Envy Manipulation at Work

Maurizio Caserta; Livio Ferrante; Francesco Reito

doi:10.1515/bejte-2019-0161

Article

Envy Manipulation at Work

Maurizio Caserta , Livio Ferrante and Francesco Reito

Published/Copyright: August 7, 2020

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From the journal The B.E. Journal of Theoretical Economics Volume 21 Issue 1

Abstract

We present a simple principal-agent model with an employer and two types of employees/workers, low and high skilled. Low-skilled workers are envious of their high-skilled peers, and incur a disutility cost whenever the latter receive a positive surplus from their labor contract. We show that: i) if the envy cost is relatively low (high), high-skilled workers obtain a payoff higher (lower) than that they receive when they are not envied; ii) if the envy cost can be manipulated (increased or reduced), high-skilled workers can take actions of envy-provocation or envy-reduction to further increase their payoff.

Keywords: envy; principal-agent model; other-regarding preferences

JEL Classification: D82; J33; M52

Corresponding author: Livio Ferrante, Department of Economics and Business, University of Catania, Corso Italia 55, 95129, Catania, Italy, E-mail: lferrante@unict.it

Appendix

(The analytical expressions are obtained using the software Mathematica–Wolfram Research)

Equilibrium with Envy

From the binding ( P C L E ) and ( I C H E ) , we obtain

w L = ( θ ‾ − c θ ‾ ) e L 2 2 ( 1 − c ) , and w H = ( θ ‾ − θ ‾ ) e L 2 + ( 1 − c ) θ ‾ e H 2 2 ( 1 − c ) .

Under the contracts ( w L , e L ) and ( w H , e H ) , the ( P C H E ) is satisfied as the payoff of H workers is positive for all c ∈ ( 0 , 1 ) . The ( I C L E ) is also satisfied, as

u L E ( w H , e H ) = w H − θ ‾ 2 e H 2 − c ⋅ u H E ( w H , e H ) = 1 2 ( θ ‾ − θ ‾ ) ( e L + e H ) ( e L − e H ) < 0 ,

as, in equilibrium, e L < e H . Using the expressions for the wages, from the maximization of the principal’s expected profit in (3), we obtain the following first-order conditions:

1 − λ − ( θ ‾ − λ θ ‾ ) e L − ( 1 − λ ) c ( 1 − θ ‾ e L ) 1 − c = 0 ; and λ ( 1 − θ ‾ e H ) = 0.

The solutions are the effort levels reported in the main text, where

d e L E d c = − ( 1 − λ ) ( θ ‾ − θ ‾ ) [ θ ‾ − [ λ + ( 1 − λ ) c ] θ ‾ ] 2 < 0.

The derivative of the high-skilled worker’s payoff with respect to c is

d u H E d c = − ( 1 − λ ) 2 { θ ‾ − 2 θ + [ λ + ( 1 − λ ) c ] θ ‾ } ( θ ‾ − θ ‾ ) 2 { θ ‾ − [ λ + ( 1 − λ ) c ] θ ‾ } 3 ,

which is positive (negative) for c < c * ( c > c * ), and equal to 0 if c = c * .

The second derivative is

d 2 u H E d c 2 = − ( 1 − λ ) 3 { 2 θ ‾ − [ 3 − λ − ( 1 − λ ) c ] θ ‾ } θ ‾ ( θ ‾ − θ ‾ ) { θ ‾ − [ λ + ( 1 − λ ) c ] θ ‾ } 4 .

The second-order condition, evaluated at c * , is

d 2 u H E d c 2 | c = c * = − ( 1 − λ ) 3 θ ¯ 16 ( θ ‾ − θ ‾ ) 2 < 0 ,

so c * corresponds to a maximum.

The second derivative is first increasing and then decreasing in c, and u H E has an inflection point at

c = ( 3 − λ ) θ ‾ − 2 θ ‾ ( 1 − λ ) θ ‾ ≡ c ˜ ,

which is lower than c * (as c * − c ˜ = ( θ ‾ − θ ‾ ) / ( 1 − λ ) θ ¯ > 0 ), lower than 1, and decreasing in λ, as d c ¯ / d λ = 2 ( θ ‾ − θ ‾ ) 2 / ( λ − 1 ) 3 θ ‾ 2 < 0 .

