Posts Tagged ‘mathematics’

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You remember the dialogue:

Queen: Slave in the magic mirror, come from the farthest space, through wind and darkness I summon thee. Speak! Let me see thy face.

Magic Mirror: What wouldst thou know, my Queen?

Queen: Magic Mirror on the wall, who is the fairest one of all?

Magic Mirror: Famed is thy beauty, Majesty. But hold, a lovely maid I see. Rags cannot hide her gentle grace. Alas, she is more fair than thee.

I was reminded of this particular snippet from Snow White and the Seven Dwarfs while reading the various defenses of contemporary macroeconomic models. Mainstream macroeconomists failed to predict the most recent economic crisis, the worst since the Great Depression of the 1930s, but, according to them everything in macroeconomics is just fine.

There’s David Andolfatto, who argues that the goal of macro models is not really prediction; it is, instead, only conditional forecasts (“IF a set of circumstances hold, THEN a number of events are likely to follow.”). So, in his view, the existing models are mostly fine—as long as they’re supplemented with some “financial market frictions” and a bit of economic history.

Mark Thoma, for his part, mostly agrees with Andolfatto but adds we need to ask the right questions.

we didn’t foresee the need to ask questions (and build models) that would be useful in a financial crisis — we were focused on models that would explain “normal times” (which is connected to the fact that we thought the Great Moderation would continue due to arrogance on behalf of economists leading to the belief that modern policy tools, particularly from the Fed, would prevent major meltdowns, financial or otherwise). That is happening now, so we’ll be much more prepared if history repeats itself, but I have to wonder what other questions we should be asking, but aren’t.

Then, of course, there’s Paul Krugman who (not for the first time) defends hydraulic Keynesianism (aka Hicksian IS/LM models)—”little equilibrium models with some real-world adjustments”—which in his view have been “stunningly successful.”

And, finally, to complete my sample from just the last couple of days, we have Noah Smith, who defends the existing macroeconomic models—because they’re models!—and chides heterodox economists for not having any alternative models to offer.

The issue, as I see it, is not whether there’s a macroeconomic model (e.g., dynamic stochastic general equilibrium, as depicted in the illustration above, or Bastard Keynesian or whatever) that can, with the appropriate external “shock,” generate a boom-and-bust cycle or zero-lower-bound case for government intervention. There’s a whole host of models that can generate such outcomes.

No, there are two real issues that are never even mentioned in these attempts to defend contemporary macroeconomic models. First, what is widely recognized to be the single most important economic problem of our time—the growing inequality in the distribution of income and wealth—doesn’t (and, in models with a single representative agent, simply can’t) play a role in either causing boom-and-bust cycles or as a result of the lopsided recovery that has come from the kinds of fiscal and monetary policies that have been used in recent years.

That’s the specific issue. And then there’s a second, more general issue: the only way you can get an economic crisis from mainstream models (of whatever stripe, using however much math) is via some kind of external shock. The biggest problem with existing models is not that they failed to predict the crisis; it’s that the only possibility of a crisis comes from an exogenous event. The key failure of mainstream macroeconomic models is to exclude from analysis the idea that the “normal” workings of capitalism generate economic crises on a regular basis—some of which are relatively mild recessions, others of which (such as we’ve seen since 2007) are full-scale depressions.  What really should be of interest are theories that generate boom-and-bust cycles based on endogenous events within capitalism itself.

With respect to both these issues, contemporary mainstream macroeconomic models have “stunningly” failed.

I imagine that’s what the slave in the magic mirror, who simply will not lie to the Queen, would say.

Treviso

Since we’re on the topic of the supposed superiority of economists, I thought I would provide a link to one of my first published articles, “The Merchant of Venice, or Marxism in the Mathematical Mode” [pdf], which appeared in the journal Rethinking Marxism.*

My basic argument is that, while mathematics has been granted the status of a special code in economic discourse (including in Marxian theory)—thus demonstrating the superiority of economists who use that special code—it is actually a set of metaphors that can be useful and harmful in turn. In other words, the use of mathematics “does not guarantee the scientificity of the theory in question; it is merely one discursive strategy among others.”

There are two interesting stories associated with this article. First, it was used against my case for tenure, by a member of the committee who (from what I have been told) was simply incensed that I would attempt to deconstruct the use of mathematics as a special language for doing economics. (Fortunately, it didn’t work and I was in fact granted tenure.)

