Multidimensional poverty index across districts in Punjab, Pakistan: estimation and rationale to consolidate with SDGs

Abstract

Multidimensional poverty index (MPI) has gotten relevance of their public policy and arguably became a strong instrument to meet sustainable development goals (SDGs). This study proposes measurement of MPI along with its components that are responsible for a change in MPI over the period from 2007 to 2018 using the Alkire–Foster method. Four rounds of multiple indicators cluster survey have been used which is only available data to account for all of the dimensions of MPI. The findings show the incidence of MPI in Punjab has downturned by an annual average of 1.1%, from 10% of the population in 2007 to 6.7% of the population in 2018. Southern Punjab is poorer than North-Central Punjab. The magnitude of poverty is lower in industrialized districts which are mainly situated in northern Punjab. Health and education dimensions need to be improved throughout the province. In the living standard dimension, cooking fuel, flooring, and sanitation need to be administered. Globally, MPI is a great poverty measurement and monitoring tool that determines the pace and progress in SDGs categorically. Due to its multidimensional structure, MPI is specifically attributed to measure the SDG # 1, 2, 3, 4, 6, 7, and 11.

Appendix

This section pertains to the methodology provided by Alkire and Foster (2011, 2011-WP 43). Poverty scores have been calculated using the following steps.

• Step 1 We have developed a matrix $X = \left[ {x_{ij} } \right]$ $\left( {i = 1,2, \cdots , n;j = 1,2, \cdots ,d} \right)$ an $n \times d$ ($n = {\text{ Number of households}}$ and $d = {\text{ Number of indicators}} = 10$) achievement matrix showing achievement of each indicator $d$ (fixed) by the household $n$. For MICS Punjab 2018, the achievement matrix is given as:

  $$X\left( {2018} \right) = \left[ {\begin{array}{*{20}c} {x_{1 1} } & \cdots & {x_{1 10} } \\ \vdots & \ddots & \vdots \\ { x_{51660 1} } & \cdots & { x_{51660 10} } \\ \end{array} } \right].$$

While for the ${\text{X}}$(2014), the number of households included is $n = 38405,$ for the ${\text{X}}\left( {2011} \right)$, number of households included is $n = 95238$ and for ${\text{X}}\left( {2007} \right)$, the number of households was $n = 91075$. Each row vector $x_{i}$ represents $i$ th household achievements, whereas each column vector $x_{*j}$ gives the distribution of $j$ th indicator achievement across the set of households. This allows poverty comparisons on three datasets of different sizes.

• Step 2 Poverty measurements can better be approximated using deprivations of HHs rather than their achievements. To do so, we transform the achievement matrix into a deprivation matrix. Let $Z_{j} > 0 \left( {Z_{1} , \cdots , Z_{10} } \right)$ denote the cutoff below which a HH is considered to be deprived in indicator $j$. We define a matrix $g^{0} = \left[ {g_{ij}^{0} } \right] = \left\{ {\begin{array}{*{20}c} {1 if x_{ij} < Z_{j} } \\ {0 otherwise} \\ \end{array} } \right.$ where $g_{ij}^{0} = 1$ states that HH $i$ is deprived in the dimension $j$ and $g_{ij}^{0} = 0$ shows no deprivation. Hence, the deprivation matrix for the year 2014 would be:

  $${\text{g}}^{0} \left( {2018} \right) = \left[ {\begin{array}{*{20}c} {g^{0} x_{1 1} } & \cdots & {g^{0} x_{1 10} } \\ \vdots & \ddots & \vdots \\ {g^{0} x_{51660 1} } & \cdots & { g^{0} x_{51660 10} } \\ \end{array} } \right].$$

The $i$ th row vector of ${\text{g}}^{0}$ is HH $i$’s deprivation vector. Similarly, matrices for ${\text{g}}^{0} \left( {2014} \right),$ ${\text{g}}^{0} \left( {2011} \right)$ and ${\text{g}}^{0} \left( {2007} \right)$ can be obtained similarly as ${\text{X}}\left( {2014} \right),$ ${\text{X}}\left( {2011} \right)$ and ${\text{X}}\left( {2007} \right)$.

