A Multi-dimensional Measure of 

Economic Well-being for the U.S.: 

The Material Condition Index

Thesia I. Garner (BLS) 

Kathleen S. Short (Census)

BLS: Division of Price and Index Number 

Census: Housing and Household Economic Statistics Division

Southern Economic Association Annual Meetings, Tampa, Florida

November 25, 2013

Disclaimer: Any opinions expressed in this presentation are those of the authors 

and do not constitute policy of the BLS or Census Bureau.
Economic Well-being: Concepts

2

Income (Y)

Consumption (C )

Wealth (W)
Recent Support for Joint Measures

 Commission on the Measurement of Economic 

Performance and Social Progress (Stiglitz , Sen, 

Fituoussi, 2009).

 Reports of OECD Expert Group on Micro Statistics on 

Household Income, Consumption and Wealth (2013)

 Integrated framework for Y, C, and W

 Analysis tools, e..g., composite multi-dimensional measures 

at micro- or household level (new field of statistics)

3
Objectives of this Research

 Test the feasibility of producing composite 

multi-dimensional measure

Material Condition Index (MCI) defined by Ruiz 

(2011)

For the U.S

 Consistently define income, consumption, and 

wealth following OECD integrated framework

4
 United States, 2011

 Garner and Short 2013

 Consumer Expenditure 

Survey

 Y, C, financial and nonfinancial W

MCI’s Produced for...

 France 1995

 Ruiz 2011

 Household Expenditure 

Survey

 Y, C, financial W

5
Materials Conditions Index (Ruiz 2011)

 Builds on work on Kolm (1977), Atkinson (1970, 

1982) and Foster and colleagues (2005, 2008, 2010)

 Atkinson’s income standard (i.e., equally distributed 

equivalent income, EDE)

 Foster and colleagues’ multidimensional measures

 Ruiz justifies method based on set of standard 

properties of aggregation functions and axioms for 

multidimensional measures

6
Materials Conditions Index (Ruiz 2011)

 Combines measures of 

 Central tendency (i.e., mean achievements) 

 Dispersion (i.e., distributions of achievements)

 Aggregation function uses nested generalized means 

to summarize achievements within dimensions and 

across dimensions to single summary index

 Question addressed: Does a joint measure modify 

picture of material living conditions relative to one 

measure alone?

7
Generalized Mean

8

 Considers whole distribution

 Aversion parameter, q, based on a utilitarian welfare 

concept

 q = 1 generalized mean reduces to arithmetic mean

 q = 0, geometric mean

 q = -1, harmonic mean

 as q decreases, greater weight placed on the lower tail of 

distribution

 Penalties applied for

 Inequality between individuals, “inter” inequality (q)

 Unbalanced achievement in dimensions, “intra” inequality (r)
Example of Distribution of Dimensions

 Penalty for unbalanced achievement, as r decreases, greater weight placed on the 9

achievements that are less

Percentage of Highest Level of Achievement

0%

10%

20%

30%

40%

50%

60%

70%

Balanced Achievements

0%

10%

20%

30%

40%

50%

60%

70%

Unbalanced Achievements
Construction of MCI

10

 Ruiz assumes log transformation of dimension 

values (Alkire and Foster 2010)

 Diminishing returns to increases in each dimension

 e.g., as income increases, diminishing returns to 

transforming income into material well-being

 People do not need excessive income, consumption or 

wealth to ensure decent levels of living

 Challenge: 0 and negative values
Construction of MCI - 2

11

 Normalize values of dimensions to ratio-scale 

measure (Alkire and Foster, 2010)

 0% lowest achievement

 100% highest achievement

 Assumption

 Comparability across Y, C, and W (e.g., 60% achievement in 

one dimension same as 60% in others)

 Result is Achievement Matrix (n x 3) with values 0 to 1

 Order of aggregation 
S- Aggregation (Specific: across 

individuals then across dimensions)

Income Consumption Wealth

YA CA WA MCIA

Individuals
I- Aggregation (Individualistic:

across dimensions then across individuals)

Income Consumption Wealth

MCIA

Individuals

MCIi

MCIi+1
Does Aggregation Matter?

14

 Do results equal for I- and S-aggregations?

 Path independence

 Nested generalized mean of curvatures q and r path 

independent only when q=r
Does Aggregation Matter?

15

 Do results equal for I- and S-aggregations?

