Are We Measuring Inequality the Wrong Way?

Are We Measuring Inequality the Wrong Way?

December 9, 2019

Originally published on Economics from the Top Down

Blair Fix

In a recent blog post called “How Not to Measure Inequality”, the anthropologist Jason Hickel argues that economists measure inequality the wrong way. Hickel thinks that standard measures of inequality (such as the Gini index), underestimate global disparities. The problem, according to Hickel, is that economists measure relative inequality. Hickel thinks we should measure absolute inequality.

What’s the difference between the two? Relative inequality is concerned with income differences relative to average income. In contrast, absolute inequality is concerned with the absolute size of income differences.

Economists almost always measure relative inequality. But Hickel thinks this is a grave error. In his words, relative inequality is a “kind of propaganda” that misrepresents the scale of global inequality. To see the true scale of global inequality, Hickel says, we should measure absolute differences in income.

I think Hickel is wrong.

But before I get to the criticism, I’ll say that Hickel’s piece is well motivated. He’s right that there are problems with our measures of inequality. When we measure inequality, we reduce a complex distribution of income to a single number. This reduction means we lose information. So a single metric never tells the whole story. In fact, it’s possible for one inequality measure can go up while another goes down. The lesson? It’s silly to rely on a single measure of inequality.

So like Hickel, I think our inequality metrics have flaws. But we disagree about what these flaws are. Hickel thinks that measuring relative (not absolute) inequality is a flaw. I disagree. In fact, I think the only sensible way to measure inequality is in relative terms. I discuss why (in painful detail) below.

If you just want the quick summary of my argument, it’s this: differences (of any kind) need context to have meaning. Suppose I told you that two numbers differ by 10. Then I ask you — is this difference ‘big’ or ‘small’? Here’s the problem. Without knowing the context, the question has no meaning. In other words, differences have meaning only in relative context. And that’s why we need to measure inequality in relative terms.

Relative inequality vs. absolute inequality

To make sense of this debate, you need to understand how relative inequality differs from absolute inequality. Let’s use a toy example to illustrate the difference.

Imagine a society with two people — Alice and Bob. We’ll look at their income in two different years. In the first year, Alice earns $2 and Bob earns $1. In the second year, Alice earns $10 and Bob earns $5.

Here is the question. Has inequality stayed the same? Or has it increased?

The answer depends on how we measure inequality.

In relative terms, inequality hasn’t changed. Why? Relative inequality looks at the ratio of incomes. In both years, Alice earns twice as much as Bob (2:1 and 10:5). Since the income ratio stays the same, we know that relative inequality stays the same.

But when we switch to absolute inequality, our results differ. In absolute terms, inequality has increased. Why? In the first year, Alice earned $1 more than Bob ($2 – $1). But in the second year, Alice earned $5 more than Bob ($10 – $5). Since the absolute difference in incomes grew, we say that absolute inequality increased.

So we have two very different ways of looking at inequality. Relative metrics look at income ratios, while absolute metrics look at income differences. Which metric should we use?

Economists have almost universally focused on relative inequality. And I think they have done so for good reason. But Jason Hickel thinks it’s dubious to focus on relative inequality. He writes:

… the relative metric begins to border on a kind of propaganda or ideology, a justification for inequality, and its dominance in economics can be seen as a kind of cultural hegemony in the Gramscian sense. (Jason Hickel)

Bold words. Let’s look at the case against relative inequality.

The case against relative inequality

As I understand it, Hickel’s charge against relative inequality is simple. He argues that what matters for the global poor is the absolute gap between themselves and the rich. So unless we shrink the absolute gap between the rich and the poor, we shouldn’t say that inequality declines.

But measures of relative inequality sometimes do just that. Relative measure can report shrinking inequality even when the rich-poor gap grows in absolute terms. Here’s Hickel:

Take for example a poor country whose average income goes from $500 to $1,000 (a 100% increase), and a rich country whose income goes from $50,000 to $75,000 (a 50% increase). The poor country’s income has grown twice as fast as the rich country’s, relative to its starting point. According to the relative metric, this is a decline in inequality … But the gap between them has nonetheless exploded, from $45,500 to $74,000. According to the absolute metric, inequality has worsened. (Jason Hickel)

On one level, I sympathize with Hickel’s argument. We don’t want to downplay the scale of global inequality. Over the last 30 years, global elites have seen their (relative) income skyrocket. And popular inequality metrics like the Gini index don’t capture this growth of top incomes. That’s why a growing number of researchers don’t use the Gini index. Instead, they look at top income shares. (If you want top income data, check out the World Inequality Database. It’s a researcher’s wonderland!)

