Do stock buybacks enrich CEOs at the expense of the economy?

Are share buybacks hurting the economy and increasing income inequality? 

Some pundits seem to think so. There was an article in Harvard Business Review, a while ago, which might have been an editorial (I can't find a byline). That followed a similar article from FiveThirtyEight, that concentrated on the economic effects. When I Googled, I came across another article from The Atlantic. I think it's a common argument ... I'm pretty sure I've seen it lots of other places, including blogs and Facebook.

They think it's a big deal, at least going by the headlines: 

-- "How stock options lead CEOs to put their own interests first" (Washington Post)

-- "Stock Buybacks Are Killing the American Economy" (The Atlantic)

-- "Profits Without Prosperity" (Harvard Business Review)

-- "Corporate America Is Enriching Shareholders at the Expense of the Economy" (FiveThirtyEight)

But ... it seems to me that neither the "hurt the economy" argument nor the "increase inequality" argument actually makes sense.

Before I start, here's a summary, in my own words, of what the three articles seem to be saying. You can check them out and see if I've captured them fairly.

"Corporations have always paid out some of their earnings in dividends to shareholders. But lately, they've been dispersing even more of their profits, by buying back their own shares on the open market. 

"This is problematic in several ways. For one, it takes money that companies would normally devote to research and expansion, and just pays it out, reducing their ability to expand the economy to benefit everyone. In addition, it artificially boosts the market price of the stock. That benefits CEOs unfairly, since their compensation includes shares of the company, and provides a perverse incentive to funnel cash to buybacks instead of expanding the business.

"Finally, it makes the rich richer, boosting the stock values for CEOs and other shareholders at the expense of the lower and middle classes."

As I said, I don't think any part of this argument actually works. The reasons are fairly straightfoward, not requiring any intricate macroeconomics.

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1. Buybacks don't increase the value of the shares

At first consideration, it seems obvious that buybacks must increase the value of your stockholdings. With fewer shares outstanding, the value of the company has to be split fewer ways, so your piece of the pie is bigger.

But, no. Your *fraction* of the pie is bigger, but the pie is reduced in size by exactly the same fraction. You break even. That *has* to be the case, otherwise it would be a way to generate free money!

Acme has one million (1 MM) shares outstanding. The company's business assets are worth $2 MM, and it has $1 MM in cash in the bank with no debt. So the company is worth $3 a share.

Now, Acme buys back 100,000 shares, 10 percent of the total. It spends $300,000 to do that. Then, it cancels the shares, leaving only 900,000.

After the buyback, the company still owns a business worth $2 million, but now only has $700,000 in the bank. Its total value is $2.7 million. Divide that by the 900,000 remaining shares, and you get ... the same $3 a share as when you started.

It's got to be that way. You can't create wealth out of thin air by market-value transactions. 

The HBR author might realize this: he or she hints that buybacks increase stock prices "in the short term," and "even if only temporarily."  I'm not sure how that would happen -- for very liquid shares, the extra demand isn't going to change the price very much. Maybe the *announcements* of buybacks could boost the shares, by signalling that the company has confidence in its future. But that's also the case for announcements of dividend increases.

One caveat: it's true the share price is higher after a buyback than a dividend, but that's not because the buyback raises the price: it's because the dividend lowers it. If the company spends the $300,000 on dividends instead of buybacks, the value of a share drops to $2.70. The shareholders still have $3 worth of value: $2.70 for the share, and 30 cents in cash from the dividend. (It's well known, and easily observed, that the change in share price actually does happen in real life.)

If the CEO chooses to spend the cash on buybacks, then, yes, the stock price will be higher than if he chose to spend it on dividends. It won't just be higher in the short term, but in the long term too. 

Are buybacks actually replacing dividends? The FiveThirtyEight article shows that both dividends and buybacks are increasing, so it's not obviously CEOs choosing to replace one with the other. 

But, sure, if the company does replace expected dividends with buybacks, the share price will indeed sit higher, and the CEO's stock options will be more valuable.

To avoid conflicts of interest, it seems like CEOs should be compensated in options that adjust for dividends paid. (As should all stock options, including the ones civilians buy. But they don't do that, probably because it's too complicated to keep track of.)  But, again: the source of the conflict is not that buybacks boost the share price, but that dividends reduce it. If you believe CEOs are enriching themselves by shady dealing, you should be demanding more dividends, not decrying buybacks.


2. The buyback money is still invested

The narratives claim that the money paid out in share buybacks is lost, that it's money that won't be used to grow the economy.

But it's NOT lost. It's just transferred from the company to the shareholders who sell their stock. 

Suppose I own 10 shares of Apple, and they do a buyback, and I sell my shares for $600 of Apple's money. That's $600 that Apple no longer has to spend on R&D, or advertising, or whatever. But, now, *I have that $600*. And I'm probably going to invest it somewhere else. 

Now, I might just buy stock in another company -- Coca-Cola, say -- from another shareholder. That just transfers money from me to the other guy -- the Coca-Cola Corporation doesn't get any of that to invest. But, then, the other guy will buy some other stock from another guy, and so on, and so on, until you finally hit one last someone who doesn't use it to buy another stock.

What will he do? Maybe he'll use the $600 to buy a computer, or something, in which case that helps the economy that way. Or, he'll donate the $600 to raise awareness of sexism, to shame bloggers who assume all CEOs and investors are "he". Or, he'll use it to pay for his kids' tuition, which is effectively an investment in human capital. 

Who's to say that these expenditures don't help the economy at least as much as Apple's would?

In fact, the investor might use the $600 to actually invest in a business, by buying into an IPO. In 2013, Twitter raised $1.8 billion in fresh money, to use to build its business. It's quite possible that my $600, which came out of Apple's bank account, eventually found its way into Twitter's.

Is that a bad thing? No, it's a very good thing. The market judged, albeit in a roundabout way, that there was more profit potential for that $600 in Twitter than in Apple. The market could be wrong, of course, but, in general, it's pretty efficient. You'd have a tough time convincing me that, at the margin, that $600 would be more profitable in Apple than in Twitter.

The economy grows the best when the R&D money goes where it will do the most good. If Consolidated Buggy Whip has a billion dollars in the bank, do you really want it to build a research laboratory where it can spend it on figuring out how to synthesize a more flexible whip handle? 

At the margin, that's probably where Apple is coming from. It makes huge, huge amounts of profit, around $43 billion last year. It spent about $7 billion on R&D. Do we really want Apple to spend six times as much on research as we think is appropriate? It seems to me that the world is much better off if that money is given back to investors to put elsewhere into the economy.

That might be part of why buyback announcements boost the stock price, if indeed they do. When Apple says it's going to buy back stock, shareholders are relieved to find out they're not going to waste that cash trying to create the iToilet or something.


3. Successful companies are not restrained by cash in the bank

According to the FiveThirtyEight article, Coca-Cola spent around $5 billion in share repurchases in 2013. But their long-term debt is almost $20 billion.

For a company like Coca-Cola, $20 billion is nothing. It's only twice their annual profit. Their credit is good -- I'm sure they could borrow another $20 billion tomorrow if they wanted to.

In other words: anytime the executives at Coke see an opportunity to expand the business, they will have no problem finding money to invest. 

If you don't believe that, if you still believe that the $5 billion buyback reduces their business options ... then, you should be equally outraged if they used that money to pay down their debt. Either way, that's $5 billion cash they no longer have handy! The only difference is, when Coca-Cola pays down debt, the $5 billion goes to the bondholders instead of the shareholders. (In effect, paying off debt is a "bond buyback".)

