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( )= (3 − ) <br />
; =1 2 6= (8)<br />
9 − <br />
Solving the above equations gives the subgame perfect equilibrium (SPE)<br />
levels of qualities, advertising, market shares and profits:<br />
= 3 ; = ; = 1 2 ; (9 − ) <br />
= , =1 2 (9)<br />
18<br />
As can be seen from (9), in equilibrium both platforms choose the same<br />
quality and ads, so that they obtain the same demand and profit. We also<br />
find that profits are increasing in . This is because platforms’ profitability<br />
of quality improvement positively depends on , whichisreflectedinthefact<br />
that equilibrium qualities are increasing in . Finally,wefind that the level<br />
of quality does not depend on the degree of substitutability.<br />
Consumer surplus () is calculated as: 5<br />
Z 1<br />
Z 1<br />
= 1 1 − 1 1 −<br />
0<br />
+ 2 (1− 1 )− 2 (1− 1 )− (1−) (10)<br />
1<br />
We now calculate social welfare ( ), defined as the sum of platforms’<br />
profits ( = 1 + 2 ) and consumer surplus (),<br />
=( − ) 2 +( 1 − 2 + +( − )( 1 − 2 )) 1 − 2 1 + 2 − +2 1 +2 2<br />
2<br />
<br />
(11)<br />
By using (9) in (11), we obtain the social welfare at the SPE in the private<br />
duopoly, which is given by:<br />
= (36 − 45) − 4(2 − 3)<br />
(12)<br />
36<br />
3 The mixed duopoly model<br />
In this section, we will assume that platform 1 is a publicly-owned firm that<br />
maximizes social welfare, while platform 2 is a private firm that maximizes its<br />
profit. Substituting (2) in (11) and maximizing the resulting welfare function<br />
5 Recall that 2 =1− 1 .<br />
6<br />
8