Thermodynamics shows US chief executives are paid nearly 130 times too much

8 Comments »Posted on Wednesday 4 November 2009 at 9:03 pm by Jacob Aron
In Mathematics

With banks being bailed out all over the place these days, many people are asking themselves why those in charge get paid such high salaries. Are CEOs really worth their million pound bonuses? Not according to Venkat Venkatasubramanian, who has calculated that US chief executives get paid nearly 130 times what they should.

As a professor of chemical engineering at Purdue University, Indiana, Venkatasubramanian seems an unlikely candidate to dictate CEO salaries. It turns out that the maths behind thermodynamics, the study of heat and energy, can also be applied to economics.

The trick is to redefine a concept called entropy. In thermodynamics entropy measures the disorder of a system. Imagine a box full of gas particles. If all the particles are packed into one corner, the system has low entropy. If they are spread out and zooming all over the place, it has high entropy. The laws of thermodynamics mean that entropy always increases over time.

What does that mean for CEO salaries? Venkatasubramanian realised that entropy could be seen as a measure of “fairness” in economics. According to the laws of supply and demand, as markets evolve salaries and the price of goods should move towards the most fair situation for everybody.

With this economic equivalent to entropy, Venkatasubramanian found that salaries should follow a lognormal distribution, a particular way of measuring the spread of data. When he compared his theory to data from income tax returns, he found that the model fit closely for the bottom 90-95% of salaries. In other words, the 5-10% at the top are getting more than their fair share.

According to the lognormal distribution, CEOs should be paid a little over 8 times more than the average employee. Looking at the salaries of 35 CEOs from top Fortune 500 companies, Venkatasubramanian discovered that their pay was on average 1,057 times what a regular employee earns – around 129 times the ideal value.

Of course, not all CEOs pull in such vast amounts. Interestingly, investment guru Warren Buffet takes a salary of \$200,000, which is about 8 times what the average employee of his company Berkshire Hathaway earns.

Venkatasubramanian points out that his figures only work for large corporations – the heads of smaller entrepreneurial start-ups will clearly be worth more than the few staff they employ. Still, he hopes that his research will be useful to governments and regulators in assessing CEO salaries, and ensuring a fair deal for all.

Venkatasubramanian, V. (2009). What is Fair Pay for Executives? An Information Theoretic Analysis of Wage Distributions Entropy, 11 (4), 766-781 DOI: 10.3390/e11040766

2. Interesting, except for the fact that the lognormal pretty much fails in most of finance. According to that distribution the market moves we saw over the last couple of years had close to zero probabilities of happening. It’s also the biggest flaw of Black Scholes.

3. Contrary to what Aron says and (apparently) what Venkatasubramanian believes, there is nothing in “the laws of supply and demand” that implies that salaries will move to a fair point. First of all, most economists would agree that there are no “laws of supply and demand.” At most there are propensities of supply curves and demand curves to have certain shapes and of excess supply and excess demand to push certain markets toward a market clearing equilibrium. But there is little reason to expect such a market clearing solution to be “fair” in any plausible sense of that term. Beside which, labor markets are particularly far removed from the markets modeled by economic theory, both in terms of meeting definitions of perfect competition and with regard to market-clearing behavior. Thus economic theory provides little basis for judging the fairness of labor market outcomes.

Furthermore, it is not at all clear from Aron’s summary why Venkatasubramanian might think that entropy has anything to do with fairness or why it has any relevance to the distribution of salaries.

4. EB: It’s Jacob, not Aron – common mistake!

Of course, not all markets are fair. That’s partly the problem being identified here – CEOs are paid too much because market conditions aren’t ideal for all. However, wouldn’t you agree that employees and employers are both free to shop around for salaries that they both deem a fair equilibrium?

Perhaps I could have gone in to more detail with the entropy analogy. If you read the paper, Venkatasubramanian suggests that unfair situations are unlikely to happen spontaneously. For example, all employees accepting lower salaries even though they are offered higher salaries. This is a low entropy state, analogous to gas molecules occupying a small corner of a box.

I don’t think that Venkatasubramanian is saying that entropy IS fairness. He’s just saying that the same maths of thermodynamics can be used in economics if we redefine the concept of entropy.

5. It seems that Venkatasubramanian is simply picking up on leverage. 90-95% of employees don’t have a large effect on the performance of other employees, and thus there is a limit to their individual value. The remaining 5%-10% (obviously including CEOs) are in roles where they can have a significant impact on the productivity of many employees, not just themselves, and it’s thus worth paying many times more to get the best performers in those roles. In effect, there are two different employment markets.

By Mike Scott on Friday 6 November, 2009 at 10:33 am

6. Why would you apply entropy to salaries, and why the lognormal distribution? The obvious statistical thermodynamic parallel to seek is a Maxwell Boltzmann distribution of individual wealth in an economy with increasing wealth.

By rtah100 on Friday 6 November, 2009 at 12:57 pm

7. CEO are employees too. Should they not “shop around”?

Performance/results are what matter. Boards who succumb to CEO salary agreements that arent tied to performance and producing results have no one to blame but themselves.

8. well, stop the buffet worshipping. he is a founder not a CEO, and every \$ of salary he doesn’t pay himself, goes into the profits of his company and ultimately his own profits. Even disregarding the fact any salary even 50-60M a year would be a drop in the bucket for a man of his wealth, he probably gets back about the same money in the pocket through increase in profitability of his firm by not paying him a big salary (i think valuation uplift and better tax rate on capital gains vs. salary income probably outweighs his salary sacrifice / leakeage of value to other shareholders, although am too lazy to try and go calculate it…)

also, the idiot savant who ran the experiment seems to have calculated in terms of gross salaries per year, where he at least could have calculated in terms of salary net of tax, and adjusted for hours worked…

By arby on Tuesday 10 November, 2009 at 5:04 pm

9. Empircally salary data follows a scaleable distribution and any attempts to model it as a cousin of the gaussian leads to these stupid remarks about ‘fairness’ – wealth is distributed according to scaleable ‘principles’ not normality, log or non log.

By Suhail on Friday 20 November, 2009 at 7:27 pm

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