• Step 3 Since each indicator has a different impact on poverty, it is better to assign them weights to show their relative significance in measuring the poverty situation. Let $W_{j} \left( {W_{1} , \ldots , W_{10} } \right)$ is the vector of weights of the respective indicators. Assigning more weight to a specific indicator suggests a high impact of that indicator in pushing HHs into poverty. The relative weights $W_{j}^{T} = \left[ {\frac{1}{6} \frac{1}{6} \frac{1}{6} \frac{1}{6} \frac{1}{18} \frac{1}{18} \frac{1}{18} \frac{1}{18} \frac{1}{18} \frac{1}{18}} \right]^{T}$ of the indicators have been assigned to the respective deprivations to measure their exact magnitude on poverty.⁷ The first four weights have been assigned to the indicators of education and health dimensions and the remaining six weights have been assigned six indicators of living standards dimension. We have obtained a weighted deprivation matrix:

  $$g^{0} \left( {W_{i} } \right)\left( {2018} \right) = \left[ {\begin{array}{*{20}c} {g^{0} \left( W \right)x_{1 1} } & \cdots & {g^{0} \left( W \right)x_{1 10} } \\ \vdots & \ddots & \vdots \\ {g^{0} \left( W \right)x_{51660 1} } & \cdots & { g^{0} \left( W \right)x_{51660 10} } \\ \end{array} } \right].$$

It shows a true picture of who is deprived in which dimension and how much weight the deprivation carries.

When from the education dimension one indicator is missing then we have assigned 1/3 weight to the second indicator. So, thus the dimension’s weight has become equal to the other dimension’s weight. The same technique we have applied for the health dimension but in the case of living standard dimension, when one indicator is missing then we have assigned 1/15 each to the remaining five indicators. So, thus living standard dimension’s weight has become equal to the other dimension’s weight.

• Step 4 The next step involves the calculation of the Weighted Deprivation Score Column Vector (WDSCV) of deprivation counts. This is called a poverty cutoff to identify the poor. As each person is allotted a deprivation score based on his or her deprivation in each indicator. So, each person’s deprivation in the household is calculated by using the weighted sum of a number of deprivations. The range of deprivation scores for each person is between 0 and 1. As the number of deprivations increases for each person causes to increase in the deprivations score. When the score reaches 1, it means that person is deprived in all the indicators, and on the other hand, 0 score depicts the person who is not deprived in any indicator. This can formally be written as:

  $$C_{i} = W_{1} I_{1} + W_{2} I_{2} + \cdots + W_{10} I_{10}$$

where $C_{i}$ is the weighted deprivation score of each person in all indicators of household 1. If the person is deprived in indicator $j$ then $I_{j} = 1$, otherwise $I_{j} = 0$. Following the same lines, we have calculated $C_{i}$ a weighted deprivation score of each person in household $n$. This can be formally be expressed in column vector of deprivation counts as:

$$C_{i} \left( {2018} \right) = \left[ {C_{1 } } \right.C_{2} \cdots \left. {C_{51660} } \right].$$

Similarly, for the years 2014, $i = 1, 2, \ldots , 38405$, 2011, $i = 1, 2, \ldots , 95238$ and for 2007, $i = 1, 2, \ldots , 91075$.

