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Chapter 11 / Production and Cost

# 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:

• 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.
• 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.