So we definitely need to choose our inequality metrics wisely. That being said, I don’t think it’s a good idea to measure inequality in absolute terms. It leads to bizarre consequences that I’ll discuss below. But more than anything, absolute inequality doesn’t make philosophical sense. Absolute differences don’t have meaning by themselves. To make sense of a difference, you need a relative context.

Differences need context to have meaning

The problem with absolute inequality is simple. Differences need context to have meaning. A difference that appears large in one context may appear small in another. So we need relative context to give meaning to an absolute difference.

To understand why context is important, let’s leave income distribution aside for a moment. Let’s look at animal size. Suppose a scientist tells you that two animals differ in size by 20g. The scientist then asks: is this difference ‘big’ or ‘small’?

What’s your answer?

You probably say, “I need to know the context”. To make sense of the mass difference, you want to know the size of the two animals. Without this relative context, you can’t decide if the mass difference is big or small.

Of mice and elephants

Suppose that the scientist tells you that the two animals are mice. One mouse weighs 10g and the other weights 30g. With this context, you know that the 20g difference is large. One mouse weights 3 times the other!

But now suppose the scientist tells you that the two animals are elephants. One elephant weights 6000kg and the other weighs 6000.02kg. In this context, the 20g difference is minuscule. One animal is larger by 0.0003% — a difference that probably wouldn’t register on the scale.

So the same difference — 20g — is enormous when it’s between two mice. But it’s insignificant when it’s between two elephants. In other words, relative context matters.

Back to income

What is true for mice and elephants is true for income. We need context to decide if an income difference is large or small. Suppose two people — Alice and Bob — have incomes that differ by $1. Is this difference large or small? It depends on the context.

Suppose that Alice earns $1.01 and Bob earns $0.01. Here the $1 difference in income is substantial. Alice earns 101 times as much as Bob. But suppose that Alice earns $1,000,001 and Bob earns $1,000,000. Here Alice earns 0.0001% more than Bob — little more than a rounding error.

Now ask yourself, which set of incomes is more equal? Like me, you probably think the latter, when Alice earns 0.0001% more than Bob. If you agree, then you’re thinking about relative inequality. Measures of relative inequality would record this 0.0001% income difference as almost perfect equality.

But not so with absolute inequality. Measures of absolute inequality record the same inequality in both situations. Why? Because the absolute difference between Alice’s and Bob’s incomes stays the same ($1). It doesn’t matter that in one case Alice earns 101 times more than Bob, and in the other case she earns 0.0001% more. When we measure absolute inequality, this relative context is irrelevant.

Hopefully the absurdity of this argument gives you some misgivings about absolute inequality. When it comes to income differences, context is everything.

Absolute inequality increases with exponential growth

Another problem with absolute inequality is that it increases with exponential growth. Advocates of absolute inequality see this as a strength. But I disagree.

To see why absolute inequality increases with exponential growth, let’s return to Alice and Bob. Suppose that in a given year, Alice earns $1 and Bob earns $2. The next year, both of their incomes double (exponential growth in action). Alice now earns $2 and Bob earns $4. Since the income difference between Alice and Bob has increased from $1 to $2, absolute inequality has increased.

This seems reasonable enough. But there’s a problem. If we allow exponential growth to increase inequality, weird things happen.

Those despotic elephants

To illustrate the bizarre consequences of mixing exponential growth and absolute inequality, let’s return to mice and elephants.

Suppose scientists want to compare the inequality of food consumption in mice and elephants. To do so, they use absolute inequality. Here’s a toy model of their data:

Mouse A = 4 kcal/day	Mouse B = 5 kcal/day
Elephant A = 40,000 kcal/day	Elephant B = 50,000 kcal/day

You can see that there’s exponential differences at work here. The elephants consume 10,000 times more energy than the mice. Now look at the absolute differences in food consumption. The two mice differ by 1 kcal per day. The two elephants differ by 10,000 kcal per day. So absolute food inequality is far greater among elephants than among mice.

Here’s the headline from the scientists’ research: “Elephants Are Despots, Mice Are Equitable, Scientists Find.”

Do you see the flaw in this headline-grabbing result? It doesn’t account for the relative context of mice and elephant food consumption. In relative context, the food consumption of the two mice differs by 20% — the same as among the two elephants. In relative terms, food inequality is the same in both species. So we have no basis for labelling elephants despots.

