Mathematical Models in Economics
I reject the usual arguments, pro and con
In my talk with John Cochrane last week, he took some positions with which I disagree, but in the interest of time I did not argue. One of those positions (I will get to the other one in a different post, but it is related) is that mathematical modeling in economics is very constructive. This issue got batted around recently on Twitter, which I found out by reading the latter part of Noah Smith’s post. Smith writes,
literary treatments of economic phenomena are usually very vague. This can cause them to function as Rorschach tests; each person can interpret Keynes or Minsky or Hayek to mean something a little closer to their desired conclusion. Mathematical modeling precludes that.
But mathematical modeling is also vague. It is vague in how the model connects to the real world. Martin Bronfenbrenner once talked about a missing “applicability theorem.” To what real-world measurements does your theory apply? The answer often rests on claims that are as vague and tenuous as any literary paper. For example, the famous Solow growth model is disconnected from the real world to the extent that the real world does not present us with K and L, which stand for capital and labor in the model. There are too many varieties of capital and labor. Piketty’s famous “r>g” (return on capital greater than economic growth) becomes hopelessly tangled in measurement controversies, particularly concerning r.
The bottom line is that clear writing in economics is rare. It is rare in articles without equations. It is rare in articles with equations.
On the anti-math side, one argument is that the world is too complex to be reduced to equations. Equations “leave out” too much. Smith writes,
On the other hand, the cost of mathematical modeling is that it’s often mathematical tractability, rather than realism, that drives the modeling choices. In other words, bank runs don’t happen exactly like Diamond & Dybvig wrote; they just chose assumptions that made the math relatively easy to solve. This is why models like the ones that one the Nobel this year shouldn’t be regarded as the way financial crises actually work, but simply as ways that they might work. Nailing down exactly how they do work is a task that will be very difficult and take a very long time.
This is true also. But it is also true of economic analysis that does not contain equations. Regardless of whether you use prose or equations, you have to simplify.
Using math can help you avoid making claims that are internally inconsistent. For example, you cannot claim that a perfectly competitive firm in long-run equilibrium is producing at a point where marginal cost is greater than the minimum of average cost. For a perfectly competitive firm, price equals marginal cost, and in the long run price equals average cost. A little calculus shows that marginal cost only equals average cost at the minimum of average cost.
Overall, what Cochrane likes about math is exactly what I don’t like. That is, when someone comes up with an elegant model, economists flock to using that model. Cochrane sees this piling on as adding to the insights of the seminal paper, proving its value. I see it as playing the game of maximizing the chance of getting your paper published, for which this strategy works really well. I see the inevitable consequence being a cumulation of a resulting literature that is ridiculous. The Overlapping Generations Model is a particularly salient example—constantly re-used without a credible application theorem.
Maybe even if you took out math, economists would still behave like lemmings, lavishing attention on one cleverly-expressed idea and overlooking its shortcomings. But my guess is that the academic conversation in economics would be much improved without the math.
A few years ago, National Affairs published an essay in which I put together my thoughts on the state of economic methodology. My conclusion:
Young economists who employ pluralistic methods to study problems are admired rather than marginalized, as they were in 1980. But economists who question the wisdom of interventionist economic policies seem headed toward the fringes of the profession.
In this respect, the barriers to effective theory in economics are different and perhaps more worrisome than was the case in 1980. The contemporary state of economic theory reflects a broader crisis in the social sciences and a deepening cleavage between the college campus and the rest of society.