# A Puzzle: Separating a Cost Increase into Price and Quantity variances/pieces

I realize this is probably a simple problem for most of you, but let's say that the price of a product you purchase increases from, say:
--\$5 in Year 1 to \$10 in Year 2, for a \$5 increase in price--
Let's say that the quantity you purchase increases from, say:
--50 widgets in Year 1 to 70 widgets in Year 2, for a 20 widget increase--
So total cost in Year 1 is \$5 multiplied by 50 widgets for a total cost of \$250.
Total cost in Year 2 is \$10 multiplied by 70 widgets for a total cost of \$700.
Year 2 cost has increased \$450 from \$250 in Year 1 to \$700 in Year 2.
So I think what would be interesting to see is a breakdown of this \$450 increase into a price change factor/contribution amount and a quantity change/contribution amount.
The question I want to answer for the price contribution amount is: what is the contribution of the price increase to the total cost increase? In my mind, this question assumes quantity is held at the Year 1 level because you're wanting to analyze change in price, and changing quantity would muddy the waters. So the answer is price increased by \$5.
Multiply this \$5 price increase by the Year 1 quantity of 50 and you get \$250.
I now want to answer the same question for the quantity contribution portion. Quantity increased by 20 widgets. The Year 1 price is \$5. 20 times \$5 is \$100. So quantity changes alone, without regard to price changes, contributed \$100 to the price increase.
So we have accounted for \$250 + \$100 = \$350 of the \$450 increase. I want obviously to add up to the total increase. What remains to be added is the increase in price multiplied by the increase in quantity. This is \$5 times 20 = \$100. \$350 + \$100 = \$450 total increase. But in this case for the \$100, I don't know of a way to break this down between quantity and price. It is a combination of both. Is there a standard method to break this out? In a report, it would seem confusing to see a price contribution, a quantity contribution, and then a mixed contribution, so I would like to somehow break this mixed method up. Any advice?
Thanks.

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