moral hazard -...
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![Page 1: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/1.jpg)
Moral Hazard
Julio Davila
2009
Julio Davila Moral Hazard
![Page 2: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/2.jpg)
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
![Page 3: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/3.jpg)
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
![Page 4: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/4.jpg)
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
![Page 5: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/5.jpg)
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
![Page 6: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/6.jpg)
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
![Page 7: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/7.jpg)
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
![Page 8: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/8.jpg)
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
![Page 9: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/9.jpg)
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
![Page 10: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/10.jpg)
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
![Page 11: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/11.jpg)
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
![Page 12: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/12.jpg)
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
![Page 13: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/13.jpg)
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
![Page 14: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/14.jpg)
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
![Page 15: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/15.jpg)
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
![Page 16: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/16.jpg)
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
![Page 17: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/17.jpg)
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
![Page 18: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/18.jpg)
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
![Page 19: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/19.jpg)
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
![Page 20: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/20.jpg)
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
![Page 21: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/21.jpg)
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
![Page 22: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/22.jpg)
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
![Page 23: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/23.jpg)
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
![Page 24: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/24.jpg)
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
![Page 25: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/25.jpg)
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
![Page 26: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/26.jpg)
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
![Page 27: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/27.jpg)
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
![Page 28: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/28.jpg)
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
![Page 29: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/29.jpg)
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
![Page 30: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/30.jpg)
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
![Page 31: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/31.jpg)
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
![Page 32: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/32.jpg)
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
![Page 33: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/33.jpg)
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
![Page 34: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/34.jpg)
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
![Page 35: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/35.jpg)
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
![Page 36: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/36.jpg)
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
![Page 37: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/37.jpg)
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
![Page 38: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/38.jpg)
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
![Page 39: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/39.jpg)
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
![Page 40: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/40.jpg)
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
![Page 41: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/41.jpg)
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
![Page 42: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/42.jpg)
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
![Page 43: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/43.jpg)
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
![Page 44: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/44.jpg)
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
![Page 45: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/45.jpg)
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
![Page 46: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/46.jpg)
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
![Page 47: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/47.jpg)
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
![Page 48: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/48.jpg)
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
![Page 49: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/49.jpg)
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
![Page 50: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/50.jpg)
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
![Page 51: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/51.jpg)
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
![Page 52: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/52.jpg)
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
![Page 53: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/53.jpg)
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
![Page 54: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/54.jpg)
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
![Page 55: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/55.jpg)
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
![Page 56: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/56.jpg)
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
![Page 57: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/57.jpg)
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
![Page 58: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/58.jpg)
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
![Page 59: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/59.jpg)
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
![Page 60: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/60.jpg)
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
![Page 61: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/61.jpg)
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
![Page 62: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/62.jpg)
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
![Page 63: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/63.jpg)
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
![Page 64: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/64.jpg)
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
![Page 65: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/65.jpg)
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
![Page 66: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/66.jpg)
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
![Page 67: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/67.jpg)
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
![Page 68: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/68.jpg)
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
![Page 69: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/69.jpg)
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
![Page 70: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/70.jpg)
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
![Page 71: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/71.jpg)
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
![Page 72: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/72.jpg)
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
![Page 73: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/73.jpg)
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
![Page 74: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/74.jpg)
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
![Page 75: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/75.jpg)
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
![Page 76: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/76.jpg)
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
![Page 77: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/77.jpg)
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
![Page 78: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/78.jpg)
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
![Page 79: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/79.jpg)
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
![Page 80: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/80.jpg)
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
![Page 81: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/81.jpg)
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
![Page 82: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/82.jpg)
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
![Page 83: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/83.jpg)
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
![Page 84: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/84.jpg)
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
![Page 85: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/85.jpg)
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
![Page 86: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/86.jpg)
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
![Page 87: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/87.jpg)
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
![Page 88: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/88.jpg)
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
![Page 89: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/89.jpg)
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
![Page 90: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/90.jpg)
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
![Page 91: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/91.jpg)
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
![Page 92: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/92.jpg)
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
![Page 93: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/93.jpg)
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
![Page 94: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/94.jpg)
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
![Page 95: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/95.jpg)
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
![Page 96: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/96.jpg)
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
![Page 97: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/97.jpg)
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
![Page 98: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/98.jpg)
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
![Page 99: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/99.jpg)
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
![Page 100: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/100.jpg)
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
![Page 101: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/101.jpg)
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
![Page 102: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/102.jpg)
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
![Page 103: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/103.jpg)
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
![Page 104: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/104.jpg)
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
![Page 105: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/105.jpg)
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
![Page 106: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/106.jpg)
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
![Page 107: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/107.jpg)
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
![Page 108: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/108.jpg)
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
![Page 109: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/109.jpg)
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
![Page 110: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/110.jpg)
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
![Page 111: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/111.jpg)
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
![Page 112: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/112.jpg)
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
![Page 113: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/113.jpg)
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
![Page 114: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/114.jpg)
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
![Page 115: Moral Hazard - cermsem.univ-paris1.frcermsem.univ-paris1.fr/davila/teaching/gradmicro2/6_moral_hazard-s… · Principal-agent problem 1two individuals: a "principal" and an "agent"](https://reader035.vdocuments.pub/reader035/viewer/2022071017/5fd138bd24191056663209b5/html5/thumbnails/115.jpg)
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