Friday July 1st, 2011 | informationliberation.com |

Mathematics can sometimes make smart people dumb. Let me explain what I mean by this. I

The danger of mathematical arguments is that a person can sometimes follow an absurd path of reasoning

One of the most common errors in applied mathematical analysis is to fail to notice when a mathematical argument

Let me give you an example of this phenomenon in action. The Australian government recently announced that it will attempt to enact legislation to impose a tax on industrial carbon-dioxide emissions, with some of the revenue being earmarked as compensation for affected consumers. At a pro-government political rally in Sydney, a young activist proudly displayed what he clearly thought to be a devastating economic argument in favor of this "carbon-pricing" scheme. See for yourself:

To those readers who have not studied neoclassical microeconomics, this is probably just a big bunch of gibberish. But to those who have, it should look quite familiar. The graph is a "utility analysis," which purports to show that imposing a tax on polluting products (which increases their price) and simultaneously giving compensation back to consumers would make them better off than they were initially — in other words, it purports to show that the Australian government's proposed scheme, or something like it, would make people better off.

This is a classic example of a mathematical analysis that proves too much. Notice, in the graph in the sign, that the two products are labeled "C" (for clean products) and "P" (for polluting products). Although they are labeled in this way, the fact that the horizontal axis represents the consumption of polluting products plays absolutely no part in the analysis. There is nothing in the graph representing the pollution that these products cause, and so the label is merely a name. The letter "P" is nothing more than an algebraic symbol, one that could just as easily stand for pies, pastries, printers, pizzas, polka lessons, picture frames, pole dancing, ponies, popcorn, pool tables, poppy-seed muffins, pornography, postcards, potatoes, potpourri, poultry, pumpkins, puppies, pudding, or any other good or service (including goods and services that don't start with the letter "P").

Thus, by the exact same mathematical argument, the graph implicitly purports to show that a government can make people better off by taxing

In fact, the analysis in the graph could be taken further than this. Why stop taxing there? Repeating the same analysis, the government could increase the happiness of their subject population further still by imposing a tax-and-compensation scheme on the polluting goods, and

But wait a minute. You don't need to be a mathematician, or an economist, to figure out that there is something funny going on here. Either some step in the analysis or some starting assumption must be faulty. In a moment I will explain what this is, but really, this exercise is largely academic. The point here is that the conclusion from the analysis is so absurd that something in the analysis must obviously be wrong, even if we are unable to pinpoint exactly what it is. It proves far too much.

Suppose that this young fellow had eschewed mathematical explanation in this instance, and instead simply stated his argument verbally: "If you have two types of goods (let's call them C and P) and the government taxes one of those goods (say, good P) and then pays consumers of that good compensation, then those consumers will be better off than they were to start with." A question would immediately spring to the listener's mind: How much compensation is needed for this to happen? And in particular, is the revenue from the tax enough to cover it? Isn't this important in deciding whether this argument is a valid reason to support the tax? In verbal form, these questions would present a serious challenge to the analyst, and an opportunity for him to discover a serious flaw in his assumptions.

In fact, these questions are the key to the flaw in the analysis. Notice that in the second step listed on the sign, the consumer is given compensation that allows him to afford the same bundle of goods that he initially started with. Since the price of the polluting products has increased, this means that the cost of the compensation being paid in the analysis is equal to the amount of polluting products initially being consumed, multiplied by the increase in price from the tax. (In mathematical parlance, this is t × P0, where 0 < t < 1 is the price increase due to the tax.)

Can the government afford this, using the revenue it extracts indirectly from these consumers? Well, let's start by being as generous as we possibly can to the argument, by invoking some fanciful assumptions in its favor. Let's assume — contrary to every sensible understanding of government — that the tax-and-compensation scheme can be enacted and administered without any costs at all. In this case, the net revenue taken from the consumers would be equal to the gross takings, which is equal to the amount of polluting products being consumed after the imposition of the tax, multiplied by the increase in price. (In mathematical parlance, this is t × P1, where 0 < t < 1 is the price increase due to the tax.)

See a problem? The gross revenue taken from consumers uses the actual consumption level

In fact, using the exact kind of mathematical model being used in the sign, it can actually be shown that the amount of compensation required to fully compensate a consumer for a price rise (called the "compensating variation"), just to make them as well off as they started, is

Obviously, the situation becomes much worse if we make more realistic assumptions about the administrative costs of the scheme, since this reduces the net revenue available for payment as compensation. In reality, a taxation scheme of this kind would require very large amounts of money for the government to create and administer, and would also impose compliance costs on the taxpayers. The situation also becomes worse for the consumer if he receives only part of the tax revenue in compensation, rather than the full amount. There would also be disparities in the compensation between consumers, so that some would be worse off, even if others got a large amount. Possible rent-seeking behavior and other economic issues could make the situation worse still, until a very grim picture of the scheme starts to emerge.

In the sign in the picture, the compensation required to get to the blue utility curve (making the consumer better off) would cost more than the gross revenue from the tax. In fact, even the compensation required to get back up to the black utility curve (making the consumer as well off as they were before the tax) would cost more than the gross revenue from the tax. Add administration costs for the scheme to this, and other realistic issues, and now you need to come up with an awful lot of extra money that is nowhere to be seen.

