## Are there unhelpful mathematical models of economic phenomena?

Take your bog-standard first-year economics story of why money (sea shells, coins, notes, bank statements) exist. Money, you will be told, is a means of exchange, a store of value, and a unit of accounting, thoughts going back to David Hume (18th century) and earlier.

When explaining the idea of exchange to students you say things like ‘you can’t exchange a hundredth of a sheep for a loaf of bread so you want something to represent the value of a hundredth of a sheep, and in any case it’s a long slog to the market carrying a sheep around’.

When explaining the idea of a store of value you say things like ‘You would like to be able to consume things when you are old without working when you are old. That means you need to save up wealth in the form of something that doesn’t perish. Sheep perish, gold does not’; and when explaining the unit of value idea you say things like ‘we all think of the value of things in terms of a numeraire, such as that milk costs 1 dollar per liter and flour 2 dollars a kilo. None of us think in terms of 1 liter of milk being worth half a kilo of flour. Given many different products, it is more convenient to think of the value of each of them in terms of something you can compare across these goods. Money performs that role and you will find that even when the unit of money changes (such as moves from the Deutschmark to the Euro) that people will continue to calculate everything back in terms of the old money for many years’.
Simple stories, no? And most students will ‘get the point’ of each of these three stories. They will see the difficulties of exchange with lumpy goods that cannot easily be stored and exchanged, and they will see the point of being able to save up for a later date and that requires some form of storable money.

Simple though these arguments are, you will be hard-pressed to find mathematical models of them that anyone would recognise as remotely capturing these verbal arguments. It tells you something about the limits of mathematical models to think through why recognisable models of money do not exist. So bear with me as I take you through the actual difficulties of modelling money and how those difficulties end up as unhelpful advise from theoretical economists to policy makers.Think of the actual difficulties involved in modelling the story of money as a medium of exchange. Before even thinking about money, you have to start from a model with exchange. This means you need to model the production of more than one good and you must build in a reason, like comparative advantage, why individuals do not simply produce all the goods they need by themselves. For realism you would want the goods to be lumpy, perishable, and to require long-term investments. After all, sheep herding and crop-growing do not happen overnight and neither sheep nor apples can meaningfully be stored for very long or exchanged in halves.
You immediately hit your first mathematical snag right there: if production is lumpy (you can’t produce half-apples), then you won’t get the simple outcome that someone will spend all his time on what he is best at. An individual could optimally spend his time by producing one sheep and two apples even though he has a comparative advantage in sheep, simply because he can’t make exactly two sheep. If you want lumpiness in your model, you thus would have to solve the problem of how a person would optimally allocate a fixed amount of time over lumpy investment projects. This is known in the Operations Research literature as the knap-sack problem (in which you need to decide which lumpy goods to put into a knapsack of particular size) and it is known to be an ‘NP-hard’ problem. Simply put, you know of such problems that there is a single optimal solution but it may take a long time to actually find it. Solving just that knapsack problem for a single individual is already something that may take a computer years if you choose the bundle of potential goods to be large enough, and there will be cases in which you will find that even with comparative advantage the sheep herders may grow enough apples to not need exchange.
How do you solve that snag, which incidentally arises in all models of production? The reality is that you don’t because solving just that one leaves you with a model in which you can solve little else and in which you are not assured of any real impetus for exchange. Hence you ‘simplify reality’. You thus presume that there is no such thing as a lumpy good and that people spend their time producing a ‘continuous’ amount of goods, say, 3.271 sheep or 14.231 apples. Without lumpiness, people will specialise in making one thing and have a reason to trade. Note that you thus have already given up on describing the most intuitive reasons for having money around: you can no longer meaningfully talk about the difficulties of exchanging a hundredth of a sheep for half an apple since you now have presumed a world in which you produce sheep in hundredths and apples in halves.
Moving on, the next modelling problem you hit is that it must be the case that different individuals happen to want what the other produces, a ‘coincidence of wants’. Indeed, you want some kind of place (a market) where people come to exchange what they have produced. In model-land you must answer every counter-factual. You must thus have a reason why traders would use money instead of giving each other credit or just exchanging bundles of good (since goods are now not lumpy, you can just go to the market with your 2/3 sheep and exchange it in one big free-for-all for all the goods you need). Such thoughts may sound absurd to you, but working them through has occupied really good mathematicians for years. It is in fact nigh impossible to solve models in which people do not know exactly beforehand what will happen in a market.

You see, as soon as you say that a person does not know beforehand what other people have produced and at what prices they might trade, you are in the world of limited information and in the world where it is possible that people make mistakes (go to the market empty handed, produce the wrong things, etc.). You are then in the business of having to specify how people form expectations about what others would do and what prices they would trade at.

