ECON 4643: Development Economics

Lecture 3: Module 1-Poverty

Author

Harounan Kazianga
Oklahoma State University
Spring 2024

Code

if (!require("pacman")) install.packages("pacman")

Loading required package: pacman

Code

pacman::p_load(haven, DescTools, knitr, kableExtra, ineq, ggplot2, formatdown)

Reading

ED Ch 6
Banerjee, A. and E. Duflo (2007). “The Economic Lives of the Poor”, The Journal of Economic Perspectives. 21(1) pp 141167 (available at the library)
Schultz, T. W. 1979. “Nobel Lecture: The Economics of Being Poor”, Journal of Political Economy. 88(4) pp 639-651 (available at the library)
Foster, Greer, Thorbeke (1984), “A Class of Decomposable Poverty Measures”, Econometrica, 1984

Introduction

Why care about poverty?
- Intrinsic interest
- Functional implications for development: human capital, incentives in production, etc.

Conceptual Issues

Poverty as an absolute concept (in contrast to inequality)
We will call someone poor whose measure of well-being is under a certain threshold, the poverty line (in contrast: relative deprivation)
What measure:
income, consumption basket, food consumption, wealth, capabilities?
Static versus dynamic measurement (temporary vs. chronic)
Household or individual level (importance of intra-household inequality)

Poverty Measures: Principles

Starting Point: Poverty line
We call someone poor if her income (or other measure) is below the poverty line
But how to make comparisons among the poor?
This gives rise to the following principles:

Three Principles of Poverty Measures

Monotonicity Principle:
a reduction in the income of a poor household must increase poverty
Transfer Principle:
a transfer of income from a poor household to any richer household must increase poverty
Transfer Sensitivity Principle:
if a transfer of income takes place from a poor household with income $y_{i}$ to a poor household with income $y_{i}+d, d>0,$ then the magnitude of the increase in poverty must be smaller for larger $y$

Poverty Measures

The poverty line is $z$
- Suppose we are interested in income poverty. Individuals with $\text{income} < z$ are poor, and those with $\text{income} \ge z$ are non-poor.

The Head Count

Head Count Ratio: \[ HCR=\frac{HC}{n} \]

where $HC$ is the number of individuals with income $y_{i}$, such that $y_{i}<Z$, and $n$ is the size of the population.

The Poverty Gap Ratio

Poverty Gap Ratio (or Poverty Index Ratio):

\[ PGR=\frac{\sum_{y_{i}<z}(z-y_{i})}{n\mu} \]

This is the average distance below the poverty line as a proportion of the poverty line across the population, where the non-poor are considered to have a zero poverty gap.

Illustration: HCR and PGR

We generate $10$ income observations between $\$10$ and $\$120$, and set the poverty line at $z=\$45$. We calculate and report the headcount ratio, and the poverty gap ratio.

Code

# Set seed for reproducibility
set.seed(123)
# Generate 10 observations with income values from 10 to 120
income <- seq(10, 120, length.out = 10)
#income <- format_decimal(income, 2)
# Set the poverty line
poverty_line <- 45
# Calculate the headcount ratio
headcount_ratio <- sum(income < poverty_line) / length(income)
# Calculate the poverty gap ratio
poverty_gap <- ifelse(income < poverty_line, poverty_line - income, 0)
poverty_gap_ratio <- sum(poverty_gap) / (length(income) * poverty_line)

# Display the results
#print(income)
#print(headcount_ratio)
#print(poverty_gap_ratio)

The income distribution is: 10, 22.2222, 34.4444, 46.6667, 58.8889, 71.1111, 83.3333, 95.5556, 107.7778, 120
Using the headcount ratio the poverty rate is 0.3
Using the poverty gap ratio, the poverty rate is 0.1519

