Consider a pop music concert. For reasons that we do not necessarily have to go into, pop music stars do not always want to charge the highest possible (single-concert profit-maximizing) prices for tickets to their concerts. In fact ticket prices are often so “low” (I still find them rather expensive) that many more people would like to go to the concert (at these prices) than there are tickets. The economic term for this is that tickets are being “rationed”. What is the result of such rationing?

If tickets are sold offline in a single “brick-and-mortar”, as people like to say, ticket booth, then we get long queues and people starting to queue at 2am of the morning of the day ticket sales begin or they even get there earlier and camp out with sleeping bags. If the selling is done online, then you have about one second in which you can buy your ticket, with many people with a slower internet connection missing out. Is the final ticket allocation in such cases of rationing Pareto-efficient? Think about it.

When people say that markets are efficient then they mean the notion of Pareto efficiency I provided in a previous post: An allocation is Pareto efficient if there is no other allocation that is a Pareto improvement. An allocation is a Pareto improvement over another allocation if the former is at least as good as the latter for everyone involved and strictly better for at least one person. As we saw, Pareto efficiency has nothing to do with fairness. If I have everything there is to be had in the world and I want to have all this stuff then this is Pareto efficient. Because any other allocation would require me to give up something and, as I do not like to do this, this other allocation is not a Pareto improvement because I am not as happy as before.

This is a joke that I heard many times and once on a big stage at the 2014 annual meeting of the Verein für Socialpolitik where some supposedly important person from a supposedly important central bank (if I recall correctly) used it as a criticism of current economic methodology (as this person understood it) and generalizing it to mean it as a criticism of any economic methodology that uses math (if I understood this person correctly).

We will come back to the idea that people typically like more money over less (all else equal) – see last lecture, but let me now turn to another important idea, trade. Let me first talk about trade without money. I will (re)introduce money into trade a little bit later, but one key idea behind trade does not require money.

I don’t know how this is done in other areas in the world, but in Graz there seem to be certain specific norms that one should follow when you host a kid’s birthday party. You invite roughly as many children as your child’s age in years. Children bring presents, but each child also goes home from the party with some little bag of goodies. As concerned parents we do not want to give the children too many sweets so we give them little presents such as little Lego or Playmobil figures or a car or something like this. We did this twice this year (we have two kids) and in both cases the first thing that happens after the kids finally find the treasure (there is often a sort of treasure hunt) is this: the kids start to trade. So, what is going on when kids are trading their presents?

To fix ideas consider a particular example with three kids. I call them Eva, Fritz, and Maria. They each receive one figure, Eva a pirate, Franz a nurse, and Maria a ghost. Assume that we are able to see inside their heads and see the kids’ preferences over these three figures. These are given in the following table.

Please take a moment or two to think about which of these kids, in your opinion, would be interested in trading their presents.

Are you done? The answer is that actually every pair of them would potentially be happy to trade presents. Let me run through all possible sequences of trades.

Suppose first that Eva and Franz trade. This leads to Eva receiving her favorite figure, the nurse, and Franz receiving his favorite figure, the pirate. Maria is still left with her ghost. So what has happened? This is what we call a Pareto improvement. A Pareto improvement is a change in allocation that keeps everyone at least as happy as before and makes at least one person strictly happier.

A quick aside: The term Pareto improvement is named after a certain Vilfredo Pareto. He is no longer with us. You can google his name if you like. My policy with names in this course is this: I try to avoid them. I generally try to avoid talking about famous dead people, because I want you to judge concepts, ideas, and results for themselves. I don’t want you to “believe” in concepts, especially not just because the concept came from some famous dead person. This does not mean that you wouldn’t benefit from a course on the history of economic thought. One of the key things you learn in such a course, in my opinion, is to see how the times influence and constrain economic thinking. This allows you to appreciate how certain older “models” need to be adapted to fit modern times, but also how certain, perhaps forgotten, ideas from older “models” might still have relevance today.

