# Inequality-adjusted GDP and median income

Dec 11 JDN 2459925

There are many problems with GDP as a measure of a nation’s prosperity. For one, GDP ignores natural resources and ecological degradation; so a tree is only counted in GDP once it is cut down. For another, it doesn’t value unpaid work, so caring for a child only increases GDP if you are a paid nanny rather than the child’s parents.

But one of the most obvious problems is the use of an average to evaluate overall prosperity, without considering the level of inequality.

Consider two countries. In Alphania, everyone has an income of about \$50,000. In Betavia, 99% of people have an income of \$1,000 and 1% have an income of \$10 million. What is the per-capita GDP of each country? Alphania’s is \$50,000 of course; but Betavia’s is \$100,990. Does it really make sense to say that Betavia is a more prosperous country? Maybe it has more wealth overall, but its huge inequality means that it is really not at a high level of development. It honestly sounds like an awful place to live.

A much more sensible measure would be something like median income: How much does a typical person have? In Alphania this is still \$50,000; but in Betavia it is only \$1,000.

Yet even this leaves out most of the actual distribution; by definition a median is only determined by what is the 50th percentile. We could vary all other incomes a great deal without changing the median.

A better measure would be some sort of inequality-adjusted per-capita GDP, which rescales GDP based on the level of inequality in a country. But we would need a good way of making that adjustment.

I contend that the most sensible way would be to adopt some kind of model of marginal utility of income, and then figure out what income would correspond to the overall average level of utility.

In other words, average over the level of happiness that people in a country get from their income, and then figure out what level of income would correspond to that level of happiness. If we magically gave everyone the same amount of money, how much would they need to get in order for the average happiness in the country to remain the same?

This is clearly going to be less than the average level of income, because marginal utility of income is decreasing; a dollar is not worth as much in real terms to a rich person as it is to a poor person. So if we could somehow redistribute all income evenly while keeping the average the same, that would actually increase overall happiness (though, for many reasons, we can’t simply do that).

For example, suppose that utility of income is logarithmic: U = ln(I).

This means that the marginal utility of an additional dollar is inversely proportional to how many dollars you already have: U'(I) = 1/I.

It also means that a 1% gain or loss in your income feels about the same regardless of how much income you have: ln((1+r)Y) = ln(Y) + ln(1+r). This seems like a quite reasonable, maybe even a bit conservative, assumption; I suspect that losing 1% of your income actually hurts more when you are poor than when you are rich.

Then the inequality adjusted GDP Y is a value such that ln(Y) is equal to the overall average level of utility: E[U] = ln(Y), so Y = exp(E[U]).

This sounds like a very difficult thing to calculate. But fortunately, the distribution of actual income seems to quite closely follow a log-normal distribution. This means that when we take the logarithm of income to get utility, we just get back a very nice, convenient normal distribution!

In fact, it turns out that for a log-normal distribution, the following holds: exp(E[ln(Y)]) = median(Y)

The income which corresponds to the average utility turns out to simply be the median income! We went looking for a better measure than median income, and ended up finding out that median income was the right measure all along.

This wouldn’t hold for most other distributions; and since real-world economies don’t perfectly follow a log-normal distribution, a more precise estimate would need to be adjusted accordingly. But the approximation is quite good for most countries we have good data on, so even for the ones we don’t, median income is likely a very good estimate.

The ranking of countries by median income isn’t radically different from the ranking by per-capita GDP; rich countries are still rich and poor countries are still poor. But it is different enough to matter.

Luxembourg is in 1st place on both lists. Scandinavian countries and the US are in the top 10 in both cases. So it’s fair to say that #ScandinaviaIsBetter for real, and the US really is so rich that our higher inequality doesn’t make our median income lower than the rest of the First World.

But some countries are quite different. Ireland looks quite good in per-capita GDP, but quite bad in median income. This is because a lot of the GDP in Ireland is actually profits by corporations that are only nominally headquartered in Ireland and don’t actually employ very many people there.

The comparison between the US, the UK, and Canada seems particularly instructive. If you look at per-capita GDP PPP, the US looks much richer at \$75,000 compared to Canada’s \$57,800 (a difference of 29% or 26 log points). But if you look at median personal income, they are nearly equal: \$19,300 in the US and \$18,600 in Canada (3.7% or 3.7 log points).

On the other hand, in per-capita GDP PPP, the UK looks close to Canada at \$55,800 (3.6% or 3.6 lp); but in median income it is dramatically worse, at only \$14,800 (26% or 23 lp). So Canada and the UK have similar overall levels of wealth, but life for a typical Canadian is much better than life for a typical Briton because of the higher inequality in Britain. And the US has more wealth than Canada, but it doesn’t meaningfully improve the lifestyle of a typical American relative to a typical Canadian.

# The Rent is Too Damn High

Housing prices are on the rise again, but they’re still well below what they were at the peak of the 2008 bubble. It may be that we have not learned from our mistakes and another bubble is coming, but I don’t think it has hit us just yet. Meanwhile, rent prices have barely budged, and the portion of our population who pay more than 35% of their income on rent has risen to 44%.

Economists typically assess the “fair market value” of a house based upon its rental rate for so-called “housing services”—the actual benefits of living in a house. But to use the rental rate is to do what Larry Summers called “ketchup economics”; 40-ounce bottles of ketchup sell for exactly twice what 20-ounce bottles do, therefore the ketchup market is fair and efficient. (In fact even this is not true, since ketchup is sold under bulk pricing. This reminds me of a rather amusing situation I recently encountered at the grocery store: The price of individual 12-packs of Coke was \$3, but you could buy sets of five for \$10 each. This meant that buying five was cheaper in total—not just per unit—than buying four. The only way to draw that budget constraint is with a periodic discontinuity; it makes a sawtooth across your graph. We never talk about that sort of budget constraint in neoclassical economics, yet there it was in front of me.)

When we value houses by their rental rate, we’re doing ketchup economics. We’re ignoring the fact that the rent is too damn highpeople should not have to pay as much as they do in order to get housing in this country, particularly housing in or near major cities. When 44% of Americans are forced to spend over a third of their income just fulfilling the basic need of shelter, something is wrong. Only 60% of the price of a house is the actual cost to build it; another 20% is just the land. If that sounds reasonable to you, you’ve just become inured to our absurd land prices. The US has over 3 hectares per person of land; that’s 7.7 acres. A family of 3 should be able to claim—on average—9 hectares, or 23 acres. The price of a typical 0.5-acre lot for a family home should be negligible; it’s only 2% of your portion of America’s land.

And as for the argument that land near major cities should be more expensive? No, it shouldn’t; it’s land. What should be more expensive near major cities are buildings, and only then because they’re bigger buildings—even per unit it probably is about equal or even an economy of scale. There’s a classic argument that you’re paying to have infrastructure and be near places of work: The former is ignoring the fact that we pay taxes and utilities for that infrastructure; and the latter is implicitly assuming that it’s normal for our land ownership to be so monopolistic. In a competitive market, the price is driven by the cost, not by the value; the extra value you get from living near a city is supposed to go into your consumer surplus (the personal equivalent of profit—but in utility, not in dollars), not into the owner’s profit. And actually that marginal benefit is supposed to be driven to zero by the effect of overcrowding—though Krugman’s Nobel-winning work was about why that doesn’t necessarily happen and therefore we get Shanghai.

There’s also a more technical argument to be had here about the elasticity of land supply and demand; since both are so inelastic, we actually end up in the very disturbing scenario in which even a small shift in either one can throw prices all over the place, even if we are at market-clearing equilibrium. Markets just don’t work very well for inelastic goods; and if right now you’re thinking “Doesn’t that mean markets won’t work well for things like water, food, and medicine?” you’re exactly right and have learned well, Grasshopper.

So, the rent is too damn high. This naturally raises three questions:

1. Why is the rent so high?
2. What happens to our economy as a result?
3. What can we do about it?

Let’s start with 1. Naturally, conservatives are going to blame regulation; here’s Business Insider doing exactly that in San Francisco and New York City respectively. Actually, they have a point here. Zoning laws are supposed to keep industrial pollution away from our homes, not keep people from building bigger buildings to fit more residents. All these arguments about the “feel” of the city or “visual appeal” should be immediately compared to the fact that they are making people homeless. So 200 people should live on the street so you can have the skyline look the way you always remember it? I won’t say what I’d really like to; I’m trying to keep this blog rated PG.

Similarly, rent-control is a terrible way to solve the homelessness problem; you’re created a segregated market with a price ceiling, and that’s going to create a shortage and raise prices in the other part of the market. The result is good for anyone who can get the rent-control and bad for everyone else. (The Cato study Business Insider cites does make one rather aggravating error; the distribution in a non-rent-controlled market isn’t normal, it’s lognormal. You can see that at a glance by the presence of those extremely high rents on the right side of the graph.)

Most people respond by saying, “Okay, but what do we do for people who can’t afford the regular rent? Do we just make them homeless!?” I wouldn’t be surprised if the Cato Institute or Business Insider were okay with that—but I’m definitely not. So what would I do? Give them money. The solution to poverty has been staring us in the face for centuries, but we refuse to accept it. Poor people don’t have enough money, so give them money. Skeptical? Here are some direct experimental studies showing that unconditional cash transfers are one of the most effective anti-poverty measures. The only kind of anti-poverty program I’ve seen that has a better track record is medical aid. People are sick? Give them medicine. People are poor? Give them money. Yes, it’s that simple. People just don’t want to believe it; they might have to pay a bit more in taxes.

So yes, regulations are actually part of the problem. But they are clearly not the whole problem, and in my opinion not even the most important part. The most important part is monopolization. There’s a map that Occupy Wall Street likes to send around saying “What if our land were as unequal as our money?” But here’s the thing: IT IS. Indeed, the correlation between land ownership and wealth is astonishingly high; to a first approximation, your wealth is a constant factor times the land you own.

Remember how I said that the average American holds 7.7 acres or 3 hectares? (Especially in economics, averages can be quite deceiving. Bill Gates and I are on average billionaires. In fact, I guarantee that Bill Gates and you are on average billionaires; it doesn’t even matter how much wealth you have, it’ll still be true.)

Well, here are some decidedly above-average landowners:

1. John Malone, 2.2 million acres or 9,000 km^2
2. Ted Turner, 2 million acres or 8,100 km^2
3. The Emmerson Family, 1.9 million acres or 7,700 km^2
4. Brad Kelley, 1.5 million acres or 6,100 km^2
1. The Pingree Family, 800,000 acres or 3,200 km^2
1. The Ford Family, 600,000 acres or 2,400 km^2
1. The Briscoe Family, 560,000 acres or 2,270 km^2
2. W.T. Wagonner Estate, 535,000 acres or 2,170 km^2

I think you get the idea. Here are two more of particular note:

1. Jeff Bezos, 290,000 acres or 1,170 km^2
1. Koch Family, 239,000 acres or 970 km^2

Yes, that is the Jeff Bezos of Amazon.com and the Koch Family who are trying to purchase control of our political system.

Interpolating the ones I couldn’t easily find data on, I estimate that these 102 landowners (there were ties in the top 100) hold a total of 30 million acres, of the 940 million acres in the United States. This means that 3% of the land is owned by—wait for it—0.000,03% of the population. To put it another way, if we confiscated the land of 102 people and split it all up into 0.5-acre family home lots, we could house 60 million households—roughly half the number of households in the nation. To be fair, some of it isn’t suitable for housing; but a good portion of it is. Figure even 1% is usable; that’s still enough for 600,000 households—which is to say every homeless person in America.

One thing you may also have noticed is how often the word “family” comes up. Using Openoffice Calc (it’s like Excel, but free!) I went through the whole top 100 list and counted the number of times “family” comes up; it’s 49 out of 100. Include “heirs” and “estate” and the number goes up to 66. That doesn’t mean they share with their immediate family; it says “family” when it’s been handed down for at least one generation. This means that almost two-thirds of these super-wealthy landowners inherited their holdings. This isn’t the American Dream of self-made millionaires; this is a landed gentry. We claim to be a capitalist society; but if you look at who owns our land and how it’s passed down, it doesn’t look like capitalism. It looks like feudalism.

Indeed, the very concept of rent is basically feudalist. Instead of owning the land we live on, we have to constantly pay someone else—usually someone quite rich—for the right to live there. Stop paying, and they can call the government to have us forced out. We are serfs by another name. In a truly efficient capitalist market with the kind of frictionless credit system neoclassicists imagine, you wouldn’t pay rent, you’d always pay a mortgage. The only time you’d be paying for housing without building equity would be when you stay at a hotel. If you’re going to live there more than a month, you should be building equity. And if you do want to move before your mortgage ends? No problem; sell it to the next tenant, paying off your mortgage and giving you that equity back—instead of all that rent, which is now in someone else’s pocket.

Because of this extreme inequality in land distribution, the top landholders can charge the rest of us monopolistic prices—thus making even more profits and buying even more land—and we have little choice but to pay what they demand. Because shelter is such a fundamental need, we are willing to pay just about whatever we have in order to secure it; so that’s what they charge us.

On to question 2. What happens to our economy as a result of this high rent?

In a word: 2009. Because our real estate market is so completely out of whack with any notion of efficient and fair pricing, it has become a free-for-all of speculation by so-called “investors”. (I hate that term; real investment is roads paved, factories built, children taught. What “investors” do is actually arbitrage. We are the investors, not them.)

A big part of this was also the deregulation of derivatives, particularly the baffling and insane “Commodity Futures Modernization Act of 2000” that basically banned regulation of derivatives—it was a law against making laws. Because of this bankers—or should I say banksters—were able to create ludicrously huge amounts of derivatives, as well as structure and repackage them in ways that would deceive their buyers into underestimating the risks. As a result there are now over a quadrillion dollars—yes, with a Q, sounds like a made-up number, \$2e15—in nominal value of outstanding derivatives.

Because this is of course about 20 times as much as there is actual money in the entire world, sustaining this nominal value requires enormous amounts of what’s called leverage—which is to say, debt. When you “leverage” a stock purchase, for example, what you’re doing is buying the stock on a loan (a generally rather low-interest loan called “margin”), then when you sell the stock you pay back the loan. The “leverage” is the ratio between the size of the loan and the amount of actual capital you have to spend. This can theoretically give you quite large returns; for instance if you have \$2000 in your stock account and you leverage 10 to 1, you can buy \$20,000 worth of stock. If that stock then rises to \$21,000—that’s only 5%, so it’s pretty likely this will happen—then you sell it and pay back the loan. For this example I’ll assume you pay 1% interest on your margin. In that case you would start with \$2000 and end up with \$2800; that’s a 40% return. A typical return from buying stock in cash is more like 7%, so even with interest you’re making almost 6 times as much. It sounds like such a deal!

But there is a catch: If that stock goes down and you have to sell it before it goes back up, you need to come up with the money to pay back your loan. Say it went down 5% instead of up; you now have \$19,000 from selling it, but you owe \$20,200 in debt with interest. Your \$2000 is already gone, so you now have to come up with an additional \$1,200 just to pay back your margin. Your return on \$2000 is now negative—and huge: -160%. If you had bought the stock in cash, your return would only have been -5% and you’d still have \$1900.

My example is for a 10 to 1 leverage, which is considered conservative. More typical leverages are 15 or 20; and some have gotten as high as 50 or even 70. This can lead to huge returns—or huge losses.

But okay, suppose we rein in the derivatives market and leverage gets back down to more reasonable levels. What damage is done by high real estate prices per se?

Well, basically it means that too much of our economy’s effort is going toward real estate. There is what we call deadweight loss, the loss of value that results from an inefficiency in the market. Money that people should be spending on other things—like cars, or clothes, or TVs—is instead being spent on real estate. Those products aren’t getting sold. People who would have had jobs making those products aren’t getting hired. Even when it’s not triggering global financial crises, a market distortion as large as our real estate system is a drain on the economy.

