## 经济代写|宏观经济学代写Macroeconomics代考|EC4505

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## 经济代写|宏观经济学代写Macroeconomics代考|Le Corbeiller and the Early Reception of Relaxation

When preparing together the program of the first meeting of the Econometric Society scheduled to take place in Lausanne, Frisch and Divisia both agreed to ask Le Corbeiller to give a conference on relaxation oscillations. At that time, Le Corbeiller-who was a close friend of Divisi ${ }^{25}$ —-was already something of an expert of relaxation oscillations, ${ }^{26}$ and he agreed to meet the members of the nascent Econometric Society in Lausanne in 1931.

In his communication, Le Corbeiller (1933) introduced the economists to the engineer’s approach of the theory of oscillations, and considered the conditions for which different devices-a clock showing harmonic movements or a pile drivermay show self-sustained oscillations. Like Van der Pol and Hamburger, he started by considering the working of a simple pendulum. When frictions are negligible, Le Corbeiller noted that the movements will be described by a regular sinusoidal curve, and thus concluded that for a device like a pendulum to keep swinging, no energy must be dissipated. This required, in the presence of friction, that a source of energy should be added which may exactly compensate for the energy dissipated. In the case of a clock, this was made possible by the addition of weights which, as long as they had not gone down, maintained unchanged the swinging of the pendulum. Thus, contrary to Fisher, Le Corbeiller seriously considered the possibility that the escapement mechanism could be used as a useful model for economic fluctuations.
In a system of relaxation equations, Le Corbeiller argued that the problem was completely different. In that case, the system will keep oscillating precisely because the source of energy acting on it is periodically and suddenly dissipated (the “discharge”). This happened every time a device was characterized by a period of charge and a period of discharge of energy as it is the case for the Wimshurst machine ${ }^{27}$ or for a pile driver. In the latter case, a heavy mass is pulled up along a pile by a steam engine acting on a piston. Once it has reached a given height, a tap is opened which lets the steam escape and the mass falls on the head of the pile: “If therefore we consider on the one hand a clock, on the other hand a pile driver, we see that in both cases there is a source of energy which maintains a periodic movement by providing the energy dissipated during each period” (Le Corbeiller, 1933: 330).

In the second part of his presentation, Le Corbeiller focused on forced oscillations and considered the interaction between two systems, a small system $A$ and a large system $B$ assumed to oscillate sinusoidally. For instance, if an engine $(B)$ is placed on a table $(A)$, and the engine is doing $n$ turns per minute, then one may expect the table to vibrate with the frequency $n$. Now, if, due to its mass and the elasticity of its legs, the natural frequency of the table is equal to $n$, in accordance with the “well-known resonance phenomena,” (1933: 330) the table will oscillate with a great amplitude and may even break down after some time. If system $B$ is not only the engine but the whole electric plant producing energy, the oscillations will really be forced as the table does not react back on the plant. If, however, both systems are of the same size, “it is highly probable that the final movement of the whole system does not admit a unique period. We are therefore in a much more complicated case than that of forced oscillations” (1933: 331). But Le Corbeiller thought that the economy was precisely in this case, where different systems interacted and reacted together, which would be a problem “of extreme difficulty” (1933: 331).

## 经济代写|宏观经济学代写Macroeconomics代考|Coming (Almost) Full Circle

By a twist of fate, Le Corbeiller, who had not shown at first an overly enthusiastic interest for the use in economics of relaxation oscillations, was one of the crucial links between these early works and the revival of nonlinear limit cycle models in the 1950s. Indeed, while Le Corbeiller refrained from advocating for a specific approach in the early 1930s, he was the one who eventually allowed the path pioneered by Hamburger to be finally taken by Goodwin (1951). Goodwin, who had taught physics during the war, returned to economics at the end of the conflict and met Le Corbeiller, then a physicist at Harvard. Following his advice, as he made it clear in the opening pages of his article, he discovered the literature on relaxation oscillations: “My debt to Professor Le Corbeiller is very great, not only for the original stimulation to search for the essential nonlinearities, but also for his patient insistence, in the face of the many difficulties which turned up, that this type of analysis must somehow be worked out” (Goodwin, 1951: 2).

Goodwin placed himself in the context of the work of Hicks (1950) on the trade cycle, which hinged upon some nonlinearities, but also the work of Tinbergen, who had published during the war a paper working out the effects of high and low limits on an accelerator-multiplier model (Tinbergen, 1943) ${ }^{33}$ In this little known paper (translated in Tinbergen 1959), Tinbergen explored the implications of multiple equilibria in a Keynesian cross diagram and showed how, depending on the slope of the expenditure curve, different types of equilibrium could be defined. In fact, Tinbergen’s work was framed in the context of what he called “saturation,” an idea which he linked to Hamburger’s use of relaxation oscillation as a device limiting the amplitude of oscillations (Tinbergen, 1938: 34 , and Chaps. 5 and 6 below).

Two decades after Hamburger’s first paper, macroeconomics had now established itself as an autonomous field of research structured around various models. Nevertheless, the issues remained similar. Goodwin began his paper by reminding the reader of the “dilemma” between explosive or damped cycles shown by a linear model, rejecting the middle case of undamped fluctuations as a “classroom” solution without any economic content (Goodwin, 1951: 1). However, he pointed out two potential ways out of this dilemma, using external forces: the “synchronized systems” such as a clock and its escapement mechanism (a clear reference to Le Corbeiller’s own distinction), or the random impulses approach developed by Frisch after the debate on relaxation oscillations (1951: 2). Although Frisch’s approach had eclipsed the other way out, Goodwin put them back on the same level, and proposed to follow relaxation oscillations as a way out of the dilemma.

# 宏观经济学代考

## 经济代写|宏观经济学代写Macroeconomics代考|Coming (Almost) Full Circle

Goodwin 将自己置身于 Hicks (1950) 关于贸易周期的工作的背景下，这取决于一些非线性，还有 Tinbergen 的工作，他在战争期间发表了一篇论文，研究了高限和低限对贸易周期的影响加速器倍增器模型（Tinbergen，1943 年）33在这篇鲜为人知的论文（Tinbergen 1959 年译）中，Tinbergen 在凯恩斯交叉图中探讨了多重均衡的含义，并展示了如何根据支出曲线的斜率定义不同类型的均衡。事实上，Tinbergen 的工作是在他所谓的“饱和”的背景下进行的，他将这一想法与 Hamburger 使用张弛振荡作为限制振荡幅度的装置联系起来（Tinbergen，1938：34，以及第 5 章和第 6 章）以下）。

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