Managerial Compensation with Keeping up with the …Managerial Compensation with Keeping up with the Joneses Agents (Temporary Title) Extremely Preliminary: please do not circulate

Managerial Compensation with Keeping up with the Joneses Agents(Temporary Title)

Extremely Preliminary: please do not circulate without permission.

Peter M. DeMarzo

Stanford University and NBER

Ron Kaniel

University of Rochester, and CEPR

November 8, 2013

1

Preamble

The objective of our work is to understand the equilibrium contracting implications of having

managers with Keeping up with the Joneses incentives. The lack of empirical evidence for relative

performance pay is some what of a puzzle. The puzzle is exacerbated by the fact that empirically

it has been shown that not only doesn’t compensation decline with peer firms performance, it in

fact increases in peer performance.

We show that assuming managers have Keeping up with the Joneses incentives allows a simple

rationalization for this evidence.

Our analysis focuses on three related models:

1. a setting with a single principle employing multiple managers.

2. a setting with multiple principle-agent pairs, in which each principle is responsible for writing

the contract of her agent. Furthermore, in determining the optimal contract of a given firm it

is assumed the effort choice of managers of other firms are taken as given, and consequently

so is their income distribution.

3. similar to the second, but relaxes the assumption that other firms managers’ actions are

taken as given when determining a firm’s contract. Specifically, in determining a manager’s

contract the firm realizes that since managers have Keeping up with the Joneses preferences

its compensation structure will impact effort choices of peer managers, which in turn will

affect the effort choice of its manager.

In addition to showing our basic point, our setting allows us to derive testable implications on

a few dimensions:

• We compare the pay performance and relative pay performance sensitivity of a division man-

ager within a conglomerate to that of manager managing a stand alone firm that is similar

to the conglomerate division.

• We analyze how do equilibrium contracts vary as one varies the degree of competition in

the relevant industry. In our simple setting we use the number of firms as the proxy for

competition.

• We derive testable predictions as to how contract structure changes as one varies output

correlation across firms and the riskiness of the firms operations.

This draft contains the solution of the model, the different propositions, and a comprehensive

set of comparative statics analysis. These results constitute the basis for the paper. The draft also

contains a list of related papers, at this point without any associated discussion.

The paper itself will be written soon.....

2

1 n + 1 Agents

qi is the output of agent i, which depends on the agent’s effort choice.

qi = ai + εi (1)

where εi ∼ N (0, σ2) have a pairwise correlation of ρ ≥ 0.

Let

qi− =1

n

∑

j 6=iqj

Contracts

wi = mi + xiqi + yiqi− (2)

where,

wi− =1

n

∑

j 6=iwj

εi− =1

n

∑

j 6=iεj

So that

Cov[ε1, εi−] = ρσ2

V ar[εi−] =1

n2(nσ2 + n(n− 1)ρσ2) =

1 + (n− 1)ρ

n

and agents i’s utility functions become

−E[e−2λ(wi−δwi−−ψ(ai))]

observe that

wi = mi + xi(ai + εi) + yi1

n

∑

j 6=i(aj + εj)

3

wi− =1

n

∑

j 6=i(mj + xjqj + yj qj−)

= mi− +1

n

∑

j 6=ixj(aj + εj) + ασxi− +

1

n

∑

j 6=iyj(aj− + εj−)

= mi− + +1

n

∑

j 6=ixj(aj + εj) +

1

n

∑

j 6=iyj(aj− + εj−)

Consider a symmetric equilibria, and solve for the reaction function of the contract for a given

principle-agent pair , taking all the other contracts as given.

W.l.o.g we solve for the reaction function of principle-agent pair 1, and to facilitate a simple

comparison to the case with only two principle agent pair we denote the contracts of other agents

by x2, y2 and the corresponding agent action by a2.

1.1 Agents Optimization

Agents have keeping up with the Joneses exert effort ai to maximize

maxai

−E[e−2λ(wi−δwi−−ψ(ai))],

where ψ(a) = 12k a

2.

subject to the IR constraint

−E[e−2λ(wi−δwi−−ψ(ai))] ≥ u

Throughout we make the following natural assumption

Assumption 1

−1 < δ < 1

Using the symmetry we can simplify to

w1 = m1 + x1(a1 + ε1) + y1(a2 + ε1−)

w1− = m2 + x2a2 + x2ε1− + y2

(

1

n(a1 + ε1) +

n − 1

n(a2 + ε1−)

)

4

Which implies that

w1 − δw1− = m1 − δm2 +

(

x1 −δy2

n

)

a1 +

(

y1 − δ

(

x2 +(n− 1)y2

n

))

a2

+

(

x1 −δy2

n

)

ε1 +

(

y1 − δ

(

x2 +(n− 1)y2

n

))

ε1−

So that

E[w1 − δw1−] = m1 − δm2 +

(

x1 −δy2

n

)

a1 +

(

y1 − δ

(

x2 +(n− 1)y2

n

))

a2

and

V ar[w1 − δw1−] =

[

(

x1 −δy2

n

)2

+ 2

(

x1 −δy2

n

)(

y1 − δ

(

x2 +(n− 1)y2

n

))

ρ

+

(

y1 − δ

(

x2 +(n− 1)y2

n

))2 1 + (n − 1)ρ

n

]

σ2

Using the MGF for normally distributed random variables, the agent maximization can be

written as

maxa1

[

2λ (E[w1 − δw1−] − ψ(a1))−1

24λ2V ar[w1 − δw1−]

]

or written slightly differently as

maxa1

2λ [E[w1 − δw1−] − ψ(a1) − 2λV ar[w1 − δw1−]]

Taking the other agents action as given, and eliminating terms that are independent of the

agents action a1, the maximization reduces to

maxa1

[

(x1 −δy2

n)a1 −

1

2ka2

1

]

with a solution

a∗1 = k

(

x1 −δy2

n

)

So that an agent’s action depends both on the sensitivity of his payoff to his output and the

sensitivity of other agents payoffs to his output.

