Page 46 - 36-1
P. 46
Ambiguity Increases and Insurance Deductibles
Appendix E: Proof of Proposition 5
Proposi
f
ndix
o
5
tion
pe
p
E
oof
: Pr
Appendix E: Proof of Proposition 5
A
ndix
f
Proposi
: Pr
pe
p
oof
o
A
E
Appendix E: Proof of Proposition 5 tion 5 ∗ ∗
Assume that the SOC holds. From FOC (9), we know that ≤ if and only if
Assume that the SOC holds. From FOC (9), we know that ≤ if and only if
A s s u m e t h a t t h e SO C ho lds. Fr om FOC (9 ), we kn ow that � ∗ ∗ ≤ � ∗ ∗ if and onl y if
� �
� �
ow that
m
Assume that the SOC holds. From FOC (9), we know that ≤ if and only if if and only if
a
lds. Fr
u
s
h
t
h
SO
e
A
s
t
C ho
t
), we kn
e
om FOC (9
∗
∗
∗
∗
∗
≤
� �
�
�
�
�
�
∗
∗ ∗
� � � � ( )� ( )
�
�
�
�
� � ( )� ( ) ( )
�
∗ ∗
∗
∗
�
� � ( )�
�
� �
� �
� �
� �
�
∗
� ∗
� �
� ∗ ( )�
�
� � � � � ( )� ( ) ( ) ∗
�
�
�
∗ �
� 1� � ( )��1 ( )� ≤0. (E1)
�
�
�
�
� � � ∗
(���)��� ��� � �� �
.
(E1)
∗
∗ ∗
∗ ∗
� �
� �
∗ ∗
(
(
1�
)��1
≤0
�
)�
�
� � � ∗ ∗ � ∗ ∗ 1� � ( )��1 ( )� ≤0. (E1)
� �
� �
� �
� �
The above equation can be rewritt � � ∗en as ∗ 1� � ( )��1 ( )� ≤0. ≤0. (E1) (E1)
(���)��� ��� �
(���)��� ��� � ��
��
∗
∗
�
∗
∗
�
�
∗
∗ �
(
1�
)��1
(
� �
�
)�
�
�
�
�
�
��
�
∗
∗
(���)��� ���
(���)��� ��� � �� � ��
on c
n be rewritt
The above equation can be rewritten as
The abo
en as
a
v
e equati
�
�
� ( )�� ( , )�1 ( , )� ( , )�1 ( , )��
The above equation can be rewritten as
∗
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�
∗
∗
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∗
∗
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∗
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�
The above equation can be rewritten as �
� ( )�� ( , )�1 ( , )� ( , )�1 ( , )�� � ( )�� ( , )�1 ( , )� ( , )�1 ( , )��
∗ ∗
∗ ∗
� �
∗ ∗
� �
� �
∗ ∗
∗ ∗
� �
∗ ∗
� �
� �
� �
� �
� �
� �
� �
� �
� �
� �
� �
� ( )�� ( , )�1 ( , )� ( , )�1 ( , )�� ( )�� ( , )�1 ( , )� ( , )�1 ( , )��
∗
∗
∗
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∗
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� ∗
�
� � �
∗ �
∗
∗
�
� � �
∗
� � �
∗
∗
∗
∗
� ( )) � ( , )�1 ( , )� ( , )�1 ( , )�� ≤ 0,
� � � �
∗
∗
∗
(
�
+ �
�
� � �
�
��
�
�
�
�
�
�
�
�
�
� � ∗ ∗ � � � � � � �
∗ ∗
� �
∗ ∗
∗ ∗
� �
� �
�
� �
+ � � ( ( )) � ( , )�1 ( , )� ( , )�1 ( , )�� ≤ 0, , )� ( , )�1 ( , )�� ≤ 0,
� (
( ( )) � ( , )�1
� �
� �
� �
� �
∗
� �
� �
� �
+ � ∗
( ( )) � ( , )�1 ( , )� ( , )�1 ( , )�� ≤ 0, (E2)
∗
�
� ∗
�
� � � � � � ∗ ∗ � ( , )� ( , )�1 ( , )�� ≤ 0,
∗ �
∗
+ � + � ( ( )) � ( , )�1 � � �
� � � � � � � �� � � � � � � � (E2)
(E2)
which is obtained by multiplying FOC (9) by �1 ( , )� and Equation (E1) by
(E2) (E2)
∗
�
�
�
which is obtained by multiplying FOC (9) by �1 ( , )� and Equation (E1) by ( , )� and Equation (E1) by
�
�
∗ ∗
�1 ( , )�, subtracting the former product from the latter product, and finally integrating by
which is obtained by multiplying FOC (9) by �1
�
∗
� �
� �
which is � ∗ ∗ obtained by multiplying FOC (9) by �1 ( , )� and Equation (E1) by ( , )� and Equation (E1) by
�
∗ �
�
∗
�1 ( , )�, subtracting the former product from the latter product, and finally integrating by ( , )�, subtracting the former product from the latter product, and finally integrating by
which is obtained by multiplying FOC (9) by �1
�
�
�
�
��
parts.
