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"What is not obvious, until you think about it, is that the ten young men would prefer the lottery. It gives higher expected utility."

It is not obvious for a simple reason. The preference for the lottery does not follow from the higher expected utility.

Min: ? Yes it does, doesn't it? Give the 10 young men a vote, on which of the two methods they will use to choose a soldier. Each will vote for the method with the higher expected utility.

Nice Rowe: "What's all this got to do with CEO pay? . . .

"But I've sometimes wondered, in my own life, whether I should try for a higher-paying job. And I've thought, 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."

One difference between you and the typical CEO is that for the typical CEO compensation is not so much about consumption as about keeping score. Am I doing better (getting more) than the other guy, and by how much?

Nick Rowe: "Give the 10 young men a vote, on which of the two methods they will use to choose a soldier. Each will vote for the method with the higher expected utility."

The simplification of expected utility is mathematically convenient, but human values are multi-dimensional. That means that choices are only partly ordered. If people made choices according to expected utility, they would be perfectly ordered. The fact, confirmed by experiments, that people's choices are not perfectly ordered does not mean that people are irrational. It means that expected utility is an over-simplification.

Min: OK. I'm assuming that simplification.

And yes, my own outlet for my competitive instincts is now blogging!

Is there really a scarcity of candidates who want to be CEO? (I'll admit I don't, but all the same I'll bet there's a large class of people for whom the power and the glory of being CEO provides a utility bonus outweighing the costs.)

I think there's a lot to Nick's argument here, but I think it hinges on scarcity driven by the "have the abilities needed" requirement. It seems to me that problems with measuring ability/performance, barriers to entry of the "CEO supply" market, and plain old cronyism are a big part of the story, too.

Matt: yep, that's the right question. Or, to change it slightly, has the scarcity of candidates increased over time?

Of course, the average person would very likely want to swap places with a CEO. But if you have the ability to be a CEO earning hundreds of millions, you will also have the ability to earn hundreds of thousands in a job you like better. Or, you've already earned a few millions, and want to sail round the world, or something. If you keep working, and climbing up, you get a much bigger yacht, but don't have time to sail it.

Both these hypotheses ignore something obvious: the psychic benefits of being CEO. In the business context, this just means that a person gets a substantial amount of utility from the power and prestige related to being in charge. For number #1, I doubt there ever was a de facto lottery system for CEO, because there would always have been many people who wanted it for the psychic benefits. For number #2, even if though there is diminishing marginal utility for consumption, there is not diminishing marginal utility for prestige. In fact, purely based on conjecture, I would bet that it is exponential. Being middle management hardly changes the amount of prestige you have, being CEO of a Fortune 500 company does drastically.

And it's likely that these psychic benefits vastly outweigh the effect of diminishing marginal utility to consumption. How do we know that? Because otherwise, we would expect to see the effects the Rowe is describing at much lower levels of income. Research shows that utility to income drops precipitously after 75,000, so if diverging income was caused by a lack of demand for those types of jobs, then the divergence would be occurring at a much lower percentile of income.


Starting to see how this works with your last comment, Nick. So if, say, most CEO candidates are wealth-satisficers (they'll quit when they have enough wealth to retire and have everything they want), then increasing executive pay reduces the years-as-CEO per candidate, which tightens the pool of candidates, which leads to increasing executive pay... This becomes a vicious cycle as long as increasing pay drives people out of the candidate pool faster than it attracts new candidates into the pool.

AFG: you could well be right. If you are right, though, it makes you wonder why they bother paying them so much. There's still competition between firms for top candidates of course. But maybe a firm could compete more cheaply by spending part of the salary on a few prestigous perks instead.

PFunkLange: Yep I didn't go into that "multiplier" dynamic you are talking about, but I think it's there. the more pay they have already earned, the more you need to pay them to keep them working.

Hmmmm. I wonder if CEOs have more kids than others? They are working for their heirs. And CEO pay has gone up over time because people have fewer kids than they did in the past?

