<p>One concept fairly established in economics is the idea of generalized Lorenz dominance.</p>
<p>The <a href="https://en.wikipedia.org/wiki/Lorenz_curve" rel="nofollow noreferrer">Lorenz curve</a> plots percentiles of the population on the x axis and the cumulative percentiles of income on the Y axis. If the point (30,10) is on the curve, then this means that the bottom 30% of the population have 10% of the income of the population.</p>
<p>This allows us to compare the inequality of two countries by comparing how low or high their Lorenz curve is. If curve A is everywhere above or equal to curve B, then A is weakly less unequal than B. However, even the poorest person in B may still earn more than A.</p>
<p>The generalized curve multiplies the Y axis with the average income of society. Now we can not only compare A and B in terms of inequality, but also by how well individuals are off in these societies. Curve A being everywhere above or equal to curve B means that every percentile of society A is weakly better off than every matching percentile in society B.</p>
<p>Shorrocks showed that under certain conditions a parameterized family of social welfare functions is higher in society A relative to society B for all parameters if and only if the generalized Lorenz curve of A is higher than B at all points.</p>
<p>Shorrocks, Anthony F. &quot;Ranking income distributions.&quot; Economica (1983): 3-17.</p>

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2015-11-19T04:18:16.267Z
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HRSE
<p>At first, you could consider simply putting weights on GDP per capita and the GINI coefficient for your measure of social welfare ($W$).</p>

<p>$$W = Y^\alpha (1-G)^{1-\alpha}$$</p>

<p>So social welfare depends both on income inequality and GDP. A few problems arise with this naive model.</p>

<ul>
<li>What is the correct weight to put on $\alpha$?</li>
<li>Perfect forced income inequality as well as perfect forced income equality hurts the economy, and this sort of measure would not take this into account.</li>
<li>As GDP increase, one might intuitively think that income inequality increases, since usually GDP increases with large scale technological shocks, which have high variance over time, and it increases slowly with labor productivity, which has very low variance over time in comparison.</li>
</ul>

<p>So what are some ways of solving these issues?</p>

<p>1.) You could try to guess what politicans treat $\alpha$ as by looking at the GINI and GDP per capita over time and find the $\alpha$ that maximizes discounted welfare over that time period, though that makes the assumption that on average, competing/bargaining for constituent interests maximizes social welfare.</p>

<p>2.) The GINI coefficient is measured as $\frac{A}{A+B}$, relative to the Lorenz curve, so you could put weights on A and B as well to show that perfect equality/perfect inequality is not preferable, but then that is a whole other subjective animal. The idea is that even long term equality or inequality can affect long term GDP growth, even if current GDP looks okay.</p>

<p>3.) For the last point, I don't have any particular recommendations of dealing with it, except maybe accounting for lag values.</p>

<p>That would be my general guess for approaching this kind of question anyhow. Take it for what you will.</p>

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2015-11-18T20:30:40.250Z
Could please you add some references to support your second claim? I mean the part about the forced version hurting the economy. (I think I get the intiution, but I would like to see some framework.)
Giskard
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2015-11-18T20:34:25.120Z
Actually this measure can easily take such effects (forced equality hurting the economy) into account: You only have to look at change in $W$. If $Y$ decreases too much, $W$ will decrease as well. Nice measure!
Giskard
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2015-11-18T20:42:07.163Z
Yeah, I was mostly riding on intuition in this measure, but I was also partly making an argument that too much equality/inequality hurts the economy independent of current GDP. Too much equality might cause long term problems with the growth rate of GDP, but I'm not sure how I'd account for that. I should probably edit my answer with a small note of that. Otherwise, glad my answer proved illuminating. It was a very interesting question.
Kitsune Cavalry
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2015-11-19T03:51:22.187Z
Your functional form seems to lack any sort of foundation. Why should it be Cobb-Douglas preferences over the Gini Coefficient? Why not quasilinear? Why not a general CES function?
HRSE
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2015-11-19T04:46:26.253Z
I'd argue against quasilinearity because to have any social welfare, you need both actual output and, arguably, non-dictatorship (one person owns ALL national output). Cobb-Douglass was mainly chosen for ease of use, though I think CES could definitely work.
Kitsune Cavalry
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Kitsune Cavalry
<p>A frequently used measure of economic progress is GDP growth. (There are alternative economic indicators as well, such as the Human Development Index.) But there are problems with GDP growth. For example population growth also causes the GDP to grow, because more people produce and consume more goods. You could correct for this by usind GDP/capita growth instead. </p>

