An Early Beginnning
After finishing my Maths degree and teaching for a while I decided to join the REAL World and got a job as an assembler programmer working deep in the bowels of an IBM mainframe. (Some might say I that I hadn't entered the real world at all.) Anyway, I decided to improve my education and undertook the study of Operations Research (aka Management Science). This is the application of mathematical techniques to business problems. The next year I thought that it would also help if I studied Economics so I enrolled in Economics 101 and then the fun started.
My text was by Paul Samuelson and I think everyone since 1950 has pretty much used the same book for their start in economics. (They are still using it today!) The information seemed to be about supply, demand and prices. I was a little concerned at the beginning because the curves that Samuelson used just seemed to appear on the page without adequate information. No data was collected or examined. No theory was developed about the shape of these curves or what equations might fit these curves. They were just hypothesised from common sense.
That is, it seemed reasonable to say that as supply increased the price would go down. If price went down then demand would go up. If you observe fresh vegetables being auctioned each morning this relationship is quite clear.
Unfortunately, at this juncture the economics lectures shifted to examining curves of supply and demand from the point of view of applying the mathematical technique of calculus to them. At the time I thought this was a bit of a leap since we had no idea of the mathematical form of these curves. My patience was further stretched when most of the lectures seemed to devolve into the teaching of elementary calculus. Pressure back at work then forced me to drop Econ 101 and great was my relief to finally retreat from academia.
All these years later when I started to think about economics again I began to realise that something was wrong with Paul Samuelson's approach. That it was not scientific was clear. We did not have a clear statement of the facts nor any idea of just what we were dealing with. We just had sketches of curves and nothing more. Was I missing something? Would all be revealed when I did econometrics (economic tricks as some would say)?
Or are we adrift in an artificial economic construct which is not grounded in the truth?
Sep 30, 2010, 13:19
Elementary Calculus is about working out the slope of graphs. Much like an engineer might work out slope up a road. Horizontal straight lines have zero slope and vertical lines have infinite slope. Slope is how much the line goes up for a given distance across.
However, when we look at a curve the problem is a bit more difficult. What we do in calculus is examine the slope of a straight line which just touches the curve at one point. Then we can say that is the slope of the curve at that point. If we go along the curve a bit it might get steeper and the slope increase. Calculus allows us to measure this change.
Notice that it is slope at a point and a mathematical point is a very small point indeed. Since it has zero dimensions - no width, no height and no depth - it is vanishingly small. This makes calculus a bit tricky.
Calculus only works on curves that are smooth and continuous. The mathematical meaning of these terms is much the same as the everyday common sense meaning. Continuous means that there must be no breaks in the curve and no points where the curve suddenly takes a different turn. Smooth means that no matter how much we magnify up the curve it will still appear to be exactly the same shape as before. Even if we take a very very small section and magnify it up a million million times the result will just look like the curve we started with.
Now when I look at graphs showing how supply and demand and price are related I find that they are neither smooth nor continuous.
Looking at what happens at the Stock Exchange we see that sometimes the picture looks like a saw tooth and sometimes it looks a bit like steps. If we think more carefully about what is happening we see that the price of a share is constant over a period of time and then suddenly (instantaneously(?)) some shares are sold and the price changes from, say, $1.47 to $1.49. Later it changes to $1.53 and then $1.50. Little steps up and down.
If we develop a mathematical function for this effect then we know it won't be smooth and continuous and it won't be much use using calculus as an analytical tool.
Sep 29, 2010, 13:59
Physics which is the scientific study of the physical REAL World also uses calculus to explain how different things work. However when we use it we must still be careful of the scale at which we are working and ask ourselves if the curves we derive are smooth and continuous.
If we are examining water and measuring how it cools we would notice that as boiling water cools down to room temperature it loses its heat at a slower and slower rate. If we graph this curve would would we get a smooth and continuous line? Would calculus be an appropriate tool for analysis?
If we take one litre of water we notice that it contains a very large number of molecules. To get a sense of this scale we know that there are roughly 100 billion stars in a galaxy and about 100 billion galaxies in the observable universe. That is 10,000 billion billion stars in the universe. Now the number of molecules in a litre of water is 33 million billion billion.
This means that there are 3300 times as many molecules in our jar of water as there are stars in the known universe - and all of those molecules are identical.
So when we are examining our curve derived from this water experiment we can expect that it will be smooth and continuous. So calculus works well and produces realistic results.
Physicists continued to use calculus as they examined and developed theories about smaller and smaller things until they got down to this molecular scale and they found that calculus was no longer an appropriate tool to use.
They had to invent a new sort of mathematics to suit these new and very different problems considering how things worked inside an atom. This new tool was called Quantum Mechanics.