No idea about the CEO application but the original exercise is brilliant. Did you just think of it?

Micro would be so much more interesting if microeconomists took it seriously, instead of just repeating their boring mantra that trade makes everyone better off.

... altho, I'm sure that CEOs don't have more kids than others. Trying to make bequests just an extension of individual utility maximization doesn't really work. For one thing, people reproduce sexually.

JW: Bergstrom thought it up, not me. Jonathan Pincus told me about the paper in Adelaide 15 years ago. I hadn't actually read it until today. The idea is simple and lovely. It's the best of micro. It's a very general idea. It explains why you should insure against loss of consumption, rather than loss of utility. It explains why tort laws that require full compensation are generally not efficient. It's an idea that's been running through my mind ever since.

OK. Here is a simple soldier example to illustrate the desirability of the lottery. Suppose that we have 8 potential soldiers, only one of whom well actually be a soldier. Suppose that they are otherwise identical in all relevant ways. Suppose that the value to them of their wages is proportional to the log base 2 of the dollar amount (Bernouilli's moral value of money, assuming that they each start from zero wealth). (The actual argument is more general, OC.) Suppose that their total wages are the same in all scenarios. (This is reasonable, since we can assume that the total real economic output is the same.)

Let the total wages = $32.

In scenario #1 each person receives $4, with a value of 2. The total value is 16. In scenario #2 each non-soldier receives $2 and the soldier receives $18. The value for each non-soldier is 1 and the value for the soldier is 4.17. The total value is only 11.17, rather less than 16. Any deviation from equal wages reduces their overall value.

Now let's eliminate the prior equality condition. Suppose that there are 7 people each with wages of $20, for a value of 4.32, and one with wages of $4, for a value of 2, before a soldier is chosen. Total wages = $144. Afterward the soldier is chosen, each person pays a flat tax proportional to their wages, so that the soldier receives 4 times his wages before he becomes a soldier. (That presumably makes each person indifferent to becoming a soldier.)

In scenario #1 the person with pre-choice wages of $4 becomes the soldier and receives $16, for a value of 4. The tax on each of the others is $2, so they each receive $18. The moral value of each of their wages is 4.17. The total value of the wages is 33.19.

In scenario #2 one of the others becomes the soldier and receives $80, for a value of 6.32. The non-soldiers with high wages each receive $10.32, for a value of 3.37, and the other non-soldier receives $2.08, for a value of 1.06. The total value of the wages in 27.58.

Given the personal indifference between scenarios #1 and #2, we assume that scenario #2 is 7 times as likely as scenario #1 in the volunteer scenario, scenario #3. That yields an expected total value of wages in the volunteer scenario of 28.28.

Assuming that the lottery scenario yields the same wages as before, we get a total value of wages of 32.25, rather better than the volunteer scenario.

OC, the best of these different scenarios, in terms of the total value of wages, is the one in which the low wage person becomes the soldier. The problem, if there is one, with the volunteer scenario lies with paying the high wage non-soldiers so much money. And, indeed, real life armies do not pay their soldiers with regard to what they would earn as non-soldiers, or with regard to their wealth. Such a volunteer army may be perceived as unfair or unpatriotic, even if it improves the general welfare in terms of wages. As mentioned above, human values are multi-dimensional.

Moi: "In scenario #1 the person with pre-choice wages of $4 becomes the soldier and receives $16, for a value of 4. The tax on each of the others is $2, so they each receive $18. The moral value of each of their wages is 4.17. The total value of the wages is 33.19."

Oops! Simple arithmetical mistake as I was fooling around with different scenarios. OC, each of the others pays $0.29 less tax. Recalculation will not change the general picture. ;)

Nick - great post, really enjoyed it.

I don't think that either of your two stories entirely explain CEO pay.

The story I'd tell would be one of extremely weak corporate governance structures combined with the extraordinary difficulty of measuring CEO performance.