The "good for the economy" argument isn't actually about buybacks -- it's about investment. If buybacks are bad, it's not because they're buybacks specifically; it's because they're something other than necessary investment.

It's as if people are buying Cadillac Escalades instead of saving for retirement. The problem isn't Escalades, specifically. The problem is that people aren't using the money for retirement savings. Banning Escalades won't help, if people just don't like saving. They'll just spend the money on Lexuses instead.

Is investment actually dropping? The FiveThirtyEight article thinks so -- it shows that companies' investment-to-payout ratio is dropping over time. But, so what? Why divide investment by payouts? Companies could just be getting rid of excess cash that they don't know what to do with (which they also get criticized for -- "sitting on cash"). Looking at Apple ... their capital expenditures went from 12 cents a share in 2007, to $1.55 in 2014 (adjusted for the change in shares outstanding). A thirteen-fold increase in research and development doesn't suggest that they're scrimping on necessary investment.


4. Companies offset their buybacks by issuing new shares

As I mentioned, the FiveThirtyEight article notes that Coke bought back $5 billion in shares in 2013. But, looking at Value Line's report (.pdf), it looks like, between 2012 and 2013, outstanding shares only dropped by about half that amount.

Which means ... even while buying back and retiring $5 billion in old shares, Coca-Cola must have, at the same time, been issuing $2.5 billion in *new* shares. 

I don't know why or how. Maybe they issued them to award to employees as stock options. In that case, the money is part of employee compensation. Even if the shares went to the CEO, if they didn't issue those shares, they'd have to pay the equivalent in cash.

So if you're going to criticize Coca-Cola for wasting valuable cash buying shares, you also have to praise it, in an exactly offsetting way, for *saving* valuable cash by paying employees in shares instead. Don't you?

I suppose you could say, yes, they did the right thing by saving cash, but they could do more of the right thing by not buying back shares! But: the two are equal. If you're going to criticize Coca-Cola for buying back shares, you have to criticize other companies that pay their CEOs exclusively in cash. 

But the HBR article actually gets it backwards. It *criticizes* companies that pay their CEOs in shares!

Suppose Coca-Cola is buying back (say) a million shares for $40 MM, which is presumably bad. Then, they give those shares to the employees, which is also presumably bad. Instead, the Harvard article says, they should take the $40 MM, and give it to the employees directly. 

But that's exactly the same thing! Either way, Coca-Cola has the same amount of cash at the end. It's just that in one case, the original shareholders have shares and the CEO has cash. The other way, the original shareholders have the cash and the CEO has the shares.

What difference does that make to the economy or the company? Little to none.


5. Inequality is barely affected, if at all

Suppose a typical CEO makes about $40 million. And suppose half of that is in stock. And suppose, generously, that the CEO can increase the realized value of his shares by 5 percent by allegedly manipulating the price with share buybacks.

You're talking about $1 million in manipulation. 

How much does that affect inequality? Hardly at all. The top 1% of earners in the United States are, by definition, around 3 million people. That includes children ... let's suppose the official statistics use only 2 million people.

The Fortune 500 companies are, at most, 500 CEOs. Let's include other executives and companies, to get, say, 4,000 people. 

That's still only one-fifth of one percent of the "one percenters."

The average annual income of the top 1% is around $717,000. Multiply that by two million people, and you get total income of around $1.4 trillion.

After the CEOs finish manipulating the stock price, the 4,000 executives earn an extra $4 billion overall. So the income of the top 1% goes from

$1,400,000,000,000

to 

$1,404,000,000,000

That's an increase of less than one-third of one percent. Well, yes, technically, that does "contribute" to inequality, but by such a negligible amount that it's hardly worth mentioning. 

And that .00333 percent is still probably an overstatement:

1. We used very generous assumptions about how CEOs capitalize on stock price changes. 

2. When the board offers the CEO stock options, both parties are aware of the benefits of the CEO being able to time the announcements. Without that benefit, pay would probably have to increase (for the same reason you have to pay a baseball player more if you don't give him a no-trade clause). So, much of this alleged benefit is not truly affecting overall compensation.

3. Price manipulation is a zero-sum game. If the executives win, someone loses. Who loses? The investors who buy the executives' shares when they sell. Who are those investors? Mostly the well-off. Some of the buyers might be pension funds for line workers, or some such, but I'd bet most of the buyers are upper middle class, at least. 

We know for sure it isn't the poorest who lose out, because they don't have pension funds or stocks. So it's probably the top 1 percent getting richer on the backs of the top 10 percent.

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Here's one argument that *does* hold up, in a way: the claim that buybacks increase earnings per share (EPS).

Let's go back to the Acme example. Suppose, originally, they have $200,000 in earnings: $190,000 from the business, and $10,000 from interest on the $1 MM in the bank. With a million shares outstanding, EPS is 20 cents.

Now, they spend $300K to buy back 100,000 shares. Afterwards, their earnings will be $197,000 instead of $200,000. With only 900,000 shares remaining outstanding, EPS will jump from 20 cents to 21.89 cents.

Does that mean the CEO artificially increased EPS? I would argue: no. He did increase EPS, but not "artificially."

Before the buyback, Acme had a million dollars in the bank, earning only 1 percent interest. On the other hand, an investment in Acme itself would earn almost 7 percent (20 cents on the $3 share price). Why not switch the 1-percent investment for a 7-percent investment? 

It's a *real* improvement, not an artificial one. If Acme doesn't actually need the cash for business purposes, the buyback benefits all investors. It's the same logic that says that when you save for retirement, you get a better return in stocks than in cash. It might be right for Acme for the same reason it's right for you.

Does the improvement in EPS boost the share price? Probably not much -- the stock market is probably efficient enough that investors would have seen the cash in the bank, and adjusted their expectations (and stock price) accordingly. A small boost might arise if the buybacks are larger, or earlier, than expected, but hardly enough to make the CEO any more fabulously wealthy than he'd be without them.

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There's another reason companies might buy back shares -- to defer tax for their shareholders.

Suppose Coca-Cola has money sitting around. They can pay $40 to me as a dividend. If they do, I pay tax on that -- say, $12. So, now, I have $12 less in value than before. The value of my stock dropped by $40, and I only have $28 in after-tax cash to compensate.

Instead of paying a dividend, Coke could use the $40 to buy back a share. In that case, I pay no tax, and the value of my account doesn't drop.  

Actually, the buybacks are just deferring my taxes, not eliminating them. When I sell my shares, my capital gain will be $40 more after the buyback than it would have been if Coke had issued a dividend instead. As one of the linked articles notes, the US tax rate on capital gains is roughly the same as on dividends. So, the total amount is a wash -- it's just the timing that changes.

Maybe that tax deferral bothers you. Maybe you think the companies are doing something unfair, and exploiting a loophole. I don't agree; for one thing, I think taxing corporate profits, and also dividends, is double taxation, a hidden, inefficient and sometimes unfair way to raise revenues. (Companies already have to pay corporate income tax on earnings, regardless of whether they use it for buybacks, dividends, reinvestment, or cash hoards.)

You might disagree with me on that point.  If you do, then why aren't you upset at companies who don't pay dividends at all? If share buybacks are a loophole because they defer taxes, then retained earnings must be a bigger loophole, because they defer even *more* taxes!

