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Moral hazard and incentives in a decentralized planning environment
JOURNAL
OF COMPARATIVE
ECONOMICS
(1986)
l&91-105
Moral Hazard and Incentives in a Decentralized Planning Environment’ PAK-WAI LIU Department of Economics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong Received June 20, 1984; revised June 18, 1985
Liu, Pak-Wai-Moral Hazard and Incentives in a Decentralized Planning Environment The relationship between the center and the socialist manager in a decentralized planning environment is modeled in this article as one between principal and agent. A class of incentive contracts is examined in which, besides a sharing rule, the center stipulates an outcome target the manager must achieve; otherwise he will be paid a fixed payment. It is shown that the optimal incentive target is the lower bound of all feasible targets, while the optimal sharing rule is strictly locally convex at the optimal target. J. Comp. Econ., June 1986,10(2), pp. 91-105. The Chinese University of Hong Kong, Shatin, N.T., Hong Kong. o 1986 Academic mess, IIIC. Journal of Economic Literature Classification Numbers: 026,021,052.
I. INTRODUCTION
The nature of the problem of the socialist planner in motivating and rewarding socialist managers in a decentralized planning environment bears much similarity to the problem of motivating branch or divisional managers of a capitalist corporation. An important issue in both cases is moral hazard which arises as a result of an asymmetry in information between the manager and the center. The manager, who is presumably motivated by self-interest, generally prefers to exert less effort, ceteris paribus, while the center is indifferent to the manager’s effort per se. On the other hand, the manager has an informational advantage in that he has private information on his effort that is presumably unobservable to the center. Besides an asymmetry in information, there may also be divergent attitudes toward risk. The manager is likely to be risk averse while the center is presumed to be risk neutral. The ’ This article was written while the author was a visiting scholar at the Harvard-Yenching Institute, Harvard University. Comments of two anonymous referees and the Editor am gratefully acknowledged.
91
0147-5967186 $3.00 Copyii&t Q 1986 by Acadcmii Pres, Inc. All rights of reproduction in any form reserved.
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problem, therefore, is to find an optimal contractual arrangement between the center and the manager that will provide incentives as well as share risk so as to induce the manager, in maximizing his own expected utility, to maximize the center’s expected benefit function also. Recent literature on incentives for socialist managers has dealt with different aspects of the problem of incentives and information. For instance, there has been considerable work on the new Soviet incentive model, the elicitation of truthful information on productive capacity by allowing managers to choose their own targets, and incentive schemes that induce socialist managers to produce at the socially desirable output level.* The point of departure from the previous literature on socialist incentives is the focus that this paper places on the need to motivate socialist managers to exert effort as a result of asymmetric information about their effort. In so doing, we will not deal with issues such as the informational requirements of the center in target-setting, selfselection of targets by the managers, and resource allocation by the center. This article draws on recent advances in the literature on the agency problem and models the relationship between the center and the socialist manager as a principal-agent relationship.3 Bergson (1978) was the first to pose the relationship between the center and the socialist manager as a principal-agent problem, but he did not provide an analysis of the moral hazard and incentive issues that are the primary interest of a principal-agent problem. In decentralized planning, the socialist planner, the principal, delegates decision-making authority to a large number of independent socialist managers, the agents, who are presumably motivated by self-interest in making production decisions. The socialist planner, who is unable to monitor the effort of a large number of socialist managers in any effective way, must coordinate their activities through contractual arrangements. For analytical purposes, it is useful to conceptualize the mode of remuneration of the socialist manager as an explicit or implicit contractual arrangement between him and the center. This article analyzes a typical class of contracts between the center and the manager that can be characterized by the triple parameters of an outcome target 2, a sharing rule s(x), and a fixed payment k. The target is based on the outcome x that can be measured in terms of profit, revenue, or output. If the socialist manager fulfills or overfulfrlls the target, he will be given a bonus according to a sharing rule that specifies how the above-target outcome will be shared between the ’ For work on the new Soviet incentive model, see Weitzman (1976, 1980) and Holmstrom (1982). Ronin (1976) and Bonin and Marcus ( 1979) analyzed the problem of eliciting truthful information on productive capacity by allowing socialist managers to settheir own targets. Incentive schemes that induce socialist managers to produce at the socially desirable output level have been studied by Domar (1974) and Tam (1979). 3 Recent literature on the agency problem includes Ross (1973), Harris and Raviv (I 978, 1979), Holmstrijm ( 1979), and Shaveh ( 1979).
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manager (enterprise) and the center. Otherwise, he will be paid a fixed payment. This payment system can be characterized by S(x) =
k
if
x < 2,
s(x)
if
x 2 2.
It is well known from the agency literature that the optimal contract between a risk neutral principal and a risk averse agent is a general sharing rule s(x) that Pareto-dominates a fixed payment (Shavell, 1979, pp. 59-62). Hence the sharing rule under consideration here is a constrained version of the general sharing rule extensively analyzed in the agency literature in that it specifies a target and a fixed payment. In the degenerate case where the target 2 is set at zero, the constrained sharing rule will reduce to the general sharing rule. The analysis of the constrained instead of the general sharing rule in this article is motivated by the consideration that it is in the spirit of the socialist planning literature. A performance target, either set by the socialist planner or selfimposed by the socialist manager, is important to socialist planning. Besides serving the purpose of motivating the manager to exert more effort by rewarding him for his above-target performance with a bonus payment, the target is also important to the center in its planning since it informs the center of the volume of resources that will be available for its intersectoral or inter&m allocation plan. In our subsequent analysis of socialist planning we will therefore assume that the target is nonzero. In this model the target is set by the planner. Therefore, this article does not address the issue of self-selection of targets by the managers and the “ratchet effect,” nor does it deal with the dynamic problem of target revision by the planner. Under the principal-agent relationship obviously if the manager’s effort is public information, the target should be set by the planner in terms of a stipulated effort level which the manager must expend. This results in a forcing contract and a first-best solution of the type analyzed by Harris and Raviv (1978, 1979). More often than not, effort is private information to the manager and is unobservable by the planner so that it cannot form the basis of a contract. An alternative is to utilize an imperfect monitor or proxy for effort if the planner is able to formulate and observe such a proxy variable. One natural candidate of an imperfect monitor of the agent’s effort is the outcome itself which is positively correlated with effort expended. It also has the advantage that it is generally observable without cost by both parties. Intuitively the outcome target, when appropriately set, will augment incentives provided by the sharing rule. If, on the other hand, it is set too high so that the manager’s probability of fulfilling the target approaches zero, it can become counterproductive because the manager may decide it is not worth his effort to work
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hard since there is almost certainty of an underfulfillment of the target. In order to avert the tendency of the manager to choose to exert zero effort and collect the fixed payment each period, the contract carries with it an implicit threat of dismissal if the manager repeatedly fails to achieve the target. Our purpose is to characterize the optimal incentive contract within the constrained class of contracts specified and analyze the relationship between the three contract parameters. This article represents the first effort to extend the results of the agency problem to the analysis of socialist planning. It extends the work of Keren (1972) who analyzed the problem of target-setting to induce maximum effort, and the work of Bergson (1978) who posed the problem between the manager and the planner as one of risk sharing between agent and principal. In neither of their works has there been a characterization of the optimal incentive contract as an outcome target, a sharing rule, and a fixed payment, nor a treatment of the problem of moral hazard. A principal-agent model applied to the problem of motivating the socialist manager is presented in Section 2. A characterization of the optimal incentive contract with a target-sharing rule-fixed payment triple is contained in Section 3. Section 4 discusses the supply of effort of the manager in response to the optimal contract and the probability of failure to achieve the target. The conclusion and a discussion on the possibility of further research are given in Section 5. 2. THE
MODEL
Let x: A X 8 - R, be a monetary outcome function where A is the set of actions (effort) and 8 is the set of all states of nature. Uncertainty in the monetary outcome may arise as a result of a number of factors such as unpredictable weather, supply uncertainty, and machine breakdowns, all of which are included in the set of all states of nature. For any effort of the socialist manager (agent) e E A, and the state of nature 0 E 8, the monetary outcome is given by x = x(e, 8). It is a stochastic variable with an induced probability density given e denoted byf(xle), and a cumulative density F(x]e). Bothfand F are assumed to be continuously differentiable. It is further assumed that more effort expended increases the outcome in every state of nature with diminishing returns, so that x, > 0 and x,, < 0, with the subscript(s) as usual denoting the partial derivative. An increase in e shifts the distribution of x to the right in the sense of first-order stochastic dominance.4 This implies that Fkxle) =G0 with the strict inequality holding at least for some x. Further let the support of the distribution of x given e be denoted by [a(e), b(e)]. It should be clear that both a and b are functions of e since an increase in e shifts the 4 For a discussion of the ordering the uncertain prospects and first-order stochastic dominance, refer to Hadar and Russell (1969).
