Contracting with Disagreement about Performance Evaluation and Compensation

Transcription

1 Contracting with Disagreement about Performance Evaluation and Compensation Anqi Li Department of Economics Washington University in St. Louis March 2015

2 Motivation Organizational frictions from disagreements about employee performance evaluation and compensation: Self-serving bias: Babcock et al. (1996); Subtle information: Bracken et al. (2001); Intertemporal comparison: Endlich (2000); Interpersonal comparison: Frank (1984, 1985). Expectation: Milgrom and Roberts (1992).

3 Preview Modeling: a general framework for analyzing the organizational implications of various kinds of disagreements about employee performance evaluation and compensation. Result: seemingly rigid policies are robust tools for providing incentives and managing disagreements simultaneously: Long-term performance appraisal; compressed compensation scheme, efficiency wage, seniority-based promotion, etc.

4 Agenda 1 Baseline Model Setup Static case Dynamic case 2 Optimal features 3 Extensions 4 Literature 5 Conclusion

5 Agenda 1 Baseline Model Setup Static case Dynamic case 2 Optimal features 3 Extensions 4 Literature 5 Conclusion

6 Setup Players: a principal and an agent: risk-neutral, zero outside option, no discounting over t = 1,, T Production technology: in period t, the agent privately exerts an effort a t A = {0, 1} that incurs a cost c(a t ) to himself and generates an expected payoff a t to the principal: 1 > c(1) = c > c(0) = 0; a = 1 is efficient. Monitoring technology: at the end of period t, a signal X t is publicly realized where 1 > P(X t = H a t = 1) = p(1) > P(X t = H a t = 0) = p(0) > 0. Incentive contract w T R: baseline wage; { } w T, {ψ t,t ( )} T t=1 : ψ t,t : Ω t R: period-t incentive payment.

7 Disagreement Point Disagreement point S t Ω: a non-contractible signal about the period-t performance evaluation that the agent thinks he deserves: Self-serving bias: S t H; Subtle (private) information; Intertemporal comparison: S t F ( a s, X s, s t); Interpersonal comparison: S i,t F i ( a i,t, a i,t, X i,t, X i,t ). Expectation: S t = F 1 (Q a = 1) Monitoring and Disagreement technology I(T ) = {{X } } 1, S 1,, X T, S T : a T : a T A T

8 Disagreement Cost Period-t disagreement cost If ψ t,t (S t ) > ψ t,t (X t ), then each player i = p, a bears a disagreement cost Λ i (χ t (X t, S t )) 0 where Assumptions: χ t ( X t, S t) = [ ψ t,t ( S t ) ψ t,t ( X t )] + The principal s participation is essential; Asymmetry between gain and loss; Disagreement about internal matters; contract as a frame; (For the time being) no anticipatory conflicts.

9 Agenda 1 Baseline Model Setup Static case Dynamic case 2 Optimal features 3 Extensions 4 Literature 5 Conclusion

10 Static Case: Tension Between Incentive and Disagreement Observation 1. The optimal contract that elicits the low effort satisfies w = 0 and ψ(x) 0 and incurs no disagreement cost. Observation 2. The optimal contract that elicits the high effort solves min 1 c ρ(1)λ p(ψ(h) ψ(l)), w,ψ( ) s.t. ψ(h) ψ(l) ρ(1) ρ(0) p(1) p(0) Λ a(ψ(h) ψ(l)) c p(1) p(0) and w + pψ(h) + (1 p)ψ(l) ρ(1)λ a (ψ(h) ψ(l)) c where ρ(a) = P(S = H, X = L a; I(1))

11 Static Case: Tension Between Incentive and Disagreement Assumption 1. z ρ(1) ρ(0) p(1) p(0) Λ a(z) single-crosses Lemma 1. Under Assumption 1, c p(1) p(0) from below at z > 0 (i) The optimal profit from eliciting the high effort is 1 c ρ(1)λ a (z ) (ii) High effort is more profitable than low effort if and only if 1 c ρ(1)λ a (z )

12 Agenda 1 Baseline Model Setup Static case Dynamic case 2 Optimal features 3 Extensions 4 Literature 5 Conclusion

13 Efficiency Wage Contract A fixed wage w T at each t = 1,, T. At t = T, test if ϕ T (X T ) = 1 T T X t > E[X a = 1] b T t=1 where b T = T 1 2 +ε for some arbitrary ε ( 0, 1 ) 2 If > then the agent passes and earns ψ T ; If then he fails and earns ψ T : ψ T ψ T = B T = αct for some arbitrary α > 1

14 Graphical Illustration Average Output; a=1 Average Output; a=0 E[Y a=1] b T E[Y a=1]+b T E[Y a=1] T

15 Main Result Define λ i,t = Λ i (B T ) T κ = [(p(1) p(0))(h L)] 1. ( ) for each i = p, a, µ T = exp 2Tb2 T and (H L) 2 Assumption 2. max{λ a,t µ T, b T } (1 + λ p,t ) 0 as T.

