Page 35 - 33-2

NTU Management Review Vol. 33 No. 2 Aug. 2023

Appendix

Proof of Proposition 1. Using backward induction, we solve the second stage first.

The first- and second-order derivatives of ( ) with respect to are
�
�
�
�
� � �
�� (� � ) � � (� � ) = − ,
�
�
= ( ) − and
� � � �
�� � ��
�
� �
respectively. Since , we have � � (� � ) < ; the function is concave. Due to
�
�� �
�
�
concavity, the optimal solution must satisfy �� (� � ) = . By solving the equation, we
�
�� �
obtain as stated in (1). Similarly, those derivatives of ( ) with respect to
�
�
�
�
�
�
are
� � �
�� (� � ) = ( ( − ) ) − and � � (� � ) = − .
�
�
�� � � � � �� �
�
� �
Since , we have � � (� � ) < ; the function is concave. Due to concavity, the
�
�� �
�
�
optimal solution must satisfy �� (� � ) = . By solving the equation, we obtain as
�
�
�
�� �
stated in (1).
Since �� � � = � � (����) and �� � � = � � ���(���)�� , we have increases
�
� � �
�� � �� �
� � � (����) �� � � � ���(���)��
��
in . Since � = and � = , we have increases
�
�
�� � � �� � � �
�
in . Since �� � = � � � � and �� � � = � � � � , we have increases in .
�
�
�� � �� � �
Since �� � � = � � � � � and �� � � = � � � � (���) , we have increases in . Since
�
�� � �� � �
27