The difference between the equilibrium payoff of H types and their payoff when c = 0 is positive for all c ∈ ( 0 , c ¯ ] , where

c ¯ = [ ( 2 − λ ) θ ‾ − θ ‾ ] ( θ ‾ − λ θ ‾ ) ( 1 − λ ) 2 θ ‾ 2

is larger than c * (as c ¯ − c * = − [ ( 2 − λ ) θ ‾ − θ ‾ ] ( θ ‾ − θ ‾ ) / ( 1 − λ ) 2 θ ‾ 2 < 0 ), and lower than 1 (as 1 − c ¯ = ( θ ‾ − θ ‾ ) 2 / ( 1 − λ ) 2 θ ‾ 2 < 0 ).

The derivative of the principal’s expected profit with respect to c is

d E [ π ] E d c = − ( 1 − λ ) 2 ( θ ‾ − θ ‾ ) 2 { θ ‾ − [ λ + ( 1 − λ ) c ] θ ‾ } 2 < 0.

The second derivative is

d 2 E [ π ] E d c 2 = − ( 1 − λ ) 3 θ ‾ ( θ ‾ − θ ‾ ) { θ ‾ − [ λ + ( 1 − λ ) c ] θ ‾ } 3 < 0 ,

so envy has an increasingly negative effect on the expected profit of the principal.

Equilibrium with Envy-Provocation

From the binding ( P C L E P ) and ( I C H E P ), we obtain

w L = θ ‾ e L 2 − ( 1 + μ ) c ( 2 μ δ + θ ‾ e L 2 ) 2 [ 1 − ( 1 + μ ) c ] , and w H = θ ‾ e L 2 + ( θ ‾ − θ ‾ ) e L 2 − ( 1 + μ ) c ( 2 μ δ + θ ‾ e H 2 ) 2 [ 1 − ( 1 + μ ) c ] .

Under the contracts ( w L , e L ) and ( w H , e H ) , the ( P C H E ) is satisfied as the payoff of H workers is positive for all c ∈ ( 0 , 1 ) . The ( I C L E ) is also satisfied, as

u L E P ( w H , e H ) = w H − θ ‾ 2 e H 2 − c ( 1 + μ ) u H E P ( w H , e H ) = 1 2 ( θ ‾ − θ ‾ ) ( e L + e H ) ( e L − e H ) < 0.

The first-order conditions of the principal’s maximization are:

1 − λ − ( θ ‾ − λ θ ‾ ) e L − ( 1 − λ ) ( 1 + μ ) c ( 1 − θ ‾ e L ) 1 − c = 0 ; and λ ( 1 − θ ‾ e H ) = 0.

The equilibrium effort levels are those reported in the text, where

d e L E P d μ = − ( 1 − λ ) c ( θ ‾ − θ ‾ ) { θ ‾ − [ λ + ( 1 − λ ) ( 1 + μ ) c ] θ ‾ } 2 < 0.

By substituting the equilibrium effort levels into the utility function of H workers, we have

u H E P ( w H E P , e H F B ) = ( 1 − λ ) 2 [ 1 − ( 1 + μ ) c ] ( θ ‾ − θ ‾ ) 2 { θ ‾ − [ λ + ( 1 − λ ) c ( 1 + μ ) ] θ ‾ } 2 − δ μ 1 − ( 1 + μ ) c .

Using the optimal manipulation activity, μ E P , reported in the text, the equilibrium payoff of H workers is

u H E P ( w H E P , e H F B ) = ( 1 − λ ) c [ ( 8 δ θ ‾ − 1 ) θ ‾ + θ ‾ ] − 8 δ [ ( 2 − λ ) θ ‾ − θ ‾ ] θ ‾ 8 c ( θ ‾ − θ ‾ ) θ ¯ ≡ u H E P .

The derivative of the high-skilled worker’s payoff with respect to c is

d u H E P d c = δ [ ( 2 − λ ) θ ‾ − θ ‾ ] c 2 ( θ ‾ − θ ‾ ) > 0.

where the term ( 2 − λ ) θ ¯ − θ ¯ is positive when c * is positive, that is when the assumption that θ ¯ < 2 θ ¯ holds.

