Problem $1.$ Suppose two firms compete in micro$-$chip industry. Each period firm $1$ produces $\backslash ($ q$\_\{1\}\backslash )$ chips and firm two produces $\backslash ($ q$\_\{2\}\backslash )$ chips and the firms face a demand curve of $\backslash ($ P$=1000-20$ Q $\backslash ),$ where $\backslash ($ Q$=$q$\_\{1\}+$q$\_\{2\}\backslash ).$ Both firms have a constant marginal cost of $\backslash (\backslash$$ $40\backslash )$ per chip, $\backslash ($ C$\backslash$left$($q$\_\{$i$\}\backslash$right$)=40$ q$\_\{$i$\}\backslash ).$

$1.$ What are the static Nash equilibrium strategies for this market? What are profits for a single period in this case? $(5$ points$)$

$2.$ Suppose the two firms agree to maximize joint profits rather than individual profits and share the proceeds equally. How many chips does each firm agree to make? What are firms profits for a single period? $(5$ points$)$

$3.$ Suppose the firms have agreed to maximize joint profits, but while firm $2$ produces according to the agreement, firm $1$ decides to cheat and maximize individual profits instead. How many chips does firm $1$ decide to produce? What are profits for each firm? $(5$ points$)$

$4.$ Suppose the discount rate is $\backslash (\backslash$beta$=\mathrm{.}9\backslash ),$ and the industry lasts for only $10$ periods. Describe the equilibrium strategies for this game. $(5$ points$)$