Changing in Poverty

Code

# Increase poverty in the hypothetical population
income1 <- income
## 1. income fall below 45 for 3 individuals with high income (>45)
### 1.a. Identify observations higher than 45
high_income_indices <- which(income1 > 45)
### 1.b Modify the first 3 observations found to be strictly smaller than 45
#### I choose a value just 44 to ensure they meet the criteria, but feel free to explore with other values
new_y <- 44
income1[high_income_indices[1:3]] <- new_y

# The modified income data is now stored in 'income1'
# print(income1)

# Calculate the headcount ratio
hcr1 <- sum(income1 < poverty_line) / length(income1)
# Calculate the poverty gap ratio
gap1 <- ifelse(income1 < poverty_line, poverty_line - income1, 0)
pgr1 <- sum(gap1) / (length(income1) * poverty_line)

## 2. Income falls for the poor 
income2 <- income
low_income_indices <- which(income2 < 45)
### 1.b Modify the first 3 observations found to be strictly smaller than 45
#### I choose a value just 44 to ensure they meet the criteria, but feel free to explore with other values
new_y2 <- 8
income2[low_income_indices[1:3]] <- new_y2

# The modified income data is now stored in 'income1'
#print(income2)

# Calculate the headcount ratio
hcr2 <- sum(income2 < poverty_line) / length(income2)
# Calculate the poverty gap ratio
gap2 <- ifelse(income2 < poverty_line, poverty_line - income2, 0)
pgr2 <- sum(gap2) / (length(income2) * poverty_line)

Suppose we want to increase poverty in this population. There two ways we can proceed:

Suppose that income fall below $\$45$ (to 44) for three individuals. The resulting income distribution is 10, 22.2222, 34.4444, 44, 44, 44, 83.3333, 95.5556, 107.7778, 120

The new headcount ratio is 0.6, and the new poverty gap ratio is 0.1585
In this case both the headcount ratio and the poverty gap ratio show that poverty has increased in this hypothetical population

Now, there is no change in income above $\$45$. However, for three poor individuals, income falls to 8.

The new headcount ratio is 0.3, and the new poverty gap ratio is 0.2467
The poverty gap ratio shows that poverty has increased, but the headcount ratio indicates that poverty has not changed.

This illustrates one key limitation of the headcount ratio. It is insensitive to decrease in income of the poor, i.e. to the poor getting poorer. Which principles do you think this measures violates?

The FGT Class of Poverty Measures

Foster-Greer-Thorbeke propose a general class of poverty measures
Most poverty measures (e.g., the HCR, PGR) can be represented as special cases of FGT poverty measures

\[ P_{\alpha}(z)=\frac{1}{n}\sum_{i=1}^{n}\left[ ^{1}\{y_{i}\leq z\}\left( \frac{z-y_{i}}{z}\right) ^{\alpha}\right] \]

For example, if $\alpha=0$, we obtain the headcount ratio, HCR
Verify that if $\alpha = 1$, you have the income gap ratio
We will write a R function call “fgt” that we calculate the poverty measure, given the population income, the poverty rate and $\alpha$.

Code

# Define the FGT poverty measure function
fgt <- function(y, z, alpha) {
  # Ensure y is a numeric vector
  if (!is.numeric(y)) {
    stop("Income data must be numeric.")
  }
  
  # Ensure z (poverty line) is numeric and positive
  if (!is.numeric(z) || z <= 0) {
    stop("Poverty line must be a positive numeric value.")
  }
  
  # Ensure alpha is numeric and non-negative
  if (!is.numeric(alpha) || alpha < 0) {
    stop("Alpha must be a non-negative numeric value.")
  }
  
  # Calculate the FGT index
  N <- length(y)
  poverty_gaps <- ifelse(y < z, ((z - y) / z)^alpha, 0)
  fgt_index <- sum(poverty_gaps) / N
  
  return(fgt_index)
}

Code

# Define the poverty line
z <- 45

hcr <-  fgt(income, z, 0)
hcr1 <- fgt(income1, z, 0)
hcr2 <- fgt(income2, z, 0)

pgr <-  fgt(income, z, 1)
pgr1 <- fgt(income1, z, 1)
pgr2 <- fgt(income2, z, 1)


fgt_version = data.frame(
  Measures = c("HCR", "PGR"),
  "income" =  c(hcr,  pgr),
  "income1" = c(hcr1, pgr1),
  "income2" = c(hcr2, pgr2)

