Affluence (richness) indices

Description

This package allows to compute the affluence indices (average affluence gap, income share of the top p %, richness headcount ratio, concave and convex measures of affluence) and to construct the confidence intervals for the affluence indices. The affluence line is defined by the user as multiple of the income median. This package also allows to compute the Medeiros's affluence line which is set as a multiple (defined by the user) of the income median. Additionally, this package allows also to compute some standard inequality and polarization measures: the Gini coefficient, the Palma index, the Wolfson polarization index. All measures may be calculated with weighted data.

Author(s)

Alicja Wolny-Dominiak, Anna Saczewska-Piotrowska
Maintainer: Alicja Wolny-Dominiak

References

1. Alichi A., Kantenga K., Sole J. (2016) Income polarization in the United States. IMF Working Paper, WP/16/121.
2. Brzezinski M. (2010) Income affluence in Poland. Social Indicators Research, 99, pp. 285-299.
https://link.springer.com/article/10.1007/s11205-010-9580-0

3. Cobham A., Sumner A.(2013). Is it all about the tails? The Palma measure of income inequality. Working Paper No. 343, Center for Global Development.
4. Creedy J. (2015). A note on computing the Gini inequality measure with weighted data. Workin Paper No. 3, Victoria University of Wellington.
5. Lerman R.I., Yitzhaki S. (1989) Improving the accuracy of estimates of Gini coefficients. Journal of Econometrics, 42(1), pp. 43-47.
doi:10.1016/0304-4076(89)90074-2

6. Medeiros M. (2006) The rich and the poor: the construction of an affluence line from the poverty line. Social Indicators Research, 78(1), pp. 1-18.
https://link.springer.com/article/10.1007/s11205-005-7156-1

7. Peichl A., Schaefer T., Scheicher C. (2008) Measuring richness and poverty - A micro data application to Europe and Germany. IZA Discussion Paper No. 3790, Institute for the Study of Labor (IZA).
8. Saczewska-Piotrowska A. (2015) Identification of determinants of income richness using logistic regression model. Zarzadzanie i Finanse. Journal of Management and Finance, 4, Part 2, pp. 241-259 (in Polish).
9. Wolfson M.C. (1994) When inequalities diverge, The American Economic Review, 84, pp. 353-358. https://www.jstor.org/stable/2117858

Palma index

Description

Computes the Palma index (also known as S90/S40 ratio)

Usage

S90S40(x, weight)

Arguments

Details

The Palma index is the ratio between the income share of the top 10\% and the bottom 40\%. The weighted Palma index (with weights w_1,w_2,...,w_n) is given by:

S90/S40 = \frac{\sum_{i=1}^n{x_iw_i}\boldsymbol{1}_{x_i>q_{w(0.9)}}}{\sum_{i=1}^n{x_iw_i}\boldsymbol{1}_{x_i \leq q_{w(0.4)}}},

where x_i is an income of individual i, n is the number of individuals, q_{w(0.9)},q_{w(0.4)} are the 0.9 and 0.4 quantiles, respectively, \boldsymbol{1}_{(\cdot)} denotes the indicator function, which is equal to 1 when its argument is true and 0 otherwise.

Value

Author(s)

Anna Saczewska-Piotrowska, Alicja Wolny-Dominiak

References

Cobham A., Sumner A.(2013). Is it all about the tails? The Palma measure of income inequality. Working Paper No. 343, Center for Global Development

Equivalised income

Description

The database contains information about equivalised income of households.

Usage

data("affluence")

Format

A data frame with 2000 observations on the following 4 variables.

income

a numeric vector (equivalised income of households; equivalisation using modified OECD scale)

education

a numeric vector (education of the household's head: 1=tertiary, 2=secondary, 3=basic vocational, 4=low)

age

a numeric vector (age of the household's head: 1=less than 35, 2=35-44, 3=45-59, 4=60 and more)

sex

a numeric vector (sex of the household's head: 0=male, 1=female)

hs_size

vector of weights

Source

Based on Council for Social Monitoring (2016). Integrated database.http://www.diagnoza.com [11.09.2016].

Examples

data(affluence)
names(affluence)

Bootstrap standard error 1

Description

Calculates the bootstrap standard errors.