But back to Pareto improvements. Is there another possibility of trade after Eva and Franz trade? My answer would be no. Why? Because Eva and Franz already have the thing they like best and so why would they trade? So I would say that there are no more trades after the first one. When I asked students in the class, they said that they thought that Franz and Maria could possibly trade. This would give Maria her most preferred figure, the pirate, and would give Franz his second most preferred figure, the ghost. When I asked them why they think Franz would be willing to give up his pirate for the ghost, students said that maybe he likes Maria. This is an interesting point. This means that my depiction of preferences in the above table is not complete (or I do not take into account that as they are friends, Franz and Maria will probably meet again and can follow an unwritten contract between them in which it says that Franz gives Maria her favorite toy and in return Maria will give Franz some smarties in school the next day). If I wanted to capture a situation in which Franz cares about what figure Maria gets, I would have to change Franz’s preferences. His preferences would then not only be over which figure he gets himself but over the allocation of figures for him and Maria. This is not impossible to do and of course quite relevant in some cases. We would then say that allocating figures has externalities. One child getting a better figure according to her own preferences affects some other child’s preferences directly. This is a situation we come back to later in the course and will be the beginning of a discussion of “market failures”.

Assuming for now that the three children only care about what figure they themselves get, we have that after Eva and Franz trade there are no more trades.

What about the other trading options? For instance, Eva and Maria could trade, giving Eva the ghost and Maria the pirate, and then Eva could trade her ghost with Franz’s nurse. There are quite a few options of trade sequences. Any trade leads to a Pareto improvement and trading stops when there are no more possible Pareto improvements. Such a final situation is called Pareto efficient. We will encounter Pareto efficiency throughout the course. Whenever we say that a market under some assumptions is efficient we mean Pareto efficient. It is one of the basic and most helpful concepts in economics, but also one of the most misunderstood, I feel. It is my goal here to explain exactly what Pareto efficiency means and what it does not mean. One thing we already see in this example is that Pareto efficiency has little, if anything, to do with what one would call fairness. If, for instance, Eva and Franz trade, we get a Pareto efficient allocation but Maria is not very happy: she still has her worst figure, while the other two got their best figures. This doesn’t seem particularly fair. I will keep repeating this point that Pareto efficiency has nothing to do with fairness throughout the course in various different contexts.

By the way, the different sequences of (bilateral) trade in the example led to three different possible Pareto efficient allocations. In terms of the kids’ preference ranking these allocations can be expressed in this table:

Can we compare these further? You might think that the second allocation, the 1-2-1 allocation (meaning Eva gets her favorite figure, the nurse, Franz his second favorite, the ghost, and Maria her favorite, the pirate) is better than the other two. But this is not clear. It could be that Franz really would like the pirate much much more than the ghost, while both Eva and Maria don’t care that much about which figure they get. Of course it could also be that Franz cares little about all this, and Eva and Maria care a lot. So we cannot really so easily compare two Pareto efficient allocations. This will be more interesting but also more radical, when we introduce money into trade, which I do in the fourth lecture.

We can use this example to talk a little bit about free trade agreements. On this topic see also my blog post on the Wachau. Suppose that two of the three children, Franz and Maria, currently have a free trade agreement, but that Eva is not part of that agreement. Would both members of the “Franz-Maria free trade zone” like to add Eva into this zone? The answer here is at least “not necessarily” and actually probably “no”. Why not? In the Franz-Maria zone trade takes place between the two of them and Franz gets the ghost and Maria the nurse – both get their second favorite figures. Suppose that before they trade they wonder whether they should let Eva participate in their trading. Maria might well be against this, as she might be justifiably worried that Franz will then trade with Eva and not with her. After all, Eva has Franz’s favorite figure, the pirate, and Franz has Eva’s favorite figure, the nurse. But if Franz and Eva trade, Maria will be stuck with her least favorite, the ghost, whereas if she only had Franz to trade with she would at least get the preferred nurse. This strikes me as not so different from the current situation in the US in which the US steel producers (Maria) are not so happy about cheaper foreign steel imports (Eva). Of course steel users in the US (Franz) would prefer the cheaper foreign steel.

