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p. 100. Accounting: A Very Short Introduction (2014).

I understand the green sentence beneath, but not the fractional formula for BEP containing CPU in red? What's the intuition for the latter?

Contribution

The ‘contribution’ is the amount that a particular product line helps towards covering fixed costs and making a profit. The contribution is measured as: Sales − Variable costs. This can also be calculated per unit of production. The contribution per unit (CPU) is:

$\color{red}{CPU = \text{Selling price per unit} − \text{Variable cost per unit}}$

That is, taking the selling price for one unit of production, and then deducting all the variable costs for that one unit of production, gives the contribution per unit.

Break-even point (BEP)

$\color{forestgreen}{\text{The break-even point is the volume of production at which the firm makes zero profit.}}$ Therefore, the break-even point is the level at which all costs (including fixed costs) can be paid for. The formula is:

$BEP = \dfrac{\text{Total fixed costs}}{\color{red}{\text{Contribution per unit (CPU)}}}$

So, above the break-even point, any extra items sold will generate profit. Every unit that is produced and sold in excess of the break-even point will generate profit of the amount of the CPU. The total profit made is:

Units in excess of break-even point × CPU

If the firm does not reach break-even point, this implies that it has not yet covered fixed costs and will therefore make a loss. The size of the loss will be the number of units under the break-even point, multiplied by the CPU.

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The break-even point can be derived using the formula given above, or by drawing up a chart such as Figure 15.

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Intuitively, the breakeven point is where the volume of sales results in the gross margin (based on the difference between selling price and variable costs per item) exactly covering fixed costs

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Very easy proof:

we know that

$π = 0$ equalas to $TR - TC = 0$

by definition, total revenue (TR) is

$TR=Q*p$

i.e. the price that I get per unit produced whereas total cost (TC) is

$TC= FC+VC*Q$

where $FC$ are the total fixed costs and the total variable costs, $VC*Q$, is the value of the variable cost per unit times the number of unit produced. Substituting in the profit expression we have

$p*Q-FC-VC*Q=0$

and isolating Q, which is the amount of unit that implies the BEP we have

$Q_{BEP}=(FC)/(p-VC)$

which is exactly your formula. What does this formula tell us? Nothing more that, if I produce such that my total revenue is equal to my total cost I get profit equals to zero. It is just another way of expressing the same thing.

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  • $\begingroup$ I think you mean $AVC$, not $VC$. $\endgroup$
    – Giskard
    Dec 22 '18 at 9:12
  • $\begingroup$ Yes, I've used VC because I generally indicate the total cost with TVC. $\endgroup$ Dec 22 '18 at 9:20
  • $\begingroup$ And why do you do that? Also, how do you denote variable cost if you denote average variable cost with VC? $\endgroup$
    – Giskard
    Dec 22 '18 at 9:26
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    $\begingroup$ Already told you, TVC. $\endgroup$ Dec 22 '18 at 9:30
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    $\begingroup$ In the text I named the total cost TC, in my comment to you we are talking of variable cost, plus in TVC what does the "V" stand for? Come on... $\endgroup$ Dec 22 '18 at 10:39

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