Span Econ Rev (2007) 9:153–158 DOI 10.1007/s10108-006-9021-z R E G U L A R A RT I C L E
Linear contracts as incentives: a puzzle Óscar Gutiérrez
Published online: 17 November 2006 © Springer-Verlag 2006
Abstract This paper reexamines the linear schedule of compensation as a tool for providing incentives to managers when contractible output is a function of costly effort and a random shock. Two puzzling situations compatible with linear schemes of compensation are presented. First, if the model parameters are such that the optimal participation on output is below 50%, the variable compensation turns out to have a negative effect on manager’s utility. Second, if it is below 25%, linear incentives allow situations in which larger utilities are reached by means of smaller rewards. Keywords Moral hazard · Linear schedules · Certainty equivalent JEL Classification J33 · M40
1 Introduction Linear payment schedules have got a wide acceptance in providing incentives to managers of firms. The simplicity of this kind of contractual arrangements makes their use widespread in actual retribution practices as well as in academic research. Some notable examples of the use of linear incentives in the agency literature include Ferstham and Judd (1987), Haubrich (1994), Aggarwal and Samwick (1999a,b), Prendergast (1999), Raith (2003) or Casadesús (2004). Furthermore, such linear arrangement is the second-best solution under some
Ó. Gutiérrez (B) Dept. d’Economia de l’Empresa, Universitat Autònoma de Barcelona, Bellaterra, Barcelona, Spain e-mail:
[email protected]
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particular circumstances, as shown by Holmstrom and Milgrom (1987) in an important paper. In spite of their undeniable success in both practical and theoretical fields, linear schedules are compatible with counterintuitive and puzzling situations whose existence seems difficult to support conceptually and empirically. This paper deals with such situations. First, we show that if the manager’s participation rate is below 50% of the output that he influences, the variable part of the compensation provides the manager a negative net income. The manager bears a disutility that exceeds the (expected) value of his monetary participation on output, and hence the fixed component of the salary must be larger than the reservation utility. It follows then that the variable fee induces the manager to work just to partially avoid such costly effect. Second, if the commission rate is below 25%, we show that both the fixed salary and the commission rate decrease with risk. Then, it may well happen that a manager with better outside opportunities than another — the former gets a higher reservation utility than the latter — is paid through a worse contract, i.e., a lower fixed salary and a lower commission rate. Then, the agent’s payment (i.e., the actual monetary reward observed in practice) of the manager with better outside opportunities is in expectation lower than the payment of the one with worse outside opportunities. The paper is organized as follows. The second section presents the standard linear model. In the third section, the main results of the paper are derived. The fourth section concludes. 2 The model A standard formulation of the moral hazard problem in its continuous version is stated as follows: there is a risk-neutral principal who employs a risk-averse agent whose preferences are described by the separable utility function U(W, a) = − exp(−rW + kra2 /2), where W is the agent’s wealth, r represents a risk-aversion coefficient, a is the (unobservable) manager’s effort and k is a positive constant (personal effort has then a monetary cost). The level of output is given by a scaled Brownian motion with a drift specified by the agent’s effort, i.e., output (per period) is simply described by y = a+ε : the agent’s effort a plus a random disturbance normally distributed as ε ∼N(0, σ ). The agent’s marginal productivity ∂y/∂a is set equal to one without loss of generality (otherwise, we can scale up the problem and redefine the cost-of-effort-parameter k so as to set the productivity equal to 1). Holmstrom and Milgrom (1987) show that under such circumstances it is optimal to reward the agent by means of a linear contract based on observable aggregates: such a contract attains the optimal trade-off between risk-sharing and effort-induction from the principal’s point of view. If there is only one observable signal y reflecting the agent’s effort (output, profits, . . .) the principal is not better off with another payment schedule than by paying the agent a fixed fee plus a fraction of the contractible output, β + αy. To characterize the (second-best) optimal contract, we just need to derive the values α and β in equilibrium. Given that personal effort has a non-random monetary
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cost, the expected utility of the agent, E(− exp(−r(β + αy)+kra2/2)), is a monotone (increasing) transformation of CEA ≡ β + αa − rα 2 σ 2 /2 − ka2 /2. Then, the optimization problem to be solved by the principal can be formulated as: max{a − β − αa} α,β
⎧ 2 r 2 2 1 ⎨ β + αa − 2 α σ − 2 ka ≥ U s. to ⎩ a ∈ arg max β + α aˆ − 2r α 2 σ 2 − 12 kˆa2 .
