In the third lecture, after a review of the second lecture, I talk about (bilateral) trade and more general exchange, efficiency, and fairness. I do this in the context of a kids’ birthday party and follow to some extent chapter 3 of Ariel Rubinstein’s “Economic Fables”. 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, I ask the students what is going on when kids are trading their presents.
I considered to choose ‘A Non-Equilibrium Approach’ as a subtitle of my dissertation thesis. About at the same time a colleague of mine stated that ‘disequilibrium economics’ are a ‘logic implausibility’ as an equilibrium in economics is not much more than a consistency condition – different to the notion in physics where it mainly refers to a state where the described system is at rest. I have to disagree with this maybe unintentional attempt to whitewash a bunch of approaches which are – as probably every other approach – criticized for good reason.
Just think of basic micro or macro and the definition of a market or an economy in equilibrium. There the term is not used to describe consistency in the derivation of the outcome, but mainly refers its characteristics – for example that supply and demand are balanced. Go further in the curriculum and think of an equilibrium in game theory. While it is also derived in a way which is consistent with the stated assumptions, its description states more than that – for example that it is a combination of strategies for which no individual has an incentive to unilaterally deviate.
Therefore, equilibrium approaches in my opinion go beyond detecting an outcome that is logically implied by assumptions and step-by-step analytics. They also tend to presume an outcome of a certain type and thereby risk the neglect of other outcomes, strategies, behaviour, and thereby even whole issues that may be highly relevant in reality.
In case my concern is not clear, a discussion of Rubinstein’s famous e-mail game may help. The e-mail game may be described as the following: A couple wants to meet and prefers being together over being separated. However, if it rains they prefer to meet inside, otherwise the prefer to meet outside. Whether it rains or not is determined by nature and only one person, let’s assume the woman, knows the weather for sure. If it will rain, she sends an e-mail to the man. Every received e-mail is read and automatically triggers a response, but every e-mail also gets lost with a certain small probability. That means that the e-mail conversation may last for a long time and even forever, but the probability for the latter case tends to be zero.
Because of the small but nevertheless positive probability for an e-mail to get lost, both parties will never know for sure how many e-mails have been sent. The woman knows whether she sent an e-mail or not, but she is confused about the state where one or two e-mails were sent (captured by the partition Pw). While it may be that the second e-mail – sent by automatic response from the man’s account – got lost, it also may be the case that her e-mail did not pass through in the first place. The moment the second e-mail passes through, the third e-mail is triggered automatically and she can distinguish that state from the ones before. However, she again cannot distinguish between the state of three and four e-mails sent – because if she would know about the fourth e-mail, she would have automatically sent the fifth, being in another state. The man faces a similar incompleteness of information (captured by the partition Pm). He in turn is confused about whether none or one e-mail was sent, just like he is confused about whether two or three e-mails were sent and so on.
Rubinstein thereby shows that the strictly formal approach does not lead to an equilibrium in which they meet outside in the nearby game even if there is a high probability for the information to pass on. In fact, the formal result of the game described above is that none of the two will risk to go outside as there is no state (described in terms of e-mails sent) about whose appearance exists common knowledge. However, the example not only shows how easy simple games may get complicated in formal term, but also shows how misleading the strictly formal conclusion can be with regard to an underlying issue. It was about a couple who wants to meet, inside on rainy days, outside otherwise. They both know their preferences. They differ only in the information they have – first about the state of nature and second about how many e-mails are sent. The second issue however should not be the one of primary interest. Instead a social scientist and therefore economist should just ask: how many e-mails have to be sent that they both know that they both know about the weather and therefore human beings of these days will coordinate for the preferred equilibrium.
One e-mail sent just states that it is rainy and the woman knows about it. Two e-mails sent means that the man received this important information, but the women does not know that yet. Three e-mails sent means that the woman knows that the man knows. Four e-mails sent means that the man now knows that the woman knows that he knows. Five e-mails mean that the woman now knows that the man knows that the women knows that the man knows. At the latest after the sixth and seventh e-mail both know that they reached the aspired situation where both know that they both know.
While they can never be sure that their last e-mail passed through, they reach a state where human beings of these and thereby the economic agents of interest will not care about it. Agents may differ with regard to the number of e-mails they require in order to believe in a successful coordination, but I claim that there are not much of them who require more than the five to seven e-mails.
So, while the formal equilibrium approach provides some insights in favour of a theoretical statement about mutual and common knowledge, it risks to draw too much attention towards the wrong issue or at least away from non-equilibrium outcomes that may be highly probable in reality. I think that this is a general issue of equilibrium economics, which are worthwhile and helpful in many regards, but always have to be done as well as interpreted with caution.
The final bit of the second lecture is an introduction to 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. Students may want to google what bonds and stocks are if they 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.
In this part of the second lecture I turn to another area in which the absence of arbitrage – due to people preferring more money over less – implies severe restrictions: sports betting. I begin by giving the students potentially fictional betting odds on three football (soccer) games, given in the following table.