The Competitive Firm's Supply Curve

A competitive firm is a price taker: it can sell as much as it likes at the market price $p$, and nothing above it. The Shutdown Rule told it when to stay open; with a cost function and Where Average Cost Is Lowest we can derive its whole supply curve.

Key idea

A price taker maximizes $\pi(q) = pq - C(q)$. It produces where $$p = MC(q)$$ on the rising part of marginal cost, as long as $p$ is at least the minimum of average variable cost. Below that price it produces nothing. Its short-run supply curve is the part of $MC$ above minimum $AVC$.

Why it works

First-order condition: $\pi'(q) = p - C'(q) = 0$. Second-order condition: $\pi''(q) = -C''(q) < 0$, so marginal cost must be rising; where MC is falling, $p = MC$ marks a profit minimum. Then compare with shutting down: producing gives $pq - VC(q) - F$, shutting down gives $-F$ (the fixed cost is owed either way in the short run). Producing is better exactly when $pq \ge VC(q)$, that is, $p \ge AVC(q)$.

$C(q) = 100 + 10q - q^2 + 0.1q^3$. The thick part of MC, above minimum AVC (\$7.50 at $q = 5$), is the supply curve. At \$20 the firm makes 10 units and just breaks even.

Worked example: A simple cost function

$C(q) = 50 + q^2$, and the price is \$20.

$MC = 2q$, so $p = MC$ gives $q = 10$.
$AVC = q$ is always below $MC$, so any positive price keeps it producing. Supply: $q = p/2$.
Profit: $200 - 150 = 50$.

Answer: $q = 10$.

Worked example: A cubic cost function

$C(q) = 100 + 10q - q^2 + 0.1q^3$ at $p = 20$.

$MC = 10 - 2q + 0.3q^2 = 20$, so $0.3q^2 - 2q - 10 = 0$.
$q = \dfrac{2 \pm \sqrt{4 + 12}}{0.6} = \dfrac{2 \pm 4}{0.6}$, so $q = 10$ (the other root is negative).
Revenue 200, cost $100 + 100 - 100 + 100 = 200$: zero economic profit. \$20 is the bottom of ATC.

Answer: $q = 10$, breaking even.

Worked example: Below the shutdown price

Same firm, but the price falls to \$7.

Minimum $AVC = 10 - q + 0.1q^2$ is at $q = 5$: \$7.50.
\$7 is below that, so every unit produced loses money on top of the fixed cost. Output: 0, loss \$100 (the fixed cost).

Answer: It shuts down.

Market supply in the short run is the horizontal sum of the firms' supply curves, the same adding-up as Adding Up to Market Demand: with 50 firms that each supply $q = p/2$, the market supplies $Q = 25p$. Each firm's curve starts at its own shutdown price, so the market curve has a step or kink wherever a group of firms enters.

Common mistake

Taking the whole MC curve as the supply curve. The falling part of MC is where $p = MC$ gives a profit minimum, and the part below minimum AVC is where the firm would rather shut down. Check: at your quantity, MC must be rising and the price must be at least AVC.

The Competitive Firm's Supply Curve

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