 Path independence

 Nested generalized mean of curvatures q and r path 

independent only when q=r

 More flexible form of index desirable

 allows for greater concern for inequality between 

individuals within dimensions (q) and less for correlation 

across dimensions ( r) 

q r ≠
U.S. Consumer Expenditure 

Survey, Interview

 Collection period: 2009 Q2 - 2012 Q1

 Y and C cover same 12 months, W at end of period

 Sample: 14,948 unique consumer units

 Adjustments to Y, C, and W

 2011 $U.S.

 Modified OECD equivalence scale 

 Results population weighted, not adjusted for 

attrition across quarters

 No standard errors

 Results PRELIMINARY 16
Variables

 Adjusted disposable income 

 Consumption expenditures

 Wealth = assets – liabilities

 All consistently defined using OECD 

framework

 New for U.S. measures of economic wellbeing: vacation properties and time shares

17
Results Presented by

18

Income

Consumption

Wealth

Jointly (MCI)

 Overall, income deciles, housing tenure

 Aversion to inequality and un-balancement in 

achievement

 Aggregation order

 Weighting dimensions
Data source: U.S. Consumer Expenditure Interview Survey, based on values in 2011 dollars U.S. 19

0 0.1 0.2 0.3 0.4 0.5 0.6

Wealth (q=1)

Income (q=1)

Consumption (q=1)

Normalized Means 

Normalized Arithmetic Means of Economic WellBeing Dimensions
20

0 0.1 0.2 0.3 0.4 0.5 0.6

MCI (q=r=-3)

MCI (q=r=-1)

MCI (q=r=1)

Wealth (q=1)

Income (q=1)

Consumption (q=1)

Normalized Means and Index

Normalized Arithmetic Dimension Means and MCI 

under Different Aversions to Inequality

MCI =MCI I S

Neutral aversion

Medium aversion

Strong aversion

Data source: U.S. Consumer Expenditure Interview Survey, based on values in 2011 dollars U.S.
Neutral Aversion to Inequality (q=r=1) 

by Income Deciles and Housing Tenure

MCII=MCIS (path independent)

-0.10

0.10

0.30

0.50

0.70

Decile 1

Decile 3

Decile 5

Decile 7

Decile 9

Decile 10

Own w/ 

mortgage

Own w/o 

mortgage or ...

Rent paid

Rent no pay/ 

std housing

Income

Consumption

Wealth

MCI
MCI with Medium Aversion to Inequality (q=r=-1) 

by Income Deciles and Housing Tenure

Neutral aversion for dimensions, MCII=MCIS (path independent)

-0.10

0.10

0.30

0.50

0.70

Decile 1

Decile 3

Decile 5

Decile 7

Decile 9

Decile 10

Own w/ 

mortgage

Own w/o 

mortgage or ...

Rent paid

Rent no pay/ 

std housing

Income

Consumption

Wealth

MCI
MCI with Strong Aversion to Inequality (q=r=-3)

by Income Deciles and Housing Tenure

Neutral aversion for dimensions, MCII=MCIS (path independent)

-0.10

0.10

0.30

0.50

0.70

Decile 1

Decile 3

Decile 5

Decile 7

Decile 9

Decile 10

Own w/ 

mortgage

Own w/o 

mortgage or ...

Rent paid

Rent no pay/ 

std housing

Income

Consumption

Wealth

MCI
0 0.1 0.2 0.3 0.4 0.5 0.6

Strong

Medium

Neutral

Path Indep,

I=S Agg

Material Condition Index

MCI, Aversion and Path Dependency

Data source: U.S. Consumer Expenditure Interview Survey, based on values in 2011 dollars U.S. 24
0 0.1 0.2 0.3 0.4 0.5 0.6

Strong

Medium

Neutral

Path Indep,

I=S Agg

Material Condition Index

MCI, Aversion and Path Dependency

25

0 0.1 0.2 0.3 0.4 0.5 0.6

Strong

Medium

Neutral

Strong

Medium

Neutral

Path Dep, SAgg, r=0.5 Path Indep, I=S Agg

Material Condition Index

Data source: U.S. Consumer Expenditure Interview Survey, based on values in 2011 dollars U.S.
0 0.1 0.2 0.3 0.4 0.5 0.6

