# 11.4 Short-Run Total Costs

As we discussed in Chapter 2, it makes sense to consider how production functions operate in the **short run**, when some inputs are fixed, and in the **long run**, when all inputs may be varied. For this chapter we will consider the problem of a firm operating in the short run; in Chapter 18, we will turn to the firm’s behavior in the long run.

Of course, firms generally have many inputs, which might be varied according to a range of time frames: a steel plant might be able to increase or decrease its labor force quickly but change the size of its physical factory only very slowly, while a Silicon Valley startup may be able to scale up virtual servers in an instant but find it difficult to hire software developers in a tight labor market. We’ll follow the convention of treating “labor” as variable and “capital” as fixed, but you should be clear that these are simply metaphors.

If output is a function of just two inputs, labor and capital, and if capital is fixed, then the labor required to produce $q$ units of output is found by inverting the production function to solve for $L$ as a function of $q$. With the production function $q = \sqrt{LK}$, if $K$ is fixed at some $\overline K$, then we can solve for the amount of labor required to produce $q$ units of output, given $K = \overline K$:
\(\begin{aligned}
q &= \sqrt{L \overline K}\\
q^2 &= L \overline K\\
L^c(q | \overline K) &= \frac{q^2}{\overline K}
\end{aligned}\)
We call $L^c(q | \overline K)$ the **short-run conditional demand for labor**: that is the labor required to produce $q$ units of output when capital is fixed. Of course, the “short-run conditional demand for capital” is just $K^c(q) = \overline K$: that is, capital is just fixed at $\overline K$.

For example, suppose $\overline K = 32$. Then in order to produce $q = 16$ units of output, you would need $L = 16^2/32 = 8$ units of labor; to produce $q = 32$ units of ouptut, you would need $L = 32^2/32 = 32$ units of labor; and to produce $q = 48$ units of output, you would need $L = 48^2/32 = 72$ units of labor. Notice that this is exactly the same as the short-run labor requirement function we used in Chapter 3 when plotting the PPF. In the context of the firm, we call this function the firm’s **conditional demand for labor** when capital is fixed at $\overline K$.

We can visualize this in two ways: an isoquant diagram, and a production function diagram. In an isoquant diagram, we can connect all the points used to produce different levels of output. We call this an **expansion path**, indicating that it shows the various combination of inputs used as the firm *expands* its production. It’s easy to see that the **short-run expansion path** will just be a horizontal line at with a height at $\overline K$:

In the lower graph, we have the short-run production function showing output as a function of labor. Note that as the firm acquires more capital ($\overline K$ increases), the amount of labor required to produce any given quantity of output decreases, shifting the short-run production function to the left. Another way of thinking about this is that it increases the slope of the production function – i.e., the $MP_L$ — at any level of $L$.

## Short-Run Costs

If capital is fixed at $\overline K$, then our conditional labor demand will be $L^c(q | \overline K)$, so the cost of producing $q$ units of output when facing input prices $w$ and $r$ is just the cost of the required labor plus the cost of capital:
\(c(q) = wL^c(q | \overline K) + r \overline K\)
For example, with our Cobb-Douglas production function $f(L,K) = \sqrt{LK}$, the labor required to produce $q$ units of output is
\(L(q | \overline K) = \frac{q^2}{\overline K}\)
so the total cost of production in the short run is
\(\begin{aligned}
c(q) &= wL(q | \overline K) + r\overline{K}\\
&= w\frac{q^2}{\overline K} + r\overline{K}
\end{aligned}\)
For example, suppose $w = 8$, $r = 2$, and $\overline K = 32$. Then this becomes
\(c(q) = \frac{8q^2}{32} + 2 \times 32 = \tfrac{1}{4}q^2 + 64\)
We might think of this as the **total cost** of producing $q$ units of output, which we’ll sometimes write $TC(q)$ to distinguish it from other kinds of costs. As you can see, this total cost function has two terms: $r \overline K$, which doesn’t vary with $q$, and $wL(q|\overline K)$, which does vary with $q$. We call the costs which do not vary with $q$ **fixed costs** (a constant $F$), and those that do **variable costs** (a function $VC(q)$). In the specific case we looked at above, the firm has fixed costs of $F = 64$, and variable costs of $VC(q) = {1 \over 4}q^2$:

- Total costs: $TC(q) = 64 + \tfrac{1}{4}q^2$
- Fixed costs: $F = 64$
- Variable costs: $VC(q) = \tfrac{1}{4}q^2$

We can plot these three cost curves in a graph with **output** on the horizontal axis and **dollars** on the vertical axis:

There are a few things to note about this diagram:

- When $q = 0$, variable costs are zero but fixed costs are still incurred; so the vertical intercept of the firm’s short-run total cost curve is its fixed cost.
- The TC and VC curves are always separated by F.
- An increase in $w$ does not affect the fixed costs, but increases how quickly costs increase.
- An increase in $r$ increases the fixed costs, but doesn’t change how quickly costs increase.
- An increase in $\overline K$ increases the fixed costs
*and*decreases how quickly costs increase. In other words, if you have a larger factory, you have more up-front costs but lower costs of increasing production. This is bad if you’re only producing a few units, but good if you want to produce a large quantity.

Having established the *total costs* of production, let’s now turn to the *unit costs*: specifically, the average and marginal costs of production.