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More on models

George Selgin wrote a post on expressing ideas in words rather than mathematics. Here are my two cents of commentary.

Math is useful when we want to know “how much?” How much is the U.S. economy growing? How much is the value of the dollar changing against a trade-weighted basket of other currencies? How much more revenue would income taxes raise if their rates were doubled?

The math that is useful for answering such questions is often nothing more advanced than what college-bound high school students learn, combined with some accounting identities. Deirde (formerly Donald) McCloskey and Arjo Klamer have made the case that accounting rather than higher mathematics is in fact the master metaphor of economics, because so much of applied economics is about what to count and how to account for it.

At bottom, a model is a device for isolating and examining what we consider to be important features of a situation. A model need not be a forest of equations. A verbal description can be a model. So can a balance sheet. So can a historical case—that’s why we say, for instance, “Bank X’s lending practices were a model of good risk management.”

A verbal model is more appropriate than a mathematical one in cases where generality is more important than extreme precision. A small change in the assumptions need not lead to a big change in the conclusions, as can happen with mathematical models (another favorite theme of Deirdre McCloskey).  Economists continue to rely heavily on verbal models in practice. But rather than following Alfred Marshall’s habit of favoring verbal exposition even for results derived from mathematical inquiry, they express insights that were originally verbal only in mathematics. Readers waste much time mentally translating the math back into words.



  1. I have posted this figure on the Where's My Model? thread but as it is still being moderated I will repost it here: Goodwin Simulation – US Economy 1913-2100

    The Goodwin Model has an undeservedly bad reputation. Goodwin himself, a Harvard Marxist, even gave up on it as "useless". If you look into its elements, it is easy to see where the errors were made.

    The basic assumption connecting output to money is Q = ms (Actually Q = m*sigma, but I use s instead because Greek characters may not show up on your site) Q is real output, m is real money and s is the output/capital ratio. For some reason, the post-Keynesians who tried to use this model always assumed that s was a constant. However, the Quantity Theory of Money also connects output and money as pQ = mV or, Q = mV/p.

    Comparing the two expressions for Q, it can be inferred that

    s = V/p

    The 2 equations that make up the Goodwin Model are:

    (1) u'/u = Ph(v)- a'/a – p'/p

    (2) v'/v = (1-u)s – a'/a – n'/n – d

    Here, u is the wages share of output = wL/pQ = wL/paL = w/pa

    p= price level, a = labor productivity, w = nominal wages

    L is workforce, v is employment percentage, n is population,

    d is capital depreciation and Ph(v) is a general exponential

    function which gives the Phillips curve. The Phillips curve has a well established empirical basis.

    The difference between my model and previous efforts is that I use

    s = V/p, which is to say that the output/capital ratio is a

    function of the price level.

    The simulation on the above chart shows the response of the economy to a constant 4% inflation rate since 1940 except for the period 1979-83 which is shown as a zero inflation period.

    The model shows what I had expected. ie., that inflation destroys the wage share and the collapse arrives right on schedule.

    I have often thought that Austrians should use models. I have read enough 40,000 essays with nary a chart or equation to come to that conclusion. The Austrians have a sound theory but can't communicate with anyone!

    On my blog I am working on a lengthy post entitled "A Dynamic Austrian Macro Model" which I am slowly piecing together. I have not had any comments so far from any serious person. The only "serious" people to look at it have been Keynesians who dogmatically reject pQ = mV.

  2. All this trashing of maths is bugging me. It seems to me that you economists don't realize that there is math other than calculus and statistics, but math is more than numbers! You decry that math cannot deal effectively with more general situations than those which may be described numerically, but honestly, to me, you only betray your limited scope of math knowledge. The verbal model you prefer is still math! Just written in imprecise forms that are subject to (mis)interpretation, with assertions made that are not rigorously proven. Not being able to rigorously show that your conclusions follow from your assumptions is not a strength. Sorry, I think I'm taking this too personally. 🙂

    1. There's a paper by Karl Menger, Carl Menger's son and a mathematician, about this topic which really explains much better than I can.

      Menger, K., 1973. Austrian Marginalism and Mathematical Economics. Carl Menger and the Austrian School of Economics, p.38-60, edited by Hicks, J.R. &Weber, W.

    2. Economics is about human behavior, so most of it should be understandable verbally, because that is how we usually communicate with each other. You did not try to reply to me with a bunch of mathematical symbols. I don't think you would have considered such a reply more precise and rigorous. What constitutes rigor depends on the context. Sometimes it involves mathematics; more often it does not.

      1. I can say "good A is preferred to good B", or I can use symbols "A>B". Both statements have the same level of rigor. Both statements are mathematical statements about an economic concept. You prefer the first statement because you don't have to translate back. That's fine. One should write in a way that is comprehensible to one's intended audience. I argue that mathematics is useful for economics, not that symbolic language is always better than verbal language. They're both mathematical.

        Rigor is the absolute certainty that every logical step follows from the previous one. For instance, the statement "Economics is about human behavior, so most of it should be understandable verbally, because that is how we usually communicate with each other" is not rigorous, which is fine, but does not convince me.

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