Producer Theory If asked to find a supply function for a fir

Producer Theory:
If asked to find a supply function for a firm in the long run, the easy for doing it is to compare AC to MC. In some cases though, that does not give the correct answer, and the profit function must be used. Can you advise in what cases this is true? When should I use a profit funciton, (and not the MC=AC shortcut), in order to find the supply curve of a 2 input firm in the long run?

Solution

MC = AC gives wrong answer if Price (P) is below Long Run Average Cost (LRAC) curve. This is because when Price is below minimum of LRAC, then firm will not produce at all even though setting MC = AC says firm will supply something, thus giving false supply curve of the firm. Also just to inform, long run supply can be wither upward sloping, downward sloping or horizontal.

In two input firm, horizontal addition of optimal output will give the market supply curve.