Inequality in Multidimensional Well-Being in the United States

Inequality in Multidimensional Well-being in the United States

Shatakshee Dhongde Georgia Institute of Technology, [email protected] Prasanta K. Pattanaik University of California Riverside, [email protected] Yongsheng Xu

Georgia State University, [email protected]

Abstract

We study inequality of multidimensional well-being of individuals in the United States in the last decade. We consider three attributes of well-being, namely, income, health, and education. Using data from the Current Population Survey we compute multidimensional well- being of every individual. We then use certain well-known measures to estimate well-being inequality at three points of time, namely, 2010, 2014 and 2018. We find that: (i) inequality of well-being in the United States increased over the period; (ii) the inference about changes in inequality in each of the attribute based on a composite measure is different from that based on well-being inequality; and (ii) our result is robust to alternate inequality measures and alternate weights attached to attributes.

Keywords: multidimensional well-being, inequality, United States

JEL classification codes: D39, D63, I31

Starting in 2010, the United Nations has annually published estimates of a global

multidimensional poverty index. Since then, there has been a rapid growth in the literature on

the measurement of multidimensional poverty over the last decade (for a review, see Pattanaik and Xu (2018), Dotter and Klasen (2017), and Aaberge and Brandolini (2015)). Just as income poverty and the distribution of income are related yet two distinct notions, so are multidimensional poverty and the distribution of multidimensional well-being. In this paper, we analyze the change in inequality in the distribution of multidimensional well-being among individuals in the United States over the period 2010-2018.

The issue of distributive inequality typically comes up when one is engaged in ethical evaluation of alternative social states or social policies. In undertaking our study of inequality in the

United States in that normative context, we start with the intuitive position that, ultimately, the inequality that we should be concerned with is the inequality in the distribution of individual well-being in the society under consideration (see Sen (1980) for an incisive discussion of the question, “Equality of what?” ). How should one measure inequality in the distribution of individual well-being when individual well-being is envisaged in a multidimensional fashion?

The analytical literature on the measurement of inequality in a multidimensional framework is vast (see, among others, Kolm (1977), Atkinson and Bourguignon (1982), Decancq and Lugo

(2012), and Tsui (1995, 1999) for a sample of contributions). In the empirical literature on multidimensional well-being and inequality, one can discern two distinct approaches. In one of these approaches (we call it the Type I approach to the measurement of inequality), one first considers the individuals’ achievements along each specific dimension of individual well-being and measures inequality in those dimension-specific achievements. Next, one takes the set of

inequality indices along the different dimensions and aggregates them in some fashion in order

to arrive at an overall measure of inequality in the society. This approach is the basis of the

inequality-adjusted Human Development Index (UNDP, various years), which is a subclass of

multidimensional generalized Gini indices studied in Gajdos and Weymark (2005). The second

approach, which we shall call the Type II approach to the measurement of inequality, first

aggregates each individual’s achievements on all dimensions to get the overall well-being of

that individual and then uses some inequality index to measure inequality in the individuals’ overall well-being levels (see Maasoumi (1986, 1989), Decancq et al. (2009), Decancq and Lugo

(2012), Bosmans et al. (2015), and Seth (2013)). In the Type II approach, a variety of unidimensional inequality indices have been used. For example, Justino (2012) uses the Theil indices to study distributions of household well-being in Vietnam, Decancq et al. (2009) use

Atkinson family of inequality measures to study well-being inequality across countries, Decancq and Lugo (2012) use generalized Gini coefficients to illustrate change in multidimensional inequality in the Russian Federation, and Decancq et al. (2017) use generalized entropy class of inequality measures to study inequality in preference-adjusted multidimensional well-being in

Russia for the period 1995-2005.

We show both analytically and empirically that the Type I and Type II approaches can provide very different results when measuring inequality in a multidimensional framework (see Sections

2.3 and 4.1). The difference is because the Type I approach ignores the impact of the association between different dimensions on inequality of well-being and it does not capture the overlap of achievements in different dimensions such as income, health and education, experienced by an individual. The problem is analogous to the problems discussed by Pattanaik

4 et al. (2012) and Pattanaik and Xu (2018) in the context of the measurement of well-being and deprivation in a multidimensional framework. This is not to deny that the assessment of inequality of the individuals’ achievements along each separate dimension can give us very useful information. In fact, in Section 4, we look at information on dimensional inequalities to gain some insight into the results regarding the change in well-being inequality in the United

States that we derive using the Type II approach.

Kolm (1977) and Atkinson and Bourguignon (1982) pioneered the development of frameworks for multidimensional inequality indices. Following their work, several studies generalized properties of univariate inequality indices in a multidimensional framework (see Lugo (2007), for a review). An important distinction, however, is that this strand of literature does not aggregate well-being of multiple attributes at the individual level. In this paper, we use the Type

II approach to the measurement of inequality in well-being in the United States in the last decade.

The plan of the paper is as follows. In Section 2, we outline the conceptual framework that we use and explain why we chose to use the Type II approach rather than the Type I approach to measuring inequality. In Section 3, we detail the data we compile on multiple dimensions of well-being. In Section 4, we present estimates of well-being inequality in the United States.

Section 5 concludes.

