Dynamics of GDP

Can the laws of physics be used to gain insight into economics?

That is a difficult question but could be the most important one in the history of economics. 

How could Newton's laws be applied to economics when no actual forces or masses of physical bodies is involved? The answer can best be understood when you realize that forces also do not exist. No one has ever found one lying around on the ground. Masses are also rather mystical quantities that on the atomic level hardly exist at all. An atom is 99.999... percent vacuum space. Also, and this is really going to stick in your craw, numbers don't really exist either. No one ever found one of them lying around on the ground. So mathematics and physics are both abstractions, ways of thinking about the natural world that lends it to analysis. 

So if they are founded on such fuzzy thinking, how did Newton come up with his so-called laws? After all, they do work really well. 

The first thing to remember about Newton (and this is not taught in physics classes and was actually covered up for centuries) is that he was first and foremost a practicing alchemist. He wrote many more words on alchemy than on physics. Also, he was extraordinarily religious. Ever read his essay on the end of times? He thought 2060 was a reasonable year. He also took a lot of heat for his theory of gravity that depended on action at a distance, which everyone knew was occult thinking. Yes, Newton doesn't have a theory of gravity. It just writes an equation for it. He didn't address how it works. 

If they are so steeped in mysticism, why do Newton's laws work so well?

It's because when we apply the laws, we do so to suitably well-defined masses. But the laws can also be adapted for application to fluids, gases and plasmas. In doing so, we get answers that enable us to understand how things work. We can do the same for economics. 

Other aspects of Newton's laws of motion involve position, velocity and acceleration. Economists use these terms all the time when describing an economy. Position is just the state of any economic parameter, e.g., GDP. Velocity is how rapidly GDP is changing. And acceleration is how fast the velocity of GDP is changing. 

Another thing to realize is that we haven't as yet defined the domain over which this economic system is to be modeled. Is it a family? City? State? Nation? This is no different than defining the dynamics of a rigid body, whether it is a rock, a rocket or a satellite. For our purposes, we will choose a nation, or rather a national economy, and even more specificaly, we will wish to predict the dynamics of a national Gross Domestic Product, GDP. 

Using these concepts, we can then adapt Newton's laws of motion to this segment of economics. We just have to define our systems (our abstractions) so that the application makes sense. In doing so, we will be able to define the forces that are at work within this economic system's parameters and how changing conditions will affect them. 

How might we go about doing that?

An economic system is not something that can be isolated — as can a missile, for instance — for analysis. But then a missile also has all sorts of forces acting on it: wind, thrust, aerodynamics, gravity, etc. However, we can define all economic parameters, idealize them, and analyze the idealized situation to see how it responds to different economic forces. The essential thing to realize about this method is that it is possible to idealize an economy and analyze that idealized model. That is what engineers do when they simulate the motion of a missile launch. They construct an abstract, idealized model and analyze that. They can't apply Newton's laws to the missile; therefore, they have to apply them to the analysis to the model. In doing so, we are in familiar territory. The big problem with an economic is how to model it so that the laws of physics can be applied to it. 

Okay then. What is the first step?

The first step in any model, and this is particularly true of a new discipline, is to start simple. Bare bones. Don't start with a full up economy with all its sophistication. We might want to start with the most obvious parameter that could represent what in the physical world we call mass. And GDP is the first thing that comes to mind. But GDP is "the monetary value of final goods and services—that are bought by the final user—produced in a country in a given period of time." In other words, GDP is a velocity: the rate of a given quantity over a period of time. It is dollars/year. The most basic parameter of an economy is its instantaneous value — the amount a country is worth, W. When we speak of a countries rate of increase of wealth, WD, we are then speaking of GDP. So GDP is the velocity of wealth. When we speak of the rate of increase of GDP we are actually talking about the acceleration of wealth, WDD. So the most basic quantity concerning an economy is wealth as its associated parameters GDP (velocity) and GDP growth rate (acceleration). 

We have now defined one parameter with three aspects:

1.  W = Wealth, W, money, dollars.
2. WD = Wealth rate of increase, Wealth velocity or dollars/second (GDP).
3. WDD = Wealth acceleration, dollars/second/second (GDP Growth Rate). 

With these parameters defined, we have a cursory dynamics of economics, and we can see that they are all terms with which any economist would be familiar.

Anyone familiar with calculus would also recognize these concepts and how they relate. They would realize that they could use the techniques of integration to get velocity from acceleration and get wealth, W, by integrating WD (GDP). And yet we haven't resorted to magic or alchemy. (Or have we?) But we also do not have a model of the economic system. We have only defined some parameters. 

So what does a model of an economic system even mean or look like?

Those with even a remote acquaintance with physics would realize that to predict motion you have to have a model of the system. And that means, taking our cue from Newton, we have to define the forces acting on the system. Newton said, in essence, that the acceleration of a body is proportional to the sum of forces acting on it. Or in the iconic mathematical equation:

a = F/m ; where / denotes division.

where a = acceleration, F = summation of forces, and m = mass. 

Rewriting this in standard format:

F = m * a ; where * denotes multiplication.

Or for an economic system:

F = p * WDD ; where p denotes something equivalent to mass, which would be the total size of the economy, in this case population. 

Here we go.

So what are the forces? Or, in terms that come directly from any economist: What are the market forces acting on the economy? That is where we get into modeling the economic system. Modeling, in its most true essence, is defining the nature of the forces that act within an economic system. What is most important is to realize that these forces work to either speed up or slow down an economy. Some forces occur naturally, others are imposed on the economy to force a result.

We know that interest rates affect an economy. Specifically, interest rates are used by a central bank (The Federal Reserve in the US) to either slow down an economy that is overheating or to speed up one that is stagnant or in a downturn. 

Now we have to find the forces within the economy that act on national wealth to increase or decrease it. 

The parameters to model would be the forces that affect GDP. 

Now that we have these quantities defined, we can write equations for them. 

When we write Newton's equation of motion for a spring-mass system with spring constant k and mass m, we get:

Fs = - k * x, where x = spring stretch.

The equation of motion for the mass is:

F = m * a = - k * x

All spring mass systems have some damping that retards motion. This damping force is frequently presented as proportional to the velocity. Fd = - c * v.

Combining these, we get:

m * a = - k * x - c * v

For an economic system, displacement (x in the spring-mass example) is represented by GDP (we'll use g) as it changes through time. Velocity (v or x') is the rate of change of GDP (g'). Acceleration, a, becomes the rate of change of g', (g''). Summing the economic forces and setting them equal to the GDP acceleration, we get:

m * g'' = - c * g' - (d-s) * g ; or

g'' + c/m * g' + (d-s)/m * g = 0

This is a second order differential equation in GDP (g) as a function of time.

Where

F = (Demand - Supply) * GDP
m is related to population, the buying public
a is the acceleration of GDP, the rate of change of growth

Spring-mass systems also have damping and that quantity is related to the velocity of the mass or rate of change of distance and is negative. The damping in an economic system is the interest rate. The interest rate is adjusted to control the rate of GDP growth (GDP velocity).

FD = - damping * GDPvelocity = Gv

The women and the kiosk, as the embodiment of economics, are an element of GDP. They represent the tension that lies at the heart of economics. This is the tension in the spring-mass system.