Malawi

Markets value rich people more

pnrj core principles, critique of neoclassical economics, inequality , , , , , ,

Feb 26, JDN 2457811

Competitive markets are optimal at maximizing utility, as long as you value rich people more.

That is literally a theorem in neoclassical economics. I had previously thought that this was something most economists didn’t realize; I had delusions of grandeur that maybe I could finally convince them that this is the case. But no, it turns out this is actually a well-known finding; it’s just that somehow nobody seems to care. Or if they do care, they never talk about it. For all the thousands of papers and articles about the distortions created by minimum wage and capital gains tax, you’d think someone could spare the time to talk about the vastly larger fundamental distortions created by the structure of the market itself.

It’s not as if this is something completely hopeless we could never deal with. A basic income would go a long way toward correcting this distortion, especially if coupled with highly progressive taxes. By creating a hard floor and a soft ceiling on income, you can reduce the inequality that makes these distortions so large.

The basics of the theorem are quite straightforward, so I think it’s worth explaining them here. It’s extremely general; it applies anywhere that goods are allocated by market prices and different individuals have wildly different amounts of wealth.

Suppose that each person has a certain amount of wealth W to spend. Person 1 has W1, person 2 has W2, and so on. They all have some amount of happiness, defined by a utility function, which I’ll assume is only dependent on wealth; this is a massive oversimplification of course, but it wouldn’t substantially change my conclusions to include other factors—it would just make everything more complicated. (In fact, including altruistic motives would make the whole argument stronger, not weaker.) Thus I can write each person’s utility as a function U(W). The rate of change of this utility as wealth increases, the marginal utility of wealth, is denoted U'(W).

By the law of diminishing marginal utility, the marginal utility of wealth U'(W) is decreasing. That is, the more wealth you have, the less each new dollar is worth to you.

Now suppose people are buying goods. Each good C provides some amount of marginal utility U'(C) to the person who buys it. This can vary across individuals; some people like Pepsi, others Coke. This marginal utility is also decreasing; a house is worth a lot more to you if you are living in the street than if you already have a mansion. Ideally we would want the goods to go to the people who want them the most—but as you’ll see in a moment, markets systematically fail to do this.

If people are making their purchases rationally, each person’s willingness-to-pay P for a given good C will be equal to their marginal utility of that good, divided by their marginal utility of wealth:

P = U'(C)/U'(W)

Now consider this from the perspective of society as a whole. If you wanted to maximize utility, you’d equalize marginal utility across individuals (by the Extreme Value Theorem). The idea is that if marginal utility is higher for one person, you should give that person more, because the benefit of what you give them will be larger that way; and if marginal utility is lower for another person, you should give that person less, because the benefit of what you give them will be smaller. When everyone is equal, you are at the maximum.

But market prices don’t actually do this. Instead they equalize over willingness-to-pay. So if you’ve got two individuals 1 and 2, instead of having this:

U'(C1) = U'(C2)

you have this:

P1 = P2

which translates to:

U'(C1)/U'(W1) = U'(C2)/U'(W2)

If the marginal utilities were the same, U'(W1) = U'(W2), we’d be fine; these would give the same results. But that would only happen if W1 = W2, that is, if the two individuals had the same amount of wealth.

Now suppose we were instead maximizing weighted utility, where each person gets a weighting factor A based on how “important” they are or something. If your A is higher, your utility matters more. If we maximized this new weighted utility, we would end up like this:

A1*U'(C1) = A2*U'(C2)

Because person 1’s utility counts for more, their marginal utility also counts for more. This seems very strange; why are we valuing some people more than others? On what grounds?

Yet this is effectively what we’ve already done by using market prices.
Just set:
A = 1/U'(W)

Since marginal utility of wealth is decreasing, 1/U'(W) is higher precisely when W is higher.

How much higher? Well, that depends on the utility function. The two utility functions I find most plausible are logarithmic and harmonic. (Actually I think both apply, one to other-directed spending and the other to self-directed spending.)

If utility is logarithmic:

U = ln(W)

Then marginal utility is inversely proportional:

U'(W) = 1/W

In that case, your value as a human being, as spoken by the One True Market, is precisely equal to your wealth:

A = 1/U'(W) = W

If utility is harmonic, matters are even more severe.

