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

�
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( ) ( ( ) ) ( ) ( ( ) )
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( ) = � .
�

� � �
Since ( ) is a quadratic function of and the coefficient of is � � >0,
� � �
�
�
�
� �
the parabola opens upward. As a result, we know that the maximum of ( ) exists
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either when is the minimum or the maximum, namely =0 or = . We then
plug in =0 and = into respectively and get
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�
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�
�
( ) ( )
�
�
� �
�
( ) =
�
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�
�
�
�
( ) ( )
� � � � �
(0) =
� �

� �
�
�
�
� � �(� �� ) >0 , ( ) > (0) , and we obtain as
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�
Since ( ) (0) =
� � � �
�
stated in (3). Moreover, since 0≤ ≤ and = , the proof is complete. Q.E.D.
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� �
Proof of Proposition 4. First, we will find out under structure L. We have � � � =
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� �
��
�
0, the Hessian matrix of being
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�

� �
� �,

� �
and the determinant of the Hessian matrix being . As we assume that
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� �
�
� �
, the determinant is positive, and thus is jointly concave in and .
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� � � � �
Due to concavity, the optimal solution must satisfy �� � � =0 and �� � � =0, where
�� � �� �
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