What Does Buy One Get One Half Off Means

30 Aug 2016

So what does Buy One Get One Half Off really mean? This actually can be shown in very simple math. If you are not interested in the math, here is the short version: IT MEANS AT LEAST 25% IN PROFIT.

Before we start, let me put down a few logical assumptions:

• The selling price of any item $i$ constitutes of cost and profit, $S_i = C_i + P_i$
• Seller is selling any item $i$ with positive cost and for profit, so $C_i > 0$ and $P_i > 0$
• Seller is selling all items at the same profit margin (ratio of profit and cost) $\alpha$, $\forall$ item $i$, $\alpha C_i = P_i$, and $\alpha > 0$
• If a buyer buys two items with different prices, the buyer pays for the full price of the higher priced item
• With buy one get one half off, seller is still making money

With these assumptions, we can begin:

Let’s take two items: item $i$ and item $j$, without loss of generality, let’s assume item $i$ is at least as expensive as item $j$, $S_i \geq S_j$.

The price we are paying for these two items is $S_i + \frac{1}{2}S_j$, which is $C_i + P_i + \frac{1}{2}(C_j + P_j)$.

In order for the seller to make money, this amount should at least cover the cost of the two items, $C_i + C_j$.

So we have an inequality, $S_i + \frac{1}{2}S_j \geq C_i + C_j$, which is $C_i + P_i + \frac{1}{2}(C_j + P_j) \geq C_i + C_j$.

By eliminating and rearranging terms, we have $P_i + \frac{1}{2}P_j \geq \frac{1}{2}C_j$.

Remember that we made an assumption earlier that $\forall$ item $i$, $\alpha C_i = P_i$, so we can substitute $C_j$ with $\frac{P_j}{\alpha}$, and have $P_i + \frac{1}{2}P_j \geq \frac{1}{2\alpha}P_j$

Rearranging the terms once more, we have $\alpha \geq \frac{1}{2\frac{P_i}{P_j} + 1}$.

Because we assumed $S_i \geq S_j$, we have $P_i \geq P_j$, so $\frac{P_i}{P_j} \geq 1$, and the right hand side of the above inequality follows $\frac{1}{2\frac{P_i}{P_j} + 1} \leq \frac{1}{3}$.

As you can see that the maximum is achieved when $P_i = P_j$, which is $S_i = S_j$, which basically means we bought two identical items.

To make money in any given scenario, the seller should set the profit margin $\alpha$ to cover the worst case. So we reached the final conclusion:

$\alpha \geq \frac{1}{3}$. If we reconsider the concept of profit margin as the ratio of profit $P_i$ versus selling price $S_i$, we are looking at the minimum profit of 25% out of the selling price.