(1)
aˆ
The first inequality in (1) represents the participation constraint and ensures that the agent receives a remuneration at least equal to his opportunity cost (market salary). The second inequality represents the incentive constraint, and can be calculated from the first-order condition (the maximum value coincides with the unique stationary value). The well-known solution to this problem is:1 α ∗ = (1 + krσ 2 )−1 ,
β∗ = U +
(α ∗ )2 (krσ 2 − 1), 2k
and
a∗ = α ∗/k.
(2)
If one of the assumptions is not satisfied (the effort is not exerted continuously, output is not continuously observed, the perturbation is non-normal, or the preferences are not in the CARA class) the contract derived is not (second-best) optimal.2 Then, contract (2) should only be seen as optimal among the linear contracts. 3 Main results We begin this section by showing that if the model parameters are such that the commission rate is below 50%, the fixed payment of the salary exceeds the reservation utility, so the variable compensation “carries a cost” and the effort exerted by the manager just helps him to partially reduce such a cost. Proposition 1 If the agent is optimally compensated through a linear schedule β ∗ + α ∗ y with α ∗< 1/2, the fixed salary exceeds the manager’s reservation utility. Proof If α ∗ = (1 + krσ 2 )−1 < 1/2 holds, then krσ 2 > 1, so β∗ = U +
(α ∗ )2 (krσ 2 − 1) > U. 2k
1 It is often assumed that the reservation utility is set equal to zero (without loss of generality). 2 There exist other conditions under which a linear payment can be optimal from the principal’s
point of view. For example, if output has a gamma distribution and the utility function is logarithmic (see Hemmer et al., 2000). In this case, the level of effort a is decided at the beginning of the relationship, and not continuously.
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It can be argued that this fact does not really mean a problem, because an agent can rationally prefer a smaller share of a gamble that has a positive probability of a negative payoff.3 However, we find surprising that the variable part of the salary provides a negative utility. Thus, the agent “has to bear” a variable fee whose overall effect represents a cost, and his actions only serve to partially reduce the corresponding disutility. In our opinion, this challenges the economic intuition of the incentives problem. For example, Raith (2003) uses linear incentives to analyze the effect of strategic competition on managerial incentives, and takes for granted that the fixed payment is smaller than the reservation utility (normalized to zero in his case), which in principle is what one would expect (Raith, op. cit., p. 1428). However we have shown that it is true only when α ∗≥ 1/2. An interesting implication of Proposition 1 is that if the fixed portion of the salary is paid before the agent takes the action, he will have strong incentives to take the money and run as the expected variable pay is negative.4 Next, we analyze what happens when the model parameters are such that the commission rate of the optimal schedule is lower than 25%. The following lemma will be useful below. Lemma If the agent is optimally compensated through a linear schedule β ∗ +α ∗ y with α ∗ < 1/4, both α ∗ and β ∗ decrease with σ 2 . Proof α ∗ always decreases with σ 2 ; see (2). On the other side, we know from krσ 2 −1 2 (2) that β ∗ = U + 2k(1+krσ 2 )2 . Denote krσ by Y. It is easy to verify that the sign ∗ of ∂β /∂Y is the opposite of (Y − 3)(Y + 1), which is positive whenever Y > 3 (noting that Y is positive by definition). Hence, ∂β ∗ /∂σ 2 is negative whenever α ∗ = 1/(1+Y) < 1/4. Let us now consider a situation where two managers 1 and 2 are equally risk and effort averse. Manager 1 demands a reservation utility U 1 to develop his actions in a firm whose output uncertainty is measured by σ1 ,while manager 2 demands a reservation utility U 2 3 holds and that the difference U 1 − U 2 is bounded 2 −1 2 −1 krσ krσ 1 2 1 above by 2k 2 2 − 2 2 , i.e., the difference between the reserva(1+krσ2 )
