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Accessible Unlicensed Requires Authentication Published by De Gruyter August 7, 2020

Envy Manipulation at Work

Maurizio Caserta, Livio Ferrante ORCID logo and Francesco Reito

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.

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:

Appendix

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

Equilibrium with Envy

From the binding (PCLE) and (ICHE), we obtain

wL=(θcθ)eL22(1c),andwH=(θθ)eL2+(1c)θeH22(1c).

Under the contracts (wL,eL) and (wH,eH), the (PCHE) is satisfied as the payoff of H workers is positive for all c(0,1). The (ICLE) is also satisfied, as

uLE(wH,eH)=wHθ2eH2cuHE(wH,eH)=12(θθ)(eL+eH)(eLeH)<0,

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

1λ(θλθ)eL(1λ)c(1θeL)1c=0;and λ(1θeH)=0.

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

deLEdc=(1λ)(θθ)[θ[λ+(1λ)c]θ]2<0.

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

duHEdc=(1λ)2{θ2θ+[λ+(1λ)c]θ}(θθ)2{θ[λ+(1λ)c]θ}3,

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

The second derivative is

d2uHEdc2=(1λ)3{2θ[3λ(1λ)c]θ}θ(θθ){θ[λ+(1λ)c]θ}4.

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

d2uHEdc2|c=c*=(1λ)3θ¯16(θθ)2<0,

so c* corresponds to a maximum.

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

c=(3λ)θ2θ(1λ)θc˜,

which is lower than c* (as c*c˜=(θθ)/(1λ)θ¯>0), lower than 1, and decreasing in λ, as dc¯/dλ=2(θθ)2/(λ1)3θ2<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¯=[(2λ)θθ](θλθ)(1λ)2θ2

is larger than c* (as c¯c*=[(2λ)θθ](θθ)/(1λ)2θ2<0), and lower than 1 (as 1c¯=(θθ)2/(1λ)2θ2<0).

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

dE[π]Edc=(1λ)2(θθ)2{θ[λ+(1λ)c]θ}2<0.

The second derivative is

d2E[π]Edc2=(1λ)3θ(θθ){θ[λ+(1λ)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 (PCLEP) and (ICHEP), we obtain

wL=θeL2(1+μ)c(2μδ+θeL2)2[1(1+μ)c],and wH=θeL2+(θθ)eL2(1+μ)c(2μδ+θeH2)2[1(1+μ)c].

Under the contracts (wL,eL) and (wH,eH), the (PCHE) is satisfied as the payoff of H workers is positive for all c(0,1). The (ICLE) is also satisfied, as

uLEP(wH,eH)=wHθ2eH2c(1+μ)uHEP(wH,eH)=12(θθ)(eL+eH)(eLeH)<0.

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

1λ(θλθ)eL(1λ)(1+μ)c(1θeL)1c=0;and λ(1θeH)=0.

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

deLEPdμ=(1λ)c(θθ){θ[λ+(1λ)(1+μ)c]θ}2<0.

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

uHEP(wHEP,eHFB)=(1λ)2[1(1+μ)c](θθ)2{θ[λ+(1λ)c(1+μ)]θ}2δμ1(1+μ)c.

Using the optimal manipulation activity, μEP, reported in the text, the equilibrium payoff of H workers is

uHEP(wHEP,eHFB)=(1λ)c[(8δθ1)θ+θ]8δ[(2λ)θθ]θ8c(θθ)θ¯uHEP.

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

duHEPdc=δ[(2λ)θθ]c2(θθ)>0.

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

The second derivative with respect to c is

d2uHEPdc2=2δ[(2λ)θθ]c3(θθ)<0,

so uHEP is first increasing and then decreasing in c, and has an inflection point at

c=(3λ)θ2θ(1λ)θc˜,

which is always larger than c* (as c*c˜=(θθ)/(1λ)θ>0), lower than 1, and decreasing in λ, as dc¯/dλ=2(θ¯θ)2/(λ1)3θ2<0.

By comparing the H worker’s payoffs with and without envy manipulation, we obtain that uHEP>uHEc(cMINEP,cMAXEP), where:

cMINEP=(16δθλ+3)θ¯[λ(16δθ1)+2]θθ2(θθ)θ22(32δθλ+2)θθ+[(2λ)2+64λδθ]θ22(1λ)[θ+(8δθ1)θ]θ;
cMAXEP=(16δθλ+3)θ[λ(16δθ1)+2]θθ2+(θθ)θ22(32δθλ+2)θθ+[[(2λ)2+64λδθ]θ22(1λ)[θ¯+(8δθ¯1)θ¯]θ¯.

The set (cMINEP,cMAXEP) is decreasing in the manipulation cost borne by each H worker, and disappears if

δ>(1λ)c{[2λ(1λ)c]θ¯θ¯}(θ¯θ¯)8{θ¯[λ+(1λ)c]θ¯}2θ¯δMAXEP.

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

dE[π]EPdc=δ[(2λ)θ¯θ¯]2(1λ)c2θ¯(θ¯θ¯)<0.

The second derivative is

d2E[π]EPdc2=2δ[(2λ)θ¯θ¯]2(1λ)c3θ¯(θ¯θ¯)>0,

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

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

It can be shown that c^ is above 0 and below cMINEP

Equilibrium with Envy-Reduction

From the binding (PCLER) and (ICHER),

wL=θ¯eL2(1+μ)c(2μδ+θ¯eL2)2[1(1+μ)c],and wH=θ¯eL2+(θ¯θ¯)eL2(1+μ)c(2μδ+θ¯eH2)2[1(1+μ)c].

Under the contracts (wL,eL) and (wH,eH), the (PCHE) is satisfied since the payoff of H workers is positive for all c(0,1). The (ICLE) is also satisfied, as

uLER(wH,eH)=wHθ¯2eH2c(1+μ)uHER(wH,eH)=12(θ¯θ¯)(eL+eH)(eLeH)<0.

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

1λ(θ¯λθ)eL(1λ)(1+μ)c(1θeL)1c=0;and λ(1θeH)=0.

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

deLERdμ=(1λ)c(θ¯θ¯){(1+μ)θ¯[(1λ)c+λ(1+μ)]θ¯}2>0.

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

uHER(wHER,eHFB)=(1λ)2(1+μ)(1+μc)(θ¯θ¯)2{(1+μ)(θ¯μθ¯)(1λ)cθ¯}2δμ(1+μ)1+μc.

At the optimal manipulation activity, μER, the equilibrium payoff of H workers is

uHER(wHER,eHFB)=(1λ){8δ[2(1λ)cλ]θ¯3+(3λ)θ¯θ¯(2+8δθ¯λ)θ¯2θ¯2}8[(2λ)θ¯θ¯](θ¯θ¯)θ¯uHER.

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

duHERdc=(1λ)2δθ2[(2λ)θθ¯](θ¯θ)<0.

By comparing the H worker’s payoffs with and without envy manipulation, we obtain that uHERuHEc[cMAXER,1], where

cMAXER=(1λ){(16δθλ+3)θθ[λ(16δθ1)+2]θ2θ¯2}(1λ)2[(2λ)θθ][(64δθλ+2)θθ](θθ)2θ216(1λ)2δθ4.

The threshold cMAXER is increasing in the manipulation cost, δ, and is equal to 1 when

δ>(2λ)θθ8θ2δMAXER.

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

dE[π]ERdc=(1λ)δθθθ>0,

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

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Received: 2019-10-30
Accepted: 2020-05-11
Published Online: 2020-08-07

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