Suppose that two laptop manufacturers, Dell and Hewlett Packard $($denoted D and HP$,$ respectively$),$ compete in output, where output is defined as daily number of laptops sold. The inverse daily laptop demand function is P$($Q$)=600–0\mathrm{.}25$Q$,$ where P is the price of a laptop, Q $=$ qD $+$ qHP is total daily laptop demand and qi denotes the daily production of company i $($i $=$ D$,$HP$).$

Assume that both companies have a constant marginal production cost of $120$ and that they have complete and perfect information.

$($a$)$ Derive and illustrate the output reaction functions of the companies.

$($b$)$ What is the Nash equilibrium outcome if the companies simultaneously choose their output levels?

$($c$)$ Suppose that Dell chooses its output first. Hewlett Packard observes Dell$’$s choice and then makes its own output decision. What is the subgame perfect Nash equilibrium in outcome and strategies?

$($d$)$ Explain the reason for any differences in parts $($b$)$ and $($c$)$ and illustrate on the diagram in $($a$).$