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with respect to 1 , we obtain the reaction function of the publicly-owned<br />
platform 1:<br />
1 ( 2 )= 2 + − 1<br />
(2 − 1) ( 1 − 2 + ) (13)<br />
In order to guarantee the second order condition of social welfare maximization<br />
by platform 1, we assume that 12. Noticethatplatform2’s<br />
reaction function is the same that the one in the previous section since it<br />
continues to be a private firm. Thus, from (5) and (13) we can calculate<br />
the second stage NE levels of advertising and market shares in the mixed<br />
duopoly:<br />
1 = (4 − 3) − ( 1 − 2 )<br />
; 2 = (3 − 2) − ( 1 − 2 )<br />
(2 − 1)<br />
(2 − 1)<br />
1 = ( 1 − 2 + )<br />
2(2 − 1)<br />
; 2 = (3 − 2) − ( 1 − 2 )<br />
;<br />
2(2 − 1)<br />
(14)<br />
By substituting the NE values of market shares and advertising of platforms<br />
1 and 2 into expressions (3) and (11) we obtain the profits functions<br />
and the welfare at the first stage of the game, in terms of the qualities:<br />
1 = 2 ((4−3)−( 1 − 2 ))( 1 − 2 +)<br />
− 2 2(2−1) 2 1<br />
2<br />
; 2 = ((3−2)−( 1− 2 )) 2<br />
− 2 2(2−1) 2 2<br />
2<br />
;<br />
= 4(−1)((3−2)−( 1− 2 ))+ 2 ( 1 − 2 +) 2<br />
+ <br />
4(2−1)<br />
2 − − (15)<br />
2 1<br />
2 2<br />
− 2 2<br />
2<br />
<br />
By maximizing the social welfare with respect to 1 ,weobtainthereaction<br />
function of the publicly-owned platform, 1 ( 2 ),and,bymaximizing<br />
platform 2’s profit with respect to 2 ,wefind the reaction function of the<br />
private platform, 2 ( 1 ). Those functions are given by:<br />
1 ( 2 )= 2 + 2 2 − 2<br />
; <br />
2 − 4 + 2 2 ( 1 )=− 22 − 3 3 + 3 1<br />
(16)<br />
+4 2 − 4 − 3<br />
Thus, from (16) we calculate the SPE levels of quality, advertising, market<br />
shares, platform 2’s profit and welfare in the mixed duopoly:<br />
1 =<br />
2 =<br />
9 2 − 12 3 +4 4 − 2 +2 5<br />
2 +24 2 − 16 3 − 12 + 2 − 6 3 +8 ; 4<br />
2 5 − 12 4 +14 3 − 4 2<br />
2 +24 2 − 16 3 − 12 + 2 − 6 3 +8 4<br />
7<br />
9