Pareto Efficiency: Why we need it—and why it’s not enough

JDN 2456914 PDT 11:45.

I already briefly mentioned the concept in an earlier post, but Pareto-efficiency is so fundamental to both ethics and economics I decided I would spent some more time on explaining exactly what it’s about.

This is the core idea: A system is Pareto-efficient if you can’t make anyone better off without also making someone else worse off. It is Pareto-inefficient if the opposite is true, and you could improve someone’s situation without hurting anyone else.

Improving someone’s situation without harming anyone else is called a Pareto-improvement. A system is Pareto-efficient if and only if there are no possible Pareto-improvements.

Zero-sum games are always Pareto-efficient. If the game is about how we distribute the same $10 between two people, any dollar I get is a dollar you don’t get, so no matter what we do, we can’t make either of us better off without harming the other. You may have ideas about what the fair or right solution is—and I’ll get back to that shortly—but all possible distributions are Pareto-efficient.

Where Pareto-efficiency gets interesting is in nonzero-sum games. The most famous and most important such game is the so-called Prisoner’s Dilemma; I don’t like the standard story to set up the game, so I’m going to give you my own. Two corporations, Alphacomp and Betatech, make PCs. The computers they make are of basically the same quality and neither is a big brand name, so very few customers are going to choose on anything except price. Combining labor, materials, equipment and so on, each PC costs each company $300 to manufacture a new PC, and most customers are willing to buy a PC as long as it’s no more than $1000. Suppose there are 1000 customers buying. Now the question is, what price do they set? They would both make the most profit if they set the price at $1000, because customers would still buy and they’d make $700 on each unit, each making $350,000. But now suppose Alphacomp sets a price at $1000; Betatech could undercut them by making the price $999 and sell twice as many PCs, making $699,000. And then Alphacomp could respond by setting the price at $998, and so on. The only stable end result if they are both selfish profit-maximizers—the Nash equilibrium—is when the price they both set is $301, meaning each company only profits $1 per PC, making $1000. Indeed, this result is what we call in economics perfect competition. This is great for consumers, but not so great for the companies.

If you focus on the most important choice, $1000 versus $999—to collude or to compete—we can set up a table of how much each company would profit by making that choice (a payoff matrix or normal form game in game theory jargon).

A: $999 A: $1000
B: $999 A:$349k

B:$349k

A:$0

B:$699k

B: $1000 A:$699k

B:$0

A:$350k

B:$350k

Obviously the choice that makes both companies best-off is for both companies to make the price $1000; that is Pareto-efficient. But it’s also Pareto-efficient for Alphacomp to choose $999 and the other one to choose $1000, because then they sell twice as many computers. We have made someone worse off—Betatech—but it’s still Pareto-efficient because we couldn’t give Betatech back what they lost without taking some of what Alphacomp gained.

There’s only one option that’s not Pareto-efficient: If both companies charge $999, they could both have made more money if they’d charged $1000 instead. The problem is, that’s not the Nash equilibrium; the stable state is the one where they set the price lower.

This means that only case that isn’t Pareto-efficient is the one that the system will naturally trend toward if both compal selfish profit-maximizers. (And while most human beings are nothing like that, most corporations actually get pretty close. They aren’t infinite, but they’re huge; they aren’t identical, but they’re very similar; and they basically are psychopaths.)

In jargon, we say the Nash equilibrium of a Prisoner’s Dilemma is Pareto-inefficient. That one sentence is basically why John Nash was such a big deal; up until that point, everyone had assumed that if everyone acted in their own self-interest, the end result would have to be Pareto-efficient; Nash proved that this isn’t true at all. Everyone acting in their own self-interest can doom us all.

It’s not hard to see why Pareto-efficiency would be a good thing: if we can make someone better off without hurting anyone else, why wouldn’t we? What’s harder for most people—and even most economists—to understand is that just because an outcome is Pareto-efficient, that doesn’t mean it’s good.

I think this is easiest to see in zero-sum games, so let’s go back to my little game of distributing the same $10. Let’s say it’s all within my power to choose—this is called the ultimatum game. If I take $9 for myself and only give you $1, is that Pareto-efficient? It sure is; for me to give you any more, I’d have to lose some for myself. But is it fair? Obviously not! The fair option is for me to go fifty-fifty, $5 and $5; and maybe you’d forgive me if I went sixty-forty, $6 and $4. But if I take $9 and only offer you $1, you know you’re getting a raw deal.

