Real life growth driver analysis problem
I am getting two Answers via two Methods. I would like to know which is Correct & why the other is Wrong.
To really simplify this problem let's say there is a farm that produce and sell apples. Both the price and quantity sold increase from one month to another.
I want to quantify the contributions of each component to the growth. In other words, how much of the $44 came from the price increase and how much came from the quantity increase?
Data for the problem
Sold in May: 10 apples
Sold in Aug: 12 apples
Price in May: $4
Price in Aug: $7
Revenue in May: 10*4= $40
Revenue in Aug: 12*7= $84
Revenue growth = $44
The logical approach (does not work)
Through trial and error I've got to a solution that gives me the right solution but I can't wrap my head around the logic of it.
May Revenue: 10 apples * 4$ = $40
Keeping quantity fixed (Price change): 10 apples * $7 = $70
Keeping price fixed (Quantity change): 12 apples * $4 = $48
Isolated Price growth: 70-40 = $30
Isolated Quantity Growth: 48-40 = $8
The problem is that 8 and 30 does not sum to the growth of 44, so I'm missing something here.
The approach that works (but not logical to me)
Using the same method as in the previous approach I'm altering but not with May as the base.
May Revenue: 10 apples * $4 = $40
Keeping quantity fixed (Price change): 10 apples * $7 = $70
Aug Revenue: 12 apples * $7 = $84
Keeping price fixed (Quantity change): 10 apples * $7 = $70
Isolated Price growth: 70-40 = $30
Isolated Quantity Growth: 84-70 = $14
This solution gives me the exact difference that I'm looking for ($44) but I don't understand how, and does it make sense?
Can someone math savvy please explain to me how I should make sense of this?
For me it makes sense to use May as the base and assume everything equal except one parameter at a time, but it doesn't give me the right answer.
What is going Wrong here :
Issue with your "not-working-approach" is that you are not accounting for the Combined Contribution.
Here is my analogy to highlight that :
When we have $(a+b)$ & want to square that , it will not match squaring the Individual Parts $a^2$ & $b^2$ & adding. We are leaving out the Combined Part $2ab$.
In more general terms :
When we want to multiply $(a+b)$ & $(c+d)$ , it is not enough to multiply each Part to get $ac+bd$ , we are leaving out the Cross Parts $ad$ & $bc$.
That is why you are getting smaller total.
Correct Way :
Let $(10)(4)=40$ & $(10+2)(4+3)=84$ be the revenues in the 2 months.
Subtract the two ( after multiplication ) to get :
$(10)(4)+(10)(3)+(2)(4)+\color{red}{(2)(3)} - (10)(4)$
$+(10)(3)+(2)(4)+\color{red}{(2)(3)}$
$+30+8+\color{red}{(2)(3)}$
$+38+\color{red}{(2)(3)}$
The highlighting indicates the Combined Part which you left out earlier.
We should Share that too between the two.
You can take it Proportionally $2:3$ which gives $2.4+3.6$ , or Equally $3+3$ or Some other way to Share.
I might take it Equally . which is easy , to get $30+3=33$ & $8+3=11$ & $33+11=44$.
Better Way :
We should look to get Percentage Contributions here , $P_1$ , Percentage of revenue growth due to Quantity & $P_2$ , Percentage of revenue growth due to Price , such that $P_1+P+2=100$.
In your Case , my tweak to the "working-approach" gives $33+11=44$ , then that is $100 \times 33/44 = 75\%$ & $100 \times 11/44 = 25\%$ , where $75\%+25\%=100\%$.