Steps for

$($Cournot Model$)$ A single good is produced by two firms. The marginal cost for both firms to produce one unit of the good is $20.$ All the output is sold at a single price, which is determined by the demand for the good and the total output of the firms. The demand curve is given by:

$P=100-Q.$

Where $P$ is price and $Q$ is the total quantity of the industry $(q1+q2).$ If these firms choose simultaneously how many units to produce in order to maximize profits, find the NE of this game, the market price, and profits.

$2.($Cartel Duopoly Model$)$ Suppose the two firms above collude by agreeing that each will produce an amount $q_c=q_1+q_2,$ and they have some way to enforce the agreement. What should they choose for $q_c?$ What are the profits of the two firms? Compare this result with the Cournot duopoly profits.

Suppose firm $1$ reneges on its promise in the previous part but firm $2$ does not. What should $1$ choose for $q_1?$ What are firm $1\text{'}$s profits, and what are firm $2\text{'}$s profits?

Suppose firm $2$ finds out what firm $1$ is going to do in the previous part, and chooses $q2$ to maximize profits, given what firm $1$ is going to do$.$ What is $q2$ now? What do you think happens if they go back and forth this way forever?

$3.($Stackelberg Model$)$ Suppose the two firms above choose quantity, but one firm moves first and the second firm observes firm $1\text{'}$s choice. What is the SPNE of this game? Compare these results with the results of the Cournot model: What do you conclude?

$4.($Stackelberg Triopoly Model$)$ Suppose there are now three firms competing in the market, and they move sequentially. The demand function remains the same $(100-$ total output$).$ What is the SPNE of this game?

$5.($Bertrand Duopoly Model$)$ Suppose the two firms move simultaneously and choose price as opposed to quantity. Customers all go to the lowest$-$price firm, and the firms split the market if they choose the same price. What is the Nash equilibrium of this game?

$6.($Cournot and Stackelberg with quadratic cost function$)$ The firms have the same cost function now, and the function is: TC $($total cost$)=q_i^2.$ Find the equilibrium in the Cournot model and the Stackelberg model.