Lecture 15: Costs in the Short Run and the Long Run

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Note: The material in this lecture is meant to largely be a review of Econ 1, so we’ll go kind of quickly. If you’re a little rusty on some of these things, you might want to review your notes from Econ 1 (or high school econ).

In the last lecture we derived a firm’s conditional demands for labor and capital from its cost-minimization problem. In this lecture we’ll talk about how a firm’s costs vary as it scales up production: that is, we’d like to derive the firm’s cost function, $c(q)$, which relates the quantity it wants to produce ($q$) with the cost of producing that quantity.

A critical question to ask is whether a firm can scale all its inputs at a moment’s notice. It might be easy to hire and fire workers on a day-to-day basis, but time-consuming to build a new factory. To account for the fact that different inputs may take different amounts of time to scale, we’ll introduce the concept of the short run and the long run. In the short run, we’ll assume that some factors of production are fixed; in the long run, we’ll assume that all factors of production can be adjusted.

Total, average, and marginal costs

Before we turn to deriving cost functions, let’s develop some tools with which to analyze those costs.

Total costs are measured in dollars: for example, the total cost of flying 200 passengers on a plane from San Francisco to New York might be €100,000.

We can also look at unit costs: that is, costs per unit of output. These are (naturally) measured in dollars per unit of output. There are two types of unit costs we might be interested in:

Average costs: This is just the total cost divided by the number of units. For example, if if costs €100,000 to fly 200 passengers from SFO to NYC, the average cost per passenger is $AC = {TC \over q} = { €100,000 \over 200 \text{ passengers} } = €500 \text{ per passenger}$
Marginal costs: This is the cost to increase production by one unit: in other words, the derivative of the total cost. It may be very different from the average cost! For example, adding a 201st passenger on a flight surely doesn’t increase the cost by an additional €500 – it probably only adds a few dollars of cost.

Because the marginal and average costs are both measured in dollars per unit, in common parlance they may be mistakenly used interchangeably. 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 (TC) 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:

See interactive graph online here.

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 (i.e., if the cost of producing zero units isn’t zero), 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.

Now that we’ve established the meaning of total, average, and marginal costs, let’s look at how a firm’s production function is related to its costs in the long run and the short run.

Long-Run Costs

In the long run, the firm can vary all of its inputs. This means that its costs may be derived from the conditional demands for labor and capital that we developed last time, $L^c(w,r,q)$ and $K^c(w,r,q)$. In particular, the long-run cost of producing $q$ units of output when facing input prices $w$ and $r$ is just the cost of the labor plus the cost of capital: $c(w,r,q) = wL^c(w,r,q) + rK^c(w,r,q)$ Students in 50Q can notice that this is exactly the same as an expenditure function from consumer theory.

Example: Cobb-Douglas Production Function

For the production function $f(L,K) = \sqrt{LK}$, we found that the conditional demand functions were $\begin{aligned} L^c(q) &= \sqrt \frac{r}{w} \times q\\ K^c(q) &= \sqrt \frac{w}{r} \times q \end{aligned}$ Given these, we can find the long-run total cost of producing $q$ units, for general $w$ and $r$, when we can vary both labor and capital: $\begin{aligned} c^{LR}(q) &= wL^c(q) + rK^c(q)\\ &= w \times \left(\sqrt \frac{r}{w} \times q\right) + r \times \left(\sqrt{\frac{w}{r}}\times q\right)\\ &= 2\sqrt{rw}q \end{aligned}$ For example, if $w = 8$ and $r = 2$, the total cost of producing $q$ units of output would be $c^{LR}(q) = (2\sqrt{8 \times 2})q = 8q$. Note that because this particular production function exhibits constant returns to scale, the long-run cost function is linear: doubling output doubles inputs, and therefore doubles total cost.

You can use the graph below to see how the cost-minimizing combination of inputs and the total cost of producing a certain amount $q$ varies with $w$, $r$, and $q$:

See interactive graph online here.

Notice that this tangency condition is exactly the same as the expansion path!

Returns to scale and long-run cost functions

Let’s now see how returns to scale and the long-run total cost function are related, and also see what some preliminary analysis of average and marginal costs reveals. To do so, let’s stick with the Cobb-Douglas production function, but let’s choose a really simple one: $f(L,K) = L^aK^a$ For example, if $a = {1 \over 2}$, we have $f(L,K) = L^{1 \over 2}K^{1 \over 2}$ and if $a = 1$, then we have $f(L,K) = LK$ Let’s now derive the firm’s total costs. To make things super simple, let’s assume $w = r = 1$. If we follow the procedure above (you can trust me here, or verify it yourself!) we end up with the total cost function $c(q) = 2q^{1 \over 2a}$ The average and marginal costs for this total cost function are $\begin{aligned} AC(q) &= {c(q) \over q} = \frac{2q^{1 \over 2a } }{q} = 2q^{ {1 \over 2a} - 1}\\ MC(q) &= {dc(q) \over dq} = {1 \over a}q^{ {1 \over 2a} - 1} \end{aligned}$ Let’s think about returns to scale and total costs for various levels of $a$:

When $a < {1 \over 2}$, by the above argument, the production function exhibits decreasing returns to scale. We can also see that it has increasingly steep total costs: for example, if $a = {1 \over 4}$, we have $\begin{aligned}c(q) &= 2q^2\\ AC(q) &= 2q\\ MC(q)&= 4q \end{aligned}$ So both our average and marginal costs of production are increasing as we scale up. Intuitively, if you have decreasing returns to scale, as you scale up, your inputs become less and less productive, so you need to add more and more of them to increase production by the same percentage.
When $a = {1 \over 2}$, likewise, we have constant returns to scale and a linear long-run cost function:$\begin{aligned}c(q) &= 2q\\ AC(q) &= 2\\ MC(q)&= 2 \end{aligned}$ In this case every unit has the same average and marginal cost, so doubling production exactly doubles your cost.
Finally, when $a > {1 \over 2}$, we have increasing returns to scale, meaning that each additional unit become cheaper in the long run. For example, if $a = 1$, we have $\begin{aligned}c(q) &= 2\sqrt{q}\\ AC(q) &= {2 \over \sqrt{2}}\\ MC(q)&= {1 \over \sqrt{q}} \end{aligned}$So both average costs and marginal costs are decreasing.

Summary of long-run costs

That was a lot of math, and your head might be swimming a little right now. So let’s just reiterate the most important parts to take away from the above analysis:

The firm’s long-run total cost function is calculated by plugging the conditional demand functions into the expression $c(w,r,q) = wL(w,r,q) + rK(w,r,q)$
If a production function exhibits decreasing returns to scale, its inputs become less productive as it scales up, so it has increasing unit costs.
If a production function exhibits constant returns to scale, its inputs’ productivity is constant as it scales up, so it has constant unit costs.
If a production function exhibits increasing returns to scale, its inputs become more productive as it scales up, so it has decreasing unit costs.

Short-Run Costs

If there are only two factors of production, and one of them is fixed, the short-run production function becomes a univariate function: that is, $f(L, \overline K)$ may be written as $f(L | K = \overline K)$. We can interpret this as the function defined by taking a vertical slice of the production function at $K = \overline K$. For example, if $f(L,K) = \sqrt{LK}$, then the short-run production function may be seen in the diagram below:

See interactive graph online here.

We might also just write $f(L)$ having substituted $\overline K$ in: for example, if $f(L,K) = \sqrt{LK}$, then if $\overline K = 100$ we could just write this as the univariate function $f(L) = 10 \sqrt{L}$ This is shown in the right-hand graph above.

Scaling and the marginal product of labor

A critical attribute of a production function is how the marginal product changes as inputs are added. In this case, we can ask how adding labor affects the $MP_L$. One can imagine that some labor-intensive production processes are tiring, and produce less output for each additional hour of labor. This is reflected in a decreasing marginal product of labor, as shown in the diagram below:

See interactive graph online here.

On the other hand, a different production technology might exhibit different characteristics. If a firm has giant machines that can work for hours at a time with little need for human oversight, the output they can produce for each hour of labor may be fairly constant: for example, 100 units per hour. This production process would be described by the linear production function $f(L) = 100L$, with $MP_L = 100$.

Total costs in the short 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 $\overline K$:

See interactive graph online here.

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 $L$.

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:

See interactive graph online here.

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.

Short-run average costs

Let’s now look at the unit costs associated with a short-run production function. Because we have three functions (total, fixed, and variable), we have three relevant average cost functions:

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:

See interactive graph online here.

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.

Short-run 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 $c(q) = wL(q) + r\overline K$ 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.

See interactive graph online here.

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 short-run and long-run costs

Let’s conclude by thinking about the relationship between short-run and long-run costs.

Recall the difference between the short run and the long run: in the short run, some input is fixed, while in the long run all inputs are variable. Intuitively, this means that in the long run you can produce any quantity at its cost-minimizing combination of outputs (i.e. its lowest possible cost), while in the short run you are “stuck” with some fixed amount of one good.

It follows that for any quantity $q$, the long-run cost of producing $q$ units of output must be no greater than the cost in the short run. As a consequence, we say that the long-run total cost is the lower envelope of the short-run total cost.

However, it’s equal to one particular short-run cost: the cost if you were lucky enough to be “stuck” with just the right cost-minimizing amount of your fixed input to produce that amount of output. Visually, we can see this if we look at the short-run and long-run expansion paths. At the point where they cross, the quantity of output produced by that combination of labor and capital has the same cost in the short run and the long run – in other words, you are “stuck” with just the right quantity of capital.

At every other quantity, the short-run cost is higher than the long-run cost. But at that quantity, the two are equal.

We’ll spend a fair amount of time looking at the following graph in lecture…play around with it for now, but plan on really understanding what’s going on in class.

See interactive graph online here.

Summary

This was a lot of stuff! Hopefully most of it was review. Let’s finish by considering the big-picture:

A firm’s production function determines the shape of its cost curves.
A production function’s returns to scale determines the shape of its long-run cost curves.
Assuming capital is the fixed input, a production functions marginal product of labor determines the shape of its short-run cost curves; specifically, its $MP_L$ is inversely related to its $MC$.
A firm’s long-run total and average cost curves represent the lower envelope of the short-run total and average costs curves.

What you should be able to do after this lecture:

Determine whether a production function exhibits decreasing, increasing, or constant returns to scale
Given a production function, derive a firm’s long-run total, average, and marginal costs of production
Given a production function and a fixed level of capital, derive a firm’s short-run fixed, variable, and total costs; and its various unit cost curves (AFC, AVC, ATC, MC)
Plot the long-run and short-run expansion paths, and find the quantity at which they cross.

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