The second derivative with respect to c is

d 2 u H E P d c 2 = − 2 δ [ ( 2 − λ ) θ ‾ − θ ‾ ] c 3 ( θ ‾ − θ ‾ ) < 0 ,

so u H E P is first increasing and then decreasing in c, and has an inflection point at

c = ( 3 − λ ) θ ‾ − 2 θ ‾ ( 1 − λ ) θ ‾ ≡ c ˜ ,

which is always larger than c * (as c * − c ˜ = ( θ ‾ − θ ‾ ) / ( 1 − λ ) θ ‾ > 0 ), lower than 1, and decreasing in λ, as d c ¯ / d λ = 2 ( θ ¯ − θ ‾ ) 2 / ( λ − 1 ) 3 θ ‾ 2 < 0 .

By comparing the H worker’s payoffs with and without envy manipulation, we obtain that u H E P > u H E ⇔ c ∈ ( c M I N E P , c M A X E P ) , where:

c M I N E P = ( 16 δ θ ‾ − λ + 3 ) θ ¯ − [ λ ( 16 δ θ ‾ − 1 ) + 2 ] θ ‾ − θ ‾ 2 − ( θ ‾ − θ ‾ ) θ ‾ 2 − 2 ( 32 δ θ ‾ − λ + 2 ) θ ‾ θ ‾ + [ ( 2 − λ ) 2 + 64 λ δ θ ‾ ] θ ‾ 2 2 ( 1 − λ ) [ θ ‾ + ( 8 δ θ ‾ − 1 ) θ ‾ ] θ ‾ ;

c M A X E P = ( 16 δ θ ‾ − λ + 3 ) θ ‾ − [ λ ( 16 δ θ ‾ − 1 ) + 2 ] θ ‾ − θ ‾ 2 + ( θ ‾ − θ ‾ ) θ ‾ 2 − 2 ( 32 δ θ ‾ − λ + 2 ) θ ‾ θ ‾ + [ [ ( 2 − λ ) 2 + 64 λ δ θ ‾ ] θ ‾ 2 2 ( 1 − λ ) [ θ ¯ + ( 8 δ θ ¯ − 1 ) θ ¯ ] θ ¯ .

The set ( c M I N E P , c M A X E P ) is decreasing in the manipulation cost borne by each H worker, and disappears if

δ > ( 1 − λ ) c { [ 2 − λ − ( 1 − λ ) c ] θ ¯ − θ ¯ } ( θ ¯ − θ ¯ ) 8 { θ ¯ − [ λ + ( 1 − λ ) c ] θ ¯ } 2 θ ¯ ≡ δ M A X E P .

The derivative of the principal’s expected profit with respect to c is

d E [ π ] E P d c = − δ [ ( 2 − λ ) θ ¯ − θ ¯ ] 2 ( 1 − λ ) c 2 θ ¯ ( θ ¯ − θ ¯ ) < 0.

The second derivative is

d 2 E [ π ] E P d c 2 = 2 δ [ ( 2 − λ ) θ ¯ − θ ¯ ] 2 ( 1 − λ ) c 3 θ ¯ ( θ ¯ − θ ¯ ) > 0 ,

so envy-provocation has a decreasing negative effect on the expected profit of the principal. We obtain that E [ π ] E P < E [ π ] E for each c ∈ ( c ^ , c * ) , where

c ^ = 4 δ [ ( 2 − λ ) θ ¯ − θ ¯ ] ( θ ¯ − λ θ ¯ ) ( 1 − λ ) [ 4 ( 2 − λ ) δ θ ¯ 2 + ( 1 − λ ) θ ¯ − ( 1 − λ + 4 δ θ ¯ ) θ ¯ ] .

It can be shown that c ^ is above 0 and below c M I N E P

Equilibrium with Envy-Reduction

From the binding ( P C L E R ) and ( I C H E R ),

w L = θ ¯ e L 2 − ( 1 + μ ) c ( 2 μ δ + θ ¯ e L 2 ) 2 [ 1 − ( 1 + μ ) c ] , and w H = θ ¯ e L 2 + ( θ ¯ − θ ¯ ) e L 2 − ( 1 + μ ) c ( 2 μ δ + θ ¯ e H 2 ) 2 [ 1 − ( 1 + μ ) c ] .