�1
� �
� �
�1 ( , )�, subtracting the former product from the latter product, and finally integrating by ( , )�, subtracting the former product from the latter product, and finally integrating by
∗
∗ �
�
�1
parts.
part
s.
��
�
�
Let us define ( , ) ( , ) ( , ) and ( , ) ( , ) ( , ) for all
∗
∗
parts. parts. � �� � � � � ∗ ∗ � � � � � � � � � � � � � ∗ ∗ � � � �
Let us define ( , ) ( , ) ( , ) and ( , ) ( , ) ( , ) for all ( , ) ( , ) ( , ) and ( , ) ( , ) ( , ) for all
�
�
Let us define
∗
� ��
� � �
� � �
� �
� �
� �
0, . Since (0, ) 0, � � � (0, ) ( , ), where or . Accordingly, Equation
� �
∗ ∗ ) and ( , ) ( , ) ( , ) for all ( , ) and ( , ) ( , ) ( , ) for all
�
�
�
∗
∗ �
�
�
∗ �
∗
�
�
�
�
�
�
�
�
0, . Since (0, ) 0, (0, ) ( , ), where or . Accordingly, Equation (0, ) 0, (0, ) ( , ), where or . Accordingly, Equation
Let us define ( , ) ( , ) ( ,( , ) ( , )
Let us define
�
� �
�
��
�
(E2) becomes
�
�
��
�
�
��
�
� �
�
�
�
�
0, . Since
� �
� �
� �
� �
0, . Since (0, ) 0, (0, ) ( , ), where or . Accordingly, Equation (0, ) 0, (0, ) ( , ), where or . Accordingly, Equation
�
�
�
∗ �
�
�
∗
(E2
omes
ec
(E2) becomes
0, . Since
b
)
� �
�
�
�
�
�
�
(E2) becomes
� ( )�� (0, )�1 (0 , )� (0, )�1 (0, )��
(E2) becomes � � ∗ � ∗ � � � � � � � �
� ( )�� (0, )�1 (0 , )� (0, )�1 (0, )�� (0, )�1 (0 , )� (0, )�1 (0, )��
� �
� �
� �
� �
∗ ∗ ∗
� �
∗ ∗
� ( )��
�
� �
� �
� �
� � �
� �
� �
� ( )�� (0, )�1 (0 , )� (0, )�1 (0, )�� (0, )�1 (0 , )� (0, )�1 (0, )��
�
�
∗
∗ �
� ∗
∗ ∗ ∗
∗
+
� � � ( )�� � � � � � � � � � � � � � � � � � � �
� ( ( )) �� (0, ) ( , )��1 (0, )� �
�
�
�
� � �
�
� �
∗ ∗
∗
� �
∗
�
+ � ( ( )) �� (0, ) ( , ) � � � �
+ � ( ( )) �� (0, ) ( , )��1 (0, )���1 (0, )�
� �
� �
� �
� �
� �
� �
�
� (0, ) ( , )��1 (0, )�� ≤ 0 �
∗
�
∗
� �
�
�
+ � � � ( ( )) �� (0, ) ( , )��1 (0, )���1 (0, )�
+ � ( ( ))
� � �� (0, ) ( , )
�
�
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�
� �
�
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� �
� �
� (
� � � (0, ) ( , )��1 (0, )�� ≤ 0 0, ) ( , )��1 (0, )�� ≤ 0
∗ � � � � � �
� (0, ) ( , )��1 (0, )�� ≤ 0 0, ) ( , )��1 (0, )�� ≤ 0
� � �� ∗
�
� �
�
�
� (
Furthermore, because � � ∗ ∗ ( ( )) (0, ) � � � �
�
� �
�
�
�
�
Furthermore, because � � � � � � ( ( )) (0, ), where or , by the definitions of
( ( )) (0, ) ( ( )) (0, )