)

kable(fgt_version, format = "html", caption = "Poverty Measures Using FGT", digits = 2, full_width = F, position = "float_top", 
      format.args = list(big.mark = "," , scientific = FALSE), col.names = c("Poverty Measure", "Income Dist 0", "Income Dist 1", "Income Dist 2" ) )

Poverty Measures Using FGT
Poverty Measure	Income Dist 0	Income Dist 1	Income Dist 2
HCR	0.30	0.60	0.30
PGR	0.15	0.16	0.25

The measures are identical to the ones we calculated earlier.
In practice, you do not need to write your own “fgt” function. The package “ineq” we used earlier has a function for calculating poverty measures.
- Syntax: pov(x, z, parameter = , type = “Foster”) , where $x$ is the variable (income), $z$ is the poverty line, parameter take the value of $\alpha$, and ‘type = “Foster”’ is for the FGT class of poverty measures.

Code

fgt1 <- pov(income, z, parameter=1, type = "Foster")
fgt1 <- round(fgt1, digits = 2 )

If we use pov(income, z, parameter=1, type = “Foster”), we get 0.3. You are encouraged to explore with different income distributions and values of the poverty line.

The Parameter $\alpha$

For $y_{i}\leq z$, $z-y_{i}$ is the income shortfall, $\frac{z-y_{i}}{z}$ the relative income shortfall
- For $\alpha=1$, we obtain $P_{1}$the poverty depth measure
For $y_{i}\leq z$, $\left( \frac{z-y_{i}}{z}\right)^{2}$ is called relative poverty severity

-For $\alpha=2$, we obtain the poverty-severity measure
You might have noticed this already: $\alpha$ indicates the weight attached to how far the poor fall from the poverty line
- For example, when $\alpha = 0$, we care only whether someone is poor or not, and it does not matter how power they are.

The FGT and the Principles of Poverty Measures

Theorem
- $P_{\alpha}$ satisfies the Monotonicity Principle for $\alpha>0$
- $P_{\alpha}$ satisfies the Transfer Principle for $\alpha>1$
- $P_{\alpha}$ satisfies the Transfer Sensitivity Principle for $\alpha>2$

Decomposability of the FGT Poverty Measures

The FGT can be decomposed to examine how different social groups contribute to overall poverty
Suppose we are given information on poverty in $J$ different subgroups (each with population $n_{j}$) of a population of size $n$
We aggregate subgroup poverty into total poverty

\[ P_{\alpha}^{total}=\sum_{j=1}^{J}\frac{n_{j}}{n}P_{\alpha}^{j}% \]

Illustration

We will create a hypothetical population combining the three income distributions we have used so far
- we have three groups, $A, B, C$
- Furtheremore, we know that $n_0 = n_1 = n_2 = 10$, and $n = 30$.
Let set $\alpha = 0$, the headcount ratio \[ P^{total} = \frac{n_A}{30}P^{1}_{0} + \frac{n_B}{30}P^{2}_{0} + \frac{n_C}{30}P^{3}_{0} \]

Code

# create new income data (groups3)

groups3 <- data.frame(
  income = c(income, income1, income2),
  group = c(rep("A", length(income)), rep("B", length(income1)), rep("C", length(income2)) )
)


# alpha = 0, the headcount ratio
alpha <- 0 


# Calculate overall poverty measure for the 3 groups dataset
overall_fgt <- fgt(groups3$income, z, alpha)

# Calculate poverty measure for each group separately and use decomposability property
groups <- unique(groups3$group)
decomposed_fgt <- 0
total_population <- nrow(groups3)

for (group in groups) {
  group_data <- groups3[groups3$group == group, ]
  group_fgt <- fgt(group_data$income, z, alpha)
  decomposed_fgt <- decomposed_fgt + (nrow(group_data) / total_population) * group_fgt
}

For $\alpha$ = 0, applying the fgt function to the 3 groups (30 observations), we find that the poverty index is 0.4. We find the identical value using the decomposability property, i.e. 0.4.

