Moral Hazard

Julio Davila

2009

Julio Davila Moral Hazard

Principal-agent problem

1 two individuals: a ”principal” and an ”agent”

2 the principal gets a surplus π if the agent makes some effort e

3 π is unknown and its density f (π, e) depends on the effort e

4 e ∈ {e, e} with e < e, and F (π, e) < F (π, e) for all π

so that ∫E (π | e) < E (π | e)

∫

Julio Davila Moral Hazard

Principal-agent problem

1 two individuals: a ”principal” and an ”agent”

2 the principal gets a surplus π if the agent makes some effort e

3 π is unknown and its density f (π, e) depends on the effort e

4 e ∈ {e, e} with e < e, and F (π, e) < F (π, e) for all π

so that ∫E (π | e) < E (π | e)

∫

Julio Davila Moral Hazard

Principal-agent problem

1 two individuals: a ”principal” and an ”agent”

2 the principal gets a surplus π if the agent makes some effort e

3 π is unknown and its density f (π, e) depends on the effort e

4 e ∈ {e, e} with e < e, and F (π, e) < F (π, e) for all π

so that ∫E (π | e) < E (π | e)

∫

Julio Davila Moral Hazard

Principal-agent problem

1 two individuals: a ”principal” and an ”agent”

2 the principal gets a surplus π if the agent makes some effort e

3 π is unknown and its density f (π, e) depends on the effort e

4 e ∈ {e, e} with e < e, and F (π, e) < F (π, e) for all π

so that ∫E (π | e) < E (π | e)

∫

Julio Davila Moral Hazard

Principal-agent problem

1 two individuals: a ”principal” and an ”agent”

2 the principal gets a surplus π if the agent makes some effort e

3 π is unknown and its density f (π, e) depends on the effort e

4 e ∈ {e, e} with e < e, and F (π, e) < F (π, e) for all π

so that ∫E (π | e) < E (π | e)

∫

Julio Davila Moral Hazard

Principal-agent problem

1 two individuals: a ”principal” and an ”agent”

2 the principal gets a surplus π if the agent makes some effort e

3 π is unknown and its density f (π, e) depends on the effort e

4 e ∈ {e, e} with e < e, and F (π, e) < F (π, e) for all π

so that ∫πf (π, e)dπ <

∫πf (π, e)dπ

Julio Davila Moral Hazard

Principal-agent problem

1 two individuals: a ”principal” and an ”agent”

2 the principal gets a surplus π if the agent makes some effort e

3 π is unknown and its density f (π, e) depends on the effort e

4 e ∈ {e, e} with e < e, and F (π, e) < F (π, e) for all π

so that

πF (π, e)

]ππ

+

∫F (π, e)dπ <

∫F (π, e)dπ + πF (π, e)

]ππ

Julio Davila Moral Hazard

Principal-agent problem

1 two individuals: a ”principal” and an ”agent”

2 the principal gets a surplus π if the agent makes some effort e

3 π is unknown and its density f (π, e) depends on the effort e

4 e ∈ {e, e} with e < e, and F (π, e) < F (π, e) for all π

so that

πF (π, e)

]ππ

+

∫F (π, e)dπ <

∫F (π, e)dπ+πF (π, e)

]ππ

Julio Davila Moral Hazard

Principal-agent problem

1 two individuals: a ”principal” and an ”agent”

2 the principal gets a surplus π if the agent makes some effort e

3 π is unknown and its density f (π, e) depends on the effort e

4 e ∈ {e, e} with e < e, and F (π, e) < F (π, e) for all π

so that ∫E (π | e) < E (π | e)

∫

I the principal wants the agent to choose e

I the agent wants to choose e

what would the principal offer to compensate the agent forchoosing e instead?

Julio Davila Moral Hazard

Principal-agent problem

1 two individuals: a ”principal” and an ”agent”

2 the principal gets a surplus π if the agent makes some effort e

3 π is unknown and its density f (π, e) depends on the effort e

4 e ∈ {e, e} with e < e, and F (π, e) < F (π, e) for all π

so that ∫E (π | e) < E (π | e)

∫I the principal wants the agent to choose e

I the agent wants to choose e

what would the principal offer to compensate the agent forchoosing e instead?

Julio Davila Moral Hazard

Principal-agent problem

1 two individuals: a ”principal” and an ”agent”

2 the principal gets a surplus π if the agent makes some effort e

3 π is unknown and its density f (π, e) depends on the effort e

4 e ∈ {e, e} with e < e, and F (π, e) < F (π, e) for all π

so that ∫E (π | e) < E (π | e)

∫I the principal wants the agent to choose e

I the agent wants to choose e

what would the principal offer to compensate the agent forchoosing e instead?