<p>But GDP/capita growth still does not take into account the distribution of wealth. Perhaps 90% of the population are doing worse than last year but 10% are doing so much better that on average there is still growth. You could argue that this is still 'better' overall, but I would argue that in a democratic setting most people would probably not vote for an economic policy that clearly and openly supports such an income shift. (If you are thinking of cases where this has happened and will happen: I am afraid I don't care about the political side of this argument. You are welcome to post about it over at politics SE.)</p>

<p>Are there indicators that capture both distribution and growth of income?
<br> A simple example with its own faults would be median GDP/capita growth. Something better (and slightly more complicated) could probably be constructed using the GINI index.
<br> Alternatively: Are there arguments against measuring something other than aggregate or average GDP?</p>

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2015-11-18T09:42:58.057Z
The problem is that inequality per se is not bad. I remember a story how in 19th century Vilfredo Pareto was measuring wealth across societies and found that universally about 10-30% always had more than the rest. He then invented the famous Pareto distribution to describe it. Some inequality in societies is very much desirable to reward greater effort, entrepreneurship, superior skills, etc. The problem arises when there is too much inequality. But where is the threshold?
Arthur Tarasov
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2015-11-18T10:05:30.613Z
@ArthurTarasov I was not trying to imply that absolute (or any kind of ) equality is desirable. If there is a measure that takes growth & exact distribution of income into account you can probably modify it in a way to measure distance from some 'ideal' income distribution. Ideally this would be left as an input parameter because as you note there is no agreement on what would be ideal.
Giskard
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2015-11-18T18:46:52.487Z
Just observing the median (or any nth percentile of) income over time will go a long way toward addressing distributional concerns. Even taking a simple average of growth rates at each income quintile (or decile, or whatever), or applying some subjective weighting scheme (maybe weighting quintiles based on how far short they are from the $70k-ish number that shows up in the literature as being "enough" income) could do it.
dismalscience
<gdp><inequality>
Measure combining growth and distribution of income
Giskard
# Measure combining growth and distribution of income

(Asked by: username_0 on 2015-11-18)

A frequently used measure of economic progress is GDP growth. (There are alternative economic indicators as well, such as the Human Development Index.) But there are problems with GDP growth. For example population growth also causes the GDP to grow, because more people produce and consume more goods. You could correct for this by usind GDP/capita growth instead.

But GDP/capita growth still does not take into account the distribution of wealth. Perhaps 90% of the population are doing worse than last year but 10% are doing so much better that on average there is still growth. You could argue that this is still 'better' overall, but I would argue that in a democratic setting most people would probably not vote for an economic policy that clearly and openly supports such an income shift. (If you are thinking of cases where this has happened and will happen: I am afraid I don't care about the political side of this argument. You are welcome to post about it over at politics SE.)

Are there indicators that capture both distribution and growth of income?
  
 A simple example with its own faults would be median GDP/capita growth. Something better (and slightly more complicated) could probably be constructed using the GINI index.
  
 Alternatively: Are there arguments against measuring something other than aggregate or average GDP?

**Tags:** gdp, inequality

> **Comments:**
> **username_1**: The problem is that inequality per se is not bad. I remember a story how in 19th century Vilfredo Pareto was measuring wealth across societies and found that universally about 10-30% always had more than the rest. He then invented the famous Pareto distribution to describe it. Some inequality in societies is very much desirable to reward greater effort, entrepreneurship, superior skills, etc. The problem arises when there is too much inequality. But where is the threshold?  
> **username_0**: @username_1 I was not trying to imply that absolute (or any kind of ) equality is desirable. If there is a measure that takes growth & exact distribution of income into account you can probably modify it in a way to measure distance from some 'ideal' income distribution. Ideally this would be left as an input parameter because as you note there is no agreement on what would be ideal.  
> **username_2**: Just observing the median (or any nth percentile of) income over time will go a long way toward addressing distributional concerns. Even taking a simple average of growth rates at each income quintile (or decile, or whatever), or applying some subjective weighting scheme (maybe weighting quintiles based on how far short they are from the $70k-ish number that shows up in the literature as being "enough" income) could do it.  

