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

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.

• What happens next depends entirely on what's going on with inflation in general. If money supply growth and inflation are very low but a company takes it upon itself to give its workers an above inflation pay rise and raise its prices by more than inflation then the likely consequence is that their sales will fall and their total profitability will shrink... they may then have to lay off workers or undo the price rise or the management may have to take a pay cut. I'm saying all of this to highlight the nativity of the popular belief that wage rises are a fundamental driver of inflation.
– Mick
Aug 9, 2021 at 14:12

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:

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

• My only issue with this explanation (which is otherwise perfect - and I will upvote it) is why do you do $0.8 \times 1.05$. My guess was that if Wages were denoted as $W$, then our new Wage costs are $1.05W$, which of the total costs is $0.8 \times 1.05 W$ – is this intuitively correct? Aug 8, 2021 at 20:59
• @vik1245 wage costs are part of total costs. If wage cots are 80 per cent of total cost then you can just multiply total cost by 0.8 to get to wage cost, introducing new variable W in such case would be redundant and inefficient and would serve no point
– 1muflon1
Aug 8, 2021 at 21:00
• Ah yes the variable W was just illustrative, but your point agrees with mine so I'm satisfied that I've understood that now! Appreciate the response! Aug 8, 2021 at 21:02
– 1muflon1
Aug 8, 2021 at 21:02
• Yep I've read your answer and it now makes perfect sense! I've accepted it as the answer! Thanks very much! Aug 8, 2021 at 21:05

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 $$r$$

• Hi James, I like the derivation you have included in the algebraic form! I'm struggling to follow it though on screen – is it possible if you could format it in LaTeX or a mathematical package? If not, I can try and edit it instead if that's ok for other viewers since I enjoy the algebraic derivation you have provided! Aug 8, 2021 at 21:01
• Done. Hope that helps. Aug 8, 2021 at 21:05
• Thank you very much James! I've just seen the updated answer, and I think it is an excellent response! I've upvoted it for the benefit of other readers. Appreciate the help! - vik. Aug 8, 2021 at 21:06