Suppose two firms $($Firm $1$ and Firm $2)$ are competing against each other in a $($duopoly$)$ market. Suppose further

that the market demand is given by the equation $P=700-30(q1+q2),$ where $P$ is the price of the good, $q1$ is

the quantity sold by firm $1,$ and $q2$ is the quantity sold by firm $2.$ Assume that the total cost of production for

firm $1$ is $100q1,$ and $100q2$ for firm $2($meaning that the additional costs to serve one customer is $$100$ for each

firm$).$

Assuming each firm moves simultaneously, find the Nash$-$Cournot equilibrium level of output for each firm.

Explain the steps you took to arrive at this solution as follows:

a$)$ Define the profit equation for one of the firms and explain how you calculated this.

b$)$ Define the Best Response equation for the firms and explain the steps you took to find this.

c$)$ Explain what you need to do with those Best Response equations to find the Nash$-$Cournot equilibrium

level of output for both firm $1$ and firm $2.$ Then provide the Nash$-$Cournot equilibrium levels of output

for each firm.

Assuming each firm moves sequentially, find the Nash$-$Stackelberg equilibrium level of output for each

firm. Explain the steps you took to arrive at this solution as follows:

a$)$ Define the profit equation for the firm that moves first and explain how you calculated this. $($Helpful

hint: assume the second player's Best Response is $q2=10-\frac{1}{2}q1).$

b$)$ Define profit for the firm that moves first. Explain how you calculated this.

c$)$ Calculate the profit maximizing level of output for this first$-$moving firm $($this is their Best Response$).$

Explain the steps you took to calculate this.

d$)$ Explain what you need to do to get the second$-$moving firm's Best Response. Then calculate this profit

maximizing level of output for the second$-$moving firm. $($Combined with the output you gave in part $c,$

these together are the Nash$-$Stackelberg equilibrium.$)$