Break-even analysis

Why it matters

Chemical plants rarely run at exactly their design rate: demand falls, feedstock is short, or a competitor cuts prices. Break-even analysis tells management the lowest production rate, or the lowest selling price, at which the plant still covers its costs, and how much profit changes with output. It is the quickest way to see how risky a project is.

Key ideas

Fixed and variable costs. Over the normal operating range,

• fixed costs (depreciation, local taxes, insurance, rent, most salaries, plant overhead and administration) do not change with output;
• variable costs (raw materials, utilities, catalysts and chemicals, royalties per unit, packaging, freight) rise in proportion to output;
• semi-variable items (maintenance, supervision, laboratory) are split into a fixed part and a variable part.

Break-even chart. Plot annual cost and revenue against annual production (or % of capacity). Total cost is a straight line starting at the fixed cost with slope v (variable cost per unit); revenue is a line through the origin with slope s (selling price). Where they cross is the break-even point: below it the plant loses money, above it profit grows at a rate of (s − v) per unit.

Contribution. The difference (s − v) is the contribution margin per unit — what each unit contributes towards fixed costs and then profit. The break-even output is fixed cost divided by contribution. The ratio (s − v)/s is the contribution (profit-volume) ratio, used to express break-even in rupees of sales.

Margin of safety is how far actual (or planned) output is above break-even, usually as a fraction of actual output. A small margin of safety means a small drop in sales turns profit into loss.

Shut-down point. In the short run a plant should keep running as long as the price covers variable cost, because fixed costs are incurred anyway. Below that price it loses less by shutting down. Operating between the shut-down point and the break-even point reduces the loss but does not make a profit.

Effect of changes. Higher fixed costs (a more capital-intensive design) raise the break-even output; a lower price or higher variable cost reduces contribution and also raises it. Capital-intensive plants with low variable costs have high break-even points but profit rises quickly above them — high operating leverage.

Target profit. To earn a required profit, the output must cover fixed costs plus that profit before tax. If the target is an after-tax profit, divide it by (1 − t) first.

Limits. Linear break-even assumes constant price and unit variable cost at all outputs; in reality price may fall to sell more, and efficiency falls at low rates. Near full capacity unit costs may rise (overtime, debottlenecking). Treat the result as an estimate.

Formulas

Total cost: C = F + v·Q Revenue: S = s·Q Break-even output: Q_BE = F / (s − v) Break-even sales: S_BE = F / [(s − v)/s] Break-even as fraction of capacity: Q_BE / Q_cap Gross profit at output Q: P_g = (s − v)·Q − F Output for target after-tax profit P_n: Q = [F + P_n / (1 − t)] / (s − v) Margin of safety = (Q − Q_BE) / Q Shut-down price: s = v

Symbols: F annual fixed cost (₹/yr), v variable cost per unit (₹/kg or ₹/t), s selling price per unit (same unit), Q annual output (kg/yr or t/yr), Q_cap design capacity, t income-tax rate (decimal), P_n net profit after tax (₹/yr). Use consistent units: if F is in ₹/yr and s, v in ₹/kg, Q is in kg/yr.

Worked examples

Example 1 (standard). A plant of capacity 10 000 t/yr has fixed costs of ₹6 crore per year, variable cost ₹30/kg and selling price ₹45/kg. Find the break-even output, the break-even percentage of capacity, and the margin of safety at 8000 t/yr.

1. Contribution = s − v = 45 − 30 = ₹15/kg = ₹15 000/t.
2. Q_BE = F/(s − v) = 6 × 10⁷ / 15 000 = 4000 t/yr.
3. Break-even = 4000/10 000 = 40% of capacity.
4. Margin of safety at 8000 t/yr = (8000 − 4000)/8000 = 0.50.
5. Q_BE = 4000 t/yr (40% of capacity); margin of safety 50%. Profit at 8000 t/yr = 15 000 × 8000 − 6 × 10⁷ = ₹6 crore/yr before tax.

Example 2 (GATE level). For the same plant, (a) what output gives a net profit of ₹3 crore/yr after 30% tax? (b) If competition cuts the price to ₹40/kg, what is the new break-even output, and what is the lowest price at which the plant should keep running in the short term?

1. (a) Required gross profit = P_n/(1 − t) = 3/0.7 = ₹4.286 crore/yr.
2. Q = (F + 4.286 × 10⁷)/(s − v) = (6 + 4.286) × 10⁷ / 15 000 = 6857 t/yr (68.6% of capacity).
3. (b) New contribution = 40 − 30 = ₹10/kg = ₹10 000/t; Q_BE = 6 × 10⁷/10 000 = 6000 t/yr (60% of capacity).
4. Shut-down price = variable cost = ₹30/kg.
5. (a) 6857 t/yr; (b) break-even rises to 6000 t/yr; keep running while the price exceeds ₹30/kg. A one-third cut in contribution raised break-even by 50%.

Common mistakes

• Mixing ₹/kg with t/yr (1 t = 1000 kg) or crore with lakh.
• Including depreciation in variable costs, or raw materials in fixed costs.
• Using a target after-tax profit directly without dividing by (1 − t).
• Confusing the break-even point (covers all costs) with the shut-down point (covers variable cost only).
• Expressing break-even in sales rupees by multiplying F by the contribution ratio instead of dividing.

For GATE CH

Expect direct numericals: break-even output or capacity fraction from fixed cost, unit variable cost and price; output for a target profit; and the effect of a change in price or cost. Occasionally a semi-variable cost must be split first. Practise unit conversions between ₹/kg and t/yr.

Quick check

1. F = ₹2 crore/yr, s = ₹50/kg, v = ₹40/kg. Break-even output in t/yr?
2. What happens to the break-even point if fixed costs rise by 20%?
3. Define contribution margin.
4. Variable cost ₹25/kg; price falls to ₹24/kg. Should the plant keep running in the short term?

Answers: 1. 2 × 10⁷/10 000 = 2000 t/yr. 2. It rises by 20% (Q_BE ∝ F). 3. Selling price minus variable cost per unit. 4. No — the price does not cover variable cost, so running increases the loss.