You are then also in the business of working out whether there are perhaps multiple equilibria (i.e. different configurations of the whole economy) and the issue of how people who don’t know each other could actually coordinate on a particular configuration. You then for instance have to contend with the possibility that nobody shows up at the market because they expect nobody else to show up. You have to contend with the possibility that you get the wrong prices, under which there is no specialisation at all.

You have to contend with the problem that the only people to show up wouldn’t want to trade with each other because they have produced the same thing and you have to figure out how a group of people would actually arrive at a price (or prices). Each of these sub-problems is considered exceptionally hard by theorists: only under very specific mathematical assumptions can you be absolutely guaranteed that the problems above do not occur.
Hence, what do you do? Well, again, the reality is that you assume away all these problems. You simply make those assumptions that guarantee you that everyone who produces something is ‘magically’ matched up with someone else who has something they want to trade with. Also, you now presume the existence of some kind of all-powerful benevolent entity, say god. You need such an entity to do away with elements in your model you cannot model but need anyway, such as how prices arise before any exchange takes place (if prices change during exchange one gets into exceptionally complicated dynamics where you need to start talking about the expectations that people have of possible price paths). So you invent a god that takes care of such issues. God, in his first incarnation as a Walrasian auctioneer, announces the prices at which everyone is willing to trade, whereby everyone believes god and acts accordingly. God, now in his second role as a benevolent and completely trusted government, then also provides a means of exchange that is not perishable, i.e. money.

Usually, a third sleight of hand is needed to get a workable model and that is to have a situation in which there is no such thing as a mistake because there is no such thing as expectations that are incorrect. This of course basically presumes away the original problem you were starting out to model, but that is an almost inevitable casualty of the wish to have a tractable economic model.

What kind of models of money do we end up with? To my taste, the best that mathematicians have come up with is the story that some sheep producers have a craving for eating apples in the night, but they are themselves just innately incapable of producing apples and their sheep always die at the end of the day (i.e. they must be eaten before the end of the day. New ones are only born at the start of the next day). This means that the sheep herder must sell his sheep during the day to the apple maker whilst buying the apples during the night (apples also perish at the end of each half day so he can’t trade during the day). In a modelling sense, that ensures you the ‘coincidence of wants’ you need to have a role for exchange and ensures that sheep herders and apple farmers cannot just trade their produce. By assuming that they not trust each other, but that they do trust the provider of money, you ensure that they do not just trade promises but use money for their trades. Within this kind of basic set-up you can even introduce monetary policy in the form of allowing god to hand out more money to specific groups or to reduce the value of the money in circulation. Whole ‘policy edifices’ have been built upon the basic structure of sheep herders having cravings for apples in the night. For those who are interested, I am talking about the model by Lagos and Wright (2005) and the many extensions on their basic idea.
Now, anyone in his right mind would laugh out loud at the story above as it comes nowhere close to the historical stories told about why we have money and what its role is in the economy: big historical problems in the emergence of money concerned the fact that there was no trusted government, and the value of money had a lot to do with the actual costs of information and transportation, costs that the story talked about above had to assume away. Yet the story of apples and sheep above, believe it or not, is one of the dominant stories told in ‘micro-founded’ monetary economics. It is in that kind of model-economy that they talk about money, credit, banks, regulation, etc. If it weren’t for the fact that it is deemed cutting-edge research, you would have to cry.
I hope you will take my word for it that the problems of generating models in which money exists because of savings and as a numeraire good are equally hard to set up and hence such models don’t exist at all as far as I know.
The value of the actual models of money are mainly as proof of concept, i.e. that you can think of a micro-model in which money emerges and where you can base the emergence of money on at least one of the underlying micro-motivations you think are important for the existence of money (the advantage of having a more varied consumption bundle). It is not the model you would have wanted but at least you can have it in the back of your mind as an example of the micro-mechanisms that are relevant.
The problem with the monetary model talked about above is that it fits so poorly. It hardly fits the many historical examples we know of the emergence of money, nor does it capture the problems we face today when thinking about money markets (trust in the institutions, the incentive problems inside organisations, the investment problem). Hence it is singularly unsuitable to use as a mental laboratory for the policy problems of today, or even as a descriptive model of the actual roles of money in our economy.

The problem of poor fit carries over to unhelpful advise: despite the fact that it is such a poor fit to reality, it is the only ‘game in town’ when it comes to micro-models of money. A most unfortunate and destructive phenomenon then appears, which is that the only game in town becomes the truth to a whole set of people making their careers on the back of it.

All the potential advantages of models become a disadvantage when a poorly-fitting model is taken too seriously. One potential advantage of models is that they can be the codification of previous knowledge and as such a good model is a quick way of conveying a lot of knowledge to the next generation who don’t have to learn what reasons went into the construction of the model in the first place.