Usefullness

Sometimes, we are interested in how much specific groups (e.g., regions, ethnic groups, etc.) contribute to overall poverty.

Code

# Initialize a list to store group-specific FGT measures and populations
group_fgt_list <- list()

# Calculate FGT for each group
groups <- unique(groups3$group)
for (group in groups) {
  group_data <- groups3[groups3$group == group,]
  group_fgt <- fgt(group_data$income, z, alpha)
  group_fgt_list[[group]] <- list(fgt = group_fgt, population = nrow(group_data))
}


total_population <- nrow(groups3)
total_fgt <- sum(sapply(group_fgt_list, function(x) x$population / total_population * x$fgt))

# Calculate and print each group's contribution
for (group in names(group_fgt_list)) {
  group_contribution <- group_fgt_list[[group]]$population / total_population * group_fgt_list[[group]]$fgt / total_fgt * 100
  cat(group, "contribution to total poverty index:", group_contribution, "%\n")
}

A contribution to total poverty index: 25 %
B contribution to total poverty index: 50 %
C contribution to total poverty index: 25 %

For $\alpha = 0$ (the headcount ratio), the total poverty index is 0.4, and the contribution of each group to total poverty is as shown above. In this case, most poverty comes from $\text{Group B}$. What if $\alpha = 1$ (poverty gap index)

Code

# Initialize a list to store group-specific FGT measures and populations
alpha = 1
group_fgt_list <- list()

# Calculate FGT for each group
groups <- unique(groups3$group)
for (group in groups) {
  group_data <- groups3[groups3$group == group,]
  group_fgt <- fgt(group_data$income, z, alpha)
  group_fgt_list[[group]] <- list(fgt = group_fgt, population = nrow(group_data))
}


total_population <- nrow(groups3)
total_fgt <- sum(sapply(group_fgt_list, function(x) x$population / total_population * x$fgt))

# Calculate and print each group's contribution
for (group in names(group_fgt_list)) {
  group_contribution <- group_fgt_list[[group]]$population / total_population * group_fgt_list[[group]]$fgt / total_fgt * 100
  cat(group, "contribution to total poverty index:", group_contribution, "%\n")
}

A contribution to total poverty index: 27.26 %
B contribution to total poverty index: 28.46 %
C contribution to total poverty index: 44.28 %

The total poverty index is 0.1857, and the contribution of each group to total poverty is as shown above. In this case, most poverty comes from $\text{Group C}$.

Application: Poverty in Tanzania

Let know move to real world data. We use the Tanzania’s data to illustrate the poverty measures. Different poverty lines can be used. I convert the expenditures per adult equivalent per day in USD. We use the international poverty line of $\$1.9$ per day per adult equivalent. Notice that for this exercise, I use the exchange rate method. The actual poverty rates would be lower with purchasing power parity method.

Code

# Load, clean and filter the data
library(haven)
library(DescTools)
data <- read_dta("C:/Users/harouna/OneDrive - Oklahoma A and M System/OSU/Spring2024/ECON4643_Spring2024/Data/Assignments/Assignment1/Assignment1_Data.dta")
## filter out observations with 0 expenditures
data <- subset(data, expmR_pae > 0 & year == 2009)
## Winsorize hhexpensesR at the 2nd and 98th percentiles


data$area <- factor(data$area, 
                    levels = c(1, 2, 3, 4),
                    labels = c("Dar es Salaam", "Rest of urban", "Rural", "Zanzibar"))

