Krugman and maths

Krugman periodically goes into bat for maths in economics and invariably trivialises the concerns of critics. He says that maths helps focus arguments and weed out error.  Too right it does. And so do words. So shouldn’t we be using each to their best effect and not give ourselves over to the scientism of associating mathematically worked out theory with science and verbal reasoning to some lower form.

Here’s a passage from Hicks writing in 1939 about the choice as to whether or not to assume perfect competition in one’s modelling. He’s showing an awareness of the tradeoffs involved in modelling – between realism, formality and usefulness.  This is the kind of sensibility that Marshall cultivated at the turn of the 20th century.

1 2here must be something to stop the indefinite expansion of the firm, but it can just as well be stopped by the limitation of the market as by rising marginal costs. . . . It is, I believe, only possible to salvage anything from this wreck – and it must be remembered that the threatened wreckage is that of the greater part of economic theory – if we can assume that the markets confronting most . . . firms . . . do not differ very greatly from perfectly competitive markets. . . . At least this get-away seems well worth trying. We must be aware, however, that we are taking a dangerous step, and probably limiting to a serious extent the problems with which our subsequent analysis will be fitted to deal. Personally, however, I doubt if most of the problems we shall have to exclude for this reason are capable of much useful analysis by the methods of economic theory.

Given the canonical state of this passage and the book they come from, and the fact that it was these concerns that hung over trade theory and justified the almost ubiquitous assumption of perfect competition in formal models (along with the rider that scale economies were important and should not be ignored, but couldn’t be easily handled using formal theoretic methods) you’d think that the new agenda of strategic trade theory might have addressed Hicks’ concerns. That is you’d think that, in constructing the new field, its pioneers would have addressed this question – would have explicitly argued that what they were doing was not, now a waste of time. Certainly there were new techniques, but you’d hope they’d give some thought to their likely utility. I know this is very verbal and all, not conveyed in an equation, but that’s what you’d expect if their energies were to be deployed rigorously. The alternative is what I call the mountaineer’s approach to the utilisation of theory.

Q: Why did you rebuild trade theory around the Dixit-Stiglitz model?

A: Because it was there.

As Krugman says, economists just ignored scale economies in trade because they were hard to model and so it was unfashionable to try. In explaining how useful maths is in economics Krugman says this:

Take the centerpiece of my early career, the work on increasing returns and trade. The models I and others used were, in a way, typical of economics: clearly untrue assumptions (symmetric constant elasticity of substitution preferences; symmetric costs across products!), and involved a fair bit of work to arrive at what sounds in retrospect like a fairly obvious point: even similar countries will end up specializing in different products, and because there are increasing returns in many sectors, this will produce gains from specialization and trade. But this point was only obvious in retrospect. People in trade were not saying anything like this until the New Trade Theory models came along and clarified our thinking and language. Trust me, I was there, and went through a number of seminar experiences in which I had to bring an uncomprehending audience through until they saw the light.

Krugman’s stuff on new approaches to trade started getting published in the late 1970s but the new fad had taken off by the mid 1980s. In 1988 the Economist breathlessly reported the new fad as “truly ground breaking stuff . . . asking, to what extent does the presence of imperfect competition invalidate the case (hitherto thought to be rock solid) for free trade”.

Apparently no-one knew about economies of scale. Well except Linder who, in 1961 wrote up the way in which car firms and others were marketing the same products for general consumption in their home markets and niche products for luxury consumption in export markets.

The factor proportions account is only one of many. . . . It is important to realise this, or we shall not be able to get off the side track onto which we have, in effect, been shunted.

So Linder’s complaining about the side track onto which economics had been shunted – the same thing Krugman complained about in the late 70s, but he’s doing it nearly twenty years before and offering some useful insights into where one might take the discussion to address the problem.

Except Raymond Vernon who became well known for his ‘product cycle’ model of trade in which the leading edge parts of the production system would be pioneered in rich countries and then get gradually commoditised and shifted to production in lower cost less developed countries. Sound familiar? He was writing this stuff in the early 1970s in books that were best sellers.

Except two distinguished antipodean economists Peter Drysdale and Peter Elkan (from New Zealand) who thought about these problems and proposed policies based around them – Peter Drysdale proposed some kind of arrangement with Japan which would enable both Australia and Japan to specialise within the auto industry – with  Australia producing larger cars and Japan smaller ones.

Here’s Peter Gray trying to make some impact on the discipline in 1988, though he’s summarising the approach he laid out in a book in 1976.

To be able to embrace all of the various forces which act on IIT 3 in a single model which yields a precise solution, is impossible because the variables are too numerous and their effects can be inconsistent across cases and through time. Even if it were possible to produce a definitive treatment of IIT, the model would be so general and would not be operational in the sense of either being subject to empirical test or useful for policy prescription. . . . Such is the complexity of IIT that our understanding of this phenomenon will be better served by . . . a general ‘loose and untidy’ model than by a selectively precise or rigorous one.

