Page 37 - 33-2

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

As we assume that in Assumption 1, the determinant is positive, and
� �
thus is jointly concave in and . Due to concavity, the optimal solution
�
�
�
�
� �� �
must satisfy � =0 and � =0, where
��
�� � �� �
�
�
� �
�
� �
= − −
�
�
�
�
1−
�
�
� �
−
�
� �
[ − 1−
�
�
�

�
� �
[ − 1−
�
�
�
�
−
�
and
� � �
1− 1−
� � � �
= − −
�
�
�
�
1− 1−
�
� �
�
−
�
� �
1− [ −
�
�
�

�
�
�
1− [ − 1−
�
�
�
�
− .
�
By solving the system of equations, we obtain and as stated in (2). Given
�
�
� �
Assumption 1, and are both positive and thus feasible.
�
�
� �
The derivatives of with respect to is
�
�
�
� � � � �
8 4
� � � � �.
= �
�
�
�
� �
�
� � �
�
� 4 − 4 − �
�
� �
� �
29