Code

z <- 1.9
#z <- 36482

hcrT <-  fgt(data$expmR_pae, z, 0)
hcr1 <- fgt(data$expmR_pae[data$area=="Dar es Salaam"], z, 0)
hcr2 <- fgt(data$expmR_pae[data$area=="Rest of urban"], z, 0)
hcr3 <- fgt(data$expmR_pae[data$area=="Rural"], z, 0)
hcr4 <- fgt(data$expmR_pae[data$area=="Zanzibar"], z, 0)

pgrT <-  fgt(data$expmR_pae, z, 1)
pgr1 <- fgt(data$expmR_pae[data$area=="Dar es Salaam"], z, 1)
pgr2 <- fgt(data$expmR_pae[data$area=="Rest of urban"], z, 1)
pgr3 <- fgt(data$expmR_pae[data$area=="Rural"], z, 1)
pgr4 <- fgt(data$expmR_pae[data$area=="Zanzibar"], z, 1)


psrT <-  fgt(data$expmR_pae, z, 2)
psr1 <- fgt(data$expmR_pae[data$area=="Dar es Salaam"], z, 2)
psr2 <- fgt(data$expmR_pae[data$area=="Rest of urban"], z, 2)
psr3 <- fgt(data$expmR_pae[data$area=="Rural"], z, 2)
psr4 <- fgt(data$expmR_pae[data$area=="Zanzibar"], z, 2)


poverty= data.frame(
   Measures  = c("HCR", "PGR", "Squared PGR"),
  "Tanzania" = c(hcrT,  pgrT, psrT),
   "Area1"   = c(hcr1,  pgr1, psr1),
   "Area2"   = c(hcr2,  pgr2, psr2),
   "Area3"   = c(hcr3,  pgr3, psr3),
   "Area4"   = c(hcr4,  pgr4, psr4)
)

kable(poverty, format = "html", caption = "Poverty Indices in Tanzania", digits = 2, full_width = F, position = "float_top", 
      format.args = list(big.mark = "," , scientific = FALSE), col.names = 
        c("Poverty Measure", "Tanzania", "Dar es Salaam", "Rest of urban", "Rural", "Zanzibar" ) )

Poverty Indices in Tanzania
Poverty Measure	Tanzania	Dar es Salaam	Rest of urban	Rural	Zanzibar
HCR	0.75	0.32	0.67	0.89	0.83
PGR	0.37	0.10	0.28	0.47	0.41
Squared PGR	0.21	0.05	0.15	0.28	0.24

Discuss the Table

Each Region’s Contribution to Total Poverty

Headcount Ratio

Code

# Initialize a list to store group-specific FGT measures and populations
alpha = 0
group_fgt_list <- list()

# Calculate FGT for each group
groups <- unique(data$area)
for (area in groups) {
  group_data <- data[data$area == area,]
  group_fgt <- fgt(group_data$expmR_pae, z, alpha)
  group_fgt_list[[area]] <- list(fgt = group_fgt, population = nrow(group_data))
}


total_population <- nrow(data)
total_fgt <- sum(sapply(group_fgt_list, function(x) x$population / total_population * x$fgt))

# Calculate and print each group's contribution
for (group in names(group_fgt_list)) {
  group_contribution <- group_fgt_list[[group]]$population / total_population * group_fgt_list[[group]]$fgt / total_fgt * 100
  cat(group, "contribution to total poverty index:", group_contribution, "%\n")
}

Rural contribution to total poverty index: 63.54 %
Rest of urban contribution to total poverty index: 13.11 %
Dar es Salaam contribution to total poverty index: 7.146 %
Zanzibar contribution to total poverty index: 16.2 %

Poverty Gap Ratio

Code

# Initialize a list to store group-specific FGT measures and populations
alpha = 1
group_fgt_list <- list()

# Calculate FGT for each group
groups <- unique(data$area)
for (area in groups) {
  group_data <- data[data$area == area,]
  group_fgt <- fgt(group_data$expmR_pae, z, alpha)
  group_fgt_list[[area]] <- list(fgt = group_fgt, population = nrow(group_data))
}


total_population <- nrow(data)
total_fgt <- sum(sapply(group_fgt_list, function(x) x$population / total_population * x$fgt))