Second, I disappointed not a few literary scholars who came to one of my seminars on the article expecting a discussion of Shakespeare’s play. The joke is that the title refers to the Treviso Arithmetic, which was written by an anonymous author in 1478 in Treviso, a commercial town annexed to the Venetian Republic in 1339, and is considered to be the first book on mathematics ever published in the West.

The problem that begins in the middle of the left-hand page of the illustration above is the following:

Two merchants, Sebastiano and Jacomo, enter into partnership. Sebastiano put in 350 ducats on the first day of January, 1472; Jacomo put in 500 ducats, 14 grossi on the first day of July, 1472. On the first day of January, 1474 they find that they have gained 622 ducats. Required is the share of each.

 

*A scholar overseas, without access to the journal, asked me to send him a copy of the article. That’s the reason I now have a pdf file of the article on hand.

Piketty-US-scale-factor

The storm unleashed by Chris Giles’s takedown (follow the links) of Thomas Piketty for the Financial Times (with responses now by Piketty himself, Neil Irwin, Simon Wren-Lewis, Steven Pressman, and others) reminds me of two stories.

First, there’s the story of a seminar by Hollis Chenery, one of the pioneers of economy-wide development planning models, at Yale University in the early 1970s. One of the participants in the seminar, who later was one of my professors in graduate school, offered Chenery a large sum of money to put together the appropriate matrix of data—and then an even larger sum of money not to invert the matrix. The point: there are so many mistakes, assumptions, and elements of pure guesswork involved in compiling any set of economic data, it is a fundamental mistake to presume the correct economic policy or strategy can be devised—and then offered as objective and accurate “expert” advice—by simply running the model.

Second, a friend in graduate school, who already had a Ph.D. in mathematics, took it upon himself to work through the mathematics presented in the tenth edition of Paul Samuelson’s famous Foundations of Economic Analysis. He told me he was amazed to find more than one hundred mistakes in the book, even after so many editions. The point of this example: lots of errors are made—and then repeated by authors and overlooked by readers—even in the most famous writings of economists. And the errors committed in Samuelons’ Foundations certainly didn’t stop the mathematization of mainstream economics in the postwar period.

As for Piketty, my view is, first, we need to give him credit for making all of his data, mistakes and all, freely available on-line. Second, even if in one or another country, during one or another period of time, the distribution of wealth has not become more unequal, the fact remains that the distribution of wealth is and remains profoundly and grotesquely unequal. Even Giles can’t dispute that point. And finally, I can only imagine what the reaction would be if Piketty had actually collected data not on wealth, but on capital in the twenty-first century, and had attempted to calculate changes in the rate of exploitation over time.

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Every time there’s a controversy in economics, the problem of mathematics seems to be at the center of the discussion. That’s because, in economics, discussing the role of mathematics is inextricably related to issues of science, epistemology, and methodology, which are themselves rarely discussed but are implicit in pretty much all of these discussions.

In short, we’re never very far from physics-envy.

That’s certainly the starting point of Noah Smith, who compares the mathematics of physics (the supposed “language of nature”) with that of macroeconomics (in which it serves to “signal intelligence”). And then, of course, Paul Krugman rides to the rescue, arguing that mathematical models, when “used properly,. . .help you think clearly, in a way that unaided words can’t.”

Uh, OK. “Used properly” is the operative clause there. The real question is, what is the proper use of mathematics in economics? And, what is the proper way of thinking about the proper use of mathematics? That’s where all the issues of science, epistemology, and methodology come to the fore.

As it turns out, one of the first articles I ever published, “The Merchant of Venice, or Marxism in the Mathematical Mode,” was on that very subject. I had mostly ignored mathematics during my undergraduate years but then, in graduate school and especially when I began to conduct the research for my dissertation (on mathematical planning models), I realized I wanted both to learn the econmath and to learn how to think about the econmath. Ironically, I ended up teaching “Mathematics for Economists” to first-year doctoral students for over a decade (it was basically a course in linear algebra and multivariate calculus, in which students also had to write a paper on the history and/or methodology of the mathematization of economics).

The argument I made in my dissertation and later in the “Merchant of Venice” article was that economists (mainstream economists especially, but also not a small number of heterodox economists, including Marxists) treated mathematics as a special language or code. They considered it special either in the sense that it was the language of nature (and therefore overprivileged) or a neutral medium for thinking and expressing ideas (and therefore underprivileged). Either way, it was considered special.