Julio Davila Moral Hazard

Principal-agent problem

1 two individuals: a ”principal” and an ”agent”

2 the principal gets a surplus π if the agent makes some effort e

3 π is unknown and its density f (π, e) depends on the effort e

4 e ∈ {e, e} with e < e, and F (π, e) < F (π, e) for all π

so that ∫E (π | e) < E (π | e)

∫I the principal wants the agent to choose e

I the agent wants to choose e

what would the principal offer to compensate the agent forchoosing e instead?

Julio Davila Moral Hazard

Principal-agent problem: observable effort

if both π, e can be observed

max maxe,w

∫[π − w(π, e)]f (π, e)dπ∫

u(w(π, e))f (π, e)dπ ≥ e

FOC−f (π, e) + λeu

′(w(π, e))f (π, e) = 0

Julio Davila Moral Hazard

Principal-agent problem: observable effort

if both π, e can be observed

maxe

maxw

∫[π − w(π, e)]f (π, e)dπ∫

u(w(π, e))f (π, e)dπ ≥ e

FOC−f (π, e) + λeu

′(w(π, e))f (π, e) = 0

Julio Davila Moral Hazard

Principal-agent problem: observable effort

if both π, e can be observed, for each e

maxe

maxw

∫[π − w(π, e)]f (π, e)dπ∫

u(w(π, e))f (π, e)dπ ≥ e

FOC−f (π, e) + λeu

′(w(π, e))f (π, e) = 0

Julio Davila Moral Hazard

Principal-agent problem: observable effort

if both π, e can be observed, for each e

maxe

maxw

∫[π − w(π, e)]f (π, e)dπ∫

u(w(π, e))f (π, e)dπ ≥ e

FOC−f (π, e) + λeu

′(w(π, e))f (π, e) = 0

Julio Davila Moral Hazard

Principal-agent problem: observable effort

if both π, e can be observed, for each e

maxe

maxw

∫[π − w(π, e)]f (π, e)dπ∫

u(w(π, e))f (π, e)dπ ≥ e

FOC1

λe= u′(w(π, e))

Julio Davila Moral Hazard

Principal-agent problem: observable effort

if both π, e can be observed, for each e

maxe

maxw

∫[π − w(π, e)]f (π, e)dπ∫

u(w(π, e))f (π, e)dπ ≥ e

FOC1

λe= u′(w(π, e))

Julio Davila Moral Hazard

Principal-agent problem: observable effort

if both π, e can be observed, for each e

maxe

maxw

∫[π − w(π, e)]f (π, e)dπ∫

u(w(π, e))f (π, e)dπ ≥ e

FOC1

λe= u′(w(π, e))

Julio Davila Moral Hazard

Principal-agent problem: observable effort

if both π, e can be observed, for each e

maxe

maxw

∫[π − w(π, e)]f (π, e)dπ∫

u(w(π, e))f (π, e)dπ ≥ e

FOC1

λe= u′(w(π, e))

Julio Davila Moral Hazard

Principal-agent problem: observable effort

if both π, e can be observed, for each e

maxe

maxw

∫[π − w(π, e)]f (π, e)dπ∫

u(w(π, e))f (π, e)dπ = e

FOC1

λe= u′(w(π, e))

Julio Davila Moral Hazard

Principal-agent problem: observable effort

if both π, e can be observed, for each e

maxe

maxw

∫[π − w(π, e)]f (π, e)dπ

u(w(e))

∫f (π, e)dπ = e

FOC1

λe= u′(w(π, e))

Julio Davila Moral Hazard

Principal-agent problem: observable effort

if both π, e can be observed, for each e

maxe

maxw

∫[π − w(π, e)]f (π, e)dπ∫

u(w(e)) = e

FOC1

λe= u′(w(π, e))

Julio Davila Moral Hazard

Principal-agent problem: observable effort

if both π, e can be observed, for each e

maxe

maxw

∫[π − w(π, e)]f (π, e)dπ∫

w(e) = u−1(e)

FOC1

λe= u′(w(π, e))

Julio Davila Moral Hazard

Principal-agent problem: observable effort

if both π, e can be observed

maxe

maxw

∫[π − u−1(e)]f (π, e)dπ∫

w(e) = u−1(v(e))

FOC1

λe= u′(w(π, e))

Julio Davila Moral Hazard

Principal-agent problem: observable effort

if both π, e can be observed

maxe

maxw

{∫πf (π, e)dπ − u−1(e)

}∫

w(e) = u−1(v(e))

FOC1

λe= u′(w(π, e))

Julio Davila Moral Hazard

Principal-agent problem: observable effort

if both π, e can be observed

e∗ = arg maxe

{∫πf (π, e)dπ − u−1(e)

}∫

w∗ = u−1(e∗)

FOC1

λe= u′(w(π, e))

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed

max maxe,w

∫[π − w(π, e)]f (π, e)dπ∫

u(w(π, e))f (π, e)dπ ≥ e

FOC1

λe= u′(w(π, e))