Keep in mind, though, the deferral from buybacks is not quite as big as it looks. When the company buys the shares, the sellers realize a capital gain immediately. If the stock has skyrocketed recently, the total tax the IRS collects after the buyout could, in theory, be a significant portion the amount it would have collected off the dividend. (For instance, if all the selling shareholders had originally bought Coca-Cola stock for a penny, the entire buyback (less one cent) would be taxed, just as the entire dividend would have been.)

There's another benefit: when Coca-Cola buys shares, it buys them from willing sellers, who are in a position to accept their capital gains tax burden right now. That's the main advantage, as I see it: the immediate tax burden winds up falling on "volunteers," those who are able and willing to absorb it right now.

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In my view, buybacks have little to do with greedy CEOs trying to enrich themselves, and they have negligible effect on the economy compared to traditional dividends. They're just the most tax-efficient way for companies to return value to their owners.




UPDATE: Finance writer Michael Mauboussin explains buybacks in more detail in a FAQ here.  (Mauboussin has also written about sports and luck.)

Does inequality make NBA teams lose?

I

Some people believe that income inequality can hurt group performance. They think that people work better together when employees are more likely to see themselves as equal.

I don't know if that's true or not. But it's a coherent hypothesis, that makes sense in terms of cause and effect.

On the other hand, here's something that doesn't make sense: the idea that when salaries are more unequal, the result is that the total becomes lower. That doesn't work, right? You can tell the CEO, "if you paid your people more equally last year, the company would have done better." But you can't tell the CEO, "if you paid your people more equally last year, they'd have collectively taken home more money."  

Because, the relationship between total pay and individual pay is already known. The total is just the sum of the individuals. Equality can't possibly cause any additional pay, beyond adding up the amounts.

It would be like saying, "You shouldn't carry $50 bills and $1 bills in the same wallet. If you reduced inequality by carrying only $5 bills and $10 bills, you'd have more money."   

That would be silly.

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Well, that's almost exactly what's happening in a recent NBA study, by the same poverty researcher who wrote the baseball inequality article I posted about three weeks ago.

The author looked up individual player Win Shares (WS) for the 2013-14 season. He measured Win Share inequality within each team by calculating the Gini Coefficient for the population of players. He then ran a regression to predict team wins from player inequality. He found a strong negative correlation, -0.43. 

In other words, the more equal teams won significantly more games. 

The author suggests this might be evidence of the benefits of equality. On the more equal teams, the better performance might have been created by the "psychological and motivational benefits" of the weaker players having "better opportunity to develop and showcase their skills."

But ... no, that doesn't make sense, for exactly the same reason as the $10 bill example. 

Win Shares is really just a breakdown and allocation of actual team wins. The formulas take the number of games a team won, and apportions that total among the players. In other words, the team totals equal the sum of the individual totals, the same way the total amount in your wallet equals the sum of the individual bills. (*)

Last year, the Spurs won 62 games, while the 76ers won only 19. That can't have anything to do with equality. It's due to the fact that the Spurs had 62 win dollars in their wallets -- say, eleven $5 bills and seven $1 bills -- while the 76ers had only 19 win dollars -- say, a $10 bill and nine $1 bills. 

It's true that the 76ers players' Win Shares were more concentrated among their best players. In fact, the top 5 percent of their players accounted for more than half the team total. But that doesn't matter. They had 19 wins total because they had a total of 19 wins individually.

If you want the Philadelphia 76ers to win 50 games this year, find players who add up to 50 Win Shares. It doesn't matter if you find ten guys with 5 WS each, or one guy with 30 WS and ten guys with 2 WS. 

In fairness to the author, he does explicitly say that the correlation does not necessarily imply causation here. But the point is: he doesn't realize that he's looking at a relationship where correlation CANNOT POSSIBLY imply causation.

And that's what I found so interesting about the study. At first reading, it looks like such a strong finding, that equality may cause teams to win more ... but after a bit of thought, it turns out it's logically impossible!

The only other time I remember seeing that kind of logical impossibility was that study "proving" that listening to children's music makes you older, by retroactively changing your year of birth. And that one had been created deliberately to make a point.

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II

As an aside, another thing I found interesting: in his baseball article, the author argued against unequal pay for baseball players because, he believed, pay seemed to have so little to do with actual merit. But, here, by measuring inequality in Win Shares instead of dollars, he seems to be arguing against inequality of merit itself!

Well, that may be just a tiny bit unfair. Reading between the lines, I think the author thinks Win Shares are much more heavily based on opportunity than they actually are. He writes, "maybe top teams, by virtue of their abundance of success, are more willing to share the glory ... Lack of opportunity, by contrast, can lead to despair and diminished performance."

But, actually, the author never demonstrates that the bad teams have more inequality of opportunity (playing time). I suspect that they don't.

In any case, we can see that the 76ers high Gini isn't much caused by differences in opportunity. Even limiting the analysis to "regulars," players with 1,000 minutes or more, the effect remains. On the 76ers, the top two players had 53 percent of the regulars' total Win Shares. On the Spurs, it was only 29 percent.

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III

So why is it that the unequal teams tend to be worse? I think it's a combination of (a) the way the Gini coefficient measures inequality, and (b) the mechanism by which NBA performance creates wins. 

Suppose that on a good team, the five regulars have field goal percentages (FG%) of 59, 57, 55, 53, and 51 percent, respectively. On a bad team, the five players are at 49, 47, 45, 43, and 41 percent.

If you measure inequality on the two teams by variance, it comes out equal: a standard deviation of 2.8 on each team. But if you measure it by Gini coefficient, or a similar calculation of "proportion of total wealth," they're different. 

On the good team, the total percentage points add up to 275. The top player, with 59, has 21.5 percent of the total.

On the bad team, the total percentage points add up to 225. The top player, with 49, has 21.8 percent of the total. So, the bad team is equal by SD, but less equal by "percent of total."

The Gini is more than just the top player, of course ... the formula it involves every member of the dataset. Using an online calculator, I found:

The Gini of the good team is 0.029. 
The Gini of the bad  team is 0.036.

So, by Gini, the bad team is less equal than the good team. (A higher Gini means less equality.)

Why does this happen, that the Gini is higher but the variance is the same? Because of the way the two measures differ. Variance stays the same when you *add* the same amount to every player. But not the Gini. The Gini stays the same when you *multiply* every player by the same amount. 

If you *add* a to every player instead of multiplying, the Gini drops. (And, if you *subtract* a positive number from every player, it increases.)

That's often what you want to have happen -- for incomes, say. If I make $50K and you make $10K, we're very unequal. But if you give both of us a $100K raise, now we're at $150K and $110K -- much more equal, intuitively.

The Gini confirms that. Before our raise, the Gini is 0.33. Afterwards, it's 0.08. (But if we use the variance instead, we look the same both ways.)

But for Win Shares, is the Gini-type of inequality really what we want? Are two players with 7 WS and 6 WS, respectively, really that much more equal, in an intuitive basketball sense, than two players with 2 WS and 1 WS? What about two players at 0.002 and 0.2 wins? In that case, one player has 100 times the wins of the other. But does "100 times" really give a proper impression of how different they are?

I don't think so. I think it's just an artifact of the way performance translates to wins.

What's wins? It's performance above replacement value. (Well, actually, WS is measuring above zero value, which is lower, but I'll call it "replacement value" anyway since the logic is the same.)  

So, to get wins, you start with performance, and subtract a constant. As we saw, when you subtract the same positive number from every player, the Gini goes up. It's a "negative raise" that makes employees less equal.