Also, by substituting in the a∗1, a∗2 for a1, a2 we get

E[w1 − δw1−] = m1 − δm2 + k

(

x1 −δy2

n

)2

+ k

(

y1 − δ

(

x2 +(n− 1)y2

n

))

(

x2 −δ( y1

n+ (n−1)y2

n)

n

)

5

1.2 Single Principle with n + 1 Agents

Optimal contract is given by

Proposition 1 Let

Ro = (n+ δ)

(

(1 + (n− 1)ρ) + (1− ρ)(1 + nρ)2σ2λ

k

)

The optimal equilibrium contract is given by

xo =(1− δρ) + (n− 1)(1− δ)(1 + nρ)

Ro

yo =n((δ − ρ)− (n− 1)ρ(1 − δ))

Ro

ao =k(1 − δ)(n+ δ)(1 + (n− 1)ρ)

Ro

Comment: for xo and yo, k, σ and λ enter together through the term σ2λk

, which impacts the

scaling term Ro.

Lemma 1 1. yo

• yo < 0 iff δ < nρ1+(n−1)ρ

.

• increases in δ, and decreases in ρ,

• varying σ2, λ, and k:

– ρ < 1: if δ > nρ1+(n−1)ρ decreases in σ2 and λ, and increases in k. Otherwise,

increases in σ2 and λ, and decreases in k.

– ρ = 1: independent of σ2, λ, and k.

2. xo

• xo > 0 Let δ1 =(n+n2) 2σ2λ

k

1+(n+n2) 2σ2λk

and δ2 = (n−1)n(n−1)n+1

• decreases in δ,

• varying ρ

– δ ≤ 0: increases in ρ.

– δ > 0: If δ >(n+n2) 2σ2λ

k

1+(n+n2) 2σ2λk

decreases in ρ; otherwise u-shaped in ρ


– ρ < 1: decreases in σ2 and λ, and increases in k.


6

3. xo > yo iff δ <1+(2n−1)ρ

(n−1)(1−ρ)n

+1+(2n−1)ρ(note: from the condition if either n = 1 or ρ = 0 then

xo > yo)

4. ao

• ao > 0

• decreases in δ, and increases in ρ

• varying σ2, and λ:

– ρ < 1 : decreases in σ2 and λ,

– ρ = 1 : independent of σ2, λ.

• increases in k

5. yo

xoincreases in δ, decreases in ρ, and is independent of σ2, λ, k.

6. mo

• increases (decreases) in δ if 2σ2λk

< (>)1−n ρ

1−ρ

1+nρ . Note that the condition implies that at

ρ = 1 always decreases in δ. Also, in the region that the cutoff1−n ρ

1−ρ

1+nρ is positive the

cutoff is decreasing in ρ.

• increases in ρ


– ρ < 1: increases (decreases) in σ2 and λ if 2σ2λk

< (>)3−n ρ

1−ρ

1+nρ ,

– ρ = 1: independent of σ2 and λ.

• increases in k if η2λk

≥ 1 or ρ > 11+n , and decreases in k if σ2λ

k< 1 and ρ sufficiently

small.

7. principle’s welfare

• decreases in δ, increases in ρ


– ρ < 1: decreases in σ2 and λ,

– ρ = 1: independent of σ2 and λ.

• increases in k

Lemma 2 As a function of n

1. yo:

7

• 0 < ρ < 1:

0 ≤ δ < ρ, decreasing

ρ < δ, decreasing for n sufficiently large

12 (1−

√3) < δ < 0, if ρ > 1

(1−δ+δ2) increasing

12 (1−

√3) < δ < 0, if ρ < 1

(1−δ+δ2) decreasing for n sufficiently large

12 (1−

√3) < δ < −1

3 , ifρ >(1+2δ−2δ2)

δ2increasing

• ρ = 0 (ρ = 1): increasing (decreasing).

2. xo:

• 0 < ρ < 1:

δ < − 1−ρk

2σ2λ+1−ρ , decreasing for n sufficiently large

δ > − 1−ρk

2σ2λ+1−ρ , increasing for n sufficiently large

δ = 0, increasing

• ρ = 0 (ρ = 1): decreasing (increasing).

3. yo

xo:

• ρ > 0:

ρ > δ2

2+δ−2δ2, decreasing

ρ < δ2

2+δ−2δ2, tent-shape

• ρ = 0: increasing.

4. a0

• 0 < ρ < 1: increasing

• ρ = 0 and ρ = 1: independent.

5. mo

• 0 < ρ < 1: increasing


6. principle’s welfare per agent hired

• 0 < ρ < 1: increasing;


Proof. to be added

8

1.2.1 Limit as n→ ∞

Corollary 1 The optimal equilibrium contract is given by

xo =1− δ

1 + (1− ρ) 2σ2λk

yo = −xoao = kxo

mo =k(1− δ)

2

1

1 + (1− ρ) 2σ2λk

Principle’s Welfare =k(1− δ)

2

1

1 + (1− ρ) 2σ2λk

1.2.2 wi = mi + ui(qi − q) + viq

uo = xo

vo = xo + yo =(1 − ρ)(n+ δ)

Ro=

1 − ρ

(1 + (n− 1)ρ) + (1− ρ)(1 + nρ) 2σ2λk

1.2.3 The δ = 0 benchmark

Setting δ = 0 in Proposition 1 yields

Corollary 2 When δ = 0

Ro = n

(

(1 + (n− 1)ρ) + (1 − ρ)(1 + nρ)2σ2λ

k

)

,

and the optimal equilibrium contract is given by

xo =n2ρ+ n

Ro

yo =−n2ρ

Ro

ao =kn(1 + (n− 1)ρ)

Ro

comment: The comparative statics for this case are embedded as part of the comparative

statics in Lemmas 1 and 2.

