## Wednesday, August 5, 2009

### Is free ice cream free?

This week is "Free Week" on This Young Economist. That is, every post this week pertains to the notion of "free." If you haven't caught it, I'm giving away a free lunch for this week's poll. Vote on the poll and sign up for your chance at a free lunch. [link here: if you want to enter, there's a form] If anything else, it will be fun. And yes, I really am giving away \$5 (I'll put everyone's entries into a numbered list, and then use Random.org's True Random Number Generator to select the winner). Now, onto today's post.

In May, Jodi Beggs wrote an interesting article about rationing free goods. A local Baskin Robbins gave away free ice cream. Understandably, lots of people showed up, among them Jodi Beggs who snapped these pictures.

Beggs concluded, "that's a damn long line for a \$4 item." Beggs then gave the standard opportunity cost story: If you value your time at \$10 per hour -- as you would if you had a job that paid \$10 per hour -- each hour in line would cost \$10 because that's what you could otherwise earn while waiting in line.

In this post, I want to take this reasoning further:

If everyone values their time at \$10 per hour, the equilibrium wait time for a "free" \$4 cone would have to be 0.4 hours or 24 minutes. Then, how many people should we expect to see standing in line? That depends on the efficiency of the Baskin Robbins workers:
• What if each person adds two minutes to the wait time? Then, we'd expect the line to have 12 people in it.
• What if each person adds 30 seconds to the wait time? Then, we should see 48 people standing in line.
This is an important insight into the effect of line efficiency. In the absence of economic reasoning, we might expect customers to spend less time in line if the workers process each customer more quickly. But, if that's the case, more people will queue up, and it takes longer to get through the line.

This idea came from an economist named Yoram Barzel in a 1974 academic paper entitled "A Theory of Rationing by Waiting," which was published in The Journal of Law and Economics. There's a whole lot more to the Barzel paper: namely, he covers the case where people have different costs of waiting in line, the case where people can buy as much as they want when they get to the front of the line, and other cases.

But, without going into the messy heterogeneous details, the point is: Depending on how long it took for the Baskin Robbins employees to dispense each ice cream cone, that horrendously long line actually might be the "right" length.

1. I just want to point out that I was merely passing through and not specificially there for the ice cream. :)

There is one caveat with your reasoning- to reach the conclusion that a person would wait in the line for 24 minutes requires an implicit assumption that he values the ice cream at a minimum of \$4, which I am not convinced is the case, since I doubt that these people waiting in line actually buy ice cream from Ben and Jerry's on a regular basis. (My evidence comes from the fact that there is not usually a line at the store and the fact that these people do not appear to be obese.)

2. Interesting point.

Differences in opportunity cost could explain why different people would "queue up." That's all part of the messy details in the Barzel paper.

That said, I was just trying to convey the Barzel reasoning that how long you wait for a "free" good does not much depend on how quickly they get each person through the line.

But, you're right: Heterogeneity in the market can change the value of the equilibrium price when we switch to paying with time, especially if the market has a bunch of low opportunity cost people.

Here's how: Suppose skinny people really enjoy talking to one another, and they would be having their conversation elsewhere if there wasn't a "free" good to be rationed. In this case, there might not be much cost to talking while standing in line because they would be talking with one another elsewhere...

As such, the line could get quite long (and full of skinny people), which could deter the regulars from showing up. If the skinny ones were asked to pay in cash, they might not pay that much (say 50 cents), but because they value an alternative use of their time at such a low rate, they're willing to be the first in line.

So, you can get skinny people at the front of the line, and the regulars could just wait for another day.

Regardless, it still doesn't matter in this story whether the workers are fast at dispensing ice cream. Line length will respond to worker efficiency to keep the wait time in equilibrium.

Oh, and I just realized that you (econgirl) were referring to a Ben and Jerry's, and I cited it as "Baskin Robbins." Sorry for the miscitation. I always get confused between the two chains... maybe because I don't eat much ice cream....

3. eat more ice cream.

4. Hey Tony

Your \$10.00/hr value on time only works if the person is giving up \$10.00/hr that particular hour which is not likely the case or they wouldn't be there.

