$2.$ Let's consider an symmetric Cournot example with $3$ firms. You are free to use any of the formula from the slides, but please indicate which formula you are using.

Suppose that inverse demand is $\backslash ($ P$($q$)=100-$q$\_\{1\}-$q$\_\{2\}-$q$\_\{3\}\backslash ).$ The constant pre$-$merger marginal costs of the firms a re $\backslash ($ c$\_\{1\}=20,$ c$\_\{2\}=20\backslash )$ and $\backslash ($ c$\_\{3\}=20\backslash )$ and there a re no fi xed co sts$..$ As sume that these marginal costs are such that all three firms have positive o utput in equilibrium.

$($a$)$ Suppose that firms $1$ and $2$ merge with no synergy. Is the merger profitable?

$($b$)$ Now suppose that the firms play an infinitely repeated game, setting quantities each period $($i$.$e$.,$ the same model we considered for collusion$).$ If they collude, each firm will produce an equal share of the joint monopoly output. For what discount factors can the firms support collusion without the merger? $($Assume that they don't anticipate the merger$).$ For what discount factors can the firms support collusion with the merger? Does this provide a possible alternative explanation for why the merger might be profitable?