Page 43 - 36-1

NTU Management Review Vol. 36 No. 1 Apr. 2026
Appendix D: Proof of Proposition 4

Assume that the second-order condition (SOC) holds. Denote as the optimal deductible level
Appendix D: Proof of Proposition 4 ∗ �
Appendix D: Proof of Proposition 4
Appendix D: Proof of Proposition 4 )= ; ) and ; )= ; ) . Thus,
∗
∗
∗
∗
when . By Definition 3, ;
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Assume that the second-order condition (SOC) holds. Denote as the optimal deductible level
∗
integrating the left-hand side of FOC (8) by parts yields that ≤ if and only if
∗
�
∗
Assume that the second-order condition (SOC) holds. Denote as the optimal deductible level
�
∗
�
Assume that the second-order condition (SOC) holds. Denote as the optimal deductible level
when . By Definition 3, ; )= ; ) and � ; )= ; ) . Thus,
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∗ . By Definition
∗ ; ) and
∗ ; )= ; ) . Thus,
when . By Definition 3, ; )= ; ) and ; )= ; ) . Thus,
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� − ; )�� ×
∗
when
∗;
� if and only if
∗ 3,
∗)=
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∗
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1+ )�1− ; )��1 + 1 + )�1
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integrating the left-hand side of FOC (8) by parts yields that ≤
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∗ ≤ if and only if
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integrating the left-hand side of FOC (8) by parts yields that ≤ if and only if
∗
∗
∗
�� integrating the left-hand side of FOC (8) by parts yields that
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� − − )� × � ; )− ; )) + 1− ) ; ) −
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1+ )�1− ; )��1 + 1 + )�1 − ; )�� × �
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∗
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∗ .
∗ − ; )�� ×
1+ )�1− ; )��1 + 1 + )�1 − ; )�� × ∗
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; ))�� ≤
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�� 1+ )�1− ; )��1 + 1 + )�1
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� � �
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∗ �
�� � − � − )� × � ; )− ; )) + 1− ) ; ) −
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� ∗ ∗
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� � �� �� � − ∗ − )� × � ; )− ; )) + 1− ) ; ) − (D1)
�� � − − )� × � ; )− ; )) + 1− ) ; ) − �
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�; ))�� ≤ . � � � �� � �
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; ))�� ≤ .
; ))�� ≤ . (D1)
We first prove that for all > , < , and 1 , Equation (D1) holds for all
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if and only if
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� (D1) (D1)
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We first prove that for all > , < , and 1 , Equation (D1) holds for all
We first prove that
� ; )− ; )� + 1− ) ; )− ; ) ≥ ,
(D2)

�
��
if and only if � � for all > , < , and 1 , Equation (D1) holds for all
��
∗
We first prove that for all > , < , and 1 , Equation (D1) holds for all
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∗ if and only if
if and only if
∗
�
for all 1 and , as follows. � � (D2)
� ; )− ; )� + 1− ) ; )− ; ) ≥ ,
�
∗
� �
�
∗ )� + 1− ) ; )− ; ) ≥ ,

The if part: Since 1 + ) 1− ; )) > and < , if Condition (D2) holds for all (D2)
� ; )− ; )� + 1− ) ; )− ; ) ≥ ,
(D2)
∗
� ; )− ;
��
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�
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�
� �
for all 1 and , as follows. ��
∗
1 and , then, for all > , < , and 1 , Equation (D1) holds for all
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∗ , as follows.
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∗
for all 1 and , as follows. )) > and < , if Condition (D2) holds for all
for all 1 and
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∗
∗
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The if part: Since 1 +
.
� ) 1− ;
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� ∗ ∗ �� and < , if Condition (D2) holds for all
The if part: Since 1 + ) 1− ; )) > and < , if Condition (D2) holds for all
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��; )) >
The if part: Since 1 + ) 1−
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∗ �
∗
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1 and , then, for all �> , < , and 1 , Equation (D1) holds for all
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The only if part: Suppose, by contradiction, that there exists such that � ; ) −
∗ , then, for all > , < , and 1 , Equation (D1) holds for all
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∗
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��
1 and , then, for all > , < , and 1 , Equation (D1) holds for all .
�

∗ 1 and
��

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; � � )� + 1 − ) ; )− ; ) < for all 1 . Because of continuity, the
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The only if � ∗ ∗ . .

∗
part: Suppose, by contradiction, that there exists such that � ; ) −
above condition also holds for all − + ⊂ , where is positive and
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∗
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� 1 . Because of continuity, the
The only if part: Suppose, by contradiction, that there exists
∗ such that � ; ) −
The only if part: Suppose, by contradiction, that there exists such that � ; ) − � �
; )� + 1 − ) ; )− ; ) < for all
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�
arbitrarily small. We define and as follows:
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� �
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∗ 1 . Because of continuity, the
; )� + 1 − ) ; )− ; ) < for all 1 . Because of continuity, the
; )� + 1 − ) ; )− ; ) < for all
above
� condition also holds for all − + ⊂ , where is positive and
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∗

� , where is positive and

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above condition
� also
above condition also holds for all − + ⊂ , where is positive and
∗ �� ∗
= { − + | � � ; � − � ;
�+ ⊂
� � holds for all −
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arbitrarily small. We define and as follows: �
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� and as follows:
arbitrarily small. We define and as follows:
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arbitrarily small. We define
∗ ≥ },

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= { − + | � � ; � − � ; ��
�

+ 1− ) ; ) − ; )
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∗ | � � ; � − � ; ��
= { − + | � � ; � − � ; �� �
∗
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= { − +
� + ⊂ | ; ) − ; )
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∗
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= { [ � − � � � �
+ 1− ) ; ) − ; ) ≥ }, � �
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+ 1− ) ; ) − ; ) ≥ },

+ 1− ) ; ) − ; ) ≥ },
� ) − ; ) < },

= { [ − + ⊂ | ; ) − ; )
+ 1 − ) ; � � � � � � ∗ � � �
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∗ ⊂ | ; ) − ; )
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= { [ − + ⊂ | ; ) − ; )
∗

= { [ − +
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for all 1 . Furthermore, we define and as follows: � � �
+ 1 − ) ; ) − ; ) < },
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− ) ; ) − ; ) < },

+ 1 − ) ; ) − ; ) < },

+ 1
� 1− ) ; ) − ; ) � ,
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= � � � ; )− ; )�
� +
�
for all 1 . Furthermore, we define and as follows: �
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� and as follows:
for all 1 . Furthermore, we define and as follows:
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for all 1 . Furthermore, we define
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� = � � � � ; � − � ; �� + 1− ) ; ) − ; ) � .

= � � � ; )− ; )� + 1− ) ; ) − ; ) � ,

�

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= � � � ; )− ; )� + 1− ) ; ) − ; ) � ,

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= � � � ; )− ; )� + 1− ) ; ) − ; ) � ,

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� �
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= � � � � ; � − � ; �� + 1− ) ; ) − ; ) � .
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Note that ≥ � and < by the definitions of and . Consider an individual with
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� � � � − � ; �� + 1− ) ; ) − ; ) � .

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utility defined as = � � � � ; � � �
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= � � � � ; � − � ; �� + 1− ) ; ) − ; ) � . �
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Note that ≥ and < by the definitions of and . Consider an individual with
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� and < by the definitions of and . Consider an individual with
Note that ≥ and < by the definitions of and . Consider an individual with
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utility defined as
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Note that ≥
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utility defined as
utility defined as
� �
35