MORAL
HAZARD
AND INCENTIVES
95
distribution of x to the right. The following boundary conditions are imposed on the distribution: fld4l4
= flWl4
= 0,
for all e.
It is essential to assume that 8 is unobservable to both the center and the manager. This is because if 8 is ex post observable, then after a certain outcome x is realized the manager’s effort can be computed from x = x(e, 0) provided that the outcome function is invertible. In that case the optimal contract is simply a forcing contract that stipulates a target effort level as mentioned earlier and it is enforceable, thus eliminating the need of an outcome target. It is further assumed that the center and the socialist manager share the same ex ante subjective belief of the probability distribution of 0. The planner is assumed to be risk neutral.’ The benefit of the action of a socialist manager to the community, aside from the manager himself, is the outcome minus the share paid to the manager, since it represents the amount of the outcome that can be distributed to the rest of the community. In this article, therefore, it is assumed that the planner, being risk neutral, maximizes the expected net outcome that will be available for distribution after compensating the manager.‘j In the principal-agent framework, it is represented by the principal maximizing his expected share of the outcome. It is assumed that the manager is paid only after effort has been expended and the outcome realized. If the outcome falls short of the target 2, he will be paid a fixed payment k. Given e, the target 2 is bounded by the closed interval [u(e), b(e)]. On the other hand, if the target is fulfilled or overfulfilled, the manager will be paid according to a sharing rule s(x). In contrast to most of the previous studies on socialist incentives that take the bonus as a fixed ’ By assuming that the socialist planner is risk neutral, the central issue is no longer risk sharing but instead provision of incentives. In this case the value of a sharing rule that depends on outcome is mainly in providing incentives. The assumption of risk neutrality of the socialist planner can be justified on the basis of the Arrow-Lind (1980) results on public investment. Set Bergson ( 1978) for a discussion. 6 The characterization of the objective function of the planner as expected output minus the bonus paid to the manager can be found in the work of Keren (1972), Bergson (1978), and Bonin and Marcus ( 1979). In our model, it is not appropriate to characterize the planner as maximizing expected output since this may entail an optimal contract in which the manager will be given the entire or close to the entire share of the output to induce him to exert the maximum effort in order to maximize the expected output, thus leaving no or little output for distribution to the rest of the community. This will quite likely be the case if the disutility of effort is relatively small. Hence expected output is not an appropriate maximand because of distributional considerations of the planner. Here it is assumed that the planner does not just aim for the highest level of expected output but is concerned with how much of the output can be distributed, and hence the income distribution between the manager and the rest of the community. This point was brought to my attention by the Editor.
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payment or as a linear function of output, in our model the sharing rule s(x) can take any general functional form of a weak monotonic function of x. Weak monotonicity of the sharing rule is a natural assumption since it ensures that proper incentives are provided because the manager will receive at least as large a payment when outcome increases.’ Let S denote the set of bounded, continuously differentiable monotonic sharing rules. Clearly S is defined only in the domain of x E [2, b(e)], since if x < f, the manager will be paid k instead. Let C = (2, s( . ), k} be the class of all contracts characterized by 2, s( . ), and k. The socialist planner’s problem is to solve the problem max f,s( f VW
i [S
(x - WW) d’e)
do + J”” f
(x - &WW
h]
.
The first term in the maximand is the expected net revenue of the center when the manager fails to achieve the target. The second term refers to the case when the target is fulfilled or overfulfilled. If the planner can observe e, he can specify the level of effort to be delivered by the manager in the contract. In this case we have a forcing contract with a first-best solution. But here we assume that e is private to the manager and unobservable to the planner. Then the planner can find only an optimal contract (A?*, s*( . ), k*) E C and because of the presence of moral hazard the solution to the problem subject to the constraints to be specified can be only second-best.* The problem can be rewritten as max [J”” (x - k)flxle) do - Jb(’ (s(x) - k)f(xle) c~x]. P Ls(.),k,e n(e)
(1)
To ensure that proper incentives are provided, the fixed payment that the manager receives when he underfulfills the target must be no greater than the payment he receives when he achieves the target. Hence the following constraint on the sharing rule, the target, and the fixed payment must be satisfied:
s(i) 3 k.
(2)
7 Milgrom ( I98 1) has shown that for an incentive contract with a sharing rule, monotonicity of the sharing rule is assured if the family of conditional density functions of x given e has the monotonic likelihood ratio property (MLRP). This means that a greater outcome is evidence of a greater effort by the manager, and hence the sharing rule should be increasing in outcome to provide the correct incentives. MLRP is a property that many families of commonly used density functions possess. * The solution of a principal-agent problem in which moral hazard is present due to an asymmetry in information on effort is only second-best This was discussedby Shaveg (I 979), Holmstriim (1979), and Harris and Raviv (1979).
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HAZARD
AND INCENTIVES
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The ith manager is assumed to have a von Neuman-Morgenstem utility function Wi which is defined over his wealth and effort. For simplicity of notation the subscript i will be dropped in the following analysis. To simplify the analysis further W is restricted to be separable so that in the case of successful achievement of the target, it can be written as a sum of two components, one depending on the payment and the other on effort, as9 WW, 4 = W(x)) - W where U R, - [0, J3] and K IF?,- [0, B’] with B and B’ as lower bounds.‘O Both U and I/ are assumed to be continuously differentiable. Furthermore U is strictly increasing and strictly concave and V is strictly increasing. This formulation takes into account the assumption that effort entails disutility to the manager. The manager’s expected utility is given by w(e, 2, s, k) =
x [U(k) - JWlf(xl4 dx
sa(e)
+
b(e) sf
[W(x)) - Wlf(xl4 do
where the first term on the right-hand side refers to the case when the manager expends effort e but fails to achieve the target, and the second term refers to the case when the manager fulfills or overfulfills the target. The socialist planner’s maximization program is constrained by a participation constraint of the manager that requires that the manager’s expected utility must be no less than a given minimum level IV0 which can be interpreted as the expected utility of being a socialist worker, an alternative to being a socialist manager. This will also be the alternative employment of the manager if he is dismissed by the center. The participation constraint is b(e)[U(s(x)) - U(k)]f(xle) s2
dx 2 W. - U(k) + V(e).
(3)
The planner’s program is further constrained by the manager’s choice of effort among all feasible actions in A, given the 2, s(a), and k triple. This constraint is the result of the unobservability of effort and the manager being delegated the decision over his choice of effort. It is assumed that the outcome of the negotiation or the hierarchical process between the center and the g This is also the formulation of the agent’s utility function used by Holmstr6m ( 1979), Grossman and Hart (1983), Christensen (198 l), and Green and Stokey (1983). lo The manager’s utility function is bounded from above to avoid the sort of problems discussed by Mirrlees ( 1975).
