Percentage increase of the price of a good if there is a pay raise

Question

I have a question about the percentage increase of the price of a good if there is a pay raise given to the firm's workers.

I am reading economics for my own personal interest so I apologise if there are any evident gaps in my knowledge that may require further attention.

I was reading an extract from an economics textbook and the following passage is provided:

Firms will try to pass on increases in their costs to customers. For example, if a firm gives a 5% pay rise to its workers, and wages account for 80% of its costs, then it will need to increase prices by 4% (80% of 5%) to maintain its profit margins.

For proof, this passage is also provided in image form (without the mathematical conversion of percentages):

I'm confused by how the calculation of $80 \% \times 5\%$ results in the percentage rise of the good's price in order to maintain the original profit margins.

No attempt at the calculation or any intuition was provided (which sadly appears to be a common theme amongst elementary economics textbooks).

I attempted to use the gross margin equation:

$Gross Margin = \frac{Revenue - COGS}{Revenue}$

but this did not help me in any way.

I was wondering how the calculation above was derived and where the intuition comes from.

Answer 1

1. The textbook likely talks about net profit not gross profit

2. If we start by assuming original net profit margin was 10%:

$$0.1 = \frac{TR-TC}{TR} \tag{*}$$

where TR is total revenue and TC total cost, if 80% of TC increases by 5% we have the following change for TC:

$$TC(0.2 + 0.8\cdot 1.05) \tag{**}$$

Next, if we want to keep net profit margin constant we need to figure out what factor 'x' will preserve the equality given by (*) if TC changes by (**) so we have

$$0.1 = \frac{TRx-TC(0.2 + 0.8\cdot 1.05)}{TRx} \\ 0.1 = 1-\frac{1.04TC}{TRx}$$

Now its trivial to see that in order for the equality to hold $x=1.04$ which is equivalent of 4% increase. If $x=1.04$ then:

$$ 0.1 = 1-\frac{1.04TC}{TR1.04} \\ 0.1 = 1-\frac{TC}{TR} \\ 0.1 =\frac{TR-TC}{TR}$$

Answer 2

That seems correct. For a concrete example, suppose this company has a wages bill of £80,000 other costs of £20,000. And a 10% profit margin. So its takings are £110,000

If it gives a 5% pay rise, its wage costs increase to £84000, other costs remain the same, so to maintain a 10% profit margin it must have taking of

$1.1×(84000+20000) = 114400$

And that is indeed a 4% increase in takings

$$\frac{114400-110000}{110000} ×100\% =4\% $$

You can work this algebraically. If:

• the profit margin is a $k$,
• the wage bill is $w$,
• other costs is $s$ and
• the pay rise is $r$%,

then the takings must rise by:

$$\frac{k(w(1+r/100) + s) - k(w+s)}{ k(w+s)} × 100\%$$

That cancels to $\frac{w}{w+s} × r$%

So if wages ($w$) make up 80% of total costs ($w+s$) And if an $r$% pay rise is offered, the takings must increase by 80% of $r$