Usage

boot.sd1(x, weight,  kp, nsim, boot.index = c("r.hc", "r.is"), gamma)

Arguments

Details

The function uses quantile method of calculating bootstrap confidence intervals.

Value

Author(s)

Alicja Wolny-Dominiak

References

Brzezinski M. (2010) Income affluence in Poland. Social Indicators Research, 99, pp. 285-299.

Examples

data(affluence)
affluence$weight <- rep(1, nrow(affluence))
boot.sd1(affluence$income, affluence$weight, 0.9, 10, "r.is", 0.95)
boot.sd1(affluence$income, affluence$weight, 2, 10, "r.hc", 0.95)

Bootstrap standard error 1

Description

The estimation of bootstrap standard error of affluence index in subpopulation.

Usage

boot.sd1.sub(x.sub, x, weight.sub, weight, kp, nsim, boot.index=c("r.hc", "r.is"), gamma)

Arguments

Details

The function uses quantile method of calculating bootstrap confidence intervals.

Value

Author(s)

Alicja Wolny-Dominiak

References

Brzezinski M. (2010) Income affluence in Poland. Social Indicators Research, 99, pp. 285-299.

Examples

data(affluence)
affluence$weight <- rep(1, nrow(affluence))
aff.sub <- subset(affluence, education == 2)

boot.sd1.sub(aff.sub$income, affluence$income, aff.sub$weight, affluence$weight, 
             0.9, 10, "r.is", 0.95)

boot.sd1.sub(aff.sub$income, affluence$income, aff.sub$weight, affluence$weight, 
             0.9, 10, "r.hc", 0.95)

Bootstrap standard error 2

Description

Calculates the bootstrap standard errors.

Usage

boot.sd2(x, weight, k, alpha, nsim, boot.index = c("r.cha", "r.fgt"), gamma)

Arguments

Details

The function uses quantile method of calculating bootstrap confidence intervals.

Value

Author(s)

Alicja Wolny-Dominiak

References

Brzezinski M. (2010) Income affluence in Poland. Social Indicators Research, 99, pp. 285-299.

Examples

data(affluence)
boot.sd2(affluence$income, weight = NULL, 2, 2, 10, "r.cha", 0.95)
boot.sd2(affluence$income, weight = NULL, 2, 2, 10, "r.fgt", 0.95)

Bootstrap standard error 2

Description

Calculates the bootstrap standard errors in subpopulation.

Usage

boot.sd2.sub(x.sub,x,weight.sub,weight,k,alpha,nsim,boot.index=c("r.cha","r.fgt"),gamma)

Arguments

Details

The function uses quantile method of calculating bootstrap confidence intervals.

Value

Author(s)

Alicja Wolny-Dominiak

References

Brzezinski M. (2010) Income affluence in Poland. Social Indicators Research, 99, pp. 285-299.

Examples

data(affluence)
affluence$weight <- rep(1, nrow(affluence))
aff.sub <- subset(affluence, education == 2)

x <- aff.sub$income
boot.sd2.sub(x, affluence$income, aff.sub$weight, affluence$weight, 2, 2, 10, "r.cha", 0.95)
boot.sd2.sub(x, affluence$income, aff.sub$weight, affluence$weight, 2, 2, 10, "r.fgt", 0.95)

Gini coefficient

Description

Computes the Gini coefficient.

Usage

gini.w(x, weight)

Arguments

Details

The Gini coefficient is the most popular measure of income inequality. The formula taking into account the weights of income w_1,w_2,...,w_n is given by:

G_w = \frac{\sum_{i=1}^nw_i\sum_{j=1}^n w_j|x_i-x_j|}{2(\sum_{i=1}^nw_i)^2\mu_w},

where x_i,x_j are incomes of individuals i and j, respectively, n is the number of individuals, \mu_w is the mean income. The Gini coefficient ranges between 1 (perfect equality) and 1 (perfect inequality).

Value

Author(s)

Alicja Wolny-Dominiak, Anna Saczewska-Piotrowska

References

1. Creedy J. (2015). A note on computing the Gini inequality measure with weighted data. Workin Paper No. 3, Victoria University of Wellington.
2. Lerman R.I., Yitzhaki S. (1989) Improving the accuracy of estimates of Gini coefficients. Journal of Econometrics, 42(1), pp. 43-47.

