17 Cost Curves
For more information on these topics, see Varian Chapter 21: Cost Curves.
17.1 Classwork 17: Cost Curves
Marginal costs and average costs. Let a firm’s costs be given by \(C(q) = q^2 + 10 q + 100\).
Marginal cost (MC) measures the change in total cost for a small change in output. Mathematically, it’s the derivative of the total cost function with respect to output: \(\frac{dC}{dq}\). The marginal cost curve is typically increasing eventually: it may have a J-shape from initial efficiency gains, but then it increases and doesn’t stop increasing as firms encounter resource constraints or diminishing returns. Calculate this firm’s marginal cost function and visualize it on a plot.
If the firm’s total revenue is given by \(TR = 20 q\), use the \(MR = MC\) rule to show that when the firm profit maximizes, it produces \(q = 5\).
Average cost (AC) is the cost per unit of output. It’s calculated by dividing total cost by the quantity produced: \(AC = \frac{C}{q}\). A firm’s cost function is made up of fixed costs which must be paid whether or not the firm produces anything (rent), and variable costs which are 0 if \(q=0\), and which increase with the firm’s output. The shape of the average cost curve is typically U-shaped because of declining average fixed costs and eventually increasing average variable costs. Average fixed costs decline as q increases: if rent is $2K per month and you produce 10 units, your average fixed cost is $200 per unit. But if you produce 1000 units, that’s an average fixed cost of just $2 per unit. Average variable costs eventually increase as output increases as firms reach their capacities. Calculate and plot \(AC\) for this firm to verify its U-shape. Explain why the firm’s fixed costs are $100 and their variable costs are \(q^2 + 10q\).
Zoom in on your plot from part c to show that \(AC\) reaches a minimum at \(q = 10\) and \(AC = 30\).
Show that \(MC = AC\) when \(AC\) reaches its minimum. This is a relationship that will always hold, and here’s the intuition about why: say I’m grading exams and I’m keeping a running average: 90, then 85, then 83. If my running average is 83 a second time, what do you know about the test I must have just graded? You know it had to have been an 83 too. In the cost function example, say average cost is decreasing, then it’s equal to 83 twice in a row, and then it’s increasing. When AC is the same thing twice in a row (like next to its minimum), you know the MC was equal to AC.
Long-Run vs. Short-Run Costs. Firms have different cost functions in the long-run versus in the short-run. In the short-run, at least one factor of production is fixed (e.g. factory size). The long-run on the other hand, is defined as the length of time it takes for every factor of production to become variable, at least in the sense that the firm can go out of business and produce 0 units at a cost of $0. So in the long-run, the firm can choose the lowest cost factory size for its output level. Suppose a firm has two possible plant sizes: small and large. The cost functions are:
- Small plant: \(C(q) = q^2 + 5q + 100\)
- Large plant: \(C(q) = 0.5q^2 + 10q + 200\)
Plot the cost curves in different colors. If the firm has a low level of output, the small plant will be the lowest cost choice. But if the firm’s level of output is large, the large plant will be the lowest cost choice. Show algebraically that the transition happens at \(q = 20\).
Visualizing the long-run average cost curve: the firm will choose the plant size with the smallest average cost given its output level \(y\), so the long-run average cost curve is the bold line that is the lower envelope of short-run average cost curves associated with each plant size.
17.2 Practice Questions
Do these questions after you’ve completed Classwork 17.