II. THE CONCEPTUAL FRAMEWORK

2. 1. The notation

Let = { , , … , } ( 2) be the finite set of functionings or attributes which people value

1 2 𝑚𝑚 in their𝐹𝐹 lives𝑓𝑓 𝑓𝑓 and which𝑓𝑓 𝑚𝑚 constitute≥ the dimensions of individual well-being.i Let =

{1, 2, … , }. Attributes can take cardinal values (income) or ordinal values (health𝑀𝑀 status or

educational𝑚𝑚 attainment). For every , let be the (finite) set of discrete values that can

𝑗𝑗 𝑗𝑗 take, the number of elements in being𝑗𝑗 ∈ 𝑀𝑀 denoted𝑉𝑉 by ( ) 2. Let denote × . Thus,𝑓𝑓 if

𝑗𝑗 𝑗𝑗∈𝑀𝑀 𝑗𝑗 𝑗𝑗 denotes the attribute of health, then𝑉𝑉 may take values𝑟𝑟 𝑗𝑗 such≥ as poor𝑉𝑉 health, fair health,𝑉𝑉 good 𝑓𝑓

𝑓𝑓𝑗𝑗 health, etc. Let × = ( ) × be an achievement matrix with denoting individual ’s

𝐴𝐴𝑛𝑛 𝑚𝑚 𝑎𝑎𝑖𝑖𝑖𝑖 𝑛𝑛 𝑚𝑚 𝑎𝑎𝑖𝑖𝑖𝑖 𝑖𝑖 achievement in dimension . We use • = ( , … , ) to denote individual ’s achievement

𝑓𝑓𝑗𝑗 𝑎𝑎𝑖𝑖 𝑎𝑎𝑖𝑖1 𝑎𝑎𝑖𝑖𝑖𝑖 𝑖𝑖 vector along all the dimensions and • = ( , … , ) to denote the achievement vector of all

𝑗𝑗 1𝑗𝑗 𝑛𝑛𝑛𝑛 the individuals along the dimension 𝑎𝑎 . 𝑎𝑎 𝑎𝑎

𝑓𝑓𝑗𝑗 For each , there is a linear ordering (“at least as high as”) defined over , with >

𝑗𝑗 𝑗𝑗 𝑗𝑗 (“higher than”)𝑗𝑗 ∈ 𝑀𝑀 being the asymmetric factor�≥ of� . For every and every 𝑉𝑉 , let� �

𝑗𝑗 𝑗𝑗 𝑗𝑗 # : > /( ( ) 1). Thus, �the≥ �numbers in {𝑗𝑗 ∈ 𝑀𝑀: } are𝑎𝑎 ∈derived𝑉𝑉 by ′ ′ 𝑗𝑗 𝑗𝑗 𝑗𝑗 𝑗𝑗 𝑗𝑗 𝑗𝑗 𝑗𝑗 𝑗𝑗 𝑗𝑗 taking𝑠𝑠�𝑎𝑎 � ≡the �usual𝑎𝑎 ∈ 𝑉𝑉“Borda𝑎𝑎 � scores”,�𝑎𝑎 � 𝑟𝑟 defined𝑗𝑗 − in terms of the linear ordering𝑠𝑠�𝑎𝑎 � 𝑎𝑎 ∈,𝑉𝑉 of the different

𝑗𝑗 elements in , and then taking a suitable positive linear transformation� of≥ these� numbers so

𝑗𝑗 that the score𝑉𝑉 for the lowest (resp. highest) ranking element of becomes 0 (resp. 1). Note

𝑗𝑗 that under the usual definition, # : > + 1 is the 𝑉𝑉Borda score of an element of ′ ′ 𝑗𝑗 𝑗𝑗 𝑗𝑗 𝑗𝑗 𝑗𝑗 𝑗𝑗 given the linear ordering over�𝑎𝑎 ∈ 𝑉𝑉. Let𝑎𝑎 � (resp.�𝑎𝑎 � ) denote such that for all 𝑎𝑎 , ∗ 𝑗𝑗 𝑗𝑗 𝑗𝑗 ∗ 𝑉𝑉 (resp. ) is the -least�≥ �(resp. 𝑉𝑉 -greatest)𝑎𝑎 element𝑎𝑎 of 𝑎𝑎. ∈ 𝑉𝑉 𝑗𝑗 ∈ 𝑀𝑀 ∗ 𝑎𝑎∗𝑗𝑗 𝑎𝑎𝑗𝑗 �≥𝑗𝑗� �≥𝑗𝑗� 6 𝑉𝑉𝑗𝑗

2.2. The individual well-being function and inequality of well-being

Let = {1, 2, … , } denote the society under consideration. A real-valued individual well-being

function𝑁𝑁 is a function𝑛𝑛 : [0, ). Thus, for all and , the well-being of when ’s

achieved functioning bundle𝑉𝑉 → is∞ is given by (𝑖𝑖 )∈. 𝑁𝑁Note that,𝑎𝑎 ∈ 𝑉𝑉by definition, the individual𝑖𝑖 𝑖𝑖well-

being function is invariant with𝑎𝑎 respect to 𝜌𝜌individuals.𝑎𝑎 ii For each achievement matrix, , determines the𝜌𝜌 vector, ( ( •), … , ( •)), of individual well-being levels and then we use𝐴𝐴 𝜌𝜌 a

1 𝑛𝑛 real-valued well-being inequality𝜌𝜌 𝑎𝑎 measure𝜌𝜌 𝑎𝑎 , , which specifies the degree of well-being

inequality corresponding to ( ( •), … , ( 𝐻𝐻 •)). Thus, for every achievement matrix, , the

1 𝑛𝑛 𝜌𝜌 𝑎𝑎 𝜌𝜌 𝑎𝑎 𝐴𝐴 inequality of well-being is given by ( ) = ( •), … , ( •) .