Examples

data(affluence)
gini.w(affluence$income, affluence$hs_size)

Medeiros's affluence line

Description

Computes the Medeiros's affluence line.

Usage

line.med(x, weight, k)

Arguments

Details

The Medeiros's affluence line is based on the concept of poverty gap related to a given poverty line (in the package this line is set as a defined by the user multiple of the median income). Based on the determined poverty gap, there is calculated the point where the income of the richest should be reduced in order to make possible enough transfers to cover this gap and eliminate poverty. The calculated point of income may be also presented as the multiple of the median income.

Value

Author(s)

Anna Saczewska-Piotrowska, Alicja Wolny-Dominiak

References

Medeiros M. (2006) The rich and the poor: The construction of an affluence line from the poverty line. Social Indicators Research, 78(1), pp. 1-18.

Examples

data(affluence)
line.med(affluence$income, affluence$hs_size, 0.6)

Wolfson polarization index

Description

Computes the Wolfson polarization index.

Usage

polar.aff(x, weight)

Arguments

Details

Standard inequality measures do not give any information about polarization. A more polarized income distribution is one that has relatively fewer middle income class and more low- and/or high-income households (Alichi et al. 2016). Low income class is very often identified with poverty and high-income class with richness. One of the measures of polarization is the Wolfson polarization index (Wolfson 1994). Weighted version of this index is given by:

P_w= 2 \left( 2T-G_w \right) \frac{\mu_w}{\rho_w},

where T is the difference between 0.5 and the income share of bottom half of the population, G_w is the Gini coefficient, \mu_w is the mean income, \rho_w is the median income.

Value

Author(s)

Alicja Wolny-Dominiak, Anna Saczewska-Piotrowska

References

1. Alichi A., Kantenga K., Sole J. (2016) Income polarization in the United States. IMF Working Paper, WP/16/121.
2. Wolfson M.C. (1994) When inequalities diverge, The American Economic Review, 84, pp. 353-358.

Examples

data(affluence)
polar.aff(affluence$income, weight = NULL)

Concave measure of affluence

Description

Computes the measure of affluence analogous to the poverty index of Chakravarty (1983).

Usage

r.cha(x, weight, k, beta)

Arguments

Details

Peichl et. al (2008) defined an affluence index. Weighted index (with weights w_1,w_2,...,w_n) is given by:

R^{CHA}_{\beta}(\boldsymbol{x},\boldsymbol{w},\rho_w) = \frac{\sum_{i=1}^n(1-(\frac{\rho_w}{x_i})^\beta)\boldsymbol{1}_{x_i > \rho_w}w_i}{\sum_{i=1}^n{w_i}}, \beta > 0,

where x_i is an income of individual i, n is the number of individuals, \rho_w is the richness line, \boldsymbol{1}_{(\cdot)} denotes the indicator function, which is equal to 1 when its argument is true and 0 otherwise. Index satisfies transfer axiom T1 (concave): a richness index should increase when a rank-preserving progressive transfer between two rich individuals takes place.

Value

Author(s)

Alicja Wolny-Dominiak, Anna Saczewska-Piotrowska

References

1. Chakravarty S.R. (1983) A new index of poverty. Mathematical Social Sciences, 6, pp. 307-313.
2. Peichl A., Schaefer T., Scheicher C. (2008) Measuring richness and poverty - A micro data application to Europe and Germany. IZA Discussion Paper No. 3790, Institute for the Study of Labor (IZA).

Examples

data(affluence)
r.cha(affluence$income, weight = NULL, 2, 2)

Concave measure of affluence in subpopulation

Description

Computes the measure of affluence in subpopulation analogous to the poverty index of Chakravarty(1983).

Usage

r.cha.sub(x.sub, x, weight.sub, weight, k, beta)

Arguments

Value

Author(s)

Alicja Wolny-Dominiak, Anna Saczewska-Piotrowska

References

1. Chakravarty S.R. (1983) A new index of poverty. Mathematical Social Sciences, 6, pp. 307-313.
2. Peichl A., Schaefer T., Scheicher C. (2008) Measuring richness and poverty - A micro data application to Europe and Germany. IZA Discussion Paper No. 3790, Institute for the Study of Labor (IZA).