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed

max maxe,w

∫[π − w(π, e)]f (π, e)dπ∫

u(w(π, e))f (π, e)dπ ≥ e

FOC1

λe= u′(w(π, e))

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed

maxe

maxw

∫[π − w(π, e)]f (π, e)dπ∫

u(w(π, e))f (π, e)dπ ≥ e

FOC−f (π, e) + λeu

′(w(π, e))f (π, e) = 0

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxe

maxw

∫[π − w(π, e)]f (π, e)dπ∫

u(w(π, e))f (π, e)dπ ≥ e

FOC−f (π, e) + λeu

′(w(π, e))f (π, e) = 0

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxe

maxw

∫[π − w(π, e)]f (π, e)dπ∫

u(w(π, e))f (π, e)dπ ≥ e

FOC−f (π, e) + λeu

′(w(π, e))f (π, e) = 0

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxe

maxw

∫[π − w(π, e)]f (π, e)dπ∫

u(w(π, e))f (π, e)dπ ≥ e

FOC1

λe= u′(w(π, e))!!!

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxe

maxw

∫[π − w(π, e)]f (π, e)dπ∫

u(w(π, e))f (π, e)dπ ≥ e

FOC1

λe= u′(w(π, e))!!!

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxe

maxw

∫[π − w(π, e)]f (π, e)dπ∫

u(w(π, e))f (π, e)dπ ≥ e

FOC1

λe= u′(w(π, e))!!!

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxe

maxw

∫[π − w(π, e)]f (π, e)dπ∫

u(w(π, e))f (π, e)dπ ≥ e

FOC1

λe= u′(w(π, e))!!!

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxe

maxw

∫[π − w(π, e)]f (π, e)dπ

u(w)

∫f (π, e)dπ = e

FOC1

λe= u′(w(π, e))

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxe

maxw

∫[π − w(π, e)]f (π, e)dπ∫

u(w) = e

FOC1

λe= u′(w(π, e))

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxe

maxw

∫[π − w(π, e)]f (π, e)dπ∫

w = u−1(e)

FOC1

λe= u′(w(π, e))

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxw

∫[π − w(π)]f (π, e)dπ∫

u(w(π))f (π, e)dπ ≥ e

e ∈ arg maxe′

{∫u(w(π))f (π, e ′)dπ − e ′

}

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxw

∫[π − w(π)]f (π, e)dπ∫

u(w(π))f (π, e)dπ ≥ e

e ∈ arg maxe′

{∫u(w(π))f (π, e ′)dπ − e ′

}

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxw

∫[π − w(π)]f (π, e)dπ∫

u(w(π))f (π, e)dπ ≥ e

∫u(w(π))f (π, e)dπ − e ≥

∫u(w(π))f (π, e ′)dπ − e ′

FOC

f (π, e) = λeu′(w(π))f (π, e) + µeu

′(w(π))[f (π, e)− f (π, e ′)

]

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxw

∫[π − w(π)]f (π, e)dπ∫

u(w(π))f (π, e)dπ ≥ e

∫u(w(π))f (π, e)dπ − e ≥

∫u(w(π))f (π, e ′)dπ − e ′

FOC

f (π, e) = λeu′(w(π))f (π, e) + µeu

′(w(π))[f (π, e)− f (π, e ′)

]

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxw

∫[π − w(π)]f (π, e)dπ∫

u(w(π))f (π, e)dπ ≥ e

∫u(w(π))f (π, e)dπ − e ≥

∫u(w(π))f (π, e ′)dπ − e ′

FOC1

u′(w(π))= λe + µe

[1− f (π, e ′)

f (π, e)

]

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxw

∫[π − w(π)]f (π, e)dπ∫

u(w(π))f (π, e)dπ ≥ e

∫u(w(π))f (π, e)dπ − e ≥

∫u(w(π))f (π, e ′)dπ − e ′

FOC1

u′(w(π))= λe + µe

[1− f (π, e ′)

f (π, e)

]with λe = 0?

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxw

∫[π − w(π)]f (π, e)dπ∫

u(w(π))f (π, e)dπ ≥ e

∫u(w(π))f (π, e)dπ − e ≥

∫u(w(π))f (π, e ′)dπ − e ′

FOC1

u′(w(π))= λe+ µe

[1− f (π, e ′)

f (π, e)

]with λe = 0?