9

1.3 n+1 Principle-Agent Pairs when actions of other agents are taken as given

1.3.1 Reaction function of Principle 1

Proposition 2 Taking the contract of all other agents as given, the optimal contract that principle

1 gives to agent 1 is given by

xreact1g =n(1 + (n− 1)ρ) + y2δ(1− ρ)(1 + nρ) 2σ2λ

k

n((1 + (n− 1)ρ) + (1− ρ)(1 + nρ) 2σ2λk

)

yreact1g = − nρ

(1 + (n− 1)ρ) + (1 − ρ)(1 + nρ) 2σ2λk

+ x2δ

+y2δ

(

1 − 1

n+

ρ

(1 + (n− 1)ρ) + (1 − ρ)(1 + nρ) 2σ2λk

)

immediate observations:

• xreact1g is independent of x2

• xreact1g increases in y2 iff δ > 0

• yreact1g increases in x2 and y2 iff δ > 0

comment: when ρ = 0 reaction function reduces to

xreact1g =n+ y2δ

2σ2λk

n(1 + 2σ2λk

)

yreact1g = x2δ + y2δ

(

1 − 1

n

)

1.3.2 Equilibrium Contracts

Proposition 3 Let

Rg = (1 − δ)Ro + δ(δ − ρ− (n− 1)ρ(1− δ))


xg =(1 − δρ) + (n− 1)(1− δ)(1 + nρ)

Rg

yg =n((δ − ρ) − (n− 1)ρ(1− δ))

Rg

ag =k(1 − δ)(n+ δ)(1 + (n− 1)ρ)

Rg

10

• Comparing the solution to the one with one principle it is easy to see that xg, yg, and ag differ

from their counterparts xo, yo, and ao in that the scaling term Rg replaces the scaling term

Ro.

• A direct calculation shows that Rg > 0, and that Rg < Ro iff δ > 0.

Lemma 3 When δ = 0 the equilibrium coincides with the case where we have one principle with

multiple agents.

Proof. By inspection.

Lemma 4

1. yg

• yg < 0 iff δ < nρ1+(n−1)ρ

; comment:same sign as yo.

• increases in δ, and decreases in ρ; at ρ = 1 independent of δ.


– for ρ < 1: if δ > nρ1+(n−1)ρ decreases in σ2 and λ, and increases in k. Otherwise,

increases in σ2 and λ, and decreases in k.

– for ρ = 1: independent of σ2, λ, and k.

– comment: same comparative statics as in one principle case.

2. xg

• xg > 0

• varying δ

– 0 ≤ ρ < 1 : u-shaped in δ, where the δ at which the minimum is attained is at δ = 0

when ρ = 0 and increases in ρ.

– ρ = 1 : independent of δ.

• tent shaped in ρ, where the peek is at ρ∗ which solves δ = 1

1+(1−ρ)2

(nρ(2+(n−1)ρ))

. At δ = 0 ρ∗ is

0 (i.e., xg decreases in ρ). Also, ρ∗ increases in δ and decreases in n.


– ρ < 1: decreases in σ2 and λ, and increases in k.


– comment: same comparative statics as in one principle case.

3. xg > yg iff δ < 1

1+(1− 1

n )

(2nρ

1−ρ+1)

. For n = 1 xg > yg. The cutoff is u-shaped in n, and for n > 1

the cutoff is increasing in ρ

11

4. ag

• ag > 0; at the limit as δ → 1 ag → 0.

• varying δ:

ρ = 1, increasing in δ

0 ≤ ρ < 1, tent-shape in δ(the max occurs at a δ∗ > 0, where δ∗ increases in ρ)

• increases in ρ

• varying σ2, and λ

– ρ < 1: decreasing in σ2, and λ;

– ρ = 1: independent of σ2, λ.

– comment: same as in one principle case.

• increasing in k, comment: same as in one principle case.

5.yg

xg= yo

xo.

6. mg (seems too messy to get full analytical characterization)

• varying δ

– ρ < 1 and n ≥ 2

2σ2λk >

1−(1+n)ρ(1−ρ)(1+nρ) , tent-shaped, for δ ≤ 0 subregion increasing.

otherwise, u-shaped, for δ ≤ 0 subregion decreasing.

– ρ < 1 and n = 1:

∗ similar characterization as the n ≥ 2 for the δ ≤ 0 range. f in addition 2σ2λk

>ρ

(1−ρ2) or 14

ρ(1−ρ2)

> 2σ2λk

(in the mid range derivative may switch sign

more than once-seems messy to check analytically)

– ρ = 1: increasing

• varying ρ: when n = 1 a sufficient condition for it to increase in ρ is δ > −0.215(seems

messy to sign for region δ < −0.215 or for general n)


– ρ < 1: if ρ ≥ 3n(1−δ)+δ(3−2δ)

n(3+n)+(3−n−n2)δ−2(1+n)δ2decreasing, otherwise tent-shaped.

– ρ = 1: independent.

• varying k : if ρ > 11+n increases, otherwise tent-shaped.

7. principle’s welfare (seems too messy to get full analytical characterization)

12

• varying δ,

– ρ = 1 : If n ≥ 7 decreases, otherwise tent-shape where max is at a δ ∗ (n) < 0.

• varying ρ

– ρ = 0 : decreasing in ρ

– ρ = 1 : increasing in ρ iff σ2λk

(1 − δ)(n + δ)(n − 2(1 + n)δ) − δ2 > 0 (note that a

sufficient condition for it to be decreasing is δ > n2(1+n)

).

• varying σ2 and λ

– for ρ = 1: independent of σ2 and λ.

– 1n+1 < ρ < 1: there exist thresholds 1−ρ

2 < δ1 <12 < δ2 < 1 such that

δ < δ1, decreasing

δ1 < δ < 12 , tent-shaped

12 < δ < δ2, increasing

δ2 < δ, u-shaped

– ρ < 1n+1 : there exist thresholds 1

2 ≤ δ1 ≤ δ2 < 1 such that

δ < 12 , decreasing

12 < δ < δ1, u-shaped

δ1 < δ < δ2, increasing

δ2 < δ, u-shaped

For n ≤ 5, there exists a ρ∗(n) < 1n+1 such that for ρ ≤ ρ∗(n) 1

2 = δ1 = δ2; for

n > 5 12 < δ1 < δ2

• varying k:

Let δ∗ = (n−nρ+n2ρ)(−1+(n+1)ρ+2(n−nρ+n2ρ))

– ρ < 1n+1 :

δ < 12 , increases

12 < δ < δ∗, u-shape

δ∗ < δ, decreases

– ρ > 1n+1 :

δ∗ > δ, increases

δ∗ < δ < 12 , tent-shape

δ > 12 , decreases

13

– ρ = 1 : decreases in k iff δ > n2n+1

Lemma 5 As a function of n

1. yg:

• 0 < ρ < 1:

0 < δ < ρ, decreasing

ρ < δ, either decreasing or tent shape, ρ sufficiently small tent-shape

δ < 0, ρ or δsufficiently small, tent shape

• ρ = 0 (ρ = 1): increasing (decreasing)

2. xg:

• 0 < ρ < 1: Let δ1 =3ρ2−ρ

√8+24ρ+17ρ2

2(1+3ρ+ρ2)< 0(> −1), δ2 =

3ρ2+ρ√

8+24ρ+17ρ2

2(1+3ρ+ρ2)≤ 1.

