Should You Raise Prices? Should You Lower Them? Derivation of the Formula For Planning Price Changes

Here's how I used algebra to derive the formula that calculates the breakeven sales volume for a prospective change in product prices.

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Here's how I created the formula that calculates the breakeven sales volume for a prospective change in product prices.The article, Should You Raise Prices? Or Should You Lower Them? These Excel Charts Can Help You Answer Those Questions, introduces the Price-Change formula. Before you use the formula, it’s a good idea to assure yourself that it’s accurate. So here’s how I derived it…

Here are the variables used in the formula:

Variables used in the formula to decide whether to raise prices or lower them.

Our gross profits before and after the price change are equal to our price minus our product cost, multiplied by our unit sales volume:

The gross profits before and after a change in prices

We’re looking for the gross-profit stay-even point, the point where gross profits after the price change equal gross profits before the price change. That is, we want:

We're looking for the stay-even point, where profits before the price change will equal the profits after it.

We can rearrange this formula to arrive at:

We rearrange the formula so that it looks like this.

We can add zero (P0 – P0) to the bottom-left and zero (V0 – V0) to the top right, giving us:

We add zero (P0 - P0) to the bottom-left and zero (V0 - V0) to the top right

Now we can multiply the left side by 1, in the form of (1/P0) / (1/P0). And we can simplify the right side, to return:

We multiply the left side by 1, in the form of (1/P0) / (1/P0), and we simplify the right side.

The top-left term is our current profit margin. The bottom-left terms are our current profit margin minus our price change as a percentage of the current price. The right-hand term is 1 + our percentage change in volume after the price change. Moving the 1 to the other side of the equation, and simplifying, gives us the Stay-Even Formula for price changes:

The Stay-Even Price-Change Formula, which tells us the point where we can change prices and leave gross margin unchanged.

Keep in mind that this formula doesn’t address other costs that might be affected by your planned price change.