Costs, scale of production, break-even

The four costs

0264 asks you to calculate these, not only classify them, and publishes the formulas.

Total variable cost = variable cost per unit x number of units
Variable cost per unit = total variable cost / number of units
Total cost = total fixed costs + total variable costs
Average cost = total cost / number of units

Worked example. A cafe has fixed costs of $1000 a week, total variable costs of $20000 a week, and makes and sells 5000 desserts a week.

Variable cost per unit = total variable cost / number of units = 20000 / 5000 = $4.00

Total cost = total fixed costs + total variable costs = 1000 + 20000 = $21000

Average cost = total cost / number of units = 21000 / 5000 = $4.20

The gap between $4.00 and $4.20 is the fixed cost spread across 5000 desserts. Do the total cost line first and the average cost from it, and set the working out like this, because a correct method with a slip in the arithmetic still earns the method mark.

Reading and completing a cost table

A factory pays $12000 a month in fixed costs, and each unit costs $6 in materials and wages.

Every row obeys the same two rules: total cost is fixed plus variable, and average cost is total cost divided by output. To fill a blank cell, find which rule the row already gives you enough for.

Average cost falls from $18.00 to $9.00 while the cost of making each unit has not changed at all. The same $12000 is shared among more units. Keep that separate from economies of scale, which go further and make the inputs themselves cheaper.

Using cost data to make decisions

Which product to make. Compare what each contributes, not what each sells for. Product A sells for $20 with a variable cost of $12, contributing 20 - 12 = $8 a unit. Product B sells for $30 with a variable cost of $24, contributing 30 - 24 = $6 a unit. A looks better until the volumes appear. If A sells 5000 units it contributes 5000 x 8 = $40000, while B selling 8000 units contributes 8000 x 6 = $48000, so B is the better product to make despite its smaller contribution per unit.

Whether to continue or stop production. A product selling at $9 with a variable cost of $7 contributes $2 a unit towards fixed costs. Even if that will not cover its share of them, stopping does not remove the rent, so the business loses $2 on every unit it stops making. Once price falls below variable cost, every unit loses money and stopping is right.

What price to set. Cost-plus pricing starts from average cost and adds a mark-up. At an output of 3000 units in the table above, average cost is $10.00, and a mark-up of 30 per cent gives 10 x 1.3 = $13.00. It only works if the business can sell that quantity at that price.

Choosing suppliers. A cheaper supplier lowers the variable cost per unit, raising contribution on every unit sold. Against that, weigh reliability, because a late delivery stops production, and quality, because faulty inputs become faulty output.

Economies of scale

Economies of scale are falls in average cost that come from producing on a larger scale.

Diseconomies of scale

Diseconomies of scale are rises in average cost caused by a business growing too large to run well.

All four are about people rather than machinery, which is why growth by merger or takeover so often runs into them.

Break-even

Contribution per unit is what each unit sold leaves towards fixed costs once its own variable cost is paid.

Contribution per unit = selling price per unit - variable cost per unit
Break-even output = fixed costs / contribution per unit
Margin of safety = actual number of sales - break-even number of sales

Break-even output is the number of units at which total revenue exactly equals total costs, so there is neither profit nor loss.

Worked example. A business sells a product for $15. The variable cost is $9 a unit. Fixed costs are $54000 a year. It expects to sell 12000 units.

Contribution per unit = selling price per unit - variable cost per unit = 15 - 9 = $6

Break-even output = fixed costs / contribution per unit = 54000 / 6 = 9000 units

Margin of safety = actual number of sales - break-even number of sales = 12000 - 9000 = 3000 units

Profit = (12000 x 6) - 54000 = 72000 - 54000 = $18000

The margin of safety answers "how far can sales fall before we make a loss". Here they can drop by 3000 units, a quarter of what is expected. Give it in units, not as a sum of money.

Reading and completing a break-even chart

Output goes along the horizontal axis, and costs and revenue in $ go up the vertical axis. Three straight lines are drawn on it.

• The fixed cost line is horizontal at $54000, because fixed costs are the same at every output.
• The total cost line starts on the vertical axis at $54000, because fixed costs are payable even at zero output, and slopes up. At 12000 units it reaches 54000 + (12000 x 9) = $162000.
• The total revenue line starts at the origin, because nothing sold earns nothing. At 12000 units it reaches 12000 x 15 = $180000.

The break-even point is where total revenue crosses total cost, at 9000 units. Check it: revenue there is 9000 x 15 = $135000 and total cost is 54000 + (9000 x 9) = $135000.

To complete or amend a chart you need only two points for each line, because all three are straight, and one point is always given: the fixed cost line at output zero for total cost, and the origin for revenue. Read break-even output down to the horizontal axis, never across to the vertical one. The gap between the revenue and total cost lines is profit to the right of break-even and loss to the left, and the margin of safety is the horizontal distance from break-even to actual sales.

What moves the break-even point

A price rise looks like the easy answer, and in the arithmetic it is. In the market it usually is not, because fewer units sell at the higher price, which the chart cannot show.

Limitations of break-even analysis

• It assumes everything produced is sold, so no unsold inventory.
• It assumes the selling price is the same at every output, when selling far more usually means charging less.
• It assumes the variable cost per unit never changes, when bulk discounts lower it and overtime raises it.
• Fixed costs are only fixed within a range. A larger factory or another machine steps them up.
• The lines are straight, which simplifies how real costs and revenue behave.
• The data is a forecast, so a wrong estimate gives a confident answer that is wrong.
• It says nothing about whether the business can actually sell the break-even quantity, nor about quality, competitors or cash flow.

Use these as evaluation, not as a list. "The break-even output of 9000 units assumes every unit is sold, and in a market this competitive that is the assumption most likely to fail" is a judgement.

Common mistakes

• Calling a cost fixed or variable by habit. Wages are variable when paid per unit and fixed when paid as a monthly salary.
• Dividing total variable cost by output and calling the answer average cost. That is variable cost per unit.
• Working out contribution as selling price minus total cost instead of minus variable cost.
• Dividing fixed costs by the selling price rather than by contribution per unit.
• Giving break-even output or margin of safety in dollars. Both are in units.
• Saying fixed costs fall as output rises. Fixed cost per unit falls; total fixed cost does not move.
• Reading the break-even point off the vertical axis, which gives revenue rather than output.
• Saying a rise in fixed costs changes contribution per unit. It changes break-even output only.

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