Break-Even Analysis

What is break-even?

The break-even point is where total revenue equals total cost — the quantity at which profit is exactly zero. Below it you lose money; above it you start earning profit.

The setup

• Revenue: $R(Q) = P \cdot Q$
• Total cost: $TC(Q) = FC + VC \cdot Q$
• Profit: $\pi(Q) = R(Q) - TC(Q) = (P - VC) \cdot Q - FC$

where $P$ = price per unit, $VC$ = variable cost per unit, $FC$ = fixed cost.

The quantity $(P - VC)$ is called the contribution margin per unit — each unit sold contributes this amount toward covering fixed costs.

Break-even formula

Set profit to zero: $0 = (P - VC) \cdot Q - FC \implies \boxed{Q^* = \dfrac{FC}{P - VC}}$

Step-by-step solution

Setup: $FC = 10{,}000$, $VC = 15$, $P = 25$.

Step 1 — Contribution margin per unit: $P - VC = 25 - 15 = 10$

Step 2 — Divide fixed cost by contribution margin: $Q^* = \dfrac{10{,}000}{10} = \boxed{1000 \text{ units}}$

Step 3 — Confirm:

• Revenue at $Q = 1000$: $25 \cdot 1000 = 25{,}000$
• Cost at $Q = 1000$: $10{,}000 + 15 \cdot 1000 = 25{,}000$
• Profit: $0$ ✓

Break-even in dollars

To hit break-even revenue: $R^* = P \cdot Q^* = 25 \cdot 1000 = 25{,}000$

Or directly: $R^* = \dfrac{FC}{1 - VC/P} = \dfrac{FC}{\text{contribution margin ratio}}$

Sensitivity: what if something changes?

• *Raise price to $30**:$Q^ = 10{,}000/(30 - 15) = 667$units. A$5 price bump drops break-even by 33%.
• *Cut VC to $12**:$Q^ = 10{,}000/(25 - 12) = 769$ units.
• *Fixed costs rise to $15K**:$Q^ = 15{,}000/10 = 1500$ units — 50% more sales needed.

Small margin moves have big break-even effects — this is the operating-leverage story.

Target profit

To earn a target profit $\pi_T$: $Q_T = \dfrac{FC + \pi_T}{P - VC}$

E.g., to earn $5,000/month:$Q_T = (10{,}000 + 5{,}000)/10 = 1500$ units.

Margin of safety

If actual sales are $Q$: $\text{Margin of safety} = \dfrac{Q - Q^*}{Q} \cdot 100\%$

Tells you how far sales can drop before you start losing money. High margin = low risk.

Common mistakes

• Subtracting $VC$ from $P$ in the wrong direction. The contribution margin is (price − variable), positive. If it's negative, you can never break even.
• Mixing total variable cost with per-unit variable cost. The formula uses per-unit VC.
• Ignoring capacity constraints. The formula may tell you to sell 10,000 units but your factory caps at 5,000.

Try it in the visualization

Revenue line and total-cost line cross at $Q^*$, with the profit region shaded green above the intersection and loss region red below. Slide $P$, $VC$, or $FC$ and watch $Q^*$ shift in real time.