Suppose the average FG% is 50 percent. Suppose that 40 percent is "replacement level" that leads to exactly zero wins, the level at which a team is so bad it will never win a game. Conversely, 60 percent is the level at which a team is so good it will never lose a game. 

If the relationship is linear, it's easy to convert player FG% to Win Shares. Actually, I'll convert to "wins per 100 games," because the "out of 100" scale is easier to follow.

On the good team we talked about earlier, the players had FG% of 59, 57, 55, 53, and 51. That corresponds to W100 of 95, 85, 75, 65, and 55.

On the bad team, the players' FG% of 49, 47, 45, 43, and 41 translate to W100s of 45, 35, 25, 15, and 5.

See what happens? The FG% looks a lot more equal than the wins. On the bad team, the best player was only 20 percent better than the worst player in field goal percentage (49 vs. 41). But in wins ... he's 800 percent better! (45/5.)  On the good team, though, there's still enough performance after subtracting that the numbers look reasonably equal. 

The actual Gini coefficients:

FG%:  The Gini of the good team is 0.029. 
FG%:  The Gini of the bad  team is 0.036.

Wins: The Gini of the good team is 0.11.
Wins: The Gini of the bad  team is 0.32.

That's just how the math works. The Gini coefficient is very sensitive to where you put your "zero". If you measure zero as 0 FG%, inequality looks low. If you measure zero as zero wins (say, FG% of 40 percent), inequality looks higher. If you measure zero as replacement level (say, FG% of 43 percent), inequality looks even higher. And if you measure zero as an NBA average team (say, FG% of 50 percent), it's even more unequal -- the top half of the teams have 100 percent of the wins! (**) 

The higher the threshold that you call zero, the greater the inequality. 

In baseball, a player hitting .304 has only about 1% more hits than a player hitting .301. But he has 300% more "hits above .300". 

In the economy, the top 10% of families may have (say) 45% of the income -- but probably close to 100% of the new Ferraris. 

And a real-life example: In the NHL, over the last ten seasons, the Black Hawks have 13% more standings points than the Maple Leafs -- but 500% more playoff appearances.

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One last analogy:

Take a bunch of middle-class workers, and tax them $40,000 each. They become much more unequal, right? Instead of making, say, $50K to $80K, they now take home $10K to $40K. There's a much bigger difference, now, in what they can afford relative to each other.

But if you tax the same $40K away from a bunch of doctors, it matters less. They may have ranged from $200K to $300K, and now it's $160K to $260K. They're a bit less equal than before, but you hardly notice.

Measuring after the $40K tax is measuring "income above $40K," which is like measuring "FG% above replacement level of 40%" -- which is like measuring Win Shares.

So that's why bad teams in the NBA appear more unequal than the good teams -- because "Wins" are what's left of "Performance" after you levy a hefty replacement-level tax. Most of the players on the good teams stay middle-class after paying the tax -- but on the bad teams, while some stay middle class, more of the others drop into poverty.

It has nothing to do with the social effects of equality or inequality.  It's just an artifact of how the Gini Coefficient and basketball interact.


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* Actually, there's a bit of wiggle room in the particular version of WS the author used, the version from basketball-reference.com. That version doesn't add up perfectly, but it promises to be close, certainly close enough that it doesn't make a difference to this argument. 

** That's if you give the bottom teams zero. If you give them a negative, the Gini actually winds up at infinity. (The overall total has to be zero relative to the average, and you can't divide by zero.)  

Do baseball salaries have "precious little" to do with ability?

Could MLB player salaries be almost completely unrelated to performance? 

That's the claim of social-science researcher Mike Cassidy, in a recent post at the online magazine "US News and World Report."

It argues an "economics lesson from America's favorite pastime." Specifically: How can it be true that high salaries are earned by merit in America, when it's not even the case in baseball -- one of the few fields in which we have an objective record of employee performance?

The problem is, though, that baseball players *are* paid according to ability. The author's own data shows that, despite his claims to the contrary. 

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Cassidy starts by charting the 20 highest-paid players in baseball last year, from Alex Rodriguez ($29 million) to Ryan Howard ($20 million). He notes that only two of the twenty ranked in the top 35 in Wins Above Replacement (WAR). The players in his list average only 2.2 WAR. That's not exceptional: it's "about what you need to be an everyday starter."  

It sounds indeed like those players were overpaid. But it's not quite so conclusive as it seems.

WAR is a measure of bulk contribution, not a rate stat. So it depends heavily on playing time. A player who misses most of the season will have a WAR near zero. 

In 2013, Mark Teixeira played only 15 games with a wrist injury before undergoing surgery and losing the rest of the season. He hit only .151 in those games, which would explain his negative (-0.2) WAR. However, it's only 53 AB -- even if Texeira had hit .351, his WAR would still have been close to zero. 

A-Rod missed most of the year with hip problems. Roy Halladay pitched only 62 innings as he struggled with shoulder and back problems, and retired at the end of the season.

If we take out those three, the remaining 17 players average out to around 2.6 WAR, at an average salary of $22 million. It works out to about $8.4 million per win. That's still expensive -- well above the presumed willingness-to-pay of $5 to $6 million per expected win.

If we *don't* take out those three, it's about $10 million per win. Even more expensive, but hardly suggestive of a wide disconnect between pay and performance. At best, it suggests that the one year's group of highest-paid players performed worse than anticipated, but still better than their lower-paid peers.

Furthermore: as the author acknowledges, many of these players have back-loaded contracts, where they are "underpaid" relative to their expected year's talent earlier in the contract, and "overpaid" relative to their expected year's talent later in the contract. 

Even a contract at a constant salary is back-loaded in terms of talent, since older players tend to decline in value as they age. I'm sure the Yankees didn't expect Alex Rodriguez to perform at age 37 nearly as well as he did at 33, even though his salary was comparable ($28MM to $33MM).

All things considered, the top-20 data is very good evidence of a strong link between pay and performance in baseball. Not as strong as I would have expected, but still pretty strong.

-------

As further evidence that pay is divorced from performance, the author notes that, even limiting the analysis to players who have free-agent status, "performance explains just 13 percent of salary."  It's not just a one-year fluke. For each of the past 30 years, the r-squared has consistently hovered in a narrow band between 10 and 20 percent.

That sounds damning, but, as is often the case, it's based on a misinterpretation of what the r-squared means. 

Taking the square root of .13 gives a correlation of .36. That's not too bad: it means that 36 percent of a player's salary (above or below average) is reflected in (above- or  below-average) performance.

Still, you do have to regress salary almost 64 percent to the mean to get performance. Doesn't that show that almost two-thirds of a player's salary is unrelated to merit?

No. It shows most of a player's salary is unrelated to *performance,* not that it's unrelated to *merit*. Performance is based on merit, but with lots of randomness piled on top that tends to dilute the relationship.

You might underestimate the amount of randomness relative to talent, especially if you're still thinking of those top-20 players. But most players in MLB are not far from the league minimum, both in salary and talent.

According to the article, the 358 lowest-paid players in baseball in 2013 made an average $534,000 each. 

With a league minimum of $500,000, those 358 players must be clustered very tightly together in pay. And the range of their talent is probably also fairly narrow. But the range of their performance will be wide, since they'll vary in how much playing time they get, and whether they have a lucky or unlucky year. 

For those 358 players alone, the correlation between pay and performance is going to be very close to zero, even if pay and talent correlate perfectly. (Actually, the author's numbers are based on only players with 6+ seasons in MLB, so it's a smaller sample size than 358 -- but the logic is the same.)