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

let ∫f (π, e)dπ = 1 =

∫f (π, e ′)dπ

assume, for all π,f (π, e) ≥ f (π, e ′)

and for some π ∈ [π′, π′],

f (π, e) > f (π, e ′)

then ∫[f (π, e)− f (π, e ′)]dπ = 0

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

let ∫f (π, e)dπ = 1 =

∫f (π, e ′)dπ

assume, for all π,f (π, e) ≥ f (π, e ′)

and for some π ∈ [π′, π′],

f (π, e) > f (π, e ′)

then ∫[f (π, e)− f (π, e ′)]dπ = 0

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

let ∫f (π, e)dπ = 1 =

∫f (π, e ′)dπ

assume, for all π,f (π, e) ≥ f (π, e ′)

and for some π ∈ [π′, π′],

f (π, e) > f (π, e ′)

then ∫[f (π, e)− f (π, e ′)]dπ = 0

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

let ∫f (π, e)dπ = 1 =

∫f (π, e ′)dπ

assume, for all π,f (π, e) ≥ f (π, e ′)

and for some π ∈ [π′, π′],

f (π, e) > f (π, e ′)

then∫ π′

...[f (π, e)− f (π, e ′)]dπ +

∫ π′

π′[f (π, e)− f (π, e ′)]dπ+∫ ...

π′[f (π, e)− f (π, e ′)]dπ = 0

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

let ∫f (π, e)dπ = 1 =

∫f (π, e ′)dπ

assume, for all π,f (π, e) ≥ f (π, e ′)

and for some π ∈ [π′, π′],

f (π, e) > f (π, e ′)

then∫ π′

...[f (π, e)− f (π, e ′)]dπ +

∫ π′

π′[f (π, e)− f (π, e ′)]dπ+∫ ...

π′[f (π, e)− f (π, e ′)]dπ>0!!

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

let ∫f (π, e)dπ = 1 =

∫f (π, e ′)dπ

then, for almost all π,f (π, e) = f (π, e ′)

and henceE (π | e) = E (π | e ′)!!

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

let ∫f (π, e)dπ = 1 =

∫f (π, e ′)dπ

then, for almost all π,f (π, e) = f (π, e ′)

and henceE (π | e) = E (π | e ′)!!

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

let ∫f (π, e)dπ = 1 =

∫f (π, e ′)dπ

then, for all π in some [π′, π′],

f (π, e) < f (π, e ′)

and henceE (π | e) = E (π | e ′)!!

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxw

∫[π − w(π)]f (π, e)dπ∫

u(w(π))f (π, e)dπ ≥ e

∫u(w(π))f (π, e)dπ − e ≥

∫u(w(π))f (π, e ′)dπ − e ′

FOC1

u′(w(π))= λe+ µe

[1− f (π, e ′)

f (π, e)

]with λe = 0?

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxw

∫[π − w(π)]f (π, e)dπ∫

u(w(π))f (π, e)dπ ≥ e

∫u(w(π))f (π, e)dπ − e ≥

∫u(w(π))f (π, e ′)dπ − e ′

FOC1

u′(w(π))<0, for all π in some [π′, π′]!!

with λe = 0?

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxw

∫[π − w(π)]f (π, e)dπ∫

u(w(π))f (π, e)dπ ≥ e

∫u(w(π))f (π, e)dπ − e ≥

∫u(w(π))f (π, e ′)dπ − e ′

FOC1

u′(w(π))= λe + µe

[1− f (π, e ′)

f (π, e)

]with λe > 0

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxw

∫[π − w(π)]f (π, e)dπ∫

u(w(π))f (π, e)dπ ≥ e

∫u(w(π))f (π, e)dπ − e ≥

∫u(w(π))f (π, e ′)dπ − e ′

FOC1

u′(w(π))= λe + µe

[1− f (π, e ′)

f (π, e)

]with λe > 0, and µe = 0?

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxw

∫[π − w(π)]f (π, e)dπ∫

u(w(π))f (π, e)dπ ≥ e

∫u(w(π))f (π, e)dπ − e ≥

∫u(w(π))f (π, e ′)dπ − e ′

FOC1

u′(w(π))= λe +µe

[1− f (π, e ′)

f (π, e)

]with λe > 0, and µe = 0?

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxw

∫[π − w(π)]f (π, e)dπ∫

u(w(π))f (π, e)dπ ≥ e

∫u(w(π))f (π, e)dπ − e ≥

∫u(w(π))f (π, e ′)dπ − e ′

FOC1

u′(w(π))= λe +µe

[1− f (π, e ′)

f (π, e)

]with λe > 0, and µe = 0?

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxw

∫[π − w(π)]f (π, e)dπ∫

u(w(π))f (π, e)dπ ≥ e

∫u(w(π))f (π, e)dπ − e ≥

∫u(w(π))f (π, e ′)dπ − e ′

FOC1

u′(w(π))= λe +µe

[1− f (π, e ′)

f (π, e)

]with λe > 0, and µe = 0?

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxw

∫[π − w(π)]f (π, e)dπ∫

u(w(π))f (π, e)dπ ≥ e

u(w(π))

∫f (π, e)dπ − e ≥ u(w(π))

∫f (π, e ′)dπ − e ′

FOC1

u′(w(π))= λe +µe

[1− f (π, e ′)

f (π, e)

]with λe > 0, and µe = 0?

