A mathematical model on the optimality of coupons

So it's been too long since I posted a pedantic over-analysis of mundane topics like last names and lawnmowers. Several of the women in our ward have a hobby/skill/amazing talent. They coupon shop, but it's on a level I could not have imagined even existed. They subscribe to multiple copies of the Sunday paper and then they cross reference their arsenal against grocery forums that determine the optimal store location to spend manufacturers coupons. Some stores double coupons, sometimes only on certain days, and most stores have weekly sales. In essence the secret sauce is to take the universe of potential coupons, which is large, intersected with the universe of goods on sale at coupon doubling institutions which is also large, and you end up with a small but very price-potent set of groceries.

You can bug Alison if you want to know the secrets to doing this, I wasn't at that particular enrichment meeting. What I want to point out is that doing all this could, indirectly, still make producers and grocery stores rich. I think on instinct we might consider this intense kind of coupon shopping as almost amoral, taking them to the cleaners when this is not necessarily the case.

Suppose there exists a consumer which possess the qualities "rich" and "good taste" (alternatively "spendthrift" and "fickle"). Depending on the day and their personal preferences they will be willing to spend $3 on either item A or B. Luckily item A and B only cost the grocer $2.00 to buy and the farmer $1.00 to produce. It would appear there should be plenty of profit to spread around and make all three happy, the trouble is, the "good taste" consumer is not willing to go with whatever is there, they want the one they want or their not buying. If the grocer supplies both, this costs $4.00 so she would lose if she served Mr. choosy and so no deal.

Now introduce another consumer possessing "bargain shopper" and "flexible". This person is willing to pay $1.00 for either item. Notice that this does absolutely nothing. If the store prices the items at $1.00 it will lose money on both sales, so still no A or B for sale. Now, suppose that the firm lowers the price after "rich" buys A or B. At $1 + $3 the store is just breaking even on $2 + $2. So the strategy is to put on sale items for which you have excess inventory. Notice that "Bargain shopper" paid the grocery store less than they paid for the item, yet the fact that they were willing to buy made the $3 sale possible. So just because the store is losing money on each and every item you bought does not mean they are not making money from you indirectly. "Bargain shopper" has sold the ability to buy whatever, whenever and is stuck with whatever "rich" leaves.

We can even get coupons in here. Suppose the grocery, rather than lowering the price just delays purchasing more inventory. Now the producer is in trouble. Since they make $1.00 on every sale, they can afford to offer $1.00 off coupons and still survive. If the price is $3.00, $1.00 is not enough to get the "bargain shopper" to buy. However if the store does coupon doubling then $3.00 becomes $1.00 at the register, inducing "Bargain" to buy. The store turns in the coupon for $1.00 netting for both goods A and B $3 + $2 - $2 - $2 = $1. So thanks to coupon doubling the grocer makes money even though they sold the item to you at a loss.

So next time you thoroughly abuse your local grocer store, rest assured that your cheapskate business may still be worth every penny.

Exercise for the reader: Assume the grocery store cannot put goods on sale, but does double coupons. Determine the price and coupon that splits the profit on the not preferred item equally.