This now becomes a disadvantage: the new generation that looks to write papers ‘on money’ need know nothing about the history of money or its uses today but only need know the dominant model, which turns into a disadvantage because that new generation will come up with twists and extensions of something that is innately unsuitable to answer any interesting question. Yet that new generation will be blissfully ignorant of the uselessness of what they are doing because they, unlike the originators of the first models on money, will lack the historical database in their heads of what actually goes on. They are simply proving their worth by being more acquainted with the mathematical ins and outs of these models than anyone else and that is what supplies them their daily dinner, not whether the model is useful to anyone else.

Another potential advantage of a good model is that you can make consistent statements instead of waffling on incoherently. One real advantage of model-land is that it is fairly easy to spot someone who is not capable of understanding models. This advantage also becomes a disadvantage in a model that fits poorly because you will see a great proliferation of consistent statements that are based on poor abstractions of real phenomena. You might term this the proliferation of ‘precisely wrong’ statements.

And it is a cop-out to say that these precisely wrong statements are not intended to be taken literally: despite being mere models, the adherents deliberately use words that convey its supposed usefulness, such as monetary policy, government, banks, etc. The pretense of usefulness pervades each paper and each grant proposal using these models. Worse still, that modelling community is a group with a big incentive to pretend that the assumptions made for convenience are ‘actually true’, i.e. it is a constituency of individuals with an incentive to presume there is no such thing as transaction costs or a trust problem when it comes to money. When such people become important they will poo poo those who make different assumptions and force them to first invest in their models. In short, a poor model that is taken seriously becomes a part of the problem.

Would you also have the same problems if monetary economics were mainly based on a set of historical case studies and an awareness of the problems faced today by economic actors? Unlikely, because you then at least have set up an ultimate goal of the discipline, which is to understand how the world came to be as it is and to help economic actors shape their world to their advantage, i.e. you are grounding your discipline in historical reality and real world problems. Having said this, one should not be blind to the disadvantage of a more verbal discipline though. The disadvantage is that when knowledge consists of a collection of examples and lessons, there is more room for the wafflers of this world to ply their trade, and there are millions of eager wafflers around.

Are there any good economic models you might ask? I believe there are and my prime example would be Industrial Organisation models of competition and market interaction. These are the Cournot models, Stackleberg models, models of complementary investments in vertical markets, oligopoly models, models of the internet as a platform, etc. The nice thing about these models is that the motivations they presume of their actors (pure greed) are pretty well spot-on and that it is not that hard in reality to see what kind of market interaction is happening, i.e. which of the I/O models to use.

Though it is hard to measure for a statistician, it is not so hard to spot as a human whether, say, the oil companies are engaging in collusion or not. It is not hard to spot a cartel, or the basic information structure of a market, nor is it hard to spot the structure of investment complementarities. In short, I/O models can do a remarkably good job of describing the particular aspects of reality one can optimally intervene in, which is of course why they are so central to the work of regulation authorities and why, for instance, auction design on the internet is done by mathematically schooled geeks. They need to know nothing of the history of auctions to nevertheless be damned good designers of auctions as long as they understand the models and have learned to spot the market patterns around them.

There are thus good models out there and the groups of disconnected geeks working on extending them are, often to their own surprise, doing something useful with their lives. We wouldn’t want to go back to waffling in those areas. The problem is thus not the existence of mathematical models per se, but rather that there are aspects of economic reality where the best we can do is a bad model.

Is money the only area where we can do no better than bad models that are worse than useless when they are taken seriously? Alas, no. What goes for money goes for many economic phenomena. To have an economic model where growth is driven by specialisation (which is what most historical economists believed was the engine of growth) has so far been beyond us, which is why we have ended up with these ridiculous representative agent models. What the pragmatists believe is true about specialisation can’t be modelled by the best minds in math econ land (this is not to say there are no models of specialisation, simply none that get close to illuminating the path-dependence, trust, and institutions that sustain it). Satisfactory ‘des-equilibrium’ models of recessions also simply don’t exist. Models of human behaviour drawing upon more than two of the known ‘irrationalities in our make-up’ are also too hard to solve. The list goes on and on: if one insists on consistent mathematical theorising from ‘micro-foundations’, nearly all of the big drivers of economic growth and economic institutions are beyond our ability to model even remotely realistically.

Mathematical models are hence in many areas a problem because they fit poorly but nevertheless live a life of their own, taking up valuable mental time of smart people, leading individuals to think about the wrong problems, leading people to think in terms of the wrong assumptions, motivating statisticians to measure the wrong things, and divorcing their discipline from reality.

Suppose you believe all this, but nevertheless want to make progress in disciplines by doing proper science, differentiating yourself from the wafflers. What is ‘proper science’ in an area where we cannot make much mathematical headway and hence where we can be reasonably certain that every grand story we tell (in maths or in words) has inconsistent parts to it? That’s the subject of a future blog….