Optimal efficiency-wage contracts with subjective evaluation Jimmy Chan

Bingyong Zheng

Shanghai University of Finance & Economics

Feb. 03, 2010

Introduction

1

Only 1–5% of US workers receive performance pay in the form of piece rate and commissions (McLeod and Parent 1999)

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Introduction

1

Only 1–5% of US workers receive performance pay in the form of piece rate and commissions (McLeod and Parent 1999)

2

More common are incentives (e.g., bonus, promotion and termination) that are based on subjective evaluations and embedded in a long-term relationship.

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Why subjective evaluation

1

Hard to separate individual contribution

2

Dysfunctional incentive from objective performance measures: rewarding A while hoping for B (Kerr 1975, AMJ) Examples (Baker, Gibbons and Murphy 1994, QJE) At H.J. Heinz company, division manager received bonus only if earning increased from the prior year. The managers delivered consistent earning growth by manipulating the timing of shipments to customers and by prepaying for services not yet received, both at some cost to the firm. In 1992, Sears abolished the commission plan in its auto-repair shop.... Mechanics misled customers into authorizing unnecessary repairs...

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Handling the incentive problems

1

Bonus + Sabotage (MacLeod 2003)

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Handling the incentive problems

1

Bonus + Sabotage (MacLeod 2003) Principal pays a bonus when performance is good.

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Handling the incentive problems

1

Bonus + Sabotage (MacLeod 2003) Principal pays a bonus when performance is good. Agent harms Principal when he thinks own performance is good but receives no bonus.

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Handling the incentive problems

1

Bonus + Sabotage (MacLeod 2003) Principal pays a bonus when performance is good. Agent harms Principal when he thinks own performance is good but receives no bonus. Efficiency loss occurs when the two disagree on performance: no bonus and sabotage.

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Handling the incentive problems

1

Bonus + Sabotage (MacLeod 2003) Principal pays a bonus when performance is good. Agent harms Principal when he thinks own performance is good but receives no bonus. Efficiency loss occurs when the two disagree on performance: no bonus and sabotage.

2

Efficiency wage (Levin 2003, Fuchs 2007)

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Handling the incentive problems

1

Bonus + Sabotage (MacLeod 2003) Principal pays a bonus when performance is good. Agent harms Principal when he thinks own performance is good but receives no bonus. Efficiency loss occurs when the two disagree on performance: no bonus and sabotage.

2

Efficiency wage (Levin 2003, Fuchs 2007) Principal pays a fixed wage.

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Handling the incentive problems

1

Bonus + Sabotage (MacLeod 2003) Principal pays a bonus when performance is good. Agent harms Principal when he thinks own performance is good but receives no bonus. Efficiency loss occurs when the two disagree on performance: no bonus and sabotage.

2

Efficiency wage (Levin 2003, Fuchs 2007) Principal pays a fixed wage. Agent is fired when evaluation is bad.

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Handling the incentive problems

1

Bonus + Sabotage (MacLeod 2003) Principal pays a bonus when performance is good. Agent harms Principal when he thinks own performance is good but receives no bonus. Efficiency loss occurs when the two disagree on performance: no bonus and sabotage.

2

Efficiency wage (Levin 2003, Fuchs 2007) Principal pays a fixed wage. Agent is fired when evaluation is bad. Efficiency loss occurs when Principal fires Agent by mistakes.

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Brief introduction of the model

Study a T-period contracting problem between risk-neutral Principal and Agent. Each period the Principal privately evaluates Agent’s performance. Agent’s self-evaluation may be positively correlated with Principal’s Incentives are provided through money burning at the end T periods. Principal wants to motivate Agent to work and to minimize money-burning cost.

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Main findings

It is optimal to link punishment across periods When correlation is very low, punish only when bad evaluation observed every period When correlation not very low, punishment is asymmetric: Punish deteriorating performance: punish if future performance is bad even if performance were good in the past Reward improvement: Agent can make up for poor performance in the past by performing better in future

Punishment cost increases in the correlation between evaluations.

