*Preface: The inscribed square problem, also known as the square peg problem or the Toeplitz’ conjecture, is an unsolved question in geometry: Does every plane simple closed curve contain all four vertices of some square? The problem was proposed by Otto Toeplitz in 1911. As of 2017, the general case remains open.*

**Math in the Workplace (Segment II)**

*Credit: Jacob Dietz, CPA*

**Pricing based on Costs When Costs Decrease**

Assume the situation is the same as the above example, except John discovers that his cost of sales decreased by 5%. He decides to recalculate his prices.

He uses $66.50 as the cost, and he calculates a sales price of $95 ($66.50 divided by .7.) His sales price is $95, his cost of sales is $66.5 (70%), his gross profit is $28.5 (30%) and his overhead is $15 (15.8%) and his net profit is $13.5 (14.2%)

What just happened? Because John calculates his sales to earn a 30% gross profit, there was no change in the gross profit percentage, but there was a change in the gross profit dollars. 30% of $95 is $1.50 less than 30% of $100. John decreased both his gross profit and net profit by $1.50, and he dropped his net income percentage to about 14.2%.

**Takeaways from Pricing**

How can John use his knowledge about the effects on profitability of price changes and cost changes? If John’s costs increase, he may not be happy at first. If he can increase his prices using the same divisor (.7 in John’s case) on the new costs, however, John’s net profit can increase if he keeps selling the same quantity.

On the other hand, let’s assume that John faces fierce competition on price, and he cannot keep the same divisor of .7. In that case, he may be able to maintain the same net profit. For example, assume that his costs went up 5%, from $70 to $73.50. If John kept his normal divisor, he would then divide $73.5 by .7 and sell it for $105. That would lead to the $16.50 profit instead of the $15 profit. If John could not raise the price $5, however, perhaps he could raise the price by $3.50 to $103.50. That would keep his net profit at $15. The calculation is the $103.50 sales price less $73.50 cost of sales less $15 overhead leaves a $15 profit.

At first glance, John may decide he makes enough profit at the $15 net profit even if he does not face withering competition on price. He may decide only to raise the price to $103.50 instead of $105 because he will still come out the same at $15.

Or will he come out the same? Even though his net profit would still be the same, his cash position may not be the same. He is making the same net profit as before the cost increase, but his cost of sales jumped. If John carries significant inventory, then that additional cost of sales may hurt his cash balance. Each item that use to cost $70 now costs $3.50 more, or $73.50.