Total cost

A firm’s total cost at a given output level is the sum of fixed costs (independent of output) and variable costs (scaling with output):

$C_T(q)=C_F+C_V(q)$

For most engineering modelling, $C_V$ is approximated as linear in $q$: $C_V\approx v\cdot q$, where $v$ is the per-unit variable cost. So total cost is

$C_T(q)=C_F+v\cdot q.$

This is the equation of a straight line on a (quantity, cost) plot, with $y$-intercept $C_F$ (the fixed cost paid even at zero output) and slope $v$ (the cost per additional unit).

Pair this with the total-revenue line $TR(q)=p\cdot q$ for selling price $p$, and you can solve for the break-even quantity at which $TR=C_T$:

$p\cdot q^{\ast }=C_F+v\cdot q^{\ast }⟹q^{\ast }=\frac{C_F}{p-v}.$

The denominator $p-v$ is the contribution margin per unit — how much each unit contributes toward covering fixed cost once variable cost is paid. The break-even quantity is exactly fixed cost divided by contribution margin per unit.

In reality the linear approximation breaks at the extremes. Bulk-purchase discounts make the variable-cost curve concave at moderate $q$. Past capacity, overtime and rushed shipping bend it back up (convex). For most decision-making the linear form is accurate enough; for capacity-planning decisions, the non-linear regime matters.

See Fixed cost, Variable cost, Break-even analysis, Supply and demand for the surrounding microeconomic picture.