See Also

r.cha

Examples

data(affluence)
r.cha(affluence$income, weight = NULL, 2, 2)

Convex measure of affluence

Description

Computes the measure of affluence analogous to the convex version of Foster, Greer and Thorbecke (1984) family of poverty indices.

Usage

r.fgt(x, weight, k, alpha)

Arguments

Details

Peichl et. al (2008) defined an affluence index. Weighted index (with weights w_1,w_2,...,w_n) is given by:

R_{\alpha}^{FGT,T2}(\mathbf{x},\mathbf{w},\rho_w)=\frac{\sum_{i=1}^{n} \left( \frac{x_i - \rho_w}{\rho_w}\right)^{\alpha}\mathbf{1}_{x_i>\rho_w}w_i}{\sum_{i=1}^{n}w_i},\alpha>1,

where x_i is an income of individual i, n is the number of individuals, \rho_w is the richness line, \boldsymbol{1}_{(\cdot)} denotes the indicator function, which is equal to 1 when its argument is true and 0 otherwise. Index satisfies transfer axiom T2 (convex): a richness index should decrease when a rank-preserving progressive transfer between two rich individuals takes place.

Value

Author(s)

Alicja Wolny-Dominiak, Anna Saczewska-Piotrowska

References

1. Foster J.E., Greer J., Thorbecke E. (1984) A class of decomposable poverty measures. Econometrica, 52, pp. 761-766.
2. Peichl A., Schaefer T., Scheicher C. (2008) Measuring richness and poverty - A micro data application to Europe and Germany. IZA Discussion Paper No. 3790, Institute for the Study of Labor (IZA).

Examples

data(affluence)
r.fgt(affluence$income, weight = NULL, 2, 1)

Convex measure of affluence in subpopulation

Description

Computes the measure of affluence in subpopulation analogous to the convex version of Foster, Greer and Thorbecke (1984) family of poverty indices.

Usage

r.fgt.sub(x.sub, x, weight.sub, weight, k, alpha)

Arguments

Value

Author(s)

Alicja Wolny-Dominiak, Anna Saczewska-Piotrowska

References

1. Foster J.E., Greer J., Thorbecke E. (1984) A class of decomposable poverty measures. Econometrica, 52, pp. 761-766.
2. Peichl A., Schaefer T., Scheicher C. (2008) Measuring richness and poverty - A micro data application to Europe and Germany. IZA Discussion Paper No. 3790, Institute for the Study of Labor (IZA).

See Also

r.fgt

Examples

data(affluence)
affluence$weight <- rep(1, nrow(affluence))
aff.sub <- subset(affluence, education == 2)
r.fgt.sub(aff.sub$income, affluence$income, aff.sub$weight, affluence$weight, 2, 1)

Richness headcount ratio

Description

Computes the richness headcount ratio.

Usage

r.hc(x, weight, k)

Arguments

Details

Richness headcount ratio is a proportion of the population with incomes above the affluence line. Weighted version (with weights w_1,w_2,...,w_n) of this ratio is given by:

R^{HC}(\boldsymbol{x},\boldsymbol{w},\rho_w) = \frac{\sum_{i=1}^n \boldsymbol{1}_{x_i > \rho_w}w_i}{\sum_{i=1}^n{w_i}},

where x_i is an income of individual i, n is the number of individuals, \rho_w is the richness line,\boldsymbol{1}_{(\cdot)} denotes the indicator function, which is equal to 1 when its argument is true and 0 otherwise.

Value

Author(s)

Alicja Wolny-Dominiak, Anna Saczewska-Piotrowska

References

1. Brzezinski M. (2010) Income affluence in Poland. Social Indicators Research, 99, pp. 285-299.
2. Saczewska-Piotrowska A. (2015) Identification of determinants of income richness using logistic regression model. Zarzadzanie i Finanse. Journal of Management and Finance, 4, Part 2, pp. 241-259 (in Polish).