After all, how many dollars worth of share do you own, either directly or indirectly? How often have you voted the shares you own? (I've voted the shares I own once). And if you - who are probably much more informed and more concerned than a lot of other shareholders - aren't voting your shares, who is governing corporations? Sure, big pension funds like Ontario Teachers exert some control, and there's a bit coming from corporate boards - but not a lot.

If you have a lot of influence over your own salary, and you're the type of driven personality that makes it into the executive suite, you're probably going to pay yourself as much as you possibly can.

Hi Nick:
I think one of the reasons someone eventually always becomes Department Chair rather than have the Dean get an external chair is because an external chair will not necessarily be from your discipline. Your department might be "punished" by having someone from another department made your chair. Generally, the optimal chair is the one department member who is the most efficient at dealing with administrative work. It reduces the burden on the chair and allows for the department to run efficiently.

Is the peer bencghmarking story not compelling for reasons I don't see? Seems to me to pretty much explain c suite salariy setting behaviour pretty well, especially since th comp commitees are outright telling us that this is what they are doing.

Patrick: "Is the peer bencghmarking story not compelling for reasons I don't see?"

One thing that the peer benchmarking story has going for it is that the timing is right. That is, executive pay really exploded just about the time that companies started being required to disclose executive salaries in order to be listed in various stock exchanges. Classic case of best intentions backfiring: something designed to rein in executive pay ended up ratcheting it up, because every CEO wants to be paid 15% more than the average executive.

Ontario and BC have public sector salary disclosure legislation. It's very hard to know whether or not this has led to an increase in public sector pay - I don't think anyone's managed to work out a way of measuring this, though some people have thought about doing so. I do know anecdotally that some academics have used the public sector salary disclosure information when negotiating salary increases ("My c.v. is just as good as so-and-so's c.v. and he's paid $15,000 more than me.")

Though the factors that I mentioned - the difficulty of measuring performance and the lack of good corporate governance - are two factors that make the benchmarking phenomenon happen.

It's very different from in professional sports, where the salary explosion essentially reflects an explosion in the profitability of professional sports. Everyone - except ordinary Canadian families who can no longer afford to take their kids to hockey games - is doing just fine out of that one.

Thanks Frances!

"I don't think that either of your two stories entirely explain CEO pay."

hey, if my story explained 10% of it I reckon i would be ahead of the game!

Livio: "Your department might be "punished" by having someone from another department made your chair."

Yep. When I was associate dean, one of my "jobs" (OK, I was never actually *given* this job, but decided to do it anyway) was to be the default parachute chair of any department that couldn't get its act together to find its own chair. So I showed up in the Biz Skool one day, wearing my oldest jacket, the one repaired with duct tape, loudly telling all the administrators that I was going to be the new Director if they didn't choose one from themselves. They found someone very quickly. Repeat performance on Social Work, loudly proclaiming my views on how the economic perspective trumped anything they could come up with. Same result. God those were fun times.

It's more when you can't find anyone to be Dean that they end up looking outside, for a true volunteer.

Patrick: there are lots of compelling stories for *high* CEO pay. But it's a little harder to explain *increasing* CEO pay. I'm not sure my story is any good. I'm just throwing it out there.

There's one big difference between CEO's and department chairs: expected duration of the job. An academic department chair at an institution can expect to remain chair until he or she chooses to leave. The institution is expected to live on.

That's not the case in the corporate world. Until 1980 or so corporations were thought of as institutions with longevity. In practice it worked too, with few mergers and bankruptcies. CEO's were meritocrats. Think of AT&T as the epitome of a private, corporate institution.

That changed in 1980 with the rise of the leveraged buyout and the hostile takeover. Suddenly management was not entrenched and could be replaced in a few months. Instead of capping your career with the CEO title and then retiring, collecting the big pay through tenure and your pension, you had to collect the same expected value over a much shorter period and arrange a golden parachute for your retirement.