𝑍𝑍 𝐴𝐴 𝐻𝐻�𝜌𝜌 𝑎𝑎1 𝜌𝜌 𝑎𝑎𝑛𝑛 � We shall use an individual well-being function satisfying (1) below (See Bossert et al. (2013),

Dhongde et al. (2016), Pattanaik and Xu (2019)𝜌𝜌 and Dhongde et al. (2019) for properties of a more general class of ρ functions satisfying (1)) :

(1) there exist positive constants ( ) with = 1, such that, for every ,

𝑗𝑗 𝑗𝑗∈𝑀𝑀 𝑗𝑗 ( )= [ ( ) ] for some𝑤𝑤 positive𝑗𝑗 ∈ 𝑀𝑀 . ∑ 𝑤𝑤 𝑎𝑎 ∈ 𝑉𝑉 γ 𝑗𝑗∈𝑀𝑀 𝑗𝑗 𝑗𝑗 ( 𝜌𝜌) figuring𝑎𝑎 ∑ in (1)𝑤𝑤 can𝑠𝑠 𝑎𝑎 be interpreted as reflecting𝛾𝛾 the relative importance attached to attribute𝑗𝑗 ∈ 𝑀𝑀 . One can, then, interpret [ ( ) ] as an individual’s index of overall

𝑗𝑗 𝑗𝑗∈𝑀𝑀 𝑗𝑗 𝑗𝑗 achievement𝑓𝑓 (IOA) in terms of the different∑ attributes𝑤𝑤 𝑠𝑠 𝑎𝑎 when her achievement vector is given by

. Given the well-being functions in (1), where an individual’s well-being is an increasing

𝑎𝑎function∈ 𝑉𝑉 of her IOA, should we just consider the distribution of individual IOA rather than the

distribution of individual well-being? Obviously, if = 1, it will not make any difference

whether we measure inequality in the distribution 𝛾𝛾of individual IOA or well-being. But, if 1,

𝛾𝛾 ≠ 7 the assessment of inequality in the society will, in general, differ, depending on whether we consider inequality of individual IOAs or well-being. When this happens, we believe that, it is inequality in the distribution of individual well-being, which should be taken into account in the context of social welfare judgments. In our empirical analysis, we shall assume that = , so 1 𝛾𝛾 2 that, as the individual’s IOA increases, her well-being increases but at a diminishing rate.

Once we derive the well-being levels, ( ), ( ), … , ( ), of the different individuals, we

1 2 𝑛𝑛 can use a widely accepted inequality measure,𝜌𝜌 𝑎𝑎 𝜌𝜌 such𝑎𝑎 as 𝜌𝜌the𝑎𝑎 Gini index to measure inequality in the distribution of individual well-being.

2.3. The Type I approach to measuring multidimensional inequality and its limitations

In our empirical analysis in the next section, we use the procedure outlined in Section 2.2. It obviously comes within what we have called the Type II approach to the measurement of well- being inequality. It may be useful to discuss a little further why we have chosen not to adopt the Type I approach to measuring inequality of well-being though we do believe that it may give us otherwise useful information. The intuitive difficulty involved in the Type I approach to measuring inequality has been noted earlier by Decancq et al. (2017). Analogous problems in the context of the measurement of deprivation and the standard of living have been discussed by Dutta et al. (2003).

Consider the Type I approach. In this approach, we first choose, for every , a real-valued inequality measure , which, for every vector, • , of individual achievements𝑗𝑗 ∈ along𝑀𝑀 dimension

𝜗𝜗𝑗𝑗 𝑐𝑐 𝑗𝑗 specifies a real number • as the index of inequality in the individuals’ achievements in

𝑗𝑗 𝑗𝑗 𝑗𝑗 terms𝑓𝑓 of . Next, given an𝜗𝜗 achievement�𝑐𝑐 � matrix, , one aggregates the dimensional

𝑓𝑓𝑗𝑗 𝐶𝐶 8

inequalities, ( • ), ( • ), … , ( • ) so as to reach a real number that serves as the index

1 1 2 2 𝑚𝑚 𝑚𝑚 of inequality 𝜗𝜗reflected𝑐𝑐 𝜗𝜗 in 𝑐𝑐the entire𝜗𝜗 achievement𝑐𝑐 matrix, . Let be the set of all -tuple of

𝐴𝐴 𝐾𝐾 𝑚𝑚 real numbers ( , … , ) such that, for all , = • for some achievement matrix .

