THE EXPERT WITNESS: Allegorical Economics - Using the Fibonacci series in storytelling (part three)


By John F. Sase, Ph.D.
Gerard J. Senick,
general editor
Julie Gale Sase,

“Advertising is to a genuine article what manure is to land--it largely increases the product.”
—P.T. Barnum, 19th Century American Entrepreneur

Last month, we continued to explore Allegorical Economics by delving into the source of all economic understanding—ourselves as human storytellers. Economists and Attorneys tell stories in both the classroom and the courtroom. Attorneys need to condense client backgrounds and their respective cases in order to evoke understanding and empathy from jurors. Therefore, we began to address the use of numbers in the stories that we tell in our professional venues. This month, we continue this topic by focusing on how to simplify more complex mathematical relationships and graphs for a wider audience by using Economic logic and some simple algebraic tools such as the Fibonacci Series and related graphs. We will do this by using three examples from current media, two books and a film.

The Challenge
In my profession as an Economist, I (Dr. Sase) am called upon to explain advanced Economic measurements by using graphs like the Bid-Rent Profile, which I have presented below. For this research, I rely upon Negative-Exponential, Quadratic, and Cubic functions, which sometimes need to be doubly integrated for three-dimensional spaces such as cities. Simply by uttering the preceding sentence, I would expect to lose the attention of half of my audience or readership. However, I have discovered a way to illustrate higher forms of mathematical measurement by using simple arithmetic—a little trick that I learned from the ancient Greek polymaths Pythagoras and Plato.

The Fibonacci Series
Many scientists regard the Italian mathematician Fibonacci (aka Leonardo of Pisa or son of Bonacci) as the greatest European in his field during the Middle Ages. Born in Pisa, Italy, during the late 12th Century, Fibonacci received a North African education under the Moors, largely due to the professional focus of his father, a customs officer and trader in the Mediterranean Region. As a young adult, Fibonacci travelled extensively around the Mediterranean coast. We believe that he met with many Arabic merchants and learned of their systems of doing arithmetic during his business trips. From these experiences, Fibonacci began to understand the many advantages provided by the more ancient system of mathematics that is evidenced by the structure of the Giza complex in Egypt and by the many temples in India and beyond.

The arithmetic sequence that we commonly call the Fibonacci Series begins with the numbers “zero” and “one” (0, 1). In many languages throughout the ages, philosophers have explained that these numbers form the foundation of the universe and that the root of all of creation is binary, as represented in the values of zero and one. We interpret this pair of symbols through the concepts of no and yes, off and on, nothing and all, and many other binary relationships. However, this pair of numbers is the seed for a series from which we can derive a universe of mathematical relationships. Starting with this binary pair makes the entire process of expansion transparent.

The sequence emerges through the sum of the two largest numbers in the sequence. We start by adding together our binary pair of numbers to obtain a third number that is also a “one”: 0 + 1 = 1. Now we have the pair of two “ones” that is necessary for further expansion of the series. We continue to sum the pair of largest numbers, which sit furthest to the right. At this point, we have a set of three numbers (0, 1, 1). By taking the sum of the second and third values, we obtain the fourth number: 1 + 1 = 2 and the expanded set of four values (0, 1, 1, 2). Again, we sum the two largest numbers located furthest to the right to obtain the fifth number 1 + 2 = 3 and the enlarged set (0, 1, 1, 2, 3). From here, the process is ongoing, such that 2 + 3 = 5 and the set (0, 1, 1, 2, 3, 5) is produced. Next, we have 3 + 5 = 8 for the set (0, 1, 1, 2, 3, 5, 8) and then 5 + 8 = 13 in the set (0, 1, 1, 2, 3, 5, 8, 13). Finally, for our immediate purposes, we take the sum of the two values furthest to the right, such that 8 + 13 = 21, producing the set (0, 1, 1, 2, 3, 5, 8, 13, 21). Hey! We are done.

The preceding graph includes the full set through 21 (0, 1, 1, 2, 3, 5, 8, 13, 21). Alternately, the function represented by the Fibonacci Series herein is a Cubic Function in which y = 0 + 1.29x - .45 x2 + .08x3 with an R2 greater than 99%. In certain cases, we may delete the first two values (0, 1), move the remaining values leftward so that the y-intercept may equal 1. The function represented by this altered series would be a Quadratic Function for which y = 1 - .57 x + .63x2 with R2 = 99%. This Fibonacci Series can be expanded easily, depending on need. Expansion requires adding 13 and 21 from the set (0, 1, 1, 2, 3, 5, 8, 13, 21) in order to get the next number of 34. Then, 21 and 34 can be added to render the next value and so on toward infinity.

Working with the Fibonacci Series
By arranging these nine values on a square, we produce a curved line that we use to illustrate practical events. We can rotate and flip our line to get a total sixteen basic lines that can be truncated or expanded as needed. These variations allow us to model an airplane taking off and then accelerating steeply upward, a toboggan sliding down a run, and a car driving up a steep hill that gradually levels off near the top. In business, we also can approximate the change in Average Fixed Cost: as the quantity produced increases, the Average Fixed Cost decreases. Likewise, we can model Variable Costs, which rise at an increasing rate as production increases.

The following composite diagram contains the sixteen rotated and flipped variations of our Fibonacci Series discussed. In addition, the diagram includes four straight lines that may be used to represent Supply and Demand, to discuss the differences within the spectrum of elasticity, and to address other topics in the field of Economics. The four straight lines reflect the ratios of the two binary values that form the base of the Fibonacci Series. The two diagonals are generated by the ratio of 1:1 and the horizontal and vertical lines are generated by the ratios of 0:1 and 1:0, respectively. With suitable expansion or contraction, this set of twenty lines within the square can be used to illustrate all of the relationships found in Economic Principles.

In the following example, we model book sales as they increase at an accelerating rate over a short period of time as we bring the Fibonacci Series into the world of contemporary publishing. The truncated Series used in the graph above provides a simple illustration of how the sales of hot best-sellers take off quickly when controversy-marketing techniques are applied. The concept and the measurement tool are simple. The question of why should be obvious (cha-ching). The rapid increase reflects promotion and sales strategies that effectively use “controversy” as a marketing tool.

Two well-known examples of this type of a marketing campaign have relied upon controversy generated on Twitter and Facebook in order to achieve a rapid spike in sales. These books are “Fifty Shades of Grey” by English author E. L. James (Vintage Books, 2012) and the recent “Fire and Fury: Inside the Trump White House” by Michael Wolff (Henry Holt and Company, 2018).

However, their stories are ones for another time. Our takeaway for attorneys this month is that, by using Economics and mathematical tools such as the Fibonacci Series, one can dig more easily through the muck within their cases in order to find some simple truth and possible answers to questions that arise during disciplined investigation.
Dr. John F. Sase teaches Economics at Wayne State University and has practiced Forensic and Investigative Economics for twenty years. He earned a combined M.A. in Economics and an MBA at the University of Detroit, followed by a Ph.D. in Economics from Wayne State University. He is a graduate of the University of Detroit Jesuit High School (

Gerard J. Senick is a freelance writer, editor, and musician. He earned his degree in English at the University of Detroit and was a supervisory editor at Gale Research Company (now Cengage) for over twenty years. Currently, he edits books for publication (

Julie G. Sase is a copyeditor, parent coach, and empath. She earned her degree in English at Marygrove College and her graduate certificate in Parent Coaching from Seattle Pacific University. Ms. Sase coaches clients, writes articles, and copyedits (