Logistic Growth Model
Given the strong response to my last tutorial on doubling times, I'm happy to offer another tutorial on logistic growth models. The mathematics gets more involved, but I have attached an interesting worksheet to illustrate how these models work. It's not really my intention to propose a realistic model of Tesla's long-term growth, but to illustrate the mathematical modeling choices one might consider toward such an end. Moreover worksheet models can be a wonderful tool to gain insight and to understand the implictions of simple assumptions.
To gain an appreciation for the logistic growth model, we need to review the exponential or geometric growth model and recognize how unrealistic it can be. The basic idea of the exponential model is that growth is always proportional to current size and that rate of growth does not change. So let X(t) represent a quantity that grows exponentially over time. Let dX(t) represent the change in X over a short period of time, dX(t) = X(t+1) - X(t). So we can represent growth as
dX(t) = b*X(t) or equivalently dX/X = b
We note a discrete solution to this is
X(t) = X(0) * (1+b)[SUP]t
[/SUP]
And the continuous solution is
X(t) = X(0) * exp( b*t )
Such a model may be suitable for say the total number of cars sold globally each year. The global growth rate is around, say, 3%. It can vary a bit from year to year and certainly from one reagion to another, but globally over longer stretches of time its pretty reasonable. One reason why it can be reliable is that it is infact a low growth rate and is in line with real growth in the global economy. On the other hand, China has seen double divit growth in its economy and is one of the hottest auto markets. Will this be sustainable for several decades? Or is it possible the China may reach some lev of saturation and the auto market begin to slow down? So the most fundamental problem with the exponential growth model is that the growth rate is assumed to persist indefinitely. This may be reasonable for the global auto market, but it is not realistic for a market or company in unsustainably high growth.
So we turn to the logistic growth model. Here we suppose that X has a maximal growth rate X is really close to zero, but that this rate diminishes as gets closer to its maximal potential. A standard model has X take values from 0 to 1, where 1 represents it full long run magnitude. Specifically,
dX = b*X*(1 - X) or dX/X = b*(1 - X)
So we see that when X is close to 0 this is nearly the exponential model. In the initial phase of growth we see nearly exponential growth. How ever as growth accumulates 1-X becomes an increasingly small number and the growth rate declines. When X = 1/2, then the growth rate is just half of the maximal rate. This is the inflection point. Past this point the rate of growth slows down so quick that the curve begins to level out. This curve is also called an S curve. It starts out flat, curves up, curves down, then levels out.
This standard model has a nice continuous solution, the so-called logistic function:
X(t) = exp( a + b*t )/(1 + exp( a + b*t ))
This is a two parameter model. The parameter b is the maximal growth rate, and a is used to set your initial value, specifically
X(0) = exp(a)/(1+exp(a)) so a = log( X(0)/(1 - X(0) )
So if you know where your starting and the rate of growth in the expontial growth phase, you can readily use this model. Suppose your long run view is that Tesla will max out at 10million cars. It is starting at 35,000 in 2014. So our initial value is 0.0035 assuming t is the number of years past 2014. Also lets suppose the maximum growth rate is 50% per year. So,
a = log( 0.0035/ 0.9965 ) = -5.65
b = 0.50
To determine where this puts us in 2020, t=6, compute
a+bt = -5.65 + 0.5*6 = -2.65
X(6) = exp(-2.65)/(1+exp(-2.65)) = 0.0660
Thus, by 2020 this model put Tesla at 660,000 cars, 6.6% of the way to 10 million. You can easily put this formula into a spreadsheet and play around with parameters until you get to something that comes close to your expectation. Personally, I'm not buying 660,000 cars, so I'll want to play with this more.
In the attacted worksheet, I want to show how to that this modeling framework further. In particular, I am not comfortable with the assumption that the is some maximum number of cars for Tesla to grow into. Specifically, the auto market will continue to grow at a few percent each year for the forseeable future. Tesla could ultimately capture 10% market share and keep growing with the industry. So how can we integrate both a long run industry growth rate with a logistic transition into 10% market share. I propose the following logistic grow model.
dX / X = (b - g)* ( 1 - M(t)/M ) + g
Here b is my maximum growth rate for Tesla, g is the long run growth rate of the global auto maket, M is the long run market share for Tesla and M(t) is the market share at time t. So the first term of the growth model is logistic growth upto hitting the long run market share, and the second term is the long run growth rate.
In my worksheet, I assume that in 2014 the global market is 70 million cars (anyone have a better number?) and g = 3%. I assume Tesla will deliver 35k this year and will ultimatel gain 10% market share. The tricky part is getting the right value for b. I find that b = 57% gets Tesla to 505k cars in 2020 and just over 10M by 2030. Moreover, the next couple of years look about right. So if nothing else, this model basically fits my expectations for the next 20 years.
I invite you to play with the model and find scenarios you find interesting. Let me know what you find, or if you have any comments, questions or suggestions. I hope you find this modelling framework and tool useful.
