Anyone who peruses my work on free banking—or my other writings for that matter—will notice that I’m not especially inclined to express my ideas mathematically. To put the matter more positively: I prefer plain English. The preference has if anything grown more marked over time. While writing Good Money, for example, I at one point let an equation slip into the first, expository chapter. But as soon as it occurred to me that a sentence would serve as well, while being a damn sight prettier to look at, out went the symbols. Likewise, when Liberty Fund decided to put The Theory of Free Banking in their Online Library of Liberty, I asked them to take out the few equations (and the one figure) found in the print version, as if they were so many blemishes (which, indeed, they were).

To resist using equations isn’t a strategy calculated to make life easy for an academic economist today. Yet it isn’t entirely for want of ability to do otherwise that I‘ve resorted to it. In fact I like math and was pretty darn good at it once upon a time. I just happen to think it wildly overrated as a means for “doing” economics—that is, for communicating ideas concerning how an economy works. For whatever its champions may think, mathematics is a language, and as such is a fit device for economic analysis only to the extent that the symbols it consists of are more capable of accurately conveying meaning than words themselves are. Of course mathematical expressions have their advantages: most obviously they tend to be less ambiguous than verbal ones; and it’s relatively easy to combine and manipulate bunches of them so as to ferret out implications or inconsistencies that might not otherwise be evident. Some ideas--Newton's laws of motion naturally come to mind--can't readily be stated at all, let alone figured out, using ordinary language. But math (by which I mean mainly algebra and calculus) has disadvantages also, the most obvious of which is that its limited grammatical repertoire simply doesn’t allow it to convey many subtleties of meaning of which properly-handled words are capable. Despite what some mathematical economists seem to suppose, it isn’t merely owing to a general lack of facility with algebra that people mainly communicate using words: it’s because you can say lots of things that way that you can’t possibly say with equations. Or maybe you can sort of say it with equations, but it’s dorky to even try.

Which brings me to free banking. Might greater resort to formal modeling enhance the theory? Of course it might. Like I said, equations can reveal things that mere words obscure. When Carl Christ reviewed *The Theory of Free Banking* for George Mason’s Market Process newsletter, he did so by writing down a formal model by which to examine my verbal reasoning. That reasoning was all O.K., according to Christ, except that his formal analysis made explicit an assumption that was only implicit in my own discussion, to wit: that by an increase in the volume of real activity I had in mind an increase in the number of transactions per period, rather than an increase in average transaction size. (In the former case reserve scale economies are realized, whereas in the latter they aren’t.) This is just the sort of thing math is good for, and it is why it’s worth trying to concoct formal models. Indeed, I later made Christ’s model the basis for one of my own, rare forays into mathematical economics, my 1994 paper “Free Banking and Monetary Control.”

On the other hand Christ’s review also convinced me that I was capable of doing reasonably “rigorous” economics using “mere” words; and I personally believe that, by making use of metaphors and other devices that have no exact mathematical counterparts (though equations are themselves metaphors of a sort), I added more to my theory’s intuitive appeal than I sacrificed in rigor.

The point is that both mathematical economics and the verbal kind have their place; neither is intrinsically better than the other; and each can serve as a useful test of the other. A formal model can reveal deficiencies or omissions in a verbal argument; but a few well-chosen words are just as capable of exposing an absurd argument or false assumption lurking in some seemingly innocent equation. The claim that “it takes a model to beat a model” would be just plain goofy were it not so effectively employed by mathematical economists anxious to insulate their work from criticisms by persons who know less math—but perhaps more economics—than they do.

Finally, I come to Larry Sechrest’s (1993) Free Banking, much of which is devoted to offering a formal interpretation of my own verbal theory. Larry’s book came up in comments on a previous post; and having made a brief remark upon it there I offered to expand upon it in light of a quick rereading. My recollection had been that Larry misinterpreted some of my arguments; in fact the misinterpretations are minor and are mainly confined to Larry’s own “verbal” exposition, as when he states (p. 14) that I argue “that a bank’s demand for reserves depends not only on the total volume of transactions but also on the frequency of those transactions.” (He presumably ought to have written “total volume of bank money outstanding and its turnover” or something like that.) But Larry’s formalization, unlike Carl Christ’s, contains no microeconomic refinements or revelations: it merely restates in symbols the comparative static conclusions I reach using words. If some find Larry’s approach more compelling (and it appears that some do), that’s lovely. But let’s not be guilty, as so many mathematical economists seem to be, of confusing a difference in rhetoric with a difference in rigor.