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Contribution

Show the Abreut et al (1991) rule is optimal for low correlation case: Fuchs (2007) works only the zero correlation case; Determine the lower bound of efficiency loss for any maximum-effort contract and for any correlation of evaluations; Find an optimal T-period contract.

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Model

There are T periods. In period 0, the Principal offers the Agent a contract. If the Agent rejects, then the game ends, and each party gets zero. If the Agent accepts, then he is hired for the next T periods. Each period, the Agent chooses et ∈ {0, 1} 1 > c = c(1) > c(0) = 0 Output is stochastic, with expected output equals et .

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Model (cont.)

Principal observes yt ∈ {H, L}, Agent observes st ∈ {G , B} History: ht ≡ e t−1 × s t−1 Strategy: σ ≡ (σ1 , . . . , σT ), σt : H t → {0, 1} Assumption 1: π(H|1) = p > π(H|0) = q Assumption 2: π(H|1, G ) > max{π(H|1, B), π(H|0, G ), π(H|0, B)}

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Model (cont.) An efficiency wage contract contains three parts: (B, W , Z (y T )) Fixed payment: B, W Punishment: Z (y T ) ≤ W

Payoffs for the Principal and Agent T

F (y , B) = −B − W +

T X

δ t−1 et

t=1

v (W , Z , σ) = B + W − Z (y T ) −

T X

δ t−1 c(et )|σ

t=1

B, W and Z are present value evaluated at t = 1.

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Optimality

Principal’s problem max E

B,W ,Z ,σ

s.t.

−B − W +

T X

δ

t−1

!

et |σ ,

t=1

σ ∈ arg max v (B, W , Z T , σ), v (B, W , Z T , σ) ≥ 0.

Contract induces maximum effort if σt (ht ) = 1. Optimal maximum-effort contract is the one that minimizes E [Z (y T )] subject to σt (ht ) = 1.

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Optimal Punishment Rule: T=1 1

One period contract

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Optimal Punishment Rule: T=1 1

One period contract

Proposition 1 When T = 1, any contract that motivates the Agent to work has a minimum money-burning cost C (Z 1 ) = (1−p)c p−q .

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Optimal Punishment Rule: T=1 1

One period contract

Proposition 1 When T = 1, any contract that motivates the Agent to work has a minimum money-burning cost C (Z 1 ) = (1−p)c p−q . 2

Minimizing the expected money-burning loss, C (Z 1 ) ≡ pZ 1 (H) + (1 − p)Z 1 (L),

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Optimal Punishment Rule: T=1 1

One period contract

Proposition 1 When T = 1, any contract that motivates the Agent to work has a minimum money-burning cost C (Z 1 ) = (1−p)c p−q . 2

Minimizing the expected money-burning loss, C (Z 1 ) ≡ pZ 1 (H) + (1 − p)Z 1 (L),

3

Subject to constraint: −[pZ 1 (H) + (1 − p)Z 1 (L)] − c ≥ −[qZ 1 (H) + (1 − q)Z 1 (L)]

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Optimal Punishment Rule: T=1 1

One period contract

Proposition 1 When T = 1, any contract that motivates the Agent to work has a minimum money-burning cost C (Z 1 ) = (1−p)c p−q . 2

Minimizing the expected money-burning loss, C (Z 1 ) ≡ pZ 1 (H) + (1 − p)Z 1 (L),

3

Subject to constraint: −[pZ 1 (H) + (1 − p)Z 1 (L)] − c ≥ −[qZ 1 (H) + (1 − q)Z 1 (L)]

4

Optimal rule Z 1 (H) = 0 and Z 1 (L) =

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Low correlation

1

Let ρ = ρ = 0)

π(L|1)−π(L|1,G ) , π(L|1)

correlation coefficient (Fuchs 2007,

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Low correlation

π(L|1)−π(L|1,G ) , π(L|1)

correlation coefficient (Fuchs 2007,

1

Let ρ = ρ = 0)