This combined with a focus on "shareholder value" meant that CEO pay rose to encourage CEO's to increase market value. So high pay for short-term value collection morphed into permanently high pay. Then of course there is the trend that CEO's were to be rewarded not for just growing the company but by cutting costs and trimming the workforce. So CEO pay rose even when workers were laid off, something not seen previously. This was shareholder friendly but image deadly.

The increase in corporate takeovers was a long time in coming. The first hostile takeover of the modern era was the Robert Young takeover of the New York Central Railroad in 1957. The trend of the 1960's and 1970's to creating conglomerates was created a backlash on Wall Street in the 1980's. Conglomerates make CEO's the allocaters of capital, a role that Wall Street felt was its own.

All of these forces merged to end the era of stable, institutional corporations and entrenched management.

One obvious solution is to formalize the CEO lottery. Offer each of n candidates an equal share of CEO pay, if they commit to the loser doing the job. This solves the problem with decreasing marginal utility of consumption, so total pay ought to be lower.

Having a lottery as an entry mechanism would also allow for better matching of compensation incentives. In addition to working as a CEO, candidates would also commit to posting a surety, whch is partially forfeited if their company drops in value as a resut of their actions. An individual candidate would not participate in such a scheme, because their ex-ante compensation must be set so very high in the first place; but a lottery participant would.

One relevant idea is that marginal utility of consumption (hence, risk neutrality) does not vary over "small" differences in utility (where the utility function can be assumed to be linear) but it does vary if the change involves a large fraction of overall utility. So the idea works for the CEO and soldier example, not so much for department chairs.

anon: "One obvious solution is to formalize the CEO lottery. Offer each of n candidates an equal share of CEO pay, if they commit to the loser doing the job. This solves the problem with decreasing marginal utility of consumption, so total pay ought to be lower."

neat! Probably difficult to enforce the deal though.

Nick: Fair enough. If I understand the argument correctly, the idea is that having comp committees decide to pay at or above peer median creates an upward wage spiral.

I'm with Min: any analysis predicated on summing utiliy is suspect. Cardinal utility is thoroughly discredited at this time,

Luckily hypthosis number 2 does not require me to believe that the volunteer army is inferior. It calls on a different concept entirely: satiation.

Come to think of it, option 1 does not require utility either (your utility story is nice gloss). That hypothesis requires only coercion or some kind of honor culture that mean people really volunteer.

"It's very different from in professional sports, where the salary explosion essentially reflects an explosion in the profitability of professional sports."

I'm not sure that that's true. In some sports, certainly baseball, maybe hockey, the salary explosion followed a breakdown in employer side market power and the increased ability of players to play one club off against the others. It didn't neccesarily relate to profitability, which is why the Nordiques, jets, and Expos folded shop and why half the NHL teams south of the border are in a perpetual state of impoverishment. In contrast, we haven't see the same salary explosion in the NFL, not because it isn't highly profitable (it is, more so than the other leagues), but because the league has been much more aggresive in dealing with its players (and perhaps, because the nature of the game (i.e., wearing a quater-ton of helmets, masks and pads) is such that the players lack the bargaining power of a Jordan, a Gretzky or a Griffey).

Maybe that's the same story that Determinant's telling - the breakdown of the corporate norm of the CEO being a "company man" allows "star" CEOs to play one company off against another (compared with the breakdown of club owner collusion).

In fact, the problesm for shareholders and club owners are in some ways quite similar. Just like shareholders and CEOs, every sports fan can tell a story about a multi-million dollar multi-year, signing by their favourite team that turned out to be a disaster. It's easier to measure performance in sports (but it can't be done perfectly - there is a stochastic element to player performance (luck, injuries) and other external factors (quality of teammates, coaching, stadium, etc) whose impacts can only be measured imperfectly), but performance can only be measures restrospectively where players are generally compensated prospectively. In that respect player compensation is unlike CEOs, whose compensation scheme tends to be retrospective. While it may be harder to explicitly measure the performance of CEOs vs. baseball players, the problem facing their employers is the same, at the time they make the compensation decision, they can't precisely measure performance.

