As we continue building up the model of firm behavior, we now introduce firm costs. Understanding how firms measure their costs of production is a key component of understanding how firms decide how many workers to hire, what price to charge, or how much output to produce.
One key notion will motivate our analysis of costs: production drives costs. Firms incur costs according to what firms pay for their inputs, so there is an essential link between firms’ inputs and their costs. In general, a firm’s total cost (TC) is the total amount paid out for the firm to acquire its inputs and produce its output. Since we differentiate between a firm’s fixed inputs and variable inputs, it is often convenient to split its costs into its total fixed cost (TFC) - the cost of the firm’s fixed inputs - and its total variable cost (TVC) - the cost of the firm’s variable inputs. Moreover, since all inputs are either fixed or variable, it must be true that total cost \(=\) total fixed cost \(+\) total variable cost, or
\begin{equation*}
TC = TFC + TVC
\end{equation*}
If we extend the simple production model above, where capital \(K\) is the firm’s fixed input, and labor \(L\) is the firm’s variable input, then \(TFC\) is the firm’s cost of capital and \(TVC\) is the firm’s cost of labor. This can be measured with more precision. For example, we use the wage rate (\(w\)) to denote what the firm pays for each unit of labor \(L\text{.}\) Even though the wage rate sounds like it should be measured like an hourly wage rate, it can scale to the unit of labor. If each unit of \(L\) is an hour of labor, then \(w\) is the cost per hour of labor; if each unit of \(L\) is one employee, then \(w\) can represent the annual salary of that employee. For the moment, take each unit of \(L\) to be an hour or labor the firm purchases. If the wage rate is \(w = 12\text{,}\) meaning each worker is paid $12/hour, then the firm’s total variable cost can be expressed as
\begin{equation*}
TVC = 12L
\end{equation*}
Hiring one unit of \(L\) costs (12)(1) = 12, hiring two units costs (12)(2) = 24, and so on. In general, then, we can write
\begin{equation*}
TVC = wL
\end{equation*}
Similarly, we use the rental rate (\(r\)) to measure the cost the firm pays for each unit of capital. If each unit of capital is a pencil, then \(r\) would be the cost per pencil. If capital is the physical location of a restaurant or a factory, then \(r\) would measure the rent paid by the firm to use the space. Therefore, just like with variable cost, the firm’s total fixed cost can be expressed as
\begin{equation*}
TFC = rK
\end{equation*}
This gives an alternate expression for the firm’s total cost: \(TC = rK + wL\text{.}\) One benefit of this \(rK + wL\) expression for cost is that it is equally useful in both short-run and long-run analyses, since it lays out the impact of changing any conceivable input. 1 In most markets, firms do not have any influence over the cost per input they purchase. Therefore, both \(w\) and \(r\) are constants. 2
In the short run, the firm’s fixed input \(K\) is constant. Since \(r\) is constant as well, then in the short run, \(TFC = r\bar{K}\) is constant. This makes sense: in the short run, since firms cannot change their fixed inputs, they cannot change their fixed costs. Imagine a restaurant who is renting a space. If this physical space is the restaurant’s capital, and the restaurant signs a year-long lease, then at a given moment, the restaurant is bound to the lease and cannot change its capital by just moving to another location. This means that the restaurant is responsible for paying its rent in the short run, regardless of how many hours it is open or how many meals it serves. Fixed costs are unavoidable in the short run.
Let’s look at an example of firm cost measurement, extending the short-run production function from above. Here, per usual, labor is the variable input, while capital is fixed as \(\bar{K} = 4\text{.}\) This gives the following production function:
\(K\)
\(L\)
\(q\)
4
0
0
4
1
10
4
2
18
4
3
24
4
4
26
For this example, the rental rate on capital will be \(r = 20\text{,}\) and the wage rate will be \(w = 10\text{.}\) Using the definitions above, we can then calculate the firm’s total fixed cost, total variable cost, and total cost:
\(K\)
\(L\)
\(q\)
\(TFC = rK\)
\(TVC = wL\)
\(TC = TFC + TVC\)
4
0
0
80
0
80
4
1
10
80
10
90
4
2
18
80
20
100
4
3
24
80
30
110
4
4
26
80
40
120
Short-run costs given short-run production function.
Notice, first, that as \(q\) increases, \(TFC\) stays the same. Since \(K\) cannot change, and \(r\) is constant, \(TFC\) does not change. Additionally, notice that if the firm produces no output, it still has a cost of 80. This is due to its fixed inputs remaining fixed: fixed cost is unavoidable in the short run. Moreover, notice that as the firm wants to produce more, its costs increase. This happens because the firm needs to hire more labor to produce more, and hiring more labor increases the firm’s variable cost - and ultimately its total cost. One way to think of this fact is to acknowledge that the firm cannot create additional output for free!
Since our aim is to examine how firms choose an optimal quantity of output to produce in order to maximize their profit, we want to see how costs vary when output varies. By using output \(q\) as our independent variable, we will get the insight we need. As we saw above, short-run TFC does not depend on the number of units of output a firm produces, since it cannot be changed. This means that TFC is not a function of \(q\text{.}\) However, if a firm aims to produce more output, it will need to acquire more labor, which will increase both its TVC and its TC. Each of these is, therefore, a function of \(q\text{:}\)
This expression of the definition of \(TC\) is occasionally used to emphasize that (1) total cost changes when the firm changes the level of output it produces; (2) since TFC is unchanging, this change in total cost is entirely driven by a change in the firm’s total variable cost.
But exactly how much does it cost for the firm to produce additional output? This measure - the change in the firm’s total cost given some change in output - is known as the firm’s marginal cost. Marginal cost is very useful to calculate, since it provides the firm with a measure of how costly it is to keep producing. Like other “marginal” terms in economics, marginal cost is a rate of change. Specifically, it gives the rate of change in total cost given a change in the firm’s output:
\begin{equation*}
MC = \frac{\Delta TC}{\Delta q}
\end{equation*}
We can extend the table above by including the firm’s marginal cost, measuring the change in cost and the change in output between rows. As we discussed above, the firm’s total fixed cost does not change in the short run. The entire change in total cost is driven by changes in total variable cost. That is, \(\Delta TC = \Delta TVC\text{.}\) Therefore, it is equivalent in the short run to express the firm’s marginal cost as \(MC = \frac{\Delta TVC}{\Delta q}\text{.}\)
\(K\)
\(L\)
\(q\)
\(TFC\)
\(TVC\)
\(TC\)
\(MC = \frac{\Delta TC}{\Delta q}\)
4
0
0
80
0
80
-
4
1
10
80
10
90
\(\frac{10}{10} = 1\)
4
2
18
80
20
100
\(\frac{10}{8} = 1.25\)
4
3
24
80
30
110
\(\frac{10}{6} = 1.67\)
4
4
26
80
40
120
\(\frac{10}{2} = 5\)
Short-run costs and marginal cost given short-run production function.
Now, to visualize firm costs. It is useful to express cost as a function of the amount of output a firm produces, so it is important when visualizing cost to represent cost as the dependent variable and output as the independent variable. Using our table above as an example, what do costs look like?
Total fixed cost does not change when output changes, so the \(TFC\) function is constant (horizontal). What about \(TVC\text{?}\) Certainly, as the firm produces more, the firm’s total variable cost increases, so the relationship should be an increasing one. But even if the curve is upward sloping, what is its shape? At first glance, the table might suggest that the curve is linear, since the \(TVC\) goes up by 10 each row. But be careful! Our independent variable is not labor (which goes up by one every row), but output, which goes up by different amounts as more workers are hired. If we graph the relationship between \(TVC\) and \(q\text{,}\) we actually see this shape:
Figure3.2.1.Given the table above, \(TVC\) is increasing and convex.
Since \(TC = TFC + TVC\text{,}\) adding the two curves together - one horizontal and one convex - will generate the total cost curve. Adding a constant number to \(TVC\) gives \(TC\) the exact same convex shape as \(TVC\text{,}\) just shifted up so its y-intercept is equal to the \(TFC\text{.}\)
Figure3.2.2.Given the table above, \(TFC\) is constant, while \(TVC\) and \(TC\) are increasing and convex. \(TC\) is a parallel upward shift of \(TVC\text{,}\) shifted up by a vertical distance equal to \(TFC\text{.}\)
\(TC\) is increasing and convex. It is increasing because additional output will cost the firm more in the form of additional labor hired. But why is the shape convex? First, let’s interpret. Convexity here shows that as the firm produces more, not only does cost increase, but it increases at an increasing rate. Additional units of output get more and more costly!
But why does additional output get more costly? As the firm wants to produce more output, it needs more labor. But as it produces more, its labor gets less and less productive because of diminishing marginal product of labor! Therefore, the firm needs to hire even more workers to produce high quantities, and since it pays each worker the same wage rate, its variable cost starts to skyrocket. \(TC\) is increasing and convex, growing steeper and steeper as \(q\) increases precisely because crowding out occurs.
This can be seen by interpreting the marginal cost as the slope of the \(TVC\) and \(TC\) curves. Slope is about \(\frac{rise}{run}\text{,}\) and since \(MC = \frac{\Delta TC}{\Delta q} = \frac{\Delta TVC}{\Delta q}\text{,}\) marginal cost gives the change in \(y\) variable over the change in \(x\) variable for both \(TC\) and \(TVC\) curves. So, when \(TC\) is convex, this means the curve has an increasing slope, or, equivalently, increasing marginal cost. But this is essentially what we just said! As the firm produces more, its added cost to produce additional units (\(MC\)) increases.
Figure3.2.3.The convexity of \(TVC\) (and \(TC\)) implies that its slope - and therefore the firm’s marginal cost - increases as \(q\) increases.
The impact of production on cost is not an accident. This is what we mean when we emphasize how production drives cost: whenever the production function is concave, the (total) cost function is convex. Both shapes are driven by diminishing marginal product of labor! Crowding out in the production process drives cost up sharply under assumption P2.
Figure3.2.4.Concave production leads to convex cost.
Production also drives cost under assumption P2B. While here the relationship is more nuanced, the underlying idea is the same. If production exhibits eventually diminishing marginal product of labor, the production function is first convex (specialization) then eventually concave (crowding out). When translated to costs, this gives firm costs first as concave (specialization) then eventually convex (crowding out). The intuition is similar to assumption P2. Under specialization, since hiring labor increases productivity, firm costs are increasing but slowly - at a decreasing rate. Once specialization ends and crowding out begins, workers start to get less and less productive, and firm costs begin to increase more steeply - at an increasing rate. Production shape and cost shape are inversely related!
Figure3.2.5.Eventually concave production leads to eventually convex cost. Inflection point occurs at \(\hat{q}\text{.}\)
This relationship can also be expressed in terms of marginal product of labor and marginal cost. Whenever the firm experiences diminishing \(MP_L\text{,}\) the firm will experience increasing \(MC\text{:}\) as crowding out renders workers less productive at the margin, added units of output will get increasingly costly. If assumption P2B is at play, we can similarly state that whenever there is increasing \(MP_L\text{,}\) the firm will experience decreasing \(MC\text{.}\)
Assumption P1
Assumption P2
Assumption P2B
productive \(L\)
diminishing \(MP_L\)
eventually diminishing \(MP_L\)
Production \(f(L)\) is ...
increasing
concave
convex then concave
Total cost \(TC\) is ...
increasing
convex
concave then convex
Marginal product of labor \(MP_L\) is ...
positive
decreasing
increasing then decreasing
Marginal cost \(MC\) is ...
positive
increasing
decreasing then increasing
Figure3.2.6.Two shapes of MC: P2 (left) and P2B (right). \(\hat{q}\) represents the inflection point under eventually diminishing marginal product of labor.
Just as with production, economists can generalize their analysis of cost by using actual mathematical functions for cost functions. The firm could, for example, have a total cost function expressed as a function of output such as:
But we have seen the cost function expressed as \(TC = rK + wL\text{,}\) where in the short run, since \(\bar{K}\) is fixed, the total cost function is
This allows for a translation. The constant part of the function \(TC(q) = 20 + 3q^2\) lines up with the firm’s total fixed cost, since as a constant value, it is the part of the firm’s cost that does not change when output changes. This is the exact definition of TFC! This leaves the \(3q^2\) as the firm’s total variable cost. Since this particular function for TVC is increasing and convex, the firm must be experiencing diminishing marginal product of labor. After all, production drives cost! 3
Figure3.2.7.TFC, TVC(q), and TC(q) for the total cost function \(TC(q) = 20 + 3q^2\text{.}\)
We can also capture a firm’s marginal cost using a mathematical function. In the above example, a firm with a total cost function of \(TC(q) = 20 + 3q^2\) would have a marginal cost function of \(MC(q) = 6q\text{.}\) 4 This expresses diminishing marginal product: as the firm produces more units, additional units add more and more to
The relationship between the shape of production (in the form of assumptions P2 and P2B) and the shape of cost curves is the cornerstone of firm theory thus far. The simple-seeming assumptions on marginal product of labor shape not only the firm’s production processes, but their total variable cost, total cost, and marginal cost curves. This relationship lays the foundation for our analysis of profit maximization, which is up next.
In Depth3.2.8.Average Total Cost.
For some applications of firm theory, it can be useful to compute a firm’s average total cost, or ATC. A firm’s ATC is its cost per unit of output: how much does it cost on average to produce one unit of output? The calculation for computing ATC is a simple average:
\begin{equation*}
ATC = \frac{\text{ total cost } }{\text{ units of output } } = \frac{TC}{q}
\end{equation*}
For example, a firm who could produce 20 units of output at a total cost of $500 would have an average cost of
On average, a unit costs $25 to produce. Now, of course, not every unit costs exactly $25 to produce - it’s an average! Since marginal cost changes as production changes, we know the first unit, the second unit, and the twelfth unit will all likely cost different amounts to produce.
So, what does ATC look like? And what factors determine this shape? First, the presence of fixed costs will likely cause ATC to start off at a high value. Even if the producing one unit of output doesn’t cost much, it is carrying the weight of its variable cost plus the entire fixed cost. As the next few units are produced, the burden of the fixed cost is spread across those units, and on average, the cost per unit drops. However, eventually, as diminishing marginal product of labor takes hold, additional units will get so increasingly costly to produce that the high marginal cost will pull the average cost up.
The combination of these forces gives us an ATC curve that is U-shaped. We can see this using the numerical example above. If \(TC(q) = 20 + 3q^2\text{,}\) for example, then
The graph below gives the U-shaped curve for ATC. The first downward-sloping segment of the curve captures the phenomenon of fixed cost spreading over additional units before marginal costs increase too drastically. The second upward-sloping segment of the curve reflects the higher and higher marginal costs as additional units are produced under diminishing marginal product of labor.
Left: Total cost and total variable cost. Right: Average total cost.
Besides calculating algebraically, how can the ATC curve be derived? To see this, let’s return to the total cost curve under diminishing marginal product of labor. Since ATC is defined as \(ATC = \frac{TC}{q}\text{,}\) we can write this as
While it might seem pointless, consider the cost graph below. If we draw a line from the origin out to the curve - known as a secant line - then \(TC - 0\) represents the change in the y-value (cost) along that line, while \(q - 0\) represents the change in x-value (quantity) along that line. This ratio of change is just a slope! That is, \(ATC\) at any point can be seen as the slope of the secant line from the origin to the curve at that point!
As the secant lines move from left to right on the cost curve, notice that the secant lines first get flatter (highlighted in blue in the graph below), reach a flattest secant line, then begin to get steeper again (highlighted in red in the graph below). This progression of slopes gives us the shape of ATC: first decreasing, then reaching a minimum, then increasing again. We can track the U-shape directly through the slopes of secants to the total cost curve!
Left: The secant line is drawn from the origin to the TC curve, and its slope is ATC. Right: As \(q\) increases, the secant lines get flatter and ATC decreases (blue), then eventually, secant lines get steeper and ATC increases (red).
This highlights an important relationship between average total cost and marginal cost. If marginal cost is the slope of TC at any point on the cost curve, and \(ATC\) is the slope of the secant to any point on the cost curve, we can get a direct comparison on the size of each. On the left side of the graph, at an output level like \(q_1\text{,}\) the TC curve is flatter than the secant line, indicating that \(MC \lt ATC\text{.}\) On the right side, at an output level like \(q_3\text{,}\) the TC curve is steeper than the secant line, showing that \(MC > ATC\text{.}\) Finally, notice that there is exactly one point, here indicated by \(q_2\text{,}\) where the secant line has the same slope as total cost itself: at \(q_2\text{,}\)\(MC = ATC\text{.}\)
We can now interpret this relationship between marginal cost and average total cost in three regions of output:
For low levels of output, \(MC \lt ATC\text{.}\) Additional units cost less to produce than the average unit. This pulls the average cost per unit downward.
For high levels of output, \(MC > ATC\text{.}\) Additional units cost more to produce than the average unit. This pulls the average cost per unit upward.
At the lowest point on \(ATC\text{,}\)\(MC = ATC\text{.}\) The lowest per unit cost occurs exactly where the per unit cost equals the marginal cost.
This should make some intuitive sense. When the average cost per unit is decreasing, it must be because additional units of output cost less than the average unit to produce - this pulls the average down!
Left: Compare the slope of the secant line (blue) with the slope of total cost (tangent lines in red). Right: The relationship between marginal cost and average total cost.
