1. a. A competitive industry is characterized by two types of firms (low-cost and high-cost) producing an identical good, y. There are m identical low-cost firms with cost functions: CL(yi) = Ly where i = 1, ..., m; L> 0. There are n identical high-cost firms with cost functions: CH(yi) = Hy where j = 1, ..., n; H>0, H > L. The industry demand is: X(p) = a - bp where a > 0, b>0. (i) State two main assumptions of the competitive model. With specific reference, explain how these two assumptions identified are incorporated into the objective problem facing each firm. (3 marks) Assuming firms behave optimally, derive the equilibrium price in this industry and the optimal output of each type of firm. (ii) (7 marks) (i) State and prove two properties of the profit function. (6 marks) (ii) A competitive firm produces a single output (y) from two inputs (x1 and x2), according to the production function below: f(x, x) = xx where a 1/3, b=1/6 Inputs x1 and x2 cost w and w2, respectively, and the firm is able to sell each unit of output produced at market price p. The firm wants to maximize profits. Determine the firm's factor demands for x1 and x2. (9 marks) 3:6: b