$3)$ Market demand for a good is given by the function

Q $=250$ P$.$

There are two competing firms in this market, and the marginal cost of production

is $4$ dollars $($constant and identical between the two firms$).$ Assume that the firms

choose quantities to produce and the price is determined by the total supply via the

demand function $($Cournot competition$).$

$($a$)$ Find the best response functions for the firms $($the optimal quantity to produce

as a function of the quantity produced by the other firm$).$

$($b$)$ What is the Nash equilibrium $($quantities produced and resulting price$)$ of this

Cournot game?

$($c$)$ What is the profit of each firm in the Nash equilibrium? What is consumer

surplus? How large is the deadweight loss?

$($d$)$ What is the backward induction solution of the game if the firms move sequen$-$

tially, with firm $1$ moving first and firm $2$ second $($Stackelberg competition$)?$

$($e$)$ What is the Nash equilibrium if the two firms compete by setting prices $($Bertrand

competition$)$ instead of quantities?