Actually as the game is often played, you have the choice the say, “Forget it; if that’s your offer, we both get nothing.” In that case the game is nonzero-sum, and the choice you’ve just taken is not Pareto-efficient! Neoclassicists are typically baffled at the fact that you would turn down that free $1, paltry as it may be; but I’m not baffled at all, and I’d probably do the same thing in your place. You’re willing to pay that $1 to punish me for being so stingy. And indeed, if you allow this punishment option, guess what? People aren’t as stingy! If you play the game without the rejection option, people typically take about $7 and give about $3 (still fairer than the $9/$1, you may notice; most people aren’t psychopaths), but if you allow it, people typically take about $6 and give about $4. Now, these are pretty small sums of money, so it’s a fair question what people might do if $100,000 were on the table and they were offered $10,000. But that doesn’t mean people aren’t willing to stand up for fairness; it just means that they’re only willing to go so far. They’ll take a $1 hit to punish someone for being unfair, but that $10,000 hit is just too much. I suppose this means most of us do what Guess Who told us: “You can sell your soul, but don’t you sell it too cheap!”

Now, let’s move on to the more complicated—and more realistic—scenario of a nonzero-sum game. In fact, let’s make the “game” a real-world situation. Suppose Congress is debating a bill that would introduce a 70% marginal income tax on the top 1% to fund a basic income. (Please, can we debate that, instead of proposing a balanced-budget amendment that would cripple US fiscal policy indefinitely and lead to a permanent depression?)

This tax would raise about 14% of GDP in revenue, or about $2.4 trillion a year (yes, really). It would then provide, for every man, woman and child in America, a $7000 per year income, no questions asked. For a family of four, that would be $28,000, which is bound to make their lives better.

But of course it would also take a lot of money from the top 1%; Mitt Romney would only make $6 million a year instead of $20 million, and Bill Gates would have to settle for $2.4 billion a year instead of $8 billion. Since it’s the whole top 1%, it would also hurt a lot of people with more moderate high incomes, like your average neurosurgeon or Paul Krugman, who each make about $500,000 year. About $100,000 of that is above the cutoff for the top 1%, so they’d each have to pay about $70,000 more than they currently do in taxes; so if they were paying $175,000 they’re now paying $245,000. Once taking home $325,000, now only $255,000. (Probably not as big a difference as you thought, right? Most people do not seem to understand how marginal tax rates work, as evinced by “Joe the Plumber” who thought that if he made $250,001 he would be taxed at the top rate on the whole amount—no, just that last $1.)

You can even suppose that it would hurt the economy as a whole, though in fact there’s no evidence of that—we had tax rates like this in the 1960s and our economy did just fine. The basic income itself would inject so much spending into the economy that we might actually see more growth. But okay, for the sake of argument let’s suppose it also drops our per-capita GDP by 5%, from $53,000 to $50,300; that really doesn’t sound so bad, and any bigger drop than that is a totally unreasonable estimate based on prejudice rather than data. For the same tax rate might have to drop the basic income a bit too, say $6600 instead of $7000.

So, this is not a Pareto-improvement; we’re making some people better off, but others worse off. In fact, the way economists usually estimate Pareto-efficiency based on so-called “economic welfare”, they really just count up the total number of dollars and divide by the number of people and call it a day; so if we lose 5% in GDP they would register this as a Pareto-loss. (Yes, that’s a ridiculous way to do it for obvious reasons—$1 to Mitt Romney isn’t worth as much as it is to you and me—but it’s still how it’s usually done.)

But does that mean that it’s a bad idea? Not at all. In fact, if you assume that the real value—the utility—of a dollar decreases exponentially with each dollar you have, this policy could almost double the total happiness in US society. If you use a logarithm instead, it’s not quite as impressive; it’s only about a 20% improvement in total happiness—in other words, “only” making as much difference to the happiness of Americans from 2014 to 2015 as the entire period of economic growth from 1900 to 2000.

If right now you’re thinking, “Wow! Why aren’t we doing that?” that’s good, because I’ve been thinking the same thing for years. And maybe if we keep talking about it enough we can get people to start voting on it and actually make it happen.

But in order to make things like that happen, we must first get past the idea that Pareto-efficiency is the only thing that matters in moral decisions. And once again, that means overcoming the standard modes of thinking in neoclassical economics.

Something strange happened to economics in about 1950. Before that, economists from Marx to Smith to Keynes were always talking about differences in utility, marginal utility of wealth, how to maximize utility. But then economists stopped being comfortable talking about happiness, deciding (for reasons I still do not quite grasp) that it was “unscientific”, so they eschewed all discussion of the subject. Since we still needed to know why people choose what they do, a new framework was created revolving around “preferences”, which are a simple binary relation—you either prefer it or you don’t, you can’t like it “a lot more” or “a little more”—that is supposedly more measurable and therefore more “scientific”. But under this framework, there’s no way to say that giving a dollar to a homeless person makes a bigger difference to them than giving the same dollar to Mitt Romney, because a “bigger difference” is something you’ve defined out of existence. All you can say is that each would prefer to receive the dollar, and that both Mitt Romney and the homeless person would, given the choice, prefer to be Mitt Romney. While both of these things are true, it does seem to be kind of missing the point, doesn’t it?

There are stirrings of returning to actual talk about measuring actual (“cardinal”) utility, but still preferences (so-called “ordinal utility”) are the dominant framework. And in this framework, there’s really only one way to evaluate a situation as good or bad, and that’s Pareto-efficiency.

Actually, that’s not quite right; John Rawls cleverly came up with a way around this problem, by using the idea of “maximin”—maximize the minimum. Since each would prefer to be Romney, given the chance, we can say that the homeless person is worse off than Mitt Romney, and therefore say that it’s better to make the homeless person better off. We can’t say how much better, but at least we can say that it’s better, because we’re raising the floor instead of the ceiling. This is certainly a dramatic improvement, and on these grounds alone you can argue for the basic income—your floor is now explicitly set at the $6600 per year of the basic income.

But is that really all we can say? Think about how you make your own decisions; do you only speak in terms of strict preferences? I like Coke more than Pepsi; I like massages better than being stabbed. If preference theory is right, then there is no greater distance in the latter case than the former, because this whole notion of “distance” is unscientific. I guess we could expand the preference over groups of goods (baskets as they are generally called), and say that I prefer the set “drink Pepsi and get a massage” to the set “drink Coke and get stabbed”, which is certainly true. But do we really want to have to define that for every single possible combination of things that might happen to me? Suppose there are 1000 things that could happen to me at any given time, which is surely conservative. In that case there are 2^1000 = 10^300 possible combinations. If I were really just reading off a table of unrelated preference relations, there wouldn’t be room in my brain—or my planet—to store it, nor enough time in the history of the universe to read it. Even imposing rational constraints like transitivity doesn’t shrink the set anywhere near small enough—at best maybe now it’s 10^20, well done; now I theoretically could make one decision every billion years or so. At some point doesn’t it become a lot more parsimonious—dare I say, more scientific—to think that I am using some more organized measure than that? It certainly feels like I am; even if couldn’t exactly quantify it, I can definitely say that some differences in my happiness are large and others are small. The mild annoyance of drinking Pepsi instead of Coke will melt away in the massage, but no amount of Coke deliciousness is going to overcome the agony of being stabbed.

And indeed if you give people surveys and ask them how much they like things or how strongly they feel about things, they have no problem giving you answers out of 5 stars or on a scale from 1 to 10. Very few survey participants ever write in the comments box: “I was unable to take this survey because cardinal utility does not exist and I can only express binary preferences.” A few do write 1s and 10s on everything, but even those are fairly rare. This “cardinal utility” that supposedly doesn’t exist is the entire basis of the scoring system on Netflix and Amazon. In fact, if you use cardinal utility in voting, it is mathematically provable that you have the best possible voting system, which may have something to do with why Netflix and Amazon like it. (That’s another big “Why aren’t we doing this already?”)

If you can actually measure utility in this way, then there’s really not much reason to worry about Pareto-efficiency. If you just maximize utility, you’ll automatically get a Pareto-efficient result; but the converse is not true because there are plenty of Pareto-efficient scenarios that don’t maximize utility. Thinking back to our ultimatum game, all options are Pareto-efficient, but you can actually prove that the $5/$5 choice is the utility-maximizing one, if the two players have the same amount of wealth to start with. (Admittedly for those small amounts there isn’t much difference; but that’s also not too surprising, since $5 isn’t going to change anybody’s life.) And if they don’t—suppose I’m rich and you’re poor and we play the game—well, maybe I should give you more, precisely because we both know you need it more.

Perhaps even more significant, you can move from a Pareto-inefficient scenario to a Pareto-efficient one and make things worse in terms of utility. The scenario in which the top 1% are as wealthy as they can possibly be and the rest of us live on scraps may in fact be Pareto-efficient; but that doesn’t mean any of us should be interested in moving toward it (though sadly, we kind of are). If you’re only measuring in terms of Pareto-efficiency, your attempts at improvement can actually make things worse. It’s not that the concept is totally wrong; Pareto-efficiency is, other things equal, good; but other things are never equal.

So that’s Pareto-efficiency—and why you really shouldn’t care about it that much.

5 thoughts on “Pareto Efficiency: Why we need it—and why it’s not enough

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