# The Spectre of Math

## February 22, 2012

### Exponential growth (and CEO pay)

Filed under: Economics,Mathematics,Politics — jlebl @ 11:17 pm

So CEO salary has increased by approximately 9.7% adjusted for inflation every year between 1990-2005 [1] (that is approx 300% increase over that time, so 4 times what they had in 1990). Anyway, that has a doubling time of $\frac{\log 2}{\log 1.097} \approx 7.5$ years. Now median CEO (among the top 200) made approximately 10 million a year in compensation in 2010 [2]. In 2009 there were about 8.3 trillion dollars in existence [3]. Anyway, approximately a CEO makes a 1 millionth of the money in the world, or in other words, if we had a million CEOs we’d exhaust our money supply. It takes about $\log_2 1000000 \approx 20$ doublings to get a million. Hence in $20\times 7.5 = 150$ years one CEO will make all the money in the world. And this is all inflation adjusted.

But we don’t have to go so far to get into trouble. Now we did talk about the top 200, so when would the top 200 make all the money in the world. Well that requires only $\log_2 \frac{1000000}{200} \approx 12.3$ doublings so $12.3 \times 7.5 \approx 92$ years. OK, so in less than 100 years, the top 200 CEOs will suck out all the money in the universe.

Anyway, the problem is the following: The companies are not rewarding an individual CEO for good performance. They are rewarding all future CEOs. The thing is, that there is no “starting salary.” A CEO that just started is (statistically) making about the same as the one who’s been around for quite a while. If you would start all CEOs at a base salary, then one particular CEOs salary could rise at 10% a year because he’d be with the company only a fixed number of years, the problem would be manageable. Now to whatever extent there is anything like a “starting salary” the increase an individual CEO makes is even higher than 10% a year. Essentially the starting salary is increasing at 10% a year.

Let’s look at even a more realistic example of how quickly do we get into trouble. The CEO salary can easily be even 1% of the revenue for the company [4]. In fact some small private colleges are paying 1% of their budgets to their university president, a group where similar thing has happened. Well, now think about this doubling. If it is 1% now, it will be 2% in 7.5 years, 4% in 15 years, 8% in 22.5 years, 16% in 30 years, 32% in 37.5 years, 64% in 45 years, and we get 100% at less than 50 years. So in less than 50 years the entire revenue would have to support the CEO. Now you say, well but the revenue is also growing. Not so fast, the 10% pay increase is overall, that includes companies that did badly and those that did well. One would think that the growth of revenue on average (including failed companies) is not that much more than inflation. And this is adjusted for inflation. In any case CEO is definitely growing a lot faster than the economy (and hence your average revenue), and hence you hit the wall sooner or later. Even if we lob off another 2% to adjust for growth, the doubling time for CEO pay is still 9 years.

Of course a problem would appear a lot earlier than in 50 years. So it’s not only the “rich get richer” and “things are not fair” argument. This state of affairs is actually unsustainable even in relatively short period of time (within our lifetimes). I think people don’t understand that exponential growth is really really fast. That’s why pyramid schemes never work. It’s why Ponzi schemes usually fail far quicker than the perpetrator hoped. 10% increase a year does not seem like much (just like 10% return on investment doesn’t seem like that terribly much).

[1] http://www2.ucsc.edu/whorulesamerica/power/wealth.html (better sources exist, but I am too lazy to search further).

1. Are you jealous? To each his own, instead of being jealous, work more, earn more, save more. Be happy with what you have, a life, you may have been aborted.

Comment by BO — April 25, 2012 @ 12:21 am

• Well, “to each his own” makes about as much sense as “ugubum foo blargh blah.” Did you read the post?

Let me do a simpler calculation on a 3rd grade level: Suppose you run a lemonade stand and make $10 dollars a day and you make. You give$1 to your mom (the CEO) the first day, and then increase it by 10% every day as reward for buying such good lemons. That doesn’t seem too bad of a raise right? The 25th day you would have to pay about $10.83 cents, that is 83 cents that you don’t have. No matter what you think about your mom deserving it. It’s not a sustainable model. If you started giving your mom$9 each day (or even \$9.99) but didn’t increase, that might be a lot, but you can keep doing that as long as you please.

So it’s not about fairness or being jeleaous. It’s about the math biting you in the ass. You can’t make the math go away by talking about “to each his own” and not being jeleous.

By the way, what does this have to do with abortion?

Comment by jlebl — April 25, 2012 @ 1:42 am

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