Solving Cost Minimization for every output gives the cost function: the least cost of producing each $q$. From it come the cost measures of Short-Run Costs, now with the Constant, Power and Sum Rules doing the work.
Key idea
$C(q)$ is the least cost of producing $q$. In the short run $C(q) = F + VC(q)$, with fixed cost $F = C(0)$. Then $$MC(q) = C'(q), \qquad AC(q) = \frac{C(q)}{q}, \qquad AVC(q) = \frac{VC(q)}{q}.$$
Why it works
Marginal cost is the extra cost of a little more output: the slope of total cost, $\lim \Delta C/\Delta q$. Fixed cost doesn't change with $q$, so its derivative is zero: a rent, a license fee or a lump-sum tax moves $AC$ but leaves $MC$ alone. Where the cost function comes from production: with $Q = \sqrt{KL}$ and capital fixed at $\bar K$, making $q$ needs $L = q^2/\bar K$, so $$C(q) = r\bar K + \frac{w}{\bar K}q^2, \qquad MC(q) = \frac{2w}{\bar K}q.$$ Marginal cost rises because labor's marginal product falls.
Worked example: Reading a cost function
$C(q) = 100 + 10q - q^2 + 0.1q^3$. At $q = 8$, find $MC$, $AVC$ and $AC$.
- $MC(q) = 10 - 2q + 0.3q^2$, so $MC(8) = 10 - 16 + 19.2 = 13.2$.
- $AVC(q) = 10 - q + 0.1q^2$, so $AVC(8) = 8.4$.
- $AC(8) = \dfrac{100}{8} + 8.4 = 12.5 + 8.4 = 20.9$.
Answer: $MC = 13.2$, $AVC = 8.4$, $AC = 20.9$.
Worked example: Cost from a production function
$Q = \sqrt{KL}$, $\bar K = 4$, $w = 2$, $r = 5$. Find $C(q)$ and $MC(q)$.
- Labor needed: $\sqrt{4L} = q$, so $L = q^2/4$.
- $C(q) = 5(4) + 2\cdot\dfrac{q^2}{4} = 20 + \dfrac{q^2}{2}$.
- $MC(q) = q$. The \$20 of capital is fixed and drops out.
Answer: $C(q) = 20 + q^2/2$, $MC(q) = q$.
Worked example: A long-run cost function
With $Q = \sqrt{KL}$ and both inputs free, the cheapest way to make $q$ costs $C(q) = 2q\sqrt{wr}$. With $w = 9$ and $r = 4$, find $MC$ and $AC$.
- $C(q) = 2q\sqrt{36} = 12q$.
- $MC = 12$ and $AC = 12q/q = 12$, at every output.
- Constant returns to scale with fixed input prices give a flat cost per unit: every unit costs the same as the last.
Answer: $MC = AC = 12$.
- $MC$ decides how much to produce: compare it with price or marginal revenue.
- $AVC$ decides whether to produce at all in the short run (the shutdown rule).
- $AC$ decides whether the firm earns a profit, and whether firms enter or leave in the long run.
Common mistake
Differentiating average cost to get marginal cost, or letting fixed cost appear in marginal cost. $MC$ is the derivative of total cost; the derivative of $AC$ tells you where $AC$ is rising or falling, which is a different question (see Where Average Cost Is Lowest). Check: change the fixed cost in your formula. If your MC changes, something is wrong.