June 22nd, 2011 3:16 pm

I think it is quite difficult to describe the functioning of an economy in mathematical terms because an economy is simply a group of people trading with each other, and each trade is based on each party's personal determination of value. Just as it would be somewhat silly to mathematically quantify the beauty of a painting, it is somewhat equally silly the quantify economic transactions. Each exchange is based on the opinions of individual traders regarding the beauty, value, or desirability of the assets they are swapping, and the value of anything is no more quantifiable than the beauty of the Mona Lisa. Value is always relative to the time, place, and opinions of market participants, and how each of them has ranked their current holdings into a "scale of values," and actively compares what they hold to what others are offering in trade.

A bottle of water may trade for a dollar at the corner store, but we cannot place the value of water at one dollar per liter and use it in mathematical calculations. The bottle's value will be remarkably higher if it is moved to a desert, and remarkably lower if it is sold near Evian, France. It has not fixed or quantifiable value outside of actual trades. If it is traded for a dollar between two men, it is only worth a dollar in that very moment, according to the man who traded a dollar for it. It is worth less than a dollar to the man who traded it away, and may be worth more or less than a dollar to the man who purchased it if the weather changes.

So why does anyone ever attempt to quantify basic economics? It seems like a fool's errand.

June 22nd, 2011 3:47 pm

StoneG, I don't think mathematics incapable of dealing with the subjective theory of value--though doing so is admittedly tricky. In any event I think it a mistake to dismiss mathematical economics altogether. As I say, it has it's place. The problem is that it has become a fetish within the profession, to the point where economics has started to be more about "models" than about any actual economy. Mathematical models, in other words, have ceased to be mere tools used by economists to study this or that real-world economic problem, becoming instead economists' preferred objects of study!

June 22nd, 2011 3:53 pm

There is a very good way to treat subjective value described by J. Huston McCulloch in his paper, "The Austrian Theory of the Marginal Use and of Marginal Utitlity."

June 22nd, 2011 4:47 pm

Agreed regarding the strange focus on models, but I don't see how millions of people making personal decisions to trade what they own to others who are making the same personal choices, based on myriad factors, including emotion, can be quantified in a meaningful way.

It seems useful to help describe how an economy works using math, in the same way that you might draw a picture to illustrate basic concepts like supply and demand, but to use it for prediction and manipulation of the economy seems like a silly idea.

June 22nd, 2011 3:50 pm

Hi Professor Selgin, thanks for taking the time to respond to my previous comments in this post. I'm a mathematician, not an economist and have a kind of amateur's enthusiasm about economics (and freedom). I also appreciate that "natural language" can be just as rigorous as "symbolic language" so don't think mathematicians aren't aware of this. There's a famous quote of V.I. Arnold, "It is almost impossible for me to read contemporary mathematicians who, instead of saying "Petya washed his hands," write simply: There is a t1 Petya(t1) belongs to the set of dirty hands, and a t2, t1 < t2 ≤ 0, such that the image of t2 under the above-mentioned mapping belongs to the complement of the set defined in the preceding sentence." Obviously, mathematical language can and has been used in ways that effectively obfuscate the underlying ideas.

But it seems to me that the free-market economists whom I deeply respect do a whole lot of hand wringing about the use of mathematics in economics. Mathematics, at its best, elucidates ideas. This is what I got out of Larry Sechrest's model, a (to me) clear way of understanding what's going on. I don't think economists should go out of their way to eschew mathematics, when it is such a useful tool. And I think the previous commenter, StoneGlasgow's ideas about why mathematics cannot possibly be used to describe economics are ridiculous. For instance, he explains that because value is subjective, it can't be quantified. So, don't quantify it! Treat it as an ordered set depending on the individual and time and place. There's more to mathematics than calculus and statistics! I think most economists are not exposed to the great breadth of math that they could potentially use and think that because some math has been misapplied, other math cannot be correctly applied. It is this attitude that I find prevalent and perplexing among economists I very much admire.

June 22nd, 2011 4:45 pm

I think we are very much in accord, Eitan. At some level the preference for mathematical over verbal modes of expression is itself a matter of subjective value. Moreover from what little contact I've had with genuine mathematicians I have no reason to doubt what you say regarding their awareness of the the relative merits of words and mathematical language. Yet I get quite a different impression from many economists, who seem to believe that doing economics well is simply a matter of mastering some mathematical modeling techniques, and (by implication) that one can dispense with knowledge of other kinds, including (economic) history and the history of economic thought. For them, the lack of a formal model is tantamount to the lack of any argument whatsoever. As I've suggested, I believe this attitude reflects their training. After all, if one never learns anything save how to write models, it is self-serving to insist that models are all one needs to know!

