Question: Suppose that demand for rollerblades is given by D(p) = A - p. The cost function for all firms is C(y) = wy2 + f , where f is a fixed set-up cost. The marginal cost of production is MC(y) = 2wy. Assume that the industry is perfectly competitive.
(a) Find a competitive firm's supply function. If there are n firms in the industry, what is industry supply?
(b) If there are n firms in the industry, find expressions for the competitive equilibrium price and quantity. What is the equation for how much each firm produces? What is the equation for the profit of each firm? [Hint: Your answer should be 4 algebraic equations that express the endogenous variables (price, quantity, firm supply, and firm profit) as a function of the exogenous variables (A, n, f , and w).]
(c) Suppose A = 100, w = $4, f = $100, and n = 2. Using the equations you derived in part (b), what is the equilibrium price and quantity? Firm supply and profits? Using two diagrams, graph this competitive equilibrium. In one diagram illustrate the market equilibrium. In the second, show the equilibrium position of a representative firm. On this second diagram make sure you indicate the profit-maximizing output of a firm as well as the profit earned.
(d) Is the equilibrium you found in part (c) a short-run or long-run equilibrium? Why? If the industry is not in long-run equilibrium, explain the adjustment process that will occur.
(e) For the parameter values given in part (c), find the long-run competitive equilibrium. On the two diagrams from part (c), indicate the long-run equilibrium. What is the long-run equilibrium number of firms?