δ1 < δ < δ2, increasing

otherwise , u-shaped

comment: δ1 is decreasing in ρ and δ2 is increasing in ρ.

• ρ = 0 (ρ = 1): negative (positive).

3.yg

xg: the same as in the one principle case.

4. ag

• δ < 0: decreasing

• 0 < ρ < 1:

if δ < 2σ2λ(1−ρ)k

for large enough n increasing.

δ > ρ, increasing

kσ2λ

small , increasing

δ < 2ρ2

1+2ρ−ρ2 and kσ2λ

large , u-shaped

If δ >2σ2λ(1−ρ)

k for large enough n decreasing.

δ > 2ρ2

1+2ρ−ρ2 , u-shaped

kσ2λ

small , u-shaped

otherwise , decreasing

14

• ρ = 0 and ρ = 1: increasing (decreasing)

Proof. to be added


Corollary 3 A principle’s reaction function is given by

xreact1g =1

1 + (1− ρ) 2σ2λk

yreact1g = − 1

1 + (1 − ρ) 2σ2λk

+ (x2g + y2g)δ


xg =1

1 + (1− ρ) 2σ2λk

yg = −xgag = kxg

mg =k

2(1− δ)

1

1 + (1− ρ) 2σ2λk

Principle’s Welfare =k(1 − 2δ)

2(1− δ)

1

1 + (1− ρ) 2σ2λk

1.3.4 wi = mi + ui(qi − q) + viq

ug = xg

vg = xg + yg =(n+ δ)(1− ρ)

Rg

1.3.5 comparing across contract types

Lemma 6

1. yg > yo iff δ(1−ρ)ρ(1−δ) > n (comment: for n = 1 condition reduces to δ > ρ)

2. xg > xo iff δ > 0

3. ag > ao iff δ > 0

4. mg > mo iff δ(

2σ2λk

− 1−(n+1)ρ(1−ρ)(1+nρ)

)

> 0

15

5. Principle’s welfare per-agent hired is higher when one principle hires all agents, as opposed

to n + 1 principle-agent pairs.

Proof. To be inserted.

16

1.4 n + 1 Principle-Agent Pairs


Proposition 4 Taking the contract of the other n agents as given the optimal contract that prin-

ciple 1 gives to agent 1 is given

xreact1m =n(1 + (n− 1)ρ) − x2δ

2ρ+ y2

(

δ(1 − ρ)(1 + nρ) 2σ2λk − (1 − 1

n )δ2ρ)

n(

1 + (n− 1)ρ+ (1 − ρ)(1 + nρ) 2σ2λk

)

yreact1m =−n2ρ+ x2δ

(

kδ2λσ2 + n+ δ + (n− 1)nρ+ n(1 − ρ)(1 + nρ) 2σ2λ

k

)

n(

1 + (n− 1)ρ+ (1 − ρ)(1 + nρ) 2σ2λk

)

+y2δ

(

(1− 1n)δ k

2λσ2 + (1 − 1n )(n+ δ) + (1 + (n− 1)n)ρ+ (n− 1)(1− ρ)(1 + nρ) 2σ2λ

k

)

n(

1 + (n− 1)ρ+ (1− ρ)(1 + nρ) 2σ2λk

)

immediate observations

• yreact1m increasing in both x2, and y2,

• xreact1m decreasing in x2

• xreact1m is increasing in y2 iff k2σ2λ

<(1−ρ)(1+nρ)

(1− 1n

)δρ. For δ < 0 decreasing in y2. When δ > 0 and

ρ > 0 the RHS of this inequality decreases in δ and ρ, and is u-shaped as a function of n.

• comparing to the reaction function for the case with actions of others taken as given:

xreact1m = xreact1g − ρδ2(

x2 + y2(1 − 1n))

n(

1 + (n− 1)ρ+ (1 − ρ)(1 + nρ) 2σ2λk

)

yreact1m = yreact1g +

(

1 +k

2λσ2

)

δ2(

x2 + y2(1 − 1n))

n(

1 + (n− 1)ρ+ (1− ρ)(1 + nρ) 2σ2λk

)

17

Comment: in the case ρ = 0 the reaction function reduces to:

xreact1m =n+ y2δ

2σ2λk

n(

1 + 2σ2λk

)

= xreact1g

yreact1m =x2δ

(

kδ2λσ2 + n + δ + n2σ2λ

k

)

n(

1 + 2σ2λk

) +y2δ(

k(n−1)δ2λσ2 + (n − 1)(n+ δ) + n+ (n− 1)n2σ2λ

k

)

n2(

1 + 2σ2λk

)

= yreact1g + (1 +k

2λσ2)δ2(

x2 + y2(1 − 1n))

n(

1 + 2σ2λk

) = yreact1g

1 + δ21 + k

2λσ2

n(

1 + 2σ2λk

)


To find the equilibrium contracts we impose the condition x1 = x2 and y1 = y2 to get

Optimal contract is given by

Proposition 5 Let

Rm = (1 − δ)Ro − δ2(1− 1

n)k

2λσ2+δ(1 − δ)(δ − n2ρ)

n


xm =(1 − δρ) + (n− 1)(1− δ)(1 + nρ) − δ2(1− 1

n) k2σ2λ

Rm

ym =n((δ − ρ)− (n− 1)ρ(1 − δ)) + δ2 k

2σ2λ

Rm

am =k(

(1− δ)(n+ δ)(1 + (n− 1)ρ)− δ2(1 − 1−δn

) k2σ2λ

)

Rm

Lemma 7 When δ = 0 the equilibrium coincides with the case where we have one principle with

multiple agents.

