the sequence of actions suggested by the principal. The incentives, however, are restricted by the inability of the parties to commit to a long-term relationship. It is here where the dynamics of the reservation utilities enters the relationship by reshaping the set of possible self-enforcing, incentive-compatible contracts.

In order to be able to characterize the optimal contract in such a setting, I construct a reduced stationary representation of the model in line with the dynamic insurance literature. The representation benefits from Green (1987)- the notion of temporary incentive compatibility, Spear and Srivastava (1987)- the recursive formulation of the problem with the agent's expected discounted utility taken as the state variable, and Phelan (1995)- the recursive structure with limited commitment, but is closest to Wang (1997) as far as the recursive form is concerned. Unlike Wang (1997), however, I formally introduce limited commitment on both sides and provide a rigorous treatment of its effect on the structure of the reduced computable version of the model. A parallel research by Aseff (2004) uses a similar general formulation2, but via a transformation due to Grossman and Hart (1983) constructs a dual, cost-minimizing recursive form closer to Phelan (1995) in order to solve for the optimal contract. Such a procedure, however, exogenously imposes the optimality of a certain action on every possible contingency.

After existence is proved, the general form of the model is reduced to a more tractable, recursive form where the state is given by the agent's (promised) expected discounted utility. On a different dimension, the state space includes the set of possible truncated histories in order to account for their influence3 on the reservation utilities. This recursive formulation does not rely on the first-order approach and is not based on Lagrange multipliers [cf. Marcet and Marimon, (1998)]. In fact, all I need is continuity of the momentary utilities. I first consider an auxiliary version where the participation of the principal is not guaranteed. The solution of this problem can be computed through standard dynamic programming methods once the state space is determined. Following the approach of Abreu, Pearce and Stacchetti (1990), the state space is shown to be the fixed point of a set operator and can be obtained through successive iteration on this operator until convergence. Given the solution of the auxiliary problem, I resort to a procedure outlined by Rustichini (1998) in order to solve for the optimal incentive compatible, two-side participation guaranteed supercontract. This is achieved by severely punishing the principal for any violation of his/her participation constraint. The procedure allows of recovering the subspace of agent's expected discounted utilities supportable by a self-enforcing incentive-compatible contract.

Next, I consider the case of executive pay. Since Jensen and Murphy (1990)'s seminal paper, there has been a big debate about the effectiveness of the observed compensation schemes in inducing the proper incentives while providing insurance to risk-averse managers. Empirical surveys and recipes abound.4 The most important question, however, is how the optimal compensation scheme should actually look like. In a dynamic model of adverse selection, Thomas and Worrall (1990) demonstrated that a legally enforceable contract would have the borrower's utility converging to minus infinity with probability one. Phelan (1995) showed that in a dynamic insurance setting characterized by one-sided commitment, there exists a non-degenerate long-run distribution of consumption. While the agency literature has mainly focused on deriving contracts inducing optimal effort, the participation constraints have largely been ignored. Some notable exceptions are Sleet and Yeltekin (2001) and Spear and Wang (2005) who concentrate on contract terminations and Cao and Wang (2008) who endogenize agent's reservation utility.

I consider the dynamic interaction between the shareholders of a corporation (treated as a risk-neutral principal) and its CEO (a risk-averse agent) in a hidden-action setting. Since I am interested in the long term dynamics of the contract and the resulting wealth distribution, I focus on long-term self-enforcing schemes that are incentive compatible. Limited commitment is assumed on both parties in the sense that both the shareholders and the CEO can commit only to short-term (single-period) contracts. This assumption is intended to reflect legal issues on the enforcement of long-term contracts. Furthermore, in my treatment of limited commitment, I allow for correlation between reservation utilities and the (finitely truncated) history of profits. This extension directly affects the set of possible endogenous utilities, but also permits the analysis of some interesting dynamic effects. For example, if the outside offer for the manager is positively correlated with current profit (due to, say, a belief on part of the outside employers that the firm's performance reveals information about the quality/type of the manager), we may expect that he/she would be motivated to increase the probability of high profits in the future (by choosing a higher level of effort). At the same time, the risk-averse managers would like to smooth consumption across states, which may require that their participation constraint does not bind for lower profit realizations. Moreover, it may become increasingly more difficult to motivate richer CEOs, especially when the shareholders face some borrowing constraints, which may lead to the suboptimality of inducing high effort for such CEOs.