# Calculate and print each group's contribution
for (group in names(group_fgt_list)) {
  group_contribution <- group_fgt_list[[group]]$population / total_population * group_fgt_list[[group]]$fgt / total_fgt * 100
  cat(group, "contribution to total poverty index:", group_contribution, "%\n")
}

Rural contribution to total poverty index: 67.8 %
Rest of urban contribution to total poverty index: 11.13 %
Dar es Salaam contribution to total poverty index: 4.731 %
Zanzibar contribution to total poverty index: 16.34 %

Squared Poverty Gap Ratio

Code

# Initialize a list to store group-specific FGT measures and populations
alpha = 2
group_fgt_list <- list()

# Calculate FGT for each group
groups <- unique(data$area)
for (area in groups) {
  group_data <- data[data$area == area,]
  group_fgt <- fgt(group_data$expmR_pae, z, alpha)
  group_fgt_list[[area]] <- list(fgt = group_fgt, population = nrow(group_data))
}


total_population <- nrow(data)
total_fgt <- sum(sapply(group_fgt_list, function(x) x$population / total_population * x$fgt))

# Calculate and print each group's contribution
for (group in names(group_fgt_list)) {
  group_contribution <- group_fgt_list[[group]]$population / total_population * group_fgt_list[[group]]$fgt / total_fgt * 100
  cat(group, "contribution to total poverty index:", group_contribution, "%\n")
}

Rural contribution to total poverty index: 69.48 %
Rest of urban contribution to total poverty index: 10.19 %
Dar es Salaam contribution to total poverty index: 3.776 %
Zanzibar contribution to total poverty index: 16.55 %

Discussion

Vulnerability

Transitory and Chronic Poverty

Poverty is a dynamic condition: individuals can move into and out of poverty, stay poor or non-poor permanently. Alongside the never poor, individuals can be categorized into four groups based on how vulnerable they are from one period to the next (e.g., from one year to the next).

Transitory or temporary poor: individuals with an average income above the poverty line but who may occasionally fall into poverty due to low vulnerability and unlikely year-to-year poverty experiences.
Chronic poor: individuals who are consistently below the poverty line but may occasionally rise above it, facing a high likelihood of experiencing poverty annually due to their vulnerability (also know as temporary poor)
Persistent poor: individuals who always live below the poverty line, facing the highest vulnerability with a certainty of being in poverty every year.
Non-poor: individuals who are always above the poverty line.

Jalan and Ravallion (2000)

Code

Jalan = data.frame(
  Category = c("never poor", "transitory poor", "chronic poor", "persistent poor"),
  "Percentage" =  c(41, 36, 18, 5)
  
)

kable(Jalan, format = "html", digits = 2, full_width = F, position = "float_top", 
      format.args = list(big.mark = "," , scientific = FALSE), col.names = c("Category", "Percentage" ) )

Table 1: Poverty dynamics in China between 1985 and 1990

Category	Percentage
never poor	41
transitory poor	36
chronic poor	18
persistent poor	5

Reading

Introduction

Conceptual Issues

Poverty Measures: Principles

Three Principles of Poverty Measures

Poverty Measures

The Head Count

The Poverty Gap Ratio

Illustration: HCR and PGR

Changing in Poverty

The FGT Class of Poverty Measures

The Parameter \(\alpha\)

The FGT and the Principles of Poverty Measures

Decomposability of the FGT Poverty Measures

Illustration

Usefullness

Application: Poverty in Tanzania

Discuss the Table

Each Region’s Contribution to Total Poverty

Headcount Ratio

Poverty Gap Ratio

Squared Poverty Gap Ratio

Discussion

Vulnerability

Transitory and Chronic Poverty

Jalan and Ravallion (2000)