1 𝑚𝑚 𝑗𝑗 𝑗𝑗 𝑗𝑗 Consider the class𝑦𝑦 of all𝑦𝑦 real-valued functions𝑗𝑗 ∈ defined𝑀𝑀 𝑦𝑦 on𝜗𝜗 �.𝑐𝑐 One� needs to choose a function 𝐶𝐶 from this class to aggregate the dimensional inequality indices.𝐾𝐾 Once such a function is 𝐺𝐺 chosen, the overall inequality for any achievement matrix is given by ( ) = 𝐺𝐺

𝐶𝐶 𝑌𝑌 𝐶𝐶 ( • ), ( • ), … , ( • ) .

𝐺𝐺�𝜗𝜗1 𝑐𝑐 1 𝜗𝜗2 𝑐𝑐 2 𝜗𝜗𝑚𝑚 𝑐𝑐 𝑚𝑚 � When we are engaged in ethical evaluation of social states, inequality of well-being distribution

is an important consideration and, to assess inequality of well-being, it makes sense to find first

the well-being of each individual and then to assess inequality in the distribution of individual

well-being, i.e., to use a Type II inequality measure (. ) defined in terms of a suitably chosen

individual well-being function (. ) and a suitably chosen𝑍𝑍 measure, , of well-being inequality,

as described in Section 2.2. But𝜌𝜌 an interesting question arises here. 𝐻𝐻 Suppose we have a

“satisfactory” Type II inequality measure (. ). Can we possibly find a Type I inequality measure

(. ) such that, for all achievement matric𝑍𝑍es, and , [ ( ) ( ) if and only if ( )

𝑌𝑌( )]? If this can be done, then, for most practical𝐴𝐴 𝐵𝐵purposes,𝑌𝑌 𝐴𝐴 ≥ it 𝑌𝑌would𝐵𝐵 not matter whether𝑍𝑍 𝐴𝐴 ≥ we

use𝑍𝑍 𝐵𝐵 the Type II measure (. ) of inequality of individual well-being, or we use a Type I measure

(. ) of inequality that will𝑍𝑍 give us the same ranking of different possible achievement matrices

𝑌𝑌as the ranking that we get by using (. ). This, however, cannot be done, given some very mild

assumptions. Thus, under some very𝑍𝑍 mild assumptions, every Type I measure of inequality will

deviate from the ranking of some achievement matrices given by the Type II measure of well-

being, namely (. ). A detailed and general study of this problem needs separate investigation and is beyond the𝑍𝑍 scope of this paper. But a simple example will illustrate the basic point.

Example 1. Let = 2 and let = 2. Suppose, for each , there are only two possible

values, 1 (satisfactory)𝑁𝑁 and 0 (unsatisfactory),𝑀𝑀 that an individual’s𝑗𝑗 ∈ 𝑀𝑀 achievement in terms of

attribute can take. Suppose we have a Type II measure of well-being inequality (. ) such

𝑗𝑗 that, for every𝑓𝑓 achievement matrix , ( ) = ( , ), ( ) , where:𝑍𝑍

𝐶𝐶 𝑍𝑍 𝐴𝐴 𝐻𝐻�𝜌𝜌 𝑐𝑐11 𝑐𝑐12 𝜌𝜌 𝑐𝑐21𝑐𝑐22 � (2) is increasing in the individual’s achievement in each dimension;

(3) 𝜌𝜌 is such that, for all achievement matrices and , if ( •) > ( •) > ( •) and

1 1 2 𝐻𝐻( •) > ( •) > ( •), then ( ) > (𝐴𝐴 ). 𝐵𝐵 𝜌𝜌 𝑎𝑎 𝜌𝜌 𝑏𝑏 𝜌𝜌 𝑎𝑎

1 2 2 () just embodies𝜌𝜌 𝑎𝑎 the𝜌𝜌 𝑏𝑏 intuition𝜌𝜌 that𝑎𝑎 each attribute𝑍𝑍 𝐴𝐴 is𝑍𝑍 a𝐵𝐵 positivel2y valued attribute. (3) is an

intuitively compelling property for a measure of well-being inequality in our two-person

society.

Now consider a Type I measure (.) such that, for every achievement matrix , ( ) =

𝑌𝑌 𝐶𝐶 𝑌𝑌 𝐶𝐶 ( • ), ( • ) , where the functions and ( = 1, 2) have the interpretations as given

1 1 2 2 𝑗𝑗 𝐺𝐺earlier�𝜗𝜗 𝑐𝑐 in this𝜗𝜗 section.𝑐𝑐 � The only restriction 𝐺𝐺that we𝜗𝜗 postulate𝑗𝑗 here is the following property of

“anonymity” (capturing the idea that, for each dimension, two achievement vectors having the

same distribution yield the same inequality index) for the functions and :

𝜗𝜗1 𝜗𝜗2 (4) for every {1, 2} and any two vectors, • and • , of individual achievements along

𝑗𝑗 ∈ 𝑐𝑐 𝑗𝑗 𝑑𝑑 𝑗𝑗 dimension , if = and = , then • = ( • ).

