# Why intuitively does Break-even point $= \dfrac{\text{Total fixed costs}}{\text{Selling price per unit} − \text{Variable cost per unit}}$?

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.

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

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

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.

• I think you mean $AVC$, not $VC$. – Giskard Dec 22 '18 at 9:12
• Yes, I've used VC because I generally indicate the total cost with TVC. – Kolmogorovwannabe Dec 22 '18 at 9:20
• And why do you do that? Also, how do you denote variable cost if you denote average variable cost with VC? – Giskard Dec 22 '18 at 9:26
• Already told you, TVC. – Kolmogorovwannabe Dec 22 '18 at 9:30
• 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... – Kolmogorovwannabe Dec 22 '18 at 10:39