"low opportunity cost people" is a good way of saying it. During that particular hour they have little opportunity cost so what the heck wait in line for an ice cream you don't have to pay money for.

Fletch

5. Thanks Fletch,

I was thinking that too, but I was trying to tell the simplest story.

On the other hand, a proponent of competitive markets would disagree with that assessment. Here's why:

If people choose their actions optimally, one consequence is that the marginal value of a unit of time is equal for each moment in time.

If it isn't, scale back the time you spend on low marginal value activities (MV will increase), and scale up the time you spend on high marginal value activities (MV will decrease).

Then, if the wage rate is \$10 per hour, people will allocate time in their other activities such that no matter what time of day it is, they value it at the same as earning \$10 per hour at work.

This reasoning assumes that you'd be able to adjust your time spent on various activities continuously. That's certainly not true, but to the extent that people are constrained by time, there should be some truth to it.

Therefore, the \$10 per hour figure isn't such a bad approximation after all.

6. Good thought Tony.

Taking it another step you say "This reasoning assumes that you'd be able to adjust your time spent on various activities continuously. That's certainly not true,"

So if a major assumption in that reasoning is "certainly not true" says something about the reasoning. If there is only "some truth" to the reasoning leads me to think there is little real world use for such reasoning.

Fletch

7. Fletch,

1. To be clear, the reason why I only claim there is "some truth" to the \$10/hour being the right number is not logical, it's mathematical.

When we write the equations out, the opportunity cost of the last hour is exactly equal to the person's wage rate if you can divide your time continously into activities.

But, when you have to chunk your time up, the last unit of time might not be exactly worth \$10, but it will be worth at least \$10 (and probably as close to \$10 as the increments allow). Therefore, the approximation isn't too bad on mathematical grounds.

2. Simple, useful explanations of the world have to abstract from many of the details of the real world. Otherwise, understanding what's going on gets too difficult.

For that reason, we make assumptions that have "some truth," but hopefully capture the essence of people's behavior in the moment. There's use to seeing the implications of a simple explanation (maybe the messy details don't matter).

Then again, maybe they do. At any rate, seeing the implications of the simple explanation (even if it isn't based on correct hypotheses) can really help you understand what's driving the phenomenon you're studying.

On a more philosophical note, we don't make assumptions because we believe them. We make them because (a) they're useful, and (b) if we think carefully, we can learn about the importance of the things we assume away.

8. Tony

This is getting interesting.

You said. ""1. To be clear, the reason why I only claim there is "some truth" to the \$10/hour being the right number is not logical, it's mathematical.""

"Not logical it's mathematical" is the part that got my attention. My first thought is that math is pure logic. no opinions, emotions, options just a strictly defined set of observable relationships. Then I decided to look up "logic and mathematics." Big mistake. I find that my observation that math is pure logic isn't universally held. (ha!) In your example we are talking about human behavior which I would observe as closer to pure " not-logic." People aren't logical. "When dealing with people, let us remember we are not dealing with creatures of logic. We are dealing with creatures of emotion, creatures bustling with prejudices and motivated by pride and vanity." As Dale Carnegie observed,

I guess my point is that if it isn't logical it can't be mathematical. Or if it is mathematical it must be logical and people are neither. I might say that people are logical in their thinking, but they begin with illogical premises.

Now that I think about it, never mind. I think I'll just get some ice cream and go to bed.

Fletch

9. Fletch,

"I guess my point is that if it isn't logical it can't be mathematical."

That's right, and it's a big reason that economists translate their logic into math. To clarify, I really meant to say that the "some truth" statement has more to do with annoying arithmetic than the logic behind the math.

As for the Dale Carnegie quote, that's also a great point: pride and vanity are important for many things. Carnegie's "How to Win Friends and Influence People" is a great example of how one can play to people's pride and vanity.

That said, it's true that people aren't logical, but for reasons stated above, it's useful to ask "What if they are logical?" If nothing else, the conclusions from that exercise give us a *logical* starting point.