Problem $6$

Consider a Cournot duopoly where firm $1$ faces cost of the form $c_1(y_1)=y_1$ and firm $2$ faces

cost of the form $c_2(y_2)=2y_2.$ The inverse demand function is given by $p=10-2Y$ where

$Y=y_1+y_2.$ The two firms have realized that competition decreases their profit and they

decide to form a cartel.

$($a$)$ State the maximization problem of the cartel.

$($b$)$ Derive the first order conditions and use them to find the optimal levels of production for the

two firms under the cartel agreement. Do you get an answer? Discuss your result. How much

will the cartel produce?

$($c$)$ Find the market quantity, the market price, and the cartel's profits. Discuss how the firms

will split the profits.