My alternative view was that mathematics was just one language among many, and therefore one set of metaphors among many. And like all metaphors, it served at one and the same time to enable and disable particular kinds of ideas. Therefore, we need to both write mathematical models and to erase them in order to produce new ideas.*

But that’s not how most economists think about mathematical models. And when they do think and write about them, they tend to invoke one or another argument for mathematics as a special code. They also tend to forget about all the other uses of mathematics in economics—not only as a signalling device but as a hammer to bludgeon all other approaches out of existence.

It’s the tool that is often used, in economics, to separate science from non-science—which, of course, if you say it quickly, becomes nonsense.

*That argument, concerning the not-so-special status of mathematics, so incensed one of my colleagues he attempted to derail my tenure case. Fortunately, another of my colleagues forced him to back down and I ended up receiving a unanimous recommendation.

Whose dream?

Posted: 14 February 2012 in Uncategorized
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Many years ago, I was invited to give a talk, “Whose Dream? The Role of Mathematics in Radical Economics,” at American University.*

I was reminded of that event on reading Ian Stewart’s essay on the Black-Scholes equation and the financial crash.

Black-Scholes underpinned massive economic growth. By 2007, the international financial system was trading derivatives valued at one quadrillion dollars per year. This is 10 times the total worth, adjusted for inflation, of all products made by the world’s manufacturing industries over the last century. The downside was the invention of ever-more complex financial instruments whose value and risk were increasingly opaque. So companies hired mathematically talented analysts to develop similar formulas, telling them how much those new instruments were worth and how risky they were. Then, disastrously, they forgot to ask how reliable the answers would be if market conditions changed.

And that’s exactly what happened years earlier with Long-Term Capital Management. The board of directors of LTCM included Myron Scholes and Robert C. Merton, who shared the 1997 Nobel Prize in Economics for a “new method to determine the value of derivatives.” Their dream of conquering uncertainty, and of making tons of money in the process, is told in the Trillion Dollar Bet.

 

* This was before we used the internet to arrange such seminars. So, the title of my talk, conveyed by telephone, appeared on posters all over campus as “Who’s Dream?” Needless to say, the participants in the seminar wanted to know what Dr. Who had to do with economics.

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Many economists are obsessed with mathematics. That’s especially true among neoclassical economists but not unknown, as I have discovered, among Marxian and other heterodox economists.

Clearly, Krugman’s critique of mathematical elegance has struck a chord:

As I see it, the economics profession went astray because economists, as a group, mistook beauty, clad in impressive-looking mathematics, for truth.

The latest to have his feathers ruffled and to proclaim that mathematics is the special code of economics is Gregory Mankiw:

mathematics is, fundamentally, the language of logic. Modern research into Keynes’s theories—I have conducted such research myself—tries to put his ideas into mathematical form precisely to figure out whether they logically cohere. It turns out that the task is not easy.

No, mathematics is one among many languages, one set of metaphors among others. But for economists like Mankiw, mathematics has a unique, special status in the way economic analysis is carried out. It’s the blunt weapon used to question the validity of Keynes’s theory and to bludgeon other not-necessarily-mathematical approaches to economics.

That’s what I argued in “The Merchant of Venice, or Marxism in the Mathematical Mode,” an essay published in 1988 (during the period when I taught graduate Mathematical Methods for Economics). When mathematics is assumed to be the special code of economics, it is either underprivileged or overprivileged. Underprivileged, if it is seen to be a neutral medium, and thus does not affect the results of economic analysis. It is the “language of logic,” which all economists should be forced to use. Overpriviledged, if it is taken to be the language of the book of nature, which means that only using mathematical formalisms and models provides access to reality.

Underprivileged or overprivileged, empiricism or rationalism—in both cases, mathematics becomes a special code.

The goal of questioning the fetishism of mathematics is, of course, not to argue against the use of mathematical forms of discourse but to recognize that, like all sets of metaphors, mathematics enlightens and obscures, it needs to be written down and erased, at the same time.

Treviso

And it all began with the Treviso Arithmetic, the earliest known printed mathematics book in the West, an anonymous textbook in commercial arithmetic written in vernacular Venetian and published in Treviso, Italy in 1478.