---

## 2 answers

### Answer by username_3 on 2015-11-19. Score: 2 | Accepted answer

One concept fairly established in economics is the idea of generalized Lorenz dominance.

The [Lorenz curve](https://en.wikipedia.org/wiki/Lorenz_curve) plots percentiles of the population on the x axis and the cumulative percentiles of income on the Y axis. If the point (30,10) is on the curve, then this means that the bottom 30% of the population have 10% of the income of the population.

This allows us to compare the inequality of two countries by comparing how low or high their Lorenz curve is. If curve A is everywhere above or equal to curve B, then A is weakly less unequal than B. However, even the poorest person in B may still earn more than A.

The generalized curve multiplies the Y axis with the average income of society. Now we can not only compare A and B in terms of inequality, but also by how well individuals are off in these societies. Curve A being everywhere above or equal to curve B means that every percentile of society A is weakly better off than every matching percentile in society B.

Shorrocks showed that under certain conditions a parameterized family of social welfare functions is higher in society A relative to society B for all parameters if and only if the generalized Lorenz curve of A is higher than B at all points.

Shorrocks, Anthony F. "Ranking income distributions." Economica (1983): 3-17.


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### Answer by username_4 on 2015-11-18. Score: 2

At first, you could consider simply putting weights on GDP per capita and the GINI coefficient for your measure of social welfare ($W$).

$$W = Y^\alpha (1-G)^{1-\alpha}$$

So social welfare depends both on income inequality and GDP. A few problems arise with this naive model.

* What is the correct weight to put on $\alpha$?
* Perfect forced income inequality as well as perfect forced income equality hurts the economy, and this sort of measure would not take this into account.
* As GDP increase, one might intuitively think that income inequality increases, since usually GDP increases with large scale technological shocks, which have high variance over time, and it increases slowly with labor productivity, which has very low variance over time in comparison.

So what are some ways of solving these issues?

1.) You could try to guess what politicans treat $\alpha$ as by looking at the GINI and GDP per capita over time and find the $\alpha$ that maximizes discounted welfare over that time period, though that makes the assumption that on average, competing/bargaining for constituent interests maximizes social welfare.

2.) The GINI coefficient is measured as $\frac{A}{A+B}$, relative to the Lorenz curve, so you could put weights on A and B as well to show that perfect equality/perfect inequality is not preferable, but then that is a whole other subjective animal. The idea is that even long term equality or inequality can affect long term GDP growth, even if current GDP looks okay.

3.) For the last point, I don't have any particular recommendations of dealing with it, except maybe accounting for lag values.

That would be my general guess for approaching this kind of question anyhow. Take it for what you will.

> **Comments:**
> **username_0**: Could please you add some references to support your second claim? I mean the part about the forced version hurting the economy. (I think I get the intiution, but I would like to see some framework.)  
> **username_0**: Actually this measure can easily take such effects (forced equality hurting the economy) into account: You only have to look at change in $W$. If $Y$ decreases too much, $W$ will decrease as well. Nice measure!  
> **username_4**: Yeah, I was mostly riding on intuition in this measure, but I was also partly making an argument that too much equality/inequality hurts the economy independent of current GDP. Too much equality might cause long term problems with the growth rate of GDP, but I'm not sure how I'd account for that. I should probably edit my answer with a small note of that. Otherwise, glad my answer proved illuminating. It was a very interesting question.  
> **username_3**: Your functional form seems to lack any sort of foundation. Why should it be Cobb-Douglas preferences over the Gini Coefficient? Why not quasilinear? Why not a general CES function?  
> **username_4**: I'd argue against quasilinearity because to have any social welfare, you need both actual output and, arguably, non-dictatorship (one person owns ALL national output). Cobb-Douglass was mainly chosen for ease of use, though I think CES could definitely work.  

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