Strong

Medium

Neutral

Strong

Medium

Neutral

Strong

Medium

Neutral

Path Dep, IAgg, r=0.5 Path Dep, S- Agg, r=0.5 Path Indep, I=S Agg

Material Condition Index

0 0.1 0.2 0.3 0.4 0.5 0.6

Strong

Medium

Neutral

Path Indep,

I=S Agg

Material Condition Index

MCI, Aversion and Path Dependency

26

0 0.1 0.2 0.3 0.4 0.5 0.6

Strong

Medium

Neutral

Strong

Medium

Neutral

Path Dep, SAgg, r=0.5 Path Indep, I=S Agg

Material Condition Index

Data source: U.S. Consumer Expenditure Interview Survey, based on values in 2011 dollars U.S.
MCI’s Based on Income, Consumption and 

Wealth vs. Income and Consumption (q=r)

27

0 0.1 0.2 0.3 0.4 0.5

YCW, strong

YCW, medium

YCW, neutral

Data source: U.S. Consumer Expenditure Interview Survey, based on values in 2011 dollars U.S.
MCI’s Based on Income, Consumption and 

Wealth vs. Income and Consumption (q=r)

28

0 0.1 0.2 0.3 0.4 0.5

YC, strong

YC, medium

YC, neutral

YCW, strong

YCW, medium

YCW, neutral

Data source: U.S. Consumer Expenditure Interview Survey, based on values in 2011 dollars U.S.
Summary

 Goals met

 Produced Y, C, and W measures using OECD Framework 

with U.S. data, with limits

 Applied Ruiz method to produce MCI, with caveats

 Preliminary results

 MCI provides different picture of economic well-being than 

does any dimension alone

 What matters

– Aversions to inequality and un-balancement

– Aggregation

– Dimensions 

29
Future Directions

 Relax assumptions for construction of MCI

Normalization

Treatment of negatives and extreme values

 Refine definition of dimensions

Expenses associated with owned housing

Imputations

– Vehicles

– Income taxes

– More social transfers in-kind

30
Contact Information

Thesia I. Garner

Division of Price and Index Number 

Research

Office of Prices and Living Conditions

www.bls.gov

202-691-6576

garner.thesia@bls.gov
Numerical Example: 

Arithmetic Mean

32

( )

3 1 2

6 2 5

= ,, = 7 3 10

7 5 14

10 5 24

X ICW

   

 

 

 

     

3 01 0 2 0

10 0 5 0 24 0

6 02 05 0 0.3 0.2 0.1

10 0 5 0 24 0 0.6 0.4 0.2

7 0 3 0 10 0 = = 0.7 0.6 0.4

10 0 5 0 24 0 0.7 1 0.6 7 0 5 0 14 0

1 1 1 10 0 5 0 24 0

10 0 5 0 24 0

10 0 5 0 24 0

N X

  −− −   −− −

−−−  

−− −  

  −− −   −− −  

−− −     −− −  

−− −

  −− −

0.3 0.2 0.1 0.2

0.6 0.4 0.2 0.4

0.7 0.6 0.4 0.6 0.6

0.7 1 0.6 0.8

1 1 1 1

   →

   →

→ →

→

   →

0.3 0.2 0.1

0.6 0.4 0.2

0.7 0.6 0.4

0.7 1 0.6

1 1 1

   

 

 

 

     

↓ ↓ ↓

(0.6 0.7 0.5)

0.6

↓

I-aggregation 

S-aggregation 
Numerical Example: 

I- Aggregation, Curvature: q=r=-2

33

↓ ↓ ↓

( )

( )

( )

( )

( )

1

2 2 22

1

2 222

1

2 222

1

2 22 2

1

2 222

1 0.3 0.2 0.1

3

1

0.3 0.2 0.1 0.6 0.4 0.2

3

0.6 0.4 0.2

1 0.7 0.6 0.4 0.7 0.6 0.4

3 0.7 1 0.6

1 1 1 1 0.7 1 0.6

3

1 111

3

− − −−

− −−−

− −−−

− −− −

− −−−

  + +    

 

→ + +        

 →

   → + +    

→        →   + +    

  + +    

   

 

(0.1 0.3 0.5 0.7 1 ) = 0.3

5

1

1

0.7

0.5

0.3

0.1

2

1

2 2 2 2 2 − − − − − −





 