The distorted real estate market in particular also has another effect: It keeps the middle class from building wealth. We have to spend so much on our homes that we don’t have any left for stocks or bonds; as a result we earn a very low return on investment—inflation-adjusted it’s only about 0.2%. So meanwhile the rich are getting 4% on bonds, or 7% on stocks, or even 50% or 100% on highly-leveraged derivatives. In fact, it’s worse than that, because we’re also paying those rich people 20% on our credit cards. (Or even worse, 400% on payday loans. Four hundred percent. You typically pay a similar rate on overdraft fees—that \$17.5 billion has to come from somewhere—but fortunately it’s usually not for long.)

Most people aren’t numerate enough to really appreciate how compound interest works—and banks are counting on that. 7%, 20%, what’s the difference really? 3 times as much? And if you had 50%, that would be about 7 times as much? Not exactly, no. Say you start with \$1000 in each of these accounts. After 20 years, how much do you have in the 7% account? \$3,869.68. Not too shabby, but what about that 20% account? \$38,337.60—almost ten times as much. And if you managed to maintain a 50% return, how much would you have? \$3,325,256.73—over \$3.3 million, almost one thousand times as much.

The problem, I think, is people tend to think linearly; it’s hard to think exponentially. But there’s a really nice heuristic you can use, which is actually quite accurate: Divide the percentage into 69, and that is the time it will take to double. So 3% would take 69/3 = 23 years to double. 7% would take 69/7 = 10 years to double. 35% would take 69/35 = 2 years to double. And 400% would take 69/400 = 0.17 years (about 1/6, so 2 months) to double. These doublings are cumulative: If you double twice you’ve gone up 4 times; if you double 10 times you’ve gone up 1000 times. (For those who are a bit more numerate, this heuristic comes from the fact that 69 ~ 100*ln(2).)

Since returns are so much higher on other forms of wealth (not gold, by the way; don’t be fooled) than on homes, and those returns get compounded over time, this differential translates into ever-increasing inequality of wealth. This is what Piketty is talking about when he says r > g; r is the return on capital, and g is the growth rate of the economy. Stocks are at r, but homes are near g (actually less). By forcing you to spend your wealth on a house, they are also preventing you from increasing that wealth.

Finally, time for question 3. What should we do to fix this? Again, it’s simple: Take the land from the rich. (See how I love simple solutions?) Institute a 99% property tax on all land holdings over, say, 1000 acres. No real family farmer of the pastoral sort (as opposed to heir of an international agribusiness) would be affected.

I’m sure a lot of people will think this sounds unfair: “How dare you just… just… take people’s stuff! You… socialist!” But I ask you: On what basis was it theirs to begin with? Remember, we’re talking about land. We’re not talking about a product like a car, something they actually made (or rather administrated the manufacturing of). We’re not even talking about ideas or services, which raise their own quite complicated issues. These are chunks of the Earth; they were there a billion years before you and they will probably still be there a billion years hence.

That land was probably bought with money that they obtained through monopolistic pricing. Even worse, whom was it bought from? Ultimately it had to be bought from the people who stole it—literally stole, at the point of a gun—from the indigenous population. On what basis was it theirs to sell? And even the indigenous population may not have obtained it fairly; they weren’t the noble savages many imagine them to be, but had complex societies with equally complex political alliances and histories of intertribal warfare. A good portion of the land that any given tribe claims as their own was likely stolen from some other tribe long ago.

It’s honestly pretty bizarre that we buy and sell land; I think it would be valuable to think about how else we might distribute land that didn’t involve the absurdity of owning chunks of the planet. I can’t think of a good alternative system right now, so okay, maybe as a pragmatic matter the economy just works most efficiently if people can buy and sell land. But since it is a pragmatic justification—and not some kind of “fundamental natural right” ala Robert Nozick—then we are free as a society—particularly a democratic society—to make ad hoc adjustments in that pragmatic system as is necessary to make people’s lives better. So let’s take all the land, because the rent is too damn high.