Econometrics and the Right Amount of Inequality
January 31, 2011 Inequality

The question of the “right amount” of inequality came up in a discussion with a student a few days ago. The student agreed that perfect income equality would be undesirable because theory and experience both show that most of us would end up following the Soviet mantra, “We pretend to work and they pretend to pay us.” However, the student argued, that clearly there can be “too much” inequality.

I do not wish to grant that latter point, maybe I’ll address it in the future. Instead I’d like to address what he said as justification. He correctly observed that, “well, income inequality is much higher in the really poor countries around the world.” He’s right. Standard measures of income inequality in places like Zimbabwe would indicate larger dispersions of income. And the student followed, “so, if we allow inequality of income to increase in America … we might become just like Zimbabwe.”  He even said that he had empirical evidence to support this! What he did was take standard income inequality measures (or so he tells me) and ran a regression of income on these measures in each country across the world for which he could secure data. His finding: a negative relationship between income inequality and income levels.

Voila! Science wins the day, and the conclusion follows: the U.S. must therefore aggressively pursue policies to limit income inequality so that we don’t morph into despotic third world nation status. Again, to remind readers – I do not wish to address the underlying questions of what kinds of inequality matter, whether measured inequality is actually increasing in the U.S., and most important whether government policy actually can do anything to improve measures of inequality. Ignore all of that. This is a simple epistemological point.

For the sake of simplicity, assume we can measure the inequality that matters, and that if you have “too much” we assign your country a value of 1 for “having inequality” and if you don’t have “too much” we assign your country a value of 0 for “not having too much.” Then, we collect this data on the actual level of income (or income per capita) in each country, and on the right hand side we include controls for the factors that might be expected to affect income such as education, the level of the capital stock, the size of the population, geographic location, and a few other factors. On the right hand side also includes the variable for “too much inequality” which again takes a value of 1 for the “bad places” and zero for the “good” ones.

Suppose in the cross-section your dependent variable is income per capita (measured in actual $1,000s of dollars and running a simple OLS model) and that your regression result tells you that the coefficient on inequality is -15.  Let me tell you how this coefficient was interpreted by the student (and this is the most common mistake I see in metrics classes and even in professional journals) and then demonstrate what that coefficient actually says.

The student took this coefficient from the cross-sectional regression to mean that, “if a country goes from having equality to inequality, then its per capita income will drop by $15,000.” Now that sounds pretty dramatic. But it is incomprehensibly wrong. So what does that coefficient tell us? Remember, we are doing an analysis looking at how differences in inequality across countries at a single point in time are related to differences in income per capita across countries at a single point in time.

Properly interpreted, here is what that -15 coefficient means. It says that if we have two otherwise identical countries (in terms of education levels, capital stock, location, climate, etc.) and that if the only observable difference between them was that one was unequal and one was equal, the unequal one would have $15,000 lower income per capita.

Can you see the difference between the two interpretations? The first assumes some kind of causality that you simply cannot extract from the given data. From a cross-sectional regression you cannot infer that if a current country changes that it will end up looking like another country. The latter interpretation appreciates this, but it also leaves open the possibility (likely) that we really don’t know what causes income per capita to change within a particular country.

To illustrate the point, let me show you a well known result in the labor economics literature. When labor economists (I was formerly one) estimate wage regressions in the cross-section, they routinely put in a geographic location dummy variable as an explanatory variable. Invariably, when the variable “lives in the South” (in the U.S.) is included in the regressions, we see a rather large negative coefficient. What this tells us is that workers in the South on average have lower wages than otherwise similar workers in the North. We can imagine why this is the case. But what this does not tell us is this … “if Wintercow relocates his family from Rochester to Atlanta, his wage will fall dramatically.” Indeed, in regressions done in the time series like this, it is not uncommon to observe a positive coefficient on the variable South.

The point of course is that while such cross-sectional regressions are interesting, they have limited use in explaining what may happen within a particular country when inequality changes.

"1" Comment
  1. A great “What is not seen” moment.

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