 → + + + +

































→

→

→

→

→
One-dimensional Index

 Assuming (x1,..., xn) is the distribution of income, for 

example, over n units of observations, the 

generalized mean of curvature q of this distribution

34

( )

1

1

=1

1

,..., = 0

n q

q

qn i

i

xx x q

n

μ

  ∀ ≠     ∑

1

1

= for q = 0

n

n

i

i

x

=

∏

 q = 1 the generalized mean simply reduces to the arithmetic mean

 q = 0, this case is the geometric mean

 q = -1, the harmonic mean

 as q decreases, greater weight on lower end of income (e.g.,) 

distribution so generalized mean refers this

(Foster, Lopez-Calva, and Szekely 2005)
Examples of 

Multi-dimensional Index

 S-Aggregation 

 I-Aggregation

35

( ) ( )

1

1

=1 =1

1 1

= < 1, < 1, 0, 0

r r

m n

q q S j

i

j i

W X x q r q r

m n

          ∀ ∀ ≠≠          

∑ ∑

( ) ( )

1

1

=1 =1

1 1

= < 1, < 1, 0, 0

q q

n m r r I j

i

i j

W X x q r q r

n m

            ∀ ∀ ≠≠          

∑ ∑
Example: MCIPI

*

 When q=r=1, MCI=0.6

 When q=r=-2, MCI=0.3

 Reduction in MCI result of discounting for 2 

forms of inequality

Within dimensions between individuals, q

Between dimensions, r

* PI = path independent assumption 36
MCI: Flexible Form q =1, r=0.5

Neutral aversion to inequality across individuals; weak aversion across dimensions 37
MCI: Flexible Form q =-1, r=0.5

Medium aversion to inequality across individuals; weak aversion across dimensions 38
MCI: Flexible Form q =-3, r=0.5

39 Strong aversion to inequality across individuals; weak aversion across dimensions
Ways to Measure: Levels or Means in Bar 

Graph e.g., GPD Per Capita or Median Household Income

40

Source; http://en.wikipedia.org/wiki/File:GDPPC_vs_Median_Income.svg
Ways to Measure: Distributions and 

Inequality as an Index, e.g., Gini Coefficient

41

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Source: Fisher, Jonathan and David S. Johnson, “Measuring the Trends in Inequality of 

Individuals and Families: Income and Consumption,” AEA Session, January 2013.

Lower Gini =

more

equal the distribution
Neutral Aversion to Inequality (q=r=1) 

by Income Deciles

MCII=MCIS (path independent)
MCI with Medium Aversion to Inequality (q=r=-1) 

by Income Deciles

Neutral aversion for dimensions, MCII=MCIS (path independent)
MCI with Strong Aversion to Inequality (q=r=-3)

by Income Deciles

Neutral aversion for dimensions, MCII=MCIS (path independent)
Neutral Aversion to Inequality (q=r=1) 

by Income Deciles and Housing Tenure

MCII=MCIS (path independent)

-0.10

0.10

0.30

0.50

0.70

Decile 1

Decile 2

Decile 3

Decile 4

Decile 5

Decile 6

Decile 7

Decile 8

Decile 9

Decile 10

Own with 

mortgage

Own without 

mortgage or ...

Rented w/ 

payment

Rented no pay 

and student ...

Income

Consumption

Wealth

MCI
MCI with Medium Aversion to Inequality (q=r=-1) 

by Income Deciles and Housing Tenure

Neutral aversion for dimensions, MCII=MCIS (path independent)

-0.10

0.10

0.30

0.50

0.70

Decile 1

Decile 2

Decile 3

Decile 4

Decile 5

Decile 6

Decile 7

Decile 8

Decile 9

Decile 10

Own with 

mortgage

Own without 

mortgage or ...

Rented w/ 

payment

Rented no pay 

and student ...

Income

Consumption

Wealth

MCI
MCI with Strong Aversion to Inequality (q=r=-3)

by Income Deciles and Housing Tenure

Neutral aversion for dimensions, MCII=MCIS (path independent)

-0.10

0.10

0.30

0.50

0.70

Decile 1

Decile 2

Decile 3

Decile 4

Decile 5

Decile 6

Decile 7

Decile 8

Decile 9

Decile 10

Own with 

mortgage

Own without 

mortgage or ...

Rented w/ 

payment

Rented no pay 

and student ...

Income

Consumption

Wealth

MCI