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxw

∫[π − w(π)]f (π, e)dπ∫

u(w(π))f (π, e)dπ ≥ e

∫u(w(π))− e ≥ u(w(π))− e ′

∫FOC

1

u′(w(π))= λe +µe

[1− f (π, e ′)

f (π, e)

]with λe > 0, and µe = 0?

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxw

∫[π − w(π)]f (π, e)dπ∫

u(w(π))f (π, e)dπ ≥ e

∫− e ≥ −e ′

∫FOC

1

u′(w(π))= λe +µe

[1− f (π, e ′)

f (π, e)

]with λe > 0, and µe = 0?

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxw

∫[π − w(π)]f (π, e)dπ∫

u(w(π))f (π, e)dπ ≥ e

∫e ≤ e ′

∫FOC

1

u′(w(π))= λe +µe

[1− f (π, e ′)

f (π, e)

]with λe > 0, and µe = 0?

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxw

∫[π − w(π)]f (π, e)dπ∫

u(w(π))f (π, e)dπ ≥ e

∫e ≤ e

∫FOC

1

u′(w(π))= λe +µe

[1− f (π, e ′)

f (π, e)

]with λe > 0, and µe = 0

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxw

∫[π − w(π)]f (π, e)dπ∫

u(w(π))f (π, e)dπ = e

∫e ≤ e

FOC1

u′(w(π))= λe +µe

[1− f (π, e ′)

f (π, e)

]with λe > 0, and µe = 0

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxw

∫[π − w(π)]f (π, e)dπ∫

w = u−1(e)

∫e ≤ e

FOC1

u′(w(π))= λe +µe

[1− f (π, e ′)

f (π, e)

]with λe > 0, and µe = 0

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxw

∫[π − w(π)]f (π, e)dπ∫

u(w(π))f (π, e)dπ ≥ e

∫e ≤ e

∫FOC

1

u′(w(π))= λe +µe

[1− f (π, e ′)

f (π, e)

]with λe > 0, and µe = 0

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

maxw

∫[π − w(π)]f (π, e)dπ∫

u(w(π))f (π, e)dπ ≥ e

∫u(w(π))f (π, e)dπ − e ≥

∫u(w(π))f (π, e)dπ − e

FOC1

u′(w(π))= λe + µe

[1− f (π, e)

f (π, e)

]with λe > 0, and µe > 0

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

let π, π′ be such that π′ is relatively likelier if e

λe + µe

[1− f (π′, e)

f (π′, e)

]<[1− f (π, e)

f (π, e)

]µe + λe

FOC1

u′(w(π))= λe + µe

[1− f (π, e)

f (π, e)

]with λe > 0, and µe > 0

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

let π, π′ be such that π′ is relatively likelier if e

λe + µe

[1− f (π′, e)

f (π′, e)

]<[1− f (π, e)

f (π, e)

]µe + λe

FOC1

u′(w(π))= λe + µe

[1− f (π, e)

f (π, e)

]with λe > 0, and µe > 0

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

let π, π′ be such that π′ is relatively likelier if e, i.e.

λe + µe

[1− f (π′, e)

f (π′, e)

]<[1− f (π, e)

f (π, e)

]µe + λe

FOC1

u′(w(π))= λe + µe

[1− f (π, e)

f (π, e)

]with λe > 0, and µe > 0

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

let π, π′ be such that π′ is relatively likelier if e, i.e.

λe + µe

[1− f (π′, e)

f (π′, e)

]>[1− f (π, e)

f (π, e)

]µe + λe

FOC1

u′(w(π))= λe + µe

[1− f (π, e)

f (π, e)

]with λe > 0, and µe > 0

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

let π, π′ be such that π′ is relatively likelier if e, i.e.

λe + µe

[1− f (π′, e)

f (π′, e)

]>[1− f (π, e)

f (π, e)

]µe + λe

FOC1

u′(w(π))= λe + µe

[1− f (π, e)

f (π, e)

]with λe > 0, and µe > 0

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

let π, π′ be such that π′ is relatively likelier if e, i.e.

λe+µe

[1− f (π′, e)

f (π′, e)

]>[1− f (π, e)

f (π, e)

]µe+λe

FOC1

u′(w(π))= λe + µe

[1− f (π, e)

f (π, e)

]with λe > 0, and µe > 0

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

let π, π′ be such that π′ is relatively likelier if e, i.e.

λe + µe

[1− f (π′, e)

f (π′, e)

]>[1− f (π, e)

f (π, e)

]µe + λe

FOC1

u′(w(π))= λe + µe

[1− f (π, e)

f (π, e)

]with λe > 0, and µe > 0

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

let π, π′ be such that π′ is relatively likelier if e, i.e.