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manager is such that the former will specify a three-parameter contract while the latter will expend effort contingent on the contract. That is, the manager’s effort function is given by e(& s( * ), k) and he will expend effort to maximize his expected utility subject to the contract parameters stipulated by the center instead of maximizing over all feasible contracts in C.” It is assumed that the planner has knowledge of the manager’s decision rule so that he can find an optimal triple which maximizes his own expected net revenue in (1). Hence at the optimum so defined the manager maximizes his expected utility subject to the parameters of (Z*, s*( - ), k*). This constraint is represented by e E argmax “) [U@(x)) - U(k)]f(xle) e s2
dx + U(k) - V(e), given 2, s( * ), and k.
In general there may be multiple values of e that satisfy this constraint. The contract is assumed to be cooperative in case of multiple solutions so that at the optimum the manager agrees to choose the greatest effort among the levels to which he is indifferent. Hence, given the parameters of the contract, the manager’s optimal supply of effort is defined. The constraint can now be represented by the first-order condition w,(e, 2, s, k) = 0. Alternatively, si
“’ [U@(x)) - U(k)]fe(xle) dx = V(e).
(4)
The ex ante Pareto-optimal contract in the class of C is obtained by choosing (a*, s*( . ), k*) to maximize (1) subject to (2), (3), and (4). The expectation in each case is taken over the commonly shared and subjectively held conditional probability distribution of x. 3. OPTIMAL
TARGET
AND
SHARING
RULE
In solving the planner’s program of maximizing (1) subject to (2), (3), and (4) as discussed in Section 1, we preclude the consideration of comer solutions in which the target and the fixed payment may be set equal to zero, effectively leaving only the sharing rule in the contractual arrangement; comer solutions ii If the manager is allowed to maximize over ah feasible contracts in C, there may be a problem that his global maximum in expect4 utility may not be attained at the constrained maximum expected net revenue of the planner. This problem of a double maximum has been discussed by Mirrlees (1975) in an unpublished paper. By constraining the manager to maximize subject to the contract parameters stipulated by the planner, his constrained maximum in expected utility, though not necessarily a global maximum over all feasible contracts, is attainable when the planner has solved his problem. This makes the problem amenable to solution. The assumption is reasonable in view of an asymmetric bargaining power between the central and the so&list managem assuming that there is no coalition of the managers.
MORAL HAZARD AND INCENTIVES
99
will obviate the need for further analysis of the optimal target and its relation to the other two parameters which is the aim of this article. We will, therefore, consider only the interior solution which is assumed to exist. The optimal target J?*, sharing rule s*( * ), and fixed payment k* are characterized by the following marginal conditions for almost all x12: (s*(x^*) - k*) - [U(s*(x^*)) - U(k*)][A
+ pfs]
+ $$
1
+ 1 _ ;i*,e)
[-I
= 0,
+ XU’(k*)]F(x^*Je) + rcU’(k*)F,(2*le)
- 7= 0
= 0,
(5) (6)
(7)
where 7, X, and p are Lagrangian multipliers for constraints (2), (3), and (4), respectively. The Kuhn-Tucker conditions require that 1, X > 0. Further characterizations of (Z*, P( . ), k*) are given by the following two propositions. PROPOSITION 1. The optimal target 2* is the lower bound of all feasible targets when the optimal sharing rule is s*(e) and the optimaljixed payment is k*. That is, 2* = g*.
Proof: Given the monotonic sharing rule s, and the fixed payment k, the lower bound of the feasible targets 2 is defined by s(Z) = k, and all feasible targets must satisfy R 2 $. First, it will be shown that if the multiplier 7 = 0, there can be no solution, so that 9 must be strictly positive. Assume q = 0. That is, constraint (2) is not binding at the optimum so that P(xI*) > k*. Since (6) holds for all x except at the end points of the support, it can be evaluated at the optimal target i*, so that (5) and (6) become s*(x^*) - k* = x + /G*le) &?*le) U(s*(x^*)) - U(k*)
’
1 = x + pfi~~*14 u’(s*(x^*)) fG*le)
(5’)
*
Now (5’) and (6’) imply that v(s*(x”*))
= WYx^*N - V*) s*(xI*) - k*
’
But clearly this is impossible under the strict inequality s*(x^*) > k*, because U is strictly concave. We conclude that for a solution to exist, q must be strictly positive, implying from the Kuhn-Tucker condition that constraint ‘* A solution cannot exist at the support of the distribution because of the boundary conditions: Aa(e) = BW)le) = 0.
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(2) is binding at the optimum. Therefore, s*(x”*) = k*, but this is simply the definition of a lower bound of the feasible targets given the optimal sharing rule s*( * ) and fixed payment k*. Hence f* = g*. Q.E.D. Besides characterizing the optimal target, the optimal sharing rule can also be characterized by rewriting (5) after substituting in the constraint s*(xI*) = k*, Tjs*‘(a*) o -= (8) m*14 . But q > 0 implies s*‘(xI*) = 0; that is, at the optimal target 3?* = g*, the optimal sharing rule is flat. Now s* E 8’ is weakly monotonic so that s*‘(x) >, 0 for all x. While the equality holds at the optimal target i*, it is argued that only the strict inequality holds in the right neighborhood of the optimal target, H~I = {xix - Z* = e, e > O}. This is because if the equality holds for all x E Ht. it will imply that the optimal sharing rule will be a fixed payment to the manager in the neighborhood of Z*. However, a fixed payment has been shown by Shave11 (1979, pp. 59-62) to be a Pareto inferior sharing rule when the agent is risk averse and there is moral hazard. ‘Hence the equality is precluded in HiI. Since s*’ = 0 at g* and s*’ > 0 for all x E H2*, the optimal sharing rule is locally strictly convex at the optimal target f*. This proves PROFQSITION 2. The optimal sharing rule andfixed payment must satisfy s*(P) = k* with s*‘(P) = 0. It is locally strictly convex at P. This result is illustrated in Fig. 1 which graphs the optimal sharing rule in the right neighborhood of ?*. It is, of course, undefined for all x < z?* since at the optimum, 1* = g*. It is interesting to note that the optimal contract provides continuity in the payment to the manager in the sense that he will get a fixed k* for failure to achieve the target, but when the target is exactly fulfilled, he gets s*(xI*) which equals k*. However, there are incentives for the manager to fulfill not just the target but to overfulfill the target since the optimal sharing rule slopes upward convexly. The manager’s payment increases more, the more the outcome exceeds the target. 4. MANAGERS
SUPPLY
OF EFFORT
It is instructive to analyze the effort response of the socialist manager to the optimal contract. The expected utility of the manager is w(e, 3, s, k). Let us assume that the solution of the optimal contract from (5), (6), and (7) yields an expected utility of w(e, 1*, s*, k*) which exceeds the utility of the zero effort choice, U(k*) - V(0). This obviates the consideration of a situation
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FIG. 1.
wherein the manager, seeing that there is too small a probability of exceeding the outcome target x?*, chooses to exert no effort at all and to collect just the fixed payment k*. The choice of zero effort is a degenerate case of the solution and will not be further considered. Suppose w(e, 22*, s*, k*) > U(k*) - V(O), then the supply of effort of the manager, given the contract parameters, can be derived by solving (4). In general it can be characterized by e = e($ s( * ), k). The supply of effort is therefore dependent on the conditional density function of x. Assume that the effort supply function is continuously differentiable; then differentiating the first-order condition w,(e, 2, s, k) = 0, we have
dw,-- w,ei di
+ wd = 0.