𝑗𝑗 1𝑗𝑗 2𝑗𝑗 2𝑗𝑗 1𝑗𝑗 𝑗𝑗 𝑗𝑗 𝑗𝑗 𝑗𝑗 To see that the inequality𝑓𝑓 𝑑𝑑 ranking𝑐𝑐 of achievement𝑑𝑑 𝑐𝑐 matrices𝜗𝜗 �𝑐𝑐 � implied𝜗𝜗 𝑑𝑑 by (. ) does not coincide

with that implied by (. ), consider the following two achievement matrices,𝑌𝑌 and . The rows

𝑍𝑍 10 𝐴𝐴 𝐵𝐵

1 1 show the individuals and the columns show the attributes. Matrix A is given by and 0 0

1 0 � � Matrix B is given by . Note that = so that ( ) = ( ). By (4), ( ) = 0 1 • • • • • � � 𝑎𝑎 1 𝑏𝑏 1 𝜗𝜗1 𝑎𝑎 1 𝜗𝜗1 𝑏𝑏 1 𝜗𝜗2 𝑎𝑎 2 ( • ). Note that • and • have the same distribution. Since is a function, it follows that

𝜗𝜗2 𝑏𝑏 2 𝑎𝑎 2 𝑏𝑏 2 𝐺𝐺 ( ) = ( • ), ( • ) = ( • ), ( • ) = ( ). By (3), ( •) > ( •) >

𝑌𝑌 𝐴𝐴 𝐺𝐺�𝜗𝜗1 𝑎𝑎 1 𝜗𝜗2 𝑎𝑎 2 � 𝐺𝐺�𝜗𝜗1 𝑏𝑏 1 𝜗𝜗2 𝑏𝑏 2 � 𝑌𝑌 𝐵𝐵 𝜌𝜌 𝑎𝑎1 𝜌𝜌 𝑏𝑏1 ( •) and ( •) > ( •) > ( •). Hence, by (4), we have ( ) > ( ). Thus, we have

2 1 2 2 𝜌𝜌(𝑎𝑎 ) = ( 𝜌𝜌), 𝑎𝑎but ( 𝜌𝜌) >𝑏𝑏 ( )𝜌𝜌. 𝑎𝑎 𝑍𝑍 𝐴𝐴 𝑍𝑍 𝐵𝐵

𝑌𝑌 𝐴𝐴 𝑌𝑌 𝐵𝐵 𝑍𝑍 𝐴𝐴 𝑍𝑍 𝐵𝐵

III. DATA

We use data from the Current Population Survey (CPS), a household survey conducted jointly

by the Census Bureau and the Bureau of Labor Statistics for over 50 years. The Annual Social

and Economic (ASEC) Supplement of the CPS is used to estimate the official poverty measure

(OPM) as well as the supplemental poverty measure (SPM). The CPS is a nationally

representative household survey that collects individual and household level data on income

sources, and other indicators such as work experience, poverty, health insurance coverage, and

education. We use data from person records for three years, namely, 2010, 2014 and 2018.

Micro data for CPS-ASEC rounds is compiled from the National Bureau of Economics Research

(NBER) data files. We choose 2011 March CPS-ASEC round since that is the first year when SPM adjusted income is available with NBER. We include in our analysis all individuals, including the elderly and the children and use full sample weights given in the data file.

3.1 Well-being Attributes

We choose three basic well-being attributes, namely, income, education and health. These attributes are almost always included in most multidimensional poverty measures, in the

United States (e.g. Dhongde and Haveman, 2017) and other high income countries such as

Germany (e.g. Nowak and Scheicher, 2017), Australia (e.g. Martinez Jr. and Peralez, 2017) and in the European Union (e.g. Weziak-Bialowolska, 2016). Furthermore, the Commission on the

Measurement of Economic Performance and Social Progress (Stiglitz et. al. 2009) recommended eight key attributes, including standard of living, education and health that should be taken into account simultaneously to define multi-dimensional well-being particularly in high-income

countries.

We use data on income adjusted for the Supplemental Poverty Measure (SPM). The SPM is an

alternative poverty measure estimated since 2011 and overcomes many of the drawbacks of

the OPM. For instance, the SPM thresholds are based on a broad measure of necessary

expenditures—food, clothing, shelter, and utilities—and are based on recent, annually updated

expenditure data (see, Fox (2020) for the latest SPM estimates).We use the SPM adjusted

income since these take into account geographic differences in the cost of living and use

equivalence scales to adjust for households with different sizes and compositions (See Bridges

and Gesumaria, 2015, Garner and Gudrais, 2018). The SPM income measure is cash income plus

in-kind government benefits (such as food stamps and housing subsidies) minus

nondiscretionary expenses such as taxes and medical out-of-pocket expenses.

The data provide income categories for individual’s SPM income. We thus have 41 income categories; the lowest category has individuals with annual incomes less than $5,000. The income categories proceed in increments of $5,000 and the highest category is income greater than or equal to $200,000. Table 1 shows some of the income categories and the proportion of individuals belonging to these categories. Individuals’ incomes in 2010 were largely affected by the Great recession. As the economy recovered, median incomes increased from $45,000-

$49,999 in 2010 to $55,000-$59,999 in 2018. Over time, the proportion of individuals in top income categories increased and that in the bottom brackets decreased.

Table 1. Attributes, Categories and Proportion of Population

No. of Percent Pop. Percent Pop. Percent Pop.

