Given the total cost function $TC(Q)=Q^3-2Q^2+8Q$ for a wheat farmer in perfect competition, find the functions for $TFC$, $TVC$, $AVC$, $AFC$, $ATC$, and $MC$.

What you are being asked to extract from $TC(Q)$

A total cost function can be split into a fixed part (cost that does not change with output) and a variable part (cost that changes with output): $$TC(Q)=TFC+TVC(Q).$$ Then the average costs are found by dividing by $Q$, and marginal cost is the derivative of $TC$ with respect to $Q$.

Separating fixed and variable costs

Given: $$TC(Q)=Q^3-2Q^2+8Q.$$ The fixed cost is the part of $TC$ that does not depend on $Q$, meaning the constant term.

• There is no constant term here, so: $$TFC=0.$$
• Therefore: $$TVC(Q)=TC(Q)-TFC=Q^3-2Q^2+8Q.$$

Computing average costs: $AVC$, $AFC$, and $ATC$

Average variable cost: $$AVC(Q)=\frac{TVC(Q)}{Q}=\frac{Q^3-2Q^2+8Q}{Q}=Q^2-2Q+8.$$ Average fixed cost: $$AFC(Q)=\frac{TFC}{Q}=\frac{0}{Q}=0.$$ Average total cost: $$ATC(Q)=\frac{TC(Q)}{Q}=\frac{Q^3-2Q^2+8Q}{Q}=Q^2-2Q+8.$$ Here $ATC=AVC$ because $AFC=0$.

Finding marginal cost $MC$

Marginal cost is the slope of the total cost curve: $$MC(Q)=\frac{dTC}{dQ}.$$ Differentiate term by term: $$\frac{d}{dQ}(Q^3-2Q^2+8Q)=3Q^2-4Q+8.$$ So: $$MC(Q)=3Q^2-4Q+8.$$