When you add in the rest of the players, and the correlation rises to 0.36 ... that's pretty good evidence that there's a strong link between pay and performance overall. And when you take into account that there's also significant randomness in the performances of the highly-paid players, it must be that the link between pay and *merit* is even higher.

------

The author has demonstrated the "low r-squared" fallacy -- the idea that if the number looks low enough, the relationship must be weak enough to dismiss. As I have argued many times, that's not necessarily the case. Without context or argument, the "13 percent" figure could mean anything at all.

In fact, here's a situation where you have an r-squared much lower than .13, but a strong relationship between pay and performance.

Suppose that player salary were somehow exactly proportional to performance. That is, at the end of the season, the r-squared turned out to be 100 percent, instead of 13 percent. (Or some number close enough to 100 percent to satisfy the author.)

In baseball, as in life, people don't perform exactly the same every day. Some days, Mike Trout will be the highest-paid player in baseball, but he'll still wind up going 0-for-4 with three strikeouts.

So even if the correlation between season pay and season performance is 100% perfect, the correlation between *single game* pay and *single game* performance will be lower.

How much lower?  I ran a test with real data. I compiled batter stats for every game of the 2008 season, and ran a regression between the player's on-base percentage (OBP) for that single game, versus his OBP for the season. 

The correlation was .016. That's an r-squared of .000265.

The r-squared of .13 the article found between pay and performance is almost *five hundred times* as large as the one I found between pay and performance. 

Even though my r-squared is tiny, we can agree that Mike Trout is still paid on merit, right? It would be hard to argue that there was a fundamental inequity in MLB pay practices for April 11, just because Mike Trout didn't produce that day.

Well, I suppose, on a technicality, you could argue that pay isn't based on merit for a game, but *is* based on merit for a season. But if you make that argument for game vs. season, you can make the same argument for season vs. expectation, or season vs. career. 

The r-squared might be only 13 percent for a single season, but higher for groups of seasons. Furthermore, if you could play the same season a million times over, luck would even out, performance would converge on merit, and the r-squared would move much closer to 100%.

And the article provides evidence of that! When the author repeated his regression by using the average of three seasons instead of one, the r-squared doubled -- now explaining "just a quarter of pay." An r-squared of 0.25 is a correlation of 0.5 -- half of performance now reflected in salary.

Half is a lot, considering the amount of luck in batting records, and taking into account that luck is much more important than talent for the bunch of players clustered at the bottom of the salary scale. 

Again, the article's own evidence is enough to refute its argument.

-------

I think we can quantify the amount of luck in a batter's season WAR. 

A couple of years ago, I calculated the theoretical SD of a team's season Linear Weights batting line that's due to luck. It came out to 31.9 runs. 

Assuming a regular player gets one-ninth of a team's plate appearances, his own SD would be 1/3 the team's (the square root of 1/9). So, that's about 10.6 runs. Let's call it 10 runs, or 1.0 WAR. 

That one-win figure, though, counts only the kind of luck that results from over- or undershooting talent. It doesn't consider injuries, suspensions, or sudden unexpected changes in talent itself. 

Going back to the top 20 players in the chart ... we saw that three of those had injuries. Another three, it appears, had sudden drops in ability after they were signed (Vernon Wells, Tim Lincecum, and Barry Zito). 

Removing those six players from the list (which might be unfair selective sampling, but never mind for now), the remainder averaged 3.4 WAR. That's about $6.4 million per win -- very close to the consensus number. It would be even lower if we adjusted for back-loaded contracts.

At an SD of 1 WAR per player, the SD of the average of 14 players is 0.27 WAR. Actually, that's the minimum; it would be higher if any of the 14 were less than full-time. Also, the list includes starting pitchers -- I don't know if the luck SD of 1 win is reasonable for starters as well as batters, but I suspect it's close enough.

So, let's go with 0.27. We'll add and subtract 2 SD -- 0.54 --from the observed average of 3.4. That gives us a confidence interval of 2.9 to 3.9 WAR.

At 3.9 WAR, we get $5.6 million per win: almost exactly the amount sabermetricians (and probably front offices) have calculated based on the assumption that teams want to pay exactly what the talent is worth.

That is: it appears the results are not statistically significantly different from a pure "pay for performance" situation.

------

When the US News article talks about luck, it's different from the kind of luck I'm calculating here. The author isn't actually complaining that the overpaid players got unlucky and underperformed their pay. Instead, he believes that the highly-paid players were overpaid for their true ability, because they were "lucky" enough to fool everyone by having a career year at exactly the right time:

"In America, we tend to think of income as a reward for skill and hard work. ...

"But baseball shows us this view of the world is demonstrably flawed. Pay has preciously little to do with performance. Instead, being a top earner means having a good season immediately preceding free agency in a year where desperate, rich teams are willing to award outsized long-term contracts. ... 

"In other words, while ability and effort matter, it’s also about good luck."

Paraphrased, I think he's saying something like: "I've shown that pay is barely related to performance. Why, then, are some players paid huge sums of money, while others make the minimum?  It can't be merit. It must be that some players have a lucky year at a lucky time, and GMs don't realize the player doesn't deserve the money."

In other words: baseball executives are not capable of evaluating players well enough to realize that they're throwing away millions of dollars unnecessarily.   

The article gives no evidence to support that; and, furthermore, it appears that the author himself doesn't try, himself, to evaluate players and factor out luck. Otherwise, he wouldn't say this:

"But among average players, salaries vary enormously. For every Francisco Cervelli (Yankees catcher, $523,000 salary, 0.8 WAR), there is a CC Sabathia (Yankees pitcher, $24.7 million salary, 0.3 WAR). Both contribute about the same to the Yankees’ success (or lack thereof), but Sabathia earns roughly 50 times more."

Does he really believe that Sabathia and Cervelli should have been paid as equal talents?  Isn't it obvious that their 2013 records are similar only because of luck and circumstance?

Francisco Cervelli earned his +0.8 WAR in 61 plate appearances. That's about one-and-a-half SDs above +0.3, his then-career average per 61 PA.

Sabathia's salary took a jump after the 2009 season, at a time where he was averaging around 4 WAR per season. From 2010 to 2012, he actually improved that trend, creating +15.6 WAR total in those three years. It wasn't until 2013 that he suddenly lost effectiveness, dropping to 0.3 as reported. 

So it's not that Sabathia was just lucky to be in the right place at the right time. It's that he was an excellent player before and after signing his contract, but he suffered some kind of unexpected setback as he aged. (Too, his contract was structured to defer some of his peak years' value to his declining years.)

And it's not that Cervelli was unlucky to be in the wrong place at the wrong time, unable to find a "desperate" team otherwise willing to pay him $20 million. He's just a player recognized as not that much better than replacement, who had a good season in 2013 -- a "season" of 61 plate appearances where he was somewhat lucky.

-------

In his bio, the author is described as "a policy associate at The Century Foundation working on issues of income inequality." That's really what the article is getting at: income inequality. The argument that MLB pay is divorced from performance is there to support the broader argument that inequality of income is caused by highly-paid employees who don't deserve it.

Here's his argument summarized in his own words:

"The first thing to appreciate is just how unequal baseball is. During the 2013 season, the eight players in baseball's 'top 1 percent' took home $197 million, or $6 million more than the 358 lowest-paid players combined. The typical, or 'median,' major league player would need to play 20 seasons to earn as much as a top player makes in one. ...