So there’s the challenge – a methodological challenge – rigorous for all its wordiness. It’s a claim that if you try to model imperfect competition, you won’t do much more than spend a lot of effort proving the obvious. How did that claim turn out? Well Krugman confirms that he was right. So when Krugman says “Trust me no-one was talking about scale economies as an engine of trade and the policy implications before I built models of it” what he really means is that no-one in polite society (amongst economists being published in the very best journals) was talking about it. They were in an echo chamber. The fact that people were screaming the irrelevance of their own models from the rooftops was water off a duck’s back. The papers kept rolling in all their irrelevance. And now, with the benefit of hindsight, comfortable in the conclusion that the new theory laboured to demonstrate the obvious, it’s still water of Krugman’s back.

To make my point as clearly as possible, there are a lot of things going on in an economy. And they’re changing over time. You have to make decisions about what’s important and what’s not. And it’s virtually always the case that work that is useful shows some aptitude for a range of different sources of insight – including – yes – formal modelling but also including clearly observable facts that are not greatly illuminated by building a model of them – or if they are, can still be important inputs into one’s thinking while one is waiting for the modelling.  Krugman’s economic commentary is as good an example of what I’m talking about as I can think of.

Krugman initially had trouble having his early work on trade and monopolistic competition published. Referees were unconvinced that an “understanding of these issues would be helped by writing down formal models”. These comments are now offered as a suitable topic for mirth by Krugman and others. The referees’ concerns might not justify refusing publication of Krugman’s clever models – particularly the model of trade and monopolistic competition to which these comments applied. But their concerns have, according to Krugman’s own account of it, been largely vindicated. Gray was right. The blind eye that Krugman turns to the profession’s distain for obvious facts in the case of trade theory is all pretty ironic given his current laments about the way in which modern macro has been blind to some of the most obvious facts of experience and the comprehensiveness of its amnesia regarding the strides made in macroeconomics since the 1930s.

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10 years ago


Does anyone actually use economics these days? I mean, what’s the point of it, and can you get a job doing it?

Seems to me an economics degree is even less useful than a law degree.

10 years ago
Reply to  SJ

If nothing else you could become a macro trader.

10 years ago
Reply to  SJ

Can’t manage to paste the link , but Noah Smith has a good argument why econ Phd’s are superior to others.

I’d add that if you do an undergrad with as many econometric options as possible, then at least you learn stats and the economic metadata. Obviously Krugman and Nicholas are thinking solely in terms of macro models, but this bypasses the more practical applications of economic math; namely, the statistical description of individual markets, and macro time series. Lest we forget that correlations can be important even if the causal mechanisms are in the eye of the beholder.

Richard Tsukamasa Green

I’ve been struggling to find a suitable metaphor. Maths is meant to be like corrective lenses, in that they make you see things clearer. But the way the profession does it is trains you to wear glasses that give ridiculous and heavy myopia. Then, you can become celebrated by developing a set of super lenses to wear over the conventional lenses that let you see almost, but not quite, as normally as someone not wearing any lenses at all – with the bonus that you have a large, heavy and “rigourous” apparatus on your head that signals your status to all the people still wearing the false myopia glasses..

SJ – This kind of silly formalism is (thankfully) largely relegated to post graduate schools. Undergraduate degrees tend to stick with simpler models combined with verbal reasoning – as well as some basic statistics. All of which is useful and employers do seem to like, if only because the graduates can understand the terminology of each other. That said, because it makes sense and is widely used in actual policy discussion, it really wouldn’t be hard for a autodidact to pick up.

Post graduate though is usually pure signalling and most valued in sectors most divorced from the real world (academia, multilateral organisations). Very rarely do I meet a non-academic Masters or PhD who applies what they learned (provided they had undergrad).

James Rice
10 years ago

To quote a film from the 90s…

Policy maker: We want the truth!
Economist: Economics can’t handle the truth! (in a mathematically tractable way)

derrida derider
derrida derider
10 years ago

Yes, its true that lots of important things in the world can’t be properly captured by formal modelling and we should pay more attention to those things. And yes, its also true that mathematical esoterica is often used just as a signal to show the user is a true member of the priesthood and not one of the laity (or worse, an infidel). Though of course economics is far from the only field where esoterica of various sorts are used for signalling group membership rather than for advancing the field – humans are peculiar creatures.

And yet I think Krugman is basically right. If it can possibly be rigorously modelled, it should be because we can have more faith in a rigorous model than in an unrigorous one. Maths uses utterly unambiguous terms where logical errors between them are immediately obvious, whereas none of that is true of spoken and written words. The alternative to a logically rigorous model with clear and explicit assumptions is never no modelling, but an heuristic model with ambiguous and unexamined implicit assumptions. Humans think about the world in models, even if they think they don’t.

And the other point Krugman makes is spot on too. A good model provides a framework for measurement – it generates hypotheses that tells you what things you should be looking at. A model will never tell you what is true, but it does tells you what might be true.

Paul Frijters
Paul Frijters
10 years ago

the siren call of ‘rigor’ is very strong. I have never seen a rigorous economic model myself but keep hearing others speak of their wonders.

10 years ago

Mark Thoma quoting Krugman’s The Real Trouble With Economics: Sociology and his own thoughts.

Robin Hanson Math: Useful & Over-Used