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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
                             ∗
                                  �
                                                                        �
                                     ∗
                      ∗
               �
                                                  ∗
                                                               ∗
                                                           �
                                                                            ∗
                                               �
                                                              �
                                                           �
                                                                           �
                                     �
                      �
                                                  �
                             �
                                                                        �
                                  �
             The above equation can be rewritten as �
                  �                   (     )��     (     ,     )�1        (      ,     )�        (     ,     )�1        (     ,     )��      �                   (     )��     (     ,     )�1        (      ,     )�        (     ,     )�1        (     ,     )��
                                                  ∗ ∗
                                     ∗ ∗
                                                                        �   �
                                                                            ∗ ∗
                                                           �   �
                                  �   �
                             ∗ ∗
                                                               ∗ ∗
               � �
                      ∗ ∗
                                               �   �
                                                                        � �
                                                           � �
                                                              � �
                                               � �
                             � �
                                  � �
                      � �
                                     � �
                                                                           � �
                                                  � �
                  �                   (     )��     (     ,     )�1        (      ,     )�        (     ,     )�1        (     ,     )��       (     )��     (     ,     )�1        (      ,     )�        (     ,     )�1        (     ,     )��
                  ∗
                                                       ∗
                                     ∗
                                                                        �
                                                                   ∗
                   �
                                  �   ∗
                                                           �
                                     � �   �
                                                  ∗ �
                                          ∗
                                                               ∗
                                               �
                                                              � �   �
                                                                                ∗
                                                                        � �   �
                      ∗
                          ∗
                                                                            ∗
                             ∗
                             �    (     )) �     (    ,     )�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,     )        (    ,     )
                                                                                �
                                               �
                                                   �
                                                                           �
                                                               �
                                                         �
                                                                  � �
                                            �
                                                                 �   �
                                                     �   �
                                           �   �
                                           �     (
                              �   �        �     (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   �   �  � ∗
                                          ∗
                                                                �
                                                                   ∗
                                  ∗
                         �
                                                       ∗
                                                    �
                                       �
                  �
              �
                              �
                       (    ,
                                                                �
                                                    �
                                          �
                                                       �
                                       �
                                  �
                               �
                   �
                          �
               �
                            �           (     )��     (     ,     )�1         (     ,     )�        (     ,     )�1        (     ,     )�� ,     )�1         (     ,     )�        (     ,     )�1        (     ,     )��
                                       �   �
                                                       ∗ ∗
                                          ∗ ∗
                                                                �   �
                                                    �   �
                                                                             �   �
                         � �
                                  ∗ ∗
                                                                                ∗ ∗
                                                                   ∗ ∗
                            �           (     )��     (    
                                                    � �
                                  � �
                                                                   � �
                                                                             � �
                                                                � �
                                                                                � �
                                                       � �
                                          � �
                                       � �
                            �           (     )��     (     ,     )�1         (     ,     )�        (     ,     )�1        (     ,     )�� ,     )�1         (     ,     )�        (     ,     )�1        (     ,     )��
                              ∗
                                                                             �
                                                                                     ∗
                                                                                ∗ �
                             �
                                                                        ∗
                                  ∗
                         �
                                               ∗
                                                            ∗
                                                    �
                                                                �
                                          ∗ �
                                       �   ∗
                                                                   ∗ �
                                                       ∗ �
                             � � �           (     )��     (    
                                
                                       �
                                                                                    �
                                                                       �
                                                                �
                                                       � ∗ �
                                                                                ��
                                                            �
                                                                   ��
                                  �
                                      �
                                          ��
                                              ∗ �
                              ∗ ∗ ��
                                                                           �
                                                    �
                                                    �
                                                               �
                                                                             � ∗
                             � � �      �                  (     )� ��     (     ,     )        (    ,     )��1        (     ,     )�
                             �
                              �               �     �  �       �            �  �
                                                               �   �
                                                                           �
                                �� ��
                                                       ∗ ∗
                                   ∗
                                                    �   �
                                              ∗ ∗
                                                                               ∗ ∗
                              ∗
                             �      �                  (     )� ��     (     ,     )        (    ,     )��1        (     ,     )� �
                             �      �                  (     )� ��     (     ,     )        (    ,     )��1       
                            �
                                                                           � (     ,     )
                                              � �
                                                               � �
                                                    � �
                                                                            � �
                                                       � �
                             � �
                                                                               � �
                                  � �
                                         �     (     ,     )        (    ,     )��1        (     ,     )��         ≤ 0  
                                                                     ∗
                                                                               ∗ �
                                                  ∗
                                                               �
                                          �
                                                            ∗
                                              ∗ ∗
                                                                   �
                                                                           �
                                                                                   ∗
                                                                  �
                                                       ∗ �
                                     ��
                                ��
                                                    �   �
                              � � �      �                  (     )� ��     (     ,     )        (    ,     )��1        (     ,     )� ,     )�
                                                    � � ��     (     ,     )        (    ,     )��1        (    
                                 �      �                  (     )�
                                                                               ��
                                                            �
                                                                     �
                                                       ��
                                                                                   �
                                                  �
                                                                            �
                                              � �
                                          �
                                                               �
                                                                  � �
                                         �     (     ,     )        (    ,     )��1        (     ,     )��         ≤ 0   ,     )        (    ,     )��1        (     ,     )��         ≤ 0  
                                                                     ∗ ∗
                                              ∗ ∗
                                                                  �   �
                                          �   �
                                                     �   �
                            �    �       �     (     � �  � �     � �  � �
                                          � �
                                         �     (     ,     )        (    ,     )��1        (     ,     )��         ≤ 0   ,     )        (    ,     )��1        (     ,     )��         ≤ 0     (E3)
                                                                  �
                                              ∗ �
                                                          �
                                                                          ∗
                                          �
                                                  ∗
                                                                     ∗ �
                                                     �
                                              �     (    
                                                     �
                                          �
                                                  �
                                             ��
                                                                     ��
                                                                          �
                                                                  �
                                                          �
                                                                                            (E3)
                                                                                            (E3)
                                                                                            (E3) (E3)


                                                      38
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