U(W) = 1-1/W

Marginal utility goes as the inverse square of wealth:

U'(W) = 1/W^2

And thus your value, according to the market, is equal to the square of your wealth:

A = 1/U'(W) = W^2

What are we really saying here? Hopefully no one actually believes that Bill Gates is really morally worth 400 trillion times as much as a starving child in Malawi, as the calculation from harmonic utility would imply. (Bill Gates himself certainly doesn’t!) Even the logarithmic utility estimate saying that he’s worth 20 million times as much is pretty hard to believe.

But implicitly, the market “believes” that, because when it decides how to allocate resources, something that is worth 1 microQALY to Bill Gates (about the value a nickel dropped on the floor to you or I) but worth 20 QALY (twenty years of life!) to the Malawian child, will in either case be priced at $8,000, and since the child doesn’t have $8,000, it will probably go to Mr. Gates. Perhaps a middle-class American could purchase it, provided it was worth some 0.3 QALY to them.

Now consider that this is happening in every transaction, for every good, in every market. Goods are not being sold to the people who get the most value out of them; they are being sold to the people who have the most money.

And suddenly, the entire edifice of “market efficiency” comes crashing down like a house of cards. A global market that quite efficiently maximizes willingness-to-pay is so thoroughly out of whack when it comes to actually maximizing utility that massive redistribution of wealth could enormously increase human welfare, even if it turned out to cut our total output in half—if utility is harmonic, even if it cut our total output to one-tenth its current value.

The only way to escape this is to argue that marginal utility of wealth is not decreasing, or at least decreasing very, very slowly. Suppose for instance that utility goes as the 0.9 power of wealth:

U(W) = W^0.9

Then marginal utility goes as the -0.1 power of wealth:

U'(W) = 0.9 W^(-0.1)

On this scale, Bill Gates is only worth about 5 times as much as the Malawian child, which in his particular case might actually be too small—if a trolley is about to kill either Bill Gates or 5 Malawian children, I think I save Bill Gates, because he’ll go on to save many more than 5 Malawian children. (Of course, substitute Donald Trump or Charles Koch and I’d let the trolley run over him without a second thought if even a single child is at stake, so it’s not actually a function of wealth.) In any case, a 5 to 1 range across the whole range of human wealth is really not that big a deal. It would introduce some distortions, but not enough to justify any redistribution that would meaningfully reduce overall output.

Of course, that commits you to saying that $1 to a Malawian child is only worth about $1.50 to you or I and $5 to Bill Gates. If you can truly believe this, then perhaps you can sleep at night accepting the outcomes of neoclassical economics. But can you, really, believe that? If you had the choice between an intervention that would give $100 to each of 10,000 children in Malawi, and another that would give $50,000 to each of 100 billionaires, would you really choose the billionaires? Do you really think that the world would be better off if you did?

We don’t have precise measurements of marginal utility of wealth, unfortunately. At the moment, I think logarithmic utility is the safest assumption; it’s about the slowest decrease that is consistent with the data we have and it is very intuitive and mathematically tractable. Perhaps I’m wrong and the decrease is even slower than that, say W^(-0.5) (then the market only values billionaires as worth thousands of times as much as starving children). But there’s no way you can go as far as it would take to justify our current distribution of wealth. W^(-0.1) is simply not a plausible value.

And this means that free markets, left to their own devices, will systematically fail to maximize human welfare. We need redistribution—a lot of redistribution. Don’t take my word for it; the math says so.

Meanwhile, we’ve been ending world hunger.

pnrj development economics , , , , , , , , , , , ,

JDN 2457303 EDT 19:56

As reported in The Washington Post and Fortune, the World Bank recently released a report showing that for the first time on record—possibly the first time in human history—global extreme poverty has fallen below 10% of the population. Based on a standard of living of $1.90 per day at 2011 purchasing power parity—that’s about $700 per year, a bit less than the average income in Malawi.

The UN World Millennium Development Goal set in 1990 was to cut extreme poverty in half by 2015; in fact we have cut it by more than two-thirds, reducing it from 37% of the world’s population in 1990 to 9.6% today. This is an estimate, based upon models of what’s going on in countries where we don’t have reliable data; ever the cautious scientists, the World Bank prefers to focus on the most recent fully reliable data, which says that we reduced extreme poverty to 12.7% in 2012 and therefore achieved the Millennium Development Goal.