• Step 5 After measuring deprivation, we have set and apply the poverty cutoff to identify the multidimensionally poor persons. This is the second cutoff of the methodology suggested by Alkire and Foster (2011). If the deprivation score of a person is equal or greater than the poverty cutoff, then the person is considered multidimensionally poor. Formally, $k$ is the threshold poverty cutoff at the level of 33%. Then, a household would be considered poor if

  $$C_{i} \ge k\quad {\text{MPI}}\,{\text{Poor}}$$
  $$C_{i} < k\quad {\text{MPI}}\,{\text{Non}} - {\text{poor}}{.}$$

• Step 6 After calculating the status of households as MPI poor or non-poor, we have derived a censored weighted deprivation matrix replacing non-poor as zero in the weighted deprivation matrix.

  $$g^{0} \left( k \right) = 1, \,{\text{if}}\,C_{i} \ge k\,\left( {{\text{said}}\,{\text{to}}\,{\text{be}}\,{\text{deprived}}\,{\text{and}}\,{\text{poor}}} \right)$$
  $$g^{0} \left( k \right) = 0,\, if\,C_{i} < k\,\left( {{\text{said}}\,{\text{to}}\,{\text{be}}\,{\text{deprived}}\,{\text{or}}\,{\text{not}}\,{\text{but}}\,{\text{non - poor}}} \right)$$

Hence, the censored deprivation matrix for the year 2018 would be:

$$g^{0} \left( k \right)\left( {2018} \right) = \left[ {\begin{array}{*{20}c} {g^{0} \left( k \right)x_{1 1} } & \cdots & {g^{0} \left( k \right)x_{1 10} } \\ \vdots & \ddots & \vdots \\ { g^{0} \left( k \right)x_{51660 1} } & \cdots & { g^{0} \left( k \right)x_{51660 10 } } \\ \end{array} } \right].$$

Similarly, censored deprivation matrices for the years 2014, 2011, and 2007 can be obtained by changing household numbers.

• Step 7 To distinguish the original deprivation score from the censored one, we use the notation $C_{i} \left( k \right)$ that defines the deprivation score of the poor. When

$C_{i} \ge k$, then $C_{i} \left( k \right) = C_{i}$.

but if

$C_{i} < k$, then $C_{i} \left( k \right) = 0$.

It tells that if the deprivation score crosses the threshold of poverty cutoff $k$, then the deprivation score of the poor remains constant. On the other hand, if the deprivation score of the household is less than the cutoff, the household would be considered non-poor and its deprivation score would be considered zero. This explains poor households along with their deprivation scores. Now, we can draw the censored weighted deprivation count vector for the year 2018,

$$C_{i} \left( k \right)\left( {2018} \right) = \left[ {C_{i} \left( k \right) C_{2} \left( k \right) \cdots C_{51660} \left( k \right)} \right].$$

For the other years, censored weighted deprivation count vector can be drawn similarly.

• Step 8 The censored weighted deprivation count vector tells the number of households that are multidimensionally poor. From this, we can find the proportion of the incidence of the households who experience multiple deprivations. This proportion is called headcount ratio

  $$H = \frac{q}{n}$$

here $q$ is the number of multidimensionally poor persons and $n$ is the total population. For the year 2018, the $H$ would be calculated as $H\left( {2018} \right) = \frac{{q\left( {2018} \right)}}{51660}$ and $H$ for the years 2014, 2011, and 2007 can be calculated by changing the total population and number of deprived households.

• Step 9 To calculate MPI, we need $H$ as well as $A$ where $A$ shows the intensity of the deprivation of households and can be defined as the average proportion of weighted deprivation. This can be calculated as:

  $$A = \frac{{\mathop \sum \nolimits_{i = 1}^{n} C_{i} \left( k \right)}}{q}$$

  where $C_{i} \left( k \right)$ is the censored deprivation score of individual $i$ and $q$ is the number of multidimensionally poor households. To calculate $A$ for the year 2018, we use $A\left( {2018} \right) = \frac{{\mathop \sum \nolimits_{i = 1}^{51660} C_{i} \left( k \right)}}{q}$. Similarly, $A$ can be calculated for other years. Computation of MPI.

• Step 10 Multidimensional poverty index which is the product of headcount ratio (H) and average intensity (A) can be computed as:

  $${\text{MPI}} = H \times A.$$

See Figs.

Fig. 1

Multidimensional poor population (2018)

1,

Fig. 2

Multidimensional poor population (2014)

2, and

Fig. 3

Multidimensional poor population (2011)

3.