Optimal Incentive Contract with Costly and Flexible Monitoring

Transcription

1 Optimal Incentive Contract with Costly and Flexible Monitoring Anqi Li 1 Ming Yang 2 1 Department of Economics, Washington University in St. Louis 2 Fuqua School of Business, Duke University May 2016

2 Motivation Firm s choice of monitoring technology has a significant impact on employee productivity. With the recent advent of IT, human resource managers have more flexibility in deciding What kinds of information to consider in performance appraisals; What types of feedback to give to employees; Whether employees should be evaluated on individual or group basis.

3 Motivation (Cont d) Standard agency models limit the principal s choice over monitoring technologies. Hard to reconcile model predictions with stylized facts: 1 Aggregation of performance information into simple grades; 2 Coexistence of individual and group-based incentive schemes among firms with similar production technologies.

4 Preview A principal-agent model with costly and flexible monitoring: Flexibility: the principal can implement any partitional monitoring technology; Cost: increase in the fineness of the partition. Characterize the optimal monitoring technology through the trade-off between the compensation cost and the monitoring cost. Explain stylized facts: 1 Aggregation of various performance information into simple grades; 2 Coexistence of individual and group-based incentive schemes among firms with similar production technologies.

5 Agenda 1 Baseline model 2 Extensions 3 Conclusion

6 Agenda 1 Baseline model 2 Extensions 3 Conclusion

7 Setup A risk-neutral principal and a risk-averse agent. The agent s payoff u(w) c(a): Consumption w 0: u(0) = 0, u > 0, u < 0; Effort a {0, 1}: c(1) = c > c(0) = 0. Each effort a generates a probability space (Ω, Σ, P a ), where P a has a well-defined probability density function p a. The principal s goal: induce high effort.

8 Incentive Contract A pair of monitoring technology P and wage scheme w : P R + : P: a finite partition of Ω whose elements belong to Σ; w( ): maps each A P to a w(a) 0. P = (A 1,, A N ) and a induce p = (p 1,, p N ), where p n = P a (A n ).

9 Incentive Contract (Cont d) Timeline: 1 Principle commits to P, w( ) ; 2 The agent privately exerts a {0, 1}; 3 Nature draws ω Ω according to P a ; 4 A(ω) P is publicly realized; 5 The principal pays w(a(ω)).

10 Implementation Cost Implementation cost: 1 w(a)p a (A) + H(P; a). A P A P w(a)p a(a): compensation cost; 2 H(P; a): monitoring cost, including the cost of Collecting information; Conducting performance appraisals Communicating the result to the agent.

11 Monitoring Cost Assumption 1. H depends only on p and satisfies the following properties: (i) H(p 1,, p N ) = H(p π(1),, p π(n) ) for all permutation π of {1,, N}; (ii) H(p 1, p N ) > H(p 1, p 1,, p N) for all p 1, p 1 > 0 such that p 1 + p 1 = p 1. The monitoring cost is Invariant to the labeling of monitoring outcomes; Increasing in the fineness of the partition. E.g.: H = f ( P ) or N n=1 p n log(p n ).

12 Main Tradeoff As P becomes finer: Inducing high effort becomes cheaper (due to risk aversion); Monitoring cost increases; The optimal monitoring technology balances the trade-off between the compensation cost and the monitoring cost.

13 z-value Define a random variable Z : Ω R by For each A Σ, define z(a) = 1 Z = 1 p 0. p }{{} 1 likelihood ratio P 0 (A) P 1 (A) }{{} average likelihood ratio = E [Z A, a = 1]. A contract is incentive compatible for the agent if u(w(a))z(a)p 1 (A) c. A P (IC) z-value contains all the useful information for deterring shirking.

14 Optimal Incentive Contract The optimal contract P, w ( ) solves min w(a)p 1 (A) + H(P; 1), P,w( ) A P s.t. (IC) and (LL).

15 Benchmark: Exogenous Monitoring Standard agency models take P as exogenously given and solve for min w(a)p 1 (A), s.t. (IC) and (LL) w:p R + A P Denote this solution by w ( ; P). Lemma 1. For any given P, λ > 0 such that u (w (A; P)) = 1 λz(a) only if w (A; P) > 0. if and

16 Z-convexity For each A Σ, let Z(A) denote the image of A under Z. Definition 1. A set A Σ is Z-convex if for all ω, ω A, { ω Ω : s (0, 1) s.t. Z(ω) = (1 s) Z(ω ) + s Z(ω ) } A. When Z(Ω) R is connected, the Z-convexity of A reduces to the convexity of Z(A).

17 Optimal Endogenous Monitoring Technology Theorem 1. Under Assumption 1, for any P = {A 1,, A N }, (i) z(a) z(a ) for all A, A P ; (ii) Any A P is Z-convex; (iii) In case Z(Ω) is connected, there exists {ẑ n } N n=0 each A n contains {ω : Z(ω) (ẑ n 1, ẑ n )}. such that

18 Implications Aggregate (potentially) high-dimension performance information into a coarse, single-dimensional grade; A theory on contract complexity that respects the sufficient statistics principle; The distributions of the optimal performance measure satisfy the strong MLRP with respect to z (A z A if z(a) z(a )). Application: design and implementation of multi-source feedback systems.

19 Agenda 1 Baseline model 2 Extensions 3 Conclusion Multiple actions Multiple agents

20 Multiple Actions The agent s action space A is finite, and the principal wants to induce a A. Define D = A {a }. For each a D, define For each A P, define Z a = 1 p a. p a z a (A) = 1 P a(a) P a (A). A contract is incentive compatible if for each a D, u(w(a))p a (A)z a (A) c(a ) c(a). (IC a ) A P

21 Z-convexity Take any profile {λ a } a D of non-negative reals. Define Z = a D λ a Z a. Definition 2. A set A Σ is Z-convex if for all ω, ω A, { } ω Ω : s (0, 1) s.t. Z(ω) = (1 s) Z(ω ) + s Z(ω ) A.

