Euler's Method and Business Dynamics

Leonhard Paul Euler
Leonhard Paul Euler was a Suisse born mathematician, physicist and astronomer, who lived from 1707 – 1783. He is considered to be the preeminent mathematician of the 18th century. Euler is renowned for most of today’s mathematical notation and terminology, especially for mathematical analysis (mathematical function), but also for his works concerning mechanics, optics and astronomy. Born in Basel, Switzerland, he met Johann Bernoulli as an adolescent. At that time, Bernoulli was considered Europe’s foremost mathematician, and he influenced Euler significantly. Euler worked in St. Petersburg, Russia, and in Berlin, Germany, for the most part of his life, as a physicist and mathematician. He wrote countless articles and his collected works fill approximately 80 quarto volumes. He basically worked in all areas of mathematics, geometry, algebra, trigonometry, calculus, but also in continuum physics, lunar theory, and various other areas of physics. Euler died on September 18, 1783 in St. Petersburg, Russia.

Euler’s Method
As per definition, the Euler Method is a so-called numerical procedure, foreseen to solve ordinary differential equations with one given, known and determined value A0. Despite the point A0, which as said is known, all other points are unknown. We assume an unknown curve that starts at a definite point (A0). The mathematical problem is to define the unknown curve. In order to define this curve, we use a differential equation as a formula. The slope of the tangent line to the curve can be computed at basically any point on the curve, based on the one determined position A0. In other words, the basis is A0 as the one and only clear defined point on an – so far – unknown curve. Through a differential equation we will be able to determine as many other points on the curve as we need to finally find (or to determine) the polygonal curve. However, this curve is not the original curve we are looking for, but – if we want to determine it that way – just the model, which is able to come as close as to the “real”, but yet unknown curve as possible. After having determined several other points on the polygonal curve (see A0, A1, A2, A3, A4 and so on), the model curve should not be too far off the real curve. However, the step sizes (between the various A-points) should be rather small, which would guarantee that the polygonal curve and the real one would not be too far off.

Euler’s Error
Compared to other higher-order techniques, such as linear multi-step and the Runge-Kutta method, Euler’s Method is rather understandable but less accurate. Assuming that f(t) and y(t) are known exactly at a given time t(0), then the approximate solution as per the Euler Method has to be time t(0) plus the step size – or h. As we know from the above explanation, the polygonal curve can only be seen as a model of the real curve. However, we want to come as close as possible. Euler’s Method assumes a certain inaccuracy on the polygonal curve. By using h for each and every step on the polygonal curve, we know from the beginning that we have to deal with a certain inaccuracy, expressed in h. This inaccuracy will be seen in each and every step, and, it is proportional. Thus, the dominant and already assumed error per step is proportional to h2. This means that we have to expect the total error at the end of the determined and fixed time proportional to the error per step per number of steps, hence proportional to h. In simple ordinary wording: If the step size h is fairly off at the very beginning of the polygonal curve, it will be even more off at the very end of the polygonal curve. Compared to the real curve, the “model curve” is fairly inaccurate.

Euler’s Method and Business Dynamics
Leonhard Euler’s method can easily be applied to business dynamics – by taking into consideration that his method is not that accurate. One of the contexts would be that of exponential growth in terms of Modes of Behavior. As we know, exponential growth arises from positive, self-reinforcing feedback. The larger the quantity, the greater is the so-called net increase. If we use the growth of population for instance, we can assume that if the population grows, the net birth rate will grow as well. The net birth rate will add on to the population and will finally lead to still more births – hence an “ever accelerating spiral”. The Euler Method can show this development very clearly, by determining the size of the population (A0) on a curve. The determination of the growth rate remains a numerical procedure, the development of the growth rate can be shown as an exponential growth rate, like in the figure above (see also Thomas R. Malthus: “Simple exponential growth will result in unlimited population density”). However, the Euler Method is usually used as a rather simple example of numerical integrations, such as the one described above (population growth as an ever-accelerated spiral). Since we know that a population growth rate is limited to a certain extent, both Malthus’ and Euler’s Method can only be seen as models that narrow down the complexity of a market, a population, or a business development that cannot be foreseen and easily understood and explained.

Sources:
Lehr- und Uebungsbuch Mathematik (Harry Deutsch)
Business Dynamics (John D. Sterman)
Human Population under Limited Growth (Thomas R. Malthus)