16 Analytical Performance Bound Theorem 1. Fix ε and α. For each T N and I(T ), every BNE σ T satisfies (i) P (F σ T ; I(T )) π T + 2µ T where π T ((α + 2) + λ a,t /c) µ T + 2κb T (1 2µ T ) α 1 + 2κb T (ii) The per-period expected profit is at least (1 π T 2µ T )(1 2κb T ) }{{} Lower bound for the principal s expected payoff c + (αc + λ a,t )µ T }{{} Upper bound for the expected payment λ p,t P(F σ T ; I(T )) }{{} Upper bound for the principal s disagreement cost

17 Asymptotic Efficiency Theorem 1 (continued). Take an arbitrary sequence {I(T )} T =1. Under Assumption 2, as T, 1 P(F σ T ; I(T )) O(max{λ a,t µ T, b T }); 2 The per-period expected profit is 1 c O(max{λ a,t µ T, b T }(1 + λ p,t )).

18 Intuition: Bound the Principal s Disagreement Cost Disagreement arises from the feeling of being underpaid the prob. of disagreement the prob. of failure for all I(T ). Thus 1 [ ( T E Λ p χ T (X )) ] T, S T σt ; I(T ) λ p,t P(Failure σt ; I(T ))

19 Intuition: Bound the Prob. of Failure Limit the gain from two types of deviations: Concentration of measure: pass by sheer luck; Informativeness: fine-tune the effort choice with past outcomes. Lemma 2 (Concentration of Measure). ( ) P ϕ T (X T ) E[ϕ T (X T ) a T ] > b T a T < 2µ T for all T N and a T A T. Lemma 3 (Informativeness). ϕ T (X T ) E[ϕ T (X T ) a T ] < b T, ϕ T (X T ) > E[X a = 1] b T implies 1 T Tt=1 a t 1 2κb T and 1 T Tt=1 c(a t ) c(1 2κb T ).

20 Agenda 1 Baseline Model 2 Optimal features 3 Extensions 4 Literature 5 Conclusion

21 Long Term Performance Appraisal Short-term bonus contract {w T, ψ t,t (X t )} T t=1. Assumption 3. S t depends only on (a t, X t ). Corollary 1. Under Assumption 3, the per-period expected disagreement cost incurred by the short-term bonus contract is bounded below by [ 1 T T E 1 ψt,t (H) ψ t,t (L) zt t=1 ρ t (1)Λ(z t ) σ T ; I(T ) ]

22 Compressed Compensation Scheme Review contract: {w T, D T, Ψ T } w T : baseline wage; D T {1, 2,, T }: dates for performance appraisals; Ψ T = {ψ t,t : Ω t t +1 R : t D T } where t = max{s < t : s D T {0}}. A max-min game between the principal, the adversary and the agent: The principal commits to {w T, D T, Ψ T } first. The adversary then specifies I(T ) to minimize the principal s expected profit. The agent observes I(T ) and makes the optimal effort choice.

23 The Principal s Problem Let λ a,t = 0 and write the principal s problem as follows: max min E {w T,D T,Ψ T } I(T ) [ T s.t. I(T ), σt m arg max E σ T [ T and Tw T + E X t ψ t,t (Xt t ) Λ p(χ T (X T, S T )) σt m ; I(T ) t=1 [ T ) c(at ) ] σ T ( ψ t,t X t t t=1 ( ψ t,t X t t ) c(at ) σt m t=1 ] 0 ]

24 Compressed Payment Scheme For each Q [0, 1] and t = 1,, T, define ψ m t,t (Q) = max { x R : P(ψm t,t (X t t ) x σm T ; } Im (T )) Q P(ψt,T m (X t t ) < x σm T ; Im (T )) Q Proposition 1. For each T N and Q [0, 1], 1 T E t D T Λ p 1 c Theorem 1 (ii) ( [ψt,t m (Q) ψt,t m ( X t ) ) t ] + σt m ; I m (T )

25 Implication The optimal compensation scheme exhibits local bunching for many disagreement technologies of interest: Self-serving bias: Q = 1; Average payment of many colleagues: Q = 1 2 ; Expectation: Q [0, 1]. Compression becomes increasingly global as the set of feasible disagreement technologies enlarges.

26 Testable Implications Compressed compensation scheme is common if employees Have heterogeneous views about how they should be evaluated and compensated because of the distinct socio-economic backgrounds and (or) prior experiences; Have access to the information that facilitates disagreement formation, e.g., colleagues pay. Can effectively penalize the principal if they are retained in the firm.

27 Distinguish Our Theory from Alternative Explanations for Compressed Compensation Schemes Second-order risk aversion: e.g., Spear and Srivastava (1987): Disclose information that facilitates disagreement formation while fixing the agent s risk attitude our contract becomes more profitable than the short-term bonus contract; First-order risk aversion our contract is more profitable than the short-term bonus contract even if the outcome volatility is small. Subjective evaluation, e.g., Fuchs (2007), Maestri (2012): The implication of self-assessment.

