calculate break-even output

4.4.3 Break‑even analysis

Objective

To calculate the break‑even output (the number of units that must be sold to cover all costs) and to interpret the result in a business context, as required by the Cambridge IGCSE Business Studies (0450) syllabus.

Key terminology & symbol key

Classification of costs (syllabus requirement 4.2)

• Fixed costs (FC) – unchanged regardless of output (e.g., rent, salaries).
• Variable costs (VC) – change in direct proportion to output (e.g., raw material per unit).
• Total cost (TC) – FC + VC × Q.
• Average cost (AC) – TC ÷ Q. Example: If TC = £12 000 for 2 000 units, AC = £6 per unit.
• Marginal cost (MC) – cost of producing one extra unit. With linear costs MC = VC, but in reality MC may vary.

Economies and diseconomies of scale (syllabus requirement 4.2)

When output changes, the average cost per unit may fall (economies) or rise (diseconomies). Simple IGCSE‑level examples:

• Economies of scale: Buying raw material in bulk reduces the per‑unit cost (VC falls) → Q_BE falls.
• Diseconomies of scale: A very large workforce creates coordination problems, increasing overhead per unit (VC rises) → Q_BE rises.

In break‑even analysis we assume costs are linear (constant VC and FC), representing a short‑run situation where economies/diseconomies are not yet evident.

Contribution and the break‑even formula

Each unit sold contributes C = SP − VC towards covering fixed costs. Setting total contribution equal to FC gives the break‑even output:

Derivation (for completeness):

1. TR = SP × Q
2. TC = FC + VC × Q
3. Set TR = TC → SP·Q = FC + VC·Q
4. SP·Q − VC·Q = FC → (SP − VC)·Q = FC
5. Q = FC/(SP − VC)

Step‑by‑step calculation of Q_BE

1. Identify total fixed costs (FC) for the period.
2. Determine the variable cost per unit (VC).
3. Find the selling price per unit (SP).
4. Calculate contribution per unit: C = SP − VC.
5. Divide FC by C → Q_BE = FC ÷ C.

Constructing a break‑even chart (syllabus requirement 4.2)

1. Draw axes: horizontal = output (Q), vertical = money (£).
2. Plot the Total Revenue line:
   • Starts at the origin (0, 0) because when Q = 0, TR = 0.
   • Slope = SP (price per unit).
3. Plot the Total Cost line:
   • Starts at FC on the vertical axis (when Q = 0, TC = FC).
   • Slope = VC.
4. The intersection point is the break‑even output Q_BE.
5. Shade the area above the TC line and below the TR line to the right of Q_BE (profit); shade the opposite area to the left (loss).

Margin of safety (syllabus requirement 4.2)

The margin of safety shows how far current or projected sales can fall before the business reaches the break‑even point.

These formulas correspond exactly to the syllabus wording “margin of safety (units)” and “margin of safety (%)”.

Limitations of break‑even analysis (syllabus requirement 4.2)

• Assumes a constant selling price per unit – ignores discounts, promotions, or price changes.
• Assumes a constant variable cost per unit – in reality VC may fall (economies) or rise (diseconomies) as output changes.
• Ignores inventory changes; production and sales are treated as occurring simultaneously.
• Only relevant in the short‑run where fixed costs truly remain fixed.
• Excludes external factors such as taxes, interest, or exchange‑rate movements that can alter FC or VC.

Exam‑relevant checklist (AO4) – when answering evaluation questions, consider:

1. Assumption of linear (constant) costs.
2. Assumption of a single product and a single price.
3. Short‑run focus (fixed costs truly fixed).
4. Exclusion of inventory, tax, interest and exchange‑rate effects.

Worked example – custom notebooks

1. Contribution per notebook: C = 10 − 4 = £6.
2. Break‑even output: Q_BE = 12 000 ÷ 6 = 2 000 notebooks.
3. If the firm expects to sell 2 500 notebooks:
   • MoS = 2 500 − 2 000 = 500 units
   • MoS % = (500 ÷ 2 500) × 100 = 20 %

Interpretation

• Sales < 2 000 → loss.
• Sales = 2 000 → break‑even (profit = £0).
• Sales > 2 000 → profit = (Units − 2 000) × £6.

Link to financial statements (syllabus requirement 5)

Knowing Q_BE allows a simple income‑statement to be prepared.

Although constructing a full statement is not examined, understanding the link demonstrates how break‑even analysis feeds directly into profitability calculations.

External influences on the break‑even point (syllabus requirement 6)

• Tax changes (syllabus 6.2): Higher corporation tax can increase business rates (a fixed cost), shifting Q_BE rightward.
• Interest rates (syllabus 6.2): More expensive borrowing raises interest expense, a component of FC, therefore Q_BE rises.
• Exchange‑rate movements (syllabus 6.3): A weaker domestic currency raises the cost of imported raw materials → VC increases → Q_BE rises.
• Legal / regulatory changes (syllabus 6.2): New health‑and‑safety requirements may add a fixed compliance cost, again moving the break‑even point to the right.

Evaluation – usefulness of break‑even analysis (AO4)

Break‑even analysis is a quick‑decision tool that highlights the relationship between cost, price and volume in a single, easy‑to‑interpret figure. It is especially useful for:

• Setting sales targets for new products.
• Assessing the impact of price changes or cost reductions.
• Communicating financial risk to managers and investors.

However, its usefulness is limited by the assumptions listed above. In dynamic markets where prices fluctuate, economies of scale appear, or inventory levels change, more sophisticated techniques (e.g., marginal analysis, budgeting, cash‑flow forecasting) give a more realistic picture. Therefore, students should treat break‑even analysis as a *first approximation* and always consider its limitations before making strategic decisions.

Practice questions

1. A bakery has fixed costs of £8 500 per month. The variable cost per cake is £3 and each cake sells for £9. Calculate the break‑even output in cakes.
2. A retailer reduces the selling price of a gadget from £25 to £20 during a promotion. Fixed costs are £15 000 and variable cost per gadget is £12.
   • What is the new break‑even output?
   • How many additional units must be sold to achieve a profit of £5 000 at the reduced price?
3. Explain how an increase in fixed costs affects the break‑even point, assuming selling price and variable cost per unit remain unchanged.
4. Using the notebook data above, calculate the profit if 2 800 notebooks are sold. Then state the margin of safety (both units and %).
5. Evaluate the reliability of break‑even analysis for a company that expects a seasonal rise in demand and plans to introduce a bulk discount.

Data‑response question (AO3 & AO4)

The chart below shows Total Revenue (TR) and Total Cost (TC) for a start‑up producing mobile‑phone accessories. The axes are labelled “Units produced and sold” (horizontal) and “£ (£ thousands)” (vertical). The two lines intersect at 1 200 units.

• Identify the break‑even output from the chart.
• Calculate the margin of safety if the company expects to sell 1 600 units.
• Interpret what the chart tells the manager about the risk of loss if sales fall below 1 200 units.
• Discuss two limitations of using this chart for long‑term planning.

Suggested diagram