11.5 Short-Run Unit Costs

Thus far we’ve been analyzing the total costs (in dollars) of producing some quantity $q$ of output. Of equal importance is analyzing a firm’s unit costs (in dollars per unit of output): specifically, its average and marginal costs.

Average costs

Given a cost in dollars (total, fixed, or variable), the average cost is just that cost divided by the number of units produced. So, for example, if it costs a total of $€96$ to produce 8 units, the average cost per unit would be $€96 \div 8$ units = $€12$ per unit; and if that $€96$ was comprised of $€64$ of fixed costs and $€32$ of variable costs, the average fixed costs would be $€64 \div 8$ units = $€8$ per unit, and the average variable costs would be $€32 \div 8$ units = $€4$ per unit.

The average cost functions are therefore just the cost functions divided by $q$:

• Average total cost: $ATC(q) = {TC(q) \over q}$
• Average fixed cost: $AFC(q) = {F \over q}$
• Average variable cost: $AVC(q) = {VC(q) \over q}$

In the specific case we looked at above, the firm has fixed costs of $F = 64$, and variable costs of $VC(q) = {1 \over 4}q^2$, these would be: \(\begin{aligned} TC(q) = 64 + \frac{q^2}{4} &\Rightarrow ATC(q) = {64 \over q} + {q \over 4}\\ F = 64 &\Rightarrow AFC(q) = {64 \over q}\\ VC(q) = \frac{q^2}{4} &\Rightarrow AVC(q) = {q \over 4} \end{aligned}\) We can plot these three cost curves in a graph with quantity on the horizontal axis and dollars on the vertical axis:

Things to note about these diagrams:

• When $q = 0$, AFC is infinite; as $q$ increases, AFC approaches zero.
• Therefore, since ATC = AFC + AVC, ATC starts out being very close to AFC, but as $q$ increases, ATC asympotitcally approaches AVC.
• If (as in this case) AVC is increasing for all values of $q$, and the firm has nonzero fixed costs, the ATC curve will have a U shape.

Marginal Cost

The marginal cost is the cost of producing an additional unit of output; mathematically, it’s just the derivative of the total cost function. Note that if we write the total cost as

then the marginal cost is \(MC = {dc \over dq} = w \times {dL \over dq}\) Since $L(q)$ is just the inverse of the production function, we can rewrite this as \(MC = w \times {1 \over dq/dL} = w \times {1 \over MP_L}\) In other words, the marginal cost is inversely related to the marginal product of labor. This makes sense: the more productive labor is, the less costly it is to produce each unit of output.

Importantly, this means that a diminishing marginal product of labor will be related to an increasing marginal cost, as shown in the diagrams below. The left-hand diagram shows the production function; $L$ is on the horizontal axis, and $q$ is on the vertical axis. The right-hand diagram shows the total cost curve: $q$ is on the horizontal axis here, while the cost in dollars is on the vertical axis.

Things to note:

• as quantity (and labor) increase, the $MP_L$ decreases (each additional unit of labor is less productive) and the $MC$ increases (each additional unit of output is more expensive).
• an increase in $w$ increases the $MC$ without affecting the $MP_L$
• an increase in $r$ only affects fixed costs, so doesn’t affect $MC$ (or $MP_L$).
• an increase in $\overline K$ increases $MP_L$ and therefore decreases $MC$.

Relationship between average and marginal cost

Both the marginal and average costs are costs per unit, and in common parlance may be mistakenly used interchangeably. For example, when thinking about drug prices, we often hear that pharmaceutical companies mark up their price far above their cost of production. However, that generally refers to the marginal cost of producing an additional dose of the drug. When a pharmaceutical company looks at its overall costs, it includes all the costs of researching the successful drugs, plus the costs of researching a range of drugs which turned out not to be successful. So its average costs (total costs of production divided by doses actually produced) may be much greater than its marginal costs (the physical cost of producing one additional dose).

There is an explicit relationship between average and marginal costs, though: specifically, that marginal costs “pull” average costs toward them: that is, if $MC < AC$ then AC will be decreasing, and if $MC > AC$ then AC will be increasing. There are three ways to show why this is the case: intuitively, mathematically, and graphically.

• Intuitively, think about your own GPA. This is an average of all your grades to date. Suppose you have less than a perfect straight-A average, but above a C average. Now think about the grade you get in this class as a “marginal” grade. If you get an A in this class, it will pull your GPA up; if you get C in this class, it will pull your GPA down. In other words, when you introduce a new data point into a set, if it’s lower than the existing average, the new average will be lower than before; and if it’s higher than the existing average, the new average will be higher than before.
• Mathematically, the average cost is given by\(AC(q) = {c(q) \over q}\)by the quotient rule, the derivative of the average is\(AC^\prime(q) = {c^\prime(q) q - c(q) \over q^2} = {MC - AC \over q}\)Therefore this is positive when $MC > AC$, and negative when $MC < AC$.
• Graphically, if we think of any point $(q, c(q))$ along a total cost curve, the MC may be represented as the slope of the curve at that point, and the AC may be represented as the slope of a ray extending from the origin to that point. Therefore, if the slope of the TC curve is less than the AC, moving to the right along the curve will pull the AC ray down; while if the slope of the TC curve is greater than the AC, moving to the right along the curve will pull the AC ray up:

This relationship between marginal and average costs helps us to understand the “U” shape of many average cost curves. Specifically, if there are any fixed costs, then the $AC$ curve starts off infinitely large; and if MC is constantly increasing, then there will be some quantity at which $MC = AC$. To the left of that point, we’ll have $MC < AC$ and to the right of that point we’ll have $MC > AC$; therefore at the point where $MC = AC$, the average cost will be minimized.