And that ties into Patrick's "peer effect" story. In baseball, where salaries are generally public knowledge, every agent acting into negotiations (or binding arbitratioN) on behalf of a particular player with a list of what every other similar player received, so that, at the end of the day, salaries are set by the owners who are the more reckless and who grossly overpay for so-so players.

I don't see why anyone would not want to be CEO since it gives them the power to direct others to doing the work. The motivation they can offer is their replacement if they don't deliver, or promotion within the organization. CEOs may choose to work hard to avoid boredom or to exercise their power but they really have no need to, all they need to do is select and delegate and replace anyone that doesn't measure up. CEOs may be workaholics or may be chosen for being workaholics, but there is no reason they need be workaholics. They would need to be capable of evaluating their subordinates, detecting BS, and motivating them, and this would require knowledge and skill and selecting a team that also does so, but I think the hours implied are not because they are necessary, but because they are enjoyable and they are especially enjoyable because it is not necessary to do anything they do not wish to. Poaching would increase salaries but not because of any unpleasantness or effort required. Salaries merely become status objects, the highest status those $1 salaries that provide for everything in gains.

"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. "

I might be missing something and I didnt read the paper, but why would someone who is risk-averse choose the "highest expected utility" option every time? Extreme risk aversion might push someone to choose the "highest minimum utility" option, which is the volunteer army.

Isnt this exactly like me choosing to get 1 million instead of tossing a coin and getting either $3 million or 0$ ?

Lord: "I don't see why anyone would not want to be CEO since it gives them the power to direct others to doing the work."

A lot of people/professors don't want to spend their time doing that. (OK, the departmental chair doesn't have quite the power of a CEO, but...).

SimonC: "Isnt this exactly like me choosing to get 1 million instead of tossing a coin and getting either $3 million or 0$ ?"

Maximising expected utility is not the same as maximising expected payoff. If you have diminishing marginal utility, then you are risk-averse, and may well chose the certain $1 million precisely because it gives you higher expected utility than a coin toss of $0 and $3 million.

I've done a little bit of research on executive compensation for a project at work. My impression of why executive pay is so high is that the market for a CEO is a lot like the market for real estate. In a bull market for real estate, an exogenous increase in demand for property can lead to a cycle of price increases, as actual sales lead to higher valued appraisals.

In the market for CEOs, the buyers are private investors in the company, not the company itself (which is controlled by management, not investors agents: the board. Through the board of directors, they want the company to select a CEO who they think will be a good steward of their invested capital. Outside advisors frequently consult for boards of directors on the level of executive compensation. These outside advisors act like appraisors in the real estate market. Executive compensation in public companies is frequently set in comparison to what is paid at a set of comparably sized companies in the same industry, not unlike property values being determined by sales of comparable properties in the same geographic market.

I think executive compensation is so high because of this ratchet effect in which exisiting executive compensation is used to determine compensation in companies in the same industry. Another factor is that private investors have a high tolerance for high absolute levels of executive compensation. Executive pay scales roughly with the size of the company in terms of revenue and assets. Investors are willing to accept higher pay at larger companies. Compensation of $5 million for a CEO of a company with $1 billion in assets is not atypical. From a investor's perspective, that is like paying only a 0.5% management fee (ignoring the rest of the executive team), which is pretty good compared to the management fees that investors pay for asset/portfolio management.

Yes, but I'm still not sure why "everyone would obviously pick the lottery" if the volunteer army provides a higher minimum utility despite an higher average utility.

Again, what about risk aversion?

I'm going through a severe case of "this seems so obviously wrong that I cant be right", so I'll go read the damn paper :)

Yes, I had the terminology wrong (payoffs vs utility), but at first glance it seems that there would be many utility functions that would make the volunteer army better. For example, what if I am so risk adverse that I only value the lowest expected payoff?