Proof. By inspection.

The following can be shown for Rm:

• When n = 1 Rm > 0

• For k2λσ2 ≤ 1, δ <= 2(1−ρ)

3−2ρ is a sufficient condition forRm > 0. Specifically, the sufficient

condition holds for δ ≤ 0.

• holding other parameters fixed, there exist a threshold H > 0 so that Rm > 0 iff k2λσ2 < H

18

The comparative statics for the case n = 1 appears in detail in Lemma 11; in the subsection

that provides the results for two principle agent pairs.

The following following lemma provides characterizations for n ≥ 2.

Comments Regarding Lemma 8:

1. the lemma below provides partial analytical characterizations, some more partial than others,

as in general it is too messy to get full analytical characterizations for this case.

2. the reported results ignore jumps at the discontinuity points

3. unless specified otherwise, thresholds are parameter dependent (i.e., when varying a specific

parameter, the thresholds potentially depend on all the other parameters)

4. keeping in mind that both δ and ρ have a finite support, statements regarding kσ2λ

being

sufficiently small or large, for a given δ or ρ, can typically be extended to hold for the whole

support of δ or ρ.1

Lemma 8 Assume n ≥ 2

1. ym

• varying δ

δ ≥ 0, increasing

δ < 0,

1 > ρ, for each δ there exist a threshold T > 0 such that increasing iff kσ2λ

< T

ρ = 1, decreasing

• varying ρ

δ > 0,

kσ2λ

sufficiently small, decreasing

intermediate levels of kσ2λ

, u-shape

kσ2λ

sufficiently large, increasing

δ < 0, decreasing

1The support of δ is the open interval (−1, 1). In some cases the relevant functions blow up at the boundary ofthe support. Thus, in some cases, one may not extended the local statement regarding k

σ2λto the whole support,

but instead the precise statement should be that it is up to ε from the boundary, where ε can be chosen as small aswe want.

19

• varying σ2, λ, and k

ρ < 1,

δ > nρ1−ρ+nρ , increasing in k

σ2λ

otherwise, u-shaped in kσ2λ

ρ = 1, increasing

2. xm

• varying δ

– given δ for kσ2λ

sufficiently large locally decreasing

– there exist a threshold M(n, ρ) such that for δ > (<)M for kσ2λ

sufficiently small

locally increasing (decreasing)

– ρ > 0: increasing in the neighborhood of δ = 0

• varying ρ

– δ > 0

kσ2λ


kσ2λ

sufficiently small, u-shaped

comment: For δ >n(n−1)

1+n(n−1) there is a cutoff between the two regions above. When

δ <n(n−1)

1+n(n−1) , have not been able to rule out a middle regions where increasing-

decreasing-increasing.

– δ < 0: there exists a cutoff 0 > M(n) > −1 such that

δ < M(n), increasing for small values of ρ

δ > M(n),

kσ2λ

sufficiently small, increasing for small values of ρ

an intermediate range of kσ2λ

, u-shaped

kσ2λ


comment: in the two cases where have shown that increasing for small values of ρ

could either be increasing throughout or increasing-decreasing-increasing; have not

been able to rule out the later thus far or find a numerical example for which it holds.

– for ρ sufficiently large increasing.

• varying σ2, λ, and k

δ < 1− n+ nρ, tent-shaped in kσ2λ

nρ1−ρ+nρ > δ > 1 − n+ nρ, increasing-decreasing-increasing in k

σ2λ

δ > nρ1−ρ+nρ , tent-shaped in k

σ2λ

20

3. am

• varying δ

– ρ > 0:

∗ increasing in the neighborhood of δ = 0

∗ kσ2λ

sufficiently large: increasing.

– ρ:

∗ δ > 0 : exist a threshold M such that decreasing iff kσ2λ

< M

∗ δ < 0 : exist threshold M1 > M2 such that decreasing iff M1 >kσ2λ

> M2

• varying ρ

δ < 0, exist a threshold M such that if kσ2λ

> M increasing, and otherwise u-shaped

δ > 0,

kσ2λ

sufficiently small, increasing

exists a midrange of kσ2λ

, tent-shaped

kσ2λ

sufficiently large, decreasing

comment: For δ > 0 have not been able to rule out a range of kσ2λ

for which increasing-

decreasing-increasing. Although, from running some numerical calculations it seems such

a range does not exist.

For any ρ there exists a threshold such that when kσ2λ

is below the threshold locally

increasing and when above it locally decreasing.


δ < 0, decreasing

δ > 0, tent-shaped

• varying k for kσ2λ

sufficiently small or sufficiently large increasing. comment: numeri-

cally, can produce cases where increasing throughout, as well as cases where increasing-

decreasing-increasing.

4. ym

xm

• varying δ: increasing iff 2n(1− ρ)(1 + nρ) + kσ2λ

δ(2 − δ) > 0

comments: 1)inequality always holds for δ > 0 2)either increasing throughout or tent-

shape.

– ρ = 1 : increasing iff δ > 0

– ρ = 0 : increasing iff 2n+ kσ2λ

δ > 0

21

• varying ρ : increasing iff δ3

2n(1−δ)(n+δ)> k

σ2λ

comment: there exists a cutoff 0 < M < 1, that depends on n and kσ2λ

, such that

increasing iff δ > M .

• varying σ2 λ, and k: increasing in kσ2λ

5. mm:

• varying δ: for any given δ for kσ2λ

sufficiently small or sufficiently large increasing.

• varying ρ:

– ρ < 1 :for any given δ, for kσ2λ

sufficiently small increasing

– ρ = 1 :for any given δ

δ > 0 or n ≥ 3, kσ2λ


n = 2, a cutoff M < 0 such that

δ < M, kσ2λ

sufficiently small decreasing

δ > M, kσ2λ


• varying σ2, and λ: for any given δ, for kσ2λ

sufficiently small decreasing.

• varying k:for any given δ, for kσ2λ

sufficiently small or sufficiently large increasing.

