## Wednesday, June 1, 2011

### Cumulative growth, exponentials, CEOs, and social planners: a recipe for disaster

Most people don't understand cumulative growth, and that's a serious problem. For companies and for societies.

Suppose you have a metric for knowledge; perhaps it's a weighed sum of patents, diseases cured, technological barriers removed, and so on. (Details of the metric itself don't matter for this post.) And, as you look back at the performance of this metric over time, you notice that around half of all knowledge was created in the past ten years.

If you were the CEO of a company with this kind of knowledge development, you might be tempted to reduce investment in R&D and use the money saved there to enhance reported earnings. Going a little wider, if you were a social planner, you might argue that, given this spurt of new knowledge, perhaps the socially responsible thing to do would be to redirect research funds into social purposes.

This is a woefully myopic way to look at the value of knowledge. Knowledge builds upon itself (and across different fields of endeavor), so its growth can be described by models like, for example:

$m(t) = 1.0717735 \times m(t-1)$

where $m$ is the metric and $t$ is time in years.

This particular equation creates a doubling of $m$ every ten years. In other words, it creates the circumstances described above: at any point in time, half the knowledge will have been created in the previous ten years. Obviously there would be a lot of other factors in a better model, but let's keep things simple.

Don't let the linear appearance of that formula fool you into thinking there's a linear process going on here: this is a special case of an auto-regressive model and makes $m(t)$ an exponential function of $t$.

The interesting thing about exponentials is that they are hard for most people to process, leading to bad decisions. That's terrible if the people making the decisions are CEOs or social planners.

Let's say that we normalize our metric of technology so that the value today is  $m(2011) = 100$. Ten years ago, the metric had a value of  $m(2001)=50$ (because half the technology was created in the last ten years).

In ten years, the value will be $m(2021)= 200$. And looking back from 2021, our ten years older selves will again notice that half of all gains (200-100) took place in the previous ten years (2011-2021).

In twenty years, the number will be $m(2031)=400$, And looking back from 2031, our twenty years older selves will again notice that half of all gains (400-200) took place in the previous ten years (2021-2031). Similarly, $m(2041) = 800$, $m(2051)=1600$ and so on.

Suppose our well-meaning [CEO |social planner] in 2011 doesn't stop the investment on new research totally, but just halves it. Most [stockholders | voters] understand that this will slow down growth but their idea of by how much is very far off the reality.

The new evolution of technology, starting at 2011 becomes (with an abuse of notation, since the metric doesn't change, it's the process that changes, but we want a way to separate them from above):

$\hat m(t) = 1.03588675 \times \hat m(t-1)$

After ten years, with this new growth rate we'll have gone from  $\hat m(2011) = 100$  to  $\hat m(2021)= 142$. Ten years later we'll have  $\hat m(2031)=202$; also  $\hat m(2041)=288$  and  $\hat m(2051)=410$.

Here's the depressing arithmetic (since we normalized the metric at 100 for 2011, these numbers show growth lost as a percentage of 2011 knowledge):

$m(2021) - \hat m(2021) = 58$
$m(2031) - \hat m(2031) = 198$
$m(2041) - \hat m(2041) = 512$
$m(2051) - \hat m(2051) = 1190$

That's right: by 2031, the knowledge that doesn't get created is almost twice the total knowledge available in 2011; and things get acceleratingly worse with time.

Another way of understanding the 2031 number is to consider that if this policy had been implemented in 1991, we'd only have one-third of the knowledge we have in 2011. (Think iPods but no iPhones or iPads. Or no cure for two-thirds of currently curable diseases.)

When a [CEO | social planner] starts talking about focusing on [results | current social ills] what they are really talking about is trading-off an enormous fraction of future growth for relatively small immediate gains in [stock options value | electoral wins].

Some companies get a short spurt of good earnings quarters from a hatchet CEO coming in and reorganizing the company to exploit its extant knowledge without creating any new one; this is soon followed by an exodus of the people who created most of the knowledge and the sale of the company piece by piece.

Governments everywhere, and across the political spectrum, are choosing to slow down the future to ameliorate something today, oblivious to the fact that if they wait a little bit, growth alone will take care of the amelioration.

But betting on the future doesn't get votes. Ending the Space Shuttle program without an alternative space vehicle apparently does.

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NOTES:

1. This post is an elaboration of a post in my personal blog, inspired by a video by astrophysicist and science popularizer extraordinaire Neil DeGrasse Tyson.

2. The coefficients used in the equations are not estimates, they are just illustrations, as are the calculations based on those equations and the elucidating examples based on those calculations.

3. Yes, it's a overly simplistic model for actual policy analysis (corporate or social); the point is to illustrate the power of exponential growth and what happens due to seemingly small changes in its growth rate that result from policy choices.

4. There are other applications of the rationale in this post; I'm focused on myopia in research and development at corporate and – to a lesser extent – societal level, because that's part of what I research. Other applications are left as exercises for the reader.