Categories 2010 2014 2018 SPM Adjusted Income 1. Less than $5,000 3.0 3.3 2.6 2. $5000 to $9,999 2.9 2.7 1.8 3. $10,000 to $14,499 5.0 4.7 3.6 …. ………………………….. …… …… …… 11. $50,000 to $54,499 5.2 5.3 4.8 12. $55,000 to $59,999 4.4 4.8 4.5 …. ………………………….. …… …… …… 21. $100,000 to $104,999 1.6 1.1 2.0 22. $105,000 to $109,999 1.4 1.1 1.8 …. ………………………….. …… …… …… 31. $150,000 to $154,999 0.4 0.5 0.7 32. $155,000 to $159,999 0.4 0.6 0.6 …. ………………………….. …… …… …… 39. $190,000 to $194,999 0.2 0.2 0.3 40. $195,000 to $199,999 0.1 0.2 0.3 41. $200,000 and over 1.5 2.5 4.4 Educational Attainment 1. Less than high school 12.2 11.2 9.6 2. High school diploma or equivalent 28.4 27.5 26.2

3. Some college but no degree 19.4 18.9 17.5 4. Associate degree 9.8 10.1 10.4 5. Bachelor’s degree 19.1 20.1 22.2 6. Masters, Professional, 11.2 12.3 14.1 Doctorate degree Self-ranked Health Status 1. Poor 3.5 3.2 2.9 2. Fair 8.3 8.2 8.3 3. Good 24.6 24.4 25.0 4. Very Good 31.4 31.1 32.0 5. Excellent 32.2 33.1 31.8

Notes: About 1% individuals in the survey have negative SPM income and 0.2% individuals have zero income. Children under the age of 18 are assigned maximum educational attainment within the household.

The CPS has data on individual’s educational attainment. Between 2010 and 2018, the proportion of individuals with lower attainment levels (less than a degree) decreased, and those with higher attainment increased. For example, compared to the 11% in 2010, 14% individuals had a masters, professional or doctorate degree in 2018. In fact, there is a first-order stochastic dominance observed; the cumulative distribution in 2018 dominates the 2014 distribution which in turn dominates the 2010 distribution. Unlike incomes, the distribution in educational attainment became more equal over time.

There are five categories for self-assessed health given in the CPS. The survey collects data on individual’s overall health status by asking them to rank their health on a scale of 1 to 5. Unlike income and education, we do not see significant change in the proportion of individuals belonging to each category, except that the proportion of individuals reporting poor health declined from 3.5% in 2010 to 2.9% in 2018. A majority (about 64%) chose health status as very good or excellent during the period.

3.2. Set-up for a Benchmark Well-being Index

As seen in Table 1, income has 41 categories, education has six categories and health has five

categories. We transform the categorical data for each attribute into scores ranging from 0 to 1.

Thus, income categories are transformed as 0, , , … ,1 , educational attainment levels are 1 2 � 40 40 � transformed as 0, , , , , 1 and the five categories of health status are transformed as 1 2 3 4 � 5 5 5 5 � 0, , , , 1 . The IOA of an individual is given by ( ) where ( ) is a 1 2 3 � 4 4 4 � �∑𝑗𝑗∈𝑀𝑀 𝑤𝑤𝑗𝑗𝑠𝑠 𝑎𝑎𝑗𝑗 � 𝑤𝑤𝑗𝑗 𝑗𝑗 ∈ 𝑀𝑀 positive constant denoting the weight of an attribute such that = 1. In the benchmark

∑𝑗𝑗∈𝑀𝑀 𝑤𝑤𝑗𝑗 case, we assign equal weights to all three attributes, so that = , = 1,2,3 and assume = 1 𝑤𝑤𝑗𝑗 3 𝑗𝑗 𝛾𝛾 in the well-being function of an individual given by ( )= [ ( ) ] . We vary weights 1 γ 2 𝜌𝜌 𝑎𝑎 ∑𝑗𝑗∈𝑀𝑀 𝑤𝑤𝑗𝑗𝑠𝑠 𝑎𝑎𝑗𝑗 assigned to each of the attributes to test the sensitivity of the results.

IV. INEQUALITY IN MULTIDIMENSIONAL WELL-BEING IN THE U.S.

4.1. Inequality using Type I and Type II Approaches

In Table 2, we estimate multiple inequality indices (expressed in percentage terms) for each attribute (normalized between 0 and 1) and using both Type I and Type II approaches. The Type

I approach calculates an average of the inequality indices for each of the attributes. The Type II approach, on the other hand, calculates the inequality indices using the multidimensional well-

being of each individual.

The Gini index is the most commonly used measure of inequality. However, it is most sensitive to differences in attributes near the mode of the distribution. Hence we also estimate inequality measures which tend to be sensitive to tails of the distribution.iii In addition to the

Gini index, we also estimate the Generalized Entropy index (GE1), also known as the Theil index and the Atkinson (A (0.5)) index.

The different inequality measures in Table 2 show similar trends. Inequality in income was higher than that in education and health. The percentage change in inequality between 2010 and 2018 in all three inequality measures shows that inequality in income increased and that in education decreased, inequality in health status declined slightly or remained constant. The total percentage change in inequality between 2010 and 2018 highlights the difference between Type I and Type II approaches. Between 2010 and 2018, inequality as measured by the

Type I approach decreased slightly. However, inequality in multidimensional well-being, that is, inequality as measured by Type II approach, in fact increased.

Table 2: Inequality in Attributes and in Multidimensional Well-being

Income Education Health Type I Type II (Average) (Well-being) 2010 36.419 27.294 14.890 26.201 11.113 2014 37.277 26.973 14.899 26.383 12.121 2018 36.856 26.257 14.907 26.007 12.079 % Chg. Gini 1.2 -3.8 0.1 -0.7 8.7 2010 21.771 12.156 4.268 12.732 2.140 2014 22.91 11.91 4.270 13.030 2.551 2018 21.978 11.389 4.230 12.532 2.495 % Chg. GE(1) 1.0 -6.3 -0.9 -1.6 16.6 2010 10.935 6.289 2.302 6.509 1.137 2014 11.443 6.183 2.302 6.643 1.365

2018 11.180 5.947 2.273 6.467 1.325 % Chg. A(0.5) 2.2 -5.4 -1.3 -0.6 16.5 Notes: The percentage change is the total change in the inequality indices between 2010 and 2018.

4.2. Inequality in Multidimensional Well-being Using Alternate Weights on Attributes

Inequality in multidimensional well-being in Table 2 is estimated by assigning equal weights to

all three attributes. We now consider what happens when we change the weights on these

attributes and re-estimate inequality in multidimensional well-being. We also vary values of

( = 1, 2) and find that our results are robust to these changes.

𝛾𝛾 𝛾𝛾 In Table 3, we use three alternate weighting schemes and estimate the Gini index. In the first

panel, we assign zero weight to one attribute and equal (1/2) weights to the remaining two

attributes. For example, when we assign zero weight to income, we see that inequality in

multidimensional well-being (i.e., inequality in the Type II approach) in fact decreases by 3.1%.

A rise in income inequality was a contributing factor in increasing inequality in well-being.

Hence, when we drop income, inequality in well-being decreases. On the other hand, in the third panel, when we increase the weight on income to 1/2 and assign 1/4 weight to the other two attributes, inequality in multidimensional well-being increases by 14.3%. Equal weights are placed on all attributes in the second panel and this shows the benchmark Gini values replicated from Table 2. Recall from Table 2 that inequality in education declined over time.

Therefore, when we place a greater weight on education, well-being inequality increases by the least percentage (4.2%) and if we remove education from the well-being index, well-being inequality increases significantly (11.6%).

Table 3: Gini Indices of Multidimensional Well-being using Alternate Weights on Attributes

Zero Zero Zero Greater Greater Greater Equal weight weight weight weight weight weight weight on all on on on on on on attributes Income Education Health Income Education Health 2010 12.515 10.668 17.33 11.113 11.230 12.983 10.493 2014 12.365 11.628 19.175 12.121 12.681 13.781 11.145 2018 12.13 11.909 18.313 12.079 12.832 13.528 11.098 % Chg. -3.1 11.6 5.7 8.7 14.3 4.2 5.8 2010-2018 Notes: The percentage change is the total change in the Gini index between 2010 and 2018. Zero weight for income implies education and health have weight equal to 1/2 respectively. Equal weights is the benchmark case when each attribute receives 1/3 weight. Higher weight for income implies income receives 1/2 weight and education and health receive 1/4 weight.

4.3. Inequality in the Conditional Distribution of Attributes

We analyze the distribution of each of the attributes to understand how an increase in (Type II) inequality may have come about. In Table 4, we provide the median values of each of the three attributes (since these are categorical variables) and the well-being index by income deciles.

That is, we summarize the distribution of education, health and well-being, conditional on income categories.

Between 2010 and 2018, median incomes increased for each decile, though growth in income was much higher in the top decile compared to the bottom decile. Median incomes in the lowest decile increased from income category 2 to 3, i.e. from $5000-$9999 to $10,000-

$14,999. On the other hand, median incomes in the top decile increased from income category,

28 to 39, i.e. from $135,000-$139,999 to $190,000 to $194,999.

In 2010, median education in each income decile was 3 (some college but no degree). In 2018, however, median education increased among higher income deciles. In income deciles 7 and 8, median education increased from 3 to 4 (associate degree) and in deciles 9 and 10, it increased to 5 (bachelor’s degree). Thus, conditional on incomes, the distribution of educational attainment became more unequal, although the unconditional distribution of educational attainment became more equal over time (as seen in Table 2). This overlap of income and educational attainment and the associated rise in inequality is captured by the distribution of the well-being index. There is not much variance in the median values of health status; most income deciles had a median value of 4 (very good health).

If we simply compare the range of the distribution of the three attributes, we find that in 2010, the bottom decile’s median attribute levels were (2, 3, 4) and the top decile’s levels were (28, 3,

4). Thus, in 2010, the only difference between bottom and top deciles was the difference in median incomes; they had similar median education and health categories. By 2018, not only had the difference in the median incomes increased, there was also a gap in educational attainment and health status for the bottom (3, 2, 3) and top deciles (39, 5, 4).