6. principle’s welfare:

• varying δ

– ρ < 1 :

kσ2λ


δ > 1− n√2n−1

, for kσ2λ

sufficiently large decreasing

δ < 1− n√2n−1

, for kσ2λ

sufficiently large increasing

n ≥ 8, for kσ2λ


comment: 0 > 1 − n√2n−1

, and for n ≥ 8 −1 > 1 − n√2n−1

.

– ρ = 1 :

kσ2λ

sufficiently large, same as for ρ < 1 comp-stat

δ > 0 and n ≥ 3, for kσ2λ


• varying ρ

22

– 1 > ρ > 0 :

δ > 12 , for k

σ2λsufficiently small decreasing

δ < 12 , for k

σ2λsufficiently small increasing

δ > −1+2n−√

1−2n+2n3

2n , for kσ2λ

sufficiently large increasing

δ < −1+2n−√

1−2n+2n3

2n , for kσ2λ


comment: 0 > −1+2n−√

1−2n+2n3

2n , is monotone in n and for n > 7 −1 > −1+2n−√

1−2n+2n3

2n

– ρ = 0 :

kσ2λ


kσ2λ

sufficiently large, same as for 1 > ρ > 0 comp-stat

– ρ = 1 :

kσ2λ

sufficiently small, exists an 12 >M > 0 where decreasing iff δ > M

kσ2λ

sufficiently large, same as for 1 > ρ > 0 comp-stat


– ρ < 1 :{

kσ2λ

sufficiently small, decreasing iff δ < 12

– ρ = 1 :

kσ2λ

sufficiently large, decreasing

kσ2λ

sufficiently small, increasing

• varying k

– ρ < 1 :

kσ2λ

sufficiently large, increasing iff δ < n−12n−1

kσ2λ

sufficiently small, increasing iff δ < 12

– ρ = 1 :

kσ2λ

sufficiently large, increasing iff δ < n−12n−1

kσ2λ

sufficiently small, increasing iff (n2δ + 2(−1 + δ)δ2 + n3(−1 + 2δ)) < 0

comment: A sufficient condition for (n2δ + 2(−1 + δ)δ2 + n3(−1 + 2δ)) < 0 is

δ < 0, and for (n2δ + 2(−1 + δ)δ2 + n3(−1 + 2δ)) > 0 is δ > 12 .

For comparative statics with respect to n we can get a partial analytical characterization.

23

Lemma 9 As a function of n, holding other parameters fixed

1. ym:

• for n sufficiently large

ρ > 0, decreasing

ρ = 0, increasing iff k2

4σ4λ2 < (1 − δ)

• for kσ2λ

sufficiently large: increasing

• for kσ2λ

sufficiently small: there exists a threshold ρ∗ > 0 such that increasing iff ρ < ρ∗

2. xm:


ρ > 0, increasing

ρ = 0, decreasing

• for kσ2λ

sufficiently large: increasing iff δ > ρ

• for kσ2λ

sufficiently small: increasing iffnρ(1−δ)

1−ρ > δ > −nρ1+ρ+nρ (comment: at ρ = 0

decreasing)

3. ym

xm:


ρ > 0, decreasing

ρ = 0, increasing iff 1 > k2σ2λ

• for kσ2λ

sufficiently large: increasing

• for kσ2λ

sufficiently small: assuming n ≥ 2 there exists a threshold ρ∗ such that increasing

in n iff ρ < ρ∗

4. am


ρ > 0, increasing iff k2σ2λ

δ < (1− ρ)

ρ = 0, decreasing

• for kσ2λ

sufficiently large: increasing iff δ < 0

24

• for kσ2λ

sufficiently small:

1 > ρ > 0, increasing

ρ = 0, increasing

ρ = 1 and n ≥ 2, decreasing

5. mm


ρ > 0, increasing

ρ = 0, decreasing

• for kσ2λ

sufficiently large, and assuming n ≥ 2 : increasing

• for kσ2λ

sufficiently small, and assuming n ≥ 2:

1 > ρ > 0, increasing iff 1 − 2δ > 0

ρ = 0, decreasing

ρ = 1, increasing

6. principle’s welfare


ρ > 0, increasing

ρ = 0, decreasing

• for kσ2λ

sufficiently large, and assuming n ≥ 2:

ρ > 0, increasing

ρ = 0, decreasing

• for kσ2λ

sufficiently small and assuming n ≥ 2:

ρ > 0, increasing

ρ = 0, increasing

Proof. to be added

25


identical to the n+ 1 principle agent pairs when actions are observable.

1.4.4 wi = mi + ui(qi − q) + viq

um = xm

vm = xm + ym =n(n+ δ)(1− ρ) + kδ2

2σ2λ

Rm


The following can be shown:

• comparing Rm to Ro

– δ > 0 ⇒ Rm < Ro

– for n = 1: δ < 0 ⇒ Rm > Ro

– for n > 1 and δ < 0: k2λσ2 <

12 ⇒ Rm > Ro, when k

2λσ2 sufficiently large Rm < Ro

• comparing Rm to Rg

– Rm = Rg − δ2((

1 − 1n

) (

1 + k2λσ2

)

+(

δn− ρ))

– Rm < Rg iff δ > nρ− (n− 1)(

1 + k2λσ2

)

∗ for n = 1 Rm < Rg iff δ > ρ

The optimal equilibrium contract can also be written as

xm =Ro

Rmxo −

δ2(1 − 1n ) k

2σ2λ

Rm=Rg

Rmxg −

δ2(1− 1n) k

2σ2λ

Rm

ym =Ro

Rmyo +

δ2 k2σ2λ

Rm=

Rg

Rmyg +

δ2 k2σ2λ

Rm

am =Ro

Rmao −

k(

δ2(1− 1−δn

) k2σ2λ

)

Rm=Rg

Rmag −

k(

δ2(1 − 1−δn

) k2σ2λ

)

Rm

Lemma 10

1. ym > yg iff Rm < 0

2. xm > xg iff δ(1− ρ) < nρ(1 − δ) and Rm > 0, or δ(1 − ρ) > nρ(1− δ) and Rm < 0

26

3. am < ag iff −δ(1−δ)(n+δ)ρ

> 2σ2λk

and Rm > 0, or −δ(1−δ)(n+δ)ρ

< 2σ2λk

and Rm < 0 (comment:

note that for δ > 0 am < ag iff Rm < 0)