"But ... pay has preciously little to do with performance. ...

"In other words, while ability and effort matter, it’s also about good luck. And if that’s true of a domain where every aspect of performance is meticulously measured, scrutinized and endlessly debated, how much more true is it of our society in general?

"We end up with CEOs that make 300 times the average worker and 45 million poor people in a country with $17 trillion in GDP. And we accept it as fair."

Paraphrasing again, the argument seems to be: "Salary inequality in baseball is bad because it's caused by teams rewarding ability that isn't really there. If baseball players were paid according to performance instead of circumstance, those disturbing levels of inequality would drop substantially, and the top 1% would no longer dominate."

It sounds reasonable, but it's totally backwards. If the correlation between pay and performance were higher, players' pay would become MORE unequal.

Suppose salaries were based directly on WAR. At the end of the season, the teams pay every free agent $6 million dollars for every win above zero, plus the $500,000 minimum. (That's roughly what they're paying now, on expectation. Since expected wins equal actual wins, that would keep the overall MLB free-agent payroll roughly the same.)

Well, if they did that the top salary would take a huge, huge jump.

Among the top 20 in the chart, the top two WAR figures are 7.5 (Miguel Cabrera) and 7.3 (Cliff Lee. 

Under the new salary scale, both players would get sharp increases. Cabrera would jump to $45 million, and Lee to $44 million. The highest salary in MLB would go to Carlos Gomez, whose 2013 season was worth 8.9 WAR (4.6 of that from defense). Under the new system, Gomez would earn some $53 million. 

Under pay-for-performance, it would take only around 4.8 WAR to earn more than the current real-life highest salary, A-Rod's $29.4 million. In 2013, that would have been accomplished by 32 players. 

Carlos Gomez's salary would exceed the real-life A-Rod by 82 percent. Meanwhile, replacement players would still be making the minimum $500K. And Barry Zito, with his negative 2.6 wins, would *owe* the Giants $15 million. 

Clearly, inequality would increase, not decrease, if the connection between pay and performance became stronger. 

Mathematically, that *has* to happen. When luck is involved, and applies equally to everyone, the set of outcomes always have a wider range than the set of talents. As usual,

var(outcomes) = var(talent) + var(luck)

Since var(luck) is always positive, outcomes always have a wider range than expectations based on talent. 

In fairness to the author, he doesn't think teams are paid by talent. As we saw, he believes teams pay by misinterpreting random circumstances, a "right place right time" or "team likes me" kind of good luck. 

If that's really happening, and you eliminate it by basing pay directly on measurable performance, then, yes, it's indeed possible for inequality to go down. If Francisco Cervelli were being paid $100 million per season, because he was Brian Cashman's favorite, then instituting straight pay-by-performance would lower the top salary from $100 million to $53 million, and inequality would decrease.

But, as we saw, that's not the case: the real-life top salaries are much lower than the "pay-by-performance" top salaries. That means that teams aren't systematically overpaying. Or, at least, that they're not overpaying by anything near as much as 82 percent.

-------

Imagine an alternate universe in which players have always been paid under the "new" system, $6 million per WAR. In that universe, as we have seen, the ratio between the top and median salaries is much higher than it is now, maybe 50 times instead of 20.

Then, someone comes along and presents a case for more equality:

"MLB salaries aren't as fair as they could be. They're based on outcomes, where they should be based on talent. Francisco Cervelli gets credit for 0.8 wins in 61 PA, even though we know he's not that good, and he just happened to guess right on a couple of pitches. 

"Players should be paid based on their established and expected performance, by selling their services to the highest bidder, before the season starts. That eliminates luck from the picture, and salaries will be based more on merit. The salary ratio will drop from 50 to 20, the range will compress, and the top players will earn only what they merit, not what they produce by luck."

Isn't THAT the situation that you'd expect someone to advocate if they were concerned about (a) rewarding on merit, (b) not rewarding on luck, and (c) reducing inequality of salaries?

Why, then, is this author advocating a move in the exact opposite direction?




(Hat tip: T.M.)

Income inequality and the Fed report

The New York Yankees are struggling. Why don't they sign Reggie Jackson? Sure, he's 68 years old, but he'd still be a productive hitter if the Yankees signed him today.

Why do I say that? Because if you look at the data, you'll see that players' production doesn't decline over time. In 1974, the Oakland A's hit .247. In 2013, they hit .254. Their hitting was just as good -- actually, better -- even thirty-nine years later!

So how can you argue that players don't age gracefully?

------

It's obvious what's wrong with that argument: the 2013 Oakland A's aren't the same players as the 1974 Oakland A's. The team got better, but the individual players got worse -- much, much worse. Comparing the two teams doesn't tell us anything at all about aging.

The problem is ridiculously easy to see here. But it's less obvious in most articles I've seen that discuss trends in income inequality, even though it's *exactly the same flaw*.

Recently, the US Federal Reserve ("The Fed") published their regular report on the country's income distribution (.pdf). Here's a New York Times article reporting on it, which says, 

"For the most affluent 10 percent of American families, average incomes rose by 10 percent from 2010 to 2013."

Well, that's not right. The Fed didn't actually study how family income changed over time. Instead, they looked at one random sample of families in 2010, and a *different* random sample of families in 2013.  

The confusion stems from how they gave the two groups the same name. Instead of "Oakland A's," they called them "Top 10 Percent". But those are different families in the two groups.

Take the top decile both years, and call it the "Washington R's." What the Fed report says is that the 2013 R's hit for an average 10 points higher than the 2010 R's. But that does NOT mean that the average 2010 R family gained 10 points. In fact, it's theoretically possible that the 2010 R's all got poorer, just like the 1974 Oakland A's all got worse. 

In one sense, the effect is stronger in the Fed survey than in MLB. If you're a .320 hitter who drops to .260 while playing for the A's, Billy Beane might still keep you on the team. But if you're a member of the 2010 R's, but wind up earning only an middle-class wage in 2013, the Fed *must* demote you to the minor-league M's, because you're not allowed to stay on the R's unless you're still top 10 percent. 

The Fed showed that the Rs, as a team, had a higher income in 2013 than 2010. The individual Rs? They might have improved, or they might have declined. There's no way of knowing from this data alone.

-----

So that quote from the New York Times is not justified. In fact, if even one family dropped out of the top decile from 2010 to 2013, you can prove, mathematically, that the statement must be false.

That has nothing to do with any other assumptions about wealth or inequality in general. It's true regardless, as a mathematical fact. 

Could it just be bad wording on the part of the Fed and the Times, that they understand this but just said it wrong? I don't think so. It sure seems like the Times writer believes the numbers apply to individuals. For instance, he also wrote, 

"There is growing evidence that inequality may be weighing on economic growth by keeping money disproportionately in the hands of those who already have so much they are less inclined to spend it."

The phrase "already have so much" implies the author thinks they're the same people, doesn't it? Change the context a bit. "Lottery winners picked up 10 percent higher jackpots in 2013 than 2010, keeping winnings disproportionately in the hands of those who already won so much."  

That would be an absurd thing to say for someone who realizes that the jackpot winners of 2013 are not necessarily the same people as the jackpot winners of 2010.

Anyway, I shouldn't fault the Times writer too much ... he's just accepting the incorrect statements he found in the Fed paper. 

And I don't think any of the misstatements are deliberate. I suspect that the Fed writers were sometimes careless in their phrasing, and sometimes genuinely thought that "team" declines/increases implied family declines/increases. 