Off to read and open a bottle of beer.

SimonC: don't read the article. Read the basics on expected utility and risk aversion. Look at the diagram on this Wiki:


Nick: read both, all is good. For some reason I had a brain fart and thought that expected utility didnt have to be E(U) = prob1*utility1+ prob2*utility2 but could also be E(U) = sqrt(prob1*utility1+ prob2*utility2), which doesnt make any sense.


do you seriously think there are a shortage of people willing to be CEO? I think there is a shortage of people who boards trust to do the job. The problem is risk aversion in very large organisations (in order to get the chance to do the job you already have to have shown you can do the job - and very few people have shown that).

Reason: "do you seriously think there are a shortage of people willing to be CEO? I think there is a shortage of people who boards trust to do the job."

Isn't that just really six of one, a half dozen of the other. There is a shortage of people willing to undertake the efforts to put themselves in a position where they could be a CEO. That's like saying there no shortage of people who want to be economists, there's just a shortage of people willing to get PHDs in economics.

reason: Maybe 90% of the population would love to swap places with a CEO. But we are not talking about the population. We are talking about that very small percentage of the population who have the ability and experience to get chosen to be CEO. And those people aren't like the rest of the population. They already have lots of money and good jobs. They can write their own job descriptions, if they want to. They can retire in comfort now, if they want to. Of that small percentage, has the proportion who are willing to be CEO declined over time? I don't know.

But yes, I agree with your general point. Boards have already massively narrowed down the number of people they will ask. And those people will be very rich already, compared to the rest of the population.

Ah. Bob's beaten me to it.

An alternate hypothesis:

Academic and technical positions can be very attractive, but there are only so many and their value is capped at a comfortable but moderate level, but in business all of management is doing what the CEO does but on a more limited scale. Now CEOs can be more demanding such that no one in the organization wants to move up, and this is often the case with insecure ones who view them as rivals, but any lack of candidates indicates poor CEO management. So the question would be whether the battle for CEO has become more intense such that many have gone from leaders to dictators, from fostering talent to suppressing it or from encouraging it to draining it. Perhaps in the constant demand for results, it has. That is my impression from those uninterested in advancement in management. How long can the short run gains can offset the long term degradation? Desperation may increase the willingness of boards to up the stakes with CEO salaries looking for magic to win the lotto.

Bergstrom's paper has a lot of applications. Consider replacing "farmer" and "soldier" with "male" and "female". Each sex is sort of a draft lottery. But the optimal wage structure (as viewed, e.g., by a person who is considering having a child (of unpredictable sex)) will be the one where the derivative of utility relative to consumption is the same for both sexes. That is, if one sex is cursed with a negative utility differential, it is disoptimizing to have the wage structure compensate for it.

One last comment. I hope I thought this one through.

I just wanted to raise a flag to say that the draft lottery always yields greater or equal expected utility than the volunteer army if it equalizes *marginal utility*, not consumption like I understood from "But it is unfair ex post, because they all get the same consumption but one has a nastier job."

Take any utility function that increases the marginal utility of consumption, such as U = sqrt(consumption - A) , where A=1 for soldiers and 0 for farmers. I didnt make this up, it is example 1.2 from Bergstrom's paper.

Then if we have 10 farmers with 100 units to consume:

Volunteer army with equal utility : soldiers get 10.9, farmers get 9.9, E(U) = U(farmers = U (soldiers) = 3.1464
Draft army with equal consumption : soldiers get 10, farmers get 10. U(farmers) = 3.1622 U(soldiers)=3, E(U) = 3.1460
Draft army with equal marginal utility : same results as volunteer army.

Also, if we are in a world with such a utility function, I wonder if it might be easier to pick wages that achieve equal utility (slowly raise the pay of the soldiers until you have enough) than wages to achieve equal marginal utility.

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