4. ym

xmvs

yg

xg

• 2σ2λk

< n−1n

12n2ρ+(2n−1)(1−ρ) : there exists a cutoffs −1 < δ1 < 0 < δ2 such that ym

xm>

yg

xg

iff δ1 < δ < δ2

• n−1n

12n2ρ+(2n−1)(1−ρ) <

2σ2λk

< n−1n

11−ρ : there exists a cutoff 0 < δ2 such that ym

xm>

yg

xg

iff δ < δ2

• n−1n

11−ρ <

2σ2λk : ym

xm>

yg

xg

5. mm vs mg: when kσ2λ

sufficiently small

n >= 3, mm > mg

n = 2 and δ > 2(1−√

11)5 , mm > mg

n = 2 and δ <2(1−

√11)

5 , a cutoff ρ∗ such that mm > mgiffρ > ρ∗

n = 1, mm > mg iff δ > 0

when kσ2λ

sufficiently large mm < mg.

(comment: messy to try to get analytical characterization for the whole parameter space)

6. Principle’s welfare per-agent hired: for kσ2λ

sufficiently large Welfarem < Welfareg, and

when it is sufficiently small Welfarem < Welfareg iff δ < 0;


27

2 Two Agents

The solution for the case with two agents is obtained by setting n = 1

2.1 Single Principle with Two Agents

Corollary 4 Let

Ro = (1 + δ)

(

1 + (1 − ρ2)2σ2λ

k

)


xo =1− δρ

Ro

yo =δ − ρ

Ro

ao =k(1− δ2)

Ro

Comment: note that the condition δ < nρ1+(n−1)ρ in Lemma 1 becomes δ < ρ, for n = 1.

2.1.1 wi = mi + ui(qi − qj) + viqj

uo = xo =1− δρ

Ro

vo = xo + yo =(1 + δ)(1− ρ)

Ro= uo +

δ − ρ

Ro

28

2.2 2 Principle-Agent Pairs when actions of other agents are taken as given


Corollary 5 Taking the contract of all other agents as given, the optimal contract that principle

1 gives to agent 1 is given by

xreact1g =1 + y2δ(1− ρ2) 2σ2λ

k

1 + (1− ρ2) 2σ2λk

yreact1g = − ρ

1 + (1 − ρ2) 2σ2λk

+ x2δ + y2δ

(

ρ

1 + (1 − ρ2) 2σ2λk

)


Corollary 6 Let

Rg = (1 − δ)Ro + δ(δ − ρ)


xg =1 − δρ

Rg

yg =δ − ρ

Rg

ag =k(1 − δ2)

Rg

Proposition 6 The equilibrium is unique.

Proof. The reaction function in Corollary 5 can be written as v1 = D+Av2, where vi = (xi, yi)T .

Any equilibrium must satisfy v2 = D+A(D+Av2) or equivalently D+AD = (I−A2)v2. A direct

calculation of the relevant determinants shows thatI −A2 is non-singular.

29

2.3 Two Principle-Agent Pairs


Corollary 7 Taking the contract of agent 2 as given, the optimal contract that principle 1 gives to

agent 1 is given by

xreact1m =1 − x2δ

2ρ+ y2δ(1 − ρ2) 2σ2λk

1 + (1 − ρ2) 2σ2λk

= xreact1g − ρx2δ

2

1 + (1 − ρ2) 2σ2λk

yreact1m =−ρ+ x2δ

(

δ k2σ2λ

+ (1 + δ))

+ y2δρ

1 + (1 − ρ2) 2σ2λk

= yreact1g +

(

1 +k

2σ2λ

)

x2δ2

1 + (1− ρ2) 2σ2λk

immediate observations

• yreact1m for ρ > 0 increasing x2 and y2; for ρ = 0 constant in x2 and increasing in y2.

• xreact1m for ρ > 0 decreasing in x2 and increasing in y2; for ρ = 0 decreasing in x2 and constant

in y2.

• comparing to the reaction function for the case with actions of others taken as given:

xreact1m = xreact1g − δ2ρx2

(

1 + (1− ρ2) 2σ2λk

)

yreact1m = yreact1g + δ2x2(1 + k

2λσ2 )(

1 + (1 − ρ2) 2σ2λk

)

in contrast to the case with n > 1, the deviations from xreact1g and yreact1g do not depend on y2.

Observe that when δ = 0 the reaction contract does not depend on the other agent’s contract;

when ρ = 0 the reaction xreact1m depends on y2, but not on x2; when ρ = 1 the reaction xreact1m depends

on x2, but not on y2; if both δ = 0 and ρ = 0 then yreact1m = 0.


Corollary 8 Let

Rm = (1− δ)Ro + δ(1− δ)(δ − ρ)

30


xm =1 − δρ

Rm

ym =

(

δ − ρ+ kδ2

2σ2λ

)

Rm

am =k(

1 − δ2 − δ kδ2

2σ2λ

)

Rm

Proposition 7 The equilibrium is unique.

Proof. Similar to the proof of Proposition 6, with a bit more work required to show that the

relevant matrix is non-singular.

Note that δ < 1 ⇒ Rm > 0

Lemma 11 Let 1 > δ∗1 be the solution to δ3

1−δ2 = 2σ2λk

, δ∗2 be the solution to δ3

1+δ = 2σ2λk

+(

2σ2λk

)2,(

comment: both δ∗1 and δ∗2 are increasing in 2σ2λk

)

1. ym

• ym < 0 iff σ2λk

(

−1 −√

1 + 2ρkσ2λ

)

< δ < σ2λk

(

−1 +√

1 + 2ρkσ2λ

)

< 1

(comment:σ2λk

(

−1 −√

1 + 2ρkσ2λ

)

> −1 iff 12(1+ρ) >

σ2λk

)

• varying δ:

– 1 > ρ:

∗ there exists a δ < 0 such that for δ > δ increases;

∗ when σ2λk< 1 decreasing for δ < −1

22726 ;

∗ There exists a cutoff, that depends on ρ, so that for kσ2λ

below the cutoff increas-

ing throughout.

