Quantifying inequality
Why quantify inequality?
- Track progress towards equity goals
- Understand how inequality differs in different regions
- Use models to estimate the effects of different policies on inequality
Why not quantify inequality?
- Inequality is not necessarily a tangible thing
- For instance, a lot of the effects of gentrification may be due to cultural changes rather than residential displacement; people no longer feeling welcome, not leaving outright [@rayle_investigating_2015]
- This is real, but it’s not (easily) quantifiable
Measures of income inequality: the P90/P10 metric
- The P90/P10 ratio is the ratio of the 90th percentile of income to the 10th
- Higher values mean more inequality
- What do you suppose the P90/P10 ratio is for the US? 5.4
- It was 4.5 in 1990
How does the US compare to other countries?
Measures of inequality: the Lorenz curve
- The P90/P10 ratio only looks at two points on the income distribution
- A Lorenz curve is a plot that shows the entire income distribution
- On the x axis it shows the percentile of income, and the y axis shows the percent of income earned by people making below that percentile of income
- In statistical terms, it is a cumulative distribution
The Lorenz curve for wealth the US
Measures of income inequality: the Gini coefficient
- The Gini coefficient is a single number based on the Lorenz curve that measures inequality
- If income were perfectly equally distributed, the Lorenz curve would be a straight line
- The Gini coefficient is the proportion of the area under that straight line that is above the Lorenz curve (between 0 and 1, sometimes between 0 and 100)
- The further the Lorenz curve is from perfect equality, the higher the Gini coefficient
The Gini index in the US
How does the US compare to other countries?
Data: CIA World Factbook
Measures of social mobility
[R]ates of absolute upward income mobility in the United States have fallen sharply since 1940. . . . [u]nder the current distribution of GDP, we would need real GDP growth rates above 6% per year to return to the rates of absolute mobility seen in the 1940s. Intuitively, because a large fraction of GDP goes to a small number of high income earners today, higher GDP growth does not substantially increase the number of children who earn more than their parents.
@chetty_fading_2017
Measures of social mobility
Percent of children earning more than their parents, by year of birth [@chetty_fading_2017]
Measures of segregation
- The US remains highly segregated
- Planners and sociologists have devised a number of measures of segregation
- Formal definitions here come from @forest_measures_2005
Schelling’s model of spatial segregation
- This model demonstrates how small individual preferences for neighborhood composition can lead to significant segregation
- Developed by Thomas Schelling, best known for his work on mutually assured destruction [@schelling_micromotives_1978]
- Most user-friendly implementation: the Parable of the Polygons
Schelling’s model of spatial segregation: demonstration
Measures of segregation: the dissimilarity index
- The dissimilarity index measures, for two groups, the proportion of either group that would have to move to create a completely unsegregated pattern
- Ranges from 0 to 1; 0 is no segregation
- If places were completely unsegregated, the percentage of each group in an area would be equal to the percentage of each other group
- e.g. if a tract is 5% of the population of the area, it should have 5% of the white population and 5% percent of the black population
- This is true even if the population sizes differ
The dissimilarity index in math
\[ D = \frac{1}{2} \sum\limits_{i=1}^{n}\left|\frac{g_{1i}}{G_1} - \frac{g_{2i}}{G_2}\right| \]
where \(g_{1i}\) is number of people in area \(i\) that are in group 1, \(G_1\) is the total number of people in group one, with \(g_i2\) and \(G_2\) the same for group two
Measures of segregation and the modifiable areal unit problem
The dissimilarity index in North Carolina, for white alone and Black alone,
- by block: 0.64
- by block group: 0.54
- by tract: 0.51
- by county: 0.33
Measures of segregation: the interaction/exposure index
- The average proportion of people in group 2 in an area, weighted by the number in group 1
- Put another way, what percentage of the area where a particular Black person lives is white, or vice-versa?
- Unlike the dissimilarity index, it is asymmetrical—the percent of the area where a Black person lives that is white ≠ the person of the area where a white person lives that is Black
- You may see this referred to as the probability that a person of one race interacts with a person of another, but that makes a lot of assumptions about social norms in the community
The interaction/exposure index in North Carolina
- What do you think it is? (at the block level)
- NC is 60% white, 20% black, 89% non-Hispanic, 11% Hispanic
- For whites interacting with Blacks: 0.11
- For Blacks interacting with whites: 0.33
- For non-Hispanic interacting with Hispanic: 0.09
- For Hispanic interacting with non-Hispanic: 0.72
Measures of segregation: entropy
- Entropy is a measure of segregation of two or more groups
- It is based on the concept of entropy from physics
- Unlike the other metrics, each area is assigned an entropy value:
\[ E_i = -\sum\limits_{g=1}^{G} p_{ig}~\mathrm{ln}~p_{ig} \]
where \(G\) is the number of groups, and \(p_{ig}\) the the proportion of area \(i\) that is group \(g\); \(0~\mathrm{ln}~0\) is defined to equal 0
(note: entropy won’t be on the final)
Measures of segregation: entropy
- Entropy can be useful because it’s defined at the area level, so you can see what neighborhoods are more or less integrated
- When all groups are equal, the maximum value of entropy is \(\mathrm{ln}~G\)
- Entropy is highest when groups are evenly divided, lowest when segregated
- If the overall proportions of groups are not equal, maximum entropy is never achievable
- Entropy of a smaller area is often compared to entropy of a larger area for this reason
Block level entropy: Orange County
- For non-Hispanic white, non-Hispanic Black, non-Hispanic Asian, and Hispanic
Land use entropy
- Sometimes entropy is used to measure how mixed land uses are in models
- I haven’t personally found it to be very predictive in my own work
References
This work by Matthew Bhagat-Conway is licensed under a Creative Commons Attribution 4.0 International License.
Social mobility and the lottery effect