To get some idea of the scale of things and the reality of atoms have a look at:
If you made a scale model of an atom with a nucleus the size of a pea, the electrons would zing around in a space larger than a major sports stadium! An atom is mostly empty space. This is not very well understood. So when we imagine atoms colliding and bouncing around like billiard balls we should realise that our model is very crude. They are not like that at all.
It is no wonder that our use of calculus to study matter on an atomic scale fails to work. Our atom is mostly space. One pea and a few specks of electrons some where in a stadium!
From this we learn that calculus is a valuable tool for use in some physical circumstances where analysis shows that a cause and the resulting effect give rise to smooth and continuous graphs - and where we can make valid assumptions about the shape of the curve and deduce a particular mathematical relationship between the cause and the effect.
This works in our REAL World where we are dealing with homogeneous objects consisting of a large number of identical particles. Unfortunately, in economics we are dealing with discrete actions of individual humans and these actions are different from each other in very many ways.
Calculus is just not an appropriate tool to use.
Sep 29, 2010, 08:23
Supply and Demand Mathematics
Fortunately for us, Management Scientists have spent a lot time and effort working on the demand and supply of goods. The branch of Management Science that describes this function is Inventory Control. The aim of Inventory Control is to optimise the supply of goods with the demand for those goods so as to minimise the overall cost.
For a full description of this body of knowledge you might like to look at (http://books.google.co.nz/books?id=yWk8SKak7bYC&printsec=frontcover&dq=inventory+control&source=bl&ots=QXIhnurzmI&sig=ybOeAiZsMslNdsbL0gUoST7op54&hl=en&ei=4sefTOqwBYeosAOz7ezVAQ&sa=X&oi=book_result&ct=result&resnum=2&ved=0CC4Q6AEwAQ#v=onepage&q&f=false)
A quick run through the chapter headings will reassure you that indeed we do not use any calculus techniques.
The two sides of Inventory Control Techniques involve measuring the demand for each item from previous sales records and then using various forecasting techniques for predicting how sales will go in the future.
We then analyse the supply chain and determine how much to order and when to place the order so that it will arrive before we run out of stock.
We then incorporate all the costs involved with this process and the cost of storing the inventory and work out the how and when for ordering goods depending on what level of service we wish to provide to our customer. We can't guarantee 100% availability as this requires an infinite stock but we aim for something like providing the item 95% of the time.
As you can imagine items come in a very wide range (like washers and Grand Pianos) and each has to be treated differently. Fortunately this problem is easily solved by computers which can handle this vast array of data.
So theoretically we can understand, in detail, the supply and demand of every good in any country. All we need is to provide inventory control software on a computer to every business and connect them up over the web. This of course begs the question, "Is it useful to add up all the supply and demand for all goods sold for the whole country?" Is some sort of average rate useful as a statistic? This warrants further thought!
It is also interesting to note at this point that all this mathematical calculation revolves around the probability of sales continuing at a certain rate with due allowance for expected variations and the probability of obtaining supplies in the future as we did in the past.
The future is uncertain but we can work usefully with this uncertainty because of Probability Theory.
Calculus is not a suitable mathematical tool.
Sep 28, 2010, 11:18
Paul Samuelson's Model
If we Google Paul Samuelson we discover that he was very interested in thermodynamics and the application of its mathematics to the problems of equilibrium in economic systems.
And in particular in the paper written by American mathematical-engineer Willard Gibbs - Equilibrium of Heterogeneous Substances.
He went on to win a Nobel Prize for his efforts in this direction.
When we think about the fundamentals of economics - being the sum of a number of human actions, maybe thousands or millions, over long periods - it is hard to imagine any similarity with the behaviour of systems consisting of many billion billions of molecules, all of which are identical and which all behave in a similar fashion in relation to heating and cooling.
A further search of the web brings up this reference:
When we read this we see that not only are a number of people skeptical about Samuelson's use of the Thermodynamic model, Samuelson himself has doubts and speaks about Social Scientists trying to imitate Physicists with their mathematical precision.
If we look back today we can see that trying to describe economics by using the differential calculus equations of thermodynamics is useless sophistry.
Thousands of students have been misinformed over the years by Samuelson's efforts and the unthinking acceptance by non-mathematicians of any esoteric equation that looks good!
So, what does this mean for us?
At this juncture I think we need to be very skeptical about the use of mathematics in the study of economics. Before I would accept any mathematical result I would not only wish to check the mathematical reasoning I would want to examine very carefully, the basic economic premises upon which the theory we are examining was constructed.
Fortunately we can understand economics without learning anything about mathematics. Certainly the Austrian economists developed a complete theory covering all human action without one mathematical equation. And this body of knowledge very clearly explains why the world is in such economic turmoil at the moment (2010) and provides a solution to our economic problems.
Sep 27, 2010, 12:44