Hence
e= t
wd _ [W(3) - WM(~le) WC%
W,
(9)
Since w, < 0 due to the second-order condition in the maximization of w and C@(Z)) 2 U(k) owing to the weak inequality s(x) > k of (2), sgn e2 = sgn f,(ile). The effort response of the manager to the target therefore in general depends on the nature of the distribution function of x given e. However, given the optimal contract of (Z*, s*(. ), k*), ei will equal zero since s*(xI*) = k* according to Proposition 1. We can further show that under the optimal contract
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since s*‘(xI*) = 0 according to Proposition 2. Hence ei = ee = 0. When the contract parameters are optimally chosen, a marginal increase in target will not induce more effort. It should be clear that the optimal target set by the planner will not necessarily induce maximum effort from the manager because the planner is constrained by the fact that the manager will optimally choose his own effort to maximize his expected utility as specified in (4). Similarly, we can analyze the effect of the other two contract parameters on the manager’s supply of effort. For the sharing rule, we have
that is,
The inequality follows from the concavity of U and the equality s
ib(“L(xle) dx = -F,(ile).
But w, < 0 and F,(Zle) -K 0. Therefore es > 0. Hence an increase in share will always induce more effort from the manager. This result holds in general and not only at the point of the optimal contract. Finally, for fixed payment we have wek
-
U’WXxl4
ek=--= W,
WC,
<
o
(11)
An increase in the fixed payment will always induce less effort. Again this result holds in general. At the optimum the probability that the realized outcome falls short of the target when the manager has expended effort e given the optimal contract is f’ P(x < 2*) = f(xle) dx = F(?*le). (12) sde) This probability depends on the shape of the conditional distribution of x. Besides the shape of the distribution, it also depends on how hard the manager is willing to work given the optimal contract. Due to first-order stochastic dominance, F&?*le) G 0, implying that at the optimum an increase in effort expended will weakly decrease the probability of failure to achieve the target. The implicit penalty of dismissal for underfulfilling the target set in the contract will deter the manager from exerting zero effort and simply collecting the
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fixed payment even if, according to his subjective evaluation, the probability of his fulfilling the target is miniscule. This together with the optimal sharing rule will motivate the manager to expend more effort to try to overfulfill the target. 5. CONCLUSION In this article a class of principal-agent contracts that stipulates an outcome target with a threat of dismissal is modeled to analyze the problem of remunerating socialist managers in a decentralized planning environment. It is shown that the optimal contract in this class should set a target which is the lower bound of all feasible targets, while the sharing rule and fixed payment must satisfy s*(Z*) = k*. The sharing rule is strictly locally convex at the optimal target. It should be pointed out that the class of contracts that has been analyzed is not the general class in that the sharing rule is restricted by the presence of a target, which is essential for the planning of the center. In effect the optimal contract within this class specifies a fixed payment for the region of low output and a sharing rule for the region of high output. But the agency literature has shown that the Pareto-optimal contract should be a general sharing rule and not a fixed payment system, meaning that the contract should not specify a target at all. We infer that in our model efficiency in production could be improved if the output target were abolished or set to zero and each socialist manager were confronted with a general sharing rule. This, however, would entail further decentralization of economic activities because the center would no longer be able to rely on output targets to make plans, and thus it would have to make more use of the market mechanism to allocate resources. On the other hand, if the output target cannot be abolished and the economy cannot be further decentralized for one reason or another, our results suggest that the target should at least be set as low as possible so as to improve efficiency. ’ 3 In a socialist economy in which there is decentralized planning, the center plans to engage managers who are willing to work hard to produce a large outcome. A contract with a specified target, bonus for overfulfilling of the target, and dismissal for failure provides a useful screening mechanism for the planner when effort is unobservable. Suppose socialist managers differ only in their preference toward effort. If the planner has knowledge of the contract parameters to effort map of each manager, he will be able to rank the managers according to this map and engage each manager with an individual contract until the vacancies are filled. In this way his expected net revenue will be maximized. However, if the number of vacancies to be filled I3 This point was suggested to me by a referee.
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is large and the transaction cost in individual contracting is considerable, or if the center has no knowledge of the contract parameters to effort map of each manager, an individualized optimal contract for each manager will not be feasible. In practice the planner who engages a large number of managers operating in the same stochastic environment may have to offer an identical contract to all managers. This contract is stipulated on the basis of the planner’s expectation of a manager’s effort response given the contract, and this in turn is based on the planner’s past experience with the managers. If the managers are ex ante identical except for their preference for effort, a Spence-type signal on individual preference for effort is not available so that screening at hiring is not possible. Since the center has no information on each manager’s preference for effort, it will engage managers until the vacancies are filled. However, after they are hired, managers who are less willing to expend effort will have a smaller probability of fulfilling the target, and over time they will be screened out, leaving those who are willing to work the hardest. As a result, while prehiring screening may not be possible, a contract with an outcome target provides a mechanism of posthiring screening to the planner even though effort is unobservable. Extensions and elaborations of the present model in several directions are possible. First, the target may be implicit rather than explicit in some contractual arrangements. In that case since there is no explicit agreement, the center and the manager may have different expectations concerning the level of the target. An analysis of this type of contractual arrangement would be a useful extension. Second, it is possible in the present model that a manager may have worked very hard and still falls short of the target and is dismissed due to the realization of a very unfavorable state of nature. This is possible because the state of nature is assumed to be unobservable ex post so that the contract cannot be made contingent on it. If it is assumed that although states of nature are unobservable, the manager is better informed than the center in that he has signals concerning the states of nature that he may transmit to the center, then the model can be extended by allowing a revision of the target contingent on the information transmitted by the manager. In this case the issue of truthful revelation of this information and the effect of the manager’s improved information on the planner’s expected payoff arises. REFERENCES Arrow, Kenneth J., and Lind, Robert C., “Uncertainly and the Evaluation of Public Investments Decisions.” Amer. Econ. Rev. 60, 3:364-378, June 1980. Bergson, Abram, “Managerial Risks and Rewards in Public Enterprises.” J. Comp. Econ. 2, 3: 21 l-225, Sept. 1978. Bonin, John P., “On the Design of Managerial Incentive Structures in a Decentralized Planning Environment.” Amer. Econ. Rev. 66,4:682-687, Sept. 1976.