The current paper is the first to look at how shocks on the reservation utilities may affect the parties to a dynamic contractual relationship. In particular, we investigate whether the optimal contract insures the manager against variability in the value of his/her outside options. We build up the intuition behind the possible effect of such an insurance on the manager's utility in the short and the long run and relate it to the properties of the limiting distribution.

The estimation is conducted in three steps. First, the state space of an auxiliary problem that does not require the participation of the principal but binds the wage from above is recovered as the limit of a generalized Bellman operator. Second, the aforementioned auxiliary problem is solved by a standard recursive procedure. Third, the optimal recursive contract and its state space are recovered by severely punishing the principal for each violation of his/her participation constraint.

In order to estimate the model, I parameterize it following the calibration of Aseff (2004) and Aseff and Santos (2005) based on the results of Hall and Liebman (1998) and Margiotta and Miller (2000).

Regarding the numerical computation, one point deserves special attention. In computing the endogenous state space we are iterating on sets and therefore need to represent them efficiently. For the class of infinitely repeated games with perfect monitoring, Judd, Yeltekin and Conklin (2003) are able to construct inner and outer convex polytope approximations based on the convexification of the equilibrium value set through a public randomization device. The algorithm I use may be of independent interest since it does not rely on the convexity of the underlying set. The main idea is to discretize the guess for the equilibrium set elementwise, extract small open balls around the gridpoints unfeasible with respect to the (non-updated) guess and use the remaining set, i.e., the guess less the extracted intervals, as a new guess for the equilibrium set. The procedure stops if the structure of the representations of two successive guesses coincides5 and the suitably defined difference between the representations is less than some prespecified tolerance level.

I derive the state space under constant reservation utilities. Then, I consider a single-period history dependence and show theoretically that if the manager's reservation utilities are sufficiently dispersed, his/her participation constraint does not bind under the worst case scenario, which is also observed when the manager can essentially commit when his/her outside option is at its lowest value. In other words, the minimum utility the CEO can be promised for initial histories characterized by lower reservation utility is generally boosted by higher reservation utilities for other states. Alternatively put, the optimal contract provides the CEO with some insurance against fluctuations in the value of his/her outside options, which ultimately smooths his/her consumption across (initial history) states. In case of positive correlation between firm's profits and manager's reservation utilities, this translates into the participation constraint of the manager being non-binding in states characterized by low profits. Computing the model actually shows that utility promises close to the reservation level are possible only under the manager's best-case scenarios when his/her reservation utility is the highest (i.e., when the highest profit has been observed).

The numerical results suggest that with a loose upper bound on wages, the optimal contract can support extremely high values for the expected discounted utility of the CEO when the participation of the principal is not guaranteed. However, when solving for the self-enforcing contract, these values naturally disappear since they violate principal's participation constraint. Exerting effort appears to be the predominant strategy for the principal, but shirking may still be optimal when the agent is rich enough. The optimal wage scheme and the future utility of the manager tend to grow in both current utility and future profit. Intuitively, both current and future compensation are used to induce poor and mid-range managers to work hard, while rich managers prove too difficult to motivate. The latter shirk and while they may face some fluctuations in their current income stream in case of binding credit constraints on part of the firm, their lifetime utility remains relatively flat.

Simulations suggest that CEO's utility weakly increases in the long run. In particular, agents who start rich tend to keep their utility level while those who start poor get richer in time. The increase is most pronounced for managers with initial utilities below the highest reservation utility. These managers first have their utilities pushed well above their reservation level. Then, the principal motivates them to work hard by rewarding success through continuation utilities while providing insurance through flatter wages. In this way, the probability of success and, therefore, of a higher reservation utility tomorrow increases which rises the manager's expected continuation utility. The long term distribution of manager's utility is non-degenerate and depends on the initial utility promise but not directly on the relevant initial history at least as far as short initial histories are concerned.

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1 See, for example, Fernandes and Phelan (2000), Ligon, Thomas and Worrall (2000), Wang (2000), Phelan and Stacchetti (2001), Sleet and Yeltekin (2001), Ligon, Thomas and Worrall (2002), Ray (2002), Thomas and Worrall (2002), Doepke and Townsend (2004), Jarque (2005), Abraham and Pavoni (2008).

2 His benchmark model is a full-commitment one, but he considers limited commitment on part of the agent as an extension.

3 The relationship between the history of observables and the reservation utilities is predetermined since the reservation utilities are exogenous to the problem.

4 See Murphy (1999) and Jensen and Murphy (2004) for a review.

5 Namely, if the representations have the same number of closed sets element by element.

EXECUTIVE COMPENSATION IN A DYNAMIC MODEL