2

Optimal contract: low correlation

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Low correlation

π(L|1)−π(L|1,G ) , π(L|1)

correlation coefficient (Fuchs 2007,

1

Let ρ = ρ = 0)

2

Optimal contract: low correlation

Proposition 2 When ρ ≤ 1 − δ, it is efficient to induce maximum effort through the strategy(   1 c if ∀ t, ytT = L p−q (1−p)T −1 b T (y T ) = Z 0 otherwise T b with money-burning cost C (Z ) = (1−p)c . p−q

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Low correlation (cont.)

Intuition for the zero correlation case (ρ = 0): Agent faces a static problem; most hard to deter single shirking; No incentive to shirk at t = 1, no incentive for single shirking; Expected punishment is convex in the number of shirking.

Per period money-burning cost: 1 (1 − p)c . T −1 1 + δ + ... + δ p−q

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Correlation not low: ρ > 1 − δ

Punishing only when “L” is observed every period is not optimal Learning problem: what would the Agent do if he learns that he probably has passed the Principal’s test?

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Two-period example: ρ > 1 − δ

1

The expected loss C (Z 2 ): p 2 Z 2 (HH) + p(1 − p)[Z 2 (HL) + Z 2 (LH)] + (1 − p)2 Z 2 (LL).

2

Incentive constraints p[Z 2 (LH) − Z 2 (HH)] + (1 − p)[Z 2 (LL) − Z 2 (HL)] ≥

c , p−q

π(H|1, G )[Z 2 (HL)−Z 2 (HH)]+π(L|1, G )[Z 2 (LL)−Z 2 (LH)] ≥ 3

Other constraints are not binding.

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δc p−q

Two-period example (cont.)

1

The optimal rule   π(H|1, G ) c δ+ , Z (LL) = p−q 1−p   c π(L|1, G ) 2 Z (HL) = δ− , p−q 1−p 2

2

2

Z (LH) = Z (HH) = 0.

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Two-period example (cont.)

1

The optimal rule   π(H|1, G ) c δ+ , Z (LL) = p−q 1−p   c π(L|1, G ) 2 Z (HL) = δ− , p−q 1−p 2

2

2

Z (LH) = Z (HH) = 0. 2

The expected efficiency loss 2

C (Z ) =

(1 − p)c (1 − p)c (δ + ρ) = δC (Z 1 ) + ρ . p−q p−q

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A backward induction approach T=1 H

 



 



ZH

  P PP P

L PPP PP P

PP

ZL

One-period contract: ZH = 0, ZL = ZL − ZH ≥

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c , p−q

c . p−q

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A backward induction approach (cont.) T=2 

  H

Z 1 (H) Z 2 (HH)

X HX X XX  L XXXZ 1 (L) Z 2 (HL)   HH H Z 1 (H) Z 2 (LH) H   H  L HX X X XX

L

XXXZ 1 (L) Z 2 (LL)

(1 − p)[Z 2 (LL) − Z 2 (HL)] ≥

c , p−q

δZ 1 (L) = π(H|1, G )Z 2 (HL) + π(L|1, G )Z 2 (LL).

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A backward induction approach (cont.) The backward induction gives the same result as Lagrangian method The efficiency loss: 2

C (Z ) = δC (Z 1 ) + ρ

(1 − p)c , p−q

Optimal rule: δZ 1 (L) =

δc p−q 2

2

=π(H|1, G )Z (HL) + π(L|1, G )Z (LL)     cπ(H|1, G ) π(L|1, G ) cπ(L|1, G ) π(H|1, G ) = δ− + δ+ . p−q 1−p p−q 1−p

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Main result: δ > 1 − ρ Efficiency-loss Lemma 1 Suppose Z T induces maximum effort in a T -period contracting game. Then C (Z T ) ≥ δC (Z T −1 ) +

