13.2 The Supply of Output

As we showed in the last chapter, for a price-taking firm, the total revenue functions is \(r(q) = p \times q\) Therefore the firm’s marginal revenue is equal to the market price: \(MR(q) = p\) Therefore, the profit-maximizing condition $MR = MC$ simply becomes $p = MC$; that is, the firm will maximize its profit when price equals marginal cost.

We solved the firm’s profit-maximizing problem in the last chapter for a specific price. Now, what we’d like to do is to solve it as a function of the price; this gives the firm’s supply function.

To see how to do this, let’s continue with our example of a firm with the production function $f(L,K) = \sqrt{LK}$, which has capital fixed at $\overline K = 32$. Let’s keep $r = 2$ as the price of capital, but let $w$ be a variable. Since the firm needs $L(q) = q^2/32$ units of labor to produce $q$ units of output, it has the short-run cost function \(c(q) = wL(q) + r\overline K = w \times \frac{q^2}{32} + 64\) and therefore the marginal cost \(MC(q) = c^\prime(q) = w \times {dL \over dq} = w \times {q \over 16}\) In the previous section we showed that when $p = 12$ and $w = 8$, the optimal output is $q = 24$. We did this by setting $MC = 12$ and solving for $q$. More generally, we can set the marginal cost equal to a generic price $p$ and solve for $q$ as a function of $p$ and $w$: \(\begin{aligned} MC(q) &= p\\ w \times \frac{q}{16} &= p\\ q^\star(p) &= {16p \over w} \end{aligned}\)

For example, if we have $w = 8$, the firm’s optimal supply is $q^\star(p) = 2p$.

The diagram below shows how the firm optimally responds to each price. Try dragging the price up and down; the quantity will adjust automatically to the profit-maximizing quantity.

Some things to notice:

• When $p > 8$, the firm can run a profit because there is a region for which $TR > TC$.
• When $p = 8$, the firm has at most zero profit; $TR = TC$ (and therefore $AR = AC$) at $q = 16$.
• When $p < 8$, the firm cannot run a profit, because the entire $TR$ line lies below the $TC$ curve (and the $AR$ line lies below the minimum of the $AC$ curve).

One might ask why a firm would ever run a loss: why produce anything at all, if you’re going to have a negative profit? The answer is that the the firm must incur its fixed costs, at least in the short run; so for all intents and purposes those are sunk costs in the short run. When deciding whether or not to produce, the firm must therefore ask if they can at least make more revenue than their variable costs. If they can, then they should produce a positive output in the short run; if they can’t, they should shut down.

In the example above, the firm’s total revenue is always greater than their total variable costs, so it’s always better to produce at the point where $P = MC$ than to shut down in the short run. However, this isn’t always the case. In the last part of this chapter, we’ll look at the case of a firm with a more complex cost function, and derive the supply decision with a bit more nuance; but for now, let’s stick with this simple example and examine the firm’s optimal choice of labor.