Still, some of the statements, in both places, are clearly not justified by the data and should not have made it into print.

------

I've read articles in the past that made a similar point, that individuals and families might be improving significantly, even though the data appears to give the impression that their group is falling behind. 

It's not hard to think of an example of how that might be possible. 

Imagine that everyone gets richer every year. During the boom, immigration grows the population by 25 percent every year, and the new arrivals all start at $10 per hour.

What happens? 

(a) the lowest bottom 20 percent of every year earn the same amount; but 
(b) everyone gets richer every year

That is: *everyone* is better off *every year*, even though the data may make it falsely appear that the poor are stagnating.

(Note: the words "rich" and "poor" are defined as "high wealth" and "low wealth," but in this post, I'm also going to [mis]use them to mean "high income" and "low income."  It should be obvious from the context which one I mean.)

-------

Now, even if you agree with everything I've said so far, you could still have other reasons to be concerned about the Fed report. For me, the me, the most important fact is the discovery that 2013's poor (bottom quintile) have 8 percent less income than 2010's poor. 

You can't conclude that any particular family dropped, but you *can* conclude that, even if they're different people, the bottom families of 2013 are worse off than the bottom families of 2010. That's real, and that's something you could certainly be concerned about. 

But, many people, like the New York Times writer, aren't just concerned about the poorer families -- they worry about how "income inequality" compares them to the richer ones. They're uncomfortable with the growing distance between top and bottom, even in good times where the "rising tide" lifts everyone's income. For them, even if every individual is made better off, it's the inequality that bothers them, not the absolute levels of income, or even now fast overall income is growing. If the "Washington R's" gain 20 percent, but the "Oakland P's" gain only 5 percent ... for them, that's something to correct.

They might say something like,

"It's nice that the overall pie is growing, and it's nice that the "P's" are getting more money than they used to. But, still, every year, it seems like the high-income "team" is getting bigger increases than the low-income "team". There must be something wrong with a system where, years ago, the top-to-bottom ratio used to be 5-to-1, but now it's 10-to-1 or 15-to-1 or higher."

"Clearly, the rich are getting richer faster than the poor are getting richer. There must be something wrong with a system that benefits the rich so much while the poor don't keep up."

Rebutting that argument is the main point of this post. Here's what I'm going to try to convince you:

Even when the rich/poor ratio increases over time, that does NOT necessarily imply that the rich are getting more benefit than the poor. 

That is: *even if inequality is a bad thing*, it could still be that the changes in the income distribution have benefited the poor more than the rich.

I can even go further: even if ALL the benefits of increased income go to the poor, it's STILL possible for the rich/poor inequality gap to grow. The government could freeze the income of every worker in the top half, and increase the income of every worker in the bottom half. And even after that, the rich/poor income gap might still be *higher*.

-------

It seems that can't be possible. If everyone's income grows at the same rate, the ratio has to stay the same, right? If rich to poor is $200K / $20K one year, and rich and poor both double equally, you get $400K / $40K, and the ratio of 10:1 doesn't change. Mathematically, R/P has to equal xR/xP.

So if benefits that are equal keep the ratio equal, benefits that favor the poor have to change the ratio in favor of the poor. No? 

No, not necessarily. For instance:

Suppose that in 2017, the ratio between rich and poor is 1.25. In 2018, the ratio between rich and poor is 1.60. Pundits say, "this is because the system only benefited the rich!"

But it could be that the pundits have it 100% backwards, and the system actually only favored the poor. 

How? Here's one way. 

There are two groups, with equal numbers of people in each group. In 2017, everyone in the bottom group made $40K, and everyone in the top group made $50K. That's how the ratio between rich group and poor group was 1.25.

The government instituted a program to help the poor, the bottom group. Within a year, the income of the poor doubled, from $40K to $80K, while the top group stagnated at $50K. 

So, in 2018, the richest half of the population earned $80K, and the poorest half earned $50K. That's how inequality increased, from 1.25 to 1.60, only from helping the poor!

------

What happened? How did our intuition go wrong? For the same reason as before: we didn't immediately realize that the groups were different people in different years. The 2017 rich aren't the same as the 2018 rich.

When the pundits argued "the system only benefited the rich," whom did they mean? The "old" 2017 rich, or the "new" 2018 rich? Without specifying, the statement is ambiguous. So ambiguous, in fact, that it almost has no meaning.

What really happened is that the system benefited the old poor, who happen to be the new rich. It failed to benefit the old rich, which happen to be new poor.

Inequality increased from 1.25 to 1.60, but it's meaningless to say the increase benefited the "rich". Which rich? Obviously, it didn't benefit the "old rich."

But, isn't it true to say that the increase benefited the new rich? 

It's true, but it doesn't tell us much -- it's true by definition! In retrospect, ANY change will have benefited the "new rich" more than the "new poor."  If you used to be relatively poor, but now you're relatively rich, you must have benefited more than average. So when you say increasing inequality favors the "new rich," you're really saying "increasing inequality favors those who benefited the most from increasing inequality."  

These examples sound absurd, but they're exact illustrations of what's happening:

-- You have a program to help disadvantaged students go to medical school. Ten years later, you follow up, and they're all earning six-figure incomes as doctors. "Damn!" you say. "It turns out that in retrospect, we only helped the rich!"

-- Or, you do a study of people who won the lottery jackpot last year, and find that most of them are rich, in the top 5%. "Damn!" you say. "Lotteries are just a subsidy for the rich!"

-- Or, you do a study of people who were treated for cancer 10 years ago, and you find most of them are healthy. "Damn!" you say. We wasted cancer treatments on healthy patients!

It makes no sense at all to regret a sequence of events on the grounds that, in retrospect, it helped the people with better outcomes more than it helped the people with worse outcomes. Because, that's EVERY sequence of events!

If you want to complain that increasing inequality is disproportionately benefiting well-off people, that can make sense only if you mean it's those who were well off *before* the increase. But the Fed data doesn't give you any way of knowing whether that's true. It might be happening; it might not be happening. But the Fed data can't prove it either way.

----

Here's an example that's a little more realistic.

Suppose that in 2010, there are five income quintiles, where people earn $20K, $40K, $60K, $80K, and $100K, respectively. I'll call them "Poor," "Lower Class," "Middle Class," "Upper Class," and "Rich", for short. We'll measure inequality by the R/P ratio, which is 5 (100 divided by 20).

Using three representative people in each group, here's what the distribution looks like:


2010 group, 2010 income
------------------------
P    L    M    U    R
------------------------
20   40   60   80   100
20   40   60   80   100
20   40   60   80   100
------------------------
R/P ratio: 5


From 2010 to 2013, people's incomes change, for the usual reasons -- school, life events, luck, shocks to the economy, whatever. In each group, it turns out that one-third of people make double what they did before, one third experience no change, and one third see their incomes drop in half. 

Overall, that means incomes have grown by 16.7%: the average of +100%, 0%, and -50%. Workers have 1/6 more income, overall. But the change gets spread unevenly, since life is unpredictable.

Here are the 2013 incomes, but still based on the 2010 grouping. The top row are the people who dropped, the middle row are the status quo, and the bottom row are the ones who doubled.