(1+krσ1 )
tion utilities is not too large. It is important to note thatthe last quantity is ∂ Y−1 positive because krσ12 > krσ22 > 3 (by assumption) and ∂Y is negative (1+Y)2 when Y > 3. Then, we obtain the following result: Proposition 2 The expected salary of the manager with better outside opportunities, manager 1, is smaller than the expected salary of manager 2: β1∗ + α1∗ E(y∗ ) < β2∗ + α2∗ E(y∗ ). Proof It is easy to show that the contract of manager 1 consists of a lower fixed salary and a lower commission rate than those of the contract of manager 2: inequality α1∗ < α2∗ is immediate from σ2 < σ1 . Given that U 1 − U 2 is bounded krσ12 −1 krσ22 −1 1 ∗ ∗ − above by 2k 2 2 2 2 , inequality β1 < β2 also holds. (1+krσ2 )
(1+krσ1 )
Finally, the expected incomes of the managers are equal to βi∗ + αi∗ E(y∗ ) = βi∗ + (αi∗ )2 /k, (i = 1, 2), a decreasing function of σi2 when commission rates are below 25%, so β1∗ + α1∗ E(y∗ ) < β2∗ + α2∗ E(y∗ ). The key fact that allows manager 1’s participation constraint to be satisfied when his compensation is lower than that of manager 2 is that he winds up exerting less effort. Therefore, cost of effort is lower for the manager with better outside opportunities. We see this fact (the manager with better outside opportunities receives a smaller fixed payment and a smaller commission rate on output than the other one) as counterintuitive. Whether it has an empirical support or not remains as an open issue. 4 Conclusions This paper shows that linear schedules of incentives are compatible with puzzling situations which seem difficult to reconcile with economic reasoning and managerial practice. In particular, we have shown that if the manager’s participation on output is below 50%, the manager receives a fixed payment above his reservation utility, which means that the variable compensation carries a net disutility in the manager’s certainty equivalent. Furthermore, we have shown that if the participation on output is below 25%, a manager may be paid through a contract with a smaller fixed payment and a smaller commission rate than those of a manager with worse outside opportunities, obtaining a smaller payment in expectation. An explanation for the existence of such unrealistic situations seems to be related with the exponential utility assumption, which could become inadequate in analyzing the incentive problem. As is well known, the exponential specification of preferences ignores wealth effects. In the Holmstrom–Milgrom framework, this drawback is avoided by the requirement of that “risk tolerances are large compared to the range of single period profits and to marginal costs
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of control” (op.cit., p. 322). In fact, if the risk tolerance (1/r) is high enough, α ∗ rises above 25% and the commented situation would not emerge. It is not usual to find such high piece rates in practice, though. In fact, the relationship pay/performance documented in practice is well below 25% (Jensen and Murphy 1990, Hall and Liebman 1998, Aggarwal and Samwick 1999a). We have shown how the linear incentives model can lead to paradoxes which leave open questions about the Holmstrom–Milgrom model and its applicability to real agency settings as a tool to analyze some managerial practices. Although the model may provide an inadequate framework to explain the relationship between CEO’s pay and firm performance, it seems sensible for situations like franchises, where agents have a high participation on output. Acknowledgments The author gratefully acknowledges Vicente Salas, Fran Ruiz and Carlos Ocaña for comments made on earlier versions of the paper; M. Paz Espinosa (co-editor) and two anonymous referees for helpful suggestions; and Eduardo Rodes, Jorge Rosell and Gema Pastor for fruitful discussions. He also thanks the financial support from Project MCYT – DGI/FEDER SEJ2004 - 07530-C04-03.
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