## A different concept

The last post here is about the concept of a horizontal Lorenz curve and Gini coefficient. I’m trying to keep these short, so I want to use this to go into some detail about the implications of differing levels of each. This is a solutions based blog, and this will eventually build up to a possible solution about how income inequality can be solved. Before I get to that solution however, I need to get some of the background ideas established.

First, I’m going to go into some detail about the vertical Gini coefficient, and to do that, I need to talk a little bit amount diminishing marginal returns to income. Here’s a simple way to think about that. Take two cases. Imagine you have no money at all. You can’t afford food, housing, water, healthcare, anything. Now say someone comes along and gives you \$10,000. By how much will your happiness increase? Now, take the case that you have a billion dollars, and your every desire is fulfilled. Say, again, someone gives you \$10,000. How much happier are you? If you’re a relatively normal person, you’re happiness will increase more in the first case than the second case. \$10,000 is a relatively small income increase for a billionaire. This is simply what is meant by diminishing marginal returns.

Now let’s think about diminishing marginal returns in the case of income inequality. Assume that there are two people in the economy, and one has a billion dollars, and one has zero dollars. The goal here is to find the optimal level of happiness. If you take \$10,000 away from the billionaire, and give it to the person with nothing, the person will nothing’s happiness will increase more than the billionaire’s happiness will decrease. If you repeat this process you will find that in aggregate, the highest level of happiness comes when both people have equal incomes. This is the basic story from welfare economics about inequality, and the one I will address in this post. I will note however, that you don’t actually need welfare economics to justify a more equal society. Here is a post that makes just that point.

Now, I will make the opposite point. Assume again you have two people in an economy. The first person has an IQ of 300 and is very skilled. The second is an average person with an IQ of 100. Say that the person with the 300 IQ works 80 hours a week, with a very high wage because their labor is so highly values, and that the second person only works 40 hours a week. Say that taxes are distortionary(your tax is a percent of your wage). If you increase taxes on the high IQ person, you are, in effect, decreasing their wage, which gives them less incentive to work. If you use this to decrease taxes on the other person, you are, in effect, increasing their wage, and giving them more incentive to work. Thus you will have more hours worked by the low productivity worker, and fewer hours worked by the high productivity worker, making less overall money available to the economy. If you keep redistributing income toward the low productivity worker, you will eventually have low enough aggregate happiness that even without equal income distribution, the society would have been better off under the original assumptions.

Both of these examples are vastly oversimplified, but they show a point, that in general, some income inequality is good, as it improves aggregate output. At some point, however, a high level of income inequality becomes sub optimal. Here is another post that goes through that more rigorously.  The show that their is a bowed out curve that shows the tradeoff between aggregate output per person, and the vertical Gini coefficient. If aggregate output per capita decreases as equality increases, and aggregate utility increases as equality increases, and as output per capita increases with diminishing marginal returns, there will be some optimal point of inequality where there is maximum aggregate utility.

The reasoning for the idea that output per capita decreases as equality increases relies on the fact that some workers have superior ability to others. People are not all equal in their ability to produce goods, when taken as a whole. This implies that the vertical Gini coefficient should be somewhere between zero and one. Across race however, there is no reason to believe that on average, a person of one race is superior to another in potential. There may exist differences in human capital due to societal factors that allow for people of one class better access to educational institutions, but does not change the idea.

That concept may need some justification. I will show here that different levels of education between two groups with equal potential is sub optimal. There is one additional assumption needed to justify the claim, which is not hard to justify. We must also assume that the cost of educating people increases as you increase the number of people. Take for example our two people of differing IQs from earlier. If you spend one hour teaching each person mathematics, who will learn more? I don’t think it’s unjustified to argue that the 300 IQ person will learn more. This means per unit of knowledge, it is cheaper to educate the higher IQ person. If you spend all of your time educating the higher IQ person, so there is no more time in the day, in order to increase the education level you must spend time educating the lower IQ person. This means your marginal cost will increase. Note that this is vastly simplified, and isn’t in regard at all to moral issues involving equal access to education.

Now let’s say you have two different groups of people, and each make up 50% of the population. If each has an equal distribution of high IQ people, then the optimal solution here would be to educate all of the high IQ people from both groups more than the low IQ people. If you have a fixed amount of time to spend on education, and educate all of one group first, then start with the high IQ section of the other group, you could have had a higher level of aggregate education by spending less time on the low IQ section of the first group, and more on the higher IQ section of the second. For this reason, as a society you gain less aggregate education(and thus less aggregate productivity and output) if you are systematically biased in your human capital production for one group over another.

What does this mean for the idea of a horizontal Gini coefficient? We start with the fact that across racial groups, there is equal potential. Next, we assume that at the optimum, there are similar levels of education across group members with similar costs of education, thus equal potential for production. If there is equal ability horizontally, then there is no reason for one group to make a smaller percent of aggregate income than their aggregate share of the population. This means we should see perfect equality among the horizontal Gini coefficient. It also means that if there is not perfect equality among the horizontal Gini coefficient, by the examples above, we will be better off as a society by moving to a point of greater equality.

As I said, this isn’t actually a solution to a problem, but it is important for the formation of a solution to income inequality. Both horizontal and vertical inequality are important for optimal utility among a society. Furthermore, how these are measured, and their optimal levels, are different, and this must be taken into account when considering the implications of any policy change.