1

u′(w(π′))>

1

u′(w(π))

FOC1

u′(w(π))= λe + µe

[1− f (π, e)

f (π, e)

]with λe > 0, and µe > 0

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

let π, π′ be such that π′ is relatively likelier if e, i.e.

1

u′(w(π′))u′(w(π)) > u′(w(π′))

1

u′(w(π))

FOC1

u′(w(π))= λe + µe

[1− f (π, e)

f (π, e)

]with λe > 0, and µe > 0

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

if only π, e can be observed, for each e

let π, π′ be such that π′ is relatively likelier if e, then

1

u′(w(π′))u′(w(π)) < u′(w(π′))

1

u′(w(π))

FOC1

u′(w(π))= λe + µe

[1− f (π, e)

f (π, e)

]with λe > 0, and µe > 0

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

• if only π, e can be observed, for each e

let π, π′ be such that π′ is relatively likelier if e, then

1

u′(w(π′))u′(w(π)) < u′(w(π′))

1

u′(w(π))

• if only π, e can be observed, for each e

w(π) = u−1(e)

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

• if only π, e can be observed, for each e

let π, π′ be such that π′ is relatively likelier if e, then

1

u′(w(π′))u′(w(π)) < u′(w(π′))

1

u′(w(π))

• if only π, e can be observed, for each e

w(π) = u−1(e)

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

• if only π, e can be observed, for each e

let π, π′ be such that π < π′ , then(?)

1

u′(w(π′))u′(w(π)) < u′(w(π′))

1

u′(w(π))

• if only π, e can be observed, for each e

w(π) = u−1(e)

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

• if only π, e can be observed, for each e

let π, π′ be such that π < π′ , then

1

u′(w(π′))u′(w(π)) < u′(w(π′))

1

u′(w(π))

only if f (π,e)f (π,e) is decreasing in π

• if only π, e can be observed, for each e

w(π) = u−1(e)

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

• if only π, e can be observed, for each e

let π, π′ be such that π < π′ , then

1

u′(w(π′))u′(w(π)) < u′(w(π′))

1

u′(w(π))

only if f (π,e)f (π,e) is decreasing in π

• if only π, e can be observed, for each e

w(π) = u−1(e)

Julio Davila Moral Hazard

Principal-agent problem: non observable effort

• if only π, e can be observed, for each e

let π, π′ be such that π < π′ , then

1

u′(w(π′))u′(w(π)) < u′(w(π′))

1

u′(w(π))

only if f (π,e)f (π,e) is decreasing in π (Monotone LR Property)

• if only π, e can be observed, for each e

w(π) = u−1(e)

Julio Davila Moral Hazard

Principal-agent problem

1 if e is not observable, implementing e costs w = u−1(e)

if e is not observable, implementing e costs w = u−1(e)

2 if e is not observable, implementing e costs w = u−1(e)

if e is not observable, implementing e costs w ∈ RΠ such that∫u(w(π))f (π, e)dπ = e

if e is optimal, it may not be implemented if non observablebecause of a too high expected cost

Julio Davila Moral Hazard

Principal-agent problem

1 if e is not observable, implementing e costs w = u−1(e)

if e is not observable, implementing e costs w = u−1(e)

2 if e is not observable, implementing e costs w = u−1(e)

if e is not observable, implementing e costs w ∈ RΠ such that∫u(w(π))f (π, e)dπ = e

if e is optimal, it may not be implemented if non observablebecause of a too high expected cost

Julio Davila Moral Hazard

Principal-agent problem

1 if e is not observable, implementing e costs w = u−1(e)

if e is not observable, implementing e costs w = u−1(e)

2 if e is not observable, implementing e costs w = u−1(e)

if e is not observable, implementing e costs w ∈ RΠ such that∫u(w(π))f (π, e)dπ = e

if e is optimal, it may not be implemented if non observablebecause of a too high expected cost

Julio Davila Moral Hazard

Principal-agent problem

1 if e is not observable, implementing e costs w = u−1(e)

if e is not observable, implementing e costs w = u−1(e)

2 if e is not observable, implementing e costs w = u−1(e)

if e is not observable, implementing e costs w ∈ RΠ such that∫u(w(π))f (π, e)dπ = e

if e is optimal, it may not be implemented if non observablebecause of a too high expected cost

Julio Davila Moral Hazard

Principal-agent problem: risk-averse agent

1 if e is not observable, implementing e costs w = u−1(e)

if e is not observable, implementing e costs w = u−1(e)

2 if e is not observable, implementing e costs w = u−1(e)

if e is not observable, implementing e costs w ∈ RΠ such that

u

(∫w(π)f (π, e)dπ

)> e

if e is optimal, it may not be implemented if non observablebecause of a too high expected cost

Julio Davila Moral Hazard

Principal-agent problem: risk-averse agent

1 if e is not observable, implementing e costs w = u−1(e)

if e is not observable, implementing e costs w = u−1(e)

