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Optimal Advertorial Allocation and Contract Design of a Multichannel Networks Company on Video Sharing
Platforms

�
According to Assumption 1, ≥0 and . Thus, we have 4 −
� � �
�
��
>0 and thus � >0, which means that increase in . Similarly, the
� �
�
�
� �
�� � � �
derivatives of with respect to is
�
�
�
� � � � �
8 4
� � � � �.
= �
�
�
�
�
�
� �
� � �
� (4 − ) ( ( − ) ) 4 − ( ( − ) ) �
�
� �
� �
�
According to Assumption 1, ≥0 and . Thus, we have 4 −
� � �
�
>0 and get � >0, which means that increase in . Q.E.D.
��
� �
�
�
�
� �
�
�� �
Proof of Proposition 3. Using backward induction, we solve the second stage first.
The first- and second-order derivatives of ( ) with respect to are
�
�
�
�
� � �
�� (� � ) � � (� � )
�
�
= ( ) − and = − ,
� � �
�� � ��
�
� �
respectively. Since >0, we have � � (� � ) 0; the function is concave. Due to
�
�� �
�
�
concavity, the optimal solution must satisfy �� (� � ) =0. By solving the equation, we
�
�� �
� � � (����) �
obtain = . Similarly, those derivatives of ( ) with respect to are
� � � �
�
�
� �
�� (� � ) � � (� � )
� = ( ( − ) ) − and � = − .
�� � � � �� � �
� �
Since >0, we have � � (� � ) 0; the function is concave. Due to concavity, the
�
�� �
�
�
�� (� � )
optimal solution must satisfy � =0. By solving the equation, we obtain =
�
�
�� �
� � (��(���)�)
. We then plug in and into and get
�
�
�
� � � �
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