June 23rd, 2011 1:06 am

I'm obviously not familiar with the state of the field of economics and culture of economists. I basically read only old classics like Adam Smith, and free market economists like the Austrian school. So from what I read, there is hardly any mathematical modeling and there are many polemics against mathematics (like Rothbard's), where they inveigh against its use on some rather silly grounds like: economics is more complicated than physics. But from what you and they say, in the broader economics community, the opposite tendency is prevalent, so I can understand your frustration. I downloaded your paper "Free Banking and Monetary Control" and look forward to reading it. :-)

June 22nd, 2011 5:02 pm

Eitan, while I admit to being mathematically illiterate, how would we benefit by listing the value of water as dependent on the time, place, and person? I'm not sure I understand. It doesn't seem helpful to state the same thing in numbers.

It's like mathematically modeling a painting. Its quality will depend on the artist, the time, place, materials used, weather, what he had for breakfast, and will similarly depend on the same things in those who view the painting. Is there a purpose to extrapolating everything that could possibly affect the value of the painting? To me it seems more useful to simply state that its value cannot be readily determined except when it trades hands, and then its value is only seen in a moment in time and space.

If it is traded for a thousand pounds of cheese, we can only say that Bob valued the painting at one thousand pounds of cheese at 3:00 PM on June 22nd, and would need to list everything Bob already owns, how long it takes him to make everything, what he knows to be available for trade in the marketplace, and how Bob ranks all of these things according to his wants and needs, and then repeat the process for the painter. In the end we are left with a model that won't be able to predict anything and will have no other purpose than to quantify the opinions and emotions of two men who traded cheese for art.

June 23rd, 2011 12:56 am

Hi StoneGlasgow, I think I was a little too harsh when I said your reasoning against the use of math was "ridiculous". Let me explain how I think about subjective value, and why it is useful for me. Say that I have a set of wants, W. It includes lots of things like satiating hunger, quaffing thirst, satisfying intellectual curiosity, etc. I don't need to write down all the things that are in it, or use any numbers at all, I just say that I have a set of wants, W. Now taking an action, I choose based on my preferences for which wants I'd like to satisfy. So that means for any two subsets u,v of W, I can put them in order, u>v, meaning I prefer to satisfy the wants in u to satisfying the wants in v. This is mathematical thinking to me, and of course it requires no symbols, but I just don't see what's so frightening about using symbols u>v. I'm not "quantifying" anything in the sense of using numbers, calculus or statistics. I find this way of thinking helps me clarify relations between concepts, like wants, utilities, marginal utilities, etc.

June 22nd, 2011 5:41 pm

Hi,

As a non-economist, non-math guy I appreciate articles written in plain English. When academics are talking amongst themselves, laying down some sick econ math beats is fine and to be expected. However I'm out in the non-academia world talking to others like myself and the math just is not relevant to those conversations. So having easy to understand material to work with makes my job a whole lot easier.

This is why I bugged you for so many details on how note producing banks actually operated. The more I know of that the easier it is to answer the inevitable questions that arise.

It's also important to make clear to the people I talk too that the folks doing free-banking were not looking to achieve any sort of macro results. Yet they created far less societal problems and many more benefits then those who are working just at the macro level. I've found that this is a serious mental stumbling block for some people and having a clearly written, easy to peruse history of what happened has made it easier help me get people over that block in a way that using models would not do.

So thanks.

I should say that I have it easier in one way in off-line discussions because I'm not talking to 100%ists or greenbackers, just regular folks who don't have any firm ideology about banking other than they want their money to be safe.

Warren

June 22nd, 2011 5:47 pm

"The point is that both mathematical economics and the verbal kind have their place; neither is intrinsically better than the other;"

I think it depends on the underlying complexity of the problems economists are trying to solve. What 'verbal economics' here seems to me to mean is that it involves human brains computing on whatever it is that is up there that they use to understand language, so the answer to "Where's my model" would be "It's all in my head'. I think people have reasonably become skeptical of this sort of answer, and there are many cases where this has been inadequate for successful problem solving. As impressive as minds are they are still posses significant bottlenecks and cannot in general be reprogrammed for specific modeling tasks. This is clear in the natural sciences where simulation of nature by general purpose computers is done as a matter of course and there is no debate on whether or not one method is better than another.

