More Economics-Based Pizza Shenanigans!

Warning! There might be major flaws in my reasoning. If so, please point them out in the comments.
What would happen if a group of people were to contribute money to a fund to buy pizza, and there was an agreement to split the pizza equally? (People eating pizza together can obviously divide it up according to how much money each person contributed, or by how hungry each person is, but we shall use the simplification of equal distribution, for illustrative purposes.) This is the flip-side of my Joey and Chandler Theorem, where the costs of each participant were the mean of the group. In this case, the benefits for each participant (slices of pizza) will be the mean of the group.

The main point of this article is to show that if a group of friends are pitching in for pizza, there will be certain effects on everybody's demand functions. Lets say there are two friends, Rachel and Monica, who pitch in for pizza together. (I will be using subscripts r and m for the two participants, and P and Q for price and quantity). The total cost for Rachel will be P*Qr. To avoid confusion, remember that there will be a difference here between quantity ordered and quantity consumed, because of the agreement to share. The total cost for Monica will be P*Qm. However the total quantity consumed for each participant will be equal to:
(Qr+Qm)/2
Let's give Rachel the following demand function for pizza:
P = 5 - Q
Since her Q will now be the mean of both her and Monica's quantity of order, her demand function will now be:
P = 5 - (Qr + Qm)/2
Notice two things about Rachel's new demand function:
First, her demand now partially depends upon Monica's order, the constant Qm/2. Secondly, the slope of Rachel's demand function has changed. It was =-1, now it is:

DP/DQr = -1/2
Notice that Rachel's demand has just become more elastic. To understand this intuitively, lets think of what would happen if the price of a slice of pizza were to increase by $0.50. Rational consumers would consume less. Now what if the price of half a slice of pizza were to increase by $0.50? Rational consumers would consume even less than in the first situation. What if the price of a slice of pizza were to decrease by $0.50? Rational consumers would consume more. What if the price of half a slice of pizza were to decrease by $0.50? Rational consumers would consume even more. This is what is happening with Rachel's demand function. For every slice of pizza she orders, she will pay the full price, but only get to consume half of it. Thus she will be more sensitive to changes in price.

Let's say that pizza is sold by the slice for the price (P) of $1. Let's also say that Rachel will contribute money to buy pizza before Monica does.
Solving for Qr, Rachel's order will be:
Qr = 8 - Qm
But this function only applies if Rachel knows how much Monica will order. If Rachel puts her money down first, she will have to use her expected value of what Monica's order will be: E(Qm)r. The person who puts his/her money down first clearly sets the tone. Let's say Monica has an identical demand function to Rachel, and that Rachel expects Monica to order 3 slices of pizza. Let's see what will happen.
Rachel will order:
Qr = 8 - E(Qm)r
Qr = 8 - 3 = 5
So, Rachel will order 5 slices of pizza, but what will Monica do now that she knows what Rachel ordered, and does not have to depend on an expected value? Monica, with her identical demand function to Rachel, will order:
Qm = 8 - 5 = 3
So, Monica will order 3 slices of pizza. But both Monica and Rachel will be splitting the pizza equally, eating 4 slices each. So, Rachel ends up paying for one of Monica's slices.

What if Rachel realizes that she got ripped off? The next time the two order pizza she has a different strategy. She immediately tells Monica that she can only afford 3 slices. What would happen? Monica, knowing that Rachel will only be contributing 3 slices to the pizza pool, to maximize her utility from the transaction, will now be forced to order...
Qm = 8 - Qr
Qm = 8 - 3
Qm = 5
...5 slices of pizza, just as Rachel had ordered the time before. Because she gave a low-balled contribution to the pizza pool, Rachel has cheated Monica out of $1. This method of low-balling has a risk, though. If Rachel misjudges how hungry Monica is, she might end up with less pizza than would maximize her utility.
In a situation like this, where costs are borne independently, but benefits are shared, the expected or actual value of what one's friend, or friends will order does something to one's demand curve. If friend A knows, or is otherwise convinced that friend B will order a higher quantity of pizza than friend A would order independently, to take advantage of this, friend A will shift her demand curve inward, and order less pizza than if she had simply been ordering by herself. And if friend A knows or is otherwise convinced that friend B will order less than friend A would order independently, to make the most of this unfortunate situation, friend A's demand curve will shift outward. For some reason the latter rule, to me seems less intuitive. But, the demand curves, with their shifting intercepts, do not lie. Or do they?
Let's get away from this algebra, and into some real life. No, the demand curves do not lie, but they are incomplete. For one thing, we must realize that these demand curves only represent the behavior of people who would choose to enter into a benefit sharing agreement. In reality, there are many other alternatives to splitting pizza with people, like getting pizza on your own, and I can imagine that many would find that arrangement more desirable than sharing. Also the price of pizza is not the only cost one pays for when pizza is ordered, and the quantity of pizza is not the only benefit. There are psychological costs and benefits as well. For example, if one does not pitch in enough money for the pizza, one might be stigmatized as a cheapskate, jerk or loser. If accountants could look into the minds of people ordering pizza in this way, they could make debits for "being called a cheapskate expense", "guilt over cheating one's friends expense" or credits for "perceived generosity revenue". But the human mind is hard to penetrate. Nonetheless we should understand that forces other than price and quantity are at work here, which is why people do not necessarily act in the ways shown on the graphs. Economic models (much like fashion models) do not have wrinkles, but real life is full of them.

Quantity Discount or Not? (This Article Could Save You Money.)

