Question

Why is it mathematically correct that the Marginal Cost curve crosses the average total cost curve at its lowest point? Show your work using graphs

Answer 1

Marginal cost refers to the change in total cost when an extra unit is produced; and on the other hand the average cost is computed as total cost divided with the number of produced units. When average cost begins to increases the marginal cost will be exceeding the average cost. When average cost remaining constant/ unchanged, then MC = AC; and it occurs when falling AC would reach to its lowest point. Now when the average cost starts decreasing the marginal cost will be less than AC. Thus we can reach to a conclusion that the marginal cost curve always intersects the average total cost curve at its lowest point because the marginal cost in the production of an additional next output unit will always impact the average total cost. Consequently average total cost will decline as long as average total cost is higher than the marginal cost.

Mathematically, TC = TFC + TVC

As, TFC remains constant thus dFC = 0
It gives, dTC=dVC
So MC = dTC/dQ = dVC/ dQ

Thus the AVC and the ATC curve falls and rises with a new entry MC.

Graph: Enclosed. In the enclosed graph, AVC decreases because MC is the additional cost for the next unit produced; and thus the next unit costs less than the average, thus will be pulling the average down. In the graph geometrically it is on the left. Now when MC exceeds the AVC, it will push the average up so AVC must be increasing. When the marginal unit costs exceed the average, the average will be increasing.