## The stealth price increases at the grocery store

One particular snack containing almonds, cashews and cranberries that I buy at a local grocery store is now in a reduced size of 14 ounces per pack (previously a 16-ounce pack). I am getting 2 ounces less with the same price. This is a stealth price increase. In this tough economic time, this must be a favorite way for manufacturers to raise prices. Breakfast cereal boxes seem to be getting smaller. It will be an interesting exercise to find out what the percentage price increases are for these items with reduced sizes. For example, what is the percentage price increase for the snack pack that I buy (going from 16-ounce to 14-ounce)?

Since the amount I pay at the cash register is the same whether it is 16-ounce or 14-ounce, I must look at the unit price (price per ounce). The old unit price is $0.28 (=$4.49/16) and the new unit price is $0.32 (=$4.49/14). Thus they are charging me 4 cents more per ounce by giving me two ounces less almonds and cashews. The percentage increase in price is 14.29% (0.04/0.28=0.1429). As a percentage, this is a hefty price increase!

The following is a general formula for finding percentage price increase.

$\displaystyle (1) \ \ \ \ \ \ \ \ \ \ \ \text{Percentage Price Increase}=\frac{\text{Dollar Price Increase}}{\text{Old Dollar Price}}$

There is another way to compute the percentage increase. Observe that 0.32/0.28=1.1429. Thus the ratio of the new price to the old price contains the information of the percentage increase. All we need to do is to subtract 1 from this ratio. We have the following formula, which is equivalent to the above formula.

$\displaystyle (2) \ \ \ \ \ \ \ \ \ \ \ \text{Percentage Price Increase}=\frac{\text{New Price}}{\text{Old Price}}-1$

It turns out that in the situation of price increase by reducing weight while charging the same price, calculating the percentage increase does not require the price amount. In our example, the percentage price increase does not depend on the price being $4.49. All we need to do is to compute the ratio of the old weight to the reduced weight. This ratio contains the information of the percentage price increase. Simply subtract 1 from this ratio, we obtain the percentage increase. In our example, 16/14=1.1429. $\displaystyle (3) \ \ \ \ \ \ \ \ \ \ \ \text{Percentage Price Increase}=\frac{\text{Old Weight}}{\text{Reduced Weight}}-1$ Formulas (1) and (2) are general formulas that work for all situations. Formula (3) can only be used for calculating percentage increases as a result of reducing the weight of the product (while charging the same dollar amount). Formula (3) works because the price of the product (e.g.$4.49 in our example) is canceled out in the derivation of Formula (2).

Formula (3) can be handy for displaying scenarios (e.g. a manufacturer may want to determine the percentage increases for different sizes of reduction in weight). The following matrix displays the percentage increases for various sizes of reduction of a 16-ounces (or any other unit of measurement for weight).

Reducing a 16-unit package:

$\displaystyle \begin{pmatrix} \text{Weight Reduction}&\text{Percentage Increase} \\{\text{ }}&\text{ }&\text{ } \\\text{1}&\text{6.67 percent} \\\text{2}&\text{14.29 percent} \\\text{3}&\text{23.08 percent} \\\text{4}&\text{33.33 percent} \\\text{5}&\text{45.45 percent} \\\text{6}&\text{60.00 percent} \\\text{7}&\text{77.78 percent} \\\text{8}&\text{100.00 percent}\end{pmatrix}$

Obviously, the more the reduction in weight, the higher the percentage increase in price. I can see that a company could very well look at a matrix such as the one above to see what they can get away with. Reducing a 16-ounce package by one ounce only gives a 6.67% increase, probably not high enough from a manufacturer’s perspective. By reducing too much, the increase will no longer be stealth. So they probably would like to target a reduction that is not so conspicuous and yet one that can achieve a significant percent increase. In the case of our example of the snack packs of almonds and cashews, reducing the package by 2 ounces may not be too noticeable while acheiving a double digit percent price increase. That was a winner for the manufacturer!

If a manufacturer is bold enough to reduce the 16-ounce package to an 8-ounce package while charging the same price, the price increase would be 100%! You would to pay twice as much to get the same 16-ounce of the product! I am sure no manufacturer in their right mind would go for this.

One final observation. For a fixed amount of reduction in weight (say 2 ounces), the larger the original weight, the smaller the percentage price increase. For example, if the original weight is 24 ounces, a reduction of 2 ounces would only acheive a 9.09% increase. See the following matrix. Each percentage is calculated using Formula (3). For example, for a reduction of 2 ounces from a 24-ounce package, the percentage increase in price is 24/22 – 1 = 0.0909.

Reducing a 24-unit package:

$\displaystyle \begin{pmatrix} \text{Weight Reduction}&\text{Percentage Increase} \\{\text{ }}&\text{ }&\text{ } \\\text{1}&\text{4.35 percent} \\\text{2}&\text{9.09 percent} \\\text{3}&\text{14.29 percent} \\\text{4}&\text{20.00 percent} \\\text{5}&\text{26.32 percent} \\\text{6}&\text{33.33 percent} \\\text{7}&\text{41.18 percent} \\\text{8}&\text{50.00 percent}\end{pmatrix}$