Relative deprivation: a new derivation and application

ABSTRACT

We derive and express the Yitzhaki index of relative deprivation as a function of mean and median household income within a reference group and one’s income, under the assumption that individual income follows the log-normal distribution or the Pareto distribution. We also suggest a way to calculate annual US regional mean and median household income in each reference group using aggregate income and household size data. Therefore, unlike prior literature that utilizes Decennial Census data due to its big sample size, our suggested methodology allows annual calculation of relative deprivation at different levels of geography (state or county). In an application of the derived index presented to illustrate its feasibility, we analyse the relationship between relative deprivation and mental health, finding suggestive evidence that increases in relative deprivation negatively affect mental health in the United States.

KEYWORDS:

• Yitzhaki index
• relative deprivation
• income distribution
• mental health

JEL CLASSIFICATION:

• C16
• I10
• I32

Acknowledgments

We would like to thank Tom Mroz, Charles Courtemanche, Rusty Tchernis, Pierre Nguimkeu, and one anonymous reviewer for their valuable comments. Any errors are, of course, our own.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ In a similar way, the Gini index, the most commonly used measure of income inequality calculated using individual income data within a reference group was derived and expressed by Crow and Shimizu (1987) as a function of the ratio of mean to median household income, under the assumption that individual income is log-normally distributed. This new approach in calculating the Gini index has been applied in the literature such as income inequality and crime in the United States (Kelly 2000; Brush 2007).

² Our annual measure of average household size is derived as follows, given differential availability from the Census and the ACS over time. For example, in our application using 2001–2012 data that will be explained in the next section, for 2000, we use average household size from the 2000 decennial Census. For 2001–2004, we assign average household size via a linear interpolation of our average household size in 2000 and the 5 years estimates for average household size taken from the 2005–2009 ACS. For 2005–2009, we use 5 years estimates for average household size from the 2005–2009 ACS. For 2010, 2011, and 2012, we use the 5 years estimates for average household size from the 2006–2010, 2007–2011, and 2008–2012 ACS respectively.

³ For detailed information regarding data inputs and methodologies of the estimates of ‘per capita personal income’ and ‘median household income’, please see the BEA (https://www.bea.gov/system/files/methodologies/LAPI2018.pdf) and the SAIPE websites (https://www.census.gov/programs-surveys/saipe/technical-documentation/methodology.html), respectively.

⁴ We use aggregate external income data to calculate the index rather than income data from the BRFSS. This is because the BRFSS does not provide a sufficient number of observations of income to credibly calculate the index particularly for sub-state (county) levels of geography. Overall about 61% of counties from the BRFSS provide less than 100 observations while about 20% of counties provide less than 30 observations.

⁵ Our state-level dataset consists of 4,257,228 adult individuals, while our county-level dataset consists of the sub-sample of 3,639,327 adult individuals whose county identifiers are available. Our descriptive statistics shows that, for example, the average number of mentally unhealthy days is 3.5 days and the largest share of race (education) group is white (college graduate) in our state-level sample. More details on descriptive statistics are available upon request.