2010 group, 2010 income
------------------------
P    L    M     U     R
------------------------
10   20   30    40    50
20   40   60    80   100
40   80  120   160   200
------------------------
R/P ratio: 5


You can easily calculate that every 2010 group got, on average, the same 16.7% increase. So, since life treated the groups equally, the 2010 rich/2010 poor ratio is still 5. In chart form:


2010 group, % change 2010-2013
------------------------------
 P     L     M     U     R  
------------------------------
+17%  +17%  +17%  +17%  +17%


But the Fed doesn't have any of those numbers, because it doesn't know which 2010 group the 2013 earners fell into. It just takes the 2013 data, and mixes it into brand new groups based on 2013 income:


2013 group, 2013 income
-------------------------
P    L     M     U     R  
-------------------------
10   30    40    80   120
20   40    50    80   160
20   40    60   100   200
-------------------------
R/P ratio: 9.6


What does the Fed find? Much more inequality in 2013 than in 2010. The ratio between rich and poor is 9.6 -- almost double what it was! 

The Fed method will also see that the bottom three groups are earning less than the corresponding group earned three years previous.  Only the top two groups, the "upper class" and "rich," are higher. Here are the changes between each new group and the corresponding old group:


Perceived change 2010-2013
--------------------------
 P    L    M    U    R  
--------------------------
-17% -8%  -8%  +8%  +60%


If you don't think about what's going on, you might be alarmed. You might conclude that none of the economy's growth benefited the lowest 60 percent at all -- that all the benefits accrued to the well off! 

But, that's not right: as we saw, the benefits accrued equally. And, as we saw, the "R" group ALWAYS has to be high, by definition, since it's selectively comprised of those who benefited the most!

In effect, comparing the 2010 sample to the 2013 sample is a subtle "cheat," creating an illusion that can be used (perhaps unwittingly) to falsely exaggerate the differences. When the poor improve their lot, the method moves them to another group, and winds up ignoring that they benefited. 

For instance, when a $30K earner moves to $90K, a $90K earner moves to $120K, and a $120K earner drops to $30K, the Fed method makes it look like they all benefited equally, at zero. In reality, the "poor" gained and the "rich" declined -- the $30K earner grew 200%, the $90K earner grew 33%, and the $120K earner dropped by -75%. 

No matter how you choose the numbers, as long as there is any movement between groups, the method will invariably overestimate how much the "rich" benefited, and underestimate how much the "poor" benefited. It never works the other way.

--------

One last example.

This time, let's institute a policy that does something special for the disadvantaged groups, to try to make society more equal. For everyone in the P and L group in 2010, we institute a program that will double their eventual 2013 income. Starting with the same 20/40/60/80/100 distribution for 2010, here's what we see after the 2013 doubling:


2010 group, 2013 income
-----------------------
P     L    M    U    R  
-----------------------
20    40   30   40   50
40    80   60   80  100
80   160  120  160  200
-----------------------
R/P ratio: 2.5


Based on the 2010 classes, we've cut the rich/poor ratio in half! But, as usual, the Fed doesn't know the 2010 classes, so they sort the data this way:


2013 group, 2013 income
-----------------------
P    L    M    U     R  
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20    40  60    80  160
30    40  80   100  160
40    50  80   120  200
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R/P ratio: 5.8


Inequality has jumped from 5.0 to 5.8. That's even after we made a very, very serious attempt to lower it, doubling the incomes of the previous poorest 40 percent of the population!

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There's an easy, obvious mathematical explanation of why this happens.

When you look at income inequality, you're basically looking at the variance of the income distribution. But, changes from year-to-year are not equal, so they have their own built-in variance.

If the changes in income are independent of where you started -- that is, if the system treats rich and poor equally, in terms of unpredictability -- then

var(next year) = var(this year) + var(changes)

Which means, as long as rich and poor are equal in how their incomes change, inequality HAS TO INCREASE. 

Take 100 people, start them with perfect equality, $1 each. 

Every day, they roll a pair of dice. They multiply their money by the amount of the roll, then divide by 7. 

Obviously, on Day 2, equality disappears: some people will have $12/7, while others will have only $2/7. The third day, they'll be even more unequal. The fourth day, even more so. Eventually, some of them will be filthy, filthy rich, having more money than exists on the planet, while others will have trillionths of a dollar, or less.

That's just the arithmetic of variation. Increasing inequality is what happens naturally, not just in incomes, but in everything -- everything where things change independently of each other and independently over time. 

What if you want to fight nature, and keep inequality from growing? You have to arrange for year-to-year changes to benefit the poor more than the rich. That effect has to be large -- as we saw earlier, doubling the income of the 40 poorest percent wasn't enough. (It was a contrived example, but, still, it sure *seemed* like it should have been enough!)

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How much do you have to tilt the playing field in favor of the poor? Thinking out loud, scrawling equations ... I didn't double-check, so try this yourself because I may have screwed up ... but here's what I got:

Without independence, 

var(next year) = var(this year) + var(changes) + 2 cov(this year, changes)

Solving on the back of my envelope ... if I've done it right, using logarithm of income and some rough assumptions ... I get that the correlation between this year's income and the change to next year's income has to be around -0.25.

My scrawls say that if you're in the top 2.5% of income, your next-year change has to be in the bottom 30%. And if you're in the bottom 2.5%, your next-year change has to be in the top 30%. 

That seems really tough to do. In a typical year that the economy grows normally, what percentage of incomes in the Fed survey would be lower than last year's? If it's 30 percent, then ... to keep inequality constant, just ONE of the things you need to do is make sure high-income people, on average, never earn more this year than last year.

You'd almost have to repeal compound interest!

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I don't mean to imply that increasing inequality is *completely* just the result of normal variation. There are lots of other factors. Progressive taxation creates a small effect on equality. Increased savings while the economy grows contributes to inequality. A growing population means that inequality increases where bestselling authors have a larger market. And so on. 

But the point is: because increasing inequality happens naturally, you can't conclude anything just from *the fact that there's an increase*. At the very least, you have to back out the natural effects if you want to really explain what's going on. You have to do some math, and some arguing. 

The argument, "Inequality is growing -- therefore, we must be unfairly favoring the rich" is not a valid one. It is true that inequality is growing. And it *might* be true that we are unfairly favoring the rich. But, the one doesn't necessarily follow from the other. 

It's like saying, "Philadelphia was warmer in June than April; therefore, global warming must be happening."

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Again, I'm not trying to argue that inequality is a good thing, or that you shouldn't be concerned about it. Rather, I'm arguing that increasing inequality does NOT tell you anything reliable about who benefits from the "system" or how much (if at all) the increase favors the rich over the poor.

I am arguing that, even if you think increasing inequality is a bad thing, the following are still, objectively, true:

-- increasing inequality is a natural mathematical consequence of variation;
-- it is not necessarily the result of any deliberate government policy;
-- it does not necessarily disproportionately favor the rich or hurt the poor;
-- there is no way to know which individuals it favors just from the Fed data;
-- the natural forces that cause inequality to increase are very strong;
-- natural inequality growth may be so strong that it will persist even after successful attempts to benefit the poor generously and significantly;
-- the poor could be gaining relative to the rich even while measured inequality increases.

As for the Fed study itself,

-- the Fed statistics do not measure income changes for any family or specific group of families;
-- the Fed statistics that measure distributional income changes for percentile groups are a biased, exaggerated estimate of the income changes for the average family starting in that percentile;
-- It is impossible to tell, from the Fed's numbers, how the poor are faring relative to the rich.

Finally, and most importantly,

-- all of these statements follow necessarily from basic logic and math -- and do not require any other arguments from politics, economics, compassion, greed, fairness, or partisanship.