� �
� �
�� ��
�
∗ ∗
∗
∗
�
Furthermore, because � ∗
( ( )) �� � ∗ � �� ∗ � � � � �
�
� ∗
�
�
�
∗ �
Furthermore, because � ( ( )) (0, ) ( ( )) (0, )
Furthermore, because �
( ( )) ( ( )) (0, ), where or , by the definitions of ( ( )) ( ( )) (0, ), where or , by the definitions of
�
�
∗ ∗
� �
∗ ∗
∗ ∗
� �
� �
� � �
�
� � ( , ), the above equation can be rearranged as
�
�
( , ) and � � � � � �
( ( )) ( ( )) (0, ), where or , by the definitions of ( ( )) ( ( )) (0, ), where or , by the definitions of
∗
∗ �
∗
∗
� ∗
∗
�
�
�
�
�
�
( , ) and ( , ), the above equation can be rearranged as ( , ) and ( , ), the above equation can be rearranged as
� �
�
�
� � �
�
�
� �
�
� �
� �
� � ( )�� ( , )�1 ( , )� ( , )�1 ( , )��
) and ( , ), the above equation can be rearrange
( , ) and ( , ), the above equation can be rearranged as �d as � � � ∗
∗
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∗
∗
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∗
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�
( ,
�
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�
� ( )�� ( , )�1 ( , )� ( , )�1 ( , )�� , )�1 ( , )� ( , )�1 ( , )��
� �
∗ ∗
∗ ∗
� �
� �
� �
� �
∗ ∗
∗ ∗
∗ ∗
� ( )�� (
� �
� �
� �
� �
� �
� �
� �
� �
� �
� ( )�� ( , )�1 ( , )� ( , )�1 ( , )�� , )�1 ( , )� ( , )�1 ( , )��
∗
�
∗
∗ �
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∗
∗
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∗
∗
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∗ �
� ∗
∗ �
∗ �
� � � ( )�� (
�
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� ∗ �
��
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∗ �
∗ ∗ ��
�
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� ∗
� � � � ( )� �� ( , ) ( , )��1 ( , )�
�
� � � � � � �
� �
�
�� ��
∗ ∗
∗
� �
∗ ∗
∗ ∗
∗
� � ( )� �� ( , ) ( , )��1 ( , )� �
� � ( )� �� ( , ) ( , )��1
�
� ( , )
� �
� �
� �
� �
� �
� �
� �
� �
� ( , ) ( , )��1 ( , )�� ≤ 0
∗
∗ �
∗
�
�
∗
∗ ∗
�
�
∗
�
∗ �
��
��
� �
� � � � ( )� �� ( , ) ( , )��1 ( , )� , )�
� � �� ( , ) ( , )��1 (
� � ( )�
��
�
�
��
�
�
�
� �
�
�
� �
� ( , ) ( , )��1 ( , )�� ≤ 0 , ) ( , )��1 ( , )�� ≤ 0
∗ ∗
∗ ∗
� �
� �
� �
� � � ( � � � � � � � �
� �
� ( , ) ( , )��1 ( , )�� ≤ 0 , ) ( , )��1 ( , )�� ≤ 0 (E3)
�
∗ �
�
∗
�
∗
∗ �
�
� (
�
�
�
��
��
�
�
�
(E3)
(E3)
(E3) (E3)
38