At a certain chain drugstore (which shall not be named), about a year ago, I came across an amazing quantity discount: ten containers of oatmeal for $10. I was immediately sure that I wanted to buy these ten items, but not sure that I could carry them all home, as I had travelled there on foot. But the temptation of cheap oatmeal was too great, and I chose to lug all of them to the cash register and brave the walk home, overloaded with oats.

But at the cash register, something shocked me. As the cashier scanned each container of oatmeal, the green text on the monitor facing me read: "oatmeal - sale $1.00", implying that, in order to obtain the discount, I didn't have to buy ten items in the first place! I could have bought three of them for three dollars, or five of them for five dollars. This quantity discount had not been a quantity discount at all!

I felt cheated and betrayed. But there is a silver lining here for me: now whenever I shop there, I find whatever deep "quantity" discounts they offer, and only buy one item. This saves significant amounts of money on the reasonable (and easier to carry) quantities I buy.

I did not, and still do not understand why this retailer (and others, as I later discovered) would offer a false quantity discount.

I have several theories. But for now let's just break the situation down so we can understand it better. Here I want to logically demonstrate why under all circumstances, the offering of a false quantity discount provides less revenue for a vendor than offering a real quantity discount. Let's say that a hypothetical retailer offers a 10 for $10 dollar oatmeal deal, but in reality the retailer has simply lowered the price of oatmeal from $2 to $1. In this situation there are five distinct groups of consumers (identified by number):

1. There are consumers who would buy 10+ items, and won't notice at the cash register, or on their receipts, that they didn't really need to buy the 10 items to get a discount.

2. There are consumers who would buy 10+ items, and will notice that they didn't really need to buy the 10 items to get a discount.

3. There are consumers who would buy fewer than 10 items, who won't notice that they didn't really need to buy 10 items to get a discount.

4. There are consumers who would buy fewer than 10 items, who will notice that they didn't really need to buy 10 items to get a discount.

5. There are consumers (like me, now that I have learned) who already know that the quantity discount is false, and will thus buy with full knowledge of what the price will be at the cash register.

Groups 2 and 4, the crafty consumers who take notice of what they have been charged, will each be given a surprise at the cash register. For group 4 it will be a pleasant surprise, and for group 2 it will be an unpleasant one.

Let's say there is someone named Steve, from group 4, who comes to the store, sees the oatmeal, is not willing to buy 10 units, and instead buys 3 units. Steve expects to be paying a total of $6 for his oatmeal, but at the cash register, he will notice the unit price is only $1. Steve will thus receive a pleasant surprise of $3.

Now what if someone named Jane from group 2 comes to the store. She sees the quantity discount and is blown away. She excitedly grabs 10 containers, and rushes to the cash register. Jane expects to be saving $10 as a result of her thrifty bulk-buying behavior, but at the cash register she will find that she did not need to buy in bulk to get the discount. This will be an unpleasant surprise.

If this had been a real quantity discount, the firm could be capturing an extra $3 from Steve, because he would have bought the oatmeal for $6 anyway. They would not be capturing anything more from Jane, but she would not have had that unpleasant surprise at the cash register, a surprise that may affect her opinion of the retailer in the future. Also, keep in mind that any customer from group 2 or 4 automatically transfers to group 5 when they find out that the firm's quantity discounts are bogus. And it is very important to note that the retailer could also be capturing extra money from each consumer from group 3, just as it could have captured surplus from Steve and the rest of group 4. Just like Steve, the consumers in group three weren't going to buy 10+ items in the first place and were happy with the price without the discount. But when they get to the cash register, they will be charged less than they were willing to pay. Simply put, the retailer is giving away money on every transaction with a group 3 or 4 person!

Discounts are only useful if consumers know about them. If prices fall in the woods, and there's no one around to hear them, they do not make a sound. A discount not known by the customer causes the firm to lose revenue on every transaction. At best, such a discount works as a pleasant surprise to the customer, who may then want to go back to the same store. The best case scenario for the firm seems to be when a consumer is from group 1. But in this scenario, the same result would be achieved if the firm had offered a real quantity discount.

So, in the face of all of this lost revenue and possible consumer frustration, why doesn't this retailer offer real quantity discounts? Maybe the firm wishes to create a subgroup of customers who know that the quantity discounts are false (group 5), thereby receiving the loyalty of this group of insiders (this is the far-fetched conspiracy theory explanation). Maybe there would be high costs involved in changing the computer system of the company's cash registers to be able to process quantity discounts (a more realistic explanation). Another possible explanation is managerial incompetence. But I try not to be quick to jump to that conclusion, however tempting it may be to make myself feel smarter than successful business managers. I'd like to think there is a reason for this policy, even a bad reason. Nonetheless, this company is losing money every time a customer from group 3 or 4 comes to the cash register. If any readers have theories or inside information on why this is, please let me know, I am very eager to solve this mystery. While its perhaps not rare that a company would mislead its customers, it is rare that a company would choose not to take money from its customers. I'm sure you can understand why I am confused.

(Follow Up to this story 1/6/11 by Alex Trenchard-Smith.

In the years since I wrote this article, I have noticed several more grocery and drugstore chains that follow the same policy of false quantity discounts. So as a shopper, definitely check your receipt to see if you really need to buy the recommended quantity. Since then I've also gained some knowledge of computer programming and reached the conclusion that companies probably do this because it's too complex to deal with Point Of Sale software that needs to be updated all the time with required quantities. They find it is more cost effective to mislead their customers than to add/and or utilize some simple if-else statements in their POS software.)