Assume ten identical young men. One of them is needed to do a dangerous job which creates disutility for the person doing it. Assume they have diminishing marginal utility of consumption. Assume (though a weaker assumption can get the same results) that the utility function is separable in consumption and the type of job.
There are two ways to allocate resources:
1. Volunteer Army. The nine compensate the one at exactly the right level so that all are indifferent between doing and not doing the dangerous job.
2. Conscript Lottery. The ten hold a lottery, and the one who draws the shortest straw does the job.
The volunteer army is fair ex post. The one who volunteers gets the same level of utility as the other nine. The extra consumption just compensates him for the extra danger.
The lottery is fair ex ante, since each has an equal chance of losing the lottery. But it is unfair ex post, because they all get the same consumption but one has a nastier job.
That's obvious. What is not obvious, until you think about it, is that the ten young men would prefer the lottery. It gives higher expected utility. That's the result of Theodore Bergstrom's minor classic "Soldiers of Fortune" paper (pdf).
The intuition is straightforward. Think about the problem from the Utilitarian perspective, of maximising the sum of the ten utilities. This requires equalising the marginal utility of consumption for all ten men. Which, assuming a separable utility function, means equalising consumption. The volunteer army gives the soldier higher consumption, and so lower marginal utility of consumption, so does not maximise total utility. By having a lottery, the ten men place themselves behind the Veil of Ignorance and implement the Utilitarian maximum.
For simplicity, I assumed a separable utility function, so that the marginal utility of consumption is independent of whether or not you did the job. This is unlikely to be true. If the job is dangerous, and you die, your marginal utility of consumption is zero, so money doesn't matter to you at all, unless you care about your heirs. If the job also involves working long hours, the marginal utility of consumption might also be less, if consumption and leisure are complements. Owning a Lotus is no fun if you never have time to drive it.
If we assume, as may be reasonable, that taking the job reduces the marginal utility of consumption, that strengthens the advantages of the lottery over the volunteer army. It also means they would actually prefer a lottery where the soldier has lower consumption than those who stays home. The loser pays the winners, as well as risking his life, in the most efficient lottery.
Of course, if the ten young men are not identical, there will be offsetting advantages of a volunteer army over a lottery. The volunteer army may act as a sorting mechanism to ensure that those with a comparative advantage become the soldiers.
A second minor classic on the same subject is Marc Bilodeau and Al Slivinski's paper "Toilet cleaning and department chairing: volunteering a public service." (See Frances' post here.) The chair of a university department gets a few extra thousand dollars salary. But most professors (at least in economics) don't want to do the job. The extra consumption isn't enough to compensate for the disutility of being chair. Marc and Al model a waiting game, where each waits for someone else to volunteer. In equilibrium, the least patient professor volunteers immediately. But she's not really a volunteer. She would be better off if someone else did the job. She only "volunteers" because she knows that one of the professors will have to do the job, and that because she's the least patient, she knows she will cry "uncle" first in the waiting game anyway. It's really a lottery, and she happened to draw the short straw by being born slightly less patient than the others.
If all professors are identical, and have diminishing job-separable marginal utility of consumption, it's more efficient to have chairs/soldiers chosen by lottery, than by compensating them enough to get a true volunteer chair/army.
The lottery is more or less enforceable for departmental chairs. It's a small numbers game, and tenured professors know it's got to be one of them, and they are unlikely to emigrate from the job to dodge the draft. They can't easily free ride. But in other contexts you have to have a volunteer army, even if a lottery would be more efficient, simply because a lottery would not be enforceable.
What's all this got to do with CEO pay? Maybe absolutely nothing. Everything that follows is purely speculation on my part. I have no evidence to support either of these hypotheses.
But I've sometimes wondered, in my own life, whether I should try for a higher-paying job. Even if I had the ability to do the job, would I really want it? I've got a very good job already, with good salary. My marginal utility of consumption diminishes rapidly. Sure, I would prefer having an Aston Martin or Lotus to an MX6, but not by that much. Would it really be worth the extra hours and hassle? The faster my marginal utility of consumption diminishes, the more money you would have to pay me to get me to volunteer. As I got richer and approached satiation, It would take an offer of all the supercars in the world to get me to volunteer for a job I didn't want to do.
Why has CEO pay increased?
Hypothesis One. Being CEO used to be more of a lottery, and is now more of a volunteer army. Managers used to stay in the same firm for life, like tenured professors. When you were chosen to be CEO you did it, like when it's your turn to be departmental chair. Some wanted the job, some didn't, but you did it anyway because you were the best person for the job, and someone has to be CEO. And now, with higher mobility between firms, it's only a volunteer army that is willing to do the job. It's only those who really want the money who are willing to do the job, and even they don't want the money that much, so you have to offer them massive amounts of money to make them want it. And because the supply is smaller, the price will be higher, and those few who do want the CEO's job may earn massive rents.
If you know that the Dean will hire a chair from outside the department if nobody inside "volunteers" to play the lottery, nobody will volunteer, because they know the Dean can always find someone from outside. And so the Dean goes outside, and has to pay a much higher salary to get a true volunteer.
Hypothesis 2. We are richer and nearer to satiation than we were in the past. (Increasing relative risk aversion??) So the same percentage pay premium brings forth a smaller supply of qualified candidates than it did in the past. So the equilibrium pay is higher. Your salary now needs to go up one hundred-fold, instead of ten-fold, to get the marginal candidate to throw his hat into the ring. At the margin, CEOs get so much higher pay precisely because they don't need it. Each extra doubling of dollars means less to us now. The more of us who think the money isn't worth it means even more money for those who think it is.
Again, this is purely conjectural, and mostly based on a sample of one - myself. There are lots of other hypotheses from the demand side of the market. This is not my area of economics.
But if I'm right, or even partly right, high CEO pay is inefficient ex ante, just like the volunteer army. It only exists because the more efficient lottery is unenforceable. You can always get a CEO from outside. And those who are willing to be CEO, and have the abilities needed, will take the best offer from any firm in the world.