When is MC is above, below, and equal to AC?

In the standard diagram with a U-shaped average cost curve, the marginal cost curve intersects the average cost at the latter's minimum.  The MC curve is below the AC curve for quantities below that intersection and above the AC curve for quantities above that intersection.  There are three explanation for these features of the diagram.  Which explanation appeals to you?

Answer 1:  Numerical Example.  The first approach looks at what happens when the average cost changes from 10 to 9, 10, or 11 as the quantity increases from 100 to 101.  These changes in average cost are exaggerated for a 1% increase in quantity, but the numbers for the three cases illustrate the character of the changes.
 

For these exaggerated numbers, the marginal cost is negative for the case on the left.  The cost of producing the extra unit is far more than offset by the decrease in the average cost of the first 100 units.  The case on the right illustrates the opposite possibility.  Producing one more unit increases the cost for the first 100 units so that the extra unit has a marginal cost of 111.  In both cases, producing the marginal unit must have some effect on the efficiency of producing the first 100 units. 

If the average cost does not change because the AC curve is flat, then MC = AC.  This is why the MC curve passes through the minimum of the AC curve.

Answer 2:  Algebra.  Let AC_n denote the average cost for a quantity of n.  Look at what happens when the average cost changes from AC₁₀₀ to AC₁₀₁ as Q changes from 100 to 101.

MC = 101 · AC₁₀₁ - 100 · AC₁₀₀

MC = AC₁₀₁ + 100 · (AC₁₀₁ - AC₁₀₀)

If AC is increasing ( AC₁₀₁ - AC₁₀₀ > 0 ), then MC > AC₁₀₁.  If AC is decreasing ( AC₁₀₁ - AC₁₀₀ < 0 ), then MC < AC₁₀₁.  If AC is constant ( AC₁₀₁ - AC₁₀₀ = 0 ), then MC = AC₁₀₁.

The two terms in the second equation have intuitive meanings.  The marginal cost is the sum of the average cost at 101 units AC₁₀₁ and the second term.  That term is the effect of the change in AC on the cost of producing the first 100 units.  The third set of numbers for Answer 1 illustrates this.  The 101st unit costs the average of 11 plus an additional 100, which is the added cost of producing the first 100 units.

Answer 3:  Calculus.  The first equality is the definition of MC.  The second equality is the definition of AC. 

MC = dTC / dQ = d(Q · AC) / dQ

Apply freshman calculus.

MC = AC + Q · dAC / dQ

If dAC / dQ > 0, then MC > AC.  If dAC / dQ < 0, then MC < AC.  If dAC / dQ = 0, then MC = AC.

Answers 2 and 3 really say the same thing.  Does the calculus version look more sophisticated?

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