ρ(1 − p)c . p−q

As C (Z 1 ) = (1 − p)c/(p − q), applying the rule repeatedly: ! T −1 X (1 − p)c δ t−1 . C (Z T ) = δ T −1 + ρ p−q t=1

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Proof 1

For all y T −1 (Here y T −1 ≡ (y2 , . . . , yT )) Z T −1 (y T −1 ) ≡

1 [π(H|1, G )Z T (H ◦y T −1 )+π(L|1, G )Z T (L◦y T −1 )] δ (*)

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Proof 1

For all y T −1 (Here y T −1 ≡ (y2 , . . . , yT )) Z T −1 (y T −1 ) ≡

2

1 [π(H|1, G )Z T (H ◦y T −1 )+π(L|1, G )Z T (L◦y T −1 )] δ (*)

Z T must satisfy IC at t = 1 X µ ˜(y T −1 )[Z T (L ◦ y T −1 ) − Z T (H ◦ y T −1 )] ≥ y T −1

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c . p−q

(**)

Proof 1

For all y T −1 (Here y T −1 ≡ (y2 , . . . , yT )) Z T −1 (y T −1 ) ≡

2

1 [π(H|1, G )Z T (H ◦y T −1 )+π(L|1, G )Z T (L◦y T −1 )] δ (*)

Z T must satisfy IC at t = 1 X µ ˜(y T −1 )[Z T (L ◦ y T −1 ) − Z T (H ◦ y T −1 )] ≥ y T −1

3

c . p−q

(**)

Combining conditions (*) and (**) X
µ ˜(y T −1 ) π(H|1)Z T (H ◦ y T −1 ) + π(L|1)Z T (L ◦ y T −1 ) C (Z T ) = y T −1

≥ δC (Z T −1 ) + ρ

(1 − p)c . p−q

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Optimal rule Corollary 1 When ρ > 1 − δ, it is efficient to induce maximum effort through: 1 1 Z (L) = c/(p − q), Z (H) = 0; and T

Z (y T ) ≡    π(H|1, G ) c T −1 T −1   δZ (L )+ if y T = L ◦ LT −1 ,   (1 − p)T −1  p − q   c π(L|1, G ) T −1 T −1 (L )− δZ if y T = H ◦ LT −1 ,  T −1  (1 − p) p − q    T −1 T T 6= LT −1 , (y−1 ) if y−1 δZ

  T The money burning cost C Z = C (Z T ). Bingyong Zheng

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Implications 1

Z

T

depends only on the last time H is observed.

T

Z (y T ) = 0

if yT = H T −˜t

X δ T −1−t (1 − p)c Z (y ) = (δ + ρ − 1) , ˜t ≡ max t|yt = H p−q (1 − p)t t=1 " # T −1 T −1−t X δ 1 (1 − p)c T T + (δ + ρ − 1) Z (L ) = p−q (1 − p)T (1 − p)t T

T

t=1

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Implications 1

Z

T

depends only on the last time H is observed.

T

Z (y T ) = 0

if yT = H T −˜t

X δ T −1−t (1 − p)c Z (y ) = (δ + ρ − 1) , ˜t ≡ max t|yt = H p−q (1 − p)t t=1 " # T −1 T −1−t X δ 1 (1 − p)c T T + (δ + ρ − 1) Z (L ) = p−q (1 − p)T (1 − p)t T

T

t=1

2

Z

T

b T as ρ → 1 − δ. converges to Z

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Implications 1

Z

T

depends only on the last time H is observed.

T

Z (y T ) = 0

if yT = H T −˜t

X δ T −1−t (1 − p)c Z (y ) = (δ + ρ − 1) , ˜t ≡ max t|yt = H p−q (1 − p)t t=1 " # T −1 T −1−t X δ 1 (1 − p)c T T + (δ + ρ − 1) Z (L ) = p−q (1 − p)T (1 − p)t T

T

t=1

2 3

T b T as ρ → 1 − δ. Z converges to Z Minimum efficiency loss increasing in ρ,

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Implications 1

Z

T

depends only on the last time H is observed.