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Bonin, John P., and Marcus, Alan J., “Information, Motivation and Control in Decentralized Planning: The Case of Discretionary Managerial Behavior.” J. Camp. Econ. 3,3:235-253, Sept. 1979. Christensen, John, “Communication in Agencies.” Bell J. Econ. 12,2:66 l-674, Autumn 198 1. Domar, Evsey D., “On the Optimal Compensation of a Socialist Manager.” Quart. J. Econ. 88, l:l-18, Feb. 1974. Green, Jerry R., and Stokey, Nancy L., “A Comparison of Tournaments and Contracts.” J. Polit. Econ. 91,3:349-364, June 1983. Grossman, Sanford J., and Hart, Oliver D., “An Analysis of the Principal-Agent Problem.” Econometrica 51, 1:7-45, Jan. 1983. Hadar, Josef, and Russell, William R., “Rules for Ordering Uncertain Prospects.” Amer. Econ. Rev. 59, 1:25-34, May 1969. Harris, Milton, and Raviv, Artur, “Some Results on Incentive Contracts with Application to Education and Employment, Health Insurance, and Law Enforcement.” Amer. Econ. Rev. 68, 1:20-30, Mar. 1978. Harris, Milton, and Raviv Artur, “Optimal Incentive Contracts with Imperfect Information.” J. Econ. Theory 20,2:231-259, Apr. 1979. Holmstriim, Bengt, “Moral Hazard and Observability.” BefZ J. Econ. 10, 1:74-91, Spring 1979. Holmstrijm Bengt, “Moral Hazard in Teams.” Bell J. Econ. 13, 2:324-340, Autumn 1982. Holmstriim Bengt, ‘Design of Incentive Schemes and the New Soviet Incentive Model.” Europ. Econ. Rev. 17,2:127-148, Feb. 1982. Keren, Michael, “On the Tautness of Plans.” Rev. Econ. Stud. 39, 12Ck469-486, Oct. 1972. Milgrom, Paul R., “Good News and Bad News: Representation Theorems and Applications.” Bell .I. Econ. 12, 2:380-391, Autumn 1981. Mirrlees, James A., “Notes on Welfare Economics, Information and Uncertainty.” In M. Balch, D. McFadden, and S. Y. Wu eds., Essays on Economic Behavior under Uncertainty, pp. 243-258. Amsterdam: North-Holland, 1974. Mirrlees, James A., “The Theory of Moral Hazard and Unobservable Behaviour, Part I.” Mimeo, Cambridge Univ., 1975. Ross, Stephen A., “The Economic Theory of Agency: The Principal’s Problem.” Amer. Econ. Rev. 63,2:134-139, May 1973. Shavell, Steven, ‘Risk Sharing and Incentives in the Principal and Agent Relationship.” Bell J. Econ. 10, 1:55-73, Spring 1979. Tam, Mo-yin S., “On Incentive Structures of a Socialist Economy.” J. Comp. Econ. 3, 3:277284, Sept. 1979. Weitzman, Martin L., “The New Soviet Incentive Model.” Bell J. Econ. 7, 1:25 l-257, Spring 1976. Weitzman, Martin L., “The ‘Ratchet Principle’ and Performance Incentives.” Bell J. Econ. 11, 1:302-308, Spring 1980.
OF COMPARATIVE
ECONOMICS
(1986)
l&91-105
Moral Hazard and Incentives in a Decentralized Planning Environment’ PAK-WAI LIU Department of Economics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong Received June 20, 1984; revised June 18, 1985
Liu, Pak-Wai-Moral Hazard and Incentives in a Decentralized Planning Environment The relationship between the center and the socialist manager in a decentralized planning environment is modeled in this article as one between principal and agent. A class of incentive contracts is examined in which, besides a sharing rule, the center stipulates an outcome target the manager must achieve; otherwise he will be paid a fixed payment. It is shown that the optimal incentive target is the lower bound of all feasible targets, while the optimal sharing rule is strictly locally convex at the optimal target. J. Comp. Econ., June 1986,10(2), pp. 91-105. The Chinese University of Hong Kong, Shatin, N.T., Hong Kong. o 1986 Academic mess, IIIC. Journal of Economic Literature Classification Numbers: 026,021,052.
I. INTRODUCTION
The nature of the problem of the socialist planner in motivating and rewarding socialist managers in a decentralized planning environment bears much similarity to the problem of motivating branch or divisional managers of a capitalist corporation. An important issue in both cases is moral hazard which arises as a result of an asymmetry in information between the manager and the center. The manager, who is presumably motivated by self-interest, generally prefers to exert less effort, ceteris paribus, while the center is indifferent to the manager’s effort per se. On the other hand, the manager has an informational advantage in that he has private information on his effort that is presumably unobservable to the center. Besides an asymmetry in information, there may also be divergent attitudes toward risk. The manager is likely to be risk averse while the center is presumed to be risk neutral. The ’ This article was written while the author was a visiting scholar at the Harvard-Yenching Institute, Harvard University. Comments of two anonymous referees and the Editor am gratefully acknowledged.
91
0147-5967186 $3.00 Copyii&t Q 1986 by Acadcmii Pres, Inc. All rights of reproduction in any form reserved.
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problem, therefore, is to find an optimal contractual arrangement between the center and the manager that will provide incentives as well as share risk so as to induce the manager, in maximizing his own expected utility, to maximize the center’s expected benefit function also. Recent literature on incentives for socialist managers has dealt with different aspects of the problem of incentives and information. For instance, there has been considerable work on the new Soviet incentive model, the elicitation of truthful information on productive capacity by allowing managers to choose their own targets, and incentive schemes that induce socialist managers to produce at the socially desirable output level.* The point of departure from the previous literature on socialist incentives is the focus that this paper places on the need to motivate socialist managers to exert effort as a result of asymmetric information about their effort. In so doing, we will not deal with issues such as the informational requirements of the center in target-setting, selfselection of targets by the managers, and resource allocation by the center. This article draws on recent advances in the literature on the agency problem and models the relationship between the center and the socialist manager as a principal-agent relationship.3 Bergson (1978) was the first to pose the relationship between the center and the socialist manager as a principal-agent problem, but he did not provide an analysis of the moral hazard and incentive issues that are the primary interest of a principal-agent problem. In decentralized planning, the socialist planner, the principal, delegates decision-making authority to a large number of independent socialist managers, the agents, who are presumably motivated by self-interest in making production decisions. The socialist planner, who is unable to monitor the effort of a large number of socialist managers in any effective way, must coordinate their activities through contractual arrangements. For analytical purposes, it is useful to conceptualize the mode of remuneration of the socialist manager as an explicit or implicit contractual arrangement between him and the center. This article analyzes a typical class of contracts between the center and the manager that can be characterized by the triple parameters of an outcome target 2, a sharing rule s(x), and a fixed payment k. The target is based on the outcome x that can be measured in terms of profit, revenue, or output. If the socialist manager fulfills or overfulfrlls the target, he will be given a bonus according to a sharing rule that specifies how the above-target outcome will be shared between the ’ For work on the new Soviet incentive model, see Weitzman (1976, 1980) and Holmstrom (1982). Ronin (1976) and Bonin and Marcus ( 1979) analyzed the problem of eliciting truthful information on productive capacity by allowing socialist managers to settheir own targets. Incentive schemes that induce socialist managers to produce at the socially desirable output level have been studied by Domar (1974) and Tam (1979). 3 Recent literature on the agency problem includes Ross (1973), Harris and Raviv (I 978, 1979), Holmstrijm ( 1979), and Shaveh ( 1979).
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manager (enterprise) and the center. Otherwise, he will be paid a fixed payment. This payment system can be characterized by S(x) =
k
if
x < 2,
s(x)
if
x 2 2.
