Suppose that the long-run total cost curve for each firm is given by TC = 500Q - 20Q2 + Q3, where Q is the quantity of the product. Also suppose there is free entry and exit. To find the quantity where ATC is minimized, the firm would need to solve the following equation for Q:
It would be impossible to do this without more information.
500 - 40Q + 3Q2 = 500 - 20Q + Q2.
500 - 40Q + 3Q2 = 500Q - 20Q2 + Q3.
500Q - 20Q2 + Q3 = 500 - 20Q + Q2.