Under the contracts ( w L , e L ) and ( w H , e H ) , the ( P C H E ) is satisfied since the payoff of H workers is positive for all c ∈ ( 0 , 1 ) . The ( I C L E ) is also satisfied, as

u L E R ( w H , e H ) = w H − θ ¯ 2 e H 2 − c ( 1 + μ ) u H E R ( w H , e H ) = 1 2 ( θ ¯ − θ ¯ ) ( e L + e H ) ( e L − e H ) < 0.

The first-order conditions of the principal’s maximization are:

1 − λ − ( θ ¯ − λ θ ‾ ) e L − ( 1 − λ ) ( 1 + μ ) c ( 1 − θ ‾ e L ) 1 − c = 0 ; and λ ( 1 − θ ‾ e H ) = 0.

The equilibrium effort levels are reported in the main text (we do not report here the equilibrium wages). We have that

d e L E R d μ = ( 1 − λ ) c ( θ ¯ − θ ¯ ) { ( 1 + μ ) θ ¯ − [ ( 1 − λ ) c + λ ( 1 + μ ) ] θ ¯ } 2 > 0.

By substituting the equilibrium effort levels into the utility function of H workers, we obtain

u H E R ( w H E R , e H F B ) = ( 1 − λ ) 2 ( 1 + μ ) ( 1 + μ − c ) ( θ ¯ − θ ¯ ) 2 { ( 1 + μ ) ( θ ¯ − μ θ ¯ ) − ( 1 − λ ) c θ ¯ } 2 − δ μ ( 1 + μ ) 1 + μ − c .

At the optimal manipulation activity, μ E R , the equilibrium payoff of H workers is

u H E R ( w H E R , e H F B ) = ( 1 − λ ) { 8 δ [ 2 − ( 1 − λ ) c − λ ] θ ¯ 3 + ( 3 − λ ) θ ¯ θ ¯ − ( 2 + 8 δ θ ¯ − λ ) θ ¯ 2 − θ ¯ 2 } 8 [ ( 2 − λ ) θ ¯ − θ ¯ ] ( θ ¯ − θ ¯ ) θ ¯ ≡ u H E R .

The derivative of the high-skilled worker’s payoff with respect to c is

d u H E R d c = − ( 1 − λ ) 2 δ θ ‾ 2 [ ( 2 − λ ) θ ‾ − θ ¯ ] ( θ ¯ − θ ‾ ) < 0.

By comparing the H worker’s payoffs with and without envy manipulation, we obtain that u H E R ≥ u H E ⇔ c ∈ [ c M A X E R , 1 ] , where

c M A X E R = ( 1 − λ ) { ( 16 δ θ ‾ − λ + 3 ) θ ‾ θ ‾ − [ λ ( 16 δ θ ‾ − 1 ) + 2 ] θ ‾ 2 − θ ¯ 2 } − ( 1 − λ ) 2 [ ( 2 − λ ) θ ‾ − θ ‾ ] − [ ( 64 δ θ ‾ − λ + 2 ) θ ‾ − θ ‾ ] ( θ ‾ − θ ‾ ) 2 θ ‾ 2 16 ( 1 − λ ) 2 δ θ ‾ 4 .

The threshold c M A X E R is increasing in the manipulation cost, δ, and is equal to 1 when

δ > ( 2 − λ ) θ ‾ − θ ‾ 8 θ ‾ 2 ≡ δ M A X E R .

The derivative of the principal’s expected profit with respect to c is

d E [ π ] E R d c = ( 1 − λ ) δ θ ‾ θ ‾ − θ ‾ > 0 ,

and reaches a minimum, E [ π ] E R = ( 1 + λ ) / 4 θ ‾ , when c = c * . Thus, for all c > c * , E [ π ] E R > E [ π ] E .

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Received: 2019-10-30

Accepted: 2020-05-11

Published Online: 2020-08-07

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https://doi.org/10.1515/bejte-2019-0161

Keywords for this article

envy; principal-agent model; other-regarding preferences