28 Agenda 1 Baseline Model 2 Optimal features 3 Extensions Anticipatory conflicts Promotion systme design Endogenous disagreement point formation General model and simulation results 4 Literature 5 Conclusion

29 Anticipatory Conflict Fix the aformentioned contract and take an arbitrary effort strategy σ T. At the end of t = 1,, T, a conflict state variable C t {0, 1} is publicly realized. If C t = 1, then conflict arises and each player i bears a disagreement cost Λ i (h t ; σ T ).

30 Anticipatory Conflict Assumption 4. (i) For all t, h t, σ T, (a) Λ a (h t ; σ T ) ˆΛ ( E [ χ T ( X T, S T ) h t ; σ T ]) for some weakly concave function ˆΛ : R + R + ; (b) There exists β (0, 1) s.t. Λ p (h t ; σ T ) βλ (h t ; σ T ). (ii) The two terms below vanish as T : ρ T = 1 T E [ T m T = ˆΛ (B T µ T ) E t=1 [ 1 T ] T C t 1 T t=1 ( Cov (ˆΛ (E[χ ) ) ) ] T X T, S T h t ; 1 T ], C t h t, 1 T 1 T

31 Anticipatory Conflict Proposition 2. Under Assumption 4, (i) P(F σ T ; I(T )) 2µ T + π T where π T satisfies π T (α + 2)µ T + 2κb T (1 2µ T ) + (m T + ρ T )/c α 1 + 2b T (ii) The per-period expected profit is bounded below by (1 π T 2µ T )(1 2κb T ) }{{} Lower bound for the principal s expected payoff c(1 + αµ T ) + m T + ρ T }{{} Upper bound for the expected compensation β 1 β [(α + 2)µ T + π T + 2κb T (1 π T 2µ T ) + (m T + ρ T )/c] } {{ } Upper bound for the principal s disagreement cost

32 Promotion System Design in the Shadow of Interpersonal Comparisons Two junior positions with no payment and one senior position. Once every T periods, the senior position becomes vacant and is filled by either an internal junior agent or an external candidate. The second option costs K T O(T ). The junior labor market is competitive and replacement is costless. Each agent can work at the junior position for at most 2T periods. From a junior agent s viewpoint, promotion conveys a benefit B T = 2αcT, α > 1.

33 Merit-Based Promotion Define merit-based promotion policies {ρ 1, ρ 2 } by s.t. ρ i : Ω 2T [0, 1], i = 1, 2, ) ρ i (x1 T, x2 T 1 x1 T, x2 T Ω T (Feasibility) i and ρ 1 ( x T, x T ) = ρ 2 (x T, x T ) x T, x T Ω T (Anonymity) If one agent wins and the other loses then the social disagreement cost is at least Λ(B T ct ).

34 Seniority-Based Promotion Label the two junior positions as the entry-level position and the junior-level position. An agent works at the entry level for T periods; moves to the junior level if 1 T Tt=1 X t > E[X a = 1] b T and gets fired otherwise. Works for another T periods before he is promoted or fired. Agents do not envy the outcome of colleagues at different organizational hierarchies (even if they do, the prob. is small).

35 Merit-Based Promotion vs. Seniority-Based Promotion Proposition 3. (i) The per-period efficiency loss of the merit-based policy is at least K T (1 i P(ρ i > ρ i ) σt ) 1 T +Λ(B T ct ) i P(ρ i > ρ i ) σt ] +efficiency loss due to shirking (ii) The per-period efficiency loss of the seniority-based policy is at most K T O(b T )/T + O(b T ) O(b T ) (iii) When T is large, (ii) is smaller than (i) if Λ(B T ct ) B T Θ(b T ) as T.

36 Simulation Poisson signals Ω = {0, 1}. General production technology and sophisticated preferences over risks and time. Treat one month as a period and assume that: Standard annual discount rate 5%; λ a,t = 0, ε =.45, α = 2, 3; Conservative estimate of the informativeness parameter: c T [ 1 15, 1 8 ]. Estimated prob. of failure is below 10% after 5 months. Attain near-efficiency over a relatively short time span under reasonable assumptions about (1) the production technology w(a ) c(a ) (2) the informativeness parameter w T and (3) the disagreement technology (λ a,t, λ p,t ).

37 Literature Contracting with reference-dependent preference: Herweg et al. (2010), Eliaz and Spiegler (2014). Contracting with social preference: Akerlof (1982), Moldovanu et al. (2007), Dubey and Geanakoplos (2010). Hart and Moore (2008) Reference dependence and social preference: Kahneman and Tversky (1979), Koszegi and Rabin (2006, 2007), Fehr and Schmidt (1999), Bolton and Ockenfels (2000), Fehr and Gächter (2000). Efficiency in agency games and mechanisms: Rubinstein and Yaari (1983), Radner (1981, 1985), Jackson and Sonnenschein (2007), Li (2014 (a)).

38 Summary Formalize various kinds of disagreements about perfomrance evaluation and compensation; examine their organizational implications. Open questions: Contract renegotiation; Learning the disagreement point process.