Why Marginal cost curve crosses the Average Total Cost curve and the Average Variable Cost curve at their minimum point.

Refer to figure 1 given below.

                                                                        fig 1

All the data is taken from the chart in the blog on "Why the profit is maximised when the marginal cost curve crosses the marginal revenue curve"

Facts to consider

1. Average total cost curve is a plot of the average total costs for all the different quantities starting from zero production. The X axis is the quantity, and the Y axis is the ATC. This curve first slopes downward, then reaches a bottom and then slopes upward.

2. Average variable cost curve has the same characteristics of the ATC curve, but always stays below the ATC curve at all the points. The difference between the values of the curves is the fixed cost.

3. Since all the three curves has quantity in the x axis, all of them can be drawn together. The MC curve also first slopes downward, reaches a bottom and then starts sloping upward. The MC curve first crosses the AVC curve and then the ATC curve at their respective bottoms. Refer to a standard text on the problem.

TC       :                   Total cost
dTC     :                  Change in total cost
VC      :                   Variable cost
ATC    :                  Average total cost; AC is the average cost
dATC  :                  Change in ATC
AVC   :                  Average variable cost; VC is the variable cost
dAVC :                   Change in AVC
FC      :                   Fixed Cost.
Q        :                  Quantity.
dQ      :                  Change in Quantity.
MC     :                  Marginal cost.

MC    =                 dTC/dQ
TC     =                  VC + FC
dTC   =                  dVC+dFC

But since FC is the fixed cost and by definition does not change, dFC = 0

So dTC=dVC

So MC = dVC/dQ

            This means that marginal cost, which is the change in total cost per unit of quantity is actualy caused by the change in variable cost only. Now here the question is why the marginal cost curve crosses the Average total cost curve and the Average variable cost curve at their respective bottoms.

             Consider the average of height in a class. Whenever a new student joins, the average changes. If the height of the new student is the same as the present average, then the average does not change, if it is less the average decreases, if it is more the average increases. 

           
             Similarly, the marginal cost is the next new comer. Marginal cost is the next change in the cost (per unit). This next change, if it is more than the average, then the average will increase or vice versa.

             Since MC = dTC/dQ = dVC/ dQ. The new comer is same for both the averages TC and VC.

So, both the AVC and the ATC curve decreases and increases due to the new comer entry MC.

            Now, in the portion of the curves, where they have a downward slope, before they hit their bottoms, the marginal cost values are lesser than than the respective values of the average curves. So the lower MC values drags the averages down. Now when the MC curve, which hits the bottom of its own, slopes upward and first crosses the AVC curve. At the crossing point MC value = AVC value. And after that the MC values drags the AVC curve upwards. This is also what happens to the ATC curve.

            Note that MC is the new comer for both the the averages as we proved earlier.