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