# What Is Marginal Average Cost?

To calculate the marginal average cost, we need to first calculate the average cost,

$latex \displaystyle \overline{TC}\left( Q \right)=\frac{TC\left( Q \right)}{Q}$

where TC(Q) is the total cost to produce Q units. Once we have the the average cost function, the marginal average cost is simply its derivative,

$latex \displaystyle \text{Marginal Average Cost}=\overline{TC}{{\,}^{\prime }}\left( Q \right)$

Example Suppose the total cost (in thousands of dollars) to produce Q units is

$latex \displaystyle TC\left( Q \right)=\frac{9Q-5}{7Q+2}$

a. Find the the average cost of producing 40 units.

b. Find the average cost function $latex \displaystyle \overline{TC}\left( Q \right)$.

c. Find the marginal average cost function $latex \displaystyle \overline{TC}{{\,}^{\prime }}\left( Q \right)$.

d. Find and interpret $latex \displaystyle \overline{TC}{{\,}^{\prime }}\left( 40 \right)$.

The marginal average cost is simply the slope of the tangent line to the average cost$latex \displaystyle \overline{TC}{{\,}^{\prime }}\left( Q \right)$. The slope has vertical units of thousands of dollars per unit and horizontal units of units. So the rate has units of thousands of dollars per unit per unit. This means that if the production were to increase by one, the average cost would drop by 0.0007701 thousand dollars per unit or 0.7701 dollars per unit. This means that the average cost is decreasing…probably a good thing for the bottom line.