Relevant here is another criticism of orthodox mathematical economics which is complexity economics. If I understand it correctly, it starts from the intuition that the class of models that many economists consider is just 'too simple' which is why they are currently failing to describe reality. This is very similar to Eitan's suggestion that economists just need to use a wider range of math out there.

I personally don't have much of a clue as to how complex economics models need to be, or whether or not the ideal computing substrate to model economies is a human brain or a silicon wafer. It would seem that we all have some capability to agree to a set of rules and constraints, and these could be rationally designed with simplicity in mind. On the other hand people often like to bend and break the rules so a realistic model of a potential economic arrangement would have to account for this capability too.

June 22nd, 2011 5:49 pm

As you say, math has its place, etc.

But it also has to be said that:

a) the increasing prevalence of math in economics over the last 60 or 70 years did not permit us to avoid the recent and ongoing financial unpleasantness;

b) the issues at the heart of debates over the current crisis mustn't be resolvable through math (or else we probably wouldn't be having them since they would have already been resolved given the amount of math flying around) and in any case appear to be more fundamental/methodological or "pre-model";

c) related to b), math focuses everyone's attention on the stuff in the equation, not on the stuff absent from the equation; and

d) the very precision of math means that there will be times when it carries less meaning per unit than words can and when it therefore is going to be much more cumbersome.

More frivolously, it's hard to convey a sense of personality or a personal style through algebra or calculus. Prose more readily permits these aspects of the scholar to be expressed. This makes life more interesting. (Is there such a thing as "a turn of phrase" in math?)

June 23rd, 2011 1:14 am

Yes, when I read math papers, personal styles come out very distinctly.

June 23rd, 2011 11:05 am

The purely formal character of mathematical inference avoids semantic confusion that often arises in natural language. This always bugs me about Austrian economics. They give lip service to formal logic, but most of their "deductions" involve semantic analysis. They analyse concepts like action, trade, and money to unpack what they "really" mean, and then proceed to construct proofs. But the point of formal logic is that it's variables don't mean anything: its proofs do not depend on any understanding of what terms mean. The Austrian method is susceptible to the illusion that one is deducing from axioms, when one is actually just slowly altering the meaning of their terms. These kind of errors seem more difficult to make in purely mathematical constructions.

Austrian epistemology reminds me of a solution someone once proposed to the problem of induction. The author mentioned Hume's argument that there is no guarantee that the thistles in a nearby field won't produce figs next year. He then claimed this argument was inconsistent with the law of identity. Apparently, we can discover the true essence of entities by some kind of unbiased reflection. Once that is accomplished, we can then induce that the thistles in the nearby field will continue to produce purple flowers instead of figs.

Putting aside the matter that such "induction" is actually a deduction, the author failed to grasp that this solution doesn't solve the problem at all, but merely shifts the question being asked. If we define thistles to be plants that do not produce figs, then it follows that the nearby thistles will not produce figs. However, the empirical question is then whether the plants in the nearby field are thistles, after all, or some other plant which we haven't divined the essence of yet which sometimes produces purple flowers and other times produces figs.

I can't help but feel that Austrian economists often commit this type of fallacy. Unfortunately, it is a fallacy that goes all too easily unnoticed when we restrict ourselves to purely natural language analysis.

June 23rd, 2011 11:46 am

Lee, with all due respect to Hume, if he begins by noting that the plants in the nearby field are "thistles," and we do in fact insist on the usual definition of a thistle, then I think we have every right to assume either that (1) those plants must not bear figs or (2) that Hume has mislead us. Had he merely said that the plants "appear to be thistles," then we could reasonably allow that they might end up doing something quite un-thistle like!

Of course in so far as thistles are capable of surprising us in a manner that defies any attempt to define them, then there's no point trying to engage in deductive reasoning with reference to them. But the whole point of Hume's "problem" is that empirics are just as futile in this case. What good does it do to affirm that the thistles haven't in fact produced figs, when they might yet do so? The bottom line is, if you want to say anything at all about thistles, or anything else, you have got to set Hume's conundrum aside. It remains useful as a reminder that empirically-based knowledge is always subject to revision. But otherwise we trudge along as if things really do have stable "identities," and I think we are right to do so.

June 23rd, 2011 12:24 pm

Well, you kind of missed my point. The author suggested that we could discover universal empirical truths by this kind of method. In my view, it amounts to little more than slightly redefining the meaning of terms, like saying: "it's obvious that all swans are white, because a swan is, among other things, a white bird." Defining swans as white birds doesn't resolve the empirical issue; it just shifts the matter to whether swans as a species of white birds actually exist.

