Problem $3.(20$ points$)$ Consider the following strategic interaction between Microsoft and Apple. Microsoft, as the leader, selects its output $\backslash ($ q$\_\{\backslash$mathrm$\{$M$\}\}\backslash )$ first. Apple, as the follower, observes Microsoft's choice of $\backslash ($ q$\_\{\backslash$mathrm$\{$M$\}\}\backslash )$ and then decides its own output $\backslash ($ q$\_\{\backslash$mathrm$\{$A$\}\}\backslash ).$

The market price is determined by the industry demand curve:

$\backslash [$

P$=200-2$ q$\_\{\backslash$mathrm$\{$M$\}\}-2$ q$\_\{\backslash$mathrm$\{$A$\}\}$

$\backslash ]$

Both Microsoft and Apple have zero fixed costs and a constant marginal cost of $40.$

$1.$ Derive the equation for Apple's best response function. $(5$ points$)$

$2.$ Determine the equilibrium output for each firm in the leader$-$follower game. Demonstrate that this equilibrium lies on Apple's best$-$response function. To illustrate, plot Apple's best$-$response function on a graph, clearly indicating its vertical intercept, horizontal intercept, and slope. Finally, calculate Microsoft's profits in this equilibrium. $(5$ points$)$

$3.$ Assume that Microsoft and Apple now choose their outputs simultaneously, rather than sequentially. Determine the Cournot equilibrium outputs for each firm, as well as the resulting industry price. Plot each firm's best$-$response function and indicate the Cournot equilibrium on a graph $($Please round up to two decimal places$).(5$ points$)$

$4.$ Based on the results, compare the outcomes of this Cournot competition to the sequential game. Specifically, analyze and discuss which firm benefits and which firm is disadvantaged by the change in the structure of the game. $(5$ points$)$