In fact, regardless of the findings of a utility analysis of this kind, there is one overriding economic argument against a coercive scheme such as the one being proposed. If it

If one were a supporter of a carbon-dioxide-emissions tax (I am not) then I doubt one would be too pleased with the above argument being presented in its favor if it were expressed in verbal form. Yet, add some mathematical bells and whistles to this absurdity, and you get a sign that was described by one sympathetic observer as the "Best Sign" at the rally.[6] In fact, not only is the analysis in the sign flawed, but when it is done properly, it actually leads to the exact

Aside from the above instance where this argument is made in mathematical form, I do not recall ever hearing a single advocate of a carbon-dioxide-emissions tax make the asinine assertion that tax-and-compensation schemes of this kind would increase the happiness of consumers regardless of the good being taxed. They are not quite that silly. Almost all arguments in favor of taxation schemes of this kind are based on completely different reasoning from this, usually using "negative externality" arguments that assert actual pollution problems. These arguments cannot really be captured in a single consumer-utility graph, since they involve assertions of interactions between the actions of one consumer and the preferences of another. The mathematical argument presented in the picture above is therefore not an advancement of the pro-tax position. It actually does a serious disservice to this position by presenting an incorrect and very ill-considered justification for it.

This shows the particular danger of getting bamboozled by applied mathematical analysis, to the extent that absurd premises slip through the net undetected. It allows a person to make the dumbest argument possible for a particular proposition, while maintaining a supreme measure of confidence, and indeed cockiness, in his own position.

When doing applied mathematical analysis we need to be careful not to fall into this trap. Though mathematics is a specialized discipline, beyond the understanding of many people, a sound analysis in applied mathematics should generally be translatable into a sound verbal argument, at least in a heuristic form. Its arguments are progressions from premises to conclusions based on logic, and hence, if you cannot explain the structure of your argument and its premises (at least in heuristic terms) to people without much mathematical training, you probably do not have a broad enough understanding of the structure of the argument to warrant reliance on it.

I have not shown this example simply to demonstrate the dangers of having inept economics students present their ham-fisted policy analysis in public. It is actually demonstrative of a wider point regarding the use and abuse of mathematical arguments: mathematics cannot do scientific problems for you. All that mathematics can do is to allow you to state problems in quantitative form and find the logical consequences of various assumptions about the problem you are trying to solve. A mathematical argument shows that certain premises lead logically to certain conclusions. But it does not guarantee that those premises bear any resemblance to reality. Whether or not they do is an important matter, deserving the utmost consideration.

Mathematics is meant to

The argument presented in the sign above hinges on the fact that it hides any discussion of the amount of revenue needed for the compensation payment that is assumed to be made. It does not compare this amount to the actual amount of revenue taken from consumers due to the price rise, and as soon as this issue is considered, we see that the argument presented in the sign is either wrong or at the very least highly misleading. Actually, the real purpose of the sign above is not to convince

It is an appeal to authority, with the authority in this case being a bunch of fancy graphical work. Like so many purported scientific justifications of government power, it is based on false premises and/or shoddy logic, masquerading as bona fide scientific analysis. It is the voice of a pretentious elite saying, We couldn't possibly explain our reasoning to you in a way that you could understand, so just defer to our clearly superior intelligence, bitches. (Note: mathematics can sometimes make smart people dumb, but it cannot make them pretentious mediocrities; they do that on their own.)

When mathematical arguments prove too much, it is often as a result of faulty assumptions. If an applied mathematical argument leads to a conclusion that is highly counterintuitive, or if the form of argument can be deployed just as effectively to prove other conclusions that are highly counterintuitive, then this is good reason to further scrutinize the assumptions made in the argument.

Mathematics is a fascinating and powerful discipline, and one that I love a great deal. Enjoy it to the extent that you are able. But, as Ayn Rand used to say, check your premises!

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Ben O'Neill is a lecturer in statistics at the University of New South Wales (ADFA) in Canberra, Australia. He has formerly practiced as a lawyer and as a political adviser in Canberra. He is a Templeton Fellow at the Independent Institute, where he won first prize in the 2009 Sir John Templeton Fellowship essay contest. Send him mail. See Ben O'Neill's article archives.

Notes

[1] In my own teaching, I like to keep students on their toes by occasionally presenting them with a flawed statistical argument that leads to a conclusion that is quite obviously absurd. My favorite practice is to give them statistical questions that invite them to conflate correlation and cause — leading to some obviously absurd conclusions — and then see if they notice the absurdity of the conclusion they are getting, rather than plowing ahead blindly with their equations.

[2] A warning against acceptance of this kind of argument is captured in the Latin maxim,

[3] Picture taken from http://twitpic.com/57awlj. I have cropped out the young man's face, since it is not my intention to embarrass him. Scrutiny of his sign is mainly for the purposes of showing a more general problem pertaining to attempts at applied mathematical analysis, though it is certainly worthy of criticism, especially for its rude and pretentious demeanor.

[4] Assuming that the utility function in the analysis is strictly quasi-concave (a common assumption in neoclassical microeconomic analysis), infinite repetition of the scheme in the graph would lead to an infinite series of strictly positive utility changes. Some further assumptions about the utility function would then be needed to ensure that this series diverges to infinity, so that utility (and the budget constraint and consumptions sets) can be boundlessly increased. In particular, a homothetic utility function is a sufficient but not necessary condition for this result.

[5] For a demonstration of this, using standard neoclassical utility analysis (as is used in the sign), see Example 3.I.1 of Mas-Colell, A., Whinston, M.D. and Green, J.R. (1995)

[6] See picture caption at http://twitpic.com/57awlj