22 Optimal Endogenous Monitoring Technology Corollary 1. Under Assumption 1, for any P = {A 1,, A N }, there exist non-negative reals {λ a } a D with max a D λ a > 0 such that (i) Any A P is Z-convex; (ii) In case Z(Ω) is connected, { there exists {ẑ } n } N n=0 such that each A n contains ω : Z(ω) (ẑ n 1, ẑ n ).

23 Implications Balanced scorecard (Kaplan and Norton (1992)): Z a : performance in aspect a; λ a : weight assigned to aspect a. {λ a } a D determines how resources are allocated across the monitoring of various deviations. When λ a is big: The boundary of A P varies sensitively with Z a ; Many resources are spent on the monitoring of deviation a.

24 Application: Multiple Tasks The agent can exert either the high effort (a i = 1) or the low effort (a i = 0) in each of two tasks i = 1, 2. Each a = a 1 a 2 generates (Ω 1 Ω 2, Σ 1 Σ 2, P a1 P a2 ). The principal wants to induce a = 11.

25 Multiple Tasks A special case of the multi-action problem: A = {11, 10, 01, 00}, a = 11, D = A {11}; Signal independence = Z 00 = Z 10 + Z 01 Z 10 Z 01. The optimal monitoring technology conducts overall evaluations that aggregate performance information across tasks.

26 The Optimal Monitoring Technology Z10 W>0 W=0 Z01 Figure: Optimal bi-partition when λ 00 is slack

27 The Optimal Monitoring Technology (Cont d) Z10 W>0 W=0 Z01 Figure: Optimal bi-partition when only λ 00 is binding

28 In Progress (λ 00 + λ 01 )/(λ 00 + λ 10 ) captures how resources are allocated across the monitorings of task 1 and task 2. How does this ratio depend on: Whether tasks are substitutes or complements? Detectabilities of one-step deviations and joint deviation?

29 Agenda 1 Baseline model 2 Extensions 3 Conclusion Multiple actions Multiple Agents

30 Background Holmstrom (1979) s sufficient principle suggests that in the absence of technological linkage or statistical correlation, agents should be evaluated on an individual basis. Standard theory uses technological linkage to justify group reward, and common productivity shock to explain tournament. Hard to reconcile with recent empirical findings, that individual and group-based compensation schemes coexist among firms with similar production technologies.

31 Multiple Agents Each of two agents i = 1, 2 can exert either the high effort (a i = 1) or the low effort (a i = 0). Each a i independently generates (Ω, Σ, P ai ). Each a = a 1 a 2 generates (Ω Ω, Σ Σ, P a1 P a2 ). The principal wants both agents to exert the high effort.

32 Incentive Contract An incentive contract consists of 1 P: a finite partition of Ω Ω; 2 (w 1, w 2 ) : P R 2 +. Implementation cost: [w 1 (A) + w 2 (A)] P 11 (A) + µ H(P; a) A P where µ is the marginal cost of monitoring.

33 z-value For each a = {10, 01}, define For each A P, define Z a = 1 p a p 11. z a (A) = 1 P a(a) P 11 (A). A contract is incentive compatible if for each i, u(w i (A)) P 11 (A) z ai =0,a i =1(A) c i A P

34 Z-Convexity Define Z = (Z 10, Z 01 ). Definition 1. A set A Σ Σ is Z-convex if for all ω, ω A, { ω : s (0, 1) s.t. Z ( ω) = s Z ( ω ) + (1 s) Z ( ω )} A.

35 Optimal Monitoring Technology Theorem 2. Under Assumption 1, (i) any A P is Z-convex; (ii) When Z(Ω Ω) is connected and the distribution of Z is atomless, the boundaries of each A P consist of straight line segments in the space of (Z 10, Z 01 ).

36 Bi-Partition: Tournament z10 W1=0 W2>0 W1>0 W2=0 z10

37 Bi-Partition: Group Evalaution z10 W1=0 W2=0 W1>0 W2>0 z10

38 Quad-Partition: Individual Evaluation z10 W1=0 W2>0 W1>0 W2>0 W1=0 W2=0 W1>0 W2=0 z10

39 Special Case: H(P; a) = f ( P ) For each N N, define C(N) = min (w 1 (A) + w 2 (A))P 11 (A), P: P =N A P s.t. (IC) and (LL).

40 Optimal Multi-Agent Contract Implementation cost C(4)+ µ f(4) C(3)+ µ f(3) C(2) C(3) C(2)+ µ f(2) C(4) µ(3) µ(2) µ

41 Implications Explain the use of different multi-agent contracts by factors that affect µ: IT, labor market regulation, access to advanced managerial practices; Human capital share of production, product market competition. Individual evaluation is sub-optimal when µ is large.

42 Literature Review Moral hazard: Single-agent: Holmstrom (1979); Multi-agent: Holmstrom (1982), Mookherjee (1984); Multi-task: Holmstrom and Milgrom (1991). Heterogeneity in managerial practices: Evidence: Bloom and Van Reenen (2010), Bloom, Sadun and Van Reenen (2012), Gibbons and Henderson (2012). Theory: Chassang (2010), Halac and Prat (working paper).

43 Conclusion A principal-agent model with costly and flexible monitoring. Endogenize the choice of monitoring technology through the trade-off between compensation cost and monitoring cost. Explain stylized facts: 1 Information aggregation; overall performance evaluation. 2 Use of different incentive schemes among firms with similar production technologies.