Let me make another point. Suppose that instead of Franz we have Fritz at the party and suppose that Eva’s present is the ghost, Fritz’ is the pirate, and Maria’s is the nurse. Eva and Maria have the same preferences as before but Fritz’ are different from Franz’. The situation is given in this table.

Now, who would like to trade? Please think about it for a moment. Actually no bilateral trade is feasible, but they could all trade together. You could imagine that the three children sit around a round table, Fritz to the right of Eva, Maria to the right of Fritz, and Eva to the right of Maria. Then they could all trade by handing their respective present to the child sitting to the right of them. Everyone should agree to this and all are happier than before. So what am I saying with this example? Bilateral trade, no matter how often this is repeated, is not sufficient to lead to a Pareto efficient allocation. Sometimes more people need to get together at the same time. Later in the course, when I talk about money, we will see that this latter case is probably much more common.

What I have to say next is very much taken from chapter 3 of Ariel Rubinstein’s “Economic Fables” and is a nice way to demonstrate that Pareto efficiency cannot be everything that we want from a good system of allocating things to people who want things. Let me go back to the first example of Eva, Franz, and Maria. Suppose that we now hand out presents with some alternative allocation protocols. Suppose, for instance, that I – the father of the birthday child and party organizer – put the kids in some arbitrary (or perhaps not even arbitrary – perhaps showing favoritism) order and tell the kids that they are allowed to choose one and only one figure in the order that I put them in. Suppose the order is Eva, Franz, and then Maria. Then Eva takes the nurse, Franz is lucky and his favorite, the pirate, is still available and so he takes the pirate, while Maria, finally, saw both of her more preferred figures disappear before it is her turn to take the only remaining and her least favorite figure, the ghost. Note that this also leads to a Pareto efficient allocation. Now change the sequence and you will see that in every case you will end up with some Pareto efficient allocation. Actually it is very easy to see that this procedure always leads to a Pareto efficient allocation. But different sequences will lead to different allocations. This was also true in the trade example. Different initial allocations (the presents) lead to different final Pareto efficient allocations. Instead of me, the party organizer, determining this order, we could have used any other way to come up with some order. We could have let the kids fight, or do a race, or play a rock-scissors-paper tournament. There are many possibilities.

One potential problem with, say, kids fighting over things is that such a system does not provide particularly strong incentives for especially weak kids to acquire something in the first place, if they know that it will just be taken away from them again. How incentives matter in the design of economic systems is the topic of another lecture, though.

As a last example of the application of the idea that the world is probably free of easy arbitrage opportunities I here provide a brief introduction of the idea of financial engineering. Assuming the absence of arbitrage is all one needs to price financial derivatives. A financial derivative, perhaps a bit narrowly defined, is a financial product – that is a risky investment possibility – with payoffs that depend exclusively on other “basic” financial products such as bonds and stocks. You may want to google what bonds and stocks are if you do not yet know. For our purposes all we need to know is that a stock of a company has a value or price that substantially varies over time. The future price of a stock is uncertain today and this uncertainty can be quite large.

Let me turn to another area in which the absence of arbitrage – due to people preferring more money over less – implies severe restrictions: sports betting. Consider a set of potentially fictional betting odds for three football (soccer) games, given in the following table.

How should you read this table? Each column represents one football game between some teams A and B. For each football game you can bet on the event that team A (the home team) wins the game, that the game ends in a draw – coded as x – and that team B (the away team) wins. The number in a cell represents the betting odds for the respective event. For instance the number 1,1 represents the betting odds on the home team (team A) winning in the first of the three games. What does this number mean? If you place 1 Euro on team A winning and team A actually ends up winning you receive 1 Euro and 10 cents back. So in the event of team A winning you gain 10 Euro-cents. If the game ends in a draw or in team B winning you get nothing back, so you simply lose your Euro.