Most of this effect comes from one very big country: China. Over 750 million people in China saw their standard of living rise above the extreme poverty level in the last 30 years.
The slowest reduction in poverty has been in Africa, specifically Sub-Saharan Africa, where extreme poverty has barely budged, from 53% in 1981 to 47% in 2011. But some particular countries in Africa have done better; thanks to good governance—including better free speech protection than the United States, shame on us—Botswana has reduced their extreme poverty rate from over 50% in 1965 to 19% today.

A lot of World Bank officials have been focusing on the fact that there is still much to be done; 10% in extreme poverty is still 10% too many, and even once everyone is above $1.90 per day that still leaves a lot of people at $3 per day and $4 per day which is still pretty darn poor. The project of global development won’t really seem complete until everyone in the world lives above not just the global poverty line, but something more like a First World poverty line, with a home to live in, a doctor to see, a school to attend, clean water, flush toilets, electricity, and probably even a smartphone with Internet access. (If the latter seems extravagant, let me remind you that more people in the world have smartphones than have flush toilets, because #weliveinthefuture.)

Pace the Heritage Foundation, the fact that what we call poverty in America typically includes having a refrigerator, a microwave, and a car doesn’t mean it isn’t actually poverty; it simply means that poverty in the First World isn’t nearly as bad as poverty in the Third World. (After all, over 9% of children in the US live in households with low food security, and 1% live in households with very low food security; hunger in America isn’t as bad as hunger in Malawi, but it’s still hunger.) Maybe it even means we should focus on the Third World, though that argument isn’t as strong as it might appear; to eliminate poverty in the US, all we’d need to do is pass a law that implements a basic income. To eliminate poverty worldwide, we’d need a global project of economic and political reforms to change how hundreds of countries are governed.

Yet, this focus on what we haven’t accomplished (as though we were going to cut funding to the UN Development Program because we’re done now or something) is not only disheartening, it’s unreasonable. We have accomplished something truly spectacular.

We are now on the verge of solving on one of the great problems of human existence, a problem so deep, so ancient, and so fundamental that it’s practically a cliche: We say “end world hunger” in the same breath as “cure cancer” (which doesn’t even make sense) or “conquer death” (which is not as far off as you may think). Yet, in a very real sense, we are on the verge of ending world hunger.

While most people have been focused on other things, from a narcissistic billionaire running for President to the uniquely American tragedy of mass shootings, development economists have been focused on one thing: Conquering global poverty. What this report means is that now, at last, victory is within our grasp.

Development economists are unsung heroes; without their research, their field work, and their advice and pressure to policymakers, we would never have gotten this far. It was development economists who made the UN Millennium Development Goals, and development economists who began to achieve them.

Yet perhaps there is an even more unsung hero in all of this: Capitalism.

I often have a lot of criticisms of capitalism, at least as it operates in the real world; yet it was in the real world that extreme poverty was just brought down below 10%, and it was done primarily by capitalism. I know a lot of people who think that we need to tear down this whole system and replace it with something fundamentally different, but the kind of progress we are making in global development tells me that we need nothing of the sort. We do need to make changes in policy, but they are small changes, simple changes—many of them could be made with the passing of a few simple laws. Capitalism is not fundamentally broken; on the contrary, it is the fundamentals of capitalism that have brought humanity for the first time within arm’s reach of ending world hunger. We need to fix the system at the edges, not throw it away.

Recall that I said most of the poverty reduction occurred in China. What has China been doing lately? They’ve been opening to world trade—that “free trade” stuff I talked about before. They’ve been cutting tariffs. They’ve been privatizing industries. They’ve been letting unprofitable businesses fail so that new ones can rise in their place. They have, in short, been making themselves more capitalist. Building schools, factories, and yes, even sweatshops is what has made China’s rise out of poverty possible. They are still doing many things wrong—not least their authoritarian government, which is now gamifying oppression in truly cyberpunk fashion—but they are doing a few very important things right.

World hunger is on the way out. And I can think of no better reason to celebrate.