2 if e is not observable, implementing e costs w = u−1(e)

if e is not observable, implementing e costs w ∈ RΠ such that∫w(π)f (π, e)dπ > u−1(e)

if e is optimal, it may not be implemented if non observablebecause of a too high expected cost

Julio Davila Moral Hazard

Principal-agent problem: risk-averse agent

∫πf (π, e)dπ−

∫w(π)u−1(e)dπ<

∫πf (π, e)dπ − u−1(e)

∧∫πf (π, e)dπ−

∫w(π)f (π, e)dπ<

∫πf (π, e)dπ − u−1(e)

if e is optimal, it may not be implemented if non observablebecause of a too high expected cost

Julio Davila Moral Hazard

Principal-agent problem: risk-averse agent

∫πf (π, e)dπ −

∫w(π)u−1(e)dπ<

∫πf (π, e)dπ − u−1(e)

∨∫πf (π, e)dπ −

∫w(π)f (π, e)dπ<

∫πf (π, e)dπ − u−1(e)

if e is optimal, it may not be implemented if non observablebecause of a too high expected cost

Julio Davila Moral Hazard

Principal-agent problem: risk-averse agent

∫πf (π, e)dπ −

∫w(π)u−1(e)dπ<

∫πf (π, e)dπ − u−1(e)

∨∫πf (π, e)dπ −

∫w(π)f (π, e)dπ<

∫πf (π, e)dπ − u−1(e)

if e is optimal, it may not be implemented if non observablebecause of a too high expected cost

Julio Davila Moral Hazard

Principal-agent problem: risk-averse agent

∫πf (π, e)dπ −

∫w(π)u−1(e)dπ<

∫πf (π, e)dπ − u−1(e)

∨∫πf (π, e)dπ −

∫w(π)f (π, e)dπ<

∫πf (π, e)dπ − u−1(e)

if e is optimal, it may not be implemented if non observablebecause of a too high expected cost

Julio Davila Moral Hazard

Principal-agent problem: risk-averse agent

∫πf (π, e)dπ −

∫w(π)u−1(e)dπ <

∫πf (π, e)dπ − u−1(e)

∨∫πf (π, e)dπ −

∫w(π)f (π, e)dπ <

∫πf (π, e)dπ − u−1(e)

if e is optimal, it may not be implemented if non observablebecause of a too high expected cost

Julio Davila Moral Hazard

Principal-agent problem: risk-averse agent

∫πf (π, e)dπ −

∫w(π)u−1(e)dπ <

∫πf (π, e)dπ − u−1(e)

∨∫πf (π, e)dπ −

∫w(π)f (π, e)dπ <

∫πf (π, e)dπ − u−1(e)

if e is optimal, it can be implemented (even if non observable)at the same cost u−1(e)

Julio Davila Moral Hazard

Principal-agent problem: risk-averse agent

∫πf (π, e)dπ −

∫w(π)u−1(e)dπ <

∫πf (π, e)dπ − u−1(e)

∨∫πf (π, e)dπ −

∫w(π)f (π, e)dπ <

∫πf (π, e)dπ − u−1(e)

if e is optimal, it can be implemented (even if non observable)at the same cost u−1(e)

Julio Davila Moral Hazard

Principal-agent problem: risk-averse agent

∫πf (π, e)dπ −

∫w(π)u−1(e)dπ >

∫πf (π, e)dπ − u−1(e)

∨∫πf (π, e)dπ −

∫w(π)f (π, e)dπ <

∫πf (π, e)dπ − u−1(e)

if e is optimal, it may not be implemented if non observablebecause of a too high expected cost

Julio Davila Moral Hazard

Principal-agent problem: risk-averse agent

∫πf (π, e)dπ −

∫w(π)u−1(e)dπ >

∫πf (π, e)dπ − u−1(e)

∨∫πf (π, e)dπ −

∫w(π)f (π, e)dπ ?

∫πf (π, e)dπ − u−1(e)

if e is optimal, it may not be implemented if non observablebecause of a too high expected cost

Julio Davila Moral Hazard

Principal-agent problem: risk-averse agent

∫πf (π, e)dπ −

∫w(π)u−1(e)dπ >

∫πf (π, e)dπ − u−1(e)

∨∫πf (π, e)dπ −

∫w(π)f (π, e)dπ >

∫πf (π, e)dπ − u−1(e)

if e is optimal, it may still be implemented if non observablebut at a higher expected cost

∫w(π)f (π, e)dπ

Julio Davila Moral Hazard

Principal-agent problem: risk-averse agent

∫πf (π, e)dπ −

∫w(π)u−1(e)dπ >

∫πf (π, e)dπ − u−1(e)

∨∫πf (π, e)dπ −

∫w(π)f (π, e)dπ >

∫πf (π, e)dπ − u−1(e)

if e is optimal, it may still be implemented if non observablebut at a higher expected cost

