05 Sep 2017

## The Fun Stuff on Liberty Classroom

I was checking the Student Dashboard for my History of Economic Thought classes on Tom Woods’ Liberty Classroom, and came across a fun question prompted by my lecture on utility and welfare theory. Just to show you the romping good time, I’ll reproduce the question and answer below. Sign up today and join the fun!

Q: OK, I give up. How do you move from Pareto suboptimality to Pareto optimality, and still make some people worse off? I keep trying to figure this out and it just makes my head spin.

A: Heh it’s a fun one, isn’t it? It’s easier to do on a blackboard, but here goes:

Picture a 2-person, 2-good economy. It’s Xavier and Yolinda, and they have apples and bananas. There is no production, just a total of 10 apples and 10 bananas, that can be split between them.

Assume their preferences are such that each person always wants more of either good, but also prefers variety.

Now suppose Xavier has 10 apples and 10 bananas, while Yolinda has 0 apples and 0 bananas. This is Pareto optimal. You can’t make Xavier happier because he has all the goods already. And if you make Yolinda happier, you necessarily hurt Xavier. So since it’s impossible to make one person better off without hurting the other, the original allocation is Pareto optimal.

Now consider a different allocation, where Xavier start with 9 apples and 1 banana, while Yolinda has 1 apple and 9 bananas. It’s plausible that this is *not* Pareto optimal, because I said they like variety. E.g. we can imagine their preferences are such that if Xavier trades away 4 of his starting apples in exchange for 4 bananas from Yolinda, that they are both getting more utility (or end up with a preferable combination of goods).

So, I hope you find it plausible that this 2nd allocation I’ve described–where Xavier starts with 9 apples and 1 banana, while Yolinda starts with 1 apple and 9 bananas–is Pareto suboptimal.

Now, imagine we move from this 2nd allocation to the 1st allocation (the one where Xavier has everything). We’ve moved from a Pareto suboptimal to a Pareto optimal allocation, and yet in doing so we made Yolinda worse off, because now she has nothing.

Go ahead, admit it. You thought you’d have to go to Steve Landsburg’s blog to get mathematical econ like this. You thought I’d just say, “GOVERNMENT BAD. APPLES GOOD.”

#### 13 Responses to “The Fun Stuff on Liberty Classroom”

1. Tel says:

There seems to be a sleight of hand.

I mean suppose Xavier simply shoots Yolinda and takes all her stuff. The final situation has only one person remaining and we can ask him; he says, “Yup, this is optimal”… but I don’t think it’s supposed to work like that.

I admit I haven’t listened to the original lecture yet, so no doubt I am missing some context here.

• Bob Murphy says:

You’re right, that’s not how it works Tel. If Xavier shoots Yolinda then her utility goes down (presumably) so she is worse off. The criterion doesn’t rely on the person’s ability to enunciate the displeasure.

Here’s the definition.

• Darien says:

If I’m grabbing this correctly, the “catch” is that we have to continue to consider Yolinda, even though the “common sense” reaction is to stop doing so because she’s dead. If we start comparing a pre-murder system of Xavier and Yolinda to a post-murder system containing only Xavier, it’s apples and oranges. Is that more or less it?

Also, it’s nice to see that we’ve moved from the days of Alice and Bob into a more socially-just, diverse set of example actors. 😉

• Tel says:

But Bob, you allowed Xavier to take all Yolinda’s fruit. She must have been complaining about that, to say the least.

I was simply adding injury to insult.

• Bob Murphy says:

Xavier didn’t take her fruit, the Walrasian auctioneer did.

2. RPLong says:

Would this example also work?

Xavier has two cans of beans and two can openers. Yolinda has one can of beans and zero can openers. This allocation is not Pareto-optimal. We can get pretty close to a Pareto optimum by reallocating one of Xavier’s can openers to Yolinda so that she can open her can of beans, but doing so makes Xavier worse off because now he has to wash can openers twice as often as he previously had to. But the marginal utility gain to Yolinda for having one can opener is much greater than the utility loss to Xavier for only having one can opener instead of two.

• Gienon says:

Let’s assume the cans are open.

• RPLong says:

Haha! I walked right into that!

3. Harold says:

Isn’t this why we tend to use Kaldor Hicks instead of Pareto? Pareto is a slightly absurd standard. One person can dictate to the whole world. As RPlong says ” But the marginal utility gain to Yolinda for having one can opener is much greater than the utility loss to Xavier for only having one can opener instead of two.” This would make the exchange K-H improvement but not Pareto because the exchange has left Xavier worse off.

However, “We’ve moved from a Pareto suboptimal to a Pareto optimal allocation, and yet in doing so we made Yolinda worse off, because now she has nothing.”

This is an interesting paradox. There are other distributions that are Pareto optimal as well. Some distribution around 5 apples and bananas each, or indeed Yolinda having all the fruit. We can go from one Pareto optimum to another and still make someone worse off. So if there are are several Pareto optima, is there a way to discover the “best” Pareto optimum?

• Bob Murphy says:

Harold wrote:

if there are are several Pareto optima,

Right, the main thing I was trying to get across to my students (when I taught this at Hillsdale years ago) was that in a typical model, there are an infinite number of Pareto optimal points. I wanted them to not think of it as “the government should move us to THE optimum position.” And then when I was driving home that point, I realized the cute little possibility that I enshrined in this question.

4. Craw says:

Cute. The change is not a Pareto improvement though. This illustrates that you cannot just talk about Pareto optimally, you have to also consider how you get there.

• Bob Murphy says:

Right.

• Harold says:

“The change is not a Pareto improvement though.” I see – this is what I missed when I called it a paradox.