It is well known from the agency literature that the optimal contract between a risk neutral principal and a risk averse agent is a general sharing rule s(x) that Pareto-dominates a fixed payment (Shavell, 1979, pp. 59-62). Hence the sharing rule under consideration here is a constrained version of the general sharing rule extensively analyzed in the agency literature in that it specifies a target and a fixed payment. In the degenerate case where the target 2 is set at zero, the constrained sharing rule will reduce to the general sharing rule. The analysis of the constrained instead of the general sharing rule in this article is motivated by the consideration that it is in the spirit of the socialist planning literature. A performance target, either set by the socialist planner or selfimposed by the socialist manager, is important to socialist planning. Besides serving the purpose of motivating the manager to exert more effort by rewarding him for his above-target performance with a bonus payment, the target is also important to the center in its planning since it informs the center of the volume of resources that will be available for its intersectoral or inter&m allocation plan. In our subsequent analysis of socialist planning we will therefore assume that the target is nonzero. In this model the target is set by the planner. Therefore, this article does not address the issue of self-selection of targets by the managers and the “ratchet effect,” nor does it deal with the dynamic problem of target revision by the planner. Under the principal-agent relationship obviously if the manager’s effort is public information, the target should be set by the planner in terms of a stipulated effort level which the manager must expend. This results in a forcing contract and a first-best solution of the type analyzed by Harris and Raviv (1978, 1979). More often than not, effort is private information to the manager and is unobservable by the planner so that it cannot form the basis of a contract. An alternative is to utilize an imperfect monitor or proxy for effort if the planner is able to formulate and observe such a proxy variable. One natural candidate of an imperfect monitor of the agent’s effort is the outcome itself which is positively correlated with effort expended. It also has the advantage that it is generally observable without cost by both parties. Intuitively the outcome target, when appropriately set, will augment incentives provided by the sharing rule. If, on the other hand, it is set too high so that the manager’s probability of fulfilling the target approaches zero, it can become counterproductive because the manager may decide it is not worth his effort to work
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hard since there is almost certainty of an underfulfillment of the target. In order to avert the tendency of the manager to choose to exert zero effort and collect the fixed payment each period, the contract carries with it an implicit threat of dismissal if the manager repeatedly fails to achieve the target. Our purpose is to characterize the optimal incentive contract within the constrained class of contracts specified and analyze the relationship between the three contract parameters. This article represents the first effort to extend the results of the agency problem to the analysis of socialist planning. It extends the work of Keren (1972) who analyzed the problem of target-setting to induce maximum effort, and the work of Bergson (1978) who posed the problem between the manager and the planner as one of risk sharing between agent and principal. In neither of their works has there been a characterization of the optimal incentive contract as an outcome target, a sharing rule, and a fixed payment, nor a treatment of the problem of moral hazard. A principal-agent model applied to the problem of motivating the socialist manager is presented in Section 2. A characterization of the optimal incentive contract with a target-sharing rule-fixed payment triple is contained in Section 3. Section 4 discusses the supply of effort of the manager in response to the optimal contract and the probability of failure to achieve the target. The conclusion and a discussion on the possibility of further research are given in Section 5. 2. THE
MODEL
Let x: A X 8 - R, be a monetary outcome function where A is the set of actions (effort) and 8 is the set of all states of nature. Uncertainty in the monetary outcome may arise as a result of a number of factors such as unpredictable weather, supply uncertainty, and machine breakdowns, all of which are included in the set of all states of nature. For any effort of the socialist manager (agent) e E A, and the state of nature 0 E 8, the monetary outcome is given by x = x(e, 8). It is a stochastic variable with an induced probability density given e denoted byf(xle), and a cumulative density F(x]e). Bothfand F are assumed to be continuously differentiable. It is further assumed that more effort expended increases the outcome in every state of nature with diminishing returns, so that x, > 0 and x,, < 0, with the subscript(s) as usual denoting the partial derivative. An increase in e shifts the distribution of x to the right in the sense of first-order stochastic dominance.4 This implies that Fkxle) =G0 with the strict inequality holding at least for some x. Further let the support of the distribution of x given e be denoted by [a(e), b(e)]. It should be clear that both a and b are functions of e since an increase in e shifts the 4 For a discussion of the ordering the uncertain prospects and first-order stochastic dominance, refer to Hadar and Russell (1969).
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HAZARD
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95
distribution of x to the right. The following boundary conditions are imposed on the distribution: fld4l4
= flWl4
= 0,
for all e.
It is essential to assume that 8 is unobservable to both the center and the manager. This is because if 8 is ex post observable, then after a certain outcome x is realized the manager’s effort can be computed from x = x(e, 0) provided that the outcome function is invertible. In that case the optimal contract is simply a forcing contract that stipulates a target effort level as mentioned earlier and it is enforceable, thus eliminating the need of an outcome target. It is further assumed that the center and the socialist manager share the same ex ante subjective belief of the probability distribution of 0. The planner is assumed to be risk neutral.’ The benefit of the action of a socialist manager to the community, aside from the manager himself, is the outcome minus the share paid to the manager, since it represents the amount of the outcome that can be distributed to the rest of the community. In this article, therefore, it is assumed that the planner, being risk neutral, maximizes the expected net outcome that will be available for distribution after compensating the manager.‘j In the principal-agent framework, it is represented by the principal maximizing his expected share of the outcome. It is assumed that the manager is paid only after effort has been expended and the outcome realized. If the outcome falls short of the target 2, he will be paid a fixed payment k. Given e, the target 2 is bounded by the closed interval [u(e), b(e)]. On the other hand, if the target is fulfilled or overfulfilled, the manager will be paid according to a sharing rule s(x). In contrast to most of the previous studies on socialist incentives that take the bonus as a fixed ’ By assuming that the socialist planner is risk neutral, the central issue is no longer risk sharing but instead provision of incentives. In this case the value of a sharing rule that depends on outcome is mainly in providing incentives. The assumption of risk neutrality of the socialist planner can be justified on the basis of the Arrow-Lind (1980) results on public investment. Set Bergson ( 1978) for a discussion. 6 The characterization of the objective function of the planner as expected output minus the bonus paid to the manager can be found in the work of Keren (1972), Bergson (1978), and Bonin and Marcus ( 1979). In our model, it is not appropriate to characterize the planner as maximizing expected output since this may entail an optimal contract in which the manager will be given the entire or close to the entire share of the output to induce him to exert the maximum effort in order to maximize the expected output, thus leaving no or little output for distribution to the rest of the community. This will quite likely be the case if the disutility of effort is relatively small. Hence expected output is not an appropriate maximand because of distributional considerations of the planner. Here it is assumed that the planner does not just aim for the highest level of expected output but is concerned with how much of the output can be distributed, and hence the income distribution between the manager and the rest of the community. This point was brought to my attention by the Editor.
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payment or as a linear function of output, in our model the sharing rule s(x) can take any general functional form of a weak monotonic function of x. Weak monotonicity of the sharing rule is a natural assumption since it ensures that proper incentives are provided because the manager will receive at least as large a payment when outcome increases.’ Let S denote the set of bounded, continuously differentiable monotonic sharing rules. Clearly S is defined only in the domain of x E [2, b(e)], since if x < f, the manager will be paid k instead. Let C = (2, s( . ), k} be the class of all contracts characterized by 2, s( . ), and k. The socialist planner’s problem is to solve the problem max f,s( f VW
i [S
(x - WW) d’e)
do + J”” f
(x - &WW
h]
.
The first term in the maximand is the expected net revenue of the center when the manager fails to achieve the target. The second term refers to the case when the target is fulfilled or overfulfilled. If the planner can observe e, he can specify the level of effort to be delivered by the manager in the contract. In this case we have a forcing contract with a first-best solution. But here we assume that e is private to the manager and unobservable to the planner. Then the planner can find only an optimal contract (A?*, s*( . ), k*) E C and because of the presence of moral hazard the solution to the problem subject to the constraints to be specified can be only second-best.* The problem can be rewritten as max [J”” (x - k)flxle) do - Jb(’ (s(x) - k)f(xle) c~x]. P Ls(.),k,e n(e)
(1)
To ensure that proper incentives are provided, the fixed payment that the manager receives when he underfulfills the target must be no greater than the payment he receives when he achieves the target. Hence the following constraint on the sharing rule, the target, and the fixed payment must be satisfied:
s(i) 3 k.
(2)
7 Milgrom ( I98 1) has shown that for an incentive contract with a sharing rule, monotonicity of the sharing rule is assured if the family of conditional density functions of x given e has the monotonic likelihood ratio property (MLRP). This means that a greater outcome is evidence of a greater effort by the manager, and hence the sharing rule should be increasing in outcome to provide the correct incentives. MLRP is a property that many families of commonly used density functions possess. * The solution of a principal-agent problem in which moral hazard is present due to an asymmetry in information on effort is only second-best This was discussedby Shaveg (I 979), Holmstriim (1979), and Harris and Raviv (1979).