Examples

data(affluence)
affluence$weight <- rep(1, nrow(affluence))
r.hc(affluence$income, affluence$weight, 3)

Richness headcount ratio in subpopulation

Description

Computes the richness headcount ratio in subpopulation.

Usage

r.hc.sub(x.sub, x, weight.sub, weight, k)

Arguments

Value

Author(s)

Alicja Wolny-Dominiak, Anna Saczewska-Piotrowska

References

1. Brzezinski M. (2010) Income affluence in Poland. Social Indicators Research, 99, pp. 285-299.
2. Saczewska-Piotrowska A. (2015) Identification of determinants of income richness using logistic regression model. Zarzadzanie i Finanse. Journal of Management and Finance, 4, Part 2, pp. 241-259 (in Polish).

See Also

r.hc

Examples

data(affluence)
affluence$weight <- rep(1, nrow(affluence))
aff.sub <- subset(affluence, education == 2)
r.hc.sub(aff.sub$income, affluence$income, aff.sub$weight, affluence$weight, 3)

Income share of the top p %

Description

Computes the income share of the top p %.

Usage

r.is(x, weight, p)

Arguments

Details

The most popular measure of richness which takes a form (with weights w_1,w_2,...,w_n):

R^{IS}(\boldsymbol{x},\boldsymbol{w},p) = \frac{\sum_{i=1}^n{x_iw_i}\boldsymbol{1}_{x_i>q_{w(1-p)}}}{\sum_{i=1}^n{x_iw_i}},

where q_{w(1-p)} is the (1-p) quantile of the population and \boldsymbol{1}_{(\cdot)} denotes the indicator function, which is equal to 1 when its argument is true and 0 otherwise. There is always p % of rich individualsa in the population.

Value

Author(s)

Alicja Wolny-Dominiak, Anna Saczewska-Piotrowska

References

Brzezinski M. (2010) Income affluence in Poland. Social Indicators Research, 99, pp. 285-299.

Examples

data(affluence)
r.is(affluence$income, weight = NULL, 0.9)

Income share of the top p % in subpopulation

Description

Computes income share of the top p % in subpopulation.

Usage

r.is.sub(x.sub, x, weight.sub, weight, p)

Arguments

Value

Author(s)

Alicja Wolny-Dominiak, Anna Saczewska-Piotrowska

References

Brzezinski M. (2010) Income affluence in Poland. Social Indicators Research, 99, pp. 285-299.

See Also

r.is

Examples

data(affluence)
affluence$weight <- rep(1, nrow(affluence))
aff.sub <- subset(affluence, education == 2)
r.is.sub(aff.sub$income, affluence$income, aff.sub$weight, affluence$weight, 0.9)

Average affluence gap

Description

Computes the average affluence gap of population.

Usage

r.med(x, weight, k)

Arguments

Details

Medeiros (2006) defined an average affluence gap. Weighted gap (with weights w_1,w_2,...,w_n) is given by:

R^{Me}= \frac{\sum_{i=1}^n \max\{x_i-\rho_w,0\}w_i}{\sum_{i=1}^nw_i},

where x_i is an income of individual i, n is the number of individuals, \rho_w is the richness line. Medeiros' index is not standarized and is an absolute measure of richness.

Value

Author(s)

Alicja Wolny-Dominiak, Anna Saczewska-Piotrowska

References

Medeiros M. (2006) The rich and the poor: the construction of an affluence line from the poverty line. Social Indicators Research, 78, pp. 1-18.

Examples

data(affluence)
r.med(affluence$income, weight = NULL, 2)

Average affluence gap in subpopulation

Description

Computes the average affluence gap in subpopulation.

Usage

r.med.sub(x.sub, x, weight.sub, weight, k)

Arguments

Value

Author(s)

Alicja Wolny-Dominiak

References

Medeiros M. (2006) The rich and the poor: the construction of an affluence line from the poverty line. Social Indicators Research, 78, pp. 1-18.

See Also

r.med

Examples

data(affluence)
affluence$weight <- rep(1, nrow(affluence))
aff.sub <- subset(affluence, education == 2)
r.med.sub(aff.sub$income, affluence$income, aff.sub$weight, affluence$weight, 2)