Why does the marginal cost curve always intersect with the average total cost curve at its lowest point?
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The reason for this is that the marginal cost is part of the average total cost. Therefore, a change in the marginal cost of making the next unit of output will affect the average total cost.
We know that the marginal cost (MC) curve is upward sloping when it intersects with the average total cost (ATC) curve. So now let us think about why the MC curve must intersect with the ATC curve at the ATC's minimum.
When the MC is less than the ATC, each new unit of output lowers the the ATC. This is mathematically necessary. If you have an average and you add another number that is lower than the average to it and then you take the new average, it has to go down because the number that was added to it was lower. This means that for as long as MC is less than ATC, ATC will go down with each extra unit made.
At some point, MC rises to the point that it equals ATC and the curves intersect. Now let's look at what happens from there. MC keeps rising. It is now greater than ATC. This means that ATC has to go up because every new unit produced increases ATC.
When we put this all together, it means that ATC decreases as MC rises to intersect with it. After they intersect, they both rise. This happens because MC is part of ATC.
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