This rise in inequality is measured by the distribution of the multidimensional well-being index by income deciles. Over a period of eight years, the median multidimensional well-being index decreased in the three lowest income deciles and increased in the remaining deciles. In fact well-being increased by more than 10% in the top two income deciles.

Table 4: Distribution of Attributes and Multidimensional Well-being by Income Deciles

2010 2018 % Change in Decile Income Educ. Health WB. Income Educ. Health WB. WB. 1 2 3 4 0.619 3 2 3 0.563 -9.1 2 5 3 4 0.639 6 2 4 0.599 -6.3 3 7 3 4 0.658 8 2 4 0.626 -4.9 4 8 3 4 0.665 9 3 4 0.677 1.9 5 9 3 4 0.683 11 3 4 0.695 1.8 6 11 3 4 0.695 13 3 4 0.719 3.4 7 13 3 4 0.707 16 4 4 0.758 7.2 8 16 3 4 0.725 19 4 4 0.791 9.1 9 20 3 4 0.758 25 5 4 0.837 10.3 10 28 3 4 0.816 39 5 4 0.922 12.9 Note: Table shows median values of income, education, health and well-being.

Suppose, instead of conditioning the distribution by income deciles, we divide individuals in

quintiles by educational attainment (see Table 5). Between 2010 and 2018, there was no change in the distribution of attributes in the lowest quintile (2, 9, 4) with a median educational attainment of less than high school. As educational levels increased we find that median incomes increased significantly more in 2018 than in 2010. For instance, with education level of bachelor’s degree (4th quintile), income in 2010 was between $45,000 and $49,000 whereas that in 2018 was between $55,000 and $59,000. In fact, in 2010, median incomes were the same for the 4th and 5th quintile. But in 2018, the 5th quintile had educational attainment of

Masters, Professional or Doctorate degree and median incomes between $95,000 and $99,000.

Table 5: Distribution of Attributes and Multidimensional Well-being by Education Quintiles

2010 2018 % Change Quintile Educ. Income Health WB. Educ. Income Health WB. in WB. 1 2 9 4 0.592 2 9 4 0.592 0.0 2 3 10 4 0.683 3 11 4 0.689 0.9 3 4 10 4 0.736 4 12 4 0.735 -0.2

4 5 11 4 0.796 5 16 4 0.816 2.6 5 6 11 4 0.842 6 20 4 0.880 4.6 Note: Table shows median values of income, education, health and well-being.

Thus, we see distinctly how the Type II approach to measuring inequality can differ from the

Type I approach. The distribution of individuals’ well-being index captures inequality in the joint distribution of attributes which is missed when we measure inequality in the marginal distribution of each attribute separately.

V. CONCLUSIONS

In the last decade multidimensional poverty has been widely used to measure deprivation in the quality of life often not captured by income poverty measures. In this paper, we assessed inequality in multidimensional well-being. We distinguished between two alternative approaches to measure such inequality. The Type I approach calculates an average of inequality indices for the separate attributes. The Type II approach, on the other hand, calculates an inequality index based on the multidimensional well-being of individuals.

Using data on three basic attributes of quality of life, namely income, education and health, we studied changes in the inequality of multidimensional well-being of individuals in the United

States in the last decade. We found that between 2010 and 2018, income inequality had increased, inequality in educational attainment had decreased and inequality in health status had hardly changed. The Type I approach showed that multidimensional inequality had slightly decreased but the Type II approach showed that, in fact, inequality in multidimensional well- being had increased. This is consistently observed in different inequality measures. Our analysis revealed that over time, the joint distribution of income and educational attainment had

21 become more unequal, although the unconditional distribution of educational attainment became more equal. This was reflected in the Type II approach to inequality since the multidimensional well-being index captured the overlap of attributes experienced by an individual; the Type I approach failed to measure this overlap. The paper, thus, highlights the importance of measuring multidimensional well-being and its distribution in order to get a comprehensive picture of changes in the quality of life over time.

i Sen (1987) defines the term, “functionings” are the “doings” and “beings” that people value. Income, which we use as one of the three attributes, is not a functioning in Sen’s sense. Instead it can be viewed as a proxy for many different functionings taken together, such as shelter from the elements, being well-nourished, social standing, etc. We shall, however, ignore in this paper fine conceptual distinctions between functionings and proxies for functionings and use the terms “functionings”, “attributes”, and “dimensions of well-being” interchangeably. ii The property that the individual well-being function ρ is invariant with respect to individuals can be interpreted either in terms of Sen’s (1987) ‘standard-evaluation’ (as opposed to what Sen (1987) calls ‘self-evaluation’) or in terms of the values advocated by the social scientist in the course of social deliberation. See, however, Decancq et al. (2017) for an approach to measuring multidimensional inequality in well-being where they assume that each individual’s well-being depends on her achievement vector as well as an individual- specific parameter reflecting her preferences. iii In a generalized entropy index GE(t), when t>0, the index is sensitive to the top of the distribution, and when t<0, the index is sensitive to the bottom of the distribution. In the Atkinson’s index A(e), the more positive e > 0 is, the more sensitive A(e) is to differences at the bottom of the distribution.

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