∗ ρ = 0: if σ2λk> 1 increasing throughout.

– ρ = 1: u-shaped (minimum at δ = 0)

• varying ρ

δ ≤ δ∗1, decreasing

δ∗1 < δ < Min[1, δ∗2], u-shapaed

otherwise, increasing

• varying σ, λ, and k:

– δ > ρ: increasing in σ and λ, and decreasing in k.

31

– 1 > ρ > δ: u-shaped in σ and λ, and tent-shaped in k

– ρ = 1: decreasing in σ and λ, and increasing in k.

2. xm

• xm > 0

• varying δ:

– 1 > ρ

k2σ2λ

<2(1−ρ2)

3+ρ , u-shaped

otherwise, increasing-decreasing-increasing

– ρ = 1 : tent-shaped

(comment: at δ = 0 decreasing when ρ > 0 and flat when ρ = 0)

• varying ρ

0 > δ > −2σ2λk, increasing

δ > 0 or − 2σ2λk > δ,

δ2

(2(1−δ)) >2σ2λk , decreasing

otherwise, u-shape

• decreasing in σ and λ, and increasing in k

3. xm − ym

Let δ1 = (1 + ρ)σ2λk

(

−1 −√

1 + 21+ρ

kσ2λ

)

, δ2 = (1 + ρ)σ2λk

(

−1 +√

1 + 21+ρ

kσ2λ

)

< 1 then

xm − ym > 0 iff δ1 < δ < δ2. (comment: δ1 < −1 iff 14(1+ρ) >

σ2λk

)

4. ym

xm

• varying δ: if k2σ2λ

> 1−ρ22+ρ increasing, otherwise u-shaped (comment: when u-shaped the

minimum occurs a negative δ, apart for where ρ = 0 in which case it occurs at δ = 0)

• varying ρ:

δ < δ∗1, decreasing

δ = δ∗1, independent

otherwise, increasing

• For δ 6= 0 decreasing in σ, and λ and increasing in k; at δ = 0 independent of σ, λ and

k.

5. am

• am > 0 iff δ < δ∗1

32

• varying δ: tent-shaped

• varying ρ

δ < 0,

δ2(1+δ)

k2σ2λ

< −1, decreasing

otherwise, tent-shaped

δ > 0,

δ < δ∗1, increasing

δ∗1 < δ, decreasing

• varying σ and λ: if δ ≤ 0 decreasing, otherwise, tent-shaped.

• varying k: if δ ≤ 0 increasing, otherwise tent-shaped.

(note that for δ > 0 tent-shaped both in k and in σ and λ)

6. mm(seems too messy to get a comprehensive analytical characterization)

• varying δ: for a given δ, for kσ2λ

sufficiently large or sufficiently small locally increasing

in δ.

• varying ρ: for a given ρ, for kσ2λ

sufficiently large or sufficiently small locally increasing

in ρ.

• varying σ, λ: for σ (λ) sufficiently small if δ > 0 (δ < 0) decreasing (increasing), and

when sufficiently large decreasing.

• varying k: for k sufficiently close to zero increasing and sufficiently large if δ > 0 (δ < 0)

increasing (decreasing)

7. principle’s welfare (seems too messy to get a comprehensive analytical characteri-

zation)

• varying δ: for a given δ, for kσ2λ

sufficiently small decreasing in δ and for k sufficiently

large if δ > 0 (δ < 0) locally decreasing (increasing) in δ.

• varying ρ: for a given ρ, for kσ2λ

sufficiently small increasing in ρ iff 1− 2δ < 0, for kσ2λ

sufficiently large locally increasing in ρ.

• varying σ and λ: for σ (λ) sufficiently small increasing, and when σ (λ) sufficiently

large decreasing iff 1 − 2δ > 0.

• varying k: for k sufficiently close to zero increasing iff 1− 2δ > 0 and for k sufficiently

large decreasing.

Proof. To be added.

Note that when δ = 0 ym < 0 and am > 0. Also, when ρ = 0 ym > 0.

33

2.3.3 wi = mi + ui(qi − qj) + viqj

um = xm =1− δρ

Rm

vm = xm + ym =

(

(1 + δ)(1− ρ) + kδ2

2σ2λ

)

Rm


comparing the equilibrium contracts in the case with one principle and two agents to the case with

two principle agent pairs we get

Lemma 12

1. ym vs yo

• δ < 0: ym > yo

• δ > 0:

– kσ2λ

sufficiently large, holding other parameters fixed, ym > yo

– kσ2λ

sufficiently small holding other parameters fixed, ym > yo iff δ > ρ

– for δ sufficiently large, holding other parameters fixed, ym > yo

– ρ = 0: ym > yo

– ρ = 1 : there exist a threshold δ > 0 such that ym > yo iff δ > δ

2. xm > xo iff δ > 0

3. am vs ao

• δ > 0:there exist a threshold ¯δ > 0 such that am > ao iff δ <¯δ

• Taking other parameters as fixed, there exists a threshold T such that am > ao iff kσ2λ

< T

• δ > 0: a sufficient condition for am < aoiskσ2λ

< 1

4. ym

xm> yo

xo

5. mm vs mo: when kσ2λ

sufficiently small mm−mo > 0 and when sufficiently large mm−mo has

the same sign as δ. (comment: messy to try to get characterization for the whole parameter

space)

34


sufficiently small or sufficiently large or if δ

sufficiently negative then welfare is higher when one principle hires all agents, as opposed to

n + 1 principle-agent pairs. (comment: numerically it seems that this is always true, but

analytically it is very messy to try to prove for the whole parameter space)


comparing the equilibrium contracts between observable and non-observable contracts we get

Lemma 13

1. ym > yg

2. xm > xg iff δ > ρ

3. am < ag

4. ym

xm>

yg

xg

5. mm vs mg: when kσ2λ

sufficiently large mm > mg iff δ > 0, and when sufficiently small

mm < mg. (comment: messy to try to get characterization for the whole parameter space)


sufficiently large Welfarem < Welfareg, and

when it is sufficiently small Welfarem < Welfareg iff δ < 0;


35

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37

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