7.1 Polypoly or perfect competition

In most cases, the marginal cost curve represents the short-term supply curve. With rising marginal costs, profit is maximized at P=GK, i.e. the company chooses the produced quantity q for the given market price P in such a way that the marginal costs GK exactly match the price. Like this, profit is maximized. If the quantity were larger, the marginal costs - the costs of these additional units - would be higher than the price, and the additional units would generate losses. If the quantity were smaller, the marginal costs - the savings for units not produced - would be below the price, and the possible profit would not be made. If the price is higher than the total average cost TDK, this difference per unit is the profit. The marginal cost curve GK always intersects the average variable cost curve VDK and the average total cost curve TDK at their minimum points.
Proof:
Total costs $K\left(q\right)$
Marginal costs $\mathit{GK}=K^{\prime }\left(q\right)$
Average total cost curve
$\mathit{TDK}=\frac{K\left(q\right)}{q}$
Thus $\mathit{TD}{K}^{\prime }=\frac{d\frac{K\left(q\right)}{q}}{\mathit{dq}}=\frac{K^{\prime }\left(q\right)q-K\left(q\right)}{q^2}$ is valid due to the quotient rule
At the minimum $\mathit{TD}{K}^{\prime }=0$ applies and therefore $K^{\prime }\left(q\right)q-K\left(q\right)=0$ or $K^{\prime }\left(q\right)=\frac{K\left(q\right)}{q}$. In other words, at the minimum $\mathit{TDK}$, $\mathit{GK}=\mathit{TDK}$ applies.
The calculation is similar for the average variable cost curve VDK instead of the average total cost curve TDK, since the difference is a constant.
$V\mathit{DK}=\frac{K\left(q\right)-K_{\mathit{fix}}}{q}$
$VD{K}^{\prime }=\frac{d\frac{K\left(q\right)-K_{\mathit{fix}}}{q}}{\mathit{dq}}=\frac{K^{\prime }\left(q\right)q-\left(K\left(q\right)-K_{\mathit{fix}}\right)}{q^2}$
$K^{\prime }\left(q\right)=\frac{K\left(q\right)-K_{\mathit{fix}}}{q}$