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HAZARD
AND INCENTIVES
91
The ith manager is assumed to have a von Neuman-Morgenstem utility function Wi which is defined over his wealth and effort. For simplicity of notation the subscript i will be dropped in the following analysis. To simplify the analysis further W is restricted to be separable so that in the case of successful achievement of the target, it can be written as a sum of two components, one depending on the payment and the other on effort, as9 WW, 4 = W(x)) - W where U R, - [0, J3] and K IF?,- [0, B’] with B and B’ as lower bounds.‘O Both U and I/ are assumed to be continuously differentiable. Furthermore U is strictly increasing and strictly concave and V is strictly increasing. This formulation takes into account the assumption that effort entails disutility to the manager. The manager’s expected utility is given by w(e, 2, s, k) =
x [U(k) - JWlf(xl4 dx
sa(e)
+
b(e) sf
[W(x)) - Wlf(xl4 do
where the first term on the right-hand side refers to the case when the manager expends effort e but fails to achieve the target, and the second term refers to the case when the manager fulfills or overfulfills the target. The socialist planner’s maximization program is constrained by a participation constraint of the manager that requires that the manager’s expected utility must be no less than a given minimum level IV0 which can be interpreted as the expected utility of being a socialist worker, an alternative to being a socialist manager. This will also be the alternative employment of the manager if he is dismissed by the center. The participation constraint is b(e)[U(s(x)) - U(k)]f(xle) s2
dx 2 W. - U(k) + V(e).
(3)
The planner’s program is further constrained by the manager’s choice of effort among all feasible actions in A, given the 2, s(a), and k triple. This constraint is the result of the unobservability of effort and the manager being delegated the decision over his choice of effort. It is assumed that the outcome of the negotiation or the hierarchical process between the center and the g This is also the formulation of the agent’s utility function used by Holmstr6m ( 1979), Grossman and Hart (1983), Christensen (198 l), and Green and Stokey (1983). lo The manager’s utility function is bounded from above to avoid the sort of problems discussed by Mirrlees ( 1975).
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manager is such that the former will specify a three-parameter contract while the latter will expend effort contingent on the contract. That is, the manager’s effort function is given by e(& s( * ), k) and he will expend effort to maximize his expected utility subject to the contract parameters stipulated by the center instead of maximizing over all feasible contracts in C.” It is assumed that the planner has knowledge of the manager’s decision rule so that he can find an optimal triple which maximizes his own expected net revenue in (1). Hence at the optimum so defined the manager maximizes his expected utility subject to the parameters of (Z*, s*( - ), k*). This constraint is represented by e E argmax “) [U@(x)) - U(k)]f(xle) e s2
dx + U(k) - V(e), given 2, s( * ), and k.
In general there may be multiple values of e that satisfy this constraint. The contract is assumed to be cooperative in case of multiple solutions so that at the optimum the manager agrees to choose the greatest effort among the levels to which he is indifferent. Hence, given the parameters of the contract, the manager’s optimal supply of effort is defined. The constraint can now be represented by the first-order condition w,(e, 2, s, k) = 0. Alternatively, si
“’ [U@(x)) - U(k)]fe(xle) dx = V(e).
(4)
The ex ante Pareto-optimal contract in the class of C is obtained by choosing (a*, s*( . ), k*) to maximize (1) subject to (2), (3), and (4). The expectation in each case is taken over the commonly shared and subjectively held conditional probability distribution of x. 3. OPTIMAL
TARGET
AND
SHARING
RULE
In solving the planner’s program of maximizing (1) subject to (2), (3), and (4) as discussed in Section 1, we preclude the consideration of comer solutions in which the target and the fixed payment may be set equal to zero, effectively leaving only the sharing rule in the contractual arrangement; comer solutions ii If the manager is allowed to maximize over ah feasible contracts in C, there may be a problem that his global maximum in expect4 utility may not be attained at the constrained maximum expected net revenue of the planner. This problem of a double maximum has been discussed by Mirrlees (1975) in an unpublished paper. By constraining the manager to maximize subject to the contract parameters stipulated by the planner, his constrained maximum in expected utility, though not necessarily a global maximum over all feasible contracts, is attainable when the planner has solved his problem. This makes the problem amenable to solution. The assumption is reasonable in view of an asymmetric bargaining power between the central and the so&list managem assuming that there is no coalition of the managers.
MORAL HAZARD AND INCENTIVES
99
will obviate the need for further analysis of the optimal target and its relation to the other two parameters which is the aim of this article. We will, therefore, consider only the interior solution which is assumed to exist. The optimal target J?*, sharing rule s*( * ), and fixed payment k* are characterized by the following marginal conditions for almost all x12: (s*(x^*) - k*) - [U(s*(x^*)) - U(k*)][A
+ pfs]
+ $$
1
+ 1 _ ;i*,e)
[-I
= 0,
+ XU’(k*)]F(x^*Je) + rcU’(k*)F,(2*le)
- 7= 0
= 0,
(5) (6)
(7)
where 7, X, and p are Lagrangian multipliers for constraints (2), (3), and (4), respectively. The Kuhn-Tucker conditions require that 1, X > 0. Further characterizations of (Z*, P( . ), k*) are given by the following two propositions. PROPOSITION 1. The optimal target 2* is the lower bound of all feasible targets when the optimal sharing rule is s*(e) and the optimaljixed payment is k*. That is, 2* = g*.
Proof: Given the monotonic sharing rule s, and the fixed payment k, the lower bound of the feasible targets 2 is defined by s(Z) = k, and all feasible targets must satisfy R 2 $. First, it will be shown that if the multiplier 7 = 0, there can be no solution, so that 9 must be strictly positive. Assume q = 0. That is, constraint (2) is not binding at the optimum so that P(xI*) > k*. Since (6) holds for all x except at the end points of the support, it can be evaluated at the optimal target i*, so that (5) and (6) become s*(x^*) - k* = x + /G*le) &?*le) U(s*(x^*)) - U(k*)
’
1 = x + pfi~~*14 u’(s*(x^*)) fG*le)
(5’)
*
Now (5’) and (6’) imply that v(s*(x”*))
= WYx^*N - V*) s*(xI*) - k*
’
But clearly this is impossible under the strict inequality s*(x^*) > k*, because U is strictly concave. We conclude that for a solution to exist, q must be strictly positive, implying from the Kuhn-Tucker condition that constraint ‘* A solution cannot exist at the support of the distribution because of the boundary conditions: Aa(e) = BW)le) = 0.
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(2) is binding at the optimum. Therefore, s*(x”*) = k*, but this is simply the definition of a lower bound of the feasible targets given the optimal sharing rule s*( * ) and fixed payment k*. Hence f* = g*. Q.E.D. Besides characterizing the optimal target, the optimal sharing rule can also be characterized by rewriting (5) after substituting in the constraint s*(xI*) = k*, Tjs*‘(a*) o -= (8) m*14 . But q > 0 implies s*‘(xI*) = 0; that is, at the optimal target 3?* = g*, the optimal sharing rule is flat. Now s* E 8’ is weakly monotonic so that s*‘(x) >, 0 for all x. While the equality holds at the optimal target i*, it is argued that only the strict inequality holds in the right neighborhood of the optimal target, H~I = {xix - Z* = e, e > O}. This is because if the equality holds for all x E Ht. it will imply that the optimal sharing rule will be a fixed payment to the manager in the neighborhood of Z*. However, a fixed payment has been shown by Shave11 (1979, pp. 59-62) to be a Pareto inferior sharing rule when the agent is risk averse and there is moral hazard. ‘Hence the equality is precluded in HiI. Since s*’ = 0 at g* and s*’ > 0 for all x E H2*, the optimal sharing rule is locally strictly convex at the optimal target f*. This proves PROFQSITION 2. The optimal sharing rule andfixed payment must satisfy s*(P) = k* with s*‘(P) = 0. It is locally strictly convex at P. This result is illustrated in Fig. 1 which graphs the optimal sharing rule in the right neighborhood of ?*. It is, of course, undefined for all x < z?* since at the optimum, 1* = g*. It is interesting to note that the optimal contract provides continuity in the payment to the manager in the sense that he will get a fixed k* for failure to achieve the target, but when the target is exactly fulfilled, he gets s*(xI*) which equals k*. However, there are incentives for the manager to fulfill not just the target but to overfulfill the target since the optimal sharing rule slopes upward convexly. The manager’s payment increases more, the more the outcome exceeds the target. 4. MANAGERS
SUPPLY
OF EFFORT
It is instructive to analyze the effort response of the socialist manager to the optimal contract. The expected utility of the manager is w(e, 3, s, k). Let us assume that the solution of the optimal contract from (5), (6), and (7) yields an expected utility of w(e, 1*, s*, k*) which exceeds the utility of the zero effort choice, U(k*) - V(0). This obviates the consideration of a situation
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HAZARD
AND INCENTIVES
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FIG. 1.
wherein the manager, seeing that there is too small a probability of exceeding the outcome target x?*, chooses to exert no effort at all and to collect just the fixed payment k*. The choice of zero effort is a degenerate case of the solution and will not be further considered. Suppose w(e, 22*, s*, k*) > U(k*) - V(O), then the supply of effort of the manager, given the contract parameters, can be derived by solving (4). In general it can be characterized by e = e($ s( * ), k). The supply of effort is therefore dependent on the conditional density function of x. Assume that the effort supply function is continuously differentiable; then differentiating the first-order condition w,(e, 2, s, k) = 0, we have
dw,-- w,ei di
+ wd = 0.
Hence
e= t
wd _ [W(3) - WM(~le) WC%
W,
(9)
Since w, < 0 due to the second-order condition in the maximization of w and C@(Z)) 2 U(k) owing to the weak inequality s(x) > k of (2), sgn e2 = sgn f,(ile). The effort response of the manager to the target therefore in general depends on the nature of the distribution function of x given e. However, given the optimal contract of (Z*, s*(. ), k*), ei will equal zero since s*(xI*) = k* according to Proposition 1. We can further show that under the optimal contract
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since s*‘(xI*) = 0 according to Proposition 2. Hence ei = ee = 0. When the contract parameters are optimally chosen, a marginal increase in target will not induce more effort. It should be clear that the optimal target set by the planner will not necessarily induce maximum effort from the manager because the planner is constrained by the fact that the manager will optimally choose his own effort to maximize his expected utility as specified in (4). Similarly, we can analyze the effect of the other two contract parameters on the manager’s supply of effort. For the sharing rule, we have
that is,
The inequality follows from the concavity of U and the equality s
ib(“L(xle) dx = -F,(ile).
But w, < 0 and F,(Zle) -K 0. Therefore es > 0. Hence an increase in share will always induce more effort from the manager. This result holds in general and not only at the point of the optimal contract. Finally, for fixed payment we have wek
-
U’WXxl4
ek=--= W,
WC,
<
o
(11)
An increase in the fixed payment will always induce less effort. Again this result holds in general. At the optimum the probability that the realized outcome falls short of the target when the manager has expended effort e given the optimal contract is f’ P(x < 2*) = f(xle) dx = F(?*le). (12) sde) This probability depends on the shape of the conditional distribution of x. Besides the shape of the distribution, it also depends on how hard the manager is willing to work given the optimal contract. Due to first-order stochastic dominance, F&?*le) G 0, implying that at the optimum an increase in effort expended will weakly decrease the probability of failure to achieve the target. The implicit penalty of dismissal for underfulfilling the target set in the contract will deter the manager from exerting zero effort and simply collecting the
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fixed payment even if, according to his subjective evaluation, the probability of his fulfilling the target is miniscule. This together with the optimal sharing rule will motivate the manager to expend more effort to try to overfulfill the target. 5. CONCLUSION In this article a class of principal-agent contracts that stipulates an outcome target with a threat of dismissal is modeled to analyze the problem of remunerating socialist managers in a decentralized planning environment. It is shown that the optimal contract in this class should set a target which is the lower bound of all feasible targets, while the sharing rule and fixed payment must satisfy s*(Z*) = k*. The sharing rule is strictly locally convex at the optimal target. It should be pointed out that the class of contracts that has been analyzed is not the general class in that the sharing rule is restricted by the presence of a target, which is essential for the planning of the center. In effect the optimal contract within this class specifies a fixed payment for the region of low output and a sharing rule for the region of high output. But the agency literature has shown that the Pareto-optimal contract should be a general sharing rule and not a fixed payment system, meaning that the contract should not specify a target at all. We infer that in our model efficiency in production could be improved if the output target were abolished or set to zero and each socialist manager were confronted with a general sharing rule. This, however, would entail further decentralization of economic activities because the center would no longer be able to rely on output targets to make plans, and thus it would have to make more use of the market mechanism to allocate resources. On the other hand, if the output target cannot be abolished and the economy cannot be further decentralized for one reason or another, our results suggest that the target should at least be set as low as possible so as to improve efficiency. ’ 3 In a socialist economy in which there is decentralized planning, the center plans to engage managers who are willing to work hard to produce a large outcome. A contract with a specified target, bonus for overfulfilling of the target, and dismissal for failure provides a useful screening mechanism for the planner when effort is unobservable. Suppose socialist managers differ only in their preference toward effort. If the planner has knowledge of the contract parameters to effort map of each manager, he will be able to rank the managers according to this map and engage each manager with an individual contract until the vacancies are filled. In this way his expected net revenue will be maximized. However, if the number of vacancies to be filled I3 This point was suggested to me by a referee.
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is large and the transaction cost in individual contracting is considerable, or if the center has no knowledge of the contract parameters to effort map of each manager, an individualized optimal contract for each manager will not be feasible. In practice the planner who engages a large number of managers operating in the same stochastic environment may have to offer an identical contract to all managers. This contract is stipulated on the basis of the planner’s expectation of a manager’s effort response given the contract, and this in turn is based on the planner’s past experience with the managers. If the managers are ex ante identical except for their preference for effort, a Spence-type signal on individual preference for effort is not available so that screening at hiring is not possible. Since the center has no information on each manager’s preference for effort, it will engage managers until the vacancies are filled. However, after they are hired, managers who are less willing to expend effort will have a smaller probability of fulfilling the target, and over time they will be screened out, leaving those who are willing to work the hardest. As a result, while prehiring screening may not be possible, a contract with an outcome target provides a mechanism of posthiring screening to the planner even though effort is unobservable. Extensions and elaborations of the present model in several directions are possible. First, the target may be implicit rather than explicit in some contractual arrangements. In that case since there is no explicit agreement, the center and the manager may have different expectations concerning the level of the target. An analysis of this type of contractual arrangement would be a useful extension. Second, it is possible in the present model that a manager may have worked very hard and still falls short of the target and is dismissed due to the realization of a very unfavorable state of nature. This is possible because the state of nature is assumed to be unobservable ex post so that the contract cannot be made contingent on it. If it is assumed that although states of nature are unobservable, the manager is better informed than the center in that he has signals concerning the states of nature that he may transmit to the center, then the model can be extended by allowing a revision of the target contingent on the information transmitted by the manager. In this case the issue of truthful revelation of this information and the effect of the manager’s improved information on the planner’s expected payoff arises. REFERENCES Arrow, Kenneth J., and Lind, Robert C., “Uncertainly and the Evaluation of Public Investments Decisions.” Amer. Econ. Rev. 60, 3:364-378, June 1980. Bergson, Abram, “Managerial Risks and Rewards in Public Enterprises.” J. Comp. Econ. 2, 3: 21 l-225, Sept. 1978. Bonin, John P., “On the Design of Managerial Incentive Structures in a Decentralized Planning Environment.” Amer. Econ. Rev. 66,4:682-687, Sept. 1976.
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