∫w(π)f (π, e)dπ (inefficiency)

Julio Davila Moral Hazard

Principal-agent problem: risk-averse agent

∫πf (π, e)dπ −

∫w(π)u−1(e)dπ >

∫πf (π, e)dπ − u−1(e)

∨∫πf (π, e)dπ −

∫w(π)f (π, e)dπ <

∫πf (π, e)dπ − u−1(e)

if e is optimal, it may not be implemented if non observablebecause of a too high expected cost

∫w(π)f (π, e)dπ

Julio Davila Moral Hazard

Principal-agent problem: risk-averse agent

∫πf (π, e)dπ −

∫w(π)u−1(e)dπ >

∫πf (π, e)dπ − u−1(e)

∨∫πf (π, e)dπ −

∫w(π)f (π, e)dπ <

∫πf (π, e)dπ − u−1(e)

if e is optimal, it may not be implemented if non observablebecause of a too high expected cost

∫w(π)f (π, e)dπ (ineff.)

Julio Davila Moral Hazard

Principal-agent problem

1 if e is not observable, implementing e costs w = u−1(e)

if e is not observable, implementing e costs w = u−1(e)

2 if e is not observable, implementing e costs w = u−1(e)

if e is not observable, implementing e costs w ∈ RΠ such that∫u(w(π))f (π, e)dπ = e

if e is optimal, it may not be implemented if non observablebecause of a too high expected cost

Julio Davila Moral Hazard

Principal-agent problem: risk-neutral agent

1 if e is not observable, implementing e costs w = u−1(e)

if e is not observable, implementing e costs w = u−1(e)

2 if e is not observable, implementing e costs w = u−1(e)

if e is not observable, implementing e costs w ∈ RΠ such that

u

(∫w(π)f (π, e)dπ

)= e

if e is optimal, it may not be implemented if non observablebecause of a too high expected cost

Julio Davila Moral Hazard

Principal-agent problem: risk-neutral agent

1 if e is not observable, implementing e costs w = u−1(e)

if e is not observable, implementing e costs w = u−1(e)

2 if e is not observable, implementing e costs w = u−1(e)

if e is not observable, implementing e costs w ∈ RΠ such that∫w(π)f (π, e)dπ = u−1(e)

if e is optimal, it may not be implemented if non observablebecause of a too high expected cost

Julio Davila Moral Hazard

Principal-agent problem: risk-neutral agent

∫πf (π, e)dπ−

∫w(π)u−1(e)dπ<

∫πf (π, e)dπ − u−1(e)

q∫πf (π, e)dπ−

∫w(π)f (π, e)dπ<

∫πf (π, e)dπ − u−1(e)

if e is optimal, it may not be implemented if non observablebecause of a too high expected cost

Julio Davila Moral Hazard

Principal-agent problem: risk-neutral agent

∫πf (π, e)dπ −

∫w(π)u−1(e)dπ <

∫πf (π, e)dπ − u−1(e)

q∫πf (π, e)dπ −

∫w(π)f (π, e)dπ <

∫πf (π, e)dπ − u−1(e)

if e is optimal, it may not be implemented if non observablebecause of a too high expected cost

Julio Davila Moral Hazard

Principal-agent problem: risk-neutral agent

∫πf (π, e)dπ −

∫w(π)u−1(e)dπ <

∫πf (π, e)dπ − u−1(e)

q∫πf (π, e)dπ −

∫w(π)f (π, e)dπ <

∫πf (π, e)dπ − u−1(e)

if e is optimal, it can be implemented (even if non observable)at the same cost u−1(e)

Julio Davila Moral Hazard

Principal-agent problem: risk-neutral agent

∫πf (π, e)dπ −

∫w(π)u−1(e)dπ <

∫πf (π, e)dπ − u−1(e)

q∫πf (π, e)dπ −

∫w(π)f (π, e)dπ <

∫πf (π, e)dπ − u−1(e)

if e is optimal, it can be implemented (even if non observable)at the same cost u−1(e)

Julio Davila Moral Hazard

Principal-agent problem: risk-neutral agent

∫πf (π, e)dπ −

∫w(π)u−1(e)dπ >

∫πf (π, e)dπ − u−1(e)

q∫πf (π, e)dπ −

∫w(π)f (π, e)dπ ?

∫πf (π, e)dπ − u−1(e)

if e is optimal, it may not be implemented if non observablebecause of a too high expected cost

Julio Davila Moral Hazard

Principal-agent problem: risk-neutral agent

∫πf (π, e)dπ −

∫w(π)u−1(e)dπ >

∫πf (π, e)dπ − u−1(e)

q∫πf (π, e)dπ −

∫w(π)f (π, e)dπ >

∫πf (π, e)dπ − u−1(e)

if e is optimal, it can be implemented (even if non observable)at the same (expected) cost u−1(e)

Julio Davila Moral Hazard