Here’s the lesson: absolute metrics will find that inequality increases with size — large things will tend to be more unequal than small things. If this doesn’t make sense with animals (and I hope it doesn’t to you), then it doesn’t make sense with people.

The alternative is to measure inequality in relative terms. The most basic property of relative inequality is that it is preserved by uniform exponential growth. So if I take two incomes and double them, inequality won’t change. This, I think, is a sensible property to have. It means we won’t label elephants despots.

Absolute inequality depends on the choice of unit

Another problem with absolute inequality is that it is affected by the choice of unit. I’ll get to why this is — but first I’ll show why relative inequality avoids this problem.

With relative inequality, we are concerned with ratios. Suppose we want to measure the size inequality of two mice. One weighs 10g and the other weighs 30g. Size inequality between the mice is determined by the ratio:

Since the units are the same, we can reduce this to a unitless ratio:

So with relative inequality, units don’t matter.

When we switch to absolute inequality, this nice property goes away. Unlike with ratios, units don’t cancel when we take differences. Absolute inequality among our mice is:

So absolute inequality is affected by units. The mass difference of our mice is 20g, 0.02kg and 20,000mg.

Now, with mass differences it’s trivial to convert units. So we have no real problem. But when we switch to measuring income inequality, things get more complicated. Suppose I want to compare absolute inequality in the US to absolute inequality in Ghana. Our US measure has units of $US. The Ghana measure has units of Ghanaian cedi. How do we convert to a common unit?

One option would be to use the official exchange rate. But this rate changes day to day (even hour to hour). As the exchange rate changes, so does our comparison of inequality

Another option is to use purchasing power to convert units. In both countries, we measure the price of the same basket of goods. This basket becomes our standard unit. But then we have to ask — which commodities go in the basket? Including or excluding different commodities will affect how we convert $US to Ghanaian cedi (and vice versa).

The point is that when it comes to incomes, unit conversion is non-trivial. This raises all sorts of problems of how we should compare measures of absolute inequality in different countries. In contrast, relative inequality has none of these problems. Why? Because relative inequality is independent of the unit of measure.

Absolute inequality is affected by inflation

A further problem with absolute inequality is that we need to adjust for inflation for it to make sense.

To understand why, let’s return to Alice and Bob. Suppose that Alice earns $1 and Bob earns $2. In the next year, both incomes double. Alice earns $2 and Bob earns $4. Has inequality increased?

Maybe. Maybe not.

Our measure of absolute inequality depends on our measure of inflation. Suppose that the growth of Alice’s and Bob’s income is caused entirely by inflation. Yes, nominal incomes have grown. But there has been no change in consumption patterns. So logically, there can be no change in inequality. To ensure that absolute inequality stays the same, we need to adjust for inflation. We use ‘real’ incomes. In the second year, Alice’s ‘real’ income remains $1 and Bob’s ‘real’ income remains $2$. So absolute inequality remains the same.

But what if the growth of Alice’s and Bob’s income is 50% due to inflation, and 50% due to the growth of ‘real’ income. Then things are different. Then inequality has increased. In year 2, Alice’s real income is $1.50 and Bob’s is $3. The difference in their incomes has grown by $0.50.

So to compare absolute inequality across time, we need to adjust for inflation. But this is no small task. In fact, there are a host of problems with measuring inflation. Jonathan Nitzan, Shimshon Bichler, and I have discussed these problems here. In short, we argue that measures of inflation are inherently subjective. Worse still, existing methods are deeply tied to (dubious) neoclassical theory.

So absolute inequality gets bogged down in the quagmire of measuring inflation. In contrast, relative inequality has no such problems. Why? Because relative inequality is determined by the ratios of nominal incomes. This, I think, is a beautiful thing. It means we can easily compare inequality in the Roman Empire to inequality in the modern US. To measure inequality, all we need are nominal incomes.

Confusing income inequality with income redistribution

A big part of the argument for measuring absolute inequality involves a simple misunderstanding. Advocates for absolute inequality confuse income inequality with income redistribution. I’ll let Hickel speak for himself here. Then I’ll tell you what I think is wrong with his argument.

When we look at inequality from the perspective of the poor … it becomes clear that the relative [inequality] metric is inappropriate as a tool for assessing distribution. Certainly if our objective is to end poverty, this is the conclusion we must draw, as an additional dollar going needlessly to the rich could have been used to reduce poverty, and yet was not. The absolute metric allows us to see this effect, by giving equal weight to each dollar. From the perspective of the poor, an additional dollar going to the rich is a dollar that could have gone to them – and indeed by rights should have gone to them – thus improving their lives by a corresponding amount, and yet was needlessly frittered away instead on a latte. (Jason Hickel)

I agree with almost everything that Hickel says here. But I don’t think it’s a compelling argument for measuring absolute inequality. It’s a compelling argument for measuring the relative scale of redistribution. Any dollar that goes to the rich is a dollar that could have gone to the poor. It’s money that could have been redistributed to decrease inequality. We can and should measure the redistribution required to lessen inequality (to see how far we fall short of what’s required).