T

Z (y T ) = 0

if yT = H T −˜t

X δ T −1−t (1 − p)c Z (y ) = (δ + ρ − 1) , ˜t ≡ max t|yt = H p−q (1 − p)t t=1 " # T −1 T −1−t X δ 1 (1 − p)c T T + (δ + ρ − 1) Z (L ) = p−q (1 − p)T (1 − p)t T

T

t=1

2 3

T b T as ρ → 1 − δ. Z converges to Z Minimum efficiency loss increasing in ρ, T

When ρ → 1 − δ, C (Z ) converges to a single one-period contract loss C (Z 1 ).

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Implications 1

Z

T

depends only on the last time H is observed.

T

Z (y T ) = 0

if yT = H T −˜t

X δ T −1−t (1 − p)c Z (y ) = (δ + ρ − 1) , ˜t ≡ max t|yt = H p−q (1 − p)t t=1 " # T −1 T −1−t X δ 1 (1 − p)c T T + (δ + ρ − 1) Z (L ) = p−q (1 − p)T (1 − p)t T

T

t=1

2 3

T b T as ρ → 1 − δ. Z converges to Z Minimum efficiency loss increasing in ρ, T

When ρ → 1 − δ, C (Z ) converges to a single one-period contract loss C (Z 1 ). T When ρ → 1, C (Z ) converges to T one-period contract loss. Bingyong Zheng

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Contract with infinite horizon

1

The Principal pays a fixed wage w to the Agent every period

2

The Agent exerts effort every period

3

At the end of T periods, the Principal terminates the employment contract with probability ψ(y T )

4

If the Agent is not fired, he continues working for the Principal with a clear record from period T + 1 on

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Infinite contracting game

First review contract





?

?

...... 1

T-1

T

......

...... T+1

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∞

Infinite horizon: main result

The optimal review length T is i. Increasing in δ; ii. Decreasing in q and c; iii. Decreasing in ρ if ρ > 1 − δ, but independent of ρ when ρ ≤ 1 − δ; iv. Equal to one if p = 1.

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Conclusion

Show robustness of results in Abreu et al. (1991), Fuchs (2007). Show optimal to link punishment decisions across periods: pooling information The optimal rule is not “fair” Efficiency wage contracts is more costly when ρ is large May not be optimal for the Principal to induce the Agent to exert maximum effort when both ρ and c are large

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Extension: communication 1

Allowing the Principal to communicate with the Agent does not change the outcome when correlation not very high

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Extension: communication 1

Allowing the Principal to communicate with the Agent does not change the outcome when correlation not very high

2

When T = 1, the optimal contract is c , p−q Z (H, G ) = Z (H, B) = 0. Z (L, G ) = Z (L, B) =

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Extension: communication 1

Allowing the Principal to communicate with the Agent does not change the outcome when correlation not very high

2

When T = 1, the optimal contract is c , p−q Z (H, G ) = Z (H, B) = 0. Z (L, G ) = Z (L, B) =

3

If the minimum efficiency loss in the T period contracting game is C (Z T ), the minimum loss in the T + 1 period game is δC (Z T ) +

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ρ(1 − p)c . p−q

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Extension: communication 1

Allowing the Principal to communicate with the Agent does not change the outcome when correlation not very high

2

When T = 1, the optimal contract is c , p−q Z (H, G ) = Z (H, B) = 0. Z (L, G ) = Z (L, B) =

3

If the minimum efficiency loss in the T period contracting game is C (Z T ), the minimum loss in the T + 1 period game is δC (Z T ) +

4

ρ(1 − p)c . p−q

For T > 1, allowing communication does not change the optimal rule if low correlation: π(L|0) > π(L|1, B). Bingyong Zheng

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Communication (cont.)

Allowing the Principal to communicate with the Agent may change the result if correlation is high Effective independence to get rid of learning problem: Zheng (GEB 2008), Obara (JET 2009)

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Thank You!