Take a good look at this table of betting odds and think about whether you find these odds plausible or implausible. You may want to use a calculator to aid your thinking. Take your time, this is not easy.

Are you done thinking? It turns out that two of these games are real games and one is a fake game, and that in fact the betting odds for the fake game are impossible in a world without arbitrage opportunities.

The first two games took place in the evening after the first time I gave this lecture, on Wednesday the 18th of October 2017. The first was Bayern Munich against Celtic Glasgow with odds of 1,1 on Bayern, 11 on a draw, and 21 on Celtic (I believe Bayern did win that night). The second was Benfica Lisbon against Manchester United with odds of 4,75 on Benfica, 3,6 on a draw, and 1,78 on Man U (I believe Man U did win that night). These are fine odds as we shall see below.

But let me first turn to the last game, which was not a real game. What is wrong with these odds of 1,9, 4,2, and 5? Well, with these odds one could do arbitrage. What you could do is to place Euro amounts on all three bets proportionally to the reciprocal of the odds. The reciprocal of the betting odds is what I would call the event’s “implicit probability”. The implicit probabilities are then 1/1,9 = 0,5263 for team A winning, 1/4,2 = 0,2381 on a draw, and 1/5 = 0,2 on team B winning. Suppose you take a target 100 Euros and place these on the three bets proportionally to the implicit probabilities. So you would place 52,63 Euros on team A winning, 23,81 on a draw, and 20 on team B winning. In total you would have placed 52,63+23,81+20=96,44 Euros on the three bets. Note that some of the 100 Euros remain in your wallet. Now what can you win with this betting scheme? Well, only three things can happen. Either team A wins, or there is a draw, or team B wins. How much money will you receive in these three events? If team A wins you get 1,9 Euros for every Euro placed on team A. As you have placed 52,63 Euros on team A you get 52,63*1,9 = 100 Euros back if team A wins. If there is a draw you get 4,2 Euros for every Euro placed on a draw. As you have placed 23,81 Euros on a draw you get 23,81*4,2 = 100 Euros back if there is a draw. If, finally, team B wins, you get 5 Euros for every Euro placed on team B. As you have placed 20 Euros on team B you get 20*5=100 euros back if team B wins. So no matter what happens in this game you always get 100 Euros back. But you have only placed 96,44 Euros. So you win 3,56 Euros in every possible case! You have made arbitrage! If you think 3,56 Euros is a miserly sum for all the trouble then multiply all your bets with 1000 and you win a sure 3560 Euros without risk.

Now note that you cannot do this with the two real games. In both cases the sum of the implicit probabilities exceeds one and, because of that, there are no arbitrage opportunities. You can try to find arbitrage opportunities in sports (or other) betting odds, but I doubt that you will find any.

There is a nice joke about economics that has to do with the topic of arbitrage. Here it is. Two people walk along the street, one of them is an economist. While they are walking they spot a 100 dollar bill on the pavement. The non-economist starts to scoop down to pick up the 100 dollar bill. The economist says: “Don’t bother! This can’t be a real 100 dollar bill. If it were a real 100 dollar bill, someone would already have picked it up.” My takeaway from this is twofold. If you do see a 100 dollar bill on the street, of course, do pick it up. But I wouldn’t start walking miles and miles of streets in the hope to find many 100 dollar bills waiting for me to pick them up.

The take away from the first class is that people pursue goals, that this leads to systematic patterns of behavior, and that these patterns are somewhat understandable, perhaps even somewhat predictable to an analyst. The particular goal we talked about in the first class was that people try to avoid wasting time. In the end I talked about queuing behavior that can be understood as a consequence of this goal: for example, we expect roughly equally long queues at supermarket checkout points and roughly equally fast queues in traffic jams.

Another goal that most people share is this. People, “ceteris paribus”, tend to prefer more money over less. The expression “ceteris paribus” means “all else equal”. I might be reluctant to accept extra money if this means someone is allowed to hit me on the head. But I generally will be happy to receive extra money if this does not come with any extra obligations.