I sometimes detect a similar line of thinking from Austrian economists. They "deduce" all kinds of things about acting men, government intervention, and business cycles from apparently indubitable "axioms." But these axioms are meaningful propositions, and deductions from them proceed by analysing the terms and unpacking concepts. This is not altogether objectionable, but I wonder how often they actually, and probably unconsciously, redefine the terms in the axioms and "discover" that all swans are white.

My point is that purely formal interpretations seem less susceptible to these types of errors.

June 23rd, 2011 5:43 pm

Well Lee, you would have a stronger point if definitions were merely concocted for the sake of saving the effort of empirical verification. But that's not how they work. Definitions follow agreed usage; so if indeed it were the case that "whiteness" were considered a defining characteristic of swans, then "black swan" would be an oxymoron. Of course it doesn't follow that, by merely insisting upon the accepted meaning of terms, one adds to the sum total of knowledge of the world. On the other hand, arguments that rest on departures from words' ordinary meanings can be worse than vacuous.

But as regard's some Austrians' insistence on the purely deductive nature of economics, I believe they confuse the epistemological basis of a small "core" of economic propositions with that of economics in the broader sense. Here it is they who are guilty of toying with a word's generally understood meaning.

The question of the inductive versus deductive basis of economic propositions is of course an entirely different matter from that concerning the relative merits of mathematical and verbal forms of argument. I am not so convinced as you are that the mathematical approach avoids the problem you refer to. I have read many formal demonstrations in which a conclusion was in fact "stuffed into" the model set-up, to be yanked-out later, with drum-roll, as if the mathematics demonstrated it. Have a look, for instance, at Sargent and Wallace's defense of a zero inflation rule: they show that a low-inflation rule is best only thanks to having stuck the (absolute value of) the inflation rate in the monetary policy loss function! This sort of "rabbit in the hat" economics it is in fact common fare in the math econ literature.

June 23rd, 2011 3:27 pm

Lee Kelly,

Perhaps my opinion of praxeology and Austrian economic methodology is way off base, but I'm not sure deducing economic theorems from a set of axioms based on primitive concepts of human action allows one to be an armchair economist, in which he can simply dispense with any and all empirical investigation, content with having discovered complex truths about the real world a priori. Rather I view it as a way of legitimizing our analysis (that is, giving it epistemic grounds) for interpreting the blooming, buzzing confusion we see around us. In other words it gives us a focal point (purposeful human action) by which we build the rest of our EMPIRICAL investigation of the economy. In this way economics can be called a science because it is based on primitive concepts of acting man. Now whether this is the correct way to view Mises's contribution and whether it's free of flaws, is a topic for another time.

George Selgin,

Excellent piece. My biggest gripe with math in the economics profession isn't so much the math per se, it's the fact that mathematics is disproportionately represented to the exclusion of some very important and interesting fields, e.g., history of economic thought, economic history, institutional economics, etc. I'm a methodological pluralist (like Caldwell) and think it's all important to some extent, but it seems one cannot be considered a great economist in today's profession unless he or she plays the math modeling game. That's very unfortunate.

Also, I read your book. Loved it. I'm reading Dowd's _Competition and Finance_ now. Great blog, keep up the good work.

June 23rd, 2011 6:04 pm

Thanks for your comments, Othyem. I quite agree with your point about the disproportionate emphasis on math econ (or just plain math) in grad programs. What's especially amazing is many mathematical economists' apparent inability to grasp the fact that, in most programs, the marginal value of exposure to non-mathematical aspects of the discipline is now (with the quantity of exposure at or near zero in most cases) almost certainly higher than that of the last unit of math.

Here's a simple way to confirm this: plan to go to almost any grad-student seminar involving a formal paper. Study the facts regarding the addressed topic for an hour or so ahead of time, and see if you can't make mincemeat out of the arguments being made. More often than not, doing so is child's play.

June 26th, 2011 1:03 pm

I would suggest that a good dynamic model can tell you some surprising things you may not have guessed otherwise. I need to refer to a figure to make my point: Goodwin Simulation - US Economy 1913-2100

This figure is from a Goodwin model consistent with the Quantity Theory of Money. It provides a conceptual view of the effect of 4% inflation on the wage share.

The effect of Volcker's interest rate hikes 1979-83 is shown as a period of zero inflation. The interruption of the inflation created considerable turbulence in the wage share, with large rebounds going forward. Who would have guessed that the housing bubble was a consequence of Volcker?

Avoiding mathematical models leads one to static analysis and explanations.