{"id":114,"date":"2020-12-23T16:11:07","date_gmt":"2020-12-23T16:11:07","guid":{"rendered":"https:\/\/www.economic-instability.com\/?p=114"},"modified":"2024-05-22T12:55:50","modified_gmt":"2024-05-22T12:55:50","slug":"tinbergen-1936-a-nonlinear-model-of-collapse","status":"publish","type":"post","link":"https:\/\/www.economic-instability.com\/?p=114","title":{"rendered":"Tinbergen (1936): A nonlinear model of collapse"},"content":{"rendered":"

\"\"Ach, mit offenen Armen stand ich gegen den Abgrund und atmete hinab! Hinab!<\/i><\/span><\/span><\/span><\/p>\n

Goethe, <\/span><\/span><\/span>Die Leiden des jungen Werther<\/i><\/span><\/span><\/span><\/p>\n

Oh, with open arms I stood against the abyss and breathed down! Down!<\/p>\n

Goethe, The Sorrows of Young Werther<\/em><\/p>\n

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   Wenn nun schliesslich p(2) noch gr\u00f6\u00dfer gew\u00e4hlt wird, so wird das zweite Gleichgewicht \u00fcberschritten, und die Bewegung verliert sich sozusagen im Abgrund.<\/i><\/span><\/span><\/span><\/p>\n

Tinbergen (1936: 216), \u201c\u00dcber den Wert Mathematischer Konjunkturtheorien\u201d<\/span><\/span><\/span><\/p>\n

If p(2) is finally chosen to be even greater, the second equilibrium is exceeded and the movement is lost in the abyss, so to speak.<\/p>\n

Tinbergen (1936: 216), \u2018On the Value of Mathematical Business Cycle Theories\u2019<\/p>\n

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This post is based on the results shown in Chapter 5<\/a> of Modeling Economic Instability: A History of Early Macroeconomics<\/em> (121-144) and in this<\/a> article. All comments are welcome.<\/p>\n

Jan Tinbergen is mostly remembered today for his pioneering work in building the first macroeconometric models during the second half of the 1930s, when he was working for the League of Nations. It was mainly for this work that Tinbergen was awarded the first \u201cNobel Prize\u201d in economics shared with Ragnar Frisch.<\/span><\/p>\n

In this paper, we argue that the success of this line of research has occulted the incredibly versatile works of Tinbergen, who, in articles published in German, Dutch, and even French in the early 1930s, explored remarkable ideas that were based on a specific treatment of unstable processes and multiple equilibria. At that time, Tinbergen was part of a small group of mathematical economists who, in the aftermath of the 1929 crash and during the early years of the depression, united around a common goal of building \u201cmacrodynamic\u201d models with the aim to reflect on what keep the economy moving and the policies likely to impact it. The discussions took place within the Econometric Society (created in 1931) and its journal <\/span>Econometrica <\/i>(<\/span>created in 1933).<\/span><\/p>\n

Tinbergen saw his first models, with reference to Irving Fisher\u2019s 1933 classic <\/span>Econometrica <\/i>paper<\/span> on the debt-deflation, as a way to interpret the potential of a crisis to trigger the collapse of the economy. At the end of the day, it turns out that Tinbergen managed to reach new insights which, strikingly, remain almost totally ignored in most history of macroeconomics and econometrics.<\/span><\/p>\n

The model presented in the following application can be found in several papers written by Tinbergen in the early 1930s. The core of the model is based on the following nonlinear difference equation:<\/p>\n

  <\/span>   <\/span>\"\begin{equation*} \label{eq:central_eq}<\/p>\n

Where the left hand side represents the supply, while the right hand side the total demand divided by prices to obtained the “purchasing power”. The first occurrence of this equation was in 1934, in an article written in German and published in the Zeitschrift f\u00fcr National\u00f6konomie<\/em>. But at the time, Tinbergen had not fully worked out the behavior of the full nonlinear model, although he had perceived with numerical examples and some excitement that something strange happened when shocks were too large.<\/p>\n

Two years later, he published in another paper written in German a much more detailed exploration of this equation, where \"k(t)\" is a simple function of \"p(t)\" with a lag corresponding to half the production lag \"\theta\". Tinbergen showed that there existed in general two position of equilibrium, and that it was always the bottom one that was unstable, so that if the economy was fluctuating around the high equilibrium, an external shock of too great a magnitude could cause it to go over the unstable equilibrium, and “into the abyss” as he wrote somewhat poetically. This behavior is easily reproduced in the following application, where an increase in \"p(1)\", the initial condition on prices in period 1, will eventually lead the economy to exceed the unstable equilibrium and plunge toward a zero level (supposed here to be a limit to the evolution of prices)<\/p>\n