a) The inverse demand function is P = 50 - 2Q, where Q = q1 + q2 is the total industry output, and the cost function is C = 10 + 2q, where q is the firm's output.
b) We can use the Cournot solution to find the profits, prices, and quantities for both firms. In the Cournot model, each firm chooses its output level given the output level of the other firm, assuming that the other firm's output level is fixed. The profit-maximizing output level for each firm is the one that maximizes its profit given the output level of the other firm.
Let q1 and q2 be the output levels of firms 1 and 2, respectively. Then the total industry output is Q = q1 + q2. The market price is determined by the inverse demand function, which is P = 50 - 2Q = 50 - 2(q1 + q2).
The profit of firm 1 is given by π1 = (P - MC1)q1, where MC1 is the marginal cost of production for firm 1. The marginal cost of production is the derivative of the cost function with respect to output, which is MC1 = 2.
Similarly, the profit of firm 2 is given by π2 = (P - MC2)q2, where MC2 is the marginal cost of production for firm 2. The marginal cost of production is also MC2 = 2.
The best response function for firm 1 is q1 = (1/2)(Q - q2), which gives the profit-maximizing output level for firm 1 given the output level of firm 2. Similarly, the best response function for firm 2 is q2 = (1/2)(Q - q1), which gives the profit-maximizing output level for firm 2 given the output level of firm 1.
To find the Nash equilibrium, we need to find the output levels of both firms such that neither firm has an incentive to change its output level given the output level of the other firm. This occurs when both firms are producing their best response output levels simultaneously.
Substituting the best response function for firm 2 into the best response function for firm 1, we get q1 = (1/2)(Q - (1/2)(Q - q1)), which simplifies to q1 = (1/3)Q. Similarly, substituting