In an industry with two firms, represent the outputs for these single product firms as q_{1} and q_{2} . The two firms decide to form a cartel and set their levels of output to maximise total profits for the industry. Consider the situation where the industry inverse demand function curve which the two firms face is:
P = 25-2q
where q is the total output or sales of the two firms. Next, suppose that these firms have different cost structures. In particular, the total cost functions for firm 1 and firm 2, respectively, are
C_{1}=q_{1}
C_{2} = 0.25q^{2}_{2}
Determine the price charged by the profit-maximizing cartel, the amount of industry profit earned, and the output and profit share for each firm. Ensure you have found a maximum by deriving the second-order sufficient conditions.