The problem is that Hickel is trying use a redistribution argument to justify measuring inequality in absolute terms. He seems to imply that if $1 of new income does not go to the poor, then inequality increases. But this does not follow. If the $1 of new income doesn’t go to the poor, it means we haven’t achieved the maximum reduction of inequality that was possible (assuming we only redistribute new income). Redistribution should not be confused with inequality itself.

Common sense and unconscious relativity

One of the most interesting aspects of Hickel’s argument is that he appeals to common sense. He argues that absolute inequality is the ‘common-sense’ measure of inequality. The average Joe, Hickel says, thinks about inequality in absolute terms:

[T]he absolute metric is the common-sense way of thinking about inequality. I recently conducted a small poll, with close to 200 respondents. I asked two questions: “What do you think economists mean when they say that inequality is decreasing?” and “What do you think economists mean whey they say that inequality is increasing?” Respondents were allowed to write in their own words, so as not to be led by multiple choice. Some 95% of them indicated that they think of absolute income gaps, not relative change. (Jason Hickel)

I have to be careful about commenting here, because I haven’t seen the language used by Hickel’s respondents. But I’m going to take a guess about what’s going on.

On one level, I think Hickel is correct. People report that they think about inequality in terms of absolute differences in incomes. The problem is that I think these people are unconsciously making a relative judgment to interpret these absolute differences.

Here’s an example of how unconscious relativity works. Suppose I ask you — is an elephant ‘large’ or ‘small’? Now, I’ve gone to great pains to show that without relative context, this question has no meaning. Still, most people will likely respond that an elephant is ‘large’. It appears that they are making an absolute judgment. But in reality, they’re unconsciously making a relative comparison. They take their own size as the (unconscious) frame of reference. Compared to a person, an elephant is large.

The same thing likely happens with income. Suppose someone’s income grew by $100,000. Is this a large or small increase? Most people will probably answer that it’s a large increase. Why? Because they are unconsciously comparing this increase to their own incomes. Unless you’re a millionaire, at $100K windfall is a big deal. So it appears that people think about absolute income growth. But they’re actually making an unconscious relative comparison.

It bears repeating (yet again) that differences have no meaning without context. Whether we provide the context consciously or not is irrelevant. So yes, people may report that they are concerned with absolute inequality. But it’s just because the relative context is provided unconsciously. If it was not, the absolute difference would have no meaning.

Yes, global inequality is obscene. But we don’t need absolute inequality to acknowledge this

I’ll conclude by saying that I agree with Hickel in political terms. I agree that global inequalities are obscene. And I agree that the income growth in the poorest countries is nowhere near what is needed to reduce global inequality.

I also I agree that economic growth is not an option for reducing global inequality. It’s suicidal for poor countries to ramp up their consumption to rich-country levels. The only sane option is for rich countries to reduce their consumption, meeting poor countries somewhere in between.

Lastly, I agree with Hickel that the power structure of the global economy is what prevents income convergence.

But having agreed with these points, I disagree that we should measure inequality in absolute terms. Income is, by definition, something that is relative. Income has no absolute magnitude — no constant reference to the real world. Income (and money in general) has meaning only in relative context. As such, income inequality is the last place where we should be using absolute measures. We should measure inequality in relative terms.

Metrics of relative inequality are not, as Hickel claims, “a kind of propaganda”. They are not “a kind of cultural hegemony in the Gramscian sense”. Metrics of relative inequality are consistent with how the rest of science understands dispersion. For instance, when scientists compare measurement error, they do so in relative terms.

A measure that has 1% error is better than a measure that has 10% error. It doesn’t matter what the absolute error is. Scientists know that absolute error is irrelevant when we compare things of different size. So if we want to evoke “cultural hegemony”, we need to apply it fairly. If relative inequality is a “kind of propaganda”, then relative measurement error is “propaganda” as well.

Yes, global inequalities are obscene. Yes, appealing to absolute inequality seems like a good way to accentuate this obscenity. But the reality is that measuring inequality in absolute terms is a fool’s errand. Differences have meaning only in relative context. We forget this at our own peril.