13.5 Supply with More Complex Cost Functions

In this chapter we looked at a firm with a specific kind of cost: both its marginal cost and average cost curves started at zero and rose monotonically, with the marginal cost always being higher than the average variable cost. This meant that (1) there was always a unique quantity for which $p = MC$, and (2) at that quantity, the price would be higher than the AVC; hence, even if the firm ran a loss, the size of the loss was always less than the fixed cost — that is, the firm could at least cover its variable costs in the short run. For such a firm, the supply curve simply is the marginal cost curve.

However, for other kinds of cost functions, this might not be the case. Consider a firm with the cost function \(c(q) = 81 + 16q - 2q^2 + \tfrac{1}{6}q^3\) This cost function will raise several difficulties. Let’s handle each in turn.

Complexity #1: Two quantities where $p = MC$

This firm’s marginal cost is \(MC(q) = c^\prime(q) = 16 - 4q + \tfrac{1}{2}q^2\) This is a parabola with a vertex at $(4,8)$, meaning that there is no quantity for which $p = MC$ if $p < 8$, and if $p > 8$ we get two quantities where $p = MC$: \(q = 4 \pm \sqrt{2p - 16}\) For example, if $p = 12.5$, then we get $p = MC$ at $q = 1$ and $q = 7$. Which is the profit-maximizing quantity?

The answer becomes clear if we plot the profit function: the reason $p = MC$ in two places is because the profit function is a cubic. The smaller quantity is actually a local minimum of the profit function:

Intuitively, this is a cost function representing a production process where $MP_L$ increases at first, and then decreases; so $MC$ starts out relatively high, decreases for a bit, and then increases. When $MC = p$ at $q = 1$, the marginal cost is getting lower, not higher; so additional units beyond 1 are generating positive marginal profits.

Mathematically, you can use a second-derivative test in cases like this to confirm that this is a local minimum rather than a local maximum.

Complexity #2: Not covering variable costs

The second difficulty is that the price may be so low in this case that the firm is not able to cover even its variable costs. Recall that if the firm produces no output, it must still incur its fixed costs. In this case, the firm has fixed costs of $€81$, so at $q = 0$ it runs a loss of 81.

If the price is sufficiently low, the firm may lose even more than its fixed costs if it chooses to operate: that is, it may pull in less in revenue than it has to pay in variable costs.

For example, suppose $p = 8.5$ in this case. Setting $p = MC$ we find the profit function has a local minimum at $q = 3$ and local maximum at $q = 5$. However, at $q = 5$, the firm’s revenue is $5 \times 8.5 = 42.5$, while its total cost is \(c(5) = 81 + 16 \times 5 - 2 \times 5^2 + \tfrac{1}{6}\times 5^3 = 131.83\) so its profit is $42.5 - 131.83 = -99.33$. This is worse than it could do if it simply produced nothing.

We can see this in the graph below: if $p < 10$, the total revenue line never rises above the variable cost curve, and the profit never rises above $-F = -81$:

Mathematically, a firm will not be able to cover its variable costs when \(TR(q) < VC(q)\) If we divide both sides by $q$, this occurs if $p < AVC(q)$ for all vaues of $q$; that is, if $p$ is below the minimum of the AVC curve. Looking at the lower diagram above, you can see this is the case: the lowest point on the AVC curve occurs at $€10$/unit; so if $p < 10$, the $AR < AVC$ for all levels of $q$, so the firm never covers its variable costs.

Putting it all together: the firm’s supply curve

Putting this together, this firm’s short-run supply decision will be: \(q^\star(p) = \begin{cases}0 & \text{ if } p < 10 \\ 4 + \sqrt{2p - 16} & \text{ if }p \ge 10\end{cases}\) (We generally assume that the firm will produce a positive output if they’re indifferent, but that